02 Teoría Del Electrón Libre en Sólidos

7232019 02 Teoriacutea Del Electroacuten Libre en Soacutelidos

httpslidepdfcomreaderfull02-teoria-del-electron-libre-en-solidos 171

Modelo del electroacuten libre

Marzo 2012

O MoraacutenDepartamento de Fiacutesica UniversidadNacional de colombia sede Medelliacuten



Metales

Propiedades fiacutesicas comunes

bull dureza fiacutesica gtgtbull σ and K gtgtbull densidad gtgt

bull reflectividad oacuteptica gtgt



Na (gas)

Nanuacutecleo

11 e-

1s22s22p63S1

aacutetomos libres

2 primeras capas llenasrArrestructura estable1 e- valenciararrcapa 3

r ~19 Aring

e- de conduccioacuten


core



core

solapamiento deorbitales 3S

e- 3Srarrdelocalizado

Na metaacutelico rArr rarrbcc

vecinos +proacuteximos 37 Aring

aacutetomo libre e- 3Srarr e- de valencia

soacutelido e- 3Srarr e- de conduccioacuten

reacciones quiacutemicas Na cede e-rArrNa+ Ej Na+Cl-





e- conduccioacuten rArrvalencia and ρ

Na rarre- conduccioacuten = aacutetomosK Cu Ag Au rarrmonovalentes

Be Mg Zn Ca rarrdivalentes

e-

conduccioacuten =2 aacutetomos

rArr

ρ equiv densidadrArr( ρ M )NAequivconcentracioacuten aacutetomica (N )

M equivpeso aacutetomicoN Aequiv Avogadro

N= (Z v ρ NA) M N= (Z v ρ NA) M Z v equivvalencia

grcm3 grmol)aacutetomos =aacutetomosmol cm3

rArr



Gas de e- libres

e- rarrcompleta libres excepto por φx

U vaciacuteometal

φ

φrarrconfina e- en el interior del cristalmodelo e- libre e- conduccioacuten mueven sin

colisiones (excepto reflexioacuten en la superficie)e- libresequivmoleacuteculas gas ideal

rArrgas de e- libres



fuerte rArre-rarr colisiones frecuentesrArr gas e- equivgas no ideal

alternativamente

e-

vdarr

InteraccioacutenequivU Coulomb apantallado de corto alcance

r gtgt

r gtgtrArrinteraccioacutenrarrdeacutebil

Interaccioacuten e--ioacuten e--e-deacutebil

ioacuten e-F F

e- e-

rArr

mecaacutenica cuaacutenticararrU repulsivorArr iteraccioacuten Coulomb

U netorarr seudopotencialrarrdeacutebil (ltlt en metal Alcalinos)

ioacutenU

probabilidad gt

xxrsquo e-

e-

ioacuten



Interaccioacuten e--e- deacutebil

Razones1 Principio de exclusioacuten de Pauli d dgtgt

d dgtgt

Minimizar E si dltlt rArr e--e-rarr cerca rArrU gtgt

sistema e- menor E posible rArr

2 Principio de E miacutenima



e-hueco de Fermi (deficiencia de e-)

r

Matemaacuteticamente

r asymp1 Aring valor exacto = f(concentracioacuten e-)

E ne 0

mtoconjunto e-+hueco

e-

r

e-

r

e-

r e-

r

e- r

e-r

e-r

e-r

e- r 1 2

apantallan interaccioacuten entre 1 and2 rArr

Interaccioacuten ltlt



Gas de e- =gas ordinario

gas ordinario N asymp1025 moleacuteculasm-3

gas e- N asymp1029 e- m-3

e- rarr16x10-19 c rArrgas e- rarrplasma

gas ordinario q =0 rArr gas neutro

gas e-

libres en metalequiv plasma densogas e-

libres en metalequiv plasma denso



Conductividad eleacutectrica (modelo de Drude)

L

A

I = V R J = I A EL=V R = ρL A σ = 1 ρ

Relaciones macroscoacutepicas

J = σ E rarrley de Ohm

I V 2 V 1

V=V 2-V 1



σ un fenoacutemeno microscoacutepico

colisioacuten e-rarr resto del medio

qE F minus=τ

vmF friccioacuten

~ minus

τrarr tiempo de colisioacuten

vmeE

dt

dvm

minusminus=

mrarrmasa efectiva (interaccioacuten e-rarrred)

ecuacioacuten dinaacutemica para los e-

rArrI ne0E ne0rArr

vrarr velocidad del e-



Teoriacutea cineacutetica colisioacutenrarrevento instantaneo rArr

cambio abrupto de v e-

solucioacuten en estado estacionario dv dt=0

m

E ev

minus=rArr rarrv e- en estado estacionario

e- de deriva

E

V

vd equiv v de deriva

=



vd rarr superpuesta a vr (v aleatoria)

p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist

centros de dispersioacuten

I ne0

pero v neta ne0=v d rArr

dispersioacuten



Caacutelculo de J netaCaacutelculo de J neta

(-Ne-

)rarrcargaV moviendose con vd rArr J || vd

( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus

N (vddt )A equiv e- cruzando A perp flujo

en dt rarre- recorren ℓ =vddt en direccioacuten vd rArr

N e- vdAdt equiv carga cruzando A en dt

rArr

2

m

Ne

E J

τ

σ σ =rArr=



τ equiv t de relajacioacutenequiv t entre 2 colisiones sucesivas

τ gtgt rArr e- acelerado por E durante t gt rArrvd gt

τ equiv t de relajacioacuten E aplicado durante un t 0 rArr vd0

en t E =0 rArr τ vmeE

dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s



τ equiv t entre 2 colisiones sucesivas

τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ

ℓ equiv distancia entre 2 colisiones sucesivasℓ equiv camino libre medio



107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86

)( 11 minusminusΩ mσ )(

3minusm N )(sτ ) ( smV

F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K



Metales σ m asymp 5 x 107Ω-1m-1

SC σ SC asymp 1 Ω-1m-1

Solucioacuten

Metales N asymp 1029

m-3

rarr V r asympV Fasymp106

ms-1

SC N asymp 1020 m-3 rarrV r asymp104 ms-1

Magnitudes de V r y V dV r asympV Fasymp106 ms-1

rArr== 2 )2 3( 2vmT k E B

2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C

τ ~10-14 sm asymp10-30 kg

E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule

W disipadaasympW absorbida (e- del E )

W absorbida =Fv

2

2

)( E m

nem E eeE N NFvW d esistemaabs

τ τ =

minusminus==

lowastminus



τ hArr F friccioacuten

rArr ℓ gt20 a

Origen de τ

colisioacuten e--red

P e- darrExperimento

Asmsvl r 21614 101010 asymptimesasymp= minusminusτ



e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten

iones absorben E de la ondarArr radiaacuten nueva

direccioacuten intensidad no modificadas

red perioacutedica



red regular no dispersadifracta la onda(Rx n0 e-)

excepto cuando condicioacuten Bragg se cumple

red perioacutedica de iones no dispersa e-conduccioacuten



En una red cristalina perfecta

experimentalmenteperiodicidad perfecta de la red

983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155

rArr infinequivrArrinfin=rarrinfin=rArr σ l

deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=

Comportamiento lineal

equiv300 K

rarrexist 1014

colisiones s

equivτ

1probabilidad e- sea dispersadot

Na

f li i i f i d l d



e- rarrsufre colisioneshArr imperfecciones de la red

Desviaciones de la perfeccioacuten de la red cristalina Vibraciones de la red (fonones)

Imperfecciones estaacuteticas (impurezas defectoscristalinos )

τ τ τ impureza fonon

111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ

00 cteK T imp fonon fonon ==rArrrarrrArrinfinrarrrArrrarr ρ ρ

uarruarrrArr )(T T fonon ρ

)(T fonon ρ ρ asymp

equiv+= )(T fononimp ρ ρ ρ

T T fonon prop)( ρ



Expresiones para τ i τ ph

1Colisioacuten con impurezas e-rarr

seccioacuten eficaz de dispersioacuten equivaacuterea del aacutetomo

Teoriacutea cineacutetica de gases

r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida

reemplazando τ i en rArr



Caacutelculo de τ ph

asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion

2 xion π asympΩrArr

ionion

ph

N Ω

= 1

l

xrarrequilibrio

T k B seccioacuten transversal de dispersioacuten

Calc lo de 2



Calculo de

Ioacuten equivoscilador armoacutenico

Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h

E D or= E B E k θ =h

( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr

m equivM (masa del ioacuten)



T ltltExperimento RazoacutenMEinsteinrarrmto de iones vecinos

Solucioacuten modelo de Debyeexist desviaciones regla de MattiessenEj Efecto Kondo

Dispersioacuten adicional de e- por

( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ



desviaciones regla de Mattiessen

particularidades en la estructura de bandasde e- de conduccioacuten

rArr



Capacidad caloriacutefica de los e- de conduccioacuten

cumplen leyes claacutesicas

modelo e- libre rArr e- rarrpartiacuteculas libres

1 Mecaacutenica

2 Magnetismo3 M Estadiacutestica

Colisiones en el modelo e- libre hArrQMCapacidad caloriacutefica e- de conduccioacuten

rArr

Caacutel l d l id d l iacutefi



Modelo Drude-Lorentz

TCgases partiacutecula libre en equilibrio a T T k E B2

3gt=lt

RT T k N E B A2

3

2

3=

=

molK cal RT E C e 323 ][ asymp=partpart=minus

minus+=

e ph

C C C

molK cal R R RC T 954

2

33 asymp=+=gtgtrArr

Caacutelculo de la capacidad caloriacutefica

rArrmol



Experimento

Explicacioacuten QMrarrE e-rarr cuantizada

2

3 RC e ltminus

= minusminus RC e

2

310 2

molK cal RC 63 asympasymp

eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0

Mediciones exactas rArr

1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F



T gt 0 K rArr e- son excitados

E teacutermica absorbida x 1 e-~kBT asymp25 meVltltE F

rArr

E no compartida igualmente x todos e-

tratamiento claacutesico

ne

e- con E ltlt E Frarrno absorben E

a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(

excitados E F rarrminusthere4 e

1 )( gtgtminusrArrgtgt T k E E E E BF F

T k E T k E T k E B B BF Ceee E f

)( minusminus

==

distribu Maxwell-Boltzman

Evaluacioacuten de la E teacutermica



e- dentro de kBT de E Frarrexcitados

fraccioacuten e- afectados equiv (kBT E F)

rArr

Cada e- absorbe en promedio kBT

rArr e- excitadosmol= NAkBT E F

rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F

Si E F=5 eV T =300 KrArrkBT E F=1200

E F=kBT FrArr C e-=2RT T F

E F=5 eVrArrT Fsim60000 KE F=5 eVrArrT Fsim60000 K

rArr C e- =C e- claacutesicohArrT =T F

R RC T T T

T RC eF

F e 2

322 asymp=rArr=rArr= minusminus

C e-claacutesico=32RasympRrArr



Nota C e- sim T (funcion lineal)

C red sim cte (T gtgt)C red sim T 3 (T ltlt)

Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi

e- equiv partiacuteculas libres E = (12)mv 2

Espacio de velocidades ejes v x v y v z

v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten

superficie de Fermi (E F)

SF no afectadaT pocos e- excitados



SF no afectadaT pocos e excitados

SFrArridentidad permanente independiente

SFrArrcaracteriacutestica fiacutesica real del metal

( ) )(32

3 22

2

N f E N m

E F F asymprArr= π h

N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )

C d ti id d leacute t i



Conductividad eleacutectrica

v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no

compensadosrArr I ne0 rArr

Interaccion e--red rArrSF esfeacuterica

v d =- (e-τ m )E

0

v dltlt v r rArr desplazamiento ltlt

Estimacioacuten de J




v d =- (e-τ m )E

v d v F equiv fraccioacuten de e- no compensada

N (v d v F ) equiv concentracioacuten de e-

Cada e- rarrv F rArr J =- e- N (v d v F )(v F )=-N e- v d

J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)

Modelo real de conduccioacuten eleacutectrica




Todos e-rArrmueven con v d (ltlt)

I rArrpocos e-

con v asympv F (gtgt)

2 aproximacionsrArrigual resultad

σ = N e-2 τ F m )

(1)

(2)

(2)rArr+ precisa conceptual

τ F=l F v F σ = N e-2 l F (T ) m v F

Claacutesica

Cuantica

Metal (300 K) l F asymp100 Aring

Relevancia de FS en fenoacutemenos de transporte



a a s a sp

I rArre- cercanos a FS

Transporte e-hArrpropiedades forma de FS

Modelo claacutesico y cuaacutentico rArrresultado similar

e- internosrArrirrelevantes

rArrprocedimiento claacutesicorArruacutetil

(τ = τ F)

Conductividad teacutermica en metales



Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=

K=K e-+K phMetales K phasymp10-2K e-

I e-=I e-

T 2 T 1EgtE E netarArrderecha

J netane0

(n e-gtgt)

Evaluacioacuten de K



vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k



Reemplazando valores K asymp50 calmks

σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π

σ Nuacutemero de Lorenz

L= 58x10-9 calΩ sK2

ElementoElementoElementoElemento NaNaNaNa CuCuCuCu AgAgAgAg AuAuAuAu AlAlAlAl CdCdCdCd NiNiNiNi FeFeFeFe

K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55

Igual forall metales

Di i l l d L



Discrepancia en el valor de L

1 Uso de un modelo simplificado (e- libre)

2 Tramiento simplificado para calcular σ y K

Tratamiento refinado L=L(metal)

Movimiento de e- de un metal en B



e-

e-

ω

α

ω ωω ω C= e- B m ν C = ω ωω ω C 2π =28xB GHz

m equiv m e- libre

1 Resonancia ciclotroacutenica

[B]= kG

B

Si B= 1kGrArrν C = 28 GHz equiv Microondas

αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m

e- absorben energiacutea de E

ω ωω ω = ω ωω ω C

ω

Maacutexima

hArr

e- mueven sincronizados con oem

e- absorben durante todo el cicloe- resonancia ciclotroacutenicaω ωω ω = ω ωω ω C

ne



ω ωω ω neω ωω ω C e- absorben durante parte del ciclo

resto del ciclo e- fuera de fase

Energiacutea retorna a la oem

ω ωω ω Cτ gtgt1α

ω

bien definido hArr

e- hace varios ciclos en τ ωc

ω ωω ω Cτ asymp 1



ω ωω ω C no es uacutenica

ω ωω ω Cτ gtgt1Experimentalmente

ω ωω ω C uarrrArr B ~5 Tτ uarrrArr T ~5 K

R ciclotroacutenicarArrmedicioacuten de me- en metales y SC

ω ωω ω C= e- B m Precisioacuten de m =f(precisioacuten de ω ωω ω C y B)




2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and

F L= q (vxB ) genera defleccioacuten de e-

[F L]= e- v xB direccioacuten -y

[F H]= e- E

H

direccioacuten +y

Estado estacionarioFH=FL e- EH= e- v B



Estado estacionarioF H= F L e E H= e v xB

E H= v xB equiv Campo Hall

En teacuterminos de cantidades medibles J x= N(-e-)v x E H= -(1Ne-)J xB

E H J xB= -(1Ne-)=RH equiv constante Hall

RH de metales comunes



ElementoElementoElementoElemento LiLiLiLi NaNaNaNa CuCuCuCu AgAgAgAg AuAuAuAu ZnZnZnZn CdCdCdCd AlAlAlAl

RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03

R ( ) conduccioacuten por huecos



RH (+)rArrconduccioacuten por huecos

J x = J x ( B)rArr R = R ( B) hArr metales comunes

metales de transicioacuten R = R ( B)

Efecto de F L cancelado por F H rArre- fluyen horizontalf(B)

MRne0

Emisioacuten termoioacutenica



E F

φ

vacio

metal

e-

Funcioacuten trabaj

x

E termoioacutenica choques con partiacuteculas vecinas

E fotoeleacutectrico choques con fotones

n

W

W

Funciones trabajo de algunos metales



ElementoElementoElementoElemento WWWW TaTaTaTa NiNiNiNi AgAgAgAg CsCsCsCs PtPtPtPt

φ [eV] 45 42 46 48 18 53


E de traslacioacuten teacutermica media= T k B2

3



ltlt desprender 1 e-eV K E 21073)300( minustimes=

auacuten al rojo incandescente

E

f (E)

a cada T exist e- con E ~1 eV

rArr e- viajando desde dentro acercandose a lasuperficie pueden alcanzar φ

2

T uarrrArrfraccioacuten de e- en disposicioacuten de salir



T uarrrArrfraccioacuten de e en disposicioacuten de salir(en equilibrio) equiv T k Ben

φ minusprop uarr

rArr Ej que fraccioacuten de e- de una peliacutecula de Cssobre W estaacuten en condicioacuten de abandonar lasuperficie a 300 y 1000 K

φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman

en condicioacuten de no equilibrio teacutermico



A

C

U

j

j

U

T 1

T 2gt T 1

rArrpozo infinito cuadrado paradescribir e- en metales

φ =minus F vacio E E

rArrpozo de potencialrarraltura de barrera finita

e- con P grandeperpsuperficie



pueden abandonar el metal rArr

contribuyen a j s

Evaluacioacuten de J x(T )

d nev J =Si v d es homogenea rArr

generalizando al caso v d = v d (k )

( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π

Modelo del e - libre

densidad de estados en k rarrV (2π)3



( ) ])([

2

2

min3

T k E f k dk dk dk m

e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados

x dk k v )(estados ocupados

rArr

rArr F Fermi-Dirac

x x k mv h=

rArrgtgt T k Bφ F Fermi-Diracrarr E Maxwell-Boltzman

rArr

T k E T k E B BF ee E f

)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus

minusinfininfinminus

minus 222 2 y B y k y

T mk k y edk edk h

T mk B2

2h

=α

Liacutemites de integracioacutenv yv z (-infininfin)

[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min

22 φ minusinfin minus =int h

h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h

Ecuacioacuten de Richardson-Dushman

3

24

h

Bmek A

π = A=120 Acm2K2factor universal



h

m

k E x

F

2

)(22

hle+φ e- con rarr100 P escapar del soacutelido

auacuten para el gas de e- libres

φe-

MQ

E= φ rArr P transmisioacuten = 0

( ) 2 1) ( φ π +F B E T k rArr Efecto del escaloacuten introducir

J s darr

suposicioacuten para el caacutelculosobrela superficie



expuarrrArrgtgt x B J T k φ T ltT fusion

200 400 600 800 100012000

5x107

1x108

2x108 D ExpGro1 fit of Data1_D

J s [ A

c m

2 ]

T [K]

Fallas del modelo del e - libre



RH (+) Be Zn Cd

σ = N e-2 τ F m )rArrσ ~N (1)

(2)

(3) FSrArrno esfeacuterica

Solucioacuten teoriacutea mas sofisticada

Interaccioacuten e--red



Metales

Propiedades fiacutesicas comunes

bull dureza fiacutesica gtgtbull σ and K gtgtbull densidad gtgt

bull reflectividad oacuteptica gtgt



Na (gas)

Nanuacutecleo

11 e-

1s22s22p63S1

aacutetomos libres


r ~19 Aring



core



core















e-


rArr





rArr



Gas de e- libres


U vaciacuteometal

φ







alternativamente

e-

vdarr


r gtgt



ioacuten e-F F

e- e-

rArr



ioacutenU

probabilidad gt

xxrsquo e-

e-

ioacuten





d dgtgt







r



E ne 0


e-

r

e-

r

e-

r e-

r

e- r

e-r

e-r

e-r

e- r 1 2










gas e-






L

A




I V 2 V 1

V=V 2-V 1





qE F minus=τ

vmF friccioacuten

~ minus


vmeE

dt

dvm

minusminus=



rArrI ne0E ne0rArr







m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






Na (gas)

Nanuacutecleo

11 e-

1s22s22p63S1

aacutetomos libres


r ~19 Aring



core



core















e-


rArr





rArr



Gas de e- libres


U vaciacuteometal

φ







alternativamente

e-

vdarr


r gtgt



ioacuten e-F F

e- e-

rArr



ioacutenU

probabilidad gt

xxrsquo e-

e-

ioacuten





d dgtgt







r



E ne 0


e-

r

e-

r

e-

r e-

r

e- r

e-r

e-r

e-r

e- r 1 2










gas e-






L

A




I V 2 V 1

V=V 2-V 1





qE F minus=τ

vmF friccioacuten

~ minus


vmeE

dt

dvm

minusminus=



rArrI ne0E ne0rArr







m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






core















e-


rArr





rArr



Gas de e- libres


U vaciacuteometal

φ







alternativamente

e-

vdarr


r gtgt



ioacuten e-F F

e- e-

rArr



ioacutenU

probabilidad gt

xxrsquo e-

e-

ioacuten





d dgtgt







r



E ne 0


e-

r

e-

r

e-

r e-

r

e- r

e-r

e-r

e-r

e- r 1 2










gas e-






L

A




I V 2 V 1

V=V 2-V 1





qE F minus=τ

vmF friccioacuten

~ minus


vmeE

dt

dvm

minusminus=



rArrI ne0E ne0rArr







m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)











e-


rArr





rArr



Gas de e- libres


U vaciacuteometal

φ







alternativamente

e-

vdarr


r gtgt



ioacuten e-F F

e- e-

rArr



ioacutenU

probabilidad gt

xxrsquo e-

e-

ioacuten





d dgtgt







r



E ne 0


e-

r

e-

r

e-

r e-

r

e- r

e-r

e-r

e-r

e- r 1 2










gas e-






L

A




I V 2 V 1

V=V 2-V 1





qE F minus=τ

vmF friccioacuten

~ minus


vmeE

dt

dvm

minusminus=



rArrI ne0E ne0rArr







m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






Gas de e- libres


U vaciacuteometal

φ







alternativamente

e-

vdarr


r gtgt



ioacuten e-F F

e- e-

rArr



ioacutenU

probabilidad gt

xxrsquo e-

e-

ioacuten





d dgtgt







r



E ne 0


e-

r

e-

r

e-

r e-

r

e- r

e-r

e-r

e-r

e- r 1 2










gas e-






L

A




I V 2 V 1

V=V 2-V 1





qE F minus=τ

vmF friccioacuten

~ minus


vmeE

dt

dvm

minusminus=



rArrI ne0E ne0rArr







m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







alternativamente

e-

vdarr


r gtgt



ioacuten e-F F

e- e-

rArr



ioacutenU

probabilidad gt

xxrsquo e-

e-

ioacuten





d dgtgt







r



E ne 0


e-

r

e-

r

e-

r e-

r

e- r

e-r

e-r

e-r

e- r 1 2










gas e-






L

A




I V 2 V 1

V=V 2-V 1





qE F minus=τ

vmF friccioacuten

~ minus


vmeE

dt

dvm

minusminus=



rArrI ne0E ne0rArr







m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)








d dgtgt







r



E ne 0


e-

r

e-

r

e-

r e-

r

e- r

e-r

e-r

e-r

e- r 1 2










gas e-






L

A




I V 2 V 1

V=V 2-V 1





qE F minus=τ

vmF friccioacuten

~ minus


vmeE

dt

dvm

minusminus=



rArrI ne0E ne0rArr







m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







r



E ne 0


e-

r

e-

r

e-

r e-

r

e- r

e-r

e-r

e-r

e- r 1 2










gas e-






L

A




I V 2 V 1

V=V 2-V 1





qE F minus=τ

vmF friccioacuten

~ minus


vmeE

dt

dvm

minusminus=



rArrI ne0E ne0rArr







m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)











gas e-






L

A




I V 2 V 1

V=V 2-V 1





qE F minus=τ

vmF friccioacuten

~ minus


vmeE

dt

dvm

minusminus=



rArrI ne0E ne0rArr







m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







L

A




I V 2 V 1

V=V 2-V 1





qE F minus=τ

vmF friccioacuten

~ minus


vmeE

dt

dvm

minusminus=



rArrI ne0E ne0rArr







m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)








qE F minus=τ

vmF friccioacuten

~ minus


vmeE

dt

dvm

minusminus=



rArrI ne0E ne0rArr







m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)









m

E ev


e- de deriva

E

V


=




p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







p

E =kBT vr

E =0 I =0vr gtgt vd

E ne0rarrvr exist


I ne0


dispersioacuten




(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







(-Ne-


( ) ( ) E m

e N

m

E e

Nev Ne J d

2

τ τ

=

minus

minus=minus=

minusminus




rArr

2

m

Ne

E J

τ

σ σ =rArr=







dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)










dt dvm minusminus=

τ

vm

dt

dvm minus=rArr

τ

0)(

t

d d evt v

minus

=

vdvd0

t

τsim 10-14 s




τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







τ = ℓ vr

rArr

r vm

Ne

2l

=rArrσ




107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






107x107 46 X1028 09 x10-14 13 x 106 110 47 37 12

211 25 31 11 350 31 25 12

139 13 43 085 370 21 19 11

08 11 275 08 220 18

05 085 075 160 15

588 845 27 16 420 7 7 1

621 585 41 14 570 55

455 59 29 14 410 55169 131 182 94 11 085

138 928 162 75

01

365 1806 202 116 118067 153 191 103

114 115 174 86



F )( Al )(ev E

F vF E obs E )(0

m

mm =

LiNa

K

Rb

Cs

Cu

Ag

AuZnCd

Hg

AlGa

In

~300 K





Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)








Solucioacuten


m-3


ms-1




2 1

3

=m

T k v B

r

lowastminus=m

E evd τ e- ~10-19 C


E ~10 Vm

rArrvd ~10-2 ms

vd vr ~10-8 ms

m asympm 0 5

SC



Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






Efecto Joule


W absorbida =Fv

2

2

)( E m


τ τ =

minusminus==

lowastminus




rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







rArr ℓ gt20 a

Origen de τ






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






e- ondas de materia

M cuaacutenticar

e

vm

hlowast=minusλ

dispersioacuten



red perioacutedica










983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)













983139983137983157983155983137983155983098 983158983145983138983154983137983139983145983283983150 983156983273983154983149983145983139983137983084

983145983149983152983141983154983142983141983139983139983145983151983150983141983155 983145983149983152983157983154983141983162983137983155


deg

Α210~l



s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






s14

10minus

asymprArrτ

Na

τ

ρ 1

2

Ne

m=


equiv300 K

rarrexist 1014

colisiones s

equivτ


Na

f li i i f i d l d







111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)










111+=

)(1

T f

fonon

=τ

)(11

2

2

T

fonon

Ne

m

imp

Ne

m fononimp ρ ρ ρ

τ τ +=+=rArr

)(1

T f

impu

=

τ

Mecanismos

independientes

rArr



equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






equivimp ρ

equiv fonon ρ





T T fonon prop)( ρ







r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)










r

ii

v

l=rArr τ

equivΩrArr i

ii

i N Ω

= 1

l

1=Ω iii N l

r ii

iiv N Ω

=rArrΑΩ 1

1~ 2 τ

ii N ~ ρ

i

i

i Ne

m

τ ρ 1

2

=

esfera

riacutegida





asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







asymp 983265983154983141983137 983140983141983148 983145983283983150

r

ph

phν

τ l=

1=Ω ionion ph N l

ionΩ

rarrbullminuse uarr

darr

Ωion


ionion

ph

N Ω

= 1

l

xrarrequilibrio


Calc lo de 2



Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






Calculo de


Introduciendo

sustituyendo

2 x

12

1 2

minus

=Ε =rArrT k Be

xω

α h

h


( )1

1 minusT ph

E eT

θ α ρ

( ) ( ) T T T

T T ph

E

ph E ~~ ρ θ

ρ θ rArrrArrgtgt

phion

phe N

m

τ ρ

12

= rArr







( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)









( ) T

E

eT ph

θ

ρ

minus

~~

5T T darr

Fe

FeimpCu +

Independiente

ρ ρ





rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)








rArr






1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)









1 Mecaacutenica



rArr






3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)








3gt=lt

RT T k N E B A2

3

2

3=

=


minus+=

e ph

C C C


2

33 asymp=+=gtgtrArr


rArrmol



Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






Experimento


2

3 RC e ltminus

= minusminus RC e

2

310 2


eV E F 5~

equiv

gt

lt==

F

F

T E E

E E E f

0

1)( 0


1

P E Pauli

T =0

prob el nivel con E

ocupado por 1 e-

E F

T =0 K

T ne0 K

f(E )

E

E F





rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)








rArr



ne


a 300 K



proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






proacuteximos a

DiracFermi De

E f T k E E BF

minusequiv+

= minus 1

1)(

)(




)( minusminus

==







rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)








rArr



rArr

F

B A

E

T k N E

2_ )(=rArr


kBT

E F

f(E )

E

k2_

part



F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






F

B

e

E

T Rk

T

E C

2=

part

part=minus

C e-darr en kBT E F





R RC T T T

T RC eF

F e 2







Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)








Caacutelculo exacto

F

B

e E

T Rk C

2

2π asympminus

Superficie de Fermi



Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






Superficie de Fermi



v x

v y

v F

v

esfera de FermiT =0

E F = (12)m v F2

V uacutenica

V direccioacuten








( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)









( ) )(32

3 22

2

N f E N m


N ~1023

cm3

rArrE Fasymp5 eV

16

2 1

2

102 minusasymp

= ms

m

E v F

F

h SF vF=f(T )





v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







v x

v

v xv

v xy

v yE=0 E ne 0

I =0

e- no



v d =- (e-τ m )E

0






v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







v d =- (e-τ m )E




J =(N e-2 τ F m)E

σ = N e-2 τ F m )σ = N e-2

τ F m

)






I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)








I rArrpocos e-



σ = N e-2 τ F m )

(1)

(2)



Claacutesica

Cuantica





a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






a a s a sp






(τ = τ F)




Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






Experimento

E F

T 2 T 1T 2 gtT 1

dx

dT K J Q minus=


I e-=I e-


J netane0

(n e-gtgt)




vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






vlC K v3

1

=

F F

F

B

v E

T Nk

K l

=

22

23

1 π

F

Bve E

T Rk C 2

2

π =minus

R=NkB

v rarrv Fl rarr l F

τ F=l F v F

E F = (12)mv F2

F B T Nk m

K τ π 2

2

3

1=

Por unidad de V

R l d l K 50 l k




σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







σ = N e-2 τ F m )

Le

k

T

K B equiv

=2

3

1 π




K [calmKs] 33 94 100 71 50 24 14 16

L[calΩ sK2] x10-9 58 54 56 59 47 63 37 55


Di i l l d L










e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)













e-

e-

ω

α


m equiv m e- libre


[B]= kG

B


αequivcoef Absor

F = -e(v xB ) E =0

ωc




e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






e-

e-

B

Sentildeal e m



ω

Maacutexima

hArr



ne







ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)










ω

bien definido hArr













2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)














2 Efecto Hall

x

+ + + + + +

- - - - - - - -

Bz

z

y

e-

e- J x

E Hv

Bz

E H

J x and



[F H]= e- E

H

direccioacuten +y












RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)














RH (RT)x10-10

[Vm3 AxWeber]-17 -25 -055 -084 -072 +03 +06 -03








MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)










MRne0




E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






E F

φ

vacio

metal

e-

Funcioacuten trabaj

x



n

W

W





φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







φ [eV] 45 42 46 48 18 53



3





E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)








E

f (E)



2





φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







φ minusprop uarr


φ W =136 eV = 22x10-19 J

)1000(1041

)300(1014

20

21

K eV T k

K eV T k

B

B

minus

minus

times=

times=

)1000(1051

)300(105

7

24

K e

K e

T k

T k

B

B

minusminus

minusminus

times=

times=φ

φ

rArr

Factor de Boltzman




A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






A

C

U

j

j

U

T 1

T 2gt T 1








contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







contribuyen a j s




( ) intsumgt

+gt

==

0)(

3 )(

2)(

k v

E E

x

k

x x

x

F

dk k ve

k vV

e J

φ π





( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






( ) ])([

2

2

min3


e J

xk x x z y x intint

infininfin

infinminus=

π

h

T k E

k

T mk k

x x

T mk k

z

T mk k

y x

BF

x

B x

B z B y

eek dk

edk edk m

e J

2

2 2

3

min

22

2222

4

int

intintinfin minus

infin

infinminus

minusinfin

infinminus

minus

times

=

h

hhh

π

intsum and x

ocupadosestados


rArr

rArr F Fermi-Dirac

x x k mv h=


rArr


)( minus=

IIIe

J timestimes=h



32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






32134 I I I

m J x timestimes=

π

T k E

k

T mk k

x x BF

x

B x eek dk I 2

3min

22

intinfin minus= h

21 I I =

α π α == intint

infininfinminus



T mk k y edk edk h

T mk B2

2h

=α


[(12) 2 ge E φ]



T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






T k E

m E

T mk k

x

T k E

k

T mk k

x x

BF

F

B x BF

x

B x

eedk eek dk

] 2)[(

2 2 2

2 12

22

min

22

2

1

intint

infin

+

minusinfin minus

= h

hh

φ

g

v x[(12)mv 2 ge E

F+φ ]

m

k

m

p E x

F 22

)(222

hlele+φ

rArr

rArr T k BT k E

k

T mk k

x B BF

x

B x eT mk

eedk

2

2

min


h

( ) T k

Bs BeT k

me J

2

3

4 φ π minus=h


3

24

h

Bmek A




h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






h

m

k E x

F

2

)(22



φe-

MQ



J s darr





200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)







200 400 600 800 100012000

5x107

1x108


J s [ A

c m

2 ]

T [K]




RH (+) Be Zn Cd


(2)






RH (+) Be Zn Cd


(2)




02 Teoría Del Electrón Libre en Sólidos

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Transcript of 02 Teoría Del Electrón Libre en Sólidos