TEMARIO DEL CURSO DE FUNDAMENTOS DE FÍSICA DE …yolotli.inaoep.mx/portalfiles/diodoparte2.pdf ·...

Curso propedéutico de Electrónica INAOE 2009 Dr. Pedro Rosales Quintero 41

1. Introducción a Física Electrónica

1.1 Propiedades de cristales y crecimiento de semiconductores

1.2 Átomos y electrones

1.3 Bandas de energía y portadores de carga en semiconductores

1.4 Exceso de portadores en semiconductores

2. Uniones

2.1 Fabricación de uniones p-n

2.2 Condiciones de equilibrio

2.3 Polarización de uniones en directa e inversa bajo condiciones de estado estacionario

2.4 Ruptura bajo polarización inversa

2.5 Condiciones de transitorio y a-c

2.6 Desviaciones de la teoría sencilla

2.7 Uniones metal-semiconductor

TEMARIO DEL CURSO DE FUNDAMENTOS DE FÍSICA DE SEMICONDUCTORES


Forward and Reverse Biased Junctions

The p-n junctions are used extensively in rectification i.e. allowing current to flow easily in only one direction.

Forward biasing(positive voltage on the p-side of the junction)

Reverse biased(positive voltage on the n-side of the junction) I (generation)

I (diffusion)


Ideal model

The ideal diode: (a) diode circuit symbol; (b) i-v characteristic; (c) equivalent circuit in the reverse direction; (d) equivalent circuit in the forward direction.


We assume that an applied voltage bias Vappears across the transition region (space charge region) of the junction rather than in the neutral n and p regions.¿?

Since an applied voltage changes the electrostatic potential barrier and thus the electric field within the transition region, we would expect changes in the various components of current at the junction.

Qualitative description of Current Flow at a Junction


Thermal equilibrium

The current at V=0 (equilibrium) is zero since the generation (drift) and diffusion currents cancel.

p

EC

EV

EF

n

qVo

-xp0 xn0

( ) 0)( =−= driftIdiffusionII

-

-

+

diffusion

diffusion+ drift

drift


Forward bias

p

EC

EV

EFn

The total electrostatic potential across the junction decreases by VF

nIF

+ -

q(V0-VF)

VF

Width of the depletion region decreases.

qVF

-xp xn

EFp

-

-

+

diffusion

diffusion+ drift

drift


The total current I is the diffusion current minus the absolute value of generation current, which we will now refer to as I0:

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎥

⎦

⎤⎢⎣

⎡=⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎥⎦

⎤⎢⎣⎡= 1exp1exp 00

thv

VI

kT

qVII

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡=

kT

qVII exp0

When V is positive and greater than a few kT/q (V>>3Vth), the exponential term is greater than the unity:

0II −=

When V is negative , the exponential term approaches to zero:

This negative generation current is also called the reverse saturation current


We expect the minority carrier concentration on each side of the junction vary with the applied bias because of variations in the diffusion of carrier across the junction. The equilibrium ratio of holes concentrations on each side

[ ]kTqVp

p

n

p0exp=

Then under bias V, ( )( ) ( )[ ]kTVVqxp

xp

n

p −=−

00

0 exp

Carrier Injection

If we assume low-level injection of mobile carriers, that is the majority carrier densities are essentially unchanged under bias,

So taking the ratio of the previous two equations we get,

( ) pp pxp =− 0

( ) [ ]kTqVp

xp

n

n exp0 =


As we can see, the minority carrier concentrations at the edge of the transition region are raised by the forward bias, The excess holes injected across the depletion region into the n side is given by,

( ) [ ]( )1exp0 −=−=∆ kTqVppxpp nnnn

Similarity for excess electrons injected on the p side

( ) [ ]( )1exp0 −=−−=∆ kTqVnnxnn pppp


ppp

pp

DL

L

p

x

pp

x

pD

t

p

p

τ

δδτδδδ

=

=−∂∂

=−∂∂

=∂∂

;022

2

2

2

Boundary conditions are,

( ) ( ) ( ) nnn pxpandppxp =∞→∆=== δδ 00

Solution of δp(x) is,

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡−⎟⎠⎞

⎜⎝⎛=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−∆=

p

nn

p

nn L

x

kT

qVp

L

xpxp exp1expexpδ

Minority carrier density decays with a characteristic length given by Lp

From the continuity equation in the n region


nnn

nn

DL

L

n

x

nn

x

nD

t

n

n

τ

δδτδδδ

=

=−∂∂

=−∂∂

=∂∂

;022

2

2

2

Boundary conditions are,

( ) ( ) ( ) ppp nxnandnnxn =∞→∆=== δδ 00

Solution of δn(x) is,

( ) ⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛=⎥

⎦

⎤⎢⎣

⎡−∆=

n

pp

n

pn L

x

kT

qVn

L

xpxn exp1expexpδ

Minority carrier density decays with a characteristic length given by Ln

From the continuity equation in the p region we can get:


Current flowSince we assume that the drift current is unaffected by the applied bias, we are now in a position to consider the net current by considering only the diffusion of injected holes into n side,

( ) ( )( ) ( )( )xpL

DqA

dx

xpdqADxI

p

p

npnp δδ

=−=

Where A is the cross sectional area of the junction. The total hole current injected into the n-side is proportional to the excess hole density at a particular point. The total current injected into the n-side can be obtained by evaluating the previous equa. at x=xn0,

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎥⎦

⎤⎢⎣⎡== 1exp0

kT

qVp

L

ADqXI n

p

pnp


By similar analysis, the injection of the electrons in the p side leads to an electron current at the junction

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎥⎦

⎤⎢⎣⎡−== 1exp0

kT

qVn

L

ADqXI p

n

npn

( ) ( )pnnp xIxII −+=

⎥⎥⎦

⎤

⎢⎢⎣

⎡+=

n

pn

p

nps L

nD

L

pDqAI 00

( )1−= kTeVs eII

Saturation current density

So, the total current into the diode is given by:


There is another way to calculating the total current into a diode. This is called charge control approximation.


0II −=

When V is negative, the exponential term approaches to zero:

This means that I0 is constant, but this is the ideal case.







2. Uniones










Reverse Bias BreakdownWhen a sufficiently large reverse bias is applied to a p-n junction, it breaks down and conducts a very large current. There are 2 mechanisms by which this can occur…

The tunneling effect

This is the basis for the tunnel diode and one mechanism for storing and erasing charges in non-volatile semiconductor memory (NVSM) devices

Avalanche multiplication

This effect imposes an upper limit on the reverse bias in most devices. It limits the collector voltage of a bipolar transistor and the drain voltage of a MOSFET.


The Tunnelling Effect

When a high electric field is applied to a p-n junction in the reverse direction, an electron can tunnel through the bandgap from the valence band to the conduction band.

EFn

EFp

<4Eg

EC

EV

EC

EV

E

Tunneling is the main breakdown mechanism for Si and GaAs junctions where the breakdown voltage is less than 4Eg/q

The tunnelling of electrons from the p side valence band to n side conduction band constitutes a reverse current from n to p, this is called Zener effect.


Avalanche multiplication

Junction breakdown at reverse biases in excess of 6Eg / q is dominated by

avalanche multiplication…

EFn

EFp

>6Eg

EC

EV

EC

EV

EA thermally generated electrons in the depletion region gains kinetic energy from the electric field

On collision with a lattice atom it breaks the bond and generates an electron-hole pair

These in turn acquire kinetic energy from the electric field and generate more electron-hole

pairs…and so it continues.


Critical field at breakdown versus background doping for Si and GaAs one-sided abrupt junctions.








2. Uniones










Transient and a-c ConditionsWe have considered the properties of p-n junctions under equilibrium and with steady state current. Most of the basics concepts of junction devices can be obtained from these properties, except for the important behavior of junctions under transient or ac conditions.

Since the most solid state devices are used for switching or for processing a-c signals.

We investigate the important influence of excess carries in transient and ac problems, i.e. the switching of a diode from its forward state to its reverse state is analyzed to illustrate a typical transient problem


Time Variation of Stored ChargeFor a proper solution of the transient problem, we must to use the time dependent continuity equations. We can obtain each component of the current at position xand time t from these equations, we can write:

To obtain the instantaneous density current, we can integrate both sides at time t to obtain:

For injection into a long n region from a p+ region, we can take the current at xn=0 to be all hole current, and Jp at xn=∞ to be zero. Then the total injected current, including time variations, is

t

txpq

txpq

x

txJ

p

p

∂∂

+=∂

∂−

),(),(),(

τδ

dxt

txpq

txpqxJJ

x

ppp ∫

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+=−0

),(),()()0(

τδ

nnnnp

np dxtxpt

qAdxtxpqA

txiti ∫∫∞∞

∂∂

+===00

),(),(),0()( δτ


dt

tdQtQti p

p

p )()()( +=

τThis results indicates that the hole current injected across the p+njunction is determined by two charge storage effects:

1. The usual recombination term in which the excess carrier distribution is replaced every τp seconds.

2. A charge buildup( or depletion )term.

We can solve for the stored charge as a function of time for a given current transient. For example the step turn–off transient.


⎥⎥⎦

⎤

⎢⎢⎣

⎡−=

+=

−+=

ppp

p

pp

ppp

p

tItQ

s

ISQ

ISSQSQ

ττ

τ

τ

ττ

exp)(

1)(

)()(

0

Solving the previous equation with Laplace transforms, with i(t > 0) = 0 and Qp(0)=Iτp , we obtain

As expected, the stored charge dies out exponentially from its initial value Iτpwith a time constant equal to the hole lifetime in the n material


The previous analysis does indicate clearly that the voltage across a p-n junction cannot be changed instantaneously, and that stored charge can present a problem in a diode intended for switch application.

Thus in the quasi-steady state approximation, the junction voltage varies according to

The switching process can be made still faster by purposely adding recombination centers, such as Au atoms in Silicon, to increase the recombination rate

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟⎠

⎞⎜⎜⎝

⎛−= 1expln)(

pnp

p t

pqAL

I

q

kTtv

ττ


Reverse Recovery Transient

In most switching applications a diode is switched from forward conduction to reverse biased state. The resulting stored charge transient is somewhat more complicated than for a simple turn-off transient.

Lets us to assume a p+-n junction is driven by a square wave generator that periodically switches from +E to –E. While E is positive the diode is forward biased, and in steady state the current If flows through the junction. If E is much larger than the small forward voltage of the junction, the source voltage appears almost entirely across de resistor (R), the positive i = If ≈ E/R. After the voltage generator is reversed (t > 0), the current must be initially reverse to i = Ir≈ -E/R.


The reason for this unusually large reverse current through the diode is that the stored charge.

The time (tsd) required for the stored charge to became zero is called the storage delay time. The critical parameter determining tsd is the carrier timelife (τp)

2

1

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

+= −

rf

fpsd II

Ierft τ


Capacitance of p-n junctionsThere are two types of capacitance associated with a junction:

1. The junction capacitance due to dipole in the transition region. (depletion capacitance).

2. The charge storage capacitance arising from the lagging behind of voltage as current changes. (Diffusion capacitance).

The junction depletion layer capacitance per unit area is defined as Cj=dQ/dV, where dQ is the incremental change in the depletion layer charge per unit area for an incremental change in the applied voltage dV. The depletion capacitance per unit area is given by [ ]2/ cmF

WdQW

dQ

dV

dQCj s

s

ε

ε

===


Thermal equilibrium

p

EC

EV

EF

What happens when we apply and external reverse bias across the junction?

n

qV0

-Wp Wn


Reverse bias

p

EC

EV

EF

The total electrostatic potential across the junction increases by VR

nIR

- +

q(V0+VR)

VR

qVR

-Wp Wn

EFn

EFp


Then, the width of the depletion region increases.

The depletion capacitance per unit area is the same as the standard expression for a parallel plate capacitor where the spacing between of the two plates represents the depletion layer width. In the case for a one-side abrupt junction (p+-n), we obtain

The quantity Cj is a voltage-variable capacitance, since Cjis proportional to (V0-V)1/2. The p-n junction device which makes use of the voltage variable properties of Cj is called a varactor.

[ ]VV

Nq

WCj dss

−==

02

εε

[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

da

das

NN

NN

q

VVW 02ε


The diffusion capacitanceAs a p-n diode is forward biased, the minority carrier distribution in the quasi-neutral region increases dramatically. In addition, to preserve quasi-neutrality, the majority carrier density increases by the same amount. This effect leads to an additional capacitance called the diffusion capacitance. The excess minority-carrier charge is obtained by integrating the charge density over the quasi-neutral region. In the case of a p+ - n junction :

[ ]

p

th

pond

pd

Wn

Xn

nnp

IkT

q

Vth

V

V

nAqLC

dV

QdC

dxppeAQ

τ=

⎥⎦⎤

⎢⎣⎡==

∆=

−=∆ ∫

exp

0







2. Uniones










Under some range of forward bias, We could be described by the ideal diode equation.

However, for the real p-n junctions I is given by:

Recombination and Generation in

Transition Region

( ) kTqVkTqV eIeII 00 1 ≈−=

( ) ( )]1´[

11 200

−≈

−+−=

+=

kTqV

kTqVG

kTqVD

GD

eII

eIeII

III

η

Where n varies between 1 and 2, depending of material and temperature. Since n determines the departure from the ideal diode characteristic, it is often called the ideality factor


Reverse Bias Current Density

Under reverse bias (V is negative), the current goes towards the saturation current density (I0). As we can see from the ideal diode equation:

However, for Silicon and GaAs p-n junctions under reverse bias IR

≠ I0 !, and IR >> I0 ! Then, ???

( )10 −= kTqVeII


Reverse Bias Generation Current Density

Under reverse bias, the generation current (IG) becomes to be very important when the depletion region is devoid of carries. Under thermal equilibrium the thermal generation of e-h pairs is balanced by recombination (G = R). Under reverse bias, there are very few e-h pairs in the depletion region, and the recombination rate is nearly zero. However, the thermal generation processes continuously supply the e-h pairs to depletion region. The IG can be determined from:

Then, IR = IG + I0 , and IG > I0 for Silicon and GaAs p-n

junctions under reverse bias

g

iW

g

iW

G

WqAndx

qAndxqAGI

ττ 2200

=== ∫∫



For higher forward bias voltages, the current no longer increases

exponentially with voltage. Instead, it increases linearly due to the

series resistance of the diode. This series resistance can be due to the

contact resistance between the metal and the semiconductor, due to the

resistivity of the semiconductor or due to the series resistance of the

connecting wires.

[ ])()( IRIRIVaV np +−=

I

+ V

-

+ Vj

-

RS

ID

- VR +

Ohmic Losses


Small signal equivalent circuitof a p-n junction

V’d

Id

-I 0 V’+∆ Vd d

Vd

I’d

I’+∆ Id d

gd

V+ _d

Id

gd

Rs

Cd

thd nV/'V

thd

dd e

nV

I

V

Ig 0

opd

d==ddd V'VV ∆+=


Graded junctionsThe case of linearity graded junction is shown in the next figure.


The Poisson equation for the case is

222

2 Wx

Wax

q

dx

dE

dx

d

ss

s ≤≤−−=−=−=εε

ρψ

Where G is the impurity gradient (in cm -4) and W is the depletion–layer width. We obtain the built-in potential (Vbi) and the depletion layer width (W ) after some “simple steps”.

sbi

qaWV

ε12

2

=

3

1

12⎥⎦

⎤⎢⎣

⎡=

qa

VW bisε







2. Uniones










Metal-Semiconductor Junction (Schottky barrier)

Metal n-Semiconductor

EV

EF

EC

Vacuum level

mqφ

( )χφ −mq

χqsqφ

What happens when we contact the metal and semiconductor in thermal equilibrium?


Metal n-Semiconductor

EV

EF

EC

Vacuum level

The Fermi level must be constant at thermal equilibrium but the vacuum level must be continuous...

mqφ

Bnqφ

χq

sqφ0qV


metal-semiconductor junction(Schottky barrier)

To ensure the continuity of the vacuum level, the Fermi level must move deeper into the semiconductor bandgap at the interface.

This involves electrons moving out from the semiconductor into the metal (which has so many electrons already that its Fermi level or band profile is unaffected).

As electrons move out, they leave behind uncompensated ionised donor atoms (as in a p++-n junction) and a depletion region is formed.

χφφ qqq mBn −=

The Schottky barrier height at the junction is the difference between the semiconductor conduction band and the metal Fermi level so that,


Electrons moving from the semiconductor to the metal encounter an energy barrier given by the built-in potential,

smbi qqqV φφ −=

For the junction between a metal and a p-type semiconductor the barrier height and built in voltage are given by,

( )χφφ qqEq mgBp −−=

Sketch the band diagram to satisfy yourself that this is the case

msbi qqqV φφ −=


Depletion width

Once the Schottky barrier height is known we can calculate the electric field profile, depletion width etc.

( )D

s

qN

VVW

−= 02ε

The problem for a Schottky barrier on an n-type semiconductor is identical to that for a one-sided abrupt p+-n junction (no depletion width on the metal side)…

ND is the donor concentration

V is the applied bias (positive forward, negative reverse)


Metal-Semiconductor Junction Under Bias

EV

EF

ECBnqφ biqV

EV

EF

ECBnqφ( )VVq −0

EV

EF

ECBnqφ ( )VVq −0

Zero bias

Forward bias (V is positive)Decreased barrier to electrons flow from the semiconductor to the metal

Reverse bias (V is negative)Increased barrier to electrons flow from the semiconductor to the metal …rectification


Figure 7.7. Current transport by the thermionic emission process. (a) Thermal equilibrium; (b) forward bias; and (c) reverse bias.

Current flow in a Schottky barrier


Current flow in a Schottky barrier

At thermal equilibrium the flow of electrons from the semiconductor to the metal is exactly balanced by the flow of electrons from the metal to the semiconductor so,

Since the barrier height on the metal side is not dependent on the applied bias the current from metal to semiconductor will always be the same so the net current under bias is then,

kTqCmssm

BneCNJJ φ−→→ ==

( )1−=−= −→→

kTqVkTqCsmms eeCNJJJ Bnφ

Saturation current density, Js

( )1−= kTqVs eJJ

…rectification


2exp* kTkT

AJ BS ⎥⎦

⎤⎢⎣⎡−=

φ

Thermionic Field Emission( )[ ][ ] ( ) ( )( )

⎥⎦

⎤⎢⎣

⎡ −−−

−−

⎥⎦⎤

⎢⎣⎡

−−−=

000

21

00 expcosh

*

E

EEq

kT

EEq

kT

Ek

EEVqETAJ fnCBfnCfnCB

S

φφπ

⎥⎦

⎤⎢⎣

⎡−=

000 exp

E

qJstJ B

S

φ

TFE

FE

TE

Thermionic Emission

Field Emission

Transport Mechanisms in Schottkydiodes


Typical Schottky Barriers


Ohmic Contacts


TEMARIO DEL CURSO DE FUNDAMENTOS DE FÍSICA DE …yolotli.inaoep.mx/portalfiles/diodoparte2.pdf ·...

Documents

Transcript of TEMARIO DEL CURSO DE FUNDAMENTOS DE FÍSICA DE …yolotli.inaoep.mx/portalfiles/diodoparte2.pdf ·...