Lecture 8

Semiconductor Physics VINonequilibrium Excess Carriers in

Semiconductors

Nonequilibrium conditions.

Excess electrons in the conduction band and excess holes in the valence band

Ambipolar transport（双极输运） : Excess electrons and excess holes diffuse, drift, and recombine with the same effective diffusion coefficient, drift mobility, and lifetime.

Ambipolar transport equation

Quasi-Fermi energy （准费米能级）for electrons and quasi-Fermi energy for holes

2

CARRIER GENERATION

AND RECOMBINATION

The Semiconductor in Equilibrium

3

0 0n pG G

The electrons and holes recombine in pairs, so the

recombination rates of electrons and holes are equal.0 0n pR R

The concentrations of electrons and holes are

independent of time; therefore, the generation and

recombination rates are equal:

The electrons and holes are created in pairs, so the

thermal-generation rates of electrons and holes are equal.

0 0 0 0n p n pG G R R

Under nonequilibrium conditions

When high-energy photons are incident on a

semiconductor, electrons in the valence band may be

excited into the conduction band.

=> Electron-hole pairs are generated.

=> The additional electrons and holes created are

called excess electrons and excess holes.

The generation rate of excess electrons and holes are

equal ' '

n pg g

4

Fig Creation of excess electron and hole densities by photons.

Excess electron and hole concentrations

2

0 0 inp n p n

5

0n n n

0p p p

As in the case of thermal equilibrium, an electron in

the conduction band may "fall down" into the valence

band, leading to the process of excess electron-hole

recombination.

The recombination rate of excess electrons and holes

are equal ' '

n pR R

Fig Recombination of excess carriers reestablishing thermal equilibrium6

The net rate of change in the electron concentration

The carrier concentration is dependent on time.

Since n0 and p0 are independent of time and ( ) ( )n pt t

If we consider a p-type material (p0 >> n0) under

low-level injection ( )0( )n t p

0

( )( )

n

r n

d tp t

dt

7

2( )[ ( ) ( )]r i

dn tn n t p t

dt

2

0 0

( ( ))[ ( ( ))( ( ))]r i

d n tn n n t p p t

dt

0 0( )[( ) ( ))]r n t n p n t

called the excess minority carrier lifetime

The excess carrier concentration is an exponential decay

from the initial excess concentration

The recombination rate

For the direct band-to-band recombination, the excess majority

carrier holes recombine at the same rate, so that for the p-type

material

8

0 0/( ) (0) (0)r np t t

n t n e n e

1

0 0( )n r p a constant for the low-level injection

'

0

0

( ( )) ( )( )n r

n

d n t n tR p n t

dt

' '

0

( )n p

n

n tR R

In the case of an n-type material (n0 >> p0) under low-

level injection0( )n t n

The excess minority carrier lifetime

9

' '

0

( )n p

p

p tR R

1

0 0( )p rn

In summary, ' ' ( )n pR R f n t

' ' ( )n pG G f n t

' ' time, spacen pG G f

CHARACTERISTICS OF EXCESS CARRIERS

The excess carriers behave with time and in space in

the presence of electric fields and density gradients

is of equal importance.

The excess electrons and holes diffuse and drift with

the same effective diffusion coefficient and with the

same effective mobility. This phenomenon is called

ambipolar transport.

10

• Continuity Equations

Fig Differential volume showing

x component of the hole-particle flux

A one-dimensional hole particle

flux is entering the differential element

at x and is leaving the element at x + dx.

The net increase in the

number of holes per unit time

due to x-component hole flux

the increase in the

number of holes per

unit time due to the

generation of holes

the decrease in the

number of holes per

unit time due to the

recombination of holes.

11

The net increase in the number of

holes per unit time in the differential

volume element is:

includes the thermal equilibrium carrier lifetime and the excess carrier lifetime.pt

Pp

pt

Fp pdxdydz dxdydz g dxdydz dxdydz

t x

12

The net increase in the hole concentration per unit time is

the continuity equation for holes

Similarly, the one-dimensional continuity equation for electrons is

p

p

pt

Fp pg

t x

nn

nt

Fn ng

t x

13

•Time-Dependent Diffusion Equations

The hole and electron current densities are

The hole and electron flux are

Substitute them into the continuity equation

where

2

2

( )p p p

pt

p pE p pD g

t x x

2

2

( )+ n n n

nt

n nE n nD g

t x x

p

p p p

J pF pE D

e x

nn n n

J nF nE D

e x

p p p

pJ e pE eD

x

n n n

nJ e nE eD

x

pE p EE p

x x x

14

The time-dependent diffusion equations for holes and electrons are

The thermal equilibrium concentrations, no and po are not functions

of time. For the special case of a homogeneous semiconductor, no

and po are also independent of the space coordinates.

Therefore, the time-dependent diffusion equations can be

Which describe the space and time behavior of the excess carriers

2

2( )p p p

pt

p p E p pD E p g

x x x t

2

2( )n n n

nt

n n E n nD E n g

x x x t

2

2

( ) ( ) ( )( )n n n

nt

n n E n nD E n g

x x x t

2

2

( ) ( ) ( )( )p p p

pt

p p E p pD E p g

x x x t

AMBIPOLAR TRANSPORT

15

Fig The creation of an internal electric field

as excess electrons and holes tend to separate

• With an applied electric field,

the excess holes and electrons

are created.

• This separation will

induce an internal electric

field between the two sets

of particles.

The negatively charged electrons and positively charged holes

then will drift or diffuse together with a single effective mobility

or diffusion coefficient. - called ambipolar transport.

intappE E E

16

To relate the excess electron and hole concentrations to the

internal electric field, Poisson's equation is introduced

where is the permittivity of the semiconductor materials

Since

then charge neutrality condition n p

The diffusion equations can be written as

intint

( )

s

Ee p nE

x

n pg g g n p

nt pt

n pR R R

2

2

( ) ( ) ( )( )p p

p p E pD E p g R

x x x t

2

2

( ) ( ) ( )( )n n

n n E nD E n g R

x x x t

int, E 0If p n

17

Combining the two diffusion equations yields

The ambipolar transport equation is derived as

where the ambipolar diffusion coefficient

and the ambipolar mobility is

2

2

( ) ( ) ( )' '

n n nD E g R

x x t

'n p p n

n p

nD pDD

n p

( )'

n p

n p

p n

n p

2

2

( ) ( )( ) ( )( )n p p n n p

n nnD pD p n E

x x

( )( )( ) ( )n p n p

nn p g R n p

t

18

The Einstein relation relates the mobility and diffusion coefficient

The ambipolar diffusion coefficient may be written in the form

Since both n and p contain the excess-carrier concentration,

which are dependent on time and space, the coefficient in

the ambipolar transport equation are not constants.

The ambipolar transport equation is a nonlinear differential equation.

pn

n p

e

D D kT

( )'

n p

n p

D D n pD

D n D p

2

2

( ) ( ) ( )' '

n n nD E g R

x x t

19

•Limits of Extrinsic Doping and Low Injection

In a p-type semiconductor with low-level

injection

In a n-type semiconductor with low-level

injection

The ambiploar

parameters

reduce to a

minority

carrier value,

which are

constants.

The equivalent ambipolar particle is negatively charged.

0 0

0 0

[( ) ( )]'

( ) ( )

n p

n p

D D n n p nD

D n n D p n

' n ' nD D

' pD D

( )'

n p

n p

p n

n p

' p

0 0p n0n p

0 0 0 and np n n

20

Consider the generation and recombination terms in

the ambipolar transport equation.

For electrons

For holes,

The generation rate for excess electrons must equal the

generation rate for excess holes.

The minority carrier lifetime is essentially a constant for low

injection.

'

0 0( ) ( ')n n n n n ng R g R G g R R

' ' 'n n n

n

ng R g R g

' ' 'p p p

p

pg R g R g

'

0 0( ) ( ')p p p p p pg R g R G g R R

0 0n nG R

0 0p pG R

' ' 'n pg g g

21

The ambipolar transport equation can be written in

terms of minority carrier parameters.

For an n-type semiconductor under low injection,

•The condition of charge neutrality:

The behavior of excess majority carriers is determined by the

minority carrier parameters.

2

2

0

( ) ( ) ( )'n n

n

n n n nD E g

x x t

2

2

0

( ) ( ) ( )'p p

p

p p p pD E g

x x t

For a p-type semiconductor under low injection,

p n

22

Table Common ambipolar transport equation simplification

23

Consider an infinitely large, homogeneous n-type semiconductor with zero

applied electric field. Assume that at time t = 0, a uniform concentration of

excess carriers exists in the crystal, but assume that g' = 0 for t> 0. If we assume

that the concentration of excess carriers is much smaller than the thermal-

equilibrium electron concentration, then the low injection condition applies.

Calculate the excess carrier concentration as a function of time for t > 0.

Example1

For the n-type semiconductor, we need to consider the

ambipolar transport equation for the minority carrier holes

Solution

We are assuming uniform concentration of excess holes so that

2

2

0

( ) ( ) ( )'p p

p

p p p pD E g

x x t

2 2( ) / ( ) / 0p x p x

24

So the bipolar transport equation reduces to

The solution is

where is the uniform concentration of excess

carriers that exists at time t = 0.

(0)p

From the charge-neutrality condition, we have that

So the excess electron concentration is given by

0

( )p p

p

d

dt

0/( ) (0) pt

p pt e

n p

0/( ) (0) pt

n pt e

For t>0, we are also assuming that g’=0

25

Consider an infinitely large, homogeneous n-type semiconductor

with a zero applied electric field. Assume that, for t < 0, the

semiconductor is in thermal equilibrium and that, for t>0, a

uniform generation rate exists in the crystal. Calculate the excess

carrier concentration as a function of time assuming the

condition of low injection.

Example2

The condition of a uniform generation rate and a homogeneous

semiconductor again implies that

Solution

The equation, for this case, reduces to

The solution to this differential equation is

0

( )'

p

p d pg

dt

2 2( ) / ( ) / 0p x p x

0/

0( ) ' (1 )pt

pp t g e

26

Consider a p-type semiconductor that is homogeneous and infinite

in extent. Assume a zero applied electric field. For a one-

dimensional crystal, assume that excess carriers are being generated

at x=0 only. The excess carriers being generated at x = 0 will begin

diffusing in both the +x and -x directions. Calculate the steady-state

excess carrier concentration as a function of x.

Example3

Solution

Since E=0, g’=0 for , and for steady state 0n t

Dividing by the diffusion coefficient

The parameter Ln, has the unit of length and is called the minority

carrier electron diffusion length.

2

2

0

( )0n

n

d n nD

dx

2 2

2 2 2

0

( ) ( )0

n n n

d n n d n n

dx D dx L

0x

27

The general solution to the equation is

The minority carrier electron concentration will then

decay toward zero at both x = and x =

These boundary conditions mean that B = 0 for x > 0

and A = 0 for x < 0.The solution is

The steady-state excess electron concentration decays

exponentially with distance away from the source at x = 0.

/ /( ) n nt L x L

n x Ae Be

/( ) (0) nx L

n x n e 0x

/( ) (0) nx L

n x n e 0x

28

Fig Steady-state electron and hole concentrations for the

case when excess electrons and holes are generated at x= 0.

29

Assume that a finite number of electron-hole pairs is generated

instantaneously at time t = 0 and at x = 0. But assume g' = 0 for t > 0.

Assume we have an n-type semiconductor with a constant applied

electric field equal to E0. which is applied in the +x direction.

Calculate the excess carrier concentration as a function of x and t.

Example4

Solution

The one-dimensional ambipolar transport equation is

The solution to this partial differential equation is of the form

using Laplace transform techniques

2

2

0

( ) ( ) ( )p p

p

p p p pD E

x x t

0/( , ) '( , ) pt

p x t p x t e

2

0

1/2

( )1'( , ) exp[ ]

(4 ) 4

p

p p

x E tp x t

D t D t

30

The total solution is

Excess-hole concentration versus distance

at various times for zero applied electric field

Excess-hole concentration versus

distance at various times for a

constant applied electric field.

0 2/

0

1/2

( )( , ) exp[ ]

(4 ) 4

pt

p

p p

x E tep x t

D t D t

31

•Dielectric Relaxation Time Constant

Initially, a concentration of

excess holes is not balanced by a

concentration of excess electrons.

How is charge neutrality

achieved and how fast?Fig The injection of a concentration of

holes into a small region at the surface

of an n-type semiconductor.

Poisson's equation is

The current equation, Ohm's law, is

The continuity equation, neglecting the effects of generation

and recombination, is

is the net charge density and the initial value is

J E

E

Jt

( )e p

32

Taking the divergence of Ohm's law and using Poisson's equation,

Substituting it into the continuity equation

Its solution is

where

called the dielectric relaxation time constant（介电弛豫时间）.

It means the time to charge neutrality.

d

t dt

d( / )( ) (0)

tt e

0d

dt

d

J E

33

Assume an n-type semiconductor with a donor impurity

concentration of Nd = 1016 cm-3. Calculate the dielectric relaxation

time constant.

Example

Solution

The conductivity is found as

The permittivity of silicon is

The dielectric relaxation time constant is then

19 16 1(1.6 10 )(1200)(10 ) 1.92( )n de N cm

14

0 (11.7)(8.85 10 ) /r F cm

1413(11.7)(8.85 10 )

5.39 101.92

d s

QUASI-FERMI ENERGY LEVELS

34

The thermal-equilibrium electron and hole concentrations

are functions of the Femi energy level.

If excess carriers are created in a semiconductor, we are no

longer in thermal equilibrium and the Fermi energy is

strictly no longer defined.

However, a quasi-Fermi level can be defined for nonequilibrium.

0 exp( )F Fii

E En n

kT

0 exp( )Fi F

i

E Ep n

kT

0 exp( )Fn Fii

E En n n

kT

0 exp( )Fi Fp

i

E Ep p n

kT

35

Example

Solution

Consider an n-type semiconductor at T = 300 K with carrier

concentrations of

In nonequilibrium, assume that the excess carrier concentrations

are

Calculate the quasi-Fermi energy levels.

The Fermi level for thermal equilibrium is

The quasi-Fermi level for

electrons in nonequilibrium is

The quasi-Fermi level for holes

in nonequilibrium is

0

i

( ) 0.2982F Fi

nE E kTIn eV

n

0( ) 0.2984Fn Fi

i

n nE E kTIn eV

n

0

i

( ) 0.179Fi Fp

p pE E kTIn eV

n

13 310n p cm

15 3 10 3 5 3

0 010 , 10 , 10in cm n cm and p cm

36

Thermal-equilibrium

energy-band diagram

Quasi-Fermi levels for

electrons and holes

Note: Fn FE E

Summary

The excess carrier generation rate and recombination rate.

Ambipolar transport: Excess electrons and holes do not move

independently of each other, but move together.

The ambipolar transport equation: the excess electrons and

holes diffuse and drift together with the characteristics of the

minority carrier under low injection and extrinsic doping.

Excess carrier behavior is a function of time and a function of

space.

The quasi-Fermi level for electrons and the quasi-Fermi level

for holes were defined to characterize the total electron and

hole concentrations in a semiconductor in nonequilibrium.

37

An n-type gallium arsenide semiconductor is doped

with Nd = 1016 cm-3 and Na = 0. The minority carrier

lifetime is

Calculate the steady-state increase in conductivity and

the steady-state excess carrier recombination rate if a

uniform generation rate, g' = 2 x 1021 cm-3s-1, is

incident on the semiconductor

HOMEWORK7

7

0 2 10p s