transport, scattering generation/recombinationsoktyabr/nnse508/nnse508...total electron current:...

NNSE508 / NENG452 Lecture #13

1

Lecture contents

• Transport, scattering

• Generation/recombination

Band-to-band

recombination Trap-assisted (SRH)

recombination

and generation

Auger

recombination

Ec

Et

Ev

E


2

Electron transport: General considerations

Motion in real space = thermal motion

+ drift + scattering

Mean free path:

Motion in wavevector space: inside a

band valley (unless intervalley

scattering is involved)

Current density is proportional to drift

velocity of carriers

2

A

cmdJ env

How “free” carriers react on external electric field ?

th mv


3 Electron transport: phenomenological approach

Kinetic energy per electron in 3D (from

kinetic theory of gases)

Thermal velocity

Force on a “free” electron includes

electric and “friction” with

momentum relaxation time m :

In the steady state drift velocity is

proportional to the field (m –drift

mobility):

And current density (s –conductivity)

gives Ohm’s law:

For semiconductor containing both

electron and holes:

sm enenvJ

Tkmv

B2

3

2

2

m

de

de

vme

dt

dvm

**

m

*

e

md

m

ev

2

*

cm

Vs

m

e

e

m

m

nm

een

e

m

*

2ms

he pne mms

1 2

7310

*

Bth d

k T cmv v

m s

-1

-1

3D:(Ω cm)

2D:(Ω/sq.)s

electric field)(


4

Scattering mechanisms

In low electric fields

• ionized impurities

• acoustic phonons

– Deformation potential

– Piezoelectric

In high electric fields

• optical phonons

• intervalley scattering

At high concentrations

• carrier-carrier scattering

To understand how these mechanisms affect mobility one needs to

consider dependence (E)

Mathiessen’s rule for relaxation time:

...

11111

niiiopacm


5 Conductivity effective mass

• For non-degenerate conduction band minimum (G-

point):

• Indirect conduction band minimum: effective mass

will depend on direction!

– For example: Si along [100] direction:

– Conductivity effective mass of Si [100] electrons::

• Valence band maximum: need to add up

contributions of light and heavy holes:

• Conductivity effective mass of holes

tlc mmm

21

3

11*

G mme

*

2 2

2 2 26 6

m mn ex ey ez

l t

e en nJ J J J

m m

mhhh

lh

lhhhlhp e

m

p

m

pJJJ 2

2323

2121

*

11

hhlh

hhlh

hh

hh

lh

lh

v mm

mm

m

p

m

p

pm

Effective masses and low field mobilities at room temperature:


6 Temperature dependence of mobility

Mobility in n-Si

From Yu and Cordona, 2003,

and Shur, 2003

Mobility in p-Si

Mobility in n-GaAs Relaxation time m depends on energy, and

therefore mobility depends on temperature

LO phonon scattering

Intravalley acoustic phonon

scattering + (2TA+LO)

intervaley phonon scattering


7

High field electron transport

Hot electrons transfer energy into thermal

vibrations of the crystal lattice (phonons).

Such vibrations can be modeled as harmonic

oscillations with a certain frequency, wph .

The energy levels of a harmonic oscillator

are equidistant with the energy difference

between the levels equal to

Two types of phonons scatter electrons

differently: acoustical (slow) and optical (fast):

Hence, process for a hot electron can be

represented as follows. The electron

accelerates in the electric field until it

gains enough energy to excite optical phonon:

phphE w

meVE optopt 40 w

kTE acac w


8

Intervalley scattering in high electric field

GaAs has an L valley just 0.29 eV higher than

G -valley

At high fields electrons can be transferred

into L-valleys. The conductivity will

depend on concentrations and mobilities

of electrons in both valleys:

Model for dependence of drift velocity on

electric field in GaAs

From Shur, 2003

LLNNe mms GG


9 Diffusion and drift of carriers

• If the carrier concentration is changing from point-to point, the carriers

will redistribute to equalize the concentration = Diffusion

• Diffusion is described by the first Fick’s law:

Electron diffusion current

Hole diffusion current

• If electric field (weak) is applied:

Total electron current:

Total hole current:

plays the role of an effective field

• Diffusion coefficient and diffusion current have sense only if

concentration change is small on the mean free path:

• Substituting the gradient with the effective field:

neDJ nn

peDJ pp

(Flux of particles)= -(Diffusion coef.) x (gradient of concentration)

neDneJ nnn m

peDpeJ pop m

D n

n

n

e

TkB

nn

1Tk

e

B


10 Equilibrium of drift-diffusion: Einstein relation

At equilibrium condition (no total current)

diffusion current is equal to drift current:

For nondegenerate semiconductor:

Carrier concentration is changing in the

electric field, following the potential j(r):

Gradient of electron concentration is :

And substituting, obtain Einstein relation:

0 neDne nnm

Tk

EENn

B

CFC exp

Tk

xenxn

B

)(exp)( 0

j

jTk

enx

Tk

enn

BB

)(

EF

EC

Drift flux

Diffusion flux

me

TkD B *

e

m

m

em


The rate of change of the carriers between x

and x + dx is determined by fluxes and

generation/recombination of carriers:

Continuity equation for electrons:

And similar for holes

Continuity equation

CE

x x dx

x

( )nJ x ( )nJ x dx

VE

nGnR

( )1 nJ xdx

q x

( ) ( )( , )

( , ) ( , )n nn n

J x J x dxn x tAdx A G x t R x t Adx

t q q

( , ) 1( , ) ( , ) ( , )n n n

n x tdivJ x t G x t R x t

t q

( , ) 1( , ) ( , ) ( , )p p p

p x tdivJ x t G x t R x t

t q

The continuity equation allows us to

calculate the carrier distribution in the

presence of generation and recombination

3

1,G R

cm s


0

0( )

t t

p t p Ge

12 Generation/Recombination

Band-to-band

recombination Trap-assisted (SHR)

recombination

and generation

Auger

recombination

Ec

Et

Ev

E

Recombination mechanisms • In the simplest model net recombination

rate (intrinsic = response of the

semiconductor) is proportional to the

EXCESS carrier density (n0, p0 –

equilibrium concentrations, ’s - minority

carrier lifetimes):

• Un might be negative – net generation

• At thermodynamic equilibrium, the

generation rate and the recombination

rate are equal:

• After generation, excess concentration

decays exponentially decays:

Ec

Ev

E

Ionization due to

energetic

particle/photon

Generation mechanisms

E-h pairs generation

by light

Ephoton=hn>Eg

0p p p

p

p pU R G

P

0n pU U

0n n n

n

n nU R G

3

1

cm s

Extrinsic generation

rate, not material

response


13 Band-to-band transitions in direct-bandgap

and indirect-bandgap semiconductors

• Low emission probability due to 2nd

order (phonon-assisted) process

(momentum conservation)

• Long radiative lifetime low radiative

quantum efficiency

• Si, Ge, diamond, GaP, AlAs…

• High emission probability

• Short radiative lifetime high radiative

quantum efficiency, efficient LED,

lasers…

• GaAs, InGaAs, GaN, InP, InGaSb…


14 Optical absorption

From Colinge, 2005

• Beer-Lambert absorption law

• Significant absorption at

• Indirect bandgap – low near

edge absorption, gradual

spectral slope (momentum

conservation)

• Direct bandgap – sharp edge

• Absorption spectrum depends

on density of sates in the bands

and transition probabilities.

Ephoton = hn > Eg

I(x) =I0 exp(-ax)


15

Optical absorption and non-equilibrium carriers

• Intensity [W/cm2=J/s-cm2] of light is exponentially

decreasing due to absorption

• Absorbed intensity in a layer with a thickness d:

• In case of uniform generation (low absorption) ad <<<1,

can be always applied to a thin slice

• Number of generated e-h pairs is equal to the number of

photon absorbed. When generation is uniform:

• Density of both electrons and holes increase under band-tp-

band optical excitation by the same amount

• Higher energy photons generate “hot” carries that relax

towards the band extrema

• G/R is typically slower (10-9 - 10-5 s) than energy

relaxation due to scattering (10-14 - 10-12 s)

Allows to treat electrons and holes

statistically independent i.e.

introduce separate quasi-Fermi

levels for electrons and holes.

0absI I da

I(x) =I0 exp(-ax)

0 0

d

absI I I e a

opt opt

n p

I IdG G

d h h

aa

n n

Dn = Dp

Extrinsic generation rate


16

Radiative recombination rate Rr and radiative life time r:

Radiative recombination rate Rr (B- radiative or

Bi-molecular recombination coefficient)

consists of equilibrium and non-equilibrium parts:

Two cases:

High injection rate (bimolecular recombination):

Low injection:

Radiative recombination

DD ))(( 00 ppnnBBnpRr

dt

dpR

nn

dt

dnr

r0

00 , pnn D

2nBRr D

0nn D

nBr

D

1

00

1pnB

r

in a doped material one of the

concentrations dominates

200 iBnnpnnB DD

Considering optical excitation: Dn = Dp

Might depend

on n

New non-equilibrium Rr (=Ur)

and


17

Radiative recombination

From Yu and Cordona, 2003

Radiative lifetime for low injection

• Minority carrier lifetime

For example for p-material lifetime of

injected electrons will be 0

1Bp

r

0nn D


for centers where the recombination rate is highest (i.e. for )

18

Non-radiative recombination through

recombination centers Nt: Shockley-Hall-Reed

(SHR) processes

Consider electron G/R: nt - centers occupied

with electrons ( f –Fermi distribution function)

lifetime:

At equilibrium, electron emission

:

Shockley-Hall-Reed (SHR) Recombination

2

exp expSHR

i

t i i tp i n i

np nU

E E E En n p n

kT kT

2

0

i

SHR

n n

nn

n npU

t iE E

n n nU R G

p p pU R G

_

11n n th n t t t

n

R v n N n f Es

n n t tG e N f E

_

1n

n th n tv N

s

_ exp i tn n th n i

E Ee v n

kTs

Combining

electrons and

holes

2

iSHR

n i p i

np nU

p n n n

For minority carriers (e.g.

electrons in p-type) 0p p n


19

Surface recombination in quasi-neutral p-type region

With surface recombination velocity sn [cm/s]

Auger recombination contribute at high carrier

concentrations:

In general in case of external generation G :

where lifetime is a result of various processes: non-

radiative through recombination centers, radiative,

Auger recombination:

Lifetime at high excitation conditions:

Other recombination mechanisms

, 0surf n nU s n n

n

nnG

dt

dn

0

Arnn

1111

0

322 CnnpCpnCR pnA

, ip n p n

21CnBnA

n

Uniform (photo)generation:

( ) ( ) 11

p

x

Ln n nn

n n n

s Ln x p x G e

s L

D D

x

n

n0 Ln

nG


Lecture recap

20

• Ohmic behavior results from scattering

• Drift-diffusion

• Recombination rate

• Continuity equation (1D)

neDneJ nnn m

1 nn n

JnG R

t q x

0n n n

n

n nU R G

Minority carrier lifetime

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