computational electronics. a.1 derivation of the boltzmann transport equation kinetic theory: we...

Computational Electronics


tftAddtA ,,,, rvrvvr

A.1 Derivation of the Boltzmann Transport EquationA.1 Derivation of the Boltzmann Transport Equation

Kinetic theory: We need to derive an equation for the single particle distribution function f(v,r,t) (classical) which gives the probability of finding a particle with velocity between v and v+dv and in the region r to r+dr

• We assume that v and r are given simultaneously which neglects quantum mechanical nature of particles.

• f(v,r,t) allows us to calculate ensemble averages over velocity and space (particle density, current density, energy density, etc.):


• Consider a hypervolume in phase space

j(r,v,t) is the flux density

j(r,v,t)ds is flux through hypersurface ds

• Consider the particle balance through the hyper-volume V

CollVV tn

dddttnddt

vrsrvjrvvrS

,,,,

Time rate of changeof # particles in V

Leakage through S

Time rate of changedue to collisions

tRtGddV

,,,, vrvrvr

Time rate of change due to G-R mechanisms

V

S

t,,rvjsd


• The flux density is written in terms of the time derivatives of the ‘position’ variables in 6D:

• Applying the divergence theorem in 6D

where the divergence of j is

mwithbn

mF

bnm

Fbn

mF

anvanvatnvvvvzyxj

zyx vz

vy

vx

zzyyxxzyx

Fv

rv

ˆˆˆ

ˆˆˆ,,,,,,,

V

tdddt ,,,, rvjvrsrvjS

z

z

y

y

x

xzyx v

nmF

vn

m

F

vn

mF

zn

vyn

vxn

v

j


which is written more compactly as:

• Particle balance is therefore:

Normalizing, we get the classical form of the Boltzmann transport equation:

nm

n vr Fvj

0F

vvr

RGColl

vrV t

ntn

nm

ntn

dd

RGColl

vr t

f

t

ff

mf

t

tf

Fv

vr ,,

First two terms on the rhs are the streaming terms


• For Bloch electrons in a semiconductor, we could have considered a 6D space x,y,z,kx,ky,kz where k is the wavevector and

• The semi-classical BTE for transport of Bloch electrons is therefore

k1

v Ek

RGColl

krk tf

tf

ffEt

tf

F

k1kr ,,


A.2 Collisional IntegralA.2 Collisional Integral

Assume instantaneous, single collisions which are independent of the driving force and take particles from k to k (out scattering) or from k to k (in scattering).

k

kzk

yk

xkIn scattering

Out scattering


(A) Out Scattering

where is the transition rate per particle from k to k

Distribution function is:

Take limit as t0

where the last term in the brackets accounts for the Pauli exclusions principle (degeneracy of the final state after scattering).

ttntn kkkrkr ,,,,

kk

N

tntf

,,,,

krkr

tftft

tf

OUT

,,,,,,

kr1krkr

kk


(B) In Scattering

By an analogous argument, the rate of change of the distribution function due to in scattering is:

Total rate of change of f (r,k,t) around k is a sum over all possible initial and final states k:

tftft

tf

IN

,,,,,,

kr1krkr

kk

kk

kkk

kr1kr

kr1krkr

tftf

tftft

tf

Coll

,,,,

,,,,,,

In scattering

Out scattering


(C) Boltzmann Equation with Collision Integral

The sum over final states k may be converted to an integral due to the small volume of k-space associated with each state:

The BTE becomes:

kkkk3 11k8

F1

kkkk

kkkrkk

ffffdV

ffEtf

k8 3

kd

V


A.3 Scattering Theory

What contributes to ? How do we calculate kk’?kk

Scattering Mechanisms

Defect Scattering Carrier-Carrier Scattering Lattice Scattering

CrystalDefects

Impurity Alloy

Neutral Ionized

Intravalley Intervalley

Acoustic OpticalAcoustic Optical

Nonpolar PolarDeformationpotential

Piezo-electric

Scattering Mechanisms

Defect Scattering Carrier-Carrier Scattering Lattice Scattering

CrystalDefects

Impurity Alloy

Neutral Ionized

Intravalley Intervalley

Acoustic OpticalAcoustic OpticalAcoustic Optical

Nonpolar PolarNonpolar PolarDeformationpotential

Piezo-electric


B.1 Time Evolution of Quantum States

When the Hamiltonian is time dependent, the state or the wavefunction of the system will be also time dependent. In other words, an electron will have a probability to transfer from one state (molecular orbital) to another. The transition probability can be obtained from the time-dependent Schrödinger Equation

)()(

tHt

ti

One the initial wavefunction, (0), is known, the wavefunction at a given later time can be determined. If H is time independent, we can easily find that

nn

tiEn

neat /)(


Suppose that a system has initial (t=0) Hamiltonian, H0 (time independent), and is at an initial eigenstate, k. Under an external influence, described by H’ (time dependent), the system will change state. For example, a molecule moves close to an electrode surface to feel an increasing interaction with the electrode. The combined Hamiltonian is

)('ˆ)0(ˆ)(ˆ0 tHHtH

the combined Hamiltonian should be a linear combination of the initial eigenstates, The wavefunction of the system corresponding to

nn

nk tCt )()(

From mathematical point of view, this is always possible since the initial eigenstates, n, form a complete set of basis. The physical picture is that the system under the influence of the external perturbation will end up in a different state with a probability given by |Cnk|2. The indices nk mean a transition from kth eigenstate to nth eigenstate. How fast the transition is or the transition rate is given by

2|)(| tCdt

dw nknk


B.2 Time-Dependent Perturbation Theory

Now we determine the transition rate according to the above definition. We assume that the initial state of the system is

k )0(the external perturbation, H’, is switched on at t=0. The time dependent Schrödinger Eq. is

)()'()(

0 tHHt

ti

For simplicity, we can rewrite this equation as

ntiE

nnk

netCt /)(')(

Note that Cnk’(t) is different from Cnk(t), but |Cnk’(t)|2=| Cnk(t)|2 and we can omit the prime.

.'//n

n

tiEnkn

n

tiEnk HeCedt

dCi nn


Multiplying by k’ and integrate, we obtain

nkn

tEEikk CnHkedt

dCi nk |'|'/)(' '

After considering that n are normalized orthogonal functions. Note that the initial condition becomes

0)0(

' dt

dCi kk

nknkC )0(

In general, solving above equation set is not easy, but we can obtain approximate solution using perturbation theory when H’ is small comparing to H0. Let us denote the solution in the absence of H’ as Cnk(0), we have

So Ck’k(0) is independent of time and the initial condition is

'' )0( kkkkC nknkC )0(


kHkedt

dCi tEEikk kk |'|'/)(

)1(' '

kHk |'|'

,1 /)(

0

'''

' dteHi

C tEEit

kkkkkk

We replace Cnk on the right hand side with Cnk(0)

and obtain the first order correction

in the above equation is often denoted as H’k’k and it measured the

coupling strength between the k’ and k states. Solving we have

One important case is that H’ is fixed once switched on. In this case,

/)(

11)(

'

/)('''

'

kk

tEEi

kkkk EEi

eH

itC

kk


B.3 Fermi-Golden Rule

)(||2

]/)[(

)/)((sin||

1|)(| '

2''22

'

'2

2''2

2' kkkk

t

kk

kkkkkk EEH

t

EE

tEEHtC

)(||2

'2'

'2' kkkkkk EEHw

Thus, we can obtain

So the transition rate is

We can conclude that (1) the transition rate is independent of time, (2) the transition can occur only if the final state has the same energy as the initial state. The later one reflects energy conservation. In the case when the energy levels are continuous band, the number of states near Ek’ for an interval of dEk’ is In the case

when the energy levels are continuous band, the number of states near Ek’ for an

interval of dEk’ is Ek’)dEk’ , where is the density of states. The transition rate

from k state to the states near Ek’ is then

)(||2

)( 2''2''' kkkkkkk EHdEwEw

This is Fermi Golden rule,

(23.18)


Assumptions made: (1) Long time between scattering (no multiple scattering events)(2) Neglect contribution of other c’s (Collision broadening ignored)

kk

kks

kkkk EEV

tP 22


B.4 Total Scattering Rate Calculation

• For the case when we have general matrix element (with q-dependence), the procedure for calculating the scattering rate out of state k is the following

2' '3

'

1 22' 03 1

0

1 22' 01

0

1( ) ' ' (cos )

2

1 22 ' ' (cos ) ( ) ( )

2

k'

1' ' (cos ) ( ) ( )

2

kk kkk

k k

k k

k k dk d d

k dk d M q E E

where k q

k dk d M q E E


• The integration over k’ can be converted into integration over q-wavevector and the integration over cos() together with the -function that denotes conservation of energy will put limits on the q-values: qmin and qmax for absorption and emission. The final expression that needs to be evaluated is:

• where

max

min

2

, 3( ) ( )

2

q

ab em

q

mk q M q dq

k

min max

min max

1 , 1

1 , 1

ab ab

k k

em ab

k k

q k k q k kE E

q k k q k kE E


qmin(em)

qmax(ab)

qmin(ab)

q-vector

cos()-1 +1

qmax(em)


0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5x 10

5

polar angle

Emission Process

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5x 10

5

polar angle

Absorption Process

Histogram of the polar angle for polar optical phonon emission and absorption. In accordance to the graph shown on the previous figure for emission forward scattering is more preferred for emission than for absorption.


Special Case: Constant Matrix Element

For the special case of constant matrix element, the expression for the scattering rate out of state k reduces to:

203

( ) 1k

mM kk

E

The top sign refers to absorption and the bottom sign refersTo emission. For Elastic scattering we can further simplify to get:

203

( )mM k

k


C.1 Elastic Scattering MechanismsC.1 Elastic Scattering Mechanisms

(A) Ionized Impurities scattering (Ionized donors/acceptors, substitutional impurities, charged

surface states, etc.)• The potential due to a single ionized impurity with net charge

Ze is:

• In the one electron picture, the actual potential seen by electrons is screened by the other electrons in the system.

unitsmksr

ZeVi

4

r2

0


What is Screening?

+

-

--

-

-

-

-

-

-

D - Debye screening length

Ways of treating screening:

• Thomas-Fermi Methodstatic potentials + slowly varying in space

• Mean-Field Approximation (Random Phase Approximation)time-dependent and not slowly varying in space

r

3D: 1

r

1

rexp

r

D

screeningcloud

Example:


• For the scattering rate due to impurities, we need for Fermi’s rule the matrix element between initial and final Bloch states

Since the u’s have periodicity of lattice, expand in reciprical space

• For impurity scattering, the matrix element has a 1/q type dependence which usually means G0 terms are small

rkrk1 rrkrk

ikni

ikni euVeudVnVn ,

*,,,

Grr rG

G

rkrk1kknn

iii

i UeeVedV

rGrG

G

rkrk1 rrrrr iknkn

iii

i euudeeVedV ,*

,

nnkkiknkn

ii

i IVuudeVedV

qrrrrr rkrk1,

*,


• The usual argument is that since the u’s are normalized within a unit cell (i.e. equal to 1), the Bloch overlap integral I, is approximately 1 for n=n [interband(valley)]. Therefore, for impurity scattering, the matrix element for scattering is approximately

where the scattered wavevector is:

• This is the scattering rate for a single impurity. If we assume that there are Ni impurities in the whole crystal, and that scattering is completely uncorrelated between impurities:

where ni is the impurity density (per unit volume).

kkq

volumeVqV

eZVV

scii

;2222

4222

qkrk

222

42

2222

42

sc

i

sc

ikki qV

eZn

qV

eZNV


• The total scattering rate from k to k is given from Fermi’s golden rule as:

If is the angle between k and k, then:

• Comments on the behavior of this scattering mechanism:

- Increases linearly with impurity concentration- Decreases with increasing energy (k2), favors lower T- Favors small angle scattering

- Ionized Impurity-Dominates at low temperature, or room temperature in impure samples (highly doped regions)

• Integration over all k gives the total scattering rate k :

kk222

422EE

qV

eZn

sc

iikk

coscos 122kk 222 kkkkkq

/;*

14

4

8 222

2

332

42

DDDsc

iik q

qkq

k

k

meZn


(A1) Neutral Impurities scattering• This scattering mechanism is due to unionized donors, neutral

defects; short range, point-like potential.• May be modeled as bound hydrogenic potential.• Usually not strong unless very high concentrations

(>1x1019/cm3).

(B) Alloy Disorder Scattering• This is short-range type of interaction as well.• It is calculated in the virtual crystal approximation or coherent

potential approximation.• Limits mobility of ternary and quaternay compounds,

particularly at low temperature.• The total scattering rate out of state k for this scattering

mechanism is of the form: 21

23

2

2 22

//*

EmEalloy

k


(C) Surface Roughness Scattering

• This is a short range interaction due to fluctuations of heterojunction or oxide-semiconductor interface.

• Limits mobility in MOS devices at high effective surface fields.

High-resolution transmission electron micrograph of the interface between Si and SiO2

(Goodnick et al., Phys. Rev. B 32, pp. 8171, 1985)

2.71 Å 3.84 Å

Modeling surface-roughnessscattering potential:

H'(r,z)Vo z (r) Vo z

Vo(z)(r)

random function that describes thedeviation from an atomically flat interface


• Extensive experimental studies have led to two commonly used forms for the autocovariance function.

• The power spectrum of the autocovariance function is found to be either Gaussian or exponentially correlated.

• Note that is the r.m.s of the roughness and is the rough-ness correlation length.

Commonly assumed power spectrums for the autocovariance function :

• Gaussian: SG(q)22 exp q22

4

• Exponential: SE(q)22

1 q22 2 3 2

Wave vector (Å-1)

AR model

Gaussian model=0.74 nm

Exponential model=0.94 nm

=0.24 nm

SPECTRUM OF HRTEM ROUGHNESS

Comparison of the fourth-order AR spectrum with the fits arising from the Exponential and Gaussian models(Goodnick et al., Phys. Rev. B 32, pp. 8171, 1985)


• The total scattering rate out of state k for surface-roughness scattering is of the form:

where E is a complete elliptic integral, Ndepl is the depletion charge density and Ns is the sheet electron density.

• It is interesting to note that this scattering mechanism leads to what is known as the universal mobility behavior, used in mobility models described earlier.

222223

422

11

150

k

kE

kNN

emsdepl

sc

srk .

*

deplssc

eff NNe

E

50.

Increasing substrate doping


100

200

300

400

1012 1013

experimental datauniformstep-like (low-high)retrograde (Gaussian)M

obili

ty [cm2 /V

-s]

Inversion charge density Ns [cm-2]

(aNs + bN

depl)-1

Ns

-1/3

PhononCoulomb

Interface-roughness

The Role of Interface Roughness:

D. Vasileska and D. K. Ferry, "Scaled silicon MOSFET's: Part I - Universal mobility behavior," IEEE Trans. Electron Devices 44, 577-83 (1997).


C.2 Inelastic Scattering MechanismsC.2 Inelastic Scattering Mechanisms

C.2.1 Some general considerations

• The Electron Lattice Hamiltonian is of the following form:

where Bloch states

• For the lattice Hamiltonian we have:

couplingPhononElectronH

nHamiltonialatticeHnHamiltoniaElectronH

HHHH

ep

le

eple

;

krk

kkkk ,,,,, ni

nnnne ueEH

21

qq

q

3q2q1q

nE

nnnEH

l

lllll

,

...

Second quantized representation, where nq is the number of phonons with wave-vector q, mode .


• Phonons:

The Fourier expansion in reciprocal space of the coupled vibrational motion of the lattice decouples into normal modes which look like an independent set of Harmonic oscillators with frequency

q

labels the mode index, acoustic (longitudinal, 2 transverse modes) or optical (1 longitudinal, 2 transverse)

q labels the wavevector corresponding to traveling wave solutions for individual components,

• The velocity and the occupancy of a given mode are given by:

ondistributiEinsteinBosee

nlBTk

;

/ 1

1q

q

qv qq


(1) For acoustic modes, , acoustic velocity.

(2) For optical modes, velocity approaches zero as q goes to zero.

u

q qv qq

0lim

Room temperature dispersion curves for the acoustic and the opticalbranches. Note that phonon energies range between 0 and 60-70 meV.


• The Electron-Phonon Interaction is categorized as to mode (acoustic or optical), polarization (transverse or longitudinal), and mechanism (deformation potential, polar, piezoelectric).

During scattering processes between electrons and phonon, both wavevector and energy are conserved to lowest order in the perturbation theory. This is shown diagramatically in the figures below.

Absorption: Emission:

kk E, kk E,qkk

qkk

EE

qq , qq ,

qkk

qkk

EE


• For emission, must hold, otherwise it is prohibited by conservation of energy. Therefore, there is an emission threshold in energy

• Emission: Absorption:

C.2.2 Deformation Potential Scattering

Replace Hep with the shift of the band edge energy produced by a homogeneous strain equal to the local strain at position r resulting from a lattice mode of wavevector q

(A) Acoustic deformation potential scattering• Expand E(k) in terms of the strain. For spherical constant

energy surface

21

0 kk eEEE

qk E

1qq nn 1qq nn


where:

and u is the displacement operator of the lattice

• Taking the divergence gives factor of e·q of the form:

Therefore, only longitudinal modes contribute.

vectoronpolarizati

eaeaNM

ii

,

*,,,

/

q

rqq

rqqq

21

q q

e

e2

ru

potentialnDeformatioE

constpotentialnDeformatioE

cellunitofvolumeofdilation

1

1

ru

.

modestransversefor

modesallongitudinforq

0qe

qe

q

q

,

,


• For ellipsoidal valleys (i.e. Si, Ge), shear strains may contribute to the scattering potential

Scattering Matrix Element:

Assuming then:

• At sufficient high temperature, (equipartition approximation):

zz

e

eEEEE

zz

zzud

ˆ

u

kk 0

; ezz is component of the strain tensor

,qulq

emissionlower

absorptionupper

uV

nqEV

lac 2

11q212

qqq 1

lBTknn


• Substituting and assuming linear dispersion relation, Fermi’s rule becomes

• The total scattering rate due to acoustic modes is found by integrating over all possible final states k’

where the integral over the polar and azimuthal angles just gives 4.

• For acoustic modes, the phonon energies are relatively small since

qkk2

21

qkk2 22

EEuV

TkEEEV

l

lBac

ackk

qkk0

232

21 4

8

2

EEkkd

V

uV

TkE

l

lBack

00q qas


• Integrating gives (assuming a parabolic band model)

where cl is the longitudinal elastic constant. Replacing k, using parabolic band approximation, finally leads to:

• Assumptions made in these derivations:

a) spherical parabolic bands b) equipartition (not valid at low temperatires)c) quasi-elastic process (non-dissipative) d) deformation potential Ansatz

23

21

lll

lBack uc

c

TkkEm

;

*

214

21

232 /

/*

Ec

TkEm

l

lBack


(B) Optical deformation potential scattering(Due to symmetry of CB states, forbidden for -minimas)• Assume no dispersion:

Out of phase motion of basis atoms creates a strain called the optical strain.

• This takes the form (D0 is optical deformation potential field)

The matrix element for spherical bands is given by

which is independent of q .

00q qas

orderzerothDDDVdo q000 eru ;

qkk1qkk2 00

0

20

2

nnVD

V ackk


• The total scattering rate is obtained by integrating over all k’ for both absorption and emission

where the first term in brackets is the contribution due to absorption and the second term is that due to emission

• For non-spherical valleys, replace

• The non-polar scattering rate is basically proportional to density of states

021

0

210

03

20

23

12

1

0

0

EEn

EnDm

do

dok /

//*

2123 //*ltmmm

0 Edok


(C) Intervalley scattering• May occur between equivalent or nonequivalent sets of valleys

- Intervalley scattering is important in explaining room tempera-ture mobility in multi-valley semiconductors, and the NDR ob-served (Gunn effect) in III-V compounds

- Crystal momentum conservation requires that qk where k is the vector joining the two valley minima


• Since k is large compared to k, assume and treat the scattering the same as non-polar optical scattering replacing D0 with Dij the intervalley deformation potential field, and the phonon coupling valleys i and j

• Conservation of energy also requires that the difference in initial and final valley energy be accounted for, giving

where the sum is over all the final valleys, j and

021

21

3

223

12

ijijij

ijij

j ij

ijdivk

EEEEn

EEnDm

ij

ijj

/

//

qq

ijq

ijij EEE minmin


C.2.3 Phonon Scattering in Polar Semiconductors

• Zinc-blend crystals: one atom has Z>4, other has Z<4.

• The small charge transfer leads to an effective dipole which, in turn, leads to lattice contribution to the dielectric function.

• Deformation of the lattice by phonons perturbs the dipole moment between the atoms, which results in electric field that scatters carriers.

• Polar scattering may be due to:

optical phonons => polar optical phonon scattering(very strong scattering mechanismfor compound semiconductors

suchas GaAs)

acoustic phonons => piezoelectric scattering(important at low temperatures invery pure semiconductors)


(A) Polar Optical Phonon Scattering (POP)

Scattering Potential:

Microscopic model is difficult. A simpler approach is to consider the contribution of this dipole to the polarization of the crystal and its effect on the high- and low-frequency dielectric constants.

• Consider a diatomic lattice in the long-wavelength limit (k0), for which identical atoms are displaced by a same amount.

• For optical modes, the oppositely charged ions in each primitive cell undergo oppositely directed displacements, which gives rise to nonvanishing polarization density P.

kTransverse mode:

kLongitudinal mode:


• Associated with this polarization are macroscopic electric field E and electric displacement D, related by:

D = E + P

• Assume D, E, P eik.r. Then, in the absence of free charge:

·D = ik ·D = 0 and E = ik E = 0

• Longitudinal modes: P||k => D=0, (LO)=0

• Transverse modes: Pk => E=0, (TO)=

Here, we have taken into account the contributionto the dielectric function due to valence electrons

kD or D=0 k||E or E=0


• The equations of motion of the two modes (in the k0 limit), for the relative displacement of the two atoms in the unit cell w=u1-u2 are:

Transverse mode: Longitudinal mode:

k

u1 u1 u1

u2 u2 u2

022

2

wdt

wdTO

MC

MMCTO

2112

21

2

u1 u1 u1

u2 u2 u2

EeM

wdt

wdTO

*122

2

22

TO

MEew

/*

tjEetE

)(


• The longitudinal displacement of the two atoms in the unit cell leads to a polarization dipole:

EwP 22

2 22

TO

MVNee

VN /*

*

• The existence of a finite polarization dipole modifies the dielec-tric function:

)(

)(

)(/

*

*

011

2

11

2

22

22

22

22

22

2

LO

TO

TOLO

TO

TO

MVNe

S

S

MVNe

EEEPED

Polarizationconstant


• The electric field associated with the perturbed dipole moment is obtained from the condition that, in the absence of macroscopic free charge:

P

ED PEDkD indindi and 00

• Consider only one Fourier component:

)()( qVieVqV ei

eeindind qE rqq

111

Pq Pq

eiqV

q

eiqV

)()( 2

,

)/()(

/

*

r

rqq

rqq

rqq

rqq

ii

qLO

ii

qLO

eaeaqV

ei

eaeaqNM

eVNe

iV

12

1222

212

)(0

111 2LO


• Matrix element squared for this interaction:

• Transition rate per unit time from state k to state k’:

• Total scattering rate per unit time out of state k:

qkk'

21

21

02

22 1

2N

qVe

VLO

kk ',

Scattering Rate Calculation:

021

21

02

2

02

1

2

LLO

Lkkkk

NqV

e

ωεεV

kk'

kk'

qkk'

',',

dqqddV

qkq

qkk

kkk ,,'

', )(cos)(

2

0

1

1 0

232


• Momentum and energy conservation delta-functions limit the values of q in the range [qmin,qmax]:

absorption:

emission:

• Final expression for k

)(1

)(1

max

min

kEkkq

kEkkq

LO

LO

thresholdemission,)(

)(1

)(1

max

min

LO

LO

LO

kE

kEkkq

kEkkq

112

10

102

2

LOLOLOk

kEN

kEN

k

em

)(sinh

)(sinh

*


Discussion:

1. The 1/q2 dependence of k,k’ implies that polar optical phonon scattering is anisotropic, i.e. favors small angle scattering

2. It is inelastic scattering process3. k is nearly constant at high energies4. Important for GaAs at room-temperature and II-VI compounds

(dominates over non-polar)

Scattering rate

Momentum relaxationrate

The larger momentum relaxation time is a consequence of the fact that POP scattering favors small angle scattering events that have smaller influence on the momentum relaxation.


(B) Piezoelectric scattering

• Since the polarization is proportional to the acoustic strain, we have

• Following the same arguments as for the polar optical phonon scattering, one finds that the matrix element squared for this mechanism is:

• The scattering rate, in the elastic and the equipartition approximation, is then of the form;

where qD is the screening wavevector.

uepz P

qkk

'' 21

21

22

2

qpz

qkk N

ee

VV

2

22

3 414 Ds

pzBk q

kv

ee

k

Tkmln

*


Total Electron-Phonon Scattering Rate Versus Energy:

Intrinsic Si GaAs

In both cases the electron scattering rates were calculatedby assuming non-parabolic energy bands.


1. Acoustic Phonon Scattering

2. Intervalley Phonon Scattering

3. Ionized Impurity Scattering

4. Polar Optical Phonon Scattering


5. Piezoelectric Scattering

6. Dislocation Scattering (e.g. GaN)

where n’ is the effective screening concentration

Ndis is the Line dislocation density 7. Alloy Desorder Scattering (AlxGa1-xAs)

Where: d is the lattice disorder (0≤d≤1)

Dalloy is the alloy disorder scattering potential

computational electronics. a.1 derivation of the boltzmann transport equation kinetic theory: we...

Documents

scattering slide

computational electronics

time dependent

scattering theory

collisions time rate

time independent

time derivatives

gr mechanisms v s slide