Nonequilibrium total dielectric function approach to the electron Boltzmann equation for inelastic scattering in doped polar semiconductors

10 Pages·0.18 MB·English

by  B. A. Sanborn

Nonequilibrium total dielectric function approach to the electron Boltzmann equation
for inelastic scattering in doped polar semiconductors
B.A. Sanborn
Semiconductor Electronics Division
National Institute of Standards and Technology, Gaithersburg, MD 20899
This paper describes a simple and general method for deriving the inelastic collision term in
theelectronBoltzmannequationforscatteringfrom acoupledelectron-phononsystem,andapplies
the method to the case of doped polar semiconductors. In the Born approximation, the inelastic
differentialscatteringrateWinel can beexpressedintermsofthenonequilibriumtotaldynamicdi-
electricfunction,whichincludesbothelectronicandlatticecontributions. Withintherandom-phase
approximationWinelseparatesintotwocomponents: anelectron-electroninteractioncontainingthe
5 nonequilibrium distribution function for excitations of the electron gas, and a Fr¨ohlich interaction
9 includingthephonondistributionfunctionandself-energyduetopolarizationoftheelectrons. Each
9
ofthesetwointeractionsisscreenedbyonlytheelectronicpartofthetotaldielectricfunctionwhich
1
contains the high frequency dielectric constant, unlike commonly used expressions which contain

n the static dielectric constant. Detailed balance between plasmons and electron-hole pairs in steady
a state is used to eliminate the nonequilibrium plasmon distribution from the Boltzmann equation,
J
resulting in a dynamically screened electron-electron collision term. The phonon self-energy modi-

9 fies the longitudinal optical phonon dispersion so that two hybrid normal modes contribute to the
1 electron-phonon collision term.

1
v
I. INTRODUCTION
3
8
0 The electron Boltzmann equation for inelastic scattering in solids is usually established either by making approxi-
1 mations to moregeneralequations derivedwith powerfulnonequilibrium Green’sfunction techniques [1–4], orsimply
0
by ansatz with the aid of analogy to simpler physical systems. An alternative method described here proceeds by
5
using time-dependent perturbation theory to determine the inelastic collision term from the total dynamic dielectric
9
function for the nonequilibrium coupled electron-phonon system. This paper treats the case of a system of polar
/
at optical phonons in the presence of conduction electrons and Coulomb interactions, though the method is more gen-
m erally applicable. Except for the nonequilibrium characterof the Boltzmann problem, the approachis similar to Van
Hove’s correlation function method [5] applied to doped polar semiconductors [6] to determine the inelastic lifetime
-
d of a quasiparticle excitation of an equilibrium system. In addition to providing physical insight, the total dielectric
n functionapproachis asystematic wayto include scatteringagainstcoupledelectron-phononmodesinthe Boltzmann
o
equation, so that the influence of mode coupling on mobility can be studied. Numericalresults for mobility in n-type
c
GaAs are presented in reference [7].
:
v In the Born approximation, the interaction between a conduction electron and a coupled electron-phonon system
i
X can be represented as an effective electron-electron interaction screened by the total dynamic longitudinal dielectric
function, ǫ (q,ω),whichincludescontributionsfrombothelectronsandphonons[8–10]. Inthis context,ǫ describes
r T T
a thelinearresponseofthe electron-phonongasnottoanexternallyappliedpotential,buttotheinternalpotentialdue
to the probe electron plus the induced density fluctuations of the gas itself. The effective interaction has the form of
a total density autocorrelationfunction. Unlike the lifetime problem [6], the entire electron-phonon gas is away from
equilibriuminthe dc transportcase,andthe correlationfunctionmust be evaluatedforthe nonequilibriumensemble.
It follows that the effective electron-electroninteraction is screened by the nonequilibrium total dielectric function in
thiscase. Infact,takingaccountofthenonequilibriumcharacteroftheinteractionisessentialtoobtainingthecorrect
formof the inelastic collisionterm, which contains nonequilibriumdistribution functions for the system’s excitations.
The effective electron-electron interaction can be separated into a purely electron-electron term and a screened
electron-phononinteraction. Intherandom-phaseapproximation(RPA),theelectron-phononpartcontainsaphonon
self-energy which arises from the polarization of the electron gas [10]. The self-energy correction modifies the lon-
gitudinal optical (LO) phonon dispersion in doped polar semiconductors, producing two hybrid normal modes with
phononstrengthinboth. ThispaperdeterminestheinelasticdifferentialscatteringrateWinel forthenonequilibrium
case in the RPA Born approximation. The result consists of an electron-electroncomponent containing the nonequi-
librium distribution function for excitations of the electron gas, and an electron-phonon component containing the
phonon self-energy and distribution function. Each of these two interactions is screened by only the electronic part
of the total dielectric function, which contains the high-frequency dielectric constant ǫ . Thus, the RPA Bornresult
∞
differs from commonly used expressions for screened interactions in doped polar semiconductors which contain the
1
static dielectric constant ǫ . Examples using ǫ rather than ǫ for n-type GaAs include a calculation of dynamically
0 0 ∞
screened electron-electron scattering [11], as well as static screening approximations for electron-LO phonon scatter-
ing [12] and electron-electron scattering [13]. For more strongly polar doped semiconductors (where ǫ ≫ ǫ ), it is
0 ∞
essential to treat this aspect of the screening problem correctly.
Thecollectivemodeshaveafinitelifetimefordecayintoelectron-holepairs. ThisLandaudampingprocessprovides
an indirect mechanism for single-particle electron-electron scattering. For the electronic system in steady state,
Landau damping creates a detailed balance relation between plasmons and electron-hole pairs which can be used to
eliminatethenonequilibriumplasmondistributionfromtheBoltzmannequationinfavorofthesingle-particleelectron
distribution. Intheapproximationofnocollisiondamping,theresultisanelectron-electroncollisiontermscreenedin
the dynamic RPA, containing the effects of electron-plasmon scattering. For direct gap doped semiconductors where
Umklappprocessesarenegligible,this collisiontermconservestotalelectronmomentumandcannotby itself degrade
an electrical current if energy surfaces are spherical. However, by rearranging the electron momentum distribution,
electron-plasmon and electron-electron scattering can have a significant indirect effect on the outcomes of other
scattering processes.
The low-energy hybrid mode arising from the plasmon-phonon coupling in doped polar semiconductors gives in-
creased low-energy electron-phonon scattering strength compared to the uncoupled case. Numerical calculations
including the plasmon-phononcoupling have given shorter inelastic lifetimes [6,14]and enhanced hot electronenergy
relaxation[15,16],especially at low temperatures and low doping. A simple way to treat the coupled modes is to use
the plasmon-pole approximation [17,18] on the phonon self-energy. This procedure is used here to include scatter-
ing against the hybrid modes in the electron-phonon collision term so that mode coupling can be treated with the
Boltzmann equation in a computationally efficient manner.
Thispaperisorganizedasfollows. InsectionII,Winel isderivedwithintheBornapproximationintermsofthetotal
nonequilibrium dielectric function for the coupled electron-phonon system. The RPA is invoked to separate Winel
into electron-electron and electron-phonon components. Section III deals with the electron Boltzmann equation by
eliminating the nonequilibrium plasmon distribution from the electron-electron collision term. The electron-phonon
collision term is simplified by applying the plasmon-pole approximationto the phonon self-energy.
II. INELASTIC DIFFERENTIAL SCATTERING RATE
A. Electron-phonon gas model
Consider a probe electron scattering from an interacting gas of electrons and polar optical phonons. Interactions
with impurities and acoustic phonons can be added by hand later. The probe electron is treated as a quasiparticle
with states |ki, effective mass m∗, and dispersion relation E = h¯2k2/2m∗. The Boltzmann equation relies on the
k
validity of the quasiparticle description by assuming that the energy-momentum relation remains well defined when
collisionsareincluded. Interactionswiththeelectron-phonongasproducetransitionsbetweenthequasiparticlestates,
but do not broaden them significantly.
For direct-gappolar semiconductors, localfield effects are unimportant so that electrons couple only to the macro-
scopic electric field set up by the LO phonons, and do not interact with the transverse optical (TO) phonons. The
Hamiltonian for the electron-LO phonon gas is [10]
1 1
H˜ = E c† c + ¯hω (a†a + )+ v∞ ρ†(q)ρ (q)−N , (1)
g k k,σ k,σ TO q q 2 2 q T T
Xk,σ Xq qX=6 0 h i
where
v 4πe2
v∞ = q = , (2)
q ǫ q2ǫ
∞ ∞
and where the electron and LO phonon annihilation operators are denoted by c and a , respectively, σ labels the
k q
electronspin, and ω is the TO phonon frequency. The relevantphononfrequency in H˜ is the bare one, ω . The
TO g TO
ion-ion Coulomb interactions explicitly included in H˜ shift ω to the longitudinal optical frequency ω , so that
g TO LO
only ω appears in expressions for observables [10]. The first and second sums in H˜ describe the noninteracting
LO g
systems of electrons andphonons, respectively. The last sum includes electron-electron,electron-phonon,and ion-ion
Coulomb interactions. The total density operator ρ is the summation of the density operators of the electrons and
T
LO phonons,
2
n¯h 1/2
ρ (q)=Ω−1/2ρ(q)+Zq A(q), (3)
T
2Mω
(cid:18) TO(cid:19)
where A(q) = a +a† , M is the reduced mass of the basis ions, Ze is the effective ionic charge, n is the electron
q −q
density, Ω is the system volume, and the Fourier components of the electron density operator are
ρ(q)= e−iq·ri = c†k+q,σck,σ. (4)
i k,σ
X X
Theelectronself-interactionhasbeenremovedfromH˜ bysubtractingthetermwithN,thenumberofelectronsinthe
g
system. The influence of interband electronic transitions is taken into account by screening the Coulomb interaction
v with the high-frequency dielectric constant ǫ .
q ∞
The effects of screening by conduction electrons and LO phonons are contained in the interaction term in H˜ and
g
will appearexplicitly in the totaldielectric function below. The RPA ormean-field approximationfor the interaction
termismadebyworkingwitheigenstates|nioftheHamiltonianH whichneglectsthetermsinH˜ thatarequadratic
g g
in density fluctuations ρ −hρ i [9].
T T
H |ni=E |ni (5)
g n
1
H =H˜ − v∞[ρ†(q)−hρ†(q)i][ρ (q)−hρ (q)i]. (6)
g g 2 q T T T T
q
X
H includesCoulombinteractionsattheHartreelevel. Excitationsoftheelectron-phonongasdescribedbyH include
g g
both electron-hole pair and collective excitations.
Define P(n,k→m,k−q) as the probability per unit time for the probe electron to make a transition from |ki to
|k−qi while the electron-phonon gas makes a transition from |ni to |mi, assuming that initially |ki is occupied and
|k−qiisunoccupied. TheinteractionsincludedinH producechargedensitycorrelations,resultinginaninteraction
g
potentialbetweengasexcitationsandprobeelectronwhichisscreenedandassumedtobeweak. UsingFermi’sGolden
rule (or the Born approximation) with the probe particle plane wave states |ki, the transition probability is
2π v∞ 2
P(n,k→m,k−q)= q |hm|ρ†(q)|ni|2 δ(E −E +h¯ω ) (7)
¯h Ω T n m k,k−q
(cid:18) (cid:19)
v∞ 2 ∞
= q dteiωk,k−qthn|ρ (q,t)|mihm|ρ†(q,0)|ni (8)
Ω¯h T T
(cid:18) (cid:19) Z−∞
ρ (q,t)=eiHgt/h¯ρ (q)e−iHgt/h¯, (9)
T T
where ω =h¯[k2−(k−q)2]/2m∗.
k,k−q
The inelastic differentialscattering rateWinel(k,k−q)is the probabilityper unit time foranelectronic transition
from the occupied state |ki to the unoccupied state |k−qi due to scattering from the electron-phonon gas. It is
foundbysummingP(n,k→m,k−q)overallpossiblefinalstates|miofthe gasandaveragingoveritsinitialstates
|ni. For the mobility problem, the average must be taken over the nonequilibrium gas ensemble.
B. Nonequilibrium ensemble averages
ConsiderthatanelectricpotentialU(r,t) isappliedtothe systemofprobeelectronpluselectron-phonongas. This
influence is included by adding the term
H (t)=e d3rU(r,t)ρ (r,t) (10)
ext T
Z
to the Hamiltonianofthe system. H is independent oftime insteady state,but itis usefulto requirethe potential
ext
to be switched on at a time in the distant past when the gas was in thermal equilibrium. This requirement makes it
possible to use the Kadanoff and Baym method [19] for evaluating a nonequilibrium expectation value. This method
uses the equilibrium density matrix to evaluate an ensemble averageat a time after U has been switched on by using
the interaction picture to include the dependence on H explicitly. The nonequilibrium expectation value of an
ext
operator O(r,t) is [19]
3
hO(r,t)i =hO (r,t)i=Z−1Tr e−βHgO (r,t) (11)
noneq U U
OU(r,t)=V−1(t)O(r,t)V(t), (cid:8) (cid:9) (12)
−i t
V(t)=T exp dt′H (t′) , (13)
ext
¯h
(cid:20) Z−∞ (cid:21)
whereh idenotesanaverageoverthe equilibriumpopulationoftheelectron-phonongas,β =1/kBT,Z =Tr{e−βHg}
is the partition function for the equilibrium gas, and T denotes the time ordering operator which orders operators
with earliest times to the right.
In the presence of U(r,t), the transition rate P(n,k→m,k−q) in the Born approximationis unchanged from its
equilibriumform(7)exceptthattheinfluenceofH mustbeincludedinthetimedependenceofthematrixelement.
ext
Using equations (11-13) to determine Winel(k,k−q) from P(n,k→m,k−q) for the nonequilibrium case yields
v∞ 2
Winel(k,k−q)= q NS (q,ω ), (14)
TU k,k−q
Ω¯h
(cid:18) (cid:19)
1 ∞
S (q,ω)= dteiωthρ (q,t)ρ (−q,0)i (15)
TU TU TU
N
Z−∞
ρ (q,t)=V−1(t)eiHgt/h¯ρ e−iHgt/h¯V(t), (16)
TU T
where S (q,ω) is the nonequilibrium total dynamic structure factor or spectral function for ρ , describing the
TU TU
density fluctuation excitations of the nonequilibrium electron-phonon gas. Similarly, the time-reversed rate is
v∞ 2
Winel(k−q,k)= q NS (−q,−ω ). (17)
TU k,k−q
Ω¯h
(cid:18) (cid:19)
Whenthegasexcitationsarenottoohighlydampedsothattheymayusefullyberegardedaselementaryexcitations,
andwhenthedeviationsfromequilibriumareslowlyvaryinginspaceandtimeonascaledeterminedbythefrequencies
anddecayratesoftheelementaryexcitations,itispossibletodefinenonequilibriumdistributionfunctionsandresponse
functions [1,3]. The imaginary (dissipative) part of the response function for the nonequilibrium coupled electron-
phonon system is
−1 ∞
Im[χ (q,ω)]= dteiωth[ρ (q,t),ρ (−q,0)]i (18)
TU TU TU
2h¯Ω
Z−∞
−n
= [S (q,ω)−S (−q,−ω)] (19)
TU TU
2h¯
where [ρ,ρ′] is the commutator ρρ′−ρ′ρ.
In general, there is no simple relation between a nonequilibrium spectral function and corresponding response
function like the fluctuation-dissipation theorem (FDT) for the equilibrium case:
−2h¯
S (q,ω)= N0(ω)+1 Im[χ (q,ω)], (20)
T T
n
(cid:2) (cid:3)
where N0(ω) = (eβh¯ω − 1)−1 is the equilibrium Bose-Einstein distribution function, while S and Imχ are the
T T
equilibriumformsof(15)and(18). Inparticular,sincetherelationbetweenthetime-reversednonequilibriumspectral
functions S (q,ω) and S (−q,−ω) is not known, Im[χ (q,ω)] is not determined by S (q,ω) alone. It is
TU TU TU TU
possible, of course, to define unknown dimensionless functions N>(q,ω) and N<(q,ω) such that
−2h¯
S (q,ω)= N>(q,ω)Im[χ (q,ω)] (21)
TU TU
n
−2h¯
S (−q,−ω)= N<(q,ω)Im[χ (q,ω)]. (22)
TU TU
n
Equations (19)and (21)-(22) then imply
N>(q,ω)−N<(q,ω)=1. (23)
Adopting the notation N<(q,ω)=N(q,ω), we have from (14),(21), and (23),
4
−2(v∞)2
Winel(k,k−q)= q [N(q,ω )+1] Im[χ (q,ω )]. (24)
k,k−q TU k,k−q
Ω¯h
Equivalently,theBornaproximationforWinel canbeexpressedintermsoftheimaginarypartofthetotalscreened
Coulomb interaction v /ǫ between the probe particle and the electron-phonon gas,
q TU
−2v
Winel(k,k−q)= q [N(q,ω )+1] Im ǫ−1(q,ω ) . (25)
Ω¯h k,k−q TU k,k−q
(cid:2) (cid:3)
This follows from the relations between the total susceptibility χ (q,ω), polarization P (q,ω), and dielectric
TU TU
function ǫ (q,ω) for the nonequilibrium coupled system,
TU
P (q,ω)
χ (q,ω)= TU (26)
TU 1−v∞P (q,ω)
q TU
ǫ (q,ω)=ǫ 1−v∞P (q,ω) . (27)
TU ∞ q TU
The relation (v∞)2Imχ (q,ω)=v Imǫ−1 follows fr(cid:2)om (26) and (27),(cid:3)and yields (25) from (24).
q TU q TU
The weight factor N(q,ω) in (25) plays the role of the distribution function for the nonequilibrium excitations of
the coupled electron-phonon system. Just as for the equilibrium case in equation (20), the structure factor has been
written asa product ofa function (Im[ǫ−1]) describing the strengthofinteractionswith the electronphonongas,and
T
afunctiongivingtheoccupationprobabilitiesforthegasexcitations. However,itshouldbeemphasizedthattheFDT
has not been used to derive (25) and N(q,ω) is an unknown function.
C. Separation of the total interaction
The totalnonequilibrium structure factorS defined by (15)is exactly separableinto a purely electronicpart Se
TU U
plus the remainder Sph which includes electron-phonon and ion-ion Coulomb interactions. Using the nonequilibrium
U
version of the total density operator (3),
S (q,ω)=Se(q,ω)+Sph(q,ω), (28)
TU U U
1 ∞
Se(q,ω)= dteiωthρ (q,t)ρ (−q,0)i, (29)
U ΩN U U
Z−∞
1 ∞ n¯h 1/2
Sph(q,ω)= dteiωt Zq hρ (q,t)A (−q,0)i (30)
U N 2Mω Ω U U
Z−∞ ( (cid:18) TO (cid:19) h
n¯h
+ hA (q,t)ρ (−q,0)i +(Zq)2 hA (q,t)A (−q,0)i . (31)
U U U U
2Mω
i TO (cid:27)
This section relates Se and Sph to corresponding electron and phonon parts of the response function, just as (21)
relates the total structure factor to the total response function. To do so, equation (25) must be split into electron-
electron and electron-phonon parts. Imposing the RPA for the dielectric function is a crucial step towards this goal.
The response of the coupled system in the RPA is the sum of the electron and ion response taken separately [9,20].
In this approximation, the total screened interaction v /ǫ is the sum of a purely electron-electron part and an
q TU
electron-phonon interaction that includes a phonon self-energy due to the polarization of the conduction electrons
[21].
v v∞ M2
q = q + q D (q,ω). (32)
ǫ ǫ |ǫ |2 U
TU U U
The electron-electron part of the total interaction is v∞/ǫ , where ǫ is the nonequilibrium RPA dielectric function
q U U
determined by electron-electron interactions only. The polarization for the noninteracting electron gas P(1) has the
same Lindhard form as the equilibrium case, except that the nonequilibrium electron distribution function f is used
instead of the Fermi-Dirac function [3].
ǫ (q,ω)=1−v∞P(1)(q,ω) (33)
U q
1 f(p)−f(p+q)
P(1)(q,ω)= . (34)
Ω ¯hω −¯hω−iδ
p,p+q
p,σ
X
5
The remaining part of the total interaction is a product of the screened electron-phonon matrix element M2|ǫ |−2,
q U
and the nonequilibrium phonon Green’s function D (q,ω) containing the self-energy correction M2D(0)χ to the
U q U U
noninteracting Green’s function D(0).
U
¯hω2 1 1
M2 =v LO − (35)
q q2ω ǫ ǫ
TO (cid:18) ∞ 0(cid:19)
D(0)
D (q,ω)= U . (36)
U 1−M2D(0)χ (q,ω)
q U U
> >
Now define unknown dimensionless functions N< and N< to relate the electron and phonon components of
e ph
S (q,ω) in (28) to the respective components of the imaginary part of v ǫ−1 in (32).
TU q TU
−2h¯ >
(v∞)2Se(±q,±ω)= v∞N<(q,ω)Im[ǫ−1(q,ω)] (37)
q U n q e U
−2h¯ M2 >
(v∞)2Sph(±q,±ω)= q N<(q,ω)Im[D (q,ω)]. (38)
q U n |ǫ |2 ph U
U
Finally, using (14),(28),(37), and (38),
−2
Winel(k,k−q)= v∞[N (q,ω )+1] Im ǫ−1(q,ω )
Ω¯h q e k,k−q U k,k−q
(
(cid:2) (cid:3)
M2
+ q [N (q,ω )+1]Im[D (q,ω )] . (39)
|ǫ (q,ω )|2 ph k,k−q U k,k−q
U k,k−q )
Within the RPA, the expressions (25) and (39) for Winel are equivalent. The last expression contains two unknown
functions,N andN ,playingtherolesofthenonequilibriumoccupationfunctionsfortheexcitationsoftheelectronic
e ph
and lattice components of the coupled system. Notice that N must describe a distribution with boson properties.
e
This is because the density fluctuation ρ† produces excitations of the electron gas which conserve the total number
q
of electrons, such as electron-hole pairs and plasmons, which have boson properties.
III. ELECTRON BOLTZMANN EQUATION
The time rate of change of the nonequilibrium electron distribution f(k) due to collisions is determined by the
differential scattering rate W(k,k−q); i.e., the probability per unit time for the transition |ki → |k−qi. The
depletionrateoff(k)isjustW(k,k−q)weightedbytheprobabilitiesthat|kiisoccupiedand|k−qiisunoccupied,
.
summed over all possible final states. The collision term, −{f (k)} , in the Boltzmann equation is this depletion
coll
rate minus the replenishment rate determined similarly. The electric field in the sample is F(r,t) = −∇[U(r,t)+
hU (r,t)i],wherehU iistheaverageinducedpotentialresultingfromthesystem’sresponsetotheappliedpotential
ind ind
U. Thus, for a homogeneous system under the influence of a static electric field F,
−eF ∂f(k) .
· =− f (k)
¯h ∂k coll
1 n o
= W(k,k−q)f(k)[1−f(k−q)]−W(k−q,k)f(k−q)[1−f(k)] . (40)
Ω
Xq n o
W(k,k−q)isasumofratesforelasticandinelasticprocesses,W =Wel+Winel. Usingtheseparationoftheinelastic
rateW in(39),theinelasticcontributiontothecollisiontermseparatesintoelectron-electronandelectron-phonon
inel
collision terms,
. . .
{f}inel ={f}ee +{f}ep . (41)
coll coll coll
6
A. Electron-electron collision term
The purely electronic component of Winel in (39) gives the collision term
.
−{f (k)}ee =
coll
2
− v∞ [N (q,ω )+1]Im[ǫ−1(q,ω )]f(k)[1−f(k−q)]
¯hΩ2 q e k,k−q U k,k−q
(
q
X
−N (q,ω )Im[ǫ−1(q,ω )]f(k−q)[1−f(k)] . (42)
e k,k−q U k,k−q
(cid:27)
ThenextsteptowardsputtingthecollisiontermintoausefulformistoeliminatetheunknownfunctionN ,whichgives
e
the occupationprobabilities ofthe density fluctuationexcitations ofthe electronsystem. This stepis straightforward
if an association is made between N and the nonequilibrum plasmon distribution, denoted here by N . In steady
e p
state, there exists a local detailed balance (LDB) between plasmons and electron-hole pairs which can be used to
eliminate N in favor of f in the electron-electron collision term [3,22]. The LDB condition is equivalent to the
e
FDT for the nonequilibrium ensemble [23] which applies when χ determines the response not to external forces, but
to internal fluctuation forces. It holds when the Landau damping rate γ = 2ImP(1) is large compared to the time
variation of f, so that the plasmons come into equilibrium with the electrons before f changes appreciably. For the
mobility problem, f is stationary and the condition alwaysholds exactly, except at small q values (where γ vanishes)
which do not contribute to the electron collision integralbecause of energy and momentum conservationrestrictions.
If LDB is used in combination with the “collisionless damping” approximation [3], equation (42) is equivalent to
the standard single-particle electron-electron collision term, but screened with the dynamic RPA dielectric function.
This may be shown easily for the case of weak damping, in which the plasmons are well-defined excitations, by
considering the Boltzmann equation for the nonequilibrium plasmon distribution N (q)=N (q,ω ) with ω =ω (q)
p p q q p
[22,24]. Plasmonsarerepresentedby the peaks in the spectralfunction Imǫ−1. Inthe RPA,this function containsno
U
corrections for collision broadening, so that the relaxation of N (q) is completely described by coherent absorption
p
and emission of electron-hole pairs.
N.p (q)= πvΩq2ωq [Np(q)+1]f(k+q)[1−f(k)]
(
k
X
−N (q)f(k)[1−f(k+q)] δ(h¯ω −¯hω ). (43)
p k,k+q q
)
.
In steady state, N =0. Therefore, the relation between the distributions N and f is necessarily
p p
n
N (q)Im[P(1)(q,ω )]= S(1)(q,ω ) (44)
p U q 2h¯ U q
2π
Im[P(1)(q,ω)]= [f(p)−f(p+q)] δ(h¯ω −¯hω) (45)
U Ω p,p+q
p
X
4π¯h
S(1)(q,ω)= f(p+q)[1−f(p)]δ(h¯ω −¯hω), (46)
U N p,p+q
p
X
(1) (1)
relating the response function Im[P ] and structure factor S for the noninteracting nonequilibrium electron gas.
U U
The detailed balance relation between plasmons and electron-hole pairss simplifies the electron-electron collision
term (42), if we assume N =N . Using the RPA interaction Im[ǫ−1]=v∞|ǫ |−2Im[P(1)] and (44-46) gives
e p U q U U
. 4π (v∞)2
−{f (k )}ee = q δ (E +E −E −E )δ(k +k −k −k )
1 coll ¯hΩ3 |ǫ (q,E1−E3)|2 1 2 3 4 1 2 3 4
2,3,4 U h¯
X
× f(k )f(k )[1−f(k )][1−f(k )]
1 2 3 4
n
−[1−f(k )][1−f(k )]f(k )f(k ) , (47)
1 2 3 4
o
7
when the notational changes k → k ,p → k are made, the summation variable q is changed to k = k −q, and
1 2 3 1
a summation over a delta function for momentum conservation is included. Notice that the nonequilibrium inverse
dielectric function was needed to obtain the collision term with nonequilibrium distributions for all four (incoming
andoutgoing)electrons. Theremainingfactorof|ǫ |2 inequation(47)appearsonlyinits equilibriumform|ǫ|2 ifthe
U
Boltzmann equation is linearized with respect to the field strength F, since the other factors vanish in equilibrium.
Also notice that ǫ rather than ǫ appears in the electronic dielectric function ǫ (q,ω) (see equation (33)) which
∞ 0 U
screens the interaction in the electron-electron collision term (47), unlike some previous calculations using ǫ for
0
electron-electron scattering in GaAs [11,13].
Itshouldbeemphasizedthat(47)includestheeffectofelectron-plasmonscatteringthroughtheresonanceinǫ(q,ω)
at ω =ω . For dc transport, the two-step process:
q
electron−hole pair → plasmon → electron−hole pair
is equivalent to a dynamically screened electron-electron scattering event because of the detailed balance between
plasmons and electrons. Since (47) conserves total electron momentum (in the absence of Umklapp processes), its
effect on the electrical current in materials with spherical energy surfaces is nonvanishing only because it rearranges
the momentum distribution, which influences other scattering processes. The most important difference between
using (47) and a previous treatment [25] of the effects of electron-plasmon scattering on mobility is that the earlier
work assumedan equilibrium (Bose-Einstein)distribution function for the plasmons. In the latter case,there is a net
momentum loss from the nonequilibrium electron system.
B. Electron-phonon collision term
If the phonons interactedonly with electrons by non-Umklappprocesses,they wouldreacha localdetailed balance
withtheelectronsjustastheplasmonsdo. But,infact,thephononsinteractwithotherphononsthroughanharmonic
lattice forces. The excess momentum given to the LO phonons by the nonequilibrium electron system is dissipated
primarilythroughinteractionswithacousticphonons,whichcanlosemomentumtotheenvironmentthroughumklapp
processes. Generally, mobility calculations have employed the “Bloch assumption” that the phonon system may be
treatedasifitwereinthermalequilibrium[26],validwhenanharmonicprocessesaremuchfasterthanelectron-phonon
scattering.
Adopting the model of equilibrium phonons, the electron-phonon component of Winel in (39) gives the collision
term,
.
−{f (k)}ep =
coll
2 M2
− q N0(ω )+1 ImD(q,ω )f(k)[1−f(k−q)]
¯hΩ2 |ǫ (q,ω )|2 k,k−q k,k−q
q U k,k−q (
X (cid:2) (cid:3)
−N0(ω )ImD(q,ω )f(k−q)[1−f(k)] . (48)
k,k−q k,k−q
)
The electron-phonon matrix element is screened by only the electronic part of the total dynamic dielectric function.
The long-wavelength,static limit of the RPA ǫ(q,ω) is the temperature-dependent Thomas-Fermi dielectric function
withthe high-frequencydielectricconstantǫ . Therefore,equation(48)inthislimitgivesasomewhatdifferentresult
∞
than treatments of electron-LO phonon scattering using Thomas-Fermi screening with the static dielectric constant
ǫ [12]. For the nondegenerate case, the long wavelength limit of (48) is in agreement with the original treatment of
0
screening in polar semiconductors by Ehrenreich [27].
The ion-ion Coulomb interactions included in H˜ (equation (1)) make a self-energy contribution to the phonon
g
Green’s function [10]. The real part of this contribution shifts the LO phonon energy from h¯ω to h¯ω in the
TO LO
denominator of the noninteracting Green’s function D(0) = −2ω /¯h(ω2 − ω2). Damping due to anharmonic
TO LO
interactions and disorder is neglected here.
The interacting phonon Green’s function
D(0) 2ω /¯h
TO
D(q,ω)= = (49)
1−M2D(0)χ(q,ω) ω2−ω2 −2ω M2χ(q,ω)/¯h
q LO TO q
contains the self-energy term Πep = M2D(0)χ due to the polarization of the electron gas. The real part of Πep
q
renormalizes the LO phonon frequency, giving rise to two hybrid modes of mixed electron-phonon character. The
8
coupled modes have been treated previously [18] with the finite-temperature plasmon-pole approximation for χ [17].
In this approximation, the electronic excitation spectrum which couples to the phonons is represented by a single
mode withenergy varyingbetweenh¯ω = 4πne2/m∗ǫ atsmallq andh2q2/2m∗ atlargeq. The plasmoncoupling
p ∞
strength and energy are determined by requiring fulfillment of the f-sum rule and zero-frequency Kramers-Kronig
p
relation, so that the plasmon-pole approximation for χ is [17,18]
1 ω2
χpp(q,ω)= p (50)
v∞ω2−ω˜2
q p
ω˜2 =ω2[1−ǫ−1(q,0)]−1. (51)
p p
Using the plasmon-pole susceptibility χpp in the phonon self-energy Πep gives a phonon Green’s function with poles
at the two frequencies ω and ω ,
+ −
2ω (ω2−ω˜2)
TO
D(q,ω)= (52)
¯h(ω2−ω2)(ω2−ω2)
+ −
ω2 = 1 ω2 +ω˜2± (ω2 −ω˜2)2+4ω2(ω2 −ω2 ) 1/2 . (53)
± 2 LO p LO p p LO TO
n (cid:2) (cid:3) o
In the plasmon-pole model for the coupled modes, the phonon spectral function −πIm[D(q,ω)] appearing in the
electron-phonon collision term (48) has delta function peaks at ω and ω ,
+ −
Im[D(q,ω)]=Im[D+(q,ω)+D−(q,ω)] (54)
πω (ω2−ω˜2)
Im[D±(q,ω)]=∓ TO p [δ(ω+ω )−δ(ω−ω )]. (55)
¯hω (ω2 −ω2) ± ±
± + −
At low densities, the low-energy hybrid mode has frequency ω− close to the uncoupled plasmon frequency ω˜ , but
p
has nonzero weight in the phonon spectral function determined by the factor (ω2 −ω˜2)/(ω2 −ω2). Even though
− p + −
this factor is small at low densities, the low-energy mode has an exponentially larger thermal occupation N0(ω )
−
at low temperatures than the high-energy hybrid mode or uncoupled phonon mode do. Therefore, for low densities
and temperatures, the presence of N0 in (48) suggests that the mobility calculated by including scattering from the
coupledmodesshouldbelowerthanthemobilitycalculatedbyusingtheuncoupledmodeapproximation. Inaddition,
“antiscreening” effects due to the dynamic nature of the system response can generate screened potentials that are
actuallygreaterthanintheunscreenedcase[27]. Reference[7]presentsnumericalresultsforn-typeGaAswhichshow
that including dynamic screening, mode coupling, and electron-electronscattering does significantly lower calculated
mobilities in situations where ionized impurity scattering does not dominate, as in modulation-doped structures.
IV. CONCLUSION
The total dielectric function ǫ (q,ω) lends itself to a simple and general method for deriving the inelastic collision
T
term in the electron Boltzmann equation for scattering from a coupled electron-phonon system. When the Born
approximationis valid,theinelasticdifferentialscatteringrateWinel canbe expressedintermsofthenonequilibrium
totaldielectricfunctionǫ ,whichincludesscreeningbybothelectronsandphonons. IntheRPA,Winel separatesinto
T
a purely electron-electron interaction plus an electron-phonon term describing interactions with the hybrid modes.
Theseinteractionsaredynamicallyscreenedbyonlytheelectrons. Reference[7]investigatestheeffects ofthe phonon
self-energy, dynamical screening, and electron-electron scattering on the mobility in n-type GaAs by means of a
numerical solution of the linearized Boltzmann equation.
Because the total dielectric function approachto the Boltzmann equationtreats the electrons and ions in a unified
manner,itcanbeappliedquitegenerallytoawidevarietyofcoupledsystems. Moreaccurateandcompletetreatments
[8,9] of the lattice part of ǫ−1 could be useful for more complex electron-phonon systems. Also, it would be very
T
interesting to try to include electron-impurity scattering through the ǫ (q,ω) for an imperfect crystal with local
T
phonon modes. Another logical direction to pursue would start from a dielectric function which goes beyond the
RPA to include electron exchange effects in order to derive the exchange scattering term in the Boltzmann equation.
However, a systematic approach to improving on the RPA is required, since exchange corrections can be largely
canceled by correlation terms.
9
ACKNOWLEDGMENTS
IthankP.B.Allen,D.R.Penn,G.W.Bryant,G.D.Mahan,S.DasSarma,andJ.R.Lowneyforhelpfulconversations.
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[19] This method is explained in section 6-2 of reference [1].
[20] B.B. Varga, Phys. Rev. 137, A1896 (1965).
[21] The derivation given in reference [10] for theequilibrium system also holds for thenonequilibrium case.
[22] H.W.Wyld and D.Pines, Phys. Rev. 127, 1851 (1962).
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48, 852 (1979).
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[26] J.M. Ziman, Electrons and Phonons (Clarendon Press, Oxford, 1960).
[27] H.Ehrenreich, J. Phys. Chem. Solids 8, 130 (1959).
10