A Wignerfunction Approach to Phonon Scattering

We consider the motion of a single electron under phonon scattering caused by a crystal lattice. Starting from the Fr6hlich Hamiltonian in the second quantization formalism we derive a kinetic transport model by using the Wigner transformation. Under the assumption of small electron-phonon interaction we derive asymptotically the operator representing electron-phonon scattering in the Wigner equation. We then consider some scaling limits and finally we give the connection of our result to the well known BarkerFerry equation.


INTRODUCTION
If a semiconductor is modeled as a perfect crystal, the electrons moving in the crystal are not scat- tered by the lattice atoms at all.Because of ther- mal energy the atoms do not remain stationary but each atom moves in a region of space centered at its lattice point.The strong forces which are pro- vided by the interaction of an atom with all the other atoms act on this atom when it is not at its lattice point.This leads to lattice vibrations which can be approximated by harmonic oscillations.
The independent normal modes of these oscilla- tions are called phonons which can be considered as particles (bosons, cf. [7, 9]).
Two interaction processes occur: the electron can be scattered such that either a phonon is emit- ted or a phonon is absorbed, where in both pro- cesses the total wave number remains constant.Due to these scattering events the number of phonons is not conserved.
To deal with this non-constant number of particles one uses the procedure of 2nd quantiza- tion, which was originally introduced in quantum *Corresponding author, e-mail: frommlet@math.tu-berlin.dere-mail: marko@majestix.numa.uni-linz.ac.at *e-mail: ringhofer@asu.edufield theory.So although our model is purely non- relativistic we use the formalism of field quantization [3].
For the sake of simplicity we use a one-electron model which means that we neglect electron- electron interactions and are just interested in the dynamics of one electron.We will also neglect the effects of the periodical potential of the station- ary lattice thus obtaining the so called Fr6hlich Hamiltonian.Moreover we model the phonon Hamiltonian such that it has states of thermal equilibrium.
In this paper we will start with the von-Neumann equation for the density matrix and obtain (by applying the Wigner transform) a kinetic pseudo- differential equation for the Wigner-matrix.By taking the trace over the space of the phonons we are led to a quantum-transport equation for the Wignerfunction of the electron with a scattering term which still depends on certain elements of the Wigner matrix (not only on its phonon trace).
To obtain a "closed" equation for the Wigner- function we assume that the interaction between the electrons and phonons is weak and we apply methods of asymptotic analysis (small coupling parameter).In the limit (no interaction at all) the phonons are assumed to be in a state of thermal equilibrium.Doing so we can finally derive a transport equation with a scattering term describ- ing the electron-phonon interaction.
In chapter 4 we discuss some scaling limits of the derived scattering term and in the last chapter we present a new derivation of the Barker-Ferry equation.

THE MATHEMATICAL MODEL
In the 2nd quantization formulation we consider a modified version of the Fr6hlich-Hamiltonian H--He + Hp +He-p, where He is the Hamiltonian for the free electron, Hp for the free phonons and He-p describes the interaction between the electron and the phonons.
The physical interpretation of ](n)(x, ql,..., q,)[2 is the probability of finding the electron in an infinitesimal neighbourhood of x (electron posi- tion space) and n phonons in an infinitesimal neighbourhood of q(n) (phonon momentum space).
The electron position density is given by 1 and the current density by where his the Planck-constant and m* the electron mass.
In the phonon-Hamiltonian w(q) is the real valued phonon-frequency and as a modification of the usual Hamiltonian we introduced the pseudodifferential operator Z(Dq) which describes the phonon-phonon interactions.The mathematical reason for introducing Z(Dq) is that we would like to have an orthonormal basis (ONB) of U consisting of eigenfunctions of Hp, which can be physically interpreted that the phonons are driven into states of thermal equilibrium.If we use the definition of the annihilation and creation operators then Hp can be written as + (2) 3h l= (q' q) @n)(x' q"q(n))dq' the sequel we set 27=1 c1 0. Finally the electron- phonon interaction Hamiltonian is given by ( -iqx) He-p ih F(q) aq e iqx a + e dq q (2.5) where the term with the annihilation operator models phonon absorption and the one with the creation operator models phonon emission.The real valued function F(q) describes the details of the electron-phonon interaction.Again using the + the interaction term reads definitions of aq and a q (He_p) Remark Since V, 2, F and co are real-valued easy calculations show that the Hamiltonian H is formally self-adjoint (for He-p see also [6], p. 209f, Segal quantization).If we further assume that F(p) -F(-p) then the Fockspace Schr6dinger equation is time reversible in the sense that the Hamiltonian H commutes with the symmetry operation q()--q() and conjugation.
To study the dynamics of the system we introduce the density operator p:S S which fulfills the von-Neumann equation

ihpt [H, p],
where [A, B] := AB-BA denotes the commutator of the operators A and B. The operator p is self- adjoint, positive and trace-class, therefore there exists an ONB {Ptl lE N} of eigenfunctions of p such that 1=1 where denotes the Fourier transform of the function Z. is supposed to be real valued.Here and in where #z>0 are the corresponding eigenvalues.
which determine the density operator p by It is easy to show that the eigenfunctions of p fulfill the equation 0 ih-pl Hpl (2.6) which gives the connection between the von- Neumann dynamics and the Schr6dinger picture (cf.[4]).Using (2.6) we obtain the equation 0 q(m) t) ih -t r(n'm)(x'q(n) ;y' Z# (Hpl) (m) (y,p(m),t)p}n) (x,q(n),t) /=1 p(m)(y,p(m) t)(Hpl)()(x,q() t)" 1 (2.7) which describes the dynamics of the density matrix.As a first step to obtain a transport equa- tion for the electrons we introduce xr,p ,t eVdr.
In the right hand side of (2.11) the subdiagonal elements W(n, n+ 1) are still present, which means that we do not have a closed equation for w(x, t).

WEAK ELECTRON-PHONON INTERACTION
To derive an approximating closed equation for w(x, v, t) we now assume that the electron-phonon interaction is small.Therefore we write in (2.5) e F(q) instead of F(q) with 0 < e << and treat the problem with methods of asymptotic analysis for e 40.The now e-dependent Wigner matrix is solution of the transport equation LW QpW + eQe-pW e (3.1) where we introduced the Wigner transport opera- Qp and Qe-p are now considered to act on the Wigner matrix W as defined in (2.8).For W we make the ansatz For the initial condition we assume We(t= O) wtA (3.3)where w/ w(t 0) is a given (Wigner) function of x and v and A is defined as the density matrix corresponding to the operator (cf.trp(e _OHm0 0 is the phonon operator acting on Ors).The (Hp exact definition of A will be given after we have introduced a special ONB in Lemma 3.1.The operator T describes the phonons in a state of ther- mal equilibrium, where/3 is a constant (indirectly proportional to the lattice temperature).Note that T is normalized such that trp T-1.
We make the following assumption on H " p.
(A1) w(q) and 2(q) are such that there exists an ON B of real valued eigen-functions k E N} in L 2(N3q) and eigenvalues )k E N such that h q, hw(q)bk(q) + (27r)3 3 q , (q )bc(q')dq' A(q).
Note that (A 1) holds if growth conditions on w(q) and on Z=Z(x) at x=q= are imposed (confinement of phonons).The eigenfunctions can be chosen real valued because w(q) and 2 are real valued.With this assumption we have the following LeMMa 3.1 If (A1) holds, then there exists an o ONB of s consisting of eigenfunctions of Hp.where k-(kl,... ,kn) is a multiindex and pn is the permutation group of n elements.Because of the structure of Hp it is obvious that o (n) ?) Hp (q(")) A with k A A, j=l Aj being the eigenvalue corresponding to bkj.By definition the functions br:(ql,..., q,) are invariant under p.ermutation of the arguments and therefore Or: := '/0,..., 0, On)(q(,)), 0,... ) 9Vs.Adding the vacuum state o := (1,0, 0,...) we finally find that the so constructed set {Or:} is an ONB of 9rs because (O)e is an ONB of L a(R3), cf. [5].Now we use the ONB constructed above to represent the matrix A of the initial condition (3.3).To obtain a unique representation we use only multiindices k-(kl,...,k) ordered such that k <_'" <_ kn.We thus obtain Vq5 Ors" Using the definition of the operator T we can write o(m)(p(m))A(n,m)(q(n), p(m)) dp(m) m=l where we have A(n,n)(q(n),p(n)) =Trl Z e-'(n)(P(n))b?)--k (q(n)) and from (3.3) we obtain the initial conditions W(t O) wA wl(t=0) W2(t--0)--W3(t--0) 0.
Using the first equation of (3.4) and the initial condition for W the separation ansatz W n,m W O(X, V, t)Mn,m (qn, pro) shows that M A, i.e., W --wA.A short calculation gives QpA O, therefore also Qp W O.
In the following we will use the convention that for a superscript c E {0, 1,2, 3, e} the function w will denote the phonon trace of the matrix W , i.e., w := trp W.

Remarks
Note that this notation is consistent for W because of the normalization of A, trp A 1.
The interpretation of the structure of W wA is that if there is no electron-phonon interaction at all (i.e., 0) the phonons will be in a state of thermal equilibrium which is given by the operator T.
If we take the phonon traces of Eqs.We have already used trp(Qp W) 0 and another calculation shows that trp(Qe_pW) 0. Actually this can be seen easily by taking into account the fact that A(n,m)--0 for n Cm and therefore (Qe-p wO)(n,n) 0. Vn > 0. With a similar argument one can see that trp(Qe_pW2) 0. So taking into account the initial conditions (3.5) we have found W is now calculated from

QpW
Qe-p (wea) ( which is the second equation of (3.4) with w replaced by w.If we are able to solve Eq. (3.7) for W we will have a closed equation for w which is exact up to the order e4.To do so we assume now V=const. which means Oh[V] =--O.In this simple case Eq. (3.7) becomes 0 O-t W1 -+-V x W1 QpW Qe-p (wea) which can be solved explicitely by means of a separation ansatz and the variation of the con- stants formula.Using the orthogonality properties 0 of the eigenfunctions of H p one .finallyobtains after long calculations trp(Qe-pW , Re ei(q-elx+i-.F(p)F(q)ea (e-'XD-e-r:D)}dqdpd- (3.8)   with F (p) := F(p)2 (p) and D e -i-P'qTw e x v q T, + t- e iz-P'qT w e x--v + -m, q Ol :---e-i2-P'qrw e x--v----m, q T, e i2--P'qrw e x V + 2m* q -' ) , v + --;m, p + q), t--So in the case of V=const.we obtain the transport equation for w where we have to replace D1 and D2 in the expression (3.8) by 0 Ot w + v. 7xW 210 scat nt-O(E4) (3.9)where I is the term on the right hand side of scat (3.8).

Remarks
The most important property of I is the non- scat locality in time.This expresses the fact that the scattering term has a memory of the whole his- tory of the states of the system, i.e., phonon scat- tering is nonlocal in time when a fully quantum viewpoint is taken.
An easy calculation shows that if Fj.(q) is either symmetric or antisymmetric, i.e., Fj.(q) Fj (q) or Fj. (q) Fj. (q), then Eq. (3.9) (without the o(ea)-term) is time reversible (i.e., the equation is invariant under the transformation t--t, v-v).F has such symmetry properties, for example, if F is antisymmetric and j is symmetric or antisymmetric which is the case if co and 2 are symmetric.
Using this notation we obtain So we can again solve Eq. (3.7) by the method of characteristics and by similar calculations as in the case of V =-const.we derive the transport equation ] iVv 2 kj) x--7-v7-, 2m* + (e-e-/ #f (X B2 vr, 2m, i,2 w x-r vr, v +Br, t-r dr (3.11)where the notation Tv,2 signifies that the PDO acts only on the second argument of w.

SCALING LIMITS
We shall assume in this chapter that Tr 1.For the independent variables we introduce the scaling t-u, x=7, V=Tv, q=a and for the other occuring quantities we have we(x, v, t) E AE, F(q) AF'(), / Apfc.

vm* AE
Setting u (7x/7v), "yx (h/m*'Tv) and "yv (hAe/ m*Ap) we obtain the scaled equation where we have dropped for the scaled quantities.The scaling is chosen such that cr is in- directly proportional to the strength of the electric field, is proportional to the scaling parameter of q and e is proportional to the strength of the electron-phonon interaction.

Limit 1
Taking the limit cr 0 (which means we consider strong electric fields) we formally obtain the limit- ing equation wt + v. Vxw--E.VvW

Remarks
Here we assumed that Fkj(q)= F(q)qdk;(q) is symmetric or antisymmetric.Then it follows that the operator f f maps real functions into real ones and the operator 6f lf maps real func- tions into purely imaginary ones.Note that AFffUce (3/2).Assuming that u and a are constant, r 0 obviously implies 40 and thus (e2/r) 0. If we then take the limit n-0 (small wave vectors q) the PDOs become differential operators 6f tx,Vv, kj w(x, v,t) itVx f (0, kj).VvW(X, v, t) + O(t 3) We set r (electric field strength of order one) and take first the limit 0 which leads to the approximation wt + v. VxW-E.VvW __E2/2 f COS(/kjT.)VvT 7-=0 MV,zw x+--w-, v+Er, t-r dr.
with the matrix n=0 E n+l kl kn+l n+l (e-a + e-r )G (R) G, j=l G F(q)(q) q dq.Using the equality f h({ + r)h({) d{ dr =0 we can proof easily that the scattering term in (4.3) is dissipative.We obtain another simplified scattering term if we are only interested in the equation for small times.

BARKER-FERRY EQUATION
In the physical literature the Barker-Ferry equa- tion is quite well known (see e.g.[2]).It is a transport equation for an electron in a constant electric field with a scattering term describing the electron-phonon interaction for the space homo- genous case.To derive this equation we make the assumption that the function F of the electron- phonon interaction Hamiltonian (2.5) is a random variable F F(q, a) (where a varies in an appro- priate measure space) such that F(p)F(q) R(q)6(p q) where R(q) is a real valued function and (.) denotes the average with respect to a.This means that the random variables F(p) and F(q) are uncorrelated for p q.
Repeating the asymptotic expansion of chapter 3 we note that the effect of the random interaction Hamiltonian is of order e.Therefore it is reason- able to assume that w = w(x,v,t) does not depend on a.If we define := w then taking kEN <'..<_kn n--1,2,... with Tr := 2, , e-/a and because of the <_...<kn construction of the ONB we have A(n,m)(q(n), p(m)) 0 n m.Plugging the ansatz (3.2) into Eq.(3.1) we obtain by equating the coefficients of equal powers of e: Qe_pW 1) + O(E4)where we have of course w= trpW= w+ 2 W 2 _If_ O(4).The case V= const.