Time-convolutionless generalized master equations and phonon-assisted hopping

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Time-convolutionless generalized master equations and phonon-assisted hopping

V. Čápek

To cite this version:

V. Čápek. Time-convolutionless generalized master equations and phonon-assisted hopping. Journal de Physique, 1983, 44 (7), pp.767-774. �10.1051/jphys:01983004407076700�. �jpa-00209657�

Time-convolutionless generalized

^master

equations ^and phonon-assisted hopping

V.

010Cápek

Institute of Physics ^{of Charles} University, Faculty ^of Mathematics and Physics, 12116 Prague, Czechoslovakia

(Reçu le 14 octobre 1982, accepté ^{le 18 mars} 1983)

Résumé. ²⁰¹⁴ Un nouveau formalisme de projection indépendant ^du temps ^a été récemment appliqué à l’étude des transferts électroniques induits par des phonons ^{dans des} semiconducteurs amorphes, utilisant des équations

maitresses à convolution de temps. ^{Dans ce} formalisme il a été impossible ^{de démontrer les} équations ^Marko-

viennes habituelles à tous les ordres dans la constante de couplage phonon-électron g, c’est-à-dire pour des valeurs finies de g. Le même formalisme est appliqué ^{ici à des} équations maîtresses sans convolution de temps : le résultat est le même ; ^{on montre} qu’il ^est impossible de démontrer les équations cinétiques habituelles (appliquées ^couramment

dans les théories de transfert) ^{à tous les} ordres, quoique la démonstration formelle à l’ordre deux soit élémentaire et justifiée.

Abstract. ²⁰¹⁴ A new time-independent projection ^{formalism has} recently ^been applied ^{to the} study ^of phonon-

assisted hopping ⁱⁿ amorphous semiconductors using the time-convolution master equations. ^A negative ^result

was obtained with regards ^{to the} possibility ^of deriving the usual Markoffian rate equations working exactly up ^to infinite order in the electron-phonon coupling ^{constant g, i.e.} keeping ^{finite values} of g throughout the calculations.

The formalism is applied ^{here to} the time-convolutionless master equations ^{with the same result : a} proof ^is given

that working exactly up ^to infinite order, ^no possibility ^{exists of} deriving ^{the rate} equations currently ^{used as a} starting point in theories of the phonon-assisted hopping although in second order in g, their formal derivation

seems to be at hand and straightforward.

Classification Physics ^Abstracts

72.20

1. Introduction.

Attention has

recently

been directed to the basic

principles

^{of the}

phonon-assisted hopping

^conduction

in the

mobility

gap of

amorphous

semiconductors

(namely

^{to the}

question

^{of the}

validity

of the usual rate

equation) starting

from first

principles, keeping

finite values of the

electron-phonon coupling,

^and

trying

^to avoid the usual Markoffian

approxima-

tion [1, 2]. ^{Some time} ago, ^we

developed

^a

theory

of the dc

hopping conductivity

[3, 4] which does not

rely

upon the Markoffian

approximation

in kinetic

equations,

^{and which}

keeps

the proper order of

limiting prodesses

^{in its more} advanced form

[5].

This

theory

is in nice

qualitative

^{as well as}

quantita-

tive agreement ^with

experiment [5-6].

Nevertheless,

this

approach

^{based on the Kubo}

theory

^{was later}

strongly

criticized from the

point

of view that the result for Q

formally

coincides with the

high=frequency

limit of the

low-frequency

^rate

equation

result for the

conductivity

[7, 8]. By ^{the rate}

equation (RE)

^one

means the usual Markoffian gain-loss

equation

with constant

(in time) hopping probabilities W mil’

^or

equation (1) already

linearized with respect ^{to the}

acting

^field

[7-11] ; fm(t)

is the site

occupation proba- bility

^{at the time t.} This criticism

(justified

^as

regards

a

change

of order of

limiting

processes in the

prelimi-

nary version of the

theory [3, 4])

^{was then}

^commonly accepted although

^{the need was}

immediately

^stressed

of

including higher

^{order terms}

[12]

^{in order to mach}

the more advanced forms of our

approach [5, 12, 13].

In the lowest order in the

electron-phonon coupling,

the derivation of

(1)

^is

unquestionable

and has been

performed

many times

[7,

^8, 10]. ^{In a} way, it is not

surprising

^{that one} gets (1) in the lowest

(second)

order in the

electron-phonon coupling

^{constant g}

since the second order

in g

^means

keeping just

^those

terms that are relevant in the van Hove limit

or

(io

^{and w}

being

the time scale and

frequency)

^{and in}

this limit,

general

methods of the

non-equilibrium

statistical mechanics

justify

^the

validity

of the Pauli

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004407076700

768

Master

Equation (PME,

from which (1) ^follows

immediately)

^{for a rescaled time} [14, 15]. So, ^our attention has been drawn to

higher

^{order terms in} g,

namely

^{to the}

problem

of whether it is still

possible

to

perform

the Markoffian

approximation beyond

^the

second order in g

taking

^{into account that, in}

reality,

g is

always

^finite. The latter fact means that the limit

g - 0 (if ^{taken at}

all)

^{must be}

performed

^{at the very} end, ^{after the cho - 0} limit in the dc calculations, ^unlike

the van Hove limit

(2a-b). (It

^{means that both PME}

and

(1)

^are also open ^to criticism

regarding

^{the incor-}

rect order of

limits.)

^The

general opinion

^of experts ⁱⁿ statistical

physics

^of

non-equilibrium phenomena

^is

that the Markoffian

approximation

ⁱⁿ

higher

^order

theories

might

^{then lead to} uncontrollable results [15, 16].

Anyway, general warnings

^{of this} type ^{have been}

commonly ignored

^as there has been no indication of such an uncontrollable result in

hopping

^theories.

We believe, however, ^{that we have}

recently

^found

such an

example : starting

from the Liouville

equation

(H ^and p

being

the Hamiltonian and the

density

matrix,

respectively)

^and

deriving (1) using

^{the time-}

independent projection

^method

applied

^{to the time-}

convolution Generalized Master

Equations

(TC-

GME) yields

^no

heating

^{effect on cold}

phonons

^{due to}

originally

hot electrons with a finite concentration

once the Markoffian

approximation

^{is introduced}

by

^«tour de force » and

working exactly

^{to infinite}

power of g [2]. ^This example ^{makes the}

possibility

^of

deriving

(1)

by

this method rather

illusory

^{in the}

infinite order in _g, although in the second order in _g, the

reasoning

^{seems to be sound} enough.

Recently,

however, ^some general works have _appear- ed

[15]

which allow the time-convolutionless Genera- lized Master Equation

(TCL-GME)

^{to be derived}

from

(3).

^Since (1) ^{does not} contain the time-convolu- tions either, ^{the use} of the TCL-GME method to derive

(1)

^{seems to be more} natural than the use of the TC-GME method. This

problem

is the main aim of the present

study.

However, ^{what we} get ^{in this} way is that here, ^as in the TC-GME formalism,

a) the derivation

of (1)

^is easy

keeping

^terms just ^to

the lowest

(second)

^{order in} g,

b)

ⁱⁿ

higher

orders, ^relevant

problems

arise,

c)

^a general

proof

may be

given

that, ^for

physical

reasons, ^no derivation of

(1)

from the TCL-GME is feasible when

working exactly

with all orders in _g.

In this connection, ^{one has} to realize that _any kind of the GME that is exact up ^to any finite order

in g

^is (if ^at all)

applicable just

^{in a} time interval that is limited from above

[17].

So, ^similar

heating experi-

ments cannot in fact be

applied

^to the lowest (second)

order version of GME, ^{which is a fact that} protects (1)

from this kind of criticism. On the other hand, ^one gets from it that (1) ^is

applicable

^{at most}

at just

^finite times, ^i.e.

(1)

^cannot be used in the dc theories as

independently

stressed in

[12].

2. Time-convolutionless

generalized

^master

equation

for electrons.

Let us

specify

the Hamiltonian of electrons in the

mobility

gap of amorphous semiconductors as

Here B.

^{are the site}

energies,

Wt is ^a

general

^harmonic

phonon frequency,

^{and Q is a}

normalizing

^volume.

Spin

indices are

suppressed

^{for the sake of}

simplicity.

^We

prefer

^{to work with}

standing phonon

^modes

(not necessarily

of the

plane-wave character),

^{i.e. the}

combination b.

⁺

b’

appears in

He _

ph and The Hamiltonian

(4)

may be

easily

^{rewritten as}

Here S is a formal small parameter ^{of the}

overlap

^{of the localized states in the}

mobility

gap and H is that part ^{of H} that is

responsible

for the transfer. On the other hand,

Ho may be easily diagonalized

with the result for

eigen-

functions

[4]

and

energies

Here,

{m}

⁼

{m1, m2, ... } designates

^{the set} of the electron

occupation

^numbers

(mi

^{= 0 or} 1) ^while

( p )

⁼

{

Ilk1’

Jlk2’ ... }

^{is the set of the}

^phonon occupation

^numbers

(Pk

^{= 0, 1,} 2, ...). ^It is clear that

(7a)

includes the

polaron

effect; (7c) includes both the

polaron

energy shifts (terms ^{with i} = j in the last ^term on the r.h.s.) ^and

the electron-electron interaction due to

exchange

of virtual

phonons

^{(i :0 j}

terms). Working

^with

(7a-c),

^we

introduce S as small parameter ^{of the}

problem

( g may be finite and

arbitrarily large

^{while the}

overlap

^is

always

less than 1).

Since the _power of g ^is

always higher

^{than or}

equal

^{to that of S}

(compare (6b-c)), working

^{in the} representa- tion

(7a-c)

^{means a}

partial

^{summation to} the infinite order in _g.

In

describing

^the

dynamics

^{of our} system, ^{let us start from a} time-convolutionless

identity

(see [16,18-22])

^{which is}

^mostly

^{due to Shibata, in the form}

corresponding

^to

[16]. Equation (8)

^{is an immediate}

consequence of the Liouville

equation (3)

and reduces to it when 0 ⁼ 1. In

general D

^{is an}

arbitrary (possibly

even

time-dependent [22]), projection

superoperator

obeying

^the

idempotency requirement

Further

and

Finally,

^{for an}

arbitrary

operator

A,

where the sign ^{« .:. »} always ^means the interaction

(Dirac) picture,

^i.e.

Now, ^{let us}

specify

^our

projection

superoperator ^{D. We set}

(see [23]) where p{v}

^are

^arbitrary

^{except that}

770

which ensures that

(9) applies.

^Therefore

where

is a

probability

^of

finding

the electron

configuration { m}

^{at the time t,}

irrespective

^{of the state of the}

phonon sub-system.

Therefore, ^{with the} choice

(18)

ofD, ^we _get ^from (8) the TCL-GME in the form

where

and

Up ^{to now, we have not}

specified p{v}.

^Now,

^according

^to Boltzmann, ^{let us set}

and let us assume the initial condition

One _may

easily

^{see that}

(17) implies

Physically,

(17) ^{is satisfied if for}

example

the electrons have a certain

configuration

^{at t} = to ^{and the}

phonons

are in a thermal

equilibrium.

^The

coefficients p{v}

^{in (16) are then}

^nothing

but the Boltzmann

weights

^{of different}

phonon

^states in this initial thermal

equilibrium

situation. The

phonon density

^{matrix at t} = to ^{then reads}

confirming

^the

previous

statement. As a consequence

of (16)

^and

(19),

(with fl being

introduced in

(16))

is the initial

(at t

⁼

to) phonon

temperature. Thus, the initial conditions

(17)

enable us to get rid of the initial condition term

(18)

^{as well as to} introduce the temperature ^{into the}

problem.

Now, ^{let us turn our attention to the}

WtmHn}(t, ^to)

coefficients. If

they

^were

time-independent, (14)

^would

coincide with the Pauli Master

Equation;

^{thus the}

question

^{of the time}

dependence

^{of W s is as crucial as the}

problem

^{of time}

decay

of memory functions in the TC-GME

[1, 2].

^Let us once more recall

that f

^oc _gS, ^so that to the lowest order in gS

(but partially

^to

arbitrary ordering)

Using properties

^{of our}

projection

superoperator

we may ^turn (21) into the form

Setting it back into

(13)

^we get

3. Lowest order result for ^«

hopping

^{rates ».}

The matrix elements

JC(mit)tnv)

^{have been}

⁽ⁱⁿ

^the

representation given by (7a-c))

calculated in detail in

[24].

First,

notice that in the second order in S,

W{mHn}(t, to)

^{turn to zero once the}

configurations { m }

^and

{ n }

^differ

by

^a

position

^{of more than a}

single

^electron. Therefore, ^we may write

where the

single-electron

^{transfer s} -> r is the

only

^{one in the}

change

^{of the}

many-electron configuration { n }

-->

{m }.

Still, however,

Wrs

in fact remains

dependent

^on the location of other electrons not

taking

part ^{in the} transfer s -->

r( { n }

^{- m}

}).

An

important

conclusion of

[24]

is that for such

single-electron

^transfers

where I

designates

the number

of phonons taking

part

(being

^{absorbed or}

emitted)

^{in the transfer}

{ nv }

->

{ m,u }.

Since in the

thermodynamic

limit and for extended modes, any

phonon occupation

^{number can}

change

^{at most}

by

^{± l,1 also} determines the number of

phonon

modes for which the

occupation

^number

changes [24].

It should be mentioned that all M(l) contributions are oc

S2 ,

^{but as a function}

of g, they

^{are of a different}

order.

Namely,

^{all M(l) are at least oc}

g4

S2 except ^the

single-phonon

contribution

(l =

1), ^{which is oc}

g2

^S2.

So, first, ^{let us take}

just

^this

single-phonon

contribution into account. It

yields

Realizing

that in the

thermodynamic limit

dk and that

772

we have in the second order in g the usual second order

pertubational

^result [7-11]

with

Setting (29a)

^into (24a), ^{we recover the} Pauli Master

Equation

from which the

rate-equations (1)

^{follow immedia-}

tely using

^{the definition}

So, ^{there are no}

problems

with the derivation

of (1 ) in the

^{lowest order in} g. Now, ^we

proceed

^to

higher

^orders.

Two- and

more-phonon

contributions

yield

^no

qualitative change.

Problems do arise, however, ^{when the}

pho-

nonless contribution is evaluated. It reads

It is seen at first

sight

^{that even after the}

thermodynamic

limit,

W("I.I(t, ^to)

^{has no limit (as a}

^function)

^when

t ^- to ^{-+ + oo.} Therefore,

starting

from the order

S2 g4 (or

^even

performing

the summation over powers

of g

to

infinity keeping

the second order in S

exactly,

^i.e.

working

^{in the order}

S 2),

^the

possibility

^of

deriving

^{the PME}

and the rate

equation (1)

^becomes

illusory.

One should stress that the terms

O(S’ g6)

^{on the r.h.s.}

of (31 )

^{do not}

remedy

^{the situation}

(compare

their form in

[24]).

4.

Higher

order effects in S.

Till _now, we have worked _up to the second order in S and

arbitrary

power of g. Therefore, ^{a natural} question

arises whether

higher

^{order terms in S cannot}

improve

the situation. We do not at present ^see any reliable _way of

calculating

^{the «}

hopping

^{rates »}

W{mHn}(t, ^to)

^{to any chosen} order of S.

Anyway,

^{in our}

opinion,

^{there is a}

proof

at hand

showing

^{that even if it was}

possible

^to

perform

^such calculations, ^{the rate} of convergence of the limit

(if

^{there is a} convergence ^at

all !)

^{is so} small that it does not

help

ⁱⁿ

deriving

the usual form of the rate equa- tion

(1).

First, ^{let us} accept ^{that we have a form of}

W{mHn}(t, ^to)

^{exact to any power of S before}

performing

the thermo-

dynamic

limit. In order to make conclusions as

general

^as

possible,

^{let us assume that we have also a certain}

kind of the

phonon-phonon scattering

Hamiltonian included. Let Sl be a common

normalizing

volume for both electrons and

phonons,

^{and let}

aM

^{be its}

macroscopic

part. ^{Let us} introduce the

probability

to find the electron

configuration {

^m

}M

^at sites inside the volume

SlM at the

^{time t,}

irrespective

^{of the}

configura-

tion at sites i outside

QM.

One should notice that in the limit Sl -> ^oo

(but

^with

QM remaining finite) P{m}M

^{is a}

well-behaving (intensive) quantity,

in contradistinction to

p{m}.

^{From (24), one easily gets that}

where

is the «

hopping

^{rate » for}

hops

inside the volume

Om

^and

A (-)m is a

correction to

hops

^{in the} very

vicinity

^{of the}

surface

of Om

^{or across} the surface. For

physical

^{reasons, it is} clear that

A {m}M(t, to)

is irrelevant to

establishing

^the

local

equilibrium

inside the

macroscopic

^volume

f2m,

^{so we shall}

drop

^{it from now on.}

At this stage, ^{let us}

perform

^the

thermodynamic

^limit

with

remaining

finite. Here

Ns and Ne

^{are the} number of sites and number of electrons, i.e. cs and ce ^are the site and electron concentrations,

respectively.

^The

important thing

is, ^{we assume that} cs ^as well as c. ^to be finite and nonzero, i.e. the numbers of

degrees

of freedom of the electron and

phonon

systems ^are

comparable. During

^the

limit

(35),

^the

macroscopic

^volume

Om

^is

kept

^finite.

Now, ^{let us assume that the} convergence

(if any)

^of

W{mHn}(t, ^ro)

^with

increasing

^{t is}

sufficiently quick

^{so that}

one may ^set

(after

^{a short}

period

^of initial times of the

length tl )

instead of W in

(33)

^and still,

arguing that Om is sufficiently

^{large, one may}

drop A (m)m

ⁱⁿ

^(33).

The latter _assump- tion means that the

asymptotic

^values

W are

achieved well before the

time tM

^of

establishing

^{the local}

equili-

brium in

Om.

^We

always keep t

^- to tM ^{here. Under these} conditions,

(33)

^{turns to}

By ^its structure,

(37)

^is

nothing

but the Pauli Master Equation

(PME).

From the mathematical theory ^of

the PME, it is known that it always

gives

^the

tendency

^{of the} system ^{to an}

equilibrium

which is in turn

determined uniquely

^{as the}

eigenstate

^of

with zero

eigenvalue.

^{Since the W s}

only depend

^{on the initial}

phonon

temperature

(20),

^{the same}

applies

^{to the} W’s and the

W’s

i.e. the local

equilibrium

^state

only depends

^{on the initial}

phonon

temperature

(20).

Up

^{to now, we have not}

specified

the initial conditions. For instance, ^we may choose the electron system

to be _very hot

(or

very

cold)

^as

compared

^to

phonons

^{at t =} to. Still, ^as may be verified, ^{it does not} exclude the

validity

^of

(18).

^{Due to} the finite electron concentration, ^one expects

(according

^to

general

^{laws of}

thermody- namics)

^a certain intermediate

(between

the initial temperatures

of phonons

^and electrons,

T in

^and

^Tinel)

^tempe-

rature to result in the local

equilibrium

^{state. However, we have} deduced above that the local

equilibrium

^tem-

perature

(the

^same for electrons and

phonons)

^is

always

^{the same} for any

T n depending just

^on

Tph

^{and in}

reality amalgamating

^with

T in .

^{This is a} relevant contradiction

showing

^that

W{mHn}(t, to)’s

^{do not, in}

reality,

achieve

time-independent

values in any

physically

reasonable time interval.

Consequently,

^the

validity

^{of the}

PME

(for

finite values of the

coupling,

^i.e.

beyond

^{the van Hove limit}

[14,15])

^{and the rate}

equation (1)

^remains

dubious except

possibly

for limited time intervals

[17].

Acknowledgments.

The author is indebted to the International Centre for Theoretical

Physics,

Trieste, ^{for their}

hospitality

^and

financial grant

during

^his stay ^{at the}

Workshop

in Solid State

Physics, 1982. During

^this stay, when the main ideas of this work

began

^to

jell,

the author also

profited

from discussions with Professor P. N. Butcher and Dr. L.

Binyai

^{and Dr. A. Aldea, which}

helped

^{him to}

improve

several formal

shortcomings

^{in the} arguments.

774

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