Unified Theory of Phasons and Polaritons in Charge Density Wave Systems

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Unified Theory of Phasons and Polaritons in Charge Density Wave Systems

S. Artemenko, W. Wonneberger

To cite this version:

S. Artemenko, W. Wonneberger. Unified Theory of Phasons and Polaritons in Charge Density Wave Systems. Journal de Physique I, EDP Sciences, 1996, 6 (12), pp.2079-2119. �10.1051/jp1:1996206�.

�jpa-00247299�

Unified Theory of Phasons and Polaritons in Charge Density

Wave Systems

S-N-

Artemenko (~)

and W.

Wonneberger _(~,*)

(~) ^Institute of

Radioengineenng

and Electronic8 of the Ru88ian

Academy

of

Science8, Mokhovayall,

103907 Moscow, Russia

(~)

Department

of

Physics, University

of Ulm, 89069

Ulm, Germany

~

(Received

30

September

1996, accepted 4 October

1996)

PACS.72.15.Nj

Collective _modes

je-g-,

in one-dimensional

conductors)

PACS.71.36+c Polanton8

(including photon-photon

and

photon-magneton interactions)

PACS.73.61-r _Electrical properties of

specific

thin films and

layer

8tructure8

(multilayer8,

8UperÎattlCe8, _{qUarltUm WeÎÎB,}

Wlres, aria dOtS)

Abstract. A unified approach to

long wave-length

phasons and

phason-polaritons

in _semi-

conducting

_quasi one-dimensional

charge density

_wave

(CDW)

_{systems is}

given.

Using Keldysh

kinetic

theory

in trie quasi-cla8sical approximation~ expres8ions for linear CDW _{currents and} CDW phase _are derived

microscopically taking

into account bath elastic and inelastic quasi- particle collisions. From

these,

_a dielectric tensor for CDW is constructed which _covers the fuit range of collision _cases. The dielectric tensor is built from

longitudinal

and transverse conduc-

tivity

functions and

generalized

condensate densities which

are

microscopically

defined.

They depend

Dot

only

_{on wave} number,

frequency,

and temperature but also _on elastic and ine1a8tic

collision _rates for

quasi-particles.

Trie dielectric tensor _is combined with Maxwell's equations.

This allows

a systematical

study

of

phason

and

phason-polariton

_spectra.

Especially,

trie hmits of collision dominated and collisionless quasi-particle _{gas are} considered. A detailed cornparison

with earher work is made. The

continuity

of the spectra at the wave-number

origin

is found to froid

generally.

Phasons _{in wires} and films which _cari be

investigated

by _microwave and optical

probes

are aise studied.

1. Introduction

Phasons _are trie linear excitations of trie order parameter

phase

in

charge density

wave

(CDW) systems. They

_were introduced

by

Overhauser

Ill

^for three-dimensional metals where

exchange

effects and electron-electron correlations _{can cause a} CDW

instability.

We restrict ourselves to

phasons

in quasi one-dimensional conductors where CDW _are the result of _a Peierls transition.

A _review of

CDW-physics

_{in quasi} one-dimensional conductors _is given in [2].

Theoretical

investigations

of

phasons

_{in quasi} one-dimensional systems ^have a

long history.

In their seminal paper

of19î4, Lee, Rice,

and Anderson [3] calculated the

longitudinal

acoustic

(LA) phason velocity

_ci at _zero

temperature

and for _one dimension.

They

then

pointed

out that

(*)

Author for

correspondence je-mail: wonnebergerflphysik.uni-ulm.de)

©

Les

Éditions

_{de Physique 1996}

Coulomb interactions will raise the

phason frequency

to _a

longitudinal optic (LO) frequency

uJLo at _zero

temperature.

Lee and

Fukuyama

_[4] ^{considered these Coiilomb effects for} an array of

chains, 1.e.,

^for the real three-dimensional CDW but still at _zero

temperature. They

found _a

phason spectrum

^{which is} discontinuous at the wave-number

origin

due to the

peculiar coupling

of

charge

to the

phase-gradients along

tire chains.

Rice, Lee,

^and Cross [5]

pointed

out the

importance

^of

quasi-partiales (qp)

for the

dynamics

of CDW. This raised the

question

about the detailed role of

quasi-partiales

^{and their}

^screening

effect _on

phasons.

Kurihara [6] calculated the

screening by

_qp in _a one-dimensional mortel and _{gave a} formula for the

phason dispersion

which shows _a continuous transition from _an acoustic

phason

at

high temperatures

to an

optic phason

at _zero

temperature,

^controlled

by

the

Debye screening length

of the qp. In his

treatment,

^he

neglected

_qp collisions, _{i e.,} their

dissipation.

In their

general

kinetic

approach

to CDW

dynamics

^{which took into account elastic}

qp collisions, Artemenko and Volkov I?i addressed

briefly

the

problem

of

phason dispersion.

At low

temperatures, they

found the

LO-phason

with

positive

curvature. Near the transition

temperatures, they

^calculated an acoustic

phason velocity

similar to a result in [5] and to our later

findings.

Nakane and Takada _[8] extended Kurihara's work to three dimensions and studied the relation of the

phason

to the Carlson-Goldman mode _in

superconductors [9].

^This work also

neglects

_qp collisions. It contains,

however,

the qp

plasma

mode

present

^{in this} limit and at certain intermediate temperatures. ^In a

subsequent

_paper,

Wong

and Takada

[loi

used _{a more}

general

^{Green's function}

^{approach including}

^elastic collisions.

They

also _{gave a}

prescription

how to deal with the

hydrodynamic

limit when fast inelastic

scattering produces

a local

equilibrium

^of the qp gas. But in _essence,

they

still considered the collisionless _case where

they

found _a coexistence of the usual low

temperature optical

mode with _a

longitudinal

acoustic mode.

However, strong

^Landau

damping

affects both.

Some years

later,

^{Artemenko and Volkov} [11] gave a more detailed

application

^{of their}

general

kinetic

theory

of CDW

dynamics _I?i

to the

phason problem. They

^{considered elastic}

and inelastic _qp

scattering

^{and their influence} on the

phason spectra

^{in different}

simplifying

limits.

Recently,

Virosztek and Maki [12]

applied

their thermal Green's function

theory

of CDW

land spin density

_waves:

SDW)

also _to the

problem

of collective modes

taking

into account the

long-range

^Coulomb interaction.

They

calculated

amplitude

and

phase propagators

in the

collisionless limit and

presented

_a wealth of results _on both spectra. ^{Some of them will be} mentioned later. In their

approach,

the

concept

^of

generalized

condensate

density

_[13] which not

only depends

_on

temperature

but also _on

frequency

and _wave

number, plays

_a central role.

Brazovskii

[14]

^{also took} up the

subject.

He constructed _a local

phase Lagrangian

for both CDW and SDW from

general arguments

^like chiral invariance

il 5,16].

He considered

dissipation

ma a

conductivity

function for _qp, but his

approach

^is

basically again

for the collisionless _case.

He also cnticized most of the earlier work _on mathematical _as well _as

physical grounds.

A

systematic

treatment of both elastic and inelastic

scattenng

processes and their relevance is thus desirable.

On the

experimental side,

_very few results _on

phason

spectra are available.

Here,

the method of choice is neutron

scattering.

^In

[17],

^the

overdamped pinned phase

mode coula be detected

near the transition

temperature.

^In

[18],

^{the Coulomb}

stiffening

of the

longitudinal phason velocity

_was measured.

Ii is noted that the basic

physics

of

phasons

in CDW _can be related to that of the

phase

oscillations _in

superconductors:

In both

systems, charge density perturbations

_are

possible

at low

temperatures

^{when the number of} qp is small. This allows

plasma-like

oscillations with

frequencies

above the dielectric relaxation

frequency

of _qp. In

CDW,

these _are the

LO-phasons.

In

superconductors, they

_are ^the

plasma

oscillations. In conventional

superconductors,

these

plasma

oscillations _are well above the _gap. The full

analogy is, however, preserved

in

layered superconductors

where

plasma

oscillations _in the direction

perpendicular

to the

layers

_are below the gap

[19, 20].

At

higher temperatures,

_{qp screen} the

charge density perturbations imposing charge quasi-neutrality.

This _converts LO-modes into LA-modes. In the collision dominated

case, these modes _can be

underdamped. They

then _are called the Carlson-Goldman mode in

superconductors

_[9] and the

LA.phason

in CDW.

The present work

gives

_a

comprehensive microscopically

based _treatment

oflong-wave phason

spectra

using quantum transport theory

in order to deal with the

complicated

_qp

dynamics.

We extend earlier work of Artemenko and Volkov

[7,21]

based _on the

Keldysh

method

[25] by

_ex-

plicitly including

inelastic

scattering

and _we set up the lineanzed

equations

for current densities and CDW

phase.

From

these,

_{we construct}

a

frequency

and _wave number

dependent

dielectric

tensor. In _{contrast to} the standard

approach,

where the

phason

_{spectra are} determined _from

the

poles

of the

phason propagator, they

_are here obtained _{as zeros of the} determinant _{of a} tensor

simply

related to the dielectric _tensor

by incorporating

Maxwell°s

equations.

The _com- ponents ^{of both} tensors _are controlled

by longitudinal

and transverse

conductivity

functions ai,t and

complex

valued functions

Bi

and

Bt

which

generalize

Brazovskii's B-function. All

these

quantities

_are

microscopically

defined and

depend

also _on elastic

(vf,vb)

and inelastic

lue) scattenng

rates. This allows to treat the different limits from

hydrodynamic

to col- lisionless in _a unified _way. We also

study phason-polaritons.

These have been mentioned earlier

by

Littlewood

[22]

^without

going

into any detail.

Long.wavelength phason-polantons

can

possibly

be

investigated experimentally by

_means of

electromagnetic

measurements in the microwave and far-infrared

region (cf. [23]).

In

detail,

Section 2

gives

_a theoretical

description

how the

Keldysh

method [25] is

applied

to CDW

dynamics

and how inelastic collisions _can be included into the formulation

[21]

^of CDW

transport theory.

Inelastic

scattering

_was considered

by

Eckem

[24]

in the

purely

_one-

dimensional _case. However, _a

fully

realistic treatment must be

three-dimensional, especially

when

phason-polaritons

_are to be studied. After the formulation of the basic

equations

in Section

2.1,

the inelastic collision

integral

is introduced in Section 2.2. It

brings

the inelastic collision rate _{as a new} parameter into the

theory.

In Section

2.3,

CDW current densities and the

phase equation

_are evaluated in full detail for

semiconducting

CDW in the semi-classical and pure limit when the gap is not

destroyed by impunties.

The opposite limit of _a

gapless

CDW state is less

significant

for real CDW materials. The central result of this work is the dielectric tensor constructed in Section 3 from these

equations by eliminating

the

phase. However,

these formulae

are so

complicated

that separate discussions in

hmiting

cases become necessary: In Section

4,

_we

study phasons

and

phason-polantons

at _zero

temperature

and establish the close

analogies

with standard

phonon-polariton theory.

In Section 5, _we concentrate _on the

usually neglected

collision dominated

case when inelastic

scattenng

^{is effective. This is}

normally

the

case for

temperatures

in the region

T~/2 _~

T < T~

(T~:

Peierls transition

temperature).

Analytical

results _near T~ _{are given} in Section 5.2. Section 5.3 presents numencal results for

a

simplified

mortel of the

temperature dependence

of the many parameter functions _appeanng

in the

theory.

We

clarify

the

softening

and eventual

vanishing

of the

LO-phason

and the

subsequent

_appearance of the

LA-phason

when the

temperature

is increased. Section 6 reports

analytical

results for

phason

and

polariton spectra

_in the collisionless hmit and makes contact to the work

[loi. Finally, phasons

in limite

geometry

are studied in Section 7. In _wires and

films,

trie spectra are very different from their bulk form and should be accessible to _microwave

and

optical probing.

2. Basic

Tl~eory

The CDW

system

^under

study

will be described

by

the

following

^mean-field Hamiltonian:

H

"

Hei

₊

Hph

+

Hei-ph

+

Himp

+

H~,

Hei

_"

~j _e(p)a)

ap,

P

H~h

₌

~jw(q)b)b~,

q

Hei-ph

₌

(a)~~ ^apA

+

c-c-j

+

~j

_g~

_a)~~

_{ap b~ + c-c-}

,

p.q~+Q A

=

gQ(bQ+b+~),

H;m~

=

~j

_U~

_a)~~

_ap,

p-q

u~ jjx» jj

₌

~ d3x vjx

_x»

^expjiqx),

~

Î

H~

=

~j

4l~

a)~~

_ap.

pq

In these

expressions,

A =

~exp(içs)

is the mean-field CDW order

parameter

^and

V(x)

is the

single impurity potential. Q

denotes

the CDW _wave vector associated with the Peierls

phonon.

The electric field is derived from

the electrical

potential

4l. The

potential

4l is the self consistent

potential

and accounts also for the Coulomb interaction between electrons at the RFA level.

2.1. KELDYSH METHOD _FOR CDW DYNAMICS. The

present approach

_uses the

Keldysh

technique

_[25] to set up transport

equations

for

generalized

distribution functions. Suitable moments of these then

provide equations

^{of motions for the}

quantities

of

interest,

_{i e.,} for the CDW

phase

_çS, the electric current

density

vector

j,

and the electric

charge density

_p.

For _a CDW

system,

^the

one-partiale

Green's function

iG(xi,fil x2.t2)

_%

(TC (i~(xi,ti)i~~ (x2,t2)))>

to which the desired kinetic

equations

are

related,

is

decomposed according

to

iG

(xi,

ti _x2, t2 "

~j _{iGap (xi,}

_{ti, x2,} t2 _exp

[1(apf

_xi

^flPFx2

₎₁

a,p

The momentum

representation

of the electron Green's function

Gnp

for the Peierls state set up in the

Keldysh

time domain is then

~~Îfl(Pl,tl> ^P2,t2)

#

(TC (anpf+Pi(~Î)~IÎpF+P2(~~))), ^(~)

where1, k

= 1

designates

the forward time and _i, k

= 2 the backward time

paths, respectively.

In

il ),

_pF is a vector in chain direction taken _as the _z-axis

(for simplicity

_{we assume} ^rotational

symmetry

^{around the chain}

direction),

1-e-, _pF " Pfez with

(pF(

" FFI Fermi momentum,

and _a,

fl

= +1 _are momentum indices

referring

to the two different parts ^of the nested Fermi

surface.

As in the

theory

of

superconductivity (for

_a review _see Rammer and Smith

[26] ),

the Green's function G is

represented by

4 _x4 matrices.

However,

the elements with _a

# fl signal breaking

of

continuous translation _invanance instead of gauge invanance as in the _case of

superconductors.

Following

Larkin and Ovchinnikov

[27],

^the elements of G _are

finally rearranged

to form

ÔR ÔK

~

o

ÔA ~, ^(~)

where

Ô~.~

are the retarded and advanced functions

(2

x 2

matrices) giving

the

density

of states, ^{while the}

I<eldysh

function

Ô~ _provides

_{the kinetics: The}

diagonal

parts _{give j} and _p and the

off-diagonal

_part is related to the

equation

of motion for the

complex

order parameter

À

as shown below.

In order to take into account the CDW

geometry

^of

parallel stacks,

_we

split

three-dimensional

momenta p into

longitudinal

_{components q} and _transverse parts ^k. The two

Dyson equations

then _are

(Ô0~(~l,~l.tli~2.k2,t2) É(ql,kl>tli~2>k2,t2)) @Ô(ql>kl,tl(~2,k2,t2) ₍₃₎

=

2~ô(qi q2)Sô(ki k2)à(ti t~)

₊

à, Ô(Ql, kl,

tli _~2,

k2, t2)

1

(Ô0

_~(Ql>

kl>

tli _Q2,

k2, t2) É(Ql, kl,

tli _Q2.

k2i t2))

" à.

(4)

The

product symbol

_{~g means} matrix

multiplication

in

Keldysh

_space, time

convolution,

and convolution with

respect

to momenta

including

transverse _ones. The latter _is clone

according

to

/d~k' _f> (5)

where S

=

(2~)~ lia)

_{is a measure} of the _cross section of the transverse Brillouin _zone and at is the distance between the chains. The self _energy from

scattering

_processes is denoted

É.

The inverse of the free

propagator

is

given by Ôp~

₌

Ôp~ lKeidysh,

where

Ô0

_"

~~ô

_{~~~~ ~}

_~~'~~~

_~

+

~~~~~~ ~~'~~~ ~~~~

_{~~~ ~~~}

à(tl t2).

o j-

ejq~

_p~,

ki) A~(pi

_P2.

ti) eo4l(pi

_P2,

ti) ôti

(6)

For

simplicity,

the

pretransitional

electron

spectrum

is assumed to have the form

e(p

+ pF "

+vfq

+

~(k),

where k _means

(k(.

^The transverse electron group

velocity

_is defined

by vt(k)

=

ô~(k)/ôk

with

(v))

_<

u)

because of

anisotropy.

The brackets

ii

_{mean an average over momenta on} the Fermi surface

according

to

(5).

Thus _{a term}

lvfqi+~jki)

^o

_j~

o _{v~qi +}

~(ki

appears in

(6),

in which _vfqi

Provides

_{a strong}

qi-dependence

_in

Ôp~

^{and hence} in

Ô.

_This

dependence

_can be eliminated

by setting

_q

= qi q2,

(

"

uF(qi

+

q2)/2, constructing âz(3)

(4)àz,

and

introducing

the function

provided É depends weakly

on

(.

Furthermore,

we set

ki,~

₌

kt

+

k/2

and

expand ~(k)

with

respect

to k and find the

quasi-

classical

transport equation

for the three functions

§~,~

and

§~

which form

# according

to the

prescription (2):

jj

^~

~jk~))à~§+ jàz(1

^~

-~(kt)) ((£~+eo4l)àz f §j (8)

àtl àt2

~~~~~~

~~

Î~~'~Î

+

Î~~~ ^Î

Here, àz and are the obvious extensions of

âz

and

Î

to the four-dimensional matrix space,

and _[,

]+

mean commutator and anticommutator,

respectively.

The

required

transport

quantities

are related to

§~

_ma

(cf.

^{Rammer and Smith}

[26]

^for a

detailed

discussion):

p(x, t)

=

ieo

Tr

Ô~ ix, t,

_{x, t}

),

2

j(x, t)

=

~°

(Vx ^Vx~

^Tr

^{Ô~ ix, t,}

^xi,

^t)x=x~,

4m

which _in the mixed

representation

of

§~

with

respect

to time and _space translate into

~~~'~~ ÎÎ Î ^21~

_{~~ ~~}

_{~~ ~~}

~~'~'

~~'

_~

~~ ~°~~~'~~Î

'

j(x, t)

=

~° ~~~~

de llr

(vfez

+

vtâz)§~(t,

e,

kt,

_{x =}

Ri

VF

Î

(~~)~

Î

Finally,

the

generalization

^of the gap

equation

is

required.

From the

equations

^of motion for

phonon operators,

the

equation

l~j~j2 ^t2

^~

^{~ ~} /

~~

/

~~~ ^{~~~ ~~ ~~~~~}

~ ~

for the mean-field _gap matrix

t o

À

~ "

-À*

_o '

is derived

[7,21].

As descnbed in

[21, 28], equation (8)

is solved

by performing

a kind of _gauge transformation which eliminates _q~

(in À)

and 4l form the1h.s. Then the matrices

§~.~

_are

obtained

perturbatively using

the solutions

gl'~

_{of the}

unperturbed

_case and

utilizing

the

quasi-classical approximation lu,

_{qvf, v, eo}

Îo grad 4l)

< A

((o

_{" vF}

IA: amplitude

coherence

length.

_v: elastic collision

rate).

With

(+

= +

fie

+

io)

where

(((e(

>

A)

=

@fi

sgn e,

nier<A)

=

11,

§~

is found

(§~

=

à~((+

-

Î-))

_to be

~~

eovfÉiA~ eovfÉiA

~ ~

ÎÎ (~ÎÎ _Q~VÎ)

^~~

_Î+ (~ÎÎ

_Q~

_Vi

^~~

~

l

~Î/

^~~~~

_ii i4Î~t2vi)

~?~~~

~

Ii _?i ^l~

~~~~

ii iiiÎ~~l2 _vii

^~

l~~~l~/1

il °z.

For

§~,

_{the ansatz}

à~

"

à~19

_ù ù _O

à~

with

à

= mil +

nzâz.

is useful.

Two kinetic

equations

for the _qp distribution functions _ni and _n= will be _given

taking

into account the inelastic collision

integral.

2.2. INELASTIC COLLISION INTEGRAL.

Going beyond

the _previous work

[21],

we include here

exphcitly

the

electron-phonon

collision

integral Iei-ph

from

long wave4ength phonons.

In

a one-dimensional

context,

this _was clone before

by

Eckem

[24].

In the above descnbed

quasi-classical

limit and for the usual _case of CDW with gap, when

(w,qvf, vi

< A

jg)

holds,

the

equations

for _ni and _{nz are}

given

_as

(2.5a)

and

(2.5b)

in [21] ^{with the} r-h-s- of

(2.5b) supplemented by Iei-ph.

In _{a more}

explicit form,

and for

(~(kt)(

«

A,

_{i e.,} for

semiconducting

CDW these

equations in

=

n(q, k,

_uJ;

kt, e))

_are after hnearization

l~?~Î

₊

^iiE))

~z +

~QUFnI

_"

i~°vFEi

_{+ ?~dUb}

fi~9) 1°, i10)

and

iwni

+

iqvfnz

₊

ikvtni

+

vt(e) [ni (ni)]

"

eovtÉt ^~~°

₊

_{Iei-ph. (Il)}

de

These _are the

transport equations

for _qp. The latter _are hnear combinations of electron states

with _momenta

differing by Q. Therefore,

_new

scattering

rates for _{qp appear}

~~~~

_~

le)

_~

_~ÎÎÎ)

'

~ ~~~ ~~

~

~~~

~ ~~

~~

V % Vf +

~,

2

which _are constructed from the bare elastic

scattering

rates vf

(forward scattering: scattenng

on the _same Fermi surface

sheet)

and _vb

(backward scattering: scattering

between nested Fermi surface

sheets). Here,

the

subscript

^denotes

scattering

of _qp in

longitudinal (chain)

direction and t the transverse

scattenng.

^{For A} -

o,

_vi reduces to _vb and _ut to vf + vb, the usual results for bare electrons. The

quantity

_no denotes the

equilibrium

value of _ni~

no(e)

=

tanh(e/2kBT).

The effective electric field E is defined

by

~2

~

_~~

~ ÎÎO~~Î

~

2eovF~'

^~~~~

Êt

= -ik 4l

-1~~

qçs

2eo

,

where _çS is the order

parameter phase

and 4l is the usual electrical

potential.

The

frequency dependent

terms _in

(10,

11,

12)

_are important ^{in the} collisionless lirait _w » v.

The inelastic collision

integral Iei-ph

contains the

expres8ions:

/~ deiw(e, cil (8(ei ci

_[n~~

il n~)(1+ N~~-~) n~(1

_n~~

)N~~-~

-cc

+8(e

_{ci [n~~}

il n~)N~-~, n~(1

_n~,

)(1

+

N~-~~ )])

In this

equation,

^which

clearly

describes first order rate processes between electrons and

phonons,

_n~

equals il ni)/2

and its

equilibrium

value is the Fermi distribution.

N~

is the distribution of

phonons

with energy e. The

scattering

kemel W is

given by:

W(e, ei)

_oc

g~(e ei(

^{"~ ~ ~}

e((e( A)e((ei1 A)tif(e ci).

ej(ei)

~

Here,

_g is the small _wave number limit of the relevant

electron-phonon couphng

constant and

tif

is the

phonon density

of states.

We _are interested in small deviations from

equilibrium,

_{i e.,}

Ici-ph

_can be linearized with

respect

to

ôni

_{= ni no " ni}

tallh(~).

e

The distribution functions _{ni, nz}

depend

_{on w, q,} and k _as well as on e and

kt.

The collision

integral

contains transverse averages

ii

_over the Brillouin _zone

according

to

(5).

This

suggests

^the

decomposition

ni " no +

(ôni)

+ nt, n= =

(nz)

₊

ônz,

where

(nt)

"

(ônz)

= o and leads to the linearized inelastic collision

integral

~~

~~ ^~ ~~~~ ~~

Î ÎÎ~T~~~~'~~~ _~~~~~~~~

~~~~~~~~~

ÎÎÎIÎ~ÎÎÎÎÎÎÎÎ

'

~~~~

cosh~(e/2kB~~

Wl~h

~ j~

e~j

=

W(e,fi)~~~hjje cil/2kBT)

The inelastic collision

integral (13)

can be made _more suitable for further calculations. In _a first

step,

we write

~~~~ ~~

ÎÎ

~~~

~~°~~'~~~ ÎÎÎI

2kBT Î~ÎÎÎÎ2kBT)

2kBT

ÎÎÎ~ÎÎÎ2kBT)1

Here,

_ve~ is the rate

la

=

O(1))

~~°~~'~~~

_~~~~~~~~

~~~°~~~~2kÎT~~°~~~~2/ÎT~

~

~~

~

ÎÎÎ ÎÎh(ÎÎ Î~~Î2kÎÎ)

~ ~~

ÎÎÎ SÎÎÎÎÎÎÎÎ~ÎÎ)

We _can suppress the energy

dependence

of the

scattering intensity

_ve~,

ve~(e, ci)

- ve~ = const., while still

maintaining

the basic structure of the inelastic collision

integral.

This is

strictly

correct for

high temperatures

when characteristic _{energies of order A}

are small

compared

to

kBT.

With

~~ e~°~~~ kBTcosl1(e/2kBTl'

the constitutive

equation (11) acquires

the useful form

iwni

₊

iqvfnz

₊

ikvtni

_{+ vtnt}

"

eovtEtn[ ₍₁₄₎

oo

~

-ve~

/ _dei

~

[n[(ei)(ôni(e)) n[(e)(ôni(ei))].

à

Î(Ei)

2.3.

EQUATIONS

FOR CURRENT DENSITIES _AND CDW PHASE. From _ni and _nz

(the

latter function vanishes _in

equilibrium),

the

expressions

for current densities and

phase

_can be obtained _as follows

Ji ~l Ii Il dei _inzi ~a~Jj i~ai ^Ei (là)

~~ 1

/

_~ ^e

~~

~~ ~fi~~tlÀt~'

~ ~0

f)

and

(suppressing

the

dynamics

of

A)

~~

_~

_~~~~~~ ~Î~ ^~~ ~Î~ ^~ Î

~~21e) 1

~

~~~ Î _~"~

~

iÎÎÎÎ~~~ ^~~~~'

^~~~~

In these

equations,

~ is the Thomas Fermi _wave number and

(ni ), (nz)

_are under8tood to be the

fluctuating

_parts of the

original quantities.

The "bare"

phason

velocities _{are ci} and _ct.

They

_are

_given by:

~ ~

~ ~Î _{~~' ~ ~Î ~~~'}

where the static condensate

density

N will be

given

below, The function

ci " ci

(q, T)

_in

(15)

and in ~ + 1

eiq~/~~

is found to be

(w~i

" vF~:

plasma frequency)

~~~~~~~

~~~~~ ~~~~~~~ ÎÎ(~ÎÎ~~

UÎQ~)

Î~ (~Î~

~ ÎQ~)Î

~~~~

Note that _ci is not

strictly

_a dielectric function. It describes virtual electron transitions _across the _{qp gap} and appears

only

_in, combination with the

chirally

invanant electric fields

[14].

^In

leading

order with respect to q and for A _»

kBT,

_ci reduces to

El(q, T)

_~

El(o, Tj~l(q> T)

_*

Eà(T)~l(~, T)

_~

E~1°)~l(~, °)

"

là Ii )~~fll'

~

An

important quantity

will tum out to be the bare

frequency

ofthe

longitudinal optical phason

îJLo given

by

£ôÎo

_"

CÎ(T)~~/fi

Except

at _zero

temperature

^{and for} q = o when it

equals 11

_uJq

,

it

is, however,

not the

physical frequency

of the

longitudinal optical phason

(À ^{is the} dimensionless

electron-phonon coupling

constant

responsible

for the Peierls transition and _wq the

corresponding frequency

of the

phonon

before it became

soft).

The static condensate

density

is

originally

^{defined within the}

^Keldysh approach by

~2

fiÎ # de

[~~~(E) ~~~(f)j tànll(f/(2kBT)). (18)

4

The expression

(18)

_can be

brought

into the _more convenient form:

~

ÎÎ

_~~

~~(Îe) %~

^{~~ ~°~~ ~}

^~~~

^~

^~~~~~~~~~

1

Tdln _A/dT'

^~~~~

This BCS type

expression

assumes that the chemical

potential

is at the

midgap position

e = o.

The chemical

potential

_{/1o can,} however, be shifted _away from the

midgap.

A uniform shift may appear due to electron-hole

asymmetry

^caused

by

the momentum

dependence

of the Fermi

velocity

and

by

that of the

phonon spectrum [29, 30].

^In these cases, the formulae

(18, 19)

for the condensate

density

N must be modified

by

the

replacement

_{e - e ~o.} This and the

smearing

of the

density

of states

by imperfect nesting

and

by impurities

makes the quasi-

particle density

N

dependent

_{on many}

parameters

^{of the}

crystal.

It _can thus be _more realistic to consider N _{as an}

adjustable parameter.

According

to

(15,16),

transverse averages ^must be

performed. By forming

moments of

(10)

and

(14)

with respect ^to

(ikvt)" In

₌

o,1), neglecting ((kvt )~nt),

and assuming

q~v)

< _vivt, the

following

solutions _are obtained for the

quantities in

_{~) and}

(vtnt).

VF

inzi

= eo

~ ni ^(Ve

₊

Dtk~)R i Il _{dei ni fi} ^{~~~~~ ~~~~~~j} _12°1

lvtntl

_{= eo}

) _{ni (fie}

+

Ôiq~ lF~

+

Ôt lk~ft k(kft

)1

DiqRk 121)

Ve

_~ /°°

^~~Î _ÎÀÙ^,

~

^ci

^Ôiq6k

⁺_y

^Ôtk(kft

à 1

Generalized diffusion functions have been introduced

according

to

with

(we

write

fie)

_-

(

for

simplicity)

ùi + V1

itde~/(~,

_ùt

= ut itd.

Furthermore,

effective electric fields _are defined _as

6

"

Ej ~~~Î

LÙ~ ~

~~

^{~ ~ ~~~~}

t~if

^{~ "}

~i

₊

~vb( p,

Ft

"

Et qk@

+

Ét.

Energy

relaxation is measured

by

_means of

Ve "

ver(1- Ni,

_fie

" Ve

iw, (22)

and abbreviates

vFl~/(2eo).

Finally,

the functions K and Y in

(20)

and

(21)

_are

given by

Y

= fie +

Diq~

₊

Dt k~, (23)

K

-

[

de

ni

^~

^~~~~

⁺

~~~

~~° i N _Ve

[

de

ni fi. ₁₂₄₎

With

(20)

and

(21), expressions

for the linearized current densities

(denoted by

the

prefix à)

and the

phase equation

_are obtained _as follows:

The

longitudinal

_current

density

is

The _{transverse current}

density

vector reads

ôlt ~2

=

i~Jpbt@

⁺

atik(kÉt _)/k~

₊

at~

(k~Ét k(kÉtl)/k~

₊

«itÉi. ₍₂₆₎

The

phase equation

takes _on the form:

v)(BiEi

₊

_BtEt

₊

_(w~

₊

iw~

w( c)k~) ~~ _jl

= o.

(27)

~~Q

It is

pointed

out that there _are important contributions from _qp to the

expressions

for current densities

25,26)

and to the CDW

phase (27). They

_are described

by

the conductivities _a and

by

backflow parameters b. These transport coefficients exhibit

strong

time and _space

dispersion

on the kinetic scale _{even in} the

quasi-classical

limit

lu,

_{qvf <}

A)

when the

corresponding dispersion

of the condensate _con be

neglected.

The backflow

parameter

bi descnbes the _qp

contribution to the current which is

proportional

to the CDW

velocity

_(oc

iwçs).

It contains both

an

equilibrium

part related to a decrease of the condensate

density

with

increasing temperature

and _a part related to the

ilonequihbrium

distribution function _nz. The latter is

responsible

for the contribution of the moving CDW to the Hall [31] ^{and to the} thermoelectric [32] ^effect.

For transverse coordinate

dependence,

the

analogous

contribution _is descnbed

by bt.

The

phase damping

function _~ also contains two distinct contributions: A direct

damping

due to

qp ^momentum

scattenng

^and

damping

from

dissipative

_processes due to qp ^currents induced

by

CDW oscillations.

In the

phase equation (2î),

_we have

phenomenologically

added _a pinning ^term

w(.

The latter

represents commensurability

_{pinning [3].} It also follows from the

phason

self energy due to random

impunties

in _a first order calculation

taking

into account the

metastability

of

static

phase

deformation _{on not too}

long

time scales. For weak

pinning

_{e-g-, an upper} bound

is _{t <} ti m

wp~ exp(ELR/kBT)

where

ELR

is the

pinning

_energy of _one Lee-Rice domain

[33].

Low

frequencies

are,

however,

outside the scope of the

present

work. A _more refined treatment of the influence of random

impurities

on the

frequency dependent conductivity

of

pinned

CDW

capable

to descnbe the low

frequency

relaxation modes _can be found in

[34-36].

It is also noted that in all

equations, Ei

_appears

only

_in the chiral combination with

-q~@

(cf.

[14]

).

The

transport

coefficients B in the

phase equation

_{are given}

by

~~

_~~

~ÎÎWÎ~~~~~

~~~~°~ ~~~~~~~~~~~Î'

~~~~

Bt

=

-bt jÎ

[(q~v) w~)«it

₊

qkv)auj

,

~ v~w

Conductivities _are defined

by

~2~2

OI

@

_{ai [Ve}

₍₁ Q~VÎ~II/Â)

lLÙ +

l~lk~j (~9)

~2~2

au "

~

at

lue (1- k~v)at/K)

iw +

Diq~ailatj

4~ ,

~

~~

j~2j

^~ ~~~< ^~

~~ 4~

Î

°

(2ùt

^'

and

~ ~

~lt

~

~)~ ^Q~al

^ÎVeVÎ~It

^/~

^~

~lÎ

_~~~~

The backflow

parameters

are:

bj =

vbd

_[Ve

(1 q~v)aj /K)

iw ₊

D2k~j

,

(31)

bt

"

-vbdqk(vev)at /K

+

D2)

The

phase damping

function

~(q, k; w)

in

(27)

is

given by

'i "

(

ÀL~( (Ve91 ^~

~~~t _Q~VÎVeÎI ~/~l'

_~~~~

Finally,

the

quantities

_ait, ^d and

Di,2

are defined

by

~ll_t

~

de

~0[fi'

_' ~

~

~~~~

~

ÎÎ

~~

~~ ÎÎ

^~

Î~'

_~~~~

Diffusion

type

coefficients

Di,2

_are

~~ ~l

Î~

^~~

^{~~ ~)}

^{' ~~~~}

~2 Î dfill ~~~'

F

In _{~, a new} Mass of coefficients _{appears which involves}

averages of

vile, uJ)Ôi,t

where _vi abbre- mates

vi(e,w)

=

ve/(

iw.

(36)

Î~ ~~

~Î

/3

^~~

~)r, (3~)

~ F

~' Î _{~~~~ ~}

~~

~~

~~

i _~~~Î ^~

~~

~~~'

F

For

completeness, displacement

_currents from

background

dielectric functions _must be added to the current

expressions (25,26).

Together

with the Maxwell's

equations,

the above

equations

constitute _a

complete

set of linearized transport

equations

for CDW. These

equations comprise

the

following

_cases:

ld < IV,Ve)>

II _{v~ < td < v,}

III (Ve, V) < ld,

where _{Ve is} the characteristic _energy relaxation rate due to

Iei-ph

Case

III,

_{1. e.,} the collisionless limit has been studied in the recent works

[12] though

the restriction is not mentioned

explicitly

in these papers. Case I is the

physically

most relevant _one for CDW at

high

and intermediate

temperatures.

At _very low

temperature,

_cases II and III become relevant. Case III

requires unusually

_pure CDW materials.

3. Dielectric Tensor

The calculation of the dielectric tensor is clone _as follows: In the

phase equation (2î),

the chiral electric fields _are

expressed by

the bare fields and

by

the

phase

contributions

(cf. (12)).

The

resulting equation

_is then solved for to give:

~2

=

~

(BiEi

₊

BtEt) (38)

fl The _new

frequency

fl is defined

by

fl~

= w~ ₊

iuJ~ _uJ( c)k~ cl _[(q~ w~/v)) Bi

+

qkBtj (39)

In the _next

step,

^the

phase

_çSis

completely

ehminated from the _current

equations (25, 26).

This gives hnear relations between currents

ôji, ôjt

and the electric fields

Ei

and

Et

which determine the

conductivity

tensor a~j. ^{To be} more

explicit,

_we denote the electric field component

parallel

to k _as

Eti Et2

then is the

remaining component

which is

perpendicular

to both the chain direction and the _wave vector k. We _{can count} these directions _as

1, 2,

and

3, respectively.

Using

the definition of the dielectric functions _e~j

Emn

E~

= ~

4~

ôjm

"~~

lù ^'

(40)

one obtains after _some

algebra using (28)

to

(31)

and within the above

orthogonal

reference frame:

~ ~

eu = ci + eo +

~l' _{~$ B/, 141)}

cuti = et +

~"°ti c)~~

~

~° ll~

~~~' j43j

Et2t2 " Et +

(~~)

Background

dielectric constants eo and _et have been added. Otherwise these formula _are

completely general

within the

present approach.

The formula

(41)

for the chain dielectric function _eu is similar to the

corresponding expression

derived

by

Brazovskii

[14].

^In

fact, neglecting

terms of order

q~k~

in

Bi (cf. (28)), Bi

becomes identical to Brazovskii's B-function

except

^for the backflow term bi

(which

is

negligible

in the

collisionless

case).

In

addition,

_eu

according

to

(41)

takes _on Brazovskii's form

except

^{for the}

geometrical

_wave number factor

cos~

8

= q~

/(q~

₊

kk)

which is absent here.

Instead,

the main

dependence

_on the transverse _wave vector k which is also

present

^{in the}

microscopic transport parameters

^{is via the}

nondiagonal

elements of the dielectric tensor.

The

longitudinal phason spectrum

is obtained

by solving detje~jjq,k;Mjj

₌ 0.

For _a full

picture, including polaritonic excitations,

Maxwell's

equation

1 rot E

=

j ^B,

rot B

=

~~

(ô) )

^~~~~

must be added where

jl~~~l

is an extemal current

imposed

_on the

system. Eliminating

the

magnetic

field from these

equations,

_one

gets:

~~~ ~~

~

~~~

m l~~~i QkEtii

,

j45~

~2

~~~~~~ ~~~~

iw4~

~~~~~ ~~~Î

'

j)~~~~ #

àJt2 (

~~~~ ~

~~~~~~

'

The

phason-polariton

modes then follow from the

requirement

that

(45)

has _a nontrivial solu- tion for electric fields

Ei,t

when the extemal currents

j)(~~l

vanish.

Using (40),

this

requires

the

vanishing

^{of the} determinant

det(M~j

where ^'

~2~2 fiiji

=

eu-j, (46)

Miti

# cru +

~

(~

₌

Mtii,

Mtiti

" cuti

,

~

^~

c~(q~

+

k~) ÀÎt2t2

" Et2t2~

~ù~

For ail

practical

_purposes,

however,

the

complex

expressions for the parameter functions

Bi

_ai, etc., _{appeanng in} the dielectric _{tensor must} be

simphfied.

An

important

limit is _case

I,

when the inelastic collision rate Ve is

larger

than the

frequency

_w and the diffusion _rates

Diq~

and

Dtk~.

4. Zero

Temperature Spectra

Phason-polanton

excitations _are easy ^to calculate at T _- o. Then the condensate fraction N

is unity ^and hence _Ve

- o.

Furthermore,

most qp

transport

parameters

vamsh,

_{e-g-, ait -} o,

a -

o, Bt

_- o while

Ei(q, T)

_-

fà(0)~i(q, 0)

" fà~i,

(47)

and

Bi

_- 1-

e~q~/~~

= ~.

(48)

Consequently,

the dielectric _tensor is reduced _to

~2~2

~

~

=Ù,

_{Etltl ~~~~~ ~~}

~~~ ~~ ~ £~~i

~~

^{~ ~~~~} ~~

The spectrum consists of _a free

electromagnetic

t2-mode

w~et

# c~

(q~

+

k~)

,

and

phason-polantons satisfying Mn Mtiti

~4

=

Miti Mtii

₊

q~k~j.

~ù

We set

2jo)~2

£l)o (q, o)

=

~

~

+

L°Îo /%

eà i

wLo is the usual

frequency

of the _zero

temperature longitudinal optical phason:

~

~~aj

_~ùS

=

~u~j,

~ùÎo"

_VF

@

~2

~2/6~2

~

F PI

~

where _{eà »} 1 is used [3]. ^The

spectral equation

then reads

explicitly

eo

+e~~i

(1-1°1j ^{~j ~(~j e~ ~C~j}

=

~~k~C~, j4g)

a _{~i ~p ~p2 ~p4}

with

fl~ given la

small _mass correction bas been

neglected) by:

Q~

= w~

w( c)k~ c)~q~, (50)

It is easy ^to see, that the

q-dependence

of _~ and _~i

Provides

the

LO-phason

in

q-direction

^with

a

small,

but

positive dispersion according

to

~Îo(Q)

"

ldlo

_{+ L°Î +}

CÎQ~,

a result

previously

^{obtained in}

I?i

^{and later in}

[12].

^{In the}

following,

_we

neglect

these small

dispersive

corrections and aise _{use eà} » _eo,

cl

<

c~let, c)

<

c~leà.

The

phason-polariton dispersion

^{relation is then} given

by

the

following bi-quadratic polynomial

w~ w~

uJ)o

+

w(

+ c~

~~

+

~~

)) (51)

et eà

+w)o

^~~~~ ₊

(w(+c)q~+c)k~) ~~

+

~)

c~

= o.

et et eà

It is noted that in the formal limit _{c - oo} the

singular

_{spectrum in [4] is} ^obtained.

Thus,

the

neglect

of retardation _causes the

singular

_wave number

dependence.

For k

= 0,

equation (51) predicts

two modes: _an

LO-phason

with

frequency w)o

₊

w(

and

a trivial _transverse

electromagnetic

^t1-mode w =

cq/@.

The charactenstic

phason-polariton

structure shows _up for _q

= o when the electric field

vectors of transverse t1-waves point ⁱⁿ chain direction. The

following

modes _are found: The

high frequency polantonic

mode

according

to

wj~ji

=

uJ)o

+

w(

+

c~k~ leà. (52)

It is _a transverse

electromagnetic

mode at

large

and _a

longitudinal optical phason

at small

wave

lengths.

^{The low}

^{frequency polantonic}

^{mode is}

^given by

~~ luJl

⁺

cl _k~l

C~k~

P°~~

(w)o

+

w()eà

₊ c2k2

j~~~

The latter branch has three different _regions. For small

k, c)k~

<

w(,

_one ^finds

~~°~~

eà

(w)o

+

w(

l~~'

~~~~

1.e.,

_an

electromagnetic

mode. Under the usual condition _Mo < wLo, it _can be written _as

~°Po12

j~

2~~~2jl/2'

~~~~

C© LO 0

where _em e eà is the relevant limit of the gap dielectric function _fi for _{wLo < w}

(< A)

_as

discussed in

[36].

^In

[35],

it _was shown that at low

temperatures w)oecc/w(

is the

"plateau

dielectric constant" e*. e* is the

nearly

constant value of the CDW dielectric function for fre-

quencies

above the dielectric relaxation

frequency

but below _Mo

(c f. _[37] ).

^The "true" dielectric

constant

Elu

_-

o) diverges

due to thermal

depinning,

_i-e-, at very

long

time scales the CDW

is

"gapless" [33]. However,

for T _- o, _any

frequency

o _{< w < Mo is} in the

plateau region,

i.e., e*

figures

_as the dielectric constant

e(o).

Thus the relation

~2

e~ =

° e*

(56)

~°ÎO