A general approach to charge/spin density waves electrodynamics

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Submitted on 1 Jan 1993

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electrodynamics

S. Brazovskii

To cite this version:

S. Brazovskii. A general approach to charge/spin density waves electrodynamics. Journal de Physique

I, EDP Sciences, 1993, 3 (12), pp.2417-2435. �10.1051/jp1:1993254�. �jpa-00246877�

Classification Physics Abstracts

71.45L 72,15N 75.30F

A general approach to charge /spin density

_waves

electrodynandcs

S. Brazovskii

Landau Institute for Theoretical Physics, Moscowi Russia

(Received

26 February1993, revised 13 August 1993, accepted 17 August

1993)

Abstract. We reconsider microscopic grounds for the electric field _response and for the phase dynamics

ofcharge/spin

density _waves in pure systems. We suggest transparent and free of lengthy calculations _{way to} derive the local Lagrangian valid at any temperature 0 < T < TMF and for arbitrary electronic spectrum provided it supports ^{the existence} of

the long _range DW order below the _mean field transition temperature TMF. The analysis is

based _on classification of normal carriers in two categories intrinsic and extrinsic _ones with respect to the DW _gap vicinity, and _{on a} proper treatment of perturbative and nonperturbative

(the

so-called

anomalies)

contributions. This approach results _e-g- in _a helpful relation between the "generalized condensate density" and the complex dielectric susceptibility of intrinsic _car- riers. On this basis we easily describe main properties of the DW~S both at low T and _near TMF. Separately for CDW and SDW _we discuss the spectra and the attenuation for the TO and LO modes, the low

frequency

relaxation and the reaction to _an external voltage. Our studies

cover systematically and generalize most of the previously derived results which have been used

for _pure systems or as

preliminary

steps to approaching the pinning problem. New results of

a potential experimental significance describe the TO, LO and

zero sound spectra interplay, the anomalous Landau damping of both LO and TO modes _near TMF> the relaxation rates for

narrow gap Dw~si the relation between the current and the driving electric field and between

the inherent and the observed nonlinear conductivity.

1 Introduction.

Special

electric and

dynamic properties

of

Charge

_or

Spin Density

Waves

(CDW /

SDW _or

generally DW)

_are related _to their

nearly

infinite

polarizability

and _to the

sliding iii.

These effects have since

naturally

been

interpreted

_or derived since [2] on the basis of the linear electric field _response function

e(qi w).

This function _was

explored

in many

publications

from

[2-8],

see also the latest review [9]. We refer _to [6a] ^{for the}

profound

first

principle study

and to the latest review [6b] ^{of the DW kinetic}

theory

^and available results. Nevertheless _a number of

contradictions, misinterpretations

^and even mistakes _may be encountered. In this article _we

will

suggest

a very

general

and transparent _way to derive the DW response and _to

give

_a

simple analysis

of the DW condensate

properties.

This

approach helps

to avoid

lengthy

calculations of earlier studies

by

virtue of _a manual book information _on normal electron

properties

of _a

corresponding

semiconductor _{or a} semimetal.

Before

going

onto _a

general description

_we demonstrate _{now some} internal contradictions of the

theory

^for the

simplest

_case T

= 0 when _no normal carriers _are present. In the _next

chapters

we will discuss in detail both effects of small but finite T and of _a

vicinity

of the _mean field transition temperature TMF.

At _zero temperature ^{the dielectric}

susceptibility

_e is

given by

the

commonly accepted

formula

E(~>W) " fh + fA + fcol

(I)

Here _eh

" const is the host contribution

(usually

_we will put _{eh "}

I).

The _{term ea is}

the interband contribution inherent to _a I-d _{narrow gap} semiconductor which is present _e-g-

already

^{for the dimerized} _case ^of

polyacetylene. Finally

_ec~i is the _most

intriguing

^collective part ^due to the CDW translational motion [2]. The

dissipation

and the free _carriers effects

being ignored

at T ₌ 0, equation

(I) acquires

the form:

~2 _K~f(g)

COS~ ~~~

P +

2 ~2

/u2

e(q,w)

₌ I +

@ _q~+°~i

where

~~ =

)'

=

ri~

=

) _(~

= fl~ i> C°S~ °

=

qj )'q _j

^{~~ =}

^{~l W~/»~ (3)}

~ ~ ~ ~

Here A is the half energy gap value: A

=

A(T),

ho _"

A(0), A(TMF)

_" 0j _q =

(qjj, qi);

_{up, rD>}

u are the

plasma frequency,

the

Debye length

and the Fermi

velocity

of _a parent

metal,

_e is the electron

charge (from

_{now on e}

= I, h

=

I),

_s is the unit _area per

chain,

_u is the CDW

velocity

^and fl~ is called _[2] the CDW effective _mass enhancement _ratio. The parameter o in

(2)

characterizes _an interchain

coupling strength.

The function

f(q), f(0)

= at T

=

0)

is

discussed below.

While most attention _was

payed

to the

divergent

collective mode term ec~i, the intermediate term also would be of

importance

both for CDW systems with

ho

r~

10~~

eV,

_ea

r~

10~ _and

especially

for

SDW,

where

ho

r~

10~3

eV,

_ea

r~

105

(!).

Recall that the parameters _{u, K, up}

are

always similar,

_{e-g- up}

r~ I eV.

The effect of _ea with respect to _ec~j is to shift the DW

plasma frequency (the phason

Coulomb _gap

[2-4])

WC and the inverse field

penetration depth

^r~~ _as follows

w(

=

wp/fl

_{~ wc}

"

ViAo/fl

r~ us

(4)

rD = K~~ _~

r _r~

u/Ao

_"

lo (5)

where _us is the

amplitude

mode

frequency.

So the reductions _are of the order of

ho /wp

_r~

ho lef.

This is _a substantial effect

already

for the CDW scales and it becomes _enormous for the SDW'S.

One motivation of the

presented

studies _{was an} observation that while the reduction

(4)

is correct for the

CDW,

the reduction

(5)

is wrong, and both

(4)

and

(5)

_{are wrong} for the SDW.

In other words the form

(2)

_may be valid _at

qjj ⁼ 0 while _at

w = 0 the _{term ea} has to be

cancelled. For the SDW the cancellation should

happen

^somehow for both

qjj

#

0 and _w

#

0.

The

inadequacy

of

(5)

^follows

already

from the _exact theorem of [10]. ^{The statement} was

that

independent

of the _presence of the CDW in its

ground

state, _or in _an

equilibrium

distorted state

(solitons, etc.),

the

homogeneous

_over crossection

(I,e,

at qi =

0)

static electric field is screened

similarly

to the parent

metal,

I,e.

r~

exp(-Kx)

rather than

r~

exp(-x Iii ), ii

₌

to /Vi

as it follows from

(2), (5).

Also there _are

disagreements

between

equations

^{for the} CDW

plastic

deformations and flows

(see [lla,

_b] and Refs,

therein)

and the reductions

(4)

and

(5).

An obvious contradiction _can be noticed _as follows. Whenever _we arrive at the

length

scale

to

the

crossover from the CDW to the parent metal behavior should be recovered. Then the _correct

screening length

is

expected

to be K~~

= rD rather than

to

_as in

(5).

A similar contradiction

concerns the

plasma frequency

reduction

(4)

for the SDW when

fl

₌ I and _WC

r~

ho>

but

no arguments

apply against

the reduction

(4)

for the CDW when

fl

_» I _so that _{wc <} ho-

Roughly

these

paradoxes

_can be resolved within the

original

method of [2]

by calculating

the next term in the series

expansion (a

_more detailed

description

of _[3] should be

used)

for the

function

f(q)

in

equation (I).

We expect it to

acquire

the form

f(q)

₌ i

U~q~/6Ai+ °((q/Ao)~) (6)

The coefficient of the second term in

equation (6)

is chosen to be

exactly

what _we need _to compensate ^{for the} ea term in

(2)

at qi = 0, _{w =} 0

(or

at any w,qjj ^{for the SDW} case,

u =

oo).

In the next section _we will _see that this cancellation is not

just

_an

artifact,

but it is related _{to some} basic

principles

of the DW

physics.

2 General

description.

To obtain _a detailed and

systematical description

_we suggest an almost

general microscopic

derivation of the

phase Lagrangian, following

the concepts ^of [12]. ^{It is based} on the observation that the effective scalar V and _vector A

potentials

and

eventually

the total

longitudinal

force F

experienced by

electrons under the DW

phase

deformation and under _an

applied

electric field

~~

~'

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

are

given by

the gauge and chiral invariant combinations

A, f

F:

~~ ^u~'

^~

_~~~

'

~

~"

~ ~

'

~ ~'

~~

^~~

~~"'~ ^u~'

~~~

where Ax and _{4~ are} the

original

vector and scalar electric field

potentials.

The substitution

(7)

of the electric field E for the effective force F

corresponds

to the transformation of electron

wave function _{1fi or} ~i =

(1fi+,1fi-)

to the local frame of _an

arbitrary

distorted

phase

~2 ⁼

p(x, t)

_{: 1fi = 1fi+}

exp(ikfx)

+1b-

exp(-ikfx)

_~

~i+

exP(ik~x

₊

w/2)

+

~i- exp(-(ik~x

₊

w/2)) (8)

This is _a chiral transformation related _to local and instantaneous

displacement.

In terms of components

(~b+,1fi-

^the

Schr6dinger equation

operator ^transforms

apparently

as

l~~ ~)~~~'~ _~l~

H

=

(-I»t _lAzla3

+

Ae'~a+

+ Ae'~a- ₊ 4~ao ~

°

=

(-i») _Al

_{a3 +}

hoi

₊

vao (9)

which _proves the statement

(7).

Here _{ao> a3> a+ = al} + ia2 _are the

unity

and the Pauli matrices. To be brief _we have

skipped

the interchain

hopping

terms

E-

(pi )a~

+

E+ (pi )aoj E+ (pi

=

j(E(pi

⁺

^E(pi

⁺

^Qi)

where

Qi

and _{pi are} the

perpendicular

DW _wave number and electronic moments.

Neglecting

the

perturbations

from A and _4~ in

(12)

the transformation

(8) provides

_{a qua-}

sidassical solution of the

Schr6dinger equation

at the

given

external

fields,

if the

phase #

is chosen from the

equilibrium

condition F ₌ 0. In this respect a convenience of the linearized

spectrum ^mode creates an important

physical paradox

which _may be related to

problems

of sc-called anomalies

(see

_{[8] for} relevant discussion and

references). Shortly

_{we see} that

only

the

phase

of the _wave function

(8)

is

perturbed,

while the

amplitude

_stays intact. This property

contradicts _to the requirement that at A ₌ 0 the total

density

of electrons deviates _as

6p

r~ 4~

while at A

#

_we miss the Fr6hlich effect

6p

r~

p'. Reminding

ourselves that for _an

arbitrary

spectrum at A

= 0 the

density

distortion _emerges from the _next order

quasiclassical

correction

as a factor

E(pjj

^uk ~

E(pF

+

k) E(pF)

_{PE = l1fiEl~}

~W

(ls=s(p)+w ^(lo)

This factor is _an identical

unity

for the linearized spectrum

approximation. Generally

it

provides

_a

negligible

correction

6pE

~

-4~/EF

for any

given

state but _a finite contribution

6p

=

-NF4~, NF

_"

(1/1r)3E(pF)/3pF,

_to the

integral density.

^{This finite} contribution _comes

as an

integral

effect of all states far below the Fermi _energy which makes it insensitive _to the presence of the gap. This feature is _an

important facility

for _our subsequent

analysis.

As _{we see} from

(7), (9)

the most traditional information for _a

corresponding

semiconductor

or a semimetal

(for

_an

appreciable magnitude

of _a nonnested part

E+ (pi

when electron-hole

pockets

_may

appear)

_can be used to describe the

particles

in the local frame. In this respect it is of

special importance

for _our

goals

to

distinguish

between "extrinsic" and "intrinsic" carriers

with respect to the DW

spectral

_gap

vicinity

in _{a sense} that the first _are

subjected

to the field E

solely

while the second

experience

the combined force F. A

frequent misinterpretation

and the contradictions _we discussed above _come from

treating

the intrinsic carriers, which

typically

dominate, _as extrinsic _ones. In nature the extrinsic carriers _are those from other bands nonaEected

by

the DW _or from the _same band but far

enough

^from the Kohn

anomaly

so that their 2kF

Umklapp

due to the DW is weaker than the host lattice

Umklapp scattering by impurities

and

by phonons (~).

The intrinsic carriers _are due to thermal excitations above the DW gap, to

incomplete nesting ^pockets,

to virtual _or real

optical

transitions across the

(~) The classification is similar to the _case of _a phonon bath in the presence of the CDW in [14]. Roughly speaking the _energy £ _range of extrinsic carriers is determined by the condition

£/A

>

AT(£)

> I where _T is the backscattering time due to impurities _or phonons of the host crystal. It follows meanwhile that _we cannot approach the transition closer than (TMF

T)TF

r~ I

where TF is the backscattering time at the Fermi surface. This constraint is similar [15] ^{to the} problem of gapless

superconductivity.

DW gap. The responses of these

particles

_ee and _e; with respect to E and F

correspondingly

are characterized

completely by

their

partial

dielectric function contributions _ea where _a

= e, I.

Typically

_we expect

(for ^simplicity

we assume here _{qi =}

0)

Ea = Re _e~ + 4,I ^{°a o}

lj2

^~

~

Ea ^{+ mill}

j~

^{Ma ioa}

~ll ~~ _W ^II

Here

Ap~,

_wa and

aa/41r

_are the

partial Debye screening lengths,

the

plasma frequencies

and the conductivities of

corresponding

carriers. In what follows _we shell

usually

_put the host value

as

unity: ei°I

= eh " I while for e)°~ ^at T <

ho

_we should

keep

its true

large

value at T ₌ 0:

et

_{= ea =}

w( /6A].

Now _{we are} able to write down the

Lagrangian

£

=

£(p,

_4~,

Ax ) (-£

is the time

dependent generalization

of the energy

functional).

It _can be done

by combining

arguments of the _gauge and the chiral invariance which have been

expressed by (7), (9).

~$ ("(viw)~

⁺

^~lw~ j~#~) )i

^{n "}

^n(T)

^"

^{A(T)/A(o) (12)}

The first term _{eh "} I + _ee is the host contribution for the electric field energy, which consists

of _{a vacuum} part

r~ I and of the extrinsic electrons contribution

r~ ee.

Reminding

ourselves that in terms of electric field

strength

E

= -V4~ the energy should be written with the "minus"

sign.

The second term _cc e; is the the first _nonzero

perturbative

contribution from intrinsic electrons _response to

(7).

It is calculated

by

the _same rules _as the ee-term but the

generalized

force F from

(7)

is used instead of Ex. There _{are no} other

perturbative

contributions from the Harniltonian

(9).

Nevertheless the

Lagrangian

is not

complete

since e-g- at the absence of normal carriers A; ₌ 0, _{w; =} 0 when _{e; =} const _we do not _recover

expected

DW distortion

energy r~

(p')~.

Even _more

puzzling

is that _{at A;}

#

0 _{we recover} the contribution

r~

(p')~

with

an

opposite sign!

The third is the _group of four terms in square brackets which describes the missed _contri- bution from the

potentials (7). Being nonperturbative

with respect to

(9)

this term does _not

depend

_on temperature ^and even more it is not affected

by

the _presence of the DW

(~).

For

this _reason it must coincide with the parent metal

compressibility

with respect to

potentials

of

equation (7)

where it is

originated by (lo).

We find instructive to write it _as

being

differences between contributions of effective fields V, ^{A and the bare} ones 4~,

Ax.

In such _a form this term describes the work

required

to

produce

a local distortion

(9)

_at the _presence of external fields

4~ and

Ax. Alternatively

it can be written in _a form

r~

[(p')~

_u~@~

4Ep]

which is inherent to the T

= 0 DW at the presence of _a local electric field E. Then its

physical meaning

is clear:

the variation over the local distortion

p/lrs gives

the

generalized

force F of

equation (7)

and the

equilibrium

condition F

= 0

provides

_a correct metal _{response to} the external fields _4~ and

Ax.

Finally

^{the fourth is the} group of three terms _cc q~ which _are due _{to an} interchain DW

elasticity,

to a

possible commensurability pinning

and to the CDW lattice inertia. The factor q insures that _a and _u have _no temperature

dependence,

but

q(

may

diverge

at A _- 0 while the whole

pinning

_{energy cc}

(qqo)~

_- 0.

(~) A rigorous description of nonperturbative results for CDW in terms of sc-called "anomalies"

was given ⁱⁿ [8], but the perturbative part ~J e; was not treated properly.

Strictly speaking

the

imaginary

parts ⁱⁿ

(11)

must not be

incorporated

to the

Lagrangian

formalism.

Correctly

_{one can} define

(12)

within the Matsubara time interval _to

perform

the

analytic

continuation at the level of response functions like the _one in

(lo)

below.

In

(12)

the interference of the third

nonperturbative

term with the

perturbative

_{one cc e;}

will be shown below to

provide

_a correct renormalization of the _cc

(p')~

DW elastic _energy at finite temperatures so that it vanishes at A

= 0 at _zero temperature. The

ambiguity

of

equation (2)

appears when _one writes the third _{term as}

r~

e;E~

rather than _as

r~

e;F~

like in equation

(12).

The

Lagrangian (12)

^{should be}

interpreted

_{as a} function of _two

independent

variables _p and 4~

(neglecting

the skin-effect _or

polaritons

_{[13] we can}

always

choose _a gauge with Ax ₌

0).

To find the total dielectric _response function _{e we} must exclude the internal variable _p from

(12)

with the

help

of _an

equilibrium

condition:

~~~" ~~

= 0

£(p,

4~) ~

£(4~)

=

/ d~r~(V4~)~ ⁽¹³⁾

w 7r

In Fourier

representation

_{e =}

e(q, w)

is

easily

found _to be

~2 ~Q2

~~~'~~

~ ~~ ~ q2

~

~

q2B

Q2 ~°~~~ ~~~~

~~ ~~~~~~~~~~~ ~

e(q,

_{W) = I + fe + ~;}C°S~ ° ₊

j~~~~a2

~°~~ ° ~~~~

where

B(q, w)

₌ I

e;(q, _w)q~ /K~,

^Q~ = q~ (w~

/u~

₊

i~w q( aq() (16)

The relation

(16)

between the effective condensate

density

B and the intrinsic carriers electric response e; is the _most instructive part of _our

approach.

The

representation (15)

is due _to this relation. In

(14)

the bare metal response

singularity

_cc

q~~

at q - 0, _w - qjj

#

0, cancels since

B(q=o

" I

being

transformed to the _remnant intrinsic carriers

singularity

due to _e; term in

(15).

Our formula

(14)

would _{agree at qi}

= 0 with the

expression

for

a(qjj,w)

in [5c] ^if one

considers B _{as a} real function _and redefines u~~ _~

Bu~~,

which _was

performed

in [5c] ^and

q(

~

Bq(,

_{a ~} Bo.

Being formally legal

in

preliminary

Matsubara

frequency

^formula this redefinition ascribes _some

unphysical T,

_q and _w

dependencies

to these

simple microscopic

parameters. ^{What is} more important, _{as we} will- _see

below,

is that _some essential contributions to relaxation parameters are omitted in this _way [5c] or

by

_a

typical tempting conjecture

B~ -

(B(~

to

misinterpret

the nominator in

(15)

_{as an} oscillator

strength

for the DW _response.

The formula

(12), (14), (15)

_are valid within the _mean field

(MF)

conditions at all temper-

atures T < TMF in

spite

of the

explicit

absence of the condensate

density

_ps for the Id DW

elasticity

_{ps in}

(12).

What

happens

_near TMF is that the normal carriers are almost nonaffected

by

_a small gap and their dielectric _response becomes

nearly

that of the parent metal _so that

ii

~ K, or more

precisely

Al

_" K~Pn,

Pn(T)

= I

p~(T) (17)

where _pn and _ps should coincide _a

priori

with the BCS-like normal and condensate densities for the

given

^{3d electronic} spectrum ^and

nesting

conditions. In _more details _at small

qjj and

w we have

apparently

the

singular

intraband contribution. Within the

nondissipative region

it is

given by

a standard formula:

T » »an,uJ ^» T;-~ E;(an _U~) =

'I

_<

j~/"

u~2

>1

'1

"

~) _~»'

_~~~~

~ ~ ~

where _< > means the normalized

averaging weighted

^{with the} Fermi distribution derivative

Nj

=

3NF/3E;

and ujj =

ujj(p)

^{is the}

longitudinal velocity

of _an electron at the _{momentum p} for the spectrum

E;(p).

Here

E;(p)

and N; _are the spectrum

(at

_a

given A)

and the onchain

concentration of intrinsic

electrons,

_p is their chemical

potential.

We find from

(16)-(18)

1

uj ^/u2

_up~

~ ~~ ~ ~" ~ ~

i

»lql/U~~

~' _~"

-'

^{~~ " '~ ~~~~}

We _see that at w = 0, _qjj - 0

B(qjj, 0)

_{- ps}

r~ q~ at T _-

TMF. Oppositely

_{at qjj =} 0, _{w -} 0

B(0,

_{bJ) - ps +}

Pn(l~

<

~(

~

/~~~

_{~~ ~ ~~} ~

(Upjj)~ ^{+ A~}

~

We obtain

immediately

that _at T _- TMF the second term in

(19)

dominates for

B(o,w) decreasing

_{as q}

only.

Indeed without _any calculations _we find at small

q(w/uq(

< 1

~ _{~~ ~ ~~}

/

w2

~u4

/A2 ~~i~~if~

^{" ~~ ~}

~'~~~~~~

where

(3)

^C =

(1r/4)(Ao/TMF)

r~ I. In this _{way we} suggest a very

simple

and transparent derivation for the _two different [3] qjj,w ^limits

fo

_{cc q} and

ii

cc q2

(in

notations of [5]) ^{for the} effective condensate

density.

Moreover _{we see} that B is real

only

at _{w =} 0 and at w~ _>

u~q(,

^{while the}

imaginary

part dominates almost

everywhere

in the sector q~ > 0:

)

r~

ii

_{» I at}

q <

(~(

< l

(22)

e Uq Uqjj

The second _term in

(21)

dominates

everywhere

at q~ < 0 when it is real.

The

singularity

at q - 0 in

(21)

saturates at

(uq/w(

< q when B reaches its parent ^metal limit B

= I due to the first part ^of

(12)

_so that DW is

ignored

_near the _zero sound _resonance.

The derivation

(21)

is valid if characteristic moments p and

consequently

the contributions to B

satisfy

the conditions

1>folpl~wnl(1>~°,~,

^When

i>ImB>(,~; T;~) ₍₂₃₎

~

where I is _a momentum relaxation

length,

_T; is the

scattering

time.

These effects _may be viewed _{as a}

giant

^Landau

damping

of

plasma

^{modes which} is

ampli-

fied here due to the

negligible perpendicular dispersion

of ujj for the open Fermi surfaces. A difference is that this

velocity

resonance affects also the real spectra which is due _to the trans- formation

(7)

of

phase

distortions into _an effective electric field. The outlined

approach

_saves

(3) This ratio may be large C » I for special marginal _cases of poor nesting [5e].

us from additional calculations

beyond

the conventional results for _a free electrons dielectric function.

At first

glance

one may notice another _source of

(21) being

finite _even at A

= 0. It _comes if

one takes into account the electron-electron interactions which _{seem to} deviative the metal _zero sound

velocity

_u ^from the Fermi

velocity

_uF which _enters the

quasipartide

spectrum.

Actually

it

happens

to be

nothing

but the renormalization of _u towards _uF.

Being

_{nonzero at} T > TMF this contribution has to be taken into account in the parent metal _term of

(12).

For these _or other _{reasons we} cannot accept the interaction induced _terms in [5a] as well _as their earlier version of the

Lagrangian (it

_was

essentially

corrected in

[5c, d])

and with the consequences of

[5a, b]

^{for the sound}

propagation

due to the interference of the interaction and the

pinning.

The effect of interactions

requires

_{a more} careful consideration because it looks different for the 3d MF _case and for the

rigorous

Id _case.

At the end _we should _warn

against

interpretation of

B(q,w)

_{as an} effective condensate

density

and

especially against

its

separation

_near TMF in two

asymptotic

types ^{of condensate} densities: the static

ii

_cc q~ and the

dynamic fo

_{cc q as} in [5]. This concept _may be

misleading

as in [16] ^{where the}

dynamic

value

fo

_was used in

essentially thermodynamical

calculations which _can be

hardly justified.

Moreover for

relatively

low

frequencies

of

sliding

DW oscillations

one

certainly

expects wT; < I _so that within the

dynamical hypothesis

_qjj

= 0 _one would have B = I iwa;

/w(

ce I rather then q or

q~.

What is also

discouraging

in these interpretations is

that _{as a} function of _w

B(w,

_qjj is real

only

at these _two limits

ii

at _{w =} and

fo

at

w/qjj

= oo.

In between the

imaginary

part dominates almost

everywhere

at 0 < w~ _<

u~q(.

^{This feature}

seems to have been _overseen in

previous

studies where the relation

(16)

and the

singularity

(21)

_were not

properly analyzed.

3. The transverse

optical

spectrum.

Consider the spectrum ^{of the TO mode}

given by poles

of

(14)

_or

(15)

I-e-

by

the

equation Bq~

=

Q~.

CDW _{AT LOW} T.

Apparently

_we have

w~/u~

=

aq(

₊

q((I eaq(/K~) ⁽²⁴⁾

On this occasion _we have

only

small but illustrative observation. The interband

polarizability

ea

provides

the correct

(decreasing) phase velocity

correction to match the

general shape

of the Kohn

anomaly.

CDW NEAR TMF. Above the

pinning frequency

_one

usuauy expects

to find the sound

spectrum ^with a reduced

velocity

_uqjj » w. Then the static limit is

supposed

to be

applied

when B m ps cc q~ so that the spectrum ^is

expected

not to

change qualitatively

with respect

to the low temperature _case, but the residue

(I.e.

the "effective

charge" squared)

scales _as

B~

r~ q~ [3]. Nevertheless the existence of _a sound spectrum ^{with the}

velocity

ce u < u is

questionable

in view of

large imaginary

contribution from

(21). Following (16)

and

(21)

the

spectral equation acquires

the form

w~

~

+ iw

(

₊

^I-~ @=

q(

⁺

^Q~ (25)

~

u u

~

We _see that the bare DW selfattenuation _~

acquires

_an additional part _~ ~ , +

(C/q)((q( Iv)

At _zero

restarting

force _q1

=

0,

_{go "} 0 the attenuation may be

relatively

small _so that the TO

spectrum _can exist

only

^if _q »

u/u

I-e- far

enough

below TMF and

provided

that the _mass enhancement is

really large: u/u

» I. For

Qo #

0 there is _a finite interval of nonattenuated oscillations: _qjj <

uoo

_e-g-

w(0)

₌

fioo>

u~

/fi~

₌ fl~ + I

In.

THE SDW _AT LOW TEMPERATURES. A remarkable feature of the SDW _case is that for

Q = 0, I-e- at qi = 0 and

neglecting

_own attenuation and the

pinning,

there is _an exact cancellation of intrinsic carriers contribution in

(15).

The formula

(14)

shows

clearly

that _at Q = 0 e coincides with its free electron form of the parent metal at all temperatures

(~).

In such _{a way} both contradictions encountered in the introduction for the SDW _{case are} resolved- the troublesome contribution _ea

disappears,

but for

simplest applications only.

If the SDW

perturbations

like _{qi,,, go are} considered then the

intuitively expected

SDW TA spectrum appears ^at w~ _ci

u~q(

⁺

Q(, Q(

"

q(

_{+ aq}

(.

Notice that for

Qo

" 0 the _e; contributions cancels

at all _w, qjj, so that _e-g- the

intergap absorption

at _w > 2A will not be visible unless _a

pinning

or a

perpendicular

component of

polarization provide

^{the SDW}

restoring.

THE SDW NEAR TMF. At _zero

restarting

force

Qo

_" 0 there _{are no} spectra except ^for the

one at w = uqjj ^{at all} temperatures. ^{But for}

Qo #

0 the spectrum evolves

differently

than at low temperatures ^{because the second term in}

(21)

dominates

being

real for this _case. In _a very broad range of _{w at (w(}

Iv

> (qjj (, (w(

Iv

_»

Qon (or equivalently

at 0 <

-q~

<

Q()

_we find the

spectral equation

_as

wfi

=

u~qo( (26)

so that _e-g- the

perpendicular dispersion

curvature _or the

pinning frequency

_are renormalized

as o ~ an, go ~

q~/~qo.

The spectrum

(26)

starts at finite

frequency w(0)

₌

uooq~/2

at qjj ⁼ 0 and _{converges to} Fermi

velocity

at

large

_{qjj, as} shown _at the

figure

I.

Very

close _to TMF _a similar spectrum

(26)

exists also for the CDW _{case at}

Qo #

0 when the attenuation _can be avoided _as

long

_{as u(qjj} < (w( holds for the solution of

(25).

Now the _same

equation (26)

is

applied

but at _more essential constraints to prevent

crossing

the

dissipation

borderline _w

=

uqjj.

noon

_{< (w( <}

uoo

I-e- at

Qonu/u

< (q( ^<

Qo (27)

These

inequalities

_are

compatible only

close to TMF at q <

u/u,

_so that the temperature interval

(27)

for the nonattenuated anomalous spectrum

(26)

is

complementary

to the interval q »

u/u

for the normal

weakly

^attenuated spectrum

(25).

4. The

longitudinal optical

spectrum.

Consider the LO mode spectrum

given by

_zeros of

(14)

_or

(15).

CDW AND SDW LO SPECTRA AT LOW TEMPERATURES. The LO mode spectrum ^{of the}

DW _can be written

(implicitly, keeping

in mind the

frequency dispersion

of B and

ea)

in _two

equivalent

forms

(~) This conclusion _was recently given in [5c]. ^Here we notice that the extension to q1 # 0 _or to relaxation brings the drastic effects of the _ea term back to lifer _{as we} will _see below.

~ ~ 2 2 Q2

jJq2 fig~/~~

il~ (28)

" ~

~));~~~~f~

~~ ~~ _~ ~~~~ ~~~~ ~~~~ ~~

For low q~ we can

neglect

_eh in

(28)

if cos@ is not

extremely

small

(the polarization

is not

nearly perpendicular)

which is

typically

true far

enough

from line _or

point

defects

ii ia].

Then the LO spectra _are defined

by

_any of the

equivalent equations:

B(Q~

+ q~) ₌ Q~ _or

e(Q~

+ q~) = K~

(29)

As well _as for the TO mode the LO spectrum exists

strictly speaking only

at q~ < 0 when B is real and positive. So it is defined for Q~ _{< o} which always holds for the SDW _or at Q~ _>

-q~

which is

possible

for the CDW.

Here the

approximate equality corresponds

to

neglecting

the host contribution _eh I-e- of the fourth order term

q~q~.

It _means that _{we are} interested in _a

frequency

_range below the metal

plasma

spectrum for _a

given polarization angle

@: wp(@) =

(uK/@)

^cos@.

omega

°~~9~p

$mega

Lo vq~

~@

,

/ /

~

/

~

/ /

damped /

I' /mega

,

uq~~

1/2

~l'

,

~~~

°/

' ,

'

~

TO, LO~CDW ^'

~ ~

~

~ /

Q~u

)

Q _u

° vqII

Fig. I. The qualitative spectra plots _near TMF ^{for the} SDW: LO, TO and for the CDW: TO, LO-CDW.

Formula

(28)

differs from the

corresponding

result of _[7] first

by

the factor

B/q~

which _is not

always important

at low temperatures. ^{Second there} are no

explicit q(

^{term in the r-h-s- of}

(28).

Instead the

dispersion

is extracted from the B-factor with the

opposite sign,

_cc

-q(,

with

respect to [7] and

against commonly accepted ezpectations

deduced from

superficial applications

of

(2). Indeed,

at qi = 0 we find from

(16), (28)

at T <

ho

for the CDW _case the low

frequency

solution

~2

₊

e

+ e; ~~~

2

l +

)

+ _e; ^" 1

~~/~~aq()

~~ ~~~~

This observation tells _us that

(at

₌

0)

_we deal with the top of _an

optical phonon

spectrum rather than with the bottom of _a

heavy plasmon

spectrum as one may guess from the standard formula which

definitely gives

the

positive sign

^of

q(

corrections. This bare

positive

curvature

r~

+q( /K2

_comes from the

q2q2

term in the nominator of

equation (28) being majorated by

the

large negative

contribution

r~

-q(((

from the B-factor.

Remarkably

it is

given by

the _same interband contribution

-eaq(/K2

as the TO

velocity dispersion

in

(24)

which is an inherent feature of _our

approach.

At the _same time the

perpendicular dispersion

has _a

positive

curvature as we can see from the approximate formula

(29)

^w2 _ci

w[

₊

u2(q(

+ aq

[ _q()

_so that in _a full 3d

picture

the _zero temperature "LO

gap"

is

actually

the saddle point

of

^the

phonon

spectrum.

For finite at small _qjj both kinds of carriers contribute

additively

to the Coulomb _gap

screening. E-g-

the LO mode

longitudinal velocity

is

given

_{as uL *}

u(K IA

where A2

=

if

₊

if.

This is _one of the _{rare cases} when extrinsic and intrinsic carriers _are not

distinguishable.

Certainly

the

dispersion,

attenuation and

(w,

_qjj) ^limits

uncertainty

via B-factor _come

only

due to intrinsic carriers. The

validity

^{of the} linear spectrum ^{is limited}

by

_a

requirement

^of

relatively high

concentration of residual carriers when their

plasma frequencies

_{wa are}

high enough

to

provide

the static

screening

~L~))~~^' )~~)~(~n)~~~> #n-Pnj~~"~~' _~'~~~))~~~

where _UT is the thermal

velocity

_{or a} ^{small Fermi}

velocity

^of remnant carriers.

Otherwise _at lower T when

u/u

_>

_wa/wp

the Landau attenuation is known to become that of the order of

frequency

and the LO spectrum

disappears.

The attenuation becomes weak

again only

at small _qjj and _at finite _w when _{we come} to the

dynamic regime

^{of normal carriers:}

e; m ea

w) /w~.

^The

spectral equation (29) acquires

^{the form}

Since for

the

_CDW we

expect

u < u then _there are

always ^{two solutions.} The solution of equation ₍₃₁₎

~

=

q(

+

aq( _q(, ^~

=

(~ _)~

^{+ (~}

⁽³²⁾

U U U pn

It

provides

the

strongly suppressed pinning frequency

_or the _transverse

dispersion.

The

high frequency

solution w > wc describes _an enhanced Coulomb _{gap wc} which exists both for low and intermediate _pn. Notice that the low

frequency

spectrum

(32)

also has _a saddle

point

structure so that it exists

only

at small

q(

^< aq

[

₊

q(.

SDW LO _SPECTRUM. For the SDW the linear spectrum ^due to the static

screening

does not exist which follows

formally

from

(28)

since in this _case Q~ _< o. The reduced Coulomb _{gap WC} also does not exist _{as we see} from

equation (31)

since at _{u - oo} the

r~

w~ _term

changes

the

sign.

^{The Coulomb} _gap ^solution exists almost _at the bare metal value wp(@) ^with no respect

to the SDW _presence. The low

frequency regime (31), (32)

which is due to

negative

_ea exists for the SDW for all _{pn «} 1:

W~ "

(Pn/Ps)U~(Q~

+

"Q~

_Q()

(33)

Remembering

that at qi = 0 spectra

(32), (33)

_are

actually

the

pinned

modes with

large

effective _mass

r~

ps/pn

_at small carders concentrations.

CDW _AND SDW LO _{SPECTRA NEAR} TMF. Near TMF the difference between the LO and

the TO modes _can be

easily

_seen from the second form of the LO spectrum

equation (28)

if _we notice that for the TO spectra the r-h-s- is

identically

_zero. The nominator at the r-h-s- of

(28)

is

nothing

but B which is

generally

^{small for small} _q so that LO and TO modes _are

typically

very close _near TMF. ^{These modes} can deviate from each other in two

special

_cases: first at q = o when B _- I, second _near the

high frequency plasma

_{resonance wp(@)}

given by

_zeros of the denominator. We _are not

considering

here the

dissipative regime

at very low

frequencies

uqjj ^< w <

T[~,

when B _m I,_{_(t} will be done in section 5.

We find two branches of the SDW LO spectra. The

high frequency

^branch is related

primarily

to normal carriers. It goes very close to the metal

plasma

_spectrum: ^w~ ₌

_p~w((@)

₊

u~q(.

The low

frequency

branch is related _to the order parameter distortions. It starts at the finite

frequency w(0)

m

uQov1

_{at qjj =} 0 and

approaches

the Fermi

velocity

at qjj »

Qovl.

At all qjj this branch _{goes very} close to the TO

s§ectrum (26),

at small

qjj these results _agree with

[7c, e].

For the CDW _case similar results _may hold but in _a very constrained

region

_uqjj <

uoo

when Q~ _< 0. At

larger

_qjj the lattice inertia reduces the ratio

w/qjj

^below u I-e- _to the

region

of

strong

attenuation. At

relatively large

_q > u

Iv

the conventional low

velocity

LO mode _appears from the

overdamped region

_w < uqjj.

Again

this LO mode is close to the low

speed

TO mode

(25).

The _summary for TO and LO modes _near TMF is

given schematically

in

figure

I.

Experimental

^studies of the CDW spectra have been

performed recently

_{[20, 21]} for the

high

_gap

compounds.

For these

essentially quasi

one-dimensional materials with low transition temperatures Tc <

TMF,

the low energy (r~

Tc)

defects 21r-solitons _are

supposed

to be dominant excitations at low T rather than

high

_energy (r~

ho

electrons. Since the 3d defects have been shown

[I Ii

to be

subjected

to the _same invariant

potentials

and forces

(7)

_as intrinsic

electrons,

_our low temperatures ^studies can be

applicable

to the low Tc case at T _< T~. The small _q results at T _m TMF _may need to be reconsidered for

applications

at T _m T~ <

TMF.

5. The LO mode attenuation and the low

frequency

relaxation.

Consider _now in _more details the

frequency dispersion

of

e(w) including

the DW attenuation and the

conductivity.

For the

homogeneous

_ac electric field

polarized

at _an

angle

the _response

function

(14), (15) acquires

the form

e(w,

_@) eh(@)e;w~

/w(

+ (eh(@) e;)w~

/w(

+ (eh(@) +

e;)Q~ /w(

I

fi

_Q2

/wj

+ w2

/wj

+ e;w4

/w(

^~~~~

where eh(@) =

eh/cos2@.

Notice that _e; and _eh enter

symmetrically

the _term

r~

Q2 _which characterizes the lattice contribution to the inertia and to the

damping

but _e; enters with the