Temperature dependence of the Peierls wavevector in quasi one dimensional conductors

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Temperature dependence of the Peierls wavevector in quasi one dimensional conductors

Claudine Noguera, Jean-Paul Pouget

To cite this version:

Claudine Noguera, Jean-Paul Pouget. Temperature dependence of the Peierls wavevector in quasi one dimensional conductors. Journal de Physique I, EDP Sciences, 1991, 1 (7), pp.1035-1054.

�10.1051/jp1:1991188�. �jpa-00246384�

J.

Phys.

Ifrance 1

(1991)

1035-1054 JUILLET 1991, _PAGE 1035

Classification

Physics

Abstracts

71.45L 64.70R 72.15N

Temperature dependence of the Peierls wavevector in quasi

_one

dimensional conductors

Claudine

Noguera

and Jean-Paul

Pouget

Laboratoire de

Physique

des Solides

(*),

Universitk Paris Sud, 91405

Orsay,

France

(Received19 Febrltary

1991,

accepted

27 March

1991)

Rksumk, Nous montrons que la

dkpendance

en

temp6rature

du vecteur d'onde

caractkristique

de l'instabilit6 d'onde de densitk de charge, _que

prbsente

_un

grand

nombre de conducteurs

quasi

unidimensionnels

(lD),

_peut dtre

expliqube

_{par une} th60rie de

champ

_moyen de la transition de Peierls ID, _{pourvu que} la branche de

phonon qui

porte l'anomalie de Kohn

prdsente

une pente finie, et pourvu que la relation de

dispersion

des Electrons _ne soit pas lindariske _au

voisinage

du

niveau de Fermi. Cette

approche

permet ^{d'estimer le}

signe

et l'ordre de grandeur des variations

thermiques

mesurdes. Nous montrons de

plus

_que 1e

gondolement

de la surface de Fermi d'un gaz d'klectrons _«

quasi

_» ID et les effets de ddsordre renforcent la

dkpendance thermique

du vecteur d'onde de

Peierls,

_en

jouant

essentiellement _sur la

longueur

de cohkrence

dlectronique.

Abstract, It is shown that the thermal

dependence

of the wavevector of the

charge density

wave

instability displayed by

_a large number of

quasi

one-dimensional

(lD)

conductors _can be accounted for within the framework of the _mean field theory of the ID Peierls transition, when the finite slope of the bare

phonon

branch

bearing

the Kohn

anomaly

and when _a realistic _non linearized electronic

dispersion

_are considered. This

theory

allows to estimate the

sign

and the order of

magnitude

of the thermal variations

experimentally

observed. Additional effects,

including

the

warping

of the Fermi surface of the

quasi-

ID electron gas, and disorder _are shown to enhance the thermal

dependence

of the Peierls wavevector,

mainly through

_a reduction of the

electronic coherence

length.

I. Introduction.

Theories of the one-dimensional

(lD)

electron gas

predict

the existence of

charge

_or

spin density

_wave instabilities at the critical wavevector q =

2

k~,

which value is related to the ID band

filling Ill.

In these theoretical treatments, the critical _wave vector q is constant in temperature. ^With the

systematic investigation

of _an

increasing

number of

quasi-lD

conductors

displaying charge density

_wave

(CDW~ instabilities,

it _was

surprisingly

found that q could

sizeably

_vary in temperature. ^This effect is

particularly

well established in the _case of

the blue bronzes

Ao.3Mo03 (A

=

K, Rb, Tl)

where _q decreases for

increasing temperature (T)

without _any apparent

discontinuity

in its rate of variation _at the Peierls critical

(*)

Unitd O02 associde _au CNRS.

1036 JOURNAL DE

PHYSIQUE

I M 7

temperature

(Tp) [2-7].

Its thermal

dependence

is _even enhanced in the V

doped

blue bronzes

[8],

^{which do} not exhibit _a true Peierls transition because of the

V/Mo

substitutional disorder, A q evolution of

opposite sign

has also been

reported

in orthorhombic

TaS~ [9]

and for the

upper

(qi)

CDW of

Nbse~ [10]. Again

an enhanced temperature

dependence

of the

qj ^wavevector is observed in

substitutionally

disordered

(Fej

~~

Nbj _~) Nb~sejo [11].

However,

whatever the

sign

of

dq/dT,

the

general

observation of _an increase of

[dq/dT[

for

increasing temperatures

means that this behaviour is _{a common} feature to the

quasi

ID conductors

[12].

In

addition,

the observation of _a somewhat similar temperature

dependence

of the CDW critical wavevector in the 2D metals

2H-TaS~

and

2H-Nbse~ [13],

suggests ^{that these effects could also be}

present

^{in conductors of}

higher dimensionality.

In the blue

bronze,

it has been

suggested _[4]

that the thermal evolution of q, which presents

a saturation at low

temperatures,

^{could be the}

signature

of _a commensurate-incommensurate transition,

However,

NMR measurements

[14-15]

_were unable _to confirm the commensurate nature of the low

temperature

CDW

ground

state and _a

comparison

of all the available

wavevector measurements

[2-7]

shows that _q has the same temperature

dependence

whatever the

de~iation

of its low temperature ^{saturation value}

(due

to

non-stoichiometry

for

example)

from the commensurate value

3/4

b*. In the _same

time,

it _was

recognized _[5] that,

in this

material,

_q has

basically

an activated behaviour in

temperature.

^It was thus

proposed [5]

that its decrease for

increasing

temperatures could be due to the reduction of the number of

electrons in the lD conduction

band, through

thermal excitations towards low

lying

electronic levels

existing just

above the Fermi energy

(E~) [16].

However such _an

explanation

is

specific

to the blue bronzes and _{cannot account} for the increase of q observed in the

MX~

materials

[9- l1].

It has also been

proposed [17]

^{that the}

exponential

variation of q upon

heating

could be due to the thermal

breaking

of _a

bipolaron

sublattice

originating

in _a strong

electron-phonon coupling.

The electronic structure of these

quasi-

lD conductors consists in several conduction bands

cutting

the Fermi level

[16, 18].

In the _case of two conduction bands

developing

their _own CDW

instability (I.e.

at 2

k)

and 2

k)~,

_a

simple

free energy minimization shows

that,

for _a

large

_range of parameters, ^{the critical} distorsion _occurs at _a q

value,

intermediate between

2k)

and

2k)~,

which varies in temperature

[19],

When

2k)

is close to

2kf

it has been

proposed

that the

temperature dependence

is due to the formation of _a kink lattice in the

relative

phase

of the

2k(

_and

2k(~

CDW

[20].

However these mechanisms lead to _a

semimetallic state under the Peierls

transition,

because _q differs from the _average

quantity (2 k)

_{+ 2}

_k)~)/2,

_{for which the}

_folding

_{of the conduction bands opens}

a gap at the Ferrni level

simultaneously

in the two bands. The mechanisms

proposed

in references

[19, 20]

thus cannot account for the formation of _a

semiconducting

state at the Peierls transition of

Ao_~MOO~

and

orthorhombic

TaS~

for

example

where _more

likely

_a CDW

instability

at

k(

₊

_k(~

_occurs

_[5].

Another characteristics of the

quasi-

ID conductors considered here is the

warped shape

of their Fermi surface

[16, 18]

due to finite interchain

tunneling

effects. In that _case, the CDW

instability

is

generally

unable to

destroy completely

the Fermi

surface,

and thus leaves

pockets

of carriers. When

kit

T is lower than the electronic energy associated with these

pockets,

_a better

adjustment

of the

nesting

leads to a small thermal variation of q

[21].

From the band _structure calculations of the blue bronze

[16],

such electronic _{energy can} be

estimated _to be close to 80K

[22],

well below the Peierls critical temperature

Tp=

183 K. As q has

already

reached its saturation at 80

K,

deviations to

perfect nesting

cannot

explain

the thermal variation of _q.

The purpose of this paper is to show that the thermal variation of q observed in these

quasi-

lD conductors is in fact _a

general phenomenon

which _can be accounted for within the framework of the mean field

theory

of the ID Peierls transition _{: a}

single

conduction band

M 7 TEMPERATURE DEPENDENCE OF THE PEIERLS WAVEVECTOR 1037

with _no interchain

coupling, having

however _a realistic non-linearized

dispersion

in the

vicinity

of the Fermi level. This variation has two contributions. A trivial _one

[22]

associated with the finite

slope

of the bare

phonon

branch

bearing

the Kohn

anomaly

and _a non-trivial

one which reties _on the

breaking of

the electron-hole symmetry

[Ii,

due _to the non-linearized band _structure. Furthermore _we show that it is

possible

to

predict

the

sign of

the deviation

of

_q

and to calculate the

magnitude

of the effect

experimentally

observed in the blue bronze. We present our lD model in

part 2,

^{and the} calculation of the temperature

dependence

of the Fermi level and of the electronic coherence

length

in part ^{3. In} part

4,

_we

give

_our results for the

phonon

and electron contributions to the temperature

dependence

of the Peierls

wavevector.

Finally

in part ⁵ our results _are

compared

with the

experimental findings.

2. The model.

We consider _a

purely

ID electron _gas with _a

tight binding dispersion relationship

_:

e~ =

2tcoska

(1)

which, developed

in the

vicinity

of the Fermi _energy

E~o

and of the Fermi wavevector

(± k~o)

reads

(henceforth

_we shall take h

=

I)

:

(k k~o)~

~~

~~°

_~

~~~~ ~~°~

_~

2 _m ~°~ ~

~ ~

~k = EFO

UF(k

₊

k~o)

+

~~ ~°~

for k

< o

(2)

Compared

to the usual

description

of the lD electron gas, in which e~ is linearized in the

vicinity

of

E~~

_we consider here the

quadratic

term, associated with _a finite effective _mass

m ~ 0 for electrons and _m

< 0 for holes. We will _see in the

following that,

in the presence of this term, the _mean

field theory of

the Peierls transition presents two new

characteristics,

which

are a temperature

dependence of

the Fermi level

E~

and _a temperature

dependence of

the CDW critical vector q.

In order to

display

these _two

points,

we consider the free energy ^F of the lD electron gas

moving

in _an electronic

potential

A _cos

(qx).

^This

potential

is self

consistently

created

by

_a

lattice

distortion, involving

_a bare

phonon

of energy w~, ^{and an}

electron-phonon coupling

constant g, that _{we assume to} be

independent

_{of q :}

A

i w~

A~

F

=

dA' _x

(q,

A'

)

A' ₊

~

(3)

o 2 g

q and A _are determined from

(3) by

the

implicit equations

_:

3F/3q

=

0 and

3F/3A

= 0. The q

deRvative of the free energy has two contributions

coming respectively

from the wavevector

dependence

of the electronic

susceptibility X(q, A)

and from the

dispersion

of the bare

phonon

mode w~.

Let _us

develop

the derivative of the free _energy with

respect

to q, as :

A _{d~ (q}

A,

)

i

dw~ A2

dA' ' A'+

=

Ao

+

&qA

j

(4)

o

dq

2

dq _g2

where _q

= 2

k~o+ &q.

The Peierls wavevector at the

thermodynamical equilibrium

is thus

given by

the

equation

_:

~01~ _~02

~~

A

~~/

1038 JOURNAL DE

PHYSIQUE

I M 7

The numerator

Ao

has been

decomposed

into _an electronic contribution

Ao~ of order _one in

I/m

and _a

phonon

contribution

Ao

^{of order} one in the bare

phonon velocity dw~

-,

while

~

dq

Aj,

which is

proportional

to the square of the electronic coherence

length (second

derivative of the electronic

susceptibility

with respect to

q),

is of order _zero with

respect

to these

quantities.

This

simple

fact tells that indeed _{for a} linearized electronic

dispersion

relation and in the

hypothesis

of _a flat

phonon dispersion relation,

the Peierls transition _occurs at a wavevector q

independent

of the temperature and

equal

_to 2

k~o.

_Ao~ ^and

At

_are

easily

found

once the electronic

susceptibility X(q, A)

is known.

x(q, A) depends

_upon the electronic

energies E~

and

E~~~

^{in the} presence of the Peierls _gap A

opened

at ±

q/2

and upon the

Fermi-Dirac functions

f(E~)

^and

f(E~~ _~) according

to

x

(q,

A

)

=

z~~ _(j~ j ^~~~j~~ ₍₆₎

The Fermi _energy

E~

_appears in the Fermi-Dirac function. It has to be determined at each temperature

by expressing

the conservation of the electron number. In the two

following paragraphs,

_we detail

successively

the temperature

dependence

of the Fermi energy

E~,

^{and the} calculation of A

j and

Ao~ ⁺ Ao~ ^{above and below the Peierls critical} temperature.

3.

Temperature dependence

of the Fermi energy and of the electronic coherence

length.

In _a lD electronic system, which _{possesses a}

single band,

the

equation

of conservation of the electron number N reads

(with p

=

I/k~ _7~

_:

N

=

~ ~

dk ^~~~~

(7)

w

(I

_{+ eXP}

P (Ek EF)) n(k) being

the

density

of states in the

reciprocal

_space

n(k)

=

L/w.

N is

simply equal

_to

2

k~o

_* n

(k).

A separate ^{evaluation of the}

integral

has _to be done above and below the Peierls

transition,

due to the different electronic

dispersion

relations.

In the metallic

regime E~

^is

given by equations (2).

As demonstrated in

Appendix I,

to lowest order in

pk)o/2

_m,

equation (7) yields

_:

(wk~ T)~

E~

=

E~o

₊

~

(8)

6 _mu

~

Due to the curvature of _e~ close to

EF,

there is _a

breaking

of the electron-hole symmetry which induces _a thermal variation of the _{Fermi energy.}

In the

semiconducting regime,

the electron-hole

symmetry breaking

also _occurs. The

general expression

for

E~

is derived in

Appendix

I.

By introducing

the effective Fermi

velocity

_:

and the _energy at

midgap

_:

~

~

~ _{~ ~}

&q

~

&q~

(io)

° ~° ~ 2 8 _m

M 7 TEMPERATURE DEPENDENCE OF THE PEIERLS WAVEVECTOR 1039

we find

that,

if

&q

=

0,

the Fermi _energy is

equal

to :

E~

=

E~o

₊

^~~ ~~ ^(l1)

4 _mu

~

and

that,

to lowest order in

&q,

the rate of variation of the Fermi level

position

^with

respect

to the

midgap

_energy when the Peierls wavevector is

changed,

is

equal,

when divided

by kit T,

to

~ ~~~q ~~~

~

~~~ ^pA

~ ~~

/~A ~~ ~

~~~~~

~

In this

equation, Kj(x)

is the modified Bessel function of order I. To derive these

expressions

we have assumed that

E~

lies in the _gap

(in

order that

E~

cuts either the valence _or the conduction

band,

_1~~[&q ^has to be

larger

than A this does not

correspond

to the limit

&q

_- ^{0 for which}

Eqs. (ll)

^and

(12a)

_are

calculated).

As _a matter of

comparison,

in the metallic

regime,

the _same

quantity

reads

(see Appendix I)

d(Eo EF) fIUF

(12b)

fl dq

2

Expressions (I I)

and

(12a)

_are valid to lowest orders in 2

kit Tiff,

and in

(E~ E~o)/kit

T.

The

temperature dependence

of

E~

involves the ratio of the gap width _over the effective _mass.

When the latter is

infinite, E~

remains constant in the whole

temperature

_range.

The

study

of the Fermi level

position,

above and below

Tp

^is a

prerequisite

to the evaluation

of the Peierls wavevector q. As shown in

expression (5),

the latter

depends

_upon the

electronic coherence

length

of the lD electron gas,

through

the factor

At.

We first calculate this

quantity

which is of order _zero with respect to the electron effective _mass and to the

2k~

bare

phonon velocity.

We thus _assume until the end of this section that

only

the electronic

susceptibility

contributes to the second derivative of the free energy with respect ^to q and that the electrons have _a linear

dispersion

in the

vicinity

of

E~.

Since in that _case, there is _a

perfect

electron-hole symmetry ^with respect to the

midgap

_energy

Eo,

the

energies E~

can be scaled with respect to

Eo (E(

=

E~ Eo)

so that the electronic

susceptibility depends

_{upon q}

only through

the factor _u

=

p (E~ Eo)

in the Fermi Dirac functions. The first derivative of _X

(q,

A reads

d

f (El,

u d

f (E(

~ ~, ^u

)

~~j~'~

=

p ~~~~ ^~~~ ~j

^{~~ ~~}

⁽¹³⁾

q q

~

(El El

+ q

)

As

expected,

it vanishes for

&q

₌ ^{0 due} to electron-hole symmetry. ^{The second derivative of}

X(q,

A

),

evaluated at

&q

=

0

consequently

reads _:

d~f(E(,

u =

0

) d~f(E(~

~, ^{u =}

0

~~~ ~~j

^~ #~

p

~

~~( ^~~~

^~

~j

^~~

~ ~~^~

(

l

4)

dq

q

k

(El El

+ q

Such _an

expression

is valid both in the metallic and

semiconducting regimes.

It

emphasizes

the

d(Eo E~)

importance ^{of the factor} which characterizes the rate of variation of the Fermi

dq

1040 JOURNAL DE PHYSIQUE ^{I bt} 7

level

position

with

respect

to the

midgap

_energy when the Peierls wavevector is

changed.

This

quantity,

divided

by k~ T,

has the dimension of _a

length. Consequently, (14)

is

proportional

to the square of _an electronic coherence

length

defined in the

Appendix

2. The summations

are

performed

in the

Appendix 2,

where _we show

that,

in the metallic

regime,

above the Peierls transition _:

Lu~ A~7 ( (3) Aj

₌

~ ~

(15) (k~ T)

16 _ar

while at low

temperatures

in the

semiconducting regime

_:

4Lupk~T

_A

(16)

~l

~r 2A

~~~

kB

T

Aj

is thus found _to

diverge

at low temperatures ⁱⁿ both _cases, but the

precise

temperature

dependence

is fixed

by

the available thermal excitations. In

particular,

in the

semiconducting regime,

A

j is

thermally

activated.

Furthermore,

_we have demonstrated in the

Appendix

3 that

d(Eo E~)

p

is

larger

m

purely

one-dimensional conductors than in real

quasi

one-

dq

dimensional

systems,

so that

equation (16) _represents

_{an upper} bound for

Ai

in real systems

having non-vanishing

transverse

hopping probabilities.

In the _course of the

calculation,

we have noticed that the first derivative of _x

(q,

A

)

is _zero

for _a linearized

dispersion

relation. We shall release this

assumption

in the

following

section to calculate

~~~~'~~

in order to evaluate

Ao.

dq

'

4. Electrol~ic and

phononic

contributions to the Peierls wavevector.

We evaluate in this section the first derivative of the free _energy vith

respect

to q. It involves two terms, which _are

generally

not considered in the

theory

of the Peierls transition _: the first

one comes from the

dispersion

of the

phonon mode,

involved in the lattice

distortion,

and the

second _one, of electronic

origin,

is due to the finite effective _mass of the electrons.

dw

4.I PHoNoNlc CONTRIBUTION. We shall call u~~ the

phonon velocity

_u~~ ~

= at

dq

2

k~

of the bare

phonon branch,

in which the Kohn

anomaly develops. Straightforwardly

from

equation (4)

_we obtain

v~~

A2

Ao~ ⁼

j

~

(17)

g

The

phonon

contribution to the Peierls _wavevector is thus

equal

to _:

~~

~~P~

~

~<)() ~~~(2~

~~~~

in the metallic

phase

and to :

ar ~

u~~A~ _A

u~ &q~~ = exp

(19)

8Lg~k~T kBT)

at low

temperatures

^{in the}

semiconducting

state. The variation of &q~~ is ^controlled

by

the thermal

dependence

of the _square of the electronic coherence

length

_: it is

quadratic

in the

bt 7 TEMPERATURE DEPENDENCE OF THE PEIERLS WAVEVECTOR 1041

metallic

phase

and activated in the

semiconducting phase.

Its

sign

is

opposite

to that of u~~.

4.2 ELECTRONIC CONTRIBUTION. We detail in

Appendix 4,

the calculation of the

quantity

Ao~ associated with the first derivative of the electronic

susceptibility

with respect to q, in the metallic and in the

semiconducting regimes.

As

already

stated above Ao~ is of order _one in

I/m.

We find in the metallic

regime

_:

It contains two contributions _: the first _one, which involves the

Logarithm,

_comes from the q

dependence

of the effective Fermi

velocity

_wF

(in

the denominator of the ID

density

of

states)

and turns out to be

proportional

to the electronic

susceptibility.

The second _{one comes} from the thermal deviation of

E~

from the

midgap

_energy

Eo.

^In

particular,

the term

7

( (3)/12

is due to the thermal variation of the Fermi energy.

Equation (20) requires

that

k~

T _« v~

k~, &q/2

mu _~ «

l, A/2

_u~

k~

«

I,

and

A/2 k~

T _« I. It leads to an electronic contribution to the Peierls wavevector &q~j

equal

to

~F &qel "

~'~~ ~~~ ~)~

[~

2

yU~k~~

7

((3) ~~2

^{°g ~}

l 7

((3)

~

'~~B

T 2

fi ⁽²¹⁾

The

leading

temperature

dependence

of &q~j ^is

quadratic.

It is similar to that of

&q~~, ^{because both}

dependences

_come from the square of the coherence

length. However,

the

amplitude

of the effect

depends mainly

_upon the value of the electronic

susceptibility.

Its

sign

is

opposite

to that of the effective _{mass :} q is

larger

than 2

k~o

^{if the carriers} are holes and smaller than 2

k~o

^if

they

_are electrons.

In the

semiconducting

state,

assuming

that

/

»

I,

and to lowest order in

I/m,

_we find

~ T

that _:

L(k~ T)~

_{A A} 2

ar~k~

T

~°' ~rmu( ^ark~ T~~~

₂

_k~

_{T 3 A} ^~~~~

which leads to

A2 ark~

_T

A 2

ar~k~

T

~~ ~~~~

4 _mu

)

A ~~~

2

k~

T 3 A ~~~~

2

ar~k~

T

In

equation (23),

_we have left the factor I which

gives

_a

change

of

sign

of 3 A

&q~j for

k~

T close to

A/6.

We shall _come back to this feature in the next section.

It should be noted

that,

in the present

derivation,

the

exponential

factors found in

Ao~ ^et

Aj

_are due to the thermal variation of

E~

in the gap,

through

the

quantity

d(Eo EF)/dq.

Without _a

precise study

^of the

position

of the Fermi level in the semiconduct-

ing

_{state, we} would not have recovered _a Peierls wavevector

equal

to

2k~o

at zero

temperature,

as it should.

&q~j can be put ⁱⁿ a more compact

form, by noticing that,

to the order _zero in

I/m,

the gap width is

given

at low temperatures

by

_:

~°

~

=

2

'~~~ ~exp (-

^~

(24)

do

A 2

kB

T

1042 JOURNAL DE

PHYSIQUE

I bt 7

Consequently,

to the lowest order in

k~ T/A,

_{&q~j can} be

simply

related to the

temperature dependence

of the Peierls _gap at low

temperatures, according

to _:

~

do

~ ~~

~~ ~~

8

mu) ^~~°

&q~j thus presents a low

temperature

saturation

effect,

similar to that

displayed by

the electronic gap, and _an activated behaviour with an activation energy

equal

to half of the total gap. In the

temperature

range close to the Peierls

transition,

the

analytical developments

that

we have used _{are no}

longer valid,

and _a numerical minimization of the free energy is

required,

in order to connect the activated

regime

with the T~ law.

5.

Comparison

with the

expedn~ental

data.

To summarize _our

results,

the thermal variation of the critical wavevector of the Peierls CDW

instability

has _two contributions. A first _one, &q~~, ^{has a}

sign opposite

to that of the

slope

at 2

k~

of the

phonon

branch

bearing

the Kohn

anomaly.

This contribution is

negative

in the blue

bronzes,

for which neutron

scattering

measurements

give u~~»0 [23].

The second

contribution,

_&q~j, has _a

sign opposite

to that of the effective _mass of the carriers

forming

the

CDW,

in the metallic

regime,

and for

k~

T

» 0.15 A in the

semiconducting

_one. This

sign

_can

be

obtaiqed

in

principle

from the _curvature of the conduction band at

E~. However, since,

in the

quasi-

ID conductors considered

here,

the 2

k~ instability couples

the ₊

kj

_wavevector of

one band to the

k(~

one of the other

band, [5, 16, 18],

it is not so

straightforward

to deduce

the _curvature of the «average» electronic

dispersion.

More

generally

the

sign

of

&q~j

depends

_upon the

sign

of the derivative of the

density

of state at the Fermi level. This

quantity

enters into the

expression

of the

thermopower,

for _a ID

tight binding

metal

[24].

Thermopower

measurements

performed

in

Nbse~

show that holes _are involved in the qj CDW transition

[25].

This

predicts

a

positive

_{&q~j, as}

experimentally

observed

[10] (the magnitude

and

sign

of &q~~ ^{is however not known in}

Nbse~).

In the _case of the blue

bronzes,

no clear indication _can be deduced from similar measurements because the

sign

of the

thermopower changes

in the metallic

phase [26].

However _a

negative

_{&q~j can} be inferred from the

negative

^value of the thermoelectric _power in the V

doped

bronze

[26].

In that _case, both contributions have the

sign

of the

&q experimentally

observed

[2-8].

In the metallic

regime, expressions (18)

and

(21) predict basically

_a T~

dependence

for

&q.

This _accounts

quite

well for the

dependence

of the critical wavevector, ⁱⁿ the whole

temperature

range, in the V

doped

blue bronze

[8]

^{which do} not exhibit _a true Peierls transition because of disorder. This

quadratic

thermal

dependence

is also

obeyed

above about 200 K in the

undoped

blue bronze

(below

this

temperature

a sizeable

pseudo

_gap is formed

[26]).

Let us now estimate

quantitatively

the

magnitude

of

&q predicted by

these

expressions.

In the blue

bronze,

_{one gets} from the band calculation of reference

[16]

: t 0.18 eV in

(I)

defined with _a

=

b/2 (b

=

7.55

h

_is the _true lattice parameter ^{in the lD}

direction).

This

gives,

with

k~o= 3/8b*,

_a Fermi

velocity u~=1.910~m/s

and _an effective _{mass m}

=

3.8m~

(m~ being

the _mass of the free

electron).

At _room

temperature,

^the

predicted

values of

&q~~ ^and &q~j are thus _:

* &q~~ =

4 _x 10~ b*

using

_u~~

=

2 _x

10~ m/s [23]

^and g

/j

=

26 mev

[22] ~NB

an

estimation of &q~~ =

6 _x

10~~

_{b* was} done

by

_a different

approach

in Ref.

[22]),

* q~j = 9 _x 10~ ^~ b*.

This

gives

a total contribution of

&q

₌ ^0.013

b*,

to be

compared

with _a measured value of

&q(

₌ ^0.04 _± ^0.01b*

[5].

^The order of

magnitude

is correct, and _we believe that the

bt 7 TEMPERATURE DEPENDENCE OF THE PEIERLS WAVEVECTOR lo43

difference of _a factor 3 _must not to be taken too

seriously

_:

indeed,

the presence of the _zone

boundary

not too far from

k~

_may

yield

_some

inaccuracy

in _our estimate of uF and m

using equation (I).

In addition

expression (21) gives only

the first term of the

expansion

of

&q~j ⁱⁿ

I/m.

We _now consider the behaviour of _&q in the _presence of _a Peierls _gap of total width A.

Expressions (19)

and

(23) predict

_an actived behaviour for &q with activation

energies equal

to A and

A/2, respectively.

Such _a

dependence

is indeed observed in the blue bronze

[5]

and in

Nbse~ [10].

In the blue

bronze,

it _was this activated

behaviour,

which had led _us to propose

[5]

that &q could be controlled

by

thermal excitations of carriers towards states situated at _an

energy

AEO

^above

E~.

^The fit

performed

at that time

yielded _AEO

close to 600 K which is about half the value of the Peierls _gap

(do

_~# 0.1-0.15 eV

according

to Refs.

[27, 28]).

However this

agreement

must not be taken too

seriously

because the activation energy

entering

into the

expression (23)

is

only weakly

temperature

dependent

at low

temperatures

^when &q is very small and

inaccurately

determined.

Expression (25)

is

interesting

because it scales &q~j on the temperature

dependence

of the

Peierls gap

(and

it should be noted that _a similar

scaling

law _can be obtained for

&q~~

by combining equations (19)

and

(24)).

In

particular,

this _proves that the Peierls wavevector q

begins

to deviate from 2 k~~ when the Peierls gap

(I.e.

the order

parameter)

_no

longer

saturates. Simultaneous measurements

[2, 4,

_{5] of q and} of the square of the order parameter

(I.e.

the

intensity

of _a

superlattice reflection) performed

in the blue bronze show that both

quantities

indeed

display

_a measurable thermal

dependence

at about the _same

temperature

:

Tp/2,

in agreement ^{with the}

prediction

of

expression (25).

According

to

expression (23)

_&q~j should vanish for

k~

T _~# 0.15 A.

Assuming

for the blue bronze

do

_~# 0.I eV

[27],

the

vanishing

_occurs at about 140 K for which Aye ^0.8

do [22].

Actually, taking

into account also the

phononic contribution,

the true

vanishing

of _&q should

occur for Ao~ ⁺ Ao~ =

0,

I-e- at about

T~#

130 K

using

the numerical values

quoted

before.

Below this temperature &q should

change

of

sign,

These features _are not

experimentally

observed. A

possible

_reason, which wil1be discussed in the next

paragraph,

is that in _a

quasi-

ID material such _as the blue bronze

Ao~ is overestimated

by

the present

calculation,

and that the

change

of

sign

_{of &q}

(if

it

exists)

should _occur for lower

temperatures

^where _q ^has

already

saturated at 2

k~o,

^within

experimental

_errors.

Let _us estimate the order of

magnitude

of _&q well above its

vanishing temperature.

^{In the} blue

bronze,

at 175

K,

for which A

yeAo/2 expression (19) gives

_{&q~~ ~# 8 x}

^10~~

b* and

expression (23), including

the contribution _Cl

given by (IV.16)

in the

Appendix 4, gives

&q~j ~# 6 _x 10~ ^~ b *. This leads to a calculated

&q

_~# IA _x 10 b * which is 8 times smaller than the

experimental

determination

(&q(

~# I-I

x10~~b*.

_We believe that this

discrepancy

is

~partly)

due to the _use of too

large

values of

Ao~ and A

j.

Expressions (13)

and

(14)

show that the order of

magnitude

of these

quantities

_are fixed

by p d(Eo E~)/dq.

^As

shown in the

Appendix I,

this last

quantity

has _an

exponential

temperature

dependence

which is due to the

divergence

of the

density

of _state at ±

A/2.

In

quasi-I

D

materials,

such _as the blue

bronze,

the

warping

of the Fermi

surface,

due to finite

tunneling

in the transverse

direction

(ii )

smooths the

divergence

of the

density

of states.

Thus,

_as shown in the

Appendix 3,

this reduces

p ~~~° ~~~

and

consequently

the values of

Ao

^and

Ai

^{in the}

dq

'

semiconducting regime (the

effect of

ii

^{is much weaker in the metallic}

regime

because the

density

^of state is _not

singular

ⁱⁿ the energy range of

interest).

We have not tried to calculate

d(Eo-E~)

more

quantitatively

the amount of reduction of

p

with

tilt-

dq

1044 JOURNAL DE PHYSIQUE I bt 7

Another

experimental

trend which _can be understood within the framework of _our

model,

is the enhancement of the

temperature dependence

of &q

by

disorder effects. At _room

temperature,

&q was shown to increase

by

_a factor of 2 when 2A fb of V _are substituted to the Mo in

1Co_~Mo03 [8].

^In a similar _way, the _&q at 140 K of the _qj CDW increases

by

_a factor of 4 _on

going

from

Nbse~

to

(Fei ~~Nbj _~)Nb1Seio [1Ii-

A

quantitative theory

of these effects

goes

beyond

the present

model,

^restricted to non-disordered

systems. However,

the enhancement _can be

qualitatively

understood in the metallic

regime where,

in the

expression (21)

&q~j is

basically given by

the ratio of the electronic

susceptibility X(q,

A

= 0 _over the

square of the electronic coherence

length

and where corrections due _to the

temperature dependence

of

E~

are weak. At

high temperatures,

disorder creates CDW Friedel oscillations of _mean square

amplitude (A~)

which broaden the Fermi

surface,

_as thermal fluctuations do.

By adding

these two effects in

quadrature and, assuming

that the

pseudo

_gap created

by

the disorder does not

change significantly

the

temperature dependence

of the Fermi energy,

&q~j is

basically given by

the

expression (21)

with T

replaced by

_an effective temperature of

about

T~

₊

~~~

~

[29].

In this

high

temperature

limit,

the disorder

basically

enhances

~

&q~j,

mainly

because of the decrease of the electronic coherence

length.

This effect also enhances &q~~, ^{as shown}

by

the

expression (18).

More

quantitatively

the substitution of 2A fb of V in the blue bronze leads to a decrease of the in-chain correlation

length

which has been

experimentally

measured

[8]. Using

this determination _our

approach predicts

_an enhancement of

&q by

_a factor

2.7,

while _a factor 2 was

actually

observed

[8].

6. Conclusion.

We have shown that the thermal

dependence

of the critical wavevector of the CDW

instability

observed in _a

large

number of

quasi-

ID metals _can be understood within the framework of the

mean field

theory

of the ID Peierls transition. It includes two contributions _{: one} is due to the finite

slope

of the bare

phonon

branch which bears the Kohn

anomaly

and the other _one

comes from the

breaking

of the electron-hole symmetry, ^which occurs when _a realistic _non-

linearized electronic

dispersion

is considered. The

sign

and the order of

magnitude

of the thermal variations &q of the Peierls wavevector

experimentally

observed have been accounted for

by

the

present theory.

We have found that the T

dependence

of &q is

basiGally

controlled

by

that of the electronic coherence

length

it

yields

_a

T~ dependence

above the Peierls critical

temperature Tp,

^and an activated behaviour below

Tp. Moreover,

we have shown that _a reduction of the electronic coherence

lingth

leads to an enhancement of &q. This allows _our results to be

generalized,

at least

qualitatively,

to systems

displaying

_a

warped

Fermi surface

or in which _some disorder exists. However, detailed calculations _are

required

to _assess these two effects _more

quantitatively.

The effects considered here could

qualitatively explain

_a somewhat similar thermal

dependence

of the critical wavevector of the CDW

instability

observed in systems ^of

higher

electronic

dimensionality,

such _as transition metal

chalcogenides [13],

and _even the

a-

Uranium

[32].

In

addition,

there is _{no reason}

why

finite

effqctive

_mass effect could _not drive similar _wave vector

dependence

in the _case of _a

spin density

wave

(SDIV~ transition,

stabilized

by nesting

effects. In this

respect,

it is

interesting

to notice that the thermal

dependence

of qsDw in Chromium

[33]

^{resembles that of} _qcDw ^{of the electronic}

systems

considered here.

AcknowledgJnents.

C. N. wishes to

acknowledge

_a fruitful discussion with A. H. Moudden at the very first

stage

of this work.

bt 7 TEMPERATURE DEPENDENCE OF THE PEIERLS WAVEVECTOR 1045

Appendix

I.

Temperature dependence

of the Fermi energy.

in this

appendix,

_we solve the

implicit equation (Eq. (7)

in the

text)

_:

oo

2

k~o

₌ ^dk

(I. I)

oo

(

I ₊ exp

p (Ek EF))

which

gives

the conservation of the electron number in the

system.

^A

separate

^{evaluation of} this

integral

has _to be done above and below the Peierls

transition,

due to the different

electronic

dispersion

relations.

* Metallic

regime

_:

Since,

for k _» 0 _:

"

~~ ~~~

~

~~~~

_{~~~~ ~}

~~ ~F0)~/2

m

equation (I.I)

reads _:

~~° ~k~o

_~

(

l + A exp

flu

~ ^* exp

(px~/2 _m)

^~~'~~

with A

= exp

p (E~ E~o).

To the lowest order in

I/m,

I-e- when

(E~ E~o)

«

kB

T and

k~

T« mu

),

_{one can} make the

following development

_:

°° i

p

^°°

x~

_exp

p

u

~ x

~~°

"

_

~~~

~~

(

i ₊ A exp

flu

~

x)

2 _m

_

~~~

~~

(

i ₊ exp

flu

~

x)~

^{~~ ~~}

which

implies

A ar

~

k~

T

Log

= ~

(I.5) (I

₊ A exp

flu

~

k~o)

6 _mu

~

thanks to the

integral

_:

I=j°°

^dx

^~~~~~~~~~

~g

j°°

^dx

~~~~~~~~

=

'~~ (I.6)

k~o I ₊ exp

flu

F

x)~

oo

(I

_{+ exp}

flu

F

x)~

3

(p

VF)~

The

temperature dependence

of

E~

is thus

given by

_:

E~

₌

E~o

+

( ark~ T~~/6 mu( (1.7)

provided

that

k)o/2

_{m «}

k~ T, k~

T«

u~k~~

and

k~

T« mu

I.

*

Semiconducting regime

_: We _assume that _a Peieris distortion _occurs at a wavevector q =

2k~o+ &q,

_so that _{a gap} of total width A is

opened

in the electronic

spectrum

^at

±

q/2. By introducing

the effective Fermi

velocity

_:

WF "

~F(1

₊

3~/2'~VF) (1.8)

and the energy ^at

midgap

:

Eo

=

E~o

+ u~

&q/2

₊

&q

_{~/8 m}

(1.9)

1046 JOURNAL DE

PHYSIQUE

I bt 7

the electron

dispersion

relation reads

E~ =Eo+ ~~~~~/~°~~~~~±(~/w)(2k+2k~o+&q)~+A~. ^(1.10)

For _an energy

E~

= E in the valence

band,

the inversion of this relation

yields,

to lowest order in

I/m (A/mw)

_«

1)

_:

~

4(E- Eo)~- A~ Eo

E

(2

^k + 2

k~o

₊

&q)

_~#

~

l ₊

~

(I,I I)

w~ mw~

and for _{an energy} E in the conduction band _:

(2

k ₊ 2

k~o

₊

&q

)~ ~#

~~~ ~~~ ~~

l

~

j°

(1.12)

w~ mw~

From these

relations,

the densities of states

nv~(E)

and

nc~(E)

_can be evaluated and introduced in

equation (I.I),

which thus

becomes,

in the low temperature ^limit

~~

» l _:

2

qn

(k )

_~#

j'

_dEn

v~

(E

_e^{~~~ ~~~} ₊ ^{~ ~} dEn

c~

(E )

_e ^{~ ~~ ~~~}

(I,13)

-oo Eo+

(1.13)

also _assumes that

E~

is in the Peierls gap, which is the _case for small &q

(I.e.

when u~

&q

_<

A).

After

long

but

straightforward calculations,

it is found that

~~'~~ ~~~ ~

_~~

_{~ ~~° ~~~}

~ 4

~~~i(

~~

~ ~~° ~~~

~ ^{~ fl~}

~

~° ~

(I.14)

In the low

temperature

limit

~~

» l

,

one can further _use the

asymptotic

limit of the 2

Bessel functions

Ko(x)

_~#

Kj(x)

_~# ^'~ e~~ for _x

- co, which

gives

t~

&qw~

_~# ^'~~ _exp

^~~ Sh

p (Eo E~)

₊ ^~

~

Ch

p (Eo E~) (I.15)

fl

² 4 mw~

We will need later the

expression

for

E~

at

&q

₌

0,

in the low temperature limit. It reads

th p

(Eo E~)

=

~

~

(I.16)

4 _mu

~

which

becomes,

for A _« 2 _mu

)

and

(Eo E~)

_«

k~

T_:

(E~ Eo)

=

(E~ E~o)

_~#

k~

T ~

~

(I.17)

4 _mu

~

We will also _use the

quantity ~~~~

~°~

evaluated at

&q

₌ 0. This

quantity

_represents the q

bt 7 TEMPERATURE DEPENDENCE OF THE PEIERLS WAVEVECTOR 1047

rate of variation of the Fermi _energy with

respect

to the

midgap

_energy when the wavevector of the Peierls distortion is

changed.

With the _same conditions of

validity

_as

equation (1.17),

equation (1.15) yields

:

d(Eo EF)

_UF

~ ~

j fl ~~ (I.18)

fl ~~

^~

pA

^~ arA

AKj ~

The _same

quantity

in the metallic

regime

is

simply equal

to _:

d

(Eo EF) fl

_UF

(1.19)

fl dq

2

since, then, Eo

is the

only

factor which

depends

_{upon q.}

Appendix

^II.

Calculation of the second derivative of the electrol~ic

susceptibility.

We present ^{in this}

appendix

the calculation of the second derivative of the electronic

susceptibility,

called A

j in the text

(Eq. (4)).

This

quantity

has to be evaluated in the absence of _curvature of the electronic

dispersion,

I-e- with

I/m

=

0. With the _same notation _as in the

Appendix I,

_{we can} write the difference of the Fermi Dirac functions _:

~~

P ~2j

j

F

f(E~) f(E~~~)

=

(II.I)

Ch

(u)

₊ Ch 4

vi y~

₊

A~)

with _u

=

p (Eo E~)

and _y

=

k ₊

k~o

+

&q/2.

The second derivative of

f(E~) f(E~~ _~)

with respect to u, evaluated at _u

= 0 is

equal

to

~~f~~ d~f(E ₎ ^P ~

~~2

~

~~

^~ =

~~

2

~

~i

_{Y~ +}

A~

~~

~

~ ~~

~~) ^~

^~~~'~~

Consequently, putting E(

=

E~ Eo

d~f(E(,

_u

=

0

d~f(E(

~ ~, ^{u =}

0

)

~~X ~~j

^~ ~#

p

~

~° ~~~

~

jj ^~~~

^~~~

(II.3)

dq dq

k

(El El

+q

)

~~

P

~~2

~2

~

~2

~'

~~ ^{~ ~~~~ ~~~ ~ ~°' ~~} ~

[l

^{+ Ch}

~~)

j~

~~~~~

Such _an

expression

has to be evaluated in the metallic

regime

and in the

semiconducting

_one.

In the first _case

~~~° ~~~

=

u~/2

and to lowest order in A,

Aj

reads

dq

L A 2vF 7

< (3)

A

j ~#

~ ~

(II. 5)

ar

(k~ T)

16 _ar

JOURNAL DE PHYSIQUE I T I,M 7, JUILLET 1991 42

1048 JOURNAL DE

PHYSIQUE

I bt 7

where _we have used the

integral:

°'

dx Sh

(x)

14

( (3

(II.6)

_~ x

Ch~ (x)

flr~

As

expected, At

_can be put ^{under the form} :

At

=

~ ~

«vi f( _(II.7)

involving explicitly

the electronic coherence

length [30]

@@

u~

~°

4

ark~

T ~~~'~~

d(Eo-E~)

proportional

to

p

_given

by (1,19). By

contrast, in the

semiconducting regime,

A

dq

cannot be taken _{as a} small

quantity compared

to the other

energies

of the

system. Using

the value of

p

~~~° ~~~

in this limit _:

dq

d(Eo EF)

_UF

(II.9)

fl

~ ~

~

AKj pA

~

~~

~~~~~.

~

4

LUF kB

T~~~

_~

_p

A

(II- lo)

~

~2

_A

To obtain this result _we have

replaced

in the

integral

Ch _x

by

its

asymptotic

form for

large

arguments, ^which is

only

valid in the limit of low

temperatures,

^{I-e- when}

PA

» I.

By analogy

with

(II.7) At

^{in the}

semiconducting phase

_can be put under the form _:

~~

_~

~~~~ ~~~

f(2

_{~~~ ~~~}

fITU~

d(Eo E~)

with

ii proportional

to

p given by (II.9)

_or

(1,18).

dq

Appendix

In.

Electrol~ic coherence

length

in the low

temperature

^Peierls state of _{a «}

quasi

_» one-dimensional electron gas.

In this

appendix,

_we wish to prove, on

qualitative grounds, that,

in the low

temperature

Peierls state, ^the

quantity p

~~~° ~~~

for _a

quasi-ID

conductor is smaller than that

dq

predicted by equation (1.18)

which

applies

to a pure ID conductor. The argument relies upon the

shape

of the

density

of states, which

presents strong

^Van Hove

singularities

at

±

~ in the ID case, while these

singularities

are smoothed

by

transverse

hopping

effects.

2

First let _us note that the

relationship

between

p ~~~°

~

~~~

and the

density

of states _comes q