On the dynamics of band Jahn-Teller systems with A15-structure

Physica

1lOA (1982) 373407

North-Holland

ON THE DYNAMICS

Publishing

OF BAND

WITH

Uniuersitiit

JAHN-TELLER

SYSTEMS

A15STRUCTURE

Robert Fakultiit fiir Physik,

Co.

Konstanz,

Received Revised

KRAGLER Postfach

5560, D7750

Konstanz,

Germany

10 July 1980 17 June 1981

We construct a thermodynamic model for a system of electronic d-bands coupled to the elastic lattice. For the electronic fluctuations a Debye-type of relaxation, for the lattice displacement a modified elastic equation of motion is assumed. As a result a coupling of certain acoustic and relaxational modes is found leading to a soft mode instability. For non-vanishing external fields displacement and dielectric response functions are derived. The special case for wavevector qjl[l IO] is worked out explicitly. A comparison between the present phenomenological model and a microscopic multiple-band electron-phonon transport theory, recently given by the author, reveals remarkable agreement between both approaches.

1. Introduction

High-temperature superconductors of AlS-structure such as Nb$n and V3Si exhibit a number of anomalous properties’) among which the structural phase transition and the associated softening of the elastic shear mode are of particular interest. The features which have been one of the main objectives of the theories for the martensitic transition of intermetallic AU-compounds, are: (i) The elastic softening on cooling, most pronounced for the modulus i(c,, - c12) which almost vanishes at the transition, whereas the shear modulus cU is only weakly temperature dependent and the bulk modulus f(c,, + 2c12) stays essentially constant”). (ii) The [llO] transverse acoustic (T,A) phonon with [liO] polarization which corresponds to the shear modulus $(c,,- c,?), exhibits a marked softening extending halfway to the Brillouin zone boundary4). (iii) A central peak for the same branch has been observed&) which diverges as the martensitic transition is approached from above. These and other anomalous properties of A15compounds, especially the exceedingly large normal-state specific heat, indicate an anomalous large density of states at the Fermi leve15). Most of the anomalies seem to arise from the presence of three orthogonal sets of linear chains of transition-metal atoms in the A15 structure which give rise to a planar Fermi surface6) and to a narrow peak in the electronic density of states. 0378-437 1/82/0ooooooO DO2.75 0 1982 North-Holland

R. KRAGLER

374

By exploiting to date explain According been

some

the quasi-one-dimensional ihe martensitic

to an earlier attempts”.‘?

which a sublattice ruled showed

band

transition

suggestion ascribing

structure

as being by Anderson

the transition

distortion

is associated.

However,

In addition,

Sham’s

that

only

the FIZ-optic

mode

is weakly

all theories driven’-“).

and Blount”) to a soft optic

out by experiment4b).

assuming a linear coupling between this optic mode the latter one is driven soft first. Recently, the importance of anharmonicities

nearly

electronically

there

have

phonon

with

this mechanism lattice-dynamical

has been theory”“)

temperature-dependent, mode

and,

and the acoustic

for the mode

shear

softening

has

been studied by Achar and Barsch’“). According to their model the shear mode softening arises from a subtle balance between two opposing effects, the softening caused by the electronic instability and the stiffening due to anharmonicities. In spite of this finding, anharmonicities enter, if at all, only through higher order electronic contributions in the known theories of the martensitic transition ‘5.‘6a,2’). The electronic models to date attribute the mode softening either (i) to a direct coupling of electronic degrees of freedom to the lattice dilatation (d-band Jahn-Teller effect at high-symmetry points of the Brillouin zone) or (ii) to a coupling via an optic mode which pairs the transition-metal atoms in the chains (Peierls-like charge-density-wave (CDW) driven instability). The first model based on the band Jahn-Teller effect is due to Labbe and Friedel (LF)‘). There, a ID band structure is assumed for each of the linear chains. The Fermi level is located close to the bottom of one of the empty d-bands (at the F-point in reciprocal space). The 1D density of states exhibits a l/u/E-singularity. The transition is regarded as a band Jahn-Teller effect where the triple degeneracy of the ID bands is removed by a tetragonal distortion which shifts the bands relative to each other. Subsequently, it was demonstrated by Cody, Cohen and Halloran (CCH)‘) constructing an idealized Jahn-Teller model with a step-function density of states

N(E)

- O(E),

as in the 2D case, and with the same degeneracy

splitting

like LF, that the essential point of the LF-model is a rapid variation of the density of states near the Fermi level. This feature, common even to the more refined versions’*“.’ 4 of the LF or CCH model which have been treated since then, is most unlikely for a realistic 3D band structure. In fact, band-structure calculations by Mattheiss22) have revealed that due to interchain coupling the ID character of the bands is wiped out. Moreover, since the assumption of the LF and CCH-model locating the Fermi level in the vicinity of the band edge, is highly artificial, Gorkov”) has considered the martensitic transition to be the result of a Peierls-like CDW-driven transition in the ID chains. The Fermi level is located close to the X-point where there is a band degeneracy

BAND

JAHN-TELLER

SYSTEMS

WITH

AIS-STRUCTURE

375

in the A15structure of the cubic lattice. Again, this X-point degeneracy is lifted by pairing the transition-metal atoms in the chains. There results a lowering of energy as the Peierls gap opens up. This energy gap is proportional to the amplitude of the pairing mode which, in turn, is coupled to the elastic strain, thus leading to a tetragonal distortion of the lattice. Based on the X-point Gorkov-model Bhatt and McMillan’4”) have formulated a dynamical Landau theory where the shear mode instability is caused by a CDW instability through the coupling of the CDW’s to the rlz-optic modes. The highly successful Bhatt-McMillan theory was even further improved to a two-band tight-binding model’“) based on a 3D-band structure. This model exhibits both a X-point Peierls gap (like the Gorkov-model) and a Jahn-Teller degeneracy splitting (like the LF-model) at the M-points and at the rX-saddle points, which are jointly responsible for the electronic instability. Hence, this model is a kind of hybrid model since it incorporates the essential features of the LF and Gorkov-model. Recently, it has been pointed out by Bhatt’&) that the essential results of the Gorkov-Peierls model by Bhatt and McMillan are independent of a specific microscopic mechanism and would likewise follow from a Jahn-Teller type instability too. Most of the above-mentioned theories of the martensitic transition are static ones, only a few onesg~“c,‘3*14a.16) are concerned with the dynamical aspect of the transition. The first proper dynamical band Jahn-Teller model was developed by Pyttep considering the interaction of the elastic strain with electronic density fluctuations. Separating static and dynamic strain contributions in the equations of motions provides a relation between the shift of the band edges and the static distortion. Renormalized acoustic modes are obtained by considering fluctuations about the static quantities. This procedure is, however, based on the assumption that acoustic modes do not induce any electron redistribution which restricts the theory to the collisionless limit. Strictly speaking, Pytte’s theory does, therefore, not apply to the hydrodynamic limit where the renormalization of the acoustic modes is of interest too. In a more recent treatment, in the spirit of Pytte, given by Thomas and the present author”), the elastic shear mode is coupled directly to a CDW of the same symmetry. In this respect the model by Bhatt and McMillan’4”) is more general in that it includes also r12-optic modes. Here, the shear modes are coupled to the CDW’s only by the intermediary of the optic modes. In both theories the shear mode softening originates from the temperature dependence of the coefficient of that term which incorporates the electronic variable to second order. While in ref. 13 the temperature dependence of this quantity which turns out to be the inverse of an effective density of states, is determined empirically, the Bhatt-McMillan model assumes instead the usual

R. KRAGLER

316

linear both

temperature theories

charge

dependence

make

density

equations dissipative

In the

as generalized

terms

associated

function.

characteristic transition. relaxation dominant

fluctuations.

arise

dissipation

of Landau

theory.

use of a Debye-type

This

Bhatt-McMillan

Lagrangian with

mechanism theory

equations

are expressed

involves

a relaxation

mechanism

the

of motion

the CDW’s

function

for the dissipative

As to the dynamical

of relaxation

and diverges

aspect, for

where

by means time

the

dynamical the of a

which

is

at the martensitic

Quite similarly, the model of Thomas and the author introduces a time l/y which characterizes interband relaxation processes in the hydrodynamic limit. As to the equations of motion one

assumes for the electronic equilibrium configuration

fluctuations a relaxation depending on the lattice

towards an instantaneous distortion. The distortion.

on the other hand, satisfies the usual equation of motion with an induced stress depending, in turn, on the electronic variable. In summary, both theories are able to reproduce qualitative features of the observed dispersion of the TrA phonon&) associated with the shear modulus $(c,, - cl?). As the central peak is concerned, however, both theories amount to the same conclusion

that this feature

cannot

be ascribed

to electron

dynamics:

at least

a slower, non-electronic relaxation mechanism seems to be required. It is the purpose of the present paper to extend the simple treatment of ref. 13 in a way such that the full symmetry of the AIS-structure is taken into account. Hence, in the high-temperature phase to which our phenomenological model is restricted, the cubic lattice is coupled to the system of three degenerate bands arising from the three orthogonal chains of transition-metal atoms. potential The paper is organized as follows: In section 2 the thermodynamic is constructed

in terms

of the

elastic

strains

e and

the electronic

d-band

occupation n, making use of symmetry. Coupled equations of motion for the relaxing electronic band system and the elastic lattice are derived in section 3. Section 4 gives the bare response and the secular equation for the coupling between

acoustic

and relaxational

modes.

The generalized

response

obtained

for non-zero external fields is the subject of section 5. It turns out that the exact response matrices are expressible by means of Dyson-like equations in terms of the bare quantities, obtained for the uncoupled system. Explicit calculations of the response matrices are performed in sections 6 and 7, where special attention is paid to the case of wavevector ql/[llO]. For this particular direction the secular equation for the coupled modes is investigated in some detail in section 8. Finally, in the Conclusion the phenomenological model is compared with a microscopic multiple-band transport theory recently given by the author16) revealing remarkable agreement between both theories.

BAND

2. Thermodynamic

JAHN-TELLER

SYSTEMS

WITH

AlS-STRUCTURE

377

potentials

In order to study the dynamics of a band Jahn-Teller system which describes at least some physical aspects of A15compounds in the hightemperature phase, we construct a model which involves the electronic band occupation numbers ni (i = x, y, z) and the elastic strain components iii (i, j = x, y, z) as relevant thermodynamic variables. Henceforth, the thermodynamic state of the system is defined by the set of variables {ni, eij}. The electronic band system is characterized by the occupation numbers of those d-bands only which partake in the dynamics of the present system. In particular, the quantities ni are defined as deviations from the equilibrium occupation of these relevant d-bands. Disregarding s-d scattering, the total number of d-electrons is conserved so that Ei ni = 0. On reasons of a compact notation the three components ni are condensed into n = {ni}. The state of the elastic lattice is characterized by homogeneous distortions which are specified by the six independent components of the strain tensor e. For the formulation of a dynamical theory, however, the acoustic mode amplitudes come into play. Therefore, it will turn out to be more convenient working with the elastic displacement u rather than with the elastic strain e. The latter one can be written as symmetrized gradient of u, i.e. e = Vsu = i[ Vu + ( VU)~].

(2. la)

The appending subscript S on V means that only the symmetric part of the displacement gradient Vu is to be taken, the superscript T denotes transposition. Similarly, the diagonal components of the strain tensor are designated by e D = VD,, = a(v)

- u.

(2. lb)

Here, the subscript D on V denotes the operation of projecting out only the diagonal components of e. The matrix 9(V) is a differential operator with diagonal elements defined by 9(V)ii = di whereas the off-diagonal elements 9(V), (i# j) vanish. This symbolic notation will considerably facilitate the manipulation of the equations of motion and will therefore allow us, to expose the principal steps of the derivation of eqs. (3.5) and (3.11) more clearly. Since the thermodynamic potential which will be constructed in terms of the variables n and e, has to fulfill certain symmetry requirements it is therefore advantageous to start already with symmetry-adapted variables {v,, E,} defined as linear combinations of the corresponding Cartesian quantities. As regards to the underlying symmetry considerations the details are exposed in 15)and will therefore not be given here.

378

R. KRAGLER

e,,Mi

EI =

(exx+ eys+

~2 =

(en

63 =

(2ezz - e, - e,,)/VG,

-

(2.2a)

e,,)NZ

and, similarly, vI

=

v2 =

(n, +

fly

(n, -

n,)/V?,

+

n,)/d,

(2.2b)

v3 = (2n, - n, - n,)/d6. For reasons

of generality

we admit

vi f 0, i.e. allow temporarily

violation

of

the conservation of d-electrons. The connection between symmetry-adapted and Cartesian bases is given by the transformation matrix T(,,j (cu = 1,2.3; j = X, Y, z),

(2.3)

so that

v=T-n, The with

remaining the

advantage potentials

Voigt

E =

r -e'.

off-diagonal

(2.2c) strain

components

eii are chosen

notation,

E, = 2eii (a = 9 - i ~ j) where of the symmetry-adapted coordinates is that can be put into block diagonal form.

in agreement

i # j. The obvious the thermodynamic

We now consider the four contributions to the thermodynamic potential: Firstly, there is an elastic strain energy contribution which is expanded in terms of the elastic strain tensor e. Linear terms vanish due to cubic symmetry. Anharmonic contributions are neglected; for the importance of those higher-order terms on the mode softening see for example Achar and Barschm). In explaining the salient features of the lattice dynamics of A15 compounds in the cubic phase the harmonic approximation will be sufficient. Hence, in terms of the symmetrized strains E = (E,, . . , Ed) we have IFL(e) = 2~ . % . G = i(c,, + 2c,zg

+ j(c,1

- C,z)(&

+ E:)+

&QE:

+ E: + E$.

(2.4)

V is a (6 x 6)-matrix diagonal in the extended basis E, which contains linear combinations of the bare elastic moduli ceP according to symmetry. The elastic matrix c,~ in Voigt notation is connected with the usual fourth rank

BAND

JAHN-TELLER

SYSTEMS

WITH

AlS-STRUCTURE

319

tensor of the elastic moduli due to the mapping c,~ = Cijkl where (Yis related to (ij) by CY= i = j or (Y= 9- i -j (i+ j) and /3 to (kl), respectively. In the case of cubic symmetry with only three independent elastic moduli and cM, the matrix V has three eigenvalues: There is a singlet of Cl13 cl2 Al-symmetry associated with the bulk modulus (c,, +2ci2) where the corresponding eigenvector describes a dilatation l l, a doublet E associated with the shear modulus (cl1 - clJ belonging to uniaxial distortions l2 and l3, and a triplet TT with eigenvalue cM involving the shear strains l4, l5 and E6. Due to the coupling of the elastic lattice to electronic degrees of freedom the bare elastic moduli c,~ will be renormalized and become temperature dependent. The crystal is stable against homogeneous deformations if all eigenvalues of C ,+ are positive. If one of these eigenvalues, however, decreases to zero the crystal may then distort continuously to a new structure, the symmetry of which is determined by the eigenvector of the “soft” eigenvalue. Secondly, there is an electronic contribution of the d-band system FB(v) = ;Y . ii-’

.u

= g-‘(v:

+ v: + I&

(2.5)

where 3 = 21. The bare coefficient Z can be interpreted as static dielectric response function due to &I = -Z&L with the electronic potential I_Lbeing conjugate to the occupation number n. In a microscopic formulation of the theory’&) the quantity Z represents an effective density of states

Z(T) = Ix (- ma4 k

= J( dENr(E)(-

af’/aE),

(2.6)

which depends sensitively on the band structure through Nl(E) and on the temperature T due to the Fermi distribution f:(E). Here, K = (k, I) with I referring to the band index. Indeed, in A15compounds Z = Z(T) increases when the temperature is lowered. Thirdly, we assume a deformation potential coupling between the shallow d-bands and the elastic lattice. To lowest order, the contribution due to the electron-lattice interaction turns out to be of simple structure in the symmetrized basis F&E, v) =

E

-

‘9 * u

(2.7a)

with the coupling matrix V. It is noteworthy that the shear strains ~4, ESand 6, do not couple in our model which means that only electronic fluctuations within each band are accounted for. In analogy to the lattice term, eq. (2.4), it follows from (2.7a)

R. KRAGI.ER

380

F&E,

V) = (%,, + 2!9&,u,

Following

Pytte’),

9,, = -2%,2, shears

we choose

so that

the longitudinal

the coupling

E? and E+ Then

(2.7b)

+ (Y,, ~ %,Z)(~Z~Z+ QV?). deformation

is restricted

the coupling

matrix

potential

to volume

such that

conserving

% can be written

elastic

as

%= 09, where

(2.8)

D = S,, - 9J12is the deformation

potential

which

the band edges due to an applied uniaxial strain. The matrix 9 is diagonal with ?pZZ= Y,, = I being accounts for the fact metry and, henceforth,

where

non-zero

the shift of only.

:‘P,, = 0

that an isotropic deformation E, cannot change symgives rise only to s-d electron redistributions, which

are neglected in the present paper. 9 assumes the following form?: y-1

considers

In terms

of the Cartesian

basis the matrix

.gb.y=s = j-;j

4; is a constant

matrix

(2.9) which

has all elements

equal

Hence, the matrix elements of .9 are ‘9,, = : and convince oneself that 5!*= 9 is an idempotent matrix. Finally, we add an external field term F&r,

to unity.

‘3i, = --- I. It is easy

(2. IO)

v) = v . 5’“’ + G * c?“‘.

where 5’“’ = (5;“‘. . . . &‘Yt) and 6”’ = (US”. . . . . CT?‘) are the external potential and stress components in the symmetrized basis. Because

chemical

of

5”“’= y * P ex’

(3. I I)

there is v - 5’“’ = n * pex’. As regards to the stress that a factor i is omitted tribution

to

will be useful

thermodynamic

3. Equations

potential

components so that Z

oY’ to (T;;X’we follow

for the derivation consists

the convention field con-

-Gex' = e:Ct. The external of the response

matrices

below.

The

of the sum of (2.4) to (2.7) and (2.10).

of motion

In order to study the dynamics of the present system we have to start from the equations of motion for the electronic band occupation n on the one side and the displacement field u on the other side. As to the electronic d-band system we assume that the band occupation numbers n relax towards their instantaneous equilibrium values no(e) belong-

BAND

JAHN-TELLER

ing to a distorted ansatz

which

structure

reference

accounts

SYSTEMS

configuration

for

this

of this rate equation

AIS-STRUCTURE

of the elastic

behavior

is most

WITH

lattice.

is a Debye-type

transparent

381

The simplest

relaxation.

in the symmetrized

The basis

with v and E (3.1) where

the relaxation

matrix

r is chosen

to be of the form

r=+P.

(3.2)

rll = 0, leading to ti, = 0, accounts for the fact that the d-electrons is conserved in our model. In the corresponding

total number of equations for u2

and v~, describing the relaxation of the orthorhombic and tetragonal configurations, rZ2 = Ta = y is assumed for reasons of simplicity. y plays the role of an interband relaxation rate characterizing the electron redistribution between equivalent d-bands. The rate equation (3.1), taken for granted here, can be justified on a microscopic basis’6”,b). Indeed, as has been demonstratedlti), rate equations of this type are deducible from a multipleband Boltzmann equation to which the relaxation-time approximation has been applied, however, modified in a way so that local particle-number conservation is guaranteed. The instantaneous equilibrium value V’(E) which appears in eq. (3.1) is determined

by the requirement

1

* v = - yZ(DP

where the relation The change-over 9-l

rewritten

t

(6F/6v) = 0,giving rise to (3.3)

eq. (3.1) becomes

&1+yB C

with

i.e.

* E - Zg”“‘.

IJO(e) = - ZDB Hence,

of stationarity,

where

* E + 9 . &‘“‘),

9’ = 9 has been used. to the Cartesian basis 9

turns

into

9

(3.4)

is accomplished

according

by multiplication

to eq. (2.9). Then,

eq. (3.4) is

as

$+y9).

with the matrix

n = - yZDiR( V) * u - yZ9 -

prxt,

(3.5)

R. KRAGIXR

382

As the elastic

lattice

u is of the usual the linear

is concerned

the equation

form but with an additional

coupling

between

the elastic

of motion induced

lattice

for the displacement

stress

term arising

and the electron

bands,

from

thus (3.7)

The meaning

of this extra

term is such that the deformation

takes

place

with

respect to a distorted reference configuration depending on the electronic band occupation n. The instantaneous equilibrium value e”(n) is inferred from the condition of stationarity, expression for the divergence V.c:e”(n)= As regards

(SF/Se) = 0. which of the induced stress

-DLRT(V).n-

coupling

* n = g(V)

v.$?D.2

to

the

(3.X) side of eq. (3.8) which originates

(2.7). we made use of the identity

* 9 f n = 9T(

With the help of eq. (2. lb) which strain tensor e to the displacement ing abbreviated notation

V) . n.

(3.9)

relates the diagonal u. and, moreover.

elements ef’ = v,, of the introducing the follow-

V. c : e = V. c : Vsu = (cV@ V) - U. where c is the fourth-rank equation (3.7) can be casted

Taking

the Fourier

transform

quantities are denoted end up with (WI + iys) (pw’1-

(3. IO)

tensor of the into the form

elastic

moduli,

the

cV@ VI * u=mr(v)*n+V~cr”‘.

(p $I-

u +iDCR’(q)

Christoffel

(3.1 I)

of eqs. (3.5) and (3.1 I). where

by the same

symbols

* n - -rZDLB((q) * u = - iyZ9

cq@q).

following

V.(Y”‘.

to the first term on the right-hand

from the electron-lattice

leads

which

the transformed

now depend

- y’“‘.

* n = -icP.q.

of (3.12a, b) it is expedient to rewrite For a further treatment coupled equations in a more succinct fashion by introducing (3 X 3) matrices:

on (q, w), we

(3.12a) (3.12b) this system

of

the following

A(w) = ~7 + iy$

(3.13a)

B(q) = - yZI%Uq).

(3.1%)

BAND

C(q)

JAHN-TELLER

SYSTEMS

WITH

AIS-STRUCTURE

383

(3.13c)

= iD9T(q),

wq,4=pw2+w3qq,

(3.13d)

E = iyZ.2.

(3.13e)

Henceforth,

eqs. (3.12a, b) can be combined to the following matrix equation:

A(w) Wd C(q)

D(q, A

* (:I=

(3.14)

cr~f.e”d)-

4. Bare response

According to eqs. (3.13b) and (3.13~) the matrices explicitly on the deformation potential parameter D, coupling between the elastic lattice and the electronic electronic band system is decoupled from the elastic (3.14) turns out to be simply

B(q) and C(q) depend which brings about the d-bands. For D = 0, the lattice. The solution of

@(q, w) = X0(% w) * 6U(q, WI,

(4. la)

Wq, 0) = E”(q, w) - Wq, ~1,

(4. lb)

where we have redefined the thermodynamic 6p = n,

6d = Xu,

variables (4.2)

and the conjugate fields SD = Cc=‘, &I = (iX)-’ aext - q.

(4.3)

X = q2fJop is a normalization factor with 0, being an appropriate frequency which characterizes the harmonic lattice. The bare response matrices are given by X’(q,o)=-A-‘.E=-Z

(4.4a)

P(q, w) = x*D-’ = (2&p) . (pw21 - q*fi-1.

(4.4b)

For the decoupled subsystem, the frequencies of the pure modes are given as the eigenvalues of the matrices A(w) and D(q, w). According to det A(w) = w(w + i-y)*, the response matrix X0 exhibits relaxation poles. In addition, the origin of all term terms involving the matrix 9 can be traced back to interband processes. Therefore, from the specific form of X0 one may conclude that interband transitions are responsible for the dynamical redistribution of electrons in order to bring the perturbed electronic band

R. KRAGLER

384

system

back

suggestive

into

For the response from

(4.4b).

which

equilibrium.

to interpret that

Since,

for

D = 0 one

X” as the bare dielectric

has

Sp = X0. SU, it is

response.

will be quite similar. matrix 3:” the argument mode frequencies 8’ has poles at the acoustic

are determined,

for example.

as the bare displacement

response

As to the electronic band Debye-type, the eigenvalues

by eq. (4.5b). Thus, of the harmonic

It follows w = tf1,

E:” can be interpreted

lattice.

system which exhibits a relaxation behavior of and eigenvectors of the homogeneous equation

A*n=(wl+iy!I)*n=O are easily

found

to be n “I = (I 11)/\/3,

(R,),

n “) = ( ii2)/\/6, 1 n”) = (I iO)/v2,

(R,).

w,=o: w I.3 = -iy

:

(4.6)

(RI).

with the mode R, corresponds to a uniform The eigenvector n”’ associated band occupation, which is just another way to express the number conservation of d-electrons. The remaining modes Rz and Rx, which are proper relaxation modes, describe tetragonal and orthorhombic band occupations, respectively. As the lattice

is concerned,

the familiar

Christoffel

equation

for an isotropic

solid D. u = (co21 ~ p ‘$A). is recovered for vanishing A = CK @ K only depends elastic

moduli Cjq+

tensor

c

u = 0.

(4.Sb)

external stress on the direction

and contains,

Us”. Here, cosines K

for cubic

the Christoffel matrix = q/y and on the bare

symmetry,

the elements

(C,, - (.jq)K;, (4.7)

(Cl?+

CM)KiKj

(i#

j).

For wave propagation in certain symmetry planes, such as (001) and (Oil). closed-form solutions for the frequencies of the acoustic bulk modes w = t f1, can be given=). Eigenvalues and eigenvectors for certain high-symmetry directions of the cubic crystal are summarized in table I. If the coupled D is non-zero and, henceforth, the matrices 6 and C are present, it renders possible, for vanishing external fields, to find even then closed-form solutions of the homogeneous system of coupled equations (3.14) for high-symmetry directions. As expected, some of the relaxational and the acoustic modes turn out to be mixed, while others are still unaffected by the coupling. This is particularly the case for the relaxation mode R. Whether

BAND

JAHN-TELLER

SYSTEMS

WITH

385

AlS-STRUCTURE

modes are coupled or not depends on the direction of q. Table II lists the coupled modes, the frequencies of which are always determined by a cubic secular equation of the form NA(q, o) = (w + ir)(W*- 0:) + i-yfi&

(4.8a)

= 0,

where (4.8b)

f_xA = a,ZD2q21pR:.

The frequencies 0, occurring in eq. (4.8a) are those of the harmonic lattice given in table I. The prefactor ar in (4.8b) is a numerical factor given in the third column of table II, accounting for the different renormalization of the

TABLE Frequencies

Rh and polarization

e(A)

along high-symmetry

I

of acoustic

directions

bulk

modes for propagation

of a cubic crystal

ci

e(A)

A

(1W

LA

‘I-IA ‘LA

1(010) (001) [1101

R:=~(C,,+C12+2Ckl)q21P

(1 IO)/+2

LA

n: = f (Cl1 -

(I io)hd

TIA

0: = c44q21p

cool)

TzA

R:=f(cll+2c12+4CM)q2/p

(lllph

LA

c12)q21p

11111 TIA n:,,=f(CII-C12+C44)q21~

TzA

TABLE Coupling

of relaxational

The frequencies

and acoustic

of the coupled with the quantity

i

[ t@J

[ttol 11111

Coupled

modes

II

modes

in the high-symmetry

modes are determined or specified

in the fourth

a&

ar

a1.A= 213

2ZD2/3c,,

&@TIA

ar,A= l/2

ZD'/(c,, - cd

Rs@LA

a1.A = l/6

ZD2/3(c,r

aTA = l/3

ZD’/(c,,

R, 69 TzA

(4.8)

column

R,$ LA

R~@TIA

directions.

by a cubic equation

+ Cl2 + 2644)

- c12+ CM)

R. KRAGLER

386

bare acoustic determine of the latter

modes

relaxation ones

decreases. section

A. The roots

the frequencies mode

become

coupled

normal

with

the acoustic

(w = -i-r)

damped

A detailed

of eq. (4.8a) which

of three whereas

discussion

the attenuation

of the secular

are in general

modes.

complex,

Due to the mixing

modes

(w = + 0,)

of the relaxation

equation

the mode

(4.8a) is postponed

to

8.

5. Generalized

response

Now, we extend electronic potential present.

our investigations to the case where external fields, the pex’ and the elastic stress sex’ introduced in eq. (2.10), are of CL”’ and iarrt - q one has to

In order to find solutions n and u in terms invert eq. (3.14) with the result (6

. D-’ . C ~ A)-‘.

. A-‘.

-(D-C

-(B.D

E

6)-l.

C.

(-A-‘.

‘.C-A) E)

‘.E-C’.D -(D-C.A

’

i

‘.B)-‘/

(5.1) The form

of this equation

which

yields

respect to the external fields pext and following dynamical susceptibilities: X(q, w) = (6 -6-l

g(q)

we define = (-

IIX)C’=

response

iaex’. q. suggests

. C - A)-’ . E.

E(q, w) = N2(6 - C . A ’ . 6) Moreover,

the linear

(iDlN)%(q).

u with the

(5.2a) (5.2b)

‘.

an electron-lattice

of n and to introduce

coupling

matrix (5.3)

proportional to the deformation potential parameter D. The physical meaning of the susceptibilities X and e, as given by (5.2). is elucidated, when the coupling is switched off. Then, the matrices 6 and C vanish, and hence X and E simplify to X0 and E”, see eq. (4.4). Since X0 and E” are the bare response matrices, X will be the dielectric response and Z the displacement response modified due to the coupling of the lattice to the relaxing band system. The notation of (4.2) and (4.3) allows to rewrite eq. (5.1) in a succinct form (5.4) The appearance

of the cross-terms

X. g . E” and Z . g’ . X0 in the response

BAND

JAHN-TELLER

SYSTEMS

WITH

AIS-STRUCTURE

387

matrix is a consequence of the electron-lattice coupling incorporated in the coupling matrix g. One easily convinces oneself of the symmetry relation X.g.ELXQ.g*= which guarantees the cross-symmetry of the response matrix. It is evident from eq. (5.4) that there are two contributions to the dielectric response (Q/W),

= X,

(6~/6J)~ = X0 - g - 8,

and, likewise, for the displacement (sd/sJ),

= E:,

(5Sa)

response

(Sd/SU), = E”. g+ . X.

(5.5b)

The subscripts J and U remind for the fact that the corresponding external fields are kept fixed, which means that either the elastic lattice or the electronic band system is assumed to be rigid. In full analogy to the microscopic treatment’@) the off-diagonal contributions to the response can be rewritten in such a way that the fields SJ and 6I.7 are eliminated in favor of 6du = B - 6J and Sp, = X - SU. This allows a very physical interpretation of the response in the coupled electron-lattice system: Henceforth, electronic fluctuations 6p are either caused by a variation of the external potential 6IJ = pext, directly affecting the band occupation, or result, rather indirectly, from an external stress 6J a aext - q. This gives rise to an elastic deformation du according to which the bands are shifted by means of the deformation potential coupling and respond, in turn, with a redistribution of electrons. Thus Sp=X.6LJ+X”.g.6du.

(5.6a)

Similarly, elastic fluctuations 6d are either caused by applying an external stress 6J = aext. q directly to the elastic lattice, or arise, in an indirect way, from an external potential 6U which unbalances the band occupation 6~~. Because of the coupling the lattice reacts with an elastic distortion to the electron redistribution. Therefore, 6d=8.8J+e0.g+.L$,

(5.6b)

Now, the response matrices X and 8, given by eqs. (5.2a) and (5.2b), can be redefined in terms of infinite series of X0 and Z”, respectively, leading to Dyson-like matrix equations. As regards to the dielectric response X, one rewrites eq. (5.2a) into X=[i+(-A-’

.E).C*.C-‘.C]-‘.(_A-‘.E),

(5.7a)

where use has been made of the relation 6 = E * CT which holds because of

3xX

R

KRAGLER

9 * 3 = 3. With (4.4a) and P(q, w) = ~ CT * D_’ * c = g(q). which is interpreted form of a matrix

as an electronic Dyson

E”(q. w) * g’(q), “self-energy”.

(5.8a) eq. (5.7a) is casted

into the

equation

x=x”+xO*P*x.

(5.9a)

The graphical representation of this equation is shown in fig. la. Iteration of X leads to a chain of X0’s, where each X” is connected with neighbouring ones by the intermediary of the quantity P. Strictly speaking. if we compare this equation with our microscopic transport theory”‘), P is not a self-energy but rather an irreducible scattering vertex which involves square of the electron-lattice coupling g and the bare lattice “propagator” the latter being represented by a wavy line, see fig. lb. Obviously, terminology used for the diagrammatic representation is taken from many-body theory. In an analogous fashion, the displacement response can also be rewritten into ~:-[1+o~‘.C.(~A~‘.E),C’l Using

‘.,v”D

eq. (4.4b) and defining

II(q, o) = p.Y-‘C

. (-A~

we end up with a matrix

an elastic

‘. E) . C’ = g+(q).

Dyson

equation

of eqs. (5. IO) and (5. I I) 5, defined

by eq. (5.2b).

‘,

“self-energy”

the E:“, this

(5.7b) by

X”(q. 0).

g(q).

(5.Xb)

of the form

_= = _=(I +E”.rI.E.

(5.Ob)

the graphical representation of which is given in fig. 2a. The quantity II would correspond to the irreducible polarization part involving the bare electronic fluctuation “propagator” X” which is contracted by g, see fig. 2b. Perhaps, the interpretation of X, 2, P and II in terms of propagators and self-energies seems far-fetched at this stage. However. this point will be elucidated in the Conclusion, where the present phenomenological model will be compared with our previous formulation of a multiple-band transport theory”), which amounts to the same results in the hydrodynamic limit.

(0)

Fig.

I. (a) Diagrammatic

m

illustration

terms of X0: (b) the electronic

= )x0(

+ ) XouPyi

of eq. (5.Ya) which gives the exact

“self-energy”

P. eq. (5.8a).

dielectric

re\ponae

X in

BAND

JAHN-TELLER

SYSTEMS

WITH

AIS-STRUCTURE

Fig. 2. (a) Diagrammatic illustration of eq. (5.9b) which gives the exact in terms of B’; (b) the lattice “self-energy” H, eq. (S.8b).

6. Calculation

of the displacement

displacement

(6.1)

of @ are as follows:

= - bqiqi, aq:, @ii

@ii = c -

t

matrix B is defined through eq. self-energy II times X2, see eq.

Q, = (D - C - A-’ - 6) = (pw21 - q2A) - ,yD29tT(q) - 3 (q). elements

response

response E:(q, w)

For Df 0, the exact displacement response (5.2b). Since C - A-’ - I3 is equal to the lattice (S.lOb), we obtain with (5.3) and (4.4a)

The matrix

389

(6.2)

where a =(c,,-c&-?xD2,

(6.3a)

b = (~12 + CM)- fxD2, c = pw2The calculation

c4‘$q2. of a

requires

inversion

of Q, and multiplication

with X2 =

21&p. Hence

(6.4) with matrix

elements

(ii = (C - aqf)(C -

aq:) - b*qiqi,

5ij = Eji = [b2q2k+ b(c - aq:)]qiqi,

(6.5)

where (i, j, k) is understood as a cyclic permutation of the indices (x, y, z). Because the pole structure of the displacement response 5, as given by eq. (6.4), is not easily seen for general wavevector q, we shall concentrate on the special case q = q(llO)/d% Ancillary to eq. (6.3a) we introduce the ab-

R. KRAGI,ER

190

breviations d = bq?/2. with which

e = c ~ u472,

the determinant

(Mb) of @ simplifies

to

det @““) = c(e? ~ d’). The matrix

elements

(6.6)

of 5, eq. (6.5), reduce

[X.X= tyV = ce,

5:: = ez - d’.

[AL= 5y.r= cd.

[.rL= [v: = 0.

to (6.7)

so that

(h.8)

Immediately

accessible

to

physical

interpretation

are

only

certain

linear

combinations of the matrix elements of E”‘“‘, which determine the renormalization of the bare acoustic mode frequencies due to electron-lattice coupling

=

=

20” iy (02 - .n~)+-------w+iy

ml iy-(W*- n:>+----w +iy

#‘O)(q, n,.

(&‘)a)

ZD*q” 6p

(6.9b)

ZD*q” 2P

w) = E:“I’,‘“‘( q, w) = $J+

flz and a3 are the harmonic

frequencies

(69c) of the longitudinal

mode.

of the

transverse shear-modes with polarization along (liO] and [OOI], respectively, propagating in [I IO]-direction. However, due to the coupling between the harmonic lattice and the relaxing electron bands, some of the elastic modes couple to a relaxational mode, as given in table I, and are, thus, modified. The frequencies of these renormalized modes are determined as the poles of E?‘“‘, eq. (6.9a-c). Obviously, the denominators of 5”“” and .\“O’ lead to the same cubic equation (4.8) with &A = l/6 and a r’A = l/2, respectively. The TzA-shear mode, however, does not couple to one of the relaxational modes and remains therefore unchanged. Similar results for Ep’(q, W) would be obtained for

BAND

JAHN-TELLER

SYSTEMS

WITH

AIS-STRUCTURE

391

propagation along other high-symmetry directions, such as [loo] or [Ill]. As expected, the coupling of those modes is in agreement with table II. Henceforth, details will not be given here, since the calculation is analogous to that for q(([llO]. It is of interest to study the displacement response function E$““(q, w) in two limiting cases: (i) In the high-frequency limit o * y, the coupling of the TrA-shear mode with the relaxation mode R2 is negligible (6.10a)

Poles occur at the harmonic mode frequency w = k&(q) found in the high-temperature regime. (ii) In the case of the low-frequency limit w + y the electron band occupation follows the slowly varying lattice distortion adiabatically. Hence, local equilibrium is established and the poles of (6. lob)

now occur at the renormalized frequency fi:(q, 7’) = n:(q) - Ss(q, T), where 6!(q, T) = Z(T)D2q2/2p. In the soft-mode theory the (quasi) harmonic frequency &(q) is commonly denoted by wee whereas w. substitutes the renormalized frequency &(q, T) which goes to zero and is thus called the soft-mode frequency. With the static displacement response function 8$r’o’(q) = -2&/fi:, following from (6.10b), the expression for the displacement response function, see eq. (6.9b), can be casted into a canonical formkx25)

WO’(q,w) = aY’O’(d

w2

_

fi*;

2

!jJr (q 2

o)’

(6.11)

7

This is the dynamic susceptibility of a damped harmonic oscillator with a frequency-dependent damping r2(q, w) = 6:(q, T)/(o - ir) which couples linearly through 6* to a relaxing mode with relaxation time l/y. The free oscillator has the (quasi) harmonic frequency R,(q). Due to the coupling to a “slow variable” which comprises the electronic degrees of freedom, the low-frequency susceptibility a l/n: turns out to be a l/h;. At the structural instability the temperature-dependent renormalized frequency fi2(q, To) vanishes while the harmonic frequency L?,(q) remains finite.

W?

R. KRAGI.ER

7. Calculation

of the dielectric

For the calculation

response X(q, w)

of the dielectric

response

matrix

X it is expedient

to

start from eq. (S.7a) X=[l-XO*P]

‘ax”,

(7.1)

which gives the exact response matrix X in terms of the bare response matrix X” and the electron “self-energy” P. According to eq. (5.8a), P requires the knowledge of the bare displacement response matrix z” which is easily derived

from eq. (6.4)

E”(q, 0) = with matrix

$$!$43%WI,

elements

6: given

(‘7.2) by eq. (6.5), however.

u and h substituted

(7.3a)

&I = (Cl1 - C44), h,, = (C,? + (‘4). The same considerations

hold for det a”. The index

help of eqs. (5.3) and (4.4a) the quantity

x0 * P = (D2x(w;y)/.hw(q)

* mq,

=

4q?5P,

5ij

=

5jt

+

4f5;

+

q:SL

0 denotes

X0. P can be rewritten

D = 0. With the into

w).+?‘(q) = zC(q, w),

with the abbreviation z = D*,Y(w; r)/9 det a” The matrix elements of 5 are defined as lit

by

-4qiqj5~-4qiqk5Yk

and

+

(7.4)

X(W; y) = -- Z iy/(w t iy).

2qjqk‘$$.

(7.5)

where

=

-2qitC

2qf5;

(i. j, k) are cyclic

+

qt[L

permutations

[ 1 - X” . PI-’ = [det( 1 - zg)lm’S(q,

+

sqiqjt1:

~

q&J&I:

-

q,qk[yk,

of (x, y, z). It follows

from eq. (7.4) that (7.6)

WI.

with l9ii = (I -

Zljj)(I

-

i!
-

(7.7)

Z'
(ii, j, k) cyclic) i%,

=

6ji

=

ZJij(

1 - Z{kk)+

Eq. (7.6) multiplied matrix X

Z2c&{jk.

from the right with X0 finally

x(w; Y)

‘(% w, = 3 det(, The matrix

elements

_ zl;) @(q, w).

of 0 = 36 - 42 turn out to be

yields the dielectric

response

(7.8)

BAND JAHN-TELLER

SYSTEMS

WlTH

AlS-STRUCTURE

393

@ii = 26ii - 6ij - i?iky

(i, j, k) cyclic

(7.9)

Oij = 2611 - 93ik- IYii.

Although

9 is symmetric,

this is no longer true for 0

since (6 - 2)‘=

4*6+6*2.

This rather involved expression for the dielectric response matrix X(q, w) is best appreciated by considering again the special case for q = q(llO)/~/z as was already done for the displacement response Z(q, w). First of all, =6”“‘, required for the evaluation of P, is obtained from eq. (6.8) by replacing the expressions for c, d and e by co = p(w2- n:>, do

=

Cc12

+

(7.3b)

c&2/2,

eo= pw2-

(Cl1

+

c44)q2/2,

according to which the determinant

as follows:

fl$)(w2- 0:).

@b”” = p3(02 - flf)(o’-

det

of a0 factorizes

(7.10)

Then, the matrix ~‘““’ reduces to

p3’(q,

w)

=

pq2(w2-

n:>

(7.11)

;; ( -

t 53

with 51= f [w2 -

(Cl1

52= -2[w253 =

[w2

-

(CII

+:

(Cl,

cl*+ +:

c44)(q2/2P)ll

s

c12+

%

c44m2/2pN,

(7.12)

c12)(q2/2P)l.

-

Because of (51

-

52)

=

P (w2

(5,

+

52)

=

t 53 =

-

m

(7.13) the determinant

t cm2

-

m,

of (7 - z{) turns out to be

det( 1 - z~(“~‘)

=[

(02-n:,+&3gq. [(&n3+-L&z3

Introducing

(w’the abbreviation

n:>-(02-

0:)

(7.14)

394

R. KRAGLER

k = z(w’the matrix

0;)p2q2 =

elements

x(w ; y)D2q2 9p(J

- n:)(02

(7.15)

.- n;)’

of ?%(“O’ (q, o) simplify

considerably:

19::‘“’ = 1911’“’= (1 - k<,)( 1 - k12) - i k’& 19$czl;“’ = (1 - k<,)’ - k2,$ (7.16)

S$“) = 19$‘~’ = k&( 1 - k(j) + ; k’& “‘0’ = 19.G with which

~"'0, L1

-

$j'l_lO,= 'L

the matrix

~("0, zv

elements

=

_I

2 &Al

~

kc'+

k52),

for 0”‘“’ can now be calculated

according

to

the prescription of eq. (7.9). However, in analogy to (6.9). we are only interested in certain linear combinations of @I!“’ which allow an immediate physical interpretation of the dielectric response X”“‘. Hence (m’-

n;> + ~

iy

ZD’q’ - __

w+iy

iy w +iy

(w2-fj9+

Finally, ponents

21,

I*

ZD’q’ . (w’~ 6P 1

by multiplication with x(w; r)/3 det(l of the dielectric response X”“’ are

xl”O’(q,w)

xi”O’(q,

=

x!i:‘“;‘o’(q, w) + x:‘,‘O’(q,w)

=

lixb; Y) 1- x(w ; r>(~2q2/6p)/(w2 - 0:)

w) =

X$‘O’(q,0)

~

(co- 0;) ‘. 0;)

‘_

- z&“‘“‘) the

(7.17)

relevant

com-

(7.18a)

x’,‘:“‘(q. 0)

(7.18b)

x(w; Y) = 1- ~(0;

rW2q2/2p)/(~2 - 0:)’

x$“O’(q, w) = X11’O’(q, w) = 2x’,“‘“(q. If we remember the form from eq. (7.4) by making following expressions: Pdq,

w)

=

't3:<'$)= I

w).

(7.18~)

of the electron “self-energy” P”“‘. easily obtained use of the property % . CR= Se, we are led to the

P3(q, w), (7.19)

P2(q, 0) = Q$@$ wMoreover, expression,

with given

02.

LZ,=~,,+&=$, C&=&-&=1 and in eqs. (7.18) can be casted into the form

S,-Sr,=I

the

BAND

JAHN-TELLER

SYSTEMS

WITH

AlS-STRUCTURE

395

(7.20) Eq. (7.20) clearly exhibits an enhancement of the bare dielectric response matrix X’(q, w) = x(w; y)S due to the electron-lattice coupling. Henceforth, the simple relaxation pole of x(w; y) = -Z iy/(w + ir) is modified such that due to the mixing with acoustic modes there occur altogether three poles determined again by the cubic equation (4.8) in section 4. However, when comparing the dielectric response function X’j’“‘(q, w), eq. (7.20) with the corresponding displacement response function a’:“‘(q, w), eq. (6.9~) one feature is noteworthy: as expected, #:“1”“’has a pole at w = +Rj. In contrast to that, the pure relaxation pole w = -iy in X1”” is modified due to its coupling to the LA mode with w = +R1.

8. Coupled modes For the calculation of the renormalized response matrices s and X in sections 6 and 7 we have confined us to q\j[llO]. This choice was motivated by the fact that the softening, as known from the experiments on Nb$Sn3,4), is most pronounced for the shear mode propagating in [llO] direction with polarization along [iio]. Obviously, both the displacement response #‘“‘, eq. (6.9b), and the dielectric response XP’O), eq. (7.18b), can be casted into the form W”‘(q, w) = (0 + iy)/%(q, X$“‘)(q, w) = -iyZ(w’-

w),

02,)/N?(q, w),

(8. la) (&lb)

where the common denominator Nz(q, w) is the polynomial given in eq. (4.8) specified for q1][110] and A = T,A. The zeros of Nt(q, w) = 0 which is a cubic equation in o, determine the frequencies w,(n = 1,2,3) of three coupled modes for given q and CQ.Since the subsequent discussion is focused only on the behavior of the [ 1lO]TiA-mode mixing with the relaxation mode RZ, the index A will be suppressed in the sequel. As to the interband relaxation rate y, introduced in (3.2), this parameter of the present phenomenological model characterizes the tendency of the electronic band system to relax towards equilibrium by means of redistribution processes. Therefore, y cannot be handled as a freely varying parameter such as temperature or wavevector. Only the case of strong interband coupling, i.e. l/y-+0, deserves special attention. In this limit the acoustic modes

1%

R. KRAGl.ER

w,,? = ?&(q,

T), see eq. (6.10b), are undamped

This temperature

dependence

states

Z, eq. (2.6), occurring

looked

at as a measure

relative

deviation

temperature

for

values

equation

w1.3= 2 \/fit

with temperature.

elastic

softening, shear

since

modulus

it determines

the

(c.:, - CT,) at a given

(c,, - c,?) at room temperature. cy = 0 and u = 1 it is easy to find the exact

(4.8a).

pled from the relaxation and the pure relaxation be the high-temperature In the opposite

the

of the renormalized

from its value

For the limiting the secular

but decrease

comes into play because of the effective density of in CY= ZD’/(crr ~ c,?). The quantity (Y itself can be

In the case

CY= 0 the acoustic

mode. Because the bare sound mode wJ = ~ i-y are recovered, limit.

modes

roots

of

are decou-

modes wI.? = -t R?(q) 0 is considered to

ck =

case cy = 1 one determines 7 - y-14 - i y/2,

(8.2)

(02 = 0.

The soft-mode w2 is an indication for a lattice instability in the system occurring at a transition temperature To which is determined through tv(T,J = 1 and defines the stability limit of the high-temperature cubic phase. Whether the modes wI,? are overdamped or not depends on the magnitude of fl?/y. For C&/y c 4 the root wI = -inSly is proportional to yz and could therefore be attributed to a diffusive mode with diffusion constant fi = ~$7, where ri = root ~3 = -ir[ I - (0,/-y)‘] is approximately in(C11~ c,2)/2p. The other dependent of q and hence interpreted as a relaxation mode. For intermediate values of CYail three modes are strongly affected by the coupling and mingle with each other so that a numerical treatment is required. Studying the q-dependence of the coupled modes for 0 < cy < I provides insight as to how the soundwave dispersion is modified due to the deformation potential coupling. For simplicity, a linear dispersion law &(q) = roq is assumed for the bare acoustic modes at 9 = 0. Fig. 3 shows

the motion

of the normal

plane as a function of wavevector q the direction of increasing q varying boundary fqBz = y/u{,. In addition to of the soundwave-like mode w~(RJ~;

modes

in the complex

frequency

where the arrows on the curves indicate between zero and half the Brillouin zone the o-plane also real and imaginary part cu) are given as a function of q. For four

values of the parameter CY,suitably chosen, typical situations are encountered: For cy s 8/9 the damping of the acoustic modes w,,? increases with q whereas the relaxation mode w3 becomes less and less attenuated and moves towards the origin. For cx approaching the value 8/9 the curves of the modes w,,? become pinched for q - y/2u0. Eventually, at (Y= 8/9, the frequencies of all three modes coincide for q3 = y/d/3v0 at w = -iy/3. As regards to the dispersion Re WI the deviation from linearity increases with 0~. A bump develops which becomes more and more pronounced and turns into a cusp touching the

BAND

JAHN-TELLER

SYSTEMS

WITH

AlS-STRUCTURE

391

Rew,

-lkw, l

Rew

k-+-

-Imw

-IRlW

a=1

Rew,

Rew,

t

t

a)819

-Imw,

-Imw, Rew

;T-Re’ + -Imw

a = 8/9

-Imw

a<819

Fig. 3. Motion of the coupled mode frequencies wi (i = 1,2,3) in the complex o-plane for four values of a and 0
398

q-axis

R. KRAGLER

in q3 for cy = 8/9. In contrast

smoothly

with q, exhibiting

a whole

range

for which

of q-values

all three

modes

reappear

teristic

feature

normal being

is clearly

to this remarkable

no dramatic between modes

strongly reflected

change.

are overdamped due

~ Im w, grows occurs

- cu/o,, and q2== y\/3

ql = 2y\/l

damped

behavior

If (Y> g/9, there

until for q > qZ propagating

to mode

in the dispersion.

now

- 2cu/2vo

mixing.

This

charac-

For 0 d q s q, Re W, goes

through a maximum and vanishes again for q = q,. Then, between q, there appears a gap where Re W, = 0. i.e. no propagating modes exist. stability limit (Y= 1 this gap then extends from zero to q,, = y/2vo. The in the w-plane, formed by the mode frequencies o,,? for OS q s

and qZ At the “loop” q,, has

contracted to zero as CY+ 1. A soft-mode w2 = 0 occurs in agreement with eq. (8.2), whereas the other mode W, remains overdamped for 0 s q s q,,. Only above q, damped but propagating modes CD,,? reappear. This behavior is

0.0025 Fig. 4. Dispersion of the soundwavelike value n/y = I corresponds to q = ! q~,.

mode.

Re(~,/y),

as function

of rl/y

‘* q and

(1. The

BAND

JAHN-TELLER

SYSTEMS

WITH

AlS-STRUCTURE

399

clearly demonstrated by Re wI which is different from zero only above the threshold qo. As regards to the damping, - Im wI increases continuously with q and assumes the constant value - iy/2 for q > qo. Figs. 4 and 5 summarize in perspective graphs the dependence of soundwave dispersion Re ol and the damping Im wl, respectively, on wavevector q (0 s q s { qsz) and parameter (Y(0 s (YG 1). The interesting behavior obviously occurs in the region 8/9 s (Ys 1 where the dispersion curves clearly show a gap and the damping curves develop an edge which becomes more and more pronounced as (Y+ 1. It can be concluded from this discussion of the behavior of the coupled modes, that the essential feature of the modified dispersion is a strong deviation from linear behavior over an appreciable region of the Brillouin zone when approaching the stability limit. Concomitant with this deviation there is a strong increase of sound wave damping which vanishes in the high-temperature limit.

Fig. 5. Damping

of the soundwave-like

mode,

- Im(wl/y),

as function

of O/y a q and cx.

R. KRAGLER

9. Conclusion In the foregoing studied

in some

feature

of

coupling

the

there

renormalization

sections detail

model,

it turned

is a mixing of these

the dynamics

by analyzing out

of acoustic modes

of a band

Jahn-Teller

the generalized

response.

that

because

and relaxational

with respect

of

the

system electron-lattice

modes

to temperature,

was

As a salient resulting

wavevector

in a and

relaxation rate. Recently, the same physical system has been treated by the author on a microscopic basis with recourse to multiple-band electron-phonon transport theory’6”.h). In the sequel we refer to these papers as I and II. Although the model differs considerably from the treatment of this microscopic phenomenological one, presented here, there is, nevertheless, far-reaching agreement between the results derived from both models. Therefore, a closer comparison of both models which will be the subject of this section. is well suited. In the phenomenological model the thermodynamic state characterized by the d-band occupation-number deviations

of the system n

and

the

is dis-

placement gradient V,u. In terms of these variables the thermodynamic potential is constructed subject to certain symmetry requirements. As regards to the microscopic model of ref. 16, a Hamiltonian description is used. The operators corresponding to the thermodynamic variables n and u The Wigner operator (-L desare P,,’ = aLa,, and A, = bh + bk, respectively. cribes density fluctuations between different electron bands where the index K = (k, I) comprises both wavevector k and band index I. A, denotes the usual phonon normal coordinate of mode A = (q, j) where j designates the branch index. Similar to the thermodynamic potential defined in section 2 the corresponding Hamiltonian for the electron-phonon system consists of three contributions: one for the phonons in quasi-harmonic approximation, another part for band Anharmonicities account

electrons and the as well as Coulomb

usual electron-phonon coupling term. interaction are not explicitly taken into

in this model.

As in eq. (2.10) external fields Jh and UK,, which couple linearly to A, and p,,,, respectively, serve as a mathematical device to generate with the help of functional derivative methods various correlation functions together with the corresponding integral equations defining them, and, moreover, establish relations for the generalized linear response. The phenomenological equations of motion for the electron band occupation n, eq. (3.S) and for the elastic displacement u, eq. (3. I I), which were the very starting point of our model, have their counterparts in the microscopic model. As the phonons are concerned, this is the equation of motion for the

BAND JAHN-TELLER

non-equilibrium

expectation

SYSTEMS

WITH

AIS-STRUCTURE

401

value ((A)), see ref. 16a ibidem, eq. (I. 2.29)

DO’ - ((A)) = J + g((p)) = j.

(9.1)

Here, j is an effective phonon source field incorporating the feedback of the electron bands with respect to the elastic lattice. DO’ is a differential operator which defines the bare phonon Green’s function Do. It is now a straightfoward procedure to rewrite eq. (9.1) into (3.12b). In order to do so, eq. (9.1) has to be Fourier transformed, multiplied by (2flhp)“*e(X) and subsequently summed over all phonon branches j. e(A) is the polarization vector for the phonon mode A. Then, the resulting equation will coincide with (3.12b) if the following transcriptions are made:

ui(q, ~1 c 2 (2~*f)-“*ei(A)((A,(w))),

(9.2a)

i

ndq, 0) =

c k

VfK(q,w>=

c

with A = (q, j) and K% = (k 2 i dynamical matrix cq@ q with

c&l q =

ci

(9.2b)

N(PK+KJWNh q, I).

Furthermore,

one has to identify

e(A)pR%*(A),

the

(9.3)

and relate the gradient of the external stress tensor sex’ to the phonon source field Jh - iuext(q, o) - q =

2 (2R,p)“*e(A)J*(w).

(9.4)

If for the deformation potential coupling tensor G”,,,,,,,,,occurring in the electron-phonon coupling g, eq. (II. 5.10), the restrictions G” = - 2G12 and G, = 0 are made ancillary to the symmetry requirements which also apply to the tensor of the elastic moduli, then D%(q)

(9.5)

= Ge.mmqm.

Using these definitions, eq. (9.1) can be casted into the following form: (PO21 - cq@ q) * u = x * &

(9.6)

where Ni = - iaext. q -

iDaT

- n.

(9.7)

Obviously, eq. (9.6) is identical with (3.12b). It is noteworthy that inversion of eq. (9.1) provides a relation for the linear response with respect to the

402

R. KRAGLER

effective

phonon

source

field j, eq. (I. 6.4), namely

6((A)) = Do. S.f Thus,

the equation

the linear

side are equivalent.

electrons

the analogous

procedure

In order to end up with a relaxation-type

occupation

a Bethe-Salpeter

(9.1) for ((A)) on the one side and eq. (9.8a) for

on the other

to the band

is not so obvious. d-band

of motion

response

As regards

(9.8a)

numbers equation

n, in the microscopic

model

leading of equation

to (3.12a) for the

we have to start from

(I. 3.3)

Ko = GG + GGIoKo

(9.9)

for the electron density fluctuation propagator K0 = ((ApAp)). Moreover, G = i((p)) is the usual electron Green’s function and I, is the irreducible particle-hole scattering vertex. As shown in lbh), the integral equation determining K0 can be brought into the form of a Peierls-Boltzmann equation (II. 4.1) supposed the ladder approximation is applied to the scattering vertex IO, see (I. 3.10), and linear response theory is used in order to establish a connection between the fluctuation propagator K. and the deviation affr = a((~,+, )) from the equilibrium distribution ft = (pKK), 6((p)) = (- i)Ko - 80, Here,

So

(9.8b)

is an effective

electronic

potential

defined

as

Sti = U + g - ((A)),

(9.10)

which takes into account the feedback of the elastic lattice with respect to the electron bands, similarly to j in the case of phonons. The structure of the resulting Boltzmann equation is even closer to (3.12a) if the collision term is taken within the relaxation-time approximation (II. 4.2) modified

in such a way that local electron

Finally, if we sum the resulting equation n(q, w), and take the long-wavelength relaxation-type of equation: (wl

+ i#?)

number

conservation

is guaranteed.

over k in order to get an equation for limit, we end up with the following

- 66,

- n = -iyZ2

(9.11)

where Sti = peX’+ iIN?

- II.

Here, U = cc’“’ has been used deduced from a Bethe-Salpeter (3.12a). As to the

microscopic

origin

(9.12) together with eqs. (9.2) and (9.5). Eq. (9. I I) equation (9.9), is obviously identical with of the

relaxation

rate

y and

the

effective

BAND

JAHN-TELLER

SYSTEMS

WITH

AlS-STRUCTURE

403

density of states Z both occurring in (9. ll), a comparison with the microscopic model reveals that y is associated with the interband scattering rate ylr, (l+ 1’) arising from the relaxation-time approximation. The interpretation of the quantity Z which emerges in the course of the derivation of the generalized Boltzmann equation is in agreement with eq. (2.6). Up to now we have retraced the fundamental equations of motion (3.12a) and (3.12b) of the phenomenological model on a microscopic basis. Yet, the analogies between both models go further. Concerning the generalized response it turns out that its structure is the same in both models, compare eqs. (II. C.16) and (5.4). Moreover, eqs. (5.5) and (5.6) which refer to the more physical interpretation of the response have their microscopic counterparts in eqs. (II. C.17) to (II. C.19). As to the response matrices X0, X, so and z which are defined through eqs. (4.4) and (5.2), within the microscopic model these quantities are deduced from the propagators Ko, K, Do and D, see eqs. (I. 2.19) and (I. 3.1) by suitable retardation and contraction, for example

Xdq, w) = - i W) =

Bjj'(q,

z. ’ 1g g &,ldkc,

DjjJq,

W +

k'e', qw + is),

iS),

(9.13a) (9.13b)

see eqs. (II. 2.9) and (II. 5.6). The bare displacement response E” is obtained from eq. (9.6) by functional differentiation with respect to J in agreement with the definition of the bare propagator Do = S((A))/SJ. The bare dielectric response X0 is similarly deduced from (9.11) by application of S/Sfi. This procedure concurs with the definition of the bare fluctuation propagator (- i)Ko = S((p))/So, see (9.8b). One easily convinces oneself that the expressions for E” and X0, using eqs. (9.6) and (9.11), are identical with those of eq. (4.4). In contrast to the present phenomenological model, however, the microscopic treatment is not restricted to the hydrodynamic limit o, q * v, -=zy. Eq. (9.11) is only a special case of the more general, q and o dependent kinetic equation deduced from (9.9). Henceforth, the expression for X0 obtained from the generalized Boltzmann equation (II. 4.7) c

t

w

y/se,) q.u,+iy,[(W-4’0.)~+iY~l.SP(q,W)

=-‘q

w-q-i)

(af O,/W [- q - u,l + iy?L] - SU(q, w), K +iy s

where ys = El, ylr and y = i -rrr (I+ I’), consists of two parts according to (II. 4.13). There is a diagonal contribution proportional to 1 due to intraband

404

R. KRAGLER

processes and another to interband transitions. In the hydrodynamic

off-diagonal

contribution

limit the dielectric

proportional

response

function

to 2 attributed (Sp/Sfi)

resulting

from eq. (9.14) X”(q, 0) = w

Ligq2(iQ21

+

iyw 9 (w + ir) + igq2 1

(9. IS)

exhibits an (intraband) di$usion pole w = -igq’ with diffusion constant 9 = 2)zF/3ys, (see ref. Iti) ibidem eq. (3.15a)) for the diagonal part and an (interband) relaxation pole w = -iy for the off-diagonal part. Clearly, the diagonal contribution vanishes in the limit q +O and there is full agreement with eq. (4.4a). Of course, guided by the transport equation (9.14), which results from a microscopic treatment, one could think of generalizing the simple relaxation ansatz for the electron bands by supplementing the left-hand side of eq. (3.1) with a drift term. However, this would only lead to a kinetic equation which is obtained from (9.14) when u, is replaced by ut so that the k summation drops out. The corresponding expression for X0 is of the same structure as (9.15) with the important difference that igq’ is formally replaced by -q +z)~.not being justified at all. The conclusion, one can draw from the foregoing consideration therefore is that the resulting expression (4.4a) of the phenomenological model is just correct for q < ~‘\/5y/vr where the relaxation pole is predominant. However, for larger q, where intraband diffusion takes over, one has to proceed along the lines given in ref. 16c using a more general Boltzmann equation approaches instead of the simple rate equation (3.1). Irrespective of these limitations, the matrix equation (5.9) defining the renormalized response functions X and E in terms of the bare quantities X0 and E”, have microscopic counterparts. Since X is obtained from the fluctuation propagator K by means of the prescription (9.13a), it is near at hand interpreting the iterative equation (5.9a) as the retarded version of the integral equation (I. 3.13) K = Ko + Kol,K.

(9.16a)

The renormalization of KO involves the chain scattering vertex I, = -igDog which considers the contribution of phonon chain diagrams not taken into account in the ladder approximation. Therefore, it is evident from the microscopic derivation that P = g - 2’ - g+, as given by (5.8a), does not correspond to an electron self-energy but rather to a retarded scattering vertex involving phonon chains giving rise to an enhancement of X0. Again, in the hydrodynamic limit, the microscopic expression for X, eq. (II. C.8b), is in agreement with the result (7.20) of the phenomenological model.

BAND

JAHN-TELLER

SYSTEMS

WITH

AIS-STRUCTURE

405

Similar considerations apply for 8, related to the exact phonon propagator D by (9.13b). It is therefore suggestive to interpret eq. (5.9b) as retarded phonon Dyson equation (I. 3.15) D = Do + D,,IZD.

(9.16b)

The renormalization of DO due to the phonon self-energy II = g(--i)&g, eq. (I. 3.16a), only considers the effect of electron density fluctuations according to the underlying physical restrictions of the microscopic model. Thus, making contact with the microscopic treatment, the quantity II = gt - X0 - g can be looked at as a retarded phonon self-energy, see (II. 5.7). Again, in the hydrodynamic limit the expression for E provided by the microscopic model, eq. (II. 5.15b), coincides with the result (6.9b) leading to the same cubic secular equation (4.8) resp. (II. 6.1) which determines the frequencies of the coupled modes. In conclusion, the detailed comparison of the phenomenological model, presented here, with a previous microscopic model, based on multiple-band electron-phonon transport theory, reveals a remarkable similarity in the structure of both theories. In spite of the different starting points the results of both models agree in the hydrodynamic limit, so that both formulations provide equivalent descriptions of a band Jahn-Teller system.

Acknowledgement

The author wants to express his gratitude to Prof. R. Klein for his interest in this work and for critical reading of the manuscript.

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