SOME ASPECTS OF NONLINEAR LATTICE MODELS

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SOME ASPECTS OF NONLINEAR LATTICE MODELS

H. Büttner, H. Bilz

To cite this version:

H. Büttner, H. Bilz. SOME ASPECTS OF NONLINEAR LATTICE MODELS. Journal de Physique

Colloques, 1981, 42 (C6), pp.C6-111-C6-118. �10.1051/jphyscol:1981634�. �jpa-00221570�

SOME ASPECTS OF NONLINEAR LATTICE MODELS

H . B i i t t n e r a n d H. B i l z *

Phys. Inst. Univ. Bayreuth, D-8580 Bayreuth, F.R.G.

" ~m-~Zanck-~nst . filr Festkll~perforschung,

7000 Stuttgart

Abstract.-

theoretical study of various static and dynamic so- lutions in a one-dimensional lattice with anharmonic electron-ion coupling is presented. The stability of the different comnensu- rate groundstates is investigated and both kink- and phonon-ex- citations are described. The coupling of nonlinear excitations to phonons is discussed in detail.

It was pointed out some time ago, that the microscopic origin of ferroelectricity in perovskites is due to the configurational insta-

2

-

bility of

and its honologues, which leads to a stronq anisotropic, anharmonic polarizability"2. This assumption was recently tested di- rectly by a M6Bbauer study on LiTaO and applied to various other ma- terials, like KTal-x Nb,034; SrTiO) 3

and snT.e617 in a self- consistent uescription of their phonon spectra.

discussion of the nonlinear excitations in a one-dimensional version of this nodel has been given in Ref. 8,9, and the coupling to phonons is studied in de-

6.37,

tail in Ref. 6. Extensions to two-" and three-dimensions ave also been successful.

In the following we shall review some results from the literature and present new investigations especially on the stability of the stat- ic nonlinear solutions and their coupling to phonon-excitations.

The essential feature of the model is a highly local electron-ion coupling. The instability of the transverse ferroelectric node is at-

2

tributed to a negative electron-shell ion-core coupling constant g2.

The paraelectric r.~ode above the phase transition is stabilized by fourth-or2er shell-core couplinq qq. For a discussion of the relation to other ferroelectric nodels see Ref. 8,6. In the simplest version of the nodel, a monatomic chain, there are additional nearest-neiqhbor shell-shell and core-core coupling constants f and f' (see Fig.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981634

JOURNAL DE PHYSIQUE

Fig

Part of the one-dimensional lattice model, with harmonic nearest-neiqhbor coupling between the electron shells f and the ions f'; the local elec- tron-ion coupling is described by a harmonic coup- ling g2 and a fourth-order term

The classical Hamilton-function of the chain is written for the dis- placements un and vn of the ion and the electron-shell with masses

and m respectively:

with tine corresponding equations of motion (in the adiabatic approxi- mation

where the difference displacement wn

vn - un is used and the differ- ence operator Dun

u ~ - + ^2un ~ + u ~ . - ~

The static periodic solutions on the lattice are the fixed points of the corresponding recurrence relations between wn+, (un+,) and

wn, w (un, u,-,)~. Usually one has to find these points numerically, but for low-order periods the analytic results are given in Table

Periods of higher order are more complicated. The displacement pattern

of period 2 ~ / 8 for example has two different displacements on site

W ~ m + l - - W8m+3

8m+5

and site w ~

~

8m+6 related by + ~

Table

Static commensurate solutions of (2) with different periods.

k = l

(ferroelectric)

k = 2 (antiferro- electric)

k = 3 (Periocion)

Wo 2

-g2/g4

2

( ^{92 4fr} I

wo

(-w,)2

t t t

2 92 3f

~ ~ ' 0 , wl=( 2 -w2)

t t

El

-822/4g4

E,

3f 2 E3 ) *

I ( % -

2f 2

k = 4 w =w =O w =(-w 2 92

0 2 1

k = 6 2 2 2 2

w =w =O w =w =(-w 0 3 1 2 4

=(-w 5

92

f 2

E6

) % I ( % -

JOURNAL DE PHYSIQUE

the equations

with the reduced intersite coupling l/f _r

l/f + ^I/£'.

The energy for the different commensurate states is always nega- tiv and corresponds to local minima of the Hamilton-function. For g2

and f,f'

the homogeneous ferroelectric state has always lowest energy. For g2

and f'

but f

there may be other states with lower energy. But this energy consideration is not suffi- cient to determine the stable groundstate of the chain. One has to investigate the stability of the solutions under small time-dependent perturbations and in addition to this, wether static solutions in the vicinity of the fixed points stay there under the nonlinear mapping

12' l 3 of equations (2). We rewrite these for wn as:

and linearize them by the ansatz:

with the result

For the different fixpoints we now investigate the eigenvalues of the matrix

which describes a so-called tangent space napping1 3. (The nethod is

analogues to that used by ree en el for another nonlinear mapping.)

based on some (so far unproven) mathematical assunptions, we also lookeci for numerical stability in the vicinity of the commensurate states and found agreement with the parameters regions of Table 2.

Table 2: Range of mapping stability for low order periodic solutions.

These results mean, that in certain parameter areas only one of the fixed points is stable, although its energy nay not be the abso- lutcjylowest. In the neighborhood of a stable region there may be incommensurate stable states, which can be reached by leaving the stability range of the parameters 12,14

Above these static groundstates there are linear phonon- and nonlinear static kink-like excitations. The latter ones describe, as in the usual $4-theory, domain walls between the degenerate ground- states of a certain period k. As an example, we write down the equa- tions for the kink structure in the antiferroelectric state

21, where the polarization sequenze of the ions is altered from 'up, down, up

to 'down, up, down

If we assume only small changes within the unit cell, the continuum approximation for each sublattice can be used and results in:

Period k

1 2

3 4

where wl and w2 describe the electron-ion displacements in the too sublattices, and a is the lattice constant of the original lattice.

The boundary conditions are wl

-w2 for x

+-, in contrast to the ferroelectric state. Numerical solutions show a static kink-like transitions in each sublattice. These solitary excitations will

Range of Parameters for Mapping Stability

0

5 ^g2/fr 1 2

g2/fr 5 -4 -4 5 g2/fr '

g2/f,

-2

Remarks

f or f' have to be negative

method is probably to weak

C6-116 JOURNAL DE PHYSIQUE

contribute to the statistical mechanics of the antiferroelectric state.

Similar solutions have been found above the periodon

3) ground-

state. Details are discussed in a forthcoming paper .

The static solutions discussed so far, can easily be extended to diatomic chains with one additional unpolarizable ion

. Furthernore there are also solutions of the same type for twolo and three-dimen- sional

systems.

Before describing the coupling of phonons to the periodons we cite here the corresponding tine dependent nonlinear solutions. They can be written as

wn

A(q) sin (wt-nqa+6)

un

(q) sin (wt-nqai6) + ^{C (q) sin 3} (wt-nqa+6) with the dispersion

4 2

MU^

3 (f+fl) sin (3qa/2) and the amplitudes

Xote the divergent character of C(q) at qc

0 ; 2 ~ / 3 where the fre- quency is zero. In order to avoid this unphysical static solutions one has to chose the phase 6 in such a way that sin(36)

0, which results in C(qc

O;2~/3)

and amplitudes given in Table

The coupling of phonons to these periodons is now calculated

with the ansatz

Our approxination is that used in the selfconsistent phonon treatment where higher powers of the phonon contribution wns are approxinated by their thermal averages:

g4wns = 3 3g4<w~s>T wns and 3g4w2 ns

3g 4 <w n s > ~ 2

This results in a temperature dependent coupling

Furthermore the phonons are influenced by the periodon solution w np' For low temperature we use its static value, which causes a tripling of the lattice constant:

For high temperature the periodons are approximated by a time average with

- from (11) and g2 replaced by g(T). This q-dependent coupling leads to a drastic change of the acoustical node near q

2 ~ / 3 . An application of this method to the phonon-spectra of various materials is reported in Ref. 6. Especially the paraelectric-incommensurate and the incormensurate-commensurate phase transitions in K2Se04 have been successfully analysed6".

Furthermore it was found, that the phonon anomalies in TaSe2

and NbSe3 may be described in terns of our sinple model7. This shows

the interrelation of our mode-coupling treatment to the description

of these metallic systems in terms of charge-density waves.

JOURNAL DE PHYSIQUE

References

R. Migoni, H. Bilz and

Bauerle, Phys.Rev.Lett. 37,

1155 (1976)

H.

Eilz,

Bussmann,

Benedek, H. Buttner and D. Strauch Ferroelectrics 25, 339 (1980) and to be published

M. Lohnert, G. Kaindl, G. Wortnann and

Salonon Phys.Rev.Lett., 47, 194 (1981)

D.

Rytz, U.T. HGchli and

Bilz, Phys-Rev. g , 359 (1980) A. Bussmann-Holder,

Bilz,

Bauerle and

Wagner

Zeitschrift Physik E, 353 (1981)

B.

Bilz,

Biittner,

Bussmann-IIolder,

Kress and U. Schroder, to be published

A.

Bussmann-Holder,

Bilz and

Biittner, Proc. Ferroelectric Conf. Philadelphia, 1981

B.

Biittner and

Bilz, in Recent Developments in Condensed Matter Physics, Vol. I (ed.

T. Devreese)

49, Plenum

(1 981)

H.

EBttner in Nonlinear Phenomena at Phase Transitions and Instabilities, Nato Adv. Study Inst. Geilo 1981

(ed.

Riste), Plenum (1981)

G. Eehnke and

Biittner, J. Phys. e, L113 (1981)

see also G. Behnke, thesis, Univ. Bayreuth, W.-Germany (1981) U. Schroder et. all Int. Conf. on Phonons, Bloomington

1981, Journal de Physique, to be published.

J.M.

Greene, J.Math.Phys., 20, 1183 (1979)

M.

Tabor, Adv. Chem. Phys. Vol. XLVI, p. 73 (1981) P. Bak, Phys. Rev. Lett., 46, 791 (1981)

E.

Biittner, to be published