Soft phonons and the central mode at structural phase transitions

Solid State Communications,

Vol. 13, pp. 967—970, 1973.

Pergamon Press.

Printed in Great Britain

SOFT PHONONS AND THE CENTRAL MODE AT STRUCTURAL PHASE TRANSITIONS P.F. Meier* IBM Zurich Research Laboratory, 8803 Ruschlikon-ZH, Switzerland (Received 26 May 1973 by J.L.Olsen)

Applying Landau’s quasi-particle picture to an interacting phonon system it is shown that the softening of phonon modes is connected with an enhancement of the quasi-particle interaction. This in turn leads to a critical slowing-down of fluctuations around local equilibrium which may give rise to an additional central peak in the scattering function.

DUE TO detailed measurements using EPR, neutron

Nb3 Sn where a transverse acoustic phonon becomes soft, the coupling to 2) energy densityin fluctuations is vanishing first order.(which One enters the constant ö has therefore to look for other mechanisms which may give rise to a slowing-down of scattering processes and lead to the Maxwell—Drude form (2).

and light scattering methods the study of structural phase transitions has experiments recently attracted growing interest.14 Several have been reported which show that in some materials a strong central component is observed in the scattering spectrum S in addition to the soft phonon peaks. S can be related to the imaginary part of the retarded phonons Greens function by S(k,w)

=

N(~)1m[w2

—~

2

+iwF(k,w)]

It should he stressed, however, that such a singular form for F cannot be obtained by finite-order perturbation theory. From the evaluation of the simple bubble diagram corresponding to a three-phonon process one can extract a form (2) with ~ycorresponding to the inverse lifetime of thermal phonons. But it is not correct to use the result for frequencies smaller

(1)

whereN(w) = [exp (h~!kB7) l]~ denotes the Bose distribution. Shirane and Axe5 were first to fit the experimental data showing the three-peak structure in Nb 3 Sn by equation (1) in using for the phonon self-energy r a Maxwell—Drude form —

F(k, w)

=

F0 +

‘V

.

than y since by taking only the first term of the perturbation expansion one assumes w y. 6 it was shown that a form of In a previous work the phonon self-energy which corresponds to the ~‘

(2)

—

A number of suggestions has been made to give a microscopic theoretical explanation of expression (2) (for a review see reference 4) but a satisfactory explanation is still lacking. In the hydrodynamic region the self-energy F has a pole like in equation (2) which

hydrodynamic one can also be obtained for k near QR if the transition is driven by an R-corner mode. The derivation was based on an argument put forward by Enz7 who pointed out that the almost symmetric shape of the soft phonon branch in SrTiO 3 and KMnF3 gives rise to an enhanced two-phonon density of states. Of importance for the following is that besides this pseudo-hydrodynamic form a further singular phonon self-energy was obtained arising from the quasi-particle interaction. In this communication we present general arguments which connect the softening of the phonon modes with an enhancement

corresponds to heat diffusion. The corresponding linewidth ‘y, however, is far too small to account for the observed phenomena. In addition, in cases like *

IBM Postdoctoral Fellow of the Institute for Theoretical Physics, University of Zurich, Switzerland. 967

968

STRUCTURAL PHASE TRANSITIONS

of this quasi-particle interaction. The latter is then shown to be responsible for a slowing-down of fluctuations around local equilibrium,

Vol. 13, No. 7

independent quantities (in addition to na), but we shall neglect this for the present discussion confining ourselves to temperatures above the transition point T~.

We suppose a lattice Hamiltonian H to be given which expresses the potential energy as a function of the elongations. in usual phonon dynamics one starts from the harmonic part of H and by diagonalization of the corresponding dynamic matrix one obtains the harmonic frequencies w~.The spectral function is then given by b(~2 ~2) For strongly anharmonic crystals like quantum crystals or for models describing structural phase transitions these harmonic frequencies turn out to be imaginary. Using a Hartree-like approximation to treat higher-order anharmonicities or, equivalently, by a variational procedure one then gets renormalized harmonic phonons8 corresponding to an effective harmonic HamiltonianH 0.

For phonons these concepts and the corresponding modification of the transport equation have first been introduced by Götze and Michelin a phenomenological way’°as well as in the framework of a microscopic theory11 using equilibrium Greens function and summing up the leading contributions to the hydrodynarnic singularities of the phonon self-energy. Nonequilibrium methods following Kadanoff and Bayrn have been used in reference 12 to derive the same results. Within the renormalized harmonic approximation the quasi-particle interaction obeys the following integral equation11”2 -~

V4(k, --k, k’, ---k) .Ilth’

1/2 a

W

x(k,w)

211— ~i(w2

~)

(3)

C.,.)

but with energies W~ depending on the temperature. Scattering states of these quasi-particles are produced by including effective anharmonic terms V 3 and V4, H

H0 + V3 + ~

(4)

which broaden and shift the sharp peaks of equation (3). At this stage it is convenient to introduce the concept of Landau’s quasi-particle interaction.9 Instead of considering the energy levels W~as the basic quantities one uses the corresponding occupation numbers ~ to describe the state of the system. The total energy E is expressed as a functional of n and the quasi-particle energies are given by

(5

—

4~kWk’

=

The spectral function is again given by a bfunction

A change of ~ also influences the energies c~ corresponding to a quasi-particle interaction f: ~2E

4WkO~~2

~ (k, -~,k’ ~k )f~~’ (7) ,

Inclusion of the quasi-particle interaction is inevitable to get the correct static limit of correlation functions but the numerical contribution off to dynamical quantities is usually very small. This is no longer the case near structural phase transitions or, more generally, if some phonon mode becomes strongly temperature dependent. To demonstrate this 1° of ~ and o.~with we calculate the derivatives respect to temperature 1 (8)

=

L

where ma

=

N(w~)[N(w~) + 11 /(kB 7).

(9)

Using equation (6) we have =

aT

= ~,

bn~

~T

~ ~,

afl~

J~k~ -~.

(10)

Introducing the integral operator I which acts on phonon variables ra according to

—

bfl~C5fl~—

f~.

(6)

In equilibrium the occupation numbers are given by N(o.~).Actually one has also to consider the order parameter or external strains as further

Ira = ra +~ faa’/fla’ra~ (Ii) one can formally solve the system of equations (8) and (10) which yields

Vol. 13, No.7

STRUCTURAL PHASE TRANSITIONS

=

mkr

(12)

7’

(13) (1 l)~1~T As long as the quasi-particle interaction is small we have 1 1 and the phonon energies are only weakly temperature dependent. To get soft phonons we need a strong interaction. Therefore we conclude that the softening of phonon modes is driven by an enhancement of the quasi-particle interaction. Near the transition point, aw~/aTincreases and at T~,the inverse of lwil no longer exist. Indeed, stability of the system as calculated from the second derivative of the free energy requires that I has only positive eigenvalues. One expects that on approaching r~ some eigenvalues of I become vanishingly small and will cause the softening of c~for some wave numbers. =

This pronounced influence of the quasi-particle interaction has important consequences for the

969

For 1”-’ 1 the approach to local equilibrium is governed by a mean relaxation time f, ~~Pa~~k!T. If the phonon frequencies vary strongly with tern-(18) perature we expect from equation (13) that f-I can no longer be neglected. Its effect on the transport equation (17) will then be a change of the time scale leading to an enhancement of the relaxation time to reach local equilibrium: IWi,O~“‘tPkITi.

(19)

Roughly f~’is obtained as the average value of I acting on the phonon widths F,~.The low-lying eigenvalues of! will diminish [‘~ in the region of wave vectors where also the softening of the mode occurs. Thus t~,> f and it is expected that fj increases as the mode gets softer. Assuming factorizing vertex functions I’~and

V 4 and approximating I~by I fm it has been demonstrated in reference 6 that this (weak) quasiparticle interaction gives rise to a phonon self-energy of the form (2) with ‘y’ = jif. In that case, however, ji is smaller than 1 and the resulting contribution 13 For to S(k, is only a small broad background. strongw)quasi-particle interactions equation (19) will —

fluctuations occurring in the system. For long wavelengths the non-equilibrium distribution function na(r, t) obeys the Boltzmann equation at

—

ark

~

a~ = L [n&]

ak + ak

(14)

with L being the linearized Peierls collision operator and E~(r,t) = c~+ ~wa(r, t) the space- and timedependent phonon energy. Denoting the derivation from local equilibrium N[E~(r, t)] by ~p: na(r, t)

=

N[E~(r, t)] + m(o.)a)~pk(r,t)

(15)

we can linearize the transport term and, upon considering =

at

(1 —I’)~~ at’

we get the following equation 1’~~p~(q t) —1

—~-

q Pa

=

(16)

L Ipal’

(17)

The microscopic deviation of this equation is well known11 .12 for q near zero. In reference 6 it was shown under which assumptions a similar equation for q near QR can be derived in the case of an R-corner soft mode. In terms of diagrams equation (l7)is obtained by summing up all ladder graphs as well as the chain diagrams.

provide a Maxwell—Drude form (2) with ~ ~i. For f~> f a’central peak in the scattering spectrum will arise with an intrinsic width proportional to This mode corresponds to a slow decay of fluctuations around local equilibrium. As the phonon mode becomes softer the width of the central mode ~.

gets smallerfluctuations. corresponding to a critical slowing-down of certain It is evident that in order to go beyond these qualitative arguments solution of the integral equations (7) an andimproved (17) is inevitable. Methods similar to the ones used in mode—mode coupling theory are necessary to handle the summations over the wav~ numbers. A corresponding model calculation which assumes a given soft-mode behavior and evaluates the quasi-particle interaction by use of relation (13) will be presented in a forthcoming contribution. In the low-temperature phase the above reasonings have to be modified by inclusion of the coupling to the order parameter a. Instead of equation (13) we then get

970

STRUCTURAL PHASE TRANSITIONS

=

(1

1)1

~

+K

aa

(20)

where öwa/~5a= 1K. Insofar as K behaves regularly it follows that the order parameter and the soft-mode frequency have the same critical temperature dependence. Experimental evidence for this fact is provided in SrTiO3 by comparing the EPR data of 4 with the light-scattering Muller and Berlinger’ experiments of Steigmeier and Auderset15 (see also reference 16) in the critical region. These aspects and the discussion of the coupling of the transport equation (17) to the order parameter will be discussed elsewhere. Concluding we may state that the above discussion of soft-phonon modes in the framework of interacting

1.

Vol. 13, No. 7

quasi-particles provides interesting new aspects on the dynamic behavior near phase transitions. The strong temperature dependence of some phonon modes is driven by an increasing quasi-particle interaction which in turn yields a slowing-down of fluctuations around local equilibrium. It is evident that these concepts are not restricted to structural phase transitions but may be applied to other dynamic systems as well.

Acknowledgements The author would like to thank J.D. Axe, K.A. Mi.iller, 1. Schneider, H. Thomas and U. Höchli for helpful discussions. He has also benefitted from discussions with many of the participants of the NATO Advanced Study Institute at Ustaoset, Norway. —

REFERENCES See the various contributions in Structural Phase Transitions and Soft Modes (Edited by SAMUELSON E.J. etal.) Oslo (1971), as well as the Review articles 2 and 3.

2.

FLEURYP.A.,Comm. Solid State Phys. 4,149, 167(1972,.

3. 4.

5.

PYTTE E., Comm. Solid State Phys. 5,41.57(1973). Papers and references cited by AXE J.D., BLUME M., FEDER J., KLEIN R., MUELLER K.A., SCHWABL F., SJOELANDER A. and STEIGMEIER E.F., THOMAS H., Proc. NATOAdvanced Study Institute on Anharmonic Lattices, Structural Transitions and Melting at Ustaoset, Norway (1973), (to be published). SI-IIRANE G. and AXE J.D.,Phys. Rev. Lett. 27, 1803 (1971).

6. 7.

BECK H. and MEtER P.F.,Helv. Phys. Acta, in press. ENZ C.P.,Phys. Rev. B6, 4695 (1972).

8.

See e.g. WERTHAMER N.R.,Am. J. Phys. 37, 763 (1969).

9.

LANDAU L.D.,SovietPhys. —JETP3,920 (1956); ibid. 5,101(1957).

10. 1]. 12. 13.

GOETZE W. and MICHEL K.H., Phys. Rev. 156, 963 (1967). GOETZE W. and MICHEL K.H.,Phys. Rev. 157, 738 (1967); ibid. Z. Phys. 223, 199 (1969). KLEIN R. and WEHNER R.K., Phys. Kondens. Mat. 10, 1(1969). BECK H. and MELER P.F.,Phys. Kondens. Mat. 12, 16(1970).

14.

All results of reference 6 may be immediately transferred to the case of a zone-center soft mode. The terms which account for the assumed approximate symmetry then vanish. MUELLER K.A. and BERLINGER W., Phys. Rep. Lett. 26, 13(1971).

15.

STEIGMEIER E.F. and AUDERSET H., Solid State Commun. 12, 565 (1973).

16.

HOECHLI U.T. and SCOTT J.F., Phys. Rev. Lett. 26, 1627 (1971).

Landau’s Quasiteilchen-Bild wird auf em System von wechselwirkenden Phononen angewendet um zu zeigen, dass das Soft-werden von Phononmoden von einer verstärkten Quasiteilchen-Wechselwirkung herrUhrt. Diese fUhrt auch zu einer kritischen Verlangsamung von Fluktuationen urn das lokale Gleichgewicht, was zu einem zusätzlichen zentralen Peak in der Streufunktion fOhren kann.