This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 155.97.178.73

This content was downloaded on 27/09/2014 at 08:29

Please note that terms and conditions apply.

Acoustic anomalies in Jahn-Teller coupled systems

View the table of contents for this issue, or go to the journal homepage for more

1971 J. Phys. C: Solid State Phys. 4 L317

(http://iopscience.iop.org/0022-3719/4/15/006)

Home Search Collections Journals About Contact us My IOPscience

Letters to the Editor L317

EISENBERG, D., and KAUZMANN, W., 1963, The Structure andproperties of Water (Oxford: Oxford

FLETCHER, N. H., 1970, The Chemical Physics oflce (London: Cambridge University Press). GUBANOV, A. I,, and NRAN’YAN, A. A., 1959, Fiz. tverd. Tela, 1, 1044-53, (1959, Sou. Phys.-Solid

HONJO, G., and SHIMAOKA, K., 1957, Acta Crystallogr., 10, 710-1. ONAKA, R., and TAKAHASHI, T., 1968, J. Phys. Soc. Japan, 24, 548-50. PAULING, L., 1935, J. Am. chem. Soc., 57, 2680-4. SEITZ, F., 1940, The Modern Theory of Solids (New York: McGraw-Hill) p. 441. SLATER, I. C., and KOSTER, G. F., 1954, Phys. Rev., 94, 1498-524.

University Press).

Sf., 1, 956-63).

Acoustic anomalies in Jahn-Teller coupled systems

Abstract. The theory of the coupling between the acoustic phonon modes and a two level electronic system is studied with reference to the Jahn-Teller phase transition in DyV04. The connection with the macroscopic strain is discussed, and expressions given for the coupled mode frequencies and the elastic constants. The model also accounts satisfactorily for the temperature variation of the elastic constant of KDP near the 122 K ferroelectric phase transition.

In recent work on the phase transitions in the rare earth vanadates (Elliott et al. 1971, Elliott 1971) we considered a theory of cooperative excitations which arose from coupling between the electronic crystal field states and the lattice vibrations. It is believed that this coupling causes the transition which may be regarded as a cooperative Jahn-Teller effect (Cooke et al. 1970). In that work we confined our attention to the optic phonons which had also been studied by Raman scattering, (Harley et al. 1971a, b), although the method was sufficiently general to study the coupling to acoustic modes as well. Indeed, in a very similar problem, the coupling of magnetic spins to the acoustic phonons had been considered earlier (Elliott and Parkinson 1967). In an early theory of the cooperative Jahn-Teller effects Kanamori (1960) stressed that

it was also important to consider separately the coupling to static macroscopic strains. More recently Pytte (1971) has also included static strain in similar problems and calculated the effect on the elastic constants. These theories are in principle applicable to any ‘spin’- phonon system. In our case the spin states 02 = & 1 correspond to two orientations of the magnetic quadrupoles, but the same method can be used in hydrogen-bonded ferrwlectrics (Kobayashi 1968). Gehring et al. (to be published) have also developed a theory of the elastic constants, and have attempted to separate the contribution of strain and optic mode coupling in the vanadates by stress experiments.

A macroscopic static strain can be considered as arising from the long wavelength acoustic phonons if the appropriate limit is taken as g --f 0. In this note we consider the coupling between electronic and acoustic modes in detail and compare the results with those obtained by introducing a static strain. We also discuss the relationship between the excitation frequencies and the elastic constants calculated from thermodynamic arguments.

We take the simplest model appropriate to DyV04 assuming two doublets given by us = f l and neglecting the magnetic degree of freedom within the doublet. A detailed C

L318 Letters to the Editor

treatment using a more correct Hamiltonian is given in a subsequent paper (Elliott, Hadey, Hayes and Smith, to be published). Our model Hamiltonian is

2e = X { c f ( ! z ) c k d + *Yw(q) + *Nee2 + X M Z ) ~ W { C % ) + cd-q ) } h i a

- Xveuz(Z) - xF+(Z) (1) I 1

where the terms give successively the phonon energy, the strain energy, the coupling of the electrons to the phonons, the coupling to static strain e and the initial crystal field splitting of the doublets. The strain e must be taken with appropriate symmetry; in DyV04, it corresponds to the rhombic distortion observed on ordering, that is

e = a,, = $(ezz - e d . (2) C is related to the appropiiate elastic constants by

c = 2VO(C11 - Cl2) = 2VoCo where VO is the unit cell volume.

Transforming to displaced phonon coordinates, as discussed by Elliott et al. (1971),

where

u"q) = N-1/2xuz(Z) exp {iq . R(I)j 1

the Hamiltonian is equivalent to

(3)

The Fourier transform of J is given in terms of the contributions from coupling to each phonon branch j by

J(k> = XJ#) Jdk) = K5W - x5 (7) j

where

XI is subtracted since it corresponds to the terms in {uz(l)}2 which give the constant Jahn- Teller selfenergy of a complex and do not contribute to the ordering.

On taking the equations of motion of the y and U operators and decoupling to linearize the equations, the normal mode frequencies are given by

where

is the 'spin wave' frequency which is'obtained by using only the spin terms in equation (6). As J? -+ 0, equation (9) shows that there is no coupling between these spin waves and the displaced phonons. The coupled mode frequencies given by equation (9) at low k and w ai e

&h2wH(k) = {J(0)(uz> + he) l2 + {r - J(k)(uz> 1 (1 0)

ah2+) = {J(o)(u~> + + W' - {J(W - Ka(k))(O?l (1 1)

Letters to the Editor L319

for the largely electronic mode and

for the acoustic mode j = a. When wa(k) B U&) at larger k values, up@) -+ w,(k) and we(k) -+

As k -+ 0 the behaviour of &(k) is anomalou?. Since (a(k) cc kli2 and wa(k) cc k it tends to a constant which depends on the direction of the wavevector k. This anomaly therefore occurs in equation (10) but not in (1 1). In equation (12) U@) -+ 0 in this limit so no dis- continuity occurs in wp(k). The first term in equations (IO) and (11) contains J at k = 0, and we must therefore examine the limit fuither. As 2nlk approaches wavelengths com- parable with the crystal size, the form of the boundary conditions becomes significant. If periodic boundary conditions are imposed the k = 0 mode does not exist and hence &(O) = 0. These conditions preclude the possibility of static strain-which must therefore be added separately in this description. The first terms in equations (10) and (1 1) become

(cf Elliott and Parkinson 1967).

{ J O ( O ) - Xa1(UZ) + d e ) (1 3) where Jo includes all optic branchesj. The equilibrium strain (e) = (~/C)(az) obtained by minimizing the strain energy adds a further term to the effective J. In fact this strain is proportional to lim,,,kQa(k), where k is taken in an appropriate direction, and Qa(k) is the appropriate acoustic mode coordinate. Since U@) cc k2, the term from T(e> in equation (13) is equivalent to limk,,Ka(k) for the same mode 'a' and the same direction of k. The strain will be largest and the energy gain gieatest for the mode with the strongeat coupling. In the case of DyV04, this corresponds to taking the limit when k is in the [110] direction for the mode transversely polarized in the plane perpendicular to the c axis.

We define

J'(0) = Jo(0) - Xa + ,y where x = 72/C.

th2w,2(k) = {J'(O)(U~>>~ + r {r - (J&) - xa)(u2) 1

~ ~ t t 2 ~ g k ) = { ~ y o ) ( ~ z ) 12 + r {r - J ( ~ ) + z ) 1 .

(14) Then

and

The latter expression is still not uniquely defined in the limit k = 0; when it is needed in that limit we take J(k) -+ J'(0). These results may also be interpreted without introducing the strain explicitly as indicating that J(0) which appears in equations (10) and (1 1) should be chosen in the limit k = 0 (where it is discontinuous) to take its maximum value. This corresponds to a change from periodic boundary conditions to allow further normal modes in the limit of long wavelength which correspond to uniform strain. Either description leads to the same properties.

The most striking feature of the system when acoustic mode coupling is explicitly included is the change,in the soft mode behaviour associated with the transition. If only optic mode coupling is included it is the electronic mode which becomes soft and WE(O) -+ 0 at T = To say when

with J = Jo(0). As we have seen J should be replaced by J'(0) when acoustic mode coupling is included to give the transition at TD. However we, the actual electronic mode frequency (19, no longer tends to zero in this case. Instead, the acoustic mode of polarization and k value which corresponds to the observed distortion become unstable.

This instability can also be qeen by treating the Hamiltonian (6) thermodynamically

(U") = r / J (1 6)

L320 Letters to the Editor

using molecular field theory on the spin and strain terms and decoupling the phonons (ie neglecting the commutators of y with a). The adiabatic elastic constant is found to be

while the isothermal one is

hb&(O) - ~ J ’ ( O ) ( U ~ ) ~ [ {J’(0)}2 - W ’ ] / ~ B T h%$(O) - 4(Jo(O) - xa)(~z)2[(Jf(0)}2 - W 2 ] / k ~ T CT CO

where W2 = {J’(0)}2(~+)2 + rz r

(UZ) = (UZ$~-’ tanh (”) W kBT *

Note that equation (17) has exactly the same form as (12) as k + 0. Above the transition temperature TD

The numerator goes to zero at T D (cf equation (16)) while the denominator would go to zero at TI given by equation (16) with J = Jo(0) - Xa = J’(0) - x. TI may be regarded as the transition temperature of a clamped crystal if all distortion were forbidden, but the crystal would actually form domains below T D in this situation. The difference T D - TI is small if coupling to the acoustic modes is weak. Then CS and CT differ from CO only in a narrow region around T D . When I? is small, kBTD = J’(0) and ~ B ( T D - TI) = x. In this case

CS/CO has the same value (T - TD)/ (T - TI) for T > TD, but is constant and equal to unity for T < T D .

Maris (1967) has suggested that the clastic constant Cs will only be measured at frequen- cies w < l / ~ where 7 is some relaxation time of the system. At high frequencies w > we, C -f CO, the unperturbed value. At intermediate frequencies the situation is much more complicated. These effects may be observable by Brillouin scattering. CT may be measured under appropriate conditions in static compressibility experiments.

We have performed a similar analysis for TbV04, which differs from DyV04 in that the lowest four levels of Tb3+ are a singlet, a doublet and a second singlet (Harley et al. 1971a, b, Gehring et al. 1971) whereas Dy3+ has two Kramers doublets. The thermodynamic expressions for the appropriate elastic constants CT and CS corresponding to equations (17) and (18) fall to zero at T = T D as expected; however the acoustic mode frequency analogous to (12) as calculated from the equations of motion still gives a finite velocity of sound T = T D . The different behaviour of CS and the velocity of sound arises because of the difference in the frequencies at which they are calculated. At each crossover point in the Brillouin zone between an acoustic phonon and an electronic excitation there is a change in aw/ak for the acoustic phonon mode, the size of the change depending on the strength of the coupling of the modes. Thus, in the Dy3+ case, the effective value of Cs is (17) if measured at a frequency w < we, and CO if w > we. For Tb3+, there is an electronic excitation at we = 0 above TD, corresponding to the separation of the doublet levels, with the result that the velocity of sound measured at finite frequencies should be greater than that given by CS at effectively zero frequency. In fact, detailed calculations (Elliott, Harley, Hayes and Smith, to be published) show that the elastic constant measured at finite

Letters to the Editor L321

frequencies (we, > w , where we, is the energy of the singlet-singlet electronic excitation) should fall to approximately half its high temperature value at TD, rather than to zero, as expected for DyV04.

No direct measurements of the effects proposed here are yet available on DyV04 or TbV04, although Gorodetsky, Luthi and Wanklyn (private communication) have measured the elastic constants in other directions in DyV04 where no anomaly should exist; small changes they observe on ordering are probably due to anharmonic effects neglected here. Gehring et al. (to be published) estimate that x/[Jo(0) - xa] - 1/3 in DyV04 from stress experiments on the optical spectrum. There is some evidence from Raman data that this ratio may be > 1 in TbV04.

However, the theory given above should also apply to KH2P04 (KDP) (Kobayashi 1968, Pirc et al., to be published). Here, a pronounced dip around TD (122 K) is observed in an elastic constant as measured by Brillouin scattering (Brody and Cummins 1968) and

I

130 140 Temperature (K)

Figure 1. A comparison between the theory for CS (full curve) and CT (broken curve) and the experimental measurements : 0 Brillouin scattering (Cummins and Brody 1968), and ultrasonic data from Garland and Novotny (1969). The best fit is obtained with CO = 7.0 x 1010 dyn cm-2, J ' ( o ) / k B = 134.0 K, x/kB = 3.6 K and r /kB = 67 K, SO that TI, = 122.0 K and TI = 118.0 K. (The latter value is very close to the clamped transition temperature TO = 117.7 given by Cummins and Brody, and CO is the same a~ the value given by Garland and Novotny for CB near TD. Kaminow and Damen (1968)

give a value of 71 K for r/kB).

ultrasound (Garland and Novotny 1969). In figure 1 we compare these measurements with values of CS since they are taken in a regime where WT is presumably less than unity. The calculations are made using equation (17) and choosing parameters to give the best agreement between theory and experiment. Results for CT are given for comparison. It can be seen that the agreement between the calculated value of Cs and the data is very satisfactory (although there are discrepancies in the region around 0.1 K below TD not visible in figure 1).

L322 Letters to the Editor

We would like to acknowledge helpful discussions with G. A. Gehring, R. T. Harley, M. A. Moore and E. Pytte. SRPS wishes to thank AERE, Harwell for financial support.

Department of Theoretical Physics, 12 Parks Road, Oxford Clarendon Laboratory, Parks Road, Oxford

R. J. ELLIOTT A. P. YOUNG

S . R. P. SMITH

14th September 1971

BRODY, E. M., and CUMMINS, H. Z. , 1968, Phys. Reu. Lett., 21, 1263-8. COOKE, A. H., et al., 1970, Solid St. Commun., 8, 689-92. ELLIOTT, R. J., 1971, Proc. Int. Conf. on Light Scattering in Solids, Paris, 1971, ed. M. Balkanski

ELLIOTT, R. J., et al., 1971, J. Phys. C: Solid St. Phys., 4, L179-84. ELLIOTT, R. J., and PARKINSON, J. B., 1967, Proc. Phys. Soc., 92, 1024-39. GARLAND, C. W., and NOVOTNY, D. B., 1969, Phys. Rev., 177,971-5. GEHRING, K. A., MALOZEMOFF, A. P., STAUDE, W. S., and TYTE, R. N., 1971, ed. Solid St. Commun.,

HARLEY, R. T., HAYES, W., and SMITH, S. R. P., 1971a, Solid St. Commun., 9, 515-7. - 1971b, Proc. Int. Conf. on Light Scattering in Solids, Paris, 1971, ed. M. Balkanski (Paris:

KAMINOW, 1. P., and DAMEN, T. C., 1968, Phys. Rev. Lett., 20, 1105-8. KANAMORI, J., 1960, J. appl. Phys., 31, 14S-23s. KOBAYASHI, K., 1968, J. Phys. Soc. Japan, 24,497-508. MARIS, H. J., 1967, Phil. Mag., 16, 331-7. PYTTE, E., 1971, Phys. Rev. B, 3, 3503-9.

(Paris : Flammarion), to be published.

9,511-4.

Flammarion), to be published.

On the Ising model with impurities

Abstract. A model of an Ising magnet with nonmagnetic impurities which can be reduced to an ordinary Ising model is presented. The critical properties are described and it is found that they are not readily amenable to study by series methods.

The effect of impurities on the critical behaviour of the Ising and Heisenberg spin systems is an important problem both experimentally and theoretically, but at the same time one about which little theoretical information is available. Adopting a simplified picture, we prescribe four possible methods of distributing the impurities which, we assume, are both nonmagnetic and noninteracting. They may either be fixed in position (quenched) or free to move (annealed); in each case the impurities are located either on the lattice sites, which is equivalent to saying that certain spins are missing (the dilute magnet), or on the bonds joining neighbouring pairs of sites, in which case their only effect is to prevent the spins interacting. A site or bond cannot be occupied by more than one impurity. The quenched models are probably the more interesting because they are equivalent to assuming that the spin relaxation rate greatly exceeds that of the spatial distribution-a reasonable approxi- mation for real systems. Aside from various approximation techniques (eg the Bethe approxi- mation) which, useful as they are, provide little guidance on the subject of critical properties, and in the absence of exact results, series extrapolation methods provide the only way of