>K9T$, 2014, Tom 145, bmh. 1, CTp. 153 155
© 2014
ON LOW-TEMPERATURE PROPERTIES OF UNIAXIAL DIELECTRICS WITH A SOFT OPTIC MODE
D. E. Khmelnitskii*
Cavendish Laboratory, University of Cambridge CBS OHE, Cambridge. UK
Received September 9, 2013
Dielectric properties, thermal expansion, and the rate of spin-lattice relaxation at low temperatures in uniaxial dielectrics with a soft optical mode are discussed.
DOI: 10.7868/S0044451014010155
1. A uniaxial dielectric with a charge separation among the constituent atoms has an optic phonon mode, which corresponds to oscillation of the dielectric polarization P- along the î axis. The frequency of such a mode at small wave vectors k is described by the relation
u2(k) = u2(0) + ,sk"2 + k
(1)
where
= £
4 7H
J
nij l'o
The coefficient h reflects the degree of charge separation among the constituent atoms. The parameters Zj and inj respectively represent the charge and the mass of the ion of kind j, and v0 is the volume of the elementary cell. In typical ionic crystals, the parameter k (the square of the plasma frequency of ions) is of the order of the square <J2D of the Debye frequency. It follows from Eq. (1) that as k —¥ 0, the frequency u>(k) depends on the angle between the direction of the wave vector k and i axis. This reflects the long-range character of dipole dipole interaction between polarizations at different points in space1).
2. The dipole dipole modification of the phonon spectrum presented in Eq. (1) plays an important role if the frequency ui(0) in the centre of the Brillouin zone is
E-mail: dekl2(fflcam.ac .uk 11 Strictly speaking, dispersion of the frequency oi(k) depends on the direction of the wave vector k. For simplicity, we neglect this anisotropy as not leading to a qualitative difference and replace the tensor s by a scalar coefficient s.
especially small (soft optic mode ui(0) -C u>d). In such a case, anharmonicity plays an important role, which results in a temperature dependence of the frequency ui(0) of the soft mode. As was shown in Ref. fl],
u/2(0, T) — u/2(0,0) « j (ilk)
fi.ui( k)
n(k) = < exp
T
1
n(k) -l
(2)
Calculating the integral in Eq. (2) with dispersion law (1), we obtain
u2(0,T) = u2(0,0) + .4T3
(3)
The parameter u/2(0,0) in Eq. (3) can, in principle, be negative2). This means that the crystal undergoes a displacement-type phase transition at the temperature Tc that can be found from the relation
u/2(0,Tr) = 0 = u/2(0,0) + ATf.
At T < Tc, a spontaneous polarization P along î axis appears.
Thermal dependence of the frequency of the soft mode (3) results in a modification of the Curie Weiss law for dielectric susceptibility ;\ (T),
C3
T3 - r? '
T > Tr.
(4)
The parameter Tc can be positive or negative. If Tc > 0, the dielectric susceptibility ;\(T) diverges at a finite temperature, indicating a transition into the
In the case where ai2(0,0) < 0, it is assumed in Eq. (3) that .4 T3 > |oj2(0)|.
D. E. Khmelnitskii
>K3TO, TOM 145, Bbin. 1, 2014
ferroelectric phase at T < Tc. If Tc <0, \(T) remains finite at all temperatures. The difference from the results in Ref. fl] is in appearance of a T3 dependence in Eqs. (3) and (4).
3. As was mentioned in Ref. fl], the presence of a soft optic mode in the phonon spectrum results in a modification of the thermal expansion in the temperature range
fi|u>(0,0)| < T < ujd.
Modification of this result for a uniaxial crystal goes along the same line as in the analysis of dielectric susceptibility: the thermal expansion coefficient a(T) is proportional to the derivative of the number of thermally excited optic phonons with respect to the temperature, and therefore
a(T) oc
dTjl
(Jk)
n(k)
•x T
(5)
4. As was shown in Ref. [2], the phase diagram in the temperature pressure plane (T.p) is determined by the number Ar(T) of thermally excited optic phonons at low temperatures. Since the modification of the phonon spectrum due to dipole dipole interaction leads to a reduction in AT(T), the phase diagram undergoes a change: the critical pressure pc of the ferro-electric transition depends on the temperature T as
Pr(T) -pc(0) x —T3
(6)
5. The spin lattice relaxation rate l/n(T) in dielectrics is determined by the spin flips due to Ramanlike scattering of phonons (see, e.g., Ref. [3]). At low temperatures T ■C ©d, in most of dielectrics, the leading contribution arises from the Raman scattering of acoustic phonons [4]
1
ri(T)
XT'
Here, the high power of temperature arises from the total number of thermal phonons AT oc T3 and the fourth power of the wave vector3). The situation changes if the dielectric lias a soft optic mode4). The contribution of these phonons in cubic crystals is [5]
n(T)
■ocT3, MO)<r«0fl.
(7)
An acoustic displacement -ua enters any equation not itself but in the form of its gradient
4' An optic displacement xa itself is allowed by the symmetry to enter equations.
The situation changes once again in uniaxial crystals if the soft optic mode corresponds to oscillations of the polarization along the î axis. Due to dispersion law (1),
1
n(T)
ocT4, M0)<T«8d.
(8)
6. Finally, the crystal might belong to one of piezoelectric crystalline classes (see, e.g., [G]). In this case, even for small wave vectors, we cannot separate the optic mode corresponding to oscillations of the polarization P along the î axis and the transverse acoustic mode. The Lagrangian of coupled vibrations of the polarization P and the elastic deformations
1 ( Ou, du.
un = ~
ÔTj
dr.;
in the long-wavelength limit has the form
£ = / dr x
m p2 mu/2(0) 2
M
m s
(VP)1
4
dP u
p4
xy
(9)
Here, m is the reduced IIlclSS characteristic for the mode of optic vibrations, M is the IIlclSS of all atoms in the unit cell, a characterises the long-wave dispersion of optic mode, A and //. are elastic moduli, and d is the piezoelectric coefficient. Diagonalizing quadratic form (9), we obtain the following expression for the frequency u;(k) of the mode, which does not vanish as k —¥ 0 (see Refs. [6,7]):
ul2 = ul2(0)
(1
4 m//
•sk"
4 m//
A
// k2.ky
A + 2// k4
(10)
Equation (10) shows that similarly to the influence of dipole dipole interaction, the piezoelectric coupling results in introducing, into the dependence of the optic vibration frequency, a dependence on the direction of the wave vector k, which does not vanish at k —¥ 0. In the spirit of the foregoing, this leads to the temperature dependence of the frequency u/2(0). Calculating the ail-harmonic correction, we obtain
u2(0,T) = tJ2(0,0) + .4'T4. (11)
Thermal expansion is also modified:
a(T)ocT3. (12)
>K3TO, TOM 145, Bbin. 1, 2014
On low-temperature properties
Equation (6) for the critical pressure pc(T) of the ferroelectric transition undergoes the modification
pc(T)-pc(0)x-T4, (13)
and the spin lattice relaxation rate5),
fitjj(0) —<r<8j), (14)
T\ (1 ) ifllfl
7. Cubic crystals with soft optic modes are a subject of active research for nearly 50 years. As regards to uniaxial crystals, there are very few examples of this kind. In recent experiments [8], the properties of brominated tris-sarcosine calcium chloride (TSCC) were studied. These experiments showed that TSCC has a weak charge separation, the parameter k in Eq. (1) is anomalously small, and therefore the effects of dipole dipole interaction are negligible. I am grateful to S. E. Rowley and S. F. Scott for the opportunity to familiarize myself with the results of these experiments and for the instructive discussions. I hope that this paper will help to find a material in which the mentioned effects are stronger.
8. Many years ago, in 1970 72, two graduate students in Chernogolovka, late V. L. Shneerson and I, embarked on the studies of the displacement type tran-
sitions at low temperatures. This short comment is dedicated to the memory of V. L. Shneerson.
REFERENCES
1. D. E. Klimcliiit.skii and V. L. Slmccrsoii, Sov. Physics Solid State 13, 687 (1971).
2. D. E. Klimolnitskii and V. L. Slmccrsoii, Sov. Physics Solid State 13, 2158 (1971).
3. G. E. Pake, Paramagnetic Resonance: An Introductory Monograph, W. A. Benjamin, Inc. New York (1962).
4. I. Waller, Zs. Pliys. 79, 370 (1932).
5. D. E. Klimcliiit.skii and V. L. Shneerson, JETP Lett. 16, 84 (1971).
6. L. D. Landau and E. M. Lil'sliits, Electrodynamics of Continuous Medium, § 17, Butterwortli-Heiiiemaim (1984).
7. A. P. Levanyuk and A. A. Sobyanin, JETP Lett. 11, 371 (1970).
8. J. C. Lasliley, J. H. D. Muims, M. Ecliizeii, Mazliar N. Ali, S. E. Rowley, and J. F. Scott, submitted to J. Adv. Mat.
j) The difference between Waller's T~ and T* in Ref. [o] seems very significant, while the difference between T~ and T'J, perhaps, is not so easy to detect.
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