- Email: [email protected]
3O3 at low temperatures
Physica B 305 (2001) 90–95
Non-phonon contribution to the specific heat of PbMg1/3Nb2/3O3 at low temperatures S.N. Gvasaliyaa, S.G. Lushnikova,*, Yosuke Moriyab, Hitoshi Kawajib, Tooru Atakeb a
A.F. Ioffe Physical Technical Institute, Russian Academy of Sciences, Polytechnicheskaya 26 194021, St. Petersburg, Russia b Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8503, Japan Received 22 January 2001
Abstract Measurements of specific heat of relaxor ferroelectric PbMg1/3Nb2/3O3 (PMN) in the temperature range from 2 to 300 K by adiabatic calorimetry and relaxation technique are reported. Comparison of different approaches to description of the temperature dependence of specific heat has shown that experimental data can be adequately described in terms of fractal formalism. Spectral dimension inferred from low temperature specific heat measurements was found to be in good agreement with spectral dimension estimated from inelastic neutron scattering data. r 2001 Elsevier Science B.V. All rights reserved. PACS: 63.50.+x; 63.20.e; 65.40.+g; 61.50.f Keywords: Specific heat; PbMg1/3Nb2/3O3
1. Introduction Perovskite-like relaxor ferroelectrics (relaxors) with the common formula AB0x B001x O3 have been attracting considerable attention of researchers during many years because of their wide application and a set of unique physical properties [1]. From classical perovskite, with formula ABO3 these compounds differ by the presence of B0 and B00 ions having different valences in crystallographically equivalent positions of B sublattice. Disordering of the B sublattice drastically changes *Corresponding author. Fax: +7-17-812-515-6747. E-mail address: [email protected]ffe.rssi.ru (S.G. Lushnikov).
dynamics of the crystalline lattice: for instance, anomaly of dielectric response in the vicinity of ferroelectric phase transition becomes wider and can reach huge values (eB105 ). That is why representatives of this family of compounds are referred to as compounds with gigantic electrostriction. Recent studies of relaxors by highresolution electron microscopy have revealed the presence of regions of nanometer scale with ordering of B0 and B00 cations in 1 : 1 ratio irrespective of their stoichiometric composition [2,3]. The PbMg1/3Nb2/3O3 (PMN) crystal is a wellknown model relaxor ferroelectric. A wide anomaly of the frequency-dependent dielectric response peaking in the vicinity of T% c B270 K is not accompanied by macroscopic changes in the
0921-4526/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 5 8 4 - 1
S.N. Gvasaliya et al. / Physica B 305 (2001) 90–95
structure. A structural phase transition in PMN is realized near 200 K only when an external electric field is applied to a sample [4]. High-resolution electron microscopic data reveal the presence in PMN of a regular array of ordered clusters about 2 107 cm in diameter and with Mg and Nb distributed as 1 : 1. The distance between centers of neighboring ordered clusters is about 2.5 107 cm [2]. Topology of the nanocluster structure arising in this family of compounds has not yet been studied, but it, no doubt, affects lattice dynamics. It seems likely that it is responsible for low-frequency excitations with fracton spectral dimension in the vibrational spectrum of PMN inferred from inelastic neutron scattering experimental data [5,6]. Small-angle synchrotron scattering in PMN [7] indicates that a complex structure of nanoclusters in the form of fractals can be formed in this compound Then it is quite natural to expect that excitations arising in the nanocluster or, to be more exact, in the fractal objectFfractonsFwill be present in the vibrational spectrum of PMN. A probable candidate for this role in light scattering spectra is the lowfrequency mode with the frequency of the order of nB50 cm1 [5]. A more detailed analysis encounters difficulties because the existence of selection rules for Raman light scattering in crystals limits the applicability of the fracton ideology developed for analysis of low-frequency light scattering spectra of non-crystalline condensed matter. To develop a model of realization of fractal structure in PMN crystals, it was necessary to find other manifestations of the non-phonon dynamics in the vibrational spectrum of PMN that would reflect specific features of its organization at the nanometer scale. Analysis of the behavior of specific heat at low temperatures gives a chance to solve this problem. Specific heat of PMN has been measured only down to T ¼ 7 K [8], and therefore, more precise measurements to lower temperatures were carried out for this investigation.
2. Experimental The sample of PMN was synthesized by a flux method [9]. The powders of MgO (Rare Metallic
91
Co., 99.9%) and Nb2O5 (Mitsui Mining & Smelting Co., reagent grade) were mixed in an agate mortar, and the mixture was heated in a box furnace at 1370 K for 24 h. The product, precursor MgNb2O6, was mixed with PbO (Rare Metallic Co., 99.99%) in a mass ratio of 1 : 7.5, and then put in a platinum crucible. The crucible covered with platinum plate was heated in a vertical tube furnace at 1510 K for 3 h. After that, the sample was cooled slowly down to 1210 K in 200 h, and then it was quenched to room temperature. The flux was dissolved with hot nitric acid, and the crystals of PMN were separated. The crystals were translucent with light brownish yellow color; the maximum crystal size was about 2 2 2 mm. The structure of the crystal was identified by powder X-ray diffraction. The precise specific heat measurements between 13 and 300 K were made on the sample of PMN using a homemade adiabatic calorimeter [10,11]. The working thermometer was a platinum resistance thermometer (type 5187L, H. Tinsley & Co.) calibrated at National Physical Laboratory NPL (UK) on the basis of the International Temperature Scale ITS-90. The p.i.d. adiabatic control system has 105 K/min stability for the temperature of the calorimeter vessel. The performance of the specific heat measurements was confirmed by measuring the standard reference material (SRM) 720 (synthetic sapphire) provided from the national institute of standards and technology (NIST) (USA), which showed the accuracy of 0.1% of the specific heat at 100 K. The amount of the sample of crystals of PMN used for the present mesurements was 11.1106 g (0.034162 mol), which was put in the calorimeter vessel together with small amount of helium gas (5 kPa at room temperature). The specific heat of the sample was obtained by subtracting that of the calorimeter vessel from the total specific heat. The contribution of the sample to the total specific heat including that of the calorimeter vessel was about 20% at room temperature. During the experiments, no anomalous behavior such as supercooling and/or thermal relaxation was observed. The specific heat measurements between 2 and 50 K were made by a relaxation method using a commercial apparatus (PPMS model 6000,
92
S.N. Gvasaliya et al. / Physica B 305 (2001) 90–95
Quantum Design Inc.). A single crystal sample of 14.29 mg (0.04394 mmol) was used for the measurements. The results were in good agreement with the data obtained by the adiabatic calorimetry.
3. Results and discussion The results of specific heat measurements at the range of 2–300 K are shown in Fig. 1. It is well seen that around the diffuse phase transition temperature of 270 K, there are no anomalies of the behavior of the Cp ; which is in agreement with the results of [8]. Let us consider in detail the low-temperature region (To30 K) in the temperature dependence of specific heat of relaxor ferroelectric PMN (Fig. 2), where it is easier to find the contributions into the crystalline lattice dynamics other than phonon. According to the Debye theory, specific heat of a crystal at low temperatures is mainly determined by acoustic phonons and can be described by
Fig. 2. (a) Low-temperature region of specific heat. Circles show the experimental data. The solid line corresponds to fitting through Eq. (3) in the framework of scheme a. The dashed line is the Debye contribution into specific heat; (b) lowtemperature region of specific heat. Circles show the experimental data. The solid line corresponds to fitting through Eq. (3) in the framework of scheme b. The dashed line is the Debye contribution to specific heat.
CD ¼ 9R
Fig. 1. Temperature dependence of specific heat of the PMN relaxor ferroelectric.
T YD
3 Z
ED =kT 0
ex x 4 dx; ðex 1Þ2
ð1Þ
where R ¼ 8:314 J mol1 K1 is the universal gas constant, YD the Debye temperature which is
S.N. Gvasaliya et al. / Physica B 305 (2001) 90–95
equal for PMN to YD ¼ 376 K [8], and ED ¼ kYD the Debye energy. Fig. 2 shows the contribution of acoustic phonons (Debye contribution) into specific heat of PMN (curve 3) calculated through Eq. (1). It is well seen that the contribution of acoustic phonons determining the Debye specific heat is negligibly small at To12 K, and at TB30 K it is about 5% of the measured specific heat, and therefore, Debye contribution cannot explain the results of the experiment. A similar excessive low temperature specific heat was also found, for instance, in canonical glasses. A commonly accepted method of describing the low-temperature behavior of specific heat of such glasses is formalism of 2-level systems (TLS) [12]. In this case a linear dependence of specific heat on temperature is expected, i.e., C ¼ CD þ AT: We also used this approach to describe excessive low temperature specific heat of PMN. Calculations have shown that it is impossible to treat the low temperature excessive specific heat of PMN (Fig. 2) by using the TLS formalism. Thus to correctly describe experimental dependencies of specific heat of PMN crystal at To30 K, it should be assumed that additional excitations with properties differing from those of TLS exist in vibrational spectrum of PMN. Probably, these are localized excitations with fracton dimension revealed in light and neutron scattering experiments [5,6]. To analyze specific heat of the PMN crystal in excess to the Debye specific heat, we employed a slightly modified scheme used for description of the behavior of low temperature specific heat in a number of fractal compounds [13,14]. Expression for specific heat is obtained from the general relation (2) Z qE q CV ¼ do GðoÞ nðoÞ _o ; ð2Þ ¼ qT V qT where nðoÞ is the Bose–Einstein distribution function, and GðoÞ is the density of vibrational states. According to the general approach [15] GðoÞpo2 at oooco is the phonon regime, and * GðoÞpod 1 at oco oooom is the fracton regime with spectral dimension d*:
Finally we get 3 Z Eco =kT T ex x4 dx C ¼ 9R YD ðex 1Þ2 0 Z Em =kT x d*þ1 e x * þ AT d dx; 2 x Eco =kT ðe 1Þ
93
ð3Þ
where Eco ¼ _oco is the energy for change from the Debye regime to fracton one (crossover energy), and Em ¼ _om the maximum energy at which the fracton regime is realized (the difference between CV and CP at all temperatures of interest is negligibly small, and therefore we omit the subscript). In the analysis of Eq. (3), the shape of the density of states function GðEÞ should be taken into account. For this reason, we again invoke the inelastic neutron scattering studies of density of states. Fig. 3 shows generalized density of vibrational states GðEÞ of PMN crystal at T ¼ 50 K in double logarithmic scale obtained from inelastic neutron scattering experiments (for description of the experiment and detailed treatment of results see [5,6]). The curve shown in Fig. 3 has two pronounced linear portions with different slopes, which corresponds to different power laws with different spectral dimension of the GðEÞ function in the regions of 2–5 meV and 5–7 meV. Spectral dimension d ¼ 2:8 in the region 2–5 meV is in a good agreement with the classical Debye regime, while the portion with dimension d* ¼ 1:7 corresponds to the fracton regime. These results should be taken into account when we introduce such parameters as d*; Eco ; Em in Eq. (3). Of particular importance is the problem of integration limits in Eq. (3). The choice of integration limits is governed by the kind of crossover from the phonon regime to fracton. However, at present there is no universally accepted model of this crossover. The question whether this crossover is ‘‘sharp’’ or there is the energy region where both regimes coexists remains open. This dictated the use of two schemes for calculating specific heat: a and b. Scheme a involves fitting of calculated specific heat to experimental values when all parameters A; d*; Eco ; Em are not specified. Scheme b is fitting at which parameter Em is fixed and the remaining parameters are not specified. While fixing Em ; we
94
S.N. Gvasaliya et al. / Physica B 305 (2001) 90–95
Fig. 3. The low-energy part of the density of vibrational states of the PMN relaxor ferroelectric in double logarithmic scale at T ¼ 50 K. The arrows denote the range of realization of the fracton regime.
used the estimate of the upper boundary of the fracton regime inferred from analysis of the density of state function. Fig. 2 shows the fitting results obtained by using both schemes through Eq. (3). It is evident that calculations through Eq. (3) adequately describe experimental data over the entire temperature range chosen for analysis, although curve 2 in Fig. 2a obtained through scheme a describes the experimental dependence of specific heat better. Of importance, here is the increase of the upper boundary of realization of fracton regime to 13.3 meV (Table 1, scheme a). If we rely on estimates based on light scattering data [6], it can be expected that the upper integration limit Em ¼ 13:3 meV obtained from the calculation scheme a is not overestimatedF‘‘the tail’’ of the density of states of the fracton regime extends to a similar value. However, numerical values of parameters calculated by using scheme b are in better agreement with neutron measurements (see Table 1). Spectral dimension d* ¼ 1:6 and the values of Em ¼ 7:2 and Eco ¼ 5 obtained from fitting the specific heat data nearly coincide with the spectral dimension inferred from the inelastic neutron scattering data. Note that dimension d* ¼
Table 1 Comparison of the model parameters estimated from inelastic neutron scattering data and specific heat measurements Source
A
d*
Eco (meV)
Em (meV)
Fitting scheme a Fitting scheme b Neutron data
0.060 0.062 F
1.3 1.6 1.7
2.9 5 5
13.3 7.2 7
1:3 obtained through scheme a correspond to the suggested fractal approach. The 5% excess of experimental specific heat in the region of 30 K over calculations through scheme b (Fig. 2b) can be attributed to the contribution of the low frequency transverse optical (TO) phonon into density of states and, hence, into specific heat. It is extremely difficult to estimate this contribution to the heat capacity, because the lower TO phonon has considerable dispersion over the Brillouin zone [16], and its description by Einstein oscillator is not quite correct. At the moment we believe that both approaches to treatment of temperature dependence of specific heat are applicable. Additional investigations aimed at finding the type of crossover are needed.
S.N. Gvasaliya et al. / Physica B 305 (2001) 90–95
4. Conclusions Analysis of experimental temperature dependencies of specific heat has shown that in the temperature range 2oTo30 K, lattice dynamics of relaxor ferroelectric PMN cannot be described in terms of phonon dynamics. Here the use of fractal formalism turns out to be fruitful. The value of spectral dimension d* obtained through Eq. (3) is in a good agreement with the values of d* obtained from inelastic neutron scattering data. This fact is a convincing proof that the low temperature behavior of specific heat of PMN is governed by contribution of fractons.
Acknowledgements This work was supported by RFBR grant 99-0218316. One of the authors (SNG) was supported by ISSEP grant a00-1191.
References [1] G.A. Smolensky, et al., Ferroelectrics and Related Materials, Academic Press, New York, 1981.
95
[2] C. Boulesteix, F. Varnier, A. Llebaria, E. Husson, J. Solid State Chem. 108 (1994) 141. [3] P. Julin, A. Bursill, Mod. Phys. Lett. 7 (1993) 609. [4] Z.G. Ye, H. Schmidt, Ferroelectrics 145 (1993) 83. [5] S.G. Lushnikov, S.N. Gvasaliya, I.G. Siny, Physica B 263–264 (1999) 286. [6] S.N. Gvasaliya, S.G. Lushnikov, I.L. Sashin, I.G. Siny, Crystallogr. Rep. 44 (1999) 250. (translated from Kristallografiya 44, No. 2, pp. 284–288, 1999). [7] S.B. Vakhrushev, A.A. Nabereznov, S.K. Sinha, Y.P. Feng, T. Egami, J. Phys. Chem. Solids 57 (1996) 1517. [8] B.A. Strukov, E.L. Sorokin, B.M. Rizak, N.K. Ushin, L.M. Sapozhnikova, Fiz. Tv. Tela 31 (1989) 121 (in Russian). [9] Z.-G. Ye, P. Tissto, Mater. Res. Bull. 25 (1990) 739. [10] T. Atake, H. Kawaji, A. Hamano, Y. Saito, Report Res. Lab. Eng. Mater., Tokyo Inst. Technol. 15 (1990) 13. [11] T. Tanaka, T. Atake, H. Nakayama, T. Eguchi, K. Saito, I. Ikemoto, J. Chem. Thermodyn. 26 (1994) 1231. [12] A. Heur, Microscopic view of the low-temperature anomalies in glasses, in: P. Esquinazi (Ed.), Tunneling Systems in Amorphous and Crystalline Solids, Springer Verlag, Berlin, 1998, p. 459. [13] S. Alexander, C. Laermans, R. Orbach, H.M. Rosenberg, Phys. Rev. B 28 (1983) 4615. [14] J. Kjems, Fractals and Experiments, in: A. Bunde, S. Havlin (Eds.), Fractals and Disordered Systems, Springer, Berlin, 1995, p. 303. [15] T. Nakayama, K. Yakubo, R. Orbach, Rev. Mod. Phys. 66 (1994) 381. [16] A.A. Naberezhnov, S.B. Vakhrushev, B. Dorner, D. Strauch, H. Mouden, Eur. Phys. J. B11 (1999) 13.
Non-phonon contribution to the specific heat of PbMg1/3Nb2/3O3 at low temperatures S.N. Gvasaliyaa, S.G. Lushnikova,*, Yosuke Moriyab, Hitoshi Kawajib, Tooru Atakeb a
A.F. Ioffe Physical Technical Institute, Russian Academy of Sciences, Polytechnicheskaya 26 194021, St. Petersburg, Russia b Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8503, Japan Received 22 January 2001
Abstract Measurements of specific heat of relaxor ferroelectric PbMg1/3Nb2/3O3 (PMN) in the temperature range from 2 to 300 K by adiabatic calorimetry and relaxation technique are reported. Comparison of different approaches to description of the temperature dependence of specific heat has shown that experimental data can be adequately described in terms of fractal formalism. Spectral dimension inferred from low temperature specific heat measurements was found to be in good agreement with spectral dimension estimated from inelastic neutron scattering data. r 2001 Elsevier Science B.V. All rights reserved. PACS: 63.50.+x; 63.20.e; 65.40.+g; 61.50.f Keywords: Specific heat; PbMg1/3Nb2/3O3
1. Introduction Perovskite-like relaxor ferroelectrics (relaxors) with the common formula AB0x B001x O3 have been attracting considerable attention of researchers during many years because of their wide application and a set of unique physical properties [1]. From classical perovskite, with formula ABO3 these compounds differ by the presence of B0 and B00 ions having different valences in crystallographically equivalent positions of B sublattice. Disordering of the B sublattice drastically changes *Corresponding author. Fax: +7-17-812-515-6747. E-mail address: [email protected]ffe.rssi.ru (S.G. Lushnikov).
dynamics of the crystalline lattice: for instance, anomaly of dielectric response in the vicinity of ferroelectric phase transition becomes wider and can reach huge values (eB105 ). That is why representatives of this family of compounds are referred to as compounds with gigantic electrostriction. Recent studies of relaxors by highresolution electron microscopy have revealed the presence of regions of nanometer scale with ordering of B0 and B00 cations in 1 : 1 ratio irrespective of their stoichiometric composition [2,3]. The PbMg1/3Nb2/3O3 (PMN) crystal is a wellknown model relaxor ferroelectric. A wide anomaly of the frequency-dependent dielectric response peaking in the vicinity of T% c B270 K is not accompanied by macroscopic changes in the
0921-4526/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 5 8 4 - 1
S.N. Gvasaliya et al. / Physica B 305 (2001) 90–95
structure. A structural phase transition in PMN is realized near 200 K only when an external electric field is applied to a sample [4]. High-resolution electron microscopic data reveal the presence in PMN of a regular array of ordered clusters about 2 107 cm in diameter and with Mg and Nb distributed as 1 : 1. The distance between centers of neighboring ordered clusters is about 2.5 107 cm [2]. Topology of the nanocluster structure arising in this family of compounds has not yet been studied, but it, no doubt, affects lattice dynamics. It seems likely that it is responsible for low-frequency excitations with fracton spectral dimension in the vibrational spectrum of PMN inferred from inelastic neutron scattering experimental data [5,6]. Small-angle synchrotron scattering in PMN [7] indicates that a complex structure of nanoclusters in the form of fractals can be formed in this compound Then it is quite natural to expect that excitations arising in the nanocluster or, to be more exact, in the fractal objectFfractonsFwill be present in the vibrational spectrum of PMN. A probable candidate for this role in light scattering spectra is the lowfrequency mode with the frequency of the order of nB50 cm1 [5]. A more detailed analysis encounters difficulties because the existence of selection rules for Raman light scattering in crystals limits the applicability of the fracton ideology developed for analysis of low-frequency light scattering spectra of non-crystalline condensed matter. To develop a model of realization of fractal structure in PMN crystals, it was necessary to find other manifestations of the non-phonon dynamics in the vibrational spectrum of PMN that would reflect specific features of its organization at the nanometer scale. Analysis of the behavior of specific heat at low temperatures gives a chance to solve this problem. Specific heat of PMN has been measured only down to T ¼ 7 K [8], and therefore, more precise measurements to lower temperatures were carried out for this investigation.
2. Experimental The sample of PMN was synthesized by a flux method [9]. The powders of MgO (Rare Metallic
91
Co., 99.9%) and Nb2O5 (Mitsui Mining & Smelting Co., reagent grade) were mixed in an agate mortar, and the mixture was heated in a box furnace at 1370 K for 24 h. The product, precursor MgNb2O6, was mixed with PbO (Rare Metallic Co., 99.99%) in a mass ratio of 1 : 7.5, and then put in a platinum crucible. The crucible covered with platinum plate was heated in a vertical tube furnace at 1510 K for 3 h. After that, the sample was cooled slowly down to 1210 K in 200 h, and then it was quenched to room temperature. The flux was dissolved with hot nitric acid, and the crystals of PMN were separated. The crystals were translucent with light brownish yellow color; the maximum crystal size was about 2 2 2 mm. The structure of the crystal was identified by powder X-ray diffraction. The precise specific heat measurements between 13 and 300 K were made on the sample of PMN using a homemade adiabatic calorimeter [10,11]. The working thermometer was a platinum resistance thermometer (type 5187L, H. Tinsley & Co.) calibrated at National Physical Laboratory NPL (UK) on the basis of the International Temperature Scale ITS-90. The p.i.d. adiabatic control system has 105 K/min stability for the temperature of the calorimeter vessel. The performance of the specific heat measurements was confirmed by measuring the standard reference material (SRM) 720 (synthetic sapphire) provided from the national institute of standards and technology (NIST) (USA), which showed the accuracy of 0.1% of the specific heat at 100 K. The amount of the sample of crystals of PMN used for the present mesurements was 11.1106 g (0.034162 mol), which was put in the calorimeter vessel together with small amount of helium gas (5 kPa at room temperature). The specific heat of the sample was obtained by subtracting that of the calorimeter vessel from the total specific heat. The contribution of the sample to the total specific heat including that of the calorimeter vessel was about 20% at room temperature. During the experiments, no anomalous behavior such as supercooling and/or thermal relaxation was observed. The specific heat measurements between 2 and 50 K were made by a relaxation method using a commercial apparatus (PPMS model 6000,
92
S.N. Gvasaliya et al. / Physica B 305 (2001) 90–95
Quantum Design Inc.). A single crystal sample of 14.29 mg (0.04394 mmol) was used for the measurements. The results were in good agreement with the data obtained by the adiabatic calorimetry.
3. Results and discussion The results of specific heat measurements at the range of 2–300 K are shown in Fig. 1. It is well seen that around the diffuse phase transition temperature of 270 K, there are no anomalies of the behavior of the Cp ; which is in agreement with the results of [8]. Let us consider in detail the low-temperature region (To30 K) in the temperature dependence of specific heat of relaxor ferroelectric PMN (Fig. 2), where it is easier to find the contributions into the crystalline lattice dynamics other than phonon. According to the Debye theory, specific heat of a crystal at low temperatures is mainly determined by acoustic phonons and can be described by
Fig. 2. (a) Low-temperature region of specific heat. Circles show the experimental data. The solid line corresponds to fitting through Eq. (3) in the framework of scheme a. The dashed line is the Debye contribution into specific heat; (b) lowtemperature region of specific heat. Circles show the experimental data. The solid line corresponds to fitting through Eq. (3) in the framework of scheme b. The dashed line is the Debye contribution to specific heat.
CD ¼ 9R
Fig. 1. Temperature dependence of specific heat of the PMN relaxor ferroelectric.
T YD
3 Z
ED =kT 0
ex x 4 dx; ðex 1Þ2
ð1Þ
where R ¼ 8:314 J mol1 K1 is the universal gas constant, YD the Debye temperature which is
S.N. Gvasaliya et al. / Physica B 305 (2001) 90–95
equal for PMN to YD ¼ 376 K [8], and ED ¼ kYD the Debye energy. Fig. 2 shows the contribution of acoustic phonons (Debye contribution) into specific heat of PMN (curve 3) calculated through Eq. (1). It is well seen that the contribution of acoustic phonons determining the Debye specific heat is negligibly small at To12 K, and at TB30 K it is about 5% of the measured specific heat, and therefore, Debye contribution cannot explain the results of the experiment. A similar excessive low temperature specific heat was also found, for instance, in canonical glasses. A commonly accepted method of describing the low-temperature behavior of specific heat of such glasses is formalism of 2-level systems (TLS) [12]. In this case a linear dependence of specific heat on temperature is expected, i.e., C ¼ CD þ AT: We also used this approach to describe excessive low temperature specific heat of PMN. Calculations have shown that it is impossible to treat the low temperature excessive specific heat of PMN (Fig. 2) by using the TLS formalism. Thus to correctly describe experimental dependencies of specific heat of PMN crystal at To30 K, it should be assumed that additional excitations with properties differing from those of TLS exist in vibrational spectrum of PMN. Probably, these are localized excitations with fracton dimension revealed in light and neutron scattering experiments [5,6]. To analyze specific heat of the PMN crystal in excess to the Debye specific heat, we employed a slightly modified scheme used for description of the behavior of low temperature specific heat in a number of fractal compounds [13,14]. Expression for specific heat is obtained from the general relation (2) Z qE q CV ¼ do GðoÞ nðoÞ _o ; ð2Þ ¼ qT V qT where nðoÞ is the Bose–Einstein distribution function, and GðoÞ is the density of vibrational states. According to the general approach [15] GðoÞpo2 at oooco is the phonon regime, and * GðoÞpod 1 at oco oooom is the fracton regime with spectral dimension d*:
Finally we get 3 Z Eco =kT T ex x4 dx C ¼ 9R YD ðex 1Þ2 0 Z Em =kT x d*þ1 e x * þ AT d dx; 2 x Eco =kT ðe 1Þ
93
ð3Þ
where Eco ¼ _oco is the energy for change from the Debye regime to fracton one (crossover energy), and Em ¼ _om the maximum energy at which the fracton regime is realized (the difference between CV and CP at all temperatures of interest is negligibly small, and therefore we omit the subscript). In the analysis of Eq. (3), the shape of the density of states function GðEÞ should be taken into account. For this reason, we again invoke the inelastic neutron scattering studies of density of states. Fig. 3 shows generalized density of vibrational states GðEÞ of PMN crystal at T ¼ 50 K in double logarithmic scale obtained from inelastic neutron scattering experiments (for description of the experiment and detailed treatment of results see [5,6]). The curve shown in Fig. 3 has two pronounced linear portions with different slopes, which corresponds to different power laws with different spectral dimension of the GðEÞ function in the regions of 2–5 meV and 5–7 meV. Spectral dimension d ¼ 2:8 in the region 2–5 meV is in a good agreement with the classical Debye regime, while the portion with dimension d* ¼ 1:7 corresponds to the fracton regime. These results should be taken into account when we introduce such parameters as d*; Eco ; Em in Eq. (3). Of particular importance is the problem of integration limits in Eq. (3). The choice of integration limits is governed by the kind of crossover from the phonon regime to fracton. However, at present there is no universally accepted model of this crossover. The question whether this crossover is ‘‘sharp’’ or there is the energy region where both regimes coexists remains open. This dictated the use of two schemes for calculating specific heat: a and b. Scheme a involves fitting of calculated specific heat to experimental values when all parameters A; d*; Eco ; Em are not specified. Scheme b is fitting at which parameter Em is fixed and the remaining parameters are not specified. While fixing Em ; we
94
S.N. Gvasaliya et al. / Physica B 305 (2001) 90–95
Fig. 3. The low-energy part of the density of vibrational states of the PMN relaxor ferroelectric in double logarithmic scale at T ¼ 50 K. The arrows denote the range of realization of the fracton regime.
used the estimate of the upper boundary of the fracton regime inferred from analysis of the density of state function. Fig. 2 shows the fitting results obtained by using both schemes through Eq. (3). It is evident that calculations through Eq. (3) adequately describe experimental data over the entire temperature range chosen for analysis, although curve 2 in Fig. 2a obtained through scheme a describes the experimental dependence of specific heat better. Of importance, here is the increase of the upper boundary of realization of fracton regime to 13.3 meV (Table 1, scheme a). If we rely on estimates based on light scattering data [6], it can be expected that the upper integration limit Em ¼ 13:3 meV obtained from the calculation scheme a is not overestimatedF‘‘the tail’’ of the density of states of the fracton regime extends to a similar value. However, numerical values of parameters calculated by using scheme b are in better agreement with neutron measurements (see Table 1). Spectral dimension d* ¼ 1:6 and the values of Em ¼ 7:2 and Eco ¼ 5 obtained from fitting the specific heat data nearly coincide with the spectral dimension inferred from the inelastic neutron scattering data. Note that dimension d* ¼
Table 1 Comparison of the model parameters estimated from inelastic neutron scattering data and specific heat measurements Source
A
d*
Eco (meV)
Em (meV)
Fitting scheme a Fitting scheme b Neutron data
0.060 0.062 F
1.3 1.6 1.7
2.9 5 5
13.3 7.2 7
1:3 obtained through scheme a correspond to the suggested fractal approach. The 5% excess of experimental specific heat in the region of 30 K over calculations through scheme b (Fig. 2b) can be attributed to the contribution of the low frequency transverse optical (TO) phonon into density of states and, hence, into specific heat. It is extremely difficult to estimate this contribution to the heat capacity, because the lower TO phonon has considerable dispersion over the Brillouin zone [16], and its description by Einstein oscillator is not quite correct. At the moment we believe that both approaches to treatment of temperature dependence of specific heat are applicable. Additional investigations aimed at finding the type of crossover are needed.
S.N. Gvasaliya et al. / Physica B 305 (2001) 90–95
4. Conclusions Analysis of experimental temperature dependencies of specific heat has shown that in the temperature range 2oTo30 K, lattice dynamics of relaxor ferroelectric PMN cannot be described in terms of phonon dynamics. Here the use of fractal formalism turns out to be fruitful. The value of spectral dimension d* obtained through Eq. (3) is in a good agreement with the values of d* obtained from inelastic neutron scattering data. This fact is a convincing proof that the low temperature behavior of specific heat of PMN is governed by contribution of fractons.
Acknowledgements This work was supported by RFBR grant 99-0218316. One of the authors (SNG) was supported by ISSEP grant a00-1191.
References [1] G.A. Smolensky, et al., Ferroelectrics and Related Materials, Academic Press, New York, 1981.
95
[2] C. Boulesteix, F. Varnier, A. Llebaria, E. Husson, J. Solid State Chem. 108 (1994) 141. [3] P. Julin, A. Bursill, Mod. Phys. Lett. 7 (1993) 609. [4] Z.G. Ye, H. Schmidt, Ferroelectrics 145 (1993) 83. [5] S.G. Lushnikov, S.N. Gvasaliya, I.G. Siny, Physica B 263–264 (1999) 286. [6] S.N. Gvasaliya, S.G. Lushnikov, I.L. Sashin, I.G. Siny, Crystallogr. Rep. 44 (1999) 250. (translated from Kristallografiya 44, No. 2, pp. 284–288, 1999). [7] S.B. Vakhrushev, A.A. Nabereznov, S.K. Sinha, Y.P. Feng, T. Egami, J. Phys. Chem. Solids 57 (1996) 1517. [8] B.A. Strukov, E.L. Sorokin, B.M. Rizak, N.K. Ushin, L.M. Sapozhnikova, Fiz. Tv. Tela 31 (1989) 121 (in Russian). [9] Z.-G. Ye, P. Tissto, Mater. Res. Bull. 25 (1990) 739. [10] T. Atake, H. Kawaji, A. Hamano, Y. Saito, Report Res. Lab. Eng. Mater., Tokyo Inst. Technol. 15 (1990) 13. [11] T. Tanaka, T. Atake, H. Nakayama, T. Eguchi, K. Saito, I. Ikemoto, J. Chem. Thermodyn. 26 (1994) 1231. [12] A. Heur, Microscopic view of the low-temperature anomalies in glasses, in: P. Esquinazi (Ed.), Tunneling Systems in Amorphous and Crystalline Solids, Springer Verlag, Berlin, 1998, p. 459. [13] S. Alexander, C. Laermans, R. Orbach, H.M. Rosenberg, Phys. Rev. B 28 (1983) 4615. [14] J. Kjems, Fractals and Experiments, in: A. Bunde, S. Havlin (Eds.), Fractals and Disordered Systems, Springer, Berlin, 1995, p. 303. [15] T. Nakayama, K. Yakubo, R. Orbach, Rev. Mod. Phys. 66 (1994) 381. [16] A.A. Naberezhnov, S.B. Vakhrushev, B. Dorner, D. Strauch, H. Mouden, Eur. Phys. J. B11 (1999) 13.