Thermally activated two-level systems in superionic conductors

Solid State Ionics 107 (1998) 59–65

Thermally activated two-level systems in superionic conductors N.R. Abdulchalikova, A.E. Aliev*, V.F. Krivorotov, P.K. Khabibullaev Heat Physics Department of Uzbek Academy of Sciences, 700135, Tashkent, Uzbekistan Received 25 March 1997; accepted 11 July 1997

Abstract In superionic crystals, as well as in vitreous superionic conductors, the monotonous growth (almost linear) of the specific heat C(T ) and the thermal conductivity l(T ) in a wide temperature range of superionic phase has been obtained. Such behaviour takes place in a temperature region that considerably exceeds the Debye temperatures of the investigated materials. It is shown that in the high temperature region the growth of C, and respectively l, may reflect a relaxation interaction of high frequency phonons with two-level systems (TLS). The examination of excess heat capacity DC(T ) in the framework of the TLS model allows to find a concentration of migration ions, which is in good accordance with an experimental data. Keywords: Superionic conductors; Heat capacity; Thermal conductivity; Two-level systems PACS: 65.40; 66.30.H; 66.70

1. Introduction There has been rapidly accelerating interest in superionic conductors in recent years because of their use as electrolytes in solid state batteries of high energy density. The use of alternative energy sources capable of generating electricity on intermittent bases (solar energy, wind power etc.) requires low cost, high efficiency electricity storage systems; solid state batteries are precisely this type of system. Besides, these materials have awakened widespread interest not only because of the possibility of their application, but also because of their remarkable

*Corresponding author. Fax: 17 371 276 2668; e-mail: [email protected]

physical properties. Numerous models have been proposed to account for the anomalous large ionic conductivity of the superionic conductors [1–3]. Recently, experiments in a large temperature regime performed on many superionic conductors have demonstrated the rapid continuous increase of the heat capacity CP and the thermal conductivity l at temperatures significantly exceeding the Debye temperatures uD of these materials. These features have been observed for all type of superionic conductors: 1D conductors [4,5]; 2D and 3D [6–8], and superionic glasses [9]. It is expected that these peculiarities are manifestations of the same effect. However, the fundamental physics behind this phenomenon is still not well understood. In this paper the heat capacity and the thermal conductivity behavior of several superionic conduc-

0167-2738 / 98 / $19.00  1998 Elsevier Science B.V. All rights reserved. PII S0167-2738( 97 )00395-0

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tors are examined in a wide temperature region over uD in detail to test the hypothesis that the rapid increase of CP and l at high temperatures is typical of all superionic conductors. The observed behavior is explained with the help of thermally activated two-level systems (TLS).

2. Experimental For the measurements the following superionic conductors (SIC) with a stable rigid sublattice in a wide temperature region were chosen: 1. a -LiIO 3 , Li 2 B 4 O 7 -quasi-one-dimensional lithium (Li 1 ) conducting SIC [10,11] 2. LaF 3 , CeF 3 , PrF 3 -three-dimensional fluorine (F 2) conducting SIC [12]. The heat capacity measurements were performed using a vacuum adiabatic calorimeter. The thin wire heater with negligible heat capacity (tungsten wire, d 5 12 mm) was wound directly on the single crystal samples in the form of a bar of 10 3 10 3 20 mm and heated by short duration heat impulses. A heat input to the sample was about 0.001 cal / g ? s and the heating rate was about 1 deg / min at 500 K. For the thermal conductivity measurements, the samples were obtained in the form of plates 10 3 10 3 1 mm. The measurements were performed using the transient hot wire method [4] in the temperature range 80–400 K in nitrogen atmosphere. A thin heater wire (tungsten wire, d 5 12 mm) was embedded between two identical single crystal plates, and a thermocouple was positioned on this sandwich to measure the sample temperature. To insure good contact between the heater wire and the samples this sandwich was compressed. The measurement errors for CP (1% for 100 , T , 300 K), 5% for 300 , T , 800 K) were determined by a calibration procedure on a standard (in our case single-crystal Al 2 O 3 , KCl and SiO 2 ). To calculate uD from elastic moduli the longitudinal and transverse sound velocities have been measured using the resonant and the pulse-echo methods.

3. Results and discussion The temperature dependencies of the heat capacity CP (T ) and the thermal conductivity l(T ) in single crystals LnF 3 (Ln 5 La, Ce, Pr), Li 2 B 4 O 7 , a -LiIO 3 are shown in Figs. 1–6 respectively. All measured points for LnF 3 fell on the same curve and show the almost linear increase of CP . The l(T ) behaviours are also the same with the exception of absolute values. The last ones are in good accordance with uD of these compounds. 1D SIC shows the same behaviour for CP (T ) and l(T ) at high temperatures. However, at low temperatures all these crystals become dielectric and l falls as T 21 [13]. Although in the case of superionic glasses and 1D SIC one can try to explain such unusual behaviour as structural peculiarities, for the series of LnF 3 it is quite unexpected. Numerical analysis of SICs with different structure and conductivity mechanisms shows that the increase of CP (T ) in the region of the superionic phase at

Fig. 1. Temperature dependence of the heat capacity for LaF 3 . Solid line is for the Debye heat capacity calculated from the data given in Table 2.

N.R. Abdulchalikova et al. / Solid State Ionics 107 (1998) 59 – 65

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Fig. 2. Temperature dependence of the thermal conductivity for LaF 3 .

Fig. 3. Temperature dependence of the heat capacity for Li 2 B 4 O 7 .

T . uD is a characteristic feature of superionic conductors. Table 1 shows specific heat, CP (T ) for several classical dielectric materials and superionic conductors represented as a polynomial CP 5 a 1 bT 1 cT 22 . The anomalous large values of the linear term coefficient b observed for SIC are in contrast with the Debye model of the specific heat. The quasi-1D Debye model of the heat capacity for non-interacting atom chains, to which can be attributed the linear chains of disordered Li 1 sublattice in Li 2 B 4 O 7 and a -LiIO 3 , gives a linear increase of CP (T ) in the region T , uD : Cv 5 (≠E / ≠T ) v 5 (p 2 / 3uD )Nk B T. However, above uD the heat capacity is constant, Cv 5 Nk B . The contribution of the anharmonic effect is also negligible. These calculations give CP 5 Cv (1 1 3ag T ) 5 Cv (1 1 5.8 3 10 25 T ). Here a is the cubic thermal expansion coefficient, (aLaF 5 9.7 3 10 26 ¨ K 21 ), g is the Gruneisen constant (g | 2). To consider the thermally activated ions as an attribute of the superionic phase, we propose a model incorporating the thermally activated two-level sys-

tems (TLS). It is well known that to explain the linear temperature dependence of CP (T ) observed in glasses at T , 1–2 K the most successful model has been put forward by Anderson et al. [14] and Phillips [15]. The situation has been reviewed in a beautiful article by Jackle et al. [16]. The model assumes the existence of an ensemble of TLS in glass with a flat distribution of level splitting. This also accounts for the presence of low energy excitations with approximately constant density of states in addition to phonons, as indicated by the specific heat studies in glass, which show the excess specific heat over that predicted by Debye theory. These TLS are considered to belong to localized structural defects which at very low temperature perform quantum mechanical tunnelling motions through a barrier between two possible equilibrium configurations. At high temperature, a different process which can be described as a structural relaxation via thermally activated transition overriding the barrier height of the doublewell potential becomes important. Using this model we had success in describing l(T ) in alkali

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Fig. 4. Temperature dependence of the thermal conductivity for Li 2 B 4 O 7 .

alumophosphate superionic glasses at T 5 100–400 K [9]. However, the application of the TLS model for crystalline materials is not so obvious. For a perfect crystalline arrangement the central force theory predicts that all atoms of a particular kind move in identical symmetric interatomic potential wells, each of which has a single central minimum for each atom corresponding to its equilibrium position. Of course, there are no amorphous arrangement effects in the latter. Therefore, to employ this model we should make the following notes: 1. The crystalline structure of SIC is determined by the arrangement of ions in the rigid sublattice. A disordered sublattice that apparently caused the anomalous behaviour of CP has a ‘friable’ structure. Each ion of this sublattice has available more than the equivalent number of equal positions and a transition state has a low activation energy. Two Ag 1 -ions in a -AgI have available 42 positions in a unit cell. The fluorine ion positions in the tysonite structure (LnF 3 ) are divided into three sublattices,

Fig. 5. Temperature dependence of the heat capacity for a -LiIO 3 .

Fig. 6. Temperature dependence of the thermal conductivity for a -LiIO 3 .

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Table 1 Substance

State

NaCl KCl Al 2 O 3 SiO 2 AgI AgNO 3 LnF 3 Ln5La,Ce,Pr Li 2 B 4 O 7 a -LiIO 3

CP 5a1bT 1cT 22 [J / mol?K]

Temperature region DT [K]

a

b310 3

c310 25

crystal liquid crystal liquid crystal liquid crystal crystal liquid crystal liquid

45.97 77.82 41.41 73.65 114.84 144.96 46.98 24.37 58.6 36.8 128.12

16.33 27.536 21.77 2 12.81 2 34.33 100.9 2 189.2 2

2 2 3.22 2 235.46 2 211.3 2 2 2 2

crystal crystal crystal

41.4 19.3 31.8

298–1074 1074–1500 298–1043 1044–1710 298–1800 1800–3500 298–848 273–423 831–1778 298–433 483–600

27.7 51.4 16.8

190.8 430.5 368

300–800 160–600 200–400

First six data from [22].

F(1), F(2) and F(3) (with multiplicities of 12, 4 and 2 respectively). Both F 3 -type ions are placed at the centres of both triangles formed by the lanthanum ions perpendicularly to the c-axis [17]. On the contrary, the four F 2 -type ions are not enclosed in the cationic planes, but located at apices of pyramids whose basis is constituted by the cationic triangles. The twelve associated F 1 -type ions form three by three equilateral triangles perpendicular to the c-axis between cationic planes. It seems that the presence of the F 1 anions involves, for electrostatic considerations, shifting of the F 2 anions outside the cationic planes. The upper and lower positions of F 2 are energetically equivalent and begin to be occupied at 400–420 K. These types of TLS correspond to the transverse vibrations of the anion (O atom) in glasses (A-type double-well potential [18]). This is also the case in a -LiIO 3 . The Li cations are displaced from I ions planes along the c-axis at a

˚ [19]. At T . 240 K the upper distance d 5 0.75 A and lower equivalent positions are occupied statistically. 2. The spectrum of barrier energy distribution has a discrete character. 3. The existence of a double-well potential in SIC is confirmed by Raman and neutron scattering investigations [17] and computer simulation of a potential relief [20]. Thus, the possibility of TLS application to SIC is quite justified in our opinion. Let us analyze our results on rare-earth fluorides, with LaF 3 taken as an example. The excess heat capacity, DCP , due to the TLS contribution is of course to be added with the usual lattice contribution (vibration component). To compare with experimental results we estimate the lattice contribution by an extrapolation from low temperature. The longitudinal and transverse sound velocities measured at 10 MHz (T 5 300 K) frequency and averaged in all directions,

Table 2 Substance

r [kg / m 3 ]

Vl [m / s]

Vt 1 [m / s]

Vt 2 [m / s]

N 310 228 [m 23 ]

uD [K]

LaF 3 CeF 3 PrF 3 a -LiIO 3 Li 2 B 4 O 7

5938 6120 6288 4450 2439

5040 5205 5465 4281 6200

2537 2647 2759 2309 4700

3335 2164 2266 2208 4400

7.3 7.48 7.64 7.45 2.813

391 397 403 215 303

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as well as densities necessary for calculation of the Debye specific heat, are presented in Table 2. We assume that at high temperature (T . 100 K) the transition overriding the barrier height of the doublewell potential becomes thermally activated, because a top limit of quantum mechanical tunnelling through the barrier calculated from [21] ] ]] T t 5 (Œ2h /p k B d)ŒE /m, for our materials (LnF 3 ) does not exceed 27 K. Here ˚ is the width of the potential barrier, E5 d52.5 A 0.48 eV is the height of the barrier, m is the mass of the tunnelling particles. This value does not exceed 40 K even by more precise accounting of the spectrum distribution of the barriers. For the excess heat capacity DCv (T )5Cexp 2 Cdebye due to TLS we can write DCv (T ) 5 (p 2 / 6)gk 2 T,

(1)

where g5N /DE is the density of states for the energy levels of TLS, N is the concentration, DE5 Emax 2Emin is a certain spread of potential barriers. In the framework of the hard double-well potential model in the approach of the Gauss distribution of mobile ion activation energies the excess heat capacity must linearly increase with increasing of temperature. The difference between the experimental and Debye heat capacities DCv as a function of temperature for LaF 3 is given in Fig. 7. We have converted CP to Cv using the expression CP 2Cv 5 a 2 TV /K, where a is the cubic thermal expansion coefficient, V is the molar volume, K is the isothermal compressibility. The obtained dependence is proximate to linear, as follows from (1). The interval of possible barriers DE was determined from frequent-temperature measurements of ionic conductivity. Emax 50.48 eV correspond to dc measurement or to very low frequencies (1–100 Hz). At high frequencies ( f . 100 MHz) Emin 50.1 eV. Emin has been determined both parallel by Raman scattering method. The TLS concentration calculated from (1) is N536310 27 m 23 , in good agreement with total concentration of disordered F 2 anions. In the main the l(T ) behaviour in this temperature

Fig. 7. Temperature dependence of DC, the difference between the experimental and Debye heat capacity for LaF 3 .

range is determined by Cv . It follows from the kinetic equation of the thermal conductivity, considering phonons as an ideal gas of quasi particles ] l 5 1 / 3Cvql

(2)

where Cv is the heat capacity at constant volume, q is the phonon velocity, l is the average phonon mean free path. The alterations of the second two terms of Eq. (2) at the superionic phase (265–800 K) are negligible. The sound velocity alteration does not exceed 5%. The phonon mean free path is determined by the number of defects. It is supposed that the concentration of disordered ions in the superionic state is constant. A similar consideration is fair for compositions Li 2 B 4 O 7 , a -LiIO 3 . Concentrations of TLS calculated from (1) are in good agreement with concentrations of disordered ions in lithium sublattice. Thus, we

N.R. Abdulchalikova et al. / Solid State Ionics 107 (1998) 59 – 65

find a linear increase of the heat capacity and the thermal conductivity in crystalline SIC in superionic phase. The observed temperature dependencies can be attributed to relaxation interaction between high frequency phonons with TLS.

Acknowledgements This work was supported by the Committee of Science and Technology of Uzbekistan (Grant No. 6-95).

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