ISSN 1063-7834, Physics of the Solid State, 2006, Vol. 48, No. 4, pp. 663–669. © Pleiades Publishing, Inc., 2006. Original Russian Text © N.A. Abdullaev, 2006, published in Fizika Tverdogo Tela, 2006, Vol. 48, No. 4, pp. 623–629.
SEMICONDUCTORS AND DIELECTRICS
Elastic Properties of Layered Crystals N. A. Abdullaev Institute of Physics, Academy of Sciences of Azerbaijan, pr. Dzhavida 33, Baku, AZ-1143 Azerbaijan e-mail:
[email protected] Received April 26, 2005
Abstract—The characteristic features of the elastic properties of layered crystals and their dependence on temperature and pressure are analyzed. The relations between the elastic constants of hexagonal layered crystals are given. It is shown that the anomalous behavior of the elastic constants in the temperature region of a phase transition affects both the magnitude and sign of the thermal expansion coefficients of layered crystals. From analyzing the pressure and temperature dependences of the elastic constants, it is found that the anharmonicity of the bonding forces between the layers is much greater than the anharmonicity of the intralayer forces. The contribution from thermal expansion to the variations of the elastic constants with temperature is estimated. PACS numbers: 62.20.Dc, 64.60.–i DOI: 10.1134/S1063783406040081
1. INTRODUCTION Studying the features of the physical processes in layered crystals is of interest, since such crystals can display effects characteristic of low-dimensional (twodimensional) systems. Layered crystals are characterized by the presence of two types of bonds. The bonding between the atoms inside a layer is strong (predominantly covalent with some admixture of ionic bonding), and the bonding between the layers is weak (of the van der Waals type). The most typical representatives of layered crystals are graphite single crystals and boron nitride single crystals isostructural to graphite. These crystals are the simplest in structure, and the layers are monatomic planes consisting of two-dimensional sequences of regular hexagons, whose vertices are occupied by atoms. The distance between the atoms in the layer plane is much shorter than the interlayer distance. This feature is typical of all layered crystals and apparently causes the anisotropy of the bonding forces in these crystals. In graphite, for example, the atoms inside a layer are at a distance of 1.421 Å and the distance between the layers is 3.35 Å. In boron nitride, these distances are1.446 and 3.33 Å, respectively. In other layered crystals, the structure of the layer itself is more complicated and consists of several monatomic planes. In MoS2 and PbI2, for example, a layer contains three monatomic planes (anions and cations form S–Mo–S sequences in MoS2 and I–Pb–I sequences in PbI2). In GaS and InSe, a layer contains four monatomic planes (anions and cations form S–Ga–Ga–S sequences in GaS and Se–In–In–Se sequences in InSe). In Bi2Te3 single crystals, the layers (called quintets) contain five monatomic planes, which form a Te–Bi–Te–Bi–Te sequence. There exist layered crystals with more complicated layer structures, for
example, TlGaSe2 and TlInS2, whose layers have a “seven-storied” structure. The anisotropy of bonding forces results in specific features of the phonon spectra of the layered crystals, such as the existence of low-frequency modes in which the layers move relative to each other as rigid units, low velocities of the acoustic modes propagating in the direction of weak bonding, and a quadratic dispersion law characteristic of the vibrations propagating in the layer plane with a displacement vector normal to the layers (the so-called TA⊥ mode). The above features result in specific lattice dynamics of layered crystals, which is manifested in the physical properties controlled by the phonon subsystem, such as, e.g., the heat capacity [1], thermal expansion [2], and heat conductivity [3]. 2. ELASTIC CONSTANTS OF LAYERED CRYSTALS AND THE RELATIONS BETWEEN THEM It is known that, in elasticity theory, a low deformation of a solid is described by the strain tensor 1 ∂U ∂U U ik = --- ⎛ ---------i + ---------k⎞ , 2 ⎝ ∂x k ∂x i ⎠
(1)
where Ui is a component of the displacement vector. The internal stresses caused by a deformation are described by the stress tensor σik. A component of the force acting on a unit volume is determined by
663
∂σ F i = ---------ik- . ∂x k
(2)
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Under the conditions of applicability of Hooke’s law, we have σ ik = C iklm U lm ,
(3)
where Ciklm are elastic moduli (components of the fourth-order elasticity tensor). Due to the symmetry requirements imposed on the stress and strain tensors, the number of independent constants decreases to 21. Furthermore, the symmetry of a specific crystal lattice makes it possible to reduce the number of independent elastic constants even more substantially. For example, numerous hexagonal layered crystals (graphite, boron nitride, GaS, GaSe, InSe, etc.) can be described by five elastic moduli, C xxxx = C yyyy = C 11 ,
C zzzz = C 33 ,
C xxzz = C yyzz = C 13 ,
C xxyy = C 12 ,
layer planes. Therefore, the Cauchy relations cannot be applied to layered crystals and we should use other relations between the elastic constants that follow from the stability criterion of the crystal lattice [4]. A lattice will be stable if the energy density is a positively defined quadratic form and, therefore, the energy increases with any small deformation. Let us consider the matrix of the coefficients of the quadratic form C 11 C 12 C 13 C 14 C 15 C 16 C 21 C 22 C 23 C 24 C 25 C 26 C 31 C 32 C 33 C 34 C 35 C 36
.
(5)
C 41 C 42 C 43 C 44 C 45 C 46 C 51 C 52 C 53 C 54 C 55 C 56 C 61 C 62 C 63 C 64 C 65 C 66
C xzxz = C yzyz = C 44 . The elastic properties of a hexagonal crystal in the symmetry plane (layer plane) are isotropic and are described by the elastic moduli C11 and C12, which characterize the Young’s modulus and Poisson ratio in the symmetry plane. The constant C33 determines the Young’s modulus in the perpendicular direction, and C13 is the corresponding Poisson ratio. The constant C44 describes stresses caused by displacements of the layers with respect to each other. The elastic constants are commonly determined by measuring the velocity of elastic waves in the crystals. For the three acoustic normal modes, the propagation velocity can be obtained by solving the Green–Christoffel equation C ijkl n j n l – δ ik ρv
2
= 0,
According to the well-known theorem of algebra, this quadratic form will be positively defined if the principal minors (the determinants of all matrices of successively higher ranks) are positive. In the case of hexagonal crystals, matrix (5) assumes the form C 11 C 12 C 13 0
0
0
C 12 C 11 C 13 0
0
0
C 13 C 13 C 33 0
0
0 0
0
0
0 C 44 0
0
0
0
0 C 44 0
0
0
0
0
(6)
,
0 C 66
(4)
and its principal minors are equal to C66, C44C66,
where ρ is the density and nj and nl are the directional cosines for the propagation direction. By choosing a suitable orientation of the propagation of waves in the crystal, we can calculate the elastic constants C11, C12, C33, C13, and C44. In ordinary crystals, the elastic constants obey the Cauchy relations [4]. These relations follow from the conditions of equilibrium of the crystal lattice, according to which the energy density should be minimum. However, the Cauchy relations are valid in the case where only central forces are involved in the interaction between atoms. In the layered crystals, the intralayer forces are substantially larger than the interlayer forces; therefore, it is quite probable that the interaction between more distant neighbors in the plane of a layer can be of the same order of magnitude as the nearest neighbor interaction associated with weak interlayer bonding. Due to the smallness of the interlayer central forces, it may be important to take into account the noncentral forces of interaction between the atoms in the
C 44 C 66 , C 33 C 44 C 66 , (C11C33 – C 13 ) C 44 C 66 , and (C11 –
2
2
2
2
2
2
C12)[C33(C11 + C12) – 2 C 13 ] C 44 C 66 . Using the relation C66 = (C11 – C12)/2, we find that the principal minors will be positive if the following conditions are satisfied: C 44 > 0, ( C 11 – C 12 ) > 0, (7)
2
C 11 C 33 – C 13 > 0, 2
C 33 ( C 11 + C 12 ) – 2C 13 > 0. Another important relation between the elastic constants can be obtained using the following arguments. Let a stretching stress p be applied to a hexagonal layered crystal in the x direction (the x and y axes are taken to be in the layer plane, and the z axis is normal to the layer plane and parallel to the symmetry axis). In this PHYSICS OF THE SOLID STATE
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ELASTIC PROPERTIES OF LAYERED CRYSTALS
case, according to [5], the diagonal components of the strain tensor are
11
U yy
12
33
13
C 33 /2 1 = p – ------------------------------ + ---------------------------------------------------- , 2 ( C 11 – C 12 ) ( C + C )C – 2C 2 (8) 11
12
33
13
C 13 U zz = p – ---------------------------------------------------- . 2 ( C 11 + C 12 )C 33 – 2C 13 From the law of energy conservation, it follows that the stretching stress applied to the crystal can only increase its volume. Therefore, we have ∆V ------- = U xx + U yy + U zz > 0. V
C 33 > C 13 .
(10)
Table 1 gives the room-temperature elastic moduli of some layered crystals determined from ultrasonic experiments. We see that graphite (for which C11/C33 ~ 30) has the strongest anisotropy of the elastic properties. Typically, C11/C33 ~ 2–4 in layered crystals. The anisotropy varies mainly due to the elastic constant C11, since the quantity C33 varies from crystal to crystal only slightly. This result is understandable if we consider the crystal structure. Indeed, in various layered crystals, the interlayer distances are approximately equal. For example, the interlayer distance is 3.35 Å in graphite, 3.33 Å in boron nitride, 3.81 Å in GaS, 3.84 Å in GaSe, and 4.19 Å in InSe. However, the interatomic distances inside a layer are substantially different. For example, these distances are 1.421 and 1.446 Å in graphite and in boron nitride, respectively, whereas in GaS, GaSe, and InSe these distances are 2.32 (S–Ga), 2.48 (Se–Ga), and 2.53 Å (Se–In), respectively. In Table 1, we see that the anisotropy of the elastic constants describing, according to Eq. (3), the linear bonding forces is weak in layered crystals, such as TlGaSe2 or TlInS2 (C11/C33 ~ 1.47 and 1.13 in TlGaSe2 and TlInS2, respectively), whereas the anisotropy of mechanical hardness is much stronger. Therefore, a substantial anisotropy of the mechanical hardness does not lead to an equally strong anisotropy of the linear bonding forces. In [6], the nonlinear elastic coefficients of layered KY(MoO4)2 single crystals were measured using the acoustooptical method and it was shown that the strong anisotropy of the mechanical hardness of layered crystals is a consequence of the unusually high anharmonicity of the bonding forces between layers. In Table 1, the value of the elastic modulus C44 in graphite is not determined uniquely. Indeed, the magnitude of C44 in graphite can vary in rather wide limits Vol. 48
Elastic moduli, 1010 Pa C11
C12
C graphite 106 18 GaS 15.7 3.3 GaSe 10.3 2.9 InSe 7.3 2.7 12 4.2 TiSe2 TaSe2 22.9 10.7 NbSe2 19.4 9.1 TlInS2 4.49 3.05 TlGaSe2 6.42 3.88
C13
C33
C44
1.5 1.5 1.2 3.0 – – – – –
3.7 3.6 3.4 3.6 3.9 5.4 4.2 3.99 4.37
0.018–0.035 0.8 0.9 1.2 1.4 1.9 1.8 0.3 0.5
(9)
Since C11, C12 > C13, it follows from Eqs. (8) and (9) that
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Table 1. Elastic moduli of layered crystals Crystals
C 33 /2 1 U xx = p ------------------------------ + ---------------------------------------------------- , 2 ( C 11 – C 12 ) ( C + C )C – 2C 2
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depending on the quality of the sample, which, in turn, depends on the presence of defects at the interfaces between the layers, different planar defects, etc. In [7], the effect of γ irradiation on the graphite elastic constants was studied. It was shown that the irradiation affects only the interlayer interaction (the elastic constant C44 increases considerably, and C33 slightly decreases), whereas C11, C12, and C13 remain virtually unchanged. Moreover, direct electron microscopic studies showed that the irradiation substantially decreases the number of basal dislocations and defects of layer matching. These data allowed the authors of [7] to assert that radiation annealing of dislocations and defects in graphite single crystals increases the value of C44. Thus, in more perfect samples, the value of C44 is greater. 3. ELASTIC MODULUS C13 Experimental determination of the elastic modulus C13 in crystals (in particular, using ultrasonic methods) meets with serious difficulties; therefore, its value is determined with large errors. For example, in graphite, C13 = (1.5–0.5) × 1010 Pa [7]; i.e., the quantity C13 is determined with an accuracy worse than 30%. Moreover, in layered crystals, technological difficulties exist, since it is necessary to prepare samples with oriented sides forming certain angles with the layers. Certain information on the elastic constant C13 can be obtained using the well-known relations from elasticity theory [5]. The resonance frequencies of bending vibrations of a thin rectangular plate rigidly fixed at one end are given by 2
k n d E eff f n = ------------------, 2 2πl 12ρ
(11)
where d is the thickness of the plate, l is its length, ρ is the density, and the values of kn are dependent on the excited harmonic (kn = 1.875 and 4.694 for n = 1 and 2,
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Table 2. Pressure dependence of the elastic moduli of layered crystals 1 ∂C ik ------- ---------, 10–11 Pa–1 C ik ∂P
Crystals
C GaS GaSe InSe
C11
C12
C13
C33
C44
4 8 8 11
6 14 16 16
21 – – –
26 63 56 50
8 – – –
respectively). For the chosen measurement geometry and quasi-hexagonal crystals, the effective Young’s modulus can be found to be Eeff = E/(1 – σ2) = C11 – 2
C 13 /C 33 , where E and σ are the Young’s modulus and Poisson ratio, respectively, describing deformation in the layer plane. It follows from Eq. (11) that the ine2 quality Eeff = C11 – C 13 /C 33 > 0 must be satisfied; this inequality is identical to that in (7). It should be noted that the elastic constant C13 is important; reliable knowledge of C13 is necessary, for example, for the interpretation of the thermal expansion data. According to [8], in the basal plane of a hexagonal crystal (in the layer plane), the linear thermal expansion coefficient (TEC) is C 33 C -γ α || = ---- ---------------------------------------------------V ( C + C )C – 2C 2 || 11 12 33 13
(12)
C 13 -γ , – ---------------------------------------------------2 ⊥ ( C 11 + C 12 )C 33 – 2C 13
where V is the volume, C is the heat capacity, Cik are the elastic moduli, and γ|| and γ⊥ are the average Grüneisen 2
parameters [9]. Since (C11 + C12)C33 – 2 C 13 > 0 (according to (7)), it follows from Eq. (12) that the linear TEC in the layer plane α|| can be negative in two cases: if the Grüneisen parameter γ|| is negative or if the elastic constant C13 is sufficiently large such that the second term in square brackets in Eq. (12) is greater than the first, i.e., if C33γ|| – C13γ⊥ < 0. Layered crystals, as a rule, exhibit a significant anisotropy of thermal expansion. Indeed, the linear TEC in the direction of weak bonding (normal to the layers), α⊥, assumes large positive values, whereas in the layer plane α|| is small and in many crystals is even negative over a wide temperature range. In the most strongly pronounced layered crystals (graphite and boron nitride), the width of the temperature range where the TEC α|| is negative is the greatest (0–600 K) [10, 11]. In order to explain the negative thermal expansion of graphite and boron nitride in the layer plane
under the assumption of a dominant contribution from the second term in Eq. (12), the authors of [10–12] had to use unrealistically high values of the elastic modulus C13 ~ 5 × 1010 Pa (according to [6, 13], C13 = (1.5–0.5) × 1010 Pa). This fact virtually implies that the strong expansion in the direction perpendicular to the layers is accompanied by a lateral compression (the so-called Poisson compression) that is larger than the expansion in the layer plane. In this case, however, C13 > C33 (since, according to the data from [13], C33 = 3.65 × 1010 Pa), which is in disagreement with condition (10). The authors of [2] have shown that, typically, the phonon spectra of layered crystals have so-called bending vibrations described by negative Grüneisen mode parameters γ||, i. At low temperatures, such vibrations provide the main contribution to the density of states and can result in negative weighted-average Grüneisen parameters γ|| [9]. In this case, the first term in square brackets in Eq. (12) is negative and it is no longer necessary to use unreasonably high values of the elastic modulus C13 for interpreting the experimental data on the thermal expansion. Using the relations between the elastic constants, we can reappraise critically some experimental studies of elastic constants. In a very interesting and useful article [14], the dependences of elastic constants (C11, C12, C33, C13, C44) on temperature (in the range 4 < T < 300 K) and hydrostatic pressure (P < 2 × 109 Pa) are given. Analyzing the data on the temperature dependence of the elastic constant C13 shows that C13 > C33 in the temperature range 150 < T < 250 K, in contradiction with condition (10). Furthermore, the shape of the C13(T) curve is also doubtful, since the increase in C13 (by a factor of almost 3) occurs entirely in a narrow temperature interval of 230 < T < 300 K (from C13 = 1.5 × 1010 Pa to C13 > 4 × 1010 Pa) and then, as the temperature decreases to liquid-helium temperatures, the quantity C13 even somewhat decreases. As noted above, this decrease can be due to the experimental difficulties involved in measuring C13 and to large measurement errors. 4. EFFECT OF TEMPERATURE AND PRESSURE ON THE ELASTIC CONSTANTS Knowledge of the temperature and pressure dependences of the elastic constants is important because these dependences contain information on the anharmonicity of interatomic interactions in crystals and are also necessary in calculating the thermodynamic characteristics of crystals, for example, the linear TEC [2]. Table 2 gives the relative variations in elastic constants in the region of the linear dependence of Cik(P) on hydrostatic pressure at a temperature of 300 K. From Table 2, we see that the pressure dependence is stronger for the interlayer elastic constants than for the intralayer elastic constants. Since the pressure dependence PHYSICS OF THE SOLID STATE
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667
Table 3. Relative changes in the elastic moduli of graphite with temperature Elastic modulus
Cik (5 K), 1010 Pa
Cik (300 K), 1010 Pa
∆C ik ----------- Tot. C ik
∆C ik ----------- TE C ik
∆C ik ----------- PPI C ik
η, %
C11 C33
112.6 4.07
106 3.65
0.062 0.115
0.009 0.066
0.053 0.049
15 57
of the elastic constants is entirely due to anharmonicity, it follows that the anharmonicity of bonding forces between the layers is much greater than that of intralayer forces. This statement agrees with the experimental data on the thermal expansion of layered crystals [2] (the linear TEC in the direction perpendicular to the layers is substantially greater than that in the layer plane) and with the conclusions drawn in [6]. In [14], the temperature dependence of the elastic constants of graphite was studied in the temperature range 4.2–300 K at atmospheric pressure. Typically, all elastic constants almost linearly increase with decreasing temperature and approach the temperature T = 4.2 K with an almost zero slope. This behavior agrees with the theory based on the quasi-harmonic approximation [16]. According to [16], the temperature dependence of the elastic constants is determined by the relation 0
C ij = C ij ( 1 – D ij ε ),
(13)
0
where C ij is the elastic constant for the static lattice, ε is the average oscillator energy, and Dij are constants (which are different for different crystals). For kT hω0 (where ω0 is the average Einstein frequency of atomic vibrations), we have ε = kT and, according to 0
Eq. (13), Cij ~ C ij (1 – Dij kT). For kT hω0, we have ε = hω0 and it follows from Eq. (13) that the elastic constants approach the values at T = 0 K with a zero slope. Analyzing the temperature dependence of the elastic constants of graphite [14] also shows a tendency of the interlayer elastic constants to change more rapidly than the intralayer elastic constants. For example, as the temperature decreases from 300 to 4.2 K, the quantity C11 increases by 6% and C33 by 12%. A faster growth of the interlayer elastic constants of layered crystals with decreasing temperature or with increasing pressure is not surprising. For example, in crystals of inert gases and in molecular crystals with van der Waals bonding between atoms, the elastic constants change much faster with growing pressure and temperature than in most ordinary crystals [17]. The elastic constants vary with temperature, as well as with pressure, due entirely to the anharmonicity, PHYSICS OF THE SOLID STATE
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which is caused by phonon–phonon interaction and lattice deformation due to thermal expansion: ∂C dC ik β ∂C = ⎛ ---------ik-⎞ + --- ⎛ ---------ik-⎞ . ---------⎝ ⎠ ∂T V χ ⎝ ∂P ⎠ T dT
(14)
Here, P, V, and T are the pressure, volume, and temperature, respectively; β is the volume TEC; and χ is the volume compressibility. In Eq. (14), the first term is the contribution of the phonon–phonon interaction to the total change in the elastic constant with temperature and the second term is the contribution of the deformation related to thermal expansion. From Eq. (14), we can estimate these contributions separately by taking experimental values of dC ik /dT and ∂C ik /∂P and using the equalities β = β ⊥ + 2β || ,
χ = χ ⊥ + 2χ || ,
C 11 + C 12 – 2C 13 -, χ ⊥ = ---------------------------------------------------2 ( C 11 + C 12 )C 33 – 2C 13
(15)
C 33 – C 13 -. χ || = ---------------------------------------------------2 ( C 11 + C 12 )C 33 – 2C 13 From Table 3, it is seen that the relative change of the elastic constant C33 (describing the interlayer interaction) with temperature is almost two times greater than the relative change of the intralayer constant C11 (0.115 and 0.062, respectively). In other words, just as in the case of the pressure dependence of the elastic constants, the anharmonicity of the bonding forces between the layers is much greater than that of the intralayer forces. Furthermore, the contribution from the thermal expansion to this change is much greater for the interlayer elastic constant C33 than for the intralayer constant (57% for C33 and 15% for C11). Earlier, we made a similar conclusion when studying the temperature dependence of the vibrational frequencies in a layered GaS crystal [18]. We studied the temperature dependences of the frequencies of two Raman-active modes in layered GaS. One mode corresponded to an interlayer vibration, and the other to an intralayer vibration. It was shown that the relative change in the frequency of the interlayer mode with temperature is much greater than the change in the frequency of the intralayer mode (by a factor of 4). The contribution of thermal expansion to this change is much greater for the interlayer mode than that for the intralayer mode (75 and 40%, respectively). Analyzing the available data on
ABDULLAEV
668 6.48 6.44
∆C11/C11 ~ 5%
C11(T) TlGaSe2
6.40 6.36 6.32 ∆C /C ~ 4.5% 33 33 ~ ~ 4.48 4.44 4.40 4.36 4.32 4.28 50
100
Cik, 1010 Pa
Cik, 1010 Pa
4.82 4.80
C33(T)
150
200
250
300
4.78 4.76 4.74 4.72~ 4.00~ 3.90 3.80 3.70 3.60 3.50
∆C11/C11 ~ 1.5%
TlInS2
C11(T)
C33(T)
∆C33/C33 ~ 9%
100
150
T, K
200 T, K
250
300
Fig. 1. Temperature dependences of the elastic moduli C11(T) and C33(T) of a TlGaSe2 single crystal.
Fig. 2. Temperature dependences of the elastic moduli C11(T) and C33(T) of a TlInS2 single crystal.
the temperature dependence of the optical phonon frequencies in other layered crystals (GaSe) and in molecular crystals (As4S4, C10H8 [19]) makes it possible to draw the following general conclusion: the contribution of thermal expansion to the total change in the frequency of interlayer or intermolecular vibrations is 60– 80%, whereas the relative change in the frequency of intralayer or intramolecular vibrations due to thermal expansion is only 20 to 40%. Since the temperature dependences of the vibrational frequencies in layered and molecular crystals are similar, we may conclude that the bonding forces in layered and molecular crystals have a similar nature.
TlGaSe2 crystals, the relative changes in the interlayer and intralayer elastic constants are almost equal (∆C11/C11 ~ 5%, ∆C33/C33 ~ 4.5%). In TlInS2 crystals, the interlayer elastic constant C33 varies much more strongly at phase transitions than does the intralayer elastic constant C11 (∆C11/C11 ~ 1.5%, ∆C33/C33 ~ 9%; Fig. 2). This behavior of the elastic constants near the phase transition in TlInS2 agrees with our measurements of the temperature dependence of the linear TEC [23]. In the region of the phase transition, the linear TEC that describes the expansion of the crystal in the direction of the weak bonds in TlInS2 (normally to the layers) undergoes a sharp jump, whereas no appreciable jumps of the linear TEC in the layer plane were observed. 3.0 TlGaSe2
2.5 C13, 1010 Pa
5. THE BEHAVIOR OF ELASTIC CONSTANTS OF LAYERED CRYSTALS AT PHASE TRANSITIONS The presence of weak interlayer bonds in layered crystals increases the probability of structural phase transitions (PTs). Phase transitions have been observed with varying temperature, for example, in layered crystals of TlGaSe2, TlInS2 [20], and CsDy(MoO4)2 [20]. As a rule, at structural PTs, drastic softening of the elastic constants of the crystal occurs. A number of coefficients of the thermodynamic potential, including the magnitudes of the jumps in the elastic constants, were calculated in [21] for K2SeO4 crystals in the region of the phase transition using the phenomenological theory. Careful measurements of the velocity and absorption of longitudinal and transverse ultrasonic waves propagating along the layers or normally to them were performed in [22] in the temperatures range 80–300 K. The incommensurate phase existing in the temperature range from 214 to 200 K in TlInS2 and from 120 to 107 K in TlGaSe2 was observed as a sharp softening of the elastic constants (Figs. 1, 2). As seen from Fig. 1, in the temperature region of the phase transition in
2.0 TlInS2
1.5
1.0 50
100
150
200
250
300
T, K Fig. 3. Temperature dependence of the elastic modulus C13(T) of TlGaSe2 and TlInS2 single crystals. PHYSICS OF THE SOLID STATE
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Using the data from [22] and Eq. (11), we can plot the temperature dependences of the elastic constant C13 in TlGaSe2 and TlInS2 crystals, as shown in Fig. 3. It is seen in Fig. 3 that, in contrast to TlInS2, the elastic constant C13 in TlGaSe2 crystals sharply increases in the temperature region of the phase transition, ∆C13/C13 ~ 80%. This variation allows us to explain one more difference in the temperature dependence of the linear TEC between TlGaSe2 and the isostructural TlInS2 compound. In TlGaSe2 single crystals, as the temperature decreases, a sharp increase in the linear TEC in the direction perpendicular to layers is accompanied by a sharp decrease in the linear TEC in the layer plane [24]. We believe that this behavior is related to a significant increase in the elastic constant C13 near the phase transition and to a decisive role being played by the second term in expression (12). 6. CONCLUSIONS Thus, knowledge of the elastic constants and their temperature and pressure dependences makes it possible not only to obtain information on the anisotropy and anharmonicity of the bonding forces in a layered crystal but also to understand the character of the physical processes induced by the specific behavior of the phonon subsystem. The information on the main parameters of the dynamics of the crystal lattice (the elastic constants) and on their dependences on temperature and pressure allows one not only to describe but also to calculate the physical characteristics, such as the heat capacity, thermal expansion, and heat conductivity. ACKNOWLEDGMENTS The author is grateful to R.A. Suleœmanov for numerous discussions of the work. REFERENCES 1. É. E. Anders, B. Ya. Sukharevskiœ, and L. S. Shestachenko, Fiz. Nizk. Temp. (Kiev) 5 (7), 783 (1979). 2. G. L. Belen’kiœ, R. A. Suleœmanov, N. A. Abdullaev, and V. Ya. Shteœnshraœber, Fiz. Tverd. Tela (Leningrad) 26 (12), 3560 (1984) [Sov. Phys. Solid State 26 (12), 2142 (1984)]. 3. N. A. Abdullaev, R. A. Suleœmanov, M. A. Aldzhanov, and L. N. Alieva, Fiz. Tverd. Tela (St. Petersburg) 44 (10), 1775 (2002) [Phys. Solid State 44 (10), 1859 (2002)]. 4. M. Born and Huang Kun, Dynamical Theory of Crystal Lattices (Clarendon, Oxford, 1954; Inostrannaya Literatura, Moscow, 1958).
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Translated by I. Zvyagin
