Unprecedented Integral-Free Debye Temperature Formulas: Sample Applications to Heat Capacities of ZnSe and ZnTe

Abstract

Detailed analytical and numerical analyses are performed for combinations of several complementary sets of measured heat capacities, for ZnSe and ZnTe, from the liquid-helium region up to 600 K. The isochoric (harmonic) parts of heat capacities, ℎ ( ), are described within the frame of a properly devised four-oscillator hybrid model. Additional anharmonicity-related terms are included for comprehensive numerical fittings of the isobaric heat capacities, ( ). The contributions of Debye and nonDebye type due to the low-energy acoustical phonon sections are represented here for the first time by unprecedented, integralfree formulas. Indications for weak electronic contributions to the cryogenic heat capacities are found for both materials. A novel analytical framework has been constructed for high-accuracy evaluations of Debye function integrals via a couple of integralfree formulas, consisting of Debye's conventional low-temperature series expansion in combination with an unprecedented hightemperature series representation for reciprocal values of the Debye function. The zero-temperature limits of Debye temperatures have been detected from published low-temperature ( ) data sets to be significantly lower than previously estimated, namely, 270 (±3) K for ZnSe and 220 (±2) K for ZnTe. The high-temperature limits of the "true" (harmonic lattice) Debye temperatures are found to be 317 K for ZnSe and 262 K for ZnTe.

Efficient Debye Function Interpolation Formulae: Sample Applications to Diamond

“…In contrast to the latter, we find again (in accordance with [1,3,11,22]) that Debye's original high-temperature Taylor series expansion is of little practical use in view of its rather bad convergence properties, which are caused by the alteration of signs of the respective expansion coefficients, i. e. Significant improvements of convergence properties have already been shown in previous studies [11,22] to be realizable by means of suitable transformations [23] of the given Taylor series expansions into structurally different versions that are embedded into conveniently chosen alternative (non-linear) functions. In the appendix, we briefly present two previous sample applications of this method to the high-temperature behavior of the Debye function, which resulted in the construction of an exponential series representation [22], and a derivation of an alternative Taylor series representation for reciprocal Debye function values, (𝜅 𝐷ℎ (𝑥)) −1 [11].…”

“…However, since the middle of the past century, a wealth of experimental studies on the thermal properties of non-metals (semiconductors as well as isolators) has been performed. From the numerous results of these studies, one could conclude that, as a rule, it becomes possible to simulate the measured heat capacity data using the Debye function integrals only under the assumption that the Debye temperature depends (more or less strongly) on the lattice temperature, T. The corresponding 𝛩 𝐷 (𝑇) dependencies have been found to be very pronounced for typical semiconductor materials, in particular, for Si and Ge [2,3] as well as for numerous III-V materials [4][5][6] and II-VI materials [7][8][9][10][11].…”

“…In Section 2, we give a brief overview of the derivation of the conventional low-and hightemperature approximations [11] for the Debye function integral, which are well known already from Debye's original paper [1]. We confirm that Debye's low-temperature formula [1,11,20,21] is in principle capable of providing quite adequate numerical values for the Debye function integral even at lattice temperatures comparable with the Debye temperature, provided the respective summation procedure is not truncated at a very low order of summation terms.…”

“…In Section 2, we give a brief overview of the derivation of the conventional low-and hightemperature approximations [11] for the Debye function integral, which are well known already from Debye's original paper [1]. We confirm that Debye's low-temperature formula [1,11,20,21] is in principle capable of providing quite adequate numerical values for the Debye function integral even at lattice temperatures comparable with the Debye temperature, provided the respective summation procedure is not truncated at a very low order of summation terms. In contrast to the latter, we find again (in accordance with [1,3,11,22]) that Debye's original high-temperature Taylor series expansion is of little practical use in view of its rather bad convergence properties, which are caused by the alteration of signs of the respective expansion coefficients, i. e. Significant improvements of convergence properties have already been shown in previous studies [11,22] to be realizable by means of suitable transformations [23] of the given Taylor series expansions into structurally different versions that are embedded into conveniently chosen alternative (non-linear) functions.…”

Analytic approximation of the Debye function

“…x > 0, which cannot be expressed in elementary functions. Despite of that this integral can be written as analytic expression with infinite series [1,3,41,42] or special functions (polylogarithms and the Riemann zeta function) [43], closed-form expressions approximating the Debye function are interesting for practical use in thermodynamic calculations. Many works are devoted to elaboration of simple approximations of the Debye functions with different ac-curacy [14,[44][45][46][47][48][49][50].…”