Exponential series representation for heat capacities of semiconductors and wide‐bandgap materials

Abstract

The Thirring series expansion is used as the starting point for the design of a structurally novel, dispersion‐related analytical model for descriptions of the temperature dependences of harmonic parts of isochoric heat capacities, CVh (T). The notorious problem of slow convergence of this conventional series expansion is satisfactorily resolved here by its transformation into an associated exponential series representation, which shows up markedly better convergence properties than the original Thirring expansion. Combining this unprecedented analytical CVh (T) model in a physically adequate way with the recently introduced power series expansion for contributions of lattice expansion and anharmonicity effects to the observable isobaric heat capacities, we get a novel effective tool for numerical simulations and analyses of corresponding Cp (T) data sets, which is suitable for applications to the practically important regions of intermediate to high temperatures. This model is usable, among other things, as a physically consistent substitution for various types of ad hoc chosen polynomials that are hitherto still frequently used, especially in the field of thermochemistry. In order to illustrate the considerable usefulness of the novel analytical tool, especially for the technologically important class of wide‐bandgap materials, we perform simultaneous least‐mean‐square fittings of combined sets of available high‐ and low‐temperature Cp (T) data for diamond, SiC‐3C, the III‐nitrides BN, AlN, and GaN, and the zinc chalcogenides ZnO, ZnS, and ZnSe. For a comparison with results obtained recently for Si and Ge on the basis of a three‐oscillator model, we have re‐examined the respective Cp (T) data sets. As interesting byproducts of present fittings we evaluate several even‐order moments of the respective phonon density of states spectral functions. The discussion of numerical results reveals, among other things, the basic physical cause of the obvious inappropriateness of the conventional Debye model with respect to the majority of the materials under study. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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

“…Within the frame of numerical analyses of experimental heat capacity data, however, one is continually concerned with a basic theoretical complication due to the inherent differences between the temperature dependencies of theoretical ℎ ( ) functions, on the one hand, and those of measured (isobaric) heat capacities, ( ), on the other hand. The latter are well known to be throughout somewhat higher, ( ) > ℎ ( ), than their theoretical (isochoric) counterparts pertaining to the harmonic lattice regime [4,50,54]. The respective differences, ( ) − ℎ ( ) > 0, are usually found to be very small from 0 up to temperatures of order ℎ , where the heat capacity amounts to about 50% of the Delong-Petit limiting value, ( ℎ ) ≅ ℎ ( ℎ ) = ℎ (∞)/2.…”

“…Within the frame of the harmonic lattice regime, the temperature dependencies of the isochoric heat capacities, ℎ ( ), are well known to be given by corresponding heat capacity shape functions, ℎ ( ), of the general form [4,6,45,[50][51][52][53][54][55] ℎ ( ) ℎ (∞)…”

“…On the other hand, one is usually concerned with a relatively strong monotonic increase of these differences, ( ( ) − ℎ ( ))/ > 0, at higher temperatures ( > ℎ ). These differences are generally ascribed to cumulative effects of lattice expansion and lattice anharmonicities [3,11,18,21,23,39,50,54,56,57]. We have repeatedly found within a larger series of heat capacity studies [14,15,21,50,54] that, within regions from ℎ up to temperatures of order 5 ℎ (at least), the rapid increase of these differences can be simulated in good approximation by a proportionality of type ( ) − ℎ ( ) ∝ × ( ℎ ( )) 2 ∝ × ( ℎ ( )) 2 (note that the structure of the latter is analogous to the known Nernst-Lindemann formula [2,3,58]…”

New Kinetic Theory for Absorption and Emission Rates of Radiation and New Equations for Lattice and Electronic Heat Capacities, Enthalpies and Entropies of Solids: Application to Copper

“…In this model, the ratio for the respective oscillation frequencies was taken to be 1/2. However, this model could not well describe the heat capacities in the cryogenic region T < 100 K [5].…”

“…, where Θ D (∞) is the Debye temperature at T → ∞ K. Different models based on Thirring and exponential series expansions have also been given for the intermediate to high temperature regions, respectively [5]. However, these models are complex and sets of seven or eight empirical parameters should be determined.…”