Advances in Condensed Matter Physics

Volume 2017, Article ID 9321439, 37 pages

https://doi.org/10.1155/2017/9321439

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

Institut für Physik, Technische Universität Chemnitz, 09107 Chemnitz, Germany

Correspondence should be addressed to R. Pässler; ed.ztinmehc-ut.kisyhp@relssap

Received 19 March 2017; Accepted 30 April 2017; Published 18 September 2017

Academic Editor: Oleg Derzhko

Copyright © 2017 R. Pässler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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, , 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 non-Debye type due to the low-energy acoustical phonon sections are represented here for the first time by unprecedented, integral-free 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 integral-free formulas, consisting of Debye’s conventional low-temperature series expansion in combination with an unprecedented high-temperature 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.

#### 1. Introduction

Since the invocation of the concept of apparently characteristic, material-specific temperature parameters, , within Debye’s classical paper [1] on specific heats of solids, one was concerned with a large variety of quotations of corresponding values (so-called “Debye temperatures”) within numerous thermophysical research papers, including various representative review articles [2–5] and books [6–10]. A brief inspection of the enormous data material published hitherto, however, showed readily that it is obviously not possible to find unique values for the individual materials in question, which should apply to duly broad temperature regions. In order to save, nevertheless, the apparently rather popular concept of Debye temperatures, it became thus already long ago a widely accepted custom to adopt the physically somewhat problematic notion of variable (-dependent) Debye temperatures, . Accordingly, Debye’s heat capacity model function [1] used to be employed in practice in the general form [3, 4, 6, 10–15]where the upper limits of integration are given by the ratios, , of adjustable Debye temperatures versus the respective lattice temperatures and represents the familiar Dulong-Petit limiting value for the isochoric lattice heat capacity in harmonic approximation (i.e., in particular , for binary materials).

In the course of numerical fittings of measured (isobaric) heat capacities, , on the basis of (1), it has continually been found that, in contrast with Debye’s original suggestion [1], proper simulations of such heat capacity curves can in fact be realized only by admitting rather strong -dependencies of the material-specific Debye temperatures, . This statement applies above all to the liquid-helium-hydrogen region, where the adjusted values are as a rule rapidly falling [3–5, 12–17] from their limiting levels, , to certain minimum values, , which used to be located in the vicinities of . Furthermore, towards higher temperatures, many curves show a more or less pronounced increase up to certain material-specific maxima, , the actual magnitudes of which are in many cases significantly higher than the respective levels. Such typical nonmonotonic behaviours of curves in the cryogenic region have been found in particular for Si and Ge [18–21] as well as for a large variety of III–V [11, 12, 15, 22–30], II–VI [14, 31–38], and I–VII [16, 39, 40] materials.

Globally one can thus assess that Debye’s original idea, according to which heat capacities of solids might be represented by Debye function integrals (of type (1)), whose upper limits of integration should be based on fixed, material-specific values, is largely illusionary. Actually, not a single binary or ternary material could be found hitherto, within a wealth of thermophysical researches, for which it would have been possible to simulate in adequate way the temperature dependence of measured heat capacities, from the liquid-helium region up to the vicinity of room temperature, in terms of a unique value. This means, among other things, that there is hardly a chance to come to appropriate quantitative descriptions of the temperature dependencies of heat capacities mainly via constructions of various, highly elaborate analytical models [41–44] for integrals of Debye’s type and imputing, at the same time, only some fixed (-independent) values for Debye temperatures (like = 632 K for GaN [43] or = 920 K for ZnO [44] and = 946 K for MgO [44]).

The basic cause of the occurrence of relatively deep minima, , in the cryogenic region had been indicated already many years ago by Schrödinger [2] to be obviously due to the onset of thermal activation of* short*-wavelength transverse acoustical (TA) phonons, which are quantum-theoretically manifested by pronounced peaks (singularities) in the respective material-specific phonon density of states (PDOS) spectra (see, e.g., the calculated spectra shown in [45, 46] for ZnSe and ZnTe). Such drastic deviations of physically realistic PDOS spectra from Debye’s naïve (quadratic) PDOS model function [1], , are continually confirmed to be indeed the main cause of the obviously typical, nonmonotonic (non-Debye) behaviours of curves.

Of course, when a representative set of properly determined values is actually available for one or the other material in question, one can readily calculate the respective heat capacities, either via direct numerical evaluations of the integral occurring in (1) or by interpolations between discrete values quoted in some of the more or less comprehensive or tables presented by various authors [6, 8, 47–49] (see also the list of presently obtained high-accuracy sample values given in Table 7).

Considerably more problematic and difficult, however, is a proper computational solution of the hitherto largely ignored* inverse* problem, namely, the reliable determination of effective Debye temperatures, , on the basis of measured heat capacities, . In view of the longstanding absence of corresponding integral-free formulas, it was the custom in thermophysical investigations over a long time to resort to some of the tables quoted above and to perform thus approximate estimations of the respective -ratios via interpolations between discrete -ratios quoted for the couples of the two neighbouring (or ) values. Such conventional interpolation procedures were still feasible as long as one used to be concerned only with relative small sets (limited to few dozens) of measured data points. However, in contemporary literature, one is repeatedly concerned with rather large sets (of hundreds) of data points (like the presently analyzed data sets given in [46] for ZnSe and ZnTe). With respect to the latter such clumsy point-by-point interpolation procedures appear to be largely obsolete. A practicable computational alternative, however, requires the availability of a duly elaborate analytical framework for the* inverse* relationship, which should hence enable performing straightforward evaluations of the actual -ratios directly on the basis of given heat capacity values, . It is the main purpose of the present study to develop here (within Appendix B) such a representative analytical framework of integral-free formulas and to present corresponding sample results for Debye temperatures of ZnSe and ZnTe.

Concerning the successive development of this novel analytical apparatus for Debye temperatures, we would still like to note that some partial results in form of asymptotic (approximate) Debye temperature expressions had already been published in two preceding papers, namely, in [13] for the low-temperature region (),and in [15] for the complementary range of higher temperatures (),

However, due to a lack of space, the somewhat unusual derivation procedure for this highly useful high-temperature formula (i.e., Eq. in [15]) could not be included into the former paper. A duly detailed display of the corresponding analytical derivation procedure is now made up, among other things, in Appendix B.

In Section 2 we briefly sketch the basic arguments and equations of the analytical framework for theoretical calculations of isobaric heat capacities. In Section 3 we display a reformulated,* integral-free* version of the multioscillator hybrid model [14, 21, 50]. This is based mainly on the usage of novel, unprecedented analytical formulas (derived in Appendix A) for the contributions of the continuous, long-wavelength sections of transverse acoustical (TA) phonons to the resulting lattice heat capacities. In Section 4 we adopt (in analogy to [14]) a special 4-peak Einstein oscillator constellation of the hybrid model, and we use the respective analytical heat capacity expression for careful fittings of compatible sets of low- and high-temperature data available for ZnSe and ZnTe (up to 600 K). In Section 5 we perform, on the basis of a couple of precision formulas derived in Appendix B, transformations of the experimental heat capacity data in question (including the corresponding theoretical and curves), into the respective Debye temperature representations. The results are discussed in Section 6.

#### 2. Basic Equations for Temperature Dependencies of Heat Capacities in Solids

Within the frame of the harmonic lattice regime, the temperature dependencies of the isochoric heat capacities, , are well known to be given by corresponding heat capacity shape functions, , of the general form [4, 6, 45, 50–55]where we have denoted by the energies of the individual phonons and represents the material-specific PDOS spectral function [3, 4, 19, 56]. The latter is supposed within the present context to have been normalized to unity [6, 45, 52, 53, 55]; that is, .

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 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*, , than their theoretical (isochoric) counterparts pertaining to the* harmonic* lattice regime [4, 50, 54]. The respective differences, , are usually found to be very small from 0 up to temperatures of order , where the heat capacity amounts to about 50% of the Delong-Petit limiting value, . On the other hand, one is usually concerned with a relatively strong monotonic increase of these differences, , at higher temperatures (). 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 up to temperatures of order (at least), the rapid increase of these differences can be simulated in good approximation by a proportionality of type (note that the structure of the latter is analogous to the known Nernst-Lindemann formula [2, 3, 58], , for the differences between isobaric and isochoric heat capacities). Admitting, furthermore, that at sufficiently high temperatures (up to melting points, if necessary) some additional higher-order power terms of may come into play, it appeared reasonable to simulate the -dependencies of isobaric heat capacities by a duly general algebraic expression of the form [14, 15, 21, 50, 54, 59]where the empirical Taylor series expansion coefficients, , are quantifying the material-specific weights of contributions of the individual, anharmonicity-related power terms [15, 21, 54, 57]. Furthermore we have still included into (5) an additional linear term, , which has to represent possible contributions of a degenerate electronic system [4, 6, 13, 23, 60–63] to the measured heat capacities. Such electronic contributions (“Sommerfeld terms” [4, 6, 60]) are known to be rather strong (tending to become even the dominant ones, at sufficiently low temperatures) in metals, due to large concentrations of conduction electrons. In contrast to this, such electronic contributions are as a rule rather weak in semiconductors and insulators due to the relatively low concentrations of free electrons associated with lattice imperfections (impurities). Accordingly, the actual strengths of such electronic contributions can vary considerably even between different samples of one and the same semiconductor material [4, 62]. We will find below (in Section 4) indications for some weak electronic contributions to the isobaric heat capacities measured at liquid-helium temperatures [46], for both ZnSe and ZnTe.

#### 3. Multioscillator Hybrid Model and Its Integral-Free Representation

Physically realistic, material-specific phonon density of states (PDOS) functions, (like those shown, e.g., in Figure 3 of [45] or in Figure 4 of [46], for ZnSe and ZnTe), which are determining the -dependencies of the respective harmonic heat capacity shape functions, (see (4)), are known to consist in general of numerous peaks (critical points) in combination with various continuous curve sections and/or gaps between them. Many fine details of such material-specific PDOS spectra, however, are practically not detectible via analyses of experimental heat capacity data [3, 64]. Their possible influence on thermal properties used to be strongly reduced by the thermal averaging process (in (4)), so that they are not discernable from the background of accidental experimental uncertainties (being usually of the order 1%). Consequently, it is as a rule possible to realize rather good fittings of available data on the basis of relatively simple theoretical model functions for PDOS spectra [64], provided that the latter are sufficiently general and flexible for reasonable reproductions of various* prominent* features, at least. This concerns above all the liquid-helium-hydrogen region [13, 64], where the lattice heat capacities are known to be represented in good approximation by Taylor series of* odd*-order terms [13, 16, 18, 23, 35, 39, 65]

This limiting behaviour is the automatic consequence of the well-known fact that the low-energy (TA phonon) tails of PDOS spectra for three-dimensional crystals are generally given by Taylor series [13, 16, 18, 23, 35, 39, 64, 65] of exclusively* even*-order power terms, , of the phonon energy. An incorporation of a corresponding* continuous* low-energy section into theoretical model functions is therefore indispensable for their actual applicability to the cryogenic region. At the same time it is also clear that such continuous low-energy tail sections are naturally ending (i.e., ) at the positions of the first critical points [14, 21, 50, 64], , which are usually corresponding to the first peaks of short-wavelength TA phonons (note that, for materials with zinc blende structure like ZnSe and ZnTe, the energy positions of the first critical points are corresponding to those of the L-points on the first Brillouin zone boundaries, i.e., [7, 32, 36, 45, 46, 66, 67]).

##### 3.1. General Analytical Framework of Multioscillator Hybrid Models

Less critical than the general ansatz for the region is the modelling of spectra for energies higher than the first critical points, . Numerous analyses have shown that excellent simulations of dependencies (4), from cryogenic to room temperatures, can be realized on the basis of combinations of several (macroscopic) Einstein oscillators [68], ( = 1 to ), the ordered energy positions of which, , including their actual weights, , are to be properly adjusted in the course of least-mean-square fitting processes for heat capacities. Thus one can take the normalized PDOS function in (4) to be generally representable by a multioscillator hybrid model function of the sufficiently general form [14, 21, 50]where we have denoted by ( for and 0 for ) the step function, which realizes the cut of the continuous part of the function just at the first singular point. By integrating (7) over the whole spectrum (from 0 beyond the highest Einstein oscillator), we see that the supposed normalization of the function to unity is equivalent to a normalization condition of the form [14, 21, 50]for the whole set of weighting factors. This condition is to be strictly fulfilled within any fitting process of experimental data on the basis of the model function (7).

Inserting, finally, the function (see (7)) into the integral (4) and denoting by ( = 1 to ) the characteristic Einstein temperatures corresponding to the energy positions of the individual macroscopic oscillators we obtain for the normalized (harmonic) heat capacity shape function the general expression [14, 21, 50]where the functions (being due to the two continuous components, , in (7)) are given in form of integrals of Debye and non-Debye type [14, 21]

Here we have denoted bythe ratio of the characteristic phonon temperature associated with the first (lowest) special point, , versus lattice temperature, (in analogy with the notation used within Debye’s theory, ; cf. (1) and Appendix A).

Informative byproducts of numerical fittings of measured data sets via (5), in combinations with (9) for the harmonic heat capacity shape function, (including (10), for the components), are the* moments*, , of material-specific PDOS spectral functions [3, 4, 16, 18, 19, 21, 50, 52, 56, 69–72], which are generally defined by

Inserting the hybrid model function (see (7)) into the integral for (see (12)), we obtain for these moments the model-specific expression [14, 21]

##### 3.2. Integral-Free Formulas for the Contributions of Long-Wave Acoustical (TA) Phonons

On the basis of the hitherto used original (i.e., integral) expressions (see (10)) for the two low-phonon-energy contributions, (in (9)), the corresponding least-mean-square analysis procedures performed in several preceding studies [14, 21, 50] turned out to be somewhat heavy and time-consuming. This disadvantage of the former hybrid model versions [14, 21, 50] was one of the main motivations for performing here (in Appendix A) a comprehensive analytical study aiming at the construction of more practicable, integral-free formulas for the integrals (10) of Debye and non-Debye type.

Consider first the integral (10) for the low-energy component of Debye type, (see Figure 1). A detailed analyses (in Section A.2 of Appendix A) has shown that the corresponding dependence (see (A.13)) can be represented in good approximation in terms of a proper combination (A.16) of truncated low- and high-temperature expansions, (see (A.14)) and (see (A.15)), assuming a switch between the two complementary branches just at their crossing point, = 6.5135 (cf. the upper inset of Figure 1). This means for the low-temperature branch, ,and for the high-temperature branch, , (the respective expansion coefficients of which, , are listed in Table 6).