Research Article  Open Access
R. Pässler, "Unprecedented IntegralFree Debye Temperature Formulas: Sample Applications to Heat Capacities of ZnSe and ZnTe", Advances in Condensed Matter Physics, vol. 2017, Article ID 9321439, 37 pages, 2017. https://doi.org/10.1155/2017/9321439
Unprecedented IntegralFree 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 liquidhelium region up to 600 K. The isochoric (harmonic) parts of heat capacities, , are described within the frame of a properly devised fouroscillator hybrid model. Additional anharmonicityrelated terms are included for comprehensive numerical fittings of the isobaric heat capacities, . The contributions of Debye and nonDebye type due to the lowenergy 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 highaccuracy evaluations of Debye function integrals via a couple of integralfree formulas, consisting of Debye’s conventional lowtemperature series expansion in combination with an unprecedented hightemperature series representation for reciprocal values of the Debye function. The zerotemperature limits of Debye temperatures have been detected from published lowtemperature data sets to be significantly lower than previously estimated, namely, 270 (±3) K for ZnSe and 220 (±2) K for ZnTe. The hightemperature 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, materialspecific 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 (socalled “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 DulongPetit 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 materialspecific Debye temperatures, . This statement applies above all to the liquidheliumhydrogen 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 materialspecific 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, materialspecific 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 liquidhelium 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 shortwavelength transverse acoustical (TA) phonons, which are quantumtheoretically manifested by pronounced peaks (singularities) in the respective materialspecific 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 (nonDebye) 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 highaccuracy 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 integralfree 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 pointbypoint 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 integralfree 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 lowtemperature 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 hightemperature 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, integralfree 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, longwavelength sections of transverse acoustical (TA) phonons to the resulting lattice heat capacities. In Section 4 we adopt (in analogy to [14]) a special 4peak 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 hightemperature 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 materialspecific 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 DelongPetit 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 NernstLindemann 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 higherorder 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 materialspecific weights of contributions of the individual, anharmonicityrelated 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 liquidhelium temperatures [46], for both ZnSe and ZnTe.
3. Multioscillator Hybrid Model and Its IntegralFree Representation
Physically realistic, materialspecific 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 materialspecific 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 liquidheliumhydrogen region [13, 64], where the lattice heat capacities are known to be represented in good approximation by Taylor series of oddorder terms [13, 16, 18, 23, 35, 39, 65]
This limiting behaviour is the automatic consequence of the wellknown fact that the lowenergy (TA phonon) tails of PDOS spectra for threedimensional crystals are generally given by Taylor series [13, 16, 18, 23, 35, 39, 64, 65] of exclusively evenorder power terms, , of the phonon energy. An incorporation of a corresponding continuous lowenergy 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 lowenergy 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 shortwavelength 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 Lpoints 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 leastmeansquare 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 nonDebye 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 materialspecific 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 modelspecific expression [14, 21]
3.2. IntegralFree Formulas for the Contributions of LongWave Acoustical (TA) Phonons
On the basis of the hitherto used original (i.e., integral) expressions (see (10)) for the two lowphononenergy contributions, (in (9)), the corresponding leastmeansquare analysis procedures performed in several preceding studies [14, 21, 50] turned out to be somewhat heavy and timeconsuming. 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, integralfree formulas for the integrals (10) of Debye and nonDebye type.
Consider first the integral (10) for the lowenergy 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 hightemperature 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 lowtemperature branch, ,and for the hightemperature branch, , (the respective expansion coefficients of which, , are listed in Table 6).
Concerning the integral (10) for the second lowenergy component, (see Figure 1), we have found within the frame of a detailed study (in Section A.2, of Appendix A) that the corresponding dependence of nonDebye type (see (A.17)) can be represented in good approximation in terms of the combination (A.24) of the truncated low and hightemperature expansions, (see (A.18)) and (see (A.23)), assuming again a switch between the two complementary branches at the respective crossing point, (cf. the lower inset of Figure 1). This means for the lowtemperature branch, , explicitlyand for the hightemperature branch, , (the respective expansion coefficients of which, , are listed in Table 6).
The large scale representations of the and curves in Figure 1 are suggesting apparently perfect coincidences between the exact (solid) curves and the respective approximate curve sections. At the same time we see, from the insets of Figure 1, that the approaches to the exact functions are really excellent only in the regions < 5 and > 9, whereas one is concerned in the vicinities of the crossing points, that is, at = 6.5135, for the Debyelike component, and at = 7.0962, for the nonDebye component, with certain overestimations of the exact values by the approximate formulas (up to a maximum order of 0.008%, for = 1, or of 0.015%, for = 2). Yet, such computational inaccuracies are by 1 to 2 orders of magnitude smaller than the typical uncertainties of measured heat capacities, . Thus, these computational inaccuracies are usually negligible within the frame of practical applications.
3.3. Connection with Conventional Cryogenic Heat Capacity Parameters
At sufficiently low (liquidhelium) temperatures, where , the first terms ( or ) in the respective lowtemperature expressions (see (14) and (16)) are strongly dominating. Thus the Debyelike and nonDebye components, ((see (10) for = 1 and 2, resp.), tend here to the known and asymptotes [14, 21]
The asymptote of the component is of Debye type [1], whereas the asymptote of the component is responsible for the onset of the typical nonDebye behaviour. The latter is frequently emerging already at very low temperatures (of order ), where the Einstein oscillator terms in (9) for the dependence are practically negligible (owing to their plateau behaviour, ; cf. also the curve in Figure 1). In this way we satisfy ourselves that the lowtemperature dependence of the harmonic lattice heat capacity, (see (4)), reduces, as expected, to a couple of two oddorder terms [13, 16, 18, 23, 35, 39, 65], (see (6)). The respective expansion coefficients, and , are readily seen to be given in terms of the weighting factors and , the characteristic phonon temperature associated with the first Einstein peak, , and the classical DulongPetit heat capacity value, , by the relations
Comparing, finally, the analytical expression for the limiting (cubic) asymptote implied by the present formalism, (due to (19)), with its conventional predecessor, (due to Debye’s theory [1–3, 8, 13]), we see that these two expressions are coincident when the weighting factor for the Debyelike component, , is just given by the cube [14, 21] of the ratio between the characteristic phonon temperature associated with the first Einstein peak, , and the limit value, of the conventional Debye temperature; that is,
4. Sample Analyses of Heat Capacity Data Sets for ZnS and ZnTe
For the sake of numerical analyses of given experimental heat capacity data within the frame of a multioscillator hybrid model like that displayed above, it is still necessary to specify the materialspecific constellation (in particular the total number, ) of Einstein oscillators to be taken into consideration. A relatively simple version in form of a threeoscillatorbased hybrid model [21] had been used in [50] for a large variety of III–V and II–VI materials with partly different crystal structures. Within the present (more detailed) study, for ZnSe and ZnTe, we shall consider a structurespecific constellation of four discrete oscillators (in analogy to [14]), which offers a physically adequate association with the contributions of the 6 phonon branches in zinc blende structure materials (consisting of 2 TA branches and 1 LA branch, for the lower parts, and 2 TO branches and 1 LO branch, for the upper parts, of the respective PDOS spectra [7, 32, 36, 45, 46, 66, 67]). Let us briefly sketch the basic arguments in favour of the corresponding fouroscillator model [14].
4.1. FourOscillator Constellation Chosen for ZnSe and ZnTe
In the course of numerous analyses of experimental data sets by different versions of the multioscillator hybrid model [14, 21, 50] we have repeatedly made the experience that a necessary condition for the contributions of two oscillator peaks to be actually discernible, in the course of leastmeansquare fitting processes, consists in a sufficiently large distance (of more than 10%, at least) between their respective ( versus ) energy positions. Considering under this aspect the known positions of the dominating optical ( versus ) phonon peaks [7, 32, 36, 45, 46, 66, 67] in ZnSe and ZnTe we see that their relative distances are markedly smaller than 10%, so that their individual contributions to the measured dependencies are practically indiscernible. Consequently, the contributions of the three optical phonon branches can be lumped together, being thus representable within the fouroscillator hybrid model [14] by a single oscillator with the highest energy, , and a fixed weight of
Qualitatively quite different is the state of affairs for the three acoustical phonon branches. From the theoretically calculated PDOS spectra [7, 32, 36, 45, 46, 66, 67] one can see that the distances of the dominating shortwave LA peaks from the prominent TA peaks (as well as from LO/TO peaks) are larger than 30%. The contributions of the dominating LA peaks, (= centres of gravity of the LA branches), are thus well discernible from the contributions of TA phonons as well as from those of the LO/TO phonons. This corresponds in reasonable approximation to a fixed weight offor the oscillator. Finally, with respect to the 2 TA phonon branches, we observe that the energy differences, for example, between the special points and , are larger than 20% [7, 32, 36, 45, 46, 66, 67], so that their individual contributions have good chances to be still discernible within the frame of duly careful numerical analysis procedures. This observation suggests representing the whole variety of shortwave TA phonons by two discrete model oscillators, ( = 1 and 2). Concerning their actual weights, and , we observe that, owing to the preceding fixations of the weights for the LA and LO/TO oscillators (i.e., (21) and (22)), the global normalization condition (8) for the weights of all contributions associated with the two TA phonon branches (i.e., and , in combination with and ) reduces, consequently, to a normalization condition of the TA phononrelated weighting factors of the special form [14]
Accordingly, only 3 of these 4 weighting factors are actually playing the role of independent fitting parameters; that is, the magnitude of one of them is automatically fixed by the set of the three other ones (e.g., ). In summary, within this special 4oscillator hybrid model [14], which turns out to be obviously well suited for binary materials with zinc blende structure, the normalized heat capacity shape function, (9), reduces to a parameterized function of just 7 independent parameters (consisting of the 4 discrete phonon temperatures , = 1 to 4, in combination with the three weighting factors , , and ).
Furthermore we would still like to observe that, in many cases, one knows at least approximate values for the 0 limiting values of Debye temperatures, (either as results of former analyses of cryogenic heat capacity data [13, 34] or from independent estimations on the basis of elastic constants [35, 73–77]). This preliminary knowledge allows us, if necessary, to substitute the weight, , of the Debyelike component, , by (according to (20)).
4.2. Additional Empirical Parameters Involved by the Isobaric Heat Capacities
It has already been pointed out above (in Section 2) that, for numerical simulations of measured isobaric heat capacity data, , by means of (5), it is generally necessary to envisage the inclusion of further empirical parameters (like , , and/or ) into the fitting process. Such additional parameters have been introduced above for a quantification of the deviations, , of measured values from the theoretical (harmonic) lattice heat capacities, (see (4)), due to lattice anharmonicities and possible electronic contributions. The actual inclusion (or exclusion) of one or the other additional parameter depends, of course, on the concrete constellation of the whole set of experimental data in consideration.
The crucial experimental basis of the present heat capacity analyses, for ZnSe and ZnTe, was provided by Kremer et al. [46] in form of unusually comprehensive and informative sets of data points comprising the regions from 2 K up to 277 K, for natural ZnSe, and up to about 150 K, for natural ZnTe (see the sets of and data points [46] represented by empty circles in Figures 2 and 3, resp., and cf. Figure in [46]). Yet, particularly from the inset of Figure 2 we see that the sequence of the respective data points () for ZnSe are indicative of some nonmonotonic (local minimum) behaviour in the liquidhelium region. This is at variance in particular to earlier data (+) given for the interval 1.7 K to 24.8 K by Birch (see Table of [34]), which showed an exclusively monotonic behaviour of the sequence of the respective data points (+) in the same region.
The more or less pronounced nonmonotonic behaviours of Kremer’s data points for natural ZnSe (Figure 2) and ZnTe (Figure 3), in the vicinities of 3.7 K or 2.6 K, respectively, can be ascribed to relatively weak (nevertheless not completely negligible) contributions of degenerate electronic systems to the measured dependencies (in analogy to similar observations reported for semiconductor materials like Si [61], Ge [62], several III–V materials [23], and FeGa_{3} [63]). Such electronic contributions are analytically represented within the present context by the respective linear (electronic) term, , in (5).
The subsequent leastmeansquare fitting processes are thus to be performed on the basis of (5) in combination with (9), (14) to (17), and (20). For present purposes we shall limit the sets of hightemperature data, which are to be actually included into the fitting processes, to temperatures up to 600 K (for both materials under study). Owing to these limitations we found that couples of just the first two anharmonicityrelated expansion coefficients, and (in (5)), are sufficient for good simulations of the respective hightemperature curve sections. The present leastmeansquare fitting processes are thus involving simultaneous adjustments of the (more or less strongly correlated) magnitudes of altogether 10 modelspecific parameters.
4.3. Combined Fittings of Low and HighTemperature Data and Results
Concerning the case of ZnSe (Figure 2) we have performed a simultaneous fit of the comprehensive set of data () given by Kremer et al. [46] in combination with a series of directly measured values (△) given for the region 400 K 600 K by Pashinkin and Malkova (cf. Table in [78]). The resulting total set of adjusted parameters is listed in Table 1. For comparisons of the presently analyzed data with some former ones given by various authors we have still shown in Figure 2 the fine K data points due to Birch [34] (+), the more or less strongly deviating data points due to Adachi et al. [10, 33] (◊) and Sirota et al. [79] (□), two data points given (for = 300 and 600 K) by Gadzhiev et al. [80] (●), and several smoothed values estimated by Pashinkin and Malkova [78] ().

For the case of ZnTe (Figure 3), the present fit comprises the set of data () given by Kremer et al. [46] in combination with the upper section (102 K < < 327 K) of the set of data points (□) given by Gavrichev et al. (in Table of [38]), including a set of approximate data points () due to Gadzhiev et al. (which we have redigitalized from Figure 1 shown in [80]), and a series of equidistant (smoothed) data points () available from the SGTE data review [81]. The resulting total set of adjusted parameters is listed in Table 1. For comparisons of the presently analyzed data with some former ones given by various authors we have still shown in Figure 3 the data points due to Adachi et al. [10, 33] (◊), Demidenko and Maltsev [31] (+), and the compatible sections (up to 600 K) of a series of smoothed values (□) quoted by Gavrichev et al. [82].
The fitted (continuous) and dependencies, which are implied by the materialspecific sets of fitted parameter values (listed for ZnSe and ZnTe in the upper part of Table 1), are represented by solid and dashed curves, respectively, in Figures 2 and 3. In the insets of these figures we have represented the respective magnitudes of ratios. The series of empty circles [46] as well as the corresponding solid curves, , show pronounced nonDebye behaviours (maxima) in the vicinities of = 16.9 K (for ZnSe) and = 13.6 K (for ZnTe). The levels indicate the estimated limiting values, , of the curves, which are due exclusively to the lattice contributions to the measured heat capacities; that is, (within the liquidheliumhydrogen region).
In the lower part of Table 1 we have still quoted a series of characteristic, materialspecific quantities, the actual values of which are resulting in unambiguous way from the fitted parameter sets. This concerns in particular the associated magnitudes (due to (19)) of the coefficients and (referring to the conventional cryogenic oddorder series expansion (6)), the first and secondorder moments of the PDOS model function, and (due to (13)), the respective average phonon temperatures [14, 21, 50], , the dispersion coefficients [14, 21, 50], , and the hightemperature () limiting values of the true Debye temperatures [14, 21, 50], (cf. (42) below). Furthermore, with respect to the frequently considered (thermochemical) reference temperature, 298.15 K, we have still quoted the fitted (smoothed) isobaric heat capacity values, [14, 50], and the respective entropy values and enthalpy differences defined by [14, 15, 50]
We have given in Table 2, for a series of selected values, a list of smoothed isobaric lattice heat capacity values, , including the corresponding (effective) lattice Debye temperatures, , which have been calculated on the basis of precision formulas to be presented below (in Section 5).

5. Transformation of Heat Capacities into Debye Temperatures
Expressive visualizations of typical nonDebye features of the temperature dependencies of heat capacities of solids are based above all on the results of pointbypoint transformations of isobaric heat capacity data, , into the respective (effective) Debye temperature values, . The latter are well known from Debye’s classical paper [1] to be defined in implicit way [1–4, 13–15, 56] by an integral of the form (1), where the magnitudes of the characteristic (dimensionless) variable, , are representing the upper limits of integration. The original task of calculating temperature dependencies of heat capacities within the frame of Debye’s theory [1] involved thus, primarily, the necessity of preparing good approximations for the dependence of Debye’s modelspecific heat capacity function, , on . A detailed analytical study of the corresponding function (see (A.1)) is performed in Appendix A. Crucial results of this study are given, above all, in terms of approximate algebraic expressions for the respective low and highcapacity dependencies, and , which can be, fortunately, transformed in exact way into corresponding algebraic expressions for the respective inverse dependencies, (see (B.1)) and (see (B.3)). Starting from these approximate expressions for low and highcapacity curve sections, we succeeded to construct in Appendix B an unprecedented, integralfree analytical apparatus for approximate (more or less rough) estimations up to highprecision calculations of the complete dependence. Let us display in explicit form the integralfree formulas which are actually to be used here for the transformations of the heat capacity data under study (for ZnSe and ZnTe) into the respective Debye temperatures.
5.1. Asymptotic Low and HighTemperature Formulas
The asymptote of Debye’s lowtemperature expansion [1, 3], (cf. (A.4)), was already found in [13] to be of primary importance for a proper interpretation of the behaviour in the liquidheliumhydrogen region. From the latter followed readily for the inverse function, , an asymptotic dependence of the form (see (B.1)). The respective limiting () dependence of the Debye temperature, , was thus explicitly given by the asymptotic expression [13, 15] (cf. Eq. in [13]), where the last representation of the dependence in (25) is in accordance with the general relation (cf. Eq. (6) in [13]). However, the inherent deviations (overestimations) of approximate values (due to (25)) from exact values are found to be lower than 1% only for very low magnitudes of heat capacities, 0 < 0.03, which are corresponding to values higher than 14 (cf. Table 9 and Figure 9).
A good basis for establishing an appropriate algebraic formula for the qualitatively different behaviour at considerably smaller values is provided by the unprecedented Taylor series expansion (see (B.3)), which we have shown here to follow in unambiguous way for the reciprocal version, (see (A.11) and (B.2)), of the function (see (A.10)). Accordingly, the temperature dependence of the Debye temperature, , for moderatetohigh heat capacities, is clearly confirmed to be given by the approximate algebraic expression [15]
Note that the latter formula corresponds just to Eq. () in [15], where it had already been taken into consideration for data discussions of various III–V materials. This formula is seen to provide a reasonable approximation (within possible deviations up to ±0.7%) to the exact curve throughout the range 0.05 < 1 (as shown in the upper inset of Figure 9 by the relative deviations, , of the approximate curve from the exact dependence).
5.2. Couple of HighPrecision Formulas for Effective Debye Temperatures
In view of the sample character of the numerical analyses performed within the present study for the unusually fine and comprehensive data provided by Kremer et al. [46], we have constructed in Appendix B (Section B.3) a highprecision framework for the dependence. This consists in a combination of a highly elaborate (12parameter) lowcapacity formula, (see (B.8)), with a duly modified (11parameter) highcapacity formula, (see (B.9)) (where = 0.26 represents the position of the point of crossing between the two complementary branches). The respective low and hightemperature formulas for the Debye temperature, , are thus given explicitly by the couple of analytical expressions of the formfor the interval (with expansion coefficients , , and listed in the second column of Table 8), andfor the interval (with expansion coefficients listed in the third column of Table 8). Note that we have still endowed the functions in (27) and (28) by an additional subscript “” (included in brackets) in order to indicate that precisely the same formulas can also be used, alternatively, for highaccuracy calculations of the “true” Debye temperatures, , which are visualizing the characteristic nonDebye features of the harmonic parts of lattice heat capacity curves, (see (4); Figures 4 and 5).
By means of this couple of the highaccuracy formulas (27) and (28) we have performed, first of all, the pointbypoint transformations, , for all those isobaric heat capacity data points, , with respect to which the concept of effective Debye temperatures is actually applicable, that is, for (cf. Figures 2 and 3). The respective Debye temperature values, > 0, are represented for ZnSe (in Figure 4) and for ZnTe (in Figure 5) by the same symbols as the original data points (shown in Figures 2 and 3, resp.).
Furthermore, we have transformed in the same way (i.e., via (27) or (28)) the continuous curves (solid curves shown in Figures 2 and 3), which resulted from the preceding numerical fittings in Section 4. These continuous curves (solid curves, in Figures 4 and 5) are naturally ending at those temperature points, , where the fitted curves (in Figure 2 or 3) are crossing just the classical DelongPetit heat capacity value, , that is, at about 401 K for ZnSe (cf. Figures 2 and 4) and about 323 K for ZnTe (cf. Figures 3 and 5).
5.3. Calculation of “True” (Harmonic Lattice) Debye Temperatures
Of considerable interest, particularly from theoretical points of view [3, 14, 21, 28, 29, 32, 36, 56, 72], is the determination of the “true” (related) Debye temperature curves, . The latter are defined with respect to the harmonic (isochoric) lattice heat capacities, (see (4)), by [14, 21]where the dimensionless variables are representing the respective upper boundaries of integration. Comparing (29) with (A.1) we see that the analytical connection of the upper boundaries of integration, , with the respective normalized (harmonic) heat capacity ratios, (in (29)), is just the same as the preceding connection between and (in (A.1)). Consequently, all the approximation formulas derived in Appendix B, for low and highcapacity branches of the function (cf. Figure 9), can be readily used for calculations of the dependence implied by (29). This means, in particular, that (27) and (28) can be simultaneously used (as already indicated by the inclusion of the subscript “h”) for highprecision calculations of the dependencies of the “true” Debye temperatures, , on the respective harmonic (isochoric) heat capacities, . The corresponding materialspecific curves are represented by dashed curves (for ZnSe and ZnTe in Figures 4 and 5, resp.).
With respect to this alternative use of (27) and (28), for calculations of curves, we would still like to point out that, apart from the liquidheliumhydrogen region (see below), appreciable differences between and values, similarly to the differences between the underlying and values (cf. Section 4), are encountered as a rule only in ranges of moderately low to higher temperatures, , where the heat capacities in question are higher than about 50% of the classical DulongPetit limiting value, . This region is automatically comprised by the range of validity of (28). Thus it is mostly not necessary to involve (27) into separate calculations of curve sections.
On the other hand, significant differences between and values may also occur at very low (liquidheliumhydrogen) temperatures, when we are (accidentally) concerned with nonnegligible contributions, , of a degenerate electronic system to the measured heat capacities, (cf. (5)). Consequently, in order to determine the respective temperature dependence of the curve within the liquidheliumhydrogen region, we have to insert , instead of , into the respective asymptotic (low) expression (see (25)). The monotonically decreasing sections, , of the “true” Debye temperature curves at sufficiently low temperatures, (see the insets of Figures 4 and 5), are thus adequately described by a corresponding asymptotic expression of the form [13]where and (see also [13] and cf. the insets of Figures 2 and 3). These asymptotic dependencies are represented by dotted curve sections in the insets of Figures 4 and 5.
6. Discussion
The present investigations were devoted above all to the finding of practicable solutions for longstanding computational problems which are frequently emerging within interpretations of the results of experimental heat capacity studies utilizing the somewhat troublesome concept of Debye temperatures [1, 3, 4, 6, 10–15]. The notorious computational complications are due to circumstance that the respective values are involved into the definition of the upper limits of integration, , due to the integral expression for isobaric heat capacities, (see (1)). Consequently, in view of the previous lack of integralfree formulas for dependencies on values, it was thus necessary either to calculate the individual values via numerical integration procedures (using (1)) for all the individual points in consideration or to determine the values approximately, via pointbypoint interpolations, using some of the available tables [6, 8, 47–49].
Within our detailed study of the analytical properties of the conventional dependence (in Appendix A) we succeeded to derive, first of all, an unprecedented, rapidly converging hightemperature Taylor series expansion, (see (A.11)). Combining the latter with Debye’s conventional lowtemperature expansion [1], (see (A.4)), it became possible to perform highaccuracy calculations of values (up to at least 10 significant figures; cf. Table 7) without involving any numerical integration procedure. The subsequent adoption of respective and formulas to the Debyelike component, (see (10)), of the longwave TA phonon contribution to the cryogenic heat capacity, including the derivation of analogous formulas for the associated nonDebye component, (see (10)), enabled us to display (in Section 3.2) an integralfree version of the multioscillator hybrid model [14, 21, 50]. This has been used here (in Section 4) for numerical fittings of the data sets under study.
The highaccuracy results obtained in Appendix A for the dependence provided us, among other things, with the numerical basis for the solution of another and practically even more important mathematical problem, namely, the construction of appropriate integralfree formulas for the inverse Debye function dependence, that is, the function (see Figure 9 and Table 9), which describes the functional dependence of the characteristic Debye versus lattice temperature ratios, , on the ratios, , of given heat capacities versus DulongPetit classical heat capacity limit.
We have derived (in Appendix B) a series of more or less elaborate formulas for low and highcapacity curve sections of the curve (solid curve in Figure 9). The simplest one of these approximate expressions is given in form of a junction formula, (see (B.4a)/(B.4b)), the actual use of which, however, is still connected with deviations from the exact dependence up to 0.6% (cf. the upper inset of Figure 9). Reductions of residual deviations to ±0.13%, or even to ±0.0153%, could be achieved (in Sections B.1 and B.2) via leastmeansquare fittings of the exact dependence, adopting either the still relatively simple (2parameter) formula, (see (B.5a) and (B.5b)), or the refined (9parameter) formula,