^{1,a)}

### Abstract

We have redigitized a large variety of phonon density of states (PDOS) spectra, that have been published by diferent researchers for group IV (diamond, 3C-SiC, Si, and Ge), III–V (BN,BP,BAs, BSb, AlN,AlP,AlAs,AlSb,GaN,GaP,GaAs,GaSb, InN, InP,InAs, and InSb), and II–VI materials (ZnO,ZnS,ZnSe,ZnTe,CdS, and CdTe), including calculations of their moments, , of orders , 1, 2, and 4. Notwithstanding the obvious differences in concrete shapes of spectra presented for one and the same material by different authors, the respective magnitudes of estimated moments have been found in most cases to be nearly the same (to within uncertainties of some few percent). For most materials under study, the average phonon temperatures of the lower and upper sections of PDOS spectra, and , are found to be by factors of order 0.6 lower or 1.4 higher, respectively, than the average phonon temperature, , of the total PDOS spectra. The estimated high-temperature limits of Debye temperatures, , are found to be significantly higher (by factors of order 1.4) than , implying an order-of-magnitude equality, (within differences not exceeding an order of , for all materials under study). The phonon temperatures, , that are effective in controlling the observable temperature dependences of fundamental energy gaps,, are found to be usually of the same order as the respective average phonon temperatures, . The existing differences between these two qualitatively different types of characteristic phonon temperatures are seen to be limited, for diamond, 3C-SiC, Si, Ge,AlN,GaN,GaP,GaAs,GaSb,InP,InSb,ZnS,ZnSe,ZnTe, and CdTe, to an order of . We design an exemplary way for precalculating harmonic parts of isochoric heat capacities on the basis of the estimated quadruplets of PDOS spectra moments. This novel calculation scheme is exemplified for silicon and germanium.

I. INTRODUCTION

II. ESTIMATION OF MOMENTS AND RELATED PHONON TEMPERATURES

III. DISCUSSION

A. Upper and lower phonon temperatures in proportion to their mean values

B. Assessment of dispersion coefficients

C. Limiting Debye temperatures related to the upper (optical) phonon temperatures

D. Comparison with effective phonon temperatures due to the gap shrinkage effect

E. Dispersion-related estimation of heat capacities using moments of PDOS spectra

IV. CONCLUDING REMARKS

### Key Topics

- III-V semiconductors
- 66.0
- Phonons
- 58.0
- II-VI semiconductors
- 34.0
- Heat capacity
- 19.0
- Dispersion
- 18.0

## Figures

Material-specific constellations of characteristic phonon temperatures for (a) group IV, (b) B-V and Al-V, (c) Ga-V and In-V, and (d) III–VI materials. The magnitudes of average phonon temperatures pertaining to the lower and upper sections of the PDOS spectra, (solid triangles) and (empty triangles), are shown in proportion to their mean values [Eq. (4)] (solid squares). Shown are also, for comparisons with and , the magnitudes of effective phonon temperatures (see Refs. 3–5), (empty squares), which had been detected in previous studies (see Refs. 3–5) from temperature dependences of fundamental energy gaps, and the limiting magnitudes of Debye temperatures [Eq. (7)], (empty circles), respectively.

Material-specific constellations of characteristic phonon temperatures for (a) group IV, (b) B-V and Al-V, (c) Ga-V and In-V, and (d) III–VI materials. The magnitudes of average phonon temperatures pertaining to the lower and upper sections of the PDOS spectra, (solid triangles) and (empty triangles), are shown in proportion to their mean values [Eq. (4)] (solid squares). Shown are also, for comparisons with and , the magnitudes of effective phonon temperatures (see Refs. 3–5), (empty squares), which had been detected in previous studies (see Refs. 3–5) from temperature dependences of fundamental energy gaps, and the limiting magnitudes of Debye temperatures [Eq. (7)], (empty circles), respectively.

Visualization of the roughly monotonic decrease of the magnitudes of total dispersion coefficients [Eq. (6)] (solid squares), with increasing ratio, [Eq. (6)], of the partial average phonon temperatures, and , pertaining to the lower and upper sections of PDOS spectra. The moderate deviations from strict monotonousness are due to more or less different, material-specific magnitudes of the respective partial dispersion coefficients [Eq. (A2)], and , for the lower and upper sections. The lower and upper boundaries [Eq. (A7)] for (represented by solid and dashed curves, respectively) are due to the usual limitations of and applying to almost all materials under study (except CdS). Empty squares are representing the comparable magnitudes of -related dispersion coefficients (see Refs. 3–5), .

Visualization of the roughly monotonic decrease of the magnitudes of total dispersion coefficients [Eq. (6)] (solid squares), with increasing ratio, [Eq. (6)], of the partial average phonon temperatures, and , pertaining to the lower and upper sections of PDOS spectra. The moderate deviations from strict monotonousness are due to more or less different, material-specific magnitudes of the respective partial dispersion coefficients [Eq. (A2)], and , for the lower and upper sections. The lower and upper boundaries [Eq. (A7)] for (represented by solid and dashed curves, respectively) are due to the usual limitations of and applying to almost all materials under study (except CdS). Empty squares are representing the comparable magnitudes of -related dispersion coefficients (see Refs. 3–5), .

Visualization of the relatively small deviations from unity of the ratios, [Eq. (A8)], of the limiting Debye temperatures, [Eq. (7)], vs average phonon temperatures of the upper (optical) PDOS-sections, . These ratios show a weak, nearly monotonic increase with increasing ratio [Eq. (6)]. The lower and upper boundaries [Eq. (A9)] for ratios (represented by solid and dashed curves, respectively) are due to the same limitations for partial dispersion coefficients, and , as in Fig. 2.

Visualization of the relatively small deviations from unity of the ratios, [Eq. (A8)], of the limiting Debye temperatures, [Eq. (7)], vs average phonon temperatures of the upper (optical) PDOS-sections, . These ratios show a weak, nearly monotonic increase with increasing ratio [Eq. (6)]. The lower and upper boundaries [Eq. (A9)] for ratios (represented by solid and dashed curves, respectively) are due to the same limitations for partial dispersion coefficients, and , as in Fig. 2.

Theoretical dependences [Eq. (12)], which follow from the quadruplets of values detected from PDOS spectra (see Ref. 12) for Si and Ge (cf. Tables I and II), in comparison with experimental data (see Ref. 6). Very fine fits of the measured heat capacity data follow from least-mean-square fittings on the basis of Eq. (13), in combination with Eq. (12). (The fitted thermodynamic parameter values are listed in Table II.)

Theoretical dependences [Eq. (12)], which follow from the quadruplets of values detected from PDOS spectra (see Ref. 12) for Si and Ge (cf. Tables I and II), in comparison with experimental data (see Ref. 6). Very fine fits of the measured heat capacity data follow from least-mean-square fittings on the basis of Eq. (13), in combination with Eq. (12). (The fitted thermodynamic parameter values are listed in Table II.)

## Tables

Characteristic phonon energy and temperature parameters of group IV, III–V, and II–VI materials. Listed are the average phonon energies [Eq. (A1)], and , and the respective average phonon temperatures, and {including their ratios, [Eq. (6)]}, for the lower and upper sections of the PDOS spectra under study. For convenience of notations, we have represented the total moments, [Eq. (2)], of orders , 1, 2, and 4, equivalently by respective (th-order) phonon energies [Eq. (3)], , including the respective average phonon temperatures, [Eq. (4)], dispersion coefficients, [Eq. (5)], and limiting magnitudes of Debye temperatures, [Eq. (7)]. For numerical comparisons with parameters obtained by preceding numerical analyses (Refs. 3–5) of the gap shrinkage effect for the materials under study we have also quoted the corresponding effective phonon temperatures, , dispersion coefficients, , and ratios .

Characteristic phonon energy and temperature parameters of group IV, III–V, and II–VI materials. Listed are the average phonon energies [Eq. (A1)], and , and the respective average phonon temperatures, and {including their ratios, [Eq. (6)]}, for the lower and upper sections of the PDOS spectra under study. For convenience of notations, we have represented the total moments, [Eq. (2)], of orders , 1, 2, and 4, equivalently by respective (th-order) phonon energies [Eq. (3)], , including the respective average phonon temperatures, [Eq. (4)], dispersion coefficients, [Eq. (5)], and limiting magnitudes of Debye temperatures, [Eq. (7)]. For numerical comparisons with parameters obtained by preceding numerical analyses (Refs. 3–5) of the gap shrinkage effect for the materials under study we have also quoted the corresponding effective phonon temperatures, , dispersion coefficients, , and ratios .

Three-oscillator model parameters following for Si and Ge as solutions of the system of coupled Eq. (11) for the material-specific sets of four moment-related (equivalent) phonon energies, (, 1, 2, and 4), that we have detected in Sec. II from corresponding PDOS spectra (see Ref. 12). For the sake of numerical comparisons we have listed also the thermal counterparts of these PDOS-related parameters as following from least-mean-square fittings of the experimental (isobaric) heat capacity data (see Ref. 6), , on the basis of Eqs. (12) and (13). (The fitted parameters are printed in bold).

Three-oscillator model parameters following for Si and Ge as solutions of the system of coupled Eq. (11) for the material-specific sets of four moment-related (equivalent) phonon energies, (, 1, 2, and 4), that we have detected in Sec. II from corresponding PDOS spectra (see Ref. 12). For the sake of numerical comparisons we have listed also the thermal counterparts of these PDOS-related parameters as following from least-mean-square fittings of the experimental (isobaric) heat capacity data (see Ref. 6), , on the basis of Eqs. (12) and (13). (The fitted parameters are printed in bold).

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