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Abstract
Characteristic nonDebye behaviors of lowtemperature heat capacities of GaP, GaAs, GaSb, InP, InAs, and InSb, which are manifested above all in form of nonmonotonic behaviors (local maxima) of the respective C p (T)/T 3 curves in the cryogenic region, are described by means of a refined version of a recently proposed lowtohightemperature interpolation formula of nonDebye type. Leastmeansquare fittings of representative C p (T) data sets available for these materials from several sources show excellent agreements, from the liquidhelium region up to room temperature. The results of detailed calculations of the respective materialspecific Debye temperature curves, Θ D (T), are represented in graphical form. The strong, nonmonotonic variations of Θ D (T) values confirm that it is impossible to provide reasonable numerical simulations of measured C p (T) dependences in terms of fixed Debye temperatures. We show that it is possible to describe in good approximation the complete Debye temperature curves, from the cryogenic region up to their definitive disappearance (dropping to 0) in the high temperature region, by a couple of unprecedented algebraic formulas. The task of constructing physically adequate prolongations of the lowtemperature C p (T) curves up to melting points was strongly impeded by partly rather large differences (up to an order of 10 J/(K·mol)) between the hightemperature data sets presented in different research papers and/or data reviews. Physically plausible criteria are invoked, which enabled an a priori rejection of a series of obviously unrealistic hightemperature data sets. Residual uncertainties for GaAs and InAs could be overcome by reevaluations of former enthalpy data on the basis of a novel set of properly specified fourparameter polynomial expressions applying to large regions, from moderately low temperatures up to melting points. Detailed analytical and numerical descriptions are given for the anharmonicityrelated differences of isobaric vs. isochoric (harmonic) parts of heat capacities. Relevant sets of empirical parameters and representative collections of heat capacity and Debye temperature values for all materials under study are presented in tabulated form.
I. INTRODUCTION
II. BASIC EQUATIONS OF THE NONDEBYE HEAT CAPACITY MODEL DESCRIPTION
III. HEAT CAPACITY DATA SELECTIONS AND FITTINGS FOR CUBIC IIIVMATERIALS
A. Basic criteria for hightemperature data selections
B. Fittings of compatible low and hightemperature data sets
IV. DETERMINATION AND ASSESSMENT OF VARIABLE DEBYE TEMPERATURES
A. Analytical descriptions and calculations of variable Debye temperatures
B. Correlation of cryogenic nonDebye behaviors of Debye temperatures vs. heat capacities
C. Large deviations and rejection of erroneous hightemperature data
V. DISCUSSION
A. Important role of conveniently chosen scaling temperatures
B. Efficient version of leastmeansquare fitting processes
C. Rough estimations of limiting Debye temperatures for the harmonic regime
D. Qualitatively different shapes of high temperature heat capacity curves
VI. IMPROVED HIGHTEMPERATURE POLYNOMIAL REPRESENTATION AND NOVEL RESULTS
A. Construction of an improved hightemperature polynomial representation
B. Joint fittings of heat capacity and enthalpy data for GaAs and InAs
VII. SUMMARY
Key Topics
 Data sets
 81.0
 IIIV semiconductors
 78.0
 Heat capacity
 57.0
 Cryogenics
 24.0
 Enthalpy
 19.0
Figures
Fitting of the data sets available for GaP from Ref. 54 (○), 57 (△), and 91 (□), and in combination with the data sets due to Refs. 5 (✰), and 6 (*), and 9 (◂). Shown for comparison are also the more or less strongly deviating data sets given in Ref. 4 (▪), 10 (◊), and 15 (×). C p (T) and C p (T)/T 3 curves (———, Eq. (12) ); C Vh (T)∝κ P (T) curves (—, Eqs. (3) and (10) ); asymptotic C Vh (T)/T 3 → 0 curve (·········, Eq. (8) ); Debye's C D (T)/T 3 curve for Θ D = Θ D (0), as quoted in Table III (····, Eq. (2) ). (Note that the same associations between different curve types and the underlying analytical expressions apply also to the subsequent Figs. 2 to 6 ). The deviating hightemperature C p (T) curve (––·––·––, Eq. (12) ) corresponds to data given in Ref. 10 (◊) and 15 (×).
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Fitting of the data sets available for GaP from Ref. 54 (○), 57 (△), and 91 (□), and in combination with the data sets due to Refs. 5 (✰), and 6 (*), and 9 (◂). Shown for comparison are also the more or less strongly deviating data sets given in Ref. 4 (▪), 10 (◊), and 15 (×). C p (T) and C p (T)/T 3 curves (———, Eq. (12) ); C Vh (T)∝κ P (T) curves (—, Eqs. (3) and (10) ); asymptotic C Vh (T)/T 3 → 0 curve (·········, Eq. (8) ); Debye's C D (T)/T 3 curve for Θ D = Θ D (0), as quoted in Table III (····, Eq. (2) ). (Note that the same associations between different curve types and the underlying analytical expressions apply also to the subsequent Figs. 2 to 6 ). The deviating hightemperature C p (T) curve (––·––·––, Eq. (12) ) corresponds to data given in Ref. 10 (◊) and 15 (×).
Fitting of the data sets available for GaAs from Ref. 32 (□), 52 (○), and 58 (△) in combination with the novel data set (●) resulting from the present reassessment (cf. Sec. IV ) of original enthalpy data due to Refs. 10 and 20 . Shown for comparison are also the more or less strongly deviating data sets given in Refs. 5 (✰), 6 (*), 10 (◊), 12 (×), 19 (▸), 20 (+), and 93 (▲). Possible alternative fits of the same data sets in combination with data due to Ref. 12 (×) and 20 (+) are indicated by doubledasheddotted curves.
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Fitting of the data sets available for GaAs from Ref. 32 (□), 52 (○), and 58 (△) in combination with the novel data set (●) resulting from the present reassessment (cf. Sec. IV ) of original enthalpy data due to Refs. 10 and 20 . Shown for comparison are also the more or less strongly deviating data sets given in Refs. 5 (✰), 6 (*), 10 (◊), 12 (×), 19 (▸), 20 (+), and 93 (▲). Possible alternative fits of the same data sets in combination with data due to Ref. 12 (×) and 20 (+) are indicated by doubledasheddotted curves.
Fitting of the data sets available for GaSb from Ref. 32 (□) and 52 (○), in combination with the data sets due to Ref. 4 (▪), and 6 (*). Shown for comparison is also the slightly deviating data set due to Ref. 21 (+).
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Fitting of the lowtemperature C p (T) data sets available for InP from Refs. 52 (○), 60 (△), and 94 (□) in combination with the data set due to Ref. 9 (◂) and partial sections (see the text) of the data sets due to Refs. 5 (✰), 6 (*), and 16 (×). Shown for comparison are also the markedly deviating data points quoted in Ref. 5 (✰) and 10 (◊) for the region 800 K ⩽T ⩽ 900 K (i. e. close to the phase transition point).
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Fitting of the lowtemperature C p (T) data sets available for InP from Refs. 52 (○), 60 (△), and 94 (□) in combination with the data set due to Ref. 9 (◂) and partial sections (see the text) of the data sets due to Refs. 5 (✰), 6 (*), and 16 (×). Shown for comparison are also the markedly deviating data points quoted in Ref. 5 (✰) and 10 (◊) for the region 800 K ⩽T ⩽ 900 K (i. e. close to the phase transition point).
Fitting of the data sets available for InAs from Ref. 32 (□), 52 (○), and 60 (△) in combination with the novel data set (●) resulting from the present reassessment of original enthalpy data due to Ref. 20 (cf. Sec. IV). Shown for comparison are also the more or less strongly deviating data sets given in Refs. 4 (▪), 5 (✰), 6 (*), 10 (◊), 13 (×), 19 (▸), and 20 (+). Possible alternative fits of the same constellation of data sets in combination with the upper section (800 K < T < 1200 K) of the data set due to Ref. 20 (+) or with the upper section (400 K < T < 1200 K) of the data set due to Ref. 4 (▪) are represented by doubledasheddotted curves.
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Fitting of the data sets available for InAs from Ref. 32 (□), 52 (○), and 60 (△) in combination with the novel data set (●) resulting from the present reassessment of original enthalpy data due to Ref. 20 (cf. Sec. IV). Shown for comparison are also the more or less strongly deviating data sets given in Refs. 4 (▪), 5 (✰), 6 (*), 10 (◊), 13 (×), 19 (▸), and 20 (+). Possible alternative fits of the same constellation of data sets in combination with the upper section (800 K < T < 1200 K) of the data set due to Ref. 20 (+) or with the upper section (400 K < T < 1200 K) of the data set due to Ref. 4 (▪) are represented by doubledasheddotted curves.
Fitting of the data sets available for InSb from Ref. 32 (□) and 52 (○) in combination with the nonlinear data sets due to Refs. 5 (✰) and 6 (*) and the data set due to Ref. 4 (▪). Shown for comparison is also the data set due to Ref. 21 (+).
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Effective Debye temperatures resulting (via Eq. (20) ) from and data available from various research papers and/or data reviews for GaP ( from Refs. 54 (○), 57 (△), and 91 (□), and from Refs. 5 (✰), 6 (*), 9 (◂), and 15 (×)), for GaAs ( from Refs. 32 (□), 52 (○), and 58 (△), and from Refs. 5 (✰), 6 (*), 12 (×), 19 (▸), 20 (+), and 93 (▲)), and GaSb ( from Refs. 32 (□) and 52 (○) and from Refs. 6 (*), 19 (▸), and 20 (+)). The continuous Θ D (T) curves (———) are resulting (via Eq. (20) ) from the respective isobaric heat capacity curves, C p (T) (as shown in Figs. 1–3 , respectively). Shown are also limited sections of the “true” (harmonic) Debye temperatures, Θ Dh (T) (—, due to Eq. (B1) , and approximate Θ D (T) curves (·········), which are resulting from a couple of complementary algebraic formulas (i. e. from Eq. (24) , for 0 < T < T min , or from Eq. (26) , for T min < T ⩽ T f ).
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Effective Debye temperatures resulting (via Eq. (20) ) from and data available from various research papers and/or data reviews for GaP ( from Refs. 54 (○), 57 (△), and 91 (□), and from Refs. 5 (✰), 6 (*), 9 (◂), and 15 (×)), for GaAs ( from Refs. 32 (□), 52 (○), and 58 (△), and from Refs. 5 (✰), 6 (*), 12 (×), 19 (▸), 20 (+), and 93 (▲)), and GaSb ( from Refs. 32 (□) and 52 (○) and from Refs. 6 (*), 19 (▸), and 20 (+)). The continuous Θ D (T) curves (———) are resulting (via Eq. (20) ) from the respective isobaric heat capacity curves, C p (T) (as shown in Figs. 1–3 , respectively). Shown are also limited sections of the “true” (harmonic) Debye temperatures, Θ Dh (T) (—, due to Eq. (B1) , and approximate Θ D (T) curves (·········), which are resulting from a couple of complementary algebraic formulas (i. e. from Eq. (24) , for 0 < T < T min , or from Eq. (26) , for T min < T ⩽ T f ).
Effective Debye temperatures resulting (via Eq. (20) ) from and data available from various research papers and/or data reviews for InP ( from Refs. 52 (○), 60 (△), and 94 (□), and from Refs. 5 (✰), 6 (*), 9 (◂), and 16 (×)), for InAs ( from Refs. 32 (□), 52 (○), and 60 (△), and from Refs. 4 (▪), 5 (✰), 6 (*), 13 (×), 19 (▸), 20 (+)), and for InSb ( from Refs. 32 (□) and 52 (○), and from Refs. 4 (▪), 5 (✰), and 6 (*)). (Note that the associations between different curve types and the underlying analytical expressions are the same as in Fig. 7 ).
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Effective Debye temperatures resulting (via Eq. (20) ) from and data available from various research papers and/or data reviews for InP ( from Refs. 52 (○), 60 (△), and 94 (□), and from Refs. 5 (✰), 6 (*), 9 (◂), and 16 (×)), for InAs ( from Refs. 32 (□), 52 (○), and 60 (△), and from Refs. 4 (▪), 5 (✰), 6 (*), 13 (×), 19 (▸), 20 (+)), and for InSb ( from Refs. 32 (□) and 52 (○), and from Refs. 4 (▪), 5 (✰), and 6 (*)). (Note that the associations between different curve types and the underlying analytical expressions are the same as in Fig. 7 ).
Temperature dependences of the anharmonicityrelated differences of isobaric vs. isochoric (harmonic) heat capacities, C p (T) − C Vh (T), which are resulting (via Eq. (11) , in combination with Eq. (10) ), from the parameter values quoted for the six cubic IIIV materials under study in Table I .
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Temperature dependences of the anharmonicityrelated differences of isobaric vs. isochoric (harmonic) heat capacities, C p (T) − C Vh (T), which are resulting (via Eq. (11) , in combination with Eq. (10) ), from the parameter values quoted for the six cubic IIIV materials under study in Table I .
Reproduction of upper sections of the previously determined C p (T) and C Vh (T) curves, for GaP, GaAs, and GaSb (cf. the solid and dashed curves, in Figs. 1 to 3 ), by means of the properly devised polynomials, i. e. Eq. (36) for (———) and Eq. (34) for (    ), with coefficients quoted in Table IV . (Symbols for selected data are the same as in Figs. 1 to 3 .)
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Reproduction of upper sections of the previously determined C p (T) and C Vh (T) curves, for GaP, GaAs, and GaSb (cf. the solid and dashed curves, in Figs. 1 to 3 ), by means of the properly devised polynomials, i. e. Eq. (36) for (———) and Eq. (34) for (    ), with coefficients quoted in Table IV . (Symbols for selected data are the same as in Figs. 1 to 3 .)
Reproduction of upper sections of the previously determined C p (T) and C Vh (T) curves, for InP, InAs, and InSb (cf. the solid and dashed curves, in Figs. 4 to 6 ), by means of the properly devised polynomials, i. e. Eq. (36) for (———) and Eq. (34) for ((    ), with coefficients quoted in Table IV . (Symbols for selected data are the same as in Figs. 4 to 6 .)
Click to view
Reproduction of upper sections of the previously determined C p (T) and C Vh (T) curves, for InP, InAs, and InSb (cf. the solid and dashed curves, in Figs. 4 to 6 ), by means of the properly devised polynomials, i. e. Eq. (36) for (———) and Eq. (34) for ((    ), with coefficients quoted in Table IV . (Symbols for selected data are the same as in Figs. 4 to 6 .)
Θ D (T) vs. T −2 representation of effective Debye temperatures (cf. Fig. 7 ), which are due to the data in consideration for GaP (Refs. 54 (○), 57 (△), and 91 (□)), for GaAs (Refs. 32 (□), 52 (○), and 58 (△)), and for GaSb (Refs. 32 (□) and 52 (○)). Solid curves are representing the fittings of the Θ D (T) data by means of Eq. (B2) , with the empirical parameter values quoted in Table V . Dashed curves show the corresponding hightemperature dependences of the “true” (harmonic) Debye temperatures, Θ Dh (T) (due to Eq. (B1) ).
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Θ D (T) vs. T −2 representation of effective Debye temperatures (cf. Fig. 7 ), which are due to the data in consideration for GaP (Refs. 54 (○), 57 (△), and 91 (□)), for GaAs (Refs. 32 (□), 52 (○), and 58 (△)), and for GaSb (Refs. 32 (□) and 52 (○)). Solid curves are representing the fittings of the Θ D (T) data by means of Eq. (B2) , with the empirical parameter values quoted in Table V . Dashed curves show the corresponding hightemperature dependences of the “true” (harmonic) Debye temperatures, Θ Dh (T) (due to Eq. (B1) ).
Θ D (T) vs. T −2 representation of effective Debye temperatures (cf. Fig. 8 ), which are due to the data in consideration for InP (Refs. 52 (○), 60 (△), and 94 (□)), for InAs (Refs. 32 (□), 52 (○), and 60 (△)), and for InSb (Refs. 32 (□) and 52 (○)). Solid curves are representing the fittings of these Θ D (T) data by means of Eq. (B2) , with the empirical parameter values quoted in Table V . Dashed curves show the corresponding hightemperature dependences of the “true” (harmonic) Debye temperatures, Θ Dh (T) (due to Eq. (B1) ).
Click to view
Θ D (T) vs. T −2 representation of effective Debye temperatures (cf. Fig. 8 ), which are due to the data in consideration for InP (Refs. 52 (○), 60 (△), and 94 (□)), for InAs (Refs. 32 (□), 52 (○), and 60 (△)), and for InSb (Refs. 32 (□) and 52 (○)). Solid curves are representing the fittings of these Θ D (T) data by means of Eq. (B2) , with the empirical parameter values quoted in Table V . Dashed curves show the corresponding hightemperature dependences of the “true” (harmonic) Debye temperatures, Θ Dh (T) (due to Eq. (B1) ).
Tables
Adjusted coefficients r 2(T h ) to r 8(T h ) and c 3 to c 7 due to Eq. (10) for the harmonic lattice heat capacity shape function, κ P (T), and associated anharmonicityrelated coefficients, A 1 and A 2, due to Eq. (12) . For the commonly considered thermochemical reference temperature, T r = 298.15 K, we have quoted the corresponding isobaric heat capacities, C p (T r ) (12) , entropies, S p (T r ) (17) , and enthalpy differences, ΔH p (T r ) ≡ H p (T r ) − H p (0) (17) .
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Adjusted coefficients r 2(T h ) to r 8(T h ) and c 3 to c 7 due to Eq. (10) for the harmonic lattice heat capacity shape function, κ P (T), and associated anharmonicityrelated coefficients, A 1 and A 2, due to Eq. (12) . For the commonly considered thermochemical reference temperature, T r = 298.15 K, we have quoted the corresponding isobaric heat capacities, C p (T r ) (12) , entropies, S p (T r ) (17) , and enthalpy differences, ΔH p (T r ) ≡ H p (T r ) − H p (0) (17) .
Click to view
Quantities manifesting the close correlation (Eq. (28) and (29) ) between the general increase of ρ(T) ≡ C p (T)/T 3 curves (Eq. (19) ; cf. the insets to Figs. 1 to 6 )) from relatively low T → 0 levels, ρ(0) = c 3, up to their local maxima, ρmax ≡ ρ(T max ), on the one hand, and the corresponding decrease of Debye temperature curves, Θ D (T) (Eq. (23) or (24) ; cf. Figs. 6 and 7 ), from T → 0 levels, Θ D (0), to respective local minima, Θ Dmin ≡ Θ D (T min ), on the other hand. A comparison of the magnitudes of various significant ratios between the quantities Θ D (0), Θ D (T min ), T Dmax , Θ Dmax , T f , and Θ Dh (∞) shows similarities of the shapes of the individual Θ D (T) curves.
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Quantities manifesting the close correlation (Eq. (28) and (29) ) between the general increase of ρ(T) ≡ C p (T)/T 3 curves (Eq. (19) ; cf. the insets to Figs. 1 to 6 )) from relatively low T → 0 levels, ρ(0) = c 3, up to their local maxima, ρmax ≡ ρ(T max ), on the one hand, and the corresponding decrease of Debye temperature curves, Θ D (T) (Eq. (23) or (24) ; cf. Figs. 6 and 7 ), from T → 0 levels, Θ D (0), to respective local minima, Θ Dmin ≡ Θ D (T min ), on the other hand. A comparison of the magnitudes of various significant ratios between the quantities Θ D (0), Θ D (T min ), T Dmax , Θ Dmax , T f , and Θ Dh (∞) shows similarities of the shapes of the individual Θ D (T) curves.
Click to view
Limiting values of “true” (harmonic) Debye temperatures, , and of the respective expansion coefficients involved by Eq. (B1) , a 2 to a 8, which have been determined, in combination with the empirical parameters p and a p , via fittings (see Figs. 12 and 13 ) of effective Debye temperatures, Θ D (T), by means of Eq. (B2) . Further quoted are the momentrelated evenorder phonon energies, (m = 2 to 10), which are resulting from Eqs. (B3a) to (B3e) for the corresponding moments, .
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Limiting values of “true” (harmonic) Debye temperatures, , and of the respective expansion coefficients involved by Eq. (B1) , a 2 to a 8, which have been determined, in combination with the empirical parameters p and a p , via fittings (see Figs. 12 and 13 ) of effective Debye temperatures, Θ D (T), by means of Eq. (B2) . Further quoted are the momentrelated evenorder phonon energies, (m = 2 to 10), which are resulting from Eqs. (B3a) to (B3e) for the corresponding moments, .
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Abstract
Characteristic nonDebye behaviors of lowtemperature heat capacities of GaP, GaAs, GaSb, InP, InAs, and InSb, which are manifested above all in form of nonmonotonic behaviors (local maxima) of the respective C p (T)/T 3 curves in the cryogenic region, are described by means of a refined version of a recently proposed lowtohightemperature interpolation formula of nonDebye type. Leastmeansquare fittings of representative C p (T) data sets available for these materials from several sources show excellent agreements, from the liquidhelium region up to room temperature. The results of detailed calculations of the respective materialspecific Debye temperature curves, Θ D (T), are represented in graphical form. The strong, nonmonotonic variations of Θ D (T) values confirm that it is impossible to provide reasonable numerical simulations of measured C p (T) dependences in terms of fixed Debye temperatures. We show that it is possible to describe in good approximation the complete Debye temperature curves, from the cryogenic region up to their definitive disappearance (dropping to 0) in the high temperature region, by a couple of unprecedented algebraic formulas. The task of constructing physically adequate prolongations of the lowtemperature C p (T) curves up to melting points was strongly impeded by partly rather large differences (up to an order of 10 J/(K·mol)) between the hightemperature data sets presented in different research papers and/or data reviews. Physically plausible criteria are invoked, which enabled an a priori rejection of a series of obviously unrealistic hightemperature data sets. Residual uncertainties for GaAs and InAs could be overcome by reevaluations of former enthalpy data on the basis of a novel set of properly specified fourparameter polynomial expressions applying to large regions, from moderately low temperatures up to melting points. Detailed analytical and numerical descriptions are given for the anharmonicityrelated differences of isobaric vs. isochoric (harmonic) parts of heat capacities. Relevant sets of empirical parameters and representative collections of heat capacity and Debye temperature values for all materials under study are presented in tabulated form.
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