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Non-Debye heat capacity formula refined and applied to GaP, GaAs, GaSb, InP, InAs, and InSb
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Figures

Image of FIG. 1.

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FIG. 1.

Fitting of the data sets available for GaP from Ref. (○), (△), and (□), and in combination with the data sets due to Refs. (✰), and (*), and (◂). Shown for comparison are also the more or less strongly deviating data sets given in Ref. (▪), (◊), and (×). () and ()/ curves (———, Eq. (12) ); ()∝κ() curves (—-, Eqs. (3) and (10) ); asymptotic ()/ → 0 curve (·········, Eq. (8) ); Debye's ()/ curve for Θ = Θ(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 high-temperature () curve (––·––·––, Eq. (12) ) corresponds to data given in Ref. (◊) and (×).

Image of FIG. 2.

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FIG. 2.

Fitting of the data sets available for GaAs from Ref. (□), (○), and (△) in combination with the novel data set (●) resulting from the present re-assessment (cf. Sec. IV ) of original enthalpy data due to Refs. and . Shown for comparison are also the more or less strongly deviating data sets given in Refs. (✰), (*), (◊), (×), (▸), (+), and (▲). Possible alternative fits of the same data sets in combination with data due to Ref. (×) and (+) are indicated by double-dashed-dotted curves.

Image of FIG. 3.

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FIG. 3.

Fitting of the data sets available for GaSb from Ref. (□) and (○), in combination with the data sets due to Ref. (▪), and (*). Shown for comparison is also the slightly deviating data set due to Ref. (+).

Image of FIG. 4.

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FIG. 4.

Fitting of the low-temperature () data sets available for InP from Refs. (○), (△), and (□) in combination with the data set due to Ref. (◂) and partial sections (see the text) of the data sets due to Refs. (✰), (*), and (×). Shown for comparison are also the markedly deviating data points quoted in Ref. (✰) and (◊) for the region 800 K ⩽ ⩽ 900 K (i. e. close to the phase transition point).

Image of FIG. 5.

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FIG. 5.

Fitting of the data sets available for InAs from Ref. (□), (○), and (△) in combination with the novel data set (●) resulting from the present re-assessment of original enthalpy data due to Ref. (cf. Sec. IV). Shown for comparison are also the more or less strongly deviating data sets given in Refs. (▪), (✰), (*), (◊), (×), (▸), and (+). Possible alternative fits of the same constellation of data sets in combination with the upper section (800 K < < 1200 K) of the data set due to Ref. (+) or with the upper section (400 K < < 1200 K) of the data set due to Ref. (▪) are represented by double-dashed-dotted curves.

Image of FIG. 6.

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FIG. 6.

Fitting of the data sets available for InSb from Ref. (□) and (○) in combination with the non-linear data sets due to Refs. (✰) and (*) and the data set due to Ref. (▪). Shown for comparison is also the data set due to Ref. (+).

Image of FIG. 7.

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FIG. 7.

Effective Debye temperatures resulting (via Eq. (20) ) from and data available from various research papers and/or data reviews for GaP ( from Refs. (○), (△), and (□), and from Refs. (✰), (*), (◂), and (×)), for GaAs ( from Refs. (□), (○), and (△), and from Refs. (✰), (*), (×), (▸), (+), and (▲)), and GaSb ( from Refs. (□) and (○) and from Refs. (*), (▸), and (+)). The continuous Θ() curves (———) are resulting (via Eq. (20) ) from the respective isobaric heat capacity curves, () (as shown in Figs. 1–3 , respectively). Shown are also limited sections of the “true” (harmonic) Debye temperatures, Θ() (—-, due to Eq. (B1) , and approximate Θ() curves (·········), which are resulting from a couple of complementary algebraic formulas (i. e. from Eq. (24) , for 0 < < , or from Eq. (26) , for < ).

Image of FIG. 8.

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FIG. 8.

Effective Debye temperatures resulting (via Eq. (20) ) from and data available from various research papers and/or data reviews for InP ( from Refs. (○), (△), and (□), and from Refs. (✰), (*), (◂), and (×)), for InAs ( from Refs. (□), (○), and (△), and from Refs. (▪), (✰), (*), (×), (▸), (+)), and for InSb ( from Refs. (□) and (○), and from Refs. (▪), (✰), and (*)). (Note that the associations between different curve types and the underlying analytical expressions are the same as in Fig. 7 ).

Image of FIG. 9.

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FIG. 9.

Temperature dependences of the anharmonicity-related differences of isobaric vs. isochoric (harmonic) heat capacities, () − (), which are resulting (via Eq. (11) , in combination with Eq. (10) ), from the parameter values quoted for the six cubic III-V materials under study in Table I .

Image of FIG. 10.

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FIG. 10.

Reproduction of upper sections of the previously determined () and () 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 .)

Image of FIG. 11.

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FIG. 11.

Reproduction of upper sections of the previously determined () and () 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 .)

Image of FIG. 12.

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FIG. 12.

Θ() vs. representation of effective Debye temperatures (cf. Fig. 7 ), which are due to the data in consideration for GaP (Refs. (○), (△), and (□)), for GaAs (Refs. (□), (○), and (△)), and for GaSb (Refs. (□) and (○)). Solid curves are representing the fittings of the Θ() data by means of Eq. (B2) , with the empirical parameter values quoted in Table V . Dashed curves show the corresponding high-temperature dependences of the “true” (harmonic) Debye temperatures, Θ() (due to Eq. (B1) ).

Image of FIG. 13.

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FIG. 13.

Θ() vs. representation of effective Debye temperatures (cf. Fig. 8 ), which are due to the data in consideration for InP (Refs. (○), (△), and (□)), for InAs (Refs. (□), (○), and (△)), and for InSb (Refs. (□) and (○)). Solid curves are representing the fittings of these Θ() data by means of Eq. (B2) , with the empirical parameter values quoted in Table V . Dashed curves show the corresponding high-temperature dependences of the “true” (harmonic) Debye temperatures, Θ() (due to Eq. (B1) ).

Tables

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Table I.

Adjusted coefficients ( ) to ( ) and to due to Eq. (10) for the harmonic lattice heat capacity shape function, κ(), and associated anharmonicity-related coefficients, and , due to Eq. (12) . For the commonly considered thermo-chemical reference temperature, = 298.15 K, we have quoted the corresponding isobaric heat capacities, ( ) (12) , entropies, ( ) (17) , and enthalpy differences, Δ ( ) ≡ ( ) − (0) (17) .

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Table II.

Collection of smoothed isobaric heat capacity values, (J/(mol·K)), due to Eq. (12) (in combination with Eq. (10) ), and the corresponding effective Debye temperatures, Θ (K), due to Eq. (20) .

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Table III.

Quantities manifesting the close correlation (Eq. (28) and (29) ) between the general increase of ρ() ≡ ()/ curves (Eq. (19) ; cf. the insets to Figs. 1 to 6 )) from relatively low → 0 levels, ρ(0) = , up to their local maxima, ρ ≡ ρ( ), on the one hand, and the corresponding decrease of Debye temperature curves, Θ() (Eq. (23) or (24) ; cf. Figs. 6 and 7 ), from → 0 levels, Θ(0), to respective local minima, Θ ≡ Θ( ), on the other hand. A comparison of the magnitudes of various significant ratios between the quantities Θ(0), Θ( ), , Θ, , and Θ(∞) shows similarities of the shapes of the individual Θ() curves.

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Table IV.

Adjusted magnitudes of the empirical parameters involved by the presently devised polynomial (36) , which provided an accurate reproduction (shown in Figs. 10 and 11 ) of upper sections, () > 0.8 · (∞), of the previously fitted () curves (cf. Figs. 1 to 6 ).

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Table V.

Limiting values of “true” (harmonic) Debye temperatures, , and of the respective expansion coefficients involved by Eq. (B1) , to , which have been determined, in combination with the empirical parameters and , via fittings (see Figs. 12 and 13 ) of effective Debye temperatures, Θ(), by means of Eq. (B2) . Further quoted are the moment-related even-order phonon energies, ( = 2 to 10), which are resulting from Eqs. (B3a) to (B3e) for the corresponding moments, .

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/content/aip/journal/adva/3/8/10.1063/1.4818273
2013-08-06
2014-04-21

Abstract

Characteristic non-Debye behaviors of low-temperature heat capacities of GaP, GaAs, GaSb, InP, InAs, and InSb, which are manifested above all in form of non-monotonic behaviors (local maxima) of the respective ()/ curves in the cryogenic region, are described by means of a refined version of a recently proposed low-to-high-temperature interpolation formula of non-Debye type. Least-mean-square fittings of representative () data sets available for these materials from several sources show excellent agreements, from the liquid-helium region up to room temperature. The results of detailed calculations of the respective material-specific Debye temperature curves, Θ(), are represented in graphical form. The strong, non-monotonic variations of Θ() values confirm that it is impossible to provide reasonable numerical simulations of measured () 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 low-temperature () curves up to melting points was strongly impeded by partly rather large differences (up to an order of 10 J/(K·mol)) between the high-temperature 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 high-temperature data sets. Residual uncertainties for GaAs and InAs could be overcome by re-evaluations of former enthalpy data on the basis of a novel set of properly specified four-parameter polynomial expressions applying to large regions, from moderately low temperatures up to melting points. Detailed analytical and numerical descriptions are given for the anharmonicity-related 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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Non-Debye heat capacity formula refined and applied to GaP, GaAs, GaSb, InP, InAs, and InSb
http://aip.metastore.ingenta.com/content/aip/journal/adva/3/8/10.1063/1.4818273
10.1063/1.4818273
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