Comparison of some theoretical models for fittings of the temperature dependence of the fundamental energy gap in GaAs

Comparison of some theoretical models for fittings of the temperature dependence of the fundamental energy gap in GaAs

R. R. O. Morais^I,*; I. F. L. Dias^I; J. L. Duarte^I; E. Laureto^I; S. A. Lourenço^II; E. C. F. da Silva^III; A. A. Quivy^III

^IDepartamento de Física, Universidade Estadual de Londrina – UEL, CP6001, CEP 86051-970, Londrina, Paraná-Brazil

^IIInstituto de Física, Universidade Federal de Uberlândia, CP593, CEP 38400-902, Uberlândia, MG, Brazil

^IIIInstituto de Física, Universidade de São Paulo – IFUSP, CP66318, CEP 05315-970, São Paulo, São Paulo, Brazil

ABSTRACT

In this work we report on a comparison of some theoretical models usually used to fit the dependence on temperature of the fundamental energy gap of semiconductor materials. We used in our investigations the theoretical models of Viña, Pässler-p and Pässler-ρ to fit several sets of experimental data, available in the literature for the energy gap of GaAs in the temperature range from 12 to 974 K. Performing several fittings for different values of the upper limit of the analyzed temperature range (T_max), we were able to follow in a systematic way the evolution of the fitting parameters up to the limit of high temperatures and make a comparison between the zero-point values obtained from the different models by extrapolating the linear dependence of the gaps at high T to T = 0 K and that determined by the dependence of the gap on isotope mass. Using experimental data measured by absorption spectroscopy, we observed the non-linear behavior of E_g(T) of GaAs for T > Θ_D.

Keywords: Non-linear behavior of E_g(T); Zero-point energy; Temperature dependence of the gap; Gallium arsenide

I. INTRODUCTION

The dependence on temperature of the energy gap, E_g(T), is a very important characteristic of semiconductor materials and heterostructures and of great scientific and technological interest. For a large number of semiconductor materials, the energy gap decreases with the increase of the temperature and shows different behaviors depending of the analyzed range of temperature. At low (T < 0.02 Θ_D, where Θ_D is the Debye temperature) and intermediary temperatures (T = Θ_D) the decrease of E_g(T) is non-linear. For high temperatures (T >> Θ_D) the energy gap decreases linearly with temperature [1-3]. The linear decrease of E_g(T) occurs due to the contribution of two distinct mechanisms: the electron-phonon interaction and the thermal expansion of the lattice [4-7]. In general, the contribution of the thermal expansion for the change of E_g(T) is small as compared to the contribution of the electron-phonon interaction [7, 8]. By this reason, several models proposed in the literature just take into account the electron-phonon interaction as a first approximation to the behavior of the E_g(T) [9]. In the case of GaAs, the electron-phonon interactions which lead to a decrease of the energy gap with the temperature are associated to the longitudinal-acoustic (LA) and the longitudinal-optical (LO) phonons, being the dominant contribution due to the LA phonons [9-11].

Among several models usually used to describe E_g(T), we find in the literature the distinguished models of Viña [12] (1984), Pässler-p [9] and Pässler-ρ [13] (1997). Essentially, these models differ among themselves with respect to the behavior of E_g(T) at low temperatures. The model of Viña [12] shows a plateau behavior for T < 50 K, making clear the absence of any dependence of the energy gap with temperature. The Pässler-ρ [13] model shows a quadratic dependence of E_g(T) [14]. The Pässler-p model shows a T^p [1] dependence for T < 0.02 Θ_D with p = 4. For several semiconductor materials E_g(T) still shows a T^p-dependence with the increase of temperature, but for values of p in the range from 2 up to 3.3. In the limit of high temperatures (T >> Θ_D), all the models describe a linear asymptotic behavior of the energy gap with temperature given by [13]:

where α_i = -dE_gi(T)/dT)_T→ ∞ = α_B, α_p or α_ρ· Θ_i = Θ_B, Θ_p or Θ_ρ are empirical parameters related to the effective energy of the phonons in the temperature scale (Θ_eff = ‹ε›/k_B)) associated to the models of Viña, Pässler-p and Pässler-ρ respectively.

Extrapolating the linear dependence of E_g(T) from high temperatures back to T = 0 K, it is possible to estimate the renormalization of the energy gap at zero temperature, i.e., the zero-point energy ΔE(0) defined as ΔE(0) = E(0) - E_g(0), where the non-renormalized energy gap, E(0), is the energy of the crystalline system with immobile atoms at theirs equilibrium positions, and E_g(0) is the renormalized gap at T = 0 K. For the different models, ΔE(0) can be written as:

with α_i and Θ_i as defined in Eq. (1). It is worthwhile to comment that, according to D. Rönnow et al [15], the renormalization of the gap is highly dependent on the model used to fit the temperature dependence of the energy gap. Still according to the same authors, the value of ΔE(0) estimated from the temperature dependence of E_g(T) should be used as a criterion for the applicability of the different proposed models [15]. Results from the literature show that the values of ΔE(0) for GaAs and others semiconductor materials seem to be dependent on the upper limit of the range of temperature used to perform the fits [16].

Due to the possibility to produce semiconductor materials with precise isotopic composition, nowadays it is possible to analyze the material properties as a function of the variation of the isotopic masses. At low temperatures, the change of the energy gap with the isotopic mass is proportional to M^-1/2, where M is the mean effective masses of the constituent atoms of the material [17]. In the high temperature range, the energy gap does not depend on M. For GaAs, due to its similarity with Ge and due to the fact that As has only one stable isotope, the effect of the substitution of the ⁶⁹Ga and ⁷¹Ga isotopes on the energy gap can be described by [18]:

where M_Ge = 70 amu and ΔE(0) = 53 meV. N. Garro et al [19] obtained ∂E_g/∂M_Ga = 0.39 (0.06) meV/amu for the direct energy gap of GaAs using reflectivity measurements at 10 K, and 0.43 meV/amu performing calculations with the pseudopotential method. If we use the experimental value ∂E_g/∂M_Ga = 0.39 meV/amu in Eq. (3), a value of ΔE(0) = 110 meV is obtained for GaAs. As a matter of fact, the values of ΔE(0) that are found in the literature for GaAs from estimates of isotopic substitution, linear extrapolation, and theoretical calculations are quite different [16,18,20]. According to our understanding, one important factor that must be taken into account for the analysis of these contrasting results is the lack of systematic experimental date for T > Θ_D, which hides a possible non-linear behavior of E_g(T) in the high temperature range. With respect to the issue, we notice that several estimates of ΔE(0) obtained by extrapolation procedures of the E_g(T) curve at T = 0 K are obtained with parameters determined from fits in small domains of temperature, with the upper limit of the temperature interval smaller than the Debye temperature [16]. According to Pässler [3], a precise description of E_g(T) and a reliable extrapolation procedure from high to very low temperatures can only be obtained if the experimental data set is measured up to or higher than the Debye temperature.

However, the Debye temperature of several materials is relatively high (for instance, Θ_D(C) = 1880 K [3], Θ_D(AlN) = 1020 K and Θ_D (GaN) = 870 K [21]), and there is a lack of experimental data for several materials at temperatures around Θ_D. Experimental data from Panish [22], Shen [23] and Lautenschlager [11], usually used in the literature to describe E_g(T) at high temperatures [3, 24], covers a wide temperature range but have a reduced number of data points with a large numerical dispersion, which does not allow us to infer about the non-linear behavior of E_g(T). In a relatively old work, Bludau et al [25] analyzing the dependence on temperature of the indirect energy gap of Si have mentioned that it seems that the predicted linearity of E_g(T) is not a precise description of the behavior of energy gap at high temperatures. Cardona and Thewalt [16] also make remarks about the accuracy of the results deriving from perturbation theory that keeps only up to the second order terms of the atomic displacements (proportional to < u² >) in the Hamiltonians used to describe the electron-phonon interactions. At high temperatures < u² > is proportional to T, but the inclusion of higher order terms which are not considered in the linear approximation could lead to larger values of the exponents of T. In this way, procedures based on the linear approximation together with extrapolation from high to low temperatures or only the high temperature behavior of E_g(T) can lead to incorrect values of the physical quantities as ΔE (0) or erroneous calculations of the thermal expansion contribution and the electron-phonon interaction to the change of the energy gap with temperature.

In the present work we analyzed the applicability of different theoretical models to describe to dependence of the fundamental energy gap with temperature in the range from 12 up to 957 K/974 K. To analyze the region for temperature T > 300 K we used experimental data from several researchers [11, 22, 23, 26, 27]. Performing several fittings for different values of the upper limit of the analyzed temperature range (T_max), we were able to follow in a systematic way the evolution of the fitting parameters up to the limit of high temperatures and discuss the origin of the contrasting results related to the renormalization of the zero-point energy. Moreover, based on the experimental data of Johnson et al [27] we were able to confirm evidences of non-linear behavior of the GaAs energy gap for T > 300 K.

II. THEORETICAL DETAILS

In this section, we present a short description of the theoretical expressions used to perform the fits of E_g (T) versus T. The expression proposed by Viña et al [12] is given by:

where E_g (T = 0) = E_B - a_B is the energy gap at T = 0 K; a_B represents the strength of the electron-phonon interaction; Θ_B = ħω/k_B is a characteristic temperature representing the effective mean energy of the phonons (Θ ≡ ‹ε›/k_B) on the temperature scale, which coincides with the phonon temperature in the single oscillator model, but does not have any correspondence with any peak in the phonon spectra; α_B = 2a_B/Θ_B is the slope of the curve E_g(T) versus T as T → ∞, i.e., α_B ≡ - (dE_g(T)/dT)_{T→ ∞}. The material-specific degree of phonon dispersion (dispersion coefficient), given by Δ = [(<ε²> - <ε >²)/ <ε>²]^1/2, is zero [28] in this case.

The expression proposed by Pässler [9] in the p-type model is given by:

where E_gp(0) is the energy gap at T = 0 K; α_p is the slope of E_g(T) as T→ ∞ i.e., α_p ≡ (dE_g(T)/dT)_{T→ ∞}; Θ_p is an empirical parameter associated to the effective mean energy of the phonons in the temperature scale Θ = ‹ε›/k_B and is given by the expression Θ = Θ_p [1.152 +0.145 ln(p - 1.7)], with a dispersion coefficient given by Δ = (p²-1)^-1/2, where p is an exponent of the spectral function.

The other model used in our work is the ρ-type Pässler model [13]:

where E_gρ(0) = E_g (T = 0) is the energy gap at T = 0 K; α_ρ is the slope of E_g(T) as T→ ∞, α _ρ ≡ -(dE_g(T)/dT) _{T→ ∞}; Θ_ρ is an empirical parameter related to the cutoff frequency ω₀ (Θ_ρ = ħω₀/k_B) and to the effective mean energy of the phonons in the temperature scale and obeys the relation Θ_r = Θ_ρ(1 - 0.5ρ), with a dispersion coefficient given by Δ = (2 - ρ)^-1[ρ(4 - 3ρ)/3]^1/2. The ρ-parameter controls the relative weight of each type of interaction: ρ → 1 represents the prevalence of the acoustic-phonon interaction, and ρ → 0 represents the prevalence of interaction with optical phonons.

In order to determine the fit parameters related to each theoretical model, we used a chi-squared procedure. The criterion used to obtain the best-fit parameters was calculated according to the expression:

where () indicates the experimental (adjusted) values of the energy gap of the ith-data point. The factor (n - m) is the number of degrees of freedom left after fitting n data points with m adjustable parameters in the fitting function.

III. EXPERIMENTAL DETAILS

The experimental data of the temperature dependence of the energy gap used in the present work were redigitalized from the following published data obtained with different characterization techniques: photoluminescence data from Grilli et al [26], in the range from 12 K up to 280 K / 33 data-points; Lautenschlager et al [11] (ellipsometry), from ~ 297 K to 510 K / 9 points; Shen et al [23] (absorption) from ~ 300 K to 881 K / 6 points; Panish et al [22] (absorption), from 294 K to 974 / 5 points, and Johnson et al [27] (absorption) from 330 K to 957 K / 23 points. Details of sample preparation and experimental procedure can be found in the cited references.

In order to perform a systematic analysis of the experimental results, we grouped the experimental data in two groups: I) data from 12 K to 280 K from Grilli et al [26], data from 294 K to 974 K from Lautenschlager et al [11], Shen et al [23], and Panish et al [22]; II) data from 12 K to 280 K from Grilli et al [26] and data from 330 K to 957 K from Johnson et al [27]. The experimental data from group I for T > 294 K is a very consistent set of data which allowed us to perform accurate fittings for temperatures above the room temperature but with a large dispersion in that whole temperatures range (see Fig. 1). A still more consistent data set analyzed by us consists of the absorption data measured in the Urbach-tail region by Johnson et al [27] (group II). So, according to our strategy, for the high temperature region we have one set of data with 20 data-points (group I) and another set with 23 data-points (group II) with almost equal upper limit temperatures (974 K / 957 K), which enable us to perform a systematic analysis of the applicability of the different models to describe E_g(T). From these experimental data we were also able to compare the results obtained from different data sets and to check anomalous behaviors of the obtained fit parameters.

IV. RESULTS AND DISCUSSIONS

In Fig. 1 we show the experimental data for E_g (T) from the groups I and II in the temperature range ~ 12 < T < 974 (957) K, and the fitted curves using the different models considered in the present work. The inset of this figure shows the redigitalized data of E_g(T) in the range ~ 12 K < T < ~ 100 K together with the fitted curves considering the whole analyzed domain of temperatures. We can observe in the region of low temperature the same pattern already observed in the literature [13, 29] with the Viña model showing a plateau behavior for T = 50 K and values of S² larger than that ones obtained with the other two models. The fit with the Pässler-ρ model leads to an approximately quadratic behavior of E_g(T) with a value of S² comparable to the one obtained with the Pässler-ρ model. The values of the parameters obtained from the fits according to the different models are shown in Table I. The extrapolations to high temperatures (T > 280 K) for E_g(T) (see Fig. 1) and for the adjusted parameters (see Figs. 2-4) were performed using the data of Grilli et at [26] in the temperature range 12 K < T < 280 K. As can be observed, the curves obtained by extrapolation are located above the experimental points. This fact confirms that erroneous conclusions about the behavior of E_g(T) at high T are inferred if the experimental data are within a small temperature range. The fits obtained with the different models in the full temperature range coincide in the high temperature region for both groups of experimental data.

We see in Fig. 1 that the experimental values of E_g(T) obtained by Johnson et al [27] are located, in general, below the data from group I. This fact is well evident when we observe the curves fitted to the different sets of experimental data. A possible explanation for this difference can be related to the different procedure used by Johnson et al [27] to obtain E_g(T), which is based on observation of the Urbach tail. Another possibility which should also be considered is that the data of Johnson et al [27] were obtained with the same technique, the same procedure, the same sample in the whole domain of analyzed temperatures, which can provide a more systematic behavior for E_g(T), with a smaller dispersion, differently from the others data sets.

In Figs 2(b)-(d) we show the parameters E_B, Θ_B, and a_B obtained with the model of Viña [12]. In order to compare these data with the others obtained by us, we also show E_gB(0) = E_B - a_B (Fig. 2(a)) and α_B = 2a_B/Θ_B (Fig. 2(e)). The symbols represent the values of the parameters obtained in different ranges of temperature for different values of T_max. The values of E_gB(0), determined for different values of T_max, are almost constant in the whole temperature range for the different sets of experimental data. The parameters E_B, Θ_B, and a_B obtained from group I increase in the range 12 K < T < 280 K showing a tendency to saturate for T > 350 K. The parameters E_B, Θ_B and a_B from group II show a slight tendency to increase with the increase of T_max. The values for ΔE(0) obtained from the product α_B = 2a_B/Θ_B using T_max ~ 3Θ_D (Θ_D = 360 K for GaAs) are ~ 64.1 meV and ~ 73 meV for the two experimental data groups I and II respectively (see Table I). The values of the parameters determined from extrapolations to high temperatures stay constant with the increase of T_max and the obtained curves pass well below the experimental points. Therefore, we verify that to obtain consistent extrapolations using the Viña model it is necessary to have experimental data beyond the Debye temperature. The parameters obtained with the Pässler-p model [9] are presented in Figs. 3(a)-(d). The parameter a_p defined as a_p = α_pΘ_p/2 is shown in Fig. 3(c). The values of the gap parameter E_gp(0) (Fig. 3(a)) has practically a constant value with T_max. The parameters Θ_p, α_p and a_p show a similar behavior with T_max (compare the shape of the curves in Figs. 3(b)-(d)). Concerning the results from group-II, we verified a step increase of Θ_p, α_p and a_p parameters in the range 280 K < T < 350 K and a small decrease with further increase of the temperature. The parameter p has values in the range from 2.4 up to 2.7 in the whole analyzed temperature range, which gives intermediary dispersion coefficients (0.3 < 0.4 (p = 2.7) < Δ < 0.54 (p = 2.4) < 0.577) and leads to a small difference among the results from the different groups of data in the temperature range delimited by T_max (see Table I). In general, the standard behavior of the parameters determined from the different data sets by the Pässler-p model is very similar. The curves obtained with the parameters determined from the extrapolation procedure are very close to the experimental data (see Fig. 3).

Based on our findings we can say that the extrapolations using the Pässler-p model provide results more consistent with the experimental data for T > 280 K. We also verified that the curves determined by extrapolation for T > Θ_D ( Θ_D = 360 K for GaAs) are superimposed to the experimental data in this temperature range. Larger errors in the range 280 K < T < 500 K are probably due to the dispersion of the group-I experimental data. The parameter a_ptends to ~ 58 meV ( ~ 68 meV corrected) and to 70 meV (84 meV corrected) for the two experimental data group I and II respectively (see Table I).

In Figs. 4 (a)-(e) we show the parameters obtained from the Pässler-ρ model [13]. We see in Fig. 4(a) that the values E_gρ(0) are practically constant with T_max. The parameters Θ_r, a_ρand α_ρ show similar trends with the increase of the temperature. The results of group II show an abrupt increase in the range 280 K < T < 350 K. The parameter Θ_r from group I show large error bars and oscillatory behavior around 500 K. Moreover, the tendency of all the parameters with the increase of the temperature is practically the same and does not depend on T_max. The ρ-parameter has values in the range from 0.4 up to 0.8 for groups I and II. However, its value is more concentrated in the region 0.3 < ρ < 0.6, which leads to an intermediary dispersion in the range 0.328 < Δ < 0.574, in agreement with the criterion of applicability of the Pässler model [29]. Similarly to the results of the Pässler-p model, the results obtained with Pässler-ρ are also in excellent agreement with the experimental data and are more consistent when compared to the ones obtained with the Viña model. The parameter a_ρ tends to ~ 65 meV and ~ 76 meV for the two experimental data group I and II respectively (see Table I). In general, the fitting parameters are strongly dependent of the range of temperature chose to perform the fits, i.e., the chosen value of T_max, mainly in the range 12 K < T < 450 K for the case of the Viña model, and in the range 12 K < T < 300 K for the Pässler-p and Pässler-ρ models. Therefore, the extrapolations of the fitted curves for E_g(T) from the low temperature experimental data to the high temperature regions lead to parameters with very consistent values for T > 450 K for the Viña model, and T > 360 K for the Pässler-p and Pässler-ρ models. Considering T_max = 974 K or 957 K, the Pässler-ρ model provides parameters with larger error bars. In order to correlate our findings with the GaAs density of phonons we must take into account a correction for the value of Θ_p according to the expression Θ_eff≡ < ε/k_B > = Θ_p [1.152 +0.145ln(p- 1.7)] [3]. This correction increases about 10% to 20% the value of Θ_p if we consider the range 280 K < T < 957 / 974 K. Considering the fits performed for T_max (957/974 K) the changes of the values of k_BΘ_i (i = B, p effective, ρ) due to the inclusion of the corrections vary from 22.2 meV (Θ_B from group I) up to 27.8 meV (Θ_effp from group II), as can be seen in Table I. The values of Θ_i from the two data sets are, therefore, below the cut-off energy of the LA-phonon branch (ε_LA = 28 meV). This result indicates that the decrease of the energy gap with temperature is mainly due to the interaction of electrons with LA phonons.

Complementary analysis, performed by expanding the range of temperature up to T ~ 1179 K and the models of Viña and Pässler-ρ along with the data from Beaudoin et al [30] which were obtained with a similar experimental procedure used by Johnson et al [27], show that the value of the a_i parameter does not show an enlargement tendency with the increase the expansion of the temperature range, becoming stabilized around ~ 75 meV (Viña) and ~ 79 meV (Pässler-ρ) [31]. The values obtained here are therefore different from that founded in the literature obtained extrapolating the linear dependence of the gaps at high T to T = 0 K and that determined by the dependence of the gap on isotope mass [16,18,20]. As we mention before, we consider that the lack of experimental data of E_g (T) in the high-T range in the fit models of E_g (T) is one of the relevant factors to consider for the precise description of E_g (T). In face of the dispersion of results for ΔE(0) values from both procedures (extrapolating the linear dependence of the gaps at high T to T = 0 K and the dependence of the gap on isotope mass) we inferred that or the linear dependence of E_g (T) at high temperatures (T >> Θ_D) is not reached in the analyzed temperature range or, as pointed out by Cardona and Thewalt [16], electron-phonon anharmonic effects lead to a higher-order dependence of the energy gap with temperature. Therefore, the choice of upper limit of temperature to which E_g (T) has a linear dependence with T is arbitrary, and can lead to incorrect values of ΔE(0) when Eq. 2 is used.

For instance, using the Pässler-p model [9] to describe the temperature dependence of the energy gap of GaAs in the range from 2 K to 280 K, and the values Θ_p = 226 K, α_p = 0.347 meV/K and p = 2.51, Cardona et al [18] estimated ΔE(0) = 53 meV (- 53 meV in the Cardona [16] notation) from the photoluminescence data measured by Grilli et al [26]. Using another set of experimental data with a different upper limit defining the domain of temperature (974 K ~ 2.7Θ_D), Cardona et al [18] obtained ΔE(0) = 90 meV. As can be find out from these results, the values of ΔE(0) obtained for GaAs seem to be dependent on the upper limit of the domain of temperature used to perform the fits. It is also important to say that the value of 90 meV was obtained from the fits of two sets of experimental results: (i) the data obtained by Grilli et al [26] in the range 2 K < T < 280 K and (ii) the absorption data obtained by Panish et al [22] in the high temperature region (294 K < T < 974 K). However, these data for high temperatures were recorded in a relatively large temperature range with a reduced set of data points (5 points), which prejudices the fits and the carried out analysis.

The lack of systematic experimental data for temperatures above the room temperature is an entanglement for the analysis of E_g (T) for a large number of materials. The data from Johnson et al [27] allow, a more systematic analysis of E_g (T) for GaAs. Using the finite difference method to analyze the experimental data of E_g (T) obtained by Johnson et al [27], it is possible to visualize the change of the parameter α_exp = - ΔE_gexp(T)/ΔT by performing derivatives in successive ranges of E_g (T). In Fig. 5 we show the values of α_exp as a function of the mean temperature for each interval ΔT obtained with the data of Johnson et al [27]. Therefore, α_expn is given by:

where T_n and T_n-1 are consecutive values of T, and E_g (T_n), E_g(T_n-1) are the respective experimental values of the energy gap. In general, in the range from 330 K up to 957 K, the value of α_expn decreases from -0.47 to -0.55 meV/K between two consecutive intervals. This variation of α_expn is relatively large ( ~ 17%) considering the predicted linear behavior of E_g (T) at high temperatures. The non-linear behavior of E_g (T) was already foreseen by Bludau et al [25] who used second order polynomes to fit the temperature dependence of Si indirect-energy gap. Cardona and Thewalt [16] also point out the importance of anharmonic terms (which leads to higher order terms in T) in the electron-phonon interaction to analyze the temperature dependence of the lattice parameter of Si. The experimental data of E_g (T), from the results of Johnson et al [27], even in the limit of high temperatures, show a small curvature that must be taken into account when we analyze the temperature dependence of the energy gaps. The extrapolation from the linear dependence at high temperature to low temperature tends to be done with some degree of arbitrariness, which can explain the contrasting results of ΔE(0) obtained with from the different experimental procedures. Therefore, the results of Johnson et al [27] allowed us confirm the non-linear temperature dependence of the GaAs energy gap at high temperature. So, our findings indicate that more systematic experimental studies searching the high temperature region (T >> Θ_D) must be performed in order to obtain a precise description of the dependence of the energy gap with temperature.

V. CONCLUSIONS

In this work we analyzed the temperature dependence of the GaAs energy gap using different sets of experimental data. We performed fits of E_g(T) versus T in different domains of temperature defined by different upper temperatures. We were able to make a systematic comparison of the fitting parameters obtained with three theoretical models in a large temperature range (12 K to 974 K / 957 K). The values obtained for the zero point renormalizations energy - ΔE(0) - from different experimental data groups and from different theoretical models for fitting the temperature dependence of GaAs obtained extrapolating the linear dependence of the gaps at high T to T = 0 K varying between 64.1 meV (Viña model) and 84 meV (Pässler-p model).

Using absorption experimental data measured in temperatures higher than the room temperature we were able to observe the non-linearity of E_g (T) as already reported in the literature, with a quite large value for dE_g (T)/dT in the range 280 K < T < 957 K. We believe that a more realistic description of E_g (T) can only be obtained if the effects of the non-linearity at high temperatures are included in the theoretical models describing the dependence on temperature of the energy gap.

ACKNOWLEDGEMENTS

The authors would like to thank the financial support granted by the Brazilian agencies CAPES, CNPq, Fundação Araucária and FBB.

(Received on 24 November, 2008)

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Publication Dates

• Publication in this collection
  22 Apr 2010
• Date of issue
  Mar 2010

History

• Received
  24 Nov 2008