A thermodynamic approach of self- and hetero-diffusion in GaAs: connecting point defect parameters with bulk properties

Abstract

The self- and hetero-diffusion in GaAs is investigated in terms of the cBΩ thermodynamic model, which connects point defect parameters with the bulk elastic and expansion properties. Point defect thermodynamic properties, such as activation enthalpy, activation volume, activation Gibbs free energy, activation entropy and isobaric specific heat of activation, are calculated as a function of temperature for Ga, H and various n- and p-type dopants (Si, Be, Cr, Fe and Zn) diffused in GaAs. The present calculations are in good agreement with the reported experimental results. The pressure dependence of Ga self-diffusion is also investigated and the diffusivities and activation volumes are predicted at different temperatures from ambient pressure up to 10 GPa, above which GaAs is transformed into the orthorhombic structure. The activation volumes of dopants are also estimated at high temperature (1124 K), as a function of pressure.

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1. Introduction

The drive to replace silicon (Si) with alternative materials such as III–V semiconductors is mainly due to their advantageous material properties that can include high electron mobility and lattice matching with ternary (and/or quaternary) III–V compounds.^1–5 III–V materials such as GaAs are technologically important for nanoelectronic devices, for lasers and radiation detectors and for high efficiency solar cells.^3,6 The scientific interest for GaAs is still undiminished, both from a practical and from a theoretical point of view.^7,8

For the efficient miniaturization and optimization of electronic devices, it is necessary to understand defect processes formed during the growth processes.⁹ Previous studies provided fundamental insights into the diffusion and other defect processes of III–V semiconductors.^10–25 Gallium diffusion dominates self-diffusion in GaAs.¹⁶ The diffusion of various n- and p-type dopants in GaAs has been extensively investigated in the past.^13–25 Silicon is a commonly used n-type dopant,⁶ that although is amphoteric (it can occupy both the Ga and As sites) in As-rich grown GaAs Si is an n-type dopant occupying the Ga site. Regarding acceptor doping, beryllium and zinc are significant p-type dopants in GaAs with high diffusion coefficients.²⁶

Associating the defect Gibbs free energy gⁱ (i = defect formation f, self-diffusion activation act, or migration m) with bulk properties in solids is an issue that has been considered for more than six decades.^27–30 In the cBΩ model formulated by Varotsos et al. it was established that gⁱ is proportional to the isothermal bulk modulus B and the mean volume per atom Ω.^31,32 The efficacy of the cBΩ model has been demonstrated in numerous and diverse systems including defect processes in Si, Ge, ZnO, olivine, diamond, PbF₂, UO₂, LiH, and AgI.^33–42

In the present study we investigate the temperature and pressure dependence of self- and hetero-diffusion in GaAs in the frame of the cBΩ model. Based on elastic and expansion properties of GaAs, thermodynamic calculations of point defect parameters are carried out for the first time in a III–IV semiconducting system at high pressures, giving further insights to the diffusion mechanisms of various n- and p-type dopants. The structure of the present work is organized as follows. In Section 2.1, we present briefly the cBΩ thermodynamic model providing all the necessary formulas for the estimation of the point defect thermodynamic properties as a function of temperature and/or pressure. In Section 2.2 the necessary calculations for the determination of the bulk modulus and the mean atomic volume of GaAs are given and can be employed for any kind of dopant (n- or p-type). Subsequently, the application of the model to different dopants is presented and our predictions are discussed in comparison with the reported experimental results.

2. Methodology

2.1 Point defect parameters in terms of the cBΩ model

The self- or hetero-diffusion process of a single diffusion mechanism can be described by an Arrhenius equation of the diffusion coefficients D, according to the following relation

D = fga₀²νe^−gact/k_BT (1)

where f is the correlation factor depending on the diffusion mechanism and the structure, g is a geometry factor, a₀ is the lattice constant, ν is the attempt frequency and k_B is Boltzmann's constant.

In the frame of the cBΩ thermodynamic model,^31,32 the defect Gibbs free energy of activation g^act which is the sum of the formation (g^f) and migration (g^m) energy, is related to the bulk properties of the solid via

g^act = c^actBΩ (2)

where B is the isothermal bulk modulus, Ω is the mean volume per atom and c^act is a dimensionless factor that is considered to be independent of temperature and pressure under certain experimental conditions.

Based on eqn (2), the thermodynamic properties of the point defects i.e., activation volume υ^act, activation entropy s^act and activation enthalpy h^act of diffusion are expressed exclusively by the elastic and expansion properties of the bulk, at any temperature (or pressure), through the following relations

where β is the coefficient of volume thermal expansion which is temperature and pressure dependent. In addition, the isobaric specific heat of activation c^act_P can be also derived from eqn (5), according to

In order to estimate the aforementioned point defect parameters using the cBΩ model (eqn (2)–(6)), the constant c^act should be determined. The latter may be derived from eqn (2) where at T = 0 K the activation Gibbs energy becomes equal to the activation enthalpy and thus, c^act = h^act₀/B₀Ω₀. Although the activation enthalpy at zero temperature may be calculated from first principles, the parameter c^act may also be determined from the available experimental diffusion data.³² When only a single diffusion measurement D₁ at a given temperature T₁ (or pressure) is available, the parameter c^act can be calculated from eqn (1), if the pre-exponential factor fga₀ν is roughly known or can be calculated. Then, the diffusivity D can be calculated at any temperature (or pressure), provided that the elastic and expansion data are known for this temperature (or pressure). Alternatively, in the case where diffusion coefficients are experimentally known over a broad temperature (or pressure) range,^40,41 the mean value of c^act may be derived by substituting eqn (3) in (1) and taking the natural logarithm of both sides

According to eqn (7), a linear dependence of ln D versus the quantity BΩ/k_BT indicates the validity of the cBΩ model regarding a single diffusion mechanism, while c^act arises directly from the slope of the line.

The previous considerations for the determination of the parameter c^act are subjected to the experimental uncertainties of the diffusion experimental data which in many cases might be considerable. However, by dividing eqn (4) with (5) it arises that the ratio s^act/h^act depends only on the bulk properties of the crystal, according to the following relation⁴²

The quantities h^act and s^act can be usually determined experimentally from the slope and the intercept (i.e. the pre-exponential factor) of the corresponding Arrhenius plot of diffusion coefficients (see eqn (1)) and thus eqn (8) may also serve to validate the cBΩ model.⁴²

The aforementioned procedure may be used to calculate the temperature and/or pressure dependence of the thermodynamic properties of point defects, provided that the corresponding quantities (β, B, Ω) are known as a function of temperature and/or pressure. The pressure dependence of the diffusion coefficients may give us information about the diffusion mechanism in the host material, through the estimation of the activation volume. Indeed, if we differentiate eqn (1) with respect to pressure, the following expression is derived for the activation volume³²

In eqn (9), the derivative of the second term in the right is equal to κ(γ − 2/3) where γ is the Grüneisen constant. Recalling that the compressibility of the activation volume is defined as κ^act = (−1/υ^act)(∂υ^act/∂P)_T, the activation volume which is temperature and pressure dependent, is then expressed as

If we assume to a first approximation that κ^act is independent of pressure and that κγ ≈ κ₀γ₀ (the zero subscript refers to P = 0), we may express the exponential of eqn (10) in a series expansion of pressure, and by subsequent substitution in eqn (9), we finally get

Eqn (11) describes the pressure dependence of the diffusion coefficients at a certain temperature, considering that the parameters κ₀, γ₀, κ^act and υ^act(0,T) are known at this temperature.

2.2 Determination of the bulk properties

Experimental data of the bulk modulus B and the mean atomic volume Ω are not always available as a function of temperature (and/or pressure), especially at high temperatures where the diffusion experiments are usually carried out. An extrapolation of the available data at high temperatures (or pressures) by assuming linear relations of B and Ω are not always sufficient, and may lead to misinterpretation or incorrect validation of the model.

In the case of GaAs, the temperature dependence of the mean atomic volume at ambient pressure (Ω(P = 0,T) = Ω₀(T)) may be calculated from the following relation

where T_R refers to room temperature, Ω₀(T_R) = 22.59 × 10⁻²⁹ m³ is the mean volume per atom at ambient pressure⁴³ and β is the coefficient of volume thermal expansion. According to reported experimental results⁴⁴ of the linear thermal expansion coefficient a_l (see Fig. 1(a)), a linear relation holds for temperatures T > T_θ (T_θ = 360 K, is the Debye temperature of GaAs), with intercept a₀ = 5.67 × 10⁻⁶ °C⁻¹ and slope, a₁ = 0.0015 × 10⁻⁶ °C⁻². For the isotropic GaAs, β = 3α_l, and thus eqn (12) leads to the following result

Fig. 1 (a) Experimental data of the linear thermal expansion coefficient (TEC) of GaAs and the linear fitting at T > 350 K. (b) The calculated mean atomic volume in the temperature range 350–1500 K, according to the linear fitting of the TEC. (c) The calculated temperature dependence of the isothermal bulk modulus, according to the Rose–Vinet EoS. Linear approaches of Ω(T) and B(T) are also included for comparison.

The calculated temperature dependence of the mean atomic volume, according to eqn (13), is depicted in Fig. 1(b). Although a linear relation of Ω₀(T) is usually used in GaAs,⁴⁴ a small deviation from linearity is observed, especially at the temperature range of the reported diffusion data in GaAs (T > 500 K).

The Rose–Vinet universal equation of state (EoS)⁴⁵ is used to derive the temperature dependences of the bulk modulus B(T) and its pressure derivative (∂B/∂P)_T, according to the following expressions:

andwhere B₀ refers to zero (ambient) pressure, T_r is a reference temperature and the parameters X, η₀(T) are defined as followsand

In the case of GaAs, the bulk modulus B₀ and its pressure derivative at room temperature are B₀ (300 K) = (75.0 ± 0.5) GPa and (∂B/∂P)_P=0 (300 K) = (4.5 ± 0.5), respectively.⁴⁴ The temperature dependence of the calculated B(T) is shown in Fig. 1(c). The above EoS should give more accurate results but it is not deviate considerably from the simplified linear approach of bulk modulus that is usually considered in the case of the GaAs semiconductor,⁴⁴ as it is seen in Fig. 1(c).

3. Results and discussion

In the following, we investigate the self-diffusion of Ga and the diffusion of H, n-type (Si) and p-type (Be, Fe, Cr, Zn) dopants in GaAs, in the frame of the cBΩ thermodynamic model. The reported experimental diffusion coefficients of the aforementioned elements in GaAs as a function of the quantity BΩ/k_BT are shown in Fig. 2.^{13,16,20–22,24,25} The linear correlations between ln D and BΩ/k_BT (correlation factors, R > 0.99) suggest the validity of the cBΩ model and thus enable the estimation of the parameter c^act in each case, according to eqn (7). The calculated values of c^act for each element are listed in Table 1.

Fig. 2 Experimental diffusion coefficients of self- and dopant-diffusion in GaAs as a function of the quantity BΩ/k_BT. The linear fittings correspond to the implementation of the cBΩ model, according to eqn (7).

Table 1 Calculated values with their uncertainties of the parameter c^act, activation enthalpy (h^act_calc), activation entropy (s^act_calc), activation Gibbs free energy (g^act), activation volume (υ^act) and activation isobaric specific heat c^act_P, according to the cBΩ model for Ga self-diffusion and various dopants diffused in GaAs. The corresponding experimental values of activation enthalpy and activation entropy are also given for comparison. The experimental values of activation entropy (if not reported) were derived from the experimentally determined pre-exponential factor through D₀ = fga²ν₀ exp(s/k_B)

Element [ref.]	Temperature (K)	h^actexp (eV)	s^actexp (kB units)	c^act	h^actcalc (eV)	s^actcalc (kB units)	g^act (eV)	υ^act (×10^−29 m^3)	c^actP (eV K^−1)
Ga^16	1068–1503	4.24 ± 0.06	6.7–8.0	0.384 ± 0.010	(4.16–4.19) ± 0.17	(3.21–3.47) ± 0.14	(3.86–3.74) ± 0.15	(3.18–3.28) ± 0.13	(5.48–7.44) ± 0.26
H^25	528–753	0.83	0.56	0.022 ± 0.002	0.77 ± 0.08	(0.50–0.53) ± 0.05	(0.74–0.75) ± 0.07	0.58 ± 0.06	(0.53–0.74) ± 0.06
Si^13	863–1268	2.45	−3.4	0.210 ± 0.009	(2.27–2.28) ± 0.25	(1.69–1.82) ± 0.19	(2.08–2.14) ± 0.23	(1.72–1.76) ± 0.19	(2.46–3.50) ± 0.27
Be^20	873–1278	1.95	−6.4	0.181 ± 0.012	(1.96–1.97) ± 0.23	(1.46–1.57) ± 0.18	(1.79–1.85) ± 0.22	(1.48–1.52) ± 0.18	(2.14–3.04) ± 0.31
Fe^24	978–1178	2.7	11.4	0.238 ± 0.021	(2.57–2.58) ± 0.26	(1.96–2.03) ± 0.20	(2.37–2.41) ± 0.24	(1.96–1.99) ± 0.20	(3.13–3.71) ± 0.34
Cr^21	1063–1263	3.2 ± 0.4	17.1	0.284 ± 0.018	(3.08–3.09) ± 0.22	(2.40–2.49) ± 0.17	(2.80–2.85) ± 0.20	(2.36–2.39) ± 0.17	(4.24–4.92) ± 0.32
Zn^22	948–1123	3.21	3.97	0.306 ± 0.026	(3.31–3.33) ± 0.30	(2.57–2.68) ± 0.24	(3.07–3.11) ± 0.28	(2.51–2.54) ± 0.23	(3.91–4.57) ± 0.29

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The various point defect parameters (activation volume, activation entropy, activation enthalpy, activation Gibbs free energy and activation isobaric specific heat) were calculated for all the aforementioned diffusants, according to eqn (2)–(6) and the results are shown in Fig. 3, as a function of temperature. The ranges of these values along with their calculated uncertainties are summarized in Table 1. The corresponding reported experimental values of activation enthalpy, h^act_exp and activation entropy, s^act_exp are also included in Table 1 for comparison. The values of the experimental activation entropy were indirectly extracted from the experimentally determined pre-exponential factor D₀ (except for the reported value of Ga), through the relation

where in the case of GaAs we have used the reported values: g = 1, f = 1/2, α = 5.653 Å and ν = 7.17 ×10¹² Hz.^16,22,43

Fig. 3 Calculated point defect thermodynamic parameters of self- and dopant diffusion in GaAs as a function of temperature, according to the cBΩ model (eqn (2)–(6)). (a) Activation volume, υ^act (b) activation entropy, s^act (c) activation specific heat, c^act_P, (d) activation enthalpy, h^act (e) the term Ts^act, (h^act = g^act + Ts^act) and (f) the activation Gibbs free energy, g^act.

As it is depicted in Fig. 3, the estimated values of activation Gibbs free energy slightly decrease with temperature while the corresponding term Ts^act increases, resulting to constant values of activation enthalpy (h^act = g^act + Ts^act). These values of activation enthalpy, calculated in the frame of the cBΩ model, are in very good agreement with the reported experimental activation energies, if the uncertainties are taking into consideration (see Table 1).^{13,16,20–22,24,25} Our estimations of activation entropy deviate considerably from the corresponding reported values of most elements, except for the case of H, where accurate results are obtained. Notably, as indicated in Table 1, the reported values of activation entropy that were derived indirectly from eqn (18), are either large (Fe, Cr) or have negative signs (Si, Be). However, the negative values of entropy suggest a process of concentration due to the action of additional forces opposing to diffusion process while, the large values (>10k_B) are not compatible with reported moderate values of activation enthalpies.²² At this point, we have to note that the values of activation entropy are subjected to large uncertainties due to the experimental errors of the determination of activation enthalpy in diffusion measurements. Furthermore, this discrepancy is also due to the approach of the attempt frequency with the Debye frequency as well as the approximated values of the parameters f and g that were used in eqn (18). Alternatively, we propose that reliable values of activation entropy may be derived according to eqn (8), where the reported experimental values of activation enthalpy h^act_exp should be used. In this case, the 2^nd term of eqn (8) which depends only in the bulk properties of GaAs, increases smoothly from 5.79 to 7.15 × 10⁻⁵ K⁻¹ in the temperature range 350–1500 K.

Finally, our calculated activation volumes range from 0.26Ω₀ in the case of the smallest diffusant atom (hydrogen) to 1.45Ω₀ for Ga self-diffusion, but practically all values are not affected by temperature, over the entire temperature range of the reported diffusion experiments.

To the best of our knowledge, self- or hetero-diffusion measurements in GaAs under pressure are not available in the literature. The pressure dependence of the diffusion coefficients for Ga self-diffusion was estimated according to eqn (11), for pressures up to 10 GPa above which GaAs transforms from the zinc blende to the orthorhombic structure.^8,46 The values of the self-diffusion coefficients D(0,T) at zero pressure that were used in eqn (11), correspond to the reported experimental values of Fig. (3).¹⁶ Based on eqn (3), the compressibility of the activation volume may be deduced from the following relation

where κ₀ = 1/B₀ is the bulk modulus at zero pressure which is temperature dependent (see Fig. 1(c)). Considering that −B(∂²B/∂P²)_T ≈ (∂B/∂P)_T,⁴⁷ the activation compressibility is finally simplified to

We have to note that, in the quasi-harmonic approximation, the quantity (∂B/∂P)_T has a constant value and does not depend on pressure or temperature, while it varies only slightly upon compression in a real (anharmonic) solid.³² Thus, our previous assumption for pressure independent activation compressibility is a reasonable approximation in order to derive the analytical expression (see eqn (11)) for the calculation of diffusion coefficients as a function of pressure. The Grüneisen constant which is also temperature dependent, was estimated using the Dugdale–MacDonald equation,⁴⁸ γ = [(∂B/∂P)_T − 1]/2, which yields values from 1.80 to 1.84 in the temperature range (1068–1503 K) of the reported Ga self-diffusion coefficients.

By substituting all the above parameters in eqn (11), the pressure dependence of Ga self-diffusion coefficients in GaAs at different temperatures was estimated and the results are depicted in Fig. 4(a). Our estimations of activation volume (see eqn (10)) for Ga self-diffusion as a function of pressure are also shown in Fig. 4(b). Our calculations are restricted to 1248 K, in order to avoid the change of GaAs to its liquid phase, in the high range of pressures.⁴⁶ We observe that, in all cases, the influence of pressure on the activation volume υ^act is more pronounced than that of temperature, and all the values converge to around 1.0Ω₀ at high pressures. The pressure dependence of activation volumes of n- and p-type dopants (Si, Be, Fe, Cr and Zn) diffused in GaAs at constant temperature (T = 1124 K) is shown in Fig. 5.

Fig. 4 (a) Pressure dependence of Ga self-diffusion coefficients in GaAs, according to the cBΩ model in the temperature range, 1068–1248 K. The corresponding variation of activation volumes is shown in (b). The pressure range is limited to 10 GPa, above which GaAs is transformed into the orthorhombic structure.

Fig. 5 Pressure dependence of activation volumes of n-type (Si) and p-type (Be, Fe, Cr, Zn) dopants diffused in GaAs at constant temperature (1124 K).

Generally, the value and the sign of activation volume can provide information about the kind of the diffusion mechanism i.e., vacancies (V) or self-interstitials (I), through the relation υ^act_V,I = ±Ω₀ + υ^r_V,I + υ^m_V,I. The positive sign of the mean atomic volume Ω₀ refers to vacancy and the minus to self-interstitial formation, while υ^r_V,I denotes the relaxation volume around a newly-created point defect (V or I) and υ^m_V,I is the corresponding migration volume. In the case of dopants diffused in interstitial sites, the activation volume equals to the migration volume of the interstitial diffusant, since no defect formation is required.⁴⁹ In a first approach, the positive estimated value of activation volume in Ga self-diffusion could be an evidence for vacancy-mediated diffusion with an outward (positive) or inward (negative) relaxation around the vacancies. Since the relaxation and the migration volumes are usually assumed to be smaller than Ω₀ in magnitude, the sign of Ω₀ actually determines the dominant diffusion mechanism.⁵⁰ Thus, in the case of Ga self-diffusion, an outward strong relaxation and/or a positive migration volume could result to an activation volume greater than Ω₀ and comparable with our estimation (≈1.4Ω₀). This value is consistent with vacancy self-diffusion by second-nearest-neighbor hopping, proposed by Wager²² who performed atomistic thermodynamic calculations in order to study the energetics of Ga self-diffusion in GaAs.

In order to correlate the size of each diffusant atom to the corresponding activation volume and possible diffusion mechanisms, we have plotted in Fig. 6 the activation volume (at zero pressure) as a function of the atomic volume of the diffused elements using their covalent radii. Typically, the activation volume increases with the increase of the atomic volume of the diffusants, with noticeable deviation from linearity in the cases of Ga and Fe. Small atoms such as H preferentially reside in interstitials sites resulting in small value of activation volume, consistently with the present calculation (υ^act_H ≈ 0.25Ω₀). In the case of Si atom which is much larger than H, its value of activation volume (υ^act_Si ≈ 0.77Ω₀) indicates either an interstitial mechanism of diffusion where no defect formation is needed (υ^f_I = 0) or a vacancy mechanism with a strong inward relaxation. Similar speculations could be made for the other elements diffused in GaAs, in order to explain (or predict) possible diffusion mechanisms. For example, in the cases of Zn and Be diffusion in GaAs, activation volume is higher in the case of the larger Zn atom (υ^act_Zn > υ^act_Be) implying that diffusion in interstitial sites is more preferable than in vacancies where the contribution to the formation volume should be significant. This is consistent with the interstitial-substitutional exchange mechanism (kick-out mechanism) which has been proposed by Yu et al. for Zn and Be diffusion in GaAs.²⁶

Fig. 6 Activation volume of each diffusing element with respect to the atomic volume calculated with the covalent radius of the elements. The (red) line represents the linear fit of the data.

4. Conclusions

The present work investigates the self- and dopants-diffusion in GaAs, in terms of the cBΩ thermodynamic model which connects the point defect parameters with the bulk elastic and expansion properties of the host material. Initiating from available experimental data of thermal expansion and the usage of the Rose–Vinet equation of state for GaAs, useful point defect parameters such as activation enthalpy, activation entropy, activation volume and activation isobaric specific heat were calculated as a function of temperature. Our results are in excellent agreement with the reported experimental activation enthalpies. Thus, we suggest that the activation entropies could be generally calculated from the experimentally determined activation enthalpies by using only the bulk properties of the material under investigation, in order to overcome the experimental uncertainties. The pressure dependence of Ga self-diffusion coefficients and activation volumes of Ga, H and other n- and p-type dopants were also estimated for the first time at elevated pressures and are discussed in the context of possible diffusion mechanisms reported in the literature. The potential of the model for the investigation of diffusion processes on a theoretical basis in previously studied unary semiconductors (Si and Ge) and binary semiconducting systems as in the present case may be also extended to other important III–IV semiconductors with limited diffusion experimental data and may suggest a possible universal application in different kind of solid compounds.

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