Thermodynamic analysis of semipolar GaN and AlN under metalorganic vapor phase epitaxy growth conditions

We perform an improved method of thermodynamic analysis for semipolar 11 2 ¯ 2 and 1 1 ¯ 01 surfaces of GaN and AlN to elucidate the relationship between growth conditions and semipolar and polar surfaces during metalorganic vapor phase epitaxy (MOVPE). The calculations for H2 carrier gas suggest that for both GaN and AlN the maximum temperature for growth on 11 2 ¯ 2 surfaces is higher than that for growth on 0001 surfaces. On the other hand, the maximum temperature for growth on GaN 1 1 ¯ 01 surfaces is comparable to that for growth on GaN 0001 surfaces, while the maximum temperature for the growth on AlN 0001 surfaces is higher than that for growth on AlN 1 1 ¯ 01 surfaces. These results could be used to provide favorable conditions for growth of group-III nitrides along a semipolar orientation during MOVPE.


Introduction
Group-III nitrides including GaN and AlN have attracted much attention for applications in optoelectronic devices such as laser diodes and light-emitting diodes. [1][2][3][4][5][6][7][8] In order to fabricate these devices, the growth of high-quality layers has been carried out along various orientations using epitaxial growth methods such as metalorganic vapor phase epitaxy (MOVPE), hydride vapor phase epitaxy (HVPE) and molecular beam epitaxy (MBE). In general, epitaxial growth of group-III nitrides has been performed along the polar [0001] direction. [9][10][11][12][13][14] However, it is well known that [0001]-oriented group-III nitrides exhibit a strong electronic field produced by spontaneous and piezoelectronic fields, which causes reduced radiative efficiency and redshift of optical transitions. To overcome these problems, there has been increasing interest in epitaxial growth along nonpolar and semipolar orientations. [15][16][17] Despite these experimental studies, understanding of the optimum growth conditions and their dependence on orientation is still lacking. This could be elucidated by understanding the growth reactions on a reconstructed semipolar surface, which depend on growth conditions such as partial pressure and temperature.
In our previous studies we proposed an approach for calculating the absolute surface energy of semipolar planes such as ( ) 1122 and ( ) 1101 using a wedge-shape geometry method. 18,19) This approach enabled us to directly compare the absolute surface energy in a semipolar orientation with that in a polar orientation. Therefore, our calculated results revealed the relative stability between semipolar and polar surfaces. However, these absolute surface energies do not incorporate the experimental conditions such as the temperature and partial pressure of molecules supplied during epitaxial growth. To compare the theoretical results with experimental results, we recently proposed an improved thermodynamic analysis that incorporates absolute surface energies on GaN of polar and nonpolar planes obtained by ab initio calculations. 20,21) According to a thermodynamic study, growth on GaN( ) 0001 surfaces is possible at higher temperatures than on ( ) 0001 surfaces during MOVPE. 21) As a result, the maximum growth temperature on polar surfaces determined by the thermodynamic approach enables us to compare the calculated results with experiments directly. However, the maximum temperature of semipolar group-III nitride surfaces under MOVPE growth conditions is still unclear. To discuss the relationship of growth conditions among various surface orientations, thermodynamic analysis of group-III nitrides on semipolar surfaces is required.
In this study, we perform the improved thermodynamic approach for semipolar ( ) 1122 and ( ) 1101 surfaces of group-III nitrides. The analysis is carried out for both semipolar and polar AlN surfaces, because the maximum temperature of MOVPE growth on polar AlN surfaces has not yet been clarified. In this paper, the stability of polar and semipolar surfaces such as ( ) 1122 and ( ) 1101 taking account of hydrogen atoms is determined using absolute surface energies. Furthermore, the orientation dependence of growth conditions is discussed on the basis of the temperature dependence of equilibrium partial pressure. Our calculations demonstrate characteristic features of the growth reaction on semipolar surfaces under MOVPE growth conditions. Section 2 briefly explains the computational method, presenting ab initio calculations and thermodynamic analysis. The results and discussion for semipolar GaN surfaces, polar AlN surfaces and semipolar AlN surfaces are given out in Sects. 3.1-3.3, respectively. In Sect. 3.4, the differences in growth conditions, such as maximum temperature and carrier gases, are also discussed. Finally, we summarize this study in Sect. 4.

Computational methods
The total energy calculations are performed within the planewave pseudopotential approach using the generalized gradient approximation. 22) We use norm-conserving pseudopotentials for Al, Ga and H atoms 23) and ultrasoft pseudopotentials for N atoms. 24) Ga 3d electrons are treated by partial core corrections. 25) The conjugate-gradient technique 26,27) is utilized for both the electronic structure calculations and geometric optimization. The valence wave functions are expanded by the plane-wave basis set with a cut-off energy of 30.25 Ry. We use 72 k-point sampling for the 1 × 1 surface unit, which provides sufficient accuracy for the total energy. The computations are performed using the Tokyo Ab initio Program Package. 28) The semipolar surfaces are simulated using 1 × 1, 2 × 2, ( ) c 2 2 and1 2 slab models consisting of seven bilayers and a vacuum region of approximately 10 Å. The bottom surface of the slab is passivated with fractional hydrogen atoms 29) and the lowermost four layers are fixed at ideal positions. These atomic configurations are considered according to previous calculations 30,31) and the electron counting (EC) rule. 32) The relationships between them are expressed as m m +   2 and m H gas 2 are the gas phase chemical potential of Ga, N 2 and H 2 molecules, respectively. The gas phase chemical potential of isolated atoms and molecules is given by 33) where k B is the Boltzmann constant, T is the gas temperature, g is the degree of degeneracy of the electron energy level, p is the partial pressure and z , trans z rot and z vibr are the partition functions for translational, rotational and vibrational motions, respectively.
For MOVPE growth of GaN, trimethylgallium (TMG) and ammonia are used as the Ga and N sources, respectively, and N 2 and H 2 are the carrier gases. TMG mostly decomposes into Ga atoms in the gas phase at general growth temperatures. [34][35][36] The decomposition reaction is described as where X(g) [X = (CH 3 ) 3 Ga, H 2 , Ga or CH 4 ] denotes the gas phase. Furthermore, the decomposition reaction of ammonia molecules from inlet to the growth section is expressed as Here, α is the decomposition ratio of ammonia molecules. In this study, α is set to 0.25, which is obtained by thermodynamic analysis of group-III nitrides under MOVPE growth conditions. [37][38][39] In order to discuss the relationship between growth conditions for semipolar and polar surfaces, it is necessary to estimate the equilibrium partial pressure p between the gas phase and the surface phase. The chemical reaction which occurs at the substrate surface to form GaN is expressed as According to the second law of thermodynamics, the equilibrium condition of this reaction is shown as is the standard Gibbs energy of the reaction in Eq. (5). R is the gas constant and a GaN is the activity of GaN that defines a value of 1. The assumption of a stoichiometric growth conditions is given by where Δp shows the difference between the decomposed partial pressure p′ and p. The gas-surface equilibrium partial pressures p , Ga p NH 3 and p H 2 are obtained by solving Eq. (6) under the stoichiometric growth conditions of Eqs. (7) and (8). The standard Gibbs free energy D -G surface gas Here, E slab is total energy of the slab model. n , Ga n N and n H are the number of atoms of Ga, N and H, respectively, in the slab model. As before, m , Ga m N and m H are the chemical potentials of Ga, N and H, respectively. σ pass is the semipolar surface energy of the bottom surface which is passivated with fractional H atoms. The approach to calculate σ pass is based on the use of a wedge-shaped geometry. 40,41) Details of the calculation procedure are described elsewhere. 18,19) n GaN top is the number of GaN formula units in the topmost layers, respectively. A slab is the surface area of the slab model. Figure 1 shows the surface phase diagrams of GaN( ) 1122 and ( ) 1101 for N 2 and H 2 carrier gases as functions of temperature and N/III ratio, where III denotes group-III elements such as Ga and Al. Details of the growth conditions are given in the figure. For GaN( ) 1122 surfaces, the surface reconstruction terminated with hydrogen atoms (3N-H + NH 2 ) appears over the entire range of temperature for both N 2 and H 2 carrier gases, as shown in Figs. 1(a) and 1(b). This is because the Gibbs free energy in Eq. (1) for 3N-H + NH 2 is much lower than that for other GaN( ) 1122 surface reconstructions. Furthermore, for GaN( ) 1101 surfaces, a structural change from the surface reconstruction covered by Ga atoms (Ga monolayer) to that with hydrogen atoms (2NH 2 + 5N-H) is seen in the case of both N 2 and H 2 carrier gases. However, the phase transition temperature is different depending on the carrier gas. For N 2 carrier gas, 2NH 2 + 5N-H appears at a lower temperature than 995°C with a V/III ratio (V denotes group-V element) of 2000 in Fig. 1(c), while in the case of H 2 carrier gas, the phase transition temperature is found to be 900°C at a V/III ratio of 2000 [ Fig. 1(d)].

Semipolar GaN surfaces
The temperature dependence of the equilibrium partial pressure of Ga, p Ga , with N 2 and H 2 carrier gases on semipolar GaN surfaces is shown in Fig. 2. In the case of GaN( ) 1122 growth under N 2 carrier gas in Fig. 2(a), the driving force Δp Ga (=p Ga 0 − p Ga , where p Ga 0 is the input Ga partial pressure) for GaN( ) 1122 surfaces is always positive over a wide range of temperature, similar to the case of the GaN( ) 0001 surface. 21) Moreover, Δp Ga on GaN( ) 1101 and ( ) 0001 surfaces becomes negative when the temperature is higher than approximately 1105°C [ Fig. 2(c)] and 1075°C, 21) respectively. It is thus expected that growth on GaN( ) 1122 surfaces is possible at a higher temperature than on GaN( ) 0001 surfaces, while the maximum growth temperature on GaN( ) 1101 surfaces is similar to that on GaN ( ) 0001 surfaces. It should be noted that the maximum temperature for growth on GaN( ) 0001 surfaces is higher than that for growth on semipolar surfaces.
For H 2 carrier gas, growth on semipolar GaN( ) 1122 and ( ) 1101 surfaces is inhibited around 1250°C and 1030°C, as shown in Figs. 2(b) and 2(d), respectively. Hence, we can predict that the maximum growth temperature on GaN( ) 1122 surfaces is higher than that on GaN( ) 0001 surfaces (1025°C 21) ), whereas the maximum temperature for growth on GaN( ) 1101 surfaces is comparable to that for growth on GaN( ) 0001 surfaces. 21) The maximum temperature for growth on GaN( ) 0001 surfaces 21) is found to be similar to  Figure 3 shows surface phase diagrams for polar AlN with N 2 and H 2 carrier gases as functions of temperature and V/III ratio. For N 2 carrier gas, the surface reconstruction terminated by NH 3 and NH 2 (Al-NH 3 + 3Al-NH 2 ) is found at low temperature regardless of the V/III ratio, as shown in Fig. 3(a). Since the H atoms desorb from the surface with increasing temperature, reconstructed surfaces covered by N atoms (N ad -H + Al-NH 2 and N adatom) are stabilized [ Fig. 3(a)]. On the other hand, for H 2 carrier gas the surface reconstruction with NH 3 and NH 2 (Al-NH 3 + 3Al-NH 2 ) on AlN( ) 0001 surfaces is seen even at high temperatures over a wide range of V/III ratios, as shown in Fig. 3(b). For AlN ( ) 0001 surfaces, the surface reconstructions covered by Al atoms (Al adlayer and Al adatom) appear even at high temperature over a wide range of V/III ratios for N 2 carrier gas [ Fig. 3(c)]. A structural change from surface reconstruction with an Al layer (Al adlayer) to that with hydrogen (3N-H) is recognized under H 2 carrier gas, as shown in Fig. 3(d).

Polar AlN surfaces
The temperature dependence of the equilibrium partial pressure of Al, p Al , with N 2 and H 2 carrier gases on polar AlN surfaces is shown in Fig. 4. The figure implies that the driving force Δp Al (=p Al 0 − p Al , where p Al 0 is the input Al partial pressure) of polar AlN surfaces is always positive over a wide range of temperature for N 2 carrier gas [see Figs. 4(a) and 4(c)]. Therefore, the maximum growth temperature on AlN( ) 0001 surfaces is comparable to that on AlN( ) 0001 ) 1101 surfaces. Black, green, red and blue lines represent the equilibrium Ga partial pressure of the bulk phase, the input partial pressure p Ga 0 and the lower and upper limit of p Ga , respectively. When p Ga is lower than p , Ga 0 growth proceeds smoothly. On the other hand, growth is inhibited when p Ga is higher than p .  Figure 5 shows surface phase diagrams of AlN( ) 1122 and ( ) 1101 for N 2 and H 2 carrier gases as functions of temperature and V/III ratio. Surface reconstruction terminated by hydrogen (2NH 2 + 14N-H) on AlN( ) 1122 is found at low temperatures, and reconstructions covered by Al and N {( ) c 2 2 Al ad + N ad } appear at high temperature over a wide range of V/III ratios for both N 2 and H 2 carrier gases [see Figs. 5(a) and 5(b)]. However, the phase transition temperature differs depending on the carrier gas. For N 2 carrier gas, the phase transition temperature is found to be 1410°C, while for H 2 carrier gas the Al ad + N ad surface reconstruction appears at high temperatures above 1630°C. Furthermore, for AlN( ) 1101 under N 2 carrier gas, reconstructed surfaces covered by Al atoms (Al bilayer and Al monolayer) are stable surface reconstructions over a wide temperature range, as shown in Fig. 5(c). On the other hand, for an AlN( ) 1101 surface with H 2 carrier gas, structural variation from surface reconstruction with Al layers (Al bilayer and Al monolayer) to one with hydrogen (2NH 2 + 5N-H) is seen around 1360°C with a V/III ratio of 1000, then the N desorbed surface reconstruction (3N de ) appears at an extremely high temperature [1740°C in Fig. 5(d)].

Semipolar AlN surfaces
The relationship between temperature and p Al for N 2 and H 2 carrier gases on semipolar AlN surfaces is shown in Fig. 6. For N 2 carrier gas, the Δp Al of the AlN( ) 1122 surface always becomes positive even at high temperature [ Fig. 6(a)], whereas growth on AlN( ) 1101 is interrupted at around 1650°C with a N/III ratio of 1000 [ Fig. 6(c)]. Moreover, growth on semipolar AlN( ) 1122 and ( ) 1101 surfaces for H 2 carrier gas with a V/III ratio of 1000 is suppressed around 1790°C and 1480°C, respectively, as shown Figs. 6(b) and 6(d). As a result, it is expected that for N 2 carrier gas the maximum growth temperature on an AlN( ) 1122 surface is comparable to that on an AlN( ) 0001 surface, while for H 2 carrier gas growth on AlN( ) 1122 surfaces can occur at a higher temperature than on AlN( ) 0001 surfaces. Furthermore, the maximum temperature for growth on AlN( ) 1122 surfaces is similar to that for AlN( ) 0001 surfaces regardless of the carrier gas. Growth on polar AlN surfaces is possible at higher temperatures than growth on AlN( ) 1101 surfaces for both N 2 and H 2 carrier gases. Table I shows the difference in growth conditions between GaN and AlN surfaces. The major features of the growth conditions are discussed in the following.    ) 1101 surfaces. Black, green, red and blue lines represent the equilibrium Al partial pressure of the bulk phase, input partial pressure p Al 0 and the lower and upper limit of p Al , respectively. When p Al is lower than p , Al 0 growth proceeds smoothly. On the other hand, growth is inhibited when p Al is higher than p .

Conclusions
We devised an improved thermodynamic approach for semipolar GaN and AlN surfaces under MOVPE growth conditions. According to the improved thermodynamic analysis for semipolar surfaces, we can discuss temperature and orientation dependence of p Ga as well as p Al . The calculated results for N 2 carrier gas revealed that growth on GaN( ) 1122 surfaces is possible at at higher temperatures than on a GaN( ) 0001 surface, while the maximum temperature for growth on an AlN( ) 1122 surface is similar to that for growth on an AlN( ) 0001 surface. The relationship of maximum growth temperature between ( ) 1101 and ( ) 0001 surfaces depends on the group-III element. For GaN, the maximum temperature for growth on a GaN( ) 1101 surface is comparable to that for growth on a GaN( ) 0001 surface for both N 2 and H 2 carrier gases. In contrast, for AlN growth on an AlN ( ) 0001 surface is possible at higher temperatures than growth on an AlN( ) 1101 surface regardless of the carrier gas. The maximum temperature for the growth on a ( ) 1122 surface is comparable to that for growth on a ( ) 0001 surface for both GaN and AlN. It should be noted that growth on ( ) 0001 surfaces at is possible at higher temperatures than for growth on ( ) 1101 surfaces regardless of the group-III element. In addition, for H 2 carrier gas, growth on ( ) 1122 surfaces is possible at higher temperatures than for growth on ( ) 0001 surfaces for both GaN and AlN. On the other hand, for growth on ( ) 1101 surfaces, a difference is found in growth conditions between GaN and AlN surfaces. The maximum temperature for growth on GaN( ) 1101 surfaces is comparable to that for growth on GaN( ) 0001 surfaces, whereas the maximum temperature for growth on AlN( ) 0001 surfaces is higher than that for growth on AlN( ) 1101 surfaces. Furthermore, the maximum temperature for growth on ( ) 0001 surfaces is always higher than that on semipolar ( ) 1101 surfaces for both GaN and AlN. These results suggest that our analysis can be used to understand which temperatures are favorable for growth on semipolar surfaces and the dependence on surface orientation during MOVPE.