Zinc-blende GaN: ab initio calculations

Materials Science and Engineering B50 (1997) 57 – 60

Zinc-blende GaN: ab initio calculations J.L.A. Alves a,*, H.W. Leite Alves a, C. de Oliveira a, R.D.S.C. Valada˜o a, J.R. Leite b a

DCNAT-FUNREI, CP 110, 36.300 -000 Sa˜o Joa˜o del-Rei, MG, Brazil b DFMM-IFUSP, CP 66318, 05389 -970 Sa˜o Paulo, SP, Brazil

Abstract The purpose of this paper is to contribute, on a theoretical basis, an understanding of future wide-gap device concepts and applications based on III–V nitride semiconductors. The electronic properties of zinc-blende structure GaN and their (110), (100) and (111) surfaces are investigated using ab initio calculations based on the full potential linear augmented plane-wave (FPLAPW) method within the large unit cell approach, and on the molecular Gaussian-92 code. Lattice constant, cohesive energy, bulk modulus are obtained from total energy calculations. Light-hole and heavy-hole effective masses along (100), (111) and (110) directions and electron masses at G point are extracted from band structure calculations and compared with previous ones based on pseudopotential methods. The hydrostatic pressure dependence of the GG, GX and GL energy gaps are also obtained. Comparing our band structure and ‘molecular cluster’ calculations, the relaxations of the surfaces are found to be mostly determined by local rehybridization or valence effects and are basically independent of energy band features. © 1997 Elsevier Science S.A. Keywords: Zinc-blende; Wide-gap device concepts; Molecular cluster calculations

1. Introduction

2. The computational methods

The zinc-blende GaN is believed to be better suited for n- and p-type doping than its wurtzite counterpart. Due to the isotropy of the cubic GaN lattice it may exhibit higher electron drift velocities and reduced phonon scattering giving rise to potentially higher mobility materials [1,2]. In this paper the atomic and electronic properties of zinc-blende structure GaN and their (110), (100) and (111) surfaces are investigated using ab initio calculations based on the full potential linear augmented plane-wave (FPLAPW) [3] method, within the large unit cell approach, and on the theoretical quantum chemistry Gaussian-92 code [4] associated to the molecular cluster model. The Ga-3d electron states are treated explicitly as part of the valence band states in order to take into account the hybridization between Ga-3d and N-2s states.

The band structure and the structural properties of the zinc-blende GaN were obtained by means of ab initio all electron self-consistent electronic structure calculations. The electron-gas data for the local-density approximation (LDA) for the exchange-correlation functional was taken from Ceperley and Alder [5]. The muffin-tin radii used were 1.82 a.u. for both gallium and nitrogen atoms. The number of the interstitial plane waves (PWs), as determined by Kmax =4.94 a.u., was about 330 PWs. With this choice of the cut-off wave vector the self-consistent energy bands were found to converge to within 10 − 7 eV and the total energy within 10 − 8 eV. The cut-off angular momentum was l= 8 for wave functions inside the spheres and l=4 for charge densities and potentials. The core electron states were treated fully relativistically whereas the valence electrons were treated non-relativistically. The summation over ten Monkhorst-Pack [6] special k-points in the irreducible part off the Brillouin-zone was used to replace the Brillouin-zone integrations. The Ga-3d electron-states were treated explicitly as part of the valence band states in order to take into account the hybridizations between Ga-3d and N-2s states.

* Corresponding author. Tel.: +55 32 3792483; fax: + 55 32 3792380; e-mail: [email protected] 0921-5107/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S 0 9 2 1 - 5 1 0 7 ( 9 7 ) 0 0 1 6 4 - 5

J.L.A. Al6es et al. / Materials Science and Engineering B50 (1997) 57–60

58 Table 1 Equilibrium properties of GaN ˚) ao (A

˚ 3 molecule−1) V (A

Ec (eV molecule−1)

B (Mbar)

B%

4.559

23.57 22.8a

12.0 9.9a 10.88b 10.25c

1.916 2.0a 1.98b 1.69c 2.40c 1.845d

4.6 4.4a

4.466b 4.45c 4.31c 4.556d

13.02d

3.9d

ao, lattice parameter; Ec, cohesive energy; V, volume per molecule; B, bulk modulus; B%, pressure derivative. a [11]; b [12]; c [13]; d [14].

In modelling the GaN surfaces for quantum chemistry calculations, we used the clusters: (i) surface (110): Ga7N7H16; (ii) surface (100): Ga4N5H12, N4Ga5H12; (iii) surface (111): Ga6N4H13, N6Ga4H13. Calculations were performed with the use of the Gaussian-92 program [4], using the effective core potential approximation and single-zeta basis sets [7].

3. Results and discussion Table 1 displays the bulk structural parameters as calculated by the FPLAPW method, and compares them with other previous calculations including explicitly the Ga-3d states [8,10] and not including the Ga-3d ˚ for the states [9]. We obtained the value ao =4.559 A lattice constant which compares very well with the ˚ [15]. Our results also experimental value of ao =4.55 A compare very well with the recent values calculated in

[10] using pseudopotential calculations and confirm the importance of including explicitly the Ga-3d states in the valence band in order to obtain accurate results. We also focus our attention on the electronic structure around the valence-band maximum and the conduction-band minimum, which are at the G point for the zinc-blende GaN, and link the electronic band calculations with the effective mass theory. According to our calculations there is some small anisotropy of the energy dispersion around the bottom of the conduction band. The calculated electron and hole effective masses are shown in Table 2, where comparison is made with other calculations and with available experimental data. The calculated electron effective masses 0.162 mo (D), 0.166 mo (L) and 0.166 mo (S) with an average value of 0.165 mo compares very well with the recent experimental value 0.159 0.01 mo [9]. The hydrostatic pressure dependence of the GG, GX and GL energy gaps are also obtained by means of the FLAPW method. Even if the LDA calculations underestimate the gap energies, they lead to correct predictions for their pressure variation (i.e. for the linear pressure coefficients) as has been demonstrated by Chang et al. [16]. Table 3 shows the dependence of the energy gaps on the lattice constant and the pressure. Our results confirm the experimental findings for the III–V semiconductors which have been shown to exhibit a non-linear (sublinear) dependence of the gap energy on pressure. The calculations predict direct band gap. Table 4 displays our results for the clusters XYH(X, Y= Ga, N) simulating the (100) surface. The X atoms correspond to the surface atoms, the Y atom corre-

Table 2 The electron and hole effective masses for GaN m*

mlh

mhh

D

L

S

D

L

S

D

L

S

0.162

0.166 0.13a 0.15b 0.21c

0.165

0.866 0.74a

1.946 1.82a

0.840 1.51a

0.21a

0.184 0.18a

0.194 0.19a

m*, mhh, mlh denote the electron, heavy and light-hole masses. The Greek symbols represent the k-directional dependence along the D(100), L(111) and S(110) directions. All values are in units of a free-electron mass mo. a [8]; b [9]; c [10]. Table 3 The first and second order pressure derivatives of the direct and indirect-gap energies on pressure

GG GX GL

dE/dP (meV GPa−1)

d 2E/dP2 (meV GPa−2)

˚ −1) dE/da (meV A

˚ −2) d2E/da 2 (meV A

33.08 2.21 35.45

−0.36 −0.032 −0.37

−22.96 0.11 −28.15

2.07 −0.043 2.61

GG, GX and GL label the gap from the top of the valence band at G to the conduction band at G, X and L, respectively.

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59

Table 4 Theoretical results for clusters simulating the GaN surfaces Surface

Cluster

(110)

GaNH4 Ga7N7H16

(slab) (slab)

˚) D z (A

0.18 0.18 0.01 0.31a 0.033 0.032b

v (°)

a (°)

b (°)

D R (%)

D E (eV)

17.8 15.4 0.3 19.4a 1.88 2.06b

117.7 116.2 111.3 118.1a — —

109.9 101.9 110.1 104.0a — —

−12 −10 −6.0 — — −6.5b

0.46 0.36 0.12 0.38a 0.06 —

(100)

Ga4N6H16 N4Ga6H16

0.42 −0.31

— —

91.62 —

— 125.86

+14 −8.0

0.40 0.33

(111)

Ga6N4H13 N6Ga4H13

−0.09 −0.46

— —

112.0 —

— 118.93

−1.0 −5.0

0.06 0.50

Dz, surface strain; v, surface dimmer tilt angle; a, average bond angle around the Ga atom; b, average bond angle around the N atom; DR, change in the Ga – N bond length upon relaxation; DE, energy lowering per surface atom upon relaxation. a [17]; b [18].

sponds to the subsurface atom and the H atoms are saturating hydrogens (Fig. 1a). The unrelaxed cluster corresponds to all bond angles fixed to the ideal zincblende angle (109.47°) and the bond lengths having their optimised values. The relaxed cluster corresponds to maintaining only the HYH angle fixed (109.47°), the other bond angles and the bond lengths being optimised. The average bond angle around the Y atom is always close to the ideal zinc-blende angle (109.47°). This indicates the proper hybridization of the ‘bulk’ atom.

The results for GaN(110) surface are also displayed in Table 4. The relaxed geometries were obtained by keeping the subsurface atoms (saturating hydrogens) fixed and allowing the surface Ga and N atoms move only in the yz plane (Fig. 1b). The surface saturating hydrogen atoms were compelled to move by the same amount as the respective Ga or N atom. For this surface we have three kinds of calculations: (i) using double-zeta basis set (first two lines of the Table 4); (ii) using minimal basis set (third line); and (iii) using three layers ‘slab’ calcula-

Fig. 1. Molecular clusters used to simulate the (100) surface (a), (110) surface (b) and (111) surface (c).

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tion by means of the FPLAPW method. The case (i) compares well with similar calculations by Swarts et al. [18]; in the case (ii) the basis set was chosen in order to get a better description of the NH3 molecule and the result is a very small relaxation, and is in qualitative agreement with case (iii) and with the previous ‘slab’ calculations using ab initio the Hartree – Fock method by Jaffe et al. [17]. Since, in case (ii) the 3d-electrons are not included in the valence states and in case (iii) and in [17] they are, it seems that the 3d-electrons might not be correlated with the small tilt angle v and with the relatively large surface bond contraction. Finally, the results for the GaN(111) surface were obtained by means of the clusters Ga6N4H13 and N6Ga4H13 (Fig. 1c). The relaxed clusters were obtained by keeping all atoms fixed except the top atoms which was allowed to move only in the z direction.

4. Final remarks More extensive theoretical and detailed experimental studies are needed to establish the relaxation model for the GaN surfaces. Based on the comparison of our results with the other available theoretical results we conclude that the relaxations should be mostly determined by local rehybridization or valence effects and should be basically independent of energy band features.

Acknowledgements The authors are indebted to the Brazilian agencies FAPEMIG, FAPESP and CNPq for financial support,

.

and grateful to R. de Paiva for help with the manuscript.

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