Stoichiometry and Structure of Polar Group-III Nitride Semiconductor Surfaces
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THEORY Our multicenter local-orbital formalism [14-17] is based on the density-functional theory in the local-density approximation [18-21]. We employ norm-conserving pseudopotentials [22] to describe the electron-ion interaction. The electronic wave functions and the charge density are represented by a superposition of pseudo-atomic orbitals (PAO) using the valence electron s- and p-orbitals for N, Al, and Ga, and the is-orbital for H. In particular, the total valence charge density is approximated by nin(r) =
2 Zni[Oi(r - Ri)]
,
(1)
which is used as input charge density for the Harris functional approach [23] applied to compute the total energy [14]. The occupation numbers ni of the PAO's ¢i(r - RI)
are
allowed to vary under the constraint that the number of electrons is constant. This is done by solving the Kohn-Sham equations iteratively, until self-consistency is achieved between the occupation numbers ni and the single-particle wave functions. Details of the formalism are summarized in Ref. [16]. The PAO's are constructed along with the pseudopotentials by imposing the boundary condition that the PAO's vanish beyond a specified cutoff radius r,. Confined atomic orbitals significantly improve the accuracy of the Harris functional, since they simulate the contraction of the atomic charge density observed in solid-state systems [24]. The confinement radii r, are chosen to guarantee that the energy difference between the atomic levels of the free atoms (r, -* oo) is essentially the same as that of the contracted and hence slightly excited atoms. We use r, = 5.4 (in atomic units) for gallium, r, = 5.4 for aluminum, r, = 3.8 for nitrogen, and r, = 3.7 for hydrogen to compute all interaction terms and overlap integrals. The only deviation from these values is that we employ less confined PAO's for gallium (rc`l = 5.7), aluminum (r"l = 6.0), and nitrogen (r°oul = 3.95) to compute the Coulomb integrals in the electron double-counting correction Ue, (see Ref. [14]). This choice of radii gives the best results for the lattice constants, the bulk moduli, and the phonon frequencies of zincblende-phase and wurtzite-phase GaN and AIN. Since the Ga 3d-state is not included in our minimal-basis local-orbital formalism, the structural parameters and the optical phonon frequencies calculated for GaN are underestimated by about 3 percent with respect to the experimental values, while for AIN, the deviations are smaller than 1 percent [25]. We use periodically repeated thin crystal films spanning five atomic double layers to describe the (0001) surfaces of GaN and AIN. Our investigations are restricted to surface geometries with (2 x 2) symmetry. The top layer of our slab-supercell is nominally terminated by nitrogen atoms, and the bottom layer is terminated by group-IIl atoms. In the case of ideal surfaces, the slab-supercell contains 20 nitrogen atoms and 20 gallium or aluminum atoms with one dangling bond per surface atom. Additional calculations performed for crystal films spanning nine atomic double layers show that fi
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