Ab initio calculations of excitons in AlN and Elliott’s model Robert Laskowski 1 and Niels Egede Christensen 2 1 Inst. of Materials Chemistry, Technische Universit¨ at Wien, Getreidemarkt 9/165TC, A-1060 Vienna, Austria and 2 Dept. of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark (Dated: November 29, 2005) The optical absorption and excitonic properties of wurtzite AlN are investigated by means of an ab initio approach taking into account electron-hole correlations. This is done by solving the Bethe-Salpeter equation, using the results of density functional theory calculations as a starting point. The main focus is on the calculation of excitonic spectra near the conduction band edge. The response is dominated by the exciton A formed out of excitations from valence Γ7 band. The n -2 quantum-number dependence of the energies in Elliott’s model fits rather well the ab initio calculations whereas the n -3 decay of the intensities is less obvious for the calculated oscillator strengths. PACS numbers: 71.15.Qe,71.35.Cc,78.20.Ci,78.40.Fy I. INTRODUCTION Aluminum nitride is a direct-gap semiconductor crys- tallizing in the wurtzite (WZ) structure. It is character- ized by a large band gap, more than 6 eV, a large bulk modulus and high a thermal conductivity, making it an interesting candidate for practical applications, for ex- ample in chip-scale U/V light sources and sensors. In all II-VI and III-V binary semiconductors crystallizing in the WZ structure the degeneracy of the p-like valence states at the Γ point is lifted by the combined action of the crystal field splitting and the spin-orbit coupling (SOC). The resulting structure at the top of the valence band consists of three separated levels. The lowest conduction band is non-degenerate s-like level. Optical transitions between the valence and the conduction states are com- monly referred to as A, B, and C in order of increasing energy. A detailed description of the typical band struc- tures of the wurtzite type semiconductors is given in Ref. 1, and a schematic energy-level diagram at the valence band maximum (VBM) is shown in Fig. 1. In contrast to the other binary II-VI and III-V semiconductors AlN has been predicted to have a negative crystal field splitting at the valence band edge, see Refs. 2–5 and references given therein. The uppermost valence state is of Γ 7(1)) (transition A) symmetry in the relativistic double group notation (index of the non-relativistic group is in parentheses). Because of the quite large crystal field splitting (around 0.2 eV) this state is well separated from the SOC splitted states Γ 9(5) and Γ 7(5) (B and C transitions). Due to this large splitting the band edge absorption is dominated by spectrum of exciton A related to transition between the Γ 7(1) state to the conduction band of Γ 7 symmetry. Some of the properties of excitonic states can be de- duced directly from the symmetry of the bands involved in the respective transitions. Excitonic reducible repre- sentations are given by the direct products of the irre- ducible representations of the conduction band (Γ 7 ) and VBM states (Γ 9 or Γ 7 ). The decompositions of these energy E 0 j =1/2 conduction band valence bands j =3/2 cf 7 1 5 9 7 so E g (A) E g (B) E g (C) E 0 FIG. 1: Schematic energy-level diagram of band splitting under action of the crystal-field and spin-orbit interactions in wurtzite crystals of compounds with negative crystal-field splitting an positive spin-orbit splitting. This is the situation in AlN. To the left, splitting induced only by the crystal field; to the right splitting induced only by the spin-orbit interac- tion. The combined effect is illustrated in the middle of the diagram. products into irreducible representations for WZ semi- conductors are: Γ 7 × Γ 9 → Γ 5 +Γ 6 Γ 7 × Γ 7 → Γ 5 +Γ 1 +Γ 2 The dipole transitions with symmetries Γ 6 and Γ 2 are forbidden. The excitons with symmetries Γ 5 and Γ 1 are dipole-active for light polarization perpendicular and parallel, respectively, to the hexagonal c axis. The exci- ton associated with the transition between bands Γ 7 → Γ 7 (A and C excitons) should be visible for both perpen- dicular and parallel polarizations. Whereas the excitons associated with Γ 9 → Γ 7 transition (B exciton) is only visible for light polarized perpendicular to the c axis. A similar analysis can be carried out for non-relativistic wave functions. In this case exciton A is excited only with light polarized in the c direction, B and C in per-