The treatment of defects and impurities in wavefunction-based conventional density functional theory (DFT) remains problematic in that the largest accessible supercell sizes commonly correspond to unrealistically large defect concentrations. For many systems, the properties of the defect states do depend sensitively upon the concentration; we discuss here the prototypical dilute magnetic semiconductor GaMnAs, in which substitutional Mn act as acceptors, which interact at high concentrations giving rise to ferromagnetism and metallic transport. The onset of this behavior occurs for concentrations somewhat lower than can be reached using conventional supercell DFT methods but can be easily modeled using atomic effective pseudopotentials (AEPs) . The total potential derived from DFT calculations including a single defect embedded in bulk GaAs is shown to converge rapidly with respect to supercell size, in contrast to the wave functions that tend to converge very slowly with increasing supercell sizes. This property of the potential is used to construct accurate AEPs with a moderate computational effort.
Practically, the impurity potential for Mn in GaAs is constructed by substracting, in reciprocal space, the known bulk GaAs potential from the total Kohn-Sham potential of a supercell containing a single Mn atom. The latter potential is calculated self-consistently using norm-conserving pseudopotentials for supercells containing 64, 128 and 250 atoms. The three Mn defect potentials extracted from these calculations can be used in large GaAs supercells, giving eigenvalues that differ by less than 20meV, demonstrating that the AEPs are convergend far before (i.e. for smaller supercell sizes) than the wave functions. By addressing only a small window in the eigenvalue spectrum (around the gap) it is straightforward to obtain an accurate wavefunction description of the important states, even in systems with many thousands of atoms.
Figure 1 shows the evolution of the Mn acceptor state (only the VBM and CBM at the Gamma-point are shown) as a function of supercell size: the band gap is that of the generalized gradient approximation used to construct the potentials. The acceptor state is identified as the spin-up gap state, and is shown in the GGA to coalesce with the bulk VBM in the limit of large supercell sizes. This result is in contradiction to the experimentally known acceptor level ionization energy of 113meV in the dilute limit (for instance in photoluminescence). This failing of the GGA has previously been obscured by the inability to look at sufficiently dilute systems – indeed for very high concentrations (at the left of the figure), the "acceptor level" is close to this experimental value, but corresponds here to the physically different case of the fully delocalized impurity band.
Experimental observation of the Mn acceptor level involves changing the occupation of the acceptor wavefunction; in the case of GaMnAs, the ground state is for 2/3 occupation of three degenerate (when spin-orbit is ignored) T2 symmetry wavefunctions. This corresponds to the 3d5 Mn with weakly bound hole configuration of the neutral acceptor. In PL experiments, recombination of a CBM electron with the acceptor bound hole occurs (Fig. 2); the energy of the resulting emission – the ionization energy of the acceptor – is then determined by the energy of the charged system. Interactions between the acceptor states mean that the energy change of occupying the hole state is not given simply by the acceptor level eigenvalue.
In order to understand the many-body behavior of the acceptor, a configuration-interaction (CI) treatment is applied to an eight level system consisting of the spin-split VBM and CBM states (the former comprising the acceptor states); the correlation energies and charging effects are considered explicitly for this subsystem, while interactions with the rest of the system are treated at the mean-field (GGA) level, and the interactions between the gap states are screened by an empirical screening function derived from the host material. The multi-determinental description provided by the CI methodology not only gives the correct multiplicities of gap states, but deals appropriately with the specific electron-hole interactions important for the calculation of optical properties. The selection of only a small window for treatment with CI, together with the use of empirical screening, makes this treatment entirely practical even for very extended systems.
We find however, that the delocalization found in the GGA means that the interactions between the Mn gap states become vanishingly small, such that the experimental ionization energy is not recovered, even in charged supercell calculations, which give a better description of the quasiparticle states.
In order to achieve localization of the gap states in the non-selfconsistent method, an additional non-local potential term is applied to the Mn d-states, raising them in energy toward the Fermi level. This has the desired effect of increasing the Mn-d/As-p interaction. The effect of this correction term is shown in Fig. 3 where the formation of a localized gap state is achieved with a shift of ≈3eV of the deep Mn states toward the VBM. As the state becomes localized, not only the acceptor level eigenvalue is increased, but also the Coulomb interaction between the acceptor states, leading to increased significance of the many-body effects. The relationship between the single-particle level and the CI-transition energy (which is principally determined by the Coulomb integral between the three Mn-VBM hybrid states), is shown in Fig. 3; for agreement with the 113meV PL emission, a single-particle level of ≈40meV above the VBM is appropriate.
The application of configuration-interaction to defect states, with careful correction for double counting terms arising from the initial density-functional calculation, is a powerful technique to investigate the optical properties of defect centers in semiconductors, where correlation and quasiparticle effects beyond the LDA are significant but may be attributed to a clearly defined subsystem, often states near the gap. This method is especially powerful when combined with the AEP method, because the ability to study very large systems is vital for capturing the physics of defect problems.