Ab initio parameterisation of the 14 band k·p Hamiltonian: Zincblende study

Despite continued and rapid progress in high performance computing, atomistic level device modelling is still largely out of reach, necessitating the use of quantum mechanical continuum methods, including k·p perturbation theory. The effective use of such methods requires reliable parameterisation, often obtained from experiment and ab initio calculations. A major limitation of this, the systematic tendency of ab initio density functional theory to underestimate semiconducting material energy band gaps and related properties, can be greatly improved upon by the inclusion of exact exchange, calculated within the Hartree-Fock formalism. We demonstrate that the 14 band k·p Hamiltonian can be effectively parameterised using this method, at greatly reduced cost in comparison to GW methods.


Introduction
The modelling of semiconductor devices typically requires multiscale methods, whereupon calculations made at fundamental levels of theory are used to feed more approximate, but critically less computationally expensive, methods. Ab initio density functional theory (DFT) is commonly used for atomistic level calculations, which are then used to parameterise tightbinding (TB), empirical pseudopotential (EPM), or multiband k·p Hamiltonians [1]. Kohn-Sham DFT, however, does not yield the real band structure of semiconductors. Although a good approximation, the energy band gaps are systematically underestimated in this class of materials.
Many body perturbation theory, typically Green's function methods, can be used to more accurately predict the band gaps [2], but at considerable computational expense. An efficient and reasonably accurate alternative is the use of hybrid functionals [3,4], which incorporate into the DFT exchange-correlation energy either a screened long range Coulomb term or a fraction of exact exchange calculated from Hartree-Fock theory using the Kohn-Sham orbitals.
Using this method, we calculate the necessary energy band gaps, dipole matrix elements, and effective masses at characteristic points in the first irreducible Brillouin zone (IBZ), i.e. at Γ point, to parameterise the 8 and 14 band k·p Hamiltonians. Although the 8 band k·p Hamiltonian, which has C 4v symmetry, is popular and widely used, it overlooks the correct C 2v atomistic symmetry of the zincblende (ZB) lattice [5]. In order to restore the correct symmetry it is necessary to implement the 14 band model, including couplings induced by the second conduction band (labelled as Γ 5c in fig. 1) stemming from p-bonding states of atoms in the ZB lattice [5]. We compare both to the DFT band structures in demonstration of this. We further show the importance of non-locality in hybrid DFT calculations to the accurate prediction of band parameters.

Methodology
All DFT calculations are performed using the Crystal [6,7] code, which implements localised basis sets of Gaussian type orbitals (GTOs). The calculation of exact exchange is very efficient using this basis. The hybrid PBE0 [8] and B3LYP [9] exchange correlation functionals are used, the former incorporating 25% of exact exchange energy to the latter's 20%. The IBZ of the ZB lattice unit cell is sampled using the Monkhorst-Pack scheme [10], with shrinking factors 16×16×16. Dipole matrix elements p ij are calculated from Bloch functions, u ki (r), such that: From the dipole matrix elements, Eq. 1, the Kane energies are calculated as follows: The kppw code [11], parameterised with the ab initio data, is used for all k·p calculations. Effective masses are defined by the curvature of the band dispersion at high symmetry points in the IBZ, in this case Γ: An alternative definition of the effective mass, from k·p theory, depends on the interaction between bands around the conduction and valence band edges, where i is the band for which the effective mass is being calculated:

Results and discussion
The effective mass calculated using this expression can be systematically improved by considering additional interactions from remote bands. For the methodology to be consistent, convergence of the two should be observed as further bands are considered in Eq. 4. Indeed, this is observed, as shown in Table 2. We use shortened notation: GaAs 0.0561 0.0552 0.0561 0.0561  Figure 1: Schematic of ZB band structure around Γ point. Top of valence band, Γ 5v , bottom of conduction band, Γ 1c , and second conduction band, Γ 5c . Energy gaps, E g i , and coupling parameters, p i , correspond to the notation in table 1.     The Luttinger parameters relate to the valence band edge effective masses along directions of high symmetry in the IBZ and largely determine the band curvature. They are calculated from appropriate pairs of the following system of equations, along different high symmetry directions: As the effective mass tensors are anisotropic, we obtain different Luttinger parameters along each direction, as detailed in table 1. Fig. 1 depicts the schematic band structure of the ZB structure, taken into account by the 14 band k·p Hamiltonian. The 8 band Hamiltonian (C 4v ) includes only the Γ 5v valence and Γ 1c conduction bands, coupled by the p 0 dipole interaction. The p 1 interaction is isomorphic with p 0 , however the inclusion of the p 2 interaction between Γ 5v and Γ 5c bands has the effect of reducing the symmetry to C 2v , as required to resemble the correct atomistic structure of ZB lattice.
Figs. 2 and 3 show the band structures along the <001> direction in the IBZ for GaAs and CdSe, respectively. Largely good agreement between the DFT and k·p calculations is seen up to 0.1Å −1 . In addition to observing the correct C 2v symmetry, the 14 band model shows a notable improvement in its depiction of the heavy hole band well beyond this region. Of greatest note is the inclusion of the higher conduction bands in the 14 band model.

Conclusion
We have demonstrated that the use of hybrid functionals in DFT calculations can be used to effectively and accurately parameterise multiband k·p Hamiltonians for some common ZB semiconductors. Further, we have shown that the 14 band model, by inclusion of higher conduction bands and having the correct C 2v symmetry of the ZB lattice, affords greater accuracy in the prediction of the band structure than the more widely used 8 band model, which is typically synonymous with k·p perturbation theory.