Electronic Structure Pseudopotential Calculations of Large ( - 1000 Atoms ) Si Quantum Dots

published in Advance ACS Absrracrs, February 1 , 1994. 0022-365419412098-2158$04.50/0 exists in larger quantum dots (11000 atoms) for which such methods appear impractical. Three classes of electronic structure approaches exist for such large sizes: ( i ) The Effective Mass Approximation (EMA).lo Here one replaces the microscopic quasi-periodic potential V(r) (which exists inside the material) by a constant potential, while the kinetic energy operator is replaced by an effective mass operator, derived from the parabolic expansion of the bulk band structure. The EMA treats “large systems” accurately and can be extended to incorporate multiband coupling (“band mixing”).lI However, for the sizes studied here (140 A), the parabolic band approximation might not be valid. This will be tested below. (i i) The Truncated Crystal Method ( TC).12J3 Instead of using parabolic bands, this method uses the actual dispersion relation of the bulk band structure and approximates the quantum dot wave functions by a sum of a few bulk Bloch wave functions of a single band at different wavevectors. It ignores band mixing and can be applied only to simple quantum dot geometries. It has been tested by comparison with direct calculations for 2D films13* and works very well. However, the formula of Rama Krishna and Friesner12 for OD quantum dots remains untested relative to direct calculations. This will be tested below. (iii) Direct Molecular Calculations. In this approach one diagonalizes the microscopic Hamiltonian consisting of the full kinetic energy and quasi-periodic atomic potential, thus avoiding the approximations of methods i and ii. However, because full variational ab-initio approaches to this Hamiltonian are impractical, the Hamiltonian or its basis representations are usually simplified. In the widely used tight binding (TB) model14 a small, implicit basis set (4-5 orbital per atom) is used; the function form of the basis function is, however, undermined, since only the empirically adjusted Hamiltonian matrix elements are used. The small basis suggests limited variational flexibility, particularly for the conduction bands. Furthermore, because the lack of explicit basis functions, it is difficult to correctly describe the dependence of the Hamiltonian matrix elements on the atomic geometry (e.g., on the surface structure of the quantum dot) or to compare the ?-space wave functions $(i) with the results of ab-initio calculations on smaller reference systems. Another version is the linear combination of atomic orbitals (LCAO)15 method in which explicit basis functions are used. This method is usually applied to rather small quantum dots (1100 atoms) while applications to larger systems require a drastic truncation of the basis set size.