Chapter 6
Introduction
Exact diagonalization of many-body Hamiltonian is the most accurate and expen-sive method to study the electron dynamics and correlation in a semiconductor QD [44] [46] [47] [50] [51] [53] [28] [62]. Efficient methods such as the Hartree-Fock method and density functional approach are known to give substantial errors in energy, many-body wave function, spin-polarized states, and exchange-correlation potentials etc. [38] [46] [47] [50] [52].
Most of the theoretical models for QDs are based on 2D electron gas systems with parabolic infinite confinement potential, which are justifiable for calculat-ing scalculat-ingle-particle states, addition energy spectra, and many qualitatively new features of QDs. In fact, 2D models have turned out to be surprisingly rich and difficult [28]. However, the Coulomb interaction between electrons in QDs is three dimensional by nature and is the most important effect of many-body dynamics in QDs. It has been shown in [43] [54] that 2D models often lead to an inadequate description of the Coulomb interaction as a consequence of the overestimated car-rier localization. The 3D models are found to reproduce the experimental data for a large class of QD structures where simplified 2D models may fail [41] [45]
[48] [54].
There is another issue concerning the accuracy of QD models, namely, the effective-mass approximation (EMA) which allows us to model the conduction electrons in a QD as a decoupled interacting system with one-band effective mass from their environment that consists of millions of atoms in crystalline structure.
This assumption makes numerous theoretical investigations possible within toler-able computing resources. Most of the existing models are based on the parabolic EMA. In [45], the QD model with nonparabolic EMA is shown to describe quan-titatively the capacitance-voltage and far-infrared measurement data. The model does estimate correctly the change in the effective mass due to the nonparabolic effect. Moreover, it is shown in [49] that the effect of band nonparabolicity can be very significant in the sense that the energy difference between the parabolic and nonparabolic cases is comparable with that of exchange energies in multielectronic system.
In addition to the computational complexity compounded greatly by the three dimensionality and the exact diagonalization, the nonparabolicity leads to nonlin-ear (cubic) eigenvalue problems with interior eigenvalues which are much harder to solve than that of linear eigenvalue problems [40] [25] [22] [56] [59] [24]. In this paper, we propose a numerical algorithm for exact diagonalization of the many-electron Hamiltonian of a disk-shaped 3D InAs/GaAs QD model with non-parabolic EMA. The algorithm consists of the advanced cubic Jacobi-Davidson
method developed in [25] [49], GMRES, a model reduction technique for both Schr¨odinger and Poisson equations, and a new numerical approach to the exact diagonalization.
The computation of all pair-wise Coulomb interactions is one of the limiting factors in ab initio electronic structure calculations [57]. Our model reduction technique is a consequence of cylindrical symmetry of the model. This technique allows us to calculate the Coulomb matrix elements within tolerable accuracy and computing times. The reduction technique is not meant for general purposes but for a test of our idea on the numerical exact diagonalization.
Since the states of a QD are localized in space, the plan-wave approach for 2D electron gas systems would require a very large number of Fourier components to define a localized state [43]. Another approach that has been developed to reduce the number of basis states is to construct the single-particle basis functions that are separable into an in-plane (parallel to the radial axis) and a perpendicular part. The Fock-Darwin states (associated with Laguerre polynomials) are a good choice for in-plane states because they are the eigenstates of parabolically confined electrons [42] [43] [53] [58]. The perpendicular, or subband, functions are deter-mined numerically by solving a 1D Hartree-Fock equation. The many-electron system is then solved by minimizing the total energy of all conduction electrons in the system in a function space spanned by a basis of multi-electron functions which are constructed as a direct antisymmetrized product of single-particle basis
functions. To achieve a convergence for a few-electron system, the size of the single-particle basis functions must be of the order of a million [53].
These approaches are not suitable for the model considered in this paper due to the hard-wall confinement potential and the nonparabolic EMA. We present here a fully numerical approach for the exact diagonalization of many-electron Hamiltonian. In our approach, the basis set of single-particle functions is ob-tained by solving the cubic eigenvalue problem resulting from a finite difference approximation of the single-electron QD model. The eigenvalue problem is solved by the block cubic Jacobi-Davidson method proposed in [49]. Since the single-particle spectrum depends on the hard-wall confinement potential which is finite, the size of the basis set will have an upper limit. The use of single-particle basis will thus reduce the computational cost without losing accuracy on the ground state energy of the whole system. Compared with other approaches, the trade-off of our approach is the solution of the cubic eigenvalue problem with the size of tens of thousands.
We briefly summarize the main results of our approach as follows. For a 4-electron QD with the hard-wall confinement potential taken as V0 = 0.77 eV (electron volts), we only need 12∼ 16 single-particle states to obtain a convergent ground state energy within meV (milli-electron volts) accuracy. The computa-tional complexity of calculating all pair-wise Coulomb interactions is C2Ns+ Ns
where C2Ns is the binomial coefficient of Ns (the number of single-particle states)
and 2, i.e., we need to solve the Poisson equation C2Ns+Nstimes for this particular system.
The single-electron nonparabolic EMA model is given in the following chapter.
Numerical methods for this model are briefly described in Chapter 8. We refer to [25] [24] for more mathematical details on the discretization of the model and the solution of the cubic eigenvalue problem. In Chapter 9, we introduce our exact diagonalization approach to the many-electron system. The special point to note is the construction of the basis set of multi-electron wave functions based on the single-electron functions. A simple example is given to illustrate the function sapce definition. In Chapter 10, we present the model reduction technique and numerical methods for calculating Coulomb matrix elements. An overall algorithm for the solution procedure of the many-electron system is summarized in this chapter as well. Numerical results are presented in Chapter 11. Finally, we make some concluding remarks in the last chapter.