Comparison of Iteration Schemes for the Solution of the Multidimensional Schriidinger-Poisson Equations

We present a fast and robust iterative method for obtaining self-consistent solutions to the coupled system of Schr6dinger’s and Poisson’s equations in quantum structures. A simple expression describing the dependence of the quantum electron density on the electrostatic potential is used to implement a predictorcorrector type iteration scheme for the solution of the coupled system of differential equations. This approach simplifies the software implementation of the nonlinear problem, and provides excellent convergence speed and stability. We demonstrate the algorithm by presenting an example for the calculation ofthe two-dimensional bound electron states within the cross-section of a GaAs-A1GaAs based quantum wire. For this example, six times fewer iterations are needed when our predictorcorrector approach is applied, compared to a corresponding underrelaxation algorithm.


INTRODUCTION
As electronic device dimensions approach nanometer scale, quantum effects are expected to dominate their electronic properties. Consequently there is considerable interest in the efficient numerical simulation of such structures, not only for exploring novel device architectures but also for maintaining reliability for reduced present-day devices. In this paper we present a fast and robust iterative method for solving the two-dimensional Schr6dinger.-Poisson equations and obtaining the electron states in the cross-section of a quantum wire, and we compare the efficiency of our method with the standard underrelaxation algorithm.
The physical model used to describe the quantum wire consists, in the effective mass approximation, of Schr6dinger's equation We avoid the expense of solving Schr6dinger's equation in a very large grid by using the quantum formulation for n (3) only within the region of the quantum wire itself, while in the rest of the simulation domain, where the electron density is small, we replace it by its classical expression. The hole concentration p is very small throughout the quantum wire, but needs to be included into Poisson's equation to ensure a correct description of the band bending in the substrate [1].

SOLUTION BY UNDERRELAXATION
This system of coupled nonlinear differential equations is usually solved by an iteration between Poisson's and Schr6dinger's equation [2,3]. Since a plain iteration by itself does not converge, it is necessary to underrelax in the electron density n using an adaptively determined relaxation parameter co (k. This underrelaxation approach can be briefly outlined as follows: 1. Solve nonlinear Poisson equation using the old electron density n (k-) to obtain electrostatic potential 4 (k). 2. Solve Schr/Sdinger's equation using q(k) and Vxc(n (k-) to obtain a new set of eigenpairs 3 Calculate an intermediate electron density ttin 4. Choose an appropriate relaxation parameter co (k) to obtain new density n(k, and repeat outer iteration until n becomes stationary. The weakness of this method is the inherent instability of the outer iteration which is controlled by the underrelaxation procedure only. The relaxation parameter co (k is not known in advance and needs to be dynamically readjusted during the course of the iteration. If co ( is too large, the total quantized charge fndx oscillates from one iteration step to the other without reaching convergence, while if co () is too small, too many iteration steps are necessary to achieve convergence.

SOLUTION BY A PREDICTOR-CORRECTOR TYPE APPROACH
To address this problem we have to modify the underrelaxation algorithm in a way that partially decouples both partial differential equations and damps the oscillations in the total electric charge. We can achieve this goal by incorporating a modified expression for the quantum electron density fi(4) into Poisson's equation, which approximates the implicit dependency of the quantum electron density n on the electrostatic potential q due to Schr6dinger's equation.
A suitable expression of this type is provided by quantum mechanical perturbation theory. Using its formalism one can show [4,5], that for a small perturbation of the electrostatic potential 4 the occupancies Ni within the electron density (3) become C-dependent as (5) 71"2h 2 T'-1/2 kBT In our predictor-corrector approach we utilize this result by applying it to arbitrary large changes in : We start the k-th iteration step by solving a modified Poisson equation convergence. The use of fi(k) also follows the rule that the latest predictor value should enter the corrector.
The numerical experiment also shows, that, unlike in the standard algorithm, convergence is achieved without underrelaxation, so the outer iteration can be reduced to a simple alternation between Poisson's and Schr6dinger's equation until the quantum electron density n becomes stationary: Residual(n) I]n (k) n(k-')ll 2 _< e.

NUMERICAL RESULTS
where the potential-independent quantum electron density n is replaced by the potential dependent predictor , (k-l) defined as in (6) It is essential to insert the latest predictor value fi(k) into the exchange correlation in Schr6dinger's equation (9), since the numerical experiment shows that the use of n (k-l) within Vxc is detrimental to We compared the efficiency of both iteration schemes by computing the bound states for the model device shown in Figure [2 Both Schr6dinger's equation as well as Poisson's equation are discretized by a box integration finite difference method to take material discontinuities into account. Since the quantum wire covers only a small part of the entire simulation domain, we employ a non-uniform rectangular mesh concentrated around the wire region to minimize computational cost while retaining high accuracy within the region of interest. For the model device discussed above a 99 x 88 grid was found to yield satisfactory spatial resolution and short program run-times on HP 100 workstations.
After discretization, Schr6dinger's equation becomes a sparse eigenvalue problem, which is solved by Chebyshev-Arnoldi iteration [6]. This method allows us to compute only the physically relevant lowest-energy states. The nonlinear Poisson equation is solved by Newton-Raphson iteration with inexact line search. The necessary solution of a sparse linear system at each iteration step is accomplished by a version of the preconditioned conjugate gradient method involving redblack reordering. The Fermi-Dirac integrals, which need to be evaluated repeatedly at each grid point, are computed using rational function approximations, which combine speed of execution with high accuracy [4,7]. We compare the convergence speed of the predictor-corrector method with a fast adaptive underrelaxation scheme (an adaptive nonlinear version of the standard Gauss-Seidel algorithm) developed by one of the authors [6], which uses a heuristic method to adjust the relaxation parameter co (k) Using the residual in the quantum electron density (11) as measure to quantify convergence we find that our predictor-corrector approach makes the outer iteration stable. In fact, the residual decreases by close to one order of magnitude each step (Fig. 2), which is a 6-fold increase in convergence speed compared to underrelaxation. Additionally we find that the residual decreases uniformly from one step to the next, while for the heuristic underrelaxation algorithm occasional increases are possible.
The inclusion of exchange correlation Vxc results in an additional slowdown of convergence for the underrelaxation scheme and also decreases its stability. However, our predictor-corrector method remains stable and efficient, if the last available predictor value (k+l) is inserted into Vxc as outlined above.
We verified the rapid uniform convergence of our predictor-corrector method for a wide range of temperatures, gate voltages and geometries, and found it in all cases clearly superior to adaptive underrelaxation. This result indicates a fairly wide range of applicability, and considering the ease of implementation of our method we strongly recommend its use even for existing codes.
Thesis, University of Illinois at Urbana-Champaign, United States.