Local Radial Basis Partition of Unity for Modeling Quantum Dot Semiconductor

Transcription

1 Local Radial Basis Partition of Unity for Modeling Quantum Dot Semiconductor

2 Abstract Solving the Schrodinger equation in quantum systems presents a challenging task for numerical methods due to the high order behavior and high dimension characteristics in the wave functions. This work introduces a local radial basis partition of unity approach to construct the numerical approximation for solution of the Schrodinger equations. The radial basis functions (RBFs) are shown to be appropriate basis functions for quantum systems. The proposed local RBFs maintain the exponential convergent rate of the RBFs, while substantially reducing the condition number and bandwidth of the discrete system compared to the conventional RBF discretization. The approximation of the interface jump condition for quantum dot applications has also been introduced. 1. Introduction The quantum dot semiconductor is an innovative nanoscale device with unique properties, such as discrete energy levels due to the low-dimensional quantum confinement of barriers. The traditional ab initio basis set calculation yields a full sized Hamiltonian matrix. On the other hand, the real-space calculation, such as the finite difference method and the finite element method, requires large degrees of freedom for reasonable accuracy. HP-clouds with orbital basis functions [3] has been introduced to provide a numerical solution of the atomic and molecular systems. However, the orbital basis functions for quantum dot semiconductors are not available. Radial basis functions (RBFs) have been introduced for the numerical solution ofpdes [6]. Standard RBF offers exponential convergence. However, the method suffers from the large condition numbers due to its "nonlocal" approximation. The nonlocality of RBF also limits its application to small scale problems. The reproducing kernel (RK) approximation [5], on the other hand, provides polynomial reproducibility in a "local" approximation, and the corresponding discrete system exhibits a relatively small condition number. Nonetheless, RK approximation produces only algebraic convergence. This work intends to combine the advantages of RBF and RK approximation functions under the partition of unity framework [1] to yield a local approximation that is more stable than that of RBF, while at the same time, offering a higher rate of convergence than that of the RK approximation. Further, the method remains stable in large discrete systems, and the locality in the approximation allows its application to large scale quantum mechanics problems. The proposed methods are applied to the solution of the SchrOdinger equation for quantum dot semiconductors using the spin density functional theory. 2. Basic Governing Equations in Quantum Dot Semiconductor The Kohn-Sham spin density functional approximation has been used to describe the density distribution of all electrons with different spin numbers [7]. In the model problems of this paper, the exchange-correlation potential is neglected, and the Kohn-Sham equation is given [7] as {2~' [p +~A~ (x)]'+ Va(x)}e:::: (Pa)= &;.ae~; (Pa) (I) where a is the spin number; e~~ and &;.<7are the Kohn-Sham wave function and corresponding eigen energy of the i -th electron; P<7represents the density distribution of electrons; p is the momentum operator; Aext is the vector potential of the external

3 magnetic field B; e is one electron charge, c is the speed of light; Va is the spin-dependent scalar potential that consists of a spin-independent external potential and the Zeeman energy. The electron density distributions in the quantum dot array with different number of dots and different dimensions are shown in Figure lea). The electron density distributions are altered significantly under the external magnetic field as shown in Figure I(b).Traditional basis sets or real-space calculations are inefficient in modeling the complex electron structures of the semiconductors composed of large number of electrons. ~.. (a) (b) Figure 1. Electron density in (a) pentagonal two-electron quantum dot with three different sizes; (b) eight-electron quantum dots with three different geometries and external magnetic fields. (Laboratory of Physics, Helsinki University of Technology) 3. Local Radial Basis Function Approximation Let the wave function e (x) be approximatedby the RBF approximationas N, eh(x) =Lg/(x)a/, (2) /=1 where Ns is number of source points, a/ is the expansion coefficient, and g/ (x) is the radial basis function, for example, the multiquadrics (MQ) RBF 2 2 n-{ g/(x)=(r/ +c) ',n=i,2,3,... (3) where r/ = Ilx- x/ii is the Euclidean distance, and the constant c in Eq. (3) is called the shape parameter of RBF. It has been shown [6] that there exists an exponential convergence rate if RBF is globally analytic or band-limited where 0 < 1]< I, 1]= e.xp(-8) with 8> 0, and h is the radial distance. However, RBF is a "global" (nonlocal) approximation in solving PDEs, and it yields a full matrix in the discrete equations. (4) 20i 10.' 1 RBF(inverseMa) 1:r 1- moo RK RBF (p=1) (Inverse MQ). m.. RK (p=1) 10 Ue(X)=1 g Xe [0,1] Q) 5 "! n xe[o,1].51 I.15 ' x X Figure2.Errors inthe reproductionof I-dimensionalconstantand linearpolynomialfunctionsby RBFand RKfunctions. Further, RBF cannot reproduce constant and linear functions in finite domain with finite number of source points. On the other hand, some local approximation methods such as reproducing kernel (RK) approximation [5] are constructed to reproduce.a,5" Ue(X)=X 2

4 polynomials for random point distributions in the finite domain. As shown in Figure 2, RBF exhibits errors in reproducing constant and linear functions in a finite domain [0,I], while RK function with linear basis (p = 1) yields exact reproduction of constant and linear functions using 11 random points. To localize RBF with polynomial reproducibility, we consider the following partition of unity approximation of wave function e (x) (5) e' (x) ~ t. [q>,(x)[a, + t. g; (x)d; ]]. where g; (x) denotes RBF's or other basis functions,for example, the multiquadrics \ RBF defined in Eq. (3), al and d; are the corresponding coefficients, and 'PI (x) is the reproducing kernel (RK) function [5] given as 'PI (x) = HT (O)M-1(x)H(x - x/)<1>a(x - XI)' M(x) = LH(x- XJ)HT(X -xj)<1>a(x -xj) (6) J where H is the basis function vector, and <1>a is the kernel function with local support size "a" which determines the smoothness and locality of the approximation. This approximation exactly reproduces basis functions contained in the H vector. The error bound of the local RBF approximation in Eq. (5) has been shown by the authors [2] wherewe used I'PI 100:::; Coogeneric constant, K denotes the maximal number of covers for any x in the problem domain n, and 0 < 1] < 1 as defined earlier. The exponential convergence rate still holds by local RBF. Further, the local RBF significantly reduces condition number compared to the standard RBF for the discretization of PDEs with least-squares functional, and it can be shown as follows: RBF: RK.: proposed local RBF: Condo~ O(h-s), Condo ~ O(h-2), Condo ~ O(h-3). 4. Numerical Results (1) Consider a spherically symmetric semiconductor composed of two materials GaAs and InAs, where R, = 300A.. is the radius of dot and R2 =RJ + 100A.. is the radius of matrix where the density of electron is assumed to be zero on the outer boundary. In order to construct the discontinuity in the approximation of wave function gradient along the normal direction across the interface, the interface enrichment function q,i (x(r,s)) = cp(r)rpl (s) is introduced as the localizing function for the nodes located on the interface, where rand s are the local coordinates normal to and alon,$ the interface, respectively, as shown in Figure 3(c). The function cp'(r) E C-J possesses discontinuous derivatives across the interface, and rpl(s) is a smooth kernel function along the interface as demonstrated in Figures 3(a) and (b). The localizing functions for nodes not away from the interface are constructed with polynomial reproducibility as (7) 3

5 follows L \p[(x)x~ + L 'i'ix)x~ =xa [:x/es, J:xJeS, (~ =>\p[(x) =HT(O)M-1(x)H(x-x[)- [ J:t;s, 'i'ix)h(x-xj) ] <1>a(x-x[) where S2 is the set of nodes on the interface, and the set Sl includes the rest of nodes. "- ",(n A -.. TI: ~.. (3) ~r.. r. minas,.. /'0,,,\n~.... (b)... (S) (c).. s Figure 3. Discretization of interface and different kernel functions for two perpendicular directions with different orders of smoothness Figure 4 shows the convergence of solutions for the first two energy levels by using the linear and quadratic FEM, the RK approximation with linear and quadratic bases, and the proposed local RBF (inverse MQ function localized by RK with linear basis) with Galerkin approximation. The local RBF achieves a much better solution accuracy and convergence rate. ~, ~ -1.5 ~ Q) -2 J ~ -2.5 :g w -3 ;; ~ '0-3.5 g -4 W o ff :.~;;;;;i"" _--""-:"'1:;:;;-:":,-;::"""",,,,,, 1.75 ~~~ ~~F... II'"~~~~:.:..:;::::;:::;:;:~~ A ;:;:~~~ ",.....~;////,./ I-II-' LinearFEM Quadratic FEM Q; "A" RK withlinear basis,,~.. RK with quadratic basis -+- Local RBF -5.~' ~ ~ -2 >- 2' -2.5 :g : -3 N '0-3.5 g -4 W o 'i ::::~;;;;;i"~ 1.81 :f:.~ ".' InAs '" ;~~:~::::~:: //,... Figure 4. Convergence of the first two energy levels. - R..1' (a) (b) (c) FigureS.(a)three-dimensionalQDA,(b)two-dimensionalmodel,(c)discretization. (2) Consider a quantum dot array (QDA) with three vertically aligned InAs/GaAs quantum dots embedded in a cylindrically symmetric matrix subjected to an external 4

6 magnetic field B (Figure 5(a». The SchrOdinger equation using the spin density functional theory as described in Section 2 is employed. The proposed local RBF with inverse MQ function localized by RK. with linear basis is used. In this problem, six electrons under the external magnetic field along the z direction with magnitude 15T are considered. The electron density contours of the first three energy levels are plotted in Figure 6, where the electrons with different energy levels are distributed in different dots under the external magnetic field, and the results compare well with the reference solution [4]. 40[00_u~ :00 40 : : : mi-- -- mi I -' ~: ::::::~:::::~um_ ~ fib ';;'.10 mh+muimm ;00 00 mim 000 : : :- ~ :-----_ , o r(nm) (a) :000 00: hu ~ ~----- o _ ',' : ~ ~ ~ ~----. o m m..10 ""+" +_00".20 r j r j ' : : : , ~ ~ o o (b) (c) Figure 6. Electron density contours for (a) 1 sl energy level (0.124 ev), (b) 2 nd energy level Conclusions (0.159 ev), (c) 3 "I energy level ( ev) under an external magnetic field. In this work, the RBFs are shown to be appropriate basis functions for approximating electronic density by properly adjusting the shape parameter. Nevertheless, the nonlocality of the standard RBFs renders a full matrix and ill-conditioning in the discrete systems. In this work, we formulated a localized RBF by introducing a RK.as the localizing function under the partition of unity framework. The error analysis shows that if the error of RK. is sufficiently small, the proposed method maintains the exponential convergence ofrbf, while significantly improving the conditioning of the discrete system and yielding a banded matrix. For application to quantum dot semiconductors, we introduced gradient discontinuity to the localized RBF approximation of wave function across the interface. Numerical examples demonstrate the effectiveness of the proposed method for quantum systems. Reference [I] I. Babuska and J.M. Melenk, The partition of unity method, Int. J. Numer. Meth. Eng., 40 (1997), [2] J.S. Chen, W. Hu, and H.Y. Hu, Reproducing Kernel Enhanced Local Radial Basis Collocation Method, Int. J. Numer. Meth. Eng., in press [3] J.S. Chen, W. Hu, and M.A. Puso, Orbital HP-Cloudfor Schrodinger Equation in Quantum Mechanics, Computer Methods in Applied Mechanics and Engineering, 196 (2007), [4] J.L. Liu, J.H. Chen, and O. Voskoboynikov, A modelfor semiconductor quantum dot molecule based on the current spin densityfunctional theory, Computer Physics Communications, 175(9) (2006), [5] W.K. Liu, S. Jun, and Y.F. Zhang, Reproducing Kernel Particle Methods, Int. J. Numer. Meth. FllUds,20 (1995), [6] W.R. Madych, Miscellaneous error boundsfor multiquadric and related interpolatory, Computer Math. Applic., 24(12) (1992), [7] H. Saarikoski, E. Riisiinen, S. Siljamiiki, A. Harju, MJ. Puska, and R.M. Nieminen, Electronic properties of model quantum-dot structures in zero andfinite magnetic fields, The European Physical Journal B, 26 (2002),