Solving Polynomial Eigenvalue Problems Arising in Simulations of Nanoscale Quantum Dots

Transcription

1 Solving Polynomial Eigenvalue Problems Arising in Simulations of Nanoscale Quantum Dots Weichung Wang Department of Mathematics National Taiwan University Recent Advances in Numerical Methods for Eigenvalue Problems, NCTS, Hsinchu, 2008/1/6

2 Cooperators and Funding Agencies Tsung-Min Hwang (National Taiwan Normal University) Wen-Wei Lin (National Tsing-Hua University) Jinn-Liang Liu (National University of Kaohsiung) Wei-Cheng Wang (National Tsing-Hua University) Wei-Hua Wang (University of California, Riverside) National Science Council National Center for Theoretical Science 2

3 Outline Motivation: Nano-physics and engineering 3D Models: The Schrödinger equations Geometries Cylinder, pyramid, and irregular types Effective mass Constant effective mass Non-parabolic effective mass 3

4 Outline (cont.) Challenges for the 3D quantum dot models Neither analytical nor experimental techniques provides enough useful information Lack of efficient 3D numerical simulation tools Only several interior eigenvalues are of interest 4

5 Outline (cont.) Main results: Discretization schemes Simple (finite diff./ finite-vol.); efficient (2nd order conv.) Interface conditions are incorporated Various geometries and various effective mass models Polynomial eigenvalue problem solvers Jacobi-Davidson Methods Deflation/Locking/Restart for the interior eigenpairs Heuristics for correction vectors Efficient and solve large problem (up to 32 million) Physics predictions 5

6 Motivation and Model

7 Nanometer 1 nm = 10^(-9) m Nano-scale nm A semiconductor QD 10 nm QD:hair 1:10,000 Why consider quantum effects? Small devices imply significant quantum effect. Plenty of Room at the Bottom Richard P. Feynman, 1959 Semiconductor QD 7

8 Quantum mechanism Quantum dot (0 dim.): An artificial structure that carriers are confined in all three-dimensions (carriers have 0-dim freedom) The carriers exhibit wavelike properties and discrete energy states exist To understand fundamental physics and to inspire applications Quantum wire (1 dim.): Quantum well (2 dim.): 8

9 QD cross-section in Transmission Electron Microscope Cross-sections of hetero-structure InAs/GaAs QDs by Transmission Electron Microscope [Schoenfeld, 00] 150 Å 300 Å 9

10 Nano-scale quantum dot fabrication E-beam, chemical solution, (bigger QDs) Molecular beam epitaxy (smaller QDs) 10

11 Energy levels (eigenvalue): Each chemical element is associated with a unique energy levels Confined Discrete Energy Levels Energy 11

12 Wave function (eigenvector) Wave function ϕ (eigenvector): p ( x, y, z) = ϕ( x, y, z) (The square of wave function is the probability of finding the particle at a certain location.) Thus ϕ should be normalized. 2. Superposition of state (various states exist simultaneously) ψ(x,t) = C 1 (t) ϕ 2 (x) + C 2 (t) ϕ 2 (x) 12

13 Mathematics Model: The Schrödinger equations (A single electron confined in a 3D quantum dot)

14 The Schrödinger equation for single particle Kinetic eng. Potential eng. λ: energy (eigenvalue); u(x,y,z): wave function (eigenvector); m: effective mass; V: confinement potential; ħ: reduced Plank constant; m and V are discontinuous across the heterojunction 14

15 The Schrödinger equation (cont.) Effective mass models Constant model Non-parabolic model Pl, g l, δl : the momentum, main energy gap, and spin-orbit splitting in the l-th region, respectively. E Δ c1 1 = 0.000, = 0.81, Δ 2 E c2 = 0.350, = 0.80, 1 E g1 = 0.235, P = , P 2 E g 2 = =

16 16 BenDaniel-Duke interface condition [u]=0, Dirichlet boundary condition The Schrödinger equation (cont.) + = dot R dot R R F m R F m ) ( ) ( λ λ h h 0 ),, ( = Z mtx R F θ + = btm Z btm Z Z F m Z F m ) ( 2 ) ( λ λ h h + = top Z top Z Z F m Z F m ) ( 2 ) ( λ λ h h 0,0), ( = R θ F 0 ),, ( = Z R F mtx θ

17 Structure schema of quantum dots JAP, 90-12, (2001) WHLL, JCP (2003) HW, CMA (2005) HWW, JNN (2008) HLWW, JCP (2004) WHC, CPC (2006) PRB, 54, 8743, (1996) PRB, 62, 10220, (2000) HWW, JCP, (2007) 17 APL, 82, 3758, (2003)

18 Corresponding eigenvalue problems Ax = λx A: unsymmetric Ax = λx A: symmetric Ax = λbx A: s.p.d. and B: pos. diag. Cubic: 3 2 ( λ A + λ A + λa + A0) x = 0 Quintic: ( λ A + λ A + λ A + λ A + λa + A ) 0 x = 0 18

19 Numerical schemes Large-scale eigensolver for polynomial eigenvalue problems Jacobi-Davidson methods (HLLW, NLAA, 2005) Fixed Point Methods (HLLW,MCM, 2004) 19

20 Extreme Large-scale Eigenvalue Problems

21 Polynomial eigenvalue problems General form: Constant effective mass model 21

22 Polynomial eigenvalue problems (cont.) Non-parabolic effective mass model multiply the common denominator or 22

23 Enlarged linear eigenvalue problem Cubic example Larger size, maybe larger condition number Issues on efficiency, accuracy, convergence Shifted and invert 23

24 Spectrum of the (968) eigenvalues 24

25 Cylindrical quantum dot and quantum dot array

26 Discretization Eigensolver Physics

27 Discretization scheme Seven-point finite difference 2-point finite difference at the hetrojunction 27

28 Mesh points Left: Half-mesh shifted in r-dir. No pole condition. [Lai 01] Right: Uniform/Non-uniform mesh on a r-z plane 28

29 The non-uniform meshes Non-uniform mesh Uniform mesh 29

30 Problem reduction Transform the 3D problem to a sequence of 2D problems Matrix viewpoint (WHLL 03, JCP) 7-point finite difference Block diagonalizing the coef. matrix by the Fourier matrix Reordering 30

31 Discretization Eigensolver Physics

32 The resulting (2D) eigenvalue problems Matrix polynomial eigenvalue problems Linear (τ=1) for constant effective mass Cubic (τ=3) for non-parabolic effective mass 32

33 33 Deflation scheme for success eigenvalues Explicit non-equivalence deflation Original: Deflated: ( ) A A A A = λ λ λ A λ ( ) ( ) ). ( of eigenpair )isan, where(, ~, ~, ~, ~ λ λ λ λ λ A y y A y A A y y A A A A y y A A A A A A A T T T = + = + + = = ( ) ~ ~ ~ ~ ~ A A A A = λ λ λ A λ

34 Deflation scheme (cont.) Theorem 1. Let ( λ1, y1) be a simple eigenpair of ~ [ σ ( A( λ)) \{ λ }] { } = σ ( A( λ)). 1 A( λ) with y T 1 y 1 = 1. 34

35 Deflation scheme (cont.) Theorem 2. Suppose λ λ and ( λ, y Let ~ λ2 T y2 = ( I y1 y1 ) y2. λ1 Then ( λ, ~ y ) is an eigenpair of ) is an eigenpair of ~ A( λ). A( λ). 35

36 Discretization Eigensolver Physics

37 Discretization Eigensolver Physics

38 Simplification of explict deflation Recursive explict deflation formula is replaced by representation of original matrices 38

39 Discretization Eigensolver Physics

40 Energy states spectrum ( n r, nθ, nz ) : no. of nodal lines of the wave fts in r, θ, z dir. 40

41 Wave functions gallery (I) 41

42 Wave functions gallery (II) 42

43 Discretization Eigensolver Physics

44 Dimension reduction by truncated Fourier series The solution is periodical in angle and thus can be approximated by truncated Fourier series The eq. associated with the Fourier mode 44

45 Uniform mesh Non-uniform mesh Uniform mesh 45

46 Finite volume formulas with O(h^2) for interior and exterior points O(h) for boundary points 46

47 The 2D problem To simplify the notations, use the following form The integral form 47

48 Top interface points (1/2) Cancel out by the interface condition 48

49 Discretization Eigensolver Physics

50 Numerical results Campaq AlphaServer DS20E; Dual 667MHz CPUs; 1 GB memory; Tru64 Unix ver. 5.0; Fortran 90; Domain [Liu, Voskoboynikov, Li, 01], [Schoenfeld, 00] InAs dot: diameter: 15 nm, height: 2.5 nm GaAs matrix: diameter: 75 nm, height:12.5 nm Mtarix size: 3D: 755x280x360 76,000,000 2D: 755x ,000 50

51 Energy states (eigenvalues) j-ord λ Ite. Time j-ord λ Ite. Time

52 Structure schema of a QD array Non-uniform mesh 52

53 Ground state energies Ground state energies for various spacer layer distances d0 number of quantum dots. 53

54 Bifurcation for two quantum dots. 54

55 When the bifurcation happens 55

56 Uniform mesh with 2nd order convergence rate Non-uniform mesh Uniform mesh 56

57 All bounded state energy levels 57

58 Dynamics of the wave functions 58

59 Effect of pile-up dots More interactions are found in higher energy levels 59

60 Effect of pile-up dots (cont.) More interactions are found in higher energy levels 60

61 Effect of gap distance Smaller gap distance leads to stronger interactions vs vs 61

62 Effect of different size QDs Larger dot results in more interactions 62

63 Effect of different size QDs Larger dots result in more interactions 63

64 Anti-crossing and crossing 64

65 Bifurcation for two same size quantum dots. 65

66 A close look of the crossing and anti-crossing Fine mesh (2560x4320=11,059,200) with Δr = Δz = nm 66

67 Wave functions in crossing and anti-crossing areas Crossing 2nd E.V. 3rd E.V. Anti-Crossing 2nd E.V. 3rd E.V. Top figures: R2 = nm; Bottom figures: R2 = nm (Δr = nm) 67

68 Pyramid quantum dot

69 Discretization Eigensolver Physics

70 A finite volume discretization 3D scheme: denote surface and volume average over the control element uniform mesh due to the geometric structure automatically builds in the interface condition 70

71 The 3D scheme Exterior/interior points: local truncation error O(h^2) Interface points: local truncation error O(h) Surfaces (N, W, S, E, B) Edges (NW, WS, SE, EN, NB, WB, SB, EB) Corners (NW, WS, SE, EN, Tip) 71

72 2nd order convergence rate 72

73 Discretization Eigensolver Physics

74 Efficiency on a ordinary PC Matrix size: 1,532,255 Pentium 4 (1.8GHz) and 800MB of main memory 74

75 Efficiency on a decent workstation Matrix size: 32,401,863 ( ) Intel Itanium II (1.0GHz) and 12 GB of main memory. (5000 sec ~ 1 hr 20 min; 8600 sec ~ 2 hr 20 min) 75

76 Sparsity patterns (L,M,N) = (8, 8, 6) 76

77 Matrices properties Aü AÁ, Aª, A A, A Symmetry (partial) (partial) yes (Diag. mtx.) Diagonal Dominance (yes) (partial) yes (Diag. mtx.) : except the row entries involving interface grids Asymptotic structure B is a low rank matrix. 77

78 Discretization Eigensolver Physics

79 Wave fts. assoc. w/ λá and λª 79

80 Computational results in energy states Changes of energy states in terms of truncated QD heights (Dimension: 63 x 63 x N) 80

81 Computational results in wave functions 81

82 Irregular shape quantum dot

83 Discretization Eigensolver Physics

84 Schema 84

85 Scheme 1: Curvilinear coordinate Matrix AL: Sym. Pos. Def.; 9 nonzeros each row Matrix BL: diagonal matrix 85

86 Scheme 2: The skewed coordinate Matrix AS: Sym. Pos. Def.; 9 nonzeros each row Matrix BS: diagonal matrix 86

87 Scheme 3: Mixed coordinate Symmetry preserving average Not clear how ω affects the convergence of the eigenvalue solver analytically Numerical investigation provides some clues 87

88 Effect of the symmetry average parameter 88

89 Convergence rate (cont.) 89

90 Summary of the proposed schemes 90

91 Discretization Eigensolver Physics

92 Jacobi-Davidson method for polynomial eigenvalue prob. A Jacobi-Davidson method with locking (and restarging) 92

93 Solving the correction equation Sleijpen and van der Vorst (SIMA96) 93

94 Computing the smallest positive eigenvalue 1 SSOR: M = ( D +σ L) D ( D + σu ) 94

95 Computing the successive eigenvalues 95

96 Performance of SSOR preconditioner Dimension: 1,935,090 HP workstation with 1.3GHz Intel Itanium II CPU with 24 GB memory Average timing for computing all target eigenpairs 96

97 Discretization Eigensolver Physics

98 Eigenvalue spectrum 98

99 Eigenvectors (wave functions) Ground state (λ1= ev), 1st excited (λ2= ev), and 2nd excited (λ3= ev) state energy The first three eigenvectors associated with the first three smallest positive eigenvalues. 99

100 Conclusion

101 A computational science approach Applied Mathematics Discretization scheme -- non-uniform mesh, uniform mesh -- finite-diff. on cylindrical cord. -- finite-volume on Cartisian and curvilinear coordinate Large-scale matrix computation -- matrix reduction -- nonlinear eigenvalue solver -- deflation scheme -- accelerator Science and Engineering Computer Science The 3D model: Schrödinger eqs. with constant or non-parabolic effective mass approx. Concerning issues: eng. level & wave ft. Computed results verifications, explanations, applications Practical algorithms (for a certain architecture and language) Robust & efficient implementations Numerical experiments Computational and visual results 101

102 Thank you.