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Application: Quantum dot simulations
Quantum well (2 dim.)
Quantum wire (1 dim.)
Quantum dot (0 dim.)
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Molecular beam epitaxy
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Energy levels (eigenvalues)
Confined Discrete Energy Levels Confined Discrete
Energy Levels
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TheSchr ¨odinger equationfor Semiconductor:
−∇ · (α∇u) + V u = λu,
where α =
( α−≡ 2m~2
1 inside, α+≡ 2m~2
2 outside, V =
V−= V1 inside, V+= V2 outside
~: Plank constant m`: effective mass V`: confinement potential λ: total energy Interface condition:
α−∂u
∂n ∂D
−
= α+∂u
∂n ∂D
+
Dirichletboundary conditions
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paraboliceffective mass
m` = c`, ` = 1, 2 where c1 and c2 are constant.
⇒ Standard and generalized eigenvalue problems non-paraboliceffective mass
1
m`(λ) = P`2
~2
2
λ + g`− V` + 1
λ + g`− V`+ δ`
, ` = 1, 2
⇒ Polynomial eigenvalue problems
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Numerical experiments for polynomial eigenproblems
Cubic polynomial eigenvalue problem (n = 1, 228, 150)
Pcn(λ)u ≡ (λ3Acn3 + λ2Acn2 + λAcn1 + Acn0 )u = 0.
Zbtm Ztop Zmtx
0
Rdot Rmtx
0
R Z
dot (l=1) matrix (l=2)
(a) Structure schema
0 50 100 150 200250 300 350400 450 500
0
50
100
150
200
250
300
350
400
450
500
nz = 2364
(b) Acn0 , Acn1
0 50 100150 200 250300 350 400 450500
0
50
100
150
200
250
300
350
400
450
500
nz = 548
(c) Acn2 , Acn3
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Precond. ω or ` Itno Time
SSOR(ω)
1.25 367 1,440 1.30 292 1,077 1.35 395 1,734 1.40 377 1,584
ILU(`)
5 157 867
6 85 483
7 100 570
8 102 631
Jacobi - 837 2,588
Table:The total iteration numbers and CPU times of polynomial Jacobi-Davidson method versus different preconditioners for SOneLS.
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Energy states and wave functions
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
1, (1,0,1) 11, (1,6,1)
2, (1,1,1) 12, (2,3,1)
3, (1,2,1) 13, (3,1,1)
4, (2,0,1) 14, (1,7,1)
5, (1,3,1) 15, (2,4,1)
6, (2,1,1) 16, (1,0,2)
7, (1,4,1) 17, (1,1,2)
8, (2,2,1) 18, (3,2,1)
9, (1,5,1) 19, (1,2,2)
10, (3,0,1) 20, (1,8,1)
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Quintic polynomial eigenvalue problem (n = 5, 216, 783)
Ppn(λ)u ≡ (λ5Apn5 + λ4Apn4 + λ3Apn3 + λ2Apn2 + λApn1 + Apn0 )u = 0.
Pyramid Dot Cuboid Matrix
(a) structure schema
0 50 100 150 200
0
50
100
150
200
nz = 1477
(b) Apn0 , Apn1
0 50 100 150 200
0
50
100
150
200
nz = 449
(c) Apn2 , Apn3
0 50 100 150 200
0
50
100
150
200
nz = 34
(d) Apn4 , Apn5
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Precond. ω or ` Itno Time
SSOR(ω)
1.80 98 4,087 1.85 98 3,831 1.90 101 4,028 1.95 97 4,036 ILU(`)
0 102 3,981
1 83 4,008
2 72 4,231
ICC(`)
0 130 4,662
1 99 4,394
2 94 5,093
Jacobi - 312 9,633
Table:The total iteration numbers and CPU times of polynomial Jacobi-Davidson method versus different preconditioners for SOneLS.
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Numerical experiments for linear eigenproblems
Symmetric eigenvalue problem (n = 5, 216, 783)
Ppc(λ)u ≡ (Apc− λI)u = 0 where Apcis asymmetric positivematrix.
Pyramid Dot Cuboid Matrix
(a) structure schema
0 50 100 150 200
0
50
100
150
200
nz = 1477
(b) Apc
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Precond. ω or ` Itno Time
SSOR(ω)
1.80 86 1,905 1.85 82 1,818 1.90 93 2,090 1.95 114 2,659 ILU(`)
0 141 3,273 1 106 3,196
2 89 3,284
ICC(`)
0 143 2,990 1 101 2,627
2 85 2,887
Jacobi - 388 6,342
Table:The total iteration numbers and total CPU times of
Jacobi-Davidson method versus different preconditioners for SOneLS.
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Unsymmetric eigenvalue problem (n = 1, 200, 000)
Pcc(λ)u ≡ (Acc− λI)u = 0 where A is aunsymmetric matrix.
(a) structure schema
0 50 100 150 200 250 300 350
0
50
100
150
200
250
300
350
nz = 1840
(b) Acc
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Arnoldi withKrylov Schurrestarting GMRES
LU ILU(6) SSOR(1.85)
Rstno Itno Time Itno Time Itno Time
10 10 163 12 5334 12 5527
15 5 161 6 4669 5 4264
20 3 159 4 4620 4 4745
25 2 159 3 1656 3 4825
30 2 169 2 4164 2 4436
35 2 179 2 3819 2 5128
40 1 165 1 3819 1 4043
45 1 172 1 4275 1 4527
50 1 180 1 4738 1 5017
Table:Numerical results for solving Pcc(λ)u = 0.
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Generalized eigenvalue problem (n = 1, 228, 150)
Pic(λ)u ≡ (A(ic)0 − λA(ic)1 )u = 0
where A(ic)0 issymmetric positive definiteand A(ic)1 is apositive diagonalmatrix.
Quantum dot Matrix
(a) structure schema
0 20 40 60 80 100
0
20
40
60
80
100
nz = 880
(b) Aic
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Lanczos withKrylov Schurrestarting GMRES
LU ILU(6) ICC(0)
Rstno Itno Time Itno Time Itno Time
10 12 274 18 5323 18 148803
15 5 263 8 4343 8 12284
20 4 272 5 4015 5 10956
25 3 277 4 4201 4 11848
30 2 274 3 4116 3 11783
35 2 288 3 4740 2 11875
40 1 268 3 5423 2 11877
45 1 278 2 4712 2 13410
50 1 289 1 3613 1 10262
Table: Numerical results for solving Pic(λ)u = 0.
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Resonances in rail tracks excited by high speed trains
With new ICE trains crossing Europe at speeds of up to 300 km/h, sound and vibration levels in the trains are an important issue.
Hilliges/Mehrmann/Mehl(2004) first proposed this problem on a project with company SFE GmbH in Berlin.
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Finite Element Model
(a). The rails are straight and infinitely long:
A 3D finite element discretization of the rail with linear isoparametric tetrahedron elements(Chu/Huang/Lin/Wu, JCAM, 2007) produces an infinite-dimensional system of O.D.E.:
M ¨x + D ˙x + Kx = F,
where M, D and K are block-tridiagonal matrices.
Figure:A 3D rail model.
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(b). The rail sections of track between two ties are identical:
The system is periodic and leads to a Palindromic QEP P (λ)x ≡ (λ2A>1 + λA0+ A1)x = 0, (7) where A0, A1 ∈ Cn×n, and A>0 = A0.
The coefficient matrices A0and A1in (7) depend on some ωassociated with the excitation frequency of the external force.
Eq. (7) is called a palindromic QEP problem.
Symplectic property
If λ is an eigenvalue of P (λ), then so is λ−1.
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Numerical results
10-20
10-10
100 1010 1020
10-18
10-16
10-14
10-12
10-10
10-8
| l |
Relative residuals of eigenpairs
S A_I QZ S A_II
Figure:Matrix size is303.
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Maxwell’s equations
If we design a three-dimensional photonic crystal appropriately, there appears a frequency range where no electromagnetic eigenmode exists. Frequency ranges of this kind are called photonic band gaps.
Large full band gaps allow strong photon localization with the gap and a detailed manipulation of photonic defect states.
Air Air Si
Si
Figure: Macroporous silicon photonic crystals
G X M R G
0 0.1 0.2 0.3 0.4 0.5 0.6
Frequency (λ1/2 a / 2π)
Figure: Band structure
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Governing equations
Maxwell’s equations:
∇ × H(x, t) = ε(x)∂tE(x, t),
∇ × E(x, t) = −µ0∂tH(x, t),
∇ · (εE(x, t)) = 0,
∇ · (H(x, t)) = 0.
E: electric field H: magnetic field
Eliminating H field, we obtain the second order Maxwell’s equation
∇ × ∇ × E(x, t) = −µ0ε(x)∂t2E(x, t).
Take E(x, t) = eiωtE(x) (time harmonic modes). Then˜
∇ × ∇ × ˜E(x) = µ0ω2ε(x) ˜E(x) ≡ λε(x) ˜E(x).
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Discretized∇ × h = λεeik·xeat edge can be represented as the following matrix form:
D∗G~h = λB~e,
where B is a diagonal matrix whose entries are the dielectric coefficients ε.
The discretization of (1) atedges forms the following generalized eigenvalue problem:
A~e = λB~e with A = D∗GG∗D. (3)
0 500 1000 1500 2000 2500 3000
0
500
1000
1500
2000
2500
3000
nz = 39000
Figure: Sparsity of matrix A.
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Numerical results
Example (Silicon-Air)
Consider a modified simple cubic lattice consisting of dielectric spheres with radius r on the lattice sites, each connected to its six nearest neighbors by thin circular cylinders which radius is s. The distance between the centers of neighbor dielectric spheres is a. Here, we take dielectric contrast εi/εo= 13and radius r/a = 0.345, s/a = 0.11 with εo= 1and a = 1.
2p
a
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G X M R G
0 0.1 0.2 0.3 0.4 0.5 0.6
Frequency (λ1/2 a / 2π)
Figure:Band structure computed with 100 × 100 × 100 grids using null space free JD Algorithm with θ∗= 0. The gap-midgap ratio is 0.1400.
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X M R
0 500 1000 1500 2000 2500
k vector
CPU Time (s)
K K
JDïS JDïH KS, ncv=25 KS, ncv=30 KS, ncv=35 KS, ncv=40
Figure:CPU time in seconds of KS and HJD with PS3-FFT preconditioner versus k. The dimension of the test problem is 375, 000(3 × 503).
