NUMERICAL COMPUTATION OF CUBIC EIGENVALUE PROBLEMS FOR A

NUMERICAL COMPUTATION OF CUBIC EIGENVALUEPROBLEMS FOR A SEMICONDUCTOR QUANTUM DOT MODELWITH NON-PARABOLIC EFFECTIVE MASS APPROXIMATION

TSUNG-MIN HWANG∗, WEN-WEI LIN† , JINN-LIANG LIU‡ , AND WEICHUNG WANG§

Abstract. We consider the three-dimensional Schrodinger equation simulating nanoscale semi-conductor quantum dots with non-parabolic effective mass approximation. To discretize the equation,we use non-uniform meshes with half-shifted grid points in the radial direction. The discretizationyields a very large eigenproblem that only several eigenpairs embedded in the spectrum are inter-ested. The eigenvalues and eigenvectors correspond to the energy states and wave functions of thequantum dots, respectively. Effective and efficient numerical algorithms for computing these valuesare essential for exploring their physical phenomena and related practical applications. We provideinsights into the resulting matrix structures that reduce the 3D problem to a set of independent 2Deigenproblems. The reduction results in cubic λ-matrix polynomial eigenproblems. Several numer-ical algorithms, such as the nonlinear Jacobi-Davidson method and the fixed point method basedon the linear Jacobi-Davidson method, are then proposed for the solutions of these eigenproblems.For computing the successive eigenvalues, we suggest and analyze a novel explicit non-equivalencedeflation technique with low-rank updates. Furthermore, we offer various acceleration schemes in-cluding Newton’s method to improve computational speed. All of the proposed algorithms havebeen implemented and successfully tested for solving the eigenproblems with sizes up to 76 millions.Numerical results are given to demonstrate the usefulness and efficiency of these algorithms.

Key words. quantum dot, Schrodinger equation, energy states, wave functions, cubic λ-matrixpolynomial, large scale eigenproblem, nonlinear Jacobi-Davidson method

AMS subject classifications. 65F15, 65F50, 65N22

1. Introduction. Semiconductor quantum dots (QDs) are nanoscale structuresin which the carriers are confined in all three-dimensions (3D). The carriers exhibitwavelike properties in QDs and discrete energy states exist in the structures. Thesubject has recently attracted intensive researches on exploring their physical phe-nomena and practical applications [19]. Methods like photoluminescence [14] andcapacitance-voltage [24] spectroscopy have been used to study QDs’ electronic andoptical properties. For practical applications, QDs also play an important role in op-toelectronic devices such as QD infrared photodetectors [23], QD lasers [12], memorydevice [18], and QD computing systems [5].

While QDs have being studied theoretically and experimentally, numerical meth-ods can provide useful simulation results. For example, pseudopotential and first-principal methods [35], adiabatic approximation methods [27], and multi-band enve-lope function [30] were considered to study basic QD physical characteristics. Thesenumerical methods, however, may suffer from excessive computing time-consuming orinsufficient accuracy for small size QDs [22]. Besides, little results can be acquired bycurrent computational methods for 3D QDs [13, Section 11.6]. On the other hand,various physical models that are most effective, e.g. the finite hard-wall potentialmodel, cannot be solved analytically. Numerical approximations therefore become an

∗Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan. E-mail:[email protected]

†Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan. E-mail:[email protected]

‡Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan.E-mail: [email protected]

§Department of Mathematics Education, National Tainan Teachers College, Tainan 700, Taiwan.E-mail: [email protected]

1

2 T.-M. Hwang, W.-W. Lin, J.-L. Liu, and W. Wang

essential tool [22]. We thus intend to develop sophisticated and efficient numericalalgorithms with high accuracy for investigating electronic properties over a wide rangeof QD parameters.

On the III-V combination, InAs/GaAs is the most intensely studied system [26].In this study, we thus concentrate on computing energy states and wave functions ofa single electron that is confined by an InAs QD embedded in a GaAs matrix. Ourmethodology, obviously, can be easily applied to other QD structures. For the targetmodel, we consider the effective one electronic band Hamiltonian, the BenDaniel-Duke boundary conditions [4], and non-parabolic electron effective mass dependingon energy and position [15, 33]. At the boundary of the QD, the finite hard-wall3D confinement potential is induced by real discontinuity of the conduction band.Furthermore, we assume the QD is in the form of cylindrical island. The geomet-ric assumption suitably approximates, for example, self-assembled QD fabricated bymolecular beam epitaxy.

Our main goal is to develop dedicated and delicate numerical algorithms that ef-ficiently determine the energy states (eigenvalues) and the corresponding wave func-tions (eigenvectors) of the model. Solving these eigenproblems is a challenge fromboth viewpoints of numerical linear algebra theory and implementation. It is mainlybecause the eigenproblem is sparse and inherently extreme large; on the other hand,we are only interested in a few discrete eigenvalues embedded in the interior of theenergy spectrum.

Several significant results have been achieved to solve the eigenproblems efficiently.First, to discretize the 3D Schrodinger equation, we modify a recently proposed 2Ddisk discretization scheme [20] in which the grid points are half-shifted in the radialdirection. Analyzing the induced matrix structure, we point out that the original 3Dproblem can be mathematically reduced to a set of independent 2D problems. Each ofthe 2D problems involves only the grid points located on a vertical half-plane with thenatural axis as the bounding straight line. The problem size is not only dramaticallyreduced to a computable and controllable scale; there also has no compromise onaccuracy.

In 2D domains, the induced eigenproblems are formed naturally in the rationalform, in which the unknowns exist in the denominators. We further suggest transfer-ring the eigenproblems to a cubic λ-matrix polynomial. The form of cubic eigenprob-lem benefits the computation in various aspects as follows. First, we can then usethe nonlinear Jacobi-Davidson method [3] directly to solve the cubic eigenproblemfor finding the smallest eigenvalue for each of the 2D problems. Furthermore, to findall target eigenvalues successively, we propose an explicit non-equivalence deflationtechnique with low-rank updates. The technique transfers the smallest eigenvalueto infinity, while all other eigenvalues remain unchanged. Thus the original secondsmallest eigenvalue becomes the smallest eigenvalue of the new transformed cubiceigenproblem. On the other hand, we also propose using fixed-point iteration methodto solve the cubic eigenproblem iteratively. Global convergence of the method is ver-ified by viewing the corresponding eigencurves. In each of the iterations, the largesteigenvalue is of interest. Linear Jacobi-Davidson method [7, 32] is used, since it canconverge to the extreme eigenvalues quickly.

We also address acceleration schemes that have been shown useful and effective.For example, after reasonable approximated eigenpairs have been achieved, Newton’smethod is alternately used to acquire quadratic convergence rate when sufficientlyaccurate eigenpairs are available. Various preconditioning schemes like SSOR and

Cubic eigenvalue problems for a semiconductor quantum dot model 3

btmZ

topZ

mtxZ

0

dotR mtxR0

R

Z

Fig. 2.1. Structure schema of a cylindrical quantum dot and the hetero-structure matrix.

diagonal matrix dramatically reduce the computational cost. An adaptive schemethat dynamically relaxes the stopping criteria; and the usage of the knowledge ofthe computed eigenvector in the fixed-point method can save the computation timegreatly.

All the above algorithms have been implemented to perform numerical experi-ments. The eigenvalues are shown and compared with the physical model to justifythe accuracy. Various eigenvector properties like radial symmetric and anti-symmetricare demonstrated. CPU timing comparisons among different algorithms are offeredto illustrate the characters of each algorithm. All the techniques and analysis appearto be extendable to general higher degree λ-matrix polynomial problems arising inother practical applications.

This paper is organized as follows. We first describe the Schrodinger equationand the discretization scheme of the target model in Section 2. The induced matricesleading to the cubic λ-matrix polynomial formulation are also discussed here. Todetermine the smallest eigenvalue, Section 3 proposes a nonlinear Jacobi-Davidsonmethod and a fixed point method based on linear Jacobi-Davidson method. Section 4introduces and analyzes an explicit non-equivalence deflation technique allowing usto compute the successive eigenpairs. Several acceleration schemes are presentedin Section 5 and computational results are demonstrated in Section 6. We finallyconclude the paper in Section 7.

2. Formulation of the eigenproblem. In this section, we first introduce thetarget QD model and the corresponding 3D Schrodinger equation. In Section 2.2, wediscuss our discretization scheme that leads to the discrete 3D Schrodinger equation.To solve the discrete Schrodinger equation, we compose the resulting eigenproblem inSection 2.3. The eigenproblem is further converted to the cubic λ-matrix polynomialeigenproblem in Section 2.4.

2.1. The quantum dot model. We consider a cylindrical InAs QD embeddedin the center of a cylindrical GaAs matrix with the same rotation axis. Figure 2.1illustrates the schema of the structure. In the simulation, the artificial boundary ofthe matrix are chosen far away from the QD to suitably simulate an infinite matrix.The grid points in the cylinder are specified by radial coordinate r, azimuthal angleθ, and natural axial coordinate z. We assume the bottom surface of the matrix iscentered at the origin. Furthermore, we let Zmtx be the coordinate of the matrix


top, Zbtm and Ztop be the bottom and top coordinates of the QD, respectively. Theradiuses of the dot and matrix are denoted as Rdot and Rmtx, respectively.

The 3D Schrodinger equation on the target domain can be written in cylindricalcoordinates as

−~2

2m`(λ)

[∂2F

∂r2+

1r

∂F

∂r+

1r2

∂2F

∂θ2+

∂2F

∂z2

]+ c`F = λF, (2.1)

where F = F (r, θ, z) is a wave function depending on the position, λ is the totalelectron energy, and the index ` is used to distinguish from the region of the dot(` = 1) and the matrix (` = 2). The notation ~ is the Plank constants, c` is theconfinement potential in the `th region.

Eq. (2.1) is equipped with Dirichlet boundary conditions on the top, bottom, andsidewall of the GaAs matrix, i.e.

F (r, θ, Zmtx) = F (r, θ, 0) = F (Rmtx, θ, z) = 0. (2.2)

The interface conditions

−~2m1(λ)

∂F∂r

∣∣∣R−dot

= −~2m2(λ)

∂F∂r

∣∣∣R+

dot

,

−~22m2(λ)

∂F∂z

∣∣∣Z−btm

= −~22m1(λ)

∂F∂z

∣∣∣Z+

btm

,

−~22m1(λ)

∂F∂z

∣∣∣Z−top

= −~22m2(λ)

∂F∂z

∣∣∣Z+

top

(2.3)

are applied on the interfaces of the two materials. Furthermore, we adopt a recentlyproposed non-parabolic effective mass approximation that is suitable for QDs withsmall sizes [22, 33]. The approximation assume the electronic effective mass satisfiesthe equation

1m`(λ)

=P 2

`

~2

(2

λ + g` − c`+

1λ + g` − c` + δ`

),

where P`, g`, and δ` stand for the momentum, main energy gap, and spin-orbit split-ting in the `th region, respectively. Note that this non-parabolic approximation ofthe electronic effective mass depends on the total energy. The semiconductor bandstructure parameters used in the numerical computations are c1 = 0.000, g1 = 0.235,δ1 = 0.81, P1 = 0.2875, c2 = 0.350, g2 = 1.590, δ2 = 0.80, and P2 = 0.1993.

2.2. Discretization scheme. To discretize the 3D cylindrical model (2.1), wemodify the disk discretization scheme described in [20]. In both discretization schemes,the grid points are shifted with a half mesh width in the radial direction. This settingavoids placing grid points on the natural axis in the sense that the coefficients ofthe finite difference functions at the axial axis are cancelled out. Therefore, no poleconditions need to be imposed. Besides, by using such discretization scheme, wecan mathematically transfer the resulting 3D eigenproblem into a set of independent2D eigenproblems. Only several 2D eigenproblems need to be solved to obtain acertain number of the smallest eigenvalues (electron energies). These observationsdramatically reduce the computational cost without losing the accuracy.

To generate mesh points, our discretization scheme partitions the domain in theazimuthal direction uniformly. In contrast, since the wave functions change rapidlyaround the heterojunction, the scheme partitions the radial and axial direction non-uniformly by refining the partitions around the interfaces. In other words, we use


1 ρ1

ρ2

ρ3

ρ4

Rdot

Rmtx

1

ζ1

ζ2

ζ3

ζ4

ζ5

ζ6

ζ7

Zbtm

Ztop

Zmtx

Fig. 2.2. Schema of the non-uniform discretization scheme of a 2D half plane.

fine meshes with mesh length ∆rf and ∆zf around the interfaces and we use coarsemeshes with mesh length ∆rc ( ∆rf < ∆rc) and ∆zc ( ∆zf < ∆zc) in other regions.

We summarize the discretization scheme by specifying the grid points as follows.Figure 2.2 demonstrates the discretization scheme schema for a certain azimuth. Letρ1, ρ2, ρ3, and ρ (for 0 < ρ1 ≤ ρ2 ≤ ρ3 < ρ) be the indices on the radial direction,ζ1 to ζ6 and ζ (for 0 < ζ1 ≤ ζ2 ≤ ζ3 < ζ4 ≤ ζ5 ≤ ζ6 < ζ) be the indices on the axialdirection. We have

ri =

(i− 1/2)∆rc, for 1 ≤ i ≤ ρ1,rρ1 + (i− ρ1)∆rf , for ρ1 + 1 ≤ i ≤ ρ3,rρ3 + (i− ρ3)∆rc, for ρ3 + 1 ≤ i ≤ ρ,

where

∆rc =Rdot

ρ1 + ic − 12

and ∆rf =ic∆rc

ρ2 − ρ1,

for some positive integer ic;

θj = (j − 1)∆θ, for j = 1, . . . , µ,

where ∆θ = 2π/µ; and

zk =

k∆zc, for 1 ≤ k ≤ ζ1,zζ1 + (k − ζ1)∆zf , for ζ1 + 1 ≤ k ≤ ζ3,zζ3 + (k − ζ3)∆zc, for ζ3 + 1 ≤ k ≤ ζ4,zζ4 + (k − ζ4)∆zf , for ζ4 + 1 ≤ k ≤ ζ6,zζ6 + (k − ζ6)∆zc, for ζ6 + 1 ≤ k ≤ ζ,

where

∆zc =Zbtm

ζ1 + kcand ∆zf =

kc∆zc

ζ2 − ζ1


for some positive integer kc. Note that the grid point rρ2 = Rdot, rρ = Rmtx, zζ2 =zbtm, zζ5 = ztop, and zρ = zmtx.

Based on the grid points, we use the centered seven-point finite difference methodto discretize Eq. (2.1). Let Fi,j,k be the approximated value of function F at the gridpoint (ri, θj , zk). The equation can be discretized as

−~2

2m`(λ)

(2Fi+1,j,k∆r1 − 2Fi,j,k(∆r1 + ∆r2) + 2Fi−1,j,k∆r2

(∆r1 + ∆r2)(∆r1)(∆r2)+

1ri

Fi+1,j,k − Fi−1,j,k

∆r1 + ∆r2+

1r2i

Fi,j+1,k − 2Fi,j,k + Fi,j−1,k

(∆θ)2+

2Fi,j,k+1∆z2 − 2Fi,j,k(∆z1 + ∆z2) + 2Fi,j,k−1∆z1

(∆z1 + ∆z2)(∆z1)(∆z2)

)+ c`Fi,j,k

= λFi,j,k, (2.4)

for ` = 1, 2, i = 1, . . . , ρ, j = 1, . . . , µ, and k = 1, . . . , ζ. Notations ∆r1 and∆r2 denote mesh lengths along the radial direction that are chosen from ∆rc and∆rf accordingly. Similarly, the mesh lengths along the axial direction, ∆z1 and∆z2, are equal to ∆zc or ∆zf depending on the positions of the grid points. In theheterojunction, we apply the interface conditions. On the sidewall, we have

1m1(λ)

Fρ2,j,k − Fρ2−1,j,k

∆rf=

1m2(λ)

Fρ2+1,j,k − Fρ2,j,k

∆rf, (2.5)

for j = 1, . . . , µ, k = ζ2 + 1, . . . , ζ5 − 1. On the bottom of the dot, we have

−~2

2m2(λ)Fi,j,ζ2 − Fi,j,ζ2−1

∆zf=

−~2

2m1(λ)Fi,j,ζ2+1 − Fi,j,ζ2

∆zf, (2.6)

for i = 1, . . . , ρ2, j = 1, . . . , µ. On the bottom of the dot, we have

−~2

2m1(λ)Fi,j,ζ5 − Fi,j,ζ5−1

∆zf=

−~2

2m2(λ)Fi,j,ζ5+1 − Fi,j,ζ5

∆zf, (2.7)

for i = 1, . . . , ρ2, j = 1, . . . , µ. Finally, the numerical boundary values for the matrixin the radial and axial direction can be easily obtained by the Dirichlet boundaryconditions (2.2).

2.3. The induced matrices. We now put the discretization and the finite dif-ference formulas together to form the eigenproblems to be solved. The notationdiag[v1, ..., vm] denotes a diagonal matrix constructed from v1,...,vm, where v1,...,vm

can be scalars, vectors, or matrix. The m-by-m zero and identity matrices are denotedby 0m and Im, respectively.

As our discretization is rooted in the scheme for 2D disk in [20], we first order theunknowns Fi,j,k by grouping the k values and therefore to obtain ζ groups. The kthgroup collects the unknowns with the grid points located on the kth disk. Here thekth disk is the horizontal cross-section of the cylinder passing through zk. In each ofthe disks, the unknowns are further grouped by the i values. So that the concentriccircles are gathered together and sorted in the order from the most inner circle tothe sidewall of the matrix. Finally, for each of circle, the j-indices are arranged inascendent. In short, the unknowns on the kth disk, for k = 1, . . . , ζ, are denoted by


F:,:,k, where

F:,:,k =

F1,:,k

...Fρ,:,k

with Fi,:,k =

Fi,1,k

...Fi,µ,k

for i = 1, . . . , ρ.

Assembling the discretization and equations (2.4)-(2.7), we obtain the followingρµζ-by-ρµζ expression:

T1 E1

B2 T2 E2

. . . . . . . . .Bζ−1 Tζ−1 Eζ−1

Bζ Tζ

F:,:,1

F:,:,2

...F:,:,ζ−1

F:,:,ζ

= D

F:,:,1

F:,:,2

...F:,:,ζ−1

F:,:,ζ

. (2.8)

The block matrices used in Eq. (2.8) are defined as follows. The matrices Bk, Ek,and Tk are generally defined by following forms where

Bk = diag [B1,k, · · · , Bρ,k] ∈ Rρµ×ρµ,

Ek = diag [E1,k, · · · , Eρ,k] ∈ Rρµ×ρµ,

and

Tk =

S1,k H1,k 0G2,k S2,k H2,k

. . . . . . . . .Gρ−1,k Sρ−1,k Hρ−1,k

0 Gρ,k Sρ,k

∈ Rρµ×ρµ. (2.9)

The matrices Si,k, Gi,k, Hi,k, Bi,k, Ei,k are defined according to following two cases.Case (i). If the matrices Si,k, Gi,k, Hi,k, Bi,k, Ei,k do not involve with the interface,

Si,k = − ~2

2m`(λ)

ηi,k − 2βi βi βi

βi ηi,k − 2βi βi

. . . . . . . . .βi ηi,k − 2βi βi

βi βi ηi,k − 2βi

∈ Rµ×µ,

(2.10)and

Gi,k = − ~2

2m`(λ)ϕiIµ, Hi,k = − ~2

2m`(λ)αiIµ, Bi,k = − ~2

2m`(λ)%kIµ, Ei,k = − ~2

2m`(λ)τkIµ,

where ηi,k, βi, ϕi, αi, %k, and τk are constants defined accordingly.Case (ii). If the matrices Si,k, Gi,k, Hi,k, Bi,k, Ei,k involve with the interfaces, then

we have

Si,k = (~2

2m1(λ)+

~2

2m2(λ))Iµ


Interface type Bi,k Ei,k Gi,k Hi,k

Sidewall 0µ 0µ−~2

2m1(λ)Iµ−~2

2m2(λ)Iµ

Top −~22m1(λ)Iµ

~22m2(λ)Iµ 0µ 0µ

Bottom −~22m2(λ)Iµ

~22m1(λ)Iµ 0µ 0µ

Table 2.1Possible choices of matrices Bi,k, Ei,k, Gi,k, Hi,k involving interface.

and the matrices Bi,k, Ei,k, Gi,k, Hi,k are chosen from the three cases de-scribed in Table 2.1. Finally, the block diagonal matrix D is defined as

D = diag[(λ− c2)Iρµ, · · · , (λ− c2)Iρµ, I, I , · · · , I, I , (λ− c2)Iρµ, · · · , (λ− c2)Iρµ

],

where

I = diag[0ρ2µ, (λ− c2)I(ρ−ρ2)µ

] ∈ Rρµ×ρµ,

and

I = diag[(λ− c1)I(ρ2−1)µ, 0µ, (λ− c2)I(ρ−ρ2)µ

] ∈ Rρµ×ρµ.

We now turn our intention on transferring the coefficient matrix in (2.8) into ablock diagonal matrix. The process is divided into following steps.

1. Reorder the unknowns.We first reorder the unknowns by the j values so that the unknowns involvingthe same azimuth are grouped together. Let Π1 ∈ Rρµ×ρµ be the permutationthat groups the unknowns on the same disk by j values. So that the unknownslocated on the same azimuth are grouped together. That is,

[F1,1,k, · · · , F1,µ,k, F2,1,k, · · · , F2,µ,k, · · · , Fρ,1,k, · · · , Fρ,µ,k] Π1

= [F1,1,k, · · · , Fρ,1,k, F1,2,k, · · · , Fρ,2,k, · · · , F1,µ,k, · · · , Fρ,µ,k] ,

for k = 1, ..., ζ. On the other hand, let Π2 ∈ Rρµζ×ρµζ be the permutationthat gathers the unknowns involving the same azimuth over the disks, sothat the unknowns involving the same vertical slice is grouped together. Forj = 1, ..., µ, the jth slice is the vertical half-plane that its azimuth is equal toθj and the slice has the natural axis as the bounding straight line. In short,by performing the permutations Π1 and Π2, we have

[FT

:,:,1, · · · , FT:,:,ζ

]diag [Π1, · · · , Π1] Π2 =

[FT

:,1,:, · · · , FT:,µ,:

]

where

F:,j,: = [F1,j,1, · · · , Fρ,j,1, F1,j,2, · · · , Fρ,j,2, · · · , F1,j,ζ , · · · , Fρ,j,ζ ]T

for j = 1, ..., µ.


2. Tridiagonalize matrices Tk.Si,k are symmetric circulant matrices that contain nonzeros on the tri-diagonaland the right-upper and left-lower corners. As shown in [17], these matricescan be diagonalized by applying Fourier matrix transformation, which is alsoa similar transformation. On the other hand, since the matrices Gi,k, andHi,k are all diagonal matrices, the matrices remain unchanged if we apply anorthonormal matrix and its transpose on their left and right sides. Therefore,by applying an orthonormal matrix W ∈ Rρµ×ρµ (which can be determinedby the procedure described in [17]) on Tk, we obtain a matrix with nonzeroson the main diagonal and two off-diagonals. We further perform a permu-tation Π1 to compose a new block diagonal coefficient matrix in which eachblock is tridiagonal. That is,

ΠT1 WT TkWΠ1 = diag [T1,k, · · · , Tµ,k] ,

where Ti,k are ρ-by-ρ tridiagonal matrices for i = 1, . . . , µ.3. Transfer Eq. (2.8) into block diagonal system.

Combining the reordering and tri-diagonalization schemes discussed above,the ρµζ-by-ρµζ expression (2.8) can be transformed to µ independent ρζ-by-ρζ systems with the form

TjFj = DjFj ,

where j = 1, . . . , µ. In other words, Eq. (2.8) has been transformed to thefollowing form

T1

. . .Tµ

F1

...Fµ

=

D1

. . .Dµ

F1

...Fµ

, (2.11)

where, for j = 1, ..., µ,

Tj =

Tj,1 E1

B2 Tj,2 E2

. . . . . . . . .Bζ−1 Tj,ζ−1 Eζ−1

Bζ Tj,ζ

∈ Rρζ×ρζ ,

Dj = diag [D1, · · · , Dζ ] ∈ Rρζ×ρζ ,

and

F1

...Fµ

= ΠT

2 diag [WΠ1, · · · , WΠ1]T

F:,:,1

...F :, :,ζ

.

The ρ-by-ρ diagonal matrices Bi, Ei, and Di are described in detail in [16].Furthermore, Tµ−j+1 = Tj and Dµ−j+1 = Dj for j = 2, . . . , µ/2 (or (µ−1)/2,if µ is odd).


Up to this point, we have discretized the 3D Schrodinger equation (2.1) to obtain(2.4). Solving the discrete 3D Schrodinger equations (2.4) results in a large sparseequation (2.8). By suitable ordering, permutation, and transformation, the ρµζ-by-ρµζ 3D problem can be converted into the system (2.11). This system contains µindependent 2D ρζ-by-ρζ problems.

We conclude this subsection by following note. If a regular discretization schemecontaining grid points on the natural axis is used, the matrices Si,k defined in (2.10)would contain nonzero elements on the first row and the first column. Therefore, thematrices Tk would not be able to be tridiagonalized. As the consequence, we will notbe able to transform the 3D problem to the 2D problems as shown above.

2.4. The cubic λ-matrix polynomial. The discretization scheme above leadsto the equations containing µ independent eigenproblems described in (2.11). Todetermine the eigenpairs, the eigenproblems corresponding to each slice can be for-mulated in varied forms. Obviously the most straightforward manner is to composeas the generalized eigenproblem

G(λ)F = λDF, (2.12)

where G(λ) is a ρζ-by-ρζ matrix and some elements contain λ in the denominators, Fis the vector contains the corresponding approximated eigenvector, and D is the corre-sponding diagonal matrix. Note that the coefficient matrix G(λ) contains eigenvaluesin rational form.

On the other hand, we can multiply the common denominator of (2.12) to formthe cubic λ-matrix polynomial (the cubic eigenproblem):

A(λ)F = (λ3A3 + λ2A2 + λA1 + A0)F = 0, (2.13)

where A1, A2, and A3 are independent to λ and these matrices correspond to thecommon factors λ, λ2, and λ3, respectively. Note that after the vector F ∈ Rρζ×ρζ

has being solved, we can compute the eigenvector of Eq. (2.8) by assuming all othercorresponding elements in (2.11) as zero and then inverting the dimension reductionprocess described in Section 2.3. The deduction and full description of the matricesin the eigenproblems (2.13) are rather complicated and tedious. We refer readers to[16] for detail. Here we show the sparsity patterns of the matrices A0, A1, A2, andA3 for ρ = 8, µ = 20, and ζ = 15 in Figure 2.4.

While the rational equation (2.12) and the cubic λ-matrix polynomial (2.13) aremathematically equivalent, we consider that the cubic λ-matrix polynomial (2.13)is a better formulation from the viewpoint of computation. First, we can use thenonlinear Jacobi-Davidson method (Algorithm 3.1) to solve (2.13) directly in a moreconvenient manner. Second, to solve the eigenproblem (2.13) iteratively by usingthe framework of fixed point method (Algorithm 3.2), we will then need to computethe largest eigenvalue of (2.13). By contrast, the eigenproblem (2.12) would requirethe smallest positive eigenvalue that is embedded in the interior of the spectrum.Although the linear Jacobi-Davidson can be used on both formulations, the methodusually converges to the extreme eigenvalues quicker. Third, aiming at (2.13), we canprovide an explicit non-equivalence deflation scheme for determining the successiveeigenvalues.

In the following sections, we will then focus on developing algorithms for solvingthe cubic λ-matrix polynomial (2.13). Note that we did not find superior perfor-mance while the eigenvalues were computed by the rational form (2.12) based on ournumerical experience.


0 20 40 60 80

0

10

20

30

40

50

60

70

80

90

(b) A2, A

3

0 20 40 60 80

0

10

20

30

40

50

60

70

80

90

(a) A0, A

1

Fig. 2.3. Sparsity patterns of the matrices A0 and A1 are shown in (a) and sparsity patternsof the matrices A2 and A3 are shown in (b). The red dots in the figures indicate the componentsinvolving interfaces.

3. Determining the smallest positive eigenvalue. In this section, we focuson computing the smallest positive eigenvalue of the cubic λ-matrix polynomial eigen-problem corresponding to each slice. Our main tool is the Jacobi-Davidson method.Following subsections present both the nonlinear Jacobi-Davidson method for cubicλ-matrix polynomial (Algorithm 3.1) and the fixed point method (Algorithm 3.2) thatis based on the linear Jacobi-Davidson method.

Jacobi-Davidson type methods have been used to solve linear eigenproblem [32],generalized eigenproblems, and polynomial eigenproblems [31]. The Jacobi-Davidsonstyle subspace iteration method is enhanced for the generalized eigenproblem (JDQZ)and the standard eigenproblem (JDQR) in [7]. In stead of the formulation stated in(2.13), the cubic eigenproblem can be linearized to

0 I 00 0 I

A0 A1 A2

FλFλ2F

= λ

I 0 00 I 00 0 −A3

FλFλ2F

. (3.1)

This enlarged eigenproblem can then be solved by variations of Lanczos and Arnoldimethods or linear Jacobi-Davidson method [2]. However, these methods can not berecommended unless following issues have been addressed. First, since the interestedsmallest positive eigenvalues, in general, are in the interior of the spectrum, the shift-and-invert technique should be taken for such a large sparse eigenproblem. However,it costs too much for solving the linear system. Second, convergence performance,efficiency, and accuracy might be reduced in solving the enlarged eigenproblem (3.1).

3.1. Nonlinear Jacobi-Davidson method. Algorithm 3.1 shows the nonlin-ear Jacobi-Davidson method for an arbitrary cubic eigenproblem. The algorithm ismodified directly from the Jacobi-Davidson method for quadratic eigenproblem illus-trated in [3, Section 9.2].


Algorithm 3.1. Nonlinear Jacobi-Davidson Method for Cubic Eigenproblem.

Given A(λ) = λ3A3 + λ2A2 + λA1 + A0.

(1) Choose an n-by-m orthonormal matrix V

(2) For i = 0, 1, 2, 3Compute Wi = AiV and Mi = V ∗Wi

End for

(3) Iterate until convergence(3.1) Compute the eigenpairs (θ, s) of (θ3M3 + θ2M2 + θM1 + M0)s = 0(3.2) Select the desired (target) eigenpair (θ, s) with ‖ s ‖2= 1.(3.3) Compute u = V s, p = A′(θ)u, r = A(θ)u.(3.4) If (||r||2 < ε), λ = θ, x = u, Stop

(3.5) Solve (approximately) a t ⊥ u from (I − pu∗

u∗p )A(θ)t(I − uu∗)t = −r.

(3.6) Orthogonalize t against V , v = t/||t||2.(3.7) For i = 0, 1, 2, 3

Compute wi = Aiv

Mi =[

Mi V ∗wi

v∗Wi v∗wi

], Wi = [Wi, wi]

End for(3.8) Expand V = [V, v]

In Algorithm 3.1, we need to solve the correction equation

(I − pu∗

u∗p)A(θ)(I − uu∗)t = −r (3.2)

approximately. Since the vector t is supposed to be orthogonal to the vector u,

(I − uu∗)t = t− uu∗t = t

and Eq. (3.2) becomes

(I − pu∗

u∗p)A(θ)t = −r (3.3)

as suggested by [31, 32]. While Eq. (3.3) can be solved by the Sherman-Morrisonformula, the following scheme solves for the vector t approximately and cheaply. Ap-plying the Sherman-Morrison formula, Eq. (3.3) can be rewritten as

A(θ)t = −r + εp. (3.4)

Let MA ≈ A(θ). The vectors M−1A r and M−1

A p can be computed approximately by,for example, SSOR. Since the vector is supposed to be orthogonal to the vector u,

u∗t = −u∗M−1A r + εu∗M−1

A p = 0

or

ε = −u∗M−1A r

u∗M−1A p

.


Ap Bp µp = 1 −(λ2A3 + λA2 + A1) A0 1/λp = 2 −(λA3 + A2) λA1 + A0 1/λ2

p = 3 −A3 λ2A2 + λA1 + A0 1/λ3

Table 3.1Possible choice of Ap and Bp.

Further, the approximate vector

t = −M−1A r + εM−1

A p.

We have suggested a schema for solving the cubic eigenproblem. We also highlightimplementation details and numerical experience as follows.

• We restart Step (3) in Algorithm 3.1 for every 50 iterations. To span the initialsubspace while restarting, we take the four Ritz vectors corresponding to thefour Ritz values that are closest to the target eigenvalue (which is equal tozero in our case). Numerical experience showed that the speed of convergencebecame very slow if we restarted with only the most recent approximationfor the desired eigenvector.

• To compute M−1A r and M−1

A p, we choose ω = 1 in the SSOR preconditioningscheme. That is, we set

MA = (D − ωL)D−1(D − ωU),

where A(θ) = D − L − U with D =diag(A(θ)), L and U are strictly lowerand upper triangular of A(θ).

• Although the global convergence proof of the nonlinear Jacobi-Davidson methodremains open, numerical experience demonstrated superior performance forthe cubic eigenproblems.

3.2. Fixed point method. The cubic λ-matrix polynomial (2.13) can also besolved iteratively by fixed point method. To do so, we can rewrite the equation as

Ap(λ)F = µBp(λ)F, (3.5)

where p = 1, 2, 3, and the corresponding Ap and Bp are listed in Table 3.1. Althoughother forms that are equivalent to (2.13) do exist, we did not find superior performancenumerically and thus will only concentrate to the intuitional forms listed in the table.

Algorithm 3.2. Fixed Point Method.

(1) Let i = 0, λi = 0.(2) Until (λ converges to the desired eigenvalue) do

(2.1) Solve A(λi)Fmax = µmaxB(λi)Fmax by Jacobi-Davidson,where µmax is the maximum (positive) eigenvalue.

(2.2) Update i = i + 1; λi = (µmax)−1p

(3) Output λi and F = Fmax.

Algorithm 3.2 illustrates the fixed point method solving the eigenproblem (3.5)iteratively. The global convergence of the fixed point algorithm can be viewed numer-ically from the behavior of the eigencurves. Figure 3.1 demonstrates the eigencurves


0 0.0289 0.060

100

200

300

400

500

600

700

800

900(a) Slope ratio is 0.14 for p=1

λ0 0.0289 0.06

101

102

103

104

105

106

(b) Slope ratio is 3.3 for p=2

λ0 0.0289 0.06

101

102

103

104

105

106

107

108

109

(c) Slope ratio is 279 for p=3

λ

Fig. 3.1. The eigencurves (solid lines) and the curves of λ−p (dash lines) for p = 1, 2, 3 areplotted in part (a), (b), (c), respectively. For λ = 0.0289, the slope ratio of the tangent lines inabsolute value are 0.14, 3.3, and 279, respectively.

and the graphics of y = λ−p for p = 1, 2, 3. To plot the eigencurves of Eq. (3.5) fora certain p and λ ∈ [λs, λe], we choose discretized points of λ in [λs, λe] and thensolve for µ in the corresponding eigenproblems. The sizes of Ap and Bp are 874-by-874. Matlab function polyeig was used for the experiments here. Suppose the curvey = λ−p and the eigencurve intersects at a certain point. This point is also the fixedpoint of the iteration in the fixed point method. Let s1 and s2 be the slope of thetangent lines of the curve y = λ−p and the eigencurve at the intersection point. Thenthe fixed point iteration converges provided |s2/s1| < 1. As shown in Figure 3.1, forλ = 0.0289, the ratio |s2/s1| equals to 0.14, 3.3, and 279 for p = 1, 2, 3 respectively. Inother words, the fixed point iteration converges only for p = 1. Approximated eigen-values and residuals were also computed by the fixed point method for p = 1, 2, 3 andthey are presented in Table 3.2. The numerical results coincide with the predictionsin Figure 3.1.

Based on the discussion, we will then use the formula

−(λ2A3 + λA2 + A1)F =1λ

A0F (3.6)

in the following computations. The fixed point method shown in Algorithm 3.2 useslinear Jacobi-Davidson method to compute the largest positive eigenvalue. The algo-rithm also nicely makes use of the characteristics that the Jacobi-Davidson methodconverges to extreme eigenvalues quickly and needs only inverse of the preconditioner.

While the global proof of the convergence in the Jacobi-Davidson iteration for the


p = 1 p = 2 p = 3Ite. λ Residual λ Residual λ Residual1 3.375e-2 7.650e-3 7.696e-2 1.368e-1 1.957e-1 1.001e-12 2.820e-2 1.074e-3 7.160e-2 6.036e-3 2.371e-1 3.114e-23 2.900e-2 1.590e-4 7.726e-2 6.372e-3 1.886e-1 3.958e-24 2.888e-2 2.337e-5 7.128e-2 6.729e-3 1.325e-1 3.812e-25 2.890e-2 3.438e-6 7.759e-2 7.104e-3 1.938e-1 3.917e-26 2.890e-2 5.057e-7 7.092e-2 7.503e-3 2.440e-1 3.800e-27 2.890e-2 7.439e-8 7.795e-2 7.921e-3 1.179e-1 9.427e-28 2.890e-2 1.094e-8 7.052e-2 8.367e-3 2.436e-1 8.495e-2

Table 3.2Approximated eigenvalues and residuals of the fixed point method for p = 1, 2, 3.

target problem remains open, the matrices structures are close to the ones that con-vergence can be verified. Local convergence of the Jacobi-David method have beenstudied in [25] and [31]. It was shown that Jacobi-Davidson converges locally andquadratically for a generalized eigenproblem, if the coefficient matrix is symmetricand the right hand side matrix is positive definite. Observing the matrix structuresof (3.6), we find that the matrix −[λ2A3 + λA2 + A1] is symmetric except the com-ponents involving the interface conditions, and A0 is positive definite if we ignorethe components involving the interface conditions. Since the number of these com-ponents that involve the interface conditions are relatively small, we conjecture thatthe convergence can be acquired. Our numerical tests verified the convergence inprinciple.

4. Non-equivalence deflation. Now we concentrate on how we can computethe successive eigenvalues after the first (smallest positive) eigenvalue has been ob-tained. Guo, Lin, and Wang [11] proposed a deflation scheme for large sparse quadraticeigenvalue problems. The same authors examined several deflation strategies for an-alytic nondefective matrix function in [10]. Ruhe [28] used the summation of the firstand the second smallest computed eigenvalue as the initial guess of Newton’s methodfor computing the second eigenvalue. Besides, the Jacobi-Davidson method has animplicit deflation scheme by reorthogonalizing [1, 6, 9]. Nevertheless, the followingexample extracted from [11] shows that the Algorithm 3.1 (nonlinear Jacobi-Davidsonmethod) method with implicit orthogonality deflation does not converge to the secondsmallest eigenvalue. Define

C(λ) =

(λ− 1)(λ− 2) 0 0λ− 1 (λ− 3)(λ− 4) 0

0 0 (λ− 5)(λ− 6)2

and C(λ) is a cubic λ-matrix polynomial. Clearly, the eigenpairs (1, [1, 0, 0]T ) and(2,

[1,− 1

2 , 0]T ) correspond to the smallest and second smallest eigenvalues of C(λ).

But the Algorithm 3.1 does not converge to [1,− 12 , 0]T with any appending vector v

of the form [0, ∗, ∗]T .We modify the results in [11] and develop an explicit non-equivalence deflation

technique to solve the basic deflation problem. For the explicit non-equivalence defla-tion of the cubic λ-matrix polynomial (2.13), suppose we have computed the smallest


eigenvalue λ1 and the corresponding normalized eigenvector y1 such that A(λ1)y1 = 0.Let the deflated system be

A(λ)F = (λ3A3 + λ2A2 + λA1 + A0)F = 0, (4.1)

where

A0 = A0,

A1 = A1 − (A1 + λ1A2 + λ21A3)y1y

T1 ,

A2 = A2 − (A2 + λ1A3)y1yT1 ,

A3 = A3 − (A3)y1yT1 .

Theorems 4.1 and 4.2 show the explicit non-equivalence deflation scheme.Theorem 4.1. Let (λ1, y1) be a simple eigenpair of A(λ) as in (2.13), i.e.,

A(λ1)y1 = 0, with yT1 y1 = 1. Then the transformed matrix polynomial A(λ) defined

by (4.1) has the same eigenvalues as those of A(λ) except λ1, which is replaced by ∞,i.e., (σ(A(λ))�{λ1}) ∪ {∞} = σ(A(λ)).

Proof. Using the identity

det(In + RS) = det(Im + SR) (4.2)

and the fundamental matrix theory, one can show that

det(A) = det

[A(λ)− λ

(3∑

k=1

(λ− λ1)k−1 A(k)(λ1)k!

y1yT1

)]

= det

[A(λ)− λ

λ− λ1

(3∑

k=1

(λ− λ1)k A(k)(λ1)k!

y1yT1

)]

= det[A(λ)− λ

λ− λ1A(λ)y1y

T1

]T

=−λ1

λ− λ1detA(λ), (4.3)

where A(k)(λ1) = dkA(λ)dλk

∣∣∣λ=λ1

. Furthermore, because Ay1 = 0, (∞, y1) is also an

eigenpair of A(λ). The theorem is established by combining this fact with (4.3).Theorem 4.2. Suppose λ2 6= λ1 and (λ2, y2) is an eigenpair of A(λ). Let

y2 = (In − λ2

λ1y1y

T1 )y2 ≡ Ty2. (4.4)

Then (λ2, y2) is an eigenpair of A(λ).Proof. First define

T = In − λ2

λ1y1y

T1 . (4.5)

Since the matrix T is invertible by (4.2) with inverse

In − λ2

λ2 − λ1y1y

T1 ,


one can easily check that

A(λ2)y2 = A(λ2)[In − λ2

λ2 − λ1y1y

T1

] [In − λ2

λ1y1y

T1

]y2 = 0.

This completes the proof.Theorems 4.1 and 4.2 have provided an explicit deflation scheme for determining

the eigenpairs successively. However, we note that if the two successive eigenvalues areclose to each other, e.g. λ1 ≈ λ2, the matrix T defined in (4.5) may be ill-conditionedand the transformation (4.4) may thus be inaccurate. But we note that even thoughthe transformation (4.4) is inaccurate, the explicit deflated results provide a goodinitial for iterative methods like Algorithm 3.2 and Newton’s method (Algorithm 5.1).

Now we show that the deflated scheme can be used repeatedly to determine thesuccessive eigenvalues. Define A

(1)i = Ai for i = 0, ..., 3. Using above explicit non-

equivalence deflation technique, the jth smallest positive eigenvalue λj of A(λ) isequal to the smallest positive eigenvalue of A(j)(λ) ≡ λ3A

(j)3 + λ2A

(j)2 + λA

(j)1 + A

(j)0

with

A(j)0 = A0,

A(j)1 = A

(j−1)1 − (A(j−1)

1 + λj−1A(j−1)2 + λ2

j−1A(j−1)3 )yj−1y

Tj−1,

A(j)2 = A

(j−1)2 − (A(j−1)

2 + λj−1A(j−1)3 )yj−1y

Tj−1,

A(j)3 = A

(j−1)3 − (A(j−1)

3 )yj−1yTj−1,

where (λj−1, yj−1) is the smallest positive eigenpair of A(j−1)(λ). For i = 1, 2, 3, let

y(1)i =

3∑

k=i

λk−i1 Aky1,

y(j)i =

3∑

k=i

λk−ij

[Ak −

j−1∑

`=1

y(`)k yT

`

]yj ,

for j > 1. Then

A(j+1)i = Ai −

j∑

k=1

y(k)i yT

k ,

for i = 1, 2, 3. This implies that, for j > 1,

A(j)(λ) =(λ3A3 + λ2A2 + λA1 + A0

)−j−1∑

k=1

(λ3y

(k)3 + λ2y

(k)2 + λy

(k)1

)yT

k

=(λ3A3 + λ2A2 + λA1 + A0

)− UV T ,

where

U =[

λ3y(1)3 + λ2y

(1)2 + λy

(1)1 , · · · , λ3y

(j−1)3 + λ2y

(j−1)2 + λy

(j−1)1

]

and

V =[

y1, · · · , yj−1

].


Furthermore, preconditioner M(j)A of A(j)(λ) can be taken as

[M

(j)A

]−1

= M−1A + M−1

A U(I − V T M−1A U)−1V T M−1

A ,

where MA is a preconditioner of A(λ).

5. Acceleration schemes. While the algorithms discussed above are capableof computing the demanded eigenpairs, several schemes can be applied to acceleratein the computation processes. We first present the Newton’s type algorithm appliedto the cubic λ-matrix polynomial (2.13). In Newton’s method, we need to solve theeigenproblem

A(λ)F = µdA(λ)

dλF, (5.1)

where A(λ) = λ3A3+λ2A2+λA1+A0 and dA(λ)dλ = 3λ2A3+2λA2+A1. Algorithm 5.1

illustrates Newton’s method for solving the eigenproblem.Algorithm 5.1 (Newton’s Method.).

Given A(λ) = λ3A3 + λ2A2 + λA1 + A0.(1) Let i = 0; Initialize λ0.(2) Until (λi converges to the desired eigenvalue) do

(2.1) Solve A(λi)Fmin = µmin(dA(λ)dλ

∣∣λ=λi

)Fmin by linearJacobi-Davidson method, where µmin is the smallesteigenvalue in absolute value sense and Fmin is thecorresponding eigenvector.

(2.2) Update λi+1 = λi − µmin; Let i = i + 1.(3) Output λi and Fmin.

Comparing the fixed point method in Algorithm 3.2 and Newton’s method in Al-gorithm 5.1, we notice that the trade-offs exist between the two algorithms. Newton’smethod is a locally quadratic convergent method and the fixed point method is aglobal linear method. However, in the Jacobi-Davidson iterations, Newton’s methodrequires more matrix-vector multiplications than that of the fixed point method. Also,Newton’s method computes the smallest positive eigenvalue that is embedded in theinterior of the spectrum. While the Jacobi-Davidson method is able to find thesetarget eigenvalues, Jacobi-Davidson tends to converge quicker when extreme eigen-values are required. The Jacobi-Davidson method may thus benefit the fixed pointmethod, because the fixed point method needs the largest eigenvalue in each itera-tion. Therefore, we may use the nonlinear Jacobi-Davidson method (Algorithm 3.1)or the fixed point method (Algorithm 3.2) in the beginning stage. After obtainingreasonable approximations of the eigenpair, we can then switch to Newton’s methodto accelerate the overall convergence.

In the fixed point method and Newton’s method, it is essential to use the previouscomputed eigenpair as the initial eigenpair of the Jacobi-Davidson methods. Byperforming such assignments, Jacobi-Davidson can converge very fast, especially whenthe outer iteration approaches the solution. On the other hand, we would like tomention that we may adaptively relax the stopping criteria of the eigenproblem ineach iteration of the fixed point and Newton’s method. In the beginning stage, thereis no need to apply strict stopping criteria. As the iterates approaching the solutions,


j-Ord. λ Ite. Time1-1 0.0874 1922 24451-2 0.1503 479 6611-3 0.2460 815 12311-4 0.3305 1273 20802-1 0.1102 631 7992-2 0.1932 524 7232-3 0.2972 666 10072-4 0.3385 851 13943-1 0.1387 1727 22043-2 0.2371 525 735

j-Ord. λ Ite. Time3-3 0.3454 2091 31913-4 0.3486 759 12524-1 0.1709 1246 15984-2 0.2812 613 8515-1 0.2054 435 5835-2 0.3245 566 8216-1 0.2412 913 12137-1 0.2777 533 6958-1 0.3141 632 8149-1 0.3496 1069 1362

Table 6.1Computational results of the discrete eigenvalue.

higher accuracy requirement is necessary. Similar ideas that use stopping criteriaadaptively are analyzed and adopted in solving eigenproblems and linear systems in[21] and [34], respectively.

6. Numerical results. We implemented the algorithms by Fortran 90 program-ming language for the numerical experiments. All the numerical tests were performedon a Compaq AlphaServer DS20E workstation equipped with dual 667 MHz CPUsand one gigabytes main memory. The operating system running on the machine isCompaq Tru64 UNIX version 5.0. The timing results are in seconds.

6.1. Energy states and wave functions. We assumed the model that a cylin-drical InAs QD with 15 nm in diameter and 2.5 nm in height is embedded in a cylin-drical GaAs matrix with 75 nm in diameter and 12.5 nm in height. The size of dotwas chosen to take into account of the non-parabolicity effect [22] and realistic dotsizes [29]. The discretization partition contained 755, 280, and 360 mesh points onthe radial, axial, and azimuthal direction, respectively. If the 3D problem were solveddirectly, the problem size would be 755×280×360 (about 76 millions). However, theproposed reduction algorithms allow us to solve only several 2D problems with size755 × 280 (about 211 thousands) to obtain all the discrete energy states. Note thatthe chosen grid size resulted in computed eigenvalues that at least three significantdigits remain unchanged for further refinement.

Algorithm 3.1 was implemented to generate the results in Table 6.1. The tablereports only the discrete energy states (eigenvalues) that are supposed to be less than0.35, which is the difference between the confinement potentials c1 and c2. The en-ergy states become continuous if they are greater than 0.35. The first two columnsof the table present the value of index j, the order of the smallest eigenvalues of eachslice (denoted as Ord.), and the computed eigenvalue λ. Iteration number of Step(3) in Algorithm 3.1 and overall timing for computing each single eigenvalue are alsopresented in the third and fourth columns, respectively. The iterative process wasstopped whenever the residual of (3.6) is less than 1.0× 10−8. While computing thefirst eigenvalue, the initial eigenvector was chosen to be the normalized vector in whichall entries are equal. For the deflated eigenproblems corresponding to the successiveeigenvalues, the initial spanned spaces were formed by several Ritz vectors that wereobtained from the previous eigenproblem and corresponding to the smallest Ritz val-


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)

Fig. 6.1. Spectrum of the energy states (eigenvalues). The quantum numbers corresponding tothe first 10 and 11-20 discrete energy states are listed on the left and right hand side of the spectrum,respectively.

ues. Numerical experience suggested that the convergence behavior of Algorithm 3.1is sensitive to the initial spanned space.

Focusing on the discrete energy states, Table 6.1 reports the computational resultsof the nine 2D problems associated with the slices that j = 1, ..., 9. For the 2Dproblems corresponding to the slices that j = 10, ..., 181, the smallest eigenvalues areall greater than 0.35. Namely, the corresponding energy states are all located in thecontinuous region of the energy states schema. The slices with θ equal to ∆θ and−∆θ, where 0 < ∆θ < π, have identical eigenvalues. For example, the problemsassociated with j = 2 and 360 have identical eigenvalues. Therefore, we do not needto solve the 2D problems with j = 182 to 360 to obtain the energy states distribution.

The energy states of the target model that are less than 0.36 are also summarizedin Figure 6.1. Twenty discrete energy states that are less than 0.35 can be distin-guished easily. The energy states that are greater than 0.35 are very close to eachother and therefore the region on the figure appears to be shadowed. The computa-tional results nicely match the physical property that the energy states are actuallycontinuous in the region.

We now characterize the wave functions (eigenvectors) of the QD. The absolutesquare of the wave functions represent the probability density P (r, z, θ) of finding theparticle at the position (r, z, θ). That is,

P (ri, zj , θk) = |F (ri, zj , θk)|2,

for i = 1, ..., ρ, j = 1, ..., µ, and k = 1, ..., ζ. We use quantum numbers nr, nθ, andnz to characterize the eigenvectors. For each shape of the eigenvectors, the quantum


λ ord. Q.N. j-Ord. λ1 (1,0,1) 1-1 0.08742 (1,1,1) 2-1 0.11023 (1,2,1) 3-1 0.13874 (2,0,1) 1-2 0.15035 (1,3,1) 4-1 0.17096 (2,1,1) 2-2 0.19327 (1,4,1) 5-1 0.20548 (2,2,1) 3-2 0.23719 (1,5,1) 6-1 0.241210 (3,0,1) 1-3 0.2460

λ ord. Q.N. j-Ord. λ11 (1,6,1) 7-1 0.277712 (2,3,1) 4-2 0.281213 (3,1,1) 2-3 0.297214 (1,7,1) 8-1 0.314115 (2,4,1) 5-2 0.324516 (1,0,2) 1-4 0.330517 (1,1,2) 2-4 0.338518 (3,2,1) 3-3 0.345419 (1,2,2) 3-4 0.348620 (1,8,1) 9-1 0.3496

Table 6.2Sorted discrete energy states and the corresponding quantum numbers.

numbers nr, nθ, and nz represent the numbers of nodal lines on the radial, azimuthal,and axial directions, respectively. The eigenvalue corresponding to the eigenvectorwith the set of quantum numbers (nr, nθ, nz) is denoted by λ(nr,nθ,nz). Table 6.2sorts the eigenvalues, presents the corresponding quantum number, and indicateswhich 2D eigenproblem was solved to obtain the eigenvalue.

Figures 6.2 to 6.6 illustrate the shapes of the wave functions. Each of the figurescontains eight sub-figures. The sub-figures on the left column present the wave func-tions in magnitude on the disk that is centered on the axial axis and parallel to ther-θ plane. The disk has the radius in 8.47 nm and its z-coordinate is 5.09 nm. Thesub-figures shown on the right column are the corresponding contour maps.

For those wave functions with quantum number nz = 2, the shapes are anti-symmetric with respect to the middle of the dot. For example, Figure 6.7 demonstratesthe wave functions corresponding to λ(1,1,2). The figure shows the wave functionscorresponding to the three disks that the z-coordinates are within the dot. To bedefinite, the z-coordinates of the disks are 5.09 nm (close to the bottom of the dot),6.25 nm (on the middle of the dot), and 7.41 nm (close to the top of the dot). Thewave functions evolve from one form to the one that is the reflection of the originalform with respect to the horizontal hyperplane passing through the middle of dot.

The slices with θ equal to ∆θ and −∆θ, where 0 < ∆θ < π, have identical eigen-values. The phenomenon can be easily checked by the matrix structures. For theeigenvectors corresponding to the multiple eigenvalues, they are identical with eachother in the shapes provided suitable rotation along the natural axis is performed.Figure 6.8 presents the eigenvectors with the quantum numbers (1, 1, 2). The eigen-vectors correspond to the fourth eigenvalue of the slices that j = 2 and 360. Namelyθ = 2π

360 and θ = − 2π360 . It is apparent that the eigenvectors can be duplicated by a

π/2 degree rotation.Before proceeding to numerical results on the fixed point algorithm, we note that

the nonlinear Jacobi-Davidson method for the deflated cubic eigenproblems can beaccelerated by switching to Newton’s method for solving the original eigenproblem.The idea is based on the following observations. First, the more deflation transforma-tions are performed, the more computational cost on low rank updates are needed.Besides, when the nonlinear Jacobi-Davidson method has achieved small residualsof the deflated eigensystems, the (transformed) iterates are supposed to be near the


Jacobi-Davidson Newton’sj-Ord. λ Ite. Time Ite. Time

1-2 0.1503 243 334 300 2822-2 0.1932 239 323 288 2693-2 0.2371 237 331 295 2764-2 0.2812 200 278 291 272

Table 6.3Comparison between the nonlinear Jacobi-Davidson and Newton’s iterations.

target solution. Therefore, it is reasonable to solve to the original cubic eigenproblemby Newton’s method which converges quadratically locally.

To implement the idea, we switched the deflated system to the original cubiceigenproblem whenever the residual is less than 1 × 10−4. The eigenproblem (5.1)was then solved by Newton’s method. The generalized eigenproblem therein is solvedby the linear Jacobi-Davidson method. To choose the test cases from the deflatedeigensystems, we observe from Table 6.1 that the second eigenvalue of slices withj = 1, 2, 3, 4 converged quickly. It is mainly because that the eigenproblems wereequipped with good initial spanned space in these cases. We thus performed theacceleration scheme in these cases to reduce the disturbance of the initial space.Table 6.3 compares the iteration numbers and timing results of partial iterationsusing nonlinear Jacobi-Davidson and Newton’s method. That is, only the iterationsafter the switch had performed were counted. The table shows that the performanceis comparable to each other. The one with Newton’s acceleration, however, convergeda litter quicker among the four cases.

6.2. Numerical experience on the fixed point algorithm. The cubic eigen-problem (3.6) can be solved by the fixed point method (Algorithm 3.2) and acceleratedby Newton’s method (Algorithm 5.1). Figure 6.9 presents numerical results with thetwo methods. Parts (a) and (b) of the figure respectively show the Jacobi-Davidsoniteration numbers and residual of (3.6) against outer iterations. We compare followingfour different schemes.Scheme 1. We implemented Algorithm 3.2 natively and the numerical results are

plotted by the dashed lines with squares. We modified the JDQZ package [8]as the kernel of the Jacobi-Davidson eigenproblem solver in the algorithm.The package randomly assigns initial eigenvector for each of the eigenproblemsin the outer iteration Step (2), Algorithm 3.2.

Scheme 2. In this scheme, we assigned the initial eigenvector by the approximatedeigenvector obtained from the previous iteration, except the first outer iter-ation. In which we just used a random eigenvector. The dotted lines with×-marks in Figure 6.9 present the results of this scheme.

Scheme 3. This scheme switched to Newton’s method (Algorithm 5.1) wheneverthe residual of equation (3.6) is less than 1.0× 10−3. The dash-dot lines withdiamonds demonstrate the numerical results.

Scheme 4. The stopping criteria of JDQZ of Schemes 1 to 3 are all fixed as theresidual of (3.6) is less than 1.0× 10−8. This scheme relaxed the criteria andchanged it adaptively according following heuristics.

jdqz tol ={

1.0× 10−1, if outer ite = 1,min(5.0× 10−2, residual× 0.5), if outer ite ≥ 2,


where jdqz tol is the adaptive stopping criterion, residual is the residualof the previous eigenproblem, and outer ite is the outer iteration numberin Step (2) of Algorithm 3.2. The solid lines with circles demonstrate thenumerical results.

We assumed that the QD has diameter in 10 nm and height in 5 nm. Theresulting eigenproblem has size 11865-by-11865 in the numerical experiments. All theschemes successfully converged to the first eigenpair with the residuals that are lessthan 1.0 × 10−8. Diagonal preconditioners were used in the linear Jacobi-Davidsoniterations. All fixed point method based schemes (Schemes 1, 2, and 4) took either 12or 13 outer steps, while Newton’s method based scheme (Scheme 3) took only 6 outeriterations. Other than these outcomes, we highlight following remarks by observingFigure 6.9.

• The first scheme took 42-70 JDQZ iterations to converge to the requiredstopping criterion for each eigenproblem. By assigning the approximatedeigenvectors, Scheme 2 achieved the same stopping criterion much quicker.Starting from the fifth iteration, the JDQZ iteration numbers are all less than20.

• By using the adaptive stopping criteria, the JDQZ iteration numbers inScheme 4 dramatically dropped down to the numbers that are less than 10.Although Scheme 4 took one more outer iteration to achieve the residual re-quirement, the overall performance was much better than the methods usingfixed stopping criterion.

• All three fixed point method based schemes (Schemes 1, 2, and 4) convergedlinearly as shown in the figure. Schemes 1 and 2 had almost identical conver-gence behavior. Scheme 3, on the other hand, had a little higher residual oneach outer iteration due to the relaxed stopping criteria.

• Starting from the fourth outer iteration, Scheme 3 switched to Newton’smethod and performed quadratic convergence behavior.

7. Conclusion. Energy states and wave functions characterize important physi-cal properties in developing nanoscale QDs. For the cylindrical InAs/GaAs model, wehave computed the discrete energy states (eigenvalues) and the corresponding wavefunctions (eigenvectors) by solving the eigenproblems composed by the discretized 3DSchrodinger equation.

We have discussed the discretization scheme that allows us to reduce the 3D prob-lem to a set of independent 2D subproblems in the form of cubic λ-matrix polynomial.To solve the eigenproblems, we have proposed using the nonlinear Jacobi-Davidsonmethod, fixed point method, explicit non-equivalence deflation technique, and severalacceleration schemes. All the suggested algorithms have been implemented and theirnumerical characteristics have been reported. All the discrete energy states and thecorresponding wave functions of the model involving the eigenproblems with sizes upto 76 millions were successfully identified. According to our numerical experience,nonlinear Jacobi-Davidson (Algorithm 3.1) is quite efficient on solving the large scaleeigenproblems. On the other hand, the fixed point and Newton’s methods (Algorithms3.2 and 5.1) provide an alternative for determining the eigenvalues. Numerical experi-ence shows that more sophisticated implementation and carefully chosen parameterscan improve the timing performance.

Further applications and challenges remain. Other geometric QDs like sphereand pyramid are common. QD array and stack are promising for many applications.


Numerical methods can be helpful tools for simulating these quantum structures.Moreover, profound matrix analysis is expected to benefit understanding of the eigen-structures of the discretized Schrodinger equation and convergence behavior of theJacobi-Davidson methods. Finally, we note that our cubic λ-matrix polynomial solvercan be easily extended to other applications involving cubic or higher degree λ-matrixpolynomials.

Acknowledgments. This work is partially supported by National Science Coun-cil in Taiwan. The authors would like to thank O. Voskoboynikov for motivating theirattention to the quantum dot model discussed in this paper.

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Fig. 6.2. The eigenvector corresponding to λ(1,0,1), λ(1,1,1), λ(1,2,1), and λ(2,0,1).










Fig. 6.7. The anti-symmetric wave functions corresponding to λ(1,1,2) = 0.3385.


Fig. 6.8. The wave functions corresponding to the multiple eigenvalue λ(1,1,1) = 0.1102.

0 5 10 150

10

20

30

40

50

60

70

80(a)

Outer iteration number

JDQ

Z it

erat

ion

num

ber

0 5 10 1510

−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

(b)

Outer iteration number

Res

idua

l

Use random initial eigenvectorRetain previous approx. ev.Switch to Newton"s methodAdaptive tolerance

Fig. 6.9. Iteration number and residual results of fixed point and Newton’s method based algo-rithms.

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