Energy States of Vertically Aligned Quantum Dot Array with Nonparabolic Effective Mass

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ELSEVIER

An International Journal Available online at www.sciencedirect.com

computers &

,,c,=.cE C__~o,,,=cT- mathematics

with applications Computers and Mathematics with Applications 49 (2005) 39-51

www.elsevier.com/locate/camwa

Energy States of

Vertically Aligned Quantum Dot Array with Nonparabolic Effective Mass

T S U N G - M I N H W A N G

Department of Mathematics, National Taiwan Normal University Taipei 116, Taiwan

min©math, ntnu. edu. t w

W E I C H U N G W A N G *

Department of Applied Mathematics, National University of Kaohsiung Kaohsiung 811, Taiwan

wwang~nuk, edu. t w

(Received February 200~; revised and accepted September 2003)

A b s t r a c t - - T h e electronic properties of a three-dimensional quantum dot array model formed by vertically aligned quantum dots are investigated numerically. The governing equation of the model is the SchrSdinger equation which is incorporated with a nonparabolic effective mass approximation that depends on the energy and position. Several interior eigenvalues must be identified from a large- scale high-order matrix polynomial. In this paper, we propose numerical schemes that are capable of simulating the quantum dot array model with up to 12 quantum dots on a personal computer.

K e y w o r d s - - S e m i c o n d u c t o r quantum dot array, The SchrSdinger equation, Energy levels, Cubic large-scale eigenvalue problems, Matrix reduction, Cubic Jacobi-Davidson method.

1. I N T R O D U C T I O N

Recent advances in fabrication of semiconductor q u a n t u m dot (QD) [1] have generated m a n y zero- d i m e n s i o n a l atomically h e t e r o j u n c t i o n interfaces. A m o n g the various nanoscale heterostructures, the q u a n t u m dot array (QDA) formed by multiple layers of vertically aligned QDs have a t t r a c t e d great interest. T h e s t r u c t u r e has been studied intensively in t h e o r y [2-4], e x p e r i m e n t [5-9], a n d c o m p u t a t i o n [10-13]. QDAs are further applied in the development of applications like the QD laser [14,151, light storage devices [16], QD molecule [17], a n d QD computers [18,19].

This work is partially supported by the National Science Council and the National Center for Theoretical Sciences in Taiwan.

*Author to whom all correspondence should be addressed.

The authors thank Wen-Wei Lin for many helpful discussions and Chia-Chi Chung for assistance on some computational results. The authors are grateful to the referees for carefully reading the manuscript and their helpful comments. Our gratitude also goes to the Academic Paper Editing Clinic, National Taiwan Normal University.

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40 T.-M. HWANG AND W. WANG

One of the main research topics in the area deals with the induced energy levels and associated wave functions. The ground and excited state spectrum is of basic physical interest and is crucial for designing photoelectric devices. Furthermore, wave function overlap results in inter- dot coupling and suggests the possible creation of artificial molecules [20] and a new computing concept [21]. The energy levels can be investigated by methods like photoluminescenee, pho- tocurrent, photoreflectanee, etc., [22] The information obtained by these methods, in certain circumstances, can be limited to lower energy states or low temperatures [23]. Also, relatively few research reports focus on real three-dimensional (3D) QDA, in which neither analytical tech- niques nor asymptotic analysis provides useful information. Numerical simulation becomes a very important way of studying this problem.

In this paper, we focus on numerical simulation of a single-electron QDA heterostructure with a nonparabolic effective mass approximation. This model is proposed in [24] and is used and extended in various works [25-28] and the references therein. After discretizing the governing equation of the model, large-scale eigenvalue problems must be solved to compute the energy levels (eigenvalues) and wave functions (eigenvectors). While several computational methods based on Lanczos or QR decomposition methods are available for 3D QDA, these methods can suffer from the lack of accuracy and efficiency.

In [29], various fixed-point methods for computing the ground state energy are presented for the single quantum dot model. Focusing on the QDA model, we further present numerical schemes which allow the computation of all the bounded energies while preserving the structure for fast- solver. This allows us to explore the energy bifurcation behavior induced by the various numbers of stacked quantum dots. Furthermore, our numerical schemes include a dimension reduction process, an explicit deflation scheme for computing all the desired eigenpairs successively, a preconditioning scheme, and a cubic Jacobi-Davidson eigenvalue solver. Note that similar schemes have been applied to a single cylinder [30] QD model.

The numerical schemes have been implemented on a personal computer. The codes efficiently simulate the QDA model formed by two to twelve quantum dots. Not only have all the desired eigenpairs been computed, several novel electronic properties have also been discovered for the QDA model. The computation shows that larger numbers of quantum dots or smaller spacer layer distances result in the lower ground state energies. Splitting of energies are observed for thin spacer layers. Energy bifurcations as a function of the spacer layer distance are found in lower energy states. The energy bifurcations are closely related to the size of nodal sets that is associated with the wave functions.

The rest of the paper is organized as follows. We first describe the vertically aligned QDA model in Section 2. The numerical schemes simulating the QDA model are discussed in Section 3.

Computational results and discussions of the findings regarding the electronic properties are presented in Section 4. We conclude the paper in Section 5.

2. V E R T I C A L L Y A L I G N E D Q U A N T U M D O T A R R A Y M O D E L We consider the model in which vertically aligned disk-shaped coaxial InAs QDs are embedded in a cylindrical GaAs matrix as shown in Figure 1. In the model, the QDs have identical radii Rdot and heights Hdot. Furthermore, the QDs are separated equally by do nm GaAs spacer layers.

Disk-shaped (cylindrical) dots have been studied extensively in theory [31,32], in computation [13,33], and by experiments [34-36].

Assuming an effective one electronic band Hamiltonian, energy and position dependent electron effective mass approximation, a finite height hard-wall 3D confinement potential, and the Ben Daniel-Duke boundary conditions [37], we consider the governing Schr6dinger equation of the model in cylindrical coordinates,

--h 2 [ a 2 F 1 0 F 1 02F 0 2 F ]

2me(A) [ Or 2 + -r -g-Jr + --r 2 --002 + Oz 2 j + eeF = AK (1)

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Energy States 41 Z~

r

d o ',

Hdot

i

matrix (/--2)

i Rdot l

I" "1

Rmtx

Figure 1. Structure schema of a cylindrical vertically aligned quantum dot array and the heterostructure matrix.

where h is the reduced Plank constant, A is the total electron energy, F = F(r, O, z) is the wave function, rag(A) and c~ are the electron effective mass and confinement potential in the ~th region.

Index g is used to distinguish the region of the QDs (for g = 1) from that of the matrix (for g = 2).

Since a significant effect of spin-orbit splitting in narrow gap semiconductors is expected, it is essential to consider nonparabolicity for the electron's dispersion relation, for which the effective mass depends on energy and position [24]. The effective mass is, thus,

1 )

r n ~ ( ~ ) - h 2 ~ + g ~ - c ~ + ~ + g ~ - c ~ + 5 ~ ' (2) where Pe, 9~, and 5g are the momentum, main energy gap, and spin-orbit splitting in the ~th region, respectively. For equation (1), the Ben Daniel-Duke boundary conditions are imposed on the interface of the two different materials.

h 2 OF(ri, Oi, zi) h 2 OF@t, Oi, zz)

2.~1(~) 0~_ 2.~2(~) On+

⁽³⁾

where (ri,

0I, ZI)

denotes the position on the interface of the dot and the matrix, and n+ and n_

denote the corresponding outward normal derivatives of the interface that are defined for the matrix and dot regions, respectively. Finally, Dirichtet boundary conditions,

F ( ~ . , eB, ZB) = 0, (4)

are imposed on the boundary (top, bottom, and wall) of the matrix, where (rB, OB, ZB) denotes the position on the matrix boundary.

In summary, equations (1)-(4) are used to compute the electron energy levels and the associated wave function in the system.

3. N U M E R I C A L S C H E M E S

In this section, we propose numerical schemes for solving the 3D QDA SchrSdinger equation (1).

The schemes are derived in a straightforward manner from the schemes used for a single QD in [30].

We first discretize the domain by choosing mesh points. Regular uniform mesh points are chosen in the azimuthal angle 0 coordinate. Nonuniform mesh points are used in the radial

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42 T . - M . HWANG AND W. WANG

Illl llll Illl

l l l l I I I I

ii!!

l l l I

fill Illl

I l l I I I I I I I I I

I l l l

fill

l l l l I l l l I I I I

iiii Illl Illl

I I I I I I I I I I I I

li,,,

II

Illl

IIII

Figure 2. Schema of the nonuni~rm discretization scheme of a two-dot quantum dot array over a 2D half plane. Note that fine meshes are used in the heterojunctions and a half of the mesh length is shined in the radial coordinate•

coordinate r and the n a t u r a l axial coordinate z with the following two special treatments• First, in the heterojunction area, fine meshes are used to capture the rapid change in the wave functions.

Secondly, half of the mesh length is shifted in the radial coordinate to avoid incorporating the pole condition [38,39]. See Figure 2 for details•

Based on the grid points, equation (1) is discretized by the 3D centered seven-point finite difference m e t h o d

--h 2 {Fi+l,j,k -- 2Fi,j,k 4- Fi-1 j k 1 F~+~ j k - F¢-I j k

2m~(A) < ( A t ) 2 ' ' +-r~ ' ' 2 A r ' '

(5) 1 F~,~+~,k -

2F<j,k(/,O)

2 + F~ j_~ k F~,j,k+~ - 2F~,~,k(Az)

2 + F~,j,k-a J]

+ 7 ' ' + + o~Fi,j,k = AF~,j,k,

where Fi,j,k is the a p p r o x i m a t e d value of wave function F at the grid point

(ri, O j, zk),

for g = 1, 2, i = 1 , . . . , p, j = 1 , . . . , #, and k = 1 , . . . , ¢. In the heterojunctions, two-point finite differences are applied to the interface conditions (3) of the QDs. T h e numerical b o u n d a r y values for the m a t r i x in the z- and r-direction are zeros according to the Dirichlet b o u n d a r y conditions (4).

Assembling the finite difference discretizations of equations (1) and (3) results in the following

pp¢-by-p#¢

3D eigenvalue problem,

T~(~) E~(~)

E2(A) B¢-l(A)

F,,~: ~ 1

F : 2

F:,:,¢-1 F:,:,~

: D (1) I F . : I

F:,:,¢-I L F - . ¢

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Energy States 43

w h e r e t h e b l o c k m a t r i c e s a r e d e f i n e d a c c o r d i n g l y . B y r e o r d e r i n g t h e u n k n o w n v e c t o r a n d u s i n g t h e f a s t F o u r i e r t r a n s f o r m a t i o n t o t r i d i a g o n a l i z e m a t r i c e s Tk (.k) (for k ~- 1 , . . . , ~), w e c a n r e w r i t e equation (6) as

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where Tj(A) a n d / ) j ( A ) are p¢-by-p~ matrices. In other words, equation (6) has been converted into p independent

p(-by-p(

eigenvalue problems of the form,

(A) = zsj (s)

for j = 1 , . . . , p. Each of the eigenvalue problems in the form of (8) is called a 2D eigenvalue problem, since the grid points of the unknowns in _Pj have the same 0 value. That is, these grid points are all located in the same vertical half-plane.

T h e Main Algorithm

(1) Initialize i = 0, A(0 °) = A0, A~ °) = AI, A~ °) -- A2, and A~ °) = A3.

(2) Construct the ith deflated system by equation (11).

(3) Solve A ( i ) ( £ i ) Fi : ~"i()'3~(i)''3 " J- "'i~2~(i)--2 + ),iA~ i) + A(~))Fi = 0 by the subroutine shown below.

(4) O u t p u t the ith smallest positive eigenvalue ),i and Fi.

(5) If (the next eigenpair is needed then Let i --- i + 1; goto Step (2).

end.

S u b r o u t i n e :

J a c o b i - D a v i d s o n M e t h o d for C u b i c E i g e n v a l u e P r o b l e m s Given A ( ~ ) = ~3A 3 + )~2A 2 + hA1 + Ao.

(1) Choose an n-by-ra orthonormal matrix V (2) For i = 0 , 1 , 2 , 3

C o m p u t e Wi = A~V and Mi = V * Wi End for

(3) Iterate until convergence

(3.1) Compute t h e eigenpairs (0, s) of of (03M3 + O2M2 + O M I ÷ Mo)s = 0

(3.2) Select the desired (target) eigenpair ( 0 , s) with [Is[j2 = 1.

(3.3) Computer u = Ys, p = A ' ( O ) u , r = A(@)u.

(3.4) If (NT,~ _< :), ~ = e, x = ~, Stop

(3.5) Solve (approximately) a t ± u from t = - - M A l t + e M [ l p ,

uM~lr*

where c = --U,MAlp.

(3.6) Orthogonalize t against V, u = t/]ltll 2.

(3.7) For i = 0 , 1 , 2 , 3 Computer wi = Air Mi = [ M~ V * wi ]

~wi*

, , ~ j, w~ = [W~,wd End for

(3.8) E x p a n d V = [V, v]

Figure 3. T h e cubic Jacobi-Davidson m e t h o d with explicit deflation scheme.

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Now, we multiply the common denominator of (8) to form the cubic A-matrix polynomial, A ( , ~ ) F -- ()~3A 3 JF)~2A 2 @ ~ A , -F Ao) F -- 0,

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where A0, A1, A2, and A3 are n x n real matrices that are independent of A. The cubic eigenvalue problem, then, can be solved by the cubic Jaeobi-Davidson method presented in Figure 3 to compute the smallest positive eigenvalue representing the ground state energy. To estimate the successive smallest positive eigenvalues (i.e., the excited energy states), we apply the explicit deflation scheme described in [40]. The technique transforms the smallest eigenvalue to infinity, while all other eigenvalues remain unchanged. Thus, the originally second smallest eigenvalue becomes the smallest eigenvalue of the new transformed cubic eigenvalue problem. To be precise, the deflated cubic A-matrix polynomial A(A) is defined as

-~k (,~) r ~-- ()~32~ 3 -F /~2-/~2 -Jr- "~-'J'--1 -I- ~Z~O) r = o ,

(io)

where

*do = Ao,

~-1 : A1 - (A1 T .~lA2 Jr- ~21A3 )

yly[,

i 2 = A2

-

(A2 + AIAa)

YlY:,

f~3 = Aa - ( A 3 ) Y G [ .

The second smallest eigenvalue A2 of (9), then, becomes the smallest eigenvalue of (10) and the associated eigenvector is

( I,~ A ^{- t}

Y2

where ~p is the eigenvector associated with the smallest positive eigenvalue in equation (10).

This transformed cubic eigenvalue problem can then be solved by the cubic Jacobi-Davidson method. Such deflation processes are performed repeatedly until all the desired eigenvalues (energy levels) and the associated eigenvectors (wave functions) are found. To do so, simply define A~ 1) =

Ai,

for i = 0 , . . . , 3, and then, the jth smallest positive eigenvalue )~j of A(A) is equal to the smallest positive eigenvalue of

with

A(J)(A) ~ A3A~ j)

+

A2A~ j)

+

AA? )

+

A (y),

A(o j) = Ao,

A~J) A~J-') (A~J-1) - - a(J -1 ) A2 zl(J-1)~

= -- T A y - - I ~ 2 Jr- j _ l ~ - 3

J Yj--lY;--1,

(n)

= -- T A j - - l z l 3

] Yj--IY/--1, A ? = A F 1) - (AF')) y.ly;_,,

where (Aj-1, yj-1) is the smallest positive eigenpair of A0-I)(A).

Rather than the recursive formulation as shown above, equation (11) can be represented by the original matrices Ai directly to reduce the amount of computation. To do so, we let

3

= )h Akyl,

k=i

I j-1 -

k-~ E

^(e)_^T

y}J)= Aj A k - Yk Ye YJ,

k=i

~=1

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Energy States 45

for i = 1,2, 3 and j > 1. Then,

J

(k)

T

AI j+l) = A i - ~ Yi Y k ,

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

j-1

A (#) ( A ) = (A3A3 + A2A2 + AA1 + A0) - E (A3y~k) + A2Y~k) + AY~k)) ykT

k = l

= (AaAa + A2A2 + AA1 + Ao) - U V r , where

and

U = [A3y~ 1) + A2y~ 1) + Ay~ 1), ... ,2,'3y3(J-1) _~ A2y~j-1) _~ Ay~j-1)]

V = [Yl,''" ,Yj--I]'

We finally note that Step (3.5) of the algorithm shown in Figure 3 solves for t approximately by using a preconditioner MA. In our computation, we use the SSOR preconditioning scheme by setting

M A = (D - wL) D -1 (D - w U ) ,

where A(A) = D - L - U with D =diag(A(A)), and L and U are strictly lower and upper triangular of A(A). Furthermore, the preconditioners M ~ ) of A (o)(A) are taken as

IMp)] -1 = MA 1 + MA1U(.[ - V T M A 1 U ) - I V TMA 1.

To conclude this section, we summarize the numerical schemes in Figure 3.

4. N U M E R I C A L R E S U L T S A N D D I S C U S S I O N S

We conduct numerical experiments to explore the bifurcations of the energies under the model and the performance of the proposed algorithms. In our numerical experiments, we assume that the height (//dot) and the radius (Rdot) of the QDs are 3 and 7.5nm, respectively. For the matrix, the radius (Rmt×) is assumed to be 37.5 nm. The matrix layers above the top and below the b o t t o m of the QDA are assumed to be 6 nm.

The QDs in these sizes are approximately comparable with the experimental model [41] and have a significant nonparabolic effect on the band structure [42]. The material parameters used in the experiments are taken from [42,43]: cl = 0.0000eV, gl = 0.4200eV, 51 = 0.4800eV, c2 = 0.7700eV, g2 = 1.5200eV, 52 = 0.3400eV. The values of the nonparabolicity parameter are Ep1 -- 3 m o P ~ / h 2 = 22.2eV and Ep2 = 3rnoP~/h 2 = 24.2eV, where m0 is the free electron mass. Then, we recalculated P~ and 1°22, accordingly.

4.1. B o u n d e d State Energies

The ground state energies of the QDAs are affected by the number of QDs and the spacer layer distances do. Figure 4a shows the computed ground state energies versus the number of QDs. It is clear that more QDs in the QDA results in a lower ground state energy for a fixed do.

Furthermore, for a fixed number of QDs, smaller spacer layer distances lead to lower ground state energies. Figure 4b shows the differences (in logarithm) of the energies for the QDAs containing n and n + 1 QDs, for n = 1 , . . . , 11. For various do = 1, 2, 3 nm, the ground state energies decrease

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46 T . - M . H W A N G A N D W . W A N G

0.38 0.36(

0.34 ( 0.32 0.3 ( 0J ==

0.28

0.26 0"242 3 I

~ - - ~ - o o o o o go°Sn%

~ 0 ~

d _ = 2 n m

" - o - - - - 0 - - - , _ _ _ , _ _ , _ _ _ , _ _ _ , , _ _ , _ _ _

0 .

• ' G ....

' " 0 . . . (3 . . . . . . . ~3 . . . O . . . 0 . . . g ° = 1 nmo

i i F i I I I I

4 5 6 7 8 9 10 11 12

Number of dots

10 ~

uJ 10 3

104 2 i

/ 1 i

. 0 . d o = 3 nm - o - d o = 2 nm d o = l nm

o . ~ - - ~

' " " o . . ~ - t 3 ~ ~ ~ " ~ ' ~ - - ~ _ . . . . " f 3 . . . . ~ b 0

. . . O... ~ ~ ~G~ _ ~ . . . . • 0 . . . --0-- _ ~

' 0 . . . @ . . . - ^{-- 0}

" " 0 .

I I I I I I I . . . '(~)

3 4 5 6 7 8 9 1~0 11

n

(b)

Figure 4. G r o u n d s t a t e energies for various spacer layer d i s t a n c e s do a n d n u m b e r of q u a n t u m dots.

0,8! i i ,

0.77~ . . . i

0 , 7 - ~ _ ~ - -

0.6

0.5

0.4

0.3 / ~

i

0"20 1 2 ⁱ 3 ⁱ 4 ⁱ 5 ⁱ

Spacer layer distance (nm)

Figure 5. Bifurcation for two q u a n t u m dots.

exponentially in a similar m a n n e r . T o be specific, the e n e r g y differences can be nicely fitted b y the linear least-squares lines w i t h slope -0.407.

T h e bifurcations of the b o u n d e d state energies versus spacer layer distances are plotted in Figures 5 a n d 6 for two- a n d t h r e e - Q D array, respectively.

A s the spacer layer distance decreases, n e w energy states are induced. T h e s e i n d u c e d states w e r e not f o u n d for a s y s t e m with completely separated dots (do --+ oo). Bifurcations occur in

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Energy States 47

o., - J

0 °

0 . 4 ~ ~

0.3

0.2

o.1 1 3 4 5

Spacer rayer distance (rim)

Figure 6. Bifurcation for three quantum dots.

0.46

0.44

g

~

^0.42

LU

0.4

0.38

, t i , i , ~ f

I I I

Spacer layer distance (rim)

(a)

4 I 3.5

= 2 . 5 '5

2 1 5

I I

/.2 1'., /.8 ,'6 ; 2'.2 2'., 2'.6 ;.8 3

Spacer layer distance (nrn)

(b)

Figure 7. Zoom-in of the bifurcation graph for do between 1 and 3 nm.

lower energy states. For example, in the two-QD (three-QD) array, bifurcation is found for the second and third energy states for do < 2 nm (do < 1 nm). The splitting of the energy states suggests a possible way to form artificial molecules with different electronic configurations.

We further explore the bifurcation phenomena by taking a close look at the bifurcation of the second (~2) and third (~3) energy states for the QDA containing two QDs. Figure 7a shows the zoom-in of the bifurcation graph for do between 1 and 3 nm. Figure 7b shows the so-called "area of nodal sets" versus do. The area of the nodal sets are measured by the following means. We

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scan the mesh p o i n t s l o c a t e d w i t h i n t h e QD and a t h i n layer (0.1 nm) a r o u n d t h e QD. Then, we count t h e n u m b e r of mesh p o i n t s at which t h e c o r r e s p o n d i n g values of t h e wave function are zero.

P r a c t i c a l l y , we count the n u m b e r of p o i n t s associated w i t h t h e wave functions whose absolute values are less t h a n t h e t h r e s h o l d 1 x 10 -9.

O b s e r v i n g t h e figure, we see t h a t the occurrence of t h e b i f u r c a t i o n is closely r e l a t e d t o the

"area of n o d a l sets". B o t h of t h e e n e r g y lines a n d t h e n o d a l set lines have intersections near do.

Similar results can b e found for o t h e r b i f u r c a t i o n points. We finally n o t e t h a t t h e shape of the wave functions r e m a i n u n c h a n g e d even when a b i f u r c a t i o n occurs. T h a t is, t h e wave functions a s s o c i a t e d w i t h t h e solid e n e r g y line in F i g u r e 7a are the same, even t h o u g h t h e order of energy s t a t e s changes.

T h e c o m p u t a t i o n a l results shown above are consistent w i t h t h e e x p e r i m e n t a l results [44], in the sense t h a t t h e e n e r g y decreases e x p o n e n t i a l l y as the n u m b e r of d o t s in t h e a r r a y increases.

As our n u m e r i c a l schemes are justified b y t h e results m e n t i o n e d , we further d e m o n s t r a t e the c o m p u t a t i o n a l results on b i f u r c a t i o n s of t h e energies. T h e results m a y inspire further e x p e r i m e n t s or a p p l i c a t i o n s like artificial molecules.

4.2. Algorithm Performances

We now show the t i m i n g and a c c u r a c y p e r f o r m a n c e s of t h e p r o p o s e d a l g o r i t h m s on a personal c o m p u t e r . T h e Q D A containing 3, 6, a n d 12 d o t s are considered. All n u m e r i c a l t e s t s were performed on a 1.8 MHz P e n t i u m IV c o m p u t e r w i t h 1,536 m e g a b y t e s m a i n memory. T h e o p e r a t i n g s y s t e m r u n n i n g on t h e machine is R e d H a t Linux version 9 w i t h K e r n e l 2.4.20 - 8.

In Table l a , shows t h e m a t r i x sizes of the original 3D and r e d u c e d 2D e i g e n p r o b l e m s d e s c r i b e d b y e q u a t i o n s (6) a n d (8). Table l b presents t h e c o m p u t a t i o n a l results of t h e g r o u n d s t a t e energy in different settings. Table l c d e m o n s t r a t e s t h e t i m i n g results for two different s t o p p i n g criteria in the i t e r a t i v e processes of t h e a l g o r i t h m d e s c r i b e d in F i g u r e 3. T h e processes were t e r m i n a t e d when the residual of e q u a t i o n (9) was less t h a n 1 × 10 - 4 or 1 x 10 - s .

Table l a clearly shows t h a t t h e d i m e n s i o n r e d u c t i o n is significant. T h e dimensions of t h e original 3D e i g e n p r o b l e m s are g r e a t l y reduced, b y a factor of 360, to a sequence of 2D p r o b l e m s w i t h t r a c t a b l e sizes. Note t h a t t h e d i s c r e t i z a t i o n scheme shown here leads to a 0.1 n m coarse mesh l e n g t h in b o t h t h e r and z directions, and 0.02 n m fine meshes a r o u n d t h e h e t e r o j u n c t i o n

Table 1. Performances of the algorithm.

(a) Matrix sizes of the eigen-problems described in equations (6) and (8).

3 Dots 6 Dots 12 Dots

(p,#,4) (393,360,356) 'i393,360,572) (393,360,1004) p~¢ (3D) 5 0 , 3 6 6 , 8 8 0 8 0 , 9 2 6 , 5 6 0 142,045,920

p4 (2D) 139,908 224,796 394,572

(b) Computed first eigenvalues. The notation r/denotes the residual of equation (9), which is used as the stopping criterion of Step (3) in the subroutine shown in Figure 3.

(p, #, ~) (393,360,356) (393,360,572) (393,360,1004) p$~ (3D) 5 0 , 3 6 6 , 8 8 0 8 0 , 9 2 6 , 5 6 0 142,045,920

p~ (2D) 139,908 224,796 394,572

(c) Computational timing results in minutes and seconds. The notation "2:21" stands for 2 minutes and 21 seconds.

~/---- 1 x 10 - 4 2:21 4:47 11:14

~/= 1 x 10 - s 4:01 9:01 25:15

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Energy States 49 interfaces. As shown in Tables l b and lc, the a l g o r i t h m is c a p a b l e of achieving a c c u r a t e solutions efficiently. A n a d j u s t a b l e p a r a m e t e r r] denoting t h e s t o p p i n g c r i t e r i o n of t h e residual in equa- t i o n (9) was chosen to be 1 x 10 - 4 or 1 x 10 - s . For these two s t o p p i n g criteria, we have o b t a i n e d eigenvalue results t h a t are consistent to five digits. However, we are able to gain one-half savings in t i m i n g when r / = 1 x 10 -4.

5. C O N C L U S I O N

We have n u m e r i c a l l y s t u d i e d the electronic p r o p e r t i e s of t h e ( I n A s / G a A s ) q u a n t u m dot a r r a y formed b y 2 to 12 v e r t i c a l l y aligned q u a n t u m dots. T h e s i m u l a t i o n involves a finite difference scheme a n d a v e r y large-scale 3D m a t r i x eigenvalue problem. T h e m a t r i x r e d u c t i o n scheme has b e e n used to t r a n s f o r m t h e 3D p r o b l e m into a sequence of t r a c t a b l e 2D problems. These 2D cubic p o l y n o m i a l m a t r i x eigenvalue problems are t h e n solved by t h e J a c o b i - D a v i d s o n m e t h o d . T h e p r o p o s e d n u m e r i c a l schemes are p r a c t i c a l and efficient so t h a t we can solve the large-scale eigenvalue p r o b l e m required to c o m p u t e all t h e desired e n e r g y s t a t e s a n d wave functions on an o r d i n a r y c o m p u t e r .

T h e c o m p u t a t i o n a l results show t h a t the g r o u n d s t a t e energy of t h e Q D A can be m a n i p u l a t e d by changing t h e n u m b e r of QDs or the distances of the space layers. A larger n u m b e r of t h e q u a n t u m d o t s or a smaller spacer layer distance results in lower g r o u n d s t a t e energies. C o m p a r i n g with t h e energy s t a t e s of c o m p l e t e l y s e p a r a t e d QDs, e x t r a i n d u c e d b o u n d e d energy s t a t e s can be o b t a i n e d b y decreasing the spacer layer distance. Bifurcations of the energy s t a t e s t a k e place whenever t h e order of t h e n o d a l sets associated with t h e wave functions exchange.

T h e n u m e r i c a l schemes can be easily generalized to other Q D A m o d e l s a n d t h e c o m p u t a t i o n a l results can assist in e s t i m a t i n g energy s t a t e s in s e m i c o n d u c t o r artificial molecules. F u t u r e directions include multielectronic theory, d i s t r i b u t i o n of the strain, self-consistent p o t e n t i a l , and various QD sizes and shapes. Also, while t h r e e - d i m e n s i o n a l e i g h t - b a n d effects have been considered in [45], it is w o r t h investigating the m u l t i b a n d effects of Q D A w i t h n o n p a r a b o l i c effective mass.

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