Computer Physics Communications 141 (2001) 66–72www.elsevier.com/locate/cpc

Computer simulation of electron energy levels for different shapeInAs/GaAs semiconductor quantum dots

Yiming Li a,∗, O. Voskoboynikova,b, C.P. Leea, S.M. Szeaa Department of Electronics Engineering, National Chiao Tung University, Hsinchu 300, Taiwan

b Kiev Taras Shevchenko University, 252030 Kiev, Ukraine

Accepted 13 July 2001

Abstract

A computational technique for the energy levels calculation of an electron confined by a 3D InAs quantum dot (QD)embedded in GaAs semiconductor matrix is presented. Based on the effective one electronic band Hamiltonian, the energyand position dependent electron effective mass approximation, a finite height hard-wall 3D confinement potential, and the BenDaniel–Duke boundary conditions, the problem is formulated and solved for the disk, ellipsoid, and conical-shaped InAs/GaAsQDs. To calculate the ground state and first excited state energy levels, the nonlinear 3D Schrödinger is solved with a developednonlinear iterative algorithm to obtain the final self-consistent solutions. In the iteration loops, the Schrödinger equation isdiscretized with a nonuniform mesh finite difference method, and the corresponding matrix eigenvalue problem is solvedwith the balanced and shifted QR method. The proposed computational method has a monotonically convergent property forall simulation cases. The computed results show that for different quantum dot shapes, the parabolic band approximation isapplicable only for relatively large dot volume. For the first excited states the non-parabolicity effect also has been found tobe stronger than it at ground state. The QD model and numerical method presented here provide a novel way to calculate theenergy levels of QD and it is also useful to clarify principal dependencies of QD energy states on material band parameter andQDs size for various QD shapes. 2001 Elsevier Science B.V. All rights reserved.

PACS: 73.20.Dx; 73.61.-r; 03.65.Ge; 02.60.Cb

Keywords: InAs/GaAs; Semiconductor quantum dot; Dot shapes; Electron energy levels; Computer simulation

1. Introduction

In recent years, the study of semiconductor quantumdots (QDs) has been of a great interest (see [1–4] andreferences therein). Especially, during the last decadewith modern nanotechnologies, it has become possi-ble to fabricate realistic semiconductor QDs in labo-ratories. Unique electronic characteristics of the QDs

* Corresponding author.E-mail address: ymli.ee87g@nctu.edu.tw (Y. Li).

make it possible to model atomic physics in macro-scopic systems experimentally and theoretically [5].The semiconductor QDs are very attractive in micro-and nano-optoelectronics applications; furthermore,the spectral broadening in semiconductor quantumdots caused by the nonuniformity in the QD size andshape is the major concerns in the practical laser andoptical applications [1,4,6–8]. Various experimentalresults demonstrate the InAs/GaAs quantum dots canhave diverse shapes, such as disk, ellipsoid, or coni-cal shapes with a circular top view cross section and

0010-4655/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0010-4655(01)00397-6

Y. Li et al. / Computer Physics Communications 141 (2001) 66–72 67

a large area-to-height aspect ratio [9–15]. The shapeof quantum dots is debated intensively in theoreti-cal works, since an accurate calculation of the elec-tronic structure obviously depends on the dot shape.The commonly used shapes include disk [16,17], ellip-soid [18,19], and conical shapes [20,21]. The energylevel calculation has been done using the effective-mass approximation with [16,17,20–26] and without[18,19,27,28] the coordinate dependence for the ef-fective mass. The variations in the dot size and shapecan produce an energy fluctuation significantly in thestrong confinement region. In addition, the theoret-ical modeling of quantum dot electronic propertiescan be done by different schemes. For a class of dif-ferent models [16,20,24,27,29–33], one needs an as-sumption about the electronic confinement potentialin the system. Among those used confinement poten-

tial models (the parabolic lateral potential, the infi-nite wall potential), a finite hard wall boundary poten-tial model is the most realistic. However, in this casethe nonlinear model problem cannot be solved ana-lytically and exactly, and on the other hand the com-puter simulation provides a new alternative to solvethe problem numerically. Unfortunately, most of de-veloped calculation were done within only the simpleparabolic band approximation for the electron effec-tive mass [24,27,31–33]. It will produce an error in theelectron energy level estimations for InAs/GaAs semi-conductor QDs.

In this study we calculate and compare the electronenergy spectra for three-dimensional small InAs/GaAsquantum dots of different shapes (see Fig. 1): disk,ellipsoid, and conical shape. All of them are cylin-drically symmetric (with the circular top view cross

Fig. 1. Schematic diagrams for different shapes semiconductor QD: (a) disk-shaped, (b) ellipsoid-shaped, and (c) conical-shaped QD.

68 Y. Li et al. / Computer Physics Communications 141 (2001) 66–72

section). We use the effective one electronic bandHamiltonian, the energy and position dependent ef-fective mass (non-parabolic) approximation, and theBen Daniel–Duke boundary conditions in this study.A hard-wall (of finite height) 3D confinement poten-tial induced by real discontinuity of the conductionband at the edge of the dot is also considered in thesimulation. To calculate the ground state and first ex-cited state energy levels, the nonlinear 3D Schrödingeris solved with a novel nonlinear iterative algorithmto obtain the final self-consistent solutions. The non-linear iteration scheme consists of (i) set an initialenergyE for starting simulation, (ii) compute effec-tive massm(E, r) with the energyE, (iii) solve theSchrödinger equation for the energyE, (iv) update thenewer energy and back to step (ii). The feedback it-eration will be terminated when a specified stoppingcriterion on energy is reached. In the iteration loops,the Schrödinger equation is discretized with a nonuni-form mesh finite difference method [34], and the cor-responding matrix eigenvalue problem is solved withthe balanced and shifted QR method [35]. The inverseiteration technique [36,37] is also applied to calcu-late the wave functions. The proposed computationalmethod has a monotonically convergent property forall simulation cases. The computed results show thatfor different quantum dot shapes, the parabolic bandapproximation is applicable only for large dot volume.For the first excited states the non-parabolicity effectalso has been found to be stronger than it at groundstate.

This paper is organized as follows. In Section 2, themodel and computer simulation algorithm are stated.The results and discussion are presented in Section 3.Section 4 is the conclusion.

2. Model and simulation technique

We consider semiconductor QDs in the one-bandenvelope-function formalism for electrons in whichthe effective Hamiltonian is given by [38,39]

H = − h2

2∇r

(1

m(E, r)

)∇r + V (r), (1)

where∇r stands for the spatial gradient,m(E, r), theelectron effective mass, is a function of both energyE

and position. The expression ofm(E, r) is as follows:

1

m(E, r)= P 2

h2

[2

E +Eg(r)−Ec(r)

+ 1

E +Eg(r)+ (r)−Ec(r)

], (2)

V (r) = Ec(r) is the confinement potential,Ec(r),Eg(r) and (r) denote, respectively, the positiondependent electron band edge, band gap, and thespin-orbit splitting in the valence band, andP is themomentum matrix element.

We investigate quantum dots of disk, ellipsoid, andconical shapes with the base (top view) radiusR0and heightz0 in the cylindrical coordinates(R,φ, z).Since the system is cylindrically symmetric, the wavefunction can be written as

Ψ (r)=Φ(R,z)exp(ilφ), (3)

wherel = 0,±1,±2, . . . is the electron orbital quan-tum number. It remains to be a two-dimensional prob-lem in (R, z) coordinates:

− h2

2mi(E)

(∂2

∂R2+ ∂

R∂R+ ∂2

∂z2− l2

R2

)Φi(R, z)

+ Vi(R, z)Φi(R, z)=EΦi(R, z), (4)

whereV1(R, z) = 0 (i = 1) inside andV2(R, z) = V0(i = 2) outside the dot. The boundary conditions are

Φ1(R, z)=Φ2(R, z), z= fS(R),

1

m1(E)

{∂Φ1(R, z)

∂R+ dfS

dR

∂Φ1(R, z)

∂z

}∣∣∣∣z=fS(R)

= 1

m2(E)

{∂Φ2(R, z)

∂R+ df s

dR

∂Φ2(R, z)

∂z

}∣∣∣∣z=fS(R)

,

(5)

where z = fS(R) (S is disk, ellipsoid, or conical-shaped QD) is the contour of the structure’s crosssection on the(R, z) plane. The structure shape isgenerated by the rotation of this contour around thez-axis.

Based on the fact that the electron effective mass is aspatial and energy dependent function, the Schrödingerequation is a nonlinear equation in energy. It is evidentthat the dependence relationship is not only appearingin Schrödinger equation itself but also on the bound-ary conditions of the model. To compute the final con-vergent solution of the model for different shape QDsself-consistently, a computational algorithm, as shown

Y. Li et al. / Computer Physics Communications 141 (2001) 66–72 69

Fig. 2. A solution procedure for the semiconductor QD simulation.

in Fig. 2, for such nonlinear problem is proposed.Due to the energy dependence of the electron effec-tive mass, our calculation consists of iteration loops toreach a “self-consistent” energy solution. The nonlin-ear iteration scheme consists of (i) set an initial energyE for starting simulation, (ii) compute effective massm(E, r) with the energyE, (iii) solve the Schrödingerequation for the energyE, (iv) update the newer en-ergy and back to step (ii). The feedback iteration willbe terminated when a specified stopping criterion onenergy is reached. In each iteration we use a centraldifference method with nonuniform mesh technique[34] to discretize the nonlinear Schrödinger equationfor disk, ellipsoid, and conical-shaped QDs. The dis-cretized Schrödinger equation together with its bound-ary conditions (5) leads to a matrix eigenvalue prob-lem

AX = λX, (6)

where A is the matrix rising from the discretizedSchrödinger equation and boundary conditions,X andλ are the corresponding eigenvectors (wave functions)and the eigenvalues (energy levels).

In the solution of this matrix eigenvalue problem,the QR algorithm is applied. The QR method is thedominant method for solving nonsymmetric matrixeigenvalue problem [35]. Because the matrixA inEq. (6) is an energy dependent, five diagonal, non-symmetric, and large sparse matrix, the eigenvaluesof such matrix can be very sensitive to small changesin the matrix elements [40]. Therefore, a balancingalgorithm to reduce the sensitivity of eigenvalues ofthe matrixA to small changes in the matrix elementsis used firstly [36,40]. The main idea of balancing ismaking use of similarity transformations to set corre-sponding rows and columns of the matrix have com-parable norms, thus reducing the overall norm of thematrix while leaving the eigenvalues unchanged. Then

70 Y. Li et al. / Computer Physics Communications 141 (2001) 66–72

the balanced matrixA is transformed into a simplerupper Hessenberg form [36,40]. In this transforma-tion process, the elimination method is applied to re-duce the balanced matrixA to the Hessenberg matrixform. The eigenvalues of the upper Hessenberg ma-trix are directly computed with the shifted QR method[35,36,40]. The balanced and shifted QR scheme usedfor the eigenvalues calculation is the most stable androbust method in semiconductor QD simulation. Whenan eigenvalue is found, the corresponding eigenvec-tor of this eigenvalue is calculated with the inverse it-eration method [36,37]. The fundamental idea of thismethod is to solve the linear system

(A− ζ I)y = b, (7)

whereb is a trial vector andζ is one of the computedeigenvalues of matrixA. The solutiony will be thecandidate for the eigenvector corresponding toζ. Inthe solution of Eq. (7), we set the trial vectorb to be anonzero unit vector. The exact wave function shouldvanish only at infinity physically, but to simulatethe QD within a finite region efficiently we have toset a zero value at a finite distance artificially. Thiscomputational consideration is designed only for thenumerical simulation purpose. The artificial boundaryis taken far enough so that it does not significant affectthe results. In our calculation experience, the proposed

computational method converges monotonically anda strict convergence criteria (the maximum normerror is less than 10−15 eV) on energies can bereached by only 14–15 feedback nonlinear iterativeloops.

3. Results and discussion

The energy spectrum of the dot consists of a setof discrete levels numerated by the set of numbers{n, l}, wheren is the nth solution of the problemwith the fixed l. In the calculations of the electronenergy spectra for narrow gap InAs cylindrical QDsin GaAs matrix we choose the semiconductor bandstructure parameters [41] for InAs: energy gap isE1g = 0.42 eV, spin-orbit splitting is 1 = 0.48 eV,the value of the non-parabolicity parameter isE1p =3m0P

21 /h

2 = 22.2 eV, m0 is the free electron ef-fective mass. For GaAs we choose:E2g = 1.52 eV, 2 = 0.34 eV, E2p = 24.2 eV. The band offset istaken asV0 = 0.77 eV. First of all, to test the robust-ness of the algorithm, we present the achieved con-vergence rate of the proposed nonlinear iterative al-gorithm. As shown in Fig. 3, this method computeenergy efficiently and has a good convergent rate forthe ground state energy level calculation with non-

Fig. 3. The achieved convergent rate of the proposed iterative scheme for the ground state energy level calculation.

Y. Li et al. / Computer Physics Communications 141 (2001) 66–72 71

parabolic band approximation, where the dot shapeis taken as conical-shaped QD,R0 = 10.0 nm, thedot volumeV = 750 nm3, and (n = 1, l = 0). Inaddition, we also obtain similar convergent resultsfor other simulation cases with various QD sizes andshapes.

As shown in Fig. 4, the calculated ground state{1,0} energy depending on the dot volume is pre-sented. For all calculations, the radiusR is fixed at10.0 nm. For a fixed dot volumeV , the disk-shapedQD has higher geometry aspect ratio, so it has higherenergy level for both the parabolic and non-parabolicband approximations then other shapes. For the firstexcited state, as shown in Fig. 5, we also have simi-lar comparison results. Furthermore, the non-paraboliccorrection leads to a decreasing in the state energy

when dot volume is increased. The difference betweenparabolic (whenmi(E) = mi0, i = 1,2 — the bandedge electronic effective mass) and non-parabolic ap-proximation results gains∼0.15 eV for the disk-shaped QD with the dot volumeV ≈ 750 nm3. Thisdifference can exceed known electron–electron inter-action corrections [24,27,31–33]. For dots have largevolume (V > 2000 nm3) the results from parabolicband approximation converge to the results calculatedwith non-parabolic band approximation. Fig. 5 showsthe first {1,1} excited energy states of the dot. Ascan be seen from the figure, the non-parabolic ap-proximation leads to large corrections in this case.A difference in energy between parabolic and non-parabolic estimations can gain 0.2 eV for conicalshape QD.

Fig. 4. Ground state (l = 0) energy with non-parabolic (solid line) and parabolic (dot line) band effective mass approximations for (a) disk,(b) ellipsoid, and (c) conical shape InAs/GaAs QDs.

Fig. 5. Excited state (l = 1) energy with non-parabolic (solid line) and parabolic (dot line) band effective mass approximations for (a) disk,(b) ellipsoid, and (c) conical shape InAs/GaAs QDs.

72 Y. Li et al. / Computer Physics Communications 141 (2001) 66–72

4. Conclusion

A novel computational technique for the energy lev-els calculation of an electron confined by a 3D InAsquantum dot embedded in GaAs semiconductor ma-trix has been presented. With the developed QD simu-lator, we have found that the widely used the parabolicband approximation lead to a large discrepancy in cal-culation results for electron energy states in varioussmall QDs. This parabolic approximation can be usedonly for large size QDs in the calculation of groundand first excited state energies. The modeling, numeri-cal method, and study presented here not only providea novel way to calculate the energy levels of QD butalso are useful to clarify principal dependencies of QDenergy states on material band parameter and QDs sizefor various QD shapes.

Based on the effective one electronic band Hamil-tonian, the energy and position dependent electron ef-fective mass approximation, a finite height hard-wall3D confinement potential, and the Ben Daniel–Dukeboundary conditions, the 3D model has been formu-lated and solved for the disk, ellipsoid, and conical-shaped InAs/GaAs QDs. To calculate the ground state(l = 0) and first excited state (l = 1) energy levels, thenonlinear 3D Schrödinger has been solved with a de-veloped nonlinear iterative algorithm to obtain the fi-nal self-consistent solutions. In the iteration loops, theSchrödinger equation was discretized with a nonuni-form mesh finite difference method, and the corre-sponding matrix eigenvalue problem was solved withthe balanced and shifted QR method. The proposedcomputational method presented a monotonically con-vergent property for both the ground and first excitedenergy states with various QD shapes. The computedresults show that for different QD shapes, the par-abolic band approximation is applicable only for rel-atively large dot volume. For the first excited statesthe non-parabolicity effect also has been found to bestronger than it at ground state.

Acknowledgements

This work was partially supported by the NSC ofTaiwan under contract No. NSC89-2215-E-009-055.

References

[1] D. Bimberg et al., Thin Solid Films 367 (2000) 235–249.[2] D. Bimberg, Semicond. 33 (1999) 951–955.[3] N.N. Ledentsov et al., Semicond. 32 (1998) 343–365.[4] D. Bhattacharyya et al., IEEE Sel. Top. Quan. Elec. 5 (1999)

648–657.[5] P.A. Maksym et al., Phys. Rev. Lett. 65 (1990) 108–111.[6] R. Leon et al., Phys. Rev. B 58 (1998) R4262–R4265.[7] F. Heinrichsdorff et al., J. Crystal Growth 195 (1998) 540–545.[8] M. Perret et al., Phys. Rev. B 62 (2000) 5092–5099.[9] X.Z. Liao et al., Phys. Rev. B 58 (1998) R4235–R4237.

[10] H. Gotoh et al., Appl. Phys. Lett. 72 (1998) 1341–1343.[11] Z.R. Wasilewski et al., J. Crystal Growth 201/202 (1999)

1131–1135.[12] I. Mukhametzanov et al., Appl. Phys. Lett. 75 (1999) 85–87.[13] H. Saito et al., Appl. Phys. Lett. 74 (1999) 1224–1226.[14] J.P. McCaffrey et al., J. Appl. Phys. 88 (2000) 2272–2277.[15] P.W. Fry et al., Physica E 9 (2001) 106–113.[16] G. Lamouche et al., Phys. Rev. B 51 (1995) 1950–1953.[17] F.M. Peeters et al., Phys. Rev. B 53 (1996) 1468–1474.[18] A. Wojs et al., Phys. Rev. B 54 (1996) 5604–5608.[19] A.H. Rodríguez et al., Phys. Rev. B 63 (2001) 125 319–

125 327.[20] P. Lelong et al., Solid State Comm. 98 (1996) 819–823.[21] D.M.T. Kuo et al., Phys. Rev. B 61 (2000) 11 051–11 056.[22] N. Singh et al., Int. J. Mod. Phys. 14 (2000) 1753–1766.[23] M. Grudman et al., Phys. Rev. B 52 (1995) 11 969–11 981.[24] O. Stier et al., Phys. Rev. B 59 (1999) 5688–5701.[25] M. Califano et al., Phys. Rev. B 61 (2000) 10 959–10 965.[26] M. Califano et al., J. Appl. Phys. 88 (2000) 5870–5874.[27] J. Shumway et al., Physica E 8 (2000) 260–268.[28] S.S. Li et al., J. Appl. Phys. 84 (1998) 3710–3713.[29] P.C. Sercel et al., Phys. Rev. Lett. 83 (1999) 2394–2397.[30] H. Jiang et al., Phys. Rev. B 56 (1997) 4696–4701.[31] C. Pryor, Phys. Rev. B 60 (1999) 2869–2874.[32] S. Bednarek et al., Phys. Rev. B 59 (1999) 12 042–13 036.[33] L.R.C. Fonseca et al., Phys. Rev. B 57 (1998) 4017–4026.[34] R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, 2000.[35] D.S. Watkins, J. Comp. App. Math. (2000) 67–83.[36] J.H. Wilkinson, C. Reinsch, Linear Algebra II of Handbook

for Automatic Computation, Springer-Verlag, 1971.[37] I.C.F. Ipsen, SIAM Rev. 39 (1997) 254–291.[38] G. Bastard, Wave Mechanics Applied to Semiconductor Het-

erostructures, Les Edition de Physique, Halsted Press, NewYork, 1988.

[39] S.L. Chuang, Physics of Optoelectronic Devices, John Wiley& Sons, New York, 1995.

[40] G.H. Golub et al., Matrix Computations, The Hohns HopkinsUniversity Press, 1996.

[41] M. Levinshtein et al., Handbook Series on SemiconductorParameters, World Scientific, Singapore, 1999.