Numerical Simulation of Quantum Dots

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If we regard. H := m0 k · p as a perturbation of. H0 := p2. 2m0. + V (r) the energy to second order in perturbation for the n-th band reads. En(k) = En(0) +. 2k2. 2m0.
Numerical Simulation of Quantum Dots Marta Markiewicz Joint work with Heinrich Voss Institute of Numerical Simulation, Hamburg University of Technology, 21071 Hamburg, Germany

Overview Quantum Dots are semiconductor heterostructures, in which the free carriers are confined to a small region by potential barriers in all three directions of space (3D). If the size of the region is less than the electron wavelength, the electronic states become quantized at discrete energy levels as it happens in an atom. Applications: micro and optoelectronic devices, modelling systems at the atomic level, proposed for qubit implementation in quantum computers How to model a quantum dot? Dots lie on wetting layer as a result of Stranski-Krastanov growth, a state of the art production technique using the relief of the elastic energy resulting from the large lattice mismatch between two materials. The deposited layer initially grows as a thin two dimensional (2D) wetting layer and after exceeding a critical thickness, the growth mode switches from 2D to 3D leading to the formation of a self-assembled quantum dot on the top of the wetting layer. Dots grow in large arrays rather than standing alone.

Three-dimensional STM image of an uncovered InAs quantum dot grown on GaAs(001). STM Picture Gallery, Electron Spectroscopy Group, Fritz-Haber-Institute, Berlin

Influence of a wetting layer and being a part of an array on the bounded states of the quantum dot is neglected in most simulations. Our results show that it is substantial and therefore should be accounted for in realistic models. A rational eigenvalue problem arises if the nonparabolic effective mass approximation is used. The Nonlinear Arnoldi algorithm is used to solve the rational eigenvalue problem in the structure preserving way.

Atomic resolution STM image of a stack of self assembled quantum dots 50x50 nm. PM Koenraad et al Physica E 17 (2003)

Nonparabolic effective mass approximation The electron wave function satisfies the Schr¨ odinger equation   2 p + V (r) Φ(r) = E(k)Φ(r), 2m0

(1)

where p = −i~∇ is the momentum, r is the lattice position vector and k is the wave vector of the electron. The Bloch theorem states that Φ(r) = eik·r φ(r), where φ(r) is a lattice periodic function. Inserting this into the Hamiltonian (1) we obtain  2    p ~ ~2 k 2 Hφ(r) := + V (r) + k · p φ(r) = E(k) − φ(r). (2) 2m0 m0 2m0 If we regard H 0 := n-th band reads

~ m0 k · p

as a perturbation of En (k) = En (0) +

H0 :=

p2 2m0

+ V (r)

the energy to second order in perturbation for the

~ ~2 X |k · pnn0 |2 ~2 k 2 + k · pnn + 2 , 2m0 m0 m0 0 En (0) − En0 (0) n 6=n

where pnn0 =

R

φ∗n0 pφn0 0 dr3 and the eigenvalues of H0 are computed in the middle of the Bernoullian zone (k = 0) H0 φn0 (r) = En (0)φn0 (r).

(3)

Since k = 0 is the extremal point of E(k) it follows that pnn = 0. Moreover, the 2 2 free particle term ~2mk0 can be neglected in comparison to the interband contributions, thus the energy expansion reads En (k) − En (0) =

|k · pnn0 |2 1 ~2 X ~2 . =: m20 0 En (0) − En0 (0) 2 m(En (0))

(4)

n 6=n

Using the Bloch basis functions for the conduction and valence bands and exploiting their isotropy we arrive at the effective mass of the j-th heterostructure component j ∈ {d-dot, m-matrix} Pj2 1 = 2 mj (E) ~



2 1 + E + Egj − Vj E + Egj + ∆j − Vj

 ,

(5)

where Pj is Kane’s momentum matrix element, Egj is the main energy band gap, ∆j is the spin-orbit splitting and Vj is the confinement potential. The parameters change across the heterojunction resulting in the following effective mass approximation for the heterostructure  md (E), r ∈ Ωd m(r, E) = , (6) mm (E), r ∈ Ωm which is discontinues across the interface. Mathematical models We consider the following models of a InAs quantum dot, quantum dot on wetting layer, array of two vertically aligned quantum dots and array of two vertically aligned quantum dots on wetting layers. All these structures are embedded into a GaAs matrix.

The bounded energy levels are the smallest eigenvalues of time independent Schr¨ odinger equation −∇

~2 ∇φ(r) + V (r)φ(r) = λφ(r), 2m(r, λ)

r ∈ Ωd ∪ Ωm

with effective mass m(r, λ) and confinement potential V (r) both discontinues across the heterojunction. On the interior boundary (∂Ωint ) we apply the BenDaniel–Duke boundary conditions

(7)

1 ∂φ(r) 1 ∂φ(r) = , mm ∂ n ¯ m ∂Ωint md ∂ n ¯ d ∂Ωint

(8)

where n ¯d, n ¯ m denote the outer normal of the dot and the matrix respectively. On the outer boundary (∂Ωout ) for the case of a dot without the wetting layer the homogenous Dirichlet boundary conditions are used φ(r)|∂Ωout = 0.

(9)

In presence of the wetting layer the situation is not as simple. The homogenous Dirichlet boundary conditions are still applicable on the top and bottom wall of the matrix (∂Ωh ) but not any more on its side walls (∂Ωv ), which cut through the wetting layer. The idea is to consider the quantum dot as a small perturbation of a wetting layer.

Therefore the eigenfunction on the boundaries of the wetting layer far enough from the dot should approach these of the pure wetting layer. If we consider the wetting layer to be infinite in x and y direction it becomes a quantum well (one dimensional structure along the z axis). Because of its one dimensionality the eigenfunctions of a such quantum well φ(x, y, z) = φ(z) are not dependent on the space directions x or y. Clearly, ∂φ(x, y, z) ∂φ(x, y, z) = =0 ∂x ∂y holds everywhere so especially on the boundary ∂Ωv . Therefore Neumann boundary conditions are a reasonable choice on the side walls of the matrix ∂Ωv : ( φ(r)| ∂Ωh = 0, (10) ∂φ(r) = 0. ∂n ¯m ∂Ωv

Nonlinear eigenvalue problem ¯ d ∪ Ωm and H := {ψ ∈ H 1 (Ω) : ψ = 0 on ∂Ωh }. Multiplying (7) by ψ ∈ H and integrating by parts, we arrive at Let Ω := Ω the variational form of the Schr¨ odinger equation

a(φ, ψ; λ)

:=

Z Z Z ~2 ~2 ∇φ · ∇ψ dr + ∇φ · ∇ψ dr + Vd φψ dr 2md (λ) 2mm (λ) Ωd Ωm Ωd Z Z +Vm φψ dr = λ φψ dx =: λb(φ, ψ) for every ψ ∈ H. Ωm

(11)



In a similar way as in [5] it can be shown, that the problem (11) has a countable set of positive eigenvalues 0 < λ1 ≤ λ2 ≤ · · · → ∞ of finite multiplicity. Moreover, for fixed ψ 6= 0 the real equation f (λ; ψ) := λb(ψ, ψ) − a(ψ, ψ; λ) = 0

(12)

has a unique positive solution p(ψ). Hence, equation (12) defines a so called Rayleigh functional p : H → R (which generalizes the Rayleigh quotient for linear eigenproblems), and the kth smallest eigenvalue of (11) satisfies the min–max characterization λk =

min

max

dim V =k u∈V,u6=0

p(u).

(13)

˜ of (11) is the kth smallest eigenvalue if and only if µ = 0 is the kth largest eigenvalue of the linear Moreover, an eigenvalue λ eigenvalue problem ˜ ˜ = µb(φ, ψ) for every ψ ∈ H. λb(φ, ψ) − a(φ, ψ; λ) (14)

Discretizing the Schr¨ odinger equation (7) with the boundary and interface conditions specified above by a Galerkin method (finite elements, e.g.) one gets a rational matrix eigenvalue problem of the form S(λ)x := λM x −

where

Aj =

Z

∇ψk · ∇ψ` dr



, j ∈ {d, m};

1 1 Aq x − Am x − Bx = 0, md (λ) mm (λ)

M=

Z

k,`

Ωj

ψk ψ` dr



; k,`



(15)

Z

B = Vq



Z ψk ψ` dr + Vm

Ωd

ψk ψ` dr

 k,`

Ωm

and ψk denotes a basis of the ansatz space. Ad , Am and B are symmetric and positive semi–definite, and M is positive definite, and for λ ≥ 0 the matrix ~2 ~2 Ad + Am + B 2md (λ) 2mm (λ) is positive definite. Hence, the eigenvalues of the discretized problem (15) satisfy a min–max principle as well, and it follows from the min–max characterization (13) of the nonlinear Schr¨odinger equation that the kth smallest eigenvalue of the discretized problem (15) is an upper bound of the corresponding eigenvalue of problem (7). Solving the discretized problem with a projection method How to solve the projected problem? The min-max characterization suggests the following fixedpoint iteration for computing the kth eigenpair. We can prove: • Global convergence to λ1 • If λk is a simple eigenvalue local quadratic convergence to λk • If S 0 (λ) is positive definite and xi is replaced by the kth eigenvector of S(αi )xi = κS 0 (αi ) then the convergence is even cubic. How to expand the search subspace?. We use a direction in the spirit of Neumaier’s Residual Inverse Iteration v = S(σ)−1 S(θ)x For the linear problem S(λ) = A − λB this is exactly a Cayley transform (A − σB)−1 (A − θB) = I + (σ − θ)(A − σB)−1 B. and therefore equivalent to the shift-invert Arnoldi method. If the linear system S(σ)v = S(θ)x is too expensive to solve we can choose a new direction as v = K −1 S(θ)x with

K ≈ S(σ).

For the linear problem this again corresponds to the inexact Cayley transform.

Safeguarded Iteration Require: α1 an approximation to the kth eigenvalue for i = 1, . . . until convergence do Determine an eigenvector xi corresponding to the kth largest eigenvalue of S(αi ) Solve x∗i S(αi+1 )xi = 0 for αi+1 end for

Nonlinear Arnoldi Require: m ≥ 1, V, V ∗ V = I, σ, K ≈ S(σ)−1 while m ≤ number of wanted eigenvalues do Compute mth eigenpair (µ, y) of SV (·)y := V ∗ S(·)V y u = V y and r = S(µ)u if krk/kuk <  then ACCEPT (λm , xm ) = (µ, u) m=m+1 Update σ and K ≈ S(σ)−1 if necessary RESTART if necessary end if v = Kr, v = v − V V ∗ v, v˜ = v/kvk, V = [V, v˜] Reorthogonalize if necessary Update SV (·) = V ∗ S(·)V end while

Numerical experiments We computed a band structure of the following InAs structures embedded in a cuboid GaAs matrix: • pure quantum dot without a wetting layer • combined quantum dot and wetting layer structure for two thicknesses of the wetting layer: 1nm and 2nm • pure wetting layer • array of two quantum dots for different dot spacings • array of two combined quantum dot and wetting layer structures for different spacings

Discretization: FEM with quadratic Lagrangian elements on a tetrahedral non-unifrom grid: the finest on the dot–matrix interface, fine on the dot/wetting layer and quite rough on the matrix. This choice of the grid accounts for the properties of the envelope functions. The arising rational eigenvalue problems were solved under MATLAB 7.0.4 on an Pentium D processor with 4 GByte RAM and 3.2 GHz by the Nonlinear Arnoldi method. We started the computation with 1-vector. For the preconditioner we used the incomplete LU –decomposition with the cutoff parameter 0.01. We stopped when the residual norm was smaller than 10−6 . Multiple eigenvalues, which exist in all cases due to the symmetry of the problem were detected without problems with the correct multiplicity. Presence of the wetting layer increases the number of bounded states of the QD/WL structure. Most of them are approximate quantum well eigenstates corresponding to pure wetting layer. While the ground state is localized quite well to the quantum dot, this property gets lost for excited eigenstates. The smallest 5 eigenvalues are substantially smaller than the corresponding pure quantum dot eigenvalues. Array structure causes the avoidance of energy levels. Table 1: Electronic eigenstates of QD/WL structure

dimension ∼DOF dot/WL ∼DOF matrix ∼DOF interface CPU time [s] bounded states λ1 λ2/3 λ4 λ5

pure QD 193124 66000 115000 12000 519 5 0.4162 0.5991 0.7180 0.7296

QD/WL 1 nm 156479 62500 83500 10500 428 18 0.3574 0.5147 0.6074 0.6199

QD/WL 2nm 152928 65600 77600 9600 442 31 0.3086 0.4324 0.4659 0.4796

pure WL 2nm 10903 6600 3300 990 18.4 28 0.4572 0.4715 0.4855 0.5125

Table 2: Discretization of 2QD/WL(2nm) structure with 1nm and 6nm spacing

dimension ∼DOF dot/WLs ∼DOF matrix ∼DOF interfaces

2QD [1nm] 136078 40000 87500 8500

2QD [6nm] 140715 40500 91500 8500

2QD/WL [1nm] 133560 51000 74000 8500

2QD/WL [6nm] 135613 50500 76500 8500

References [1] S.L. Chuang, Physics of Optoelectronic devices, John Wiley & Sons, New York, 1995 [2] L.R. Fonseca, J.L. Jimenez, Electronic coupling in InAs/GaAS self-assembled stacked double-quantum-dot systems, Physical Review B 58 (1998), pages 9955–9960 [3] M. Markiewicz, H. Voss, Electronic States in Three Dimensional Quantum Dot/Wetting Layer Structures, to appear in ICCSA 2006 proceeding [4] R.V.N. Melnik, M. Willatzen Modelling Coupled Motion of Eletrons in Quantum Dots with Wetting Layers, Proceedings of the 5th Internat. Conference on Modelling and Simulation of Microsystems, MSM 2002, pages 506–509, Puerto Rico, USA 2002 [5] H. Voss, Iterative projection methods for computing relevant energy states of a quantum dot, to appear in J. Comput. Phys., 2005