Computer simulation of the non-uniform and anisotropic diamagnetic shift of electronic energy levels in double quantum dot molecules

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Computer simulation of the non-uniform and anisotropic diamagnetic shift

of electronic energy levels in double quantum dot molecules

L.M. Thu

*

_{, O. Voskoboynikov}

Department of Electronics Engineering, National Chiao Tung University, 1001 Ta Hsueh Rd., Hsinchu 30010, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 5 October 2009 Accepted 18 March 2010 Available online 8 April 2010

Keywords:

Quantum dot molecules Electronic energy levels Computer simulation

a b s t r a c t

In this study we calculated the lowest energy states of electrons confined in an asymmetrical InAs/GaAs double lens-shaped quantum dot molecule in external magnetic field. Based on the effective three-dimensional one electronic-band Hamiltonian approximation (with the energy and position dependent electronic effective mass) the electronic energy states of the system were computed by non-linear iter-ative method using Comsol MultiPhysics package. Our description allows us to simulate the semiconduc-tor quantum dot molecule in arbitrary directed magnetic field. Simulation results clearly show that the diamagnetic shifts of the electronic energy levels are anisotropic and non-uniform. Therefore we demon-strate an opportunity to dynamically manipulate electronic states not only by varying the magnitude but also changing the direction of the magnetic field.

1. Introduction

Recently studies on structural, electronic, and optical properties of semiconductor quantum dots attracted much attention from experimental and theoretical points of view (see for instance[1]

and references therein). Advance in modern semiconductor tech-nologies makes it possible to fabricate vertically stacked quantum dots of high quality and uniformity[2–4]. Stacked quantum dots allow us to form artificial quantum dot molecules (QDMs). Those semiconductor nano-objects are very attractive candidates for pos-sible applications in the solid state quantum memory[5]. They are also considered as structural elements for new nano-structured metamaterials [6]. The quantum mechanical coherent coupling and forming of the molecular states in QDMs can be considered in complete analogy to real molecules. However artificial design of QDMs provides us with much wider opportunities to manipulate and reconfigure wave functions of electrons confined in QDMs. In order to control the quantum coupling between electronic quan-tum states localized in different dots one can adjust distance be-tween dots (static approach). Another possibility to control dynamically the coherent coupling between dots is the application of external electromagnetic fields (known as dynamic approach)

[7]. In this paper we simulated the electronic states of electrons conﬁned in InAs/GaAs double QDMs assembled from the quantum dots with substantially different lateral radii[8]when the external magnetic ﬁeld is applied to the system (seeFig. 1). In contrast to

most of the known calculations we perform our simulations for several distances between dots in QDMs and few directions of the applied magnetic ﬁeld.

2. Computational method

We theoretically consider the lowest energy states of electrons conﬁned in the asymmetrical double InAs/GaAs lens-shaped quan-tum dot molecule. The molecule consists of two quanquan-tum dots with substantially different radii

q

L>

q

S and heights hL< hS (L

and S stand for the ‘‘Large” and ‘‘Small” dots in the molecule). The inter-dot (base-to-base in the molecule) distance is d (see

Fig. 1). Therefore, our system is highly asymmetrical in z direction. In our simulation the uniform external magnetic field can be ap-plied in an arbitrary direction. To compute electronic energy states confined in the asymmetrical InAs/GaAs quantum dot molecule with three-dimensional hard-wall confinement potential we adopt the effective one-band Hamiltonian as in[8,9]:

^ H ¼1 2 Y r 1 mðE; rÞ Y r þ VðrÞ þ

l

B 2 gðE; rÞ

r

B ð1Þ

whereQr= i⁄rr+ eA(r) is the electron momentum operator,rris

the spatial gradient, A(r) is the vector potential for the uniform arbi-trary directed external magnetic ﬁeld B = (Bx, By, Bz) and B =

cur-lA(r), m(E, r) is the energy and position dependent electron effective mass: 1 mðE; rÞ¼ 2P2 3h2 2 E þ EgðrÞ VðrÞ þ 1 E þ EgðrÞ VðrÞ þ

DðrÞ

and

* Corresponding author. Tel.: +886 3 5712121 54239; fax: +886 3 5724361. E-mail addresses: [email protected], [email protected] (L.M. Thu).

Computational Materials Science 49 (2010) S281–S283

Contents lists available atScienceDirect

Computational Materials Science

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gðE; rÞ ¼ 2 1 m0 mðE; rÞ

DðrÞ

3ðE þ EgðrÞÞ þ 2DðrÞ

is the electronic Landé factor. In the equations above: V(r) is the electron’s conﬁnement potential, Eg(r) andD(r) stand for the

posi-tion dependent band gap and spin–orbit splitting in the valence band, P is the momentum matrix element,

r

is the vector of the Pau-li matrixes,

l

Bstands for the Bohr magneton, m0and e are the free

electron elementary mass and charge. The hard-wall conﬁnement potential can be presented as: V(r) = 0, if r is inside the dots; and V(r) = V0, if r is outside the dots (V0is the electronic band offset in

InAs/GaAs heterostructures). For the electrons conﬁned in the QDM the envelop wave functions F(r) should satisfy the Schröding-er equation

HFðrÞ ¼ EFðrÞ ð2Þ

Since we are dealing with the hard-wall conﬁnement potential, the Ben Daniel-Duke boundary conditions[9]for the wave functions in Schrödinger equation has to be imposed on the boundary between two materials. The Ben Daniel-Duke boundary conditions for elec-tron wave functions are given by

FðrÞjin¼ FðrÞjout 1 mðE; rÞ

r

rFðrÞ njin¼ 1 mðE; rÞ

r

rFðrÞ njout ð3Þ

Here subscribe ‘‘in” and ‘‘out” denote the different sides of the inter-face (InAs and GaAs respectively) and n is the outward normal unit vector at the boundary. Solving the Schrödinger equation with the Ben Daniel-Duke boundary conditions we can ﬁnd the energy states of electrons conﬁned in the QDM.

In our simulations we use realistic semiconductor material parameters and dimension of the dots in the molecule known in literature[8,10–12]. Our molecule consists of two quantum dots with substantially different radii

q

L>

q

Sand height hL< hS(L and

S stand for ‘‘Large” and ‘‘Small” dot inFig. 1). The simulations were performed for the double QDM with geometry parameters[4,8]:

q

L= 25 nm,

q

S= 9.5 nm, hL= 3 nm, hS= 4 nm, and few inter-dot

dis-tances (base-to-base) d1= 20 nm, d2= 10 nm and d3= 5 nm.

The material parameters for the InAs/GaAs quantum dot mole-cule are taken from Ref.[10]with correction for strained InAs in-side the dot: EgInAs= 0.842 eV, DInAs= 0.39 eV, meInAs= 0.044 m0.

For the GaAs surrounding matrix we take the data from [11]:

EgGaAs= 1.52 eV, DGaAs= 0.341 eV, meGaAs= 0.067 m0. Using these

parameters the band offset of conduction band can be found as: V0= 0.474 eV. The energy states of electrons conﬁned in QDM are

obtained numerically form solutions of Schrödinger equation (1) with Ben Daniel-Duke boundary conditions (3) by the non-linear iterative method[13–15]using the Comsol MultiPhysics package (www.comsol.com).

3. Simulation results and discussion

We present our calculation for few distances between quantum dots in the molecule: d1= 20 nm, d2= 10 nm and d3= 5 nm with

uniform external magnetic field directed in three directions: B{1}= (0, 0, Bz), Bf2g¼p1ffiffi₂ (0, By, Bz), and B{3}= (0, By, 0).

Combina-tion of the distance d and direcCombina-tions of magnetic ﬁeld B gives us nine conﬁgurations for our system: [di, B{j}], where i, j = 1, 2, 3.

For those conﬁgurations, we simulate the lowest conﬁned elec-tronic energy states. The results are shown in Figs. 2–4. When the distance between dots in the molecule is large enough, the tun-nel coupling between dots is weak. Therefore, in [d1, B{1}]

conﬁgu-ration the diamagnetic shifts of electronic energies become non-uniform because of the non-non-uniformity of the QDMs geometry in the z-direction. This leads to the anticrossing at BAC= 10.7 T for

electronic energy of two lowest states e1and e2(seeFig. 2a). The

details of the anticrossing were found and discussed in Ref.[8]. An important aspect is that the anticrossing manifests a redistribu-tion of the electronic wave funcredistribu-tion inside the quantum dot mole-cule: the electronic wave function of the state e1at the anticrossing

point relocates from the large dot to the small one whereas the wave function of the state e2relocates in the opposite direction.

When the magnetic ﬁeld direction changes from B{1}to B{2}the

anticrossing point shifts to a larger magnitude of the magnetic ﬁeld (Fig. 2b). When B = B{3}(magnetic ﬁeld is parallel to the xy-plane)

the anticrossing disappears (Fig. 2c). The electronic wave functions localized in the large and small dots are under strong conﬁnement in z-direction. Therefore, in this last conﬁguration ([d1, B{3}]) the

wave functions hardly can be controlled by external magnetic

0 252 0.24880 0.248 . 10.6 10.7 10.8 0.24876

E

(eV

)

0.244 e1 e2

B (T)

0 5 10 15 20 0.252

E (eV)

0.248

B (T)

0 5 10 15 20 0.252

E (eV)

0.248

B (T)

a

b

c

0 5 10 15 20

Fig. 2. Magnetic ﬁeld dependence of the lowest electronic energy levels of the double QDM for d1= 20 nm at different directions of the magnetic ﬁeld: (a) B = B{1};

(b) B = B{2}; (c) B = B{3}(inset: anticrossing region).

z

B

z

B

y

x

y

B

x

Matrix

QDM

ρ

S

h

S

d

ρ

S

h

d

h

L

ρ

_L

Fig. 1. Schematic of the asymmetrical InAs/GaAs semiconductor double quantum dot molecule in uniform arbitrary directed external magnetic ﬁeld.

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ﬁelds. So, change in the direction of magnetic ﬁeld from B{1}to B{3}

shifts the anticrossing point and gradually removes it. This means that one can manipulate electronic states in double QDMs by vary-ing the magnitude and changvary-ing direction of the magnetic ﬁeld as well.

The small distance between dots in the molecule (d3= 5 nm)

brings strong tunnel coupling between dots and leads to the strong hybridization between two electronic states e1and e2. This

generates molecular electronic states with symmetric and anti-symmetric wave function conﬁgurations along z-direction[8,12]. The diamagnetic shifts of electronic energies become uniform (even for B{1}) and this leads to the disappearance of the

anti-crossing (Fig. 3a). The system with distance d2= 10 nm presents

an intermediate case between the large and small distances. In this conﬁguration we can see a combination of the anticrossing and hybridization. It leads to a weak convergence of the elec-tronic energies and the anticrossing disappears even for B = B{1}

(seeFig. 4a). Clearly, the reduction of the distance between dots conventionally hybridizes electronic states from different dots and this ﬁnally forms typical molecular states in the QDM (see

Figs. 3 and 4).

In conclusion, we calculated the lowest electronic energy states for a realistic three-dimensional model of asymmetrical lens-shaped double InAs/GaAs quantum dot molecule in the uniform arbitrary directed external magnetic field for few distance between dots in the molecule. The simulation results clearly show that the diamagnetic shifts of the electronic energy levels are considerably different for different distances. In addition the application of the magnetic filed in various directions generates anisotropy in the diamagnetic shifts. Energy states of electrons confined in QDMs can be manipulated by adjusting the distance between dots in the molecule (statically) or application external magnetic field (dynamically). Moreover, one can dynamically manipulate elec-tronic states not only by varying the magnitude but also changing the direction of the external magnetic filed. The results can be use-ful in modeling and investigation of the magnetic properties of QDMs.

Acknowledgements

This work is supported by the National Science Council of the Republic of China under Contracts No. 97-2112-M-009-012-MY3 and No. NSC 97-2120-M-009-004, and by the Aim for the Top Uni-versity Plan of the National Chiao Tung UniUni-versity and Ministry of Education of Taiwan, ROC.

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V)

0 27

E (e

0.24 . 0 5 10 15 20 0.21

B (T)

(eV)

0 24 0.27

E

_0.21.

B (T)

0 5 10 15 20

V)

0.27

E (e

0.24 0 5 10 15 20 0.21

B (T)

a

b

c

Fig. 3. Magnetic ﬁeld dependence of the lowest electronic energy levels of the double QDM for d3= 5 nm (description of the levels and magnetic ﬁeld directions

see inFig. 2). 0.252 0.248 0 5 10 15 20

B (T)

0 252.

E (e

V)

0.248 0 5 10 15 20

B (T)

0.252

E (eV)

0.248 0 5 10 15 20

B (T)

E (e

V)

a

b

c

Fig. 4. Magnetic ﬁeld dependence of the lowest electronic energy levels of the double QDM for d2= 10 nm (description of the levels and magnetic ﬁeld directions

see inFig. 2).