Magneto-optics of layers of triple quantum dot molecules

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Phys. Status Solidi B 246, No. 4, 771– 774 (2009) / DOI 10.1002/pssb.200880595

p s s

basic solid state physics

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www.pss-b.com

physica

Magneto-optics of layers

of triple quantum dot molecules

Thu Le Minh and Oleksandr Voskoboynikov*

Department of Electronics Engineering, National Chiao Tung University, 1001 Ta Hsueh Rd, Hsinchu 30050, Taiwan Received 24 April 2008, revised 27 September 2008, accepted 28 October 2008

Published online 14 January 2009 PACS 73.21.La, 78.20.Ls, 78.67.–n

*_{Corresponding author: e-mail}_{[email protected]}_{, Phone: +886 3 5712121}_{× 54174, Fax: +886 3 7533722}

1 Introduction Modern progress in semiconductor technologies makes it possible to fabricate vertically stacked quantum dots of high quality and uniformity [1]. Those semiconductor nano-objects provide us with the possibility to manipulate and reconfigure electronic wave functions in three-dimensional space. The quantum me-chanical coherent coupling and forming of molecular states in the stacked quantum dots can be considered in complete analogy to real molecules. Quantum dot’s molecules (QDM) have attracted much interest because they are at-tractive candidates for the implementation of quantum bits [2]. So, coherent dynamic control of the electronic con-figuration in QDMs is a key target for application of semi-conductor nano-devices in quantum information technol-ogy. In the same time semiconductor nano-objects (quan-tum dots, quan(quan-tum dot molecules, etc.) are very promising construction elements for semiconductor based metamate-rials of high potential in the field of optics [3, 4]. Proper understanding of the connection between the electronic state coherent coupling in isolated nano-objects and the collective electromagnetic response from layers made from them [5] is a prerequisite to make new nano-structured metamaterials, not resembling anything in nature.

In this theoretical study we consider impact of the coherent manipulation of electronic states in the triple vertical lens-shaped circular QDM on the collective mag-neto-optical reflection from layers of those nanoobjects. The manipulation is performed for InAs/GaAs quantum dot molecules assembled from the dots with substantially different diameters and when an external magnetic field is along the system’s growth direction. The influence of the surrounding semiconducting matrix upon the polarizability of embedded nano-objects (QDM) has been investigated using a hybrid discrete/continuum model [6].

2 Theory The system to be investigated consists of a square lattice with lattice parameter a_L, composed of InAs triple quantum dot molecules (as it is shown in Fig. 1) of characteristic size a Ⰶ λ (λ is wavelength of light of the frequency ω), embedded in GaAs matrix. The optical re-sponse of a single molecule is described by means of Kramers/Heisenberg type of polarizabilities. The hybrid discrete continuum method allows us to simulate the col-lective electromagnetic response of embedded molecules [6]. The bare embedded polarizability αB (which was de-

We consider theoretically the impact of the coherent manipu-lation of electronic states in the triple vertical lens-shaped cir-cular InAs/GaAs quantum dot molecules on the collective magneto-optical reflection from layers of those nano-objects. The quantum dots have substantially different diameters in contrast to most of the known simulations. We demonstrate a possibility to drive molecular electronic ground states (to redistribute electronic wave functions within the molecule) by applying an external magnetic field along the system’s growth direction. The change in the states leads to the

redis-tribution of the intensity of the reflectance and absorbance peaks in the collective magneto-optical response of a layer of the triple quantum dot molecules. Varying the distance be-tween quantum dots within the layer one can study optically the transition from “molecular” to “atomic” configuration. We show that the changes in the quantum mechanical behav-iour of the molecules can be made observable by monitoring changes of the ellipsometric parameters Ψ and Δ in the mag-neto-optical response of layers of such nano-objects.

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772 T. L. Minh and O. Voskoboynikov: Magneto-optics of layers of triple QD molecules

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Figure 1 Schematic of magneto-optical response from a layer of

triple quantum dot molecules.

rived in [7] as a measurable quantity) we can present by the expression

( )

S 2 2 T T 2

( )

BE BE eh h e h ,e h ,e 3 _{ˆ ˆ} _ˆˆ _, 4 i j i j i j e r F F f α ω α ω = + È_Îxx +yy ˘_˚

Â

(1) where α_BES is the static part of the polarizability and f_hi,ej(ω) is the function introduced in [5], which depends on transi-tion energies E_ij=E_hi+E_ej+Eg of the resonance optical tran-sitions from hole energy levels (hi) to electron energy levels (ej) (Eg stands for the energy gap of the dot’s mate-rial). The hole – electron overlap integrals 〈F_hi| F_ej〉 should be calculated using the envelop wave functions F, and reh stands for the bulk inter-band optical matrix element.

To compute the transition energies and the wave func-tions of the InAs/GaAs QDM with the hard-wall confine-ment potential we use realistic semiconductor material pa-rameters (for instance the band offset of the InAs/GaAs strained heterostructure, corrected to the strain conditions band parameters, etc.). This allows us to simulate the mag-netic dependence of the electron energy states and wave functions in a system of very asymmetrical shape: our quantum dots have substantially different diameters and different heights in contrast to the most of known simula-tions (see insert in Fig. 1). The effective one-band Hamil-tonian for electrons (holes) [5] is taken as the following

(

)

( )

B

(

)

r r e,h e,h e,h 1 1 ˆ _, _, 2 , 2 H V g E m E μ Π Π = + r + r σ B◊ r (2) where Π_r = –iħ∇_r + eA(r) is the electron momentum

operator, ∇_r is the spatial gradient, A(r) is the vector poten-tial of the magnetic field B = curl A, me(E, r) is the energy and position dependent electron effective mass

(

)

( )

2 2 e g g 1 2 2 , 3 1 Δ P m E E E V E E V È = _{Í +} -Î ˘ + _˙ + - + _˚ r r r r r r and 0 e g Δ( ) ( , ) 2 1 ( , ) 3( ( )) 2Δ( ) m g E m E E E Ï ¸ = Ì - ₊ ₊ ˝ Ó ˛ r r r r r

is the electronic Landé factor. For the holes we use energy-independent effective mass mh and the Landé factor gh. In the equations above: V(r) is the confinement potential, Eg(r) and Δ(r) stand for the position dependent band gap and spin – orbit splitting in the valence band, P is the mo-mentum matrix element, σ is the vector of the Pauli ma-trixes, μB is the Bohr magneton, and m0 is the free electron mass. The hard-wall confinement potential we present as: V(r) = 0, if r is inside the dots; and V(r) = V0, if r is outside the dots,where V0 is the band offset in InAs/GaAs het-erostructures. In our simulation the magnetic field B is di-rected along the system z-axis, and we can treat the prob-lem in cylindrical coordinates (ρ, φ, z). The envelop wave functions can be represented as

( )

(

, exp

)

( )

,

F r =Φ ρ z ilφ (3)

where l = 0, ±1, ±2, . . . is the orbital quantum number. This leads to a two-dimensional problem in the (ρ, z) coordinates and forms a set of the quantum numbers i, j = {n, l, s}, where n is the principal quantum number, and s = ±1 refers to the orientation of the electron spin along z-axis. Finally we use the computed wave functions and energies to simulate the polarizability tensor (1).

Having ready the polarizability tensor for the isolated QDM we can define the collective electromagnetic re-sponse of the layer of QDM. For the system of the embedded nano-objects each presented by the discrete dipole strength p_k we have to solve the system of equations [7]: 1 1 BE,k k m kq q 0, q k α- ε- π -

Â

= p t p E (4)

where t_kq is the frequency dependant intercellular transfer tensor, screened by the matrix dielectric constant ε_m, E₀ is the external electric field of light. Solution of (4) deter-mines the Vlieger expressions [8] for reflection (r_ss and r_pp) and transmission (t_ss and t_pp) coefficients for the reflected electric field:

(

)

1 ss ss ss cos , 1 , k y i k r f A f t r θ -= -= + 2 pp 2 pp 2 cos sin _,

cos cos sin

cos cos _,

cos cos sin

k i k i x k i z i k i k i z i x k i z i k i f f r A f A f f A t A f A f θ θ θ θ θ θ θ θ θ θ = -- -= --

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-Phys. Status Solidi B 246, No. 4 (2009) 773

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where subscripts “ss” and “pp” refer to the light polariza-tion convenpolariza-tionally, θi is the angle of incidence,

( )

(

)

1 1 0 BE, m 0 L 3 0 0 3 m 0 0 , 2π , , 6π , 4π . u uu u u k u u L A f t f ia k N ik t V k a c α α ω ε α ε ε ω ε α ε - -= - + = = - + = =

Here N_u {u = x, y, z} and V denote the depolarization factor and the volume of the molecule respectively. The planar tensor f is defined for the two-dimensional lattice of the nano-objects as f_x= f_y= –f_z/2, f_z = 9.03362 [8].

3 Simulation results In our simulations we use real-istic semiconductor material parameters and dimensions of the dots in the molecule known in literature [9]. Our mole-cule consists of three quantum dots with substantially dif-ferent radii ρC > ρS and heights hC < hS [C and S stand for the “Central” and “Side” dots in Fig. 1]. So, the system is non-uniform in z-direction. Here we calculate the low elec-tron (upper hole) energy states of a system assembled from the lens-shaped dots with: ρC = 25 nm, ρS = 9.5 nm, hC = 3 nm, hS = 4 nm, and different inter-dot (base-to-base) distance d.

For the strained InAs inside of the dots according to the corrections done in [10] we choose: EgInAs = 0.842 eV, ΔInAs = 0.39 eV, meInAs(0) = 0.044 m0, mhInAs = 0.074 m0, ghInAs = 2.33, and εInAs = 15.2. For the GaAs matrix we take from [11]: EgGaAs = 1.52 eV, ΔGaAs = 0.341 eV, meGaAs(0) = 0.067 m0, mhGaAs = 0.5 m0, ghGaAs = 3.6, and εm = εGaAs = 13.1. The conducting band offset of the dot material is V0 = 0.474 eV. The dumping parameter γ is chosen to be 1 meV for all calculations [5]. The energy states and wave functions of the electrons and holes con-fined in the quantum dot molecule, are found by the nonlinear iterative method [12] using the Comsol Mul-tiphysics package (www.comsol.com). The static part of the polarizability tensor was calculated with the approach described in [13] also using Comsol Multiphysics package.

The tunnel coupling between dots creates molecular states in symmetric and anti-symmetric configurations. The non-uniformity of the QDM geometry in z-direction gener-ates interesting features of the lowest energy stgener-ates for electrons with l = 0 relating to the edge of the optical ab-sorption in the system. For reason of clarity we concentrate on the lowest three energy electronic states pe1 = {1, 0, +1}, p_e2= {2, 0, +1}, p_e3= {3, 0, +1}, the highest heavy hole state ph1 = {1, 0, +1}, and inter-band optical transitions be-tween them satisfying the selection rule: lh – le = 0 (see Ref. [5 – 7, 14]). The corresponding allowed transition en-ergies are shown Fig. 2(a). Increasing external magnetic field we can see the diamagnetic shift for the energies. The diamagnetic shifts are non-uniform – different slopes for

Figure 3 Wave function’s evolution in magnetic field for the

lowest three electron’s energy levels and one hole level. The bot-tom graph is sketching the dots geometry in (ρ, z)-plane. Inter-dot distance: d = 10 nm.

Figure 2 Optical transition hi → ej in the triple

QDM at the edge of the optical absorption. Inter-dot distance: d = 10 nm. (a) Transition energies as func-tions of magnetic field for three closest to the edge transitions. (b) The modulus squared hole electron overlap integrals for the same transitions.

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774 T. L. Minh and O. Voskoboynikov: Magneto-optics of layers of triple QD molecules

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Figure 4 Magnetic field dependence of the ellipsometric angles

for the layer of triple QDM, aL = 80 nm: (a) and (b) inter-dot

dis-tance d = 10 nm; (c) and (d) inter-dot distance d = 12 nm. Dashed curve: B = 0 T; Solid curve: B = 20 T.

different transitions. More importantly, it can be seen from Fig. 3 that the electronic wave functions are redistributing within the molecule. For instance, the probability density of the state pe1 changes from the on-centre concentrated con-figuration to the on-side concentrated one and the probabil-ity densprobabil-ity of the state pe3 changes in the opposite direction. In the same time, the probability density of the hole state ph1 remains concentrated on the central dot. Clearly, the mag-netic field changes the overlap integrals for the correspond-ing optical transitions as it is shown in Fig. 2(b).

This clear quantum mechanical effect obviously shows itself in the collective magneto-optical response of the layer of QDM. Here we present results of our simulation for the ellipsometric angles Ψ and Δ, which follow from the commonly used definition:

pp ss tan e .i r r Δ Ψ =

For the angle of incidence θ_i = 60°, close to the Brews-ter angle, the ellipsometric angles Ψ and Δ clearly change their profile when the magnetic field changes from zero to 20 T as it is shown in Fig. 4(a, b). We present only peaks corresponding to the transitions considered in Fig. 3. Our calculation results show that we always can distinguish them if γ = 1 meV [5–7]. We stress that Ψ-angle peak-to-peak ratio replicates the peak-to-peak-to-peak-to-peak ratio for the prob-ability density of the electronic states corresponding to the transitions (see Fig. 3). So, the ellipsometric data consists of important information on the quantum mechanical con-figuration of the molecular states in the QDM.

To demonstrate the connection between the reconfigu-ration of the probability density of the molecular states and the ellipsometric parameters we present in Fig. 4(c, d) the ellipsometric angles Ψ and Δ for the layers of QDM with

the inter-dot distance d = 12 nm. For this distance the tun-nel coupling between dots with larger distance is weaker and the molecular states gradually “dissociate” into “atomic” states of isolated dots. For this configuration the “atomic” electronic states located mainly in the central dot should play the main role in the overlap integrals. Accord-ingly, for d = 12 nm we can recognize one main peak in Ψ-ellipsometric data.

In conclusion, we studied theoretically the magneto-optical response from a layer of embedded semiconductor triple quantum dot molecules. The calculation results clearly suggest measurable values for the ellipsometric an-gles for any modern ellipsometric setup. We emphasize that the magneto-ellipsometric data reproduce important information on the quantum mechanics of the molecules. Varying magnetic field and the distance between quantum dots within the layer we can investigate optically the tran-sition from “molecular” to “atomic” behaviour of the sys-tem. The approach can be potentially useful for the design of new nano-structured metamaterials.

Acknowledgements This work is supported by the

Na-tional Science Council of the Republic of China under Contracts No. NSC 96-2112-M-009-009 and NSC 96-2120-M-009-010 and by the Ministry of Education of Taiwan under contract No. MOEATU 95W803.

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