Effect of geometry on the excitonic diamagnetic shift of nano-rings

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Phys. Status Solidi C 8, No. 2, 375 – 377 (2011) / DOI 10.1002/pssc.201000449

Effect of geometry on the excitonic

diamagnetic shift of nano-rings

L. M. Thu**, W. T. Chiu*, Ta-Chun Lin, and O. Voskoboynikov

Department of Electronics Engineering, National Chiao Tung University, 1001, Hsinchu 30010, Taiwan R.O.C.

Received 13 June 2010, accepted 6 July 2010 Published online 27 October 2010

Keywords nano-rings, exciton, diamagnetic shift

** Corresponding author: e-mail [email protected], Phone: + 886 3 5712121 ext: 54239, Fax: +886 3 5733722 ** e-mail [email protected], Phone: + 886 3 5712121 ext: 54239, Fax: +886 3 5733722

We simulate magneto-excitons confined in asymmetri-cally wobbled three-dimensional InAs/GaAs nano-rings. The wobbling asymmetry reproduces realistic experimen-tal geometry of the rings and generates an asymmetry in the side valleys of the three dimensional confinement

po-tential. Using our mapping method and the exact diago-nization approach we calculated the excitonic diamag-netic shift and found that even a small wobbling asym-metry can drastically change the diamagnetic shift coeffi-cient.

1 Introduction The single InAs/GaAs nano-ring's

(see for instance [1,2] and references therein) magneto-exitonic emission demonstrates an interesting discrepancy with the conventional theory. Recently it was found that the diamagnetic shift of a single exciton's peak is consid-erably smaller than that expected from traditional theory [3]. It was also found that a perfect in geometry wobbled symmetric nano-ring is particularly hard to achieve [2]. In this work we theoretically demonstrate the impact of the wobbling asymmetry of nano-rings on the diamagnetic shift of the single exciton's peak. To make a link to realis-tic three-dimensional shapes of the rings we use our map-ping method [4], which makes it possible to project the ring's actual geometry onto the position dependent effec-tive masses, energy gap, band offsets of electrons and holes confined in the ring. Using the exact diagonalization

method we simulate magneto-excitons and the diamagnetic shift and demonstrate that even a small wobbling asymme-try strongly effects the diamagnetic shift.

2 Simulation method We assume that the

InAs/GaAs nano-ring was grown on a substrate surface (x-y

plane) and the external magnetic field is applied in z-direction. In our simulation we map the wobbled ring ge-ometry [4] for the case when the ring's height h(x,y) is asymmetrical along x-axis (see Fig. 1(a)). We fit the height as the following [2,4]:

where the difference between parameters ain and aout repre-sents the wobbling asymmetry along x-direction, and b controls the range of the wobbling asymmetry.

On the base of (1) we can (as it was described in [4])

(

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1 _, _, exp 1 ) , ( ; , 1 exp 1 ) , ( 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 2 0 0 2 2 2 2 2 2 2 0 R y x R y x h y x y x h b y R x a h y x h R y x R y x R y x R R h y x y x h b y R x a h y x h M out M in > + γ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ₊ ₋ γ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − ξ + ⋅ ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ₋ ₊ − ⋅ + + = ≤ + γ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ₊ ₋ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ₊ ₋ − ⋅ γ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − ξ + ⋅ ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ₋ ₊ − ⋅ + + = ∞ ∞ ∞ ∞ (1)

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376 L. M. Thu et al.: Effect of geometry on the excitonic diamagnetic shift of nano-rings

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introduce a three-dimensional electronic confinement po-tential: , ) , ( 1 tanh 1 tanh 1 4 1 1 ) , , ( 1 0 ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + − ⋅ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⋅ − × Δ = a z y x h a z z E z y x Ve e (2)

where ΔEe = Vmax - Vmin is the electronic band offset of the

system, Vmax is the maximum value of the potential, Vmin is

the minimum value of the potential (inside the ring), and a controls the range on the potential boundaries of the ring.

From Ve(x,y,z) we define the mapping function M(x,y,z)

as the following [4]: . ) , , ( 1 ) , , ( e e E z y x V z y x M Δ − = (3)

The mapping function projects the ring geometry onto the position dependent parameters: electron's (hole's) effective mass - m*

e(h) (x,y,z), energy gap - Eg(x,y,z), hole’s

confine-ment potential Vh(x,y,z), and permittivity ε(x,y,z):

[

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[

1 ( , , )

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, ) , , ( ) , , ( , ) , , ( ) , , ( ) , , ( , )] , , ( 1 [ ) , , ( ) , , ( , ) , , ( 1 ) , , ( ) , , ( ) ( ) ( ) ( * ) )( ( * ) )( ( * ) ( z y x M z y x M z y x E z y x V z y x E z y x V z y x M E z y x M E z y x E z y x M m z y x M m z y x m out in in g e g h out g in g g out h e in h e h e − ⋅ ε + ⋅ ε = ε − − = − ⋅ + ⋅ = − ⋅ + ⋅ = (4) where the subscripts (in) and (out) indicate the bulk mate-rial parameters for the inside (InAs) and outside (GaAs) re-gions.

Using the parameters mapped above we perform simu-lation of the ground state excitonic energy of the ring. To obtain wave functions and energies of non-interacting elec-trons and holes (see [4] and reference therein), we solve the following Schrödinger equations

), ( ) ( 1 2 1 ˆ , ˆ ) ( ) ( * ) ( ) ( ) ( ) ( ) ( ) ( ) ( r Π r Π H Η r r eh eh h e h e h e h e h e h e h e V m E + = Ψ = Ψ (5) where Πe(h)

r = -ih∇r + (-) eA(r) is the momentum operator, ∇r represents the spatial gradient, r = {x, y, z}, A(r) is the vector potential of the magnetic field B = curl A, and e is the absolute value of the free electron charge. Having solu-tions for the non-interacting electrons and holes we then simulate the exciton ground state energy and exciton ground state wave function through the exact diagonaliza-tion method (see [5-7]). Our calculadiagonaliza-tion was running under COMSOL multiphysics package (www.comsol.com).

3 Calculation results The material and geometry

parameters for our simulation were taken from [2,3,7,8]

and adjusted to the realistic semiconductor material pa-rameters for InAs/GaAs with complex strained composition in [8]. For instance for InCGa1-CAs/GaAs nano-ring we

as-sumed that the In content inside the ring is C = 0.895. For the electrons in the conduction band we used m*

e(in) = 0.046

m0, m*e(out) = 0.067 m0, Vmin = 0.349 eV, Vmax = 0.774 eV,

ΔE = 0.425 eV (m0 is the free electron mass). For the holes we admitted m*

h(in) = 0.119 m0, m*h(out) = 0.5 m0. The band

gap parameters were taken Eg(in) = 0.913 eV (InCGa1-CAs)

and Eg(out) = 1.519 (GaAs). We also used: εin = 14.9 and εout

= 12.9. To quantify the wobbling asymmetry we introduce the following asymmetry parameter

%, 100 ⋅ − = δ − − + h h h h (6) where h+ is the maximal value of the rim height for

posi-tive x and h- represents the opposite one. The mapping

function of the rings was defined with: h0 = 2 nm, hM = 3

nm, h∞ = 0.2 nm, γ0 = 3 nm, γ∞ = 5 nm, ξ = 0.2, R = 6 nm, a

= 0.4nm, and b = 5nm. To achieve desired geometry within this mapping we choose correspondingly (see Fig 1(b): ain

= 0, and aout = 0 (δh = 0%, h+ = h- = 3.6 nm, no asymmetry);

ain = 0.123 and aout = 0.054 (δh ≈ 5.5%, h+ ≈ 3.8 nm, h- =

3.6 nm); and ain = 0.247 and aout = 0.120 (δh ≈ 11% (h+ ≈

4.0 nm, h- = 3.6 nm). Using these three different

geome-tries we map for each of them the electronic confinement potential accordingly. The potential profiles for different asymmetry parameters are shown in Fig. 2.

Figure 1 (a) Geometry of the asymmetrically wobbled InAs/GaAs nano-ring for δh ≈ 5.5%. (b) Projection of the ring height onto the x-z plane for: δh= 0% (solid curve); δh≈ 5.5% (dot-ted curve); δh≈ 11% (dashed curve).

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Phys. Status Solidi C 8, No. 2 (2011) 377

Contributed Article

Figure 2 The electronic confinement potential projected onto x-z plane for (a) δh = 0% and (b) δh ≈ 11%.

We define the diamagnetic shift of the energy of the exciton confined in the ring as the following [4]

ΔE_X =E_X(B)−E_X(0)≈d_X ⋅B2, (7) where EX(B) is the magnetic field dependent exciton’s

ground state energy and dX is the diamagnetic shift

coeffi-cient. The simulation results (averaged by the Zeeman splitting) are shown in Fig. 3. Those read: dX ≈ 10 μeV/T2

for δh = 0% and 9 μeV/T2 for δh ≈ 5.5%, and 8 μeV/T2 for

δh ≈11% The last one is close to the experimental data: 6.8

μeV/T2 _[3].

Figure 3 Diamagnetic shift of the ground state energy of the exciton confined in the ring for different wobbling asymmenries

( 0

X E

Δ presents the optical transition energy diamagnetic shift for non-interacting electrons and holes).

We found that the excitonic wave function is equally distrib-uted in both side of the ring along x-direction when δh = 0%. At the same time if δh exceeds 10% the wave function is already lo-calized in the potential valley at the positive x-side. This is a clear reason for the suppression of the diamagnetic shift since the dia-magnetic shift coefficient is defined by the wave function distri-butions.

4 Conclusion We simulated the excitonic

diamag-netic shift of the asymmetrically wobbled InAs/GaAs nano-rings. Using our mapping method and direct diagonaliza-tion approach we managed to obtain an accurate explana-tion for the experimental data on the reducexplana-tion of the dia-magnetic shift coefficient. We argue that the diadia-magnetic shift’s suppression reproduces actual asymmetry in the ring geometry.

Acknowledgements This work is supported by the

Na-tional Science Council of the Republic of China under Contracts No. NSC 97-2112-M-009-012-MY3, No. NSC 97-2120-M-009-004, No. NSC 98-2918-I-009-001 and by the Ministry of Educa-tion of Taiwan under contract No. MOEATU 95W803.

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