Dipole-allowed optical absorption in a parabolic quantum dot with two electrons

Physics Letters A 372 (2008) 4323–4326 www.elsevier.com/locate/pla

Dipole-allowed optical absorption in a parabolic quantum dot with two electrons Jinsheng Huang a,∗ , Libin b a Jieyang Vocational and Technical College, Jieyang 522000, PR China b Department of Physics, Guangzhou University, Guangzhou 510006, PR China

Received 12 March 2008; accepted 24 March 2008 Available online 28 March 2008 Communicated by V.M. Agranovich

Abstract Dipole-allowed optical absorption in a parabolic quantum dot with two electrons are studied by using the exact diagonalization techniques and the compact density-matrix approach. Numerical results are presented for typical GaAs parabolic quantum dots. The results show that the total optical absorption coefficient of two electrons in quantum dot is about five times smaller than that of one electron in quantum dot. Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved. PACS: 78.20.Ci; 78.67.-n; 42.65.-k Keywords: Optical absorption; Parabolic quantum dot; Density-matrix approach

1. Introduction With recently quick advances of modern technology, it has now become possible to produce quantum dots (QDs) by using techniques such as etching or molecular beam epitaxy, etc. Essentially, they are little islands of two-dimensional electrons which are laterally confined by an artificial potential. Alternatively, they can be thought of as artificial atoms where the confining potential replaces the potential of the nucleus. Typical dot sizes are about 100 nm and each dot typically contains between 2 and 200 electrons [1–4]. Quantum confinement of electron in quantum dot lead to the formation of discrete energy levels, and the drastic change of optical absorption spectra and many novel optical properties which is not in the bulk materials. The new, unusual properties of the low-dimensional nanometer-sized semiconductor, which promise applications mostly in far-infrared (FIR) laser amplifiers, photodetectors and highspeed electro-optical modulators, have attracted the attention * Corresponding author. Tel.: +86 663 8850 452; fax: +86 663 8859 886.

E-mail address: [email protected] (J. Huang).

of many researchers [2,5]. Among the properties, more attention had been paid to the nonlinear optical properties [6–9]. Sauvage and Boucaud discussed nonlinear optical properties of QDs [10]. Wang and Guo [11] calculated the interband optical absorptions in a parabolic quantum dot. As for more than one electron in a single dot with a parabolic confinement potential, the electron–electron interaction cannot affect the FIR absorption [12], even in the presence of magnetic fields [13–15]. Recent optical measurements of QDs indeed exhibit absorption frequencies that are independent of the number of electrons and well fitted to the single-electron absorption spectrum [16]. In this Letter, the linear, third-order nonlinear and total optical absorption coefficients of dipole-allowed optical transitions in parabolic quantum dot with two electrons are theoretically studied. In Section 2, we propose a procedure of exact diagonalization to obtain the eigenfunctions and eigenenergies of two electrons states in QD. Furthermore, a simple analytical formula for the optical absorptions are introduced using the compact density-matrix approach. In Section 3, numerical calculations on typical GaAs material are performed, and compare the results of two electrons in quantum dot with that of one electron. The results show that the total optical absorption coef-

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ficient of one electron in quantum dot is about five times bigger than that of two electrons in quantum dot. 2. Theory In the effective mass approximation, the Hamiltonian for two electrons in parabolic QD can be written as H=

 2  2  Pi 1 ∗ 2 2 e2 + ω r m e 0 i + ∗ 2me 2 r21

(1)

i=1

where Pi and ri , respectively, denote the momentum, and the position vector of the ith electrons originating from the center of the dot, m∗e is the effective mass of electron. r21 is the distance between electrons.  is the dielectric constant, and ω0 is the strength of the confinement. As the two electrons in QD are identical, so a proper picture must include the antisymmetry of the state when the electrons are exchanged. The wave function factorizes into a spatial part and a spin part, which can have either total spin S12 = 0 (singlet) or S12 = 1 (triplet). The singlet state is antisymmetric under the exchange of spins, so to retain the correct overall antisymmetry of the wave function, the spatial part must be symmetric. The triplet, by contrast, is symmetric, so the corresponding spatial part must be antisymmetric. Hence, the eigenstates are classified by the total spin S12 and the total orbital angular momentum L of the two electrons. Under the center-of-mass frame, we introduce the center-ofmass coordinate ξ = (r2 + r1 )/2 and the relative coordinate η = r21 = r2 − r1 and the Hamiltonian equation can be rewritten as H=

P2ξ

P2η e2 1 1 + Mω02 ξ 2 + + μω02 η2 + 2M 2 2μ 2 r21

(2)

where M = 2m∗e , and μ = m∗e . To obtain the eigenfunctions and eigenenergies, we diagonalized H in a model space spanned by the translational invariant 2D harmonic product bases    φ[K] = A˜ φnω1 1 (η)φnω2 2 (ξ ) L χS (3) where χS = [ζ (1)ζ (2)]S , ζ (i) is the spin state of the ith electron and the spins of two electrons are coupled to S, 1 + 2 = L is the total orbital angular momentum. The angular quantum number 1 = odd if the spin S12 = 1, and 1 = even if S12 = 0 ω (r) is a twosuch that the wave function is antisymmetried. φn dimensional harmonic oscillator state with frequencies ω, energy (2n + || + 1)h¯ ω. ω serves as an adjustable variational parameter around ω0 to minimize the eigenvalues [17]. A˜ is the antisymmetrizer. The matrix elements of H are then given by the following expressing: φ[K] |H |φ[K  ]    = 2(n1 + n2 ) + |1 | + |2 | + 2 h¯ ωδ[K][K  ] e2 + Rn1 1 (η) R   (η)η dη δ1 , δn2 ,n δ2 , , 1 2 2 4πεη n1 1

where [K] denotes the whole set of quantum numbers (n1 , 1 , n2 , 2 ) in brevity, Rn (r) is the radial part of 2D harmonic oscillator function. The dimension of the model space is constrained by 0  N = 2(n1 +n2 )+|1 |+|2 |  30. Since the whole set of eigenstates of the harmonic product basis forms a complete basis in the Hilbert space, the procedure of increasing the number of linearly independent eigenstates is converging to the exact result. The limits are set only by the capacity of the computer to diagonalize K × K Hermitean matrices. Using the compact density-matrix approach, the optical absorption coefficient α(ν, I ) can be expressed by [11,18] α(ν, I ) = α (1) (ν, I ) + α (3) (ν, I )

 |Mj k |2 σs Γj k μ =ν εR [h¯ (ν − νj k )2 + h¯ Γj2k ]

2|Mj k |4 σs Γj2k I − , ε0 nr c h¯ 3 Γjj [(ν − νj k )2 + Γj2k ]2

(4)

E −E

where νj k = k h¯ j , εR is the real part of the positivity, defined as εR = n2r ε0 , nr is the medium refractive index, c is the speed of light in free space, I is the incident optical intensity, μ is the permeability of the system, ε0 is the positivity of vacuum, σs is the electron density in the system. Γ is the phenomenological operator. Diagonal matrix element Γjj of operator Γ , which is called as relaxation rate of j th state, is the inverse of the relaxation time Tj for the state |j , namely Γjj = 1/Tj , whereas non-diagonal matrix element Γj k (j = k) is called as the relaxation rate of j th state and kth state. The matrix elements Mj k of transition dipole moment from the j th state to the kth states are evaluated from: Mj k = 2qj |ξ eiθ |k where q is the electronic charge. The states of two electrons will be denoted by 2S12 +1 L. The singlet and triplet states of the two interacting electrons with the total angular momentum L = 0 (S state) and L = 1 (P state) are denoted by 1 S, 3 S and 1 P , 3 P , respectively. The ground state is the 1 S state. The second state is the 3 P state and the energy of the 3 S state is the highest in the four states [17]. Dipole operator is independent of the electron spin. The dipole-allowed optical transitions are always from the same spin states, but the angular momenta must differ by unity [15]. We restrict our study to the transition of the 1 S state (L = 0) to the 1 P states (L = 1). Fig. 1 shows the lowest energy of the 1 P and 1 S state as a function of the variational parameter hω ¯ with the parabolic confinement strength h¯ ω0 = 20.0 meV. From Fig. 1, we can see that the appropriate values for the 1 P and 1 S state is about 78.022 meV and 57.5942 meV, respectively. 3. Numerical results and discussion In the following, we will discuss the linear, third-order and total optical absorption coefficients of two electrons in the GaAs parabolic quantum dot. The parameters used in our numerical work are adopted as:  = 12.4, m∗e = 0.067me (me

J. Huang, Libin / Physics Letters A 372 (2008) 4323–4326

Fig. 1. The energy of the 1 P and 1 S state as a function of the variational parameter h¯ ω with the parabolic confinement strength hω ¯ 0 = 20.0 meV.

Fig. 2. The linear third-order and total optical absorption coefficients versus the incident photon energy h¯ ν with the parabolic confinement strength h¯ ω0 = 150.0 meV. For two cases: (1) I = 3.0 × 109 W/m2 (solid line); (2) I = 5.0 × 109 W/m2 (dotted line).

is the free-electron mass), σs = 6 × 1022 m−3 , μ = 4π × 10−7 Hm−1 , nr = 3.2, Tj = 1 ps, 1/Γj k = 0.14 ps. Fig. 2 shows the linear, and third-order nonlinear optical absorption (α1 and α3 ) as well as the total optical absorption coefficient (α1 + α3 ) at an incident optical intensity of 3.0 × 109 W/m2 (solid line) and 5.0 × 109 W/m2 (dotted line), with the parabolic confinement strength hω ¯ 0 = 150.0 meV. From Fig. 2, we can see that the linear absorption coefficient α1 , which comes from the linear susceptibility χ (1) (ω) term, is positive, whereas the third-order nonlinear optical absorption coefficient α3 generated by the χ (3) (ω) term is negative [11]. So the total optical absorption coefficient αtotal is significantly reduced by the α3 contribution. Therefore, the third-order nonlinear optical absorption coefficient α3 should be considered when the

4325

Fig. 3. The total optical absorption coefficients α versus the incident photon energy hν ¯ for three different values of the parabolic confinement strength h¯ ω0 with I = 1.0 × 108 W/m2 . For two cases: (1) one electron in QD (dotted line); (2) two electrons in QD (solid line).

incident optical intensity I is comparatively strong, which can induce nonlinear absorption. Fig. 3 illustrates the total optical absorption coefficient αtotal as a function of the photon energy hν ¯ for three different values of parabolic confinement strength hω ¯ 0 : (a) h¯ ω0 = 20.0 meV, (b) hω ¯ 0 = 60.0 meV, (c) h¯ ω0 = 100.0 meV, with the incident optical intensity I = 1.0 × 108 W/m2 . For two cases: (1) one electron in QD (dotted line); (2) two electrons in QD (solid line). We observe three absorption peaks which appear at h¯ ν = 21.96 meV, hν ¯ = 61.6 meV, h¯ ν = 102.45 meV respectively. A very important feature is that the stronger the parabolic confinement is, the sharper the absorption peak will be and the higher the absorption peak will be. The reason is that the electronic dipolar transition matrix element will increase with the parabolic confinement potential, thus leading to the total optical absorption coefficient increase. It is similar to the case of one electron in one-dimensional quantum dots [9]. As the parabolic confinement strength increases, the absorption peak will move to the right side, which shows a confinement-strengthinduced blue shift of the resonance in the semiconductor quantum dot [19]. Meanwhile, the total optical absorption coefficient of one electron in QD (dotted line) is about five times the total optical absorption coefficient of two electrons in QD (solid line). Fig. 4 shows the total optical absorption coefficient as a function of the incident photon energy h¯ ν with the parabolic confinement strength h¯ ω0 = 20.0 meV, while the incident optical intensity I for six various values: (a) I = 0, (b) I = 3.75 × 108 W/m2 , (c) I = 6.25 × 108 W/m2 , (d) I = 8.75 × 108 W/m2 , (e) I = 11.25 × 108 W/m2 , (f) I = 13.75 × 108 W/m2 . It can been seen from Fig. 4 that the total optical absorption coefficient will reduce significantly with the incident optical intensity I increasing, and at sufficiently high-incident optical intensities the absorption will be strongly bleached. We also can see that the strong absorption saturation begins to oc-

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absorption coefficient α3 . Furthermore, the total optical absorption coefficient α will reduce when the incident optical intensity increases. The contributions to Dipole-allowed absorption coefficient of two electrons include (1) the transition dipole matrix element variations, and (2) the energy shifts (quantum-confined stark effects). Therefore, theoretical study on the nonlinear optical absorption may make a great contribution to experimental studies, may have profound consequences as regards practical application of electro-optical devices, and optical absorption saturation also has extensive application in optical communication, etc. References

Fig. 4. The total optical absorption coefficients α versus the incident photon energy h¯ ν for six different values of the incident optical intensity I with the parabolic confinement strength hω ¯ 0 = 20.0 meV.

cur at around I = 6.25 × 108 W/m2 . When the incident optical intensity I exceeds the value of I = 6.25 × 108 W/m2 , the absorption peak will be significantly split up into two peaks, which is in consequence of the absorption will be strongly bleached. In this Letter, the linear and nonlinear optical absorption coefficients (α1 and α3 ) as well as the total optical absorption coefficient (α1 + α3 ) for a nanometer-size GaAs parabolic quantum dot with two electrons are studied. The results show that the linear optical absorption coefficient α1 is not related to the incident optical intensity, whereas the incident optical intensity has great influence on the third-order nonlinear optical

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