Intersubband optical absorptions of a two-electron quantum ring

Physics Letters A 374 (2010) 1188–1191

Contents lists available at ScienceDirect

Physics Letters A www.elsevier.com/locate/pla

Intersubband optical absorptions of a two-electron quantum ring Wenfang Xie School of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 16 November 2009 Received in revised form 16 December 2009 Accepted 22 December 2009 Available online 29 December 2009 Communicated by R. Wu

a b s t r a c t In this Letter, the optical properties of a quantum ring with two electrons are studied. Its effective-mass Hamiltonian matrix was diagonalized with numerical methods, followed by the calculations of a number of optical quantities. We have found that the intersubband optical absorptions strongly depends on the ring radius, electron–electron interaction, and the incident optical intensity. We also found that the spinsinglet states are more sensitive to ring radius than the spin-triplet states. © 2009 Elsevier B.V. All rights reserved.

Keywords: Electron–electron interaction Quantum ring Semiconductor

1. Introduction Rapid advances in semiconductor technology have led to the fabrication of quantum rings (QRs) containing only a few electrons [1–3]. QRs are small semiconductor ring-shape structures in which electrons are confined in all spatial dimensions. Both the diameter and the ring width can be changed independently. Consequently discrete energy eigenvalues and charges arise, as in atomic systems. The low number of particles and dimensions of a ring makes it a new kind of artificial atoms, which exhibit a richer variety of physical phenomena than quantum dots. On moreover, the ring diameter and the potential strength of a QR can be tuned so that it can evolve from a quasi-one-dimensional to two-dimensional system. The crossover from a two-dimensional electron gas to a nanoscale QR can therefore be studied. Lorke and co-worker [2] applied self-assembly techniques to create InGaAs rings containing only a few electrons. They first observed far-infrared optical response in QRs, revealing a magnetoinduced change in the ground state from angular momentum  = 0 to  = 1, with a flux quantum piercing the interior. Recently, Kuroda and co-worker [4] investigated the optical absorptions of QRs using a droplet-epitaxial technique. Quite different from the conventional submicron mesoscopic structures, the nano-scale rings are in the true quantum limit. Due to the weak electron– electron interaction in these rings, the observed absorptions could be well explained with the model of a single-electron parabolic ring [5]. On the other hand, theoretically, few electrons in coupled concentric double QRs have been studied by Szafran and Peeters [6]. Castelano and co-worker [7] investigated the ground

state of artificial molecules made of two vertically coupled QRs by the spin density functional theory. The intersubband optical absorptions of QRs and other lowdimensional systems have attracted an enormous interest in recent years [8–16]. These results show that intersubband optical absorptions in low-dimensional semiconductors have very large optical nonlinearity. Both linear and nonlinear intersubband optical absorptions can be used for practical applications in photodetectors and high-speed electro-optical devices [17,18]. Very recently, some authors studied the linear [19] and the nonlinear [12] optical absorptions of a two-electron quantum dot. In this work, we will devote our calculations to the linear and the third-order nonlinear optical absorption coefficients of a two-electron QR. To the best of our knowledge, this problem has not been studied extensively in the literature. The variation of the ring radius will allow us to gain more insight into its linear and nonlinear optical properties. 2. Theory Usually the confinement of small, two-dimensional quantum dots can be well approximated by a harmonic potential. Correspondingly, we model QRs realized in experiments [1,2] by a potential of the form V (r ) = 12 me ω02 (r − R 0 )2 . For moderate confinement this potential can be viewed as a harmonic dot perturbed by a potential barrier in the middle. Hence the eigenfunctions of an harmonic oscillator can be used as a basis set. The system we study has two interacting electrons moving in the (x, y ) plane, and confined by the ring potential V (r ). The Hamiltonian is given by

H= E-mail address: [email protected]. 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.12.059

  p2 i i =1,2

2me

 + V (r i ) +

e2

 |r1 − r2 |

,

(1)

W.F. Xie / Physics Letters A 374 (2010) 1188–1191

where r i are position vectors originating from the center of the  i are the corresponding momentum vectors. me is the ring, and p effective mass of an electron, and  is the dielectric constant of the semiconductor material. The mean radius of the ring is R 0 , and the characteristic confinement energy is h¯ ω0 . It is convenient to introduce the center of mass and relative  = (r1 + r2 )/2, and r = r1 − r2 . The total coordinates, given by R Hamiltonian can then be re-expressed as H = H cm + H rel + H mix with

P2

H cm =

2M

+

1 2

M ω02 R 2 ,

(2)

0  H rel = H rel + H rel ,

p2

0 H rel =

1

+ μω02 r 2 ,

2μ e2

 = H rel

r

(3)

(5)

,

and





α (υ ) = υ

μ Im 0 χ (υ ) , R

(9)

where μ is the permeability of the system,  R is the real part of the permittivity. Using the compact density-matrix method, the optical absorption coefficient is given by [23]

 r 2 + 4R 2 + 4r · R

where



α (1) (υ ) =

 , r 2 + 4R 2 − 4r · R

+



α (υ ) = α (1) (υ ) + α (3) (υ , I ),



H mix = me ω02 R 20 − me ω02 R 0

can offer additional information on the fine structure and selection rules of the optical absorption [20]. Hence, the electric dipole moment of the transition M fi is not independent of the relative motion. Unlike a parabolic quantum dot, the variables of a QR is not independent. This is not satisfied with the Kohn theorem [21]. On the other hand, 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 a h¯ t [22]. Hence, we restrict our study to the transitions 1 S → 1 P and 3 P → 3 S. The susceptibility χ is related to the absorption coefficient α (υ ) by

(4)

2

1189

(6)

4π β F S σ s nr

(10)

hυ | M fi |2 δ( E fi − hυ ),

(11)

where the total and reduced masses are M = 2me and μ = me /2. The Hamiltonian has cylindrical symmetry which implies that the total orbital angular momentum L is a good quantum number. The total spin of two electrons, S is also conserved. Hence, the eigenstates of the two electrons can be labeled by (L , S). To obtain the eigenenergies and eigenstates, H is diagonalized in the model space spanned by translational invariant harmonic product states

and

Φ[LSK ] =

are the linear and the third-order nonlinear optical absorption coefficients, respectively. nr is the refractive index of the semiconductor. σs is the electron density in the QR. β F S = e 2 /(4π0 h¯ c ) is the fine structure constant where e is the electric charge of an electron. hυ denotes the photon energy. The δ function in Eqs. (10) and (11) are replaced by a narrow Lorentzian by means of

  

˜ ϕ ω ( R )ϕ ω (r ) χ S A n1 1 n2 2 L

(7)

[K ]

where ϕnω is a two-dimension harmonic oscillator wave function with a frequency ω and an energy (2n + || + 1)¯hω , and [ K ] denotes the set quantum numbers (n1 , 1 , n2 , 2 ) in brevity, χ S = [η(1)η(2)] S . In practical calculations, ω serves as an adjustable variational parameter around ω0 to minimize the ground state en˜ is a anti-symmetrizer. ergy. A Since the whole set of eigenstates of the harmonic product basis forms a complete set, increasing the number of basis functions makes the calculated results more accurate. The only limit comes from the capacity of a computer to diagonalize an N × N Hermitian matrix. Since we are interested only in the low-lying states and in the qualitative aspects, the model space adopted is neither too large to facilitate numerical calculation, nor too small to ensure the accuracy. We have extended the dimension of the basis space step by step; in each step the new results were compared with previous results until satisfactory convergence is achieved. In this Letter, the dimension of the model space is constrained by 0  N = 2(n1 + n2 ) + |1 | + |2 |  30. If N is increased by 2, the change in ground state energy is less than 0.1%. The oscillator strength is a very important physical quantity in the study of the optical properties which are related to the electronic dipole-allowed absorptions. Generally, the oscillator strength P fi is defined as

Pfi =

4me h¯ 2

E fi | M fi |2 ,

(8)

where E fi = E f − E i are the difference between the final and

 |i  is the electric dipole moment of the initial states. M fi = 2 f | R transition from i state to f state in the QR. The oscillator strength

α (3) (υ , I ) 32π 2 β F2 S σs I

| M ff − M ii |2 4 | M fi |2 nr2 Γff  [(hυ − E fi )2 − (h¯ Γfi )2 + 2 E fi ( E fi − hυ )] , × (12) ( E fi )2 + (h¯ Γfi )2

=−

δ( E fi − hυ ) =

hυ | M fi |4 δ 2 ( E fi − hυ ) 1 −

h¯ Γfi

π [(hυ − E fi )2 + (h¯ Γfi )2 ]

.

(13)

Here Γ is a phenomenological operator. The diagonal matrix elements Γff , called the relaxation rate of the f th state, is the inverse of the relaxation time T ff for the state | f , namely Γff = 1/ T ff ; the nondiagonal matrix elements Γfi ( f = i ) is called the relaxation rate of the f th state through the ith state. 3. Numerical results and discussion In our calculations we take the material parameters of GaAs: me = 0.067m0 , where m0 is the single electron bare mass,  = 12.4, nr = 3.2, T ff = 1 ps, and T fi = 1.4 ps. We assume h¯ ω0 = 3.0 meV. The adopted oscillator energy corresponds to a length   = 2h¯ /me ω0 = 27.5 nm which defines the width d = 2 of the considered rings. In addition, the electron density is taken σs = 3.0 × 1022 m−3 . The nanometer-sized GaAs QRs can be made by means of a droplet epitaxial technique [24,25]. Fig. 1 shows the variations of the absorption energies from 1 S → 1 P and 3 P → 3 S as a function of the ring radius. These E are representative of the optical gap of the system for two different spin configurations. From this figure we observed the followings: (1) The absorption energies decrease with increasing R 0 and reach a finite minimum values depending on the spin, as R 0 is increased further, the absorption energies begin to slowly increase. As is well

1190

W.F. Xie / Physics Letters A 374 (2010) 1188–1191

Fig. 1. The absorption energies from the 1 S state to the 1 P state for the spin-singlet states (solid line) and the 3 P state to the 3 S state for the spin-triplet states (dashed line), respectively, as a function of the ring radius. Parameters are taken appropriate for GaAs and h¯ ω0 = 3.0 meV.

Fig. 2. The variation of oscillator strengths from the 1 S state to the 1 P state for the spin-singlet states (solid line) and the 3 P state to the 3 S state for the spin-triplet states (dashed line), respectively, as a function of the ring radius. Parameters are taken appropriate for GaAs and h¯ ω0 = 3.0 meV.

known, the energy of the low-lying states in a confined system is determined by a competition between the single particle energy and the interacting energy. With our potential model an increase in R 0 implies an increase of ring radius and an decrease of its width. The Coulomb interaction energy decreases but single-particle energies increase. When an increase in the single-particle energies cannot be compensated for by a decrease in the Coulomb interaction energy, the absorption energies decrease with an increasing ring radius, if not they will increase. (2) The absorption energy of the spin-singlet states is higher than that of the spin-triplet states. As the ring radius increases, the difference of the absorption energies between the spin-singlet and the spin-triplet states increases. Hence, the excitation of the spin-triplet state is harder than that of the spin-singlet state. (3) The variation of the absorption energy of the spin-triplet states is slower than that of the spin-singlet states. Hence, the red or blue shift of the resonance in QRs due to a varying radius is more detectable for the spin-singlet states. In Fig. 2, we present the variations of oscillator strength from the 1 S → 1 P (solid line) and the 3 P → 3 S (dashed line), respec-

Fig. 3. Linear, the third-order nonlinear and the total optical absorption coefficients from the 1 S state to the 1 P state for the spin-singlet states as a function of the incident photon energy hυ for three different values of the ring radius. Parameters are taken appropriate for GaAs, h¯ ω0 = 3.0 meV and I = 1.0 × 103 W/m2 .

tively, as a function of the ring radius. As shown from this figure, the oscillator strength of the spin-triplet states is a monotonic function of R 0 and decreases slowly with an increasing R 0 . However, the oscillator strength of the spin-singlet states is not monotonic functions of R 0 . The oscillator strength of the spinsinglet states, at the beginning, decreases with an increasing R 0 , but it reaches a minimum value at around the ring radius, as R 0 is increased further, it begins to increase. Hence, the influence of the ring radius on intersubband optical absorptions of the spinsinglet states is stronger than that of the spin-triplet states. On the other hand, we also find that the oscillator strength of the spinsinglet states is larger than that of the spin-triplet states when R 0 < 9.8 nm. Hence, when R 0 < 9.8 nm, the intersubband optical absorptions of the spin-singlet states will be stronger than those of the spin-triplet states. However, when R 0 > 9.8 nm, the results will be opposite. In order to study the intersubband optical absorptions of a twoelectron QR, a numerical calculation has been performed for the linear α (1) (υ ), the third-order nonlinear α (3) (υ , I ) and the total α (υ , I ) optical absorption coefficients as a function of the incident photon energy hυ in the range from 0 to 10 meV for three different ring radii, i.e., R 0 = 5.0, 10.0, and 15.0 nm, respectively. Figs. 3 and 4 show the results of the spin-singlet and the spin-triplet states, respectively. In Figs. 3 and 4 the incident optical intensity I is set to be 1.0 × 103 W/m2 . From these figures we can find that the qualitative properties of the intersubband optical absorptions for the spin-singlet and the spin-triplet states are similar. However, the quantitative differences are also obvious. First, the quantum effect of the ring radius on the linear, the nonlinear and the total optical absorptions is clear. It can be clearly seen that for each R 0 , the α (1) (υ ), α (3) (υ , I ) and α (υ , I ) as a function of hυ has an prominent peak, respectively, at the same position, which occurs due to the one-photon resonance enhancement, i.e., hυ ≈ E. Secondly, it is readily seen that the large α (1) (υ ), which comes from the linear susceptibility term, is positive. Whereas α (3) (υ , I ), which is generated by the nonlinear third-order susceptibility term, is negative. So the total absorption coefficient α (υ , I ) is significantly reduced by the α (3) (υ , I ) contribution. Hence we can say that one should take into account the nonlinear (intensity-dependent) absorption coefficient near the resonance frequency (hυ E), especially at higher intensity values. Thirdly, the resonant peaks of the linear, the nonlinear and the total absorption coefficients suffer an obvious red-shift with an increasing R 0 . The physical origin is that,

W.F. Xie / Physics Letters A 374 (2010) 1188–1191

1191

Based on the computed energies and wave functions, the linear α (1) (υ ), the third-order nonlinear α (3) (υ , I ) and the total α (υ , I ) optical absorption coefficients have been investigated in detail for the ring radius. The results are presented as a function of the incident photon energy for the different values of the ring radius. We have found the intersubband optical absorptions of a two-electron QR are strongly affected by the ring radius, the electron–electron interaction, and the incident optical intensity. We have also compared the results of two different spins. It is found that the influence of the ring radius on the intersubband optical absorptions of the spin-singlet states is stronger than those of the spin-triplet states. Acknowledgement This work is financially supported by the National Natural Science Foundation of China under grant No. 10775035. Fig. 4. Linear, the third-order nonlinear and the total optical absorption coefficients the 3 P state to the 3 S state for the spin-triplet states as a function of the incident photon energy hυ for three different values of the ring radius. Parameters are taken appropriate for GaAs, h¯ ω0 = 3.0 meV and I = 1.0 × 103 W/m2 .

when R 0 < 15.0 nm, the energy difference E from the 1 S state to the 1 P state or the 3 P state to the 3 S state is decreased as an increasing R 0 (see Fig. 1). This is obviously different from the results of a two-electron quantum dot [12]. In the two-electron quantum dot, as the quantum confinement increases, the absorption coefficients will suffer an obvious blue-shift. As is well know, the quantum confinement of QRs increases as an increasing R 0 . However, we also note that the electron–electron interaction in QRs decreases as an increasing R 0 . It is the competition between the confinement energy and the electron–electron interaction energy that finally determines the feature of the absorption coefficients. Finally, it is obvious that the influence of the ring radius on the linear α (1) (υ ), the third-order nonlinear α (3) (υ , I ) and the total α (υ , I ) optical absorption coefficients of the spin-singlet states is stronger than those of the spin-triplet states. It is readily seen that these results are in good agreement with those in Fig. 2. In conclusion, we have investigated the intersubband optical absorptions of a two-electron QR. Calculations are made by using the method of numerical diagonalization of Hamiltonian matrix within the effective-mass approximation. The absorption energy and the oscillator strength for intersubband transition from the 1 S state (L = 0) to the 1 P state (L = 1) for the spin-singlet states and the 3 P state (L = 1) to the 3 S state (L = 0) for the spin-triplet states, respectively, have been calculated as a function of the ring radius.

References [1] R.J. Warburton, C. Schäflein, D. Haft, F. Bickel, K. Lorke, J.M. Garcia, W. Schoenfeld, P.M. Petroff, Nature 405 (2000) 926. [2] A. Lorke, R.J. Luyken, A.O. Govorov, J.P. Kotthaus, J.M. Garcia, P.M. Petroff, Phys. Rev. Lett. 84 (2000) 2223. [3] S.S. Li, J.B. Xia, J. Appl. Phys. 91 (2002) 3227. [4] T. Kuroda, T. Mano, T. Ochiai, S. Sanguinetti, K. Sakoda, G. Kido, N. Koguchi, Phys. Rev. B 72 (2005) 205301. [5] T. Chakraborty, P. Pietiläinen, Phys. Rev. B 50 (1994) 8460. [6] B. Szafran, F.M. Peeters, Phys. Rev. B 72 (2005) 155316. [7] L.K. Castelano, G.-Q. Hai, B. Partoens, F.M. Peeters, Phys. Rev. B 74 (2006) 045313. [8] K.X. Guo, S.W. Gu, Phys. Rev. B 47 (1993) 16322. [9] G.H. Wang, K.X. Guo, Physica B 315 (2002) 234. [10] S. Baskoutas, E. Paspalakis, A.F. Terzis, Phys. Rev. B 74 (2006) 153306. [11] I˙ . Karabulut, Ü. Atav, H. Safak, M. Tomak, Eur. Phys. J. B 55 (2007) 283. [12] J.S. Huang, B. Li, Phys. Lett. A 372 (2008) 4323. [13] I. Karabulut, S. Baskoutas, J. Appl. Phys. 103 (2008) 073512. [14] Y.U. You-Bin, Commun. Theor. Phys. 49 (2008) 1615. [15] W.F. Xie, Phys. Status Solidi B 246 (2009) 1313. [16] W.F. Xie, Opt. Commun. 282 (2009) 2604. [17] F. Caposso, K. Mohammed, A.Y. Cho, IEEE J. Quantum Electron. QE-22 (1986) 1853. [18] D.A.B. Miller, Int. J. High Speed Electron. Syst. 1 (1991) 19. [19] M. Sahin, Phys. Rev. B 77 (2008) 045317. [20] T. Chakraborty, P. Pietiläinen, Phys. Rev. Lett. 95 (2005) 136603. [21] L. Jacak, P. Hawrylak, A. Wójs, Quantum Dots, Springer-Verlag, Berlin, Heidelberg, 1998. [22] T. Chakraborty, P. Pietilainen, Phys. Rev. Lett. 95 (2005) 136603. [23] E.M. Goldys, J.J. Shi, Status Solidi B 210 (1998) 237. [24] T. Mano, N. Koguchi, J. Gryst. Growth 278 (2005) 108. [25] T. Mano, T. Kuroda, S. Sanguinetti, T. Ochiai, T. Tateno, J. Kim, T. Noda, M. Kawabe, K. Sakoda, G. Kido, N. Koguchi, Nano Lett. 5 (2005) 425.