Linear and nonlinear optical absorptions of an exciton in a quantum ring

Physica E 43 (2010) 49–53

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Linear and nonlinear optical absorptions of an exciton in a 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 in fo

abstract

Article history: Received 18 December 2009 Received in revised form 10 June 2010 Accepted 15 June 2010 Available online 22 June 2010

Optical absorptions of excitons have been studied in a quantum ring with parabolic confinement potential. Binding energies of the ground (L ¼0) and the first excited (L ¼ 1) states has been investigated as a function of the ring radius. Based on the computed energies and wave functions, the oscillator strengths have been examined, as well as the linear, third-order nonlinear and total optical absorption coefficients. The results are presented as a function of the incident photon energy. It is found that the electron–hole interaction and the ring radius have a great effect on the optical properties of excitons in a quantum ring. & 2010 Elsevier B.V. All rights reserved.

1. Introduction Rapid advances in semiconductor technology have led to the fabrication of quantum rings (QRs) containing only a few carriers [1–3]. QRs are small semiconductor ring-shape structures in which carriers are confined in all spatial dimensions. A QR structure is a particularly interesting nanostructure. Both the diameter and the ring width can be separately changed. As a consequence, discreteness of energy and charge arise, like in atomic systems. It is obvious that these man-made ‘‘artificial atoms’’ provide the experimental realization of a text-book example of many-particle physics: a finite number of quantum particles in a trap. 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 magneto-induced change in the ground state from angular momentum ‘ ¼ 0 to 1, with a flux quantum piercing the interior. These nanorings bind only a few carriers, in contrast with previously available mesoscopic rings on GaAs, [4] containing a number of carriers two or three orders of magnitude larger. The low number of particles and the dimensions of the ring make such systems a new kind of artificial atoms, which exhibit a richer variety of physical phenomena compared to quantum dots. The Aharonov–Bohm effect is one of the most distinct physical phenomena which illustrate the importance of the quantum mechanical phase. In theory, Aharonov–Bohm oscillations in QRs have been studied both analytically and numerically [5–10]. The magnetic field effects on excitons confined in the quantum rings of finite width have been studied theoretically [11,12]. Govorov and co-worker [13] studied the polarized excitons in nanorings and the optical Aharonov–Bohm effect. Recently, Kuroda and

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co-worker [14] investigated the optical transitions in QR complexes by using a droplet-epitaxial technique. Quite different from the conventional submicron mesoscopic structures, the nanoscopic rings are in the true quantum limit. The weak Coulomb interaction in these rings makes them most suitable for the observed state transitions, which can be well explained with the single-electron spectrum of a parabolic ring [15]. On the other hand, for double rings, Szafran and Peeters [16] studied fewelectron eigenstates in coupled concentric double QRs by the exact diagonalization technique. Castelano and co-worker [17] used the spin density functional theory to investigate the ground states of artificial molecules, which are made of two vertically coupled QRs. In semiconductor low-dimensional structures, the electron– hole interaction is enhanced owing to the confinement effect. Hence, quantum confinement effects of excitons have intensely been investigated such as quantum wells, quantum wires, and quantum dots, as examples of multicarrier systems. Since the pioneering work by Efros and Efros [18] and Brus [19], many theoretical studies have been devoted to exciton states in microcrystals [20–27]. It is now well-known that the confinement of carriers into dimensions of a few tens of nanometers, provides strong blueshift of the photoluminescence features from that in the original bulk material. This is a clear consequence of quantum confinement in these low-dimensional structures. The linear intersubband absorption has been studied experimentally within the conduction band of a GaAs quantum well [28,29]. A very large oscillator strength and a narrow bandwidth were observed. These suggest that intersubband optical transitions in a low-dimensional semiconductor 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 [30,31]. Recently, some authors studied the linear [32,33], and nonlinear [34–36] optical properties of

50

W. Xie / Physica E 43 (2010) 49–53

low-dimensional semiconductors. Most of them are related to the linear and the nonlinear optical properties of quantum dots. For example, Baskoutas and co-worker [34] studied the effects of excitons in nonlinear optical rectification in one-dimensional semiparabolic quantum dots. Thus, it is then intriguing to ask what happens for the electronic and the optical properties of an exciton confined by a QR. In this work, we will focus on studying the linear and nonlinear optical properties of an exciton confined in a parabolic QR by using the matrix diagonalization method. To the best of our knowledge, the nonlinear optical absorption has not been studied extensively in the literature.

a good quantum number. To obtain the eigenfunction and the eigenenergy associated with the exciton QRs, we diagonalized H. The exact diagonalization method is used in spanning the total Hamiltonian for a given basis and extract the lowest eigenvalues (energies) of the matrix generated. The better the basis describes the Hamiltonian, the faster the convergence will be. To obtain the eigenenergies and eigenstates, H is diagonalized in model space spanned by translational invariant harmonic product states

FL ¼

X

~ o ~ jo n1 ‘1 ðRÞjn2 ‘2 ðr Þ,

ð7Þ

½K

2. Theoretical model Within the effective mass approximation, the Hamiltonian of an exciton in the two-dimensional QR can be written as " 2 # X ~ e2 pi H¼ , ð1Þ þVi  ~ 2m e j r ~ r hj e i i ¼ e,h where i¼e,h correspond to the electron and hole, respectively. ~ r i ð~ p i Þ is the position vector (the momentum vector) of each particle originating from the center of the ring. And mi and e are the effective mass of each particle and the dielectric constant of the corresponding semiconductor material, respectively. Usually the confinement of small, two-dimensional quantum dots are to a very good approximation harmonic. The QRs are well approximated by using parabolic potentials, giving soft confinement barriers, appropriate to samples produced by self-assembly [2]. For moderate confinement this potential corresponds to a harmonic dot with its center removed. Obviously, the QR structure is modeled by a parabolic confinement potential, in which the wave functions expressed in terms of the center of mass and relative coordinates are used as a basis set. Hence, for parabolic confinement potential across the width of the ring, we use Vi ¼ 12mi o20 ðri r0 Þ2 ,

ð2Þ

where the mean radius of the ring is r0, and the characteristic confinement energy is ‘o0 , giving a characteristic ring width pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2‘=mi o0 for each particle. For the parabolic confinement potential it is convenient to separate the problem into center-of-mass and relative coordinates, described as used by ~ R ¼ ðme~ r ¼~ r e ~ r h, r e þ mh~ r h Þ=M, and ~ where the total reduced masses are M¼me + mh, and m ¼ me mh =M. The total Hamiltonian can then be re-expressed as H¼Hcm + Hrel + Hmix, with individual terms Hcm ¼ 0 Hrel ¼

P2 1 þ M o20 R2 , 2M 2

ð3Þ Pfi ¼

2

p 1 þ mo20 r 2 , 2m 2

ð4Þ

0 and Hrel ¼ Hrel þ Hurel , where

Hurel ¼

e2 , er

ð5Þ

and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2mh ~ m2h 2 2 2 2 ~ Hmix ¼ M o0 r0 me o0 r0 R2 þ r  Rþ 2 r 2 M M rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2m m e 2 e ~ r ~ R þ 2 r2 : mh o0 r0 R2  M M

where jo n‘ is a two-dimensional harmonic oscillator state with a frequency o and an energy ð2n þ j‘jþ 1Þ‘o, and [K] denotes the set quantum numbers ðn1 ,‘1 ,n2 ,‘2 Þ in brevity. Let fFL g denote the of basis functions including all the FL having their N smaller or equal to its upper limit Nmax. Here N ¼ 2ðn1 þ n2 Þ þ j‘1 j þ j‘2 j. Evidently, the total number, NH, of basis functions of the set is determined by Nmax. When o ¼ o0 , the basis function is an exact solution of Hamiltonian if the Coulomb interaction is removed. In practice calculation, o serves as variational parameter to minimize the eigenvalues. This would lead to a reduction of the number of basis functions. This point would be particularly noticeable if the system with smaller o0 was concerned. We will take a different o for the different states. 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 N  N Hermitean matrices. With this harmonic product basis set, matrix elements for the Coulomb interaction and mixing terms can be calculated analytically [37,38]. Since we are interested only in the low-lying states and in the qualitative aspects, the model space adopted is neither very large to facilitate numerical calculation, nor very small to ensure qualitative accuracy. This is achieved by extending the dimension of the model space step by step. In each step, the new results are compared with previous results from a smaller space, until satisfactory convergence is achieved. After the diagonalization we obtain the eigenvalues and eigenstates. 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. It would be measurable in photoluminescence experiments. Generally, the oscillator strength Pfi is defined as 4me

‘2

ð8Þ

where DEfi ¼ Ef Ei are the difference between the final and initial states. Mfi ¼ /f j~ r jiS is the electric dipole moment of the transition from i state to f state in the QR. The oscillator strength can offer additional information on the fine structure and selection rules of the optical absorption [39]. Hence, the electric dipole moment of the transition Mfi is not independent of the center-of-mass motion. Unlike a parabolic quantum dot, the variables of a QR is not independent. The susceptibility w is related to the absorption coefficient aðuÞ by

ð6Þ

aðuÞ ¼ u The Hamiltonian has cylindrical symmetry which implies that the total orbital angular momentum L is a conserved quantity, i.e.,

DEfi jMfi j2 ,

rffiffiffiffiffi m Im½e0 wðuÞ,

eR

ð9Þ

where m is the permeability of the system, and eR is the real part of the permittivity. Using the compact density-matrix method, the

W. Xie / Physica E 43 (2010) 49–53

optical absorption coefficient is given by [35] ð1Þ

17.0

ð3Þ

aðuÞ ¼ a ðuÞ þ a ðu,IÞ,

ð10Þ

4pbFS ss hujMfi j2 dðDEfi huÞ, nr

ð11Þ

and ( 2 jMff Mii j2 32p2 bFS ss I 4 2 a ðu,IÞ ¼  hujM j d ð D E huÞ 1 fi fi 2 nr Gff 4jMfi j2 ) ½ðhuDEfi Þ2 ð‘Gfi Þ2 þ 2DEfi ðDEfi huÞ ,  ðDEfi Þ2 þð‘Gfi Þ2 ð3Þ

ð12Þ

are the linear and the third-order nonlinear optical absorption coefficients, respectively. nr is the refractive index of the semiconductor. ss is the electron density in the QR. bFS ¼ e2 =ð4pe0 ‘cÞ is the fine structure constant where e is the electronic charge of an electron. hu denotes the photon energy. The d function in Eqs. (11) and (12) are replaced by a narrow Lorentzian by means of

‘Gfi

p½ðhuDEfi Þ2 þð‘Gfi Þ2 

:

15.5 15.0 L=0 L=1

14.0 0

5

10

15

20

Ring radius r0 (nm) Fig. 1. The variation of the binding energies of an exciton in a QR as a function of the ring radius r0 for the ground state (solid line) and the first excited state (dashed line), respectively.

ð13Þ

Here G is the phenomenological operator. Diagonal matrix element Gff of operator G, which is called as relaxation rate of fth state, is the inverse of the relaxation time Tff for the state jf S, namely Gff ¼ 1=Tff . Whereas, nondiagonal matrix element Gfi ðf a iÞ is called as the relaxation rate of fth state and ith state. In QRs, the selection rules ðD‘ ¼ 7 1Þ determine the fine state of the exciton after the absorption. Hence, we restrict our study to the transition of the ground state (L¼0) to the first excited state (L¼1).

16.0

14.5

1.6

Oscillator strength

dðDEfi huÞ ¼

Binding energy EB (meV)

16.5

where

að1Þ ðuÞ ¼

51

1.2

with without

0.8

3. Results and discussion The physical parameters used in our calculations are the mass me ¼0.067m0 and mh ¼0.090m0, where m0 is the single electron bare mass, the dielectric constant e ¼ 12:4, nr ¼3.2, Tff ¼1 ps, Tfi ¼1.4 ps, and ss ¼ 3:0  1024 m3 which are typical for a GaAs system [34,35]. We assume ‘o0 ¼ 3:0 meV. The adopted oscillator pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi energy corresponds to lengths de ¼ 2‘=me o0 ¼ 27:5 nm and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh ¼ 2‘=mh o0 ¼ 23:76 meV for the electron and the hole, respectively. The nanometer-sized GaAs QRs can be made by means of a droplet epitaxial technique [40,41]. The check of the accuracy of the ground state is presented in Table 1, where E are given in accord with Nmax (NH). Obviously, when Nmax ¼ 30 (NH ¼ 1496), the ratio of the difference in energy DE=E ¼ 0:0496=11:3127  0:44% is less than 0.5%. This is sufficient for our qualitative purpose. We define the binding energy of an exciton as EB ¼ E0 E,

ð14Þ

where E0 and E are the energies of an exciton without and with the Coulomb interaction in the QRs, respectively. In Fig. 1 we show the variation of the exciton binding energies EB of the ground and the first excited states in a QR as a function of the ring Table 1 Eigenenergies of the ground state of an exciton in accord with Nmax. Nmax(NH) 20(506) E(meV)

22(650)

24(819)

26(1015)

28(1240)

30(1496)

 10.6905  10.8604  11.0103  11.1437  11.2631  11.3127

‘o0 ¼ 3:0 meV is assumed.

0.4 0

5

10 Ring radius r0 (nm)

15

20

Fig. 2. The variation of oscillator strengths of an exciton with (solid line) and without (dashed line) the electron–hole interaction, respectively, in a QR as a function of the ring radius.

radius r0. It is evident from the figure that the exciton binding energies decrease monotonically as the ring radius r0 increases. We know that the confinement of QRs increases with an increasing r0 [12]. Hence, this result is different from the case of quantum dots. The binding energy increases as the quantum confinement increases. The physical origin is that, with the ring radius of QRs increasing, the Coulomb interaction between the electron and the hole decreases, leading to the decrease of the binding energy. In Fig. 2, we present the variations of oscillator strength of an exciton with (solid line) and without (dashed line) the electron– hole interaction, respectively, as a function of the ring radius. Obviously, the oscillator strength of an electron and a hole without the Coulomb interaction is larger than that of an exciton. Hence, in QRs, the intersubband optical absorptions of an electron and a hole without the Coulomb interaction will be stronger than those of the excitons. In order to study the optical absorptions of an exciton in QRs, a numerical calculation has been performed for the linear, thirdorder nonlinear and total optical absorption coefficients as a

W. Xie / Physica E 43 (2010) 49–53

Absorption coefficient α (106/m)

24

an electron and a hole Linear Nonlinear Total

20 16 12 exciton 8 4 0

an electron and a hole

-4 -8 0

2

4 hυ (meV)

6

8

Fig. 3. Linear, the third-order nonlinear and the total optical absorption coefficients of an exciton with and without the electron–hole interaction in a QR as a function of the incident photon energy hu for r0 ¼ 10.0 nm.

12 r0 = 10.0nm Absorption coefficient α (106/m)

function of the incident photon energy for r0 ¼10.0 nm and the incident optical intensity I¼1.0  105 W/m2. Fig. 3 shows the results of the electron–hole pair without and with the Coulomb interaction. The quantum effects of QRs on the linear, third-order nonlinear and total optical absorptions are clear. It can be clearly seen that the að1Þ ðuÞ, að3Þ ðu,IÞ and aðu,IÞ as a function of hu has an prominent peak, respectively, at the same position, which occurs due to the one-photon resonance enhancement, i.e., hu  DE. From Fig. 3, we note that the large að1Þ ðuÞ, which comes from the linear susceptibility term, is positive. Whereas að3Þ ðu,IÞ, which is generated by the nonlinear third-order susceptibility term, is negative. So the total absorption coefficient aðu,IÞ is significantly reduced by the að3Þ ðu,IÞ contribution. Hence we can say that one should take into account the nonlinear (intensity-dependent) absorption coefficient near the resonance frequency ðhu C DEÞ, especially at higher intensity values. On the other hand, from this figure we can find that the exciton effects on the optical absorptions in QRs. The linear, third-order nonlinear and total optical absorption coefficients of an electron and a hole without the Coulomb interaction are greater in the case with the Coulomb interaction. It is readily seen that these results are in good agreement with those in Fig. 2. We also find that another important aspect of the exciton effect is the absorption coefficients, which will suffer an obvious blue-shift by the electron–hole interaction. Hence, the optical absorption coefficients of an exciton in QRs are strongly affected by the electron–hole interaction. Fig. 4 shows the linear, third-order nonlinear and total optical absorption coefficients of an exciton in QRs as a function of the incident photon energy hu for three different ring radii, i.e., R0 ¼5.0, 10.0, and 15.0 nm, respectively. As illustrated in Fig. 4, the resonant peaks of the linear, the nonlinear and the total absorption coefficients of an exciton QR suffer an obvious redshift with an increasing r0. The physical origin is that the energy difference DE from the ground state to the first excited state is decreased as an increasing r0. On the other hand, we find that the linear and the third-order absorption coefficients decrease with an increasing r0, and hence the total absorption coefficient is significantly reduced with an increasing r0. Furtherly, in order to show better the influence of the nonlinear optical absorption and the incident optical intensity for the total absorption coefficient a, in Fig. 5, we set r0 ¼10.0 nm and plot a as a function of the incident photon energy hu for six

r0 = 5.0nm 8

r0 = 15.0nm Linear Nonlinear Total

4

0

0

2

4 hυ (meV)

6

8

Fig. 4. Linear, third-order nonlinear and total optical absorption coefficients of an exciton in a QR as a function of the incident photon energy hu for three different values of the ring radius.

r0 = 10.0nm 8 Total absorption coefficient (106/m)

52

I = 0.0W/m2 I = 1.0x105W/m2 I = 1.5x105W/m2 I = 2.0x105W/m2 I = 2.5x105W/m2 I = 3.0x105W/m2

6

4

2

0 0

2

4

6

8

hυ (meV) Fig. 5. The total absorption coefficient of an exciton in a QR as a function of the incident photon energy hu for six different values of the incident optical intensity I for r0 ¼10.0 nm.

different values of I¼0, 1.0  105, 1.5  105, 2.0  105, 2.5  105, and 3.0  105 W/m2, respectively. From this figure, it can be clearly seen that the peak of the total absorption coefficient decreases prominently with the increase of I. The absorption will be strongly bleached at sufficiently high-incident optical intensities. Fig. 5 shows that the strong absorption saturation begins to occur at around I¼2.5  105 W/m2. When the incident optical intensity exceeds this value, the absorption peak will be significantly split up into two peaks, which is in consequence of the absorption will be strongly bleached. Obviously, the total absorption coefficient, especially for the resonant peak, is strongly affected by the incident optical intensity due to the contributions of the third-order nonlinear term. Hence, the nonlinear effects should be considered for the optical absorptions of an exciton in QRs, in particular for the case with a strong incident optical intensity. In conclusion, we have applied the parabolic confining potential to a description of the exciton in semiconductor QRs. We investigated the binding energy, the oscillator strength and the intersubband optical absorptions of an exciton in QRs. The

W. Xie / Physica E 43 (2010) 49–53

binding energies of the ground (L¼0) and the first excited (L¼1) states and the oscillator strength for intersubband transition from the ground state to the first excited state have been calculated as a function of the ring radius. Based on the computed energies and wave functions, the linear, third-order nonlinear and total optical absorption coefficients have been investigated in detail for the electron–hole interaction, the ring radius and the incident optical intensity. We have found the intersubband optical absorptions of an exciton in QRs are strongly affected by the Coulomb interaction, the ring radius, and the incident optical intensity.

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