Tunneling effect on second-harmonic generation in quantum dot molecule

Accepted Manuscript Tunneling effect on second-harmonic generation in quantum dot molecule Xiancong Jiang, Kangxian Guo

PII:

S0749-6036(16)30024-6

DOI:

10.1016/j.spmi.2016.01.025

Reference:

YSPMI 4160

To appear in:

Superlattices and Microstructures

Received Date: 5 November 2015 Revised Date:

18 January 2016

Accepted Date: 20 January 2016

Please cite this article as: X. Jiang, K. Guo, Tunneling effect on second-harmonic generation in quantum dot molecule, Superlattices and Microstructures (2016), doi: 10.1016/j.spmi.2016.01.025. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Xiancong Jiang, Kangxian Guo

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Tunneling effect on second-harmonic generation in quantum dot molecule

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(Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, P.R. China)

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Abstract: The tunneling effect on second-harmonic generation (SHG) coefficient in quantum dot molecule (QDM) is investigated theoretically. By using the compact-density matrix approach and the iterative method , we obtain an analytical expression of the SHG coefficient and numerical calculations for GaAs QDM. The results show that the tunneling strength , Rabi frequency and the size of the QDM have influences on the SHG coefficient.

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Keywords: Second-harmonic generation; Quantum dot molecule; Tunneling

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*Corresponding author: e-mail: [email protected].

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Introduction

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1

With the rapid development of epitaxy techniques in the past few years,

it is possible to study the low-dimensional systems in nanoscale such as quantum well[1-7] , quantum wire[8-13] and quantum dot. For a three dimensional

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confinement of carriers, quantum dot(QD) has been attracted the attention of many researchers. In 1999, quantum information processing by using

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quantum dot spins and cavity quantum electrodynamics was studied by A. Imamoglu etal.[14-17]. In addition, much attention have been paid to the nonlinear optical effects in QD, such as optical rectification (OR), secondharmonic generation (SHG), third-harmonic generation (THG), electro-optic effect (EOE), etc[18-22]. Because of such a low-dimensional system, QD may exhibit pronounced optical nonlinearity which could lead to finding out op-

application.

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tical media with sufficiently large and quick nonlinear responses for various

With the femtosecond laser was invented, the Nonlinear Optics has de-

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veloped rapidly in the recent years. As we all know, the SHG is one of the main nonlinear optical effects. With the powerful laser, the nonlinear

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material will radiate second harmonic light whose frequency is twice larger than the incident light. However, the SHG can not be observed in a symmetric quantum system. It is because that the optical transitions between the electronic states with the same parity are not allowed. For this reason, the signal of SHG has high sensitivity in detecting the structure symmetry change. With a such prominent feature, SHG has been studied in many fields by many researchers. In 2007, the SHG in parabolic quantum dots with electric and magnetic fields has been studied by Li etal.[23]. The result shows

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that the magnitude of electric and magnetic fields have a great influence on

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the SHG coefficient and the theoretical value of the SHG can be enhanced

over 10−6 m/V by the applied fields. In 2013, Liu etal. studied the polaron effects on SHG in cylindrical quantum dots with magnetic field and obtained the two photon resonant peak of SHG with the electron-LO phonon inter-

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action(ELOPI) is about over 15 ∼ 22.5 times larger than that without the ELOPI[24]. Latter, in 2014, R.Khordad and H.Bahramiyan researched the

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SHG of modified Gaussian quantum dots under the influence of electron-LO phonon, electron-SO phonon, and electron-LO+SO phonon interations[25], Mou etal. studied the SHG in asymmetrical semi-exponential[26] and Zhai studied the SHG in asymmetrical Gaussian quantum wells[27], respectively. Recently, some researchers have taken a great interest in quantum dot

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molecule (QDM), which can be fabricated using self-assembled dot growth technology or molecule beam epitaxy combing with in situ atomic layer precise etching[28,29]. An asymmetric quantum dot molecule consists of two QDs with different band structures which are coupled by tunneling. An elec-

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tron can be excited from the valence band to the conduction band in a dot by a near resonance incident light, and the electron can tunnel to the other

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dot with the help of the external bias voltage. Since the tunneling effect is sensitive to external bias voltage, the interdot oscillations can be controlled by the applied voltage. Many works have been carried out many problems about the asymmetric QDM system, such as voltage-controlled slow light, tunneling induced transparency, exciton qubits, robust states, coherent control of tunneling[30-34]. In this paper, we study the nonlinear second-harmonic generation in an asymmetric QDM with tunneling effect. This paper is organized as follows. 3

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In Sec.2, physical system and model, Hamiltonian, relevant eigenstates and

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eigenenergies will be discussed in detail. The analytical expressions for the

nonlinear SHG are obtained by the compact-density matrix approach and the iterative method. Sec.3 is devoted to the numerical results and discussions.

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Finally, a brief conclusion will be made in the Sec.4.

Theory

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Let us consider a QDM with two QDs. A schematic representation of the Hamiltonian for our model can be seen in Fig.1[30]. In Fig.1(a), the interdot tunneling is weak because the levels are out of resonance without a gate voltage. On the contrary, Fig.1(b) shows that with a gate voltage, the conduction-band levels in different QDs get close to resonance, the tunneling

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become much stronger than before while the valence-band levels are much more out of resonance. For this reason, we can neglect the hole tunneling and the Hamiltonian of the system can be represented in Fig.1(c). |0i is

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the ground state without excitation. |1i is the direct-exciton state in one dot where an electron is excited from valence band to its conduct band with an incident pulse. By tunneling effect, an electron tunnel from one dot to

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another. We use |2i to describe such phenomenon called indirect exciton state which contains a hole in the first dot and an electron in the second dot. Using this configuration, the Hamiltonian can be written as[30] ˆ = H

X j

h ¯ ωj |jihj| + Te (|1ih2| + |2ih1|) + h ¯ Ω(eiωL t |0ih1| + e−iωL t |1ih0|), (1)

where h ¯ ωj stand for the energy of the state |ji, j = 0, 1, 2. Te stand for the electron-tunneling strength. Ω =

µ01 E 2¯ h

is one-half Rabi frequency for the

probe laser field, here µ01 is the dipole momentum matrix element and E is the 4

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iωL t Uˆ = exp[ (|1ih1| − |0ih0| + |2ih2|)], 2

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electric field amplitude, respectively. Applying the unitary transformation (2)

and using the Baker-Hausdorff lemma[35], we can obtain a time-independent version of Hamiltonian written as follows 



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−δ1 2¯hΩ 0 1  ′ ˆ H =  2¯hΩ δ1 2Te  , 2 0 2Te δ2

(3)

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where δ1 = h ¯ (ω10 − ωL ) is the detuning between the frequency of optical pulse and exciton transition, δ2 = δ1 + 2¯hω12 and ωij = ωi − ωj is the optical transition between ith and jth energy states. To solve the time-independent Hamiltonian, we can get three different eigenvalues and three corresponding eigenstates.

with

(4)

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ψn = a(En )|0i + b(En ) · a(En )|1i + c(En ) · a(En )|2i, (n = 0, 1, 2) (En − δ2 · 2¯hΩ) , G δ1 + En , b(En ) = 2¯hΩ 2Te · (δ1 + En ) c(En ) = , 2¯hΩ · (En − δ2 )

a(En ) =

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(5)

where

(6) (7)

1

G = ((En − δ2 )2 · (2¯hΩ)2 + (δ1 + En )2 · (En − δ2 )2 + (δ1 + En )2 · 4Te2 )− 2 . (8)

ψn is superposed states of the system with |0i, |1i and |2i. The QD which used to form a QDM is a quantum well quantum dot(QWQD). |0i, |1i and |2i can be written as follows[36] |0i = ϕe , 5

(9)

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(10)

′

(11)

|2i = ϕh · ϕe , with s

2 πz sin , L L

(12)

αh 1 ϕh = √ exp(− αh2 (x2 + y 2)) · π 2

s

2 πz sin , L L

(13)

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αe 1 ϕe = √ exp(− αe2 (x2 + y 2 )) · π 2

s

1 2 πz αe sin , (14) ϕe = √ exp(− αe2 (x2 + y 2 )) · π 2 R R 1 2 γz 2γz αe−h (x2 + y 2 )) · Cm z exp(− )F [−m, 2, ]. = √ exp(− αe−h π 2 m+1 m+1 (15)

Where αk =

q

mk ω0 , h ¯

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′

ϕe−h

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|1i = ϕe−h ,

mk(k=e,h,e−h) is the effective mass for electron, hole or

exciton, respectively. L(R) is the first(second) quantum dot’s well width, Cm me−h e2 , ε¯ h2

F is the confluent hypergeometric

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is the normalization constant, γ =

function. For simplicity, we only consider the z direction. The SHG coefficient in QDM can be obtained by using the compactdensity matrix approach and the iterative method. Assuming an incident

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light E(t) = E˜ exp(−iωt) + E˜ exp(iωt) is applied to the system. The evolution of the one-electron density matrix ρ is given by the time-depending

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Schr¨odinger equation[37,38,39] ∂ρij 1 = [H0 − qzE(t), ρ]ij − Γij (ρ − ρ(0) )ij , ∂t i¯h

(16)

where H0 is the Hamiltonian for this system without the incident field E(t), q is the electronic charge, ρ(0) is the unperturbed density matrix and Γij is the relaxation rate. For simplicity, we will assume that Γij = Γ0 = 1/T0 for i 6= j. Eq.(16) is solved using the usual iterative method[40,41], ρt =

X k

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ρ(k) (t),

(17)

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with =

1 1 (k+1) [H0 − ρ(k+1) ]ij − i¯hΓij ρij − [qz, ρ(k) ]ij E(t). i¯h i¯h

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(k+1)

∂ρij ∂t

(18)

˜ can be expressed as The electric polarization of the QDM depended on E(t)

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(17). We only considering the first two orders, i.e.

˜ iωt + χ(2) ˜ 2 2iωt ) + c.c. + ε0 χ(2) ˜2 P (t) = ε0 (χ(1) 2ω E e 0 E , ω Ee (2)

(2)

(19)

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where χ(1) ω , χ2ω and χ0 denote the linear, second-harmonic generation and optical rectification susceptibility, respectively.

By using the same method as[40] and [41], the SHG coefficient expression in the near-resonant regime is given as: (2)

q 3 ρs M01 M02 M12 , ε0 (¯hω − E10 − i¯hΓ0 )(2¯hω − E20 − i¯hΓ0 )

(20)

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χ2ω =

where Eij = Ei − Ej is the transition energy between the ith and the jth subbands. Mij (i, j = 0, 1, 2) is dipole matrix element, h ¯ ω is the incident photon energy and ρs is the electron density of system. M01 M02 M12 is the

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product of matrix elements and also means geometric factor.

3

Results and Discussions

In this paper, numerical calculations are carried out for GaAs QDM.

In the calculations, we have used the following parameters: m∗e = 0.067m0 , m∗h = 0.09m0 , ε0 = 8.85 × 10−12 F m−1 , Γ0 = 5 × 1012 Hz, ρ = 5 × 1024 m−3 , Te ∼ 0.01 − 0.1meV and Te ∼ 1 − 10meV for weak[42] and strong[43] tunneling regime, respectively. (2)

In Fig.2, we show the SHG χ2ω as a function of the incident photon 7

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energy h ¯ ω for three different values of the first QWQD’s well width L=3nm,

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4nm, 5nm with the second QWQD’s well width R=1nm. In Fig.3, we show

the geometric factor M01 M02 M12 as a function of well width L, also with

R=1nm. From both Fig.2 and Fig.3, a very important feature is that as the

width of the two wells turns more asymmetric, the sharpness and the peak

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intensity of the SHG turn larger. This illustrates that the SHG does derive from the system symmetry broken.

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From Eqs.(3-7,19), we obtain a function between Rabi energy h ¯ Ω and the geometric factor M01 M02 M12 which is illustrated in Fig.4. From Fig.4, we can observe that the curve is not monotonous with h ¯ Ω. What we can see in Fig.4 is that the value of geometric factor M01 M02 M12 rises rapidly between h ¯ Ω = 0.05meV and h ¯ Ω = 0.25meV . On the contrary, when h ¯ Ω is

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larger than 0.25meV, the value of geometric factor M01 M02 M12 has hardly changed. It’s readily seen that the SHG is influenced strongly by h ¯ Ω when the value of h ¯ Ω is changing from 0.05meV to 0.25meV . When the value of h ¯ Ω is larger than 0.25meV, the value of the SHG can be treated as a constant.

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The physical reason for the phenomena can be well explained. Because the Rabi frequency Ω reflects the interaction strength between the QDM and the

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incident electromagnetic field, the larger the Ω, the greater the probability for electrons could be excited to upper state. It means that the SHG will get larger with the rising value of h ¯ Ω. However, when Ω becomes a comparatively large number, the difference of different state’s probability which the electron will stay turns small while the system is becoming steady. So, the geometric factor M01 M02 M12 and the SHG can be treated as a constant. Fig.5 shows the geometric factor M01 M02 M12 as a function of the tunneling strength Te with h ¯ Ω = 0.05meV and h ¯ Ω = 10meV , respectively. As we 8

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can see in Figs.5(a) and 5(b), whatever Te belongs to weak or strong tunneling

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regime, the value of the geometric factor M01 M02 M12 decreases simultane-

ously with the increasing value of Te . At the meantime, Fig.6 shows the same

result that the peak of the SHG decreases with the increasing value of Te . The physical reason for the phenomena can be explained as follows. Accord-

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ing to the previous work, we known that the exciton effect can enhance the SHG[44]. However, with the enhancement of Te , the excited electron which

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in first dot can easily tunnel to the second dot. At the same time, the state of the system will change from direct-exciton state |1i to indirect-exciton state |2i. Therefore, the influence of exciton effect will become weak. (2)

Fig.7 shows that the SHG χ2ω as a function of δ2 with δ1 = 0 and incident photon energy h ¯ ω = 20meV . We can observe that the curve of Fig.7 is

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decreasing monotonously from δ2 = −0.7meV to δ2 = 0.7meV . The physical reason can be explained as follows. When δ2 < 0meV , the energy level of |2i is higher than |1i so that the electron-tunneling effect is hard to happen. On the contrary, the electron-tunneling effect can happen easily when

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δ2 ≥ 0meV . Therefore, the SHG will decrease with the increasing value of δ2 . At the meanwhile, it is more effectively illustrates that the SHG is influenced

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strongly by the tunneling effect.

4

Conclusion

In this paper, we have investigated the tunneling effect on the secondharmonic generation in quantum dot molecule by using the compact-density matrix approach within the effective-mass approximation and the iterative method. Based on the numerical compute, we have studied not only the in9

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fluence by the tunneling strength but also by the well width and Rabi energy

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on the second-harmonic generation. Our result shows that the theoretical (2)

value of χ2ω can reach the magnitude of 10−4m/V . But with the tunnel-

ing effect turns stronger, the peak decreases quickly. At last, we hope that these important conclusions can make a great contribution to the experimen-

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tal studies, and may open new opportunities for optical exploitation of the

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quantum-size effect in devices.

Acknowledgments: This Work is supported by the National Natural Science Foundation of China (under Grant Nos. 61178003, 61475039), Guangdong Provincial Department of Science and Technology (under Grant Nos. 2012A080304010, S2012010010115, 2012A080304005) and Guangzhou

References

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Figure captions Fig. 1. Schematic band structure and level configuration of QDM.(a)Without a gate voltage, the tunneling is weak.(b)With applied a gate voltage, the

of energy levels on Hamiltonian. (2)

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conduction-band levels are resonance. The tunneling become stronger.(c)Scheme

Fig. 2. The second-harmonic generation χ2ω as a function of the incident

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photon energy h ¯ ω, with R = 1nm, Te = 1meV , h ¯ Ω = 10meV , for three different values of L, L = 3, 4, 5nm, respectively.

Fig. 3. The geometric factor M01 M02 M12 as a function of L with R = 1nm, Te = 1meV , h ¯ Ω = 10meV , respectively.

Fig. 4. The geometric factor M01 M02 M12 as a function of h ¯ Ω with L = 3nm, R = 1nm, Te = 0.01meV , respectively.

5(a). The geometric factor M01 M02 M12 as a function of Te with

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Fig.

L = 3nm, R = 1nm, h ¯ Ω = 0.05meV , respectively. Fig.

5(b).

The geometric factor M01 M02 M12 as a function of Te with

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L = 3nm, R = 1nm, h ¯ Ω = 10meV , respectively. (2)

Fig. 6. The second-harmonic generation χ2ω as a function of the incident

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photon energy h ¯ ω, with L = 3nm, R = 1nm, h ¯ Ω = 10meV , for two different values of Te , Te = 0.05meV and Te = 10meV , respectively. (2)

Fig. 7. The SHG χ2ω as a function of δ2 , with δ1 = 0, h ¯ ω = 20meV , L = 3nm, respectively.

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Fig. 1(a)

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Fig. 1(b)

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Fig. 1(c)

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140 130

L=3nm

120

L=4nm

110

L=5nm

90 80

-7

(10 m/V)

100

70

TE D

2

(2)

60 50 40 30 20 10

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0

12

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10

14

16

18

20

(meV)

Fig. 2

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22

24

26

28

30

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13 12 R = 1nm

10 9 8 7 6

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The geometric factor M

01

M

02

M

12

(10

-27

3

m )

11

5 4 3 2 1

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0

1

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0

2

3

4

5 L (nm)

Fig. 3

19

6

7

8

9

10

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86

84 83 82 81 80

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The geometric factor M

01

M

02

M

12

(10

-29

3

m )

85

79 78 77 76

Te

0.01meV

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75

0.1

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0.0

0.2

0.3

0.4

0.5

0.6

(meV)

Fig. 4

20

0.7

0.8

0.9

1.0

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80 75

65

=0.05meV

60 55 50 45 40 35 30 25 20 15 10 5

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The geometric factor M

01

M

02

M

12

(10

-29

3

m )

70

EP

0

0.02

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0.01

0.03

0.04

0.05

0.06

Te (meV)

Fig. 5(a)

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0.07

0.08

0.09

0.10

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85

75 70

=10meV

65 60 55 50 45 40

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The geometric factor M

01

M

02

M

12

(10

-29

3

m )

80

35 30 25 20 15

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10

2

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1

3

4

5

6

Te (meV)

Fig. 5(b)

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7

8

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140 130 120

Te=1meV Te=5meV

110

Te=10meV

90 80 70 60

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2

(2)

-7

(10 m/V)

100

50 40 30 20 10

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0

10

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5

15

20

25

(meV)

Fig. 6

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30

35

40

45

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28.00

27.75

27.50

27.00

-7

(10 m/V)

27.25

26.50

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2

(2)

26.75

26.25

26.00

25.75

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25.50

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-0.6

-0.4

-0.2

0.0

meV

Fig. 7

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0.2

0.4

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ACCEPTED MANUSCRIPT 1. A GaAs asymmetric quantum dot molecule has been proposed. 2. The second-harmonic generation coefficient is obtained by using the compact-density matrix approach and the iterative method. 3. The second-harmonic generation is strongly affected by tunneling strength Te,

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Rabi frequency Ω and the size of the quantum dot.