1. Introduction

In the last two decades, much attentions have been paid to the nonlinear optical properties of quantum dots (QDs) because of its significant quantum confinement effect, compared with quantum well and quantum wire [1,2]. New low-dimensional semiconductor nanoscopic can produce novel nonlinear optical properties, which are conducive to the development of semiconductor optoelectronics devices.

The optical absorption coefficients (OAC) and refractive index changes (RIC) effect of QDs play an important role in nonlinear optical properties, especially in presence of noise. There are many experimental and theoretical works focusing on the optical properties of QDs. For instance, I. Karabulut et al. studied the OAC and RIC in spherical quantum dots: effects of impurities, electric field, size, and optical intensity [3]. M. Sahin et al. researched the nonlinear optical properties of a donor impurity in a spherical quantum dot with the electric field and different confining potential [4,5]. These researches show that the geometrical shape of QDs and the strength of external electric field make a great difference to the optical properties. Nevertheless, the electronic state and the orientation of the electric field are not analyzed clearly in this process. E. Kasapoglu et al. surveyed the OAC and RIC in triangular quantum dots considering the strength and orientation of applied electric field [6]. However, the surveys mentioned above only considered the case of regular and single quantum dots. In fact, the shape of quantum dots will deviate from the regular shape in many cases during the growth process, and there will be a variety of the same or different shape of quantum dots stacking distribution in optical materials. In both cases, the restricted potential energy of the electron is greatly deviated.

In this paper, we introduce a two-dimensional Y-shaped quantum dot system with adjustable shape in order to simulate the stacking effect caused by multiple triangular quantum dots and research the influence of the change in constrained potential energy of electron on optical properties. We study the electronic state, optical absorption coefficients, and refractive index changes with the applied electric field. The outline of this paper is as follows. In Section 2, a model for Y-shaped QDs is given and the analytical forms of the optical absorption coefficients (OAC) and refractive index changes is obtained. In Section 3, the numerical results and discussions are presented. In Section 4, a brief conclusion is acquired.

2. Theory

We discuss the system in which electrons assumed to be moving on the x-y plane are much more confined in the z-direction than in the x and y. The confining potential under an electric field is given as [7,8],

 

Fig. 1. ($a$) Three-dimensional view and contour lines of the confining potentials $V$ when $F=0$. ($b$) The contour lines of the potentials $V$ under the electric field at different direction angle $\theta$ with $\theta =0^{\circ }, 60^{\circ }, 90^{\circ },120^{\circ }$ and $F=90$ kV/cm.

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Under the effective mass approximation, the Hamiltonian, for such a system, can be expressed as [7,8]:

The energy eigenvalues and eigenfunctions of this system may be calculated by solving the Schrödinger equation corresponding to Eq. (2) with a 2D finite difference method [9].

In our work, by means of the compact density matrix approach and the iterative method, the linear and third-order nonlinear OAC are respectively expressed as [10]:

The total OAC for a vertically polarized incident light is given by [10]:

We can also obtain the linear and the third-order nonlinear RIC of the system [10,11],

The total RIC for a vertically polarized ncident light can be writed as [10,11]

In the above equation, $\sigma _\upsilon$ denotes the electron density, $\mu$ is the permeability, $n_r$ expresses the refractive index, $\varepsilon _R$ is the real part of the permittivity, $\omega$ signifies the incident photon frequency, and $I=2\varepsilon _0n_rc|\widetilde {E}|^2$ is the incident optical intensity. $M_{21}= <\psi _2|e\overrightarrow {r}|\psi _1>$, $M_{11}= <\psi _1|e\overrightarrow {r}|\psi _1>$ and $M_{22}= <\psi _2|e\overrightarrow {r}|\psi _2>$ are dipole matrix elements, $E_{21}=\hbar \omega _{21} = E_2-E_1$.

3. Results and discussions

In this section, the OAC and RIC are numerically calculated for the typical GaAs/AlAs semiconductor. The parameters used in our calculations are as follows [12]: $m^{1}=0.067m_0$ (where $m_0$ is the mass of free electron), $\sigma _\Omega =5.0\times 10^{22}$ ${\rm m}^{-3}$, $n_r$=3.2, $\varepsilon _{0}=8.85\times 10^{-12}$ Fm$^{-1}$, $\Gamma _{0}=1/0.14$ ps and $\hbar \omega _{0}=80$ meV.

3.1 Influences of QDs with an electric field $F$ on the electronic state

Fig. 2 shows the square modes of the electron ground state $\psi _1$ and the first excited state wave function $\psi _2$ with three values of $\beta$ and $\theta$ for $F=90$ kV/cm. We find that the angle $\theta$ mainly affects the distribution of the first excited state, but has little influence on the ground state for triangular quantum dots ($\beta =1$) . The distribution of the ground state and the first excited state is more affected by the angle $\theta$ for Y-shaped ($\beta =3$) and trilobal QD ($\beta =5$), and is compressed towards the angular orientation. The dipole matrix elements $M_{21}$, which reflects the mixing degree of ground state and excited state, as function of the angle $\theta$ is drawn in Fig. 3(a), with different values of $F$, $F=0, 30, 60, 90$ kV/cm, and $\beta$, $\beta =1, 3, 5$. For three kinds of quantum dots, Fig. 3(a) shows that $M_{21}$ increases with the increasing of $F$ in the case of a small angle $\theta$, but presents the opposite change in a larger angle case. $M_{21}$ will also decrease when the angle $\theta$ enhances from 0$^{\circ }$. In general, for a fixed strength and angle $\theta$ of electric field $F$, $M_{21}$ increases as $\beta$ enhances (the increasing asymmetry).

 

Fig. 2. The square modes of the electron ground state $\psi _1$ and the first excited state wave function $\psi _2$ with three values of $\beta$ and $\theta$ for $F=90$ kV/cm, respectively.

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Fig. 3. ($a$) The dipole matrix elements $M_{21}$ as function of the angle $\theta$. ($b$) The difference of subband energies $E_{21}$ as function of the angle $\theta$.

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To further research the changes in the energy levels of the system, the difference of subband energies as function of the angle $\theta$ are portrayed in Fig. 3(b). It can be seen that the energy intervals $E_{21}$ decrease with the increasing of $\beta$. Moreover, $E_{21}$ enlarges when $\theta$ increases for $\beta =1$ and $\beta =3$, but presents periodic change under a high electric field intensity with $\beta =5$.

3.2 Influences of QDs with an electric field $F$ on the optical absorption coefficients

In Fig. 4($a_{1}$-$c_2$), we plot the total absorption coefficients $\alpha (\omega ,I)$ as a function of incident photon energy. From these pictures, we find some interesting phenomena. Firstly, we find that the resonant peak of $\alpha (\omega ,I)$ decreases with the increasing strength and angle $\theta$ of electric field $F$ in case of $\theta =0$ and $F=90$ kV/cm respectively, but the $\alpha ^{(1)}(\omega )$ and $\alpha ^{(3)}(\omega ,I)$ enhance in this process which is quite different from previous studies [3–6]. Secondly, the larger the value of $\beta$, the more obviously the resonant peak of $\alpha (\omega ,I)$ is affected by the electric field. The reasons for these features can be described as follows. The total $\alpha (\omega ,I)$ are due to the linear term $\alpha ^{(1)}(\omega )$ and the third-order nonlinear term $\alpha ^{(3)}(\omega ,I)$, which are proportional to $|M_{21}|^2$ and $|M_{21}|^4$, respectively. Meantime, $M_{21}$ increases with the increasing $F$ and $\theta$, as shown in Fig. 3(a). The nonlinear term makes larger contribution to the total coefficient $\alpha (\omega ,I)$, which causes $\alpha (\omega ,I)$ to go down. The physical reason is that the overlap of different electron states enlarges when the asymmetry of QDs enlarges, so that $M_{21}$ becoming large. Thirdly, the resonant peak of $\alpha (\omega ,I)$ exhibits a red shift in Fig. 4($a_1$, $b_1$, $c_1$). The reason for this trait is attributed to the fact the energy intervals decline with the increasing $F$ in case of $\theta =0$. The physical reason for this feature is that as the strength of electric field increases associated with shape factor, the quantum confinement of the electron becomes weak, which results in the decrease of the energy internal $E_{21}$.

 

Fig. 4. The total absorption coefficients $\alpha (\omega , I)$ as a function of incident photon energy with $\beta =1, 3, 5$: $\theta =0$ for $(a_1,\, b_1,\, c_1)$; $I$=0.4 ${\rm MW}/{\rm{cm}}^{2}$ and $F=90$ kV/cm for $(a_2,\, b_2,\, c_2)$.

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Finally, it can be clearly seen that high light intensity will lead to the bleaching effect, while the increase of quantum dot asymmetry ($\beta$ increases ) will accelerate the bleaching effect, and the saturation light intensity will decrease, when comparing Fig. 4 ($a_1$, $b_1$, $c_1$). For instance, we can obviously see the bleaching effect when $I$ is equal to 0.6 ${\rm MW}/{\rm{cm}}^2$ ($\beta$=5), 0.7 ${\rm MW}/cm^{2}$ ($\beta$=3) and 0.8 ${\rm MW}/cm^{2}$ ($\beta$=1), respectively. Besides, the stronger the bleaching effect is, the more prominently the resonant peak is split up into two peaks. The reason is that when $I$ is very strong or $\beta$ enhances, the total optical absorption coefficients mainly comes from the nonlinear term.

3.3 Influences of QDs with an electric field $F$ on the refractive index changes

Fig. 5($a_1$-$c_2$) show total refractive index changes $\Delta n/n_{r}$ as a function of incident photon energy with $I=0.4\ {\rm MW}/cm^2$. From these diagrams, we can clearly find that there are some similar characteristics between the OAC and RIC under the same average confining energy condition. First, the resonant peaks of $\Delta n/n_{r}$ decrease with the increasing angle $\theta$ for $F=90$ kV/cm, and move toward the lower incident photon energy region with the increasing electric field $F$ in case of $\theta =0^{\circ }$. The degree of red shift increases with the enhancement of $\beta$. The physical origin of these features is same to the explanation for that of the OAC. Second, we also can find that the resonant peaks of $\Delta n/n_{r}$ increase obviously with the enhancement of $F$ for $\theta =0^{\circ }$, which is clearly different from that of OAC. The reason lies in the fact that although the linear $\Delta n^{(1)}/n_{r}$ and nonlinear $\Delta n^{(3)}/n_{r}$ all augment as $M_{21}$ increases in this process, the linear term makes larger contribution to the total RIC.

 

Fig. 5. Total refractive index changes $\Delta n/n_{r}$ as a function of incident photon energy with $\beta =1, 3, 5$ and $I$=0.4 ${\rm MW}/cm^{2}$: $\theta =0$ for $(a_1,\, b_1,\, c_1)$; $F=90$ kV/cm for $(a_2,\, b_2,\, c_2)$.

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However, by comparing the optical characteristics of three kinds of quantum dots, we also find two important features. One is that the optical properties of triangular quantum dots are less affected by electric field (strength or angle). On the other hand, as the angle $\theta$ increases, the corresponding frequency shift of the peak positions of $\alpha (\omega ,I)$ and $\Delta n/n_{r}$ is periodic and this phenomenon is more obvious for the trilobal QD, which can be explained by Fig. 3(b).

4. Conclusion

In this paper, we study the electronic state, OAC and RIC in regulable Y-shaped QDs with the applied electric field. Our results show that the peak positions of OAC and RIC will present a red shift whether the applied electric field or the asymmetry of QDs increases. Moreover, in the full range of direction angles of electric field, higher optical coefficients can be obtained when the angle is zero. More importantly, the resonant peak of OAC under an electric field declines with the increasing the electric field and asymmetry of QDs, but those for RIC is opposite. Therefore, the magnitude or angle of external electric field and the shape of QDs have a great influence on the OAC and RIC. The study of the shape effect of QDs with the external factors on the nonlinear optical properties plays an important role in semiconductor physics because they can simulate the real situation, and greatly modulate and optimize the performance of low-dimensional quantum devices.