On the optical Stark effect of excitons in InGaAs prolate ellipsoidal quantum dots

In this paper, we study the exciton absorption spectra in InGaAs prolate ellipsoidal quantum dots when a strong pump laser resonant with electron quantized levels is active. Our obtained results by renormalized wavefunction theory show that, under suitable conditions, the initial exciton absorption peak is split into two new peaks as the evidence of the existence of the three-level optical Stark effect of excitons. We have suggested an explanation of the origin of the effect as well as investigating the effect of pump field energy, size, and geometric shape of the quantum dots on effect characteristics. The comparison with the results obtained in the spherical quantum dots implies the important role of geometric shape of the quantum structures when we examine this effect.


Introduction
One of the low dimensional structures recently attracting much interest from researchers is the semiconductor quantum dots [1][2][3][4], which are structures that confine carriers in all three spatial dimensions. Quantum dot structures have many interesting applications, such as producing artificial atoms and molecules [5], single-electron transistors [6], and quantum dot lasers [7]. In recent years, there has seen many studies on optical properties in quantum dots with different shapes, for example cylindrical, cubic, and spherical quantum dots [8][9][10][11][12]. These studies show that optical properties of quantum dots are highly dependent on external fields and size of quantum dots. It is worth mentioning that the shape of quantum dots also makes a huge difference to the optical properties of quantum dots [13]. Therefore, quantum dots are expected to create even more breakthrough applications in the future.
The optical Stark effect of excitons is one of the unique optical properties of bulk materials in general, and low dimensional structures in particular. The optical Stark effect of excitons occurs due to the coupling of the two exciton states under the excitation of near resonant beam [14]. Scientists divided this effect into two types as follows. First, two-level optical Stark effect arises when a pump laser of strong intensity couples exciton ground state and an exciton excited state in the quantum system [14,15]. Second, three-level optical Stark effect is the result of a couple of two excited states of exciton under the control of a pump laser beam of lower intensity [16,17]. The latter has received more attention from researchers because they are more likely to occur and have better potential application. In general, optical Stark effect has potential applications in manufacturing ultrafast optical switching of future optical devices [11,18], optical modulators [19], mesoporous hybrid multifunctional system [20], and optically controlled field-effect transistors [21]. The optical Stark effect of excitons in quantum dots has also been of interest in both experimental and theoretical research [11,[22][23][24], but mainly for spherical quantum dots. The investigation of this effect in a more complicated and more practical shaped quantum dot structures, such as ellipsoidal quantum dots, has not been carried out in detail.
In this paper, we study theoretically the three-level optical Stark effect of excitons in InGaAs prolate ellipsoidal quantum dots. We applied the renormalized wavefunction method to investigate the dependence of the exciton absorption spectra on the external fields, size, and shape of quantum dot, when there exists the optical Stark effect of excitons. In addition, the effects of the pump laser energy, size, and geometric shape of quantum dots on the behavior of the effect are also clarified. The article includes the following main sections: part 2 presents model and theory, part 3 presents the main results and discussion, and conclusions are presented in part 4.

Wavefunctions and energy levels of electron and hole in prolate ellipsoidal quantum dots
In this section, we present the wavefunctions and energy levels of electron and hole in prolate ellipsoidal quantum dots. We consider a prolate ellipsoidal quantum dot with rotational symmetry around the z axis, with a and c are its semi-axes along the xOy plane and stay in the z-direction, respectively; in which x, y, z are the coordinates in Cartesian coordinate system with its origin at the ellipsoidal symmetry center (Fig. 1). Simply put, we assume that prolate ellipsoidal quantum dot is set under the effect of an infnite potential of the form [25][26][27][28][29] where   i S r  depends on parameters a and c which are semi-axes of the ellipsoidal quantum dot, as follows with c a  , and the surface of the prolate ellipsoidal quantum dot has the shape as in Fig. 1. To conveniently examine the prolate ellipsoidal quantum dots, in this study, we use the prolate spheroidal coordinates In the Eqs. (13) and (14), the indices n, l, and m are principle, orbital, and azimuthal quantum numbers, respectively. However, for ellipsoidal quantum dots, since its spherical symmetry no longer exists, so the index l in the wavefunction and energy expressions of the particle in equations (4) and (8) no longer represent the orbital quantum numbers. However, in this paper, we still use the set of indices n, l, m in Eqs. (4) and (8) is the radius of a sphere with the same volume.
In the effective mass approximation and envelope-function theory, the total wavefunction of electrons (holes) in a prolate ellipsoidal quantum dot with an infinite potential has the following form where   , ,  are the periodic Bloch functions in conduction and valence band.
As we set the zero energy at the top of the valence band, the expression of the quantized energy levels of electron and hole is then determined as follows where Eg is the bandgap of the semiconductor.

Intersubband optical transition
In this paper, we study the three-level optical Stark effect of excitons in prolate ellipsoidal quantum dots. Thus, we need to examine a three  When the electromagnetic field is not so strong, we can omit higher-order term and using some gauges and approximations, the electron-electromagnetic field interaction Hamiltonian then can be written as follows [31,33] int 0ˆx in which we set where q , 0 m , and p  are the bare mass, the charge, and the momentum of the particle, respectively; x A and x  are respectively magnitude and frequency of the laser, with x clarifying if that is a pump or probe laser; and n  is the unit vector pointing to the wave propagation direction.
The matrix element for an intersubband transition between two electron quantized levels defined as follows in which where p A and p  are magnitude and frequency of the pump laser. Equation (23) From Eq. (4), we have Choosing polarization vector along the z axis, while replacing Eq. (25) into Eq. (24), and performing some additional calculations, we have the expression of 21 V as follows *  4  21  110 100  110  100  0   1  1 *  1 *  1  1  2  2  10  10  00  00  1 1   2   , , , , From there, we have the expression of matrix element for an intersubband transition between two electron quantized levels as follows Under the effects of probe laser, matrix element for an interband transition between two quantized levels of electron and hole is 10 int 10 0 where t A and t  are magnitude and frequency of the probe laser, respectively; cv p is the polarization matrix element between conduction and valence band, and has the form of The absorption spectra of the excitons is defined using the expression of the transition rates (or the absorption probabilities in a unit of time). According to Fermi's golden rule, the expression of transition rates is defined as [34]   where fi T is the matrix element for interband transition between initial i and final f states; i E and f E are the corresponding energy levels of initial i and final f states.
From Eqs. (30) and (32), we get the expression of transition rates as follows Applying 'Lorentz line' function [35 ]   Eq. (33) can be rewritten as follows where 100 100 and  is the phenomenological linewidth of absorption peak.

b) With the presence of the pump laser
Under the effect of a strong pump laser resonating with the energy distance between the two quantized levels of the electron, the wavefunctions of electron are renormalized by the pump laser effect and can be described as where coefficients   l c t ( 0,1 l  ) are determined through the time-dependent Schrodinger equation and have the following form [12]   where 21 110 100 where 21 V is the quantity given in Eq. (26).
Therefore, the energy spectrum of electron corresponding to wavefunction given in Eq. (42) In order to defind the three-level optical Stark effect of exciton in prolate ellipsoidal quantum dots, the pump laser intensity must be significantly stronger than the probe laser. At the same time, the detuning of the pump laser to the electron levels must be much smaller than the frequency of the pump laser and band gap of active material in quantum dots The matrix element for the interband transition between the hole state and the electron superposition state specified by the renormalized wavefunction under the effect of the probe laser when the system is irradiated by a strong resonant pump laser is defined as follows (48) Next, we perform the similar calculation as in the previous section for the pump laser inactivity, the expression of transition rate for interband transition between the hole state and the electron superposition state when the system is irradiated by a resonant pump laser has following formula Applying 'Lorentz line' function, the approximated form of the transition rate with pump laser effect can be rewritten as

Results and discussion
To clarify the results obtained above, in this section, we perform the calculation for transition rate in In0.53Ga0.47As/In0.52Al0.48As prolate ellipsoidal quantum dots in two cases: meV. In addition, since this paper examines prolate ellipsoidal quantum dot structure. This is a low dimensional structure, so the length of the major axis 2c and minor axis 2a must be selected to be smaller the bulk exciton Bohr radius in the dot material In0.53Ga0.47As. Thus, we have selected the length of the semi-major axis as 25 a  Å to do the calculation in this section; while the length of the semi-minor axis will depend on the selected value of  ( c a    ).
In Figure 3, we depict the dependence of transition rate on the photon energy of the probe laser in a prolate ellipsoidal quantum dot with the value of the ellipsoidal aspect ratio of 3   in two conditions: without the pump laser effect (dashed line) and with the pump laser effect (solid line). From the figure we can see that, without the pump laser, the absorption spectra of excitons includes only one absorption peak. However, when the system is irradiated by a strong pump laser exactly resonant with the energy distance between the two quantized levels of the electron, the graph appeared two distinct equal peaks in the exciton absorption spectra. These two vertices are bilaterally symmetrical with respect to the original peak when the pump laser does not operate. Moreover, we found that the exciton absorption intensity is drastically reduced in the presence of a laser field. This reduction is the consequence of the conservation of the electronic transition rate as discussed in Ref. [11]. In short, the above results demonstrate the existence of the three-level optical Stark effect of exciton in the prolate ellipsoidal quantum dots. In addition, comparing the results found in the prolate ellipsoidal and spherical quantum dots [11], we find that they are similar. The reason is given to the fact that both prolate ellipsoidal and spherical quantum dots are quasi-zero-dimensional systems.  will also change, the more the detuning increases, the higher the height of one peak increases while the height of the other peak decreases. As the detuning increases, one absorption peak tends to move towards the original peak (which is the peak in the inactivity of the pump laser), the other is far from the position of the original peak, and this is clearly shown in Fig. 6. From   Fig. 6, we also find that the more the detuning, the more redshifted two peaks. In addition, the detuning of the pump laser also strongly influences the transition rate of two absorption peaks as shown in Fig. 7.   From Fig. 7, we see that when the pump laser detuning is increased, the transition rate of the high-energy absorption peak increases and approaches to the transition rate of original peak when the pump laser was not active, while the transition rate of low-energy absorption peak decreases and approaching to zero. At the same time, the transition rate of these two absorption peaks depends monotonously on the pump laser detuning.   , the height of the low-energy absorption peak is still clearly observed; but with 1.8   , the height of the low-energy absorption peak is very small. As the value of  decreases, the low-energy absorption peak will almost disappear, and the high-energy absorption peak will reach the original peak. In addition, the location of these two exciton absorption peaks is also strongly dependent on the amplitude of pump laser as shown in the Fig. 9. In addition, in the case 0     as presented in Figs. 5 and 8, after being split (and decreasing peak intensity), the second peak intensity increases with the detuning and tends to return as in the case of no pump laser. This phenomenon can be explained as follows. When the detuning increases, the pump laser is hard to make an intraband transition between two electron levels (indicated by thick dashed arrow in Fig. 4), that means it is hard to connect those two electron levels to become one big degeneracy electron level. Hence, under effect of strong electric field of the pump laser, the symmetry of the initial electron levels (as well as the symmetry of this big degeneracy electron level) is hard to be broken and, hence, the initial electron levels are hard to be split into sub levels. This is the reason why we see one large splitting peak near the original peak (in the case of no pump laser), while another splitting peak nearly disappears. Therefore, when we probe the exciton spectrum, the second peak intensity increases with the detuning and tends to return as in the case of no pump laser and if the detuning is large enough, we will see only one peak in the spectrum. meV. From Fig. 9, we find that when increasing the amplitude of the pump laser, the distance between the two exciton absorption peaks also increases, but these two peaks are located symmetrically on both sides of the original peak according to energy conservation law.
In other words, the optical Stark effect can be controlled using the strong pump laser. we can see two exciton absorption peaks, which again confirm the existence of the excitonic optical Stark effect in this structure. Besides, when  was increased, both absorption peaks moved quickly to the lower energy region, corresponding to a red shift in the absorption spectrum. Therefore, changing the optical properties of prolate ellipsoidal quantum dot structures will become easier to control and more flexible by changing only two parameters semi-minor axis and semi-major axis of this quantum dot structure. This is one of the advantages of ellipsoidal quantum dot structures compared to spherical quantum dots. To further investigate the properties of absorption spectra of interband transitions in prolate ellipsoidal quantum dots, we continue examining the shift of the photon energy of probe laser as a function of the ellipsoidal aspect ratio  in the case of the pump laser is not switched on ( Fig. 11). As shown in Fig. 11, we see a red shift in the absorption spectra when increasing the ellipsoidal aspect ratio  as mentioned in Figure 10. In particular, the strongest shift occurs in the range of 1.  Lastly, we compare the exciton absorption spectra in the prolate ellipsoidal quantum dot ( Fig. 12a) and the spherical quantum dot (Fig. 12b) [11] with the same volume.

Conclusion
To sum up, in this study, we have investigated the characteristics of the exciton absorption spectra in the InGaAs/InAlAs prolate ellipsoidal quantum dot through calculating the transition rate of the interband transition using the renormalized wavefunction theory. The results show that when the two quantized energy levels of electrons are coupled by a polarized pump field, one initial absorption peak of exciton is split into distinct absorption peaks as an evidence of the existence of the three-level optical Stark effect. This is the consequence of the splitting of the electron quantized levels due to the effect of the strong resonant pump laser. We also find analytical expressions for the electron splitting levels and attempt to explain the mechanism of effect generation. Another important result is that the exciton absorption spectra are not only strongly affected by the energy of the pump laser but also depend strongly on geometric shapes of quantum dots. Specifically, with the same volume, the exciton absorption spectra in the spherical quantum dots and the prolate ellipsoidal quantum dots are completely different.
Furthermore, when we increase the ellipsoidal aspect ratio of the prolate ellipsoidal quantum dots, a clear red shift in the optical absorption spectra was observed. The existence of two parameters of semi-minor axis and semi-major axis in the prolate ellipsoidal quantum dot structures makes the ability to control optical properties of the ellipsoidal quantum dots easier and more flexible than the spherical quantum dots. This is one of the advantages of the ellipsoidal quantum dot structures compared to the spherical quantum dots. We believe that the interesting features in the optical absorption spectra in the prolate ellipsoidal quantum dots when the optical Stark effect occurs have the great potential application to the development of computing devices and quantum information.