OPTICALLY DETECTED ELECTROPHONON RESONANCE AND LINEWIDTHS IN TRIANGULAR QUANTUM WELLS

. In the present paper, we study the linear optical absorption power in triangular quantum wells subjected to a laser field when electrons are scattered with longitudinal optical phonons (LO phonons). The analytic expressions are obtained for optical absorption power via electron-LO phonon scattering. The linear optically detected electron-phonon resonance (ODEPR) effect in a specific GaAs/AlAs quantum well with a triangular potential is investigated. The conditions for the ODEPR are determined on the basis of the energy conservation law. From the curves expressing the dependence of the absorption power on the photon energy, we find the ODEPR-linewidths as the profiles of the curves. The computational results show that the ODEPR-linewidths increase with temperature and decrease with the electric field.


Introduction
Optically detected electron-phonon resonances (ODEPR) and half linewidths are well-known as an effective tool for investigating the scattering mechanisms of carriers and can be used to study electron-phonon scattering processes. There have been many studies on this phenomenon in low dimensional semiconductors with different types of confined potentials [1][2][3][4][5]. However, these works mostly have dealt with quantum wells with the square shapes. Recent advances in molecular-beam epitaxy have enabled us to fabricate several quantum wells (QW) with nonsquare shapes, which are more sensitive to external fields compared with the square shape wells [6]. Non-square QW shapes can have potential profiles with steps, triangular, trapezoidal, V-groove, and hyperbolic forms.
It is well known that the quantum confinement of electrons in the triangular quantum well is much stronger than that in the square quantum well with the same width. Therefore, some novel optical properties can be expected in such a quantum system. Motivated by this idea, we consider the linear optically detected electrophonon resonance in a quantum well with 120 a triangular potential. The dependence of the ODEPR peak linewidths on the electric field and the temperature of the system is clearly displayed.
The paper is organized as follows: the model and theoretical framework are described in section 2; the results and discussions are presented in section 3; and the conclusions are given in section 4.

Triangular quantum well model and analytical results
We consider a triangular quantum well structure where electrons move freely in the  () xy plane and are confined in the z-direction with the following potential [7]   with frequency  and amplitude F is applied along the z-direction, the analytical expression of the linear absorption power due to photon absorption, accompanied by an absorption and/or emission of the phonon is given by [9,10] jos.hueuni.edu.vn Vol. 127, No. 1A, 2018 In Eq. (5), E is the energy separation between the two states,  , ff are the distribution functions of the electron in the initial and final states, respectively, is the linewidth function and can be expressed as follows [11] 2 0 2 1 1 1 m is the matrix elements of the electron-phonon interaction, which depends on the scattering mechanism; q is the coupling factor, where 0 is the permittivity of the free space; where  LO is the longitudinal phonon frequency. These analytical results can be verified by numerical computation and graphical plotting using Mathematica software.

Numerical results and discussion
In this section, we use the Mathematica software to numerically calculate the optical absorption  Figure 1 shows the dependence of the linear absorption power on the photon energy.
From the figure, we can see three resonant peaks, the appearance of which can be explained as follows:  Using the profile method, we obtain the dependence of the ODEPR linewidths on electric field F as shown in Fig. 4 and temperature T as shown in Fig. 5. It can be seen from Fig. 4 that the ODEPR-linewidths decrease with electric field F. This is probably because when the electric field increases, the confinement of the electron decreases, so that the probability of electronphonon scattering decreases, and therefore do the ODEPR linewidths. Concerning the temperature dependence of linewidths, the ODMPR linewidths increase with temperature because the probability of electron LO-phonon scattering rises (Fig. 5).

Conclusions
We have theoretically investigated the linear optical absorption power and linewidths due to confined electrons interacting with LO-phonons in a QW with the triangular potential subjected to a laser field. We have found that the absorption power magnitude and linewidths depend significantly on the change of the electric field. The resonant peaks of the absorption power shift towards higher photon energies when the electric field increases, while the linewidths decrease with the electric field. Furthermore, our results also reveal that with the increase of the temperature, the absorption power magnitude and linewidths increase, but the resonant peaks are stable with the variation of temperature.