The Impurity and Decay-Magnetic Polaron Effects in III–V Compound Gaussian Quantum Wells

Abstract

The effects of a decay magnetic field and hydrogen-like impurities on the ground-state binding energy (GSBE) and ground-state energy (GSE) of weak-coupling bound polarons in asymmetrical Gaussian potential (AGP) III–V compound quantum wells (QWs) were studied based on unitary transformation methods and linear combination operators. By numerical calculation, we found that the polarons were affected by the AGP, the decay magnetic field, Coulomb impurities, and the type of crystal, which led to a series of interesting phenomena, such as changes in the ground-state energy and the ground-state binding energy. The results obtained provide good theoretical guidance for optoelectronic devices and quantum information.

1. Introduction

In low-dimensional semiconductors, especially in quantum dots [1,2], quantum rods [3], quantum wires [4,5], and quantum wells (QWs) [6,7], there are obvious electron–phonon interactions, which produce quasi-particles called polarons. In recent years, since polarons have critical influence on the photoelectric characteristics of materials, research on polarons has become increasingly popular among researchers. Therefore, many researchers have conducted theoretical and experimental studies on the polaron effect in QWs [8]. In theoretical studies, researchers have presented a variety of methods, including effective mass approximation [9], perturbation theory [10], the variational method [11], the iterative method [12], and the compact-density-matrix approach [13], to investigate polarons in QWs. In experimental studies, researchers have applied several methods to investigate polarons in QWs. For example, Frankerl et al. [14] performed low-temperature time-resolved photoluminescence (PL) experiments and found that strong carrier decay time variations with detection photon energy prove a significant effect of carrier localization, depending primarily on the width of the QW. Gusev et al. [15] observed both topological insulator and gapless semimetal phases, based on theoretical predictions and depending on sample parameters, through investigating nonlocal and local resistance in HgTe-based double QWs. Donmez et al. [16] performed magnetotransport measurements to evaluate the influences of thermal annealing and Bi on the electronic transport characteristics of n-type modulation-doped AlGaAs/GaAsBi QWs. These research works evaluated the effects of different parameters, such as the magnetic field [17], electric field [18], hydrogen-like impurity field [19], temperature [20], and laser field [21], and found that only one or two of them had important influences on polarons in QWs.

Polaron properties are greatly affected by both a non-uniform magnetic field and hydrogen-like impurities. In fact, this is a common situation in the experiments and processing of QW devices. Therefore, studying polaron non-uniform magnetic fields and hydrogen-like impurity effects on the photoelectric properties of QWs is of great importance. However, no research has been conducted on hydrogen-like impurity effects on weak-coupling bound-magneto-polaron properties in AGCPQWs using unitary transformation and linear combination operator methods.

We investigated the AGP range and hydrogen-like impurity effects on the ground-state energy (GSE) and ground-state binding energy (GSBE) of weak-coupling bound-magneto-polarons in AGP III–V compound QWs (see Figure 1). It was found that the self-parameters of QWs, coupling-parameters, hydrogen-like impurities, and decay magnetic field are vital for polaron effects in nanostructures.

2. Theoretical Method

In this section, we considered an electron confined in an AGCPQW that moves in a GaAs semiconductor crystal and interacts with bulk longitudinal optical (LO) phonons in a decay magnetic field. According to effective mass approximations, the electron–phonon system Hamiltonian in the presence of hydrogen-like impurities at the coordinate origin could be expressed as [19]

$$\begin{array}{l}H=\frac{1}{2m}{\left(p_x-\frac{{\overline{\beta }}^2}{4}y\right)}^2+\frac{1}{2m}{\left(p_y+\frac{{\overline{\beta }}^2}{4}x\right)}^2+\frac{p_z^2}{2m}+{\displaystyle \sum _{\mathbf{q}}\hslash {\omega }_{LO}{a}_{\mathbf{q}}^{{†}_{}}}{a}_{\mathbf{q}}\\ +{\displaystyle \sum _{\mathbf{q}}\left[V_q{a}_{\mathbf{q}}\exp \left(i\mathbf{q}\cdot \mathbf{r}\right)+h.c\right]}+V\left(z\right)-\frac{\eta }{r},\end{array}$$

(1)

$$V\left(z\right)=\{\begin{array}{cc}-V_0\exp \left(-\frac{z^2}{2R^2}\right)\hfill & \hfill z\ge 0\\ \infty \hfill & \hfill z<0\end{array}$$

(2)

$${\overline{\beta }}^2=2eB/c$$

(3)

$$\begin{array}{l}{V}_q=i\left(\frac{\hslash {\omega }_{LO}}{q}\right){\left(\frac{\hslash }{2m{\omega }_{LO}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\left(\frac{4\pi \alpha }{V}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\\ \alpha =\left(\frac{e^2}{2\hslash {\omega }_{LO}}\right){\left(\frac{2m{\omega }_{LO}}{\hslash }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(\frac{1}{{\epsilon }_{\infty }}-\frac{1}{{\epsilon }_0}\right),\end{array}$$

(4)

where $\beta$ is the parameter related to the magnetic field, $m$ is the band mass of the electron, and $a_{\mathbf{q}}^{†}\left(a_{\mathbf{q}}\right)$ is the creation (annihilation) operator of LO phonons with frequency ${\omega }_{LO}$. $p^{}$ and $r^{{}^{}}$ are momentum and position vectors for electrons, respectively. $V\left(z\right)$ is the AGCP along the z direction representing the AGCPQW growth direction [22,23,24], $V_0$ is the AGCPQW barrier height, $R$ is the AGCP range, and $\eta$ is the Coulombic impurity potential strength that satisfies $\eta =e^2/{\epsilon }_0$. In addition, the decay magnetic field shown in Figure 2 can be expressed as

$$B=\frac{B_0^{}\exp \left(-\omega t\right)}{C},$$

(5)

where $B_0$ is the initial magnetic induction intensity; $\omega$ and $t$ are the decay frequency and the decay time of the magnetic field, respectively; and $C$ is a dimensionless parameter whose default value is chosen as $C=0.05$.

Employing Fourier expansion to the Coulomb-bound potential, it can be written as

$$-\frac{e^2}{{\epsilon }_{0}r}=-\frac{4\pi e^2}{{\epsilon }_0\nu }{\displaystyle \sum _{\mathbf{q}}\frac{1}{q^2}}\exp \left(-i\mathbf{q}·\mathbf{r}\right)$$

(6)

To compute the Hamiltonian, it needs to be quantized. Therefore, Equation (1) was carried out by two unitary transformations [25,26]:

$$U_1=\exp \left(-i{\displaystyle \sum _{\mathbf{q}}\hslash \mathbf{q}\cdot \mathbf{r}{a}_{\mathbf{q}}^{†}{a}_{\mathbf{q}}}\right),$$

(7)

$$U_2=\exp \left({\displaystyle \sum _{\mathbf{q}}\left(a_{\mathbf{q}}^{†}{f}_q-a_{\mathbf{q}}{f}_q^{\ast }\right)}\right),$$

(8)

where $f_q\left(f_q^{\ast }\right)$ is a variational function.

The following linear combination operator was introduced:

$$\begin{array}{l}{p}_j={\left[\frac{m\hslash \lambda }{2}\right]}^{\frac{1}{2}}\left(b_j+b_j^{†}\right),\\ r_j=i{\left[\frac{\hslash }{2m\lambda }\right]}^{\frac{1}{2}}\left(b_j-b_j^{†}\right),\end{array}$$

(9)

with $\lambda$ being the variational parameter. The system ground-state wave function was written as

$$|{\psi }_0\rangle =|0{\rangle }_a|0{\rangle }_b,$$

(10)

where ${|0\rangle }_b$ and ${|0\rangle }_a$ are the vacuum state and unperturbed zero-phonon state of the b operator, respectively. The expected value of Equation (1) with respect to $|{\psi }_0\rangle$ was written as

$$F_0\left(\lambda ,f_q\right)=\langle {\psi }_0|U_2^{-1}{U}_1^{-1}H{U}_1{U}_2|{\psi }_0\rangle$$

(11)

Performing the variation of $F_0(\lambda ,f_q)$ with respect to $\lambda$ gave

$${\lambda }^2-\frac{4\eta }{3\hslash }\sqrt{\frac{m}{\pi \hslash }}{\lambda }^{\frac{3}{2}}-\frac{V_0}{3m{R}^2}-\frac{\hslash e^2{B}_0^2\exp \left(-2\omega t\right)}{12m^2\lambda C^2}=0$$

(12)

where $\lambda$ is the weak-coupling bound-polaron vibrational frequency in AGCPQWs. The GSBE and GSE of the bound polaron (GSEPB) were stated as

$$E_0=\frac{3}{4}\hslash \lambda -\alpha \hslash {\omega }_{LO}-V_0+\frac{\hslash V_0}{4m\lambda R^2}+\frac{\hslash e^2{B}_0^2\exp (-2\omega t)}{16m^2\lambda C^2}-2\eta \sqrt{\frac{m\lambda }{\pi \hslash }}$$

(13)

$$E_b=2\alpha \hslash {\omega }_{LO}+V_0-\frac{\hslash V_0}{4m\lambda R^2}-\frac{\hslash e^2{B}_0^2\exp (-2\omega t)}{16m^2\lambda C^2}+2\eta \sqrt{\frac{m\lambda }{\pi \hslash }}$$

(14)

3. Numerical Results and Discussion

To clearly determine the influences of the AGCP range, hydrogen-like impurities, and decay magnetic field on the GSBE and GSEPB in AGPQWs, numerical calculations for GaAs, InAs, and InSb semiconductor AGPQWs were performed. The parameters applied for calculations in experiments are shown in

Table 1. Parameters related to GaAs, InAs, and InSb crystals.

Crystal	$\mathit{\alpha }$	${\mathit{\omega }}_{\mathit{L}\mathit{O}}$	$\mathit{m}/{\mathit{m}}_{\mathit{e}}$	$\mathit{m}$
InSb	0.022	5.72 × 1013 Hz	0.0138	1.26 × 10−32 kg
InAs	0.052	4.55 × 1013 Hz	0.0230	2.10 × 10−32 kg
GaAs	0.068	5.50 × 1013 Hz	0.0657	4.28 × 10−32 kg

[27].

To better discuss the aforementioned issue, different GaAs, InAs, and InSb crystal materials are presented in Figure 3, where the following physical quantities were selected: $V_0=3\mathrm{m}\mathrm{e}\mathrm{V}$, $R=1\mathrm{n}\mathrm{m}$, $\omega =0.5\mathrm{H}\mathrm{z}$, $T=0.5\mathrm{s}$, and $B_0=4\mathrm{T}$. To compare the variations of the polaron ground-state energy in different crystals due to the Coulomb field of impurities, polaron ground-state energies of GaAs, InAs, and InSb crystal materials without impurities were calculated to be 56.233, 145.862, and 234.741 meV, respectively. Next, the energy variation $\Delta E_0$ of the ground state of polarons in different materials was calculated as the Coulomb impurity potential strength changed. We found that by increasing the Coulomb potential, the variations of the ground-state energy $\Delta E_0$ of polarons in different materials decreased. From the order of the coupling constants summarized in the table, it was seen that an increase in the coupling constant increased the polaron ground-state energy, indicating that stronger coupling between electrons and phonons results in a larger coupling constant in the crystal and in turn stronger binding of electrons and lower ground-state energy.

Similarly, in Figure 4, the GSBE change rule in Gaussian potential QWs of different GaAs, InAs, and InSb crystal materials varied with the Coulomb potential. On the one hand, this indicated that higher crystal material coupling constants result in stronger electron binding, higher polaron formation ground-state energy (i.e., the rule presented in Figure 3), and larger binding energy. On the other hand, by increasing the Coulomb potential, impurities greatly affect polarons. Therefore, it was equivalent to an extra electron-binding potential, which increased the polaron GSBE.

To clearly describe the effects of the potential and width of the self-parameter potential well in weakly coupled materials on the GSBE and the ground-state energy of polarons, we simulated the polaron-forming bound-magneto-polaron process under different magnetic fields, with $V_0=3\mathrm{m}\mathrm{e}\mathrm{V}$, $\omega =0.5\mathrm{H}\mathrm{z}$, $T=0.5\mathrm{s}$, $B_0=4\mathrm{T}$, and impurity Coulomb field $\eta =30\mathrm{m}\mathrm{e}\mathrm{V}\cdot \mathrm{n}\mathrm{m}$. As shown in the Figure 5, It was found that the bound-magneto-polaron ground-state energy increased by increasing the bound potential and decreased by increasing the well width. The larger the AGCP range is, the weaker the confinement ability of the electron, and the weaker the coupling between the electron and the phonon. As the well height of the AGCP increased, the confinement strength of the electrons increased in the growth direction of the QW, facilitating polaron formation. The change law of binding energy followed the opposite pattern. This was consistent with Figure 3.

As can be seen from Figure 6, this bound magneto-polaron was affected by the changing magnetic field, which changed the GSBE and ground-state energy of the polaron. To show this change, we also adopted a GaAs crystal material. Under QW parameters of $V_0=3\mathrm{m}\mathrm{e}\mathrm{V}$, $R=1\mathrm{n}\mathrm{m}$, and Coulomb impurity field $\eta =30\mathrm{m}\mathrm{e}\mathrm{V}·\mathrm{n}\mathrm{m}$, we calculated the change rule of the bound magneto-polaron with related magnetic field parameters. The ground-state energy increased by increasing the initial magnetic field, and the negative value of the binding energy was consistent with that of the ground-state energy.

4. Conclusions

We studied the effects of the AGP range and hydrogen-like impurities on the GSBE, ground-state energy, and vibrational frequency of weak-coupling bound polarons in AGP III–V compound QWs based on unitary transformation and linear combination operator methods. The following conclusions can be drawn from numerical simulations: (1) The Coulomb impurity in the material enhances the polaron effect and then reduces the ground-state energy of the polaron. (2) The influence of Coulomb impurities on the polaron effect in different materials is obviously different, which may be caused by the different electronic–phonon coupling constant of materials. The larger the electronic–phonon coupling constant of materials, the more obvious the influence of Coulomb impurities. (3) The change in the confinement potentials has a significant influence on the polaron effect. With the increase in the confinement potentials, the electron–phonon interaction strength changes, leading to a change in the polaron energy. (4) The intensity of electron motion is enhanced by the influence of the magnetic field, resulting in an increase in the polaron energy. However, since the magnetic induction intensity of the decay field gradually decreases, the polaron energy in the decay field decreases and becomes more stable. The obtained results revealed that these factors have great effects on bound-polaron properties in AGP III–V compound QWs, which can guide us to recognize, understand, and apply the polaron effect of AGP III–V compound QWs.