Magnetic and Temperature Effects on Optical Quantum Transition Line Properties in Electron-Piezoelectric Phonon Coupled Materials Under Square Well Confinement Potential

Abstract

Despite extensive research on semiconductor materials, the influence of temperature and magnetic field on the optical quantum transitions within semiconductors remains insufficiently understood. We therefore investigated the Optical Quantum Transition Line Properties (OQTLP), including line shapes (LS) and line widths (LW), as functions of temperature and magnetic field in electron–piezoelectric-phonon-interacting systems within semiconductor materials. A theoretical framework incorporating projection-based equations and equilibrium average projection was applied to GaAs and CdS. Similarly, LW generally increases with magnetic field in a square-well confinement potential across most temperature regions. However, in high magnetic fields at low temperatures, LW decreases for GaAs. Additionally, LW increases with rising temperature. We also compare the LW and LS for transitions within intra- and inter-Landau levels to analyze the quantum transition process. The results indicate that intra-Landau level transitions primarily dominate the temperature dependence of quantum transitions in GaAs and CdS.

1. Introduction

The growing interest in semiconductors has recently led to various studies aimed at improving their material properties. In addition to conventional Si and Ge semiconductors, active research is being conducted on III–V and II–VI compound semiconductors due to their superior optical and electronic properties. Among them, GaAs and CdS have attracted increasing attention because of their unique piezoelectric properties and strong optical responses under external fields. GaAs compound semiconductors exhibit a high optical conversion efficiency of over 40%, more than twice that of Si semiconductors. Additionally, CdS is widely known as a representative material for optical sensors [1,2].

A quantum well is a structure where electron movement is confined to a single dimension, leading to quantum confinement. This confinement alters the energy states, changing them from a continuous range to discrete levels due to the limiting potential. These structures provide a distinctive opportunity to adjust both the quantum well energy levels and the carrier density by carefully managing the potential at the electrodes placed along the outer boundary [3]. The study of electron dynamics coupled with piezoelectric phonons in semiconductors has drawn significant attention from condensed matter physicists. Wurtzite (ZnO, CdS, CdSe, AgI) and sphalerite (ZnS, ZnSe, GaAs) crystals are recognized for their piezoelectric activity. Their superior acoustoelectric and optically conductive properties make them suitable for applications such as ultrasound amplifiers, luminescent materials, and photometric devices [4]. Various approaches exist to derive the scattering factors for the electron-background particle correlation response function [5,6,7,8,9,10,11,12,13,14,15,16,17,18], including the use of the projected Liouville equation method.

This study aims to understand the electron behavior phenomenon by analyzing the energy absorption rate according to the magnetic field dependence and temperature dependence of GaAs and CdS through theoretical interpretations of the electron-piezoelectric phonon correlation system [19,20]. Understanding low-dimensional electronic systems is essential, with cyclotron transitions within quasi-two-dimensional (2D) quantum wells being a key focus of recent theoretical research [21].

Quantum transport theories, developed using the projected Liouville equation method, provide an effective framework for examining scattering mechanisms in solids. This study proposes a new quantum transport theory in linear–nonlinear form, employing the projected Liouville equation method alongside the equilibrium average projection scheme [12,22]. The equilibrium average projection scheme simplifies the process of deriving generalized susceptibility and scattering mechanisms through a unified theoretical extension.

Among the diverse methods for analyzing optical conductivity related to piezoelectric properties, photon quantum transition absorption analysis is conducted in this study. The optical quantum transition line properties (OQTLP), including line shapes (LS) and line width (LW), effectively correspond to the scattering mechanism influencing carrier behavior.

In this paper, we integrate the propagator with the conventional series expansion representation into quantum transport theory. We conduct more rigorous commutator calculations than those in previous studies [22,23]. The primary distinction between this work and previous works lies in the markedly different distribution function components of the LW. Using theoretical analysis, we explore the magnetic dependence of absorbed energy (P(B)) across temperature variations in GaAs and CdS, as well as the magnetic dependence of the LW under specific external fields. To investigate the OQTLP, we compare the temperature dependence of the LW in intra-Landau- and inter-Landau-level transitions. Similarly, we compare the magnetic dependence of the LW in both intra-Landau- and inter-Landau-level transitions. In addition, we compare the magnetic dependence of the LS of intra-Landau-level transitions with that of the LS of inter-Landau-level transitions.

The novelty of this study lies in three key aspects: (1) The application of a combined theoretical framework (Projected Liouville Equation Method and Equilibrium Average Projection Scheme) to analyze OQTLP with a rigorous quantum transport approach. (2) A comparative study of intra- and inter-Landau level transitions, which has not been systematically explored in previous works. (3) A wide-range analysis of temperature and magnetic field effects, moving beyond the scope of traditional fixed-condition analyses. By integrating these elements, this study provides a deeper understanding of the quantum transition mechanisms governing optical absorption and scattering processes in GaAs and CdS, thereby contributing to the broader field of semiconductor physics.

2. Linear Response Theory in Existing Systems

A system is considered to be influenced by an oscillating electric field given by $F(t)=F_u{e}^{-i{\omega }_{u}t}{\overrightarrow{e}}_u$, where ${\omega }_u$ denotes the angular frequency and ${\overrightarrow{e}}_u$ is a unit vector aligned with the direction of the field ($u=x,y,z,$ etc.). The Hamiltonian $H(t)$ and the associated Liouville operator $L(t)$ are defined in Equations (1) and (2).

$$H(t)=H_s+H_u^{\prime }(t)=H_s+r_{u}F(t)=H_s+r_u{F}_u{e}^{-i{\omega }_{u}t}$$

(1)

$$\begin{array}{l}L(t)X=[L_s+L_u^{\prime }(t)]X\\ =[L_s+L_u^{\prime }]XF(t)\equiv [(H_s+H_u^{\prime }),X]F(t)\end{array}$$

(2)

where $H_s$ includes the Hamiltonian describing primary particles (e.g., electrons), background particles (e.g., impurities or phonons), and the interactions between them. $H_s$ and $L_s$ are not time-dependent components. The time-dependent Hamiltonian is formulated as $H_u(t)=r_u{F}_u{e}^{-i{\omega }_{u}t}$. The operator $L_u$ is linked to the response operator $r_u$ aligned with the direction $u$ and indicates that $L_u^{\prime }X\equiv [r_u,X]$ applies to any arbitrary operator $X$. The expectation value of the system’s dynamic variable $r_k(t)$ can be expressed as $r_k(t)=Tr\{r_k\rho (t)\}$. The projection operator is introduced to derive $r_k(t)$, leading to the practical response equation $P_{k}X\equiv B_{k}Tr\left\{r_{k}X\right\}$. The $r_k$ is the time-independent operator associated with $r_k(t)$. Then, $r_k(t)=(1/B_k)(P_k\rho (t))$ aligned with the projection operator, $B_k\equiv L_u^{\prime }{\rho }_s/{\mathsf{\Lambda }}_{ku}$, where ${\mathsf{\Lambda }}_{ku}$ represents the normalization factor (${\mathsf{\Lambda }}_{ku}\equiv Tr\left\{r_k{L}_u^{\prime }{\rho }_s\right\}$). The initial condition of the Quantum Statistical Mechanical system is assumed to be the equilibrium density, as expressed in Equation (3).

$${\rho }_s\equiv \rho (0)\equiv \exp (-\beta H_s)/Tr\exp (-\beta H_s)$$

(3)

This equilibrium density is determined by the system’s equilibrium energy. Additionally, ${\rho }^{\prime }$ represents the time-evolution density induced by the external field. The system’s density matrix is then expressed as $\rho (t)={\rho }_s+{\rho }^{\prime }(t)$. By a similar procedure to the equilibrium average projection scheme [23], the kinetic equations for $r_k(t)$ can be derived within the imposed external field conditions. This time-dependent function can be analyzed in the energy space ($\omega$-space) by applying the Fourier–Laplace transform to $X(t)$, as follows:

$$X\left({\omega }_u\right)\equiv Fourrier-LaplaceTransform\left[X\left(t\right)\right]\equiv {\displaystyle {\int }_0^{\infty }\exp (-i{\omega }_{u}t)X\left(t\right)dt}$$

(4)

The linear response in $\omega$-space is then obtained as follows:

$$r_k({\omega }_u)={\displaystyle \frac{-(i/\hslash ){\mathsf{\Lambda }}_{ku}{F}_u({\omega }_u)}{{\omega }_u-A_{ku}+{\mathsf{\Xi }}_{ku}({\omega }_u)}}$$

(5)

where $A_{ku}$ is the normalization factor expressed as $(-i/\hslash {\mathsf{\Lambda }}_{ku})Tr\left\{r_k{L}_s{L}_u^{\prime }{\rho }_s\right\}$) and ${\mathsf{\Xi }}_{ku}({\omega }_u)$ represents the scattering factor described as follows:

$${\mathsf{\Xi }}_{ku}({\omega }_u)\equiv {\displaystyle \frac{i}{\hslash {\mathsf{\Lambda }}_{ku}}}Tr\{r_k{L}_s{G}_k^q({\omega }_u)Q_k{L}_s{L}_u^{\prime }{\rho }_s\}$$

(6)

where $Q_k\equiv 1-P_k$ and $G_k^q({\omega }_u)$ is the propagator, as given in Equation (7).

$$G_k^q({\omega }_u)\equiv {\displaystyle \frac{1}{\hslash \omega -Q_k{L}_s}}$$

(7)

Equation (5) can be employed to analyze various systems in condensed matter physics. The response formula $r_k({\omega }_u)$ applies to the optical quantum transition system (OQTS). The LSs and LWs of optical quantum transitions can be obtained by considering a many-body system affected by a circularly polarized external field $E_{+}(t)=E_0\exp (+i\omega t)$. The system’s total Hamiltonian is obtained as $H(t)=H_s+H^{\prime }(t)=H_s+(-i/\omega )J^{+}{E}_{+}(t)$ by using the Coulomb gauge $\overrightarrow{E}(t)=-\partial \overrightarrow{A}(t)/\partial t$. The operator $J^{\pm }$ w for many-electron current consists of two components $J^{+}={\displaystyle \sum _{\beta }{j}_{\beta }^{+}{a}_{\beta +1}^{+}{a}_{\beta }}$ and $J^{-}={\displaystyle \sum _{\alpha }{(j_{\alpha }^{+})}^{*}{a}_{\alpha }^{+}{a}_{\alpha +1}}$ with $j^{\pm }=j_x\pm i{j}_y$ representing the two constituent parts of the single-electron current operator $j$. Considering the electron–phonon interaction system, the Hamiltonian is formulated as follows:

$$\begin{array}{l}{H}_s=H_e+H_P+V\\ ={\displaystyle \sum _{\beta }〈\beta \left|h_0\right|\beta 〉}{a}_{\beta }^{+}{a}_{\beta }+{\displaystyle \sum _q\hslash {\omega }_q}{b}_q^{+}{b}_q+{\displaystyle \sum _q{\displaystyle \sum _{\alpha ,\mu }{C}_{\alpha ,\mu }\left(q\right)}{a}_q^{+}{a}_{\mu }}\left(b_q+b_{-q}^{+}\right)\end{array}$$

(8)

where $H_e$, $H_P$, $V$, $a_1(a_2^{+})$, $b_1(b_2^{+})$, and $\overrightarrow{q}$ correspond to the electrons Hamiltonian, the phonon Hamiltonian, the electron–phonon interaction Hamiltonian, the annihilation operator, and the wave vector of phonon (or impurity), respectively. $C_{\alpha ,\mu }\left(q\right)$ represents the coupling matrix element of electron–phonon interaction and is expressed as $C_{\alpha ,\mu }\left(q\right)\equiv V_q<\alpha |\exp (i\overrightarrow{q}\cdot \overrightarrow{r})|\mu >$, where $V_q$, $|\alpha >$, $\overrightarrow{r}$ are the coupling coefficient of materials, the eigenstate of a single electron, and the position vector of the electron, respectively.

Applying a time-invariant magnetic field $\overrightarrow{B}={\displaystyle B_z\overrightarrow{z}}$ to a many electron system, the single-electron energy states are quantized into Landau levels. A system in which electrons are confined to infinite square well potentials in the z-direction, between $z=0$ and $z=L_z$, is also considered. A single Hamiltonian $h_e$ is defined by using Landau gauge $\overrightarrow{A}\equiv (0,B_x,0)$ as follows:

$$h_e=-{\displaystyle \frac{{\hslash }^2}{2m}}\{{\partial }_z^2+{\partial }_y^2+{\partial }_x^2+2i\left({\displaystyle \frac{eB}{\hslash }}\right)x\partial y-{\left({\displaystyle \frac{eB}{\hslash }}\right)}^2{x}^2\}+U(z)$$

(9)

We obtain the eigenstate in Q3D system as Equation (10).

$${\psi }_{N_{\alpha },n_{\alpha },k_y,k_z}(x,y,z)\equiv |\alpha >={\tilde{C}}_G{\tilde{\varphi }}_{kr}^{(plw)}(k){\tilde{\phi }}_{N_{\alpha }}(x_1){\tilde{\mathsf{\Phi }}}_{n_{\alpha }}^{(cfn)}(z)$$

(10)

where ${\tilde{\phi }}_{kr}^{(plw)}$ represents the plane wave (${\tilde{\varphi }}_{kr}^{(plw)}\equiv \exp (i{k}_{y}y)$). The explicit function ${\tilde{\varphi }}_{N_{\alpha }}(x_1)$ and the confined wave function ${\tilde{\mathsf{\Phi }}}_{n_{\alpha }}^{(cfn)}(z)$ are given as follows:

$${\tilde{\phi }}_{N_{\alpha }}(x_1)={\tilde{N}}_r\exp (-{\displaystyle \frac{x_1^2}{2l_0^2}})H_{N_{\alpha }}({\displaystyle \frac{x_1}{l_0}})$$

(11)

Here, $H_{N_{\alpha }}(x)$ is the Hermite polynomials function, $l_0=\sqrt{\hslash /eB}$ is the radius of cyclotron motion.

$${\tilde{\mathsf{\Phi }}}_{n_{\alpha }}^{(cfn)}(z)=\left(\begin{array}{l}{\displaystyle \frac{1}{\sqrt{\frac{z_0}{2}+\frac{1}{{\kappa }_{n_{\alpha }}}}}}\sin (k_{n_{\alpha }}z)\hspace{1em}\hspace{1em}\hspace{1em}(0\le z\le z_0)\\ {\displaystyle \frac{1}{\sqrt{\frac{z_0}{2}+\frac{1}{{\kappa }_{n_{\alpha }}}}}}\exp (-{\kappa }_{n_{\alpha }}(z-z_0))\hspace{1em}(z_0\le z)\end{array}\right.$$

(12)

Here, the quantization conditions ${\kappa }_{n_{\alpha }}$ and $k_{n_{\alpha }}$, corresponding to the z-direction components of the electron wave vectors $\kappa$ and $k$. These are derived by solving the formulas $\kappa =-k\cot k{z}_0$ and $\kappa +k=2m_e^{*}{U}_0/\hslash$, under the conditions $\kappa >0$ and $k>0$, where $U_0$ is the constant potential for $0<z<z_0$. The normalization factors are given by ${\tilde{C}}_G\equiv 1/\sqrt{L_y}$, ${\tilde{N}}_r\equiv 1/\sqrt{(\sqrt{\pi }{2}^{N}N!l_0)}$. These values vary in different systems. The corresponding eigenvalue is expressed as follows:

$${\epsilon }_{N_{\alpha },n_{\alpha },k_{y\alpha ,}{k}_{z\alpha }}=(N_{\alpha }+{\displaystyle \frac{1}{2}})\hslash {\omega }_0+n_{\alpha }^2{\displaystyle \frac{{\hslash }^2{\pi }^2}{2m_e^{*}{L}_{z(sys)}^2}}(N_{\alpha }=0,1,2,3,\dots n_{\alpha }=0,1,2,3,\dots )$$

(13)

where ${\omega }_0$, $m_e^{*}$, $L_{z(sys)}$ represent the angular frequency of cyclotron motion (${\omega }_0=eB/m_e^{*}$), the effective mass of electron, and the size of materials in the z-direction, respectively. The induced current operator led by the circularly polarized field and the time-independent operator linked to the expectation of dynamic variable are, respectively, expressed as $J_{+}^{(R)}\equiv {\tilde{g}}_{(sys)}{\displaystyle \sum _{\beta }\sqrt{N_{\beta }}{a}_{\beta +1}^{+}{a}_{\beta }}$ and $J_{-}^{(L)}\equiv {\tilde{g}}_{(sys)}{\displaystyle \sum _{\beta }\sqrt{(N_{\beta }+1)}{a}_{\beta }^{+}{a}_{\beta +1}}$, where ${\tilde{g}}_{(sys)}$ is given by $(-ie\hslash /m_e^{*})\sqrt{1/l_0^2}$. $J_{+}^{(R)}$ and $J_{-}^{(L)}$ represent the right and left circularly polarization current, RCPC and LCPC, respectively.

The parameter $V_q$, describing the electron–phonon coupling in piezoelectric materials within the isotropic approach, is defined in Equation (14).

$$V{(q)}^2={\displaystyle \frac{{\overline{K}}^2\hslash {\upsilon }_s{e}^2}{2\chi {\epsilon }_{0}qV}}{\displaystyle \frac{1}{q}}$$

(14)

where $\overline{K}$ and $\chi$ are the electromechanical constant and the dielectric constant, respectively. The approximation of long wavelength (${\omega }_q\approx v_{s}q$) provides accurate results for these materials. Using Kubo’s approximation, phonon energy is given by $\hslash {\omega }_q\approx \hslash v_{s}q$, with $v_s$ representing the speed of sound in the material.

3. Optical Quantum Transition Line Properties

It is assumed that the electric field $E_{+}(t)=E_0\exp (+i\omega t)$ is circularly polarized and directed along with the z-axis. The delivered absorption power is given by $P(\omega )=(E_0^2/2)\mathrm{Re}\{\sigma (\omega )\}$. It includes the real part of the optical conductivity tensor $\mathrm{Re}\{\sigma (\omega )\}$. The absorption power and the scattering factor function represent the LS and LW of optical quantum transitions. To implement the linear response model to the OQTS, we replace $r_k$ to $J_k\equiv J_{+}^{(R)}$, $L_u^{\prime }X$ to $L_u^{\prime }X\equiv (i/\omega )[J_u,X]$, and $J_u\equiv J_{-}^{(L)}$ in a circulating external field with a frequency of $\omega$. From the response model, the ohmic current is defined in Equation (15).

$$J_k({\omega }_u)=\sigma (\omega )E_u({\omega }_u)=\left[{\displaystyle \frac{-(i/\hslash ){\mathsf{\Lambda }}_{ku}}{{\omega }_u-A_{ku}^{(R)}+{\mathsf{\Xi }}_{ku}({\omega }_u)}}\right]E_u({\omega }_u)$$

(15)

The series expansion form of the operator $G_k^q({\omega }_u)$, and the projection operator can be used for a simplified expression of the scattering factor. This derivation assumes a weak interaction approximation in pair-interacting systems. The scattering factor is then given by Equation (16).

$${\mathsf{\Xi }}_{ku}({\omega }_u)\equiv {\displaystyle \frac{i}{\hslash {\mathsf{\Lambda }}_{ku}}}<L_u^{\prime }{L}_v{G}_d{L}_v{J}_k>$$

(16)

where $G_d=1/(\hslash \omega -L_d)$ is the diagonal propagator. From refs [22,23], $T{r}^{(e)}\{J_k{L}_1{L}_2\cdots \cdots L_n{L}^{\prime }{\rho }_s\}={(-1)}^{n+1}\langle L^{\prime }{L}_n\cdots \cdots L_2{L}_1{J}_k\rangle$ is obtained. The expression $\langle \cdots \rangle$ represents the ensemble average of electron states. The interacting Hamiltonian commutator is calculated by performing a thorough calculation using a moderately weak coupling interaction, as below:

$$[a_v^{+}{a}_x(b_l+b_{-l}^{+}),a_{\mu }^{+}{a}_{\alpha +1}(b_q+b_{-q}^{+})]=a_v^{+}{a}_x{a}_{\mu }^{+}{a}_{\alpha +1}({\delta }_{-l,q}+{\delta }_{l,-q})+[a_v^{+}{a}_x,a_{\mu }^{+}{a}_{\alpha +1}](b_q{b}_{-l}^{+}+b_{-q}^{+}{b}_l)]$$

(17)

Equation (17) is applied to the electron–phonon interacting system to obtain its composing matrix elements. ${\mathsf{\Xi }}_{ku}({\omega }_u)$ is expressed as ${\mathsf{\Xi }}_{ku}({\omega }_u)\equiv i{\Delta }_{total}+{\gamma }_{total}\left(\omega \right)$. It consists of the real and imaginary parts, which represent the half-width and the line shift in the response type formula. The imaginary part ${\Delta }_{total}$ is negligible because it is very small in real systems. Using continuous approximation, the absorption power and the scattering factor function can be obtained as follows:

$$P\left(\omega \right)\propto \left({\displaystyle \frac{e^2{\omega }_o^2}{{\pi }^2\hslash \omega }}\right)\left[{\displaystyle \frac{{\gamma }_{total}\left(\omega \right){\displaystyle \sum _{N_{\alpha }}{\displaystyle {\int }_{-\infty }^{\infty }d{k}_{z\alpha }}\left(N_{\alpha }+1\right)\left(f_{\alpha }-f_{\alpha +1}\right)}}{{\left(\omega -{\omega }_o\right)}^2+{\left({\gamma }_{total}\left(\omega \right)\right)}^2}}\right]$$

(18)

$$\begin{gathered} {\gamma }_{total}\left(\omega \right)\equiv \mathrm{Re}{\mathsf{\Xi }}_{ku}^{(R)}({\omega }_u)\equiv {\displaystyle \sum _{\mp }{\displaystyle \sum _{N_{\alpha }=0}}{\displaystyle \sum _{N_{\beta }=0}}{\gamma }_{\alpha ,\beta }^{(R)\mp }} \\ \begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left({\displaystyle \frac{\mathsf{\Omega }}{4\pi {\hslash }^2{\upsilon }_s}}\right)\left({\displaystyle \frac{\pi }{L_z}}(2+\delta (n_{\alpha },n_{\beta }))\right){\displaystyle \sum _{\mp }{\displaystyle \sum _{N_{\alpha }=0}}{\displaystyle \sum _{N_{\beta }=0}{\displaystyle {\int }_{-\infty }^{\infty }d{k}_{z\alpha }}{\displaystyle {\int }_{-\infty }^{\infty }d{q}_z}{Y}_{\alpha ,\beta }^{(R)\mp }}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}/{\displaystyle \sum _{N_{\alpha }=0}^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }d{k}_{z\alpha }}}\left(N_{\alpha }+1\right)\left(f_{\alpha +1}-f_{\alpha }\right)\end{array} \end{gathered}$$

(19)

The integrand factor is expressed as Equation (20).

$$\begin{gathered} Y_{\alpha ,\beta }^{(R)\mp }\equiv A_{\beta ,\alpha +1}^{\mp }{(}_{\beta ,\alpha }^{\alpha ,\beta })\left(N_{\alpha }+1\right)\left(f_{\alpha +1}-f_{\alpha }\right)-B_{\beta ,\alpha +1}^{\mp }{(}_{\beta +1,\alpha +1}^{\alpha ,\beta })\sqrt{\left(N_{\alpha }+1\right)\left(N_{\beta }+1\right)}\left(f_{\beta +1}^{\pm }-f_{\beta }^{\pm }\right) \\ -C_{\alpha ,\beta +1}^{\mp }{(}_{\alpha +1,\beta +1}^{\beta ,\alpha })\sqrt{\left(N_{\alpha }+1\right)\left(N_{\beta }+1\right)}\left(f_{\beta +1}^{\mp }-f_{\beta }^{\mp }\right)+D_{\alpha ,\beta }^{\mp }{(}_{\alpha +1,\beta }^{\beta ,\alpha +1})\left(N_{\alpha }+1\right)\left(f_{\alpha +1}-f_{\alpha }\right) \end{gathered}$$

(20)

Each term of electron–phonon interacting systems is derived as follows:

$$\begin{gathered} A_{\beta ,\alpha }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })\equiv \left\{{\tilde{S}}_{\beta ,\alpha }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })(q_z,q_{\perp 1}^{\mp \beta \alpha })\left[{\tilde{N}}_q^{\pm }(q_z,q_{\perp 1}^{\mp \beta \alpha })\pm \left(1-f_{\beta }^{\pm }\right)\right]\right.\left.+{\tilde{S}}_{\beta ,\alpha }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })(q_z,q_{\perp 2}^{\mp \beta \alpha })\left[{\tilde{N}}_q^{\pm }(q_z,q_{\perp 2}^{\mp \beta \alpha })\pm \left(1-f_{\beta }^{\pm }\right)\right]\right\} \\ {B}_{\beta ,\alpha }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })\equiv \left\{{\tilde{S}}_{\beta ,\alpha }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })(q_z,q_{\perp 1}^{\mp \beta \alpha })\left[{\tilde{N}}_q^{\pm }(q_z,q_{\perp 1}^{\mp \beta \alpha })\mp f_{\alpha +1}\right]\right.+\left.{\tilde{S}}_{\beta ,\alpha }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })(q_z,q_{\perp 2}^{\mp \beta \alpha })\left[{\tilde{N}}_q^{\pm }(q_z,q_{\perp 2}^{\mp \beta \alpha })\mp f_{\alpha +1}\right]\right\} \end{gathered}$$

(21)

$$\begin{gathered} C_{\alpha ,\beta }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })\equiv \left\{{\tilde{S}}_{\beta ,\alpha }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })(q_z,q_{\perp 3}^{\mp \alpha \beta })\left[{\tilde{N}}_q^{\pm }(q_z,q_{\perp 3}^{\mp \alpha \beta })\pm \left(1-f_{\beta +1}^{\mp }\right)\right]\right.\left.+{\tilde{S}}_{\beta ,\alpha }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })(q_z,q_{\perp 4}^{\mp \alpha \beta })\left[{\tilde{N}}_q^{\pm }(q_z,q_{\perp 4}^{\mp \alpha \beta })\pm \left(1-f_{\beta +1}^{\mp }\right)\right]\right\} \\ {D}_{\alpha ,\beta }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })\equiv \left\{{\tilde{S}}_{\beta ,\alpha }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })(q_z,q_{\perp 3}^{\mp \alpha \beta })\left[{\tilde{N}}_q^{\pm }(q_z,q_{\perp 3}^{\mp \alpha \beta })\mp f_{\beta }^{\mp }\right]\right.\left.+{\tilde{S}}_{\beta ,\alpha }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })(q_z,q_{\perp 4}^{\mp \alpha \beta })\left[{\tilde{N}}_q^{\pm }(q_z,q_{\perp 4}^{\mp \alpha \beta })\mp f_{\beta }^{\mp }\right]\right\} \end{gathered}$$

where $\tilde{S}$ is the interacting matrix, expressed as follows:

$${\tilde{S}}_{\beta ,\alpha }^{\mp }{(}_{\mu ,\nu }^{\kappa ,\eta })(q_z,q_{\perp n}^{\mp \beta \alpha })\equiv V{(q_z,q_{\perp n}^{\mp \beta \alpha })}^2{\displaystyle \frac{\sqrt{{(q_{\perp n}^{\mp \beta \alpha })}^2+q_z^2}}{|\hslash v_s{q}_{\perp n}|}}{K}_{\mu ,\nu }^{\kappa ,\eta }({\displaystyle \frac{r_0^2}{2}}{(q_{\perp n}^{\mp \beta \alpha })}^2)$$

(22)

where $q_{\perp n}^{\mp \kappa \lambda }$ ($n=1,2,3,4$) is the vertical components of phonon wave vectors. It is derived as follows:

$$\begin{gathered} q_{\perp 1}^{\mp \beta \alpha }\equiv +\sqrt{{\left({\displaystyle \frac{\pm }{\hslash v_s}}\right)}^2{\left\{{\tilde{E}}_{\beta \alpha }^{(el)}-{\tilde{E}}_{\beta \alpha }^{(cfp)}-{\tilde{E}}_1^{(k\epsilon )}\right\}}^2-q_z^2}\equiv -q_{\perp 2}^{\mp \beta \alpha } \\ {q}_{\perp 2}^{\mp \beta \alpha }\equiv +\sqrt{{\left({\displaystyle \frac{\pm }{\hslash v_s}}\right)}^2{\left\{{\tilde{E}}_{\alpha \beta }^{(el)}-{\tilde{E}}_{\alpha \beta }^{(cfp)}+{\tilde{E}}_3^{(k\epsilon )}\right\}}^2-q_z^2}\equiv -q_{\perp 4}^{\mp \beta \alpha } \end{gathered}$$

(23)

where ${\tilde{E}}_{\beta \alpha }^{(el)}\equiv \hslash \omega -(N_{\beta }-N_{\alpha })\hslash {\omega }_0$, ${\tilde{E}}_{\beta \alpha }^{(cfp)}\equiv \left[{(N_{\beta }+1)}^2-{(N_{\alpha }+1)}^2\right]\hslash {\tilde{w}}_0$, ${\tilde{E}}_1^{(k\epsilon )}\equiv {\displaystyle \frac{{\hslash }^2}{2m^{*}}}(\pm 2k_{z_{\alpha }}{q}_z+q_z^2)$, ${\tilde{E}}_3^{(k\epsilon )}\equiv {\displaystyle \frac{{\hslash }^2}{2m^{*}}}(\mp 2k_{z_{\alpha }}{q}_z+q_z^2)$, and ${\tilde{w}}_0\equiv {\tilde{\epsilon }}_0^{(sys)}/\hslash \equiv {\displaystyle \frac{\hslash {\pi }^2}{2m^{*}{L}_{z(sys)}^2}}$ are provided.

The parameter $V_q$ with $q_{\perp n}^{\mp \kappa \lambda }$ ($n=1,2,3,4$) is then obtained as follows:

$$V{(q_z,q_{\perp n}^{\mp \kappa \lambda })}^2={\displaystyle \frac{{\overline{K}}^2\hslash {\upsilon }_s{e}^2}{2\chi {\epsilon }_{0}qV}}{\displaystyle \frac{1}{\sqrt{q_z^2+{(q_{\perp n}^{\mp \kappa \lambda })}^2}}}$$

(24)

From the Fermi–Dirac distribution functions, the eigenvalues are given by Equation (25).

$${\epsilon }_{\alpha }\equiv {\epsilon }_{N_{\alpha },n_{\alpha },k_{y\alpha },k_{z\alpha }}+({\epsilon }_c-{\epsilon }_F), {\epsilon }_{\beta }^{\pm }\equiv \left[{\epsilon }_{N_{\alpha },n_{\alpha },k_{y\alpha ,}{k}_{z\alpha }}+({\epsilon }_c-{\epsilon }_F)\right]\Delta (k_{y\beta },k_{y\alpha }\pm q_y)$$

(25)

where ${\epsilon }_c$, ${\epsilon }_F(T)$, and ${\epsilon }_g(T)$ represent the conduction band minimum energy, the Fermi energy at T, and the band gap energy at T, respectively. Their relations are expressed as ${\epsilon }_c-{\epsilon }_F(T)=0.5[{\epsilon }_g(T)-\kappa T/(T+\xi )-(3/4)K_{B}T\ln (\overline{m}/m^{*})]$, where $\kappa$ and $\xi$ are characteristic constants of the material, $m^{*}$ and $\overline{m}$ denote the effective mass of an electron and the density of state effective mass of a hole, respectively. The phonon distribution parts are given by ${\tilde{N}}_q^{-}\equiv <b_q^{+}{b}_l>=n_q{\delta }_{ql}$ and ${\tilde{N}}_q^{+}\equiv <b_l{b}_q^{+}>=(n_q+1){\delta }_{ql}$, where $n_q=\left\{{[e^{\epsilon (q,T)}-1]}^{-1}\right\}$ is the Bose–Einstein distribution function. The phonon energy is expressed as Equation (26).

$$\epsilon (q,T)=\left\{{\displaystyle \frac{\hslash {\omega }_q}{K_{B}T}}\right\}=\left\{{\displaystyle \frac{\hslash v_s}{K_{B}T}}\sqrt{{(q_{\perp n}^{\mp \kappa \lambda })}^2+q_z^2}\right\}$$

(26)

In the case of $N_{\alpha }\langle N_{\beta }$ and $N_{\kappa }\langle N_{\lambda }$, the K-matrix is given by Equation (27).

$$\begin{array}{l}{K}_{\chi ,\lambda }^{\alpha ,\beta }(t)\equiv \sqrt{N_{\alpha }!/N_{\beta }!}\sqrt{N_{\chi }!/N_{\lambda }!}{(\sqrt{t})}^{(N_{\beta }-N_{\alpha })}{(\sqrt{t})}^{(N_{\lambda }-N_{\chi })}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \exp (-t)L_{N_{\alpha }}^{N_{\beta }-N_{\alpha }}(t)L_{N_{\chi }}^{N_{\lambda }-N_{\chi }}(t)\end{array}$$

(27)

where $L_n^m(t)={(n!)}^{-1}\exp (t)t^{-m}(d^n/d{t}^n)[t^{n+m}\exp (-t)]$ is the Legendre function, where $t\equiv (l_0^2{(q_{\perp n}^{\mp \kappa \lambda })}^2)/2$.

The result of Equation (18) from Equations (19)–(23) are more rigorous calculations than most previous studies [11,12,13,14]. From the first to the fourth term of Equation (20), there are contained the distribution functions of significantly different forms compared to other theories [22,23]. The distribution part of this study explains the quantum transition process well. In addition, the equilibrium average projection scheme applied in this study supports to analysis of the quantum transition of electron-piezoelectric phonon-coupled materials. Assuming weak interactions, effective transitions occur through transitions within the same level ($(\alpha ,\beta )=(0,0)$) and between adjacent levels ($(\alpha ,\beta )=(0,1)$, $(\alpha ,\beta )=(1,0)$). Within the intra- and inter-Landau-and inter-Landau-level transitions, LW ${\gamma }_{total}\left(\omega \right)$ is given as follows:

$${\gamma }_{total}\left(\omega \right)\equiv {\gamma }_{\mathrm{int}ra}+{\gamma }_{\mathrm{int}er}\equiv ({\gamma }_{0,0}^{-}+{\gamma }_{0,0}^{+})+\left[({\gamma }_{0,1}^{-}+{\gamma }_{1,0}^{-})+({\gamma }_{0,1}^{+}+{\gamma }_{1,0}^{+})\right]$$

(28)

Through double wavelength vector integration, these results become available for numerical analysis.

4. Case Study

This case study analyzes the absorption power and scattering factor function of materials in which piezoelectric potential scattering is the primary mechanism, specifically GaAs and CdS. We have summarized the material constants required for numerical analysis in

Table 1. Material constants of GaAs and CdS.

Symbol	Contents	Value
		GaAs	CdS
$m^{*}$	Effective mass of electron	0.066 m0	0.19 m0
$\stackrel{-}{m}$ $m_0$	Effective mass of hole Free-electron mass	0.40 m0 $9.1095\times {10}^{-31}\mathrm{kg}$	0.7 m0 $9.1095\times {10}^{-31}\mathrm{kg}$
ρ	Mass density	536 kg/m3	4820 kg/m3
κ	Characteristic constant	2.52 × 10−4 eV/K	8.58 × 10−4 eV/K
ξ	Characteristic constant	204	235
$\stackrel{-}{K}$	Electromechanical constant	0.1726 m/s	4.77 × 10−3 m/s
${\stackrel{-}{v}}_S$	Speed of sound	4300.5 m/s	3045 m/s
${\stackrel{\sim }{\epsilon }}_S$	Energy gap	2.56 eV	0.744 eV
$L_Z$	Length of well in the z direction	20 × 10−9 m	20 × 10−6 m

.

As the speed of sound $v_s$, the average value, ${\overline{v}}_s=(v_{sl}+v_{st})/2$ is used. Here, it is the longitudinal sound velocity $v_{sl}$ = 5614 $\mathrm{m}/\mathrm{s}$ and the transverse sound velocity $v_{st}$ = 2987 $\mathrm{m}/\mathrm{s}$ of the GaAs. And it is the longitudinal sound velocity $v_{sl}$ = 4280 $\mathrm{m}/\mathrm{s}$ and the transverse sound velocity $v_{st}$ = 1810 $\mathrm{m}/\mathrm{s}$ of the CdS. The value of ${\epsilon }_0$ is 8.85419 $\times {10}^{-12}$ ${\mathrm{C}}^2/\mathrm{N}·{\mathrm{m}}^2$. By substituting these constants into the equations mentioned above, it is available to analyze the LSs and LWs of the optical quantum transitions of GaAs and CdS [16].

Figure 1 shows the effect of the magnetic field on the LS $P(B)$ of GaAs under a specific external field with a wavelength of λ = 220 μm and temperatures of T = 10, 30, 50, 90, and 120 K. At T = 10, the LS exhibits a sharp and narrow peak near B ≈ 8.76 T, indicating strong resonance behavior with minimal thermal broadening. As T increases, the peak broadens significantly, reflecting the growing thermal interaction effects in the system. The gain factors applied to the curves for T = 30, 50, 90, and 120 K highlight the dramatic rise in absorption power with increasing T. This behavior suggests that the $P(B)$ in GaAs is strongly temperature-dependent, with higher temperatures enhancing scattering mechanisms or transition probabilities that contribute to the LS. The broadening of the peak at elevated temperatures also indicates a reduction in coherence due to increased phonon interactions.

Figure 2 illustrates the temperature dependence of the LW $\gamma (T)$ for GaAs under external fields with λ = 172, 190, 220, 394, and 513 μm. At all wavelengths, the LW increases as T rises, demonstrating a strong thermal dependence. Shorter wavelengths (λ = 172, 190, 220 μm) exhibit steeper slopes, indicating a more pronounced increase in $\gamma (T)$ with T. In contrast, longer wavelengths (λ = 394, 513 μm) show a more gradual rise in LW.

As the total LW is composed of the summation of intra-Landau-level and inter-Landau-level transitions, each component of the LW at λ = 220 μm is analyzed and illustrated in Figure 3. The total LW $\gamma {(T)}_{total}^{(GaAs)}$ increases with T, indicating strong thermal broadening effects. The inter-Landau-level component $\gamma {(T)}_{\mathrm{int}erL}^{(GaAs)}$ increases significantly with T, which likely dominates the overall thermal broadening. The intra-Landau-level component $\gamma {(T)}_{\mathrm{int}raL}^{(GaAs)}$ remains nearly constant across the temperature range, suggesting minimal thermal dependence. This behavior highlights the role of inter-Landau-level transitions in contributing to the overall LW, especially at elevated T. In contrast, the intra-Landau-level component is relatively unaffected by T. These results indicate that the intra-level transition process plays a more significant role in the scattering effect compared to the inter-level transition process.

Figure 4 further investigates the contributions of intra- and inter-Landau-level transitions to the LW and their impact on the LS in GaAs. Specifically, it compares $\gamma {(T)}_{total}$ with its components: $\gamma {(T)}_{\mathrm{int}raL}^{(GaAs)}$ and $\gamma {(T)}_{\mathrm{int}erL}^{(GaAs)}$. The total absorption power curve $P(B)$ closely matches the behavior of the intra-level contribution $P(B)$ with $\gamma {(T)}_{\mathrm{int}raL}^{(GaAs)}$, highlighting the dominant role of intra-level transitions in determining the overall broadening. The inter-level contribution $P(B)$ with $\gamma {(T)}_{\mathrm{int}erL}^{(GaAs)}$ is less significant, as its effect on $P(B)$ is noticeably weaker. This comparison underscores the dominant influence of intra-Landau-level transitions on the broadening of absorption power in GaAs at T = 30 and λ = 220 μm. The agreement between the LS components in Figure 3 and the absorption power broadening in Figure 4 demonstrates the strong correlation between line width contributions and the overall absorption process.

Figure 5 shows the LS $P(B)$ of CdS as a function of the magnetic field at λ = 220 μm and temperatures of T = 10, 30, 50, 90, and 120 K. At T = 10 K, the absorption peak is sharp and narrow near B ≈ 8.76 T, indicating minimal thermal broadening. As T rises, the peaks broaden significantly and $P(B)$ increases, with gain factors. This broadening reflects significant thermal interactions and increased scattering at higher T. The consistent peak position across all temperatures highlights the magnetic field’s dominant role in resonance. The broader peaks in CdS, compared to GaAs, suggest reduced coherence in its quantum transitions. These results emphasize the temperature dependence of scattering and the influence of phonon interactions in CdS.

Figure 6 illustrates the temperature dependence of the LW $\gamma (T)$ of CdS for various wavelengths of λ = 172, 190, 220, 394, 513 μm. At all λ, LW increases with rising T, demonstrating a strong thermal influence on the scattering mechanisms in CdS. The rate of increase becomes more pronounced at higher T, particularly for shorter wavelengths (λ = 172, 190, 220 μm). At any given T, the LW decreases with increasing λ, indicating weaker scattering effects for longer wavelengths (λ = 394, 513 μm). This trend suggests that shorter λ are more sensitive to thermal interactions, resulting in broader LW.

Figure 7 presents the effect of temperature on LW of CdS at a fixed λ = 172 μm, decomposed into its total, intra-Landau-level, and inter-Landau-level contributions. The total LW $\gamma {(T)}_{total}^{(CdS)}$ increases steadily with T, indicating the cumulative effect of intra- and inter-level transitions as thermal interactions intensify. The inter-Landau-level component $\gamma {(T)}_{\mathrm{int}erL}^{(\mathrm{CdS})}$ increases in the quantum limit at low temperatures, below 70 K. The intra-Landau-level component $\gamma {(T)}_{\mathrm{int}raL}^{(GaAs)}$ remains nearly constant as T increases in the high-temperature region. While the intra-level transitions dominate at lower temperatures due to their minimal thermal interaction, the inter-level transitions play a larger role at elevated temperatures, driving the overall broadening of the line width. This behavior highlights the dynamic nature of scattering mechanisms in CdS under varying thermal conditions.

Figure 8 compares the LS $P(B)$ of CdS at T = 30 K and λ = 220 μm, showing contributions from the total LW $\gamma {(T)}_{total}$, intra-Landau-level LW $\gamma {(T)}_{\mathrm{int}raL}^{(CdS)}$, and inter-Landau-level LW $\gamma {(T)}_{\mathrm{int}erL}^{(CdS)}$. The $\gamma {(T)}_{total}$ shows the overall profile dominated by both intra- and inter-level contributions. The total absorption peak occurs near B ≈ 8.76 T, with a broader distribution compared to individual contributions. The $\gamma {(T)}_{\mathrm{int}raL}^{(CdS)}$ contributes to a sharper and narrower peak, highlighting their dominance in determining the core structure of the absorption profile. The narrower profile indicates minimal thermal broadening for the intra-level transitions. The $\gamma {(T)}_{\mathrm{int}erL}^{(CdS)}$ shows a much broader peak, reflecting the higher thermal sensitivity of inter-level transitions.

The effect of temperature on LS and LW has shown similar trends in both GaAs and CdS. Building on this observation, Figure 9 presents a comparative analysis of the T dependence of LW for GaAs and CdS at various external fields (λ = 172, 190, 220, 394, and 513 μm). For both materials, the LW increases with rising T, demonstrating similar trends. However, at each λ, the LW of CdS is consistently higher than that of GaAs across the entire temperature range, highlighting the stronger thermal broadening effects in CdS. This suggests that CdS experiences greater phonon interaction at elevated T compared to GaAs.

Figure 10 illustrates the effect of temperature on LW for GaAs and CdS at a fixed wavelength of λ = 220 μm, with total, intra-, and inter-Landau-level transitions. The total LW of CdS is consistently higher than that of GaAs across the temperature range, reflecting stronger phonon interactions in CdS. Additionally, the intra-level contributions remain nearly temperature-independent for both materials, while the inter-level contributions exhibit a noticeable increase with T. This demonstrates that the overall temperature dependence of LW is primarily driven by the inter-level transitions for both GaAs and CdS.

Figure 11 compares the LS of GaAs and CdS, the $P{(B)}_{total}^{(GaAs)}$ and $P{(B)}_{total}^{(CdS)}$ with λ = 220 μm at T = 30 K. For GaAs, the absorption power exhibits a sharp peak near B ≈ 8.76 T, the total LW showing a broader profile compared to the intra-level component, indicating a significant contribution from inter-level transitions. Similarly, for CdS, the LS shows a peak at a similar magnetic field, but the total LW is more dominant relative to the intra-level contribution, reflecting stronger phonon interactions in CdS. However, the $P{(B)}_{total}^{(GaAs)}$ has a much larger value than the $P{(B)}_{total}^{(CdS)}$. Specifically, the value of the $P{(B)}_{total}^{(GaAs)}$ is nearly ${10}^{150}$ times than the value of $P{(B)}_{total}^{(CdS)}$. This fact confirms that GaAs exhibits much larger mobility compared to other materials. From this analysis, it is evident that thermal quantum transition properties are similar for CdS, a wurtzite-type material, and GaAs, a sphalerite structure.

In the case of the QTR theory projected from EAPS used in this study, it is advantageous in terms of simplifying the calculation because LWs and LSs can be obtained directly through EAPS. If only the EAPS theory is used, a complex process must be followed to calculate the absorbed power to obtain LWs and LSs. Therefore, the QTR theory projected from EAPS is a useful method for explaining the resonance phenomenon based on scattering effects and quantum transitions from a microscopic perspective, as demonstrated by the variations in values with changes in temperature and magnetic field.

This work provides a theoretical foundation for understanding and optimizing low-dimensional semiconductor systems, with implications for advanced optoelectronic and quantum devices.

5. Conclusions

This study investigates the Optical Quantum Transition Line Properties (OQTLP), including line shapes (LSs) and line widths (LWs), in electron–piezoelectric-phonon-coupled systems within GaAs and CdS under a square-well confinement potential. The results reveal that LSs and LWs are strongly influenced by temperature and magnetic field, with intra-Landau level transitions playing a dominant role in quantum transition processes. Specifically, while LS consistently increased with temperature and magnetic field, LW exhibited nonlinear behavior, showing a decrease in high magnetic fields at low temperatures for GaAs.

These findings highlight the critical interplay between temperature, magnetic fields, and phonon interactions in determining the quantum transition characteristics of wurtzite- and sphalerite-type structures. This work provides a theoretical foundation for understanding and optimizing low-dimensional semiconductor systems, with implications for advanced optoelectronic and quantum devices.

Future studies could further investigate other confinement potentials or external field polarizations to deepen our understanding of quantum transition mechanisms in piezoelectric materials.