1. Introduction

We are trying to apply quantum transport theory (QTR) to a system that confines electrons by squared well limiting potentials. Quantum transport theory (QTR) research is of great importance for investigating the microphenomena of many material systems. Many theories on this subject include Boltzmann’s transport theory ⁽¹⁾, Green’s functional approach ^(2-3), force-balance approach ⁽⁴⁾, Feynman’s path integration approach⁽⁵⁾, and projection operator approach ^(6-7). It can be found in the various methodologies mentioned above. Despite the fact that these methods are very reasonable, the investigation of nonlinear behavior is still limited.

However, in order to more accurately analyze the movement behavior of electrons under the influence of a strong electric field, nonlinear effects must be considered. In addition, a similar methodology by Zwanzig ⁽⁸⁾, Kenkre’s team, who used the projection operator directly into the Liouville equation, proposed a response function using Kubo’s theory using the lowest order approximation ^(8-10). This theory contains nonlinear factors, so it can be used in nonlinear systems, but because exponents are included in the theoretical formula, this term is insufficient in scalability. The study of quantum transport theory based on the projected Liouville equation method for investigating the scattering mechanisms of solids provides an immensely useful tool.

A new theory of quantum transport of linear-nonlinear form was proposed using the projected Liouville equation method with the Equilibrium Average Projection Scheme (EAPS) ⁽¹¹⁾. The advantage of using EAPS is that the quantum transfer theory can be extended to obtain generalized susceptibility and scattering factors in a single step process. Also, since the magnetic field dependence analysis of QTLW needs to calculate the absorption power of various external field wavelengths, it is very difficult to prove it by interpretation and experimentation with other theories. EAPS’ QTR theory has an advantage because it is possible to obtain QTLW directly through EAPS at various external field wavelengths. In short, the absorption power calculation is not necessary to obtain QTLW⁽¹²⁾.

In this study, we study the optical QTLS, which represents the absorption power, and the QTLW, which represents the scattering effect of the electron deformation potential phonon interaction system. The QTLS and QTLW with magnetic field dependence was analyzed in various ways through numerical calculation. The quantum transfer process was analyzed by comparing the magnetic field dependence of QTLW and the QTLS of the two transfer processes(intra-Landau level and inter-Landau level transition process). In addition, this study analyzes the magnetic field dependence in several cases. The analysis of several cases can be difficult in other theories because the absorption power calculation is required to obtain QTLW. However, you can directly obtain QTLW through the EAPS theory presented in this paper.

2. The Theory(EAPS) and System

We consider a many body system, for the brief review of the quantum transport theory(EAPS) to be applied to the system, which is subject to an oscillatory external field $f(t)=f_{l}e^{-i\omega_{l}t}\hat e_{l}$ where $\hat e_{l}$ is the unit vector in the external field direction $l(l=x,\:y,\:z,\: etc)$ and $\omega_{l}$ is the angular frequency. Then the Hamiltonian and the corresponding Liouville operator $L_(t)$ are given by

and

where $H_{s}$ and $L_{s}$ are time-independent. The timedependent Hamiltonian is defined as $H_{l}(t)=r_{1}f_{l}e^{-i\omega_{l}t}$. The Liouville operator $L_{l}$ corresponds to the response operator $r_{l}$ in the direction $l$, which implies that $L_{l}'X\equiv\left[r_{1},\:X\right]$ for an arbitrary operator $X$. $H_{s}$ is the Hamiltonian of the main particle, representing the Hamiltonian of the interaction between the main and background particles.

Consider the initial condition of the QSM (Quantum Statistic Mechanical) system as the equilibrium density $\rho_{s}\equiv\rho(0)$$\equiv\exp(-BH_{s})/Tr\exp(-BH_{s})$ determined by the equilibrium energy of the system. Here, $\rho'$ can be called the time evolution density due to the external field. For this case, the density matrix of the system is $\rho(t)=\rho_{s}+\rho'(t)$. For any dynamic variable in the k-direction, we define the expected value of the system dynamic variable as $r_{k}(t)=Trr_{k}\rho(t)$. To get $r_{k}(t)$, we define a projection operator that derives a useful response formula to $P_{k}X\equiv B_{k}Trr_{k}X$. Where $r_{k}$ is the timeindependent operator corresponding to the expectation of the dynamic variable $r_{k}(t)$.

So we can write the dynamic variable as $r_{k}(t)=(1/B_{k})(P_{k}\rho(t))$ using the direction of the projection operator $B_{k}\equiv L_{l}'\rho_{s}/\Lambda_{kl}$ with a normalization factor of $\Lambda_{kl}\equiv Tr r_{k}L_{l}'\rho_{s}$. In the procedure described in EAPS in Ref. ⁽¹¹⁾, we derive the equation of kinetic for the expected value of the dynamic variable in the existing external field, and then calculate it through the Fourier-Laplace transform of the time-dependent function as follows: It is defined as $X(\omega_{l})\equiv FLT[X(t)]\equiv\int_{0}^{\infty}\exp(-i\omega_{l}t)X(t)dt$. We obtain the linear response formula in the energy space( ω-space) as

where $A_{kl}\equiv(-i/\hbar\Lambda_{kl})Tr r_{k}L_{s}L_{l}'\rho_{s}$, the scattering factor function, $\Xi_{k l}\left(\omega_{l}\right)$ is

$Q_{k}\equiv 1-P_{k}$ and the propagator, $G_{k}^{q}(\omega_{l})$ is

Eq. (3) appears to be similar to that derived using the quantum Langevin approach, as shown by Hu and O’Connell in Ref. ⁽²⁾. However, the scattering factor function, Eq. (4) is noticeably different from the formula derived from Hu and O’Connell. We can apply the theory of Eq. (3) in various condensed matter systems. When a static magnetic field of $\vec{B}=B_{z}\hat z$ is applied to an electronic system, a single electron energy state is quantized to the Landau levels. We select an electronic system trapped in the infinite square well potential (SQWP) between and z in the z-direction. Using the Landau gauge $\vec{A}\equiv(0,\:Bx,\:0)$, we have a single electron Hamiltonian e given by

and we obtain the eigenstate of the system, as below,

where the plane wave is given by $\widetilde\Phi_{kr}^{(pl\omega)}\equiv\exp(ik_{y}y)$, and the explicit function is given by

Here $H_{N_{a}}(x)$ is the Hermite polynomials function, $l_{0}=\sqrt{\hbar /e B}$ is the radius of cyclotron motion, $\omega_{c}=e B/m_{e}^{*}$ is the cyclotron frequency, $m_{e}^{*}$ is the electron effective mass, $x_{a}=-\hbar k_{za}/e B$$=-\hbar k_{za}/m_{e}^{*}\omega_{0}$ is the center of cyclotron motion and $x_{1}\equiv x-x_{a}$. The confined wave function is

where $K_{n_{a}}$ and $k_{n_{a}}$(quantization condition for the z-direction components of the electro wave vector of $K$ and $k$) are obtained by solving the simultaneou use quations $K =-k\cot kz_{0}$ and $K +k=2m_{e}^{*}U_{0}/\hbar$, with the conditions $0< K$ and $0<k$. Here the square well confinement potential $U(z)\equiv U_{0}$ is a constant potential in the region $0<z<z_{0}$, and $U(z)\equiv\infty$ in the region $z<0,\: z_{0}<z$. The values of the normalization factors are $\widetilde C_{G}\equiv 1/\sqrt{L_{y}},\:\widetilde N_{r}\equiv 1/\sqrt{(\sqrt{\pi}2^{N}N !l_{0})}$. The values of the normalization factors are subject to change in other systems. We obtain the corresponding eigenvalue as

where $L_{z(sys)}\equiv z_{0}$ is the size of materials in z-direction.

If we consider a many body system which is subject to external fields $E_{+}(t)=E_{0}\exp(i\omega t)$ by circularly polarized oscillatory where $\omega$ is the angular frequency then, using Coulomb gauge $\vec{E}(t)=-\partial\vec{A}(t)/\partial t$, the total Hamiltonian of the system is $H(t)=H_{s}+H'(t)=H_{s}+(-i/\omega)J^{+}E_{+}(t)$. The many-electron current operators $J^{\pm}$ are defined as $J^{+}=j_{x}\pm ij_{y}$ and $J_{\beta}^{-}=\sum_{\beta}(j_{\beta}^{+})^{+}\alpha_{\beta}^{+}\alpha_{\beta +1}$ where $j^{\pm}=j_{x}\pm ij_{y}$ are two components of the single electro current operator , $j_{\beta}^{+}\equiv\sum_{\beta}\left <\beta +1\left | j^{+}\right |\beta\right > =\widetilde g_{(sys)}\sum_{\beta}\sqrt{(N_{\beta}+1)_{\beta}}$ elements of the matrix are and $j_{\beta}^{-}=(j_{\beta}^{+})^{*}\equiv\sum_{\beta}\left <\beta\left | j^{+}\right |\beta +1\right >$$=\widetilde{g}_{(sys)}\sum_{\beta}\sqrt{(N_{\beta})}$, where $\widetilde{g}_{(sys)}\equiv(-ie\hbar /m_{e}^{*})\sqrt{1/l_{0}^{2}}$. The $\widetilde{g}_{(sys)}$ can be changed in other systems and external fields. Here $\alpha_{1}(\alpha_{2}^{+})$ is the annihilation operator (creation operator) of fermions, and $|\beta >(|\alpha >)$ is the eigenstate of single electron.

We derive the system Hamiltonian of the electron-phonon interacting system as

where $H_{e}$ is Hamiltonian of the electrons, $H_{0}$ is a single electron Hamiltonian, $H_{p}$ is the phonon Hamiltonian and $V$ is the electron-phonon (or impurity) interaction Hamiltonian, the $\alpha_{1}(\alpha_{2}^{+})$ and $b_{1}(b_{2}^{+})$are the annihilation operator( or creation operator) of fermion and boson particles, and $\vec{q}$ is phonon (or impurity) wave vector.

$C_{\alpha ,\:\mu}(q)$ is the electron-phonon interaction coupling matrix element, $\vec{r}$ is the position vector of the electron and $V_{q}$ is the coupling coefficient of the materials. The electron-deformation potential phonon interaction parameter $V_{q}$ in the isotropic interaction formalism is given by $V(q)^{2}=\dfrac{\overline{K}^{2}\hbar\nu_{s}e^{2}}{2\chi\epsilon_{0}q V}q$ where $\overline{K}$ is the electromechanical constant. Also $\chi$ is the dielectric constant. Since the long wavelength approximation $\omega_{q}\approx\nu_{s}q$ is suitable for describing deformation potential materials (semiconductors), it use Kubo’s approximation for phonon energy $\hbar\omega_{q}\approx\hbar\nu_{s}q$, where $\nu_{s}$ is sound velocity in a solid.

3. The Scattering Factor Function and the Absorption Power Formula

We assume that an oscillating electric field $E(t)=E_{0}\exp(i\omega t)$ is applied along the z axis to give $P(\omega)=(E_{0}^{2}/2)Re\sigma(\omega)$ the absorption force transmitted to the system. Where “Re” represents the actual component and $\sigma(\omega)$ is the optical conductivity tensor, which is the formula coefficient. Here the scattering factor function means the optical QTLW, and the absorption power means the optical QTLS.

In order to apply the linear response formula to optical quantum transition systems in a RCPEF(right circularly polarized external field), we replace $r_{k}$ with $J_{k}\equiv J^{-},\: L'_{l}X$ with $L'_{l}X\equiv(-i/\omega)[J^{+},\:X]$ and $J_{l}\equiv J^{+}$for the current system under an oscillating of frequency ωat external field. We obtain the Ohmic right circular current from the response formula,

where is $\Lambda_{k l}^{(R)}=-\left\{\left(\frac{i}{\omega}\right) \sum_{\alpha} j_{\alpha+1, \alpha}^{+} j_{\alpha, \alpha+1}^{+}\left(f_{\alpha+1}-f_{\alpha}\right)\right\}, A_{k l}^{(R)}=i \omega_{c}$.

Using the properties of the projection operator and the conventional series expansion of the propagator $G_{k}^{q}(\omega_{l})$. In a pair-interacting system, the scattering factor is obtained in simple form using a weakly interacting system approximation, as below,

where the diagonal propagator is $G_{d}= 1/(\hbar\omega -L_{d})$. Here we use the relation ⁽¹²⁾,

where $\langle\cdots\rangle$ denotes the ensemble average of electron states.

In this work, we use a more rigorous interacting Hamiltonian commutative calculation with the MWC(moderately weak coupling) given by

The above relation is used to obtain the matrix element of the electron-phonon interaction system. The scattering factor function $\Xi_{k l}\left(\omega_{1}\right)$ is complex as $\Xi_{k l}\left(\omega_{l}\right) \equiv i \Delta_{\text {total }}+\gamma_{\text {total }}(\omega)$. In most cases, the imaginary component of scattering factor $\Delta_{\text {total }} \equiv \operatorname{Im} \Xi_{k l}\left(\omega_{l}\right)$ gives the line shift of the response type formula, and the real component of scattering factor $\gamma_{t o t a l}(\omega) \equiv \operatorname{Re} \Xi_{k l}(\omega)$ gives half the width of the response type formula.

In most cases, the imaginary component of the scattering factor $\triangle_{total}$ is ignored as a small value term in real systems. Then, through continuous approximation, we get the final derivation of the absorption force formula (or QTLS formula) as Eqs.(17) in the right circularly polarized outer field system.

where the scattering factor function (or QTLW) is given by

The integrand-factors $Y_{\alpha ,\:\beta}^{(R)\mp}$ are complex.

4. Analysis of the QTLS and the QTLW

Analyze the absorption power and line widths of germanium (Ge) by numerically calculating the theoretical results of optical QTLS and optical QTLW obtained to use the Liouville equation method projected from EAPS. It is well known that deformation-potential scattering predominates in pure germanium (Ge). The band structure of Germanium (Ge) can be approximated by an elliptical shape. The band structures of Ge can be approximated to be ellipsoidal. We use the values of $\overline{m}=0.22m_{0}$ and $m*=0.35m_{0}$ for the effective masses of Ge, where $m_{0}$ is the free-electron mass. The other physical constants of Ge are $\rho =5.36g/cm^{3}$, $\epsilon_{g}(0)=0.744e V$, $\nu_{s}=5.95\times 10^{5}cm/s$, $\xi =235K$ and $K=4.77\times 10^{-4}e V/K$. The widely accepted value of the deformation potential coupling parameter is used. We use $E_{l}=13.2 \mathrm{eV}$ for Ge. Insert these constants into the EAPS theory to get a line shape from which line width can be measured. Inserting these constants into Eqs. (17)-(18) yields the line shapes from which the line width can be measured.

In FIG. 1, the relativity frequency (Δω) dependence of the absorption power (QTLS), $P^{(Jc)}(Δω)$ of Ge, with λ=220, 394, 513μm at T=30K can be observed. From the graph of $P^{(Jc)}(Δω)$, we can see the broadening effects near the resonance peaks for various external fields. The small graph above shows the location of the resonant peak relative to the external field and the magnetic field dependence of the maximum absorbed power. The analysis of the relativity frequency(Δω) dependence of the absorption power (QTLS) reveals the magnetic field dependence of the absorption power, which is provided by the external field wavelength and system conditions.

In FIG. 2, shows the magnetic field dependence of QTLW, $γ(T)$ of Ge, at T=30, 50, 55, 60, 70, 75, 80, 90 and 120K. The results also provide a reasonable explanation for the directional properties of the electromagnetic motion given by the magnetic field direction and the state of the system. We see that the $γ^{(Jc)}(B)$ term increases as the magnetic field increases at the majority of the temperatures measured. The magnetic field dependence analysis of QTLW is very difficult in other theories or experiments because it requires calculation or observation of the absorption power at various external field wavelengths. EAPS’ QTR theory has the advantage that QTLW can be obtained directly through EAPS at various external field wavelengths. Therefore, the calculation of absorption power is not necessary to obtain QTLW ⁽¹²⁾. For RCF, we denote the total QTLW as $\gamma^{(J c)}_{\text {total }} \equiv \gamma(T)_{\text {intraL }}^{(J c)}+\gamma(T)_{\text {int } e r L}^{(J c)}$ to analyze the quantum transfer process. Where $\gamma_{\text {interL }}^{(J c)} \equiv \gamma_{\text {interL }}^{(J) e m}+\gamma_{\text {interL }}^{(J) a b}$ and $\gamma_{\text {intraL }}^{(J c)} \equiv \gamma_{\text {intraL }}^{(J c) e m}+\gamma_{\text {intraL }}^{(J c) a t}$ are the QTLWs of the total absorption transition and phonon emission processes, respectively.

Here, $\gamma_{\text {int } e r L}^{(J) e m} \equiv \gamma_{0,1}^{\left(J_{c}\right)+} \quad \gamma_{\text {int } r a L}^{(J) e m} \equiv \gamma_{0,0}^{(J c)+}, \quad \gamma_{\text {int } e r L}^{(J c) a b} \equiv \gamma_{0,1}^{(J)-}$ and $\gamma_{\text {int } r a L}^{(J c) a b} \equiv \gamma_{0,0}^{(J c)-}$ are the QTLW of the inter-level emission transition, intra-level emission transition process, the inter-level absorption transition and intra-level emission absorption, respectively.

In FIG. 3, comparisons of the magnetic field dependence of QTLW, $\gamma(B)_{total}$, $\gamma(B)_{intraL}$, and $\gamma(B)_{interL}$ of Ge, at T=30K are given. We see that the $γ(T)^{(Jc)}$$_{interL}$ term increases as the magnetic field increases. We also see that the intra-level emission transition process is a more dominant contributor to the scattering effect than the intra-level transition process, as the $γ(T)^{(Jc)}$$_{interL}$ value is closer to $γ(T)^{(GaAs)}$$_{total}$. The analysis of th e contributions of the two processes to the total scattering effect represents the characteristic of the magnetic field dependence of the scattering effect of the system. The contributions of the two processes can be also observed in various cases in various systems. In this work, our result show that values QTLW are $γ(T)^{(Jc)}$$_{interL}$ < $γ(T)^{(Jc)}$$_{interL}$ < $γ(T)^{(Jc)}$$_{total}$. The results have a similarity to the temperature dependence of the scattering effect of FIG. 3.

In FIG. 4, we also compare the QTLS, $P(B)$ of Ge with the $γ(T)_{total}$, $P(B)$ with $γ(T)_{interL}$ and $P(B)$ with $γ(T)interL$. Since the scattering effect value is inversely related to the broadening of the power absorptions, we see a good agreement between $γ(T)$ with λ=220, 394 and 513μm, at T=30K (FIG. 3) and the broadening of the power absorptions $P(B)$ (FIG. 4). We can also see from these analyses that the more dominant expansion effect in the germanium(Ge) quantum limit low temperature region is the phonon intra-level emission process. Our results also indicate that EAPS’ QTR theory has the potential to explain quantum transitions in various cases.

5. Conclusion

In conclusion, we would like to emphasize that our EAPS theory provides a simpler methodology for the analysis of the various cases being presented than competing theories, as the number of computational steps required by EAPS is reduced. The EAPS theory can expand the theory by separating the line width in each quantum transition aspect for various cases. The advantage of EAPS theory is that it allows a simple analysis of each quantum transfer process. Finally, we expect that the EAPS theory is also useful in other condensed material systems.