Abstract We theoretically explore the manipulation of a temporal electron-spin splitter by a δ-potential in an embedded magnetic-electric-barrier microstructure (EMEBM), which is constructed by patterning a ferromagnetic stripe and a Schottky-metal stripe on the top and bottom of an InAs/AlxIn1−xAs heterostructure, respectively. Spin polarization of the dwell time remains, even though a δ-potential is inserted by atomic-layer doping. Both the magnitude and sign of the spin-polarized dwell time can be manipulated by changing the weight or position of the δ-potential. Thus, a structurally controllable temporal electron-spin splitter can be obtained for spintronics device applications. Keywords：embedded magnetic-electric-barrier microstructure (EMEBM);δ-potential;dwell time;spin polarization;manipulable temporal spin splitter
PDF (547KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Gui-Xiang Liu, Ge Tang, Jian-Lin Liu, Qing-Meng Guo, Shuai-Quan Yang, Shi-Shi Xie. Manipulation of a temporal electron-spin splitter via a δ-potential in an embedded magnetic-electric-barrier microstructure. Communications in Theoretical Physics, 2021, 73(12): 125704- doi:10.1088/1572-9494/ac3327
1. Introduction
Experimentally, advanced materials-growth techniques make it possible to control high-mobility two-dimensional electron gas (2DEG) contained in the interface of a semiconductor heterostructure by an inhomogeneous magnetic field [1], e.g. depositing a small ferromagnetic (FM) stripe on the surface of a GaAs/AlxGa1−xAs heterostructure [2], leading to a magnetically modulated semiconductor microstructure (MMSM) [3]. An MMSM actually consists of magnetic materials and a semiconductor, where the former produces an inhomogeneous magnetic field that locally influences the motion of electrons in the latter. Due to low dimensionality, small size and quantum confinement, an MMSM contains some novel quantum effects [4], e.g. spin filtering [5–7], giant magnetoresistance (GMR) [8, 9] and Goos-Hänchen shift [10, 11]. These important effects can be used to develop new electronic devices, such as GMR devices [12] and spatial spin splitters [13].
More recently, transmission time for electrons through an MMSM has received considerable interest, due to its possible application as a spin-polarized source in the field of semiconductor spintronics [14]. Physically, spin polarization can be generated if the transmission time for an electron to traverse an MMSM depends on its spins, i.e. this MMSM is employed as a temporal spin splitter (TSS) [15]. About 20 years ago, Zhai et al [16] considered the transmission time for electrons across a rectangular magnetic-barrier (MB) microstructure. One year later, Xu et al [17] proposed to separate electron spins by transmission time. However, such an idea to spin inject into a semiconductor did not attract enough attention at the time, perhaps because the definition or measurement of time has been contentious in the community [18]. Electron-spin polarization realized by transmission time in MMSM has regained interest due to successful experiments [19–21] to measure the tunneling time, especially the direct measurement of the time spent by a tunneling atom within the barrier region last year [22]. Lu et al [23] first calculated the dwell time for an electron in a realistic MB microstructure with the help of the Winful theory [24]. On top of this work, the dwell time for an electron in another realistic MB microstructure and parallel magnetic-electric-barrier microstructure (PMEBM) were explored by Guo et al [25] and Lu et al [26], respectively. Chen et al [27, 28] and Zhang et al [29] studied the dwell time of an electron in parallel and antiparallel double δ-MB microstructures, respectively. These research studies demonstrated that dwell time depends on electron spins because of the interaction between spins and structural magnetic fields in MMSM. This makes electron spins separable in time dimensions, induces electron-spin polarization in semiconductors and develops a TSS for spintronics device applications.
Very recently, we investigated the dwell time for an electron in an embedded MEB microstructure (EMEBM), which can be realized experimentally by constructing an FM stripe and a Schottky-metal (SM) stripe on the top and bottom of an InAs/AlxIn1−xAs heterostructure respectively [30]. It is demonstrated that electron-spin polarization can be obtained by the spin-related dwell time, and the spin polarization ratio can reach optimum by depositing a proper SM stripe. Based on these findings, a TSS device was put forward successfully. As a controllable spin-polarized source is desired for spintronics device applications [31], in the present work, we introduce a δ-potential by atomic-layer doping [32] to investigate manipulation of this EMEBM-based TSS device, focusing on developing a controllable spin-polarized source.
2. Theory and method
The EMEBM-based TSS device [33] is plotted in figure 1(a), where an FM stripe with horizontal magnetization ($\vec{M}$) and an SM stripe applied by a negative voltage (−${V}_{g}$) are constructed [34] on the top and bottom of an InAs/AlxIn1−xAs heterostructure, respectively. Figure 1(b) is its structural model inserted by a δ-potential, $V\delta (x-{x}_{0}),$ where the FM-stripe width is $L$ and the SM stripe with a width $d$ is located at $x={x}_{s}.$ The magnetized FM stripe produced a magnetic-field profile and can be approximated as [35]$\begin{eqnarray}\vec{B}\left(x\right)={B}_{z}\left(x\right){\vec{e}}_{z}\,{\rm{with}}\,{B}_{z}\left(x\right)=B\left[\delta \left(x+L/2\right)\right]\left.-\delta \left(x-L/2\right)\right],\end{eqnarray}$and its magnetic vector potential can be expressed, in the Landau gauge, by$\begin{eqnarray}\begin{array}{lll}\vec{A}(x)&=&[0,{A}_{y}(x),0]\,{\rm{with}}\,{A}_{y}(x)\\ &=&\left\{\begin{array}{ll}0, & x\lt -L/2\\ B{{\ell }}_{B}, & -L/2\lt x\lt +L/2\\ 0, & x\gt +L/2\end{array}\right.,\end{array}\end{eqnarray}$where ${{\ell }}_{B}$ is magnetic length and $\delta (x)$ stands for Delta function.
Figure 1.
New window|Download| PPT slide Figure 1.(a) The EMEBM-based TSS device, and (b) its structural model with a δ-potential.
The Hamiltonian describing an electron in the EMEBM-based TSS device is written, in single particle effective mass approximation, by$\begin{eqnarray}\begin{array}{lll}H&=&\displaystyle \frac{{P}_{x}^{2}}{2{m}^{\ast }}+\displaystyle \frac{{[{P}_{y}+e{A}_{y}(x)]}^{2}}{2{m}^{\ast }}+\displaystyle \frac{e{g}^{\ast }{\sigma }_{z}\hslash }{4{m}_{0}}{B}_{z}(x)\\ & & +\,U\left(x\right)+V\delta \left(x-{x}_{0}\right),\end{array}\end{eqnarray}$in which ${m}^{\ast },$ ${m}_{0},$ $\vec{p}=({p}_{x},{p}_{y})$ and ${g}^{\ast }$ are the effective mass, free mass, momentum and effective g-factor, respectively, ${\sigma }_{z}=+1/-{\rm{1}}$ corresponds to a spin-up/down of electrons and the third term stands for interaction between electron spins and structural magnetic fields.
Since an electron in this TSS device is free along the y direction, its stationary Schrödinger equation, $H{\rm{\Phi }}(x,y)=E{\rm{\Phi }}(x,y),$ can be solved by ${\rm{\Phi }}(x,y)={{\rm{e}}}^{{\rm{i}}{k}_{y}y}\psi (x),$ where $E$ and ${k}_{y}$ are the incident energy and y-component of the wave vector, respectively, while the x-component of the wave function satisfies the reduced one-dimensional (1D) Schrödinger equation$\begin{eqnarray}-\displaystyle \frac{{\hslash }^{2}}{2{m}^{\ast }}\psi ^{\prime\prime} (x)+{U}_{{\rm{eff}}}(x,V,{x}_{0})\psi (x)=E\psi (x).\end{eqnarray}$with an effective potential experienced by an electron in the EMEBM-based TSS device,$\begin{eqnarray}\begin{array}{lll}{U}_{{\rm{eff}}}(x,V,{x}_{0})&=&\displaystyle \frac{{\hslash }^{2}{[{k}_{y}+e{A}_{y}(x)]}^{2}}{2{m}^{\ast }}\\ & & +\,U(x)+\displaystyle \frac{e{g}^{* }\hslash {\sigma }_{z}}{4{m}_{0}}{B}_{z}(x)+V\delta (x-{x}_{0}).\end{array}\end{eqnarray}$To facilitate solving equation (4), we represent all the physical quantities in a dimensionless form, such as $x\to x{{\ell }}_{B},$ $E\to E{E}_{0},$ ${B}_{z}(x)\to {B}_{z}(x){B}_{0}$ and $t\to t{\tau }_{0},$ where ${{\ell }}_{B}=\sqrt{\tfrac{\hslash }{e{B}_{0}}},$ ${E}_{0}=\tfrac{e\hslash {B}_{0}}{{m}^{\ast }},$ ${\tau }_{0}=\tfrac{{m}^{\ast }}{e{B}_{0}}$ and a typical magnetic field ${B}_{0}.$
Using the improved transfer matrix method (ITMM) [36], the reduced 1D Schrödinger equation (4) can be solved. In incident and outgoing regions, wave functions can be assumed as$\begin{eqnarray}\begin{array}{l}{\psi }_{{\rm{left}}}(x)=\exp ({\rm{i}}{k}_{l}x)+\gamma \exp (-{\rm{i}}{k}_{l}x),\,\,\,\,x\lt -L/2,\\ {\psi }_{{\rm{right}}}(x)=\tau \exp ({\rm{i}}{k}_{r}x),\,\,\,\,x\gt L/2,\end{array}\end{eqnarray}$where ${k}_{l}={k}_{r}=\sqrt{2E-{{k}_{y}}^{2}},$ and $\gamma $/$\tau $ is the reflection/transmission amplitude. In the device region, wave functions can be written by linear combinations of plane waves$\begin{eqnarray}\psi (x)=\left\{\begin{array}{l}{c}_{1}{{\rm{e}}}^{{\rm{i}}{k}_{1}x}+{d}_{1}{{\rm{e}}}^{-{\rm{i}}{k}_{1}x},\,-L/2\lt x\lt {x}_{0},\\ {c}_{2}{{\rm{e}}}^{{\rm{i}}{k}_{2}x}+{d}_{2}{{\rm{e}}}^{-{\rm{i}}{k}_{2}x},\,{x}_{0}\lt x\lt ({x}_{s}-d/2),\\ {c}_{3}{{\rm{e}}}^{{\rm{i}}{k}_{3}x}+{d}_{3}{{\rm{e}}}^{-{\rm{i}}{k}_{3}x},\,({x}_{s}-d/2)\lt x\lt ({x}_{s}+d/2),\\ {c}_{{\rm{4}}}{{\rm{e}}}^{{\rm{i}}{k}_{{\rm{4}}}x}+{d}_{{\rm{4}}}{{\rm{e}}}^{-{\rm{i}}{k}_{{\rm{4}}}x},({x}_{s}+d/2)\lt x\lt +L/2,\end{array}\right.\end{eqnarray}$in which ${k}_{1}={k}_{2}={k}_{4}=\sqrt{2E-{({k}_{y}+B)}^{2}},$ ${k}_{3}=\sqrt{2\left(E-U\right)-{\left({k}_{y}+B\right)}^{2}},$ and ${c}_{j}/{d}_{j}$ $\left(j=1,\,2,\,3,\,{\rm{4}}\right)$ is undetermined constants. According to continuity of wave functions at boundaries, we readily obtain transfer matrices,$\begin{eqnarray}\begin{array}{l}{M}_{1}=\left(\begin{array}{cc}\cos \,{k}_{1}{d}_{1}+\displaystyle \frac{2V}{{k}_{1}}\,\sin \,{k}_{1}{d}_{1} & -\displaystyle \frac{\sin \,{k}_{1}{d}_{1}}{{k}_{1}}\\ {k}_{1}\,\sin \,{k}_{1}{d}_{1}-2V\,\cos \,{k}_{1}{d}_{1} & \cos \,{k}_{1}{d}_{1}\end{array}\right),\\ {M}_{2}=\left(\begin{array}{cc}\cos \,{k}_{2}{d}_{2} & -\displaystyle \frac{\sin \,{k}_{2}{d}_{2}}{{k}_{2}}\\ {k}_{2}\,\sin \,{k}_{2}{d}_{2} & \cos \,{k}_{2}{d}_{2}\end{array}\right),\\ {M}_{3}=\left(\begin{array}{cc}\cos \,{k}_{3}{d}_{3} & -\displaystyle \frac{\sin \,{k}_{3}{d}_{3}}{{k}_{3}}\\ {k}_{3}\,\sin \,{k}_{3}{d}_{3} & \cos \,{k}_{3}{d}_{3}\end{array}\right),\\ {M}_{4}=\left(\begin{array}{cc}\cos \,{k}_{4}{d}_{4}-\displaystyle \frac{{\rm{\Lambda }}}{{k}_{4}}\,\sin \,{k}_{4}{d}_{4} & -\displaystyle \frac{\sin \,{k}_{4}{d}_{4}}{{k}_{4}}\\ {k}_{4}\,\sin \,{k}_{4}{d}_{4}+{\rm{\Lambda }}\,\cos \,{k}_{4}{d}_{4} & \cos \,{k}_{4}{d}_{4}\end{array}\right),\end{array}\end{eqnarray}$and the total transfer matrix,$\begin{eqnarray}M={M}_{1}\times {M}_{2}\times {M}_{3}\times {M}_{4}\equiv \left(\begin{array}{l}{m}_{11}\,\,\,\,\,\,\,\,\,{m}_{12}\\ {m}_{21}\,\,\,\,\,\,\,\,{m}_{22}\end{array}\right),\end{eqnarray}$where ${d}_{1}={x}_{0}+L/2,$ ${d}_{2}=-{x}_{0},$ ${d}_{3}=d,$ ${d}_{{\rm{4}}}=L/2-d,$ and ${\rm{\Lambda }}=0.5{g}^{\ast }{\sigma }_{z}B\left({m}^{\ast }/{m}_{0}\right).$ So, one can obtain$\begin{eqnarray}\begin{array}{l}\tau (E,{k}_{y},{\sigma }_{z},V,{x}_{0})\\ =\,\displaystyle \frac{2{k}_{l}}{({k}_{l}{m}_{11}-{\rm{\Lambda }}{k}_{r}{m}_{12}+{k}_{r}{m}_{22})+{\rm{i}}({\rm{\Lambda }}{m}_{11}+{k}_{l}{k}_{r}{m}_{12}-{m}_{21})},\\ \gamma (E,{k}_{y},{\sigma }_{z},V,{x}_{0})\\ =\,\displaystyle \frac{({k}_{l}{m}_{11}+{\rm{\Lambda }}{k}_{r}{m}_{12}-{k}_{r}{m}_{22})-{\rm{i}}({\rm{\Lambda }}{m}_{11}-{k}_{l}{k}_{r}{m}_{12}-{m}_{21})}{({k}_{l}{m}_{11}-{\rm{\Lambda }}{k}_{r}{m}_{12}+{k}_{r}{m}_{22})+{\rm{i}}({\rm{\Lambda }}{m}_{11}+{k}_{l}{k}_{r}{m}_{12}-{m}_{21})}.\end{array}\end{eqnarray}$Group delay and self-interference delay can be evaluated, respectively,$\begin{eqnarray}\begin{array}{lll}{t}_{g}\left(E,{k}_{y},{\sigma }_{z},V,{x}_{0}\right)&=&{\left|\tau \right|}^{2}\times \displaystyle \frac{\partial }{\partial E}{\rm{Arg}}(\tau )\\ & & +\,{\left|\gamma \right|}^{2}\times \displaystyle \frac{\partial }{\partial E}{\rm{Arg}}(\gamma ),\end{array}\end{eqnarray}$and$\begin{eqnarray}{t}_{i}\left(E,{k}_{y},{\sigma }_{z},V,{x}_{0}\right)=-\displaystyle \frac{\text{Im}\left(\gamma \right)}{{k}_{l}}\times \displaystyle \frac{\partial {k}_{l}}{\partial E}.\end{eqnarray}$Thus, dwell time spent by an electron in the EMEBM-based TSS device can be obtained from the Winful theory [24] by$\begin{eqnarray}\begin{array}{lll}{t}_{d}\left(E,{k}_{y},{\sigma }_{z},V,{x}_{0}\right)&=&{t}_{g}\left(E,{k}_{y},{\sigma }_{z},V,{x}_{0}\right)\\ & & +\,{t}_{i}\left(E,{k}_{y},{\sigma }_{z},V,{x}_{0}\right).\end{array}\end{eqnarray}$From the dwell time (13), the spin polarization rate describing the degree of the electron-spin polarization effect can be defined as$\begin{eqnarray}{P}_{t}\left(E,{k}_{y},V,{x}_{0}\right)=\displaystyle \frac{{t}_{d}^{\uparrow }\left(E,{k}_{y},V,{x}_{0}\right)-{t}_{d}^{\downarrow }\left(E,{k}_{y},V,{x}_{0}\right)}{{t}_{d}^{\uparrow }\left(E,{k}_{y},V,{x}_{0}\right)+{t}_{d}^{\downarrow }\left(E,{k}_{y},V,{x}_{0}\right)},\end{eqnarray}$where ${t}_{d}^{\uparrow }\left(E,{k}_{y},V,{x}_{0}\right)$ and ${t}_{d}^{\downarrow }\left(E,{k}_{y},V,{x}_{0}\right)$ are the dwell time for spin-up and spin-down electrons, respectively.
Regarding the above theory, we would like to point out the following points. The dwell time refers to the time spent by an electron in the EMEBM region, which can be measured by a Larmor clock, as in [22]. We consider the realization of spin polarization by separating electron spins in the time dimension; the spin polarization ratio, therefore, is defined by spin-dependent dwell time in equation (14), which is an experimentally immeasurable quantity as it is not involved in any kind of averaging. This work mainly explores how to manipulate electron-spin polarization via a δ-potential: only a y-component of the wave vector (${k}_{y}=0.0,$ i.e. the normal incidence) is considered for simplicity. Experimentally, a δ-potential can be introduced into the EMEBM by depositing a 'wire-like' stripe on the surface of an InAs/AlxIn1−xAs heterostructure with the help of atomic-layer doping; therefore, the position of the δ-potential can be controlled by changing the patterned position of the 'wire-like' stripe. For InAs 2DEG, spin–orbit coupling (SOC) could be comparable to or larger than the Zeeman effect; however, only the latter is involved (we will discuss the effect of SOC in subsequent work).
3. Results and discussion
For InAs (${m}^{\ast }=0.024{m}_{0}$ and ${g}^{\ast }=15.0$) and ${B}_{0}=0.1{\rm{T}},$ we have ${E}_{0}=0.48{\rm{eV}},$ ${{\ell }}_{B}=81.3{\rm{nm}}$ and ${\tau }_{{\rm{0}}}={\rm{1}}{\rm{.39ps}}.$ For simplicity, δ-potential $V\delta (x-{x}_{0})$ is assumed to be located between the MB and the EB, i.e. $-\tfrac{L}{2}\lt {x}_{0}\lt \left({x}_{s}-\tfrac{d}{2}\right)$, as indicated in figure 1(b). To highlight the influence of the δ-potential, some parameters of the EMEBM-based TSS device remain unchangeable: $B=6.0,$ $U=8.0,$ $L=3.0,$ $d=1.0,$ and ${x}_{s}=0.5,$ and the y-component of the wave vector is taken by ${k}_{y}=0.0$ (i.e. normal incidence).
First, we see if an appreciable electron-spin polarization still appears, after introducing a δ-potential into the EMEBM-based TSS device. For this reason, as a function of the incident energy, we have calculated the dwell time of an electron in this δ-doped EMEBM and the electron-spin polarization ratio, as shown in figure 2 and its inset, respectively. In this figure, the δ-potential is taken as $V=4.0$ and ${x}_{0}=-0.35,$ while the dashed and dotted curves correspond to the dwell times for spin-up and spin-down electrons, respectively. It is clear that in this δ-doped EMEBM the dwell time for a spin-up electron is obviously different from that for a spin-down electron. This discrepancy in dwell times between spin-up and spin-down electrons means that one can not only separate electron spins from the time dimension, but also such a δ-doped EMEBM can serve as a TSS device. Therefore, as indicated by the inset of figure 2, a considerable electron-spin polarization effect still appears, even if a δ-potential is inside the EMEBM-based TSS device. In fact, this spin-dependent dwell time for an electron to spend in this δ-doped EMEBM stems from the interaction between electron spins and structural magnetic fields, which has nothing to do with whether or not there is a δ-potential present.
Figure 2.
New window|Download| PPT slide Figure 2.The dwell time of an electron in δ-doped EMEBM and the spin polarization ratio.
However, the effective potential experienced by an electron in the EMEBM-based TSS device is related to the δ-potential from equation (5); therefore, the δ-potential impacts on the performance of the EMEBM-based TSS device. Next, we explore in detail the control of the δ-potential on dwell time and spin polarization for an electron in the EMEBM-based TSS device, as shown in figure 1.
In figure 3, we first present the influence of the δ-potential's weight on dwell time and its spin polarization for an electron in the EMEBM-based TSS device, where the δ-potential is fixed at ${x}_{0}=-0.35.$ Figure 3(a) gives the spin polarization ratio versus incident energy for a specific δ-potential's weight: $V=4.0$ (solid curve), $8.0$ (dashed curve) and $12.0$ (dotted curve), respectively. When changing the δ-potential's weight (its position is fixed unchangeable), the spin polarization ratio is greatly altered. For the considered energy range, the spin polarization ratio decreases slightly and its spectrum shifts towards the high-energy region with the increasing δ-potential's weight. This modulation of spin-polarized dwell time by the δ-potential's weight can be observed more evidently from figure 3(b) because, as a function of the δ-potential's weight, the spin polarization ratio is shown directly for a given incident energy: $E=15.0$ (solid line), $21.0$ (dashed line) and ${\rm{27.0}}$ (dotted line), respectively. It is true that the δ-potential's weight impacts strongly on dwell time and its spin polarization, including the magnitude and sign of the spin polarization ratio. Moreover, this influence of ${P}_{t}$ by $V$ shows an obvious dependence on incident energy $E,$ in particular for a relatively small energy (see the solid curve). Therefore, one can manipulate the EMEBM-based TSS device (see figure 1) well by changing the δ-potential's weight, which can be understood from the dependence of the effective potential experienced by an electron in the EMEBM on the weight of the δ-potential.
Figure 3.
New window|Download| PPT slide Figure 3.Spin polarization varies with (a) incident energy for concrete δ-potential's weights and (b) δ-potential's weight for fixed incident energy, respectively.
From equation (5), the effective potential (${U}_{{\rm{eff}}}$) still depends on the position (${x}_{0}$) of the δ-potential, except for its weight ($V$); therefore, the δ-potential's position also produces an impact on the EMEBM-based TSS device, as shown in figure 1. In figure 4(a), we plot variation of the spin polarization ratio (${P}_{T}$) with incident energy ($E$) for the concrete δ-potential's position: ${x}_{0}=-0.2$ (solid curve), $-{\rm{0.4}}$ (dashed curve) and $-{\rm{0.6}}$ (dotted curve), respectively, where the δ-potential's weight remains at $V=4.0.$ When moving the δ-potential's position, the spin polarization ratio is greatly changed, especially in the situation of a relatively high-energy region. This phenomenon tells us that one can also control the performance of the EMEBM-based TSS device by appropriately tuning the position of the δ-potential. The influence of the δ-potential's position on the EMEBM-based TSS device can be seen more clearly from figure 4(b), where the spin polarization ratio is presented directly as a function of the δ-potential's position for a fixed incident energy: $E=16.0$ (solid curve), $18.0$ (dashed curve) and $20.0$ (dotted curve), respectively. Indeed, both the magnitude and sign of the spin polarization ratio switch dramatically with the δ-potential's position, which makes it possible to control the EMEBM-based TSS device by changing the δ-potential's position. Besides, the modulation of the δ-potential's position to the spin polarization ratio exhibits an anisotropic characteristic with respect to the incident energy, i.e. this control effect is dissimilar at different incident energies.
Figure 4.
New window|Download| PPT slide Figure 4.Spin polarization changes with (a) incident energy for concrete δ-potential's positions and (b) δ-potential's position for fixed incident energy.
4. Conclusions
In summary, by inserting a δ-potential we theoretically study how to control dwell time and spin polarization of an electron in an EMEBM-based TSS device, which can be realized by patterning an FM stripe and an SM stripe on the top and bottom of an InAs/AlxIn1−xAs heterostructure, respectively. An obvious spin polarization effect still appears, even if a δ-potential is inserted. Both the magnitude and sign of the spin polarization ratio can be controlled by changing the weight or position of the δ-potential. Such a δ-doped EMEBM can be used as a structurally manipulable TSS, as a controllable spin-polarized source for spintronics device applications.
Acknowledgments
This work was supported by the Science and Technology Innovation Plan Project of Hunan Province in China (S2019JJQNJJ2177) and the National Natural Science Foundation of China (11864009).
,, Shuai-Quan Yang, Shi-Shi XieCollege of Science, 
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