1. Introduction
A nanorod is an elastic adjustable low-dimensional structure with an ellipsoid aspect ratio [1–5]. In the (Quantum rod) QR system, the interactions between electrons and phonons tend to form a quasiparticle structure—polaron [6, 7]. Nanorods have attracted wide attention due to their attractive applications in optoelectronic devices [8]. On the other hand, polarons can form stable-level structures, playing a key role in regulating the photoelectric properties of nanosystems [9, 10]. Therefore, polaron in nanorods has been studied and discussed in several reports. Zhu et al [11] reported investigating the behavior of carriers in rutile TiO2 nanorods photoanodes at different application potentials and surface polaron densities modulated by direct electrochemical protonation using in situ femtosecond transient absorption spectroscopy (TAS) assisted spectroelectrochemistry. Sun et al [12] reported obtaining quantitative results of the electron–phonon coupling strength in CdSe quantum dots (QDs) and rods using low-temperature scanning tunneling microscopy. Scholes et al [13] reviewed the basic characteristics of excitons in nanoscience. The contents included limiting effect, localization and delocalization, exciton binding energy, exchange interaction and fine structure of exciton, exciton-vibration coupling, and exciton dynamics. The review summarizes the current understanding of excitons in quantum dots, conjugated polymers, carbon nanotubes, and photosynthetic light-harvesting antenna complexes.
Substantial theoretical and experimental work has been done to regulate polaron-level structures in nanostructures. The polaron energy levels can be controlled by reducing the dimension of nanomaterials [14], controlling the electron-restricted intensity [15], doping [16], and so on [17–19]. However, more methods are required to be explored to obtain a more stable polaron levels regulation. Further, the influence of the external environment on energy levels cannot be ignored.
In this study, more direct means of regulating polaron levels, such as the influence of external fields, such as changing magnetic field [20], electric field [21], noise field [22], and so on, which has attracted the attention of investigators. These environmental effects are, in fact, unavoidable. Therefore, this manuscript simulates the effect of different magnetic fields on the Landau energy level of polaron in nanorods and provides a theoretical reference for polaron-related experiments.
2. Theory
The Hamiltonian of the electron–phonon interaction system (see figure 1) can be expressed as1 ), are
$\begin{eqnarray}\begin{array}{rcl}H & = & \displaystyle \frac{1}{2m}\left(p+\displaystyle \frac{{eA}}{c}\right)+\displaystyle \frac{1}{2}m{\omega }_{\rho }^{2}{\rho }^{2}+\displaystyle \frac{1}{2}m{\omega }_{z}^{2}{z}^{2}\\ & & +\displaystyle \sum _{q}{\hslash }{\omega }_{\mathrm{LO}}{a}_{q}^{\ +\ }{a}_{q}+\displaystyle \sum _{q}\left[{V}_{q}{a}_{q}\exp \left({\rm{i}}q\cdot r\right)+{\rm{h}}.{\rm{c}}.\right],\end{array}\end{eqnarray}$
m is the band mass, ωρ and ωz are the measures of the transverse and longitudinal confinement strengths of the three-dimensional anisotropic harmonic potential in the radius and the length directions of the rod.${a}_{q}^{\dagger }\left({a}_{q}\right)$ denotes the creation (annihilation) operator of the bulk LO phonons with wave vector q(qρ, qz). $p=\left({p}_{\rho }\right.,\left.{p}_{z}\right)$ and r = (ρ, z) are the momentum and position vectors of the electron, respectively. Vq and αe−ph in equation ( $\begin{eqnarray}\begin{array}{rcl}{V}_{q} & = & {\rm{i}}\left(\displaystyle \frac{{\hslash }{\omega }_{{LO}}}{q}\right){\left(\displaystyle \frac{{\hslash }}{2m{\omega }_{\mathrm{LO}}}\right)}^{\tfrac{1}{4}}{\left(\displaystyle \frac{4\pi {\alpha }_{{\rm{e}}\ -\ \mathrm{ph}}}{v}\right)}^{\tfrac{1}{2}},\\ {\alpha }_{{\rm{e}}\ -\ \mathrm{ph}} & = & \left(\displaystyle \frac{{e}^{2}}{2{\hslash }{\omega }_{\mathrm{LO}}}\right){\left(\displaystyle \frac{2m{\omega }_{\mathrm{LO}}}{{\hslash }}\right)}^{\tfrac{1}{2}}\left(\displaystyle \frac{1}{{\varepsilon }_{\infty }}-\displaystyle \frac{1}{{\varepsilon }_{0}}\right).\end{array}\end{eqnarray}$
Figure 1. Schematic diagram of the magnetopolaron in a nanorod. |
A coordinate transformation [19] was done, changing the ellipsoidal boundary into a spherical one: $x^{\prime} =x,\ y^{\prime} =y,\ z^{\prime} =z/e^{\prime} z$, where $e^{\prime} $ was the ellipsoid aspect ratio, and $\left(x^{\prime} ,y^{\prime} ,z^{\prime} \right)$ was the transformed coordinate. The electron–phonon system Hamiltonian in the new coordinate was changed to $H^{\prime} $. Using the following the first Lee-Low-Pines transformation [25] to $H^{\prime} $
$\begin{eqnarray}U=\exp \left[\displaystyle \sum _{q}({f}_{q}{a}_{q}^{\dagger }-{f}_{q}^{* }{a}_{q})\right],\end{eqnarray}$
Treating fq as a variational function, the following expression is obtained:
$\begin{eqnarray}H^{\prime\prime} ={U}^{-1}H^{\prime} U.\end{eqnarray}$
The trial ground state and the first-excited state wavefunctions of the system may be chosen as
$\begin{eqnarray}\left|{\varphi }_{0}\right\rangle =\left|0\right\rangle \left|{0}_{\mathrm{ph}}\right\rangle ={\pi }^{-\tfrac{3}{4}}{\lambda }_{0}^{\tfrac{3}{2}}\exp \left[-\displaystyle \frac{{\lambda }_{0}^{2}{r}^{2}}{2}\right]\left|{0}_{\mathrm{ph}}\right\rangle ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\left|{\varphi }_{1}\right\rangle & = & \left|1\right\rangle \left|{0}_{\mathrm{ph}}\right\rangle \\ & = & {\left(\displaystyle \frac{{\pi }^{3}}{4}\right)}^{-\tfrac{1}{4}}{\lambda }_{1}^{\tfrac{5}{2}}r\cos \theta \exp \left(-\displaystyle \frac{{\lambda }_{1}^{2}{r}^{2}}{2}\right)\left|{0}_{\mathrm{ph}}\right\rangle .\end{array}\end{eqnarray}$
The trial wave function was applied to the Hamiltonian to find the minimum expected value. The ground state and first-excited state of electron energy in the QR can be written as
$\begin{eqnarray}\begin{array}{rcl}{E}_{0} & = & \left\langle {\varphi }_{0}\right|H^{\prime\prime} \left|{\varphi }_{0}\right\rangle \\ & = & \displaystyle \frac{{{\hslash }}^{2}{\lambda }_{0}^{2}}{2m}+\displaystyle \frac{e^{\prime2}{{\hslash }}^{2}{\lambda }_{0}^{2}}{2m}+\displaystyle \frac{{e}^{2}{B}^{2}}{8{\lambda }_{0}^{2}{{mc}}^{2}}+\displaystyle \frac{m{\omega }_{\rho }^{2}}{2{\lambda }_{0}^{2}}\\ & & +\displaystyle \frac{m{\omega }_{z}^{2}}{4e^{\prime2}{\lambda }_{0}^{2}}-\displaystyle \frac{\sqrt{2}{\alpha }_{{\rm{e}}\ -\ \mathrm{ph}}}{\sqrt{\pi }}{\lambda }_{0}{\hslash }{\omega }_{\mathrm{LO}}{r}_{0}A\left(e^{\prime} \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{E}_{1} & = & \left\langle {\varphi }_{1}\right|H^{\prime\prime} \left|{\varphi }_{1}\right\rangle \\ & = & \displaystyle \frac{{{\hslash }}^{2}{\lambda }_{1}^{2}}{2m}+\displaystyle \frac{e^{\prime2}{{\hslash }}^{2}{\lambda }_{1}^{2}}{2m}+\displaystyle \frac{{e}^{2}{B}^{2}}{8{\lambda }_{1}^{2}{{mc}}^{2}}+\displaystyle \frac{m{\omega }_{\rho }^{2}}{2{\lambda }_{1}^{2}}\\ & & +\displaystyle \frac{m{\omega }_{z}^{2}}{4e^{\prime2}{\lambda }_{1}^{2}}-\displaystyle \frac{7}{8}\displaystyle \frac{{\alpha }_{{\rm{e}}\ -\ \mathrm{ph}}}{\sqrt{\pi }}\sqrt{2}{\lambda }_{1}{\hslash }{\omega }_{\mathrm{LO}}{r}_{0}A\left(e^{\prime} \right).\end{array}\end{eqnarray}$
When there are different forms of magnetic fields outside, the forms of different sagittal potential magnetic fields are as follows [26]:
(First) Case I
$\begin{eqnarray}\begin{array}{rcl}{A}_{y}\left(x\right) & = & \frac{{B}_{0}}{\alpha }-\frac{{B}_{0}}{\alpha }{{\rm{e}}}^{-\alpha x},\\ {B}_{1} & = & {B}_{0}{{\rm{e}}}^{-\alpha x}Q.\end{array}\end{eqnarray}$
(Second) Case II9 ) and (10 ) where α is the constant parameter.
$\begin{eqnarray}\begin{array}{rcl}{A}_{y}\left(x\right) & = & -\displaystyle \frac{{B}_{0}}{x+\alpha },\\ {B}_{2} & = & -\displaystyle \frac{{B}_{0}}{{\left(x+\alpha \right)}^{2}}Q.\end{array}\end{eqnarray}$
In equations ((Third) Case III
$\begin{eqnarray}\begin{array}{rcl}{A}_{y}\left(x\right) & = & 2{B}_{0}d{\tanh }_{Q}\left(x/2d\right),\\ {B}_{3} & = & {B}_{0}\displaystyle \frac{Q}{{\cosh }_{Q}^{2}\left(x/2d\right)},\end{array}\end{eqnarray}$
where B0 is constant (initial) magnetic induction intensity, d is constant. The Q is a deformation parameter, which satisfies the expression ${\sinh }_{Q}^{}\left(a\right)=\tfrac{{{\rm{e}}}^{a}-Q{{\rm{e}}}^{-a}}{2}$, ${\cosh }_{Q}^{}\left(a\right)=\tfrac{{{\rm{e}}}^{a}+Q{{\rm{e}}}^{-a}}{2}$ and $\tfrac{{\rm{d}}}{{\rm{d}}a}{\tanh }_{Q}^{}\left(a\right)=\tfrac{q}{{\cosh }_{Q}^{2}\left(a\right)}$.Since these three magnetic fields corresponded to different magnetic environments, and the nanorod was in such a magnetic environment, it was only natural that the polaron would be affected by the different magnetic fields and exhibit unique properties.
3. Numerical results
The polaron energy levels in the quantum rod were theoretically calculated using the crystal parameters αe−ph = 3.81 and ℏωLO = 21.5824 meV in an attempt to explore the effects of three magnetic fields on the polaron energy levels in the nanorod, to provide theoretical ideas for the application of quantum information related to polaron levels. The polaron energy levels depended on the changes in the three irregular magnetic fields, as shown in figures 2–4. Figure 2 shows the variation relationship of the three magnetic fields with the parameters and coordinate positions of magnetic fields. When the parameters are limited to α = 0.5, d = 0.5 the initial magnetic field B0 = 50 T is obtained. Due to the difference of vector potential and position coordinate, magnetic field intensity changes, which can simulate different magnetic field environments well, paving the way for studying the influence of changing magnetic field on polaron in materials.
Figure 2. Variation of magnetic induction intensity of the three magnetic fields. |
First, the polaron energy levels in the quantum rod were calculated. The calculations revealed that the polaron forms a stable energy level. Further, the energy level is affected by different magnetic fields. As the initial magnetic field B0 changed, the polaron levels also changed.
In order to show the influence of the magnetic field on the polaron levels, the parameters α = 0.5, d = 0.5, Q = 3, ωρ = 3 × 1013 Hz, ωz = 3 × 1013 Hz, $e^{\prime} =0.9$, r = 1 nm (in figure 3) were selected for numerical calculations. The calculations revealed that the changes in the initial magnetic field determined the change in the polaron energy level of the quantum rod. In this process, the magnetic field provides the energy of the polaron movement, promoting the intensification of the polaron movement that results in increased polaron energy. When negatively charged electrons become polarized and move in the lattice, the magnetic field changes the overall motion state, increasing the total energy of the polarization subsystem. In fact, magnetic fields with different vector potentials have different effects on the ground state energy level and the excited state energy level of polaron. Thus, the stable polaron formed by the interaction between electrons and phonons is affected in the complex magnetic field environment. Further, the magnetic field can regulate the energy state of the polaron.
Figure 3. The ground state energy and first-excited state energy vary with the magnetic field |
Figure 4. Changes of energy levels with the restricted intensity in the horizontal and horizontal directions of the quantum rod |
In order to better explore the influence of specific factors on the polaron energy level, the relevant parameters were limited to the following: α = 0.5, d = 0.5, x = 0.5, Q = 3, B0 = 50 T, $e^{\prime} =0.9$, r = 1 nm of the nanorod. The calculations revealed the relationship between the polaron energy level and the change of the horizontal and horizontal constrained intensity, indicating that the constrained intensity limits the polaron energy level and determines the level size. The trends are similar, but the magnetic field is still present.
In addition, the quantum rod’s polaron energy level varies with the magnetic field’s cyclotron frequency and polaron radius. When the parameters related to the nanorod are defined as α = 0.5, d = 0.5, x = 0.5, Q = 3, ωρ = 3 × 1013 Hz,ωz = 3 × 1013 Hz, B0 = 50 T and magnetic field, the cyclotron frequency of the magnetic field will cause the ground state energy level and the excited state energy level of the polaron to decrease first and then increase. Firstly, the magnetic field can limit the movement of electrons, which prompted in limited within the scope of electronics is hampered by more phonon, electron and phonon formation by the energy of the polaron, and increase the frequency of the magnetic field, the electron kinetic energy increases, lead to electronic kinetic energy increases, electronic overcome phonon energy cloud movement increases, eventually led to the increased level of polaron. It can be seen from figure 5 that the magnetic field environment of the three vector potentials has different influences on the polaron energy level.
Figure 5. Variation of cyclotron frequency and polaron radius between energy level and magnetic field |
4. Conclusion
Through numerical calculation, the parameters related to the initial magnetic field and the vector potential of the magnetic field were found. The parameters of the quantum rod itself could adjust the polaron energy level, confirming the existence of tunable polarons in quantum rods. Therefore, in practice, the polaron state can be controlled by adjusting the magnetic field and the parameters of the quantum rod, providing a theoretical idea for the design of low-dimensional semiconductor devices with good photoelectric performance.