The Landau-level structure of a single polaron in a nanorod under a non-uniform magnetic field

Tian-Ji Ou,Ran An,Wei Zhang,Shuang Han,Yong Sun,*,Hamid-Reza Rastegar-Sedehi,Xin-Jun Ma and Jing-Lin Xiao

1 Institute of Condensed Matter Physics,College of Physics and Electronic Information,Inner Mongolia Minzu University,Tongliao 028043,China

2 Department of Physics,College of Sciences,Jahrom University,Jahrom 74137-66171,Iran

Abstract Nanorod is a unique low-dimensional nanometer structure in which the Landau level arrangement of polaron is essential for understanding its quasiparticle system.However,the stability of the polaron level is susceptible to external factors,such as changing magnetic fields.In this manuscript,the Pekar variational method is employed to calculate the external magnetic field’s effect on the nanorod’s polaron Landau level.It was found that different magnetic fields have different effects on the polaron energy levels of the nanorod,which demonstrated that the external environment had critical effects on the polaron energy levels.This study provides a theoretical basis for regulating the interaction between electrons and phonons in low-dimensional nanomaterials.

Keywords: single polaron,energy-levels,non-uniform magnetic field,nanorod

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 TiO2nanorods 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 electronrestricted 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

An electron moves and interacts with bulk LO phonons in a polar nanorod with a three-dimensional harmonic potential.In the non-uniform magnetic field,the momentum operatorp+with the vector potentialA=(0,Ay(x),0) is considered [23,24].

The Hamiltonian of the electron–phonon interaction system (see figure 1) can be expressed as

m is the band mass,ωρand ωzare 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.denotes the creation (annihilation) operator of the bulk LO phonons with wave vector q(qρ,qz).p=(pρ,pz) and r=(ρ,z) are the momentum and position vectors of the electron,respectively.Vqand αe-phin equation (1),are

A coordinate transformation[19]was done,changing the ellipsoidal boundary into a spherical one:x′=x,y′=y,z′=ze′z,wheree′ was the ellipsoid aspect ratio,and(x′,y′,z′) was the transformed coordinate.The electron–phonon system Hamiltonian in the new coordinate was changed toH′.Using the following the first Lee-Low-Pines transformation [25] toH′

Fig.1.Schematic diagram of the magnetopolaron in a nanorod.

Treating fqas a variational function,the following expression is obtained:

The trial ground state and the first-excited state wavefunctions of the system may be chosen as

The trial wave function was applied to the Hamiltonian to find the minimum expected value.The ground state and firstexcited state of electron energy in the QR can be written as

When there are different forms of magnetic fields outside,the forms of different sagittal potential magnetic fields are as follows [26]:

(First) Case I

(Second) Case II

In equations (9) and (10) where α is the constant parameter.

(Third) Case III

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 ?ωLO=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.

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 B0changed,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=3×1013Hz,ωz=3×1013Hz,e′=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.

Fig.2.Variation of magnetic induction intensity of the three magnetic fields.

Fig.3.The ground state energy and first-excited state energy vary with the magnetic field

Fig.4.Changes of energy levels with the restricted intensity in the horizontal and horizontal directions of the quantum rod

Fig.5.Variation of cyclotron frequency and polaron radius between energy level and magnetic field

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′=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=3×1013Hz,ωz=3×1013Hz,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.

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.

Acknowledgments

Natural Science Foundation of Inner Mongolia (Nos.2020BS01001 and 2022MS01014),the Basic Scientific Research Business Projects in Colleges and Universities Directly under Inner Mongolia Autonomous Region (No.GXKY22059),and the Doctoral Scientific Research Foundation of Inner Mongolia Minzu University.(Nos.BS511 and BS625).

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