Graphene-Substrate Effects on Characteristics of Weak-Coupling Bound Magnetopolaron

Graphene has many unique properties which have made it a hotbed of scientiﬁc research in recent years. However, it is not expected intuitively that the strong eﬀects of the substrate and Coulomb doping in the center of crystal cell on the polaron in monolayer graphene. Here, the interaction energy of surface electron (hole) in the graphene and optical phonons in the substrate, which give rise to weakly coupled polarons, is analyzed in the context of the Coulomb doping. The ground-state energy of the polaron is calculated using the Lee-Low-Pine unitary transformation and linear combination operator method. It is found that the ground-state energy is an increasing function of magnetic ﬁeld strength, the bound Coulomb potential, and the cutoﬀ wavenumber. Numerical results also reveal that the ground-state energy reduces as the distance between the graphene and the substrate is increased. Moreover, the ground energy level of polaron shows the two (+) and ( − ) branches and zero-Landau energy (ground) level separation in the graphene-substrate material.


Introduction
Monolayer graphene (MG) is the archetypal example of a zeroenergy gap semiconductor. e effective mass of charge carriers near the Dirac point, a phenomenon characterized by linearly dispersive band structure, is zero. Because of its unique twodimensional structure, graphene has a wealth of novel mechanical, thermal, optical, and electrical properties. Graphene materials have extraordinary electrical conductivity, magnitudes of order stronger than steel, and excellent optical transmission, making it widely used in high-performance nanoelectronic devices, composites, field emission materials, gas sensing, and energy storage. It is expected to replace silicon as an important raw material for the next generation of semiconductor materials. As such, graphene has been an area of intense focus in solid-state research in recent years [1][2][3][4][5][6][7][8][9][10]15].
Studies have shown that polarons in graphene have a significant effect on the photoelectric and transport properties. Using linear combination operators and the LLP unitary transformation, Li et al. [11][12][13] found a regulatory mechanism of zero-Landau level splitting and that a band gap opens as a direct result of polarons. Xiao et al. [14][15][16] discussed the effects of Coulomb impurities and polar substrates on the splitting of polaron zero-Landau level in graphene. Wang et al. [17][18][19][20][21] used the LLP unitary transformation to analyse the effect of impurities on monolayer graphene's physical properties. Additionally, they calculated the effect of polarons in MG on different substrates, thereby highlighting MG's sensitivity to its local environment. Other studies [22][23][24] have used Huybrechts' linear combination operator and Pekar's variational approach to study magnetopolarons in MG under strong interactions between electrons and surface acoustic (SA) phonons. Despite the high interest in this area, there have been few studies on the properties of weakly coupled polarons in MG.
In this paper, the linear combination operator method and quadratic LLP unitary transformation are used to study substrate effects on the properties of bound magnetopolarons in MG under weak coupling of electrons and surface optical (SO) phonons.

Theory
Here, we consider MG on a substrate with the opposing side exposed to air. A uniform magnetic field is applied perpendicularly to the graphene's surface, and a bound potential created by the disordered arrangement of Coulomb impurities between the substrate and graphene is incorporated.
e Hamiltonian of such a system as shown in Figure 1, accounting for electron-SO phonon interactions can be written as where where Q is the charge of the electron, η � (k 0 − k ∞ )/[(k ∞ + 1)(k 0 + 1)] represents the dielectric constant of the substrate, k ∞ and k 0 are the high-and low-frequency dielectric constants, respectively, ε 0 is the frequency of phonons,ω so,v is the SO frequency of phonons and v � 1, 2, − (g/r) is the Coulomb potential, and g is the parameter of Coulomb potential which stands the strength of affecting the polaron. One can write the momentum and position of an electron using linear combination operators [16] given by where λ � ����� eB/2Z √ is the variational parameter and j � x, y.A Fourier series expansion is carried out to obtain the binding potential of the last term in equation (1): where A represents the area of graphene. e linear combination operators, equation (6), are substituted into equation (1), and the following LLP unitary transformation is performed: Following this transformation, the result of equation (1) can be rewritten as 2 Advances in Condensed Matter Physics Taking the wave function of the system as |0〉 is the unperturbed zero-phonon state satisfying the operation a k | 0〉 � 0. It is straightforward to show that the expected value of the polaron subsystem is e resultant eigenvalue of the electronic kinetic energy term corresponding to the zero-Landau level can be written as where β � ±1. e above formula corresponds to the band exponent of the conduction band and the valence band.
In the same way, the phonon energy, electron-phonon interaction energy, and Coulomb potential energy eigenvalues for the zero-Landau level are given by One can thus write the ground-state eigenenergy value, E 0 , of the entire system as where f * k and f k are the variational parameters. From equation (16), one can use the variation method to obtain the ground-state energy of weakly coupled bound magnetopolarons in MG: where d represents the distance z between the substrate and the graphene monolayer and k c is the cutoff wave number.

Results and Discussion
ree polar materials SiC, HfO 2 , and h − BN were selected as substrates for numerical calculation to analyse the effects of distance z, between graphene and substrates, the binding parameters g, magnetic field intensity B, and cutoff wave number k c of phonons on the ground-state energy of the weakly coupled bound magnetopolarons in MG. Table 1 details the substrate parameters used for the calculations. Figures 2 and 3 show the relationship between magnetic field strength and cutoff wave number in MG when z � 1nm and g � 0.2 for the three substrates. One can clearly observe that the original zero-Landau energy level is split into two  symmetric energy bands. e splitting is the result of the well-known Lorenz effect and polaron effect. Additionally, it is trivial to see that the substrate material has an effect on the energy level value and that the absolute value of energy increases with increasing magnetic field strength and cutoff wavenumber. One notable difference is the apparent levelling off of ground-state energy once a threshold cutoff wavenumber is reached. e magnetic field can cause the splitting degree of ground-state energy, which is consistent with the results of references [5,18,20,21]. We also calculated the effect of substrate material on the ground-state energy of graphene polaron. e greater dielectric constant of the substrate material can cause the stronger splitting of the two branches of the ground-state energy of the polaron.
is is very important for the study of the surface optical polaron of the graphene on the substrate and also provides a theoretical reference for the experiment. Figures 4 and 5 show the dependence of the ground-state energy of the weakly coupled bound magnetopolarons on the substrate distance z and Coulomb bound potential parameters. Figure 4 reveals that the absolute value of the ground-state energy decreases gradually as the substrate spacing increases, although only by ∼50 meV over 8 nm. From equation (17), it is straightforward to see that the interaction between electrons on the graphene surface and phonons on the substrate surface is weakened with increased spacing. As can be seen from Figure 4, the band near the Dirac point splits into two opposing bands with linearly increasing absolute energy E 0 . At constant distance z and cutoff wave number k c , the enhancement of g increases the Coulomb binding energy and thereby increases the absolute energy of the ground state. is phenomenon is in agreement with the results of Ref. [9][10][11]25]. However, we find that the parameter of Coulomb potential is an important factor in adjusting the polaron ground energy and causing the increase in splitting of energy level.

Conclusion
e ground-state energy of bound magnetopolarons in a single layer of graphene is calculated using the linear combination operators and LLP variational method, in the context of weakly coupling of electron and surface optical   (SO) phonon. e results show that the absolute value of the ground-state energy decreases when the magnetic field strength, bound potential strength, or cutoff wave number are increased. On the other hand, when one increases the distance between the graphene monolayer and substrate, the strength of the interaction decreases and the value of the ground-state energy is reduced. ese results provide new ideas and methods for further understanding polaron effects in graphene and provide [25] theoretical basis for the preparation of functional quantum optical devices based on graphene structures.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.