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

Authors: Zhao-Hua Ding; Yan-Bo Geng; Ying Zhao; Yong Sun; Jing-Lin Xiao
Journal: Advances in Condensed Matter Physics
 (2020)

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## Abstract

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.


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## Body

## 1. Introduction

Monolayer graphene (MG) is the archetypal example of a zero-energy gap semiconductor. The effective mass of charge carriers near the Dirac point, a phenomenon characterized by linearly dispersive band structure, is zero. Because of its unique two-dimensional 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 [1–10, 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. [11–13] found a regulatory mechanism of zero-Landau level splitting and that a band gap opens as a direct result of polarons. Xiao et al. [14–16] discussed the effects of Coulomb impurities and polar substrates on the splitting of polaron zero-Landau level in graphene. Wang et al. [17–21] 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 [22–24] 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.


## 2. 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.The Hamiltonian of such a system as shown in Figure1, accounting for electron-SO phonon interactions can be written as(1)H=He+Hph+He−ph−gr,where(2)He=VF0πx−iπyπx+iπy0,(3)πx=px−eBy2,πy=py+eBx2,(4)Hph=∑k,vℏωso,vak+ak,(5)He−ph=∑k,vMk,va−k++akeik⋅r,Mk,v=Q2ηℏωso,v2ε0k1/2e−kz,where Q is the charge of the electron, η=k0−k∞/k∞+1k0+1 represents the dielectric constant of the substrate, k∞ and k0 are the high- and low-frequency dielectric constants, respectively, ε0 is the frequency of phonons,ωso,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.Figure 1
Schematic diagram of magnetopolaron in graphene with Coulomb impurities under the substrate.One can write the momentum and position of an electron using linear combination operators [16] given by(6)pj=ℏλ2bj++bj,rj=i2λbj−bj+,where λ=eB/2ℏ 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):(7)1r=2πA∑k1kexp−ik⋅r,where A represents the area of graphene. The linear combination operators, equation (6), are substituted into equation (1), and the following LLP unitary transformation is performed:(8)U1=exp−i∑kk⋅rak+ak,U2=exp∑kfkak+−fk∗ak.Following this transformation, the result of equation (1) can be rewritten as(9)H′=U2−1U1−1HU1U2=VF0ℏλ/2bx++bx−∑kℏkxak++fk∗ak+fk−ieB/2λby−by+/2−iℏλ/2bx++bx+i∑kℏkyak++fk∗ak+fk−ieB/2λbx−bx+/2ℏλ/2bx++bx−eB/8λbx−bx+−∑kℏkxak++fk∗ak+fk−iℏλ2by++by−i∑kℏkyak++fk∗ak+fk−ieB8λby−by+0+∑k,vℏωso,vak+ak+ak+fk+fk∗ak+fk∗fk+∑k,vMk,v∗ak++fk+exp−k24λexp−∑jkjbj+2λexp∑jkjbj2λ+∑k,vMk,vak+fkexp−k24λexp∑jkjbj+2λexp−∑jkjbj2λ−g2πA∑k1kexp∑jkjbj2λexp−∑jkjbj+2λexp−k22λ2.Taking the wave function of the system as(10)ψn0=12Cn′n−10Cnn0,Cn′=1−δn,0,Cn=1+δn,0,bj+n=n+1n+1,bjn=nn−1,0 is the unperturbed zero-phonon state satisfying the operation ak0=0.It is straightforward to show that the expected value of the polaron subsystem is(11)Ee,n2=0ψnHe′2ψn0=12VF2Cn′2∑kℏ2k2fk∗fk+Cn2∑kℏ2k2fk∗fk.The resultant eigenvalue of the electronic kinetic energy term corresponding to the zero-Landau level can be written as(12)Ee,0=±Ee,n2=βVF∑kℏkfk∗fk,where β=±1. The 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(13)Eph,0=β∑k,vℏωso,vfk∗fk,(14)Ee−ph,0=βMk,v∗fk∗exp−k24λ2+Mk,vfkexp−k24λ2,(15)Er,0=−2πgA∑k1kexp−k24λ2.One can thus write the ground-state eigenenergy value,E0, of the entire system as(16)E0=β∑k,vVFℏkfk∗fk+ℏωso,vfk∗fk+Mk,v∗fk∗exp−k24λ2+Mk,vfkexp−k24λ2−2πgA1kexp−k24λ2,where fk∗ and fk 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:(17)E0=±∫0kcQ2ηℏωso,v⋅e−2kz4πε0VFℏk+ℏωso,v+2πgA1kexp−k24λ2dk,where d represents the distance z between the substrate and the graphene monolayer and kc is the cutoff wave number.


## 3. Results and Discussion

Three polar materialsSiC, HfO2, 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 kc 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.Table 1
Experimental parameters used in the numerical calculations.
Quantity (units)SiCHfO2h−BNk0ε09.722.05.1k∞ε06.55.04.1ℏωso,1meV11619101ℏωso,2meV16753195Figures2 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. The 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. The 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. The greater dielectric constant of the substrate material can cause the stronger splitting of the two branches of the ground-state energy of the polaron. This 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.Figure 2
Variable magnetic fieldB versus the ground energy E0.Figure 3
Variable cutoff wavenumberkc versus the ground energy E0.Figures4 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 E0. At constant distance z and cutoff wave number kc, the enhancement of g increases the Coulomb binding energy and thereby increases the absolute energy of the ground state. This phenomenon is in agreement with the results of Ref. [9–11, 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.Figure 4
Variable substrate distancez versus the ground energy E0.Figure 5
Variable parameter of Coulomb potentialg versus the ground energy E0.


## 4. Conclusion

The 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. The 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. These 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.



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