Magneto-transport properties of B-, Si- and N-doped graphene

Journal Pre-proof Magneto-transport properties of B-, Si- and N-doped graphene Po-Hsin Shih, Thi-Nga Do, Godfrey Gumbs, Danhong Huang, Thanh Phong Pham, Ming-Fa Lin PII:

S0008-6223(19)31335-1

DOI:

https://doi.org/10.1016/j.carbon.2019.12.088

Reference:

CARBON 14924

To appear in:

Carbon

Received Date: 16 August 2019 Revised Date:

18 December 2019

Accepted Date: 28 December 2019

Please cite this article as: P.-H. Shih, T.-N. Do, G. Gumbs, D. Huang, T.P. Pham, M.-F. Lin, Magnetotransport properties of B-, Si- and N-doped graphene, Carbon (2020), doi: https://doi.org/10.1016/ j.carbon.2019.12.088. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Author Contribution Statement Title: Magneto-transport properties of B-, Si- and N-doped graphene Po-Hsin Shih: Methodology, Software, Writing - Original Draft, Investigation Thi-Nga Do: Conceptualization, Methodology, Formal analysis, Writing - Original Draft, Investigation, Supervision Godfrey Gumbs: Conceptualization, Writing - Review & Editing Danhong Huang: Writing - Review & Editing Thanh Phong Pham: Writing - Review & Editing Ming-Fa Lin: Methodology, Resources, Funding acquisition

Magneto-transport properties of B-, Si- and N-doped graphene Po-Hsin Shiha , Thi-Nga Dob,c∗ , Godfrey Gumbsd , Danhong Huange , Thanh Phong Phamb,c , Ming-Fa Lina a Department b Laboratory

of Physics, National Cheng Kung University, Tainan, Taiwan 701

of Magnetism and Magnetic Materials, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

c Faculty

of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam

d Department

of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, New York 10065, USA

e US

Air Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force Base, New Mexico 87117, USA

January 3, 2020

Abstract The effect due to doping by B, Si, N atoms on the magneto-transport properties of graphene is investigated using the generalized tight-binding model in conjunction with the Kubo formula. The crucial electronic and transport properties are greatly diversified by different types of dopant and doping concentrations. The effect of these guest atoms includes opening a band gap, thereby giving rise to rich Landau level energy spectra and consequently a unique quantum-Hall conductivity. The Fermienergy dependent quantum-Hall effect appears as a step structure having both integer and half-integer plateaus. Doping with Si leads to an occurrence of a zero quantum-

1

Hall conductivity, unlike the plateau sequence for pristine graphene. The predicted dopant- and concentration-enriched quantum-Hall effect for doped graphene can provide useful information for magneto-transport measurements, possible technological and even metrology applications.

* Corresponding author. E-mail address: [email protected] (T. N. Do)

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1. Introduction Understanding the physics origin of the quantum-Hall effect (QHE) has led to a prolonged effort by numerous theoretical and experimental researchers. In particular, the Hall conductivity of condensed matter systems under a strong external magnetic field has received special attention [1-20]. Various behaviors of magnetic-field-quantized Landau levels (LLs) have been employed to explain the QHE. The electron filling of LLs and wave functions have been proven to connect to the integer and fractional quantum-Hall states. In this work, we will systematically explore the QHE in Si-, B-, and N-doped graphene with various doping concentrations using the tight-binding model to determine Landau energy bands. Ever since the time when graphene was experimentally discovered, its exotic QHE has attracted a lot of studies from both experimental and theoretical aspects [1-14]. Monolayer graphene shows amazing physical properties because of its single-atom thickness and hexagonal symmetry. The magnetically-quantized LLs of the isotropic Dirac cone form a distinctive energy spectrum for which the conduction and valence energy-level separations depend on both field strength and quantum number of specified LLs. Such an established √ dependence, Bz nc,v , has been verified through experiment using scanning tunneling spectroscopy (STS) [21], optical spectroscopies [22], and transport measurement [5-6]. Interestingly, it has been realized that the inter-LL transitions obey a specific selection rule associated with the equivalence of two sublattices in the crystal structure. Only the excitations between valence and conduction LLs whose quantum numbers satisfy ∆ n = |nv − nc | = 1 are allowed, resulting in exotic QHE. Magnetic transport measurements have revealed the intriguing half-integer quantum-Hall conductivity σxy = (m + 1/2)4e2 /h (m is an integer and the factor of 4 represents the spin and sublattice degeneracy) in monolayer graphene [5-6]. Such an extraordinary quantization phenomenon is related to the quantum anomaly of the n = 0 LL coming from the Dirac point [1]. Currently, scientists are in search of novel materials yielding unusual physical phenomena and having potential for device applications. By introducing graphene-based defect 3

systems, such as substituting impurities or guest atoms in a hexagonal carbon lattice, one could effectively manipulate the fundamental properties inherent to graphene. Among various pertinent guest atoms, the carbon host atoms have been successfully substituted by Si [23], B [24-25], and N [26-27] in chemical vapor deposition (CVD) or arc discharge experiments. This doping eliminates the equivalence of two sublattices, causing dramatic changes in physical properties, including the energy-gap opening and the deformation of the original Dirac cone. These features of doped graphene have been verified by a number of experimental studies [28-31]. Previous studies have demonstrated the emergence of different ionization potentials and non-uniform hopping integrals [32-33]. Specifically, for Si-, B-, and N-doped graphene, the π bonding extending on a hexagonal lattice is significantly distorted. As a result, the low-lying energy band is determined by both the dopants and C atoms. Additionally, based on this unusual band structure due to doping, the magnetically quantized LLs have been theoretically predicted to be feature-rich and unique [34]. These facts make it promising for the QHE of such special LL energy spectra to yield abundant and unique features. It is critical to mention that, though substitutional doping might reduce the mobility of graphene [35], the QHE could still be observed in doped graphene at suitable temperature and magnetic field [36]. In this work, we provide a comprehensive investigation of the traditional QHE which originates from discrete LLs formed in doped graphene in the presence of a uniform perpendicular magnetic field. The modification of crystal structure by doping is expected to enrich the quantum Hall conductivity as the significant influence of lattice geometry on the transport properties has been reported in the literature [37-38]. It is worth mentioning that this effect is distinct from the anomalous Hall effect (AHE) in the absence of an external magnetic field, which has recently become the focus of much interest [39-43]. The competition between the traditional QHE and the AHE in the presence of external magnetic field is a curious phenomenon which has gained significant attention from scientists [39-41]. Previously, it has been reported that the AHE still appears in the regime of weak magnetic field [39-41]. That is, the ordinary and anomalous Hall effects may simultaneously exist in

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the presence of an external magnetic field; which phenomenon is dominant depends on the field strength. The rest of this paper is outlined as follows. In Sec. 2, we employ the tight-binding method to solve for the eigenvalues and eigenfunctions of the magnetic Hamiltonian, and then use these results to calculate the quantum-Hall conductivity by means of the Kubo formula. This procedure enables us to identify allowed transition channels through the aforementioned magneto-electronic selection rules, leading to an understanding of Fermienergy and magnetic-field dependent quantum conductivity. Fundamental properties of doped graphene are greatly enriched by various types of dopants and their concentrations. In Sec. 3, we investigate the QHE with respect to Si, B, and N dopants for 2% and 12.5% doping ratios. Our results demonstrate a robust correlation between the unusual QHE and the dopant- and concentration-dependent rich LLs in doped graphene systems. Our theoretical predictions provide essential information for future experimental verification of the QHE in these graphene-based defect materials. Section 4 is devoted to our concluding remarks.

2. Model and Computational Methods We have employed the tight-binding method and the dynamic Kubo formula from linear response theory to carefully investigate the rich magneto-transport properties of graphene doped with Si, B, and N atoms. The tight-binding model [44] has been demonstrated to be an appropriate method to explore fundamental magnetic properties of layered materials, including the magneto-electronic, magneto-optical, quantum transport properties and magneto-Coulomb excitations in conjunction with the Kubo formula and modified randomphase approximation. This model could give results which are in good agreement with those obtained by other methods, such as the band structure from first-principles calculations [45], the Landau level energy spectra from STS measurements [46-47], magneto-optical spectra from optical spectroscopy [48], and the quantum Hall conductivity from transport measurements [49].

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The crystal structures of doped graphene for two dopant concentrations are presented in Figs. 1(a) and 1(b). For a 1:2n2 concentration doping (n is the cell multiplicity), the super cell is obtained by expanding the original unit cell of pristine graphene n times along the lattice vectors a~1 and a~2 , that is, (na~1 , na~2 ). In this work, we choose a rectangular super cell as marked by the red lines for convenience in considering the effect of a magnetic field. A single supercell comprises 2n2 sublattices of two types, Ai and Bi , in which i represents the lattice site. Both C-2pz host orbital and the guest orbitals of B − 2pz , Si − (2pz , 3pz ), and N − 2pz play the key roles in governing the crucial characteristics in the low-energy range. The Hamiltonian matrix elements include non-uniform bond lengths, site energies and nearest-neighbor hopping integrals and are written as [34]

∗ Hj+2nl−2n,j+2nl−n = Hj+2nl−n,j+2nl−2n = tj+2nl−2n,j+2nl−n f1 ∗ Hj+2nl−2n,m(j)+2nl−n+1 = Hm(j)+2nl−n+1,j+2nl−2n = tj+2nl−2n,m(j)+2nl−n+1 f2 ∗ Hj+2nl−2n,j+2n[m(l)+1]−n = Hj+2n[m(l)+1]−n,j+2nl−2n = tj+2nl−2n,j+2n[m(l)+1]−n f3

H1,1 = B/Si/N −C ,

(1)

where the hopping term is defined as

tα,β =

   γB/Si/N −C , if α or β equal 1   γC−C ,

(2)

otherwise .

In this notation, f1,2,3 is defined in terms of the wave vector ~k and the vectors connecting ~ 1,2,3 through ei~k·R~ 1,2,3 . Moreover, m(k) = k + n − 2 mod the nearest-neighbor lattice sites R n is modulo function, and j, l are integers (j, l = 1, 2, 3, ..., n). The C-C and B/Si/N-C bond lengths have been verified as being slightly different due to doping, especially the buckling effect of the guest atoms in graphene [32-33]. For Si dopant, the guest atoms are located at a distance of dSi above graphene sheet, implying a significant adjustment of the π bonding on a hexagonal lattice. The consequential nonuniform nearest-neighbor hopping integrals and site energies associated with the major 6

Figure 1: (Color online) Crystal structures of doped graphene for chosen doping concentrations of (a) 12.5% and (b) 2% dopants. The green and orange spheres denote, respectively, the host C and guest Si/B/N atoms. 2pz orbitals of the C host atoms and the minor 2pz /3pz orbitals of the guest atoms lead to extraordinary electronic properties. The parameters are appropriately fitted, as shown in Table I, so that the energy bands calculated by the tight-binding and first-principles methods agree with each other. The effect of a uniform perpendicular magnetic field on the system could be treated with R Rm the use of the vector potential-dependent Peierls phase, given by ∆Gmm0 = 2π A(r) · dr φ0 Rm0

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Table 1: Tight-binding parameters for pristine and doped graphene Parameter

Pristine graphene

Si-doped

B-doped

N-doped

bond lengths (˚ A)

b = 1.42

bC−Si = 1.7

bC−B = 1.42

bC−N = 1.42

buckled distance (˚ A)

d=0

dSi = 0.93

dB = 0

dN = 0

hopping (eV)

t =-2.7

site energy (eV)

=0

tC−Si = −1.3 tC−B = -2.17 Si =1.3

B = 2.33

tC−N = -2.1 N = -2

(φ0 = hc/e is the flux quantum), in the nearest-neighbor hopping integral [34]. Such a phase is position-dependent so that the unit cell will be extended along the armchair direction. Consequently, the magnetic Hamiltonian becomes a large matrix whose periods strongly depend on the density of guest atoms and vector potential. This magneticHamiltonian matrix can be solved efficiently by using the diagonalization method to obtain the LL energy spectrum and sub-envelope functions [44]. Interestingly, the amplitudes of the sub-envelope functions on distinct sublattices within an enlarged unit cell can be regarded as electron spatial distributions, and therefore, they are used to analyze the main features of the LL wave functions. Within the linear-response theory, the Hall conductivity is calculated from the Kubo formula, yielding [50]

σxy

hα|ˆ v x |βihβ|ˆ v y |αi ie2 ~ P P (fα − fβ ) . = 2 S (Eα − Eβ ) + Γ20 α α6=β

(3)

In this notation, |αi and Eα are the LL state and LL energy, respectively, fα,β is the FermiDirac distribution functions, S the area of a supercell, and Γ0 a broadening parameter for LL. Furthermore, the velocity operators v ˆx,y directly determine the allowable inter-LL transitions, and can be evaluated directly from the gradient approximation [51]. The defect effect in doped graphene can lead to a significant modification of quantum conductivities. We note that the inter-valley interaction is neglected in our calculations of the magnetotransport coefficients of doped graphene. In graphene [52-53], the intra-valley scattering, 8

but not that due to inter-valley, is the dominant scattering mechanism. In general, the weak inter-valley scattering only plays the role of establishing the equilibrium between the two valleys. Explicitly, the application of an external magnetic field may induce distinct chemical potentials at the two valleys, K and K0 . However, the inter-valley scattering can adjust the electron populations in the two valleys, so that the two valleys eventually reach a uniform chemical potential. For silicene [54-55], it was argued that the inter-valley scattering from nonmagnetic impurities is highly suppressed by time-reversal symmetry. Therefore, the physics should be effectively governed by single-Dirac-cone like behavior. On the other hand, one could effectively suppress the inter-valley scattering in graphene by sandwiching a monolayer graphene sheet between clean hexagonal boron nitride layers, as done in Ref. [56]. These imply the inconsequential effect of the inter-valley interaction on the magneto-transport properties of (Si, B, N)-doped graphene in our present work.

3. Results and Discussion The effect of doping for chosen dopants and concentrations remarkably diversifies the electronic properties of graphene. The substitution by guest atoms opens a direct band gap in monolayer graphene since it breaks down the lattice symmetry, as seen in Fig. 2. The magnitude of the energy gap (Eg ) is greatly affected by dopants and doping concentration, as demonstrated in Table II for pristine graphene and Si-, B-, and N-doped systems with 2% and 12.5% guest atoms. Additionally, the significant influence of doping on Fermi energy is worthy of reiteration (Table III). For pristine graphene, the Fermi energy is equal to zero (Fig. S1 of the Supplemental Material) and it remains the same in the presence of Si dopants. On the contrary, the Fermi energy shifts down to the valence band or up to the conduction band for B- and N-doped systems, respectively. These observations are attributed to the fact that a B atom acts as a p-dopant while an N atom behaviors as an n-dopant since a B/N atom has one less/more electron than a C atom. The 3D plots of energy bands with separated occupied and unoccupied states are presented in Fig. S2 of the Supplemental Material. Our numerical results are in agreement with previous theoretical

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Figure 2: (Color online) Energy bands of doped graphene with Si, B and N guest atoms for concentrations of 2% [(a)-(c)] and 12.5% [(d)-(f)]concentrations. calculations based on the first-principles method [57-59]. The prediction of the dopantand concentration-dependent band structures in doped graphene could be very useful for graphene electronic applications. In the presence of a uniform perpendicular magnetic field, the quantized LLs show unusual characteristics due to doping. Because of the opening of a band gap, the conduction and valence LL energy spectra are separated from each other, as shown in Figs. 3(a) and 3(d) for the magnetic-field-dependent energy spectra of Si-doped graphene. This separation

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Table 2: Band gap of doped graphene Eg (eV) Si-doped

B-doped

N-doped

2%

0.19

0.12

0.13

12.5%

0.74

0.68

0.68

Table 3: Fermi levels of doped graphene Fermi levels (eV) Si-doped (EF (Si)) B-doped (EF (B)) N-doped (EF (N )) 2%

0

-0.78

0.84

12.5%

0

-1.76

1.81

remains invariant with changing field strength so that it has no influence on the inter-LL transition between the valence and conduction bands. Each LL is four-fold degenerate, except for the lowest conduction and highest valence states in the vicinity of zero energy whose degeneracy is reduced by half. For Si-doped graphene, the Fermi level sits close to zero energy. Therefore, the conduction and valence states are unoccupied and occupied, respectively. On the other hand, they are either unoccupied for B or occupied for N dopants due to the shift of Fermi energy. These two doubly degenerate LLs are split from the original zero-energy LL (n = 0) of pristine graphene due to the substitution of guest atoms. The Bz -dependence of LLs near the Fermi energy in B- and N-doped systems needs a detailed explanation. There, the density of LL states is very high, as clearly displayed in Figs. 3(b) and 3(c) for 2% as well as 3(e) and 3(f) for 12.5% concentrations of B and N guest atoms. Especially for dopant density of 12.5%, the LLs exhibit small splitting at higher and deeper energy ranges, altering the state-degeneracy degree of freedom. These phenomena play a key role in the unique properties of quantum-Hall conductivity discussed in detail below. The contribution of guest and host atoms to the Hall conductivity through the velocity matrix elements in Eq. (3) could be understood based on the amplitude of LL wave functions on the sublattices, as demonstrated in Figs. 4(I) through 4(III). Our numerical calculations 11

Figure 3: (Color online) The Bz -dependent LL energies of doped graphene with Si, B and N dopants for concentrations of 2% [(a)-(c)] and 12.5% [(d)-(f)]. show that the A sublattice, which consists of guest atoms, obviously dominates the conduction bands of (Si, B)-doped systems as well as the valence band of N-doped graphene, and vice versa for the B sublattice. Especially for a Si dopant with 3pz orbital, its spatial distribution is even greater than that of the C atoms. The quantum number, which is 12

Figure 4: (Color online) The LL energies and wavefunctions of the first few levels for (I) Si, (II) B, and (III) N guest atoms with a 12.5% concentration at Bz = 20 T. The blue and red bars represent LLs localized at 1/6 and 2/6 centers, respectively. critical in understanding the selection rule of inter-LL transitions, is determined based on the nodes of the LL spatial distribution. Here, LL wave functions are only localized in some specific regions within the field-enlarged unit cell, particularly at 1/6, 2/6, 4/6 and 5/6 guiding centers for a (kx = ky = 0) state. Furthermore, LLs localized at 1/6 and 4/6 c,v (2/6 and 5/6) are equivalent in the main characteristics and classified as nc,v 1 (n2 ) group.

This classification holds true for all doped systems, regardless of dopant type and doping concentration. The doubly-degenerate conduction LL belongs to nc1 group for Si and B dopants but to nc2 group for N, while the opposite is true for the valence LL. The quantum-Hall effect of doped graphene displays unusual and specific properties that may be manipulated by the degree to which the sample is doped. The Fermi energydependent Hall conductivity quantization under an external magnetic field exhibits a step

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structure, as shown in Fig. 5 for the inverse quantum-Hall conductivity 1/σxy of three different guest atoms with 2% and 12.5% concentrations. The corresponding plots for σxy could be found from Fig. S3 of the Supplemental Material. With varying Fermi energy (EF ), a plateau is formed as one LL is kept occupied. The inter-LL transition between the occupied and unoccupied levels requires a specific selection rule where the oscillation mode of the wave functions on the sublattices must be compatible for the initial and final LL states. We find this rule as ∆n = ±1 & 0 for 2% but ∆n = ±1 for 12.5% doped graphene.

Figure 5: (Color online) The EF -dependent inverse quantum-Hall conductivity 1/σxy of Si-, B-, and N-doped graphene at Bz = 20 T. The solid (dashed) curve represents the dopant density of 2% (12.5%). The inset shows the quantum-Hall conductivity σxy of 12.5% B-doped graphene. The conductivity is quantized following a common sequence of σxy = {0; 2(2m+1)e2 /h} (m is an integer) for all dopants. It should be noticed that EF on the x-axis of Fig. 5 represents the energy shift from the original Fermi level in the absence of doping. The 14

induced band gap from doping is reflected in the discontinuity of 1/σxy near zero energy, as shown for Si guest atoms by solid and dotted red curves. This is attributed to the appearance of zero conductivity in Figs. S3(a) and S3(d) of the Supplemental Material. Meanwhile, there exist two half-integer steps of 2e2 /h height for quantum-Hall conductivity σxy near zero energy, in addition to the conventional steps of 4e2 /h. This is in contrast with that for pristine graphene where the two half-integer steps occur exactly at zero energy [5]. Furthermore, the width of this zero-energy plateau becomes a measure for the size of the band gap, which strongly depends on dopant types and doping concentrations. The quantum-Hall conductivities in the vicinity of the Fermi levels in the presence of B and N dopants deserve careful scrutiny since they could be identified in magnetotransport measurements. For sufficiently small doping concentration, e.g., 2% as shown by solid blue and green curves in Fig. 5 (also in Figs. S3(b) and S3(c)), the quantized sequence of quantum-Hall conductivity remains unchanged even at higher or lower Fermi energies. The doping effect might not be strong enough to remarkably alter the essential physical properties of the system. However, with increasing dopant density, the splitting of LLs near Fermi energies for B and N dopants will give rise to significant changes in the quantum-Hall effect. There, the wider and narrower conductivity plateaus are interspersed in the spectra, referring to the inset plot for 12.5% B-doped graphene (also in Figs. S3(e) and S3(f)), in which the narrower one is related to the splitting of degenerate LLs. Accordingly, each step is reduced by half in height, obeying a new quantized sequence of σxy = m (2e2 /h) instead of σxy = m (4e2 /h). Our theoretical predictions for doped graphene with various dopants should provide useful insight for future magneto-transport measurements, as done for monolayer and fewlayer graphene [5-6, 8-11]. Specifically, we have found the half-integer QHE as predicted for monolayer graphene by different theory groups [1-3], situated on thin graphite films [4], and verified by experimental measurements [5-6]. Such a special quantization in undoped graphene is attributed to the quantum anomaly of the n = 0 LL corresponding to the Dirac point [1]. The results obtained in this work by the well-developed theoretical framework,

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however, are related to splitting of LLs by dopant interaction and expected to be confirmed experimentally.

4. Concluding Remarks In conclusion, the generalized tight-binding model and the Kubo formula are employed to explore the interesting QHE in doped graphene for Si, B, and N guest atoms with various concentrations. The theoretical framework takes into consideration the substantial factors of the doping system, including non-uniform bond lengths, site energies, hopping integrals, and external field. This method could be extended to investigate many other condensedmatter materials. Doped graphene is an emergent 2D binary compound material, which presents diverse electronic and transport properties under magnetic quantization. The doping effect can produce a band gap of different sizes, leading to remarkable changes in the main features of LLs and thus the magneto-transport properties. The Fermi-energy dependent QHE exhibits both integer and half-integer conductivities. The step structure could be controlled by changing dopant or doping concentration, in which the latter has more significant influence. Our theoretical predictions present useful technological and device information and could be verified by magneto-transport measurements. Acknowledgments The authors thank the Ministry of Science and Technology of Taiwan (R.O.C.) under Grant No. MOST 105-2112-M-017-002-MY2. T. N. Do would like to thank the Ton Duc Thang University. D.H. would like to thank support from the AFOSR and from the DoD Lab-University Collaborative Initiative (LUCI) Program.

References [1] V.P. Gusynin, S.G. Sharapov, Unconventional Integer Quantum Hall Effect in Graphene, Phys. Rev. Lett. 95 (2005) 146801. doi:10.1103/PhysRevLett.95.146801. [2] N.M.R. Peres, F. Guinea, A.H. Castro Neto, Electronic properties of two-dimensional carbon, Annals of Physics. 321 (2006) 1559–1567. doi:10.1016/j.aop.2006.04.006. 16

[3] A. Tsukuda, H. Okunaga, D. Nakahara, K. Uchida, T. Konoike, T. Osada, Quantum Hall Transport across Monolayer-Bilayer Boundary in Graphene, J. Phys.: Conf. Ser. 334 (2011) 012038. doi:10.1088/1742-6596/334/1/012038. [4] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Electric Field Effect in Atomically Thin Carbon Films, Science. 306 (2004) 666–669. doi:10.1126/science.1102896. [5] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature. 438 (2005) 197–200. doi:10.1038/nature04233. [6] Y. Zhang, Y.-W. Tan, H.L. Stormer, P. Kim, Experimental observation of the quantum Hall effect and Berry’s phase in graphene, Nature. 438 (2005) 201–204. doi:10.1038/nature04235. [7] E.

McCann,

Effect

in

a

V.I.

Fal’ko,

Graphite

Landau-Level

Bilayer,

Phys.

Degeneracy

and

Quantum

Rev.

96

(2006)

Lett.

Hall

086805.

doi:10.1103/PhysRevLett.96.086805. [8] T. Taychatanapat, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, Quantum Hall effect and Landau-level crossing of Dirac fermions in trilayer graphene, Nature Physics. 7 (2011) 621–625. doi:10.1038/nphys2008. [9] S.H. Jhang, M.F. Craciun, S. Schmidmeier, S. Tokumitsu, S. Russo, M. Yamamoto, Y. Skourski, J. Wosnitza, S. Tarucha, J. Eroms, C. Strunk, Stacking-order dependent transport properties of trilayer graphene, Phys. Rev. B. 84 (2011) 161408. doi:10.1103/PhysRevB.84.161408. [10] L. Zhang, Y. Zhang, J. Camacho, M. Khodas, I. Zaliznyak, The experimental observation of quantum Hall effect of l=3 chiral quasiparticles in trilayer graphene, Nature Physics. 7 (2011) 953–957. doi:10.1038/nphys2104.

17

[11] A. Kumar, W. Escoffier, J.M. Poumirol, C. Faugeras, D.P. Arovas, M.M. Fogler, F. Guinea, S. Roche, M. Goiran, B. Raquet, Integer Quantum Hall Effect in Trilayer Graphene, Phys. Rev. Lett. 107 (2011) 126806. doi:10.1103/PhysRevLett.107.126806. [12] M. Koshino, E. McCann, Landau level spectra and the quantum Hall effect of multilayer graphene, Phys. Rev. B. 83 (2011) 165443. doi:10.1103/PhysRevB.83.165443. [13] S. Yuan, and

R. Rold´an,

ABC-stacked

M.I. Katsnelson,

trilayer

graphene,

Landau level spectrum of ABA-

Phys.

Rev.

B.

84

(2011)

125455.

doi:10.1103/PhysRevB.84.125455. [14] T.-N. Do, C.-P. Chang, P.-H. Shih, J.-Y. Wu, M.-F. Lin, Stacking-enriched magnetotransport properties of few-layer graphenes, Physical Chemistry Chemical Physics. 19 (2017) 29525–29533. doi:10.1039/C7CP05614A. [15] L. Li, F. Yang, G.J. Ye, Z. Zhang, Z. Zhu, W. Lou, X. Zhou, L. Li, K. Watanabe, T. Taniguchi, K. Chang, Y. Wang, X.H. Chen, Y. Zhang, Quantum Hall effect in black phosphorus two-dimensional electron system, Nature Nanotechnology. 11 (2016) 593–597. doi:10.1038/nnano.2016.42. [16] R.S. Meena, S.K. Singh, A. Pal, A. Kumar, R. Jha, K.V.R. Rao, Y. Du, X.L. Wang, V.P.S. Awana, High field (14 T) magneto transport of Sm/PrFeAsO, Journal of Applied Physics. 111 (2012) 07E323. doi:10.1063/1.3675156. [17] R. Singha, S. Roy, A. Pariari, B. Satpati, P. Mandal, Magnetotransport properties and giant anomalous Hall angle in the half-Heusler compound TbPtBi, Phys. Rev. B. 99 (2019) 035110. doi:10.1103/PhysRevB.99.035110. [18] F. Yang, Z. Zhang, N.Z. Wang, G.J. Ye, W. Lou, X. Zhou, K. Watanabe, T. Taniguchi, K. Chang, X.H. Chen, Y. Zhang, Quantum Hall Effect in ElectronDoped Black Phosphorus Field-Effect Transistors, Nano Lett. 18 (2018) 6611–6616. doi:10.1021/acs.nanolett.8b03267.

18

[19] R. Ma, S.W. Liu, W.Q. Yang, M.X. Deng, Quantum Hall effect in monolayer and bilayer black phosphorus, EPL. 119 (2017) 37005. doi:10.1209/0295-5075/119/37005. [20] J. Yin, S. Slizovskiy, Y. Cao, S. Hu, Y. Yang, I. Lobanova, B.A. Piot, S.-K. Son, S. Ozdemir, T. Taniguchi, K. Watanabe, K.S. Novoselov, F. Guinea, A.K. Geim, V. Fal’ko, A. Mishchenko, Dimensional reduction, quantum Hall effect and layer parity in graphite films, Nat. Phys. 15 (2019) 437–442. doi:10.1038/s41567-019-0427-6. [21] D.L. Miller, K.D. Kubista, G.M. Rutter, M. Ruan, W.A. de Heer, P.N. First, J.A. Stroscio, Observing the Quantization of Zero Mass Carriers in Graphene, Science. 324 (2009) 924–927. doi:10.1126/science.1171810. [22] Z. Jiang, E.A. Henriksen, L.C. Tung, Y.-J. Wang, M.E. Schwartz, M.Y. Han, P. Kim, H.L. Stormer, Infrared Spectroscopy of Landau Levels of Graphene, Phys. Rev. Lett. 98 (2007) 197403. doi:10.1103/PhysRevLett.98.197403. [23] W. Zhou, M.D. Kapetanakis, M.P. Prange, S.T. Pantelides, S.J. Pennycook, J.-C. Idrobo, Direct Determination of the Chemical Bonding of Individual Impurities in Graphene, Phys. Rev. Lett. 109 (2012) 206803. doi:10.1103/PhysRevLett.109.206803. [24] L.S. Panchakarla, K.S. Subrahmanyam, S.K. Saha, A. Govindaraj, H.R. Krishnamurthy, U.V. Waghmare, C.N.R. Rao, Synthesis, Structure, and Properties of Boron- and Nitrogen-Doped Graphene, Advanced Materials. 21 (2009) 4726–4730. doi:10.1002/adma.200901285. [25] Z.-H. Sheng, H.-L. Gao, W.-J. Bao, F.-B. Wang, X.-H. Xia, Synthesis of boron doped graphene for oxygen reduction reaction in fuel cells, Journal of Materials Chemistry. 22 (2012) 390–395. doi:10.1039/C1JM14694G. [26] L. Qu, Y. Liu, J.-B. Baek, L. Dai, Nitrogen-Doped Graphene as Efficient Metal-Free Electrocatalyst for Oxygen Reduction in Fuel Cells, ACS Nano. 4 (2010) 1321–1326. doi:10.1021/nn901850u.

19

[27] A. Yanilmaz, A. Tomak, B. Akbali, C. Bacaksiz, E. Ozceri, O. Ari, R. T.Senger, Y.Selamet, H. M.Zareie, Nitrogen doping for facile and effective modification of graphene surfaces, RSC Advances. 7 (2017) 28383–28392. doi:10.1039/C7RA03046K. [28] C.-K. Chang, S. Kataria, C.-C. Kuo, A. Ganguly, B.-Y. Wang, J.-Y. Hwang, K.J. Huang, W.-H. Yang, S.-B. Wang, C.-H. Chuang, M. Chen, C.-I. Huang, W.F. Pong, K.-J. Song, S.-J. Chang, J.-H. Guo, Y. Tai, M. Tsujimoto, S. Isoda, C.-W. Chen, L.-C. Chen, K.-H. Chen, Band Gap Engineering of Chemical Vapor Deposited Graphene by in Situ BN Doping, ACS Nano. 7 (2013) 1333–1341. https://doi.org/10.1021/nn3049158. [29] C. Zhang, L. Fu, N. Liu, M. Liu, Y. Wang, Z. Liu, Synthesis of Nitrogen-Doped Graphene Using Embedded Carbon and Nitrogen Sources, Advanced Materials. 23 (2011) 1020–1024. https://doi.org/10.1002/adma.201004110. [30] D. Wei, Y. Liu, Y. Wang, H. Zhang, L. Huang, G. Yu, Synthesis of N-Doped Graphene by Chemical Vapor Deposition and Its Electrical Properties, Nano Lett. 9 (2009) 1752– 1758. https://doi.org/10.1021/nl803279t. [31] D. Usachov, O. Vilkov, A. Gr¨ uneis, D. Haberer, A. Fedorov, V.K. Adamchuk, A.B. Preobrajenski, P. Dudin, A. Barinov, M. Oehzelt, C. Laubschat, D.V. Vyalikh, NitrogenDoped Graphene: Efficient Growth, Structure, and Electronic Properties, Nano Lett. 11 (2011) 5401–5407. https://doi.org/10.1021/nl2031037. [32] S. J.Zhang, S. S.Lin, X. Q.Li, X. Y.Liu, H. A.Wu, W. L.Xu, P. Wang, Z. Q.Wu, H. K.Zhong, Z. J.Xu, Opening the band gap of graphene through silicon doping for the improved performance of graphene/GaAs heterojunction solar cells, Nanoscale. 8 (2016) 226–232. doi:10.1039/C5NR06345K. [33] M. Shahrokhi, C. Leonard, Tuning the band gap and optical spectra of silicon-doped graphene: Many-body effects and excitonic states, Journal of Alloys and Compounds. 693 (2017) 1185–1196. doi:10.1016/j.jallcom.2016.10.101. 20

[34] P.-H. Shih, T.-N. Do, B.-L. Huang, G. Gumbs, D. Huang, M.-F. Lin, Magnetoelectronic and optical properties of Si-doped graphene, Carbon. 144 (2019) 608–614. doi:10.1016/j.carbon.2018.12.040. [35] H. Liu, Y. Liu, D. Zhu, Chemical doping of graphene, Journal of Materials Chemistry. 21 (2011) 3335–3345. https://doi.org/10.1039/C0JM02922J. [36] S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H.R. ¨ Kim, Y.I. Song, Y.-J. Kim, K.S. Kim, B. Ozyilmaz, J.-H. Ahn, B.H. Hong, S. Iijima, Roll-to-roll production of 30-inch graphene films for transparent electrodes, Nature Nanotech. 5 (2010) 574–578. https://doi.org/10.1038/nnano.2010.132. [37] Z.-Q. Fan, W.-Y. Sun, Z.-H. Zhang, X.-Q. Deng, G.-P. Tang, H.-Q. Xie, Symmetrydependent spin transport properties of a single phenalenyl or pyrene molecular device, Carbon. 122 (2017) 687–693. https://doi.org/10.1016/j.carbon.2017.07.019. [38] Z.-Q. Fan, W.-Y. Sun, X.-W. Jiang, Z.-H. Zhang, X.-Q. Deng, G.-P. Tang, H.-Q. Xie, M.-Q. Long, Redox control of magnetic transport properties of a single anthraquinone molecule with different contacted geometries, Carbon. 113 (2017) 18–25. https://doi.org/10.1016/j.carbon.2016.11.021. [39] D.

Culcer,

magnetic

A.

MacDonald,

two-dimensional

Q.

Niu,

systems,

Anomalous

Phys.

Rev.

B.

Hall 68

effect (2003)

in

para045327.

doi:10.1103/PhysRevB.68.045327. [40] Y. Deng, Y. Yu, M.Z. Shi, J. Wang, X.H. Chen, Y. Zhang, Magnetic-field-induced quantized anomalous Hall effect in intrinsic magnetic topological insulator MnBi2 Te4 , ArXiv:1904.11468 [Cond-Mat]. (2019). http://arxiv.org/abs/1904.11468 (accessed October 8, 2019). [41] B. Meng, H. Wu, Y. Qiu, C. Wang, Y. Liu, Z. Xia, S. Yuan, H. Chang, Z. Tian, Large anomalous Hall effect in ferromagnetic Weyl semimetal candidate PrAlGe, APL Materials. 7 (2019) 051110. doi:10.1063/1.5090795. 21

[42] S. Kumar, M. Pepper, S.N. Holmes, H. Montagu, Y. Gul, D.A. Ritchie, I. Farrer, Zero-Magnetic Field Fractional Quantum States, Phys. Rev. Lett. 122 (2019) 086803. doi:10.1103/PhysRevLett.122.086803. [43] G. Shavit, Y. Oreg, Fractional Conductance in Strongly Interacting 1D Systems, Phys. Rev. Lett. 123 (2019) 036803. doi:10.1103/PhysRevLett.123.036803. [44] Chen, S.-C.; Wu, J.-Y.;Lin, C.-Y.;Lin, M.-F. Theory of Magnetoelectric Properties of 2D Systems. IOP Publishing 2017, 2053-2563, ISBN: 978-0-7503-1674-3. [45] F. Zhang, B. Sahu, H. Min, A.H. MacDonald, Band structure of ABC-stacked graphene trilayers, Phys. Rev. B. 82 (2010) 035409. doi:10.1103/PhysRevB.82.035409. [46] G. Li, A. Luican, E.Y. Andrei, Scanning Tunneling Spectroscopy of Graphene on Graphite, Phys. Rev. Lett. 102 (2009) 176804. doi:10.1103/PhysRevLett.102.176804. [47] L.-J. Yin, S.-Y. Li, J.-B. Qiao, J.-C. Nie, L. He, Landau quantization in graphene monolayer, Bernal bilayer, and Bernal trilayer on graphite surface, Phys. Rev. B. 91 (2015) 115405. doi:10.1103/PhysRevB.91.115405. [48] E. A. Henriksen, P. Cadden-Zimansky, Z. Jiang, Z.Q. Li, L.-C. Tung, M.E. Schwartz, M. Takita, Y.-J. Wang, P. Kim, H.L. Stormer, Interaction-Induced Shift of the Cyclotron Resonance of Graphene Using Infrared Spectroscopy, Phys. Rev. Lett. 104 (2010) 067404. doi:10.1103/PhysRevLett.104.067404. [49] T. Taychatanapat, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, Quantum Hall effect and Landau-level crossing of Dirac fermions in trilayer graphene, Nature Phys. 7 (2011) 621–625. doi:10.1038/nphys2008. [50] P. Dutta, S.K. Maiti, S.N. Karmakar, Integer quantum Hall effect in a lattice model revisited: Kubo formalism, Journal of Applied Physics. 112 (2012) 044306. doi:10.1063/1.4748312.

22

[51] M. F. Lin, K.W.-K. Shung, Plasmons and optical properties of carbon nanotubes, Phys. Rev. B. 50 (1994) 17744–17747. doi:10.1103/PhysRevB.50.17744. [52] L. Jiang, Y. Zheng, H. Li, H. Shen, Magneto-transport properties of gapped graphene, Nanotechnology. 21 (2010) 145703. doi:10.1088/0957-4484/21/14/145703. [53] S. Xiang, V. Miseikis, L. Planat, S. Guiducci, S. Roddaro, C. Coletti, F. Beltram, S. Heun, Low-temperature quantum transport in CVD-grown single crystal graphene, Nano Res. 9 (2016) 1823–1830. doi:10.1007/s12274-016-1075-0. [54] Y.-Y. Zhang, W.-F. Tsai, K. Chang, X.-T. An, G.-P. Zhang, X.-C. Xie, S.-S. Li, Electron delocalization in gate-tunable gapless silicene, Phys. Rev. B. 88 (2013) 125431. doi:10.1103/PhysRevB.88.125431. [55] W.-F. Tsai, C.-Y. Huang, T.-R. Chang, H. Lin, H.-T. Jeng, A. Bansil, Gated silicene as a tunable source of nearly 100% spin-polarized electrons, Nat Commun. 4 (2013) 1–6. doi:10.1038/ncomms2525. [56] M. Kim, J.-H. Choi, S.-H. Lee, K. Watanabe, T. Taniguchi, S.-H. Jhi, H.-J. Lee, Valley-symmetry-preserved transport in ballistic graphene with gate-defined carrier guiding, Nature Physics. 12 (2016) 1022–1026. doi:10.1038/nphys3804. [57] Y.-C. Zhou, H.-L. Zhang, W.-Q. Deng, A 3Nrule for the electronic properties of doped graphene, Nanotechnology. 24 (2013) 225705. doi:10.1088/0957-4484/24/22/225705. [58] Y. Fujimoto, Formation, Energetics, and Electronic Properties of Graphene Monolayer and Bilayer Doped with Heteroatoms, Advances in Condensed Matter Physics. (2015). doi:10.1155/2015/571490. [59] S.S. Varghese, S. Swaminathan, K.K. Singh, V. Mittal, Energetic Stabilities, Structural and Electronic Properties of Monolayer Graphene Doped with Boron and Nitrogen Atoms, Electronics. 5 (2016) 91. doi:10.3390/electronics5040091.

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FIGURE CAPTIONS Fig. 1 - Crystal structures of doped graphene for chosen doping concentrations of (a) 12.5% and (b) 2% dopants. The green and orange spheres denote, respectively, the host C and guest Si/B/N atoms. Fig. 2 - Energy bands of doped graphene with Si, B and N guest atoms for concentrations of 2% [(a)-(c)] and 12.5% [(d)-(f)]concentrations. Fig. 3 - The Bz -dependent LL energies of doped graphene with Si, B and N dopants for concentrations of 2% [(a)-(c)] and 12.5% [(d)-(f)]. Fig. 4 - The LL energies and wavefunctions of the first few levels for (I) Si, (II) B, and (III) N guest atoms with a 12.5% concentration at Bz = 20 T. The blue and red bars represent LLs localized at 1/6 and 2/6 centers, respectively. Fig. 5 - The EF -dependent inverse quantum-Hall conductivity 1/σxy of Si-, B-, and N-doped graphene at Bz = 20 T. The solid (dashed) curve represents the dopant density of 2% (12.5%). The inset shows the quantum-Hall conductivity σxy of 12.5% B-doped graphene.

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Magneto-transport properties of B-, Si- and N-doped graphene Po-Hsin Shiha , Thi-Nga Dob,c∗ , Godfrey Gumbsd , Danhong Huange , Thanh Phong Phamb,c , Ming-Fa Lina a Department b Laboratory

of Physics, National Cheng Kung University, Tainan, Taiwan 701

of Magnetism and Magnetic Materials, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

c Faculty

of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam

d Department

of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, New York 10065, USA

e US

Air Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force Base, New Mexico 87117, USA

December 18, 2019

Abstract The effect due to doping by B, Si, N atoms on the magneto-transport properties of graphene is investigated using the generalized tight-binding model in conjunction with the Kubo formula. The crucial electronic and transport properties are greatly diversified by different types of dopant and doping concentrations. The effect of these guest atoms includes opening a band gap, thereby giving rise to rich Landau level energy spectra and consequently a unique quantum-Hall conductivity. The Fermienergy dependent quantum-Hall effect appears as a step structure having both integer and half-integer plateaus. Doping with Si leads to an occurrence of a zero quantum-

1

Hall conductivity, unlike the plateau sequence for pristine graphene. The predicted dopant- and concentration-enriched quantum-Hall effect for doped graphene can provide useful information for magneto-transport measurements, possible technological and even metrology applications.

* Corresponding author. E-mail address: [email protected] (T. N. Do)

2

1. Introduction Understanding the physics origin of the quantum-Hall effect (QHE) has led to a prolonged effort by numerous theoretical and experimental researchers. In particular, the Hall conductivity of condensed matter systems under a strong external magnetic field has received special attention [1-20]. Various behaviors of magnetic-field-quantized Landau levels (LLs) have been employed to explain the QHE. The electron filling of LLs and wave functions have been proven to connect to the integer and fractional quantum-Hall states. In this work, we will systematically explore the QHE in Si-, B-, and N-doped graphene with various doping concentrations using the tight-binding model to determine Landau energy bands. Ever since the time when graphene was experimentally discovered, its exotic QHE has attracted a lot of studies from both experimental and theoretical aspects [1-14]. Monolayer graphene shows amazing physical properties because of its single-atom thickness and hexagonal symmetry. The magnetically-quantized LLs of the isotropic Dirac cone form a distinctive energy spectrum for which the conduction and valence energy-level separations depend on both field strength and quantum number of specified LLs. Such an established √ dependence, Bz nc,v , has been verified through experiment using scanning tunneling spectroscopy (STS) [21], optical spectroscopies [22], and transport measurement [5-6]. Interestingly, it has been realized that the inter-LL transitions obey a specific selection rule associated with the equivalence of two sublattices in the crystal structure. Only the excitations between valence and conduction LLs whose quantum numbers satisfy ∆ n = |nv − nc | = 1 are allowed, resulting in exotic QHE. Magnetic transport measurements have revealed the intriguing half-integer quantum-Hall conductivity σxy = (m + 1/2)4e2 /h (m is an integer and the factor of 4 represents the spin and sublattice degeneracy) in monolayer graphene [5-6]. Such an extraordinary quantization phenomenon is related to the quantum anomaly of the n = 0 LL coming from the Dirac point [1]. Currently, scientists are in search of novel materials yielding unusual physical phenomena and having potential for device applications. By introducing graphene-based defect 3

systems, such as substituting impurities or guest atoms in a hexagonal carbon lattice, one could effectively manipulate the fundamental properties inherent to graphene. Among various pertinent guest atoms, the carbon host atoms have been successfully substituted by Si [23], B [24-25], and N [26-27] in chemical vapor deposition (CVD) or arc discharge experiments. This doping eliminates the equivalence of two sublattices, causing dramatic changes in physical properties, including the energy-gap opening and the deformation of the original Dirac cone. These features of doped graphene have been verified by a number of experimental studies [28-31]. Previous studies have demonstrated the emergence of different ionization potentials and non-uniform hopping integrals [32-33]. Specifically, for Si-, B-, and N-doped graphene, the π bonding extending on a hexagonal lattice is significantly distorted. As a result, the low-lying energy band is determined by both the dopants and C atoms. Additionally, based on this unusual band structure due to doping, the magnetically quantized LLs have been theoretically predicted to be feature-rich and unique [34]. These facts make it promising for the QHE of such special LL energy spectra to yield abundant and unique features. It is critical to mention that, though substitutional doping might reduce the mobility of graphene [35], the QHE could still be observed in doped graphene at suitable temperature and magnetic field [36]. In this work, we provide a comprehensive investigation of the traditional QHE which originates from discrete LLs formed in doped graphene in the presence of a uniform perpendicular magnetic field. The modification of crystal structure by doping is expected to enrich the quantum Hall conductivity as the significant influence of lattice geometry on the transport properties has been reported in the literature [37-38]. It is worth mentioning that this effect is distinct from the anomalous Hall effect (AHE) in the absence of an external magnetic field, which has recently become the focus of much interest [39-43]. The competition between the traditional QHE and the AHE in the presence of external magnetic field is a curious phenomenon which has gained significant attention from scientists [39-41]. Previously, it has been reported that the AHE still appears in the regime of weak magnetic field [39-41]. That is, the ordinary and anomalous Hall effects may simultaneously exist in

4

the presence of an external magnetic field; which phenomenon is dominant depends on the field strength. The rest of this paper is outlined as follows. In Sec. 2, we employ the tight-binding method to solve for the eigenvalues and eigenfunctions of the magnetic Hamiltonian, and then use these results to calculate the quantum-Hall conductivity by means of the Kubo formula. This procedure enables us to identify allowed transition channels through the aforementioned magneto-electronic selection rules, leading to an understanding of Fermienergy and magnetic-field dependent quantum conductivity. Fundamental properties of doped graphene are greatly enriched by various types of dopants and their concentrations. In Sec. 3, we investigate the QHE with respect to Si, B, and N dopants for 2% and 12.5% doping ratios. Our results demonstrate a robust correlation between the unusual QHE and the dopant- and concentration-dependent rich LLs in doped graphene systems. Our theoretical predictions provide essential information for future experimental verification of the QHE in these graphene-based defect materials. Section 4 is devoted to our concluding remarks.

2. Model and Computational Methods We have employed the tight-binding method and the dynamic Kubo formula from linear response theory to carefully investigate the rich magneto-transport properties of graphene doped with Si, B, and N atoms. The tight-binding model [44] has been demonstrated to be an appropriate method to explore fundamental magnetic properties of layered materials, including the magneto-electronic, magneto-optical, quantum transport properties and magneto-Coulomb excitations in conjunction with the Kubo formula and modified randomphase approximation. This model could give results which are in good agreement with those obtained by other methods, such as the band structure from first-principles calculations [45], the Landau level energy spectra from STS measurements [46-47], magneto-optical spectra from optical spectroscopy [48], and the quantum Hall conductivity from transport measurements [49].

5

The crystal structures of doped graphene for two dopant concentrations are presented in Figs. 1(a) and 1(b). For a 1:2n2 concentration doping (n is the cell multiplicity), the super cell is obtained by expanding the original unit cell of pristine graphene n times along the lattice vectors a~1 and a~2 , that is, (na~1 , na~2 ). In this work, we choose a rectangular super cell as marked by the red lines for convenience in considering the effect of a magnetic field. A single supercell comprises 2n2 sublattices of two types, Ai and Bi , in which i represents the lattice site. Both C-2pz host orbital and the guest orbitals of B − 2pz , Si − (2pz , 3pz ), and N − 2pz play the key roles in governing the crucial characteristics in the low-energy range. The Hamiltonian matrix elements include non-uniform bond lengths, site energies and nearest-neighbor hopping integrals and are written as [34]

∗ Hj+2nl−2n,j+2nl−n = Hj+2nl−n,j+2nl−2n = tj+2nl−2n,j+2nl−n f1 ∗ Hj+2nl−2n,m(j)+2nl−n+1 = Hm(j)+2nl−n+1,j+2nl−2n = tj+2nl−2n,m(j)+2nl−n+1 f2 ∗ Hj+2nl−2n,j+2n[m(l)+1]−n = Hj+2n[m(l)+1]−n,j+2nl−2n = tj+2nl−2n,j+2n[m(l)+1]−n f3

H1,1 = B/Si/N −C ,

(1)

where the hopping term is defined as

tα,β =

   γB/Si/N −C , if α or β equal 1   γC−C ,

(2)

otherwise .

In this notation, f1,2,3 is defined in terms of the wave vector ~k and the vectors connecting ~ 1,2,3 through ei~k·R~ 1,2,3 . Moreover, m(k) = k + n − 2 mod the nearest-neighbor lattice sites R n is modulo function, and j, l are integers (j, l = 1, 2, 3, ..., n). The C-C and B/Si/N-C bond lengths have been verified as being slightly different due to doping, especially the buckling effect of the guest atoms in graphene [32-33]. For Si dopant, the guest atoms are located at a distance of dSi above graphene sheet, implying a significant adjustment of the π bonding on a hexagonal lattice. The consequential nonuniform nearest-neighbor hopping integrals and site energies associated with the major 6

Figure 1: (Color online) Crystal structures of doped graphene for chosen doping concentrations of (a) 12.5% and (b) 2% dopants. The green and orange spheres denote, respectively, the host C and guest Si/B/N atoms. 2pz orbitals of the C host atoms and the minor 2pz /3pz orbitals of the guest atoms lead to extraordinary electronic properties. The parameters are appropriately fitted, as shown in Table I, so that the energy bands calculated by the tight-binding and first-principles methods agree with each other. The effect of a uniform perpendicular magnetic field on the system could be treated with R Rm the use of the vector potential-dependent Peierls phase, given by ∆Gmm0 = 2π A(r) · dr φ0 Rm0

7

Table 1: Tight-binding parameters for pristine and doped graphene Parameter

Pristine graphene

Si-doped

B-doped

N-doped

bond lengths (˚ A)

b = 1.42

bC−Si = 1.7

bC−B = 1.42

bC−N = 1.42

buckled distance (˚ A)

d=0

dSi = 0.93

dB = 0

dN = 0

hopping (eV)

t =-2.7

site energy (eV)

=0

tC−Si = −1.3 tC−B = -2.17 Si =1.3

B = 2.33

tC−N = -2.1 N = -2

(φ0 = hc/e is the flux quantum), in the nearest-neighbor hopping integral [34]. Such a phase is position-dependent so that the unit cell will be extended along the armchair direction. Consequently, the magnetic Hamiltonian becomes a large matrix whose periods strongly depend on the density of guest atoms and vector potential. This magneticHamiltonian matrix can be solved efficiently by using the diagonalization method to obtain the LL energy spectrum and sub-envelope functions [44]. Interestingly, the amplitudes of the sub-envelope functions on distinct sublattices within an enlarged unit cell can be regarded as electron spatial distributions, and therefore, they are used to analyze the main features of the LL wave functions. Within the linear-response theory, the Hall conductivity is calculated from the Kubo formula, yielding [50]

σxy

hα|ˆ v x |βihβ|ˆ v y |αi ie2 ~ P P (fα − fβ ) . = 2 S (Eα − Eβ ) + Γ20 α α6=β

(3)

In this notation, |αi and Eα are the LL state and LL energy, respectively, fα,β is the FermiDirac distribution functions, S the area of a supercell, and Γ0 a broadening parameter for LL. Furthermore, the velocity operators v ˆx,y directly determine the allowable inter-LL transitions, and can be evaluated directly from the gradient approximation [51]. The defect effect in doped graphene can lead to a significant modification of quantum conductivities. We note that the inter-valley interaction is neglected in our calculations of the magnetotransport coefficients of doped graphene. In graphene [52-53], the intra-valley scattering, 8

but not that due to inter-valley, is the dominant scattering mechanism. In general, the weak inter-valley scattering only plays the role of establishing the equilibrium between the two valleys. Explicitly, the application of an external magnetic field may induce distinct chemical potentials at the two valleys, K and K0 . However, the inter-valley scattering can adjust the electron populations in the two valleys, so that the two valleys eventually reach a uniform chemical potential. For silicene [54-55], it was argued that the inter-valley scattering from nonmagnetic impurities is highly suppressed by time-reversal symmetry. Therefore, the physics should be effectively governed by single-Dirac-cone like behavior. On the other hand, one could effectively suppress the inter-valley scattering in graphene by sandwiching a monolayer graphene sheet between clean hexagonal boron nitride layers, as done in Ref. [56]. These imply the inconsequential effect of the inter-valley interaction on the magneto-transport properties of (Si, B, N)-doped graphene in our present work.

3. Results and Discussion The effect of doping for chosen dopants and concentrations remarkably diversifies the electronic properties of graphene. The substitution by guest atoms opens a direct band gap in monolayer graphene since it breaks down the lattice symmetry, as seen in Fig. 2. The magnitude of the energy gap (Eg ) is greatly affected by dopants and doping concentration, as demonstrated in Table II for pristine graphene and Si-, B-, and N-doped systems with 2% and 12.5% guest atoms. Additionally, the significant influence of doping on Fermi energy is worthy of reiteration (Table III). For pristine graphene, the Fermi energy is equal to zero (Fig. S1 of the Supplemental Material) and it remains the same in the presence of Si dopants. On the contrary, the Fermi energy shifts down to the valence band or up to the conduction band for B- and N-doped systems, respectively. These observations are attributed to the fact that a B atom acts as a p-dopant while an N atom behaviors as an n-dopant since a B/N atom has one less/more electron than a C atom. The 3D plots of energy bands with separated occupied and unoccupied states are presented in Fig. S2 of the Supplemental Material. Our numerical results are in agreement with previous theoretical

9

Figure 2: (Color online) Energy bands of doped graphene with Si, B and N guest atoms for concentrations of 2% [(a)-(c)] and 12.5% [(d)-(f)]concentrations. calculations based on the first-principles method [57-59]. The prediction of the dopantand concentration-dependent band structures in doped graphene could be very useful for graphene electronic applications. In the presence of a uniform perpendicular magnetic field, the quantized LLs show unusual characteristics due to doping. Because of the opening of a band gap, the conduction and valence LL energy spectra are separated from each other, as shown in Figs. 3(a) and 3(d) for the magnetic-field-dependent energy spectra of Si-doped graphene. This separation

10

Table 2: Band gap of doped graphene Eg (eV) Si-doped

B-doped

N-doped

2%

0.19

0.12

0.13

12.5%

0.74

0.68

0.68

Table 3: Fermi levels of doped graphene Fermi levels (eV) Si-doped (EF (Si)) B-doped (EF (B)) N-doped (EF (N )) 2%

0

-0.78

0.84

12.5%

0

-1.76

1.81

remains invariant with changing field strength so that it has no influence on the inter-LL transition between the valence and conduction bands. Each LL is four-fold degenerate, except for the lowest conduction and highest valence states in the vicinity of zero energy whose degeneracy is reduced by half. For Si-doped graphene, the Fermi level sits close to zero energy. Therefore, the conduction and valence states are unoccupied and occupied, respectively. On the other hand, they are either unoccupied for B or occupied for N dopants due to the shift of Fermi energy. These two doubly degenerate LLs are split from the original zero-energy LL (n = 0) of pristine graphene due to the substitution of guest atoms. The Bz -dependence of LLs near the Fermi energy in B- and N-doped systems needs a detailed explanation. There, the density of LL states is very high, as clearly displayed in Figs. 3(b) and 3(c) for 2% as well as 3(e) and 3(f) for 12.5% concentrations of B and N guest atoms. Especially for dopant density of 12.5%, the LLs exhibit small splitting at higher and deeper energy ranges, altering the state-degeneracy degree of freedom. These phenomena play a key role in the unique properties of quantum-Hall conductivity discussed in detail below. The contribution of guest and host atoms to the Hall conductivity through the velocity matrix elements in Eq. (3) could be understood based on the amplitude of LL wave functions on the sublattices, as demonstrated in Figs. 4(I) through 4(III). Our numerical calculations 11

Figure 3: (Color online) The Bz -dependent LL energies of doped graphene with Si, B and N dopants for concentrations of 2% [(a)-(c)] and 12.5% [(d)-(f)]. show that the A sublattice, which consists of guest atoms, obviously dominates the conduction bands of (Si, B)-doped systems as well as the valence band of N-doped graphene, and vice versa for the B sublattice. Especially for a Si dopant with 3pz orbital, its spatial distribution is even greater than that of the C atoms. The quantum number, which is 12

Figure 4: (Color online) The LL energies and wavefunctions of the first few levels for (I) Si, (II) B, and (III) N guest atoms with a 12.5% concentration at Bz = 20 T. The blue and red bars represent LLs localized at 1/6 and 2/6 centers, respectively. critical in understanding the selection rule of inter-LL transitions, is determined based on the nodes of the LL spatial distribution. Here, LL wave functions are only localized in some specific regions within the field-enlarged unit cell, particularly at 1/6, 2/6, 4/6 and 5/6 guiding centers for a (kx = ky = 0) state. Furthermore, LLs localized at 1/6 and 4/6 c,v (2/6 and 5/6) are equivalent in the main characteristics and classified as nc,v 1 (n2 ) group.

This classification holds true for all doped systems, regardless of dopant type and doping concentration. The doubly-degenerate conduction LL belongs to nc1 group for Si and B dopants but to nc2 group for N, while the opposite is true for the valence LL. The quantum-Hall effect of doped graphene displays unusual and specific properties that may be manipulated by the degree to which the sample is doped. The Fermi energydependent Hall conductivity quantization under an external magnetic field exhibits a step

13

structure, as shown in Fig. 5 for the inverse quantum-Hall conductivity 1/σxy of three different guest atoms with 2% and 12.5% concentrations. The corresponding plots for σxy could be found from Fig. S3 of the Supplemental Material. With varying Fermi energy (EF ), a plateau is formed as one LL is kept occupied. The inter-LL transition between the occupied and unoccupied levels requires a specific selection rule where the oscillation mode of the wave functions on the sublattices must be compatible for the initial and final LL states. We find this rule as ∆n = ±1 & 0 for 2% but ∆n = ±1 for 12.5% doped graphene.

Figure 5: (Color online) The EF -dependent inverse quantum-Hall conductivity 1/σxy of Si-, B-, and N-doped graphene at Bz = 20 T. The solid (dashed) curve represents the dopant density of 2% (12.5%). The inset shows the quantum-Hall conductivity σxy of 12.5% B-doped graphene. The conductivity is quantized following a common sequence of σxy = {0; 2(2m+1)e2 /h} (m is an integer) for all dopants. It should be noticed that EF on the x-axis of Fig. 5 represents the energy shift from the original Fermi level in the absence of doping. The 14

induced band gap from doping is reflected in the discontinuity of 1/σxy near zero energy, as shown for Si guest atoms by solid and dotted red curves. This is attributed to the appearance of zero conductivity in Figs. S3(a) and S3(d) of the Supplemental Material. Meanwhile, there exist two half-integer steps of 2e2 /h height for quantum-Hall conductivity σxy near zero energy, in addition to the conventional steps of 4e2 /h. This is in contrast with that for pristine graphene where the two half-integer steps occur exactly at zero energy [5]. Furthermore, the width of this zero-energy plateau becomes a measure for the size of the band gap, which strongly depends on dopant types and doping concentrations. The quantum-Hall conductivities in the vicinity of the Fermi levels in the presence of B and N dopants deserve careful scrutiny since they could be identified in magnetotransport measurements. For sufficiently small doping concentration, e.g., 2% as shown by solid blue and green curves in Fig. 5 (also in Figs. S3(b) and S3(c)), the quantized sequence of quantum-Hall conductivity remains unchanged even at higher or lower Fermi energies. The doping effect might not be strong enough to remarkably alter the essential physical properties of the system. However, with increasing dopant density, the splitting of LLs near Fermi energies for B and N dopants will give rise to significant changes in the quantum-Hall effect. There, the wider and narrower conductivity plateaus are interspersed in the spectra, referring to the inset plot for 12.5% B-doped graphene (also in Figs. S3(e) and S3(f)), in which the narrower one is related to the splitting of degenerate LLs. Accordingly, each step is reduced by half in height, obeying a new quantized sequence of σxy = m (2e2 /h) instead of σxy = m (4e2 /h). Our theoretical predictions for doped graphene with various dopants should provide useful insight for future magneto-transport measurements, as done for monolayer and fewlayer graphene [5-6, 8-11]. Specifically, we have found the half-integer QHE as predicted for monolayer graphene by different theory groups [1-3], situated on thin graphite films [4], and verified by experimental measurements [5-6]. Such a special quantization in undoped graphene is attributed to the quantum anomaly of the n = 0 LL corresponding to the Dirac point [1]. The results obtained in this work by the well-developed theoretical framework,

15

however, are related to splitting of LLs by dopant interaction and expected to be confirmed experimentally.

4. Concluding Remarks In conclusion, the generalized tight-binding model and the Kubo formula are employed to explore the interesting QHE in doped graphene for Si, B, and N guest atoms with various concentrations. The theoretical framework takes into consideration the substantial factors of the doping system, including non-uniform bond lengths, site energies, hopping integrals, and external field. This method could be extended to investigate many other condensedmatter materials. Doped graphene is an emergent 2D binary compound material, which presents diverse electronic and transport properties under magnetic quantization. The doping effect can produce a band gap of different sizes, leading to remarkable changes in the main features of LLs and thus the magneto-transport properties. The Fermi-energy dependent QHE exhibits both integer and half-integer conductivities. The step structure could be controlled by changing dopant or doping concentration, in which the latter has more significant influence. Our theoretical predictions present useful technological and device information and could be verified by magneto-transport measurements. Acknowledgments The authors thank the Ministry of Science and Technology of Taiwan (R.O.C.) under Grant No. MOST 105-2112-M-017-002-MY2. T. N. Do would like to thank the Ton Duc Thang University. D.H. would like to thank support from the AFOSR and from the DoD Lab-University Collaborative Initiative (LUCI) Program.

References [1] V.P. Gusynin, S.G. Sharapov, Unconventional Integer Quantum Hall Effect in Graphene, Phys. Rev. Lett. 95 (2005) 146801. doi:10.1103/PhysRevLett.95.146801. [2] N.M.R. Peres, F. Guinea, A.H. Castro Neto, Electronic properties of two-dimensional carbon, Annals of Physics. 321 (2006) 15591567. doi:10.1016/j.aop.2006.04.006. 16

[3] A. Tsukuda, H. Okunaga, D. Nakahara, K. Uchida, T. Konoike, T. Osada, Quantum Hall Transport across Monolayer-Bilayer Boundary in Graphene, J. Phys.: Conf. Ser. 334 (2011) 012038. doi:10.1088/1742-6596/334/1/012038. [4] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Electric Field Effect in Atomically Thin Carbon Films, Science. 306 (2004) 666669. doi:10.1126/science.1102896. [5] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature. 438 (2005) 197200. doi:10.1038/nature04233. [6] Y. Zhang, Y.-W. Tan, H.L. Stormer, P. Kim, Experimental observation of the quantum Hall effect and Berrys phase in graphene, Nature. 438 (2005) 201204. doi:10.1038/nature04235. [7] E.

McCann,

Effect

in

a

V.I.

Falko,

Graphite

Landau-Level

Bilayer,

Phys.

Degeneracy Rev.

Lett.

and 96

Quantum (2006)

Hall

086805.

doi:10.1103/PhysRevLett.96.086805. [8] T. Taychatanapat, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, Quantum Hall effect and Landau-level crossing of Dirac fermions in trilayer graphene, Nature Physics. 7 (2011) 621625. doi:10.1038/nphys2008. [9] S.H. Jhang, M.F. Craciun, S. Schmidmeier, S. Tokumitsu, S. Russo, M. Yamamoto, Y. Skourski, J. Wosnitza, S. Tarucha, J. Eroms, C. Strunk, Stacking-order dependent transport properties of trilayer graphene, Phys. Rev. B. 84 (2011) 161408. doi:10.1103/PhysRevB.84.161408. [10] L. Zhang, Y. Zhang, J. Camacho, M. Khodas, I. Zaliznyak, The experimental observation of quantum Hall effect of l=3 chiral quasiparticles in trilayer graphene, Nature Physics. 7 (2011) 953957. doi:10.1038/nphys2104.

17

[11] A. Kumar, W. Escoffier, J.M. Poumirol, C. Faugeras, D.P. Arovas, M.M. Fogler, F. Guinea, S. Roche, M. Goiran, B. Raquet, Integer Quantum Hall Effect in Trilayer Graphene, Phys. Rev. Lett. 107 (2011) 126806. doi:10.1103/PhysRevLett.107.126806. [12] M. Koshino, E. McCann, Landau level spectra and the quantum Hall effect of multilayer graphene, Phys. Rev. B. 83 (2011) 165443. doi:10.1103/PhysRevB.83.165443. [13] S. Yuan, R. Roldn, M.I. Katsnelson, Landau level spectrum of ABA- and ABC-stacked trilayer graphene, Phys. Rev. B. 84 (2011) 125455. doi:10.1103/PhysRevB.84.125455. [14] T.-N. Do, C.-P. Chang, P.-H. Shih, J.-Y. Wu, M.-F. Lin, Stacking-enriched magnetotransport properties of few-layer graphenes, Physical Chemistry Chemical Physics. 19 (2017) 2952529533. doi:10.1039/C7CP05614A. [15] L. Li, F. Yang, G.J. Ye, Z. Zhang, Z. Zhu, W. Lou, X. Zhou, L. Li, K. Watanabe, T. Taniguchi, K. Chang, Y. Wang, X.H. Chen, Y. Zhang, Quantum Hall effect in black phosphorus two-dimensional electron system, Nature Nanotechnology. 11 (2016) 593597. doi:10.1038/nnano.2016.42. [16] R.S. Meena, S.K. Singh, A. Pal, A. Kumar, R. Jha, K.V.R. Rao, Y. Du, X.L. Wang, V.P.S. Awana, High field (14 T) magneto transport of Sm/PrFeAsO, Journal of Applied Physics. 111 (2012) 07E323. doi:10.1063/1.3675156. [17] R. Singha, S. Roy, A. Pariari, B. Satpati, P. Mandal, Magnetotransport properties and giant anomalous Hall angle in the half-Heusler compound TbPtBi, Phys. Rev. B. 99 (2019) 035110. doi:10.1103/PhysRevB.99.035110. [18] F. Yang, Z. Zhang, N.Z. Wang, G.J. Ye, W. Lou, X. Zhou, K. Watanabe, T. Taniguchi, K. Chang, X.H. Chen, Y. Zhang, Quantum Hall Effect in ElectronDoped Black Phosphorus Field-Effect Transistors, Nano Lett. 18 (2018) 66116616. doi:10.1021/acs.nanolett.8b03267.

18

[19] R. Ma, S.W. Liu, W.Q. Yang, M.X. Deng, Quantum Hall effect in monolayer and bilayer black phosphorus, EPL. 119 (2017) 37005. doi:10.1209/0295-5075/119/37005. [20] J. Yin, S. Slizovskiy, Y. Cao, S. Hu, Y. Yang, I. Lobanova, B.A. Piot, S.-K. Son, S. Ozdemir, T. Taniguchi, K. Watanabe, K.S. Novoselov, F. Guinea, A.K. Geim, V. Falko, A. Mishchenko, Dimensional reduction, quantum Hall effect and layer parity in graphite films, Nat. Phys. 15 (2019) 437442. doi:10.1038/s41567-019-0427-6. [21] D.L. Miller, K.D. Kubista, G.M. Rutter, M. Ruan, W.A. de Heer, P.N. First, J.A. Stroscio, Observing the Quantization of Zero Mass Carriers in Graphene, Science. 324 (2009) 924927. doi:10.1126/science.1171810. [22] Z. Jiang, E.A. Henriksen, L.C. Tung, Y.-J. Wang, M.E. Schwartz, M.Y. Han, P. Kim, H.L. Stormer, Infrared Spectroscopy of Landau Levels of Graphene, Phys. Rev. Lett. 98 (2007) 197403. doi:10.1103/PhysRevLett.98.197403. [23] W. Zhou, M.D. Kapetanakis, M.P. Prange, S.T. Pantelides, S.J. Pennycook, J.-C. Idrobo, Direct Determination of the Chemical Bonding of Individual Impurities in Graphene, Phys. Rev. Lett. 109 (2012) 206803. doi:10.1103/PhysRevLett.109.206803. [24] L.S. Panchakarla, K.S. Subrahmanyam, S.K. Saha, A. Govindaraj, H.R. Krishnamurthy, U.V. Waghmare, C.N.R. Rao, Synthesis, Structure, and Properties of Boron- and Nitrogen-Doped Graphene, Advanced Materials. 21 (2009) 47264730. doi:10.1002/adma.200901285. [25] Z.-H. Sheng, H.-L. Gao, W.-J. Bao, F.-B. Wang, X.-H. Xia, Synthesis of boron doped graphene for oxygen reduction reaction in fuel cells, Journal of Materials Chemistry. 22 (2012) 390395. doi:10.1039/C1JM14694G. [26] L. Qu, Y. Liu, J.-B. Baek, L. Dai, Nitrogen-Doped Graphene as Efficient Metal-Free Electrocatalyst for Oxygen Reduction in Fuel Cells, ACS Nano. 4 (2010) 13211326. doi:10.1021/nn901850u.

19

[27] A. Yanilmaz, A. Tomak, B. Akbali, C. Bacaksiz, E. Ozceri, O. Ari, R. T.Senger, Y. Selamet, H. M.Zareie, Nitrogen doping for facile and effective modification of graphene surfaces, RSC Advances. 7 (2017) 2838328392. doi:10.1039/C7RA03046K. [28] C.-K. Chang, S. Kataria, C.-C. Kuo, A. Ganguly, B.-Y. Wang, J.-Y. Hwang, K.J. Huang, W.-H. Yang, S.-B. Wang, C.-H. Chuang, M. Chen, C.-I. Huang, W.F. Pong, K.-J. Song, S.-J. Chang, J.-H. Guo, Y. Tai, M. Tsujimoto, S. Isoda, C.-W. Chen, L.-C. Chen, K.-H. Chen, Band Gap Engineering of Chemical Vapor Deposited Graphene by in Situ BN Doping, ACS Nano. 7 (2013) 13331341. https://doi.org/10.1021/nn3049158. [29] C. Zhang, L. Fu, N. Liu, M. Liu, Y. Wang, Z. Liu, Synthesis of Nitrogen-Doped Graphene Using Embedded Carbon and Nitrogen Sources, Advanced Materials. 23 (2011) 10201024. https://doi.org/10.1002/adma.201004110. [30] D. Wei, Y. Liu, Y. Wang, H. Zhang, L. Huang, G. Yu, Synthesis of N-Doped Graphene by Chemical Vapor Deposition and Its Electrical Properties, Nano Lett. 9 (2009) 17521758. https://doi.org/10.1021/nl803279t. [31] D. Usachov, O. Vilkov, A. Grneis, D. Haberer, A. Fedorov, V.K. Adamchuk, A.B. Preobrajenski, P. Dudin, A. Barinov, M. Oehzelt, C. Laubschat, D.V. Vyalikh, NitrogenDoped Graphene: Efficient Growth, Structure, and Electronic Properties, Nano Lett. 11 (2011) 54015407. https://doi.org/10.1021/nl2031037. [32] S. J.Zhang, S. S.Lin, X. Q.Li, X. Y.Liu, H. A.Wu, W. L.Xu, P. Wang, Z. Q.Wu, H. K.Zhong, Z. J.Xu, Opening the band gap of graphene through silicon doping for the improved performance of graphene/GaAs heterojunction solar cells, Nanoscale. 8 (2016) 226232. doi:10.1039/C5NR06345K. [33] M. Shahrokhi, C. Leonard, Tuning the band gap and optical spectra of silicon-doped graphene: Many-body effects and excitonic states, Journal of Alloys and Compounds. 693 (2017) 11851196. doi:10.1016/j.jallcom.2016.10.101. 20

[34] P.-H. Shih, T.-N. Do, B.-L. Huang, G. Gumbs, D. Huang, M.-F. Lin, Magnetoelectronic and optical properties of Si-doped graphene, Carbon. 144 (2019) 608614. doi:10.1016/j.carbon.2018.12.040. [35] H. Liu, Y. Liu, D. Zhu, Chemical doping of graphene, Journal of Materials Chemistry. 21 (2011) 33353345. https://doi.org/10.1039/C0JM02922J. [36] S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H.R. Kim, Y.I. Song, Y.-J. Kim, K.S. Kim, B. zyilmaz, J.-H. Ahn, B.H. Hong, S. Iijima, Roll-to-roll production of 30-inch graphene films for transparent electrodes, Nature Nanotech. 5 (2010) 574578. https://doi.org/10.1038/nnano.2010.132. [37] Z.-Q. Fan, W.-Y. Sun, Z.-H. Zhang, X.-Q. Deng, G.-P. Tang, H.-Q. Xie, Symmetrydependent spin transport properties of a single phenalenyl or pyrene molecular device, Carbon. 122 (2017) 687693. https://doi.org/10.1016/j.carbon.2017.07.019. [38] Z.-Q. Fan, W.-Y. Sun, X.-W. Jiang, Z.-H. Zhang, X.-Q. Deng, G.-P. Tang, H.-Q. Xie, M.-Q. Long, Redox control of magnetic transport properties of a single anthraquinone molecule with different contacted geometries, Carbon. 113 (2017) 1825. https://doi.org/10.1016/j.carbon.2016.11.021. [39] D.

Culcer,

magnetic

A.

MacDonald,

two-dimensional

Q.

Niu,

systems,

Anomalous

Phys.

Rev.

B.

Hall 68

effect (2003)

in

para045327.

doi:10.1103/PhysRevB.68.045327. [40] Y. Deng, Y. Yu, M.Z. Shi, J. Wang, X.H. Chen, Y. Zhang, Magnetic-field-induced quantized anomalous Hall effect in intrinsic magnetic topological insulator MnBi2 Te4 , ArXiv:1904.11468 [Cond-Mat]. (2019). http://arxiv.org/abs/1904.11468 (accessed October 8, 2019). [41] B. Meng, H. Wu, Y. Qiu, C. Wang, Y. Liu, Z. Xia, S. Yuan, H. Chang, Z. Tian, Large anomalous Hall effect in ferromagnetic Weyl semimetal candidate PrAlGe, APL Materials. 7 (2019) 051110. doi:10.1063/1.5090795. 21

[42] S. Kumar, M. Pepper, S.N. Holmes, H. Montagu, Y. Gul, D.A. Ritchie, I. Farrer, Zero-Magnetic Field Fractional Quantum States, Phys. Rev. Lett. 122 (2019) 086803. doi:10.1103/PhysRevLett.122.086803. [43] G. Shavit, Y. Oreg, Fractional Conductance in Strongly Interacting 1D Systems, Phys. Rev. Lett. 123 (2019) 036803. doi:10.1103/PhysRevLett.123.036803. [44] Chen, S.-C.; Wu, J.-Y.;Lin, C.-Y.;Lin, M.-F. Theory of Magnetoelectric Properties of 2D Systems. IOP Publishing 2017, 2053-2563, ISBN: 978-0-7503-1674-3. [45] F. Zhang, B. Sahu, H. Min, A.H. MacDonald, Band structure of ABC-stacked graphene trilayers, Phys. Rev. B. 82 (2010) 035409. doi:10.1103/PhysRevB.82.035409. [46] G. Li, A. Luican, E.Y. Andrei, Scanning Tunneling Spectroscopy of Graphene on Graphite, Phys. Rev. Lett. 102 (2009) 176804. doi:10.1103/PhysRevLett.102.176804. [47] L.-J. Yin, S.-Y. Li, J.-B. Qiao, J.-C. Nie, L. He, Landau quantization in graphene monolayer, Bernal bilayer, and Bernal trilayer on graphite surface, Phys. Rev. B. 91 (2015) 115405. doi:10.1103/PhysRevB.91.115405. [48] E. A. Henriksen, P. Cadden-Zimansky, Z. Jiang, Z.Q. Li, L.-C. Tung, M.E. Schwartz, M. Takita, Y.-J. Wang, P. Kim, H.L. Stormer, Interaction-Induced Shift of the Cyclotron Resonance of Graphene Using Infrared Spectroscopy, Phys. Rev. Lett. 104 (2010) 067404. doi:10.1103/PhysRevLett.104.067404. [49] T. Taychatanapat, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, Quantum Hall effect and Landau-level crossing of Dirac fermions in trilayer graphene, Nature Phys. 7 (2011) 621625. doi:10.1038/nphys2008. [50] P. Dutta, S.K. Maiti, S.N. Karmakar, Integer quantum Hall effect in a lattice model revisited: Kubo formalism, Journal of Applied Physics. 112 (2012) 044306. doi:10.1063/1.4748312.

22

[51] M. F. Lin, K.W.-K. Shung, Plasmons and optical properties of carbon nanotubes, Phys. Rev. B. 50 (1994) 1774417747. doi:10.1103/PhysRevB.50.17744. [52] L. Jiang, Y. Zheng, H. Li, H. Shen, Magneto-transport properties of gapped graphene, Nanotechnology. 21 (2010) 145703. doi:10.1088/0957-4484/21/14/145703. [53] S. Xiang, V. Miseikis, L. Planat, S. Guiducci, S. Roddaro, C. Coletti, F. Beltram, S. Heun, Low-temperature quantum transport in CVD-grown single crystal graphene, Nano Res. 9 (2016) 18231830. doi:10.1007/s12274-016-1075-0. [54] Y.-Y. Zhang, W.-F. Tsai, K. Chang, X.-T. An, G.-P. Zhang, X.-C. Xie, S.-S. Li, Electron delocalization in gate-tunable gapless silicene, Phys. Rev. B. 88 (2013) 125431. doi:10.1103/PhysRevB.88.125431. [55] W.-F. Tsai, C.-Y. Huang, T.-R. Chang, H. Lin, H.-T. Jeng, A. Bansil, Gated silicene as a tunable source of nearly 100% spin-polarized electrons, Nat Commun. 4 (2013) 16. doi:10.1038/ncomms2525. [56] M. Kim, J.-H. Choi, S.-H. Lee, K. Watanabe, T. Taniguchi, S.-H. Jhi, H.-J. Lee, Valley-symmetry-preserved transport in ballistic graphene with gate-defined carrier guiding, Nature Physics. 12 (2016) 10221026. doi:10.1038/nphys3804. [57] Y.-C. Zhou, H.-L. Zhang, W.-Q. Deng, A 3Nrule for the electronic properties of doped graphene, Nanotechnology. 24 (2013) 225705. doi:10.1088/0957-4484/24/22/225705. [58] Y. Fujimoto, Formation, Energetics, and Electronic Properties of Graphene Monolayer and Bilayer Doped with Heteroatoms, Advances in Condensed Matter Physics. (2015). doi:10.1155/2015/571490. [59] S.S. Varghese, S. Swaminathan, K.K. Singh, V. Mittal, Energetic Stabilities, Structural and Electronic Properties of Monolayer Graphene Doped with Boron and Nitrogen Atoms, Electronics. 5 (2016) 91. doi:10.3390/electronics5040091.

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FIGURE CAPTIONS Fig. 1 - Crystal structures of doped graphene for chosen doping concentrations of (a) 12.5% and (b) 2% dopants. The green and orange spheres denote, respectively, the host C and guest Si/B/N atoms. Fig. 2 - Energy bands of doped graphene with Si, B and N guest atoms for concentrations of 2% [(a)-(c)] and 12.5% [(d)-(f)]concentrations. Fig. 3 - The Bz -dependent LL energies of doped graphene with Si, B and N dopants for concentrations of 2% [(a)-(c)] and 12.5% [(d)-(f)]. Fig. 4 - The LL energies and wavefunctions of the first few levels for (I) Si, (II) B, and (III) N guest atoms with a 12.5% concentration at Bz = 20 T. The blue and red bars represent LLs localized at 1/6 and 2/6 centers, respectively. Fig. 5 - The EF -dependent inverse quantum-Hall conductivity 1/σxy of Si-, B-, and N-doped graphene at Bz = 20 T. The solid (dashed) curve represents the dopant density of 2% (12.5%). The inset shows the quantum-Hall conductivity σxy of 12.5% B-doped graphene.

24

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: