Magneto-electronic and optical properties of Si-doped graphene

Accepted Manuscript Magneto-electronic and optical properties of Si-doped graphene Po-Hsin Shih, Thi-Nga Do, Bor-Luan Huang, Godfrey Gumbs, Danhong Huang, MingFa Lin PII:

S0008-6223(18)31174-6

DOI:

https://doi.org/10.1016/j.carbon.2018.12.040

Reference:

CARBON 13743

To appear in:

Carbon

Received Date: 30 October 2018 Revised Date:

11 December 2018

Accepted Date: 13 December 2018

Please cite this article as: 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 (2019), doi: https://doi.org/10.1016/ j.carbon.2018.12.040. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.

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Magneto-electronic and optical properties of Si-doped

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graphene

Ming-Fa Lina,f †

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

b Institute c Department

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a Department

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Po-Hsin Shiha ,Thi-Nga Dob,∗ , Bor-Luan Huanga† , Godfrey Gumbsc,d , Danhong Huange ,

of Physics, Academia Sinica, Taipei, Taiwan 115

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

d Donostia

International Physics Center (DIPC), P de Manuel Lardizabal,

e US

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4, 20018 San Sebastian, Basque Country, Spain Air Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force Base,

Topology Center, National Cheng Kung University, Tainan, Taiwan 701

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f Quantum

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New Mexico 87117, USA

December 14, 2018

Abstract

The rich and unique magnetic quantization phenomena of Si-doped graphene de-

fect systems for various concentrations and configurations are fully explored by using the generalized tight-binding model. The non-uniform bond lengths, site energies and hopping integrals, as well as a uniform perpendicular magnetic field (Bz zˆ) are taken into account simultaneously. The quantized Landau levels (LLs) are classified into

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ACCEPTED MANUSCRIPT four different groups based on the probability distributions and oscillation modes. The main characteristics of the LLs are clearly reflected in the magneto-optical selection rules which cover the dominating ∆ n = |nv − nc | = 0, the coexistent ∆ n = 0 and ∆ n = 1, along with the specific ∆ n = 1. These rules for inter-LL excitations

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are attributed to the non-equivalence or equivalence of the Ai and Bi sublattices in a supercell. The spectral intensity can be controlled by oscillator strength using a

canonical momentum (vector potential) as well as by density of states using concen-

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tration and distribution of doped Si atoms.

* Corresponding author. E-mail address: [email protected] (T. N. Do) Tel: +886-6-275-7575; Fax: +886-6-74-7995.

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†Corresponding author.

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E-mail address: [email protected] (M.F. Lin)

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ACCEPTED MANUSCRIPT 1. Introduction Magnetic quantization is one of the mainstream topics in physical science, which include the rich magneto-electronic properties [1–3], magneto-optical selection rules, [4–6]

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and quantum Hall effect in few-layer graphene systems. [7–9] Disparate physical phenomena could be achieved by changing the atomic components, [10] lattice symmetries, [11, 12] the lattice geometries such as planar, buckling, rippled, and folding structures, [13–15] the stacking configurations, [16–18] number of layers, [19, 20] distinct dimensionalities, [21, 22]

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spin-orbital couplings, [2, 23] single- or multi-orbital hybridizations, [24] electric field, [25] and either uniform or non-uniform magnetic field. [2,26] In this Letter, our aim is to inves-

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tigate the interesting quantization phenomena of monolayer graphene under the influence of Si-doped defects.

Monolayer graphene presents novel physical properties, mainly as a result of its hexagonal symmetry and the single-atom thickness. The isotropic Dirac-cone structures, initiated from the K and K0 valleys (corners of the first Brillouin zone), are magnetically quantized

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into exceptional LLs, with a specific energy spectrum whose energy is proportional to the square root of both the magnetic-field strength and the quantum number of corresponding √ valence and conduction LLs, Bz nc,v . This simple relation has been verified by scan-

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ning tunneling spectroscopy (STS), [36] optical spectroscopies, [27] and transport equipment. [7] The magneto-optical absorption peaks are identified to satisfy a specific selection

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rule ∆ n = |nv − nc | = 1, directly reflecting the equivalence of the A and B sublattices. Such a rule determines the available scattering processes, leading to the unconventional half-integer Hall conductivity of σxy = (m + 1/2)4e2 /h, [7] in which m is an integer and the factor of 4 represents the spin- and sublattice-dependent degeneracy. This unusual magnetic quantization is attributed to the quantum anomaly of nc,v = 0 LLs associated with the Dirac point. The fundamental properties of graphene are efficiently modified by creating a defect environment generated by substituted impurities or guest atoms in a hexagonal carbon

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ACCEPTED MANUSCRIPT lattice. Various guest-atom-doped graphene systems are expected to present interesting physical phenomena and possess potential applications. Up to now, carbon host atoms have been successfully substituted by guest atoms of Si, [28] B, [29] and N [29, 30] through chemical vapor deposition (CVD) or arc discharge methods. These new 2D materials dis-

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play non-equivalence of the original A and B sublattices, leading to the possible existences of energy-gap engineering and the tilted Dirac cone. According to first-principles calculations, the nitrogen and boron doped graphene systems present peculiar essential physical properties which are sensitive to the doping concentration and position. [31–33] Especially

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for Si-doped graphene, [34, 35] the π bonding extending on a hexagonal lattice is distorted or even destroyed by the different ionization potentials and the non-uniform hopping in-

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tegrals. That is, there exists a greatly modified Dirac cone or a significant energy gap, together with the Si- and C-dominated low-lying band structure. The drastic changes in the energy dispersion relations, band gap, and atom-dominated wave functions play critical roles in diversifying the magnetic quantization phenomena. As far as we are aware, there have never been any experimental observation of the magneto-optical absorption spectra

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in Si-doped graphene systems. Our theoretical results should be very useful for the experi-

spectroscopies.

2. Model

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mental measurements of Landau levels by STS and magneto-absorption spectra by optical

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The complex combined effects, which arise from the distinct ionization potentials, the non-uniform hopping integrals and bond lengths on a deformed hexagonal lattice, the various Bz -induced Peierls phases, and the excitations of electromagnetic waves, are accurately included in the huge Hamiltonian matrix. Here, we take advantage of the generalized tightbinding model built from subenvelope functions on the separate sublattices, corroborated with the dynamic Kubo formula from linear response theory, to fully explore the diversified electronic and optical properties in Si-doped graphene. The details of the tight-binding Hamiltonian are provided in section I of the Supplemental Materials. In our numerical cal4

ACCEPTED MANUSCRIPT culations, an exact diagonalization method is proposed to solve for the magneto-electronic properties and magneto-optical spectra more efficiently. [43] The tight-binding model is an appropriate method to investigate fundamental magnetic properties of any 2D layered material, including the magneto-electronic properties, [43] magneto-optical and quantum

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transport properties via the dynamic and static Kubo formulas; magneto-Coulomb excitations within the modified random-phase approximation (RPA). In this paper, various kinds of LLs appearing during the variation of Si-distribution configuration and concentration are thoroughly investigated. Their main features, characterized by probability

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distributions and oscillation modes, are clearly illustrated by the distinct magneto-optical selection rules. This work opens a new research category in the fundamental properties

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of 2D layered materials. Our theoretical predictions need further experimental verifications by using appreciate measurement methods, such as, STS, [18,36–38] magneto-optical spectroscopies, [27, 39–42] and quantum transport measurements. [7] The presence of a uniform perpendicular magnetic field (Bz zˆ) significantly changes the

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physical features of the systems. The dimension of the magnetic Hamiltonian matrix is determined by the guest-atom- and vector-potential-dependent periods, in which the latter is much longer than the former, and their ratio is assumed to be an integer for convenience

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in the calculations. The vector potential, chosen as Bz xˆ y , creates a position-related Peierls R Rm A(r) · dr for φ0 = hc/e being the flux quantum, in the nearestphase of ∆Gmm0 = 2π φ0 R 0 m

neighbor hopping integral. [43] Consequently, the magnetic unit cell becomes an enlarged

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rectangular one, as indicated in Fig. 1(c). For the type I guest-atom distribution, such a unit cell covers 16RB atoms (8RB A and B atoms). Here, RB is defined as the ratio of flux quantum versus magnetic flux through each hexagon, e.g., RB = 8000 at Bz = 10 T. The Bloch wave functions under a Bz zˆ can be expressed by the linear superposition of the 16RB tight-binding functions in an enlarged unit cell. When the complex Hamiltonian is solved by the exact diagonalization method, [43] the Bz -induced LL energy spectrum and the magnetic subenvelope functions could be computed more efficiently. When Si-doped graphene is irradiated by an electromagnetic wave, the occupied valence 5

ACCEPTED MANUSCRIPT states are vertically excited to unoccupied states in the conduction band. In addition to ∆k = 0, the electric dipole perturbations require the inter-LL excitations to satisfy a new magneto-optical selection rule, i.e., ∆ n = 0. Such interesting behavior has never been discovered for other layered condensed -matter systems. According to the linear Kubo

nc ,nv

1stBZ

i 1 . E c (nc , k) − E v (nv , k) − ω − iΓ

(1)

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h ×Im

E 2 ˆ · P dk D c c E v v Ψ (n , k) Ψ (n , k) 2 (2π) me

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A(ω) ∝

XZ

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formula, the intensity of magneto-optical excitations is characterized by

ˆ P and me are, respectively, the unit vector of the electric polarization, moHere, E, mentum operator and bare electron mass. Because the direction of the planar electric field ˆ k x ˆ is chosen in the present work. The hardly affects the optical absorption spectra, E D E ˆ v v square of the velocity matrix element ( Ψc (nc , k) E·P Ψ (n , k) ) is critical to understand me

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the optical selection rules of the systems. It directly determines the available excitation channels and the spectral strength, since it is associated with the spatial distribution symmetries of the initial and final LLs. The second term in the integrand is the delta-functionlike joint density of states arising from the inter-LL transitions of (nv , k) → (nc , k), in

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which the broadening factor is Γ = 1 meV. The critical dipole factor is evaluated from the gradient approximation, as successfully utilized in carbon-related sp2 -bonding systems. [44]

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That is,

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p E ∂ D c c v v E x Ψ (n , k) H Ψ (n , k) . Ψc (nc , k) Ψv (nv , k) ∼ = me ∂kx

(2)

Equation (2) clearly indicates that the electric-dipole magneto-optical excitations are dominated by the Ai (Bi ) subenvelope functions of the initial nv LL and the Bi0 (Ai0 ) ones of the final nc LL. Since the velocity matrix element is associated with the k-dependent nearest-neighbor hopping integrals, i and i0 denote the nearest-neighbor lattice sites (de6

ACCEPTED MANUSCRIPT tails are given in section II of the Supplemental Materials). This is the source of specific magneto-optical selection rules.

3. Results and Discussion

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We consider four types of specific Si-doped graphene systems which can clearly illustrate the diversified properties. They cover (type I) 2:16 [Si:(C+Si)] concentration for both guest atoms in the same sublattice (the Si-(A1 , A6 )-sublattice distribution; red spheres in Fig.

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1(a)), (type II) 2:16 concentration for guest atoms in different sublattices (the Si-(A6 , B4 ) configuration; green spheres), (type III) 2:64 concentration with similar guest atom

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substitution as type II (Si-(A1 , A19 configuration; Fig. 1(b)), and (type IV) a pristine graphene with equivalent A and B sublattices. Types I and III (type II) present nonequivalent (equivalent) Ai and Bi sublattices in a Si-induced unit cell, while both A and B sublattices are fully equivalent for pristine graphene. For example, type I has a rectangular traditional cell comprising two Si and fourteen C atoms, which is consistent with the Landau

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gauge under Bz zˆ. It is noticed that these four types of Si-doped graphene systems exhibit highly symmetric configurations, they are relatively stable. There exist slight buckling near the guest atoms (∼ 0.93 ˚ A deviation from graphene plane) and the distinct C-C and Si-C bond length (1.42 ˚ A & 1.70 ˚ A), according to the first-principles calculations.

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[34, 35] Though this buckled structure indicates remarkable modifications of the π bonding extending on a hexagonal lattice, the non-uniform site energies and nearest-neighboring

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hopping integrals due to the major 2pz orbitals of the C host atoms and the minor 3pz orbitals of Si guest atoms are sufficient in understanding the low-lying energy bands. These parameters are optimized as Si−C =1.3 eV, γC−C =2.7 eV and γSi−C =1.3 eV, respectively, in order to reproduce the band structures from the first-principles calculations. They are valid for many different distribution configurations and concentrations of Si-doped graphene systems. Si-doped graphene exhibits unusual low-energy electronic properties. For type I (the red curves in Fig. 2(a)), the valence and conduction bands nearest to the Fermi level (EF ) 7

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(a)

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(c)

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(b)

Figure 1: (Color online) Geometric structures of Si-doped graphene systems of (a) (type I) 2:16 concentration under the Ai - (red balls) and (type II) [Ai , Bj ]-sublattice distributions (green spheres), (b) (type III) 2:64 concentration for the Ai -sublattice distribution. An 8 enlarged rectangular unit cell in Bz zˆ is presented in (c).

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(b)

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(a)

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Figure 2: (Color online) The (a)-(b) low-lying energy bands for four types of Si distributions and concentrations.

have parabolic energy dispersions separated by a direct energy gap of Eg = 0.74 eV. The electronic energy spectrum is anisotropic along the different k-directions and asymmetric about EF . Similar results are also obtained for type III of the lower-concentration system

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with a 0.26 eV band gap (the blue curves in Fig. 2(a)). An energy gap appears only if the guest atoms are situated at the same sublattices (either the Ai or Bi ). The non-uniform site energies and hopping integrals further induce partial termination of the π bonding

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(the minor localized states), as observed in the zero-field and magnetic wave functions (Figs. 3(a), 3(b); 3(d)). On the other hand for the type II distribution configuration,

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the doped graphene shows vanishing Eg (the solid curve in Fig. 2(b)). The guest-atom distribution with equal weight induces the distorted π and thus the strongly modified Dirac cone structure with an obvious shift of Dirac point, the reduced Fermi velocity, and the anisotropic energy spectrum. Regarding type IV, graphene exhibits a well-behaved Dirac cone (the dashed curve) because of the purely hexagonal symmetry. The magnetic-quantized LLs exhibit rich and unique features. The low-lying LL energy spacings, as shown for Bz = 10 T in Figs. 3(a) and 3(b), are almost uniform and have an energy gap of equivalent value with the zero-field band gap. In general, the quantum 9

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Figure 3: (Color online) The conduction and valence LL energy spectra and the corre-

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sponding probability distributions for (a)-(b) type I near the 1/6 and 2/6 localization centers at Bz = 10 T. Similar plots for (c) type II, (d) type III and (e) type IV are

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also presented. In panel (d), B1st = {B1,2,15,19,20,29 }, B2nd = {B5,14,16,23,30,32 }, B3rd = {B3,4,6,8,11,12,17,18,22,24,25,26 }, B4th = {B9,10,13,27,28,31 }, and B5th = {B7,21 } denote the B sublattices which are from nearest to the furthest from the doped Si atoms, respectively.

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ACCEPTED MANUSCRIPT number for each LL is defined from the zero points of the dominating oscillation mode. For the 12.5 % Si-Ai -sublattice graphene, the magnetic Bloch wave function arises from the subenvelope functions of the 16 tight-binding functions on the corresponding sublattices. Its spatial probability distribution of the (kx = 0, ky = 0) state is localized at (1/6 and 4/6)

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as well as (2/6 and 5/6) of an enlarged unit cell (Fig. 1(c)). Any (kx , ky ) LL states in the reduced first Brillouin zone are doubly degenerate for each spin degree of freedom. The decoration of Si guest atoms leads to the destruction of the planar inversion symmetry and thus the non-degenerate 1/6 and 2/6 LL states. According to the neighboring chemical

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environment, the original 16 sublattices could be classified into four subgroups of (A1 , A6 ), the other Ai (A2 , A3 , A4 , A5 , A7 , A8 ), the nearest-neighbor Bi (B1 , B2 , B4 , B5 , B6 , B7 ), and

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the next-nearest-neighbor Bi (B3 , B8 ). The low-energy conduction LL states are dominated by the A1 sublattice with the Si-3pz tight-binding function, so that the zero-point number of the well-behaved probability distribution could serve as a good quantum number. nc =0, 1, 2 and appears in the normal sequence. Specifically, the contributions from the B3 sublattice are small, as seen in the zero-field wave functions. The oscillation modes are characterized

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by nc for the significant sublattices except for the weak nc ± 1 B3 sublattice. On the other hand, the valence LL states mainly originate from all the Bi sublattices of the C-2pz tight-binding functions, where they have similar oscillation modes in determining nv . The

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contributions from the Ai sublattices are very small, and the number of zero points is nv − 1 or nv + 1 (see Figs. 3(a) and 3(b)). The sequences of nc and nv present good orderings,

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i.e., the crossing or anti-crossing behaviors are absent. These reliable magneto-electronic properties are very useful in understanding the rich magneto-optical excitation spectra. The spatial oscillatory modes are sensitive to the configuration distribution and concentration of the guest atoms. There exist four types of LLs, corresponding to four types of lattice geometries with different concentrations and configurations. For a very strong non-equivalence between Ai and Bi sublattices and enough high concentration (2:16 under the Si-(A1 , A6 ) configuration in Fig. 1(a)), only the significant sublattices exhibit similar oscillation modes for the low-lying valence and conduction LLs (the first kind in Figs. 11

ACCEPTED MANUSCRIPT 3(a)-3(b)). However, the enhanced equivalence (green spheres in Fig. 1(a)) and the reduced concentration (Fig. 1(b)) can create the composite behaviors related to the heavily non-equivalent Ai and Bi sublattices and the fully equivalent ones (e.g., pristine graphene). The former, with two Si atoms in the A6 and B4 sublattices, has the strikingly equivalent

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environment. All the sublattices make significant contributions to the LL wave functions, in which the difference between the zero point number is ± 1 for the Ai and Bi sublattices (the second type in Fig. 3(c)). Specifically, their spatial distributions are highly asymmetric and the localization centers deviate conspicuously from 1/6 and 2/6, directly reflecting the

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significantly tilted Dirac-cone (Fig. 2(b)). Also, a grossly distorted distribution consists of the main nc,v mode and the side nc,v ± 1 ones. The localization centers are recovered to

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their normal positions under a decrease in concentration with the Si-Ai distribution (2:64 in Fig. 1(b)). The certain Bi sublattices, which are farthest from the Si atom and possess nc ± 1 modes, become observable for the conduction LLs, and so do the Ai sublattices in the valence LLs (the third kind in Fig. 3(d)). Moreover, the wave functions in other Bi sublattices present highly asymmetric distributions for the Si-dominated LLs. Finally, pris-

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tine graphene displays well-behaved LLs around the localization centers and the difference of ± 1 in the zero-point number due to the equivalent A and B sublattices (the fourth kind in Fig. 3(e)).

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The Bz -dependent LL energy spectrum, as clearly indicated in Figs. 4(a)-4(e), possesses typical features. Crossing or anti-crossing behavior is forbidden for the low-lying LLs,

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illustrating well separated LL states and specific-mode wave functions. For the first and third kinds of LLs (Figs. 4(a), 4(b) and 4(d)), the dispersion relation is almost linear, and the LL energy spacing is uniform. Specifically, the initial valence and conduction LLs, which are, respectively, related to the 1/6 and 2/6 localization centers, remain fixed energies during variation of the field strength. They come entirely from the localized electronic states, since the magnetic wave functions vanish on both the Ai and Bi sublattices, as observed from Figs. 3(a) and 3(b). That is, the termination of π bonding appears on a guest-host mixed hexagonal lattice. A uniform perpendicular magnetic field can create 12

(b) 2/6

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(a) 1/6

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Figure 4: (Color online) The (a)-(e) Bz -dependent LL energy spectra corresponding to four

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types of lattice geometries in Fig. 3. The density of states is also shown for the type I of Si distribution configuration.

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the delta function-like prominent peaks across the Fermi level (see Figs. 4(a) and 4(b)). On √ the other hand, the second and fourth categories of LLs show the B z -dependent energy spectra except for the constant energy of the degenerate nc,v = 0 LLs. The latter has the largest energy spacing among the four descriptions of LLs because of the lowest density of states.

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Figure 5: (Color online) The magneto-optical absorption spectra of (a) type I, (b) type II, (c) type III, and (d) type IV of Si-doped graphene systems.

The main characteristics of the LLs are directly reflected in the magneto-optical absorption spectra with several delta function-like peaks, as demonstrated in Fig. 5. For 14

ACCEPTED MANUSCRIPT Si-A1 -doped graphene with 2:16 concentration (see Fig. 5(a)), the spectral intensity gradually declines with increasing frequency, whereas the energy spacing between two neighboring absorption peaks is almost uniform. Only the inter-LL transitions, which correspond to the identical quantum mode in the valence and conduction LLs, are revealed as significant

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absorption peaks. For example, the threshold frequency due to 0v → 0c is 0.743 eV very close to the energy gap. The magneto-optical selection rule, ∆ n = 0, could be thoroughly examined from the electric-dipole momentum in Eq. (2). It is mainly dominated by the specific Hamiltonian matrix elements covering the nearest-neighboring hopping integrals.

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Consequently, the effective vertical excitations depend on the subenvelope functions of the Bi /Ai & Bi+1 /Ai sublattices in the nv /nc LLs. Additionally, the significant sublattices

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present the same zero-point number. These results are responsible for a new selection rule, is never found in other condensed matter systems.

Both ∆ n = 0 and 1 come to exist together under the reduced non-equivalence of A−i and Bi sublattices, as clearly shown in Figs. 5(b) and 5(c). As for Si-(A3 , B5 )-decorated

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graphene with 2:16 concentration, two categories of inter-LL excitation channels frequently appear during the variation of frequency. The absorption peaks of ∆ n = 1 decrease quickly, while the opposite is true for those with ∆ n = 0. The former and the latter, respectively, come from the neighboring Ai and Bi sublattices with the mode difference of ± 1 and 0.

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The lower-frequency absorption peaks are dominated by ∆ n = 1, since the corresponding LLs, similar to those of graphene, are magnetically quantized from the low-lying tilted

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Dirac cone. However, with increasing energy, the enlarged range of localizattion centers and the enhanced distortion of the spatial probability (see Fig. 3(c)) create and enhance the available channels of ∆ n = 0 through the strengthened side mode. Such characteristics lead to strong competition between these two kinds of magneto-selection rules. The coexistent selection rules on the other hand present another kind of behavior for Si-Ai -doped graphene with reduced concentration, as shown in Fig. 5(c). The ∆ n = 0 channels dominate the lower-frequency absorption spectrum, since the significant sublattices possess the same oscillation modes (see Fig. 3(d)). Their peak intensities slowly 15

ACCEPTED MANUSCRIPT increase with increasing frequency, clearly indicating the significant competition or cooperation between two categories of inter-sublattice transitions. Bi → Ai and Ai → Bi (except for the furthest ones) appear under the nv → nc inter-LL transition, in which the second category is absent for a sufficiently high concentration in Fig. 5(a). Especially, the rapid

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enlargement of the ∆ n = 1 absorption peaks is due to the enhanced oscillations in the Ai sublattices of the valence LLs and the furthest Bi sublattices of the conduction LLs, as well as the strengthened side modes in other Bi sublattices. On the other hand, it is well known that graphene only allows the ∆ n = 1 absorption peaks with a uniform optical

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spectrum (Fig. 5(d)) as a result of the full equivalence of A and B sublattices. Among all the Si-doped graphene systems, the pristine one has the strongest intensity and the largest

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energy spacings. These are closely related to the smallest density of states for the isotropic Dirac cone, as indicated from a detailed comparison with the separated parabolic bands and the tilted Dirac cone (Fig. 2).

The diverse magneto-electronic properties and absorption spectra could be verified by

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STS and optical spectroscopies, respectively. Specifically, STS is a very efficient method for examining the quantized energy spectra. The tunneling differential conductance (dI/dV) is approximately proportional to the density of states, and it directly reflects the structure,

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energy, number and height of the LL peaks. The theoretical predictions have been partially √ confirmed through magnetic measurements, verifying the B z -dependent LL energies in monolayer graphene, [36] the linear Bz -dependence in bilayer AB stacking, [37] the coexis-

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tence of the square-root and linear Bz -dependencies in trilayer AB-stacked graphene, [18] and the 3D and 2D characteristics of the Landau subbands in Bernal graphite. [38] Magnetic quantization phenomena of layered systems could also be identified by magnetooptical spectroscopies. [27, 39–42] The examined phenomena are exclusive in graphenerelated systems, such as 0D LLs in few-layer graphenes and 1D Landau subbands in bulk graphite. [27, 39, 40] A lot of prominent delta function-like absorption peaks are clearly shown by the inter-LL excitations due to massless and massive Dirac fermions in monolayer [27] and AB-stacked bilayer graphenes. [39] The former and the latter absorption 16

ACCEPTED MANUSCRIPT frequencies are square-root and linearly proportional to Bz , respectively. As for the interLandau-subband excitations in Bernal graphite, one could observe a strong dependence on the wave vector kz , which characterizes both kinds of Dirac quasiparticles. [41,42] In short, the experimental examinations on four kinds of LLs and the distinct magneto-optical selec-

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tion rules could provide full information about the diversified essential properties, establish the emergent binary or ternary graphene compounds, and confirm our well-developed theoretical framework.

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4. Conclusion

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In summary, the Si-doped graphene systems, an emergent 2D binary compound materials, are worthy of systematic theoretical and experimental research and are very suitable for exploring novel physical phenomena. The rich and unique quantization phenomena are investigated by using the generalized theoretical framework which takes into account simultaneously all the critical factors of non-uniform bond lengths, site energies and hop-

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ping integrals, together with external field effects without the use of perturbation theory. This method is expected to be very useful in gaining a comprehensive understanding of the essential properties of the main-stream layered systems. Si-doped graphene with various concentrations and configurations have revealed the diverse electronic and optical proper-

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ties under magnetic quantization, being absent in other condensed-matter materials. There exist four categories of LLs, according to the probability distributions and oscillation modes

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on the distinct sublattices, and the relationships between Ai and Bi sublattices. They cover significant Bi sublattices of valence LLs Ai sublattices of conduction LLs with the same modes, the observable (Ai , Bi ) sublattices with a mode difference of ± 1, the significant deviations of localization centers and the highly asymmetric distributions composed of the central and side modes, the same modes for valence Bi and conduction Ai sublattices, the ± 1 zero-point differences between valence Ai and conduction Bi sublattices in addition to the perturbed multi-modes in most of the conduction Bi sublattices (except for the furthest ones). The oscillator-like fluctuation modes with the equivalent A and B sublattices. Such 17

ACCEPTED MANUSCRIPT LLs lead to unusual magneto-optical selection rules of the dominant ∆ n = 0, the coexistent ∆ n = 1 and 0 with strong competitions, and the specific ∆ n = 1. The interesting features of LLs correspond to various concentrations and distribution configurations. They are, SiAi -doped graphene with a sufficiently high concentration, the (Ai , Bi )-decorated graphene,

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low-concentration Ai -doped system, and the pristine one. The theoretical predictions in this work might be confirmed by suitable experimental measurements.

Acknowledgments

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This material is based upon work supported by the Air Force Office of Scientific Research (AFOSR) under award number FA2386-18-1-0120. D.H. acknowledges the support from the

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AFOSR and from the DoD Lab-University Collaborative Initiative (LUCI) Program. G.G. would like to acknowledge the support from the Air Force Research Laboratory (AFRL)

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through Grant #12530960.

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References [1] Yin, L.-J.; Bai, K.-K.; Wang, W.-X.; Li, S.-Y.; Zhang, Y.; He, L. Landau Quantization of Dirac Fermions in Graphene and Its Multilayers. Front. Phys. 2017, 12 (4), 127208.

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[2] Do, T.-N.; Shih, P.-H.; Gumbs, G.; Huang, D.; Chiu, C.-W.; Lin, M.-F. Diverse Magnetic Quantization in Bilayer Silicene. Phys. Rev. B 2018, 97 (12), 125416.

[3] Lado, J. L.; Fernndez-Rossier, J. Landau Levels in 2D Materials Using Wannier Hamil-

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tonians Obtained by First Principles. 2D Mater. 2016, 3 (3), 035023.

[4] Stepniewski, R.; Pastor, K.; Grynberg, M. Gamma 8 -Bands Anisotropy and Selection

Phys. 1980, 13 (31), 5783.

M AN U

Rules for Far-Infrared Magneto-Optical Transitions in HgTe. J. Phys. C: Solid State

[5] Gopalan, S.; Furdyna, J. K.; Rodriguez, S. Inversion Asymmetry and Magneto-Optical Selection Rules in n-Type Zinc-Blende Semiconductors. Phys. Rev. B 1985, 32 (2),

TE D

903913.

[6] Wu, J.-Y.; Chen, S.-C.; Do, T.-N.; Su, W.-P.; Gumbs, G. The Diverse Magneto-Optical Selection Rules in Bilayer Black Phosphorus. arXiv:1712.09508 [cond-mat] 2017.

EP

[7] Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Two-Dimensional Gas of Massless Dirac Fermions

AC C

in Graphene. Nature 2005, 438 (7065), 197-200. [8] Novoselov, K. S.; McCann, E.; Morozov, S. V.; Falko, V. I.; Katsnelson, M. I.; Zeitler, U.; Jiang, D.; Schedin, F.; Geim, A. K. Unconventional Quantum Hall Effect and Berry Phase of 2p in Bilayer Graphene. Nature Physics 2006, 2 (3), 177-180. [9] Do, T.-N.; Chang, C.-P.; Shih, P.-H.; Wu, J.-Y.; Lin, M.-F. Stacking-Enriched Magneto-Transport Properties of Few-Layer Graphenes. Phys. Chem. Chem. Phys. 2017, 19 (43), 29525-29533.

19

ACCEPTED MANUSCRIPT [10] John, R.; Merlin, B. Optical Properties of Graphene, Silicene, Germanene, and Stanene from IR to Far UV A First Principles Study. Journal of Physics and Chemistry of Solids 2017, 110, 307-315.

RI PT

[11] Weinberg, M.; Staarmann, C.; lschlger, C.; Simonet, J.; Sengstock, K. Breaking Inversion Symmetry in a State-Dependent Honeycomb Lattice: Artificial Graphene with Tunable Band Gap. 2D Mater. 2016, 3 (2), 024005.

[12] Do, T.-N.; Shih, P.-H.; Chang, C.-P.; Lin, C.-Y.; Lin, M.-F. Rich Magneto-Absorption

SC

Spectra of AAB-Stacked Trilayer Graphene. Phys. Chem. Chem. Phys. 2016, 18 (26),

M AN U

17597-17605.

[13] Shih, P.-H.; Chiu, C.-W.; Wu, J.-Y.; Do, T.-N.; Lin, M.-F. Coulomb Scattering Rates of Excited States in Monolayer Electron-Doped Germanene. Phys. Rev. B 2018, 97 (19), 195302.

[14] Lee, K. J.; Kim, D.; Jang, B. C.; Kim, D.-J.; Park, H.; Jung, D. Y.; Hong, W.; Kim,

TE D

T. K.; Choi, Y.-K.; Choi, S.-Y. Multilayer Graphene with a Rippled Structure as a Spacer for Improving Plasmonic Coupling. Advanced Functional Materials 2016, 26 (28), 5093-5101.

EP

[15] Liu, F.; Song, S.; Xue, D.; Zhang, H. Folded Structured Graphene Paper for High Performance Electrode Materials. Advanced Materials 2012, 24 (8), 1089-1094.

AC C

[16] Lin, C.-Y.; Do, T.-N.; Huang, Y.-K.; Lin, M.-F. Optical Properties of Graphene in Magnetic and Electric Fields. IOP Publishing 2017, 2053-2563, ISBN: 978-0-75031566-1.

[17] Huang, B.-L.; Chuu, C.-P.; Lin, M.-F. Asymmetry-Enriched Electronic and Optical Properties of Bilayer Graphene. arXiv:1805.10775 [cond-mat] 2018.

20

ACCEPTED MANUSCRIPT [18] Jhang, S. H.; Craciun, M. F.; Schmidmeier, S.; Tokumitsu, S.; Russo, S.; Yamamoto, M.; Skourski, Y.; Wosnitza, J.; Tarucha, S.; Eroms, J.; et al. Stacking-Order Dependent Transport Properties of Trilayer Graphene. Phys. Rev. B 2011, 84 (16), 161408.

RI PT

[19] Pontes, R. B.; Miwa, R. H.; da Silva, A. J. R.; Fazzio, A.; Padilha, J. E. LayerDependent Band Alignment of Few Layers of Blue Phosphorus and Their van Der Waals Heterostructures with Graphene. Phys. Rev. B 2018, 97 (23), 235419.

[20] Zhao, S.; Lv, Y.; Yang, X. Layer-Dependent Nanoscale Electrical Properties of

SC

Graphene Studied by Conductive Scanning Probe Microscopy. Nanoscale Res Lett

M AN U

2011, 6 (1), 498.

[21] Mostofizadeh, A.; Li, Y.; Song, B.; Huang, Y. Synthesis, Properties, and Applications of Low-Dimensional Carbon-Related Nanomaterials. Journal of Nanomaterials 2018, 2011, 685081.

[22] Meunier, V.; Souza Filho, A. G.; Barros, E. B.; Dresselhaus, M. S. Physical Properties

025005.

TE D

of Low-Dimensional sp2 -Based Carbon Nanostructures. Rev. Mod. Phys. 2016, 88 (2),

[23] Krasovskii, E. E. Spin-Orbit Coupling at Surfaces and 2D Materials. J. Phys.: Con-

EP

dens. Matter 2015, 27 (49), 493001.

[24] Wang, S. A Comparative First-Principles Study of Orbital Hybridization in Two-

AC C

Dimensional C, Si, and Ge. Phys. Chem. Chem. Phys. 2011, 13 (25), 11929-1938. [25] Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Electric Field Effect in Atomically Thin Carbon Films. Science 2004, 306 (5696), 666-669. [26] Park, S.; Sim, H.-S. Magnetic Edge States in Graphene in Nonuniform Magnetic Fields. Phys. Rev. B 2008, 77 (7), 075433.

21

ACCEPTED MANUSCRIPT [27] Jiang, Z.; Henriksen, E. A.; Tung, L. C.; Wang, Y.-J.; Schwartz, M. E.; Han, M. Y.; Kim, P.; Stormer, H. L. Infrared Spectroscopy of Landau Levels of Graphene. Phys. Rev. Lett. 2007, 98 (19), 197403.

RI PT

[28] Zhou, W.; Kapetanakis, M. D.; Prange, M. P.; Pantelides, S. T.; Pennycook, S. J.; Idrobo, J.-C. Direct Determination of the Chemical Bonding of Individual Impurities in Graphene. Phys. Rev. Lett. 2012, 109 (20), 206803.

[29] Panchakarla, L. S.; Subrahmanyam, K. S.; Saha, S. K.; Govindaraj, A.; Krishna-

SC

murthy, H. R.; Waghmare, U. V.; Rao, C. N. R. Synthesis, Structure, and Properties of Boron- and Nitrogen-Doped Graphene. Advanced Materials 2009, 21 (46), 4726-

M AN U

4730.

[30] Qu, L.; Liu, Y.; Baek, J.-B.; Dai, L. Nitrogen-Doped Graphene as Efficient Metal-Free Electrocatalyst for Oxygen Reduction in Fuel Cells. ACS Nano 2010, 4 (3), 1321-1326. [31] Natha, P.; Chowdhurya, S.; D.Sanyal, D.; Jana, D. Ab-initio calculation of electronic

73, 275-282.

TE D

and optical properties of nitrogen and boron doped graphene nanosheet. Carbon 2014,

[32] Natha, P.; D.Sanyal, D.; Jana, D. Semi-metallic to semiconducting transition in

EP

graphene nanosheet with site specific co-doping of boron and nitrogen. Physica E: Low-dimensional Systems and Nanostructures 2014, 56, 64-68.

AC C

[33] Chowdhurya, S.; Jana, D. Electronic and magnetic properties of modified silicene/graphene hybrid: Ab initio study. Materials Chemistry and Physics 2016, 183, 580-587.

[34] Zhang, S. J.; Lin, S. S.; Li, X. Q.; Liu, X. Y.; Wu, H. A.; Xu, W. L.; Wang, P.; Wu, Z. Q.; Zhong, H. K.; Xu, Z. J. Opening the Band Gap of Graphene through Silicon Doping for the Improved Performance of Graphene/GaAs Heterojunction Solar Cells. Nanoscale 2015, 8 (1), 226-232.

22

ACCEPTED MANUSCRIPT [35] Shahrokhi, M.; Leonard, C. Tuning the Band Gap and Optical Spectra of SiliconDoped Graphene: Many-Body Effects and Excitonic States. Journal of Alloys and Compounds 2017, 693, 1185-1196.

RI PT

[36] Miller, D. L.; Kubista, K. D.; Rutter, G. M.; Ruan, M.; Heer, W. A. de; First, P. N.; Stroscio, J. A. Observing the Quantization of Zero Mass Carriers in Graphene. Science 2009, 324 (5929), 924-927.

[37] Rutter, G. M.; Jung, S.; Klimov, N. N.; Newell, D. B.; Zhitenev, N. B.; Stroscio, J. A.

SC

Microscopic Polarization in Bilayer Graphene. Nature Physics 2011, 7 (8), 649-655.

M AN U

[38] Li, G.; Andrei, E. Y. Observation of Landau Levels of Dirac Fermions in Graphite. Nature Physics 2007, 3 (9), 623-627.

[39] Orlita, M.; Faugeras, C.; Borysiuk, J.; Baranowski, J. M.; Strupinski, W.; Sprinkle, M.; Berger, C.; de Heer, W. A.; Basko, D. M.; Martinez, G.; et al. Magneto-Optics of Bilayer Inclusions in Multilayered Epitaxial Graphene on the Carbon Face of SiC.

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Phys. Rev. B 2011, 83 (12), 125302.

[40] Plochocka, P.; Faugeras, C.; Orlita, M.; Sadowski, M. L.; Martinez, G.; Potemski, M.; Goerbig, M. O.; Fuchs, J.-N.; Berger, C.; de Heer, W. A. High-Energy Limit of

EP

Massless Dirac Fermions in Multilayer Graphene Using Magneto-Optical Transmission Spectroscopy. Phys. Rev. Lett. 2008, 100 (8), 087401.

AC C

[41] Plochocka, P.; Solane, P. Y.; Nicholas, R. J.; Schneider, J. M.; Piot, B. A.; Maude, D. K.; Portugall, O.; Rikken, G. L. J. A. Origin of Electron-Hole Asymmetry in Graphite and Graphene. Phys. Rev. B 2012, 85 (24), 245410. [42] Nicholas, R. J.; Solane, P. Y.; Portugall, O. Ultrahigh Magnetic Field Study of Layer Split Bands in Graphite. Phys. Rev. Lett. 2013, 111 (9), 096802. [43] 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. 23

ACCEPTED MANUSCRIPT [44] Lin, M. F.; Shung, K. W.-K. Plasmons and Optical Properties of Carbon Nanotubes.

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Phys. Rev. B 1994, 50 (23), 17744-17747.

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FIGURE CAPTIONS Fig. 1 - Geometric structures of Si-doped graphene systems of (a) (type I) 2:16 concentration under the Ai - (red balls) and (type II) [Ai , Bj ]-sublattice distributions (green

rectangular unit cell in Bz zˆ is presented in (c).

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spheres), (b) (type III) 2:64 concentration for the Ai -sublattice distribution. An enlarged

Fig. 2 - The (a)-(b) low-lying energy bands for four types of Si distributions and concentrations.

Fig. 3 - The conduction and valence LL energy spectra and the corresponding probabil-

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ity distributions for (a)-(b) type I near the 1/6 and 2/6 localization centers at Bz = 10 T. Similar plots for (c) type II, (d) type III and (e) type IV are also presented. In panel (d),

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B1st = {B1,2,15,19,20,29 }, B2nd = {B5,14,16,23,30,32 }, B3rd = {B3,4,6,8,11,12,17,18,22,24,25,26 }, B4th = {B9,10,13,27,28,31 }, and B5th = {B7,21 } denote the B sublattices which are from nearest to the furthest from the doped Si atoms, respectively.

Fig. 4 - The (a)-(e) Bz -dependent LL energy spectra corresponding to four types of lattice geometries in Fig. 3. The density of states is also shown for the type I of Si

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distribution configuration.

Fig. 5 - The magneto-optical absorption spectra of (a) type I, (b) type II, (c) type III,

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and (d) type IV of Si-doped graphene systems.

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