• No results found

radiation with MO transport has been explored. Finally, we have studied Faraday rotation which should be possible to be realized in the experiments.



A Chebyshev expansion

The first kind Chebyshev polynomials can be defined as,Tn(x)=cos(ncos−1(x)) in the range [−1, 1]. The recursion relations, T0(x)=1, T1(x)= xand Tn+1(x)=2xTn(x)−Tn1(x) and the orthogonality relation,

Z 1

1Tn(x)Tm(x) dx


2 (A.0.1)

are satisfied by these polynomials. The expansion of the Dirac delta in terms of Chebyshev polynomials, can be written as,





1+δn0, (A.0.2)


n()= 2Tn() π√

1−2. (A.0.3)

Also the Green‘s function can be expressed in terms of theTn(x) as,




gσ,λn ()Tn(H0)

1+δn0, (A.0.4)


gσ,λn ()=−2σieniσcos1(+iσλ)

p1−(+iσλ)2. (A.0.5)

The functiongσ,λrepresents both the retarded and the advanced Green’s function in the limit λ→0+, whereλis the finite broadening parameter. g+,0+ andg,0+ are the retarded and the advanced Green’s function respectively. Hence, the Dirac deltas and Green’s function are combinations of a polynomial ofH0 (the unperturbed Hamiltonian) and a coefficient which are functions of the frequency and the energy parameters. The trace in the conductivity can TH-2574_166121018

be written as a trace over a product of polynomials and ˆhoperators. TheΓmatrix needed in the expression of conductivity is written as,

Γnα11···,···,αnmm= Tr N



1+δn10 · · ·h˜αmTnm(H0) 1+δnm0


, (A.0.6)

whereNis the number of unit cells. The upper indices can be used for any number of indices:

α111α21· · ·α1N1 and ˜hα1 =(i¯h)N1α1. Here the coefficients of the Chebyshev expansion can be written similarly in a matrix form as,



df()∆n() (A.0.7)


Λnm(ω)=¯h Z




+∆n()gmA(/h¯−ω), (A.0.8) where f()=(1+eβ(µ))1 is the Fermi-Dirac distribution function, where βis the inverse temperature andµis the chemical potential. Hence the first-order conductivity becomes,

σαβ(ω)= −ie2c2ω









whereΩcis the volume of the unit cell.

Also, the density of states can be found in terms of Chebyshev polynomials using Eq. (A.0.2) as,

ρ()= 1

NTrδ(−H0)= 1 π√




µnTn(), (A.0.10)

whereN=dim ˆH0is the Hilbert space dimension,µndenotes the Chebyshev moments. The Chebyshev momentsµnis,

µn= 1 N


2 TrTn(H0). (A.0.11)


[1] N. K. P. R. Mello,Quantum Transport in Mesoscopic Systems: Complexity and Statistical Fluctuations. Oxford University Press, 2004.

[2] S. Datta,Electronic Transport in Mesoscopic Systems. Cambridge Studies in Semiconductor Physics and Microelectronic Engineering. Cambridge University Press, 1995.

[3] R. Landauer,Spatial variation of currents and fields due to localized scatterers in metallic conduction,IBM Journal of Research and Development1no. 3, (1957) 223–231.

[4] E. H. Hall,On a new action of the magnet on electric currents,American Journal of Mathematics2no. 3, (1879) 287–292.http://www.jstor.org/stable/2369245. [5] K. v. Klitzing, G. Dorda, and M. Pepper,New method for high-accuracy

determination of the fine-structure constant based on quantized hall resistance,Phys.

Rev. Lett.45(Aug, 1980) 494–497.


[6] D. C. Tsui, H. L. Stormer, and A. C. Gossard,Two-dimensional magnetotransport in the extreme quantum limit,Phys. Rev. Lett.48(May, 1982) 1559–1562.


[7] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs,Quantized hall conductance in a two-dimensional periodic potential,Phys. Rev. Lett.49(Aug, 1982) 405–408.https://link.aps.org/doi/10.1103/PhysRevLett.49.405.

[8] M. I. Dyakonov and V. Perel,Current-induced spin orientation of electrons in semiconductors,Physics Letters A35no. 6, (1971) 459–460.


[9] M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas,Giant magnetoresistance of (001)fe/(001)cr magnetic superlattices,Phys. Rev. Lett.61(Nov, 1988) 2472–2475.


[10] V. Cros, A. Fert, P. Sénéor, and F. Petroff,The 2007 nobel prize in physics: Albert fert and peter grünberg, inThe spin, pp. 147–157, Springer. 2009.

[11] S. Datta and B. Das,Electronic analog of the electro-optic modulator,Applied Physics Letters56no. 7, (1990) 665–667.

[12] M. Wu, J. Jiang, and M. Weng,Spin dynamics in semiconductors,Physics Reports 493no. 2-4, (Aug, 2010) 61–236.


[13] Y. A. Bychkov and E. I. Rashba,Oscillatory effects and the magnetic susceptibility of carriers in inversion layers,Journal of Physics C: Solid State Physics17no. 33, (Nov, 1984) 6039–6045.https://doi.org/10.1088/0022-3719/17/33/015. [14] J. B. Miller, D. M. Zumbühl, C. M. Marcus, Y. B. Lyanda-Geller,

D. Goldhaber-Gordon, K. Campman, and A. C. Gossard,Gate-controlled spin-orbit quantum interference effects in lateral transport,Phys. Rev. Lett.90(Feb, 2003) 076807.https://link.aps.org/doi/10.1103/PhysRevLett.90.076807. [15] C. L. Kane and E. J. Mele,Quantum spin hall effect in graphene,Phys. Rev. Lett.95

(Nov, 2005) 226801.


[16] J. Klinovaja and D. Loss,Giant spin-orbit interaction due to rotating magnetic fields in graphene nanoribbons,Phys. Rev. X3(Jan, 2013) 011008.


[17] A. Stern and N. H. Lindner,Topological quantum computation—from basic concepts to first experiments,Science339no. 6124, (2013) 1179–1184.

[18] B. A. Bernevig and S.-C. Zhang,Quantum spin hall effect,Phys. Rev. Lett.96(Mar, 2006) 106802.


[19] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang,Quantum spin hall effect and topological phase transition in hgte quantum wells,Science314no. 5806, (Dec, 2006) 1757–1761.http://dx.doi.org/10.1126/science.1133734.

[20] M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L.

Qi, and S.-C. Zhang,Quantum spin hall insulator state in hgte quantum wells, Science318no. 5851, (Nov, 2007) 766–770.


[21] L. Fu and C. L. Kane,Topological insulators with inversion symmetry,Phys. Rev. B 76(Jul, 2007) 045302.


[22] M. Z. Hasan and C. L. Kane,Colloquium: Topological insulators,Rev. Mod. Phys.

82(Nov, 2010) 3045–3067.


[23] X.-L. Qi and S.-C. Zhang,Topological insulators and superconductors,Rev. Mod.

Phys.83(Oct, 2011) 1057–1110.


[24] D. R. Cooper, B. D’Anjou, N. Ghattamaneni, B. Harack, M. Hilke, A. Horth, N. Majlis, M. Massicotte, L. Vandsburger, E. Whiteway, and V. Yu,Experimental review of graphene,ISRN Condensed Matter Physics2012(Apr, 2012) 501686.


[25] M. O. Goerbig,Electronic properties of graphene in a strong magnetic field,Rev.

Mod. Phys.83(Nov, 2011) 1193–1243.


[26] Y. Barlas, K. Yang, and A. H. MacDonald,Quantum hall effects in graphene-based two-dimensional electron systems,Nanotechnology23no. 5, (Jan, 2012) 052001.


[27] D. Miller, K. D. Kubista, G. M. Rutter, M. Ruan, W. D. de Heer, P. First, and J. Stroscio,Observing the quantization of zero mass carriers in graphene,Science 324(2009) 924 – 927.

[28] X.-L. Qi and S.-C. Zhang,The quantum spin hall effect and topological insulators, Physics Today63(2010) 1–33.

[29] C. L. Kane and E. J. Mele,Z2topological order and the quantum spin hall effect, Phys. Rev. Lett.95(Sep, 2005) 146802.


[30] J. Balakrishnan, G. K. W. Koon, M. Jaiswal, A. C. Neto, and B. Özyilmaz,Colossal enhancement of spin–orbit coupling in weakly hydrogenated graphene,Nature Physics9no. 5, (2013) 284–287.

[31] X. Hong, S.-H. Cheng, C. Herding, and J. Zhu,Colossal negative magnetoresistance in dilute fluorinated graphene,Physical Review B83no. 8, (2011) 085410.

[32] B. Yang, M. Lohmann, D. Barroso, I. Liao, Z. Lin, Y. Liu, L. Bartels, K. Watanabe, T. Taniguchi, and J. Shi,Strong electron-hole symmetric rashba spin-orbit coupling in graphene/monolayer transition metal dichalcogenide heterostructures,Physical Review B96no. 4, (2017) 041409.

[33] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim,The electronic properties of graphene,Rev. Mod. Phys.81(Jan, 2009) 109–162.


[34] P. R. Wallace,The band theory of graphite,Phys. Rev.71(May, 1947) 622–634.


[35] T. Zhang, S. Wu, R. Yang, and G. Zhang,Graphene: Nanostructure engineering and applications,Frontiers of Physics12no. 1, (Jan, 2017) 127206.


[36] B. Song, G. F. Schneider, Q. Xu, G. Pandraud, C. Dekker, and H. Zandbergen, Atomic-scale electron-beam sculpting of near-defect-free graphene nanostructures, Nano Letters11no. 6, (Jun, 2011) 2247–2250.


[37] T. Kato and R. Hatakeyama,Site- and alignment-controlled growth of graphene nanoribbons from nickel nanobars,Nature Nanotechnology7no. 10, (Oct, 2012) 651–656.https://doi.org/10.1038/nnano.2012.145.

[38] K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus,Edge state in

graphene ribbons: Nanometer size effect and edge shape dependence,Phys. Rev. B 54(Dec, 1996) 17954–17961.


[39] K. Wakabayashi, M. Fujita, H. Ajiki, and M. Sigrist,Electronic and magnetic properties of nanographite ribbons,Phys. Rev. B59(Mar, 1999) 8271–8282.


[40] V. Pardo and W. E. Pickett,Half-metallic semi-dirac-point generated by quantum confinement intio2/vo2nanostructures,Phys. Rev. Lett.102(Apr, 2009) 166803.


[41] V. Pardo and W. E. Pickett,Metal-insulator transition through a semi-dirac point in oxide nanostructures: vo2(001) layers confined within tio2,Phys. Rev. B81(Jan, 2010) 035111.https://link.aps.org/doi/10.1103/PhysRevB.81.035111. [42] P. Dietl, F. Piéchon, and G. Montambaux,New magnetic field dependence of landau

levels in a graphenelike structure,Phys. Rev. Lett.100(Jun, 2008) 236405.


[43] S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett,Tight-binding modeling and low-energy behavior of the semi-dirac point,Phys. Rev. Lett.103(Jul, 2009) 016402.


[44] G. E. Volovik,Reentrant violation of special relativity in the low-energy corner, Journal of Experimental and Theoretical Physics Letters73no. 4, (Feb, 2001) 162–165.http://dx.doi.org/10.1134/1.1368706.

[45] G. Montambaux, F. Piéchon, J.-N. Fuchs, and M. O. Goerbig,Merging of dirac points in a two-dimensional crystal,Phys. Rev. B80(Oct, 2009) 153412.


[46] K. Esaki, M. Sato, M. Kohmoto, and B. I. Halperin,Zero modes, energy gap, and edge states of anisotropic honeycomb lattice in a magnetic field,Physical Review B 80no. 12, (2009) 125405.

[47] V. M. Pereira, A. H. Castro Neto, and N. M. R. Peres,Tight-binding approach to uniaxial strain in graphene,Phys. Rev. B80(Jul, 2009) 045401.


[48] E. Zhao and A. Paramekanti,Bcs-bec crossover on the two-dimensional honeycomb lattice,Phys. Rev. Lett.97(Dec, 2006) 230404.


[49] S.-L. Zhu, B. Wang, and L.-M. Duan,Simulation and detection of dirac fermions with cold atoms in an optical lattice,Phys. Rev. Lett.98(Jun, 2007) 260402.


[50] J.-M. Hou, W.-X. Yang, and X.-J. Liu,Massless dirac fermions in a square optical lattice,Physical Review A79no. 4, (2009) 043621.

[51] L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T. Esslinger,Creating, moving and merging dirac points with a fermi gas in a tunable honeycomb lattice,Nature483 no. 7389, (Mar, 2012) 302–305.http://dx.doi.org/10.1038/nature10871. [52] S. Katayama, A. Kobayashi, and Y. Suzumura,Pressure-induced zero-gap

semiconducting state in organic conductorα-(bedt-ttf) 2i3 salt,Journal of the Physical Society of Japan75no. 5, (2006) 054705.

[53] A. Kobayashi, S. Katayama, Y. Suzumura, and H. Fukuyama,Massless fermions in organic conductor,Journal of the Physical Society of Japan76no. 3, (2007) 034711.

[54] K. L. Lee, B. Gremaud, R. Han, B.-G. Englert, C. Miniatura,et al.,Ultracold fermions in a graphene-type optical lattice,Physical Review A80no. 4, (2009) 043411.

[55] N. W. Ashcroft and N. D. Mermin,Solid state physics, cornell university, 1976.

[56] J. M. Ziman,Principles of the Theory of Solids. Cambridge university press, 1972.

[57] S. Ganguly,Quantum conductance in spin-orbit coupled devices: A focus on transport in Graphene. PhD thesis, 2017.

[58] C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James,Direct calculation of the tunneling current,Journal of Physics C: Solid State Physics4no. 8, (Jun, 1971) 916–929.https://doi.org/10.1088/0022-3719/4/8/018.

[59] D. S. Fisher and P. A. Lee,Relation between conductivity and transmission matrix, Phys. Rev. B23(Jun, 1981) 6851–6854.


[60] R. Landauer,Conductance determined by transmission: probes and quantised constriction resistance, .https://doi.org/10.1088/0953-8984/1/43/011. [61] G. Thorgilsson, G. Viktorsson, and S. Erlingsson,Recursive green’s function method

for multi-terminal nanostructures,Journal of Computational Physics261(Mar, 2014) 256–266.http://dx.doi.org/10.1016/j.jcp.2013.12.054.

[62] M. L. Sancho, J. L. Sancho, J. L. Sancho, and J. Rubio,Highly convergent schemes for the calculation of bulk and surface green functions,Journal of Physics F: Metal Physics15no. 4, (1985) 851.

[63] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal,Kwant: a software package for quantum transport,New Journal of Physics16no. 6, (Jun, 2014) 063065.http://dx.doi.org/10.1088/1367-2630/16/6/063065.

[64] A. Weiße, G. Wellein, A. Alvermann, and H. Fehske,The kernel polynomial method, Rev. Mod. Phys.78(Mar, 2006) 275–306.


[65] J. H. García, B. Uchoa, L. Covaci, and T. G. Rappoport,Adatoms and anderson localization in graphene,Physical Review B90no. 8, (Aug, 2014).


[66] J. H. García, L. Covaci, and T. G. Rappoport,Real-space calculation of the conductivity tensor for disordered topological matter,Phys. Rev. Lett.114(Mar, 2015) 116602.


[67] P. L. Chebyshev,Théorie des mécanismes connus sous le nom de parallélogrammes.

Imprimerie de l’Académie impériale des sciences, 1853.

[68] S. Butenko and P. M. Pardalos,Numerical methods and optimization: An introduction. CRC Press, 2014.

[69] J. Plemelj,Problems in the Sense of Riemann and Klein. 1964.

[70] R. Kubo,Statistical-mechanical theory of irreversible processes. i. general theory and simple applications to magnetic and conduction problems,Journal of the Physical Society of Japan12no. 6, (1957) 570–586,

https://doi.org/10.1143/JPSJ.12.570. https://doi.org/10.1143/JPSJ.12.570.

[71] H. Aoki and T. Ando,Effect of localization on the hall conductivity in the

two-dimensional system in strong magnetic fields,Solid State Communications38 no. 11, (1981) 1079–1082.https:

//www.sciencedirect.com/science/article/pii/0038109881900211. [72] G. D. Mahan,Many Particle Physics, Third Edition. Plenum, New York, 2000.

[73] R. Landauer,Electrical resistance of disordered one-dimensional lattices,The

Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics 21no. 172, (1970) 863–867,https://doi.org/10.1080/14786437008238472. https://doi.org/10.1080/14786437008238472.

[74] S. M. João, M. An ¯delkovi´c, L. Covaci, T. G. Rappoport, J. M. V. P. Lopes, and A. Ferreira,Kite: high-performance accurate modelling of electronic structure and response functions of large molecules, disordered crystals and heterostructures, Royal Society Open Science7no. 2, (Feb, 2020) 191809.


[75] K. S. Novoselov,Electric field effect in atomically thin carbon films,Science306 no. 5696, (Oct, 2004) 666–669.


[76] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim,Experimental observation of the quantum hall effect and berry’s phase in graphene,Nature438no. 7065, (Nov, 2005) 201–204.http://dx.doi.org/10.1038/nature04235.

[77] V. P. Gusynin and S. G. Sharapov,Unconventional integer quantum hall effect in graphene,Phys. Rev. Lett.95(Sep, 2005) 146801.


[78] E.-J. Kan, Z. Li, J. Yang, and J. G. Hou,Will zigzag graphene nanoribbon turn to half metal under electric field?,Applied Physics Letters91no. 24, (Dec, 2007) 243116.http://dx.doi.org/10.1063/1.2821112.

[79] X. Lin and J. Ni,Half-metallicity in graphene nanoribbons with topological line defects,Phys. Rev. B84(Aug, 2011) 075461.


[80] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim,Chiral tunnelling and the klein paradox in graphene,Nature Physics2no. 9, (Aug, 2006) 620–625.


[81] X. Du, I. Skachko, A. Barker, and E. Y. Andrei,Approaching ballistic transport in suspended graphene,Nature Nanotechnology3no. 8, (Jul, 2008) 491–495.


[82] K. I. Bolotin, K. J. Sikes, J. Hone, H. L. Stormer, and P. Kim,Temperature-dependent transport in suspended graphene,Phys. Rev. Lett.101(Aug, 2008) 096802.

https://link.aps.org/doi/10.1103/PhysRevLett.101.096802. [83] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi,Electronic transport in

two-dimensional graphene,Rev. Mod. Phys.83(May, 2011) 407–470.


[84] K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko, M. I. Katsnelson,

U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim,Unconventional quantum hall effect and berry’s phase of2πin bilayer graphene,Nature Physics2no. 3, (Feb, 2006) 177–180.http://dx.doi.org/10.1038/nphys245.

[85] S. Ghosh, W. Bao, D. L. Nika, S. Subrina, E. P. Pokatilov, C. N. Lau, and A. A.

Balandin,Dimensional crossover of thermal transport in few-layer graphene,Nature Materials9no. 7, (May, 2010) 555–558.


[86] A. A. Balandin,Thermal properties of graphene and nanostructured carbon materials,Nature Materials10no. 8, (Aug, 2011) 569–581.


[87] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and et al.,Boron nitride substrates for high-quality graphene electronics,Nature Nanotechnology5no. 10, (Aug, 2010) 722–726.http://dx.doi.org/10.1038/nnano.2010.172.

[88] M. Neek-Amal and F. M. Peeters,Nanoindentation of a circular sheet of bilayer graphene,Phys. Rev. B81(Jun, 2010) 235421.


[89] Y. Zhang, C. Wang, Y. Cheng, and Y. Xiang,Mechanical properties of bilayer graphene sheets coupled by sp3 bonding,Carbon49no. 13, (2011) 4511–4517.


//www.sciencedirect.com/science/article/pii/S0008622311004945. [90] E. McCann and V. I. Fal’ko,Landau-level degeneracy and quantum hall effect in a

graphite bilayer,Phys. Rev. Lett.96(Mar, 2006) 086805.


[91] E. V. Castro, K. S. Novoselov, S. V. Morozov, N. M. R. Peres, J. M. B. L. dos Santos, J. Nilsson, F. Guinea, A. K. Geim, and A. H. C. Neto,Biased bilayer graphene:

Semiconductor with a gap tunable by the electric field effect,Phys. Rev. Lett.99 (Nov, 2007) 216802.


[92] A. S. Rodin, A. Carvalho, and A. H. Castro Neto,Strain-induced gap modification in black phosphorus,Phys. Rev. Lett.112(May, 2014) 176801.


[93] J. Guan, Z. Zhu, and D. Tománek,Phase coexistence and metal-insulator transition in few-layer phosphorene: A computational study,Phys. Rev. Lett.113(Jul, 2014) 046804.https://link.aps.org/doi/10.1103/PhysRevLett.113.046804. [94] A. N. Rudenko, S. Yuan, and M. I. Katsnelson,Toward a realistic description of

multilayer black phosphorus: From gw approximation to large-scale tight-binding simulations,Phys. Rev. B92(Aug, 2015) 085419.


[95] C. Dutreix, E. A. Stepanov, and M. I. Katsnelson,Laser-induced topological transitions in phosphorene with inversion symmetry,Phys. Rev. B93(Jun, 2016) 241404.https://link.aps.org/doi/10.1103/PhysRevB.93.241404. [96] Y. Hasegawa, R. Konno, H. Nakano, and M. Kohmoto,Zero modes of tight-binding

electrons on the honeycomb lattice,Phys. Rev. B74(Jul, 2006) 033413.


[97] Y. Suzumura, T. Morinari, and F. Piéchon,Mechanism of dirac point inαtype organic conductor under pressure,Journal of the Physical Society of Japan82no. 2, (2013) 023708,https://doi.org/10.7566/JPSJ.82.023708.


[98] B. I. Halperin,Quantized hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,Phys. Rev. B 25(Feb, 1982) 2185–2190.


[99] R. B. Laughlin,Quantized hall conductivity in two dimensions,Phys. Rev. B23(May, 1981) 5632–5633.https://link.aps.org/doi/10.1103/PhysRevB.23.5632. [100] C. Wu, B. A. Bernevig, and S.-C. Zhang,Helical liquid and the edge of quantum spin

hall systems,Phys. Rev. Lett.96(Mar, 2006) 106401.


[101] M. Onoda and N. Nagaosa,Spin current and accumulation generated by the spin hall insulator,Phys. Rev. Lett.95(Sep, 2005) 106601.


[102] O. V. Yazyev and M. I. Katsnelson,Magnetic correlations at graphene edges: Basis for novel spintronics devices,Phys. Rev. Lett.100(Jan, 2008) 047209.


[103] K.-i. Sasaki, J. Jiang, R. Saito, S. Onari, and Y. Tanaka,Theory of superconductivity of carbon nanotubes and graphene,Journal of the Physical Society of Japan76 no. 3, (Mar, 2007) 033702.http://dx.doi.org/10.1143/JPSJ.76.033702. [104] D. Kang, B. Wang, C. Xia, and H. Li,Perfect spin filter in a tailored zigzag graphene

nanoribbon,Nanoscale Research Letters12no. 1, (May, 2017) 357.


[105] G. Zhang, X. Li, G. Wu, J. Wang, D. Culcer, E. Kaxiras, and Z. Zhang,Quantum phase transitions and topological proximity effects in graphene nanoribbon heterostructures,Nanoscale6(2014) 3259–3267.


[106] Y.-W. Son, M. L. Cohen, and S. G. Louie,Erratum: Half-metallic graphene nanoribbons,Nature446no. 7133, (Mar, 2007) 342–342.


[107] Y.-W. Son, M. L. Cohen, and S. G. Louie,Energy gaps in graphene nanoribbons, Phys. Rev. Lett.97(Nov, 2006) 216803.

https://link.aps.org/doi/10.1103/PhysRevLett.97.216803. [108] K. Wakabayashi, M. Sigrist, and M. Fujita,Spin wave mode of edge-localized

magnetic states in nanographite zigzag ribbons,Journal of the Physical Society of Japan67no. 6, (1998) 2089–2093,https://doi.org/10.1143/JPSJ.67.2089. https://doi.org/10.1143/JPSJ.67.2089.

[109] E. V. Castro, N. M. R. Peres, J. M. B. Lopes dos Santos, A. H. C. Neto, and

F. Guinea,Localized states at zigzag edges of bilayer graphene,Phys. Rev. Lett.100 (Jan, 2008) 026802.


[110] B. K. Nikoli´c,Statistical properties of eigenstates in three-dimensional mesoscopic systems with off-diagonal or diagonal disorder,Phys. Rev. B64(Jun, 2001) 014203.


[111] K. Saha,Photoinduced chern insulating states in semi-dirac materials,Phys. Rev. B 94(Aug, 2016) 081103.


[112] Q. Chen, L. Du, and G. A. Fiete,Floquet band structure of a semi-dirac system, Phys. Rev. B97(Jan, 2018) 035422.


[113] S. F. Islam and A. Saha,Driven conductance of an irradiated semi-dirac material, Phys. Rev. B98(Dec, 2018) 235424.


[114] J. Lado, N. García-Martínez, and J. Fernández-Rossier,Edge states in graphene-like systems,Synthetic Metals210(2015) 56–67.https:

//www.sciencedirect.com/science/article/pii/S0379677915300072. Reviews of Current Advances in Graphene Science and Technology.

[115] R. Seshadri, K. Sengupta, and D. Sen,Edge states, spin transport, and

impurity-induced local density of states in spin-orbit coupled graphene,Phys. Rev. B 93(Jan, 2016) 035431.


[116] S. Ganguly and S. Basu,Adatoms in graphene nanoribbons: spintronic properties and the quantum spin hall phase,Materials Research Express4no. 11, (Nov, 2017) 115004.https://doi.org/10.1088/2053-1591/aa9508.

[117] H. Xu, T. Heinzel, and I. V. Zozoulenko,Edge disorder and localization regimes in bilayer graphene nanoribbons,Phys. Rev. B80(Jul, 2009) 045308.


[118] E. Hill, A. Geim, K. Novoselov, F. Schedin, and P. Blake,Graphene spin valve devices,IEEE Transactions on Magnetics42no. 10, (Oct, 2006) 2694–2696.


[119] M. Nishioka and A. Goldman,Spin transport through multilayer graphene,Applied Physics Letters90no. 25, (2007). Funding Information: The authors were grateful to Daniel C. Frisbie for useful advice. This work was supported by the National Science Foundation through the University of Minnesota Materials Research Science and Engineering Center under Grant No. NSF/DMR-0212032.

[120] N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees,Electronic spin transport and spin precession in single graphene layers at room temperature, Nature448no. 7153, (Jul, 2007) 571–574.


[121] J. E. Moore,The birth of topological insulators,Nature464no. 7286, (Mar, 2010) 194–198.https://doi.org/10.1038/nature08916.

[122] H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, and A. H. MacDonald, Intrinsic and rashba spin-orbit interactions in graphene sheets,Phys. Rev. B74(Oct, 2006) 165310.https://link.aps.org/doi/10.1103/PhysRevB.74.165310. [123] Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, and Z. Fang,Spin-orbit gap of graphene:

First-principles calculations,Phys. Rev. B75(Jan, 2007) 041401.


[124] C. Weeks, J. Hu, J. Alicea, M. Franz, and R. Wu,Engineering a robust quantum spin hall state in graphene via adatom deposition,Phys. Rev. X1(Oct, 2011) 021001.

https://link.aps.org/doi/10.1103/PhysRevX.1.021001. [125] L. Kou, B. Yan, F. Hu, S.-C. Wu, T. O. Wehling, C. Felser, C. Chen, and

T. Frauenheim,Graphene-based topological insulator with an intrinsic bulk band gap above room temperature,Nano Letters13no. 12, (Dec, 2013) 6251–6255.


[126] J. Zhang, C. Triola, and E. Rossi,Proximity effect in graphene–topological-insulator heterostructures,Phys. Rev. Lett.112(Mar, 2014) 096802.

https://link.aps.org/doi/10.1103/PhysRevLett.112.096802. [127] K. Zollner, T. Frank, S. Irmer, M. Gmitra, D. Kochan, and J. Fabian,Spin-orbit

coupling in methyl functionalized graphene,Phys. Rev. B93(Jan, 2016) 045423.


[128] B. Lalmi, H. Oughaddou, H. Enriquez, A. Kara, S. Vizzini, B. Ealet, and B. Aufray, Epitaxial growth of a silicene sheet,Applied Physics Letters97no. 22, (Nov, 2010) 223109.http://dx.doi.org/10.1063/1.3524215.

[129] P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. Le Lay,Silicene: Compelling experimental evidence for graphenelike two-dimensional silicon,Phys. Rev. Lett.108(Apr, 2012) 155501.


[130] L. Chen, C.-C. Liu, B. Feng, X. He, P. Cheng, Z. Ding, S. Meng, Y. Yao, and K. Wu, Evidence for dirac fermions in a honeycomb lattice based on silicon,Phys. Rev. Lett.

109(Aug, 2012) 056804.

https://link.aps.org/doi/10.1103/PhysRevLett.109.056804. [131] W.-F. Tsai, C.-Y. Huang, T.-R. Chang, H. Lin, H.-T. Jeng, and A. Bansil,Gated

silicene as a tunable source of nearly 100Nature Communications4no. 1, (Feb, 2013).http://dx.doi.org/10.1038/ncomms2525.

[132] X. Wang, P. Wang, G. Bian, and T.-C. Chiang, Topological phase transitions in stanene and stanene-like systems by scaling the spin-orbit coupling,Europhysics Letters115no. 3, (Aug 2016) 5.