Nonlinear effects in topological materials

Abstract

Materials, where charge carriers have a linear energy dispersion, usually exhibit a strong nonlinear optical response in the absence of disorder scattering. This nonlinear response is particularly interesting in the terahertz frequency region. We present a theoretical and numerical investigation of charge transport and nonlinear effects, such as the high harmonic generation in topological materials including Weyl semimetals (WSMs) and a-T₃ systems. The nonlinear optical conductivity is calculated both semi-classically using the velocity operator and quantum mechanically using the density matrix. We show that the nonlinear response is strongly dependent on temperature and topological parameters, such as the Weyl point (WP) separation b and Berry phase {\phi }_{\mathrm{B}}. A finite spectral gap opening can further modify the nonlinear effects. Under certain parameters, universal behaviors of both the linear and nonlinear response can be observed. Coupled with experimentally accessible critical field values of 1{\mathrm{0}}^4-1{\mathrm{0}}^{\mathrm{5}}\mathrm{V/m}, our results provide useful information on developing nonlinear optoelectronic devices based on topological materials.

Graphical abstract

Keywords

terahertz / nonlinear effects / topological materials / Weyl semimetals (WSMs) / a-T₃ systems

References

[1] Ando Y. Topological insulator materials. Journal of the Physical Society of Japan, 2013, 82(10): 102001

[2] Otsuji T, Popov V, Ryzhii V. Active graphene plasmonics for terahertz device applications. Journal of Physics D, Applied Physics, 2014, 47(9): 094006

[3] Bansil A, Lin H, Das T. Topological band theory. Reviews of Modern Physics, 2016, 88(2): 021004

[4] Qi X L, Zhang S C. Topological insulators and superconductors. Reviews of Modern Physics, 2011, 83(4): 1057–1110

[5] Hasan M Z, Kane C L. Topological insulators. Reviews of Modern Physics, 2010, 82(4): 3045–3067

[6] Reddy D, Register L F, Carpenter G D, Banerjee S K. Graphene field-effect transistors. Journal of Physics D, Applied Physics, 2011, 44(31): 313001

[7] Sanderson M, Huang S, Bao Q, Zhang C. Optical conductivity of a commensurate graphene-topological insulator heterostructure. Journal of Physics D, Applied Physics, 2017, 50(38): 385301

[8] Wehling T, Black-Schaffer A M, Balatsky A V. Dirac materials. Advances in Physics, 2014, 63(1): 1–76

[9] Huang S C, Sanderson M, Zhang Y, Zhang C. High efficiency and non-Richardson thermionics in three dimensional Dirac materials. Applied Physics Letters, 2017, 111(18): 183902

[10] Lundgren R, Fiete G A. Electronic cooling in Weyl and Dirac semimetals. Physical Review B: Condensed Matter and Materials Physics, 2015, 92(12): 125139

[11] Burkov A A, Balents L. Weyl semimetal in a topological insulator multilayer. Physical Review Letters, 2011, 107(12): 127205

[12] Xu S Y, Belopolski I, Alidoust N, Neupane M, Bian G, Zhang C, Sankar R, Chang G, Yuan Z, Lee C C, Huang S M, Zheng H, Ma J, Sanchez D S, Wang B, Bansil A, Chou F, Shibayev P P, Lin H, Jia S, Hasan M Z. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science, 2015, 349(6248): 613–617

[13] Lv B Q, Weng H M, Fu B B, Wang X P, Miao H, Ma J, Richard P, Huang X C, Zhao L X, Chen G F, Fang Z, Dai X, Qian T, Ding H. Experimental discovery of Weyl semimetal TaAs. Physical Review X, 2015, 5(3): 031013

[14] Lu L, Wang Z, Ye D, Ran L, Fu L, Joannopoulos J D, Soljačić M. Experimental observation of Weyl points. Science, 2015, 349(6248): 622–624

[15] Xu S Y, Alidoust N, Belopolski I, Yuan Z, Bian G, Chang T R, Zheng H, Strocov V N, Sanchez D S, Chang G, Zhang C, Mou D, Wu Y, Huang L, Lee C C, Huang S M, Wang B K, Bansil A, Jeng H T, Neupert T, Kaminski A, Lin H, Jia S, Zahid Hasan M. Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide. Nature Physics, 2015, 11(9): 748–754

[16] Ouyang T, Xiao H, Tang C, Hu M, Zhong J. Anisotropic thermal transport in Weyl semimetal TaAs: a first principles calculation. Physical Chemistry Chemical Physics, 2016, 18(25): 16709

[17] Meng T, Balents L. Weyl superconductors. Physical Review B: Condensed Matter and Materials Physics, 2012, 86(5): 054504

[18] Hosur P, Parameswaran S A, Vishwanath A. Charge transport in Weyl semimetals. Physical Review Letters, 2012, 108(4): 046602

[19] Goswami P, Tewari S. Axionic field theory of (3+1)-dimensional Weyl semimetals. Physical Review B: Condensed Matter and Materials Physics, 2013, 88(24): 245107

[20] Vazifeh M M, Franz M. Electromagnetic response of Weyl semimetals. Physical Review Letters, 2013, 111(2): 027201

[21] Armitage N P, Mele E J, Vishwanath A. Weyl and Dirac semimetals in three-dimensional solids. Reviews of Modern Physics, 2018, 90(1): 015001

[22] Ashby P E, Carbotte J P. Magneto-optical conductivity of Weyl semimetals. Physical Review B: Condensed Matter and Materials Physics, 2013, 87(24): 245131

[23] Fuchs J N. Dirac fermions in graphene and analogues: magnetic field and topological properties. 2013, arXiv:1306.0380

[24] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K. The electronic properties of graphene. Reviews of Modern Physics, 2009, 81(1): 109–162

[25] Peres N M R. The transport properties of graphene: an introduction. Reviews of Modern Physics, 2010, 82(3): 2673–2700

[26] Sarma S D, Adam S, Hwang E H, Rossi E. Electronic transport in two-dimensional graphene. Reviews of Modern Physics, 2011, 83(2): 407–470

[27] Baireuther P S. Universiteit Leiden, (Doctoral dissertation). 2017

[28] Bell J S, Jackiw R. A PCAC puzzle: π⁰→gg in the s-model. Il Nuovo Cimento, 1969, 60(1): 47–61

[29] Adler S L. Axial-vector vertex in spinor electrodynamics. Physical Review, 1969, 177(5): 2426–2438

[30] Son D T, Spivak B Z. Chiral anomaly and classical negative magnetoresistance of Weyl metals. Physical Review B: Condensed Matter and Materials Physics, 2013, 88(10): 104412

[31] Landsteiner K. Anomalous transport of Weyl fermions in Weyl semimetals. Physical Review B: Condensed Matter and Materials Physics, 2014, 89(7): 075124

[32] Shekhar C, Nayak A K, Sun Y, Schmidt M, Nicklas M, Leermakers I, Zeitler U, Skourski Y, Wosnitza J, Liu Z, Chen Y, Schnelle W, Borrmann H, Grin Y, Felser C, Yan B. Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP. Nature Physics, 2015, 11(8): 645–649

[33] Wang F, Ran Y. Nearly flat band with Chern number C = 2 on the dice lattice. Physical Review B: Condensed Matter and Materials Physics, 2011, 84(24): 241103

[34] Bercioux D, Urban D F, Grabert H, Häusler H. Massless Dirac-Weyl fermions in a T₃ optical lattice. Physical Review A, 2009, 80(6): 063603

[35] Liu Z, Wang Z F, Mei J W, Wu Y S, Liu F. Flat Chern band in a two-dimensional organometallic framework. Physical Review Letters, 2013, 110(10): 106804

[36] Yamada M G, Soejima T, Tsuji N, Hirai D, Dincă M, Aoki H. First-principles design of a half-filled flat band of the kagome lattice in two-dimensional metal-organic frameworks. Physical Review B: Condensed Matter and Materials Physics, 2016, 94(8): 081102

[37] Su N, Jiang W, Wang Z, Liu F. Prediction of large gap flat Chern band in a two-dimensional metal-organic framework. Applied Physics Letters, 2018, 112(3): 033301

[38] Neupert T, Santos L, Chamon C, Mudry C. Fractional quantum Hall states at zero magnetic field. Physical Review Letters, 2011, 106(23): 236804

[39] Malcolm J D, Nicol E J. Analytic evaluation of Kane fermion magneto-optics in two and three dimensions. Physical Review B: Condensed Matter and Materials Physics, 2016, 94(22): 224305

[40] Häusler W. Flat-band conductivity properties at long-range Coulomb interactions. Physical Review B: Condensed Matter and Materials Physics, 2015, 91(4): 041102

[41] Du L, Zhou X, Fiete G A. Quadratic band touching points and flat bands in two-dimensional topological Floquet systems. Physical Review B: Condensed Matter and Materials Physics, 2017, 95(3): 035136

[42] Urban D F, Bercioux D, Wimmer W, Häusler W. Barrier transmission of Dirac-like pseudospin-one particles. Physical Review B: Condensed Matter and Materials Physics, 2011, 84(11): 115136

[43] Shen R, Shao L B, Wang B, Xing D Y. Single Dirac cone with a flat band touching on line-centered-square optical lattices. Physical Review B: Condensed Matter and Materials Physics, 2010, 81(4): 041410

[44] Illes E, Nicol E J. Klein tunneling in the α-T₃ model. Physical Review B: Condensed Matter and Materials Physics, 2017, 95(23): 235432

[45] Louvet T, Delplace P, Fedorenko A A, Carpentier D. On the origin of minimal conductivity at a band crossing. Physical Review B: Condensed Matter and Materials Physics, 2015, 92(15): 155116

[46] Illes E, Nicol E J. Magnetic properties of the α-T₃ model: magneto-optical conductivity and the Hofstadter butterfly. Physical Review B: Condensed Matter and Materials Physics, 2016, 94(12): 125435

[47] Islam S K F, Dutta P. Valley-polarized magnetoconductivity and particle-hole symmetry breaking in a periodically modulated α-T₃ lattice. Physical Review B: Condensed Matter and Materials Physics, 2017, 96(4): 045418

[48] Shareef S, Ang Y S, Zhang C. Room-temperature strong terahertz photon mixing in graphene. Journal of the Optical Society of America B, Optical Physics, 2012, 29(3): 274

[49] Wright A R, Xu X G, Cao J C, Zhang C. Strong nonlinear optical response of graphene in the terahertz regime. Applied Physics Letters, 2009, 95(7): 072101

[50] Ang Y S, Sultan S, Zhang C. Nonlinear optical spectrum of bilayer graphene in the terahertz regime. Applied Physics Letters, 2010, 97(24): 243110

[51] Ang Y S, Zhang C. Enhanced optical conductance in graphene superlattice due to anisotropic band dispersion. Journal of Physics D, Applied Physics, 2012, 45(39): 395303

[52] Mikhailov S A. Non-linear electromagnetic response of graphene. Europhysics Letters, 2007, 79(2): 27002

[53] Gong S, Zhao T, Sanderson M, Hu M, Zhong R, Chen X, Zhang P, Zhang C, Liu S. Transformation of surface plasmon polaritons to radiation in graphene in terahertz regime. Applied Physics Letters, 2015, 106(22): 223107

[54] Sanderson M, Ang Y S, Gong S, Zhao T, Hu M, Zhong R, Chen X, Zhang P, Zhang C, Liu S. Optical bistability induced by nonlinear surface plasmon polaritons in graphene in terahertz regime. Applied Physics Letters, 2015, 107(20): 203113

[55] Zuber J W, Zhao T, Gong S, Hu M, Zhong R B, Zhang C, Liu S G. Tunable strong photo-mixing in Weyl semimetals. Physical Review B: Condensed Matter and Materials Physics, 2020, 101(8): 085307

[56] Zhu C, Wang F, Meng Y, Yuan X, Xiu F, Luo H, Wang Y, Li J, Lv X, He L, Xu Y, Liu J, Zhang C, Shi Y, Zhang R, Zhu S. A robust and tuneable mid-infrared optical switch enabled by bulk Dirac fermions. Nature Communications, 2017, 8(1): 14111

[57] Huang S, Sanderson M, Tian J, Chen Q, Wang F, Zhang C. Hot carrier relaxation in three dimensional gapped Dirac semi-metals. Journal of Physics D, Applied Physics, 2018, 51(1): 015101

[58] Hwang C, Siegel D A, Mo S K, Regan W, Ismach A, Zhang Y, Zettl A, Lanzara A. Fermi velocity engineering in graphene by substrate modification. Scientific Reports, 2012, 2(1): 590

[59] Illes E, Carbotte J P, Nicol E J. Hall quantization and optical conductivity evolution with variable Berry phase in the α-T₃ model. Physical Review B: Condensed Matter and Materials Physics, 2015, 92(24): 245410

[60] Chen L, Zuber J W, Ma Z, Zhang C. Nonlinear optical response of the α-T₃ model due to the nontrivial topology of the band dispersion. Physical Review B: Condensed Matter and Materials Physics, 2019, 100(3): 035440

[61] Dóra B, Kailasvuori J, Moessner R. Lattice generalization of the Dirac equation to general spin and the role of the flat band. Physical Review B: Condensed Matter and Materials Physics, 2011, 84(19): 195422

[62] Tapasztó L, Dobrik G, Nemes-Incze P, Vertesy G, Lambin P, Biró L P. Tuning the electronic structure of graphene by ion irradiation. Physical Review B: Condensed Matter and Materials Physics, 2008, 78(23): 233407

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