Abstract

The intrinsic spin-orbit coupling in 2D staggered monolayer semiconductors is very large as compared to graphene. The large spin orbit interaction in these materials leads to the opening of a gap in the energy spectrum and spin-splitting of the bands in each valley. In this paper, we theoretically investigate the mechanical steering of beams from these spin-orbit rich, staggered 2D materials. Mechanical steering results in noticeable deviations of the reflected and transmitted ray profiles as predicted from classical laws of optics. These effects are generally called the Goos–Hänchen (GH) and Imbert-Fedorov shifts. We find that electric and magnetic field modulated giant spatial and angular GH shifts can be achieved in these materials for incident angles in the vicinity of the Brewster angle in the terahertz regime. We also determine the dependence of beam shifts on the chemical potential and find that the Brewster angle and the sign of GH shift can be controlled by varying the chemical potential. This allows the possibility of realizing spin and valley dependent optical effects that can be useful readout markers for experiments in quantum information processing, biosensing, and valleytronics, employed in the terahertz regime.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
  38. T. T. Tang, J. Li, M. Zhu, L. Luo, J. Yao, N. Li, and P. Zhang, “Realization of tunable Goos–Hänchen effect with magneto-optical effect in graphene,” Carbon 135, 29–34 (2018).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  44. Q. Li, B. Zhang, and J. Shen, “Goos–Hänchen shifts of reflected terahertz wave on a COC-air interface,” Opt. Express 21(5), 6480–6487 (2013).
    [Crossref]
  45. X. Li, P. Wang, F. Xing, X. D. Chen, Z. B. Liu, and J. G. Tian, “Experimental observation of a giant Goos–Hänchen shift in graphene using a beam splitter scanning method,” Opt. Lett. 39(19), 5574–5577 (2014).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  53. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
    [Crossref]
  54. V. P. Gusynin and S. G. Sharapov, “Unconventional Integer Quantum Hall Effect in Graphene,” Phys. Rev. Lett. 95(14), 146801 (2005).
    [Crossref]
  55. S. Grosche, M. Ornigotti, and A. Szameit, “Goos–Hänchen and Imbert-Fedorov shifts for Gaussian beams impinging on graphene-coated surfaces,” Opt. Express 23(23), 30195–30203 (2015).
    [Crossref]
  56. M. Shah and M. S. Anwar, “Magneto-optical effects in the Landau level manifold of 2D lattices with spin-orbit interaction,” Opt. Express 27(16), 23217–23233 (2019).
    [Crossref]
  57. X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise controlling of positive and negative Goos–Hänchen shifts in graphene,” Carbon 149, 604–608 (2019).
    [Crossref]

2020 (1)

S. Chen, X. Ling, W. Shu, H. Luo, and S. Wen, “Precision measurement of the optical conductivity of atomically thin crystals via photonic spin Hall effect,” Phys. Rev. Appl. 13(1), 014057 (2020).
[Crossref]

2019 (3)

G. Ye, W. Zhang, W. Wu, S. Chen, W. Shu, H. Luo, and S. Wen, “Goos–Hänchen and Imbert-Fedorov effects in Weyl semimetals,” Phys. Rev. A 99(2), 023807 (2019).
[Crossref]

M. Shah and M. S. Anwar, “Magneto-optical effects in the Landau level manifold of 2D lattices with spin-orbit interaction,” Opt. Express 27(16), 23217–23233 (2019).
[Crossref]

X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise controlling of positive and negative Goos–Hänchen shifts in graphene,” Carbon 149, 604–608 (2019).
[Crossref]

2018 (6)

A. Farmani, A. Mir, and Z. Sharifpour, “Broadly tunable and bidirectional terahertz graphene plasmonic switch based on enhanced Goos–Hänchen effect,” Appl. Surf. Sci. 453, 358–364 (2018).
[Crossref]

W. Wu, W. Zhang, S. Chen, X. Ling, W. Shu, H. Luo, S. Wen, and X. Yin, “Transitional Goos–Hänchen effect due to the topological phase transitions,” Opt. Express 26(18), 23705 (2018).
[Crossref]

A. Das and M. J. Pradhan, “Goos–Hänchen shift for Gaussian beams impinging on monolayer-MoS2-coated surfaces,” J. Opt. Soc. Am. B 35(8), 1956–1962 (2018).
[Crossref]

T. T. Tang, J. Li, M. Zhu, L. Luo, J. Yao, N. Li, and P. Zhang, “Realization of tunable Goos–Hänchen effect with magneto-optical effect in graphene,” Carbon 135, 29–34 (2018).
[Crossref]

Z. Chen, X. Chen, L. Tao, K. Chen, M. Long, X. Liu, K. Yan, R. I. Stantchev, E. Pickwell-MacPherson, and J.-B. Xu, “Graphene controlled Brewster angle device for ultra broadband terahertz modulation,” Nat. Commun. 9(1), 4909 (2018).
[Crossref]

X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
[Crossref]

2017 (5)

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos–Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Y. Huang, Z. Yu, C. Zhong, J. Fang, and Z. Dong, “Tunable lateral shifts of the reflected wave on the surface of an anisotropic chiral metamaterial,” Opt. Mater. Express 7(5), 1473–1485 (2017).
[Crossref]

A. Molle, J. Goldberger, M. Houssa, Y. Xu, S.-C. Zhang, and D. Akinwande, “Buckled two-dimensional Xene sheets,” Nat. Mater. 16(2), 163–169 (2017).
[Crossref]

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos–Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
[Crossref]

E. Azarova and G. Maksimova, “Spin- and valley-dependent Goos–Hänchen effect in silicene and gapped graphene structures,” J. Phys. Chem. Solids 100, 143–147 (2017).
[Crossref]

2016 (6)

Y. C. Fan, N. H. Shen, F. L. Zhang, Z. Y. Wei, H. Q. Li, Q. Zhao, Q. H. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically Tunable Goos–Hänchen Effect with Graphene in the Terahertz Regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[Crossref]

A. Castellanos-Gomez, “Why all the fuss about 2D semiconductors?” Nat. Photonics 10(4), 202–204 (2016).
[Crossref]

J. Zhao, H. Liu, Z. Yu, R. Quhe, S. Zhou, Y. Wang, C. C. Liu, H. Zhong, N. Han, J. Lu, Y. Yao, and K. Wu, “Rise of silicene: A competitive 2D material,” Prog. Mater. Sci. 83, 24–151 (2016).
[Crossref]

C. Grazianetti, E. Cinquanta, and A. Molle, “Two-dimensional silicon: the advent of silicene,” 2D Mater. 3(1), 012001 (2016).
[Crossref]

S. Saxena, R. P. Chaudhary, and S. Shukla, “Stanene: Atomically Thick Free-standing Layer of 2D Hexagonal Tin,” Sci. Rep. 6(1), 31073 (2016).
[Crossref]

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1(11), 16055 (2016).
[Crossref]

2015 (3)

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
[Crossref]

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pin- heiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. 92(20), 205415 (2015).
[Crossref]

S. Grosche, M. Ornigotti, and A. Szameit, “Goos–Hänchen and Imbert-Fedorov shifts for Gaussian beams impinging on graphene-coated surfaces,” Opt. Express 23(23), 30195–30203 (2015).
[Crossref]

2014 (4)

X. Li, P. Wang, F. Xing, X. D. Chen, Z. B. Liu, and J. G. Tian, “Experimental observation of a giant Goos–Hänchen shift in graphene using a beam splitter scanning method,” Opt. Lett. 39(19), 5574–5577 (2014).
[Crossref]

T. Tang, J. Qin, J. Xie, L. Deng, and L. Bi, “Magneto-optical Goos–Hänchen effect in a prism-waveguide coupling structure,” Opt. Express 22(22), 27042–27055 (2014).
[Crossref]

M. E. Davila, L. Xian, S. Cahangirov, A. Rubio, and G. L. Lay, “Germanene: a novel two-dimensional germanium allotrope akin to graphene and silicene,” New J. Phys. 16(9), 095002 (2014).
[Crossref]

S.-Y. Lee, J. Le Deunff, M. Choi, and R. Ketzmerick, “Quantum Goos–Hänchen shift and tunneling transmission at a curved step potential,” Phys. Rev. A 89(2), 022120 (2014).
[Crossref]

2013 (4)

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-Gap Quantum Spin Hall Insulators in Tin Films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref]

M. Ezawa, “Photoinduced Topological Phase Transition and a Single Dirac-Cone State in Silicene,” Phys. Rev. Lett. 110(2), 026603 (2013).
[Crossref]

Q. Li, B. Zhang, and J. Shen, “Goos–Hänchen shifts of reflected terahertz wave on a COC-air interface,” Opt. Express 21(5), 6480–6487 (2013).
[Crossref]

C. J. Tabert and E. J. Nicol, “Magneto-optical conductivity of silicene and other buckled honeycomb lattices,” Phys. Rev. B 88(8), 085434 (2013).
[Crossref]

2012 (7)

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

M. Ezawa, “Spin-valley optical selection rule and strong circular dichroism in silicene,” Phys. Rev. B 86(16), 161407 (2012).
[Crossref]

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(15), 155501 (2012).
[Crossref]

X. Zhou, X. Li, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
[Crossref]

D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled Spin and Valley Physics in Monolayers of MoS2 and Other Group-VI Dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
[Crossref]

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7(8), 490–493 (2012).
[Crossref]

I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, “Giant Goos–Hänchen Effect and Fano Resonance at Photonic Crystal Surfaces,” Phys. Rev. Lett. 108(12), 123901 (2012).
[Crossref]

2011 (4)

H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin Hall effect of light near the Brewster angle on reflection,” Phys. Rev. A 84(4), 043806 (2011).
[Crossref]

S. Longhi, G. D. Valle, and K. Staliunas, “Goos–Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
[Crossref]

F. H. Koppens, D. E. Chang, and F. Javier, “Graphene Plasmonics: A Platform for Strong Light–Matter Interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref]

Z. H. Wu, F. Zhai, F. M. Peeters, H. Q. Xu, and K. Chang, “Valley-Dependent Brewster Angles and Goos–Hänchen Effect in Strained Graphene,” Phys. Rev. Lett. 106(17), 176802 (2011).
[Crossref]

2009 (1)

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two- and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
[Crossref]

2007 (2)

D. Xiao, W. Yao, and Q. Niu, “Valley-Contrasting Physics in Graphene: Magnetic Moment and Topological Transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref]

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,” Nature 448(7153), 571–574 (2007).
[Crossref]

2006 (2)

J. He, J. Yi, and S. He, “Giant negative Goos–Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14(7), 3024–3029 (2006).
[Crossref]

X. Yin and L. Hesselink, “Goos–Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89(26), 261108 (2006).
[Crossref]

2005 (1)

V. P. Gusynin and S. G. Sharapov, “Unconventional Integer Quantum Hall Effect in Graphene,” Phys. Rev. Lett. 95(14), 146801 (2005).
[Crossref]

2004 (2)

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85(3), 372–374 (2004).
[Crossref]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref]

1989 (2)

T. Hashimoto and T. Yoshino, “Optical heterodyne sensor using the Goos–Hänchen shift,” Opt. Lett. 14(17), 913–915 (1989).
[Crossref]

A. Lakhtakia, “Would Brewster recognize today’s Brewster angle?” Opt. News 15(6), 14–18 (1989).
[Crossref]

1972 (1)

C. Imbert, “Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam,” Phys. Rev. D 5(4), 787–796 (1972).
[Crossref]

1955 (1)

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Figures (6)

Fig. 1.
Fig. 1. Schematic representation of the beam reflection at a 2D staggered material-substrate interface in the presence of an external electric and magnetic fields are shown for partial reflection (PR) and total internal reflection (TIR) conditions. The spatial and angular GH shifts for (a) PR and for (b) TIR. The incident, classically pridected and reflected beams are denoted by a, b and c respectively.
Fig. 2.
Fig. 2. Schematic representation of the allowed transitions between LL’s for three different magnetic fields in the $K$ valley at (a) 1 T (b) 3 T (c) 5 T. Blue lines represent Landau levels for spin up ($\sigma =\uparrow$) and red lines represent Landau level for spin down ($\sigma =\downarrow$). The same color scheme applies for the Landau levels transitions. (d) Longitudinal conductivity as a function of photon frequency. The parameters used are $\Delta _{z}=0.5\Delta _{so}$ and chemical potential $\mu _{F}=0$.
Fig. 3.
Fig. 3. Modulus and phase of the $s$ and $p$ polarized reflection coefficients for 2D staggered graphene-substrate system as a function of incident angle for different magnetic fields in $K$ valley for PR: (a) $R_{ss}$, (b)$R_{pp}$, (c) $\phi _{pp}$ and (d) $\phi _{ss}$. The parameters used are $\Gamma =0.2\Delta _{so}$, refractive index $n_{2}=1.84$ and chemical potential $\mu _{F}=0$.
Fig. 4.
Fig. 4. The $p$ polarized spatial and angular GH shifts for charge neutral staggered graphene-substrate system as a function of incident angle for different magnetic fields in the $K$ valley in the TI regime for PR and TIR. (a) The $p$ polarized spatial GH shifts for PR, (b) the $p$ polarized angular GH shifts for PR, (c) the $p$ polarized spatial GH shifts for TIR and (d) the $p$ polarized angular GH shifts for TIR. The dashed lines represents the values of $\theta _{B}$ and $\theta _{C}$ for the native dielectric substrate. The parameters used are identical across all figures, unless stated otherwise.
Fig. 5.
Fig. 5. (a) Longitudinal conductivity as a function of incident photon frequency and the $p$ polarized spatial and angular GH shifts for the staggered graphene-substrate system as a function of incident angle for magnetic field $B=1$ T in the $K$ valley in three distinct topological regimes and for four different chemical potentials. The incidence is external PR, while (b) and (c) show the $p$ polarized spatial and angular GH shifts with modulation of the external electric field, for the TI, VSPM and BI at a magnetic field of 1 T. (d) Schematic representation of the allowed transitions between LL’s for three different values of chemical potential $\mu _{F}$=0, 10 and 22 meV, and (e) and (f) are the $p$ polarized spatial and angular GH shifts with modulation of the chemical potential in the TI and classical regimes for a magnetic field of 1 T.
Fig. 6.
Fig. 6. The $p$ polarized spatial and angular GH shifts as a function of incident angle for both spins and valleys in the TI regime for $B=$1 T. (a) The spatial GH shifts for both spins and valleys, (b) the $p$ polarized angular GH shifts for both spins and valleys. The spatial GH shifts as a function of photon energy in the $K$ valley for different magnetic and electric fields. (c) The $p$ polarized spatial GH shifts for three different magnetic fields in the TI regime, (d) the spatial GH shifts for $B=1$ T in three distinct topological regimes. The $p$ polarized spatial and angular GH shifts as a function of incident angle in the $K$ valley for three different relative permittivities in the TI regime for $B=$1 T. (e) The $p$ polarized spatial GH shifts, (f) the $p$ polarized angular GH shifts for three different relative permittivities.

Tables (3)

Tables Icon

Table 1. Table of allowed transitions in K valley in the n = 1 , 0 , 1 subspace, for different magnetic fields in the TI regime with Δ s o = 0.5 Δ z .

Tables Icon

Table 2. Table of allowed transitions in K valley in the n = 1 , 0 , 1 subspace, for B = 1 T in three different topological regimes for Δ s o = 8 meV.

Tables Icon

Table 3. Table of allowed transitions in K valley in the n = 1 , 0 , 1 subspace, for different chemical potentials in the TI regime with Δ s o = 8 meV.

Equations (12)

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H ^ ξ σ = v F ( ξ k x τ ^ x + k y τ ^ y ) 1 2 ξ σ Δ s o τ ^ z + 1 2 Δ z τ ^ z
E ( ξ , σ , n , t ) = { t 2 v F 2 e B | n | + Δ ξ σ 2 , if n 0. ξ Δ ξ σ , if n = 0.
r p p = α + T α L + β α + T α + L + β ,
r s s = ( α T α + L + β α + T α + L + β ) ,
r s p / p s = 2 Z 0 2 μ 0 μ 1 μ 2 k 1 z k 2 z ( σ x y a n t i s y m + σ x y s y m ) Z 1 ( α + T α + L + β ) ,
α ± L = ( k 1 z ε 2 ± k 2 z ε 1 + k 1 z k 2 z σ L / ( ε 0 Ω ) ) ,
α ± T = ( k 2 z μ 1 ± k 1 z μ 2 + μ 0 μ 1 μ 2 σ T Ω ) ,
β = Z 0 2 μ 1 μ 2 k 1 z k 2 z [ ( σ x y a n t i s y m ) 2 ( σ x y s y m ) 2 ]
Re Im } ( σ x x ( Ω ) ) σ 0 = 2 v F 2 e B π × ξ , σ m , n Θ ( E n μ F ) Θ ( E m μ F ) E n E m × [ ( A m B n ) 2 δ | m | ξ , | n | + ( B m A n ) 2 δ | m | + ξ , | n | ] { F G ,
Re Im } ( σ x y ( Ω ) ) σ 0 = 2 v F 2 e B π × ξ , σ m , n ξ Θ ( E n μ F ) Θ ( E m μ F ) E n E m × [ ( A m B n ) 2 δ | m | ξ , | n | ( B m A n ) 2 δ | m | + ξ , | n | ] { -G F
Θ G H = 2 ( R p p 2 ρ p p + R p s 2 ρ p s ) 2 k 1 ( R p s 2 + R p p 2 ) Λ R + χ p p + χ p s ,
Δ G H = 2 ( R p p 2 φ p p + R p s 2 φ p s ) Λ R 2 k 1 ( R p s 2 + R p p 2 ) Λ R + χ p p + χ p s .

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