Abstract

We study theoretically optical properties of black phosphorus (BP) sheet arrays being embedded in the dielectric multilayer structure using transfer-matrix method. It is found that the dielectric multilayer structures containing BP sheet arrays can exhibit rich optical properties. In some frequency regions, strong anisotropy appears for the scattering and absorption of polarized waves. Thus, they can be used as the selective absorption materials for two kinds of polarized wave. In contrast, in some visible and near-infrared frequency regions, low absorption and strong magnetic response have been observed. This means that they can be also used as good magnetic response materials at visible and near-infrared frequencies. Our results could find applications in various control of polarized waves.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Anisotropic infrared plasmonic broadband absorber based on graphene-black phosphorus multilayers

Yijun Cai, Kai-Da Xu, Naixing Feng, Rongrong Guo, Haijun Lin, and Jinfeng Zhu
Opt. Express 27(3) 3101-3112 (2019)

Active tuning of the hybridization effects of mid-infrared surface plasmon resonance in a black phosphorus sheet array and a metal grating slit

Yan Huang, Yan Liu, Cizhe Fang, Yao Shao, Genquan Han, Jincheng Zhang, and Yue Hao
Opt. Mater. Express 10(1) 14-28 (2020)

Infrared absorber based on sandwiched two-dimensional black phosphorus metamaterials

Jiao Wang and Yannan Jiang
Opt. Express 25(5) 5206-5216 (2017)

References

  • View by:
  • |
  • |
  • |

  1. M. Mansuripur, The Principles of Magneto-Optical Recording (Cambridge University Press, 1995).
  2. H. Ebert, “Magneto-optical effects in transition metal systems,” Rep. Prog. Phys. 59(12), 1665–1735 (1996) (and references therein).
    [Crossref]
  3. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
    [Crossref] [PubMed]
  4. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
    [Crossref] [PubMed]
  5. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
    [Crossref] [PubMed]
  6. L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
    [Crossref] [PubMed]
  7. H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, “Phosphorene: an unexplored 2D semiconductor with a high hole mobility,” ACS Nano 8(4), 4033–4041 (2014).
    [Crossref] [PubMed]
  8. F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458 (2014).
    [Crossref] [PubMed]
  9. L. Kou, C. Chen, and S. C. Smith, “Phosphorene: fabrication, properties, and applications,” J. Phys. Chem. Lett. 6(14), 2794–2805 (2015).
    [Crossref] [PubMed]
  10. H. Liu, Y. Du, Y. Deng, and P. D. Ye, “Semiconducting black phosphorus: synthesis, transport properties and electronic applications,” Chem. Soc. Rev. 44(9), 2732–2743 (2015).
    [Crossref] [PubMed]
  11. S. B. Lu, L. L. Miao, Z. N. Guo, X. Qi, C. J. Zhao, H. Zhang, S. C. Wen, D. Y. Tang, and D. Y. Fan, “Broadband nonlinear optical response in multi-layer black phosphorus: an emerging infrared and mid-infrared optical material,” Opt. Express 23(9), 11183–11194 (2015).
    [Crossref] [PubMed]
  12. A. Morita, “Semiconducting black phosphorus,” Appl. Phys., A Mater. Sci. Process. 39(4), 227–242 (1986).
    [Crossref]
  13. L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
    [Crossref] [PubMed]
  14. L. Li, F. Yang, G. J. Ye, Z. Zhang, K. Watanabe, T. Taniguchi, Y. Wang, X. H. Chen, and Y. Zhang, “Quantum hall effect in black phosphorus two-dimensional electron gas” http://arxiv.org/abs/1504.07155 (2015).
  15. S. P. Koenig, R. A. Doganov, H. Schmidt, A. H. Castro Neto, and B. Ozyilmaz, “Electric field effect in ultrathin black phosphorus,” Appl. Phys. Lett. 104(10), 103106 (2014).
    [Crossref]
  16. A. S. Rodin, A. Carvalho, and A. H. Castro Neto, “Strain-induced gap modification in black phosphorus,” Phys. Rev. Lett. 112(17), 176801 (2014).
    [Crossref] [PubMed]
  17. T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
    [Crossref]
  18. A. N. Rudenko and M. I. Katsnelson, “Quasiparticle band structure and tight-binding model for single- and bilayer black phosphorus,” Phys. Rev. B 89(20), 201408 (2014).
    [Crossref]
  19. V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89(23), 235319 (2014).
    [Crossref]
  20. S. Yuan, A. N. Rudenko, and M. I. Katsnelson, “Transport and optical properties of single- and bilayer black phosphorus with defects,” Phys. Rev. B 91(11), 115436 (2015).
    [Crossref]
  21. Q. Liu, X. Zhang, L. B. Abdalla, A. Fazzio, and A. Zunger, “Switching a normal insulator into a topological insulator via electric field with application to phosphorene,” Nano Lett. 15(2), 1222–1228 (2015).
    [Crossref] [PubMed]
  22. M. Ezawa, “Topological origin of quasi-flat edge band in phosphorene,” New J. Phys. 16(11), 115004 (2014).
    [Crossref]
  23. R. Fei and L. Yang, “Strain-engineering the anisotropic electrical conductance of few-layer black phosphorus,” Nano Lett. 14(5), 2884–2889 (2014).
    [Crossref] [PubMed]
  24. J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, “High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus,” Nat. Commun. 5, 4475 (2014).
    [Crossref] [PubMed]
  25. R. Fei, A. Faghaninia, R. Soklaski, J.-A. Yan, C. Lo, and L. Yang, “Enhanced thermoelectric efficiency via orthogonal electrical and thermal conductances in phosphorene,” Nano Lett. 14(11), 6393–6399 (2014).
    [Crossref] [PubMed]
  26. P. Li and I. Appelbaum, “Electrons and holes in phosphorene,” Phys. Rev. B 90(11), 115439 (2014).
    [Crossref]
  27. A. Ziletti, A. Carvalho, D. K. Campbell, D. F. Coker, and A. H. Castro Neto, “Oxygen defects in phosphorene,” Phys. Rev. Lett. 114(4), 046801 (2015).
    [Crossref] [PubMed]
  28. X. Y. Zhou, R. Zhang, J. P. Sun, Y. L. Zou, D. Zhang, W. K. Lou, F. Cheng, G. H. Zhou, F. Zhai, and K. Chang, “Landau levels and magneto-transport property of monolayer phosphorene,” http://arxiv.org/abs/1411.4275 .
    [Crossref]
  29. Y. Jiang, R. Roldan, F. Guinea, and T. Low, “Magneto-electronic properties of multilayer black phosphorus,” http://arxiv.org/abs/1505.00175 .
  30. J. M. Pereira and M. I. Katsnelson, “Landau levels of single and bilayer phosphorene,” http://arxiv.org/abs/1504.02452 .
  31. M. Tahir, P. Vasilopoulos, and F. M. Peeters, “Magneto-optical transport properties of monolayer phosphorene,” http://arxiv.org/abs/1505.06780 .
  32. D. Cakir, C. Sevik, and F. M. Peeters, “Remarkable effect of stacking on the electronic and optical properties of few layer black phosphorus,” http://arxiv.org/abs/1506.04707 .
  33. X. Chen, Y. Wu, Z. Wu, Y. Han, S. Xu, L. Wang, W. Ye, T. Han, Y. He, Y. Cai, and N. Wang, “High quality sandwiched black phosphorus heterostructure and its quantum oscillations,” http://arxiv.org/abs/1412.1357 .
    [Crossref]
  34. V. Tayari, N. Hemsworth, I. Fakih, A. Favron, E. Gaufr’es, G. Gervais, R. Martel, and T. Szkopek, “Two-dimensional magnetotransport in a black phosphorus naked quantum well,” http://arxiv.org/abs/1412.1357 .
    [Crossref]
  35. R. Kubo, “Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems,” J. Phys. Soc. Jpn. 12(6), 570–586 (1957).
    [Crossref]
  36. A. Ishihara, Statistical Physics (Academic Press, 1971).
  37. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69(5), 742–756 (1979).
    [Crossref]
  38. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).
  39. J. Hao and L. Zhou, “Electromagnetic wave scatterings by anisotropic metamaterials: Generalized4×4, ” Phys. Rev. B 77(9), 094201 (2008).
    [Crossref]
  40. P. J. van Zwol and G. Palasantzas, “Repulsive Casimir forces between solid materials with highrefractive-index intervening liquids,” Phys. Rev. A 81(6), 062502 (2010).
    [Crossref]
  41. S. Singh, J. P. Remeika, and J. R. Potopowicz, “Nonlinear optical properties of ferroelectric lead titanate,” Appl. Phys. Lett. 20(3), 135–137 (1972).
    [Crossref]
  42. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55(10), 1205–1208 (1965).
    [Crossref]
  43. 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. B 92, 085419 (2015).

2015 (8)

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

S. Yuan, A. N. Rudenko, and M. I. Katsnelson, “Transport and optical properties of single- and bilayer black phosphorus with defects,” Phys. Rev. B 91(11), 115436 (2015).
[Crossref]

Q. Liu, X. Zhang, L. B. Abdalla, A. Fazzio, and A. Zunger, “Switching a normal insulator into a topological insulator via electric field with application to phosphorene,” Nano Lett. 15(2), 1222–1228 (2015).
[Crossref] [PubMed]

A. Ziletti, A. Carvalho, D. K. Campbell, D. F. Coker, and A. H. Castro Neto, “Oxygen defects in phosphorene,” Phys. Rev. Lett. 114(4), 046801 (2015).
[Crossref] [PubMed]

L. Kou, C. Chen, and S. C. Smith, “Phosphorene: fabrication, properties, and applications,” J. Phys. Chem. Lett. 6(14), 2794–2805 (2015).
[Crossref] [PubMed]

H. Liu, Y. Du, Y. Deng, and P. D. Ye, “Semiconducting black phosphorus: synthesis, transport properties and electronic applications,” Chem. Soc. Rev. 44(9), 2732–2743 (2015).
[Crossref] [PubMed]

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. B 92, 085419 (2015).

S. B. Lu, L. L. Miao, Z. N. Guo, X. Qi, C. J. Zhao, H. Zhang, S. C. Wen, D. Y. Tang, and D. Y. Fan, “Broadband nonlinear optical response in multi-layer black phosphorus: an emerging infrared and mid-infrared optical material,” Opt. Express 23(9), 11183–11194 (2015).
[Crossref] [PubMed]

2014 (13)

L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
[Crossref] [PubMed]

H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, “Phosphorene: an unexplored 2D semiconductor with a high hole mobility,” ACS Nano 8(4), 4033–4041 (2014).
[Crossref] [PubMed]

F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458 (2014).
[Crossref] [PubMed]

M. Ezawa, “Topological origin of quasi-flat edge band in phosphorene,” New J. Phys. 16(11), 115004 (2014).
[Crossref]

R. Fei and L. Yang, “Strain-engineering the anisotropic electrical conductance of few-layer black phosphorus,” Nano Lett. 14(5), 2884–2889 (2014).
[Crossref] [PubMed]

J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, “High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus,” Nat. Commun. 5, 4475 (2014).
[Crossref] [PubMed]

R. Fei, A. Faghaninia, R. Soklaski, J.-A. Yan, C. Lo, and L. Yang, “Enhanced thermoelectric efficiency via orthogonal electrical and thermal conductances in phosphorene,” Nano Lett. 14(11), 6393–6399 (2014).
[Crossref] [PubMed]

P. Li and I. Appelbaum, “Electrons and holes in phosphorene,” Phys. Rev. B 90(11), 115439 (2014).
[Crossref]

S. P. Koenig, R. A. Doganov, H. Schmidt, A. H. Castro Neto, and B. Ozyilmaz, “Electric field effect in ultrathin black phosphorus,” Appl. Phys. Lett. 104(10), 103106 (2014).
[Crossref]

A. S. Rodin, A. Carvalho, and A. H. Castro Neto, “Strain-induced gap modification in black phosphorus,” Phys. Rev. Lett. 112(17), 176801 (2014).
[Crossref] [PubMed]

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

A. N. Rudenko and M. I. Katsnelson, “Quasiparticle band structure and tight-binding model for single- and bilayer black phosphorus,” Phys. Rev. B 89(20), 201408 (2014).
[Crossref]

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89(23), 235319 (2014).
[Crossref]

2010 (1)

P. J. van Zwol and G. Palasantzas, “Repulsive Casimir forces between solid materials with highrefractive-index intervening liquids,” Phys. Rev. A 81(6), 062502 (2010).
[Crossref]

2008 (1)

J. Hao and L. Zhou, “Electromagnetic wave scatterings by anisotropic metamaterials: Generalized4×4, ” Phys. Rev. B 77(9), 094201 (2008).
[Crossref]

2005 (1)

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[Crossref] [PubMed]

2004 (2)

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[Crossref] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[Crossref] [PubMed]

1996 (1)

H. Ebert, “Magneto-optical effects in transition metal systems,” Rep. Prog. Phys. 59(12), 1665–1735 (1996) (and references therein).
[Crossref]

1986 (1)

A. Morita, “Semiconducting black phosphorus,” Appl. Phys., A Mater. Sci. Process. 39(4), 227–242 (1986).
[Crossref]

1979 (1)

1972 (1)

S. Singh, J. P. Remeika, and J. R. Potopowicz, “Nonlinear optical properties of ferroelectric lead titanate,” Appl. Phys. Lett. 20(3), 135–137 (1972).
[Crossref]

1965 (1)

1957 (1)

R. Kubo, “Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems,” J. Phys. Soc. Jpn. 12(6), 570–586 (1957).
[Crossref]

Abdalla, L. B.

Q. Liu, X. Zhang, L. B. Abdalla, A. Fazzio, and A. Zunger, “Switching a normal insulator into a topological insulator via electric field with application to phosphorene,” Nano Lett. 15(2), 1222–1228 (2015).
[Crossref] [PubMed]

Appelbaum, I.

P. Li and I. Appelbaum, “Electrons and holes in phosphorene,” Phys. Rev. B 90(11), 115439 (2014).
[Crossref]

Basov, D. N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[Crossref] [PubMed]

Campbell, D. K.

A. Ziletti, A. Carvalho, D. K. Campbell, D. F. Coker, and A. H. Castro Neto, “Oxygen defects in phosphorene,” Phys. Rev. Lett. 114(4), 046801 (2015).
[Crossref] [PubMed]

Carvalho, A.

A. Ziletti, A. Carvalho, D. K. Campbell, D. F. Coker, and A. H. Castro Neto, “Oxygen defects in phosphorene,” Phys. Rev. Lett. 114(4), 046801 (2015).
[Crossref] [PubMed]

A. S. Rodin, A. Carvalho, and A. H. Castro Neto, “Strain-induced gap modification in black phosphorus,” Phys. Rev. Lett. 112(17), 176801 (2014).
[Crossref] [PubMed]

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Castro Neto, A. H.

A. Ziletti, A. Carvalho, D. K. Campbell, D. F. Coker, and A. H. Castro Neto, “Oxygen defects in phosphorene,” Phys. Rev. Lett. 114(4), 046801 (2015).
[Crossref] [PubMed]

A. S. Rodin, A. Carvalho, and A. H. Castro Neto, “Strain-induced gap modification in black phosphorus,” Phys. Rev. Lett. 112(17), 176801 (2014).
[Crossref] [PubMed]

S. P. Koenig, R. A. Doganov, H. Schmidt, A. H. Castro Neto, and B. Ozyilmaz, “Electric field effect in ultrathin black phosphorus,” Appl. Phys. Lett. 104(10), 103106 (2014).
[Crossref]

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Chen, C.

L. Kou, C. Chen, and S. C. Smith, “Phosphorene: fabrication, properties, and applications,” J. Phys. Chem. Lett. 6(14), 2794–2805 (2015).
[Crossref] [PubMed]

Chen, G.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

Chen, X. H.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
[Crossref] [PubMed]

Coker, D. F.

A. Ziletti, A. Carvalho, D. K. Campbell, D. F. Coker, and A. H. Castro Neto, “Oxygen defects in phosphorene,” Phys. Rev. Lett. 114(4), 046801 (2015).
[Crossref] [PubMed]

Deng, Y.

H. Liu, Y. Du, Y. Deng, and P. D. Ye, “Semiconducting black phosphorus: synthesis, transport properties and electronic applications,” Chem. Soc. Rev. 44(9), 2732–2743 (2015).
[Crossref] [PubMed]

Doganov, R. A.

S. P. Koenig, R. A. Doganov, H. Schmidt, A. H. Castro Neto, and B. Ozyilmaz, “Electric field effect in ultrathin black phosphorus,” Appl. Phys. Lett. 104(10), 103106 (2014).
[Crossref]

Du, Y.

H. Liu, Y. Du, Y. Deng, and P. D. Ye, “Semiconducting black phosphorus: synthesis, transport properties and electronic applications,” Chem. Soc. Rev. 44(9), 2732–2743 (2015).
[Crossref] [PubMed]

Ebert, H.

H. Ebert, “Magneto-optical effects in transition metal systems,” Rep. Prog. Phys. 59(12), 1665–1735 (1996) (and references therein).
[Crossref]

Enkrich, C.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[Crossref] [PubMed]

Ezawa, M.

M. Ezawa, “Topological origin of quasi-flat edge band in phosphorene,” New J. Phys. 16(11), 115004 (2014).
[Crossref]

Faghaninia, A.

R. Fei, A. Faghaninia, R. Soklaski, J.-A. Yan, C. Lo, and L. Yang, “Enhanced thermoelectric efficiency via orthogonal electrical and thermal conductances in phosphorene,” Nano Lett. 14(11), 6393–6399 (2014).
[Crossref] [PubMed]

Fan, D. Y.

Fang, N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[Crossref] [PubMed]

Fazzio, A.

Q. Liu, X. Zhang, L. B. Abdalla, A. Fazzio, and A. Zunger, “Switching a normal insulator into a topological insulator via electric field with application to phosphorene,” Nano Lett. 15(2), 1222–1228 (2015).
[Crossref] [PubMed]

Fei, R.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

R. Fei and L. Yang, “Strain-engineering the anisotropic electrical conductance of few-layer black phosphorus,” Nano Lett. 14(5), 2884–2889 (2014).
[Crossref] [PubMed]

R. Fei, A. Faghaninia, R. Soklaski, J.-A. Yan, C. Lo, and L. Yang, “Enhanced thermoelectric efficiency via orthogonal electrical and thermal conductances in phosphorene,” Nano Lett. 14(11), 6393–6399 (2014).
[Crossref] [PubMed]

Feng, D.

L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
[Crossref] [PubMed]

Firsov, A. A.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[Crossref] [PubMed]

Ge, Q.

L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
[Crossref] [PubMed]

Geim, A. K.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[Crossref] [PubMed]

Gleeson, H. F.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[Crossref] [PubMed]

Grigorenko, A. N.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[Crossref] [PubMed]

Guo, Z. N.

Hao, J.

J. Hao and L. Zhou, “Electromagnetic wave scatterings by anisotropic metamaterials: Generalized4×4, ” Phys. Rev. B 77(9), 094201 (2008).
[Crossref]

Hu, Z.-X.

J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, “High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus,” Nat. Commun. 5, 4475 (2014).
[Crossref] [PubMed]

Ji, W.

J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, “High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus,” Nat. Commun. 5, 4475 (2014).
[Crossref] [PubMed]

Jia, Y.

F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458 (2014).
[Crossref] [PubMed]

Jiang, Y.

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Katsnelson, M. I.

S. Yuan, A. N. Rudenko, and M. I. Katsnelson, “Transport and optical properties of single- and bilayer black phosphorus with defects,” Phys. Rev. B 91(11), 115436 (2015).
[Crossref]

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. B 92, 085419 (2015).

A. N. Rudenko and M. I. Katsnelson, “Quasiparticle band structure and tight-binding model for single- and bilayer black phosphorus,” Phys. Rev. B 89(20), 201408 (2014).
[Crossref]

Khrushchev, I. Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[Crossref] [PubMed]

Koenig, S. P.

S. P. Koenig, R. A. Doganov, H. Schmidt, A. H. Castro Neto, and B. Ozyilmaz, “Electric field effect in ultrathin black phosphorus,” Appl. Phys. Lett. 104(10), 103106 (2014).
[Crossref]

Kong, X.

J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, “High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus,” Nat. Commun. 5, 4475 (2014).
[Crossref] [PubMed]

Koschny, T.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[Crossref] [PubMed]

Kou, L.

L. Kou, C. Chen, and S. C. Smith, “Phosphorene: fabrication, properties, and applications,” J. Phys. Chem. Lett. 6(14), 2794–2805 (2015).
[Crossref] [PubMed]

Kubo, R.

R. Kubo, “Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems,” J. Phys. Soc. Jpn. 12(6), 570–586 (1957).
[Crossref]

Li, L.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
[Crossref] [PubMed]

Li, P.

P. Li and I. Appelbaum, “Electrons and holes in phosphorene,” Phys. Rev. B 90(11), 115439 (2014).
[Crossref]

Liang, Y.

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89(23), 235319 (2014).
[Crossref]

Linden, S.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[Crossref] [PubMed]

Liu, H.

H. Liu, Y. Du, Y. Deng, and P. D. Ye, “Semiconducting black phosphorus: synthesis, transport properties and electronic applications,” Chem. Soc. Rev. 44(9), 2732–2743 (2015).
[Crossref] [PubMed]

H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, “Phosphorene: an unexplored 2D semiconductor with a high hole mobility,” ACS Nano 8(4), 4033–4041 (2014).
[Crossref] [PubMed]

Liu, Q.

Q. Liu, X. Zhang, L. B. Abdalla, A. Fazzio, and A. Zunger, “Switching a normal insulator into a topological insulator via electric field with application to phosphorene,” Nano Lett. 15(2), 1222–1228 (2015).
[Crossref] [PubMed]

Lo, C.

R. Fei, A. Faghaninia, R. Soklaski, J.-A. Yan, C. Lo, and L. Yang, “Enhanced thermoelectric efficiency via orthogonal electrical and thermal conductances in phosphorene,” Nano Lett. 14(11), 6393–6399 (2014).
[Crossref] [PubMed]

Low, T.

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Lu, S. B.

Luo, Z.

H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, “Phosphorene: an unexplored 2D semiconductor with a high hole mobility,” ACS Nano 8(4), 4033–4041 (2014).
[Crossref] [PubMed]

Malitson, I. H.

Miao, L. L.

Morita, A.

A. Morita, “Semiconducting black phosphorus,” Appl. Phys., A Mater. Sci. Process. 39(4), 227–242 (1986).
[Crossref]

Neal, A. T.

H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, “Phosphorene: an unexplored 2D semiconductor with a high hole mobility,” ACS Nano 8(4), 4033–4041 (2014).
[Crossref] [PubMed]

Ou, X.

L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
[Crossref] [PubMed]

Ozyilmaz, B.

S. P. Koenig, R. A. Doganov, H. Schmidt, A. H. Castro Neto, and B. Ozyilmaz, “Electric field effect in ultrathin black phosphorus,” Appl. Phys. Lett. 104(10), 103106 (2014).
[Crossref]

Padilla, W. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[Crossref] [PubMed]

Palasantzas, G.

P. J. van Zwol and G. Palasantzas, “Repulsive Casimir forces between solid materials with highrefractive-index intervening liquids,” Phys. Rev. A 81(6), 062502 (2010).
[Crossref]

Pendry, J. B.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[Crossref] [PubMed]

Petrovic, J.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[Crossref] [PubMed]

Potopowicz, J. R.

S. Singh, J. P. Remeika, and J. R. Potopowicz, “Nonlinear optical properties of ferroelectric lead titanate,” Appl. Phys. Lett. 20(3), 135–137 (1972).
[Crossref]

Qi, X.

Qiao, J.

J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, “High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus,” Nat. Commun. 5, 4475 (2014).
[Crossref] [PubMed]

Remeika, J. P.

S. Singh, J. P. Remeika, and J. R. Potopowicz, “Nonlinear optical properties of ferroelectric lead titanate,” Appl. Phys. Lett. 20(3), 135–137 (1972).
[Crossref]

Rodin, A. S.

A. S. Rodin, A. Carvalho, and A. H. Castro Neto, “Strain-induced gap modification in black phosphorus,” Phys. Rev. Lett. 112(17), 176801 (2014).
[Crossref] [PubMed]

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Rudenko, A. N.

S. Yuan, A. N. Rudenko, and M. I. Katsnelson, “Transport and optical properties of single- and bilayer black phosphorus with defects,” Phys. Rev. B 91(11), 115436 (2015).
[Crossref]

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. B 92, 085419 (2015).

A. N. Rudenko and M. I. Katsnelson, “Quasiparticle band structure and tight-binding model for single- and bilayer black phosphorus,” Phys. Rev. B 89(20), 201408 (2014).
[Crossref]

Schmidt, H.

S. P. Koenig, R. A. Doganov, H. Schmidt, A. H. Castro Neto, and B. Ozyilmaz, “Electric field effect in ultrathin black phosphorus,” Appl. Phys. Lett. 104(10), 103106 (2014).
[Crossref]

Singh, S.

S. Singh, J. P. Remeika, and J. R. Potopowicz, “Nonlinear optical properties of ferroelectric lead titanate,” Appl. Phys. Lett. 20(3), 135–137 (1972).
[Crossref]

Smith, D. R.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[Crossref] [PubMed]

Smith, S. C.

L. Kou, C. Chen, and S. C. Smith, “Phosphorene: fabrication, properties, and applications,” J. Phys. Chem. Lett. 6(14), 2794–2805 (2015).
[Crossref] [PubMed]

Soklaski, R.

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89(23), 235319 (2014).
[Crossref]

R. Fei, A. Faghaninia, R. Soklaski, J.-A. Yan, C. Lo, and L. Yang, “Enhanced thermoelectric efficiency via orthogonal electrical and thermal conductances in phosphorene,” Nano Lett. 14(11), 6393–6399 (2014).
[Crossref] [PubMed]

Soukoulis, C. M.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[Crossref] [PubMed]

Tang, D. Y.

Taniguchi, T.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

Tománek, D.

H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, “Phosphorene: an unexplored 2D semiconductor with a high hole mobility,” ACS Nano 8(4), 4033–4041 (2014).
[Crossref] [PubMed]

Tran, V.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89(23), 235319 (2014).
[Crossref]

van Zwol, P. J.

P. J. van Zwol and G. Palasantzas, “Repulsive Casimir forces between solid materials with highrefractive-index intervening liquids,” Phys. Rev. A 81(6), 062502 (2010).
[Crossref]

Vier, D. C.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[Crossref] [PubMed]

Wang, H.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458 (2014).
[Crossref] [PubMed]

Wang, J.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

Watanabe, K.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

Wegener, M.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[Crossref] [PubMed]

Wen, S. C.

Wu, H.

L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
[Crossref] [PubMed]

Xia, F.

F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458 (2014).
[Crossref] [PubMed]

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Xu, X.

H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, “Phosphorene: an unexplored 2D semiconductor with a high hole mobility,” ACS Nano 8(4), 4033–4041 (2014).
[Crossref] [PubMed]

Yan, J.-A.

R. Fei, A. Faghaninia, R. Soklaski, J.-A. Yan, C. Lo, and L. Yang, “Enhanced thermoelectric efficiency via orthogonal electrical and thermal conductances in phosphorene,” Nano Lett. 14(11), 6393–6399 (2014).
[Crossref] [PubMed]

Yang, F.

J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, “High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus,” Nat. Commun. 5, 4475 (2014).
[Crossref] [PubMed]

Yang, L.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89(23), 235319 (2014).
[Crossref]

R. Fei and L. Yang, “Strain-engineering the anisotropic electrical conductance of few-layer black phosphorus,” Nano Lett. 14(5), 2884–2889 (2014).
[Crossref] [PubMed]

R. Fei, A. Faghaninia, R. Soklaski, J.-A. Yan, C. Lo, and L. Yang, “Enhanced thermoelectric efficiency via orthogonal electrical and thermal conductances in phosphorene,” Nano Lett. 14(11), 6393–6399 (2014).
[Crossref] [PubMed]

Ye, G. J.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
[Crossref] [PubMed]

Ye, P. D.

H. Liu, Y. Du, Y. Deng, and P. D. Ye, “Semiconducting black phosphorus: synthesis, transport properties and electronic applications,” Chem. Soc. Rev. 44(9), 2732–2743 (2015).
[Crossref] [PubMed]

H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, “Phosphorene: an unexplored 2D semiconductor with a high hole mobility,” ACS Nano 8(4), 4033–4041 (2014).
[Crossref] [PubMed]

Yeh, P.

Yen, T. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[Crossref] [PubMed]

Yu, Y.

L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
[Crossref] [PubMed]

Yuan, S.

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. B 92, 085419 (2015).

S. Yuan, A. N. Rudenko, and M. I. Katsnelson, “Transport and optical properties of single- and bilayer black phosphorus with defects,” Phys. Rev. B 91(11), 115436 (2015).
[Crossref]

Zhang, H.

Zhang, X.

Q. Liu, X. Zhang, L. B. Abdalla, A. Fazzio, and A. Zunger, “Switching a normal insulator into a topological insulator via electric field with application to phosphorene,” Nano Lett. 15(2), 1222–1228 (2015).
[Crossref] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[Crossref] [PubMed]

Zhang, Y.

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
[Crossref] [PubMed]

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[Crossref] [PubMed]

Zhao, C. J.

Zhou, J.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[Crossref] [PubMed]

Zhou, L.

J. Hao and L. Zhou, “Electromagnetic wave scatterings by anisotropic metamaterials: Generalized4×4, ” Phys. Rev. B 77(9), 094201 (2008).
[Crossref]

Zhu, Z.

H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, “Phosphorene: an unexplored 2D semiconductor with a high hole mobility,” ACS Nano 8(4), 4033–4041 (2014).
[Crossref] [PubMed]

Ziletti, A.

A. Ziletti, A. Carvalho, D. K. Campbell, D. F. Coker, and A. H. Castro Neto, “Oxygen defects in phosphorene,” Phys. Rev. Lett. 114(4), 046801 (2015).
[Crossref] [PubMed]

Zunger, A.

Q. Liu, X. Zhang, L. B. Abdalla, A. Fazzio, and A. Zunger, “Switching a normal insulator into a topological insulator via electric field with application to phosphorene,” Nano Lett. 15(2), 1222–1228 (2015).
[Crossref] [PubMed]

ACS Nano (1)

H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, “Phosphorene: an unexplored 2D semiconductor with a high hole mobility,” ACS Nano 8(4), 4033–4041 (2014).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

S. P. Koenig, R. A. Doganov, H. Schmidt, A. H. Castro Neto, and B. Ozyilmaz, “Electric field effect in ultrathin black phosphorus,” Appl. Phys. Lett. 104(10), 103106 (2014).
[Crossref]

S. Singh, J. P. Remeika, and J. R. Potopowicz, “Nonlinear optical properties of ferroelectric lead titanate,” Appl. Phys. Lett. 20(3), 135–137 (1972).
[Crossref]

Appl. Phys., A Mater. Sci. Process. (1)

A. Morita, “Semiconducting black phosphorus,” Appl. Phys., A Mater. Sci. Process. 39(4), 227–242 (1986).
[Crossref]

Chem. Soc. Rev. (1)

H. Liu, Y. Du, Y. Deng, and P. D. Ye, “Semiconducting black phosphorus: synthesis, transport properties and electronic applications,” Chem. Soc. Rev. 44(9), 2732–2743 (2015).
[Crossref] [PubMed]

J. Opt. Soc. Am. (2)

J. Phys. Chem. Lett. (1)

L. Kou, C. Chen, and S. C. Smith, “Phosphorene: fabrication, properties, and applications,” J. Phys. Chem. Lett. 6(14), 2794–2805 (2015).
[Crossref] [PubMed]

J. Phys. Soc. Jpn. (1)

R. Kubo, “Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems,” J. Phys. Soc. Jpn. 12(6), 570–586 (1957).
[Crossref]

Nano Lett. (3)

R. Fei and L. Yang, “Strain-engineering the anisotropic electrical conductance of few-layer black phosphorus,” Nano Lett. 14(5), 2884–2889 (2014).
[Crossref] [PubMed]

Q. Liu, X. Zhang, L. B. Abdalla, A. Fazzio, and A. Zunger, “Switching a normal insulator into a topological insulator via electric field with application to phosphorene,” Nano Lett. 15(2), 1222–1228 (2015).
[Crossref] [PubMed]

R. Fei, A. Faghaninia, R. Soklaski, J.-A. Yan, C. Lo, and L. Yang, “Enhanced thermoelectric efficiency via orthogonal electrical and thermal conductances in phosphorene,” Nano Lett. 14(11), 6393–6399 (2014).
[Crossref] [PubMed]

Nat. Commun. (2)

J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, “High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus,” Nat. Commun. 5, 4475 (2014).
[Crossref] [PubMed]

F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458 (2014).
[Crossref] [PubMed]

Nat. Nanotechnol. (2)

L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014).
[Crossref] [PubMed]

L. Li, G. J. Ye, V. Tran, R. Fei, G. Chen, H. Wang, J. Wang, K. Watanabe, T. Taniguchi, L. Yang, X. H. Chen, and Y. Zhang, “Quantum oscillations in a two-dimensional electron gas in black phosphorus thin films,” Nat. Nanotechnol. 10(7), 608–613 (2015).
[Crossref] [PubMed]

Nature (1)

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005).
[Crossref] [PubMed]

New J. Phys. (1)

M. Ezawa, “Topological origin of quasi-flat edge band in phosphorene,” New J. Phys. 16(11), 115004 (2014).
[Crossref]

Opt. Express (1)

Phys. Rev. A (1)

P. J. van Zwol and G. Palasantzas, “Repulsive Casimir forces between solid materials with highrefractive-index intervening liquids,” Phys. Rev. A 81(6), 062502 (2010).
[Crossref]

Phys. Rev. B (7)

J. Hao and L. Zhou, “Electromagnetic wave scatterings by anisotropic metamaterials: Generalized4×4, ” Phys. Rev. B 77(9), 094201 (2008).
[Crossref]

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. B 92, 085419 (2015).

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

A. N. Rudenko and M. I. Katsnelson, “Quasiparticle band structure and tight-binding model for single- and bilayer black phosphorus,” Phys. Rev. B 89(20), 201408 (2014).
[Crossref]

V. Tran, R. Soklaski, Y. Liang, and L. Yang, “Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus,” Phys. Rev. B 89(23), 235319 (2014).
[Crossref]

S. Yuan, A. N. Rudenko, and M. I. Katsnelson, “Transport and optical properties of single- and bilayer black phosphorus with defects,” Phys. Rev. B 91(11), 115436 (2015).
[Crossref]

P. Li and I. Appelbaum, “Electrons and holes in phosphorene,” Phys. Rev. B 90(11), 115439 (2014).
[Crossref]

Phys. Rev. Lett. (2)

A. Ziletti, A. Carvalho, D. K. Campbell, D. F. Coker, and A. H. Castro Neto, “Oxygen defects in phosphorene,” Phys. Rev. Lett. 114(4), 046801 (2015).
[Crossref] [PubMed]

A. S. Rodin, A. Carvalho, and A. H. Castro Neto, “Strain-induced gap modification in black phosphorus,” Phys. Rev. Lett. 112(17), 176801 (2014).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

H. Ebert, “Magneto-optical effects in transition metal systems,” Rep. Prog. Phys. 59(12), 1665–1735 (1996) (and references therein).
[Crossref]

Science (2)

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004).
[Crossref] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[Crossref] [PubMed]

Other (11)

M. Mansuripur, The Principles of Magneto-Optical Recording (Cambridge University Press, 1995).

L. Li, F. Yang, G. J. Ye, Z. Zhang, K. Watanabe, T. Taniguchi, Y. Wang, X. H. Chen, and Y. Zhang, “Quantum hall effect in black phosphorus two-dimensional electron gas” http://arxiv.org/abs/1504.07155 (2015).

A. Ishihara, Statistical Physics (Academic Press, 1971).

X. Y. Zhou, R. Zhang, J. P. Sun, Y. L. Zou, D. Zhang, W. K. Lou, F. Cheng, G. H. Zhou, F. Zhai, and K. Chang, “Landau levels and magneto-transport property of monolayer phosphorene,” http://arxiv.org/abs/1411.4275 .
[Crossref]

Y. Jiang, R. Roldan, F. Guinea, and T. Low, “Magneto-electronic properties of multilayer black phosphorus,” http://arxiv.org/abs/1505.00175 .

J. M. Pereira and M. I. Katsnelson, “Landau levels of single and bilayer phosphorene,” http://arxiv.org/abs/1504.02452 .

M. Tahir, P. Vasilopoulos, and F. M. Peeters, “Magneto-optical transport properties of monolayer phosphorene,” http://arxiv.org/abs/1505.06780 .

D. Cakir, C. Sevik, and F. M. Peeters, “Remarkable effect of stacking on the electronic and optical properties of few layer black phosphorus,” http://arxiv.org/abs/1506.04707 .

X. Chen, Y. Wu, Z. Wu, Y. Han, S. Xu, L. Wang, W. Ye, T. Han, Y. He, Y. Cai, and N. Wang, “High quality sandwiched black phosphorus heterostructure and its quantum oscillations,” http://arxiv.org/abs/1412.1357 .
[Crossref]

V. Tayari, N. Hemsworth, I. Fakih, A. Favron, E. Gaufr’es, G. Gervais, R. Martel, and T. Szkopek, “Two-dimensional magnetotransport in a black phosphorus naked quantum well,” http://arxiv.org/abs/1412.1357 .
[Crossref]

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Schematic of a stack of N monolayer BP in the dielectric multilayer structure. The dielectric constants of N dielectric layers are denoted by ε (i) ( i=1,2,...,N ). The spacing between two adjacent BP layers is denoted by d (i) ( i=1,2,...,N ).
Fig. 2
Fig. 2 The reflections | r ss | 2 and | r sp | 2 (a), | r pp | 2 and | r ps | 2 (b) for the monolayer BP as a function of the wavelength at α= 0 0 and α= 45 0 . The corresponding transmissions | t sp | 2 and | t ps | 2 (c), | t ss | 2 and | t pp | 2 (d) as a function of the wavelength at α= 0 0 and α= 45 0 . The absorptions for S wave (e) and P wave (f) as a function of the wavelength at various incident angles.
Fig. 3
Fig. 3 The | r sp / r ss | 2 (a) and | r ps / r pp | 2 (b) for the monolayer BP as a function of the wavelength at various incident angles. The corresponding PCR (c) and β (d) as functions of the wavelength and the rotated angle θ. The real parts (e) and imaginary parts (f) of the surface conductivities ( σ xx and σ yy ) as a function of the wavelength in unit of σ 0 = e 2 /4 .
Fig. 4
Fig. 4 The reflections | r ss | 2 and | r sp | 2 (a), | r ss / r pp | 2 (b), for 14 monolayer BP sheets embedded in the one-dimensional photonic crystal composed of alternating 7 Teflon and PbTi O 3 layers, as a function of the wavelength at α= 0 0 . The corresponding transmissions | t ss | 2 and | t pp | 2 (c), | t ss / t pp | 2 (d) as a function of the wavelength at α= 0 0 . The dashed lines in (b) and (d) represent the reflection and transmission spectra for the one-dimensional photonic crystal without BPs.
Fig. 5
Fig. 5 The transmissions | t ss | 2 and | t sp | 2 (a), | t pp | 2 and | t ps | 2 (b) for the 70 monolayer BP sheets being embedded in the fused quartz as a function of the wavelength at α= 0 0 and 60 0 . Here θ= 30 0 . The corresponding reflections | r ss | 2 , | r sp | 2 , | r pp | 2 and | r ps | 2 (c), β (d) as a function of the wavelength at α= 0 0 . The corresponding absorptions for S wave (e) and P wave (f) as a function of the wavelength at α= 0 0 and 60 0 .
Fig. 6
Fig. 6 The | t sp / t ss | 2 (a) and | t ps / t pp | 2 (b) as a function of the wavelength at α= 0 0 and 60 0 . The γ (c) and PCT (d) as a function of the wavelength at α= 0 0 and 60 0 .

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

σ xx ( ω )= lim ε 0 + e β 1 ωΩ 0 e εt sinωt×2Imφ|f( H ) J x ( t )[ 1f( H ) ] J x ( t )|φdt ,
σ yy ( ω )= lim ε 0 + e β 1 ωΩ 0 e εt sinωt×2Imφ|f( H ) J y ( t )[ 1f( H ) ] J y ( t )|φdt ,
( J x J y )=( σ xx 0 0 σ yy )( E x E y ),
( J x J y )=( cosθ sinθ sinθ cosθ )( σ xx 0 0 σ yy )( cosθ sinθ sinθ cosθ )( E x E y ) =( cos 2 θ σ xx + sin 2 θ σ yy sinθcosθ( σ xx σ yy ) sinθcosθ( σ xx σ yy ) sin 2 θ σ xx + cos 2 θ σ yy )( E x E y )=( σ 11 σ 12 σ 21 σ 22 ).( E x E y )
k ( i ) ×( k ( i ) × E ( i ) )+ ω 2 μ 0 ε 0 ε ( i ) E ( i ) =0.
( ω 2 μ 0 ε 0 ε ( i ) k y 2 k z ( i )2 0 0 0 ω 2 μ 0 ε 0 ε ( i ) k z ( i )2 k y k z ( i ) 0 k y k z ( i ) ω 2 μ 0 ε 0 ε ( i ) k y 2 )( E x ( i ) E y ( i ) E z ( i ) )=0.
p 1 ( i ) =( 1 0 0 ), p 2 ( i ) =( 1 0 0 ), p 3 ( i ) =( 0 n z ( i ) n ( i ) n y n ( i ) ), p 4 ( i ) =( 0 n z ( i ) n ( i ) n y n ( i ) )
E ( i ) = σ=1 4 E 0 ( i ) exp{ i[ k σ ( i ) rωt ] } = σ=1 4 A σ ( i ) p σ ( i ) exp{ i[ k y ( i ) y+ k zσ ( i ) ( z z i )ωt ] } ,
H ( n ) = σ=1 4 H 0 ( i ) exp{ i[ k σ ( i ) rωt ] } = σ=1 4 A σ ( i ) q σ ( i ) exp{ i[ k y ( i ) y+ k zσ ( i ) ( z z i )ωt ] } , q σ ( i ) = k σ ( i ) ω μ 0 ×p.
E y ( i1 ) E y ( i ) =0, E x ( i1 ) E x ( i ) =0( n=1,2N+1 ),
H y ( i1 ) H y ( i ) = J x , H 1x ( i1 ) H 2x ( i ) = J y ( i=1,2N ),
H y ( N ) H y ( N+1 ) =0, H 1x ( N ) H 2x ( N+1 ) =0.
E 0 ( 0 ) =( E 0s ( in ) E 0s ( r ) E 0p ( in ) E 0p ( r ) )=Q E 0 ( N+1 ) =Q( E 0s t 0 E 0p t 0 ).
Q= D ( 0 ) 1 i=1 N ( D 0 ( i ) P ( i ) D ( i ) 1 ) D ( N+1 ) ,
D ( i ) =( 1 1 0 0 n z ( i ) μ 0 c n z ( i ) μ 0 c 0 0 0 0 n z ( i ) n ( i ) n z ( i ) n ( i ) 0 0 n ( i ) μ 0 c n ( i ) μ 0 c )
D 0 ( i ) =( 1 1 0 0 n z ( i ) μ 0 c + σ 11 n z ( i ) μ 0 c + σ 11 σ 12 n z ( i ) n ( i ) σ 12 n z ( i ) n ( i ) 0 0 n z ( i ) n ( i ) n z ( i ) n ( i ) σ 21 σ 21 n ( i ) μ 0 c σ 22 n z ( i ) n ( i ) n ( i ) μ 0 c σ 22 n z ( i ) n ( i ) ).
P ( i ) =( e i k z1 ( i ) d ( i ) 0 0 0 0 e i k z2 ( i ) d ( i ) 0 0 0 0 e i k z3 ( i ) d ( i ) 0 0 0 0 e i k z4 ( i ) d ( i ) ).
r ss = ( E 0s ( r ) E 0s ( in ) ) E 0p (in) =0 = Q 21 Q 33 Q 23 Q 31 Q 11 Q 33 Q 13 Q 31 , r sp = ( E 0p ( r ) E 0s ( in ) ) E 0p (in) =0 = Q 41 Q 33 Q 43 Q 31 Q 11 Q 33 Q 13 Q 31 ,
r ps = ( E 0s ( r ) E 0p ( in ) ) E 0s (in) =0 = Q 11 Q 23 Q 21 Q 13 Q 11 Q 33 Q 13 Q 31 , r pp = ( E 0p ( r ) E 0p ( in ) ) E 0s (in) =0 = Q 11 Q 43 Q 41 Q 13 Q 11 Q 33 Q 13 Q 31 ,
t ss = ( E 0s ( t ) E 0s ( in ) ) E 0p (in) =0 = Q 33 Q 11 Q 33 Q 13 Q 31 , t sp = ( E 0p ( t ) E 0s ( in ) ) E 0p (in) =0 = Q 31 Q 11 Q 33 Q 13 Q 31 ,
t ps = ( E 0s ( t ) E 0p ( in ) ) E 0s (in) =0 = Q 13 Q 11 Q 33 Q 13 Q 31 , t ps = ( E 0p ( t ) E 0p ( in ) ) E 0s (in) =0 = Q 11 Q 11 Q 33 Q 13 Q 31 .
a s =1 | r ss | 2 | r sp | 2 n z N+1 n z 0 | t ss | n z N+1 n z 0 | t sp |
a p =1 | r pp | 2 | r ps | 2 n z N+1 n z 0 | t pp | n z N+1 n z 0 | t ps |.
tan2β= 2Re[ χ ] 1 | χ | 2 ,
tan2γ= 2Re[ χ t ] 1 | χ t | 2
PCR= | r sp | 2 | r ss | 2 + | r sp | 2 and PCT= | t sp | 2 | t ss | 2 + | t sp | 2 ,
r sp = r ps = 2 μ 0 c σ 12 n z ( 2 ) n z ( 1 ) n ( 2 ) ( 1+ n z ( 2 ) n z ( 1 ) + μ 0 c σ 11 n z ( 1 ) )( n ( 1 ) n z ( 2 ) n (2) n z ( 1 ) + n ( 2 ) n ( 1 ) + μ 0 c σ 22 n z ( 2 ) n ( 1 ) n ( 2 ) ) μ 0 2 c 2 σ 12 2 n z ( 2 ) n z ( 1 ) n ( 1 ) n ( 2 ) ,
t sp = 2 μ 0 c σ 12 n ( 1 ) ( 1+ n z ( 2 ) n z ( 1 ) + μ 0 c σ 11 n z ( 1 ) )( n ( 1 ) n z ( 2 ) n ( 2 ) n z ( 1 ) + n ( 2 ) n ( 1 ) + μ 0 c σ 22 n z ( 2 ) n ( 1 ) n ( 2 ) ) μ 0 2 c 2 σ 12 2 n z ( 2 ) n z ( 1 ) n ( 1 ) n ( 2 ) ,
t ps = 2 μ 0 c σ 12 n z ( 2 ) n z ( 1 ) n ( 2 ) ( 1+ n z ( 2 ) n z ( 1 ) + μ 0 c σ 11 n z ( 1 ) )( n ( 1 ) n z ( 2 ) n ( 2 ) n z ( 1 ) + n ( 2 ) n ( 1 ) + μ 0 c σ 22 n z ( 2 ) n ( 1 ) n ( 2 ) ) μ 0 2 c 2 σ 12 2 n z ( 2 ) n z ( 1 ) n ( 1 ) n ( 2 ) .

Metrics