Abstract

We study the nonlinear optical response generated by the massless Dirac quasiparticles residing around the topologically-protected Dirac/Weyl nodal points in three-dimensional (3D) topological semimetals. Analytical expressions of third-order interband nonlinear optical conductivities are obtained based on a quantum mechanical formalism which couples 3D Dirac fermions with multiple photons. Our results reveal that the massless Dirac fermions in three dimensions retains strong optical nonlinearity in terahertz frequency regime similar to the case of the two-dimensional Dirac fermions in graphene. At room temperature, the Kerr nonlinear refractive index and the harmonic generation susceptibility are found to be n2 = 10−11 ∼ 10−8 m2W−1 and χ(3) = 10−14 ∼ 10−8 m2V−2, respectively, in the few terahertz frequency regimes, which is comparable to graphene and orders of magnitudes stronger than many nonlinear crystals. Importantly, because 3D topological Dirac/Weyl semimetals possess bulk structural advantage not found in the strictly two-dimensional graphene, greater design flexibility and improved ease-of-fabrication in terms of photonic and optoelectronic device applications can be achieved. Our finding reveals the potential of 3D topological semimetals as a viable alternative to graphene for nonlinear optics applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (5)

S. Yamashita, “Nonlinear optics in carbon nanotube, graphene, and related 2d materials,” APL Photonics 4(3), 034301 (2019).
[Crossref]

A. Singh, S. Ghosh, and A. Agarwal, “Terahertz shifted optical sideband generation in graphene,” Phys. Rev. B 99(12), 125419 (2019).
[Crossref]

K. J. Ooi, Y. Ang, Q. Zhai, D. T. Tan, L. Ang, and C. Ong, “Nonlinear plasmonics of three-dimensional dirac semimetals,” APL Photonics 4(3), 034402 (2019).
[Crossref]

K. Sonowal, A. Singh, and A. Agarwal, “Giant optical activity and kerr effect in type-i and type-ii weyl semimetals,” Phys. Rev. B 100(8), 085436 (2019).
[Crossref]

Q. Wang, X. Wu, L. Wu, and Y. Xiang, “Broadband nonlinear optical response in bi2se3-bi2te3 heterostructure and its application in all-optical switching,” AIP Adv. 9(2), 025022 (2019).
[Crossref]

2018 (5)

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, and J. Teichert, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

B. Yao, Y. Liu, S.-W. Huang, C. Choi, Z. Xie, J. F. Flores, Y. Wu, M. Yu, D.-L. Kwong, and Y. Huang, “Broadband gate-tunable terahertz plasmons in graphene heterostructures,” Nat. Photonics 12(1), 22–28 (2018).
[Crossref]

T. Jiang, D. Huang, J. Cheng, X. Fan, Z. Zhang, Y. Shan, Y. Yi, Y. Dai, L. Shi, and K. Liu, “Gate-tunable third-order nonlinear optical response of massless dirac fermions in graphene,” Nat. Photonics 12(7), 430–436 (2018).
[Crossref]

A. Thakur, K. Sadhukhan, and A. Agarwal, “Dynamic current-current susceptibility in three-dimensional dirac and weyl semimetals,” Phys. Rev. B 97(3), 035403 (2018).
[Crossref]

A. Singh, S. Ghosh, and A. Agarwal, “Nonlinear and anisotropic polarization rotation in two-dimensional dirac materials,” Phys. Rev. B 97(20), 205420 (2018).
[Crossref]

2017 (7)

A. Singh, K. I. Bolotin, S. Ghosh, and A. Agarwal, “Nonlinear optical conductivity of a generic two-band system with application to doped and gapped graphene,” Phys. Rev. B 95(15), 155421 (2017).
[Crossref]

Y. Liu, Z.-M. Yu, and S. A. Yang, “Transverse shift in andreev reflection,” Phys. Rev. B 96(12), 121101 (2017).
[Crossref]

K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille, A. Nugroho, P. Goswami, M. Ochi, M. Ikhlas, M. Nakayama, and S. Akebi, “Evidence for magnetic weyl fermions in a correlated metal,” Nat. Mater. 16(11), 1090–1095 (2017).
[Crossref]

K. J. A. Ooi and D. T. H. Tan, “Nonlinear graphene plasmonics,” Proc. R. Soc. A 473(2206), 20170433 (2017).
[Crossref]

Z. Liu, C. Zhang, and J. C. Cao, “Nonlinear optical conductivity resulting from the local energy spectrum at the M point in graphene,” Phys. Rev. B 96(3), 035206 (2017).
[Crossref]

C. Zhu, F. Wang, Y. Meng, X. Yuan, F. Xiu, H. Luo, Y. Wang, J. Li, X. Lv, and L. He, “A robust and tuneable mid-infrared optical switch enabled by bulk dirac fermions,” Nat. Commun. 8(1), 14111 (2017).
[Crossref]

K. Ooi, D. Ng, T. Wang, A. Chee, S. Ng, Q. Wang, L. Ang, A. Agarwal, L. Kimerling, and D. Tan, “Pushing the limits of cmos optical parametric amplifiers with usrn: Si 7 n 3 above the two-photon absorption edge,” Nat. Commun. 8(1), 13878 (2017).
[Crossref]

2016 (1)

O. V. Kotov and Y. E. Lozovik, “Dielectric response and novel electromagnetic modes in three-dimensional dirac semimetal films,” Phys. Rev. B 93(23), 235417 (2016).
[Crossref]

2015 (4)

L. Miao, Y. Jiang, S. Lu, B. Shi, C. Zhao, H. Zhang, and S. Wen, “Broadband ultrafast nonlinear optical response of few-layers graphene: toward the mid-infrared regime,” Photonics Res. 3(5), 214 (2015).
[Crossref]

M. Kargarian, M. Randeria, and N. Trivedi, “Theory of kerr and faraday rotations and linear dichroism in topological weyl semimetals,” Sci. Rep. 5(1), 12683 (2015).
[Crossref]

C. Shekhar, A. K. Nayak, Y. Sun, M. Schmidt, M. Nicklas, I. Leermakers, U. Zeitler, Y. Skourski, J. Wosnitza, and Z. Liu, “Extremely large magnetoresistance and ultrahigh mobility in the topological weyl semimetal candidate nbp,” Nat. Phys. 11(8), 645–649 (2015).
[Crossref]

J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M. Hirschberger, W. Wang, R. J. Cava, and N. P. Ong, “Evidence for the chiral anomaly in the dirac semimetal na3bi,” Science 350(6259), 413–416 (2015).
[Crossref]

2014 (4)

Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng, D. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain, and Y. L. Chen, “Discovery of a three-dimensional topological dirac semimetal, na3bi,” Science 343(6173), 864–867 (2014).
[Crossref]

S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Büchner, and R. J. Cava, “Experimental realization of a three-dimensional dirac semimetal,” Phys. Rev. Lett. 113(2), 027603 (2014).
[Crossref]

M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, and H. Lin, “Observation of a three-dimensional topological dirac semimetal phase in high-mobility cd 3 as 2,” Nat. Commun. 5(1), 3786 (2014).
[Crossref]

W. H. Cao and Y. S. Ang, “Effect of asymmetry on nonlinear optical response in graphene,” Europhys. Lett. 107(3), 37007 (2014).
[Crossref]

2013 (2)

W. Chen, G. Wang, S. Qin, C. Wang, J. Fang, J. Qi, X. Zhang, L. Wang, H. Jia, and S. Chang, “The nonlinear optical properties of coupling and decoupling graphene layers,” AIP Adv. 3(4), 042123 (2013).
[Crossref]

S. Lu, C. Zhao, Y. Zou, S. Chen, Y. Chen, Y. Li, H. Zhang, S. Wen, and D. Tang, “Third order nonlinear optical property of bi 2 se 3,” Opt. Express 21(2), 2072–2082 (2013).
[Crossref]

2012 (6)

H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856 (2012).
[Crossref]

S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, “Dirac semimetal in three dimensions,” Phys. Rev. Lett. 108(14), 140405 (2012).
[Crossref]

S. Shareef, Y. S. Ang, and C. Zhang, “Room-temperature strong terahertz photon mixing in graphene,” J. Opt. Soc. Am. B 29(3), 274 (2012).
[Crossref]

Y. S. Ang and C. Zhang, “Enhanced optical conductance in graphene superlattice due to anisotropic band dispersion,” J. Phys. D: Appl. Phys. 45(39), 395303 (2012).
[Crossref]

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nat. Photonics 6(8), 554–559 (2012).
[Crossref]

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

2011 (4)

Y. S. Ang and C. Zhang, “Subgap optical conductivity in semihydrogenated graphene,” Appl. Phys. Lett. 98(4), 042107 (2011).
[Crossref]

A. A. Burkov, M. D. Hook, and L. Balents, “Topological nodal semimetals,” Phys. Rev. B 84(23), 235126 (2011).
[Crossref]

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[Crossref]

H. Yang, X. Feng, Q. Wang, H. Huang, W. Chen, A. T. S. Wee, and W. Ji, “Giant two-photon absorption in bilayer graphene,” Nano Lett. 11(7), 2622–2627 (2011).
[Crossref]

2010 (3)

E. Hendry, P. J. Hale, J. Moger, A. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105(9), 097401 (2010).
[Crossref]

X. G. Xu, S. Sultan, C. Zhang, and J. C. Cao, “Nonlinear optical conductance in a graphene pn junction in the terahertz regime,” Appl. Phys. Lett. 97(1), 011907 (2010).
[Crossref]

Y. S. Ang, S. Sultan, and C. Zhang, “Nonlinear optical spectrum of bilayer graphene in the terahertz regime,” Appl. Phys. Lett. 97(24), 243110 (2010).
[Crossref]

2009 (1)

A. R. Wright, X. G. Xu, J. C. Cao, and C. Zhang, “Strong nonlinear optical response of graphene in the terahertz regime,” Appl. Phys. Lett. 95(7), 072101 (2009).
[Crossref]

Agarwal, A.

A. Singh, S. Ghosh, and A. Agarwal, “Terahertz shifted optical sideband generation in graphene,” Phys. Rev. B 99(12), 125419 (2019).
[Crossref]

K. Sonowal, A. Singh, and A. Agarwal, “Giant optical activity and kerr effect in type-i and type-ii weyl semimetals,” Phys. Rev. B 100(8), 085436 (2019).
[Crossref]

A. Singh, S. Ghosh, and A. Agarwal, “Nonlinear and anisotropic polarization rotation in two-dimensional dirac materials,” Phys. Rev. B 97(20), 205420 (2018).
[Crossref]

A. Thakur, K. Sadhukhan, and A. Agarwal, “Dynamic current-current susceptibility in three-dimensional dirac and weyl semimetals,” Phys. Rev. B 97(3), 035403 (2018).
[Crossref]

A. Singh, K. I. Bolotin, S. Ghosh, and A. Agarwal, “Nonlinear optical conductivity of a generic two-band system with application to doped and gapped graphene,” Phys. Rev. B 95(15), 155421 (2017).
[Crossref]

K. Ooi, D. Ng, T. Wang, A. Chee, S. Ng, Q. Wang, L. Ang, A. Agarwal, L. Kimerling, and D. Tan, “Pushing the limits of cmos optical parametric amplifiers with usrn: Si 7 n 3 above the two-photon absorption edge,” Nat. Commun. 8(1), 13878 (2017).
[Crossref]

Akebi, S.

K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille, A. Nugroho, P. Goswami, M. Ochi, M. Ikhlas, M. Nakayama, and S. Akebi, “Evidence for magnetic weyl fermions in a correlated metal,” Nat. Mater. 16(11), 1090–1095 (2017).
[Crossref]

Alidoust, N.

M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, and H. Lin, “Observation of a three-dimensional topological dirac semimetal phase in high-mobility cd 3 as 2,” Nat. Commun. 5(1), 3786 (2014).
[Crossref]

Ang, L.

K. J. Ooi, Y. Ang, Q. Zhai, D. T. Tan, L. Ang, and C. Ong, “Nonlinear plasmonics of three-dimensional dirac semimetals,” APL Photonics 4(3), 034402 (2019).
[Crossref]

K. Ooi, D. Ng, T. Wang, A. Chee, S. Ng, Q. Wang, L. Ang, A. Agarwal, L. Kimerling, and D. Tan, “Pushing the limits of cmos optical parametric amplifiers with usrn: Si 7 n 3 above the two-photon absorption edge,” Nat. Commun. 8(1), 13878 (2017).
[Crossref]

Ang, Y.

K. J. Ooi, Y. Ang, Q. Zhai, D. T. Tan, L. Ang, and C. Ong, “Nonlinear plasmonics of three-dimensional dirac semimetals,” APL Photonics 4(3), 034402 (2019).
[Crossref]

Ang, Y. S.

W. H. Cao and Y. S. Ang, “Effect of asymmetry on nonlinear optical response in graphene,” Europhys. Lett. 107(3), 37007 (2014).
[Crossref]

S. Shareef, Y. S. Ang, and C. Zhang, “Room-temperature strong terahertz photon mixing in graphene,” J. Opt. Soc. Am. B 29(3), 274 (2012).
[Crossref]

Y. S. Ang and C. Zhang, “Enhanced optical conductance in graphene superlattice due to anisotropic band dispersion,” J. Phys. D: Appl. Phys. 45(39), 395303 (2012).
[Crossref]

Y. S. Ang and C. Zhang, “Subgap optical conductivity in semihydrogenated graphene,” Appl. Phys. Lett. 98(4), 042107 (2011).
[Crossref]

Y. S. Ang, S. Sultan, and C. Zhang, “Nonlinear optical spectrum of bilayer graphene in the terahertz regime,” Appl. Phys. Lett. 97(24), 243110 (2010).
[Crossref]

Awari, N.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, and J. Teichert, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

Bai, X.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[Crossref]

Balents, L.

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T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nat. Photonics 6(8), 554–559 (2012).
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L. Thylen, P. Holmström, L. Wosinski, B. Jaskorzynska, M. Naruse, T. Kawazoe, M. Ohtsu, M. Yan, M. Fiorentino, and U. Westergren, “Chapter 6 - nanophotonics for low-power switches,” in Optical Fiber Telecommunications (Sixth Edition), I. P. Kaminow, T. Li, and A. E. Willner, eds. (Academic, Boston, 2013), Optics and Photonics, pp. 205–241, 6th edition ed.

Wosnitza, J.

C. Shekhar, A. K. Nayak, Y. Sun, M. Schmidt, M. Nicklas, I. Leermakers, U. Zeitler, Y. Skourski, J. Wosnitza, and Z. Liu, “Extremely large magnetoresistance and ultrahigh mobility in the topological weyl semimetal candidate nbp,” Nat. Phys. 11(8), 645–649 (2015).
[Crossref]

Wright, A. R.

A. R. Wright, X. G. Xu, J. C. Cao, and C. Zhang, “Strong nonlinear optical response of graphene in the terahertz regime,” Appl. Phys. Lett. 95(7), 072101 (2009).
[Crossref]

Wu, L.

Q. Wang, X. Wu, L. Wu, and Y. Xiang, “Broadband nonlinear optical response in bi2se3-bi2te3 heterostructure and its application in all-optical switching,” AIP Adv. 9(2), 025022 (2019).
[Crossref]

Wu, R.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[Crossref]

Wu, X.

Q. Wang, X. Wu, L. Wu, and Y. Xiang, “Broadband nonlinear optical response in bi2se3-bi2te3 heterostructure and its application in all-optical switching,” AIP Adv. 9(2), 025022 (2019).
[Crossref]

Wu, Y.

B. Yao, Y. Liu, S.-W. Huang, C. Choi, Z. Xie, J. F. Flores, Y. Wu, M. Yu, D.-L. Kwong, and Y. Huang, “Broadband gate-tunable terahertz plasmons in graphene heterostructures,” Nat. Photonics 12(1), 22–28 (2018).
[Crossref]

Xiang, Y.

Q. Wang, X. Wu, L. Wu, and Y. Xiang, “Broadband nonlinear optical response in bi2se3-bi2te3 heterostructure and its application in all-optical switching,” AIP Adv. 9(2), 025022 (2019).
[Crossref]

Xie, Z.

B. Yao, Y. Liu, S.-W. Huang, C. Choi, Z. Xie, J. F. Flores, Y. Wu, M. Yu, D.-L. Kwong, and Y. Huang, “Broadband gate-tunable terahertz plasmons in graphene heterostructures,” Nat. Photonics 12(1), 22–28 (2018).
[Crossref]

Xiong, J.

J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M. Hirschberger, W. Wang, R. J. Cava, and N. P. Ong, “Evidence for the chiral anomaly in the dirac semimetal na3bi,” Science 350(6259), 413–416 (2015).
[Crossref]

Xiu, F.

C. Zhu, F. Wang, Y. Meng, X. Yuan, F. Xiu, H. Luo, Y. Wang, J. Li, X. Lv, and L. He, “A robust and tuneable mid-infrared optical switch enabled by bulk dirac fermions,” Nat. Commun. 8(1), 14111 (2017).
[Crossref]

Xu, S.-Y.

M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, and H. Lin, “Observation of a three-dimensional topological dirac semimetal phase in high-mobility cd 3 as 2,” Nat. Commun. 5(1), 3786 (2014).
[Crossref]

Xu, X. G.

X. G. Xu, S. Sultan, C. Zhang, and J. C. Cao, “Nonlinear optical conductance in a graphene pn junction in the terahertz regime,” Appl. Phys. Lett. 97(1), 011907 (2010).
[Crossref]

A. R. Wright, X. G. Xu, J. C. Cao, and C. Zhang, “Strong nonlinear optical response of graphene in the terahertz regime,” Appl. Phys. Lett. 95(7), 072101 (2009).
[Crossref]

Yamashita, S.

S. Yamashita, “Nonlinear optics in carbon nanotube, graphene, and related 2d materials,” APL Photonics 4(3), 034301 (2019).
[Crossref]

Yan, M.

L. Thylen, P. Holmström, L. Wosinski, B. Jaskorzynska, M. Naruse, T. Kawazoe, M. Ohtsu, M. Yan, M. Fiorentino, and U. Westergren, “Chapter 6 - nanophotonics for low-power switches,” in Optical Fiber Telecommunications (Sixth Edition), I. P. Kaminow, T. Li, and A. E. Willner, eds. (Academic, Boston, 2013), Optics and Photonics, pp. 205–241, 6th edition ed.

Yan, S.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[Crossref]

Yang, H.

H. Yang, X. Feng, Q. Wang, H. Huang, W. Chen, A. T. S. Wee, and W. Ji, “Giant two-photon absorption in bilayer graphene,” Nano Lett. 11(7), 2622–2627 (2011).
[Crossref]

Yang, S. A.

Y. Liu, Z.-M. Yu, and S. A. Yang, “Transverse shift in andreev reflection,” Phys. Rev. B 96(12), 121101 (2017).
[Crossref]

Yao, B.

B. Yao, Y. Liu, S.-W. Huang, C. Choi, Z. Xie, J. F. Flores, Y. Wu, M. Yu, D.-L. Kwong, and Y. Huang, “Broadband gate-tunable terahertz plasmons in graphene heterostructures,” Nat. Photonics 12(1), 22–28 (2018).
[Crossref]

Yi, Y.

T. Jiang, D. Huang, J. Cheng, X. Fan, Z. Zhang, Y. Shan, Y. Yi, Y. Dai, L. Shi, and K. Liu, “Gate-tunable third-order nonlinear optical response of massless dirac fermions in graphene,” Nat. Photonics 12(7), 430–436 (2018).
[Crossref]

Young, S. M.

S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, “Dirac semimetal in three dimensions,” Phys. Rev. Lett. 108(14), 140405 (2012).
[Crossref]

Yu, M.

B. Yao, Y. Liu, S.-W. Huang, C. Choi, Z. Xie, J. F. Flores, Y. Wu, M. Yu, D.-L. Kwong, and Y. Huang, “Broadband gate-tunable terahertz plasmons in graphene heterostructures,” Nat. Photonics 12(1), 22–28 (2018).
[Crossref]

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nat. Photonics 6(8), 554–559 (2012).
[Crossref]

Yu, Z.-M.

Y. Liu, Z.-M. Yu, and S. A. Yang, “Transverse shift in andreev reflection,” Phys. Rev. B 96(12), 121101 (2017).
[Crossref]

Yuan, X.

C. Zhu, F. Wang, Y. Meng, X. Yuan, F. Xiu, H. Luo, Y. Wang, J. Li, X. Lv, and L. He, “A robust and tuneable mid-infrared optical switch enabled by bulk dirac fermions,” Nat. Commun. 8(1), 14111 (2017).
[Crossref]

Zabolotnyy, V.

S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Büchner, and R. J. Cava, “Experimental realization of a three-dimensional dirac semimetal,” Phys. Rev. Lett. 113(2), 027603 (2014).
[Crossref]

Zaheer, S.

S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, “Dirac semimetal in three dimensions,” Phys. Rev. Lett. 108(14), 140405 (2012).
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Zeitler, U.

C. Shekhar, A. K. Nayak, Y. Sun, M. Schmidt, M. Nicklas, I. Leermakers, U. Zeitler, Y. Skourski, J. Wosnitza, and Z. Liu, “Extremely large magnetoresistance and ultrahigh mobility in the topological weyl semimetal candidate nbp,” Nat. Phys. 11(8), 645–649 (2015).
[Crossref]

Zhai, Q.

K. J. Ooi, Y. Ang, Q. Zhai, D. T. Tan, L. Ang, and C. Ong, “Nonlinear plasmonics of three-dimensional dirac semimetals,” APL Photonics 4(3), 034402 (2019).
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Zhang, C.

Z. Liu, C. Zhang, and J. C. Cao, “Nonlinear optical conductivity resulting from the local energy spectrum at the M point in graphene,” Phys. Rev. B 96(3), 035206 (2017).
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Y. S. Ang and C. Zhang, “Enhanced optical conductance in graphene superlattice due to anisotropic band dispersion,” J. Phys. D: Appl. Phys. 45(39), 395303 (2012).
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S. Shareef, Y. S. Ang, and C. Zhang, “Room-temperature strong terahertz photon mixing in graphene,” J. Opt. Soc. Am. B 29(3), 274 (2012).
[Crossref]

Y. S. Ang and C. Zhang, “Subgap optical conductivity in semihydrogenated graphene,” Appl. Phys. Lett. 98(4), 042107 (2011).
[Crossref]

Y. S. Ang, S. Sultan, and C. Zhang, “Nonlinear optical spectrum of bilayer graphene in the terahertz regime,” Appl. Phys. Lett. 97(24), 243110 (2010).
[Crossref]

X. G. Xu, S. Sultan, C. Zhang, and J. C. Cao, “Nonlinear optical conductance in a graphene pn junction in the terahertz regime,” Appl. Phys. Lett. 97(1), 011907 (2010).
[Crossref]

A. R. Wright, X. G. Xu, J. C. Cao, and C. Zhang, “Strong nonlinear optical response of graphene in the terahertz regime,” Appl. Phys. Lett. 95(7), 072101 (2009).
[Crossref]

Zhang, H.

Zhang, X.

W. Chen, G. Wang, S. Qin, C. Wang, J. Fang, J. Qi, X. Zhang, L. Wang, H. Jia, and S. Chang, “The nonlinear optical properties of coupling and decoupling graphene layers,” AIP Adv. 3(4), 042123 (2013).
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Zhang, Y.

Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng, D. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain, and Y. L. Chen, “Discovery of a three-dimensional topological dirac semimetal, na3bi,” Science 343(6173), 864–867 (2014).
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R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[Crossref]

Zhang, Z.

T. Jiang, D. Huang, J. Cheng, X. Fan, Z. Zhang, Y. Shan, Y. Yi, Y. Dai, L. Shi, and K. Liu, “Gate-tunable third-order nonlinear optical response of massless dirac fermions in graphene,” Nat. Photonics 12(7), 430–436 (2018).
[Crossref]

Zhao, C.

L. Miao, Y. Jiang, S. Lu, B. Shi, C. Zhao, H. Zhang, and S. Wen, “Broadband ultrafast nonlinear optical response of few-layers graphene: toward the mid-infrared regime,” Photonics Res. 3(5), 214 (2015).
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S. Lu, C. Zhao, Y. Zou, S. Chen, Y. Chen, Y. Li, H. Zhang, S. Wen, and D. Tang, “Third order nonlinear optical property of bi 2 se 3,” Opt. Express 21(2), 2072–2082 (2013).
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Zhao, J.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[Crossref]

Zhou, B.

Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng, D. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain, and Y. L. Chen, “Discovery of a three-dimensional topological dirac semimetal, na3bi,” Science 343(6173), 864–867 (2014).
[Crossref]

Zhu, C.

C. Zhu, F. Wang, Y. Meng, X. Yuan, F. Xiu, H. Luo, Y. Wang, J. Li, X. Lv, and L. He, “A robust and tuneable mid-infrared optical switch enabled by bulk dirac fermions,” Nat. Commun. 8(1), 14111 (2017).
[Crossref]

Zou, Y.

AIP Adv. (2)

W. Chen, G. Wang, S. Qin, C. Wang, J. Fang, J. Qi, X. Zhang, L. Wang, H. Jia, and S. Chang, “The nonlinear optical properties of coupling and decoupling graphene layers,” AIP Adv. 3(4), 042123 (2013).
[Crossref]

Q. Wang, X. Wu, L. Wu, and Y. Xiang, “Broadband nonlinear optical response in bi2se3-bi2te3 heterostructure and its application in all-optical switching,” AIP Adv. 9(2), 025022 (2019).
[Crossref]

APL Photonics (2)

S. Yamashita, “Nonlinear optics in carbon nanotube, graphene, and related 2d materials,” APL Photonics 4(3), 034301 (2019).
[Crossref]

K. J. Ooi, Y. Ang, Q. Zhai, D. T. Tan, L. Ang, and C. Ong, “Nonlinear plasmonics of three-dimensional dirac semimetals,” APL Photonics 4(3), 034402 (2019).
[Crossref]

Appl. Phys. Lett. (4)

Y. S. Ang and C. Zhang, “Subgap optical conductivity in semihydrogenated graphene,” Appl. Phys. Lett. 98(4), 042107 (2011).
[Crossref]

A. R. Wright, X. G. Xu, J. C. Cao, and C. Zhang, “Strong nonlinear optical response of graphene in the terahertz regime,” Appl. Phys. Lett. 95(7), 072101 (2009).
[Crossref]

X. G. Xu, S. Sultan, C. Zhang, and J. C. Cao, “Nonlinear optical conductance in a graphene pn junction in the terahertz regime,” Appl. Phys. Lett. 97(1), 011907 (2010).
[Crossref]

Y. S. Ang, S. Sultan, and C. Zhang, “Nonlinear optical spectrum of bilayer graphene in the terahertz regime,” Appl. Phys. Lett. 97(24), 243110 (2010).
[Crossref]

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W. H. Cao and Y. S. Ang, “Effect of asymmetry on nonlinear optical response in graphene,” Europhys. Lett. 107(3), 37007 (2014).
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J. Opt. Soc. Am. B (1)

J. Phys. D: Appl. Phys. (1)

Y. S. Ang and C. Zhang, “Enhanced optical conductance in graphene superlattice due to anisotropic band dispersion,” J. Phys. D: Appl. Phys. 45(39), 395303 (2012).
[Crossref]

Nano Lett. (2)

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[Crossref]

H. Yang, X. Feng, Q. Wang, H. Huang, W. Chen, A. T. S. Wee, and W. Ji, “Giant two-photon absorption in bilayer graphene,” Nano Lett. 11(7), 2622–2627 (2011).
[Crossref]

Nat. Commun. (3)

M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, and H. Lin, “Observation of a three-dimensional topological dirac semimetal phase in high-mobility cd 3 as 2,” Nat. Commun. 5(1), 3786 (2014).
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C. Zhu, F. Wang, Y. Meng, X. Yuan, F. Xiu, H. Luo, Y. Wang, J. Li, X. Lv, and L. He, “A robust and tuneable mid-infrared optical switch enabled by bulk dirac fermions,” Nat. Commun. 8(1), 14111 (2017).
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K. Ooi, D. Ng, T. Wang, A. Chee, S. Ng, Q. Wang, L. Ang, A. Agarwal, L. Kimerling, and D. Tan, “Pushing the limits of cmos optical parametric amplifiers with usrn: Si 7 n 3 above the two-photon absorption edge,” Nat. Commun. 8(1), 13878 (2017).
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Nat. Mater. (1)

K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille, A. Nugroho, P. Goswami, M. Ochi, M. Ikhlas, M. Nakayama, and S. Akebi, “Evidence for magnetic weyl fermions in a correlated metal,” Nat. Mater. 16(11), 1090–1095 (2017).
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Nat. Photonics (4)

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nat. Photonics 6(8), 554–559 (2012).
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A. Grigorenko, M. Polini, and K. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
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B. Yao, Y. Liu, S.-W. Huang, C. Choi, Z. Xie, J. F. Flores, Y. Wu, M. Yu, D.-L. Kwong, and Y. Huang, “Broadband gate-tunable terahertz plasmons in graphene heterostructures,” Nat. Photonics 12(1), 22–28 (2018).
[Crossref]

T. Jiang, D. Huang, J. Cheng, X. Fan, Z. Zhang, Y. Shan, Y. Yi, Y. Dai, L. Shi, and K. Liu, “Gate-tunable third-order nonlinear optical response of massless dirac fermions in graphene,” Nat. Photonics 12(7), 430–436 (2018).
[Crossref]

Nat. Phys. (1)

C. Shekhar, A. K. Nayak, Y. Sun, M. Schmidt, M. Nicklas, I. Leermakers, U. Zeitler, Y. Skourski, J. Wosnitza, and Z. Liu, “Extremely large magnetoresistance and ultrahigh mobility in the topological weyl semimetal candidate nbp,” Nat. Phys. 11(8), 645–649 (2015).
[Crossref]

Nature (1)

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, and J. Teichert, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
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Opt. Express (1)

Opt. Lett. (1)

Photonics Res. (1)

L. Miao, Y. Jiang, S. Lu, B. Shi, C. Zhao, H. Zhang, and S. Wen, “Broadband ultrafast nonlinear optical response of few-layers graphene: toward the mid-infrared regime,” Photonics Res. 3(5), 214 (2015).
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Phys. Rev. B (9)

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S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, “Dirac semimetal in three dimensions,” Phys. Rev. Lett. 108(14), 140405 (2012).
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S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Büchner, and R. J. Cava, “Experimental realization of a three-dimensional dirac semimetal,” Phys. Rev. Lett. 113(2), 027603 (2014).
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J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M. Hirschberger, W. Wang, R. J. Cava, and N. P. Ong, “Evidence for the chiral anomaly in the dirac semimetal na3bi,” Science 350(6259), 413–416 (2015).
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L. Thylen, P. Holmström, L. Wosinski, B. Jaskorzynska, M. Naruse, T. Kawazoe, M. Ohtsu, M. Yan, M. Fiorentino, and U. Westergren, “Chapter 6 - nanophotonics for low-power switches,” in Optical Fiber Telecommunications (Sixth Edition), I. P. Kaminow, T. Li, and A. E. Willner, eds. (Academic, Boston, 2013), Optics and Photonics, pp. 205–241, 6th edition ed.

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Figures (6)

Fig. 1.
Fig. 1. Schematic illustration of linear and nonlinear optical processes in $(3+1)$ massless Dirac fermions. For simplicity, we choose $k_x=k_y=0$ plane. The first-order current is denoted by $J_1(\omega )$. The third-order terms are $J_3(\omega )$ and $J_3(3\omega )$. $J_3(\omega )$ corresponds to the Kerr process where two incoming photons are absorbed, followed by the immediate emission of a third photon. $J_3(3\omega )$ term corresponds to the simultaneous absorption of three photons that leads to high-harmonic generation.
Fig. 2.
Fig. 2. Comparison between linear interband and intraband optical conductivity at $T=300K$ and $\mu =0.05eV$. Both the real and the imaginary parts of $\chi ^{(1)}(\omega )$ along the $x$-and $z$-directions have the same frequency dependence but with different magnitude. The $\chi ^{(1)}(\omega )$ along the $x$-direction is about 15 times large than that along the $z$-direction. (a) and (b) show the real and imaginary part of $\sigma ^{(1)}(\omega )$, respectively.
Fig. 3.
Fig. 3. Third-order nonlinear susceptibility at $T = 300$ K. (a) and (c) show the real part value of $\chi ^{(3)}(\omega )$ for Kerr and HHG process, respectively. (b) and (d) shows the imaginary part corresponding to (a) and (c), respectively.
Fig. 4.
Fig. 4. The crossover effect from negative-valued to positive-valued $\chi ^{(3)}(\omega )$. (a) The red circles highlight the critical frequency at which the crossover occurs. (b) shows the $\mu$-dependence of the crossover frequency at $T=300$ K.
Fig. 5.
Fig. 5. The nonlinear refractive index along the $z$-direction at 300K. The Three type lines represent different chemical potentials condition. (a) and (b) shows the $n_2$ and $k_2$ of the HHG effect, respectively. (c) and (d) shows the $n_2$ and $k_2$ of the optical Kerr effect, respectively.
Fig. 6.
Fig. 6. The nonlinear refractive index along the $x$-direction at 300K. The Three type lines represent different chemical potentials condition. (a) and (b) shows the $n_2$ and $k_2$ of the HHG effect, respectively. (c) and (d) shows the $n_2$ and $k_2$ of the optical Kerr effect, respectively.

Equations (18)

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ε k = v x 2 k x 2 + v y 2 k y 2 + v z 2 k z 2 ,
ξ k = ( cos ( θ / 2 ) e i ϕ sin ( θ / 2 ) , )
H k ( t ) = ( P z P + A e i ω t P + + A e i ω t P z ) ,
ψ k ( t ) = n = ( α n β n ) e i n ω t e i ε k t ,
( ε k P z n ω ~ ) α n = P β n + A β n 1 ( ε k + P z n ω ~ ) β n = P + α n + A α n 1 .
J ( t ) = g e ( 2 π ) 3 N β μ ( ε k ) v ^ d 3 k = g e ( 2 π ) 3 1 v x v y v z N ( P ) v ^ d 3 P ,
J i ( t ) = g e ( 2 π ) 3 v i v j v k Λ d ϵ N β μ ( ϵ ) j Λ ( ϵ ) ϵ 2 ,
j Λ ( ϵ ) = sin ( θ ) d θ d ϕ n , m n + m = Λ ( α n β m + β n α m ) e i ( n m ) ω t ,
α n = C n { ε k e i ϕ α n 1 + [ ε k cos ( θ 2 ) ( n 1 ) ω ~ ] β n 1 } β n = C n { ε k e i ϕ β n 1 [ ε k cos ( θ 2 ) + ( n 1 ) ω ~ ] α n 1 } ,
σ i n t e r ( 1 ) = g e 2 24 π v i v j v k ω N β μ ( ω ~ / 2 ) ,
σ I ( ω ) = σ 0 v i v j v k 0 ϵ C N β μ ( ϵ ) N β μ ( ω ~ / 2 ) ω ~ 2 4 ϵ 2 ϵ d ϵ
σ i n t e r ( 3 ) ( ω ) = g e 4 v i 3 8 π 3 ω 3 16 15 N β μ ( ω ~ ) σ i n t e r ( 3 ) ( 3 ω ) = g e 4 v i 3 8 π 3 ω 3 [ 16 45 N β μ ( ω ~ ) 4 45 N β μ ( ω ~ 2 ) 3 5 N β μ ( 3 ω ~ 2 ) ] ,
ω N σ I ( n ) ( ω ) = 1 π ω N σ R ( n ) ( ω ) ω ω d ω
σ i n t r a ( 1 ) ( ω ) = g e 2 v i 6 π 2 3 v j v k τ 1 i ω τ [ μ 2 + π 2 3 ( k B T ) 2 ] , σ i n t r a ( 3 ) ( ω ) = g e 4 v i 3 5 π 2 3 v j v k τ ( 1 + ω 2 τ 2 ) ( 1 2 i ω τ ) n ¯ F , σ i n t r a ( 3 ) ( 3 ω ) = 3 g e 4 v i 3 5 π 2 3 v j v k τ ( 1 i ω τ ) ( 1 2 i ω τ ) ( 1 3 i ω τ ) n ¯ F ,
χ ( 3 ) ( ω ) = i σ ( 3 ) ( ω ) ϵ 0 ω  ,  χ ( 3 ) ( 3 ω ) = i σ ( 3 ) ( 3 ω ) 3 ϵ 0 ω .
n + i k = 1 + χ ( 1 ) ,
n 2 = 3 4 ϵ 0 c ( n 2 + k 2 ) [ χ R ( 3 ) + k n χ I ( 3 ) ] ,
k 2 = 3 4 ϵ 0 c ( n 2 + k 2 ) [ χ I ( 3 ) k n χ R ( 3 ) ] .

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