Abstract

We demonstrate that single layer graphene exhibits a strong nonlinear photon-mixing effect in the terahertz frequency regime. Up to room temperature, the third-order nonlinear current in graphene grows rapidly with increasing temperature. The third-order nonlinear current can be as strong as the linear current under a moderate electric field strength of 104V/cm. Because of the unique Dirac behavior of the graphene quasi-particles, low Fermi level and electron fillings optimizes the optical nonlinearity. Under a strong-field condition, the strong-field-induced Dirac fermion population redistribution and nonequilibrium carrier heating effects further amplify the optical nonlinearity of graphene. Our results suggest that doped graphene can potentially be utilized as a strong terahertz photon mixer in the room-temperature regime.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
    [CrossRef]
  2. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
    [CrossRef]
  3. Y. Zhang, Y. Tan, H. L. Sormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
    [CrossRef]
  4. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
    [CrossRef]
  5. T. Ando, T. Nakanishi, and R. Saito, “Berry’s phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67, 2857–2862 (1998).
    [CrossRef]
  6. A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100, 117401 (2008).
    [CrossRef]
  7. K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
    [CrossRef]
  8. A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, “Integer quantum Hall transition: an alternative approach and exact results,” Phys. Rev. B 50, 7526 (1994).
    [CrossRef]
  9. C. Zhang, L. Chen, and Z. Ma, “Orientation dependence of the optical spectra in graphene at high frequencies,” Phys. Rev. B 77, 241402(R)(2008).
    [CrossRef]
  10. Y. Zheng and T. Ando, “Hall conductivity of a two-dimensional graphite system,” Phys. Rev. B 65, 245420 (2002).
    [CrossRef]
  11. V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95, 146801 (2005).
    [CrossRef]
  12. K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
    [CrossRef]
  13. S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, “Strong suppression of weak localization in graphene,” Phys. Rev. Lett. 97, 016801 (2006).
    [CrossRef]
  14. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
    [CrossRef]
  15. 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, 072101 (2009).
    [CrossRef]
  16. S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79, 27002 (2007).
    [CrossRef]
  17. S. A. Mikhailov, and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys. Condens. Matter 20, 384204 (2008).
    [CrossRef]
  18. K. L. Ishikawa, “Nonlinear optical response of graphene in time domain,” Phys. Rev. B 82, 201402(R) (2010).
    [CrossRef]
  19. Y. S. Ang, S. Sultan, and C. Zhang, “Nonlinear optical spectrum of bilayer graphene in the terahertz regime,” Appl. Phys. Lett. 97, 243110 (2010).
    [CrossRef]
  20. Y. S. Ang and C. Zhang, “Subgap optical conductivity in semihydrogenated graphene,” Appl. Phys. Lett. 98, 042107(2011).
    [CrossRef]
  21. E. Hendry, P. J. Hale, J. Moger, and A. K. Savchenko, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010).
    [CrossRef]
  22. J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
    [CrossRef]
  23. M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
    [CrossRef]
  24. R. P. Feynman, “Forces in Molecules,” Phys. Rev. 56, 340–343 (1939).
    [CrossRef]
  25. C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
    [CrossRef]
  26. F. Gao, G. Wang, and C. Zhang, “Strong photon-mixing of terahertz waves in semiconductor quantum wells induced by Rashba spin-orbit coupling,” Nanotechnology 19, 465401 (2008).
    [CrossRef]
  27. P. A. Wolff and G. A. Pearson, “Theory of optical mixing by mobile carriers in semiconductors,” Phys. Rev. Lett. 17, 1015–1017 (1966).
    [CrossRef]
  28. H. M. Dong, W. Xu, and R. B. Tan, “Temperature relaxation and energy loss of hot carriers in graphene,” Solid State Commun. 150, 1770–1773 (2010).
    [CrossRef]
  29. D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. Heer, P. N. First, and T. B. Norris, “Ultrafast relaxation of excited Dirac fermions in epitaxial graphene using optical differential transmission spectroscopy,” Phys. Rev. Lett. 101, 157402(2008).
    [CrossRef]
  30. S. Butscher, F. Milde, M. Hirtschulz, E. Malic, and A. Knorr, “Hot electron relaxation and phonon dynamics in graphene,” Appl. Phys. Lett. 91, 203103 (2007).
    [CrossRef]
  31. W. S. Bao, S. Y. Liu, and X. L. Lei, “Hot-electron transport in graphene driven by intense terahertz fields,” Phys. Lett. A 374, 1266–1269 (2010).
    [CrossRef]
  32. G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
    [CrossRef]

2011 (3)

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

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
[CrossRef]

2010 (6)

W. S. Bao, S. Y. Liu, and X. L. Lei, “Hot-electron transport in graphene driven by intense terahertz fields,” Phys. Lett. A 374, 1266–1269 (2010).
[CrossRef]

H. M. Dong, W. Xu, and R. B. Tan, “Temperature relaxation and energy loss of hot carriers in graphene,” Solid State Commun. 150, 1770–1773 (2010).
[CrossRef]

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

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
[CrossRef]

K. L. Ishikawa, “Nonlinear optical response of graphene in time domain,” Phys. Rev. B 82, 201402(R) (2010).
[CrossRef]

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

2009 (2)

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, 072101 (2009).
[CrossRef]

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

2008 (7)

D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. Heer, P. N. First, and T. B. Norris, “Ultrafast relaxation of excited Dirac fermions in epitaxial graphene using optical differential transmission spectroscopy,” Phys. Rev. Lett. 101, 157402(2008).
[CrossRef]

F. Gao, G. Wang, and C. Zhang, “Strong photon-mixing of terahertz waves in semiconductor quantum wells induced by Rashba spin-orbit coupling,” Nanotechnology 19, 465401 (2008).
[CrossRef]

S. A. Mikhailov, and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys. Condens. Matter 20, 384204 (2008).
[CrossRef]

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[CrossRef]

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef]

C. Zhang, L. Chen, and Z. Ma, “Orientation dependence of the optical spectra in graphene at high frequencies,” Phys. Rev. B 77, 241402(R)(2008).
[CrossRef]

2007 (4)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[CrossRef]

S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79, 27002 (2007).
[CrossRef]

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

S. Butscher, F. Milde, M. Hirtschulz, E. Malic, and A. Knorr, “Hot electron relaxation and phonon dynamics in graphene,” Appl. Phys. Lett. 91, 203103 (2007).
[CrossRef]

2006 (1)

S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, “Strong suppression of weak localization in graphene,” Phys. Rev. Lett. 97, 016801 (2006).
[CrossRef]

2005 (3)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[CrossRef]

V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95, 146801 (2005).
[CrossRef]

Y. Zhang, Y. Tan, H. L. Sormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[CrossRef]

2004 (1)

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

2002 (1)

Y. Zheng and T. Ando, “Hall conductivity of a two-dimensional graphite system,” Phys. Rev. B 65, 245420 (2002).
[CrossRef]

1998 (1)

T. Ando, T. Nakanishi, and R. Saito, “Berry’s phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67, 2857–2862 (1998).
[CrossRef]

1994 (1)

A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, “Integer quantum Hall transition: an alternative approach and exact results,” Phys. Rev. B 50, 7526 (1994).
[CrossRef]

1966 (1)

P. A. Wolff and G. A. Pearson, “Theory of optical mixing by mobile carriers in semiconductors,” Phys. Rev. Lett. 17, 1015–1017 (1966).
[CrossRef]

1939 (1)

R. P. Feynman, “Forces in Molecules,” Phys. Rev. 56, 340–343 (1939).
[CrossRef]

Ando, T.

Y. Zheng and T. Ando, “Hall conductivity of a two-dimensional graphite system,” Phys. Rev. B 65, 245420 (2002).
[CrossRef]

T. Ando, T. Nakanishi, and R. Saito, “Berry’s phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67, 2857–2862 (1998).
[CrossRef]

Ang, Y. S.

Y. S. Ang and C. Zhang, “Subgap optical conductivity in semihydrogenated graphene,” Appl. Phys. Lett. 98, 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, 243110 (2010).
[CrossRef]

Bao, W. S.

W. S. Bao, S. Y. Liu, and X. L. Lei, “Hot-electron transport in graphene driven by intense terahertz fields,” Phys. Lett. A 374, 1266–1269 (2010).
[CrossRef]

Berger, C.

D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. Heer, P. N. First, and T. B. Norris, “Ultrafast relaxation of excited Dirac fermions in epitaxial graphene using optical differential transmission spectroscopy,” Phys. Rev. Lett. 101, 157402(2008).
[CrossRef]

Blau, W. J.

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Boebinger, G. S.

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

Bolotin, K. I.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[CrossRef]

Boudouris, B. W.

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Butscher, S.

S. Butscher, F. Milde, M. Hirtschulz, E. Malic, and A. Knorr, “Hot electron relaxation and phonon dynamics in graphene,” Appl. Phys. Lett. 91, 203103 (2007).
[CrossRef]

Cao, J. C.

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, 072101 (2009).
[CrossRef]

Carbone, F.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

Chen, C.

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Chen, L.

C. Zhang, L. Chen, and Z. Ma, “Orientation dependence of the optical spectra in graphene at high frequencies,” Phys. Rev. B 77, 241402(R)(2008).
[CrossRef]

Chen, Z.-L.

G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
[CrossRef]

Chua, L.-L.

G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
[CrossRef]

Cismaru, A.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
[CrossRef]

Clark, J.

G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
[CrossRef]

Coleman, J. N.

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Crommie, M. F.

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Deligeorgis, G.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
[CrossRef]

Divin, C.

D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. Heer, P. N. First, and T. B. Norris, “Ultrafast relaxation of excited Dirac fermions in epitaxial graphene using optical differential transmission spectroscopy,” Phys. Rev. Lett. 101, 157402(2008).
[CrossRef]

Dong, H. M.

H. M. Dong, W. Xu, and R. B. Tan, “Temperature relaxation and energy loss of hot carriers in graphene,” Solid State Commun. 150, 1770–1773 (2010).
[CrossRef]

Dragoman, D.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
[CrossRef]

Dragoman, M.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
[CrossRef]

Dubonos, I. V.

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

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[CrossRef]

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

Feynman, R. P.

R. P. Feynman, “Forces in Molecules,” Phys. Rev. 56, 340–343 (1939).
[CrossRef]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[CrossRef]

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

First, P. N.

D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. Heer, P. N. First, and T. B. Norris, “Ultrafast relaxation of excited Dirac fermions in epitaxial graphene using optical differential transmission spectroscopy,” Phys. Rev. Lett. 101, 157402(2008).
[CrossRef]

Fisher, M. P. A.

A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, “Integer quantum Hall transition: an alternative approach and exact results,” Phys. Rev. B 50, 7526 (1994).
[CrossRef]

Friend, R. H.

G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
[CrossRef]

Fudenberg, G.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[CrossRef]

Gao, F.

F. Gao, G. Wang, and C. Zhang, “Strong photon-mixing of terahertz waves in semiconductor quantum wells induced by Rashba spin-orbit coupling,” Nanotechnology 19, 465401 (2008).
[CrossRef]

Geim, A. K.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[CrossRef]

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, “Strong suppression of weak localization in graphene,” Phys. Rev. Lett. 97, 016801 (2006).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[CrossRef]

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

Geng, B.

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Girit, C.

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Goh, R. G. S.

G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
[CrossRef]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[CrossRef]

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

Grinstein, G.

A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, “Integer quantum Hall transition: an alternative approach and exact results,” Phys. Rev. B 50, 7526 (1994).
[CrossRef]

Gusynin, V. P.

V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95, 146801 (2005).
[CrossRef]

Hale, P. J.

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

Heer, W. A.

D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. Heer, P. N. First, and T. B. Norris, “Ultrafast relaxation of excited Dirac fermions in epitaxial graphene using optical differential transmission spectroscopy,” Phys. Rev. Lett. 101, 157402(2008).
[CrossRef]

Heinz, T. F.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef]

Hendry, E.

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

Hernandez, Y.

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Hirtschulz, M.

S. Butscher, F. Milde, M. Hirtschulz, E. Malic, and A. Knorr, “Hot electron relaxation and phonon dynamics in graphene,” Appl. Phys. Lett. 91, 203103 (2007).
[CrossRef]

Ho, P. K. H.

G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
[CrossRef]

Hone, J.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[CrossRef]

Horng, J.

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Ishikawa, K. L.

K. L. Ishikawa, “Nonlinear optical response of graphene in time domain,” Phys. Rev. B 82, 201402(R) (2010).
[CrossRef]

Jiang, D.

S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, “Strong suppression of weak localization in graphene,” Phys. Rev. Lett. 97, 016801 (2006).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[CrossRef]

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

Jiang, Z.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[CrossRef]

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

Katsnelson, M. I.

S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, “Strong suppression of weak localization in graphene,” Phys. Rev. Lett. 97, 016801 (2006).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[CrossRef]

Kim, P.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[CrossRef]

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

Y. Zhang, Y. Tan, H. L. Sormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[CrossRef]

Klima, M.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[CrossRef]

Knorr, A.

S. Butscher, F. Milde, M. Hirtschulz, E. Malic, and A. Knorr, “Hot electron relaxation and phonon dynamics in graphene,” Appl. Phys. Lett. 91, 203103 (2007).
[CrossRef]

Konstantinidis, G.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
[CrossRef]

Kuzmenko, A. B.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

Lei, X. L.

W. S. Bao, S. Y. Liu, and X. L. Lei, “Hot-electron transport in graphene driven by intense terahertz fields,” Phys. Lett. A 374, 1266–1269 (2010).
[CrossRef]

Li, X.

D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. Heer, P. N. First, and T. B. Norris, “Ultrafast relaxation of excited Dirac fermions in epitaxial graphene using optical differential transmission spectroscopy,” Phys. Rev. Lett. 101, 157402(2008).
[CrossRef]

Lim, G.-K.

G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
[CrossRef]

Liu, S. Y.

W. S. Bao, S. Y. Liu, and X. L. Lei, “Hot-electron transport in graphene driven by intense terahertz fields,” Phys. Lett. A 374, 1266–1269 (2010).
[CrossRef]

Lotya, M.

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Louie, S. G.

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Ludwig, A. W. W.

A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, “Integer quantum Hall transition: an alternative approach and exact results,” Phys. Rev. B 50, 7526 (1994).
[CrossRef]

Lui, C. H.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef]

Ma, Z.

C. Zhang, L. Chen, and Z. Ma, “Orientation dependence of the optical spectra in graphene at high frequencies,” Phys. Rev. B 77, 241402(R)(2008).
[CrossRef]

Maan, J. C.

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

Mak, K. F.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef]

Malic, E.

S. Butscher, F. Milde, M. Hirtschulz, E. Malic, and A. Knorr, “Hot electron relaxation and phonon dynamics in graphene,” Appl. Phys. Lett. 91, 203103 (2007).
[CrossRef]

Mikhailov, S. A.

S. A. Mikhailov, and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys. Condens. Matter 20, 384204 (2008).
[CrossRef]

S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79, 27002 (2007).
[CrossRef]

Milde, F.

S. Butscher, F. Milde, M. Hirtschulz, E. Malic, and A. Knorr, “Hot electron relaxation and phonon dynamics in graphene,” Appl. Phys. Lett. 91, 203103 (2007).
[CrossRef]

Misewich, J. A.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef]

Moger, J.

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

Morozov, S. V.

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, “Strong suppression of weak localization in graphene,” Phys. Rev. Lett. 97, 016801 (2006).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[CrossRef]

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

Muller, A. A.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
[CrossRef]

Nakanishi, T.

T. Ando, T. Nakanishi, and R. Saito, “Berry’s phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67, 2857–2862 (1998).
[CrossRef]

Neculoiu, D.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
[CrossRef]

Ng, W.-H.

G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
[CrossRef]

Norris, T. B.

D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. Heer, P. N. First, and T. B. Norris, “Ultrafast relaxation of excited Dirac fermions in epitaxial graphene using optical differential transmission spectroscopy,” Phys. Rev. Lett. 101, 157402(2008).
[CrossRef]

Novoselov, K. S.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[CrossRef]

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, “Strong suppression of weak localization in graphene,” Phys. Rev. Lett. 97, 016801 (2006).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[CrossRef]

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

Park, C.

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Pearson, G. A.

P. A. Wolff and G. A. Pearson, “Theory of optical mixing by mobile carriers in semiconductors,” Phys. Rev. Lett. 17, 1015–1017 (1966).
[CrossRef]

Plana, R.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
[CrossRef]

Ponomarenko, L. A.

S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, “Strong suppression of weak localization in graphene,” Phys. Rev. Lett. 97, 016801 (2006).
[CrossRef]

Saito, R.

T. Ando, T. Nakanishi, and R. Saito, “Berry’s phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67, 2857–2862 (1998).
[CrossRef]

Savchenko, A. K.

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

Schedin, F.

S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, “Strong suppression of weak localization in graphene,” Phys. Rev. Lett. 97, 016801 (2006).
[CrossRef]

Segalman, R. A.

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Sfeir, M. Y.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef]

Shankar, R.

A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, “Integer quantum Hall transition: an alternative approach and exact results,” Phys. Rev. B 50, 7526 (1994).
[CrossRef]

Sharapov, S. G.

V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95, 146801 (2005).
[CrossRef]

Sikes, K. J.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[CrossRef]

Sormer, H. L.

Y. Zhang, Y. Tan, H. L. Sormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[CrossRef]

Stormer, H. L.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[CrossRef]

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

Sultan, S.

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

Sun, D.

D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. Heer, P. N. First, and T. B. Norris, “Ultrafast relaxation of excited Dirac fermions in epitaxial graphene using optical differential transmission spectroscopy,” Phys. Rev. Lett. 101, 157402(2008).
[CrossRef]

Tan, H.-W.

G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
[CrossRef]

Tan, R. B.

H. M. Dong, W. Xu, and R. B. Tan, “Temperature relaxation and energy loss of hot carriers in graphene,” Solid State Commun. 150, 1770–1773 (2010).
[CrossRef]

Tan, Y.

Y. Zhang, Y. Tan, H. L. Sormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[CrossRef]

van der Marel, D.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

van Heumen, E.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

Wang, F.

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Wang, G.

F. Gao, G. Wang, and C. Zhang, “Strong photon-mixing of terahertz waves in semiconductor quantum wells induced by Rashba spin-orbit coupling,” Nanotechnology 19, 465401 (2008).
[CrossRef]

Wang, J.

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Wolff, P. A.

P. A. Wolff and G. A. Pearson, “Theory of optical mixing by mobile carriers in semiconductors,” Phys. Rev. Lett. 17, 1015–1017 (1966).
[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, 072101 (2009).
[CrossRef]

Wu, Y.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef]

Wu, Z.-K.

D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. Heer, P. N. First, and T. B. Norris, “Ultrafast relaxation of excited Dirac fermions in epitaxial graphene using optical differential transmission spectroscopy,” Phys. Rev. Lett. 101, 157402(2008).
[CrossRef]

Xu, W.

H. M. Dong, W. Xu, and R. B. Tan, “Temperature relaxation and energy loss of hot carriers in graphene,” Solid State Commun. 150, 1770–1773 (2010).
[CrossRef]

Xu, X. G.

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, 072101 (2009).
[CrossRef]

Zeitler, U.

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

Zettl, A.

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Zhang, C.

Y. S. Ang and C. Zhang, “Subgap optical conductivity in semihydrogenated graphene,” Appl. Phys. Lett. 98, 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, 243110 (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, 072101 (2009).
[CrossRef]

F. Gao, G. Wang, and C. Zhang, “Strong photon-mixing of terahertz waves in semiconductor quantum wells induced by Rashba spin-orbit coupling,” Nanotechnology 19, 465401 (2008).
[CrossRef]

C. Zhang, L. Chen, and Z. Ma, “Orientation dependence of the optical spectra in graphene at high frequencies,” Phys. Rev. B 77, 241402(R)(2008).
[CrossRef]

Zhang, Y.

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

Y. Zhang, Y. Tan, H. L. Sormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[CrossRef]

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

Zheng, Y.

Y. Zheng and T. Ando, “Hall conductivity of a two-dimensional graphite system,” Phys. Rev. B 65, 245420 (2002).
[CrossRef]

Ziegler, K.

S. A. Mikhailov, and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys. Condens. Matter 20, 384204 (2008).
[CrossRef]

Adv. Mater. (1)

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Appl. Phys. Lett. (5)

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
[CrossRef]

S. Butscher, F. Milde, M. Hirtschulz, E. Malic, and A. Knorr, “Hot electron relaxation and phonon dynamics in graphene,” Appl. Phys. Lett. 91, 203103 (2007).
[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, 072101 (2009).
[CrossRef]

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

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

Europhys. Lett. (1)

S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79, 27002 (2007).
[CrossRef]

J. Phys. Condens. Matter (1)

S. A. Mikhailov, and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys. Condens. Matter 20, 384204 (2008).
[CrossRef]

J. Phys. Soc. Jpn. (1)

T. Ando, T. Nakanishi, and R. Saito, “Berry’s phase and absence of back scattering in carbon nanotubes,” J. Phys. Soc. Jpn. 67, 2857–2862 (1998).
[CrossRef]

Nanotechnology (1)

F. Gao, G. Wang, and C. Zhang, “Strong photon-mixing of terahertz waves in semiconductor quantum wells induced by Rashba spin-orbit coupling,” Nanotechnology 19, 465401 (2008).
[CrossRef]

Nat. Mater. (1)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[CrossRef]

Nat. Photon. (1)

G.-K. Lim, Z.-L. Chen, J. Clark, R. G. S. Goh, W.-H. Ng, H.-W. Tan, R. H. Friend, P. K. H. Ho, and L.-L. Chua, “Giant broadband nonlinear optical absorption response in dispersed graphene single sheets,” Nat. Photon. 5, 554–560 (2011).
[CrossRef]

Nature (3)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[CrossRef]

C. Chen, C. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471, 617–620 (2011).
[CrossRef]

Y. Zhang, Y. Tan, H. L. Sormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[CrossRef]

Phys. Lett. A (1)

W. S. Bao, S. Y. Liu, and X. L. Lei, “Hot-electron transport in graphene driven by intense terahertz fields,” Phys. Lett. A 374, 1266–1269 (2010).
[CrossRef]

Phys. Rev. (1)

R. P. Feynman, “Forces in Molecules,” Phys. Rev. 56, 340–343 (1939).
[CrossRef]

Phys. Rev. B (4)

A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, “Integer quantum Hall transition: an alternative approach and exact results,” Phys. Rev. B 50, 7526 (1994).
[CrossRef]

C. Zhang, L. Chen, and Z. Ma, “Orientation dependence of the optical spectra in graphene at high frequencies,” Phys. Rev. B 77, 241402(R)(2008).
[CrossRef]

Y. Zheng and T. Ando, “Hall conductivity of a two-dimensional graphite system,” Phys. Rev. B 65, 245420 (2002).
[CrossRef]

K. L. Ishikawa, “Nonlinear optical response of graphene in time domain,” Phys. Rev. B 82, 201402(R) (2010).
[CrossRef]

Phys. Rev. Lett. (7)

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

S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, “Strong suppression of weak localization in graphene,” Phys. Rev. Lett. 97, 016801 (2006).
[CrossRef]

V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95, 146801 (2005).
[CrossRef]

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[CrossRef]

P. A. Wolff and G. A. Pearson, “Theory of optical mixing by mobile carriers in semiconductors,” Phys. Rev. Lett. 17, 1015–1017 (1966).
[CrossRef]

D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. Heer, P. N. First, and T. B. Norris, “Ultrafast relaxation of excited Dirac fermions in epitaxial graphene using optical differential transmission spectroscopy,” Phys. Rev. Lett. 101, 157402(2008).
[CrossRef]

Science (2)

K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-temperature quantum Hall effect in graphene,” Science 315, 1379 (2007).
[CrossRef]

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

Solid State Commun. (2)

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008).
[CrossRef]

H. M. Dong, W. Xu, and R. B. Tan, “Temperature relaxation and energy loss of hot carriers in graphene,” Solid State Commun. 150, 1770–1773 (2010).
[CrossRef]

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 (5)

Fig. 1.
Fig. 1.

Temperature dependence of third-order nonlinear current density for μ < 0 at ω = 1 THz .

Fig. 2.
Fig. 2.

Temperature dependence of third-order nonlinear current density for μ > 0 at ω = 1 THz .

Fig. 3.
Fig. 3.

Temperature dependence of β at ω = 1 THz . β exhibits contrasting behavior at the low- and high-temperature regimes.

Fig. 4.
Fig. 4.

Critical field of E c ( S ) at ω = 1 THz and μ = 0.1 eV . Weak-field critical field E c is also shown.

Fig. 5.
Fig. 5.

Temperature dependence of strong-field third-order nonlinear current density at ω = 1 THz . Note that T = T lattice if nonequilibrium heating is ignored and T = T hot if nonequilibrium heating is considered. Since T hot > T lattice , the nonlinear optical response is significantly stronger if carrier heating is considered.

Equations (21)

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

H ^ = v F [ 0 p p + 0 ] ,
E ( r , t ) = μ E μ exp { i ( q μ · r ω μ t ) } ,
v s ( 0 ) = s v F ( p p ) ,
v s ( 1 ) = s v F [ u p p p ( p · u p 2 ) ] ,
v s ( 2 ) = s v F [ p 2 p ( u p ) 2 + u p ( p · u p 2 ) + 3 p 2 p ( p · u p 2 ) ] ,
v s ( 3 ) = s v F [ 1 2 u p ( u p ) 2 + 3 2 p p ( u p ) 2 ( p · u p 2 ) + 3 2 u p ( p · u p 2 ) 2 5 2 p p ( p · u p 2 ) 3 ] ,
J ( i ) = e s 0 2 π μ ε ph k B T Λ d 2 p v s ( i ) f ( ε s ) ,
J T = 0 ( 1 ) = i e 2 π μ E μ exp { i ( q μ · r ω μ t ) } .
J T ( 1 ) = i e 2 π k B T ω ln [ 1 + exp ( 1 + ω k B T ) ] × μ E μ exp { i ( q μ · r ω μ t ) } ,
J T = 0 ( 3 ) = i s e 4 v F 2 8 π 2 μ μ ν ξ ( ε μ ν ξ μ ε μ ν ξ ) E μ · E ν E ξ ω μ ω ν ω ξ × exp { i [ ( q μ + q ν + q ξ ) · r ( ω μ + ω ν + ω ξ ) t ] } ,
J T ( 3 ) = i s e 4 v F 2 8 π 2 μ ν ξ E μ · E υ E ξ ω μ ω ν ω ξ d ε p ε p 2 1 1 + exp ( ε p μ k B T ) × exp { i [ ( q μ + q ν + q ξ ) · r ( ω μ + ω ν + ω ξ ) t ] } ,
E c ( ω , T = 0 K ) = 2 ω v F [ 2 μ e 2 ( μ 3 ω ) ] 1 / 2 ,
β = { k B T ω ln [ 1 + exp ( ω k B T + 1 ) ] | J T > 0 ( 3 ) | / | J T = 0 ( 3 ) | } 1 / 2 ,
χ ( 3 ) = e 4 v F 2 8 π 2 μ ( ω 3 μ ω 3 ) 1 ω 2 ω 3 ( 1 ω 1 ε 0 ) 2 .
J ( 1 S ) = J ( 1 w ) ,
J ( 2 S ) = 0 ,
J ( 3 S ) = J ( 3 w ) + J ( 3 ) ,
J ( 3 ) = J T = 0 ( 3 ) μ k B T d p p exp ( ε 0 μ k B T ) ( exp ( ε 0 μ k B T ) + 1 ) 2 ,
J T = 0 ( 3 ) = i s μ ν ξ E μ · E υ E ξ ω μ ω ν ω ξ e 4 v F 2 8 π 2 μ × exp { i [ ( q μ + q ν + q ξ ) · r ( ω μ + ω ν + ω ξ ) t ] } .
E c ( S ) ( T = 0 ) = 2 ω v F [ 2 ω e 2 ( μ 3 ω ) ] .
β ( S ) ( T ) = { k B T ω ln [ 1 + exp ( ω k B T + 1 ) ] | J T > 0 ( 3 S ) | / | J T = 0 ( 3 S ) | } 1 / 2 .

Metrics