## The Physics of ultrafast saturable absorption in graphene

Optics Express, Vol. 18, Issue 5, pp. 4564-4573 (2010)

http://dx.doi.org/10.1364/OE.18.004564

Acrobat PDF (425 KB)

### Abstract

The ultrafast saturable absorption in graphene is experimentally and theoretically investigated in the femtosecond (fs) time regime. This phenomenon is well-modeled with valence band depletion, conduction band filling and ultrafast intraband carrier thermalization. The latter is dominated by intraband carrier-carrier scattering with a scattering time of 8 ( ± 3) fs, which is far beyond the time resolution of other ultrafast techniques with hundred fs laser pulses. Our results strongly suggest that graphene is an excellent atomic layer saturable absorber.

© 2010 OSA

## 1. Introduction

1. P. R. Wallace, “The band theory of graphite,” Phys. Rev. **71**(9), 622–634 (1947). [CrossRef]

4. P. Avouris, Z. Chen, and V. Perebeinos, “Carbon-based electronics,” Nat. Nanotechnol. **2**(10), 605–615 (2007). [CrossRef]

1. P. R. Wallace, “The band theory of graphite,” Phys. Rev. **71**(9), 622–634 (1947). [CrossRef]

19. Z. H. Ni, W. Chen, X. F. Fan, J. L. Kuo, T. Yu, A. T. S. Wee, and Z. X. Shen, “Raman spectroscopy of epitaxial graphene on a SiC substrate,” Phys. Rev. B **77**(11), 115416 (2008). [CrossRef]

1. P. R. Wallace, “The band theory of graphite,” Phys. Rev. **71**(9), 622–634 (1947). [CrossRef]

2. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. **81**(1), 109–162 (2009). [CrossRef]

*π*·

*α*= 2.3% (where

*α*is the fine structure constant) [5

5. T. Ando, Y. S. Zheng, and H. Suzuura, “Dynamical Conductivity and Zero-Mode Anomaly in Honeycomb Lattices,” J. Phys. Soc. Jpn. **71**(5), 1318–1324 (2002). [CrossRef]

7. T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B **78**(8), 085432 (2008). [CrossRef]

8. A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. **100**(11), 117401 (2008). [CrossRef] [PubMed]

11. 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**(19), 196405 (2008). [CrossRef] [PubMed]

12. J. M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M. G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. **92**(4), 042116 (2008). [CrossRef]

17. Z. Liu, Y. Wang, X. Zhang, Y. Xu, Y. Chen, and J. Tian, “Nonlinear optical properties of graphene oxide in nanosecond and picosecond regimes,” Appl. Phys. Lett. **94**(2), 021902 (2009). [CrossRef]

## 2. Graphene characterization

18. A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K. S. Novoselov, S. Roth, and A. K. Geim, “Raman spectrum of graphene and graphene layers,” Phys. Rev. Lett. **97**(18), 187401 (2006). [CrossRef] [PubMed]

21. D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hierold, and L. Wirtz, “Spatially resolved Raman spectroscopy of single- and few-layer graphene,” Nano Lett. **7**(2), 238–242 (2007). [CrossRef] [PubMed]

22. J. Hass, F. Varchon, J. E. Millán-Otoya, M. Sprinkle, N. Sharma, W. A. de Heer, C. Berger, P. N. First, L. Magaud, and E. H. Conrad, “Why multilayer graphene on 4H-SiC(0001[over ]) behaves like a single sheet of graphene,” Phys. Rev. Lett. **100**(12), 125504 (2008). [CrossRef] [PubMed]

^{−1}and the two-phonon 2D peak at ~2717 cm

^{−1}are clearly identifiable. The smaller peaks at ~1520 cm

^{−1}and ~1713 cm

^{−1}originate from the 4H-SiC substrate. The former is the overtone of the SiC TO(X) phonon at 761 cm

^{−1}while the latter is a combination of the optical phonons with wave vectors near the M point at the zone edge of the SiC substrate [19

19. Z. H. Ni, W. Chen, X. F. Fan, J. L. Kuo, T. Yu, A. T. S. Wee, and Z. X. Shen, “Raman spectroscopy of epitaxial graphene on a SiC substrate,” Phys. Rev. B **77**(11), 115416 (2008). [CrossRef]

20. J. C. Burton, L. Sun, F. H. Long, Z. C. Feng, and I. T. Ferguson, “First- and second-order Raman scattering from semi-insulating 4H-SiC,” Phys. Rev. B **59**(11), 7282–7284 (1999). [CrossRef]

^{−1}) suggests good crystallinity in the graphene samples. The comparable Raman intensity of the G band and the 2D band as well as the broad 2D band (at 100 cm

^{−1}) confirms the primarily two-layer nature of the graphene sample [19

19. Z. H. Ni, W. Chen, X. F. Fan, J. L. Kuo, T. Yu, A. T. S. Wee, and Z. X. Shen, “Raman spectroscopy of epitaxial graphene on a SiC substrate,” Phys. Rev. B **77**(11), 115416 (2008). [CrossRef]

21. D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hierold, and L. Wirtz, “Spatially resolved Raman spectroscopy of single- and few-layer graphene,” Nano Lett. **7**(2), 238–242 (2007). [CrossRef] [PubMed]

## 3. Ultrafast saturable absorption investigation

23. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. **26**(4), 760–769 (1990). [CrossRef]

^{TM}regenerative amplifier that was seeded by a Coherent Mira

^{TM}oscillator. The full-width-at-half-maximum of the photon energy dispersion was 0.04 eV. The laser pulse was focused perpendicular on the sample with a lens of focal length 30 cm. The beam waist at the focus was 31 ± 2 μm and was confirmed with a standard two-photon absorption experiment on a 0.5 mm thick ZnSe bulk crystal. In our experiments, as the graphene sample was translated through the focal point along the laser propagation axis, the transmittances at different

*z*positions were recorded. The effects of the ultrafast saturable absorption and the spatio-temporal reshaping of the transmitted 800nm fs pulses in graphene were analyzed. The response from a bare SiC substrate was also measured under the same conditions and at the same pump intensities (≤ 120 GW/cm

^{2}) where negligible nonlinear absorption has been found. The nonlinear absorption from SiC substrate will be obvious at 200 GW/cm

^{2}, where, graphene still shows saturable absorption.

*ε = ћν*, where

_{F}|k|*ν*= 10

_{F}^{6}

*m/s*is the Fermi velocity) as shown in Fig. 2(a) . For epitaxially grown graphene layers on SiC substrate, the first layer is highly doped and the Fermi level is located at approximately 0.35 eV in conduction band [13

13. D. Sun, Z. K. Wu, C. Divin, X. Li, C. Berger, W. A. de 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**(15), 157402 (2008). [CrossRef] [PubMed]

*ε = ћω/2*) and holes in valence band (

*ε*= -

*ћω*/2) is created with momentum conservation. Here, the effects of triangular warping and other nonlinear effects have been found to be negligible even at visible wavelengths (i.e. up to 3.1 eV) [7

7. T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B **78**(8), 085432 (2008). [CrossRef]

10. R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science **320**(5881), 1308 (2008). [CrossRef] [PubMed]

*ћω*is the photon energy,

*D( ± ћω/2) = ћω/(πћ*are the density of states at

^{2}ν_{F}^{2})*ε = ± ћω/2*and ƒ

_{t}(ε) is the electron occupation probability at time

*t*. The Dirac fermion transition matrix

*M*

_{t}from initial state

*t*is [10

10. R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science **320**(5881), 1308 (2008). [CrossRef] [PubMed]

12. J. M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M. G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. **92**(4), 042116 (2008). [CrossRef]

15. M. Breusing, C. Ropers, and T. Elsaesser, “Ultrafast carrier dynamics in graphite,” Phys. Rev. Lett. **102**(8), 086809 (2009). [CrossRef] [PubMed]

_{t}(-

*ћω*/2) ≈1 and ƒ

_{t}(

*ћω*/2) ≈0. With the incident power

*α*but independent of the excitation wavelength and the material parameter

*ν*.

_{F}12. J. M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M. G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. **92**(4), 042116 (2008). [CrossRef]

15. M. Breusing, C. Ropers, and T. Elsaesser, “Ultrafast carrier dynamics in graphite,” Phys. Rev. Lett. **102**(8), 086809 (2009). [CrossRef] [PubMed]

*I*can be expressed as

_{t}*I*exp(

_{0}*-t*). The dynamics for VB and CB electron occupation probabilities

^{2}/τ^{2}*τ*is half-width at 1/

*e*of the maximum of laser pulse and

*τ*is the ultrafast carrier relaxation time which is dominated by intraband c-c scattering. The first and second terms on the right hand side (RHS) of Eqs. (4) and (5) represent the photon-absorption-induced electron occupation probability balance between the valence band and conduction band. The third term on the RHS of Eq. (4) (or Eq. (5)) is derived from hole (or electron) relaxation increasing (or decreasing) in the valence (or conduction) band respectively. The initial conditions prior to laser excitation are:

_{1}*Erf*[

*x*] is the Gaussian error function.

*w*is the minimum beam waist,

_{0}*λ*is the pump laser wavelength,

*z*is on-axis relative position to the focal point,

*r*is radial coordinate and

*I*

_{00}is the on-axis peak irradiance at the focal point.

*z*, the absorbed energy per pulse is:

*τ*

_{1}as the only fitting parameter. Our fitted value

*τ*

_{1}(7 ± 3 fs) agrees well with the 13±3 fs intraband carrier equilibration time reported by Breusing

*et al*. for 20 – 30 nm thick graphite that was probed using transient absorption spectroscopy with 7 fs laser pulses [15

15. M. Breusing, C. Ropers, and T. Elsaesser, “Ultrafast carrier dynamics in graphite,” Phys. Rev. Lett. **102**(8), 086809 (2009). [CrossRef] [PubMed]

*τ*

_{1}comprises contributions from both the intraband c-c scattering and the dissipative c-op coupling, i.e.

*τ*, where

_{1}= τ_{c-c}τ_{c-op}/(τ_{c-c}+ τ_{c-op})*τ*and

_{c-c}*τ*

_{c-op}refer to the lifetimes for c-c scattering and c-op coupling respectively. It is usual that

*τ*

_{c-op}»

*τ*so that

_{c-c}*τ*

_{1}≈

*τ*. A typical value of

_{c-c}*τ*is ~100 fs [15

_{c-op}**102**(8), 086809 (2009). [CrossRef] [PubMed]

*τ*is calculated to be 8 ± 3 fs. Such ultrafast dynamics in graphene is far beyond the temporal resolution of typical pump-probe experiments conducted with hundred fs pulses [12

_{c-c}**92**(4), 042116 (2008). [CrossRef]

14. R. W. Newson, J. Dean, B. Schmidt, and H. M. van Driel, “Ultrafast carrier kinetics in exfoliated graphene and thin graphite films,” Opt. Express **17**(4), 2326–2333 (2009). [CrossRef] [PubMed]

*T*

_{0}with different chemical potentials (

*μ*

_{e}and

*μ*

_{h}) according to the following conditions of particle number and energy conservation:where

*ћω*= 1.55 eV photons, the dependence of the electron temperature (

*T*

_{e}) and corresponding chemical potential (

*μ*

_{e}) on the electron density (

*N*

_{e}) are shown in Fig. 3 . The thermalized

*T*

_{e}mono decreases while the corresponding

*μ*

_{e}mono increases as

*N*

_{e}increases. For

*N*

_{e}= 1.9 × 10

^{5}/um

^{2}, a thermalized

*T*

_{e}= 4000 K will be obtained. At the threshold electron density,

*N*

_{th}= 1.5 × 10

^{5}/um

^{2}, the thermalized temperature is equal to the Fermi temperature of

*N*

_{th}. This corresponds to

*μ*

_{e}(

*N*

_{th}) = 0. For electron density

*N*

_{e}<

*N*

_{th}(in the limit of low laser pump intensity), the established temperature will be greater than the Fermi temperature of

*N*

_{e}. At this electron density, more of the fermions are in the higher excited states within the CB. This results in

*μ*

_{e}(

*N*

_{e}) < 0. One possibility is that at high temperatures, the fermion gas approaches the classical ideal gas. In addition, the threshold electron density

*N*

_{th}is dependent on the pump photon energy (

*ћω*). With

*μ*

_{e}= 0 and the conditions of Eq. (12), the following relationship can be deduced:

^{5}/um

^{2}. Beyond this upper limit, 100% transmittance for the subsequent incoming photons will be observed.

*t*is derived as:

*τ*

_{1}= 7 fs along with the same parameters as used in our experiments. Similarly, the calculated spatial profiles are shown in Fig. 4(d). For comparison, the input temporal and spatial laser profiles are plotted in Fig. 4(a) and 4(c). In Fig. 4(b) the calculated maximum transmittance occur a few fs after time zero which is taken to be the center of the temporal profile. This asymmetry in the temporal profile arises due to the two competing processes during the accumulation of carriers in the VB and CB: photo absorption of incoming photons at the same energy (within the pulse duration) and the intraband c-c scattering (that leads to the redistribution of the carriers). This causes a delay in reaching the maximum transmittance. The narrowing of the temporal profile of the transmitted pulses occurs when the pump intensity is increased from 1 to 5 GW/cm

^{2}. But this effect becomes less obvious at higher pump intensities (80 GW/cm

^{2}). This is attributed to the saturation of the saturable absorption, which is dependent on the saturation depth of the materials. From Fig. 4(d), the narrowing of the spatial profiles shows a similar trend to that of the temporal profiles. Unlike the temporal case, the transmittances of the spatial profile always show symmetrical narrowing.

*τ*

_{1}= 100 fs and all other parameters remain the same. The narrowing of the temporal profile and the subsequent saturation of the saturable absorption occurs at much lower pump intensities compared to that with

*τ*

_{1}= 7 fs. In light of the competing processes discussed earlier, the temporal transmittance profile is more asymmetrical. The transmittance also has a longer tail, which is caused by the slower intraband carrier relaxation.

*πα*= 2.3%) of graphene provides a potential saturable absorption depth (i.e. maximum change in transmittance). This large linear absorption arises from the unique properties of graphene: i.e. two dimensional massless fermions and a conical band structure. Secondly, the excited states absorption in graphene is momentum forbidden and they require the assistance of phonons. The only photon coupling process for the excited state electrons is through stimulated emission (

*ε*(

*ћω*/2) →

*ε*(-

*ћω*/2)). The nonlinear optical properties of graphene and graphene oxide were previously investigated using picosecond and nanosecond lasers on small pieces of sample dispersed in a solution [16

16. J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband Nonlinear Optical Response of Graphene Dispersions,” Adv. Mater. **21**(23), 2430–2435 (2009). [CrossRef]

17. Z. Liu, Y. Wang, X. Zhang, Y. Xu, Y. Chen, and J. Tian, “Nonlinear optical properties of graphene oxide in nanosecond and picosecond regimes,” Appl. Phys. Lett. **94**(2), 021902 (2009). [CrossRef]

^{2}. However, due to the ultrafast intraband c-c dynamics, a pump intensity as high as 1000 GW/cm

^{2}is needed to reach the maximum transmittance. For the graphene layers epitaxially grown on SiC substrate, the SiC nonlinear absorption signals will dominate in the Z-scans above 200 GW/cm

^{2}at 1.55 eV. The damage threshold of graphene is determined to be higher than 300 GW/cm

^{2}.

*I*

_{s}) is another crucial parameter [24]. The attenuation of the incident light through a single graphene layer can be described by:where

*I*

_{in}is the input light intensity,

*I*

_{T}is the transmitted light intensity,

*α*

_{0}is the linear absorption coefficient of a single layer of graphene,

*I*

_{s}is the saturation intensity of a single layer of graphene. Considering that light sequentially passes through the different graphene layers, the

*z*position dependent transmittance is calculated from the ratio of the energy of the transmitted pulse to that of the incident pulse. These energies are obtained by integrating

*I*

_{T}and

*I*

_{in}(for a Gaussian pulse) over the temporal and spatial domains. Figure 6(a) shows a typical experimental data fitted with

*I*

_{s}as the only unknown parameter. Figure 6(b) shows the obtained saturation intensities vs. pump intensities. The saturation intensity was found to be ~4( ± 1) GW/cm

^{2}for a single layer of graphene and it is independent of the pump intensity.

## 4. Conclusion

*z*-scan measurements in conjunction with theoretical calculations, the scattering time is determined to be 8 ( ± 3) fs. Such ultrafast c-c dynamics is far beyond the time resolution of other ultrafast techniques with typical hundred fs laser pulses. In addition, our results show that the spatiotemporal profiles of an intense Gaussian pulse narrow upon transmission through graphene. This narrowing can be controlled either by tuning the incident light intensity or by varying the number of layers. Our findings reveal that graphene possesses great potential for applications as passive mode-lockers, optical pulse shapers or output couplers. Graphene, with all its fascinating properties, continues to offer us new insights into the field of nonlinear optics and their applications.

## Acknowledgements

## References and links

1. | P. R. Wallace, “The band theory of graphite,” Phys. Rev. |

2. | A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. |

3. | A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. |

4. | P. Avouris, Z. Chen, and V. Perebeinos, “Carbon-based electronics,” Nat. Nanotechnol. |

5. | T. Ando, Y. S. Zheng, and H. Suzuura, “Dynamical Conductivity and Zero-Mode Anomaly in Honeycomb Lattices,” J. Phys. Soc. Jpn. |

6. | V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. |

7. | T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B |

8. | A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. |

9. | F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science |

10. | R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science |

11. | 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. |

12. | J. M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M. G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. |

13. | D. Sun, Z. K. Wu, C. Divin, X. Li, C. Berger, W. A. de 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. |

14. | R. W. Newson, J. Dean, B. Schmidt, and H. M. van Driel, “Ultrafast carrier kinetics in exfoliated graphene and thin graphite films,” Opt. Express |

15. | M. Breusing, C. Ropers, and T. Elsaesser, “Ultrafast carrier dynamics in graphite,” Phys. Rev. Lett. |

16. | J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband Nonlinear Optical Response of Graphene Dispersions,” Adv. Mater. |

17. | Z. Liu, Y. Wang, X. Zhang, Y. Xu, Y. Chen, and J. Tian, “Nonlinear optical properties of graphene oxide in nanosecond and picosecond regimes,” Appl. Phys. Lett. |

18. | A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K. S. Novoselov, S. Roth, and A. K. Geim, “Raman spectrum of graphene and graphene layers,” Phys. Rev. Lett. |

19. | Z. H. Ni, W. Chen, X. F. Fan, J. L. Kuo, T. Yu, A. T. S. Wee, and Z. X. Shen, “Raman spectroscopy of epitaxial graphene on a SiC substrate,” Phys. Rev. B |

20. | J. C. Burton, L. Sun, F. H. Long, Z. C. Feng, and I. T. Ferguson, “First- and second-order Raman scattering from semi-insulating 4H-SiC,” Phys. Rev. B |

21. | D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hierold, and L. Wirtz, “Spatially resolved Raman spectroscopy of single- and few-layer graphene,” Nano Lett. |

22. | J. Hass, F. Varchon, J. E. Millán-Otoya, M. Sprinkle, N. Sharma, W. A. de Heer, C. Berger, P. N. First, L. Magaud, and E. H. Conrad, “Why multilayer graphene on 4H-SiC(0001[over ]) behaves like a single sheet of graphene,” Phys. Rev. Lett. |

23. | M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. |

24. | Y. R. Shen, “The Principles of Nonlinear Optics,” John Wiley, New York and Chichester, |

**OCIS Codes**

(190.4400) Nonlinear optics : Nonlinear optics, materials

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

(160.4236) Materials : Nanomaterials

**ToC Category:**

Materials

**History**

Original Manuscript: January 4, 2010

Revised Manuscript: February 9, 2010

Manuscript Accepted: February 9, 2010

Published: February 19, 2010

**Virtual Issues**

February 24, 2010 *Spotlight on Optics*

**Citation**

Guichuan Xing, Hongchen Guo, Xinhai Zhang, Tze Chien Sum, and Cheng Hon Alfred Huan, "The Physics of ultrafast saturable absorption in graphene," Opt. Express **18**, 4564-4573 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4564

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### References

- P. R. Wallace, “The band theory of graphite,” Phys. Rev. 71(9), 622–634 (1947). [CrossRef]
- A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]
- A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007). [CrossRef] [PubMed]
- P. Avouris, Z. Chen, and V. Perebeinos, “Carbon-based electronics,” Nat. Nanotechnol. 2(10), 605–615 (2007). [CrossRef]
- T. Ando, Y. S. Zheng, and H. Suzuura, “Dynamical Conductivity and Zero-Mode Anomaly in Honeycomb Lattices,” J. Phys. Soc. Jpn. 71(5), 1318–1324 (2002). [CrossRef]
- V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. 96(25), 256802 (2006). [CrossRef] [PubMed]
- T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78(8), 085432 (2008). [CrossRef]
- A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100(11), 117401 (2008). [CrossRef] [PubMed]
- F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008). [CrossRef] [PubMed]
- R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320(5881), 1308 (2008). [CrossRef] [PubMed]
- 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(19), 196405 (2008). [CrossRef] [PubMed]
- J. M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M. G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. 92(4), 042116 (2008). [CrossRef]
- D. Sun, Z. K. Wu, C. Divin, X. Li, C. Berger, W. A. de 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(15), 157402 (2008). [CrossRef] [PubMed]
- R. W. Newson, J. Dean, B. Schmidt, and H. M. van Driel, “Ultrafast carrier kinetics in exfoliated graphene and thin graphite films,” Opt. Express 17(4), 2326–2333 (2009). [CrossRef] [PubMed]
- M. Breusing, C. Ropers, and T. Elsaesser, “Ultrafast carrier dynamics in graphite,” Phys. Rev. Lett. 102(8), 086809 (2009). [CrossRef] [PubMed]
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