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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 4 — Feb. 16, 2009
  • pp: 2326–2333
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Ultrafast carrier kinetics in exfoliated graphene and thin graphite films

Ryan W. Newson, Jesse Dean, Ben Schmidt, and Henry M. van Driel  »View Author Affiliations


Optics Express, Vol. 17, Issue 4, pp. 2326-2333 (2009)
http://dx.doi.org/10.1364/OE.17.002326


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Abstract

Time-resolved transmissivity and reflectivity of exfoliated graphene and thin graphite films on a 295 K SiO2/Si substrate are measured at 1300 nm following excitation by 150 fs, 800 nm pump pulses. From the extracted transient optical conductivity we identify a fast recovery time constant which increases from ~200 to 300 fs and a longer one which increases from 2.5 to 5 ps as the number of atomic layers increases from 1 to ~260. We attribute the temporal recovery to carrier cooling and recombination with the layer dependence related to substrate coupling. Results are compared with related measurements for epitaxial, multilayer graphene.

© 2009 Optical Society of America

1. Introduction

Graphene is a single two-dimensional atomic layer of carbon atoms arranged in a hexagonal lattice [1

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

,2

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

]. Electronically, as an isolated film, it is a zero band gap semiconductor with a linear dispersion relation in the vicinity of the K and K’ points of the Brillouin zone. Electrons in the vicinity of the band gap behave as zero mass Dirac fermions, unlike electrons associated with parabolic bands in, e.g., bulk semiconductors. Novel quantum and transport properties [3

3. G. W. Semenoff, “Condensed-matter simulation of a 3-dimensional anomaly,” Phys. Rev. Lett. 53, 2449–2452 (1984). [CrossRef]

,4

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

] in graphene have made this material the focus of fundamental physics [5

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

,6

6. 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] [PubMed]

] as well as a possible platform for new types of nanoelectronic devices. It has been shown theoretically [7

7. B. Partoens and F. M. Peeters, “From graphene to graphite: Electronic structure around the K point,” Phys. Rev. B 74, 075404 (2006). [CrossRef]

] that as one forms multilayer structures based on (hexagonal Bernal) ABAB-stacking, the electronic structure and many properties of graphene evolve to those of graphite by approximately 10 layers. Knowledge of ultrafast, hot carrier kinetics and how they evolve with number of layers is essential for many applications. We have therefore examined how these kinetics change by studying exfoliated thin films with 1 layer (graphene), 2 layers, etc., up to 260 layers by probing the time-resolved reflectivity and transmissivity.

Recently there have been reports on the ultrafast carrier dynamics of epitaxial graphene. Dawlaty et al. [19

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

] have studied ultrafast optical transmissivity of a few samples with between 6 and 37 layers using 85 fs, 780 nm pump and probe beams. From the nonexponential time dependence of the transmissivity recovery, they assign a fast time constant of ~70–120 fs to carrier thermalization, and a slower time constant of ~0.4–1.2 ps to carrier cooling. The dependence of the latter time constant on sample was attributed to the degree of crystalline order. More recently, D. Sun et al. [20

20. D. Sun, Z.-K. Wu, C. Divin, X. B. 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, 157402 (2008). [CrossRef] [PubMed]

] have reported time-resolved transmissivity on an epitaxial sample. The first layer was doped with an electron concentration of 9×1012 cm-2, resulting in an electron Fermi level of ~350 meV. The remaining estimated 15–20 layers were neutral. Time-resolved transmissivity was recorded at different temperatures with 150 fs probe pulses between 1.57 and 2.4 μm following a 800 nm, 150 fs pump beam. The time-resolved transmissivity is therefore a measure of the response of both doped and undoped layers. As with the work of Dawlaty et al., a non-exponential decay of the transmissivity was observed but this was postulated to be a stretched exponential decay. Possible sources for the non-exponential behavior were suggested to be hot-phonons, sample disorder and density-dependent carrier scattering.

2. Experimental

Our graphene samples were fabricated from natural flake graphite by micromechanical cleaving with transparent tape followed by application onto a 500 μm thick Si substrate with a 300 nm SiO2 surface layer. Samples used in the experiments had between 1 and 260 layers and had uniform thickness over a circular area of at least 15 μm in diameter. Samples with up to seven layers can be distinguished using an optical microscope, due to the optical interference effect. Samples between one and three layers were confirmed using Raman Spectroscopy. For the thicker samples, atomic force microscopy was used to determine the number of layers with an accuracy of ±10%. Raman spectra of the thin samples showed no defect-induced D lines [21

21. M. A. Pimenta, G. Dresselhaus, M. S. Dresselhaus, L. G. Cançado, A. Jorio, and R. Saito, “Studying disorder in graphite-based systems by Raman spectroscopy,” Phys. Chem. Chem. Phys. 91276–1291 (2007). [CrossRef] [PubMed]

]. The Si wafer was slightly n-doped, however the free electron concentration of ~1×1018 cm-3 was sufficiently low to allow for significant transmission at our probe wavelength.

For the time-resolved optical experiment, an 80 MHz Ti:sapphire oscillator provides 1.0 nJ 150 fs pump pulses at 800 nm, while simultaneously pumping an 80 MHz OPO system delivering 2.5 pJ 150 fs probe pulses at 1300 nm. The probe wavelength was chosen to be below the indirect Si band gap so that only free carrier absorption can occur. The probed electron and hole-coupled states are within the linear regime of the electron dispersion relation. After the probe’s optical delay line, the collinearly polarized pump and probe beams were focused through a 40X (numerical aperture = 0.65) objective onto the samples that were held at room temperature (295K). Samples were positioned to be at the focal plane of the probe beam and not the pump beam, which provided a pump spot diameter (~20 μm FWHM) greater than the probe spot diameter (~3 μm FWHM). This assured the probe-sampled area experienced uniform excitation. For an optical absorption of ~2%/layer in graphene and an incident pump photon flux of < 1015 cm-2, we estimate a peak carrier density per layer < 1013 cm-2. In the case of silicon, for which the absorption depth of the pump beam is ~12 μm, the estimated induced peak carrier density is ~1017 cm-3. If only optically generated electrons and holes are produced, the maximum carrier temperature, following thermalization, could be several thousand Kelvin. However, the presence of thermally activated carriers or (unintentional) doping will reduce this temperature. The pump intensity was kept well below the damage threshold of graphene, which was confirmed by the lack of both visual damage and defect-induced D lines in the Raman spectra before and after the pump-probe experiments. Samples were imaged in a confocal arrangement with a CCD camera. Transmitted and reflected probe beams were measured with biased Ge photodiodes and a lock-in amplifier. It was verified that all reflectivity and transmissivity signals were linear in probe intensities up to the maximum intensity reported here. At the probe wavelength the unexcited complex refractive indices of Si, SiO2 and graphite are 3.543+1.5×10-4 i, 1.45, and 3.27+2.54i respectively.

3. Results

For each of the graphene/graphite samples, as well as the bare SiO2/Si region immediately surrounding it, the time-resolved differential change in reflectivity (ΔR/R) and transmissivity (ΔT/T) were measured for up to 35 ps of probe delay. Samples with 0 (bare SiO2/Si), 1, 2, … 260 layers were measured; a subset of these measurement for the time delay interval -2 to 15 ps are shown in Fig. 1.

Fig. 1. Time-dependent differential reflectivity ΔR/R and transmissivity ΔT/T of graphene/ graphite samples with number of carbon layers indicated.

The SiO2/Si time-resolved reflectivity appears to be similar to that of Sabbah and Riffe [22

22. A. J. Sabbah and D. M. Riffe, “Femtosecond pump-probe study of silicon carrier dynamics,” Phys. Rev. B 66, 165217 (2002). [CrossRef]

], although for their probe wavelength of 800 nm, the measured transient optical properties are dominated by interband absorption whereas ours are dominated by free carrier absorption. The nonzero initial value (i.e., before a particular pump pulse arrives) of the reflectivity and transmissivity occurs because of some steady-state carrier accumulation effects in the silicon since the recombination time is longer than the inter-pulse separation time of ~12.5 ns. The results from bulk graphite are also in agreement with the results of Seibert et al. [23

23. K. Seibert, G. C. Cho, W. Kütt, H. Kurz, D. H. Reitze, J. I. Dadap, H. Ahn, and M. C. Downer, “Femtosecond carrier dynamics in graphite,” Phys. Rev. B , 42, 2842–2851 (1990). [CrossRef]

] after correcting for the difference in probe wavelengths. Their data was obtained with visible and near infrared probe wavelengths. As the wavelength increases, the optically probed electron and hole states are closer to the band edge and Fermi level, and the apparent relaxation time increases [23

23. K. Seibert, G. C. Cho, W. Kütt, H. Kurz, D. H. Reitze, J. I. Dadap, H. Ahn, and M. C. Downer, “Femtosecond carrier dynamics in graphite,” Phys. Rev. B , 42, 2842–2851 (1990). [CrossRef]

]. From the data in Fig. 1, one can observe that as the number of graphene layers increases, the signals becomes increasingly different from that of the bare substrate, with the Si exhibiting less and less of an effect due to increasing absorption from the carbon layers. For > ~25 layers we observe the typical behavior of bulk graphite [23

23. K. Seibert, G. C. Cho, W. Kütt, H. Kurz, D. H. Reitze, J. I. Dadap, H. Ahn, and M. C. Downer, “Femtosecond carrier dynamics in graphite,” Phys. Rev. B , 42, 2842–2851 (1990). [CrossRef]

]. Note that at the extremes of layer number, i.e. for the bare SiO2/Si and bulk graphite, the signals are very different, with Si showing a very long recombination time, while the graphite change recovers within 15 ps.

To extract the time-dependent optical properties of the carbon layers we model the entire multilayer system with a transfer matrix method. For samples with < ~100 layers, the fact that both the Si and the carbon layers are contributing to the change in reflectivity and transmissivity must be taken into account; the oxide layer is assumed to be inert. In particular the optical response of the Si needs to be isolated from the effect of the graphene/graphite. To do this, the change in the Si complex optical conductivity is first determined from the ΔR/R and ΔT/T experimental data for the bare SiO2/Si substrate. This information can then be used in our multilayer model to compute the change in the graphene/graphite optical conductivity from the ΔR/R and ΔT/T experimental data on our samples. The fact that silicon’s optical response includes a decay time much greater than that of graphite aids in determining the magnitude of the effect from silicon in our multilayer sample. In essence, for times > 20 ps we can assume the graphene/graphite material properties have returned to their quiescent values and the silicon is the only material contributing to the nonzero differential reflectivity and transmissivity. This allows us to scale the material changes accordingly to get the most accurate view of the effects occurring only in graphene/graphite.

The transfer matrix method [24

24. C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent and incoherent interference,” Appl. Opt. 41, 3978–3987 (2002). [CrossRef] [PubMed]

] is used to numerically compute the relative changes in the real and imaginary parts of the optical conductivity, ΔσRR and ΔσII respectively, from ΔR/R and ΔT/T. Each of the individual carbon layers can be represented by a transfer matrix, with ΔσR and ΔσI decreasing into the sample from the illuminated face according to the graphite pump absorption. We take each layer’s quiescent optical properties as those of bulk graphite with a thickness of 0.335 nm; such an assumption for visible and near visible light has worked well in determining the sample thickness [15

15. J. Hass, F. Varchon, J. E. Millán-Otoya, M. Sprinkle, W. A. de Heer, C. Berger, P. N. First, L. Magaud, and E. H. Conrad, “Rotational stacking and its electronic effects on graphene films grown on 4H-SiC(000 1̄),” arXiv:cond-mat/0706.2134 (2007).

,16

16. P. Blake, E. W. Hill, A. H. C. Neto, K. S. Novoselov, D. Jiang, R. Yang, T. J. Booth, and A. K. Geim, “Making graphene visible,” Appl. Phys. Lett. 91, 063124 (2007). [CrossRef]

]. When the SiO2 and Si layers are included, the total reflection and transmission amplitude coefficients can be computed and used to obtain ΔR/R and ΔT/T. At each time delay the ΔσRR and ΔσI/㧃I values are chosen to obtain the ΔR/R and ΔT/T values observed.

In this way, we use the time-resolved reflectivity and transmissivity data to deduce the time dependence of σR and σI. This dependence is shown in Fig. 2 for a select set of samples. The particular values of differential conductivity plotted are those for the top layer in our multilayer samples. The changes will, of course, be smaller for layers closer to the substrate where the pump intensity will be lower. For all our samples, the ΔσRR show a similar type of behavior, i.e., a rapid decrease during the pump pulse followed by a non-exponential recovery. However, the ΔσII data sets show different behavior as the number of layers increases. We observe that for 1–3 layers, except for some behaviour around the pulse overlap time, ΔσII is very small compared to its value for thicker samples. For the thicker samples, ΔσII shows a magnitude that is clearly non-zero and a non-exponential time dependence similar to that of ΔσRR. For the thickest samples, the behavior of both real and imaginary parts of the optical conductivity are similar to that observed previously for graphite [23

23. K. Seibert, G. C. Cho, W. Kütt, H. Kurz, D. H. Reitze, J. I. Dadap, H. Ahn, and M. C. Downer, “Femtosecond carrier dynamics in graphite,” Phys. Rev. B , 42, 2842–2851 (1990). [CrossRef]

].

Fig. 2. Extracted time-resolved ΔσRR and ΔσII for the top layer of graphene/ graphite samples with number of carbon layers indicated.

The two time constants extracted from the ΔσRR data are shown for samples with up to 260 layers in Fig. 3. Note that the error bars not only reflect experimental uncertainty due to measurements on a particular sample, but also variation in different samples with the same number of layers. The fast time constant increases, at best only slightly, from ~200 fs to ~300 fs over a few layers and thereafter apparently remains constant while the slower time constant increases from 2.5 ps to 5 ps as the sample thickness increases from 1 to ~30 layers before apparently remaining constant to ~260 layers.

Fig. 3. Extracted fast and slow time constants from time-resolved ΔσRR for our graphene/ graphite samples as a function of the number of layers.

4. Discussion

The fact that the slow decay time increases gradually with number of layers hints to transport of carriers and/or thermal energy across the interface. If one makes the commonly-used assumption of no inter-layer interactions and, more importantly, no graphite-SiO2 interaction, decay times should not increase after a few layers. However, the consideration of ballistic or diffusive transport could explain this trend. For example, in a simple decay-diffusion model (ballistic transport would yield even faster recovery) of carriers and/or carrier energy, using a decay time of 5 ps, a diffusion constant of 0.1 cm2/s, and a SiO2 interface surface recombination constant of 104 cm/s, a similar trend is recovered. The diffusion constant is based on the known value for interlayer diffusion of heat (thermal diffusivity) [30

30. J. Norley, “The role of natural graphite in electronics cooling,” (Electronics Cooling Magazine, 2001). http://www.electronics-cooling.com/articles/2001/2001_august_techbrief.php

,31

31. J. P. Holman, Heat Transfer 9th Ed. (McGraw-Hill, 2002).

]. Both the change in reflectivity and transmissivity experience decay times that increase gradually from ~2.5 ps to 5 ps in ~30 layers. While the values of these model parameters are not the point of emphasis here, the underlying mechanism associated with them does offer a possible explanation as to why this slow decay time constant increases with number of layers.

5. Summary

Acknowledgments

We thank Sida Wang for early help in making graphene samples and Joshua Folk for advice on exfoliation. We acknowledge financial support from NSERC Canada.

References and links

1.

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

2.

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

3.

G. W. Semenoff, “Condensed-matter simulation of a 3-dimensional anomaly,” Phys. Rev. Lett. 53, 2449–2452 (1984). [CrossRef]

4.

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

5.

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

6.

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] [PubMed]

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B. Partoens and F. M. Peeters, “From graphene to graphite: Electronic structure around the K point,” Phys. Rev. B 74, 075404 (2006). [CrossRef]

8.

C. Berger, Z. M. Song, T. B. Li, A. Y. Ogbazghi, R. Feng, Z. T. Dai, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Ultrathin epitaxial graphite: 2D electronic gas properties and a route toward graphene-based nanoelectronics,” J. Phys. Chem. B 108, 19912–19916 (2004). [CrossRef]

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C. Berger, Z. M. Song, X. B. Li, X. S. Wu, N. Brown, C. Naud, D. Mayou, T. B. Li, J. Haas, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, “Electronic confinement and coherence in patterned expitaxial graphene,” Science 312, 1191–1196 (2006). [CrossRef] [PubMed]

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J. Hass, R. Feng, J. E. Millán-Otoya, X. Li, M. Sprinkle, P. N. First, W. A. de Heer, E. H. Conrad, and C. Berger, “Structural properties of the multilayer graphene/4H-SiC(000 1̄) system as determined by surface x-ray diffraction,” Phys. Rev. B 75, 214109 (2007). [CrossRef]

11.

P. Darancet, N. Wipf, C. Berger, W. A. de Heer, and D. Mayou, “Quenching of quantum Hall effect and the role of undoped planes in multilayered epitaxial graphene,” arXiv:cond-mat/0711.0940 (2008).

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W. A. de Heer, C. Berger, X. S. Wu, P. N. First, E. H. Conrad, X. B. Li, T. B. Li, M. Sprinkle, J. Hass, M. L. Sadowski, M. Potemski, and G. Martinez, “Epitaxial graphene,” Solid State Commun. 143, 92–100 (2007). [CrossRef]

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M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer,“Landau level spectroscopy of ultrathin graphite layers,” Phys. Rev. Lett. 97, 266405 (2006). [CrossRef]

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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(000 1 ) behaves like a single sheet of graphene,” Phys. Rev. Lett. 100, 125504 (2008). [CrossRef] [PubMed]

15.

J. Hass, F. Varchon, J. E. Millán-Otoya, M. Sprinkle, W. A. de Heer, C. Berger, P. N. First, L. Magaud, and E. H. Conrad, “Rotational stacking and its electronic effects on graphene films grown on 4H-SiC(000 1̄),” arXiv:cond-mat/0706.2134 (2007).

16.

P. Blake, E. W. Hill, A. H. C. Neto, K. S. Novoselov, D. Jiang, R. Yang, T. J. Booth, and A. K. Geim, “Making graphene visible,” Appl. Phys. Lett. 91, 063124 (2007). [CrossRef]

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20.

D. Sun, Z.-K. Wu, C. Divin, X. B. 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, 157402 (2008). [CrossRef] [PubMed]

21.

M. A. Pimenta, G. Dresselhaus, M. S. Dresselhaus, L. G. Cançado, A. Jorio, and R. Saito, “Studying disorder in graphite-based systems by Raman spectroscopy,” Phys. Chem. Chem. Phys. 91276–1291 (2007). [CrossRef] [PubMed]

22.

A. J. Sabbah and D. M. Riffe, “Femtosecond pump-probe study of silicon carrier dynamics,” Phys. Rev. B 66, 165217 (2002). [CrossRef]

23.

K. Seibert, G. C. Cho, W. Kütt, H. Kurz, D. H. Reitze, J. I. Dadap, H. Ahn, and M. C. Downer, “Femtosecond carrier dynamics in graphite,” Phys. Rev. B , 42, 2842–2851 (1990). [CrossRef]

24.

C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent and incoherent interference,” Appl. Opt. 41, 3978–3987 (2002). [CrossRef] [PubMed]

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P. A. George, J. Strait, J. Dawlaty, S. Shivaraman, M. V. S. Chandrasekhar, F. Rana, and M. G. Spencer, “Ultrafast optical-pump terahertz-probe spectroscopy of the carrier relaxation and recombination dynamics in epitaxial graphene,” arXiv:cond-mat/0805.4647v3 (2008).

27.

F. Rana, “Graphene terahertz plasmon oscillators,” IEEE Trans. Nanotechnol. 7, 91–99 (2008). [CrossRef]

28.

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] [PubMed]

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30.

J. Norley, “The role of natural graphite in electronics cooling,” (Electronics Cooling Magazine, 2001). http://www.electronics-cooling.com/articles/2001/2001_august_techbrief.php

31.

J. P. Holman, Heat Transfer 9th Ed. (McGraw-Hill, 2002).

OCIS Codes
(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors
(160.4236) Materials : Nanomaterials

ToC Category:
Ultrafast Optics

History
Original Manuscript: December 15, 2008
Revised Manuscript: January 30, 2009
Manuscript Accepted: February 3, 2009
Published: February 5, 2009

Citation
Ryan W. Newson, Jesse Dean, Ben Schmidt, and Henry M. van Driel, "Ultrafast carrier kinetics in exfoliated graphene and thin graphite films," Opt. Express 17, 2326-2333 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2326


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References

  1. P. R. Wallace, "The band theory of graphite," Phys. Rev. 71, 622-634 (1947). [CrossRef]
  2. A. K. Geim and K. S. Novoselov, "The rise of graphene," Nat. Mater. 6, 183-191 (2007). [CrossRef] [PubMed]
  3. G. W. Semenoff, "Condensed-matter simulation of a 3-dimensional anomaly," Phys. Rev. Lett. 53, 2449-2452 (1984). [CrossRef]
  4. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, "Electric field effect in atomically thin carbon films," Science 306, 666-669 (2004). [CrossRef] [PubMed]
  5. Y. B. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, "Experimental observation of the quantum Hall effect and Berry’s phase in graphene," Nature 438, 201-204 (2005). [CrossRef] [PubMed]
  6. 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] [PubMed]
  7. B. Partoens and F. M. Peeters, "From graphene to graphite: Electronic structure around the K point," Phys. Rev. B 74, 075404 (2006). [CrossRef]
  8. C. Berger, Z. M. Song, T. B. Li, A. Y. Ogbazghi, R. Feng, Z. T. Dai, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, "Ultrathin epitaxial graphite: 2D electronic gas properties and a route toward graphene-based nanoelectronics," J. Phys. Chem. B 108, 19912-19916 (2004). [CrossRef]
  9. C. Berger, Z. M. Song, X. B. Li, X. S. Wu, N. Brown, C. Naud, D. Mayou, T. B. Li, J. Haas, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, "Electronic confinement and coherence in patterned expitaxial graphene," Science 312, 1191-1196 (2006). [CrossRef] [PubMed]
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