We report studies of cyclotron resonance in monolayer graphene. Cyclotron resonances are detected by observing changes in the photoconductive response of the sample. An electron velocity at the Dirac point of 1.093×106 m s−1 is obtained, which is the fastest velocity recorded for all known carbon materials. In addition, a significant asymmetry exists between band structure for electrons and holes, which gives rise to a 5% difference between the velocities at energies of 125 meV away from the Dirac point.
Ever since the isolation of graphene in 2004 (Novoselov et al. 2004), a vast amount of interest has been shown in this truly two-dimensional system in which a flat monolayer of carbon atoms are arranged in a honeycomb lattice. Owing to the two-dimensional nature of the material, graphene has a fascinating electronic band structure in which charge carriers behave as Dirac fermions with extremely high velocities due to the near-linear dispersion relations close to the K-point in the Brillouin zone. This results in the observation of new scientific phenomena such as chiral quantum Hall effects (Novoselov et al. 2005; Zhang et al. 2005, 2006) as well as realistic potential for applications in high-speed electronics (Geim & Novoselov 2007). Theoretically, the study of graphene began in 1947 (Wallace 1947), but it is only very recently that measurements of the electron velocities were performed on monolayers of graphene (Deacon et al. 2007; Jiang et al. 2007). Close to the K-point, the graphene dispersion relation takes the form , where c* is the velocity of charge carriers, and crosses over at the Fermi energy, indicating that graphene is a zero-gap semiconductor with symmetric bands. Applying a magnetic field to graphene leads to the formation of Landau levels (McClure 1956; figure 1) given by(1.1)where |N| is the Landau quantum index and B is the magnetic field. This allows us to make precise measurements of the electron (hole) velocity near the Fermi energy by studying the cyclotron resonance of graphene monolayers, in which a significant asymmetry between the electron and hole bands is observed, in contrast to the prediction of simple tight-binding theory (Saito et al. 1992; Reich et al. 2002).
2. Experimental details
Graphene monolayer samples were produced by micromechanical cleavage of bulk graphite onto a SiO2/Si wafer with multiple electrodes contacted onto the graphene monolayer by conventional microfabrication. The samples were characterized by studying Shubnikov–de Haas oscillations to confirm that they were single layers, as multilayer graphenes have a more complex dispersion relation (McCann & Fal'ko 2006; Novoselov et al. 2006); this process also allowed us to verify the relationship between gate voltage and carrier densities. Experiments were carried out by studying the changes of the photoconductivity for graphene samples when illuminated with infrared radiation produced by a CO2 laser, with energies between 115 and 135 meV. The typical laser power densities were approximately 3×104 Wm−2, meaning the power on the samples is roughly 5 μW. The experiments were set up in the Faraday geometry, where incident radiation is normal to the samples and parallel to magnetic field, as shown in figure 2b. Samples were immersed in liquid helium at 1.5 K, a current of I=100 nA was supplied to the samples with data collected in a two-contact configuration as this gives qualitatively similar response compared with a four-contact configuration, but much better signal-to-noise ratio. The magnitude of the photoresponse signal is related to the amount of light absorbed by the sample, and hence is directly related to the absorption coefficient, with the greatest positive signals detected at Landau-level occupancy ν=nh/eB at −3.0, −0.76, 0.88 and 3.1; 0 being the Dirac point. This demonstrates that the photoconductive signals show a derivative behaviour, with large positive signals observed at the edges of the conductance peaks, at the points where resistivity changes most rapidly with temperature and chemical potential. The peaks are assigned as 1−, 0−, 0+ and 1+ transitions, respectively, and the transitions take place from the Dirac point (N=0) to the N=−1(+1) Landau level, as holes (electrons) absorb a photon. The 1− and 1+ peaks are pure hole and electron transitions, whereas both 0− and 0+ peaks contain contributions from both transitions but with one type of charge carriers more dominant than the other.
By sweeping charge carrier density at each value of magnetic field and recording the photoconductivity at each point, we were able to identify resonant cyclotron transitions for pure and mixtures of hole- and electron-like transitions. In order to produce full resonances to accurately measure the resonance positions, traces of photoconductive signals were then taken at fixed Landau-level occupancies following the lines shown in figure 3b. A typical trace taken at laser energy of 135 meV is shown in figure 3c, with Lorentzian fitting shown as the red line.
3. Results and discussion
Evidence of cyclotron resonance can be observed easily with this set-up, large photoconductive voltage variations as high as 3% can be seen at resonance, with the data suggesting a significant difference in resonance positions for the electrons and holes. The fixed occupancy traces showing the resonances are then fitted with conventional Lorentzian lineshapes with a linear background to correct for the bolometric response caused by strong localization of the carriers at high field.
Figure 4 shows the resonance positions plotted as a function of magnetic field and immediately a clear splitting between electron- and hole-like resonances can be seen, which equation (1.1) does not predict. Fitting velocities to each of the resonances separately gives values of c*=(1.117, 1.118, 1.105, 1.069±0.004)×106 m s−1 for the 1+, 0+, 0−, 1− resonances, respectively. Nearest-neighbour tight-binding theory (Saito et al. 1998) predicts the dispersion relation of graphene in terms of the carbon–carbon interaction energy γ0 and the overlap integral s0, and close to the Dirac point, this gives the electron velocity as(3.1)where . First-principle calculations (Saito et al. 1998) typically give γ0=3.03 eV, s0=0.129 and m s−1 with other reports in the regime γ0=2.7−3.1 eV (Reich et al. 2002). Fitting the data in figure 4 to this relation gives m s−1 at the Dirac point, corresponding to γ0=3.38 eV, with s0=0.6±0.1. The unusually large fitted value for s0=0.6±0.1 reflects the large asymmetry observed for electron and hole velocities. The value for agrees with Jiang's direct cyclotron resonance results (2007) and is significantly greater than the values reported for previous studies of the Fermi velocities for graphite and multilayers of graphene sheets in metallic systems (Sadowski et al. 2006; Zhou et al. 2006; Li & Andrei 2007). Plotting the Fermi velocities as a function of the number of graphene layers, it can be seen that the Fermi velocity falls by approximately 20% between monolayer graphene and bulk graphite, as shown in figure 5, whereas the value of approximately 2.9 eV deduced from the band structure of semiconducting carbon nanotubes (Filho et al. 2004) still corresponds with values deduced from the theoretical and graphite values of γ0. This progressive increase of electron velocity as numbers of graphene layers decrease suggests that the π bonds which are normal to the graphene surface have an important role in determining the Fermi velocity, as these bonds are directly responsible for the interlayer coupling and the coupling to the SiO2 layer. A similar situation was observed in a recent report on filling carbon nanotubes with crystalline material (Li et al. 2006) in which it was suggested that the coupling between the carbon atoms and manganese telluride increases the transfer integral.
The origin of the large asymmetry between electrons and holes is still not well understood, as the tight-binding model predicts a difference of only 1% in total between the hole and electron velocities at E≈125 meV with the actual observed difference being five times larger. However, all the analysis used is based on single-particle theory and it is possible that many-body interactions could affect the quantities measured in this report. Although electron–electron interactions can be neglected for long-wavelength excitations for parabolic systems as stated by Kohn's theorem (Kohn 1961), linear systems such as single-layer graphene are predicted to show velocity renormalization effects from both electron–electron interactions (González et al. 1994) and electron–phonon coupling (Park et al. 2007).
We have successfully measured the Fermi velocity in monolayer graphene using cyclotron resonance, which is found to be considerably larger than that seen in thicker graphitic systems. We have shown that using photoconductivity gives significantly narrower linewidths than that observed in infrared absorption on large area samples (Jiang et al. 2007) which allows us to detect an asymmetry between the carrier velocity for the hole- and electron-like parts of the dispersion relation close to the Dirac point. The single-particle picture gives an adequate description of the broad outline of behaviour seen but does not provide an explanation for the asymmetry or the dependence on the number of graphene layers. These phenomena, together with observations such as the deviation of precise scaling for higher-order Landau level transitions (Jiang et al. 2007), suggest that many-body interactions may prove to be important in a full understanding of the behaviour of this system. Also the roles of spin splitting, valley splitting and excitonic interactions in this system still remain unanswered and may turn out to be very significant in providing a full description of the properties of monolayer graphene, as is the case for carbon nanotubes.
Part of this work has been supported by EuroMagNET under the EU contract RII3-CT-2004-506239 of the 6th Framework ‘Structuring the European Research Area, Research Infrastructures Action’.
One contribution of 11 to a Discussion Meeting Issue ‘Carbon-based electronics: fundamentals and device applications’.
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