Abstract
Originating from relativistic quantum field theory, Dirac fermions have been invoked recently to explain various peculiar phenomena in condensedmatter physics, including the novel quantum Hall effect in graphene^{1,2}, the magneticfielddriven metal–insulatorlike transition in graphite^{3,4}, superfluidity in ^{3}He (ref. 5) and the exotic pseudogap phase of hightemperature superconductors^{6,7}. Despite their proposed key role in those systems, direct experimental evidence of Dirac fermions has been limited. Here, we report the first direct observation of relativistic Dirac fermions with linear dispersion near the Brillouin zone (BZ) corner H, which coexist with quasiparticles that have a parabolic dispersion near another BZ corner K. In addition, we also report a large electron pocket that we attribute to defectinduced localized states. Thus, graphite presents a system in which massless Dirac fermions, quasiparticles with finite effective mass and defect states all contribute to the lowenergy electronic dynamics.
Main
For most condensedmatter systems, the physics is formulated in terms of the nonrelativistic Schrödinger equation, and the lowenergy excitations are quasiparticles with finite effective mass. For some special systems, for example, graphene/graphite, where the dispersion is expected to be linear near the Fermi energy, E_{F}, and only touch E_{F} at one point (Dirac point), the physics is described by the relativistic Dirac equation with the speed of light replaced by the Fermi velocity v_{F}. The lowenergy excitations in this case are Dirac fermions, which have zero effective mass and a vanishing density of states at the Dirac point. The Dirac fermions are proposed to be responsible for various anomalous phenomena observed in these systems^{1,2,3,8}. So far, transport experiments in graphene have shown results in agreement with the presence of Dirac fermions^{1,2}. Phase analysis of quantum oscillations in graphite has also suggested coexistence of both Dirac fermions and quasiparticles with finite effective mass^{9}. Here, we report the first direct observation of massless Dirac fermions coexisting with quasiparticles with finite effective mass in graphite, by using angleresolved photoemission spectroscopy (ARPES). ARPES provides the advantage of directly probing the electronic structure with both energy and momentum information, not accessible by any other measurement.
Figure 1a shows an ARPES intensity map measured near the Brillouin zone (BZ) corner H. The outofplane momentum, k_{z}, is 0.5c^{*}, where c^{*} is the reciprocal lattice constant (see the Methods section for the extraction of k_{z} values). Following the maximum intensity in this map, a linear Λshaped dispersion can be clearly observed. The dispersion can be better extracted by following the peak positions in the momentum distribution curves (MDCs), momentum scans at constant energies, shown in Fig. 1b. Here, two peaks in the MDCs disperse linearly and merge near E_{F}. The Fermi velocity extracted from the dispersion is 0.91±0.15×10^{6} m s^{−1}, similar to a value 1.1×10^{6} m s^{−1} reported by a magnetoresistance study of graphene^{2}. We note that at low energy near E_{F}, this linear dispersion is also observed along other cuts through the H point, with similar Fermi velocity. This linear and isotropic dispersion is in agreement with the behaviour of Dirac fermions.
Another way of probing the linear and isotropic dispersion is to study the intensity maps at constant energy. At E_{F} (Fig. 2a), the intensity map shows a small object near H. The details of this small object will be discussed later. With increasing binding energy, this object expands and shows a circular shape (Fig. 2b,c). We note that only the circular shape in the first BZ is clearly observed. This is attributed to the dipole matrix element^{10}, which suppresses or enhances the intensity in different BZs. However, taking the threefold symmetry of the sample, this circular shape in the first BZ is expected to extend to other BZs and thus the electronic structure is isotropic near H. As the energy changes to −0.9 eV, the constant energy map deviates slightly from the circular shape (see the arrow in Fig. 2d). This deviation increases with binding energy and a trigonal distortion is clearly observed at −1.2 eV (Fig. 2e). This trigonal distortion is determined by the relevant tightbinding parameters for graphite and further studies to analyse this trigonal distortion are in progress. Overall, Fig. 2 shows that from E_{F} to −0.6 eV, the electronic structure is isotropic in the k_{x}–k_{y} plane. Similarly, the Fermi velocity measured within the first BZ is 1.0×10^{6} m s^{−1} with a ≤10% variation along different directions, consistent with the circular constant energy maps shown here. Combining the results of Figs 1 and 2, we conclude that from E_{F} to −0.6 eV, the dispersion shows a conelike behaviour near each BZ corner H, similar to that expected for graphene (Fig. 2f).
To resolve the details of the lowenergy dispersion and the small object at E_{F} (Fig. 2a), we show in Fig. 3a an intensity map measured near H with lower photon energy to give better energy and momentum resolution. In the intensity map, two bands dispersing linearly towards E_{F} can be distinguished, as is also clear in the MDCs (Fig. 3b) where two peaks can be observed for all binding energies. The extracted dispersion (open circles in Fig. 3a) from MDCs (Fig. 3b) shows two bands dispersing linearly towards E_{F}, with a minimum separation of 0.020±0.004 Å^{−1} at E_{F}. This linear dispersion near the H point, as well as the isotropic electronic structure shown in Fig. 2 from E_{F} to −0.6 eV, is a basic characteristic of Dirac quasiparticles, which points to the presence of Dirac quasiparticles in the lowenergy excitations near the H point in graphite. Furthermore, from the extracted dispersions, the Dirac point is extrapolated to be 50±20 meV above E_{F}, and thus the small object observed at E_{F} is a hole pocket, in agreement with previous studies of the threedimensional band structure of graphite^{9,11}. Assuming an ellipsoidal shape for the hole pocket^{12}, the hole concentration is estimated to be 3.1±1.3×10^{18} cm^{−3}, from the 0.020 Å^{−1} separation of the peaks at E_{F}. This hole concentration is in agreement with reported values^{12,13,14}. We note that, given the current resolution of the ARPES technique, we are not able to resolve the two hole pockets at the H point reported by the other experimental probe^{15}. In fact, according to ref. 15, the difference in energy between these two hole pockets at the H point is ≤1 meV, which is beyond the current resolution of the ARPES technique. The presence of holes with Dirac fermion dispersion is further supported by the angleintegrated intensity (Fig. 3c), which is proportional to the twodimensional density of states, barring the matrix element. In this energy range, a linear behaviour, similar to that expected for Dirac fermions, is observed. In addition, the energy intersect is at ≈50 meV above E_{F}, in agreement with the Dirac point energy extrapolated from the dispersions.
Figures 1–3 show that near the H point, the lowenergy excitations in graphite are Dirac fermions characterized by linear and isotropic conelike dispersion, in agreement with transport measurements in graphite where Dirac fermions are suggested to coexist with quasiparticles that have finite effective mass^{9}. To gain direct insight into the different types of quasiparticles, ARPES can provide a unique advantage by directly measuring the effective mass as well as accessing its momentum dependence. Figure 4a shows the intensity map near another highsymmetry point in the BZ corner, the K point. The dispersion (open circles) shows a parabolic behaviour, in sharp contrast to the linear behaviour observed near the H point (Figs 1, 3). This parabolic dispersion points to the presence of quasiparticles with finite effective mass. To determine the effective mass, we first extract the lowenergy dispersion, then fit the MDC and energy distribution curve (EDC) dispersions (Fig. 3b,c) with a parabolic function. In both cases, the effective mass is determined to be 0.069±0.015m_{e}, where m_{e} is the freeelectron mass. This effective mass measured by ARPES is different from the values reported by transport measurements^{13,14,16}, 0.052 and 0.038m_{e} for electrons and holes respectively. This difference may be due to the fact that transport measurements are not momentum selective, and therefore the mass measured is the average mass over all k_{z} values. On the other hand, ARPES is momentum selective and the value for the effective mass is only for this specific k_{z} value. Taking this into account, the agreement between these measurements is reasonable. In summary, the data presented so far show that the lowenergy excitations in graphite change from massless Dirac fermions with linear dispersion near H (Figs 1–3) to quasiparticles with parabolic dispersion and finite effective mass near K (Fig. 4a).
We now discuss another interesting feature observed in graphite, that is, a large electron pocket near E_{F}. Figure 4e shows the intensity map measured in the same experimental conditions as Fig. 4d except at a different spot. In Fig. 4e, a strong and large electron pocket within 50 meV below E_{F} is the dominant feature. Weak signatures of the parabolic π band (as in Fig. 4d) can still be observed, as the MDC at E_{F} (Fig. 4f) demonstrates. Here, in addition to the two main peaks (black arrows) corresponding to the electron pocket, a central weak peak (grey arrow) corresponding to the top of the parabolic band can also be distinguished. From the separation (≈0.1 Å^{−1}) between the two main peaks at E_{F}, the electron concentration is determined to be 8.0±0.7×10^{19} cm^{−3}, which is an order of magnitude higher than the values reported^{13,14}. Moreover, from the dispersion (Fig. 4e), the effective mass is extracted to be 0.42±0.07m_{e}, which is also much larger than any mass reported by transport measurements^{13,14,16}.
This large electron pocket is observed in most of the samples measured, and thus it represents an important feature associated with graphite. We note that a similar large electron pocket has been reported recently and proposed to be associated with either edge states or dangling bonds^{12}. Here we provide detailed characterization of this large electron pocket, which is important in revealing its origin. We propose defectinduced localized states as a possible explanation for this large electron pocket, on the basis of the following reasons. First, the electron concentration and effective mass measured for this electron pocket are much larger than reported values. Second, although the parabolic π band in Fig. 4a,d is observed in all the samples measured and in different spots (averaged over ≈100 μm) within the same sample, this large electron pocket strongly depends on the spot position within the same sample. Third, scanning tunnelling microscopy shows that zigzag edges can induce a peak in the local density of states at an energy (≈−0.03 eV) similar to the electron pocket observed here^{17}. In fact, it has been shown that a low concentration of defects (for example, edge states, vacancies and so on) can induce selfdoping to the sample^{18,19}. If this interpretation is correct, then further studies on this large electron pocket may provide insights into the magnetic properties of nanographite ribbons, because it has been proposed that some defectinduced localized states are magnetic^{20,21}.
Methods
Highresolution ARPES data have been taken at Beamlines 12.0.1 and 7.0.1 of the Advanced Light Source in Lawrence Berkeley National Laboratory with photon energies from 20 to 155 eV with energy resolution from 15 to 65 meV. Data were taken from singlecrystal graphite (Kish graphite for Fig. 3a,b and natural graphite for other figures) at a temperature of 25 K. Throughout this paper, the k_{z} values are estimated using the standard freeelectron approximation of the ARPES final state^{22,23,24}. The inner potential needed for extracting the k_{z} value was determined from the periodicity of the detailed dispersion at the Brillouin zone centre ΓA using a wide range of photon energies from 34 to 155 eV. The consistency of the k_{z} values is confirmed by the degeneracy of the π bands near H and a maximum splitting near K (Fig. 5), both in agreement with band structure^{11,25,26}. Note that although this large energy range is good for discussing the splitting of the π bands, it cannot capture the deviation from linear behaviour at the K point expected at low energy (as discussed in Fig. 4).
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Acknowledgements
We thank A. Castro Neto, V. Oganesyan, A. Bill, K. McElroy, C. M. Jozwiak and D. Garcia for useful discussions and E. Domning and B. Smith for beam line 12.0.1 control software. This work was supported by the National Science Foundation through Grant No. DMR0349361, the Director, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering of the US Department of Energy under Contract No. DEAC0376SF00098 and by the Laboratory Directed Research and Development Program of Lawrence Berkeley National Laboratory under the Department of Energy Contract No. DEAC0205CH11231.
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Zhou, S., Gweon, GH., Graf, J. et al. First direct observation of Dirac fermions in graphite. Nature Phys 2, 595–599 (2006). https://doi.org/10.1038/nphys393
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DOI: https://doi.org/10.1038/nphys393
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