The low energy electronic excitations in single layer and bilayer graphite (graphene) resemble quantum-relativistic particles also known as Dirac Fermions (DF). They possess an internal degree of freedom, chirality, that leads to unusual Landau Level (LL) energy sequences in a magnetic field and profoundly alters the magneto-transport properties. One of the consequences is an anomalous Quantum-Hall effect, recently detected in both single layer and bi-layer graphene. However the underlying cause, the unusual LL sequence, was never observed. Here we report the direct observation of LL of DF by means of low temperature Scanning-Tunnelling-Spectroscopy (STS) on the surface of graphite in fields up to 12 Tesla. We find evidence of coexistence of massless and massive DF, and identify the zero-energy LL which is a unique consequence of their quantum-relativistic nature. Surprisingly these strictly two-dimensional properties emerge even on bulk graphite in samples where the interlayer coupling is weak.
Electronic properties of surface areas decoupled from graphite are studied using scanning tunnelling microscopy and spectroscopy. We show that it is possible to identify decoupled graphene monolayer, Bernal bilayer, and Bernal trilayer on graphite surface according to their tunnelling spectra in high magnetic field. The decoupled monolayer and bilayer exhibit Landau quantization of massless and massive Dirac fermions, respectively. The substrate generates a sizable band gap, ~35 meV, in the Bernal bilayer, therefore, the eightfold degenerate Landau level at the charge neutrality point is split…
The band structure of ABC-stacked N-layer graphene comprises topologically corresponding flat surface and gapped bulk subbands, as a consequence of the unique stacking configuration. In this paper, the bulk subbands are for the first times ever obtained for arbitrary N. A non-perturbative effective Hamiltonian closed in the bulk subspace is derived and used. The gapped bulk subbands are shown to evolve towards the zero energy with increasing N and in the infinite limit, they touch linearly along a circle. This outcome is a manifestation of the dimensional crossover to a three-dimensional Dirac…
The quantum Hall effect (QHE) originates from discrete Landau levels forming in a two-dimensional (2D) electron system in a magnetic field. In three dimensions (3D), the QHE is forbidden because the third dimension spreads Landau levels into multiple overlapping bands, destroying the quantisation. Here we report the QHE in graphite crystals that are up to hundreds of atomic layers thick - thickness at which graphite was believed to behave as a 3D bulk semimetal. We attribute the observation to a dimensional reduction of electron dynamics in high magnetic fields, such that the electron spectrum…
Dirac and Normal Fermions in Graphite and Graphene: Implications to the Quantum Hall Effect
在准2D高取向热解石墨（HOPG）中测量的磁阻和量子霍尔效应（QHE）的Shubnikov de Haas（SdH）振荡的光谱分析[Phys.Rev.Lett. 90，156402（2003）]揭示了两种类型的载流子：具有Berry相位0的正常（有质量）电子和具有Berry相位π的Dirac类（无质量）空穴。我们证明，最近报道的整数和半整数QHE的双层和单层石墨烯同时发生在HOPG样品。
Three dimensional (3D) topological insulators (TIs) are an important class of materials with applications in electronics, spintronics and quantum computing. With the recent development of truly bulk insulating 3D TIs, it has become possible to realize surface dominated phenomena in electrical transport measurements e.g. the quantum Hall (QH) effect of massless Dirac fermions in topological surface states (TSS). However, to realize more advanced devices and phenomena, there is a need for a platform to tune the TSS or modify them e.g. gap them by proximity with magnetic insulators, in a clean ma…
Comment on "Consistent Interpretation of the Low-Temperature Magnetotransport in Graphite Using the Slonczewski-Weiss-McClure 3D Band-Structure Calculations" (arXiv:0902.1925)
在2004年，我们已经表明，石墨中的导电性的很大一部分是由无质量的线性光谱的空穴-狄拉克费米子，与大量的正常载流子-电子共存。在最近的一封信[物理。102，166403（2009），arXiv：0902.1925] Schneider等人修改了我们的结论，指出两种类型的载体都是巨大的。由于这两个群体使用相同的方法来确定Shubnikov de Haas振荡的相位，我们在这里评论说，争议源于Schneider 2009等人对实验结果的不当处理。
The high-magnetic-field thermoelectric effect in nodal-line semimetals with straight nodal lines was investigated. Three-dimensional (3D) Dirac/Weyl semimetals exhibit constant thermoelectric Hall conductivity at the high-magnetic-field quantum limit, resulting in a boundless linear increase of Seebeck coefficient. This is known as the quantized thermoelectric Hall effect (QTHE), and is expected to lead to high-performance thermoelectricity at low temperatures. Here, in addition to Dirac/Weyl semimetals, we demonstrated that 3D semimetals with straight nodal lines can also exhibit the QTHE und…
Using low temperature scanning tunneling spectroscopy, we probe the Landau levels of the topologically protected state of Sb2Te3(0001) after in-situ cleavage of a single crystal. Landau levels are visible for magnetic fields B > 2 T at energies, which confirm the Dirac type dispersion including the zeroth Landau level. We find different Dirac velocities for the lower and the upper part of the Dirac cone in reasonable agreement with previous density functional theory data. The Dirac point deduced from the zeroth Landau level shifts by about 40 meV between different areas of the sample indicatin…
Topologically non-trivial states characterized by Berry curvature appear in a number of materials ranging from spin-orbit-coupling driven topological insulators to graphene. In multivalley conductors, such as mono- and bilayer graphene, despite a zero total Chern number for the entire Brillouin zone, Berry curvature with different signs concentrated in different valleys can affect the observable material's transport characteristics. Here we consider thin films of rhombohedral graphite, which appear to retain truly two-dimensional properties up to tens of layers of thickness and host two-dimens…
Recent transport experiments have revealed the activation of longitudinal magnetoresistance of Weyl semimetals in the quantum limit, suggesting the breakdown of chiral anomaly in a strong magnetic field. Here we provide a general mechanism for gapping the zeroth chiral Landau levels applicable for both Dirac and Weyl semimetals. Our result shows that the zeroth Landau levels anticross when the magnetic axis is perpendicular to the Dirac/Weyl node separation and when the inverse magnetic length $l_B^{-1}$ is comparable to the node separation scale $Δk$. The induced bulk gap increases rapidly be…
We calculate the electronic structure of AA'AA'...-stacked alternating twist N-layer (tNG) graphene for N = 3, 4, 5, 6, 8, 10, 20 layers and bulk alternating twist (AT) graphite systems where the lattice relaxations are modeled by means of molecular dynamics simulations. We show that the symmetric AA'AA'... stacking is energetically preferred among all interlayer sliding geometries for progressively added layers up to N=6. Lattice relaxations enhance electron-hole asymmetry, and reduce the magic angles with respect to calculations with fixed tunneling strengths that we quantify from few layers…
In a system with Dirac cones, spatial modulation in material parameters induces a pseudo magnetic field, which acts like an external magnetic field. Here, we derive a concise formula to count the pseudo Landau levels in the simplest setup for having a pseudo magnetic field. The formula is so concise that it is helpful in seeing the essence of the phenomenon, and in considering the experimental design for the pseudo magnetic field. Furthermore, it is revealed that anisotropic Dirac cones are advantageous in pseudo Landau level formation in general. The proposed setup is relatively easy to be re…
A three-dimensional weak topological insulator (WTI), being equivalent to stacked layers of two-dimensional quantum spin-Hall insulators, accommodates massless Dirac electrons on its side surface. A notable feature of WTIs is that surface states typically consist of two Dirac cones in the reciprocal space. We study the Landau quantization of Dirac electrons of WTIs in a perpendicular magnetic field. It is shown that when the magnetic length $l_B$ is much larger than the interlayer distance $a$, surface electrons are quantized into Landau levels according to the ordinary quantization rule for D…
We study the electronic and magnetic properties of multilayer quantum dots (MQDs) of graphite in the nearest-neighbor approximation of tight-binding model. We calculate the electronic density of states and orbital susceptibility of the system as function of the Fermi level location. We demonstrate that properties of MQD depend strongly on the shape of the system, on the parity of the layer number and on the form of the cluster edge. The special emphasis is given to reveal the new properties with respect to the monolayer quantum dots of graphene. The most interesting results are obtained for th…
We study a gapped graphene monolayer in a combination of uniform magnetic field and strain-induced uniform pseudomagnetic field. The presence of two fields completely removes the valley degeneracy. The resulting density of states shows a complicated behaviour that can be tuned by adjusting the strength of the fields. We analyze how these features can be observed in the sublattice, valley and full density of states. The analytical expression for the valley DOS is derived.
Anomalously sharp (delta-function-like) $n=0$ Landau level in the presence of disorder is usually considered to be a manifestation of the massless Dirac fermions in magnetic fields. This property persists even when the Dirac cone is tilted, which has been shown by Kawarabayashi et al. [Phys. Rev. B {\bf 83}, 153414 (2011)] to be a consequence of a "generalized chiral symmetry". Here we pose a question whether this property will be washed out when the tilted Dirac fermion becomes massive. Surprisingly, the levels persist to be delta-function-like, although the mass term that splits $n=0$ Landau…
This review aims at a theoretical discussion of Dirac points in two-dimensional systems. Whereas Dirac points and Dirac fermions are prominent low-energy electrons in graphene (two-dimensional graphite), research on Dirac fermions in low-energy physics has spread beyond condensed-matter systems. In these alternative systems, a large versatility in the manipulation of the relevant band parameters can be achieved. This allows for a systematic study of the motion and different possible fusions of Dirac points, which are beyond the physical limits of graphene. We introduce the basic properties of …
We study transport across p-n junctions of gapped two-dimensional semi-Dirac materials: nodal semimetals whose energy bands disperse quadratically and linearly along distinct crystal axes. The resulting electronic properties --- relevant to materials such as TiO$_2$/VO$_2$ multilayers and $α$-(BEDT-TTF)$_2$I$_3$ salts --- continuously interpolate between those of mono- and bi-layer graphene as a function of propagation angle. We demonstrate that tunneling across the junction depends on the orientation of the tunnel barrier relative to the crystalline axes, leading to strongly non-monotonic cur…
At low temperatures, graphite presents a magnetoresistance anomaly which manifests as a transition to a high-resistance state (HRS) above a certain critical magnetic field $\text{B}_\text{c}$. Such HRS is currently attributed to a c-axis charge-density-wave taking place only when the lowest Landau level is populated. By controlling the charge carrier concentration of a gated sample through its charge neutrality level (CNL), we were able to experimentally modulate the HRS in graphite for the first time. We demonstrate that the HRS is triggered both when electrons and holes are the majority carr…
