Reentrant metallic behavior of graphite in the quantum limit

Quantum-Hall physics and three dimensions

The discovery of the quantum Hall effect (QHE) in 1980 marked a turning point in condensed matter physics: given appropriate experimental conditions, the Hall conductivityσxyof a two-dimensional electron system is exactly quantized. But what happens to the QHE in three dimensions (3D)? Experiments over the past 40 years showed that some of the remarkable physics of the QHE, in particular plateau-like Hall conductivitiesσxyaccompanied by minima in the longitudinal resistivityρxx, can also be found in 3D materials. However, since typicallyρxxremains finite and a quantitative relation betweenσxyand the conductance quantume2/hcould not be established, the role of quantum Hall physics in 3D remains unsettled. Following a recent series of exciting experiments, the QHE in 3D has now returned to the center stage. Here, we summarize the leap in understanding of 3D matter in magnetic fields emerging from these experiments.

Helical Edge States and Quantum Phase Transitions in Tetralayer Graphene

Helical conductors with spin-momentum locking are promising platforms for Majorana fermions. Here we report observation of two topologically distinct phases supporting helical edge states in charge neutral Bernal-stacked tetralayer graphene in Hall bar and Corbino geometries. As the magnetic field B_{⊥} and out-of-plane displacement field D are varied, we observe a phase diagram consisting of an insulating phase and two metallic phases, with 0, 1, and 2 helical edge states, respectively. These phases are accounted for by a theoretical model that relates their conductance to spin-polarization plateaus. Transitions between them arise from a competition among interlayer hopping, electrostatic and exchange interaction energies. Our work highlights the complex competing symmetries and the rich quantum phases in few-layer graphene.

Extremely high magnetoresistance and conductivity in the type-II Weyl semimetals WP2 and MoP2

The peculiar band structure of semimetals exhibiting Dirac and Weyl crossings can lead to spectacular electronic properties such as large mobilities accompanied by extremely high magnetoresistance. In particular, two closely neighboring Weyl points of the same chirality are protected from annihilation by structural distortions or defects, thereby significantly reducing the scattering probability between them. Here we present the electronic properties of the transition metal diphosphides, WP2 and MoP2, which are type-II Weyl semimetals with robust Weyl points by transport, angle resolved photoemission spectroscopy and first principles calculations. Our single crystals of WP2 display an extremely low residual low-temperature resistivity of 3 nΩ cm accompanied by an enormous and highly anisotropic magnetoresistance above 200 million % at 63 T and 2.5 K. We observe a large suppression of charge carrier backscattering in WP2 from transport measurements. These properties are likely a consequence of the novel Weyl fermions expressed in this compound.

Charge density waves in strongly correlated electron systems

Strong electron correlations are at the heart of many physical phenomena of current interest to the condensed matter community. Here we present a survey of the mechanisms underlying such correlations in charge density wave (CDW) systems, including the current theoretical understanding and experimental evidence for CDW transitions. The focus is on emergent phenomena that result as CDWs interact with other charge or spin states, such as magnetism and superconductivity. In addition to reviewing the CDW mechanisms in 1D, 2D, and 3D systems, we pay particular attention to the prevalence of this state in two particular classes of compounds, the high temperature superconductors (cuprates) and the layered transition metal dichalcogenides. The possibilities for quantum criticality resulting from the competition between magnetic fluctuations and electronic instabilities (CDW, unconventional superconductivity) are also discussed.

Negative magnetoresistance without well-defined chirality in the Weyl semimetal TaP

Weyl semimetals (WSMs) are topological quantum states wherein the electronic bands disperse linearly around pairs of nodes with fixed chirality, the Weyl points. In WSMs, nonorthogonal electric and magnetic fields induce an exotic phenomenon known as the chiral anomaly, resulting in an unconventional negative longitudinal magnetoresistance, the chiral-magnetic effect. However, it remains an open question to which extent this effect survives when chirality is not well-defined. Here, we establish the detailed Fermi-surface topology of the recently identified WSM TaP via combined angle-resolved quantum-oscillation spectra and band-structure calculations. The Fermi surface forms banana-shaped electron and hole pockets surrounding pairs of Weyl points. Although this means that chirality is ill-defined in TaP, we observe a large negative longitudinal magnetoresistance. We show that the magnetoresistance can be affected by a magnetic field-induced inhomogeneous current distribution inside the sample.

Drastic Pressure Effect on the Extremely Large Magnetoresistance in WTe2: Quantum Oscillation Study

The quantum oscillations of the magnetoresistance under ambient and high pressure have been studied for WTe2 single crystals, in which extremely large magnetoresistance was discovered recently. By analyzing the Shubnikov-de Haas oscillations, four Fermi surfaces are identified, and two of them are found to persist to high pressure. The sizes of these two pockets are comparable, but show increasing difference with pressure. At 0.3 K and in 14.5 T, the magnetoresistance decreases drastically from 1.25×10(5)% under ambient pressure to 7.47×10(3)% under 23.6 kbar, which is likely caused by the relative change of Fermi surfaces. These results support the scenario that the perfect balance between the electron and hole populations is the origin of the extremely large magnetoresistance in WTe2.

Quantum Oscillations, Thermoelectric Coefficients, and the Fermi Surface of Semimetallic WTe2

We present a study of angle-resolved quantum oscillations of electric and thermoelectric transport coefficients in semimetallic WTe2, which has the particularity of displaying a large B(2) magnetoresistance. The Fermi surface consists of two pairs of electronlike and holelike pockets of equal volumes in a "Russian doll" structure. The carrier density, Fermi energy, mobility, and the mean-free path of the system are quantified. An additional frequency is observed above a threshold field and attributed to the magnetic breakdown across two orbits. In contrast to all other dilute metals, the Nernst signal remains linear in the magnetic field even in the high-field (ωcτ≫1) regime. Surprisingly, none of the pockets extend across the c axis of the first Brillouin zone, making the system a three-dimensional metal with moderate anisotropy in Fermi velocity, yet a large anisotropy in the mean-free path.

Anisotropic giant magnetoresistance in NbSb2

The magnetic field response of the transport properties of novel materials and then the large magnetoresistance effects are of broad importance in both science and application. We report large transverse magnetoreistance (the magnetoresistant ratio ~ 1.3 × 10(5)% in 2 K and 9 T field, and 4.3 × 10(6)% in 0.4 K and 32 T field, without saturation) and field-induced metal-semiconductor-like transition, in NbSb2 single crystal. Magnetoresistance is significantly suppressed but the metal-semiconductor-like transition persists when the current is along the ac-plane. The sign reversal of the Hall resistivity and Seebeck coefficient in the field, plus the electronic structure reveal the coexistence of a small number of holes with very high mobility and a large number of electrons with low mobility. The large MR is attributed to the change of the Fermi surface induced by the magnetic field which is related to the Dirac-like point, in addition to orbital MR expected for high mobility metals.

ELECTRONIC PROPERTIES OF GRAPHENE, FEW-LAYER GRAPHENE, AND BULK GRAPHITE UNDER VERY HIGH MAGNETIC FIELD

In the present work, we report on the magneto-transport properties of graphitic based materials (graphene, few-layer graphene, and bulk graphite) in very high magnetic field. Quantum Hall Effect (QHE) has been studied in graphitic systems in very high pulsed magnetic field (up to B = 57 T ) and at low temperature (≤ 4 K). Graphene sample shows well-defined Hall resistance plateaus at filling factors v = 2,6,10, etc. Few-layer graphene systems display clear signatures of standard and unconventional QHE. Magneto-transport studies on bulk highly oriented pyrolytic graphite show a charge density wave transition at strong enough magnetic field as well as Hall coefficient sign reversal.

High magnetic field induced charge density waves and sign reversal of the Hall coefficient in graphite

We report on the investigation of magnetic field induced charge density waves and Hall coefficient sign reversal in a quasi-two-dimensional electronic system of highly oriented pyrolytic graphite under very strong magnetic field. The change of Hall sign coefficient from negative to positive occurs at low temperature and high magnetic field just after the charge density wave transition, suggesting the role of hole-like quasi-particles in this effect. Angular dependent measurements show that the charge density wave transition and Hall sign reversal fields follow the magnetic field component along the c-axis of graphite.

Electronic properties of a biased graphene bilayer

We study, within the tight-binding approximation, the electronic properties of a graphene bilayer in the presence of an external electric field applied perpendicular to the system-a biased bilayer. The effect of the perpendicular electric field is included through a parallel plate capacitor model, with screening correction at the Hartree level. The full tight-binding description is compared with its four-band and two-band continuum approximations, and the four-band model is shown to always be a suitable approximation for the conditions realized in experiments. The model is applied to real biased bilayer devices, made out of either SiC or exfoliated graphene, and good agreement with experimental results is found, indicating that the model is capturing the key ingredients, and that a finite gap is effectively being controlled externally. Analysis of experimental results regarding the electrical noise and cyclotron resonance further suggests that the model can be seen as a good starting point for understanding the electronic properties of graphene bilayer. Also, we study the effect of electron-hole asymmetry terms, such as the second-nearest-neighbour hopping energies t' (in-plane) and γ(4) (inter-layer), and the on-site energy Δ.

Disordered electrical potential observed on the surface of SiO2 by electric field microscopy

The electrical potential on the surface of ∼300 nm thick SiO(2) grown on single-crystalline Si substrates has been characterized at ambient conditions using electric field microscopy. Our results show an inhomogeneous potential distribution with fluctuations up to ∼0.4 V within regions of 1 µm. The potential fluctuations observed at the surface of these usual dielectric holders of graphene sheets should induce strong variations in the graphene charge densities and provide a simple explanation for some of the anomalous behaviors of the transport properties of graphene.

Searching for the fractional quantum Hall effect in graphite

Measurements of basal plane longitudinal rho(b)(B) and Hall rho(H)(B) resistivities were performed on highly oriented pyrolytic graphite samples in a pulsed magnetic field up to B=50 T applied perpendicular to graphene planes, and temperatures 1.5 K<or=T<or=4.2 K. At B>30 T and for all studied samples, we observed a sign change in rho(H)(B) from electron- to holelike. For our best quality sample, the measurements revealed the enhancement in rho(b)(B) for B>34 T (T=1.8 K), presumably associated with the field-driven charge density wave or Wigner crystallization transition. In addition, well-defined plateaus in rho(H)(B) were detected in the ultraquantum limit revealing possible signatures of the fractional quantum Hall effect in graphite.

A high-magnetic-field-induced density-wave state in graphite

Whilst the realization of graphene, probably one of the best two-dimensional carrier systems to study, has attracted much research interest recently, graphite, which may be regarded as multi-layered graphene, has also been known to exhibit very interesting phenomena at high magnetic fields and low temperatures. The electron-hole system in the compensated semimetal graphite undergoes a magnetic-field-induced electronic phase transition and successive transitions, including a reentrant transition back to the normal phase, at higher magnetic fields. In this article, we review the physics of the high-magnetic-field phase of graphite and elaborate on our studies on this subject using pulsed high magnetic fields.

Influence of interlayer tunneling on the quantized Hall phases in intentionally disordered multilayer structures

Stability of the quantized Hall phases is studied in weakly coupled multilayers as a function of the interlayer correlations controlled by the interlayer tunneling and by the random variation of the well thicknesses. A strong enough interlayer disorder destroys the symmetry responsible for the quantization of the Hall conductivity, resulting in the breakdown of the quantum Hall effect. A clear difference between the dimensionalities of the metallic and insulating quantum Hall phases is demonstrated. The sharpness of the quantized Hall steps obtained in the coupled multilayers with different degrees of randomization was found consistent with the calculated interlayer tunneling energies. The observed width of the transition between the quantized Hall states in random multilayers is explained in terms of the local fluctuations of the electron density.

Localized distributions of quasi-two-dimensional electronic states near defects artificially created at graphite surfaces in magnetic fields

We measured the local density of states of a quasi two-dimensional electron system (2DES) near defects, artificially created by Ar-ion sputtering, on surfaces of highly oriented pyrolytic graphite (HOPG) with scanning tunneling spectroscopy (STS) in high magnetic fields. At valley energies of the Landau level spectrum, we found two typical localized distributions of the 2DES depending on the defects. These are new types of distributions which are not observed in the previous STS work at the HOPG surface near a point defect [Y. Niimi, Phys. Rev. Lett. 97, 236804 (2006).10.1103/PhysRevLett.97.236804]. With increasing energy, we observed gradual transformation from the localized distributions to the extended ones as expected for the integer quantum Hall state. We show that the defect potential depth is responsible for the two localized distributions from comparison with theoretical calculations.

The metal-insulator-like and insulator-metal-like behaviors in antimony

The resistivity and Hall resistivity of semimetallic antimony were measured from 2 to 300 K in magnetic fields up to 14 T. We found that in low field, the resistivity shows metallic behavior. In a modest field, the resistivity decreases to a minimum and then increases with decreasing temperature, showing a metal-insulator-like transition. In high field, the resistivity drops at low temperatures, showing an insulator-metal-like transition. The metal-insulator-like behavior can be explained by the competition of zero field resistivity and magneto-resistance, which is reciprocal to the zero field resistivity. The insulator-metal-like behavior can be explained by the imbalance of two carrier densities which changes the magneto-resistance from being reciprocal to proportional to the zero field resistivity.

Low-density ferromagnetism in biased bilayer graphene

We compute the phase diagram of a biased graphene bilayer. The existence of a ferromagnetic phase is discussed with respect to both carrier density and temperature. We find that the ferromagnetic transition is first-order, lowering the value of U relatively to the usual Stoner criterion. We show that in the ferromagnetic phase the two planes have unequal magnetization and that the electronic density is holelike in one plane and electronlike in the other.

Lattice-induced double-valley degeneracy lifting in graphene by a magnetic field

We show that the recently discovered double-valley splitting of the Landau levels in the quantum Hall effect in graphene can be explained as the perturbative orbital interaction of intravalley and intervalley microscopic orbital currents with a magnetic field. This effect is facilitated by the translationally noninvariant terms that correspond to graphene's crystallographic honeycomb symmetry but do not exist in the relativistic theory of massless Dirac fermions in quantum electrodynamics. We discuss recent data in view of these findings.

Sample-size effects in the magnetoresistance of graphite

Conduction electrons in graphite are expected to have micrometer large de Broglie wavelength as well as mean free path. A direct influence of these lengths in the electric transport properties of finite-size samples was neglected in the past. We provide a direct evidence of this effect through the size dependence of the magnetoresistance, which decreases with the sample size even for samples hundreds of micrometers large. Our findings may explain the absence of magnetoresistance in small few graphene layers samples and ask for a general revision of the experimental and theoretical work on the transport properties of this material.

Theory of the three-dimensional quantum Hall effect in graphite

We predict the existence of a three-dimensional quantum Hall effect plateau in a graphite crystal subject to a magnetic field. The plateau has a Hall conductivity quantized at 4e2/variant Planck's over 2pi 1/c0 with c0 the c-axis lattice constant. We analyze the three-dimensional Hofstadter problem of a realistic tight-binding Hamiltonian for graphite, find the gaps in the spectrum, and estimate the critical value of the magnetic field above which the Hall plateau appears. When the Fermi level is in the bulk Landau gap, Hall transport occurs through the appearance of chiral surface states. We estimate the magnetic field necessary for the appearance of the effect to be 15.4 T for electron carriers and 7.0 T for holes.

Oscillating Nernst-Ettingshausen effect in bismuth across the quantum limit

In elemental bismuth, 10(5) atoms share a single itinerant electron. Therefore, a moderate magnetic field can confine electrons to the lowest Landau level. We report on the first study of metallic thermoelectricity in this regime. The main thermoelectric response is off-diagonal with an oscillating component several times larger than the nonoscillating background. When the first Landau level attains the Fermi energy, both the Nernst and the Ettingshausen coefficients sharply peak, and the latter attains a temperature-independent maximum. These features are yet to be understood. We note a qualitative agreement with a theory invoking current-carrying edge excitations.

Nernst effect in semimetals: the effective mass and the figure of merit

We present a study of electric, thermal, and thermoelectric transport in elemental bismuth, which presents a Nernst coefficient much larger than what was found in correlated metals. We argue that this is due to the combination of an exceptionally low carrier density with a very long electronic mean-free path. The low thermomagnetic figure of merit is traced to the lightness of electrons. Heavy-electron semimetals, which keep a metallic behavior in the presence of a magnetic field, emerge as promising candidates for thermomagnetic cooling at low temperatures.

Dirac and normal fermions in graphite and graphene: implications of the quantum Hall effect

Spectral analysis of the Shubnikov-de Haas magnetoresistance oscillations and the quantum Hall effect (QHE) measured in quasi-2D highly oriented pyrolytic graphite (HOPG) [Phys. Rev. Lett. 90, 156402 (2003)] reveals two types of carriers: normal (massive) electrons with Berry phase 0 and Dirac-like (massless) holes with Berry phase pi. We demonstrate that recently reported integer- and semi-integer QHEs for bilayer and single-layer graphenes take place simultaneously in HOPG samples.

Real-space imaging of alternate localization and extension of quasi-two-dimensional electronic States at graphite surfaces in magnetic fields

We measured the local density of states (LDOS) of a quasi-two-dimensional (2D) electron system near point defects on a surface of highly oriented pyrolytic graphite with scanning tunneling microscopy and spectroscopy. Differential tunnel conductance images taken at very low temperatures and in high magnetic fields show a clear contrast between localized and extended spatial distributions of the LDOS at the valley and peak energies of the Landau level spectrum, respectively. The localized electronic state has a single circular distribution around the defects with a radius comparable to the magnetic length. The localized LDOS is in good agreement with a spatial distribution of a calculated wave function for a single electron in 2D in a Coulomb potential in magnetic fields.

Intervalley scattering, long-range disorder, and effective time-reversal symmetry breaking in graphene

We discuss the effect of certain types of static disorder, like that induced by curvature or topological defects, on the quantum correction to the conductivity in graphene. We find that when the intervalley scattering time is long or comparable to tau(phi), these defects can induce an effective time-reversal symmetry breaking of the Hamiltonian associated to each one of the two valleys in graphene. The phenomenon suppresses the magnitude of the quantum correction to the conductivity and may result in the complete absence of a low-field magnetoresistance, as recently found experimentally. Our work shows that a quantitative description of weak localization in graphene must include the analysis of new regimes, not present in conventional two-dimensional electron gases.

Electrostatic force microscopy on oriented graphite surfaces: coexistence of insulating and conducting behaviors

We present measurements of the electric potential fluctuations on the surface of highly oriented pyrolytic graphite using electrostatic force and atomic force microscopy. Micrometric domainlike potential distributions are observed even when the sample is grounded. Such potential distributions are unexpected given the good metallic conductivity of graphite because the surface should be an equipotential. Our results indicate the coexistence of regions with "metalliclike" and "insulatinglike" behaviors showing large potential fluctuations of the order of 0.25 V. In lower quality graphite, this effect is not observed. Experiments are performed in Ar and air atmospheres.

Unconventional integer quantum Hall effect in graphene

Monolayer graphite films, or graphene, have quasiparticle excitations that can be described by (2+1)-dimensional Dirac theory. We demonstrate that this produces an unconventional form of the quantized Hall conductivity sigma(xy) = -(2e2/h)(2n+1) with n = 0, 1, ..., which notably distinguishes graphene from other materials where the integer quantum Hall effect was observed. This unconventional quantization is caused by the quantum anomaly of the n=0 Landau level and was discovered in recent experiments on ultrathin graphite films.

STS observations of Landau levels at graphite surfaces

Scanning tunneling spectroscopy (STS) measurements were made on surfaces of two different kinds of graphite samples, Kish graphite and highly oriented pyrolytic graphite (HOPG), at very low temperatures and in high magnetic fields. We observed a series of peaks in the tunnel spectra associated with Landau quantization of the quasi-two-dimensional electrons and holes. A comparison with the calculated local density of states at the surface layers allows us to identify Kish graphite as bulk graphite and HOPG as graphite with a finite thickness of 40 layers. This explains the qualitative difference between the two graphites reported in the recent transport measurements which suggested the quantum-Hall effect in HOPG. This work demonstrates how powerful the combined approach between the high quality STS measurement and the first-principles calculation is in material science.

Electric field modulation of galvanomagnetic properties of mesoscopic graphite

Electric field effect devices based on mesoscopic graphite are fabricated for galvanomagnetic measurements. Strong modulation of magnetoresistance and Hall resistance as a function of the gate voltage is observed as the sample thickness approaches the screening length. Electric field dependent Landau level formation is detected from Shubnikov-de Haas oscillations. The effective mass of electron and hole carriers has been measured from the temperature dependent behavior of these oscillations.

Metal-insulator-like behavior in semimetallic bismuth and graphite

When high quality bismuth or graphite crystals are placed in a magnetic field directed along the c axis (trigonal axis for bismuth) and the temperature is lowered, the resistance increases as it does in an insulator but then saturates. We show that the combination of unusual features specific to semimetals, i.e., low carrier density, small effective mass, high purity, and an equal number of electrons and holes (compensation), gives rise to a unique ordering and spacing of three characteristic energy scales, which not only is specific to semimetals but which concomitantly provides a wide window for the observation of apparent field-induced metal-insulator behavior. Using magnetotransport and Hall measurements, the details of this unusual behavior are captured with a conventional multiband model, thus confirming the occupation by semimetals of a unique niche between conventional metals and semiconductors.

Phase analysis of quantum oscillations in graphite

The quantum de Haas-van Alphen (dHvA) and Shubnikov-de Haas oscillations measured in graphite were decomposed by pass-band filtering onto contributions from three different groups of carriers. Generalizing the theory of dHvA oscillations for 2D carriers with an arbitrary spectrum and by detecting the oscillation frequencies using a method of two-dimensional phase-frequency analysis which we developed, we identified these carriers as (i) minority holes having a 2D parabolic massive spectrum p(2)(perpendicular)/2m(perpendicular), (ii) massive majority electrons with a 3D spectrum and (iii) majority holes with a 2D Dirac-like spectrum +/-vp(perpendicular) which seems to be responsible for the unusual strongly-correlated electronic phenomena in graphite.

The Elusive Bose Metal

The conventional theory of metals is in crisis. In the past 15 years, there has been an unexpected sprouting of metallic states in low-dimensional systems, directly contradicting conventional wisdom. For example, bosons are thought to exist in one of two ground states: condensed in a superconductor or localized in an insulator. However, several experiments on thin metal-alloy films have observed that a metallic phase disrupts the direct transition between the superconductor and the insulator. We analyze the experiments on the insulator-superconductor transition and argue that the intervening metallic phase is bosonic. All relevant theoretical proposals for the Bose metal are discussed, particularly the recent idea that the metallic phase is glassy. The implications for the putative vortex-glass state in the copper oxide superconductors are examined.