Graphene is a rapidly rising star on the horizon of materials science and condensed-matter physics. This strictly two-dimensional material exhibits exceptionally high crystal and electronic quality, and, despite its short history, has already revealed a cornucopia of new physics and potential applications, which are briefly discussed here. Whereas one can be certain of the realness of applications only when commercial products appear, graphene no longer requires any further proof of its importance in terms of fundamental physics. Owing to its unusual electronic spectrum, graphene has led to the emergence of a new paradigm of 'relativistic' condensed-matter physics, where quantum relativistic phenomena, some of which are unobservable in high-energy physics, can now be mimicked and tested in table-top experiments. More generally, graphene represents a conceptually new class of materials that are only one atom thick, and, on this basis, offers new inroads into low-dimensional physics that has never ceased to surprise and continues to provide a fertile ground for applications.

The rise of graphene

This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.

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The electronic properties of graphene

Quantum electrodynamics (resulting from the merger of quantum mechanics and relativity theory) has provided a clear understanding of phenomena ranging from particle physics to cosmology and from astrophysics to quantum chemistry. The ideas underlying quantum electrodynamics also influence the theory of condensed matter, but quantum relativistic effects are usually minute in the known experimental systems that can be described accurately by the non-relativistic Schrodinger equation. Here we report an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon) in which electron transport is essentially governed by Dirac's (relativistic) equation. The charge carriers in graphene mimic relativistic particles with zero rest mass and have an effective 'speed of light' c* approximately 10(6) m s(-1). Our study reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions. In particular we have observed the following: first, graphene's conductivity never falls below a minimum value corresponding to the quantum unit of conductance, even when concentrations of charge carriers tend to zero; second, the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; and third, the cyclotron mass m(c) of massless carriers in graphene is described by E = m(c)c*2. This two-dimensional system is not only interesting in itself but also allows access to the subtle and rich physics of quantum electrodynamics in a bench-top experiment.

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Two-dimensional gas of massless Dirac fermions in graphene

Two-dimensional materials are attractive for use in next-generation nanoelectronic devices because, compared to one-dimensional materials, it is relatively easy to fabricate complex structures from them. The most widely studied two-dimensional material is graphene, both because of its rich physics and its high mobility. However, pristine graphene does not have a bandgap, a property that is essential for many applications, including transistors. Engineering a graphene bandgap increases fabrication complexity and either reduces mobilities to the level of strained silicon films or requires high voltages. Although single layers of MoS(2) have a large intrinsic bandgap of 1.8 eV (ref. 16), previously reported mobilities in the 0.5-3 cm(2) V(-1) s(-1) range are too low for practical devices. Here, we use a halfnium oxide gate dielectric to demonstrate a room-temperature single-layer MoS(2) mobility of at least 200 cm(2) V(-1) s(-1), similar to that of graphene nanoribbons, and demonstrate transistors with room-temperature current on/off ratios of 1 × 10(8) and ultralow standby power dissipation. Because monolayer MoS(2) has a direct bandgap, it can be used to construct interband tunnel FETs, which offer lower power consumption than classical transistors. Monolayer MoS(2) could also complement graphene in applications that require thin transparent semiconductors, such as optoelectronics and energy harvesting.

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Single-layer MoS2 transistors

We report the measurement of the thermal conductivity of a suspended single-layer graphene. The room temperature values of the thermal conductivity in the range ∼(4.84 ± 0.44) × 103 to (5.30 ± 0.48) × 103 W/mK were extracted for a single-layer graphene from the dependence of the Raman G peak frequency on the excitation laser power and independently measured G peak temperature coefficient. The extremely high value of the thermal conductivity suggests that graphene can outperform carbon nanotubes in heat conduction. The superb thermal conduction property of graphene is beneficial for the proposed electronic applications and establishes graphene as an excellent material for thermal management.

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Superior Thermal Conductivity of Single-Layer Graphene

Each time a vortex enters or exits a small superconductor, a different fluxoid state develops which can be characterized by its vorticity, i.e., the number of fluxoids inside. We have studied magnetization response of such individual states and found clear signatures of first and second order transitions within the states, which reveal the existence of distinct vortex phases for a fixed number of fluxoids. We attribute the transitions to the merger of individual vortices into a single giant vortex and switching between different arrays of vortices.

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Fine structure in magnetization of individual fluxoid states

We have used ballistic Hall micromagnetometry to study the magnetization of individual submicrometer nickel disks (80 nm high, $0.1--1.0\ensuremath{\mu}\mathrm{m}$ diameter). At low temperatures, hysteresis loops of the disks no longer show inversion symmetry in a magnetic field, as if the time reversal symmetry were broken. Furthermore, the magnetization of the smallest disks can be ``frozen'' in two possible states that are characterized by hysteresis loops which are each other's inverse. At temperatures below 19.5 K a magnetic field as high as 2 T cannot switch between the states, proving that it is extremely difficult to fully polarize a small ferromagnetic particle. On the other hand, at slightly higher temperatures (only $Tg19.8 \mathrm{K}),$ a field as low as 0.1 T appears to be enough to fully polarize the disks. We attribute this extraordinary behavior to the glass-liquid transition experienced by spins at the particle surface.

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Memory effects in individual submicrometer ferromagnets

We report on mesoscopic Hall sensors made from various materials and their suitability for accurate magnetization studies of submicron samples over a wide temperature range and, especially, at room temperature. Among the studied devices, the best stability and sensitivity have been found for Hall probes made from a high-concentration two-dimensional electron gas (HC-2DEG). Even at 300 K, such submicron probes can reliably resolve local changes in dc magnetic field of ≈1 G, which corresponds to a flux sensitivity of less than 0.1 φ0 (φ0=h/e is the flux quantum). The resolution increases 100 times at temperatures below 80 K. It is also much higher for the detection of ac magnetic fields because resistance fluctuations limiting the low-frequency stability of the studied devices can be eliminated. Our second choice for room-temperature Hall micromagnetometry is gold Hall probes, which can show a sensitivity of the order of 10 G. The capabilities of HC-2DEG and gold micromagnetometers are demonstrated by measuring nm-scale movements of individual domain walls in a ferromagnet.

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Submicron probes for Hall magnetometry over the extended temperature range from helium to room temperature

We report quenching of the Hall effect with increasing magnetic field confined in a micron-sized spot. Such fields were created by placing tall ferromagnetic pillars on top of a two-dimensional electron gas, which allowed us to achieve the field strength up to 0.4 T under the pillars in the absence of external field. The quenching is accompanied by an anomalous increase in resistance and occurs when the cyclotron diameter matches the size of the magnetic spot. The results are explained by a rapid increase in the number of electrons that are scattered or quasilocalized by the magnetic region.

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Scattering of ballistic electrons at a mesoscopic spot of strong magnetic field

We analyze theoretically and experimentally vortex configurations in mesoscopic superconducting squares. Our theoretical approach is based on the analytical solution of the London equation using Green's-function method. The potential-energy landscape found for each vortex configuration is then used in Langevin-type molecular-dynamics simulations to obtain stable vortex configurations. Metastable states and transitions between them and the ground state are analyzed. We present our results of the first direct visualization of vortex patterns in micrometer-sized Nb squares, using the Bitter decoration technique. We show that the filling rules for vortices in squares with increasing applied magnetic field can be formulated, although in a different manner than in disks, in terms of formation of vortex ``shells.''

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S. V. Dubonos

Papers

Fine structure in magnetization of individual fluxoid states

Memory effects in individual submicrometer ferromagnets

Submicron probes for Hall magnetometry over the extended temperature range from helium to room temperature

Scattering of ballistic electrons at a mesoscopic spot of strong magnetic field

Vortex states in mesoscopic superconducting squares: Formation of vortex shells