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

Ever since its discovery, the Berry phase has permeated through all branches of physics. Over the last three decades, it was gradually realized that the Berry phase of the electronic wave function can have a profound effect on material properties and is responsible for a spectrum of phenomena, such as ferroelectricity, orbital magnetism, various (quantum/anomalous/spin) Hall effects, and quantum charge pumping. This progress is summarized in a pedagogical manner in this review. We start with a brief summary of necessary background, followed by a detailed discussion of the Berry phase effect in a variety of solid state applications. A common thread of the review is the semiclassical formulation of electron dynamics, which is a versatile tool in the study of electron dynamics in the presence of electromagnetic fields and more general perturbations. Finally, we demonstrate a re-quantization method that converts a semiclassical theory to an effective quantum theory. It is clear that the Berry phase should be added as a basic ingredient to our understanding of basic material properties.

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Berry phase effects on electronic properties

We provide a broad review of fundamental electronic properties of two-dimensional graphene with the emphasis on density and temperature dependent carrier transport in doped or gated graphene structures. A salient feature of our review is a critical comparison between carrier transport in graphene and in two-dimensional semiconductor systems (e.g. heterostructures, quantum wells, inversion layers) so that the unique features of graphene electronic properties arising from its gap- less, massless, chiral Dirac spectrum are highlighted. Experiment and theory as well as quantum and semi-classical transport are discussed in a synergistic manner in order to provide a unified and comprehensive perspective. Although the emphasis of the review is on those aspects of graphene transport where reasonable consensus exists in the literature, open questions are discussed as well. Various physical mechanisms controlling transport are described in depth including long- range charged impurity scattering, screening, short-range defect scattering, phonon scattering, many-body effects, Klein tunneling, minimum conductivity at the Dirac point, electron-hole puddle formation, p-n junctions, localization, percolation, quantum-classical crossover, midgap states, quantum Hall effects, and other phenomena.

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Electronic transport in two-dimensional graphene

We present the science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems, targeting an evolution in technology, that might lead to impacts and benefits reaching into most areas of society. This roadmap was developed within the framework of the European Graphene Flagship and outlines the main targets and research areas as best understood at the start of this ambitious project. We provide an overview of the key aspects of graphene and related materials (GRMs), ranging from fundamental research challenges to a variety of applications in a large number of sectors, highlighting the steps necessary to take GRMs from a state of raw potential to a point where they might revolutionize multiple industries. We also define an extensive list of acronyms in an effort to standardize the nomenclature in this emerging field.

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Science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems

Owing to the fact that graphene is just one atom thick, it has been suggested that it might be possible to control its properties by subjecting it to mechanical strain. New analysis indicates not only this, but that pseudomagnetic behaviour and even zero-field quantum Hall effects could be induced in graphene under realistic amounts of strain. Among many remarkable qualities of graphene, its electronic properties attract particular interest owing to the chiral character of the charge carriers, which leads to such unusual phenomena as metallic conductivity in the limit of no carriers and the half-integer quantum Hall effect observable even at room temperature1,2,3. Because graphene is only one atom thick, it is also amenable to external influences, including mechanical deformation. The latter offers a tempting prospect of controlling graphene’s properties by strain and, recently, several reports have examined graphene under uniaxial deformation4,5,6,7,8. Although the strain can induce additional Raman features7,8, no significant changes in graphene’s band structure have been either observed or expected for realistic strains of up to ∼15% (refs 9, 10, 11). Here we show that a designed strain aligned along three main crystallographic directions induces strong gauge fields12,13,14 that effectively act as a uniform magnetic field exceeding 10 T. For a finite doping, the quantizing field results in an insulating bulk and a pair of countercirculating edge states, similar to the case of a topological insulator15,16,17,18,19,20. We suggest realistic ways of creating this quantum state and observing the pseudomagnetic quantum Hall effect. We also show that strained superlattices can be used to open significant energy gaps in graphene’s electronic spectrum.

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Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering

We show that the manifestation of quantum interference in graphene is very different from that in conventional two-dimensional systems. Because of the chiral nature of charge carriers, it is not only sensitive to inelastic, phase-breaking scattering, but also to a number of elastic scattering processes. We study weak localization in different samples and at different carrier densities, including the Dirac region, and find the characteristic rates that determine it. We show how the shape and quality of graphene flakes affect the values of the elastic and inelastic rates and discuss their physical origin.

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Weak localization in graphene flakes.

We show that quantum interference in graphene can result in antilocalization of charge carriers---an increase of the conductance, which is detected by a negative magnetoconductance. We demonstrate that depending on experimental conditions one can observe either weak localization or antilocalization of carriers in graphene. A transition from localization to antilocalization occurs when the carrier density is decreased and the temperature is increased. We show that quantum interference in graphene can survive at high temperatures, up to $T\ensuremath{\sim}200\text{ }\text{ }\mathrm{K}$, due to weak electron-phonon scattering.

https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.103.226801

Transition between electron localization and antilocalization in graphene.

The wave nature of electrons in low-dimensional structures manifests itself in conventional electrical measurements as a quantum correction to the classical conductance. This correction comes from the interference of scattered electrons which results in electron localisation and therefore a decrease of the conductance. In graphene, where the charge carriers are chiral and have an additional (Berry) phase of \pi, the quantum interference is expected to lead to anti-localisation: an increase of the conductance accompanied by negative magnetoconductance (a decrease of conductance in magnetic field). Here we observe such negative magnetoconductance which is a direct consequence of the chirality of electrons in graphene. We show that graphene is a unique two-dimensional material in that, depending on experimental conditions, it can demonstrate both localisation and anti-localisation effects. We also show that quantum interference in graphene can survive at unusually high temperatures, up to T~200 K.

Transition between electron localisation and antilocalisation in graphene

We have performed the first experimental investigation of quantum interference corrections to the conductivity of a bilayer graphene structure. A negative magnetoresistance--a signature of weak localization--is observed at different carrier densities, including the electroneutrality region. It is very different, however, from the weak localization in conventional two-dimensional systems. We show that it is controlled not only by the dephasing time, but also by different elastic processes that break the effective time-reversal symmetry and provide intervalley scattering.

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Weak localization in bilayer graphene.

We propose a method of measuring the electron temperature Te in mesoscopic conductors and demonstrate experimentally its applicability to micron-size graphene devices in the linear-response regime (TeT, the bath temperature). The method can be especially useful in case of overheating, Te > T. It is based on analysis of the correlation function of mesoscopic conductance fluctuations. Although the fluctuation amplitude strongly depends on the details of electron scattering in graphene, we show that Te extracted from the correlation function is insensitive to these details.

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F. V. Tikhonenko

Papers

Weak localization in graphene flakes.

Transition between electron localization and antilocalization in graphene.

Transition between electron localisation and antilocalisation in graphene

Weak localization in bilayer graphene.

Quantum Transport Thermometry for Electrons in Graphene