The electronic properties of graphene
Open accessAbstract
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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Non-relativistic electrons in graphene monolayer obey the massless relativistic dispersion known as the Dirac cone, leading to realization of analogies to Zitterbewegung and Klein tunneling in low-energy condensed-matter physics and atom optics.
“Recently, the non-relativistic electrons in graphene monolayer were found to obey the massless relativistic dispersion known as the Dirac cone, leading to the realization of analogies to Zitterbewegung and Klein tunneling in low-energy condensed-matter physics and atom optics 4 -19 .”
From Observation of interaction-induced phenomena of relativistic quantum mechanics (2021) · cites this paper as [12]
Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.
Experimental — accuracy not yet measuredThe first type of Dirac points occur in lattices with unbroken time-reversal symmetry, occur in pairs, and are stabilized by a symmetry of the underlying lattice.
“Dirac points in graphene1–3 or photonic graphene12 belong to the first type, which occur in lattices with unbroken T. These occur in pairs and are stabilized by a symmetry of the underlying lattice.”
From Observation of an unpaired photonic Dirac point (2020) · cites this paper as [3]
Section II.B, “Dirac fermions” (page unavailable), describes graphene’s two Dirac points at K and K′ and says their wavefunctions are related by time-reversal symmetry. The cited passages do not establish the broader claim that this type of Dirac point is stabilized by an underlying lattice symmetry, or address photonic graphene. (LLM, conf=0.85)
Experimental — accuracy not yet measuredIn isotropic graphene, c_v = c_d, the lattice is hexagonal and tunnelling is equivalent in all directions, preserved by six-fold symmetry.
“In the isotropic graphene, c v = c d , the lattice is hexagonal and the tunnelling is equivalent in all directions, which is preserved by the six-fold symmetry [1].”
From Observing movement of Dirac cones from single-photon dynamics (2021) · cites this paper as [1]
Section II.A (page unavailable) describes graphene as hexagonal and uses one hopping energy, t, for its three nearest-neighbor bonds, supporting equivalent nearest-neighbor hopping in the model. The cited passages do not establish the specific equality c_v = c_d or state that tunnelling is equivalent in all directions; the model also distinguishes nearest- from next-nearest-neighbor hopping. (LLM, conf=0.83)
Experimental — accuracy not yet measuredPhotonic Dirac points are known to occur in pairs in photonic graphene and other similar photonic crystals.
“photonic Dirac points are known to occur in pairs in “photonic graphene” and other similar photonic crystals”
From Observation of an unpaired photonic Dirac point (2020) · cites this paper as [3]
The “Dirac fermions” section describes two electronic Dirac points, K and K′, in graphene. Searches for “photonic graphene photonic crystals Dirac points pairs” and “photonic” found no passage addressing whether photonic Dirac points occur in pairs in photonic graphene or similar crystals. (LLM, conf=0.78)
Experimental — accuracy not yet measuredThe DO model is obtained from the free Dirac equation by introducing an external potential via a non-minimal coupling.
“The DO model is obtained from the free Dirac equation by the introduction of the external potential via a non-minimal coupling [57,59,63].”
From Classical simulation of relativistic quantum mechanics in periodic optical structures (2011) · cites this paper as [57]
The paper’s “B. Dirac fermions” section derives a massless Dirac-like description of electrons in graphene, but does not describe the DO model. A further search for “Dirac oscillator” and “non-minimal coupling” found no passage explaining the claimed construction; the coupling example retrieved uses minimal coupling instead. (LLM, conf=0.72)
Experimental — accuracy not yet measured
The Dirac oscillator (DO) is an exactly solvable model in relativistic quantum mechanics.
“The relativistic extension of the quantum harmonic oscillator, the so-called Dirac oscillator (DO) [57,58,59], provides a paradigmatic and exactly solvable model in relativistic quantum mechanics.”
From Classical simulation of relativistic quantum mechanics in periodic optical structures (2011) · cites this paper as [57]
The paper’s discussion of Dirac fermions in graphene uses harmonic-oscillator operators to solve a magnetic-field problem, but does not identify that problem as the Dirac oscillator. Searches for “Dirac oscillator” and its description as a relativistic extension of the quantum harmonic oscillator found no passage addressing the claimed model. The indexed passages provide section labels but no page numbers. (LLM, conf=0.76)
Experimental — accuracy not yet measuredThe analogy of Hamiltonian (2) with the Jaynes-Cummings model allows relating zitterbewegung and Rabi oscillations.
“Interestingly, the analogy of Hamiltonian (2) with the JaynesCummings model allows us to relate ZB and Rabi oscillations, as other authors have pointed out 7,29,30 .”
From Revivals, classical periodicity, and zitterbewegung of electron currents in monolayer graphene (2009) · cites this paper as [30]
The authors are able to create approximately quadratic electric potentials which, due to Klein tunneling, are nonconfining in the relativistic limit.
“We are also able to create approximately quadratic electric potentials which, due to Klein tunneling, are nonconfining in the relativistic limit [18].”
From Quantum Simulation of the Klein Paradox with Trapped Ions (2011) · cites this paper as [18]
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Graphene and graphene oxide: biofunctionalization and applications in biotechnology
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A review of chemical vapour deposition of graphene on copper
Mattevi, C., Kim, H., & Chhowalla, M. (2010) · Journal of Materials Chemistry · doi:10.1039/c0jm02126a · Cited by 1,456 · see citing works
Carbon Nanomaterials in Biosensors: Should You Use Nanotubes or Graphene?
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Graphene-based materials in electrochemistry
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Evidence of Silicene in Honeycomb Structures of Silicon on Ag(111)
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A review on mechanics and mechanical properties of 2D materials—Graphene and beyond
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High electron mobility, quantum Hall effect and anomalous optical response in atomically thin InSe
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Self-citationCarbon nanomaterials for electronics, optoelectronics, photovoltaics, and sensing
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Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride
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<i>Colloquium</i> : Andreev reflection and Klein tunneling in graphene
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Phonon-limited mobility in n -type single-layer MoS 2 from first principles
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Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy
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Dirac charge dynamics in graphene by infrared spectroscopy
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- Authors
- Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S., & Geim, A. K.
- Affiliations
- Boston University (US)
- Venue
- Reviews of Modern Physics
- Publisher
- American Physical Society (APS)
- Published
- 2009-01-14
- Volume
- 81
- Issue
- 1
- Pages
- 109-162
- ISSN
- 0034-6861 (ISSN-L)
- 1539-0756
- OpenAlex
- W2105685140
- Cited by
- 24,803
- References
- 459
- Citations by year
- 2026: 853 · 2025: 1031 · 2024: 1100 · 2023: 1204 · 2022: 1363 · 2021: 1537 · 2020: 1727 · 2019: 1637 · 2018: 1709 · 2017: 1741 · 2016: 1787 · 2015: 1711 · 2014: 1637 · 2013: 1542 · 2012: 1544
- FWCI
- 673.10
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Reviews of Modern Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Research Explorer (The University of Manchester) (submittedVersion)
- DIGITAL.CSIC (Spanish National Research Council (CSIC)) (submittedVersion, other-oa)
- arXiv (Cornell University)
- Funders
- National Science Foundation (0343790)
- Ministerio de Economía y Competitividad
- Engineering and Physical Sciences Research Council
- Division of Materials Research
- Topics
- Physical Sciences › Materials Science › Materials Chemistry › Graphene research and applications
- Physical Sciences › Materials Science › Materials Chemistry › 2D Materials and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and electron transport phenomena
- Keywords
- graphene, Dirac fermions, two-dimensional Dirac materials, quantum Hall effect, tunneling, confinement, stacking sequence, multilayer graphene, edge states, armchair and zigzag edges, nanoribbons, disorder, electron-electron interaction, electron-phonon coupling, transport properties, spectral properties
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