STM-D-1046Paper2008Published and peer-reviewed

Analytic Model for the Energy Spectrum of a Graphene Quantum Dot in a Perpendicular Magnetic Field

S. Schnez · K. Ensslin · M. Sigrist · T. Ihn

Abstract and summary · read the original at the source · arXiv non-exclusive distribution licence

In one page

Graphene is a sheet of carbon one atom thick, and its electrons behave strangely: they travel as though they had no mass at all, obeying the Dirac equation that describes fast relativistic particles rather than the Schrödinger equation that governs electrons in an ordinary metal. Schnez, Ensslin, Sigrist and Ihn, at ETH Zürich, cut a disc out of that sheet — a quantum dot about 70 nanometres across — put a magnetic field through it, and solved the problem exactly on paper. Their trick is an old one: pretend the electron becomes infinitely heavy at the rim, which pins it inside without any messy edge chemistry. The answer comes out as a clean condition on Laguerre polynomials, and it contains both limits you would want. Turn the field off and it reproduces the known result for a Dirac billiard, complete with an energy gap that confinement alone creates. Make the dot large and the graphene Landau levels appear. In between, a single ratio decides which physics rules.

Why it matters hereChapter 5 rests on the claim that relativistic behaviour can emerge from an ordinary lattice, and graphene is the cleanest table-top case of it — massless Dirac particles arising from carbon atoms sitting still. This paper is the exactly solvable version of that system: confine the Dirac electrons in a disc, add a magnetic field, and the whole spectrum comes out in closed form, so the emergent picture can be checked against a real measurement rather than argued about.

What it claims

01Starting from the free Dirac equation in cylindrical coordinates with the magnetic field in the symmetric gauge, and applying the infinite-mass boundary condition — a mass-related potential that is zero inside the dot and tends to infinity at its edge, which for circular confinement fixes the ratio of the two spinor components at the rim — the energy spectrum of a circular graphene quantum dot in a perpendicular magnetic field is determined by a single implicit condition on generalised Laguerre polynomials, labelled by a radial quantum number, an angular momentum quantum number and a valley index.Section 2, Derivation, equations 1 to 6
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02The spectrum obeys electron-hole symmetry: the energy of a state in one valley is the exact negative of the energy of the corresponding state in the other valley.Section 2, sentence following equation 6
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03In the zero-field limit the condition reduces to the relation between Bessel functions of neighbouring order derived by Berry and Mondragon for a Dirac billiard. Pairs of states are degenerate at zero magnetic field, and there is no state at both zero field and zero energy, so confinement alone opens an energy gap between the negative and positive energy states.Section 2, equation 8 and the paragraph following
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04In the opposite limit, where the dot radius is large compared with the magnetic length, the same condition returns the graphene Landau levels, proportional to the square root of the magnetic field times an integer. The ratio of the dot radius to the magnetic length is therefore the single parameter governing the crossover from confinement-dominated energies to Landau levels, and the model describes the whole transition including both limiting cases.Section 2, equations 9 to 11
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05Evaluated for a dot of radius 70 nanometres, the lowest positive-energy state lies about 4 millielectronvolts above zero at zero magnetic field, giving a gap of about 8 millielectronvolts between electron and hole states. Because the spacing to the next excited state is much smaller, the electron-hole transition should be detectable experimentally as a confinement-enhanced energy.Section 2, Numerical Results and Comparison to Experiment, paragraphs 1 and 2
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06What to watch: the model gives slopes of the single-particle levels against magnetic field of between 7 and 12 millielectronvolts per tesla, where the authors’ own transport measurements on a 70 nanometre graphene dot gave about 2.5. They name the three things that would close the gap — the real device is not a perfect circle, disorder is expected to matter in graphene nanostructures, and the experiment was not in the single-electron regime so interaction effects belong in a thorough treatment. Tight-binding or quasi-classical simulations can carry all three.Section 2, closing paragraph, and Section 3, Summary
What to watch

Read it · abstract

Abstract

We analytically calculate the energy spectrum of a circular graphene quantum dot with radius R subjected to a perpendicular magnetic field B by applying the infinite-mass boundary condition. We can retrieve well-known limits for the cases R, B going to infinity and B going to zero. Our model is capable of capturing the essential details of recent experiments. Quantitative agreement between theory and experiment is limited due to the fact that a circular dot is not close enough to the experimental geometry, that disorder plays a significant role, and that interaction effects may be relevant.

S. Schnez, K. Ensslin and T. Ihn of the Solid State Physics Laboratory and M. Sigrist of the Institute for Theoretical Physics, ETH Zürich, Analytic model of the energy spectrum of a graphene quantum dot in a perpendicular magnetic field, arXiv:0810.3216, version 2 dated 27 February 2009; published as Physical Review B 78, article 195427, 26 November 2008, with a correction at Physical Review B 95, article 039901, 2017. The manuscript is at arxiv.org/abs/0810.3216 and the published article at doi.org/10.1103/PhysRevB.78.195427.

(Abstract only — see the rights note above. On this site, the companion treatment of Dirac electrons in graphene wires and dots is at /library/stm-5579f2cb8e, graphene ripples read as a two-dimensional Ising system are at /library/stm-cdb7b34f9e, the proposal to harvest energy from those ripples is at /library/stm-b278fb0be1, the measured fluctuation-induced current from freestanding graphene is at /library/stm-8f7ed94186, and the Casimir force between graphene sheets is at /library/stm-7e76215e73.)

The way in

https://arxiv.org/abs/0810.3216SOURCE REACHED AND READ IN FULL. The manuscript, version 2 dated 27 February 2009, was fetched from arXiv as 0810.3216 and read end to end; every claim below carries its equation or section from that text. The version of record is Physical Review B volume 78, article 195427, published 26 November 2008 under the American Physical Society’s default licence, and a correction was later issued as Physical Review B volume 95, article 039901, 10 January 2017. The arXiv posting carries arXiv’s non-exclusive distribution licence, which is not a Creative Commons licence and does not permit republication, so this page holds the summary, the claims and the authors’ own abstract and sends the reader to the source. The authors are at the Solid State Physics Laboratory and the Institute for Theoretical Physics at ETH Zürich; the publisher records give initials rather than given names, so the initials are kept here.

How to cite it

S. Schnez, K. Ensslin, M. Sigrist, T. Ihn (2008) Analytic Model for the Energy Spectrum of a Graphene Quantum Dot in a Perpendicular Magnetic Field. doi:10.1103/PhysRevB.78.195427

Where it sits in the curriculum

The vacuum as a quantum fluid