Aharonov-Bohm effect and broken valley degeneracy in graphene rings

Recher, Patrik; Trauzettel, Björn; Rycerz, Adam; Blanter, Ya. M.; Beenakker, C. W. J.; Morpurgo, Alberto F.

doi:10.1103/physrevb.76.235404

Cited by 292 publications

(330 citation statements)

References 21 publications

(17 reference statements)

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“…The spectral and transport properties of Dirac electrons confined in graphene quantum dots have been investigated analytically [23][24][25] and by numerical means. [26][27][28][29] Also, the energy spectrum and conductance of Aharonov-Bohm rings have been the focus of several publications, [30][31][32] as well as superlattice effects in graphene antidot lattices 33,34 and the density of states of nanoribbon-superconductor junctions. 35 One upshot of these studies is the understanding that the confinement of charge carriers in graphene affects the coherent electron and hole dynamics considerably.…”

Section: A Graphene-based Nanostructuresmentioning

confidence: 99%

Edge effects in graphene nanostructures: From multiple reflection expansion to density of states

2011

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We study the influence of different edge types on the electronic density of states of graphene nanostructures. To this end we develop an exact expansion for the single-particle Green's function of ballistic graphene structures in terms of multiple reflections from the system boundary, which allows for a natural treatment of edge effects. We first apply this formalism to calculate the average density of states of graphene billiards. While the leading term in the corresponding Weyl expansion is proportional to the billiard area, we find that the contribution that usually scales with the total length of the system boundary differs significantly from what one finds in semiconductor-based, Schrödinger-type billiards: The latter term vanishes for armchair and infinite-mass edges and is proportional to the zigzag edge length, highlighting the prominent role of zigzag edges in graphene. We then compute analytical expressions for the density of states oscillations and energy levels within a trajectory-based semiclassical approach. We derive a Dirac version of Gutzwiller's trace formula for classically chaotic graphene billiards and further obtain semiclassical trace formulas for the density of states oscillations in regular graphene cavities. We find that edge-dependent interference of pseudospins in graphene crucially affects the quantum spectrum.

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Section: A Graphene-based Nanostructuresmentioning

confidence: 99%

Edge effects in graphene nanostructures: From multiple reflection expansion to density of states

2011

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“…With this representation, we can see that for a massless field the equation C β j (η, γa) = 0 is reduced to the one given in [29] for graphene rings described by the Dirac model with the infinite mass boundary condition on the edges. Note that to obtain an analytical approximation of the spectrum, in [29] the asymptotic form of the Hankel functions for large arguments was used. This approximation is valid for rings with the radius much larger than the width.…”

Section: Fermionic Modesmentioning

confidence: 99%

“…The first terms in the brackets of the coefficients of the functions J 2 β j (xr) and J 2 β j +ǫ j (xr) are canceled for the contributions coming from the positive-and negative-energy modes. For the remaining part we get 29) that is further simplified to…”

Section: Charge Densitymentioning

confidence: 99%

Induced fermionic charge and current densities in two-dimensional rings

2016

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For a massive quantum fermionic field, we investigate the vacuum expectation values (VEVs) of the charge and current densities induced by an external magnetic flux in a two-dimensional circular ring. Both the irreducible representations of the Clifford algebra are considered. On the ring edges the bag (infinite mass) boundary conditions are imposed for the field operator. This leads to the Casimir type effect on the vacuum characteristics. The radial current vanishes. The charge and the azimuthal current are decomposed into the boundary-free and boundary-induced contributions. Both these contributions are odd periodic functions of the magnetic flux with the period equal to the flux quantum. An important feature that distinguishes the VEVs of the charge and current densities from the VEV of the energy density, is their finiteness on the ring edges. The current density is equal to the charge density for the outer edge and has the opposite sign on the inner edge. The VEVs are peaked near the inner edge and, as functions of the field mass, exhibit quite different features for two inequivalent representations of the Clifford algebra. We show that, unlike the VEVs in the boundary-free geometry, the vacuum charge and the current in the ring are continuous functions of the magnetic flux and vanish for half-odd integer values of the flux in units of the flux quantum. Combining the results for two irreducible representations, we also investigate the induced charge and current in parity and time-reversal symmetric models. The corresponding results are applied to graphene rings with the electronic subsystem described in terms of the effective Dirac theory with the energy gap. If the energy gaps for two valleys of the graphene hexagonal lattice are the same, the charge densities corresponding to the separate valleys cancel each other, whereas the azimuthal current is doubled.

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“…(1)] is implemented with the use of the sparse-matrix solver ARPACK. 52 We note here that, unlike the continuous Dirac-Weyl equations, 16,17 both the K and K ′ valleys are automatically incorporated in the tight-binding treatment of graphene sheets and nanostructures.…”

Section: Basic Elements Of Tb Approachmentioning

confidence: 99%

Graphene flakes with defective edge terminations: Universal and topological aspects, and one-dimensional quantum behavior

2012

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Systematic tight-binding investigations of the electronic spectra (as a function of the magnetic field) are presented for trigonal graphene nanoflakes with reconstructed zigzag edges, where a succession of pentagons and heptagons, that is 5-7 defects, replaces the hexagons at the zigzag edge. For nanoflakes with such reczag defective edges, emphasis is placed on topological aspects and connections underlying the patterns dominating these spectra. The electronic spectra of trigonal graphene nanoflakes with reczag edge terminations exhibit certain unique features, in addition to those that are well known to appear for graphene dots with zigzag edge termination. These unique features include breaking of the particle-hole symmetry, and they are associated with nonlinear dispersion of the energy as a function of momentum, which may be interpreted as nonrelativistic behavior. The general topological features shared with the zigzag flakes include the appearance of energy gaps at zero and low magnetic fields due to finite size, the formation of relativistic Landau levels at high magnetic fields, and the presence between the Landau levels of edge states (the socalled Halperin states) associated with the integer quantum Hall effect. Topological regimes, unique to the reczag nanoflakes, appear within a stripe of negative energies ε b < ε < 0, and along a separate feature forming a constant-energy line outside this stripe. The ε b lower bound specifying the energy stripe is independent of size.Prominent among the patterns within the ε b < ε < 0 energy stripe is the formation of threemember braid bands, similar to those present in the spectra of narrow graphene nanorings; they are associated with Aharonov-Bohm-type oscillations, i.e., the reczag edges along the three sides of the triangle behave like a nanoring (with the corners acting as scatterers) enclosing the magnetic flux through the entire area of the graphene flake. Another prominent feature within the ε b < ε < 0 energy stripe is a subregion of Halperin-type edge states of enhanced density immediately below the zero-Landau level. Furthermore, there are features resulting from localization of the Dirac quasiparticles at the corners of the polygonal flake.A main finding concerns the limited applicability of the continuous Dirac-Weyl equation, since the latter does not reproduce the special reczag features. Due to this discrepancy between the tightbinding and continuum descriptions, one is led to the conclusion that the linearized Dirac-Weyl equation fails to capture essential nonlinear physics resulting from the introduction of a multiple topological defect in the honeycomb graphene lattice.

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Aharonov-Bohm effect and broken valley degeneracy in graphene rings

Cited by 292 publications

References 21 publications

Edge effects in graphene nanostructures: From multiple reflection expansion to density of states

Edge effects in graphene nanostructures: From multiple reflection expansion to density of states

Induced fermionic charge and current densities in two-dimensional rings

Graphene flakes with defective edge terminations: Universal and topological aspects, and one-dimensional quantum behavior

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