Energy spectrum for two-dimensional periodic potentials in a magnetic field

Kühn, Oliver; Fessatidis, Vassilios; Cui, H. L.; Selbmann, P.E.; Horing, Norman J. M.

doi:10.1103/physrevb.47.13019

Cited by 28 publications

(32 citation statements)

References 16 publications

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“…The fractal nature of the Hofstadter butterfly had captivated researchers for many years [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. The paper by Hofstadter [5] was for the energy spectrum of a periodic square lattice in the tight-binding approximation and subject to a perpendicular magnetic field.…”

Section: Introductionmentioning

confidence: 99%

“…The paper by Hofstadter [5] was for the energy spectrum of a periodic square lattice in the tight-binding approximation and subject to a perpendicular magnetic field. Ever since that time, there have been complementary calculations for the hexagonal lattice [14], the two-dimensional electron gas (2DEG) with an electrostatic periodic modulation potential [21,22,30], and even bilayer graphene where different stacking of the two types of atoms forming the sublattices was considered [31]. It has been claimed that one may be able to observe evidence of the existence of Hofstadter's butterfly in such experimentally measured quantities as density of states and conductivity of the 2DEG [2,3,33].…”

Section: Introductionmentioning

confidence: 99%

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Revealing Hofstadter spectrum for graphene in a periodic potential

Gumbs

Iurov²,

Huang³

et al. 2014

Phys. Rev. B

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We calculate the energy bands for graphene monolayers when electrons move through a periodic electrostatic potential in the presence of a uniform perpendicular magnetic field. We clearly demonstrate the quantum fractal nature of the energy bands at reasonably low magnetic fields. We present results for the energy bands as functions of both wave number and magnetic flux through the unit cells of the resulting moiré superlattice. The effects due to pseudospin coupling and Landau orbit mixing by a strong scattering potential have been exhibited. At low magnetic fields when the Landau orbits are much larger than the period of the modulation, the Landau levels are only slightly broadened. This feature is also observed at extremely high magnetic fields. The density of states has been calculated and shows a remarkable self-similarity like the energy bands. We estimate that for modulation period of 10 nm the region where the Hofstadter butterfly is revealed is B 2 T.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Revealing Hofstadter spectrum for graphene in a periodic potential

Gumbs

Iurov²,

Huang³

et al. 2014

Phys. Rev. B

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“…[21], and references given therein. Our calculation of the density of states in the k x -k y -plane perpendicular to the magphys.…”

Section: Planar Densities Of the Bloch States In Cubic Latticesmentioning

confidence: 99%

Ensembles of Electron States in Cubic Lattices Quantized in the Presence of a Magnetic Field

Olszewski

Baszczak

2002

phys. stat. sol. (b)

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PACS: 71.15.Dx; 71.70.Di; 73.43.Cd Quantization of the Landau levels and their density versus energy in cubic crystal lattices are examined for the case of weak and strong magnetic fields. Bands of the tightly-bound s-electron states are chosen as examples of the calculations. With the assumption of a constant phase taken in the quantization process, the density of the Landau levels calculated for weak magnetic fields is found practically identical with the planar density of the Bloch states derived in each lattice case. Simultaneously, the frequency of the electron gyration in the lattice due to the action of the external magnetic field becomes proportional to the reciprocal value of the density of the Landau states. For strong magnetic fields the Roth phase and its connection with the Berry phase are also examined. Repercussions of the applied theory to the quanta of the weak-field Hall conductivity are demonstrated.

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“…Surprisingly, the strong field approach is closely related to the strong potential one: if potential-induced coupling between different Landau levels is neglected, the same secular equation is obtained, but with the magnetic field replaced by its inverse. 15 Including such coupling has a profound effect on the calculated energy spectrum, 23,24 even for weak coupling strength. The resulting rearrangement of the Hofstadter butterfly has recently been confirmed by experiments.…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional Bloch electrons in perpendicular magnetic fields: An exact calculation of the Hofstadter butterfly spectrum

2013

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The problem of two-dimensional, independent electrons subject to a periodic potential and a uniform perpendicular magnetic field unveils surprisingly rich physics, as epitomized by the fractal energy spectrum known as Hofstadter's butterfly. It has hitherto been addressed using various approximations rooted in either the strong potential or the strong field limiting cases. Here, we report calculations of the full spectrum of the single-particle Schrödinger equation without further approximations. Our method is exact, up to numerical precision, for any combination of potential and uniform field strength. We first study a situation that corresponds to the strong potential limit, and compare the exact results to the predictions of a Hofstadter-like model. We then go on to analyze the evolution of the fractal spectrum from a Landau-like nearly free electron system to the Hofstadter tight-binding limit by tuning the amplitude of the modulation potential.

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Energy spectrum for two-dimensional periodic potentials in a magnetic field

Cited by 28 publications

References 16 publications

Revealing Hofstadter spectrum for graphene in a periodic potential

Revealing Hofstadter spectrum for graphene in a periodic potential

Ensembles of Electron States in Cubic Lattices Quantized in the Presence of a Magnetic Field

Two-dimensional Bloch electrons in perpendicular magnetic fields: An exact calculation of the Hofstadter butterfly spectrum

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