Theory of Shubnikov-de Haas and Quantum Hall Oscillations in Graphene under Bias and Gate Voltages
Magnetic oscillations in graphene under gate and bias voltages, measured by Tan et al.[Phys. Rev.B84, 115429 (2011)] are analyzed theoretically. The Shubnikove-de Haas (SdH) oscillations occur at the lower fields while the Quantum Hall (QH) oscillations occur at the higher fields. Both SdH and QH oscillations have the same periods: εF/ωc, where εF is the Fermi energy and ωc the cyclotron frequency. Since the phases are different by π/2, transitions between the maxima and the minima occur at some magnetic field strength. A quantum statistical theory of the SdH oscillations is developed. A distinctive feature of two-dimensional (2D) magnetic oscillations is the absence of the background. That is, the envelopes of the oscillations approach zero with zero-slope central line. The amplitude of the SdH oscillations decreases like [sinh(2π2M∗kBT/eB)]−1, where M∗ is the magnetotransport mass of the field-dressed electron distinct from the cyclotron mass m∗ of the electron. A theory of the QHE is developed in terms of the composite (c)-bosons and c-fermions. The half-integer QHE in graphene at filling factor ν = (2P + 1)/2, P = 0,±1,±2, · · · arises from the Bose-Einstein condensation of the c-bosons formed by the phonon exchange between a pair of like-charge c-fermions with two fluxons. The QH states are bound and stablized with a superconducting energy gap. They are more difficult to destroy than the SdH states. The temperature dependence of the magnetic resistance between 2 K and 50 K is interpreted, using the population change of phonons (scatterers).
