Coulomb impurity under magnetic field in graphene: A semiclassical approach

Abstract

We address the problem of a Coulomb impurity in graphene in the presence of a perpendicular uniform magnetic field. We show that the problem can be solved below the supercritical impurity magnitude within the WKB approximation. Without impurity the semiclassical energies correctly reproduce the Landau level spectrum. For a given Landau level the WKB energy depends on the absolute value of angular momentum in a way which is consistent with the exact diagonalization result. Below the supercritical impurity magnitude, the WKB solution can be expanded as a convergent series in powers of the effective fine structure constant. Relevance of our results to validity of the widely used Landau level projection approximation is discussed.

Coulomb Impurity Problem of Graphene in Strong Coupling Regime in Magnetic Fields

“…[1][2][3] This is because no complex energy solutions (resonances) are possible in the Coulomb impurity problem in magnetic fields: the effective potential does not allow resonant states since the vector potential diverges while the Coulomb potential goes to zero in the limit r → . [1][2][3][4] Zhang et al 5 have investigated this problem using the WKB method for = 0 and g < 1/2. No analytical solutions are known for general values of g, R, , and B, * Author to whom correspondence should be addressed.…”

Bound states of charges on top of graphene in a magnetic field

“…This has been calculated [19,37,38] and demonstrated experimentally [19] for one Coulomb impurity. Below we consider several impurities.…”

Landau level transitions in doped graphene in a time dependent magnetic field

“…In this case, the Landau levels involved decay for longer times due to the coupled system of differential equations of eq. (19). Is of major interest to study different initial conditions with different time-dependent magnetic fields for larger orders in λ to obtain more detailed functions for the probability amplitudes and the conditions that allow possible cycles between the lowest Landau levels.…”

“…Experimentally, the Landau levels in graphene have been observed by measuring cyclotron resonances of the electrons and holes in infrared spectroscopy experiments [11] and by measuring tunneling current in scanning tunneling spectroscopy experiments [12]. The experimental study of the n = 0 Landau level has attracted a great deal of attention, and has evolved rapidly, as better quality samples become available, and higher magnetic fields and lower temperatures were studied ( [13], [14], [15], [16]), but in particular, Landau level mixing is neglected (see [17], [18]), and this is not as clearly justified because the single particle Landau level gaps in graphene scale as √ B, which is the same as the interaction strength [19]. In turn, only optical transitions between Landau levels has been calculated.…”