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Register a new userSemiclassical quantization of electrons in magnetic fields: the generalized Peierls substitution
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A generalized Peierls substitution which takes into account a Berry phase term must be considered for the semiclassical treatment of electrons in a magnetic field. This substitution turns out to be an essential element for the correct determination of the semiclassical equations of motion as well as for the semiclassical Bohr-Sommerfeld quantization condition for energy levels. A general expression for the cross-sectional area is derived and used as an illustration for the calculation of the energy levels of Bloch and Dirac electrons.
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This paper provides a review of developments in the physics of two-dimensional electron systems in perpendicular magnetic fields.
The S-matrix theory formulation of closed-orbit theory recently proposed by Granger and Greene is extended to atoms in crossed electric and magnetic fields. We then present a semiclassical quantization of the hydrogen atom in crossed fields, which succeeds in resolving individual lines in the spectrum, but is restricted to the strongest lines of each n-manifold. By means of a detailed semiclassical analysis of the quantum spectrum, we demonstrate that it is the abundance of bifurcations of closed orbits that precludes the resolution of finer details. They necessitate the inclusion of uniform semiclassical approximations into the quantization process. Uniform approximations for the generic types of closed-orbit bifurcation are derived, and a general method for including them in a high-resolution semiclassical quantization is devised.
We carry out an explicit calculation of the vacuum polarization tensor for an effective low-energy model of monolayer graphene in the presence of a weak magnetic field of intensity $B$ perpendicularly aligned to the membrane. By expanding the quasiparticle propagator in the Schwinger proper time representation up to order $(eB)^2$, where $e$ is the unit charge, we find an explicitly transverse tensor, consistent with gauge invariance. Furthermore, assuming that graphene is radiated with monochromatic light of frequency $omega$ along the external field direction, from the modified Maxwells equations we derive the intensity of transmitted light and the angle of polarization rotation in terms of the longitudinal ($sigma_{xx}$) and transverse ($sigma_{xy}$) conductivities. Corrections to these quantities, both calculated and measured, are of order $(eB)^2/omega^4$. Our findings generalize and complement previously known results reported in literature regarding the light absorption problem in graphene from the experimental and theoretical points of view, with and without external magnetic fields.
We measure the Hall conductivity, $sigma_{xy}$, on a Corbino geometry sample of a high-mobility AlGaAs/GaAs heterostructure in a pulsed magnetic field. At a bath temperature about 80 mK, we observe well expressed plateaux in $sigma_{xy}$ at integer filling factors. In the pulsed magnetic field, the Laughlin condition of the phase coherence of the electron wave functions is strongly violated and, hence, is not crucial for $sigma_{xy}$ quantization.
The longitudinal resistivity of two dimensional (2D) electrons placed in strong magnetic field is significantly reduced by applied electric field, an effect which is studied in a broad range of magnetic fields and temperatures in GaAs quantum wells with high electron density. The data are found to be in good agreement with theory, considering the strong nonlinearity of the resistivity as result of non-uniform spectral diffusion of the 2D electrons. Inelastic processes limit the diffusion. Comparison with the theory yields the inelastic scattering time of the two dimensional electrons. In the temperature range T=2-10(K) for overlapping Landau levels, the inelastic scattering rate is found to be proportional to T^2, indicating a dominant contribution of the electron-electron scattering to the inelastic relaxation. In a strong magnetic field, the nonlinear resistivity demonstrates scaling behavior, indicating a specific regime of electron heating of well-separated Landau levels. In this regime the inelastic scattering rate is found to be proportional to T^3, suggesting the electron-phonon scattering as the dominant mechanism of the inelastic relaxation.
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