Quantum point contacts with smooth geometries: Exact versus approximate results

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Abstract
Transport through quantum point contacts (QPC’s) with various geometries close to those used experimentally is studied in the ballistic regime. We neglect impurity and temperature effects in an effort to understand the detailed status, as a function of geometry, of various popular approximations in this field, namely, the local adiabatic, the global adiabatic, and the diagonal approximation. By appeal to a combination of two well-known procedures, we are in a position to study, by exact numerical solution, continuous but rapidly varying geometries of QPC’s. Our calculations show that the diagonal approximation is unreliable (as found previously by Castaño and Kirczenow). The local adiabatic approximation (Glazman et al.) can serve as a rough estimate, but significantly underestimates the sharpness of conductance steps. The global adiabatic approximation is remarkably successful, in spite of strong mode mixing. However, this approximation can also significantly underestimate the degree of conductance quantization. The physical reason for this is given. Resonance effects in continuous QPC’s are found that confirm our physical interpretation.