Conclusions

Quantum scattering theory in the 2D half-plane can provide an alternative description of the mesoscopic conductance of non-interacting particles. It is especially useful in `open' systems (e.g. those with nearby scatterers in the reservoir regions) where the usual transverse-channel approach is inappropriate. We have considered elastic potentials in zero magnetic field, in linear response in the low temperature limit. Conductance is proportional to the transmission cross section integrated over all incident angles, Eq.(7.1). We also define a half-plane partial-wave basis applicable with the usual Landauer formula, and relate this to our transmission cross section result (Appendix L). A difference between this and previous work is the ability to treat a direct `leadless' connection to the reservoir.

We have studied the case of a single slit in a thin hard wall in detail. Its conductance (7.16) in the small-aperture limit arises entirely from its p-wave transmission coefficient, which can be found by matching to solutions of Laplace's equation (Appendix K). Using the example of a slit QPC combined with an open cavity structure, we show that an arbitrarily small QPC can carry up to a single quantum of conductance via resonant tunnelling (equal to the limit in the closed-dot resonant tunnelling case). This requires a resonance at the Fermi energy. If $n$ coincident resonances occur for different incoming channels, then $n$ conductance quanta can in theory be achieved through this same tunneling QPC, a result which we believe has not been noted previously.

We emphasize that conductance is proportional to phase-space density of the reservoir states. Therefore the universal quantum of conductance $e^2/h$ per spin in Fermi gas systems is a direct result of the uniform phase-space density (angular distribution) in a thermal occupation of the Fermi sea. This insight is supported by discussion of attempts to exceed this universal value. When the reservoir occupation differs from thermal, the conductance formula requires generalization: an angle-dependent weight is included in the cross section integral (7.25), equivalently the Landauer formula requires inclusion of the incoming ensemble (7.26). This result, and our approach in general, is relevant to the emerging field of matter-wave conductance by microfabricated structures (for instance, a quantum point contact in 3D), under general illumination by atom waves. We hope this work provides new tools for the study of coherent electron and matter-wave systems.