Abstract
Having described the transverse quantization of motion into subbands in low-dimensional mesoscopic systems, we will now see how these are utilized to describe coherent transport through devices like quantum billiards within the effective independent-electron picture. This is done within the Landauer-Büttiker theory of transport in multiterminal structures, which relates the scattering matrix of the system to its electrical conductance. After presenting the general framework, we focus on the linear response regime of transport.
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Notes
- 1.
Consider, for example, an open (1) and a closed (2) channel in a lead segment (along x and around x = 0) connecting two scatterers, where the wave function can be written as
$$\displaystyle{ \psi =\psi _{1} +\psi _{2}\;\ \ \ \ \ \psi _{1} =\chi _{1}(ae^{\text{i}kx} + be^{-\text{i}kx}),\ \ \psi _{ 1} =\chi _{2}(ce^{-\kappa x} + de^{\kappa x}),\ \ k,\kappa > 0, }$$χ 1 and χ 2 being the corresponding orthonormal transversal modes . The exponentially increasing part is here physical because of the finite extent of the segment, and originates from the state decaying into it from the right. In contrast to the probability density, where the counterpropagating waves interfere while the decaying modes do not, the total current density consists of an incoherent sum of the propagating mode currents and a coherent combination 2Im(cd ∗)κ from the decaying modes,
$$\displaystyle{ j \propto (\vert a\vert -\vert b\vert )k + (cd^{{\ast}}- d^{{\ast}}c)\text{i}\kappa, }$$which separate upon the y-integration over the orthonormal transversal wave functions. Thus, unless cd ∗ happens to be real, there is a contribution to transport from the closed channels between the scatterers.
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Morfonios, C.V., Schmelcher, P. (2017). Coherent Electronic Transport: Landauer-Büttiker Formalism. In: Control of Magnetotransport in Quantum Billiards. Lecture Notes in Physics, vol 927. Springer, Cham. https://doi.org/10.1007/978-3-319-39833-4_3
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