My question is regarding Andreev bound states and their transmission probabilities. But to make this self-contained, lets quickly recap, for which I will draw from Tosi, L., Metzger, C., Goffman, M. F., Urbina, C., Pothier, H., Park, S.,Krogstrup, P. (2018). Spin-orbit splitting of Andreev states revealed by microwave spectroscopy as I like their descriptions:

The Josephson supercurrent that flows through a weak link between two superconductors is a manifestation of the coherence of the many-body superconducting state. The link can be an insulating barrier, a small piece of normal metal, a constriction or any other type of coherent conductor. Regardless of its specific nature the supercurrent is a periodic function of the phase difference $\delta$ between the electrodes. However, the exact function is determined by the geometry and material properties of the weak link. A unifying microscopic description of the effect has been achieved in terms of the spectrum of discrete quasiparticle states that form at the weak link: the Andreev bound states (ABS).

Andreev bound states are formed from the coherent Andreev reflections that quasiparticles undergo at both ends of a weak link. Quasiparticles acquire a phase at each of these Andreev reflections and while propagating along the weak link of length $L$. Therefore, the ABS energies depends on $\delta$, on the transmission probabilities for electrons through the weak link and on the ratio $\lambda = L/\xi$ where $\xi$ is the superconducting coherence length. Assuming ballistic propagation, $\xi = \hbar v_F/\Delta$ is given in terms of the velocity $v_F$ of quasiparticles at the Fermi level within the weak link and of the energy gap $\Delta$ of the superconducting electrodes. In a short junction, defined by $L \gg \xi$, each of the $N$ conduction channels of the weak link, with transmission probability $\tau_i$, gives rise to a single spin-degenerate Andreev level at energy $E_{A,i} = \sqrt{1-\tau_i\sin^2{\delta/2}}$

Now, if I have $N$ conduction channels occupied at chemical potential $\mu$, and assume parabolic dispersion, no SOI, and no Zeeman field, there is a clear hierarchy in Fermi velocities of the aforementioned conduction channels. The 'deepest' lying band (its band minimum is furthest away from $\mu$) will have the highest $v_F$, while the outer most subband will have the lowest $v_F$. My question is then as follows. Does this then also mean that the outer most subband will always have the lowest transmission probability $\tau_N$? Is it true that $\tau_i \geq \tau_j$, for $i>j$? Can we directly relate $\tau_i$ to $v_{F,i}$, or does that require specific assumptions, and when are those not satisfied? I know at least one case in which it is true; in the (excellent, in my opinion) thesis by Bretheau he derives an expression for $\tau_i$ when the junction hosts a delta scatterer of strength $V$ such that $\tau = \frac{1}{1+\left(\frac{m_eV/\hbar^2}{k_F}\right)^2}$, where one does have this hierarchy. What about the case of a smooth barrier potential? And what if every conduction band does not have to see the same scattering potential (say due to where the wavefunction lives in the 3D structure)? Could that lead to a case where this is broken? And finally, and perhaps crucially, what about if the junction is disordered?


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