The proximity-induced gap

So far, the subsystem in contact with the semiconductor that acts as charge reservoir was assumed to be a normal metal. What happens when this metal turns into a superconductor. As discussed in chapter 2, the semiconductor will pick up superconducting correlations via the proximity effect. Physically, this can be under- stood in terms of a nonzero probability to find the electrons from the semiconductor inside the superconductor. As a result of this effect, a proximity-induced gap opens in the semiconductor spectrum at the chemical potential. The question is how is this proximity-induced gap affected by the interface-induced potential? Here, we

focus on the magnitude of the proximity-induced gap in a thin film at zero magnetic field in the presence of an interface-induced potential. Since the magnetic field and the Rashba spin-orbit coupling are not included, the induced superconducting state discussed here is topologically trivial.

The superconducting proximity effect can be formally incorporated into the the- ory as a surface self-energy term. A self-energy term is naturally obtained in the Green function formalism once we integrate out the degrees of freedom associated with the superconductor. Our simplifying working assumption is that the nonlocal contributions to the self-energy are negligible on length scales 1/kF relevant to the low-energy problem, where kF is the typical Fermi momentum of the semiconductor system. Consequently, we approximate the self-energy by a purely local contribu- tion. Note that this assumption was used in all the calculations for both TI-based and SM-based heterostructures [see, for example, Eq. (2.38)]. The information about the proximity-induced gap can be extracted from the density of states,

DOS (ω) = −1 πIm  T r  1 ω + iη − Hef f − Σ(ω)  . (4.1)

Note that Hef f in Eq. (4.1) denotes the Hamiltonian for the semiconductor and the interface-induced electrostatic potential.

The dependence of the proximity-induced gap on the filling factor and on the thickness of the semiconductor slab is shown in Fig. 4.5. First, note the different scales in panels (a) and (b). As expected, increasing the film thickness results in a lower amplitude of the wave functions at the interface and, consequently, in a smaller induced gap. Concerning the dependence on the filling factor (or, equivalently, on the chemical potential), we note that the induced gap tends to increase with increasing filling factor. However, the dependence is discontinuous and even non-monotonic in

Figure 4.5: Proximity-induced gap as a function of the filling factor for two different values of the film thickness: (a) 40 nm and (b) 200 nm. The discontinuities in the dependence of the gap on the filling factor are caused by the chemical potential crossing the bottom of different sub-bands.

the thick film case. We identify these discontinuities with filling factors corresponding to values of the chemical potential exactly at the bottom of some confinement-induced band. We note that for comparison with experiment a similar calculation has to done for the 1D case. While some details will be different, we expect the behavior in the 1D case to be qualitatively similar to these findings.

As mentioned above, the common feature seen in Fig. 4.5, i.e., the discontinuities of the proximity-induced gap as function the filling factor, is related to the chemical potential crossing the the bottom of confinement-induced sub-bands. This behavior has some similarities with that observed in TI-based systems and shown in Fig. 2.18. Note, however, that the physics is different. In the TI case it was a finite size effect: the surface-type states have finite “widths” and, consequently, experience an effective magnetic flux smaller than the nominal value corresponding to the full cross section of the nanoribbon. Since the characteristic “width” is band dependent, the degeneracy condition is not realized for all bands at exactly the same value of the magnetic field, which results in a sharp drop of the induced gap when the chemical

Figure 4.6: Induced superconducting gap as a function of the effective semiconductor- superconductor coupling.

potential is the vicinity of a nearly degenerate point. By contrast, the discontinuities in the semiconductor structure originate in the different “rigidities” of the wave functions associated with different bands. As pointed out in the context of Fig. 4.4- a, wave functions from higher energy bands tend to be less localized in the potential minimum near the interface. Consequently, when the chemical potential crosses into a new band the (minimum) wave function amplitude at the interface is realized by states from the top (new) band, hence it undergoes a discontinuous change. This conclusion has to be carefully verified in the 1D case (i.e., for semiconductor wires), possibly using a more detailed description of the SM band structure, such as an 8-band Kane-type model. Finally, we note that in Fig. 4.5-a the proximity-induced gap is nearly fixed at 0.29 eV for filling factors in the range 0.6−1.6%, which suggests that thin slabs are more favorable for realizing a robust superconducting state.

The superconducting correlations are (formally) introduced by the self-energy term. We note that the self-energy depends on the Green function of the supercon- ductor at the interface and on the effective semiconductor-superconductor coupling.

This coupling is a key parameter that determines many of the low-energy properties of the structure, including the induced gap. We have already seen that the topo- logical phase diagram shows a strong dependence on the effective coupling, see Fig. 3.4. The dependence of the induced gap on the the effective coupling parameter for a fixed potential profile is plotted in Fig. 4.6. The size of the gap increase with increasing the effective coupling. Note, however, that the dependence is not linear, which reflects the fact that the induced gap cannot be larger that the bulk SC gap regardless of the coupling strength.