Christopher Reeg
ContactDepartment of PhysicsUniversity of Basel Klingelbergstrasse 82 CH4056 Basel, Switzerland

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Example textPublications
Show all abstracts.1.  Finitesize effects in a nanowire strongly coupled to a thin superconducting shell 
Christopher Reeg, Daniel Loss, and Jelena Klinovaja. arXiv:1707.08417
We study the proximity effect in a onedimensional nanowire strongly coupled to a finite superconductor with a characteristic size which is much shorter than its coherence length. Such geometries have become increasingly relevant in recent years in the experimental search for Majorana fermions with the development of thin epitaxial Al shells which form a very strong contact with either InAs or InSb nanowires. So far, however, no theoretical treatment of the proximity effect in these systems has accounted for the finite size of the superconducting film. We show that the finitesize effects become very detrimental when the level spacing of the superconductor greatly exceeds its energy gap. Without any finetuning of the size of the superconductor (on the scale of the Fermi wavelength), the tunneling energy scale must be larger than the level spacing in order to reach the hard gap regime which is seen ubiquitously in the experiments. However, in this regime, the large tunneling energy scale induces a large shift in the effective chemical potential of the nanowire and pushes the topological phase transition to magnetic field strengths which exceed the critical field of Al.
 
2.  Lowfield topological threshold in Majorana double nanowires 
Constantin Schrade, Manisha Thakurathi, Christopher Reeg, Silas Hoffman, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 96, 035306 (2017)
A hard proximityinduced superconducting gap has recently been observed in semiconductor nanowire systems at low magnetic fields. However, in the topological regime at high magnetic fields, a soft gap emerges and represents a fundamental obstacle to topologically protected quantum information processing with Majorana bound states. Here we show that in a setup of double Rashba nanowires that are coupled to an swave superconductor and subjected to an external magnetic field along the wires, the topological threshold can be significantly reduced by the destructive interference of direct and crossedAndreev pairing in this setup, precisely down to the magnetic field regime in which current experimental technology allows for a hard superconducting gap. We also show that the resulting Majorana bound states exhibit sufficiently short localization lengths, which makes them ideal candidates for future braiding experiments.
 
3.  Transport signatures of topological superconductivity in a proximitycoupled nanowire 
Christopher Reeg and Dmitrii L. Maslov. Phys. Rev. B 95, 205439 (2017)
We study the conductance of a junction between the normal and superconducting segments of a nanowire, both of which are subject to spinorbit coupling and an external magnetic field. We directly compare the transport properties of the nanowire assuming two different models for the superconducting segment: one where we put superconductivity by hand into the wire, and one where superconductivity is induced through a tunneling junction with a bulk swave superconductor. While these two models are equivalent at low energies and at weak coupling between the nanowire and the superconductor, we show that there are several interesting qualitative differences away from these two limits. In particular, the tunneling model introduces an additional conductance peak at the energy corresponding to the bulk gap of the parent superconductor. By employing a combination of analytical methods at zero temperature and numerical methods at finite temperature, we show that the tunneling model of the proximity effect reproduces many more of the qualitative features that are seen experimentally in such a nanowire system.
 
4.  Destructive interference of direct and crossed Andreev pairing in a system of two nanowires coupled via an swave superconductor 
Christopher R. Reeg, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 96, 081301(R) (2017)
We consider a system of two onedimensional nanowires coupled via an swave superconducting strip, a geometry that is capable of supporting Kramers pairs of Majorana fermions. By performing an exact analytical diagonalization of a tunneling Hamiltonian describing the proximity effect (via a Bogoliubov transformation), we show that the excitation gap of the system varies periodically on the scale of the Fermi wavelength in the limit where the interwire separation is shorter than the superconducting coherence length. Comparing with the excitation gaps in similar geometries containing only direct pairing, where one wire is decoupled from the superconductor, or only crossed Andreev pairing, where each nanowire is considered as a spinpolarized edge of a quantum Hall state, we find that the gap is always reduced, by orders of magnitude in certain cases, when both types of pairing are present. Our analytical results are further supported by numerical calculations on a tightbinding lattice. Finally, we show that treating the proximity effect by integrating out the superconductor using the bulk Green's function does not reproduce the results of our exact diagonalization.
 
5.  Hard superconducting gap in a normal layer coupled to a superconductor 
Christopher R. Reeg and Dmitrii L. Maslov. Phys. Rev. B 94, 020501(R) (2016)
The ability to induce a sizable gap in the excitation spectrum of a normal layer placed in contact with a conventional superconductor has become increasingly important in recent years in the context of engineering a topological superconductor. The quasiclassical theory of the proximity effect shows that Andreev reflection at the superconductor/normal interface induces a nonzero pairing amplitude in the metal but does not endow it with a gap. Conversely, when the normal layer is atomically thin, the tunneling of Cooper pairs induces an excitation gap that can be as large as the bulk gap of the superconductor. We study how these two seemingly different views of the proximity effect evolve into one another as the thickness of the normal layer is changed. We show that a fully quantummechanical treatment of the problem predicts that the induced gap is always finite but falls off with the thickness of the normal layer $d$. If $d$ is less than a certain crossover scale, which is much larger than the Fermi wavelength, the induced gap is comparable to the bulk gap. As a result, a sizable excitation gap can be induced in normal layers that are much thicker than the Fermi wavelength.
 
6.  Proximityinduced triplet superconductivity in Rashba materials 
Christopher R. Reeg and Dmitrii L. Maslov. Phys. Rev. B 92, 134512 (2015)
We study a proximity junction between a conventional swave superconductor and a conductor with Rashba spinorbit coupling, with a specific focus on the spin structure of the induced pairing amplitude. We find that spintriplet pairing correlations are induced by spinorbit coupling in both one and twodimensional systems due to the lifted spin degeneracy. Additionally, this induced triplet pairing has a component with an odd frequency dependence that is robust to disorder. Our predictions are based on the solutions of the exact Gor'kov equations and are beyond the quasiclassical approximation.
 
7.  Zeroenergy bound state at the interface between an swave superconductor and a disordered normal metal with repulsive electronelectron interactions 
Christopher R. Reeg and Dmitrii L. Maslov. Phys. Rev. B 90, 024502 (2014)
In recent years, there has been a renewed interest in the proximity effect due to its role in the realization of topological superconductivity. Here, we study a superconductorâ€“normal metal proximity system with repulsive electronelectron interactions in the normal layer. It is known that in the absence of disorder or normal reflection at the superconductorâ€“normal metal interface, a zeroenergy bound state forms and is localized to the interface [FauchÃ¨re et al., Phys. Rev. Lett. 82, 3336 (1999).]. Using the quasiclassical theory of superconductivity, we investigate the lowenergy behavior of the density of states in the presence of finite disorder and an interfacial barrier. We find that as the mean free path is decreased, the zeroenergy peak in the density of states is broadened and reduced. In the quasiballistic limit, the bound state eliminates the minigap pertinent to a noninteracting normal layer and a distinct peak is observed. When the mean free path becomes comparable to the normal layer width, the zeroenergy peak is strongly suppressed and the minigap begins to develop. In the diffusive limit, the minigap is fully restored and all signatures of the bound state are eliminated. We find that an interfacial potential barrier does not change the functional form of the density of states peak but does shift this peak away from zero energy.
