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

Short CV
2016present:  Postdoc, University of Basel, under supervision of Profs. Daniel Loss and Jelena Klinovaja 
20112016:  PhD in Physics, University of Florida, under supervision of Prof. Dmitrii Maslov 
20072011:  BS in Physics and Mathematics, University of Florida, under supervision of Prof. Russ Bowers 
Publications
Show all abstracts.1.  Zeroenergy Andreev bound states from quantum dots in proximitized Rashba nanowires 
Christopher Reeg, Olesia Dmytruk, Denis Chevallier, Daniel Loss, and Jelena Klinovaja. arXiv:1810.09840
We study an analytical model of a Rashba nanowire that is partially covered by and coupled to a thin superconducting layer, where the uncovered region of the nanowire forms a quantum dot. We find that, even if there is no topological superconducting phase possible, there is a trivial Andreev bound state that becomes pinned exponentially close to zero energy as a function of magnetic field strength when the length of the quantum dot is tuned with respect to its spinorbit length such that a resonance condition of FabryPerot type is satisfied. In this case, we find that the Andreev bound state remains pinned near zero energy for Zeeman energies that exceed the characteristic spacing between Andreev bound state levels but that are smaller than the spinorbit energy of the quantum dot. Importantly, as the pinning of the Andreev bound state depends only on properties of the quantum dot, we conclude that this behavior is unrelated to topological superconductivity. To support our analytical model, we also perform a numerical simulation of a hybrid system while explicitly incorporating a thin superconducting layer, showing that all qualitative features of our analytical model are also present in the numerical results.
 
2.  Proximity effect in a twodimensional electron gas coupled to a thin superconducting layer 
Christopher Reeg, Daniel Loss, and Jelena Klinovaja. Beilstein Journal of Nanotechnology 9, 1263 (2018)
There have recently been several experiments studying induced superconductivity in semiconducting twodimensional electron gases that are strongly coupled to thin superconducting layers, as well as probing possible topological phases supporting Majorana bound states in such setups. We show that a large band shift is induced in the semiconductor by the superconductor in this geometry, thus making it challenging to realize a topological phase. Additionally, we show that while increasing the thickness of the superconducting layer reduces the magnitude of the band shift, it also leads to a more significant renormalization of the semiconducting material parameters and does not reduce the challenge of tuning into a topological phase.
 
3.  Metallization of Rashba wire by superconducting layer in the strongproximity regime 
Christopher Reeg, Daniel Loss, and Jelena Klinovaja. Phys. Rev. B 97, 165425 (2018)
Semiconducting quantum wires defined within twodimensional electron gases and strongly coupled to thin superconducting layers have been extensively explored in recent experiments as promising platforms to host Majorana bound states. We study numerically such a geometry, consisting of a quasionedimensional wire coupled to a disordered threedimensional superconducting layer. We find that, in the strongcoupling limit of a sizable proximityinduced superconducting gap, all transverse subbands of the wire are significantly shifted in energy relative to the chemical potential of the wire. For the lowest subband, this band shift is comparable in magnitude to the spacing between quantized levels that arise due to the finite thickness of the superconductor (which typically is ∼500 meV for a 10nmthick layer of Aluminum); in higher subbands, the band shift is much larger. Additionally, we show that the width of the system, which is usually much larger than the thickness, and moderate disorder within the superconductor have almost no impact on the induced gap or band shift. We provide a detailed discussion of the ramifications of our results, arguing that a huge band shift and significant renormalization of semiconducting material parameters in the strongcoupling limit make it challenging to realize a topological phase in such a setup, as the strong coupling to the superconductor essentially metallizes the semiconductor. This metallization of the semiconductor can be tested experimentally through the measurement of the band shift.
 
4.  DIII topological superconductivity with emergent timereversal symmetry 
Christopher Reeg, Constantin Schrade, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 96, 161407(R) (2017)
We find a class of topological superconductors which possess an emergent timereversal symmetry that is present only after projecting to an effective lowdimensional model. We show that a topological phase in symmetry class DIII can be realized in a noninteracting system coupled to an swave superconductor only if the physical timereversal symmetry of the system is broken, and we provide three general criteria that must be satisfied in order to have such a phase. We also provide an explicit model which realizes the class DIII topological superconductor in 1D. We show that, just as in timereversal invariant topological superconductors, the topological phase is characterized by a Kramers pair of Majorana fermions that are protected by the emergent timereversal symmetry.
 
5.  Finitesize effects in a nanowire strongly coupled to a thin superconducting shell 
Christopher Reeg, Daniel Loss, and Jelena Klinovaja. Phys. Rev. B 96, 125426 (2017)
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.
 
6.  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.
 
7.  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.
 
8.  Destructive interference of direct and crossed Andreev pairing in a system of two nanowires coupled via an swave superconductor 
Christopher 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.
 
9.  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.
 
10.  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.
 
11.  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 superconductor–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 superconductor–normal metal interface, a zeroenergy bound state forms and is localized to the interface [Fauchère 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.
 
12.  Molecular dynamics in precision deuteriomethyl branched polyethylene from solidstate deuterium NMR 
Clifford R. Bowers, Yuying Wei, Brian S. Aitken, Christopher R. Reeg, Christopher D. Akel, and Kenneth B. Wagener. Polymer 53, 2633 (2012)
Deuterium quadrupolar echo NMR was applied to precision CD3 branched polyethylene at temperatures ranging from below the glass transition up to the melting point. The CD3 branches were placed on every 15th or 21st carbon with zero variation in the branch spacing by acyclic diene metathesis polymerization chemistry. The deuterium lineshapes were simulated and fit to the experimental spectrum assuming appropriate models that approximate the motions in the amorphous and crystalline phases. Spectral contributions of each phase were isolated by T1 fitting. The fitting results comprise the isotropic reorientation correlation time distribution and axial jump angle distribution in these two phases, respectively. The mean jump angle was found to increase monotonically with temperature, approaching 35° near the melting point, consistent with previous carbon13 NMR results on this same polymer.
 
13.  Structure and thermal conductivity of Na_{1x}Ge_{3+z} 
M. Beekman, S. Stefanoski, W. WongNg, J. A. Kaduk, Q. Huang, C. Reeg, C. R. Bowers, and G. S. Nolas. J. Solid State Chem. 183, 1272 (2010)
Structural analyses as well as low temperature thermal conductivity is reported for the binary phase Na_{1−x}Ge_{3+z}. Specimens were characterized by thermal analysis, conventional and synchrotron powder Xray diffraction, neutron powder diffraction, ^{23}Na nuclear magnetic resonance spectroscopy, and electrical and thermal transport measurements. With structural characteristics qualitatively analogous to some aluminumsilicate zeolites, the crystal structure of this phase exhibits an unconventional covalently bonded tunnellike Ge framework, accommodating Na in channels of two different sizes. Observed to be nonstochiometric, Na_{1−x}Ge_{3+z} concurrently exhibits substantial structural disorder in the large channels and a low lattice thermal conductivity, of interest in the context of identifying novel low thermal conductivity intermetallics for thermoelectric applications.
