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Second Order Topological Superconductivity: Majorana and parafermion corner states

Jelena Klinovaja

Recently, a lot of interest has been raised by the generalization of conventional TIs/TSCs to so-called higher order TIs/TSCs. While a conventional d-dimensional TI/TSC exhibits (d − 1)-dimensional gapless boundary modes, a d-dimensional n-th order TI/TSC hosts gapless modes at its (d − n)-dimensional boundaries. In my talk, I will consider a Josephson junction bilayer consisting of two tunnel-coupled two-dimensional electron gas layers with Rashba spin-orbit interaction, proximitized by a top and bottom s-wave superconductor with phase difference φ close to π [1-3]. In the presence of a finite weak in-plane Zeeman field, the bilayer can be driven into a second order topological superconducting phase, hosting two Majorana corner states (MCSs). If φ=π, in a rectangular geometry, these zero-energy bound states are located at two opposite corners determined by the direction of the Zeeman field. If the phase difference φ deviates from π by a critical value, one of the two MCSs gets relocated to an adjacent corner. As the phase difference φ increases further, the system becomes trivially gapped. The obtained MCSs are robust against static and magnetic disorder.

In the second part of my talk, I will switch from non-interacting systems [4,5], in which one neglects effects of strong electron-electron interactions, to interacting systems and, thus, to exotic fractional phases. I will show that this is indeed possible and explicitly construct a two-dimensional (2D) fractional second-order TSC. I will consider a system of weakly coupled Rashba nanowires in the strong spin-orbit interaction (SOI) regime. The nanowires are arranged into two tunnel-coupled layers proximitized by a top and bottom superconductor such that the superconducting phase difference between them is π. In such a system, strong electron- electron interactions can stabilize a helical topological superconducting phase hosting Kramers partners of Z_2m parafermion edge modes, where m is an odd integer determined by the position of the chemical potential. Furthermore, upon turning on a weak in-plane magnetic field, the system is driven into a second- order topological superconducting phase hosting zero-energy Z_2m parafermion bound states localized at two opposite corners of a rectangular sample.

References

[1] Y. Volpez, D. Loss, and J. Klinovaja, Phys. Rev. Lett. 122,126402 (2019) 
[2] K. Plekhanov, M. Thakurathi, D. Loss, and J. Klinovaja, Phys. Rev. Research 1, 032013(R) (2019) 
[3] K. Plekhanov, N. Müller, Y. Volpez, D. M. Kennes, H. Schoeller, D. Loss, and J. Klinovaja, arXiv:2008.03611 
[4] K. Laubscher, D. Loss, and J. Klinovaja, Phys. Rev. Research 1, 032017(R) (2019)
[5] K. Laubscher, D. Loss, and J. Klinovaja, Phys. Rev. Research 2, 013330 (2020)

Two-dimensional Topological Superconductivity

Eun-Ah Kim

One could envision different strategies for designing topological superconductivity. One strategy would be to restrict the phase space in the kinetic energy by manipulating the band structure. Another strategy would be to restrict the pairing interaction. I will our proposals following each of these strategies. For the band-structure manipulation, I will discuss the prediction of p-wave superconductivity in p-doped TMD's with intermediate interaction. I will also discuss the competition for the surface state among multiple topological orders in FeSeTe. For the strategy of manipulating pairing interaction, I will discuss our proposal of using a metal-quantum paramagnet heterostructure.

Fingerprints of Majorana modes beyond the zero-bias conductance peak

Satoshi Ikegaya

The unambiguous detection of Majorana bound states (MBSs) in topological superconductors has been a central topic of condensed matter physics for recent years. So far, the presence of MBSs was demonstrated experimentally in a number of topologically nontrivial superconducting systems. In this connection, clear evidences of Majorana bound states are only obtained by the detection of zero-bias conductance peaks in tunneling transport measurements. In recent years, it became clear that various additional signatures of Majorana modes need to be investigated in order to complete our understanding.
In our presentation, we summarize two unambiguous fingerprints which can act as a ‘smoking gun’evidence. First we study the anomalous nonlocal conductance due to chiral Majorana edge states in a superconductor/ferromagnet hybrid as shown in Fig. 1(a). We obtain the important result that the chiral nature of the Majorana edge states causes an anomalously long-range and chirality-sensitive nonlocal transport in this device. This, in turn, enables us to identify conclusively the moving direction and further properties of the chiral Majorana edge states [1]. Secondly, we propose a novel experiment for achieving the first experimental observation of the anomalous proximity effect caused by Majorana bound states. In particular, we discuss the differential conductance of a semiconductor/superconductor hybrid as shown in Fig. 1(b), which contains a planar topological Josephson junction realized in recent experiments. The conductance spectrum changes drastically through the topological phase transition because the Majorana bound state appearing only in the topologically nontrivial phase can penetrate into the dirty normal segment and form the resonant transmission channel there [2]. In general, our results allow contrasting singlet and triplet superconductivity employing properties of Majorana modes beyond zero bias peaks.

[1] S. Ikegaya, Y. Asano, and D. Manske, Phys. Rev. Lett. 123, 207002 (2019)
[2] S. Ikegaya, S. Tamura, D. Manske, and Y. Tanaka, arXiv: 2007.12888 (2020)

Odd-frequency pairing in topological superconductors

Yukio Tanaka

It is known that odd-frequency pairing ubiquitously presents in superconductor junctions [1]. Especially, in the presence of zero energy surface Andreev bound state (ZESABS) realized in topological superconductors, the odd-frequency pairing is amplified near the surface or interface. One of the remarkable property generated by odd-frequency pairing is anomalous proximity effect in diffusive normal metal (DN) / superconductor junction where quasiparticle density of states in DN has a zero energy peak (ZEP) of LDOS due to the penetration of odd-frequency spin-triplet s-wave pairing [2,3]. It has been shown that proximity coupled nano-wire junction [4] is an idealistic system to study anomalous proximity effect due to odd-frequency triplet-s wave pairing [5].
We have further clarified the relation between induced odd-frequency pairing and the bulk quantity defined by Green’s function[6]. Odd-frequency Cooper pairs with chiral symmetry emerging at the edges are a useful physical quantity. We have shown that the odd-frequency Cooper pair amplitudes can be expressed by a winding number extended to a nonzero frequency and can be evaluated from the spectral features of the bulk. We have found that the odd-frequency Cooper pair amplitudes are classified into two categories: the amplitudes in the first category have the singular functional form proportional to 1/z (where z is a complex frequency) that reflects the presence of ZESABS, whereas the amplitudes in the second category have the regular form proportional to z.
Recently, we have found that the presence of ZESABS generates new type of thermopower.
We have shown that the thermoelectric effect in ferromagnet / superconductor junctions can be entirely dominated by ingap Andreev reflection processes. Consequently, the electric current from a temperature bias changes sign in the presence of ZESABS and resulting odd-frequency pairing [7].

[1]Y. Tanaka, M. Sato and N. Nagaosa, J. Phys. Soc. Jpn. 81 011013 (2012)
[2]Y. Tanaka and A.A. Golubov, Phys. Rev. Lett. 98 037003 (2007)
[3]Y. Tanaka, A.A. Golubov, S. Kashiwaya, and M. Ueda, Phys. Rev. Lett. 99 037005 (2007); Y. Tanaka, Y. Tanuma, and A. A. Golubov, Phys. Rev. B 96 054552 (2007)
[4]Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70 012507 (2004)
[5]R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010), Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010)
[6]Y. Asano and Y. Tanaka, Phys. Rev. B 87 104513 (2013)
[7]S. Ta mura, S. Hoshino and Y. Tanaka, Phys. Rev. B 99 , 184512 (2019)
[8]T. Savander , S. Tamura, C. Flindt, Y. Tanaka and P. Burset , arXiv: 2008.00849

Propagation and interference of d-wave superconducting pairs in graphene

Javier Villegas

Following earlier work in which we demonstrated the Klein-like tunneling of d-wave superconducting paints intro graphene [1], here we present experiments that show the longrange preparation of d-wave correlations in that material. For this, we fabricated devices that behave as a proximitized Fabry-Pérot cavitiy, where d-wave Andreev pairs interferences are produced. The interferences manifest themselves in series of pronounced conductance oscillations analogous to those produced by De Gennes-Saint James resonances in conventional superconductor/metal junctions. Their observation imply that the d-wave Andreev pairs propagate over distances of a few hundred nm in the CVD graphene used for the experiments [2]. We will end up by discussing ongoing experiments in applied magnetic field, which also produces an intriguing series of conductance oscillations in the superconducting state of the junctions.

[1] D. Perconte, F. A. Cuellar, C. Moreau-luchaire, M. Piquemal-Banci, R. Galceran, P. R. Kidambi, M.-B. Martin, S. Hofmann, R. Bernard, B. Dlubak, P. Seneor, and J. E. Villegas, Nat. Phys. 14, 25 (2018)
[2] D. Perconte, K. Seurre, V. Humbert, C. Ulysse, A. Sander, J. Trastoy, V. Zatko, F. Godel, P. R. Kidambi, S. Hofmann, X. P. Zhang, D. Bercioux, F. S. Bergeret, B. Dlubak, P. Seneor, and J. E. Villegas, Phys. Rev. Lett. 125, 87002 (2020)

On-line SPICE-SPIN+X Seminars

On-line Seminar: 25.11.2020 - 15:00 (CET)

Three-dimensional magnetic systems: the future is bright!

Claire Donnelly, University of Cambridge

Three dimensional magnetic systems promise significant opportunities for applications, for example providing higher density devices and new functionalities associated with complex topology and greater degrees of freedom [1,2]. With recent advances in both characterization and nanofabrication techniques, the experimental investigation of these complex systems is now possible, opening the door to the elucidation of new properties and rich physics.
For the characterization of 3D nanomagnetic systems, we have developed techniques to map both the three-dimensional magnetic structure, and its response to external excitations. In a first demonstration of X-ray magnetic nanotomography [3,4], we determined the complex magnetic structure within the bulk of a μm-sized soft magnetic pillar. The magnetic configuration contained vortices and antivortices, as well as Bloch point singularities [3]. With these new datasets comes a new challenge concerning the identification of such nanoscale topological objects within complex reconstructed magnetic configurations. To address this, we have recently implemented calculations of the magnetic vorticity [5,6], that make possible the location and identification of 3D magnetic solitons, leading to the first observation of nanoscale magnetic vortex rings [6].
In addition to the static magnetic structure, the dynamic response of the 3D magnetic configuration to excitations is key to our understanding of both fundamental physics, and applications. With our recent development of X-ray magnetic laminography [7,8], it is now possible to determine the magnetisation dynamics of a three-dimensional magnetic system [7].
Finally, recent advances in nanofabrication make possible the fabrication of complex 3D magnetic nanostructures [9], leading to the realisation of artificial chiral structures [10] and 3D spintronic devices [11]. These new experimental capabilities for 3D magnetic systems open the door to complex three-dimensional magnetic structures, and their dynamic behaviour.

PDF file of the talk available here

[1] Fernández-Pacheco et al., “Three-dimensional nanomagnetism” Nat. Comm. 8, 15756 (2017)
[2] Donnelly and V. Scagnoli, “Imaging three-dimensional magnetic systems with X-rays” J. Phys. D: Cond. Matt. (2019).
[3] Donnelly et al., “Three-dimensional magnetization structures revealed with X-ray vector nanotomography” Nature 547, 328 (2017).
[4] Donnelly et al., “Tomographic reconstruction of a three-dimensional magnetization vector field” New Journal of Physics 20, 083009 (2018).
[5] Cooper, “Propagating magnetic vortex rings in ferromagnets.” PRL. 82, 1554 (1999).
[6] Donnelly et al., “Experimental observation of vortex rings in a bulk magnet” Nat. Phys. (2020)
[7] Donnelly et al., “Time-resolved imaging of three-dimensional nanoscale magnetization dynamics”, Nature Nanotechnology 15, 356 (2020).
[8] Witte, et al., “From 2D STXM to 3D Imaging: Soft X-ray Laminography of Thin Specimens”, Nano Lett. 20, 1305 (2020).
[9] Skoric et al., “Layer-by-Layer Growth of Complex-Shaped Three-Dimensional Nanostructures with Focused Electron Beams” Nano Lett. 20, 184 (2020).
[10] Sanz-Hernández et al., “Artificial Double-Helix for Geometrical Control of Magnetic Chirality” ACS Nano 14, 8084 (2020).
[11] Meng et al., “Non-planar geometrical effects on the magnetoelectrical signal in a three-dimensional nanomagnetic circuit” In preparation.

Resonant p-wave oscillations of Josephson current in Nb-Bi2Te2.3Se0.7-Nb topological junctions

Alexander Golubov

Recent proposals of inducing p-wave superconductivity by sandwiching a conventional s- wave superconductor (S) with a topological insulator (TI) triggered a burst of research activity. Topological superconductors harness the inherent electron – hole symmetry of excitations in a superconductor with the helical nature of the electronic states in topological materials, that may lead to Majorana zero energy states. Various theoretical models focused on possible pairing symmetries of the proximity induced superconducting order in topological layers. Several experimental groups already realized S-TI interfaces and S-TI-S junctions, and studied the Josephson current across them. Few papers have reported unusual Shapiro steps consistent with the formation of the p-wave correlations. However, measurements that could provide the definitive evidence of the p-wave superconductivity should be phase-sensitive, to discriminate between the p- and other (s-, d-...) pairings symmetries.
In this work we report a new type of oscillations of the critical Josephson current in magnetic field observed in the Nb-Bi2Te2.3Se0.7-Nb junctions [1,2]. The ultra-short period ∼ 1 Oe of these oscillations and their sharply peaked shape reflect the resonant transmission via Andreev bound states with the ultra-fine ∼ 1 μeV interlevel spacing. We argue that the ultra- fine oscillations revealed in our S-TI-S devices is the direct consequence of the p-wave superconducting order induced at S-TI contacts.

[1] V.S. Stolyarov, D.S. Yakovlev, S.N. Kozlov, O.V. Skryabina, D.S. Lvov, A.I. Gumarov, O.V. Emelyanova, P.S. Dzhumaev, I.V. Shchetinin, R.A. Hovhannisyan, S.V. Egorov, A.M. Kokotin, W.V. Pogosov, V.V. Ryazanov, M.Yu. Kupriyanov, A.A. Golubov, D. Roditchev. COMMUNICATIONS MATERIALS 1:38 (2020)| https://doi.org/10.1038/s43246-020-0037-y | www.nature.com/commsmat.
[2] V.S. Stolyarov, D. Roditchev, V.L Gurtovoy, S.N. Kozlov, D.S. Yakovlev, O.V. Skryabina, V.M. Vinokur, A.A. Golubov, submitted to Nature Physics.

Flat Bands in Flatlands

Jeanie Lau

In a flat band system, the charge carriers’ energy-momentum relation is very weakly dispersive. The resultant large density of states and the dominance of Coulomb potential energy relative to the kinetic energy often favor the formation of strongly correlated electron states, such as ferromagnetism, nematicity, antiferromagnetism, superconductivity, and charge density waves. The advent of two-dimensional (2D) materials and their heterostructures has ushered in a new era for exploring, tuning and engineering of flat band system. Here I will present our results on transport measurements of high quality few-layer 2D material devices, including intrinsic magnetism and helical edge states in few-layer graphene, and observation of both superconductivity and the Mott-like insulating state in a tBLG device with a twist angle of ~0.93°.

Long range unconventional Josephson effect across a half metallic ferromagnet

Jacobo Santamaria

The Josephson effect results from the coupling of two superconductors across a non- superconducting spacer to yield a quantum coherent state. In ferromagnets, singlet (opposite- spin) Cooper pairs decay over very short distances, and thus Josephson coupling requires a nanometric spacer. This is unless equal-spin triplet pairs are generated which, theoretically, can couple superconductors across much longer distances. Despite many experimental hints of triplet superconductivity, long range triplet Josephson effects have remained elusive. In this talk I will discuss a micron-range Josephson coupling across the half-metallic ferromagnet
La0.7Sr0.3MnO3 combined with the high-temperature superconductor YBa2Cu3O7 in planar junctions. These display the Josephson physics’ hallmarks: critical current oscillations due to flux quantization and quantum phase locking under microwave excitation. The marriage of high- temperature quantum coherent transport and full spin polarization brings unique opportunities for the practical realization of superconducting spintronics, and enables novel strategies for quantum computing.

Topological phases combining superconductivity and magnetism

Mario Cuoco

In this talk I will present different routes to generate and manipulate topological phases due to the interplay between superconductivity and magnetism. The search for new variants of semimetals (SMs) recently highlighted the interplay of Dirac fermions physics and magnetism. Indeed, antiferromagnetic (AFM) SMs can be obtained where both time and inversion are broken while their combination is kept [1,2] or due to chiral- [2] and time-symmetry [2,3] combined with non-symmorphic transformations [2]. Here, we discuss materials, i.e. transition metal oxide systems, that can exhibit AFM-SM phase due to orbitally directional double- exchange effects [4, 2]. In this context, the impact of s-wave spin-singlet pairing on AFM-SMs with Dirac points or nodal loops at the Fermi level [5] is generally shown to convert the semimetal into various types of nodal topological superconductors. The changeover from fully gapped to gapless phases is dictated by symmetry properties of the AFM-superconducting state that set out the occurrence of a large variety of electronic topological transitions [4].
Finally, I will focus on various quantum platforms marked by spin-singlet or spin-triplet pairing interfaced with non-trivial magnetic patterns and discuss the nature of the emerging topological phases [6,7,8]. The coexistence of ferromagnetism or antiferromagnetism with spin-triplet superconductivity is also analysed and discussed with respect to relevant materials cases.

[1] P. Tang, Q. Zhou, G. Xu, and S.-C. Zhang, Nat. Phys. 12, 1100 (2016).
[2] W. Brzezicki and M. Cuoco, Phys. Rev. B 95, 155108 (2017).
[3] S. M. Young and B. J. Wieder, Phys. Rev. Lett. 118, 186401 (2017).
[4] W. Brzezicki, C. Noce, A. Romano, and M. Cuoco, Phys. Rev. Lett. 114, 247002 (2015).
[5] W. Brzezicki and M. Cuoco, Phys. Rev. B 97, 064513 (2018).
[6] M. T. Mercaldo, M. Cuoco, P. Kotetes, Phys. Rev. B 94, 140503(R) (2016).
[7] A. Romano, P. Gentile, C. Noce, I. Vekhter, M. Cuoco, Phys. Rev. Lett. 110, 267002 (2013). 8. P. Kotetes, M. T. Mercaldo, M. Cuoco, Phys. Rev. Lett. 123, 126802 (2019).
[9] M. T. Mercaldo, P. Kotetes, M. Cuoco, Phys. Rev. B 100, 104519 (2019).