Session M51: Invited Session: Topological Superconductivity in Ferromagnetic Metal Chains

Topological Superconductivity with Magnetic Atoms

Chains of magnetic impurities embedded in a conventional $s$-wave superconductor may induce the formation of a topologically non-trivial superconducting phase. If such a phase is formed along a chain, then its ends carry Majorana fermions. We investigate this possibility theoretically by developing a tight-binding Bogoliubov-de Gennes description, starting from the Shiba bound states induced by the individual magnetic impurities. While the resulting Hamiltonian has similarities with the Kitaev model for one-dimensional spinless $p$-wave superconductors, there are also important differences, most notably the long-range (power-law) nature of hopping and pairing as well as the complex hopping amplitudes. We develop an analytical theory, complemented by numerical approaches, which accounts for the electron long-range pairing and hopping along the chain [1], inhomogeneous magnetic order in the chain of embedded impurities or spin-orbit coupling in the host superconductor, and the possibility of direct electron hopping between the impurity atoms. This allows us to elucidate the domain of parameters favoring the formation of a topological phase and to find the spatial structure [2] of Majorana states appearing in that phase.\\ \\ This talk is based on joint work with F. von Oppen, Falko Pientka, and Yang Peng.\\ \\ \noindent [1] Falko Pientka, Leonid I. Glazman, and Felix von Oppen, Phys. Rev. B {\bf 88}, 155420 (2013).\\ \noindent [2] Falko Pientka, Leonid I. Glazman, and Felix von Oppen, Phys. Rev. B {\bf 89}, 180505(R) (2014).   

Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor

Majorana fermions are zero-energy excitations predicted to localize at the edge of a topological superconductor, a state of matter that can form when a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin-orbit interaction. With the goal of realizing a one-dimensional topological superconductor, we have fabricated ferromagnetic iron atomic chains on the surface of superconducting lead [1]. Using high-resolution spectroscopic imaging techniques, we show that the onset of superconductivity, which gaps the electronic density of states in the bulk of the chains, is accompanied by the appearance of zero-energy end-states. This spatially resolved signature provides strong evidence, corroborated by other observations and theoretical modeling [2], for the formation of a topological phase and edge-bound Majorana states in this system. Our results demonstrates that atomic chains are viable platform for future experiments to manipulate Majorana bound states [3] and to realize other related 1D or 2D topological superconducting phases. [1] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Science 346, 602 (2014). [2] Jian Li, Hua Chen, Ilya K. Drozdov, A. Yazdani, B. Andrei Bernevig, A.H. MacDonald, ArXiv:1410.3453 (2014). [3] Jian Li, Titus Neupert, B. Andrei Bernevig, Ali Yazdani, ArXiv:1404.4058 (2014).   

Theory of Topological Superconductivity in Ferromagnetic Metal Chains on Superconducting Substrates

Recent experiments have provided evidence that one-dimensional (1D) topological superconductivity based on transition metal atom chains formed on a superconducting substrate can be realized experimentally when the chain behaves like a ferromagnetic macrospin [1]. In this talk I will address the structural and bonding considerations which determine whether or not a particular atom chain will have magnetic and electronic properties favorable for topological superconductivity. By using a Slater-Koster tight-binding model to account for important features of transition metal electronic structure, I conclude that topological states are common for ferromagnetic chains on superconductors and that they are nearly universal when ferromagnetic transition metal chains form straight lines on superconducting substrates. The proximity induced superconducting gap on the chain is $\sim \Delta E_{so} / J$ where $\Delta$ is the $s$-wave pair-potential on the chain, $E_{so}$ is the spin-orbit splitting energy induced in the normal chain state bands by hybridization with the superconducting substrate, and $J$ is the exchange-splitting of the ferromagnetic chain $d$-bands. Because of the topological character of the 1D superconducting state, Majorana end modes appear within the gaps of finite length chains. I will specifically discuss the spatial decay length of the Majorana end modes which can be much shorter than the coherence length from the induced $p$-wave gap on the chain due to its strong coupling to the three-dimensional superconducting substrate, in agreement with experimental results [2]. Pb is a particularly favorable substrate material for ferromagnetic chain topological superconductivity because it provides both strong $s-$wave pairing and strong Rashba spin-orbit coupling, but there seems to be considerable scope to optimize the 1D topological superconductivity by varying the atomic composition and structure of the chain. [1] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Science 346, 602 (2014). [2] J. Li, H. Chen, I. K. Drozdov, A. Yazdani, B. A. Bernevig, and A. H. MacDonald, arXiv:1410.3453   

Experimental progress on Majoranas in semiconductors

Majoranas in semiconductor nanowires can be probed via various electrical measurements. Tunnel spectroscopy reveals zero-bias peaks in the differential conductance. These zero-bias peaks have a particular dependence on magnetic field (amplitude and direction) and electron density. This dependence allows to falsify many alternative theories for the observations. New challenges include a direct demonstration of topological protection, which is provided by a parity protection: How stable is the system's occupation in terms of an even or an odd number of quasi-particles? We demonstrate that the quasi-particle parity in a superconducting Cooperpair box can be stable over timescales of minutes. To demonstrate this protection for Majoranas it is crucial that the induced superconducting gap has negligible sub-gap states. To obtain such ``hard gaps'' under Majorana conditions currently forms the most important challenge. We report on progress in optimizing materials and measurement techniques.   

Fermionic and Majorana Bound States in Hybrid Nanowires With Rashba and Synthetic Spin-Orbit Interactions

I will present recent results on exotic bound states with non-Abelian braid statistics in one-dimensional condensed matter systems. Majorana fermions can emerge in a variety of setups in which either Rashba or synthetic spin-orbit interaction (SOI) is present. Here, I will discuss candidate materials such as semiconducting Rashba nanowires [1-3], graphene nanoribbons [4], atomic magnetic chains or magnetic semiconductors [5]. The topological phase in the presence of a non-uniform SOI hosts fermionic bound states inside the proximity gap [6]. They are localized at the junction between two wire sections characterized by different directions of the SOI vectors, and they coexist with Majorana bound states localized at the nanowire ends. At the same time, much effort is invested in identifying systems that host even more exotic quaiparticles, parafermions. In my talk, I will a setup consisting of two quantum wires with Rashba spin-orbit interactions coupled to an s-wave superconductor, in the presence of strong electron-electron interactions [7] that hosts time-reversal invariant parafermions. [1]~J. Klinovaja and D. Loss, Phys. Rev. B 86, 085408 (2012). [2] J. Klinovaja, P. Stano, and D. Loss, Phys. Rev. Lett. 109, 236801 (2012). [3]~D. Rainis, L. Trifunovic, J. Klinovaja, and D. Loss, Phys. Rev. B 87, 024515 (2013). [4]~J. Klinovaja and D. Loss, Phys. Rev. X 3, 011008 (2013); J. Klinovaja and D. Loss, Phys. Rev. B 88, 075404 (2013). [5] J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Phys. Rev. Lett. 111, 186805 (2013); B. Braunecker and P. Simon, Phys. Rev. Lett. 111, 147202 (2013); M. Vazifeh and M. Franz, Phys. Rev. Lett. 111, 206802 (2013). [6] J. Klinovaja and D. Loss, arXiv:1408.3366. [7] J. Klinovaja and D. Loss, Phys. Rev. B 90, 045118 (2014).