We study the Yu-Shiba-Rusinov states in materials with bulk band inversion such as iron-based topological superconductors or doped topological insulators. We show that the structure of the YSR state spectrum depends on the doping level relative to the chemical potential at which the band-inversion occurs. Moreover, we demonstrate that the transition from ferromagnetic to antiferromagnetic coupling and vice versa, which is caused by the coupling of magnetic impurities through the overlap of YSR states, is highly dependent on the doping level. Additionally, topological edge states may have a substantial impact on the YSR states, leading to a decrease in YSR state energies and the creation of new states when the magnetic impurity approaches the boundary.
Spin-orbit coupled superconductors are potentially interesting candidates for realizing topological and potentially non-Abelian states with Majorana Fermions. We argue that time-reversal broken spin-orbit coupled superconductors generically can be characterized as having sub-gap states that are bound to localized non-magnetic impurities. Such bound states, which are referred to as Shiba states, can be detected as sharp resonances in the tunneling spectrum of the spin-orbit coupled superconductors. The Shiba state resonance can be tuned using a gate-voltage or a magnetic field from being at the…
Inspired by the recent experimental observation of topological superconductivity in ferromagnetic chains, we consider a dilute 2D lattice of magnetic atoms deposited on top of a superconducting surface with a Rashba spin-orbit coupling. We show that the studied system supports a generalization of $p_x+ip_y$ superconductivity and that its topological phase diagram contains Chern numbers higher than $ξ/a$ $(\gg1)$, where $ξ$ is the superconducting coherence length and $a$ is the distance between the magnetic atoms. The signatures of nontrivial topology can be observed by STM spectroscopy in fini…
We study a chain of magnetic moments exchange coupled to a conventional three dimensional superconductor. In the normal state the chain orders into a collinear configuration, while in the superconducting phase we find that ferromagnetism is unstable to the formation of a magnetic spiral state. Beyond weak exchange coupling the spiral wavevector greatly exceeds the inverse superconducting coherence length as a result of the strong spin-spin interaction mediated through the subgap band of Yu-Shiba-Rusinov states. Moreover, the simple spin-spin exchange description breaks down as the subgap band …
Topological states of matter support quantized nondissipative responses and exotic quantum particles that cannot be accessed in common materials. The exceptional properties and application potential of topological materials have triggered a large-scale search for new realizations. Breaking away from the popular trend focusing almost exclusively on crystalline symmetries, we introduce the Shiba glass as a platform for amorphous topological quantum matter. This system consists of an ensemble of randomly distributed magnetic atoms on a superconducting surface. The collection of magnetic moments g…
Chains of magnetic adatoms on superconductors have been discussed as promising systems for realizing Majorana end states. Here, we show that dilute Yu-Shiba-Rusinov (YSR) chains are also a versatile platform for quantum magnetism and correlated electron dynamics, with widely adjustable spin values and couplings. Focusing on subgap excitations, we derive an extended $t-J$ model for dilute quantum YSR chains and use it to study the phase diagram as well as tunneling spectra. We explore the implications of quantum magnetism for the formation of a topological superconducting phase, contrasting it …
Chains of magnetic atoms placed on the surface of an s-wave superconductor with large spin-orbit coupling provide a promising platform for the realization of topological superconducting states characterized by the presence of Majorana zero-energy modes. In this work we study the properties of the one-dimensional chain of Yu-Shiba-Rusinov states induced by magnetic impurities using a realistic model for the magnetic atoms that include the presence of multiple scattering channels. These channels are mixed by the spin-orbit coupling and, via the hybridization of the Yu-Shiba-Rusinov states at dif…
Time-reversal symmetric topological superconductivity in Machida-Shibata lattices
I. Institute for Theoretical Physics, University of Hamburg（汉堡大学第一理论物理研究所）；Centre for Ultrafast Imaging（超快成像中心）；RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所交叉理论与数学科学（iTHEMS））
Recent experimental investigations of arrays of magnetic atoms deposited on top of a superconductor have opened a new chapter in the search of topological superconductivity. We generalize the microscopic model derived by Pientka et al. [Phys. Rev. B \textbf{88}, 155420 (2013)] to accommodate the effects of finite supercurrent in the host material. Previously it was discovered that helical chains with nonplanar textures are plagued by a gapless phase. We show that by employing supercurrent it is possible to tune the chain from the gapless phase to the topological gapped phase. It is also possib…
Magnetic atoms coupled to the Cooper pairs of a superconductor induce Yu-Shiba-Rusinov states (in short Shiba states). In the presence of sufficiently strong spin-orbit coupling, the bands formed by hybridization of the Shiba states in ensembles of such atoms can support low-dimensional topological superconductivity with Majorana bound states localized on the ensembles' edges. Yet, the role of spin-orbit coupling for the hybridization of Shiba states in dimers of magnetic atoms, the building blocks for such systems, is largely unexplored. Here, we reveal the evolution of hybridized multi-orbit…
Two-dimensional (2D) magnet-superconductor hybrid systems are intensively studied due to their potential for the realization of 2D topological superconductors with Majorana edge modes. It is theoretically predicted that this quantum state is ubiquitous in spin-orbit coupled ferromagnetic or skyrmionic 2D spin-lattices in proximity to an s-wave superconductor. However, recent examples suggest that the requirements for topological superconductivity are complicated by the multi-orbital nature of the magnetic components and disorder effects. Here, we investigate Fe monolayer islands grown on a sur…
When magnetic atoms are inserted inside a superconductor, the superconducting order is locally depleted as a result of the antagonistic nature of magnetism and superconductivity1. Thereby, distinctive spectral features, known as Yu-Shiba-Rusinov states, appear inside the superconducting gap2-4. The search for Yu-Shiba-Rusinov states in different materials is intense, as they can be used as building blocks to promote Majorana modes5 suitable for topological quantum computing6. Here we report the first realization of Yu-Shiba-Rusinov states in graphene, a non-superconducting 2D material, and wit…
We study the interplay between the Kondo effect and (color) superconductivity in doped Dirac metals with magnetic impurities and in quark matter with colorful impurities. We first point out that the overscreened Kondo effect arises in the normal state of these systems. Next the (color) superconducting gap is incorporated as a mean field and the phase diagram for a varying gap and temperature is constructed nonperturbatively. A rich phase structure emerges from a competition of effects unique to a multichannel system. The Kondo-screened phase is shown to disappear for a sufficiently large gap. …
The coupling of a spin to an underlying substrate is the basis for a plethora of phenomena. In the case of a metallic substrate, Kondo screening of the adatom magnetic moment can occur. As the substrate turns superconducting, an intriguing situation emerges where the pair breaking due to the adatom spins leads to Yu-Shiba-Rusinov bound states, but also intertwines with Kondo phenomena. Through scanning tunneling spectroscopy, we analyze the interdependence of Kondo screening and superconductivity. Our data obtained on single Fe adatoms on Nb(110) show that the coupling and the resulting YSR st…
The competition of the free spin state of a paramagnetic impurity on a superconductor with its screened counterpart is characterized by the energy scale of Kondo screening $k_BT_K$ compared to the superconducting pairing energy $Δ$. When the experimental temperature suppresses Kondo screening, but preserves superconductivity, i.e., when $Δ/k_B > T > T_K$, this description fails. Here, we explore this temperature range in a set of manganese phthalocyanine molecules decorated with ammonia (MnPc-NH$_3$) on Pb(111). We show that these molecules suffice the required energy conditions by exhibiting …
The combination of different exotic properties in materials paves the way for the emergence of their new potential applications. An example is the recently found coexistence of the mutually antagonistic ferromagnetism and superconductivity in hydrogenated boron-doped diamond, which promises to be an attractive system with which to explore unconventional physics. Here, we show the emergence of Yu-Shiba-Rusinov (YSR) bands with a spatial extent of tens of nanometers in ferromagnetic superconducting diamond using scanning tunneling spectroscopy. We demonstrate theoretically how a two-dimensional …
Odd-parity ground state in dilute Yu-Shiba-Rusinov dimers and chains
Freie Universität Berlin（柏林自由大学）；Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin（达勒姆复杂量子系统中心及物理系，柏林自由大学）；Institut für Experimentelle und Angewandte Physik, Christian-Albrechts-Universität zu Kiel（基尔大学实验与应用物理学研究所）；Ruprecht Haensel Laboratory, Deutsches Elektronen-Synchrotron DESY（鲁普雷希特·亨塞尔实验室，德国电子同步加速器研究所）
We propose a new mechanism for topological superconductivity based on an antiferromagnetically ordered chain of magnetic atoms on the surface of a conventional superconductor. In a weak Zeeman field, a supercurrent in the substrate generates a staggered spin-current, which converts the preexisting topologically-unprotected Shiba states into Majorana fermions (MFs). The two experimental knobs can be finely tuned providing a platform with enhanced functionality for applications. Remarkably, the electronic spin-polarization of the arising edge MF wavefunctions depends solely on the parity of the …
When the exchange interaction between the impurity spin and the spins of itinerant quasiparticles are strong or weak enough, the ground states for a magnetic impurity in a superconductor are the screened or free spins, respectively. In both cases, the lowest excited state is a bound state within the superconducting gap, known as the Yu-Shiba-Rusinov (YSR) state. The YSR state is spatially localized, energetically isolated, and fully spin-polarized, leading to applications such as functional scanning probes. While any application demands identifying whether the impurity spin is screened or free…
Recent experiments announced an observation of topological superconductivity and Majorana quasiparticles in Shiba chains, consisting of an array of magnetic atoms deposited on top of a superconductor. In this work we study helical Shiba chains and generalize the microscopic theory of subgap energy bands to a regime where the decoupled magnetic impurity energy and the hybridization of different impurity states can be significant compared to the superconducting gap of the host material. From exact solutions of the Bogoliubov-de Gennes equation we extract expressions for the topological phase bou…
Magnetic impurities have a dramatic effect on superconductivity by breaking the time-reversal symmetry and inducing so-called Yu-Shiba-Rusinov (YSR) low energy bound states within the superconducting gap. The spatial extent of YSR states is greatly enhanced in 2D systems, which should facilitate the formation of coupled states. Here, we observe YSR states on single cobalt phthalocyanine (CoPC) molecules on a 2D superconductor NbSe$_2$ using low-temperature scanning tunneling microscopy (STM) and spectroscopy (STS). We use STM lateral manipulation to create controlled CoPc dimers and demonstrat…
Recent studies of mutually interacting magnetic atoms coupled to a superconductor have gained enormous interest due to the potential realization of topological superconductivity. The Kondo exchange coupling J_K of such atoms with the electrons in the superconductor has a pair-breaking effect which produces so-called Yu-Shiba-Rusinov (YSR) states within the superconducting energy gap, whose energetic positions are intimately connected with the requirements for topological superconductivity. Here, using the tip of a scanning tunneling microscope, we artificially craft a multi-impurity Kondo syst…
Recently, it has been suggested that topological superconductivity and Majorana end states can be realized in a chain of magnetic impurities on the surface of an s-wave superconductor when the magnetic moments form a spin helix as a result of the RKKY interaction mediated by the superconducting substrate. Here, we investigate this scenario 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 model Hamiltonian has similarities with the Kitaev model for one-dimensional …
The traditional symmetries that protect superconductivity are time-reversal and inversion. Here, we examine the minimal symmetries protecting superconductivity in two dimensions and find that time-reversal symmetry and inversion symmetry are not required, and having a combination of either symmetry with a mirror operation on the basal plane is sufficient. We classify superconducting states stabilized by these two symmetries, when time-reversal and inversion symmetries are not present, and provide realistic minimal models as examples. Interestingly, several experimentally realized systems, such…
Magnetic atoms on superconductors induce localized Yu-Shiba-Rusinov (YSR) bound states. The proposal that topological superconductivity and Majorana modes can be engineered in arrays of hybridizing YSR states has led to their intense investigation. Here, we study Majorana modes emerging from bands of hybridized YSR states in artificially constructed Manganese (Mn) chains on superconducting Niobium (Nb). By controlling the chain geometry on the single atom level, we can measure the interaction-induced energy splitting of Majorana modes from both chain's ends with increasing chain length. We fin…
This work presents the emergence of superconductivity in Ir - doped Weyl semimetal T$_d$ - MoTe$_{2}$ with broken inversion symmetry. Chiral anomaly induced planar Hall effect and anisotropic magneto-resistance confirm the topological semimetallic nature of Mo$_{1-x}$Ir$_{x}$Te$_{2}$. Observation of weak anisotropic, moderately coupled type-II superconductivity in T$_d$ -Mo$_{1-x}$Ir$_{x}$Te$_{2}$ makes it a promising candidate for topological superconductor.
We analyze the phase diagram associated with a pair of magnetic impurities trapped in a superconducting host. The natural interplay between Kondo screening, superconductivity and exchange interactions leads to a rich array of competing phases, whose transitions are characterized by discontinuous changes of the total spin. Our analysis is based on a combination of numerical renormalization group techniques as well as semi-classical analytics. In addition to the expected screened and unscreened phases, we observe a new molecular doublet phase where the impurity spins are only partially screened …
We study superconductivity in spin-orbit-coupled systems in the vicinity of inversion symmetry breaking. We find that due to the presence of spin-orbit coupling, fluctuations of the incipient parity-breaking order generate an attractive pairing interaction in an odd-parity pairing channel, which competes with the s-wave pairing. We show that applying a Zeeman field suppresses the s-wave pairing and promotes the odd-parity superconducting state. Our work provides a new mechanism for odd-parity pairing and opens a route to novel topological superconductivity.
While multi-layer systems can possess global inversion centers, they can have regions with locally broken inversion symmetry. This can modify the superconducting properties of such a system. Here we analyze two dimensional multi-layer systems yielding spatially modulated antisymmetric spin-orbit coupling (ASOC) and discuss superconductivity with mixed parity order parameters. In particular, the influence of ASOC on the spin susceptibility is investigated at zero temperature. For weak inter-layer coupling we find an enhanced spin susceptibility induced by ASOC, which hints the potential importa…
In superconductors, magnetic impurities induce a pair-breaking potential for Cooper pairs, which locally affects the Bogoliubov quasiparticles and gives rise to Yu-Shiba-Rusinov (YSR or Shiba, in short) bound states in the density of states (DoS). These states carry information on the magnetic coupling strength of the impurity with the superconductor, which determines the many-body ground state properties of the system. Recently, the interest in Shiba physics was boosted by the prediction of topological superconductivity and Majorana modes in magnetically coupled chains and arrays of Shiba imp…
Superconductivity with $T_{\rm c}$ $\simeq$ 2 K was discovered in the intermetallic binary compounds Rh$_2$Ga$_9$ and Ir$_2$Ga$_9$. This is the first observation of superconductivity in the Rh-Ga and Ir-Ga binary systems. Both compounds crystallize in a distorted Co$_2$Al$_9$-type structure (monoclinic, space group: $Pc$), which lacks spatial inversion symmetry. Specific heat measurements revealed that both compounds are weak-coupling BCS superconductors having an isotropic superconducting gap. Measurements in magnetic fields indicated type-I superconductivity with a critical field $H_{\rm c}(…
By using scanning tunneling microscopy (STM) we find and characterize dispersive, energy-symmetric in-gap states in the iron-based superconductor $\mathrm{FeTe}_{0.55}\mathrm{Se}_{0.45}$, a material that exhibits signatures of topological superconductivity, and Majorana bound states at vortex cores or at impurity locations. We use a superconducting STM tip for enhanced energy resolution, which enables us to show that impurity states can be tuned through the Fermi level with varying tip-sample distance. We find that the impurity state is of the Yu-Shiba-Rusinov (YSR) type, and argue that the en…
Diode effect in microwave irradiated Josephson junctions with Yu-Shiba-Rusinov states
Institute for Theoretical Physics and Astrophysics, University of Würzburg（维尔茨堡大学理论物理与天体物理研究所）；Würzburg-Dresden Cluster of Excellence ct.qmat（维尔茨堡-德累斯顿卓越集群 ct.qmat）
Majorana fermions can be realized as quasiparticles in topological superconductors, with potential applications in topological quantum computing. Recently, lattices of magnetic adatoms deposited on the surface of s-wave superconductors - Shiba lattices - have been proposed as a new platform for topological superconductivity. These systems possess the great advantage that they are accessible via scanning-probe techniques, and thus enable the local manipulation and detection of Majorana modes. Realizing quantum bits on this basis will rely on the creation of nanoscopic Shiba lattices, so-called …
We study the subgap Yu-Shiba-Rusinov (YSR) states in the Richardson's model of a superconductor with a magnetic impurity for different electron pairing strengths from the weak-coupling Bardeen-Cooper-Schrieffer (BCS) regime to the strong-coupling Bose-Einstein-condensate (BEC) regime. We observe that the effect of the increasing pairing strength on the YSR excitation spectrum as a function of hybridisation strength and impurity on-site potential is only quantitative and, in fact, rather weak when the results are appropriately rescaled. We furthermore show that the problem is analytically solva…
A magnetic atomic impurity inside a superconductor locally distorts superconductivity. They scatter Cooper pairs as a potential with broken time-reversal symmetry, what leads to localized bound states with subgap excitation energies, named hereon Shiba states. Most conventional approaches to study Shiba states treat magnetic impurities as point scatterers with an isotropic exchange interaction, while the complex internal structure of magnetic impurities is usually neglected. Here, we show that the number and the shape of Shiba states are correlated to the spin-polarized atomic orbitals of the …
We study the superconducting state of multi-orbital spin-orbit coupled systems in the presence of an orbitally driven inversion asymmetry assuming that the inter-orbital attraction is the dominant pairing channel. Although the inversion symmetry is absent, we show that superconducting states that avoid mixing of spin-triplet and spin-singlet configurations are allowed, and remarkably, spin-triplet states that are topologically nontrivial can be stabilized in a large portion of the phase diagram. The orbital-dependent spin-triplet pairing generally leads to topological superconductivity with po…
The symmetry properties and the general overview of the superconductivity theory in the itinerant ferromagnets and in materials without space parity are presented. The basic notions of unconventional superconductivity are introduced in broad context of multiband superconductivity which is inherent property of ferromagnetic metals or metals without centre of inversion.
We show that dilute magnetic impurities in a conventional superconductor give origin to an odd-frequency component of superconductivity, manifesting itself in Yu-Shiba-Rusinov bands forming within the bulk superconducting gap. Our results are obtained in a general model solved within the dynamical mean field theory. By exploiting a disorder analysis and the limit to a single impurity, we are able to provide general expressions that can be used to extract explicitly the odd-frequency superconducting function from scanning tunneling measurements.
Local spins coupled to superconductors give rise to several emerging phenomena directly linked to the competition between Cooper pair formation and magnetic exchange. These effects are generally scrutinized using a spectroscopic approach which relies on detecting the in-gap bound modes arising from Cooper pair breaking, the so-called Yu-Shiba-Rusinov (YSR) states. However, the impact of local magnetic impurities on the superconducting order parameter remains largely unexplored. Here, we use scanning Josephson spectroscopy to directly visualize the effect of magnetic perturbations on Cooper pai…
Although multilayer systems possess global inversion symmetry, some of the layers lack local inversion symmetry because no global inversion centers are present on such layers. Such locally non-centrosymmetric systems exhibit spatially modulated Rashba spin-orbit coupling. In this study, the superconductivity in multilayer models exhibiting inhomogeneous Rashba spin-orbit coupling is investigated. We study the electronic structure, superconducting gap, and spin susceptibility in the superconducting state with mixed parity order parameters. We show the enhancement of the spin susceptibility by R…
There is presently a tremendous activity around the field of topological superconductivity and Majorana fermions. Among the many questions raised, it has become increasingly important to establish the topological or non-topological origin of features associated with Majorana fermions such as zero-bias peaks. Here, we compare in-gap features associated either with isolated magnetic impurities or with magnetic clusters strongly coupled to the atomically thin superconductor Pb/Si(111). We study this system by means of scanning tunneling microscopy and spectroscopy (STM/STS). We take advantage of …
Magnetic superconductors define a broad class of strongly correlated materials in which superconductivity may coexist with either localized or itinerant long-range magnetic order. In this work we consider a multiband model of a disordered magnetic superconductor which realizes coexistence of unconventional superconductivity and a spin-density-wave. We derive an exact $T$-matrix and compute a single particle density of states in this system. In a purely superconducting state the interband scattering potential leads to an appearance of the localized Yu-Shiba-Rusinov bound states. Our main findin…
Unlike other pnictides, SrPtAs has a hexagonal structure, containing layers with As-Pt atoms that form a honeycomb lattice. These layers lack inversion symmetry which allows for a spin-orbit coupling that we show has a dramatic effect on superconductivity in this material. In particular, for conventional s-wave superconductivity in SrPtAs, both the spin susceptibility and the paramagnetic limiting field are enhanced significantly with respect to that usually expected for s-wave superconductors. SrPtAs provides a prime example of a superconductor with locally broken inversion symmetry.
We investigate effects of ordinary nonmagnetic disorder in the bulk of a superconductor on magnetic adatom-induced Shiba states and on the proximity-induced superconductivity in a nanowire that is tunnel coupled to the bulk superconductor. Within the formalism of self-consistent Born approximation, we show that, contrary to widespread belief, the proximity-induced topological superconductivity can be adversely affected by the bulk superconducting disorder even in the absence of any disorder in the nanowire (or the superconductor-nanowire interface) when the proximity tunnel coupling is strong.…
The theory of paramagnetic limit of superconductivity in metals without inversion center is developed. There is in general the paramagnetic suppression of superconducting state. The effect is strongly dependent on field orientation in respect to crystal axes. The reason for this is that the degeneracy of electronic states with opposite momenta forming of Cooper pairs is lifted by magnetic field but for some field directions this lifting can be small or even absent.
Inversion and time reversal are essential symmetries for the structure of Cooper pairs in superconductors. The loss of one or both leads to modifications to this structure and can change the properties of the superconducting phases in profound ways. Lacking inversion, superconductivity in noncentrosymmetric materials has become an important topic, in particular, in the context of topological superconductivity as well as unusual magnetic and magneto-electric properties. Recently, crystal structures with local, but not global inversion-symmetry breaking have attracted attention, as superconducti…
