In this work we consider the influence of potential impurities deposited on top of two-dimensional chiral superconductors. As discovered recently, magnetic impurity lattices on an $s$-wave superconductor may give rise to a rich topological phase diagram. We show that similar mechanism takes place in chiral superconductors decorated by non-magnetic impurities, thus avoiding the delicate issue of magnetic ordering of adatoms. We illustrate the method by presenting the theory of potential impurity lattices embedded on chiral $p$-wave superconductors. While a prerequisite for the topological state engineering is a chiral superconductor, the proposed procedure results in vistas of nontrivial descendant phases with different Chern numbers.
A field theoretic approach is developed to investigate the profile and spectrum of sub-gap states in a superconductor subject to a weak magnetic impurity potential. Such states are found to be associated with inhomogeneous supersymmetry broken instanton configurations of the action.
In the theory of dilute magnetic impurities in superconductors, the effect of all impurity spin-components is expressed via a single magnetic scattering rate $Γ_\mathrm{m}$. In a more realistic setting, magnetic impurities are anisotropic. In this case, the spatial randomness of three spin-components of impurities gives rise to generally different scattering rates $Γ_i$ ($i=1,2,3$). We explore the effects of anisotropic magnetic impurities on the in-plane critical field in 2D superconductors. We discuss singlet, triplet and parity-mixed order parameters allowed in systems without the inversion…
We show that a chain of nonmagnetic impurities deposited on a fully gapped two- or three-dimensional superconductor can become a topological one-dimensional superconductor with protected Majorana bound states at its end. A prerequisite is that the pairing potential of the underlying superconductor breaks the spin-rotation symmetry, as it is generically the case in systems with strong spin-orbit coupling. We illustrate this mechanism for a spinless triplet-superconductor p+ip and a time-reversal symmetric Rashba superconductor with a mixture of singlet and triplet pairing. For the latter, we sh…
We analyze the Josephson current in a junction consisting of two superconductors (S) and a ferromagnetic layer (F) for arbitrary impurity concentration. In addition to non-magnetic impurities, we consider also magnetic ones and spin-orbit scattering. In the limit of weak proximity effect we solve the linearized Eilenberger equation and derive an analytical expression for the Josephson critical current valid in a broad range of parameters. This expression enables us to obtain not only known results in the dirty and clean limits but also in a intermediate region of the impurity concentration, wh…
We study odd-parity nematic superconductivity in doped topological insulators in presence of surface magnetic impurities. The peculiar surface subgap spectrum, characterized by a Majorana flatband, nodal cones and the surface states of the parent topological insulator, gives rise to an overall ferromagnetic RKKY interactions between the surface impurities. An additional coupling between the impurities and a preemptive chiral order parameter promote a surface time-reversal symmetry breaking solution at the surface of the system. We discuss the relevant scenarios and suggest to engineer surface …
One of the remarkable features of disordered d-wave superconductors is strong sensitivity of long range properties to the microscopic realization of the disorder potential. Particularly rich phenomenology is observed for the -- experimentally relevant -- case of dilute distributions of isolated impurity centers. Building on earlier diagrammatic analyses, the present paper derives and analyses a low energy effective field theory of this system. Specifically, the results of previous diagrammatic T-matrix approaches are extended into the perturbatively inaccessible low energy regimes, and the lon…
The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction of magnetic impurities in a superconductor exponentially decreases when the distance $r$ between them is larger than the superconductor's coherence length, because this interaction is mediated by quasiparticles which have a gap in their energy spectrum. At the same time, the spin-singlet superconducting condensate was always assumed to stay neutral to magnetic impurities. Due to a spin-orbit coupling (SOC), however, Cooper pairs gain an admixture of spin-triplet correlated states which provide for a link between impurity spins and an s-wave …
We present a comprehensive theoretical study of thermodynamic properties of superconductors with a dilute concentration of magnetic impurities, with focus on how the properties of the superconducting host change if the magnetic moments of the impurities order ferromagnetically. Scattering off the magnetic impurities leads to the formation of a band of Yu-Shiba-Rusinov states within the superconducting energy gap that drastically influences superconductivity. In the magnetically ordered system, the magnetization displays a sudden drop as function of impurity density or magnetic moment amplitude…
We consider the effects of magnetic impurities on the surface of superconducting Cu doped $Bi_{2}Se_{3}$ in the odd parity pairing phase which support topologically protected Majorana fermions surface states with linear spectrum. We show that a single magnetic impurity on the surface may induce a pair of in-gap localized bound states. The energy of the in-gap state is extremely sensitive to the orientation of the magnetic impurity due to the so-called Ising properties of Majorana fermions. The magnetic impurity induced spin-texture, which can be measured using spin sensitive STM, is calculated…
When subject to a weak magnetic impurity distribution, the order parameter and quasi-particle energy gap of a weakly disordered bulk s-wave superconductor are suppressed. In the Born scattering limit, recent investigations have shown that `optimal fluctuations' of the random impurity potential can lead to the nucleation of `domains' of localised states within the gap region predicted by the conventional Abrikosov-Gor'kov mean-field theory, rendering the superconducting system gapless at any finite impurity concentration. By implementing a field theoretic scheme tailored to the weakly disordere…
We propose a method to probe and control the interactions between an ensemble of magnetic impurities in a superconductor via microwave radiation. Our method relies upon the presence of sub-gap Yu-Shiba-Rusinov (YSR) states associated with the impurities. Depending on the sign of the detuning, radiation generates either a ferro- or antiferromagnetic contribution to the exchange interaction. This contribution can bias the statistics of the random exchange constants stemming from the RKKY interaction. Moreover, by measuring the microwave response at the YSR resonance, one gains information about …
Bound states in superconductors are expected to exhibit a spatially resolved electron-hole asymmetry which is the hallmark of their quantum nature. This asymmetry manifests as oscillations at the Fermi wavelength, which is usually tiny and thus washed out by thermal broadening or by scattering at defects. Here we demonstrate theoretically and confirm experimentally that, when coupled to magnetic impurities, bound states in a vortex core exhibit an emergent axial electron-hole asymmetry on a much longer scale, set by the coherence length. We study vortices in 2H-NbSe$_2$ and in 2H-NbSe$_{1.8}$S…
We calculate frequency and temperature dependence of the anomalous ac Hall conductivity induced by impurity scattering in a chiral px+ipy superconductor, such as Sr2RuO4, with spontaneous time-reversal-symmetry breaking in the absence of an external magnetic field. We consider two models of disorder, Gaussian and non-Gaussian, characterized by the second and third moments of the random impurity potential, respectively. Within both models, we find that the anomalous Hall conductivity has a finite real value at zero frequency, exhibits singularities at the threshold of photon absorption across t…
We present a theory of the subgap tails of the density of states in a diffusive superconductor containing magnetic impurities. We show that the subgap tails have two contributions: one arising from mesoscopic gap fluctuations, previously discussed by Lamacraft and Simons, and the other associated to the long-wave fluctuations of the concentration of magnetic impurities. We study the latter both in small superconducting grains and in bulk systems [$d=1,2,3$], and establish the dimensionless parameter that controls which of the two contributions dominates the subgap tails. We observe that these …
When subject to a weak magnetic impurity potential, the order parameter and quasi-particle energy gap of a bulk singlet superconductor are suppressed. According to the conventional mean-field theory of Abrikosov and Gor'kov, the integrity of the energy gap is maintained up to a critical concentration of magnetic impurities. In this paper, a field theoretic approach is developed to critically analyze the validity of the mean field theory. Using the supersymmetry technique we find a spatially homogeneous saddle-point that reproduces the Abrikosov-Gor'kov theory, and identify instanton contributi…
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…
The emergence of a topological superconducting state in van der Waals heterostructures provides a new platform for exploring novel strategies to control topological superconductors. In particular, impurities in van der Waals heterostructures, generically featuring a moire pattern, can potentially lead to the unique interplay between atomic and moire length scales, a feature absent in generic topological superconductors. Here we address the impact of non-magnetic impurities on a topological moire superconductor, both in the weak and strong regime, considering both periodic arrays and single imp…
Magnetic impurities on a superconductor induce sub-gap Yu-Shiba-Rusinov (YSR) bound states, localized at the impurity site and fading away from it for distances up to several nanometers. In this article, we present a theoretical method to calculate the spatial distribution of the YSR spectrum of a two-dimensional superconductor with arbitrary Fermi contours (FCs) in the presence of magnetic impurities. Based on the Green's Function (GF) formalism, we obtain a general analytical expression by approximating an arbitrary contour shape to a regular polygon. This method allows us to show the connec…
We investigate the persistent currents, spin-polarized local density of states, and spectral functions of topological superconductors constructed by placing ferromagnetic impurities on top of an s-wave superconductor with Rashba spin-orbit interaction. We solve self-consistently for the superconducting order parameter and investigate both two-dimensional blocks and one-dimensional wires of ferromagnetic impurities, with the magnetic moments pointing both perpendicular and parallel to the surface. We find that the topologically protected edge states of ferromagnetic blocks give rise to spin-pol…
We address the question of whether individual nonmagnetic impurities can induce zero-energy states in time-reversal-invariant topological superconductors, and define a class of symmetries which guarantee the existence of such states for a specific value of the impurity strength. These symmetries allow the definition of a position-space topological Z_2 invariant, which is related to the standard bulk topological Z_2 invariant. Our general results are applied to the time-reversal-invariant p-wave phase of the doped Kitaev-Heisenberg model, where we demonstrate how a lattice of impurities can dri…
We investigate the effect of single and multiple impurities on the Zeeman-localized, spin polarized bound states in dilute magnetic semiconductor hybrid system. Such bound states appear whenever a dilute magnetic semiconductor showing giant Zeeman effect is exposed to an external magnetic field showing nanoscale inhomogeneity. We consider the specific example of a superconductor-dilute magnetic semiconductor hybrid, calculate the energy spectrum and the wave functions of the bound states in the presence of a single impurity, and monitor the evolution of the bound state as a function of the imp…
We theoretically study bound states generated by magnetic impurities within conventional $s$-wave superconductors, both analytically and numerically. In determining the effect of the hybridization of two such bound states on the energy spectrum as a function of magnetic exchange coupling, relative angle of magnetization, and distance between impurities, we find that quantum phase transitions can be modulated by each of these parameters. Accompanying such transitions, there is a change in the preferred spin configuration of the impurities. Although the interaction between the impurity spins is …
The performance of a quantum computation system is investigated, with qubits represented by magnetic impurities in coupled quantum dots filled with two electrons. Magnetic impurities are electrically manipulated by electrons. The dominant noise source is the electron mediated indirect coupling between magnetic impurities and host spin bath. As a result of the electron mediated coupling, both noise properties and the time needed for elementary gate operations, depend on controllable system parameters, such as size and geometry of the quantum dot, and external electric and magnetic fields. We fi…
Impurities embedded in electronic systems induce bound states which under certain circumstances can hybridize and lead to impurity bands. Doping of insulators with impurities has been identified as a promising route towards engineering electronic topological states of matter. In this paper we show how to realize tuneable Chern insulators starting from a three dimensional topological insulator whose surface is gapped and intentionally doped with magnetic impurities. The main advantage of the protocol is that it is robust, and in particular not very sensitive to the impurity configuration. We ex…
Hybrid semiconductor-superconductor quantum dot devices are tunable physical realizations of quantum impurity models for a magnetic impurity in a superconducting host. The binding energy of the localized sub-gap Shiba states is set by the gate voltages and external magnetic field. In this work we discuss the effects of the Zeeman spin splitting which is generically present both in the quantum dot and in the (thin-film) superconductor. The unequal $g$-factors in semiconductor and superconductor materials result in respective Zeeman splittings of different magnitude. We consider both classical a…
