Yu-Shiba-Rusinov States in Superconductors

Updated 7 July 2026

YSR states are impurity-induced bound states in superconductors formed when magnetic exchange interactions break Cooper pairs.
They exhibit pronounced particle-hole asymmetry and spin-resolved features observable in tunneling spectroscopy, enabling insights into many-body ground states.
Extended models show that hybridization in impurity arrays produces complex spatial patterns that reflect dimensionality, electronic structure, and host characteristics.

Yu-Shiba-Rusinov (YSR) states are impurity-induced bound states that appear inside the superconducting energy gap when a localized magnetic moment interacts with a superconductor. In a clean superconductor, the low-energy excitations are gapped by the superconducting order parameter $\Delta$ , so for sufficiently low temperature $k_B T < \Delta$ there are no quasiparticle states within that gap; a magnetic impurity breaks Cooper pairs through exchange scattering and can trap subgap quasiparticle states at discrete energies. In tunneling spectroscopy, these states appear as pairs of narrow peaks inside the superconducting gap, often with pronounced particle-hole asymmetry, and their zero-energy crossings are tied to changes in the many-body ground state. Recent work has extended this framework from single adatoms in conventional $s$ -wave superconductors to molecular spins, diluted arrays, proximitized graphene, superconductors with nontrivial $Z_2$ bands, Ising superconductors, dirty films, and even a charge-exciton analogue in an exciton condensate (Wang et al., 2020, Kwon et al., 19 Dec 2025).

1. Canonical mechanism and single-impurity energetics

In the standard picture, a magnetic impurity in an $s$ -wave superconductor exchange-couples to the superconducting quasiparticles and locally breaks Cooper pairs. The resulting impurity-bound excitation is localized, energetically isolated, and appears inside the interval $(-\Delta,\Delta)$ . A widely quoted single-impurity expression is

$E_{\mathrm{YSR}}=\pm \Delta \frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi \rho_0 J,$

or, in equivalent notation, $\alpha \propto \nu_0 J S$ (Ptok et al., 2017, Trivini et al., 2024). In the ordinary case, increasing the exchange coupling drives the YSR level toward zero energy; the zero-energy crossing signals a change in the fermion parity of the impurity-bound state (Chiu et al., 2021).

Potential scattering is an equally important ingredient when particle-hole asymmetry is discussed. In the Green-function treatment for Mn on $\beta$ -Bi $_2$ Pd $k_B T < \Delta$ 0\beta=\pi \nu_0 U $k_B T < \Delta$ 1 $k_B T < \Delta$ 2 $k_B T < \Delta$ 3V_m $k_B T < \Delta$ 4J_m $k_B T < \Delta$ 5\alpha_m² = 1+\beta_m^{2 $</sup> (<a href="/papers/2105.06651" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Oppen et al., 2021</a>).</p> <p>The basic competition is not merely between a classical local field and a gap parameter. Quantum-impurity descriptions emphasize that the relevant many-body sectors differ in parity and spin. In the screened-spin regime, the ground state is a singlet and the lowest excited state is a doublet; in the free-spin regime, the ground state is a doublet and the lowest excited state is a singlet. The subgap peaks at$ k_B T < \Delta$6 can therefore look superficially similar at zero field even when the underlying many-body assignment is different (Machida et al., 2022). This distinction becomes central in spin-resolved and Zeeman-resolved experiments.}

2. Spin structure, particle-hole partners, and Zeeman diagnostics

YSR states are intrinsically spin-resolved objects. Spin-polarized scanning tunneling spectroscopy on Fe adatoms deposited on the surface of $k_B T < \Delta$7 showed that finite-energy YSR states at about $k_B T < \Delta$8 come in pairs with opposite spin polarization, exactly as expected theoretically: when the tip and impurity magnetic moments are parallel, one member of the YSR pair is enhanced; when they are antiparallel, the opposite member is enhanced; reversing the tip magnetization reverses the contrast (Wang et al., 2020). This directly confirms the prediction that the electron-like and hole-like components of a YSR pair carry opposite spin signatures.

The same experiment also sharpened the distinction between YSR and Majorana interpretations. Some Fe adatoms produced zero-bias peaks rather than finite-energy YSR pairs, but the zero-bias signal showed no measurable spin dependence: changing the tip polarization or flipping the Fe atom’s moment did not alter the peak within experimental error. The absence of a spin signature was therefore taken as evidence against Majorana bound states associated with Fe adatoms on $k_B T < \Delta$9 and in favor of a pair of YSR states accidentally tuned very close to zero energy (Wang et al., 2020).

Magnetic-field response provides a separate diagnostic of the many-body ground state. Ultra-low-temperature STM/STS of Fe-decorated superconducting Nb tips showed that the YSR peak either splits or shifts in a magnetic field. A split peak identifies a screened-spin ground state because the excited doublet is Zeeman split; a shifted peak identifies a free-spin ground state because the Zeeman-split ground-state doublet leaves only one thermally occupied branch visible in tunneling (Machida et al., 2022). The same work used this splitting-versus-shifting criterion as an unambiguous ground-state diagnostic.

A related but distinct route to spin polarization is provided by a spin-split superconducting host. In a superconductor proximitized by a ferromagnetic insulator, the homogeneous exchange field $s$0 produces finite YSR polarization even at zero external magnetic field. In the quantum single-site treatment, the normalized polarization scales as

$s$1

so any finite $s$2 produces finite polarization, whereas the classical approach predicts fully polarized YSR excitations even in the absence of exchange and external magnetic field (Skurativska et al., 2023). This contrast has become a useful benchmark for distinguishing quantum from classical impurity descriptions.

3. Spatial structure, dimensionality, and the breakdown of the point-scatterer picture

The textbook asymptotic form for a point-like magnetic impurity in an isotropic three-dimensional superconductor is

$s$3

with particle-like and hole-like phase shifts $s$4 (Etzkorn et al., 2018). This form captures a single centered scatterer, Friedel-like oscillations at the Fermi-wavelength scale, and a sizable particle-hole phase shift that should track the YSR energy. It remains a useful reference, but a large fraction of recent YSR phenomenology is defined by systematic departures from it.

A clear example is Cu-phthalocyanine on V(100), where each molecule generates several YSR states at different energies and with distinctly different spatial intensity patterns. The strongest YSR intensity is not on the Cu center, but close to one of the pyrrolic N atoms, and the YSR signal remains detectable well beyond the molecular footprint, with clear oscillations of the intensities without strong particle-hole scattering phase differences. The interpretation is that the molecule must be treated as an extended scatterer with internal orbital structure rather than as a point impurity (Etzkorn et al., 2018).

Fe adatoms on $s$5 show a related dimensional crossover. The finite-energy YSR peaks extend laterally over several nanometers, whereas nearby features at $s$6 decay over only about $s$7. The longer decay is consistent with a YSR state mediated by the superconducting condensate, while the shorter decay is attributed to the magnetic dipole field of the impurity suppressing superconductivity locally. For the YSR state itself, the observed decay is more consistent with an effective $s$8 form, $s$9, rather than the $Z_2$0 $Z_2$1 form, suggesting that the relevant physics may be dominated by surface states (Wang et al., 2020).

Lattice geometry and reduced dimensionality produce their own characteristic textures. In a triangular-lattice model for 2H-NbSe$Z_2$2, single-impurity YSR states form a robust 6-fold star-shaped pattern, and the long-range oscillating particle-hole asymmetry is mainly attributed to reduced dimensionality, triangular-lattice geometry, and Fermi-surface structure rather than to weak in-plane spin-orbit coupling (Ptok et al., 2017). In superconducting graphene induced by Pb islands, the YSR signal at a graphene grain boundary extends more than 20 nm away from the boundary and shows a $Z_2$3-type modulation, which the authors attribute to intervalley scattering and the sublattice-sensitive response of graphene (Río et al., 2020). Taken together, these results establish that the real-space structure of a YSR state is a probe of dimensionality, Fermi-surface anisotropy, orbital content, and impurity extent.

4. Hybridization, arrays, and many-body collective YSR systems

When several magnetic impurities are brought close enough that their YSR wavefunctions overlap, the single-impurity picture gives way to hybridized collective modes. On the surface of $Z_2$4-Bi$Z_2$5Pd, diluted arrays of Mn atoms show a progression from a single YSR pair for an isolated atom, to split YSR peaks in dimers, to larger splittings in clusters and chains, and to multiple collective in-gap resonances in a $Z_2$6 Mn square array. The splitting size increases with the number of atoms, and the orientation of the structure along $Z_2$7, $Z_2$8, and $Z_2$9 modulates both the splitting and the particle-hole asymmetry because the coupling is shaped by the strongly anisotropic, approximately square-like Fermi contour. In the 25-atom array, three in-gap YSR peaks labeled $s$0, $s$1, and $s$2 were resolved, and their spatial distribution reflects a chiral LDOS (Trivini et al., 2024).

Molecular systems show that collective YSR physics need not rely on weak substrate-mediated exchange alone. Fe-tetraphenyl porphyrin on Pb(111) exhibits two pairs of subgap peaks that are interpreted as two distinct YSR channels coupled by intramolecular Hund’s-like exchange. Spectra on pyrrole and Fe sites show the same YSR peaks but the amplitude of their particle and hole components is reversed, and this identical particle-hole asymmetry pattern for both $s$3 and $s$4 is explained by two spin-hosting orbitals with opposite potential scattering signs that are coupled strongly (Rubio-Verdú et al., 2020). The result is a many-body YSR state spread across a molecular platform rather than a set of independent local channels.

Inter-impurity exchange can dominate the YSR spectrum even when the impurity-host Kondo scale remains large. In a deliberately engineered Fe cluster on oxygen-reconstructed Ta(100), the YSR energy correlates much better with the exchange-splitting gap $s$5 than with the fitted Kondo temperature $s$6. Numerical renormalization group calculations reproduce a crossover from two decoupled Kondo singlets at small antiferromagnetic $s$7 to a local antiferromagnetic dimer at large $s$8, and the YSR state shows a characteristic cross-over in its energetic position and particle-hole asymmetry across that transition (Kamlapure et al., 2019).

Symmetric clusters reveal an additional organizing principle. For equidistant magnetic adatom clusters, the low-energy theory predicts two sectors: spin-polarized dispersive states whose energies depend only on the net cluster moment $s$9, and spin-unpolarized pinned states whose wave functions satisfy $(-\Delta,\Delta)$0 and decouple from the net moment. In a three-adatom equilateral cluster, the pinned levels occur at

$(-\Delta,\Delta)$1

and remain stable against thermal fluctuations of the spin texture as long as the equal-distance geometry is preserved (Körber et al., 2018). This establishes that YSR hybridization can generate not only Shiba bands and bonding-antibonding splittings, but also symmetry-protected pinned subgap states.

5. Realistic impurity structure, anisotropy, and control paradigms

A major theme of recent theory is that real transition-metal impurities are not well captured by either a classical fixed spin or a spin-$(-\Delta,\Delta)$2 one-channel impurity. The zero-bandwidth model developed for “real metals” incorporates higher impurity spins, multiple exchange channels, crystal or ligand-field effects, and magnetic anisotropy, while deliberately neglecting Kondo renormalizations of the exchange couplings. Its single-channel limit is most relevant for transition-metal impurities embedded into metallic coordination complexes on superconducting substrates, while the multi-channel case models transition-metal adatoms (Oppen et al., 2021). Within this framework, the number, degeneracy, and anisotropy splitting of YSR lines are set by channel symmetry and impurity multiplet structure.

Single magnetic molecules provide an explicit realization of this complexity. Numerical renormalization group calculations for a molecule with one active orbital, a large core spin $(-\Delta,\Delta)$3, exchange coupling $(-\Delta,\Delta)$4, and uniaxial anisotropy $(-\Delta,\Delta)$5 show that the critical Coulomb interaction for the singlet/doublet transition decreases in the presence of exchange coupling for both ferro- and antiferromagnetic cases. The number of YSR states increases to two pairs, and, depending on the sign and strength of the anisotropy field, up to three pairs of YSR states can emerge within the gap (Pradhan et al., 2020). The key point is that additional internal spin degrees of freedom create multiple subgap transitions, some of which may have zero spectral weight in single-particle tunneling.

Control protocols now extend beyond static exchange coupling. If a bosonic mode couples to the tunneling between an Anderson impurity and a superconducting host, a $(-\Delta,\Delta)$6-wave conduction-band channel opens in addition to the usual even-parity channel, implying an additional pair of odd-parity YSR states. In the Schrieffer-Wolff description, the even channel involves even numbers of virtual bosons and the odd channel involves odd numbers, so the exchange couplings depend sensitively on the bosonic occupation or on the amplitude and frequency of a classical Floquet drive (Müller et al., 2022). This is a direct route to phononic or photonic control of YSR energies, spin polarizations, and ground-state parity.

Device-oriented proposals push this logic further. A quantum dot embedded between two small superconducting islands can host two spin-singlet YSR states with different spatial charge distributions; in a symmetric device these are the even and odd combinations of left-channel and right-channel singlets. Because the transition dipole and quadrupole matrix elements between the two states are nonzero, electric-field pulses on the gate electrode can coherently manipulate the pair as a qubit basis (Pavešić et al., 2021). In a complementary direction, the Richardson-model study of the BCS-BEC crossover found that the effect of increasing pairing strength on the YSR excitation spectrum is only quantitative and rather weak after suitable rescaling, and that the deep BEC limit is exactly solvable and equivalent to a zero-bandwidth effective BCS mean-field Hamiltonian (Žitko et al., 2022). This suggests that a small number of effective degrees of freedom can remain predictive even far from the weak-coupling BCS limit.

6. Topological bands, unconventional hosts, and disorder-broadened YSR physics

In superconductors with nontrivial $(-\Delta,\Delta)$ 7 bands, YSR spectroscopy probes not only impurity exchange but also the underlying topological band structure. For a magnetic impurity coupled to a superconducting topological-insulator model, the in-gap spectrum can show two zero-energy crossings as the exchange coupling is increased: one originates from the superconducting topological surface Dirac cone and the second from the bulk $(-\Delta,\Delta)$ 8 bands near the surface. Finite chemical potential and in-plane exchange components hybridize the two YSR states and gap out a finite-energy crossing, producing the experimentally relevant “bend back” of the in-gap branch (Chiu et al., 2021). This directly demonstrates that a surface-only model is incomplete for $(-\Delta,\Delta)$ 9.

Graphene-based realizations broaden the class of YSR hosts. In proximized multilayer graphene with nanometer-scale Pb islands, sharp in-gap features localized at graphene grain boundaries were identified as the first observation of YSR states in graphene and as the first observation of YSR states in a chemically pure system. The interpretation is that the grain boundary hosts a local magnetic moment, superconductivity is induced by proximity from Pb, and the YSR states vanish above $E_{\mathrm{YSR}}=\pm \Delta \frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi \rho_0 J,$ 0 (Río et al., 2020). Hydrogenated graphene goes further: individual chemisorbed hydrogen atoms act as paramagnetic centers, and proximity-induced superconductivity produces “unconventional” YSR states even though pristine graphene has vanishing DOS at the Dirac point. In this setting the low-coupling behavior is approximately linear,

$E_{\mathrm{YSR}}=\pm \Delta \frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi \rho_0 J,$ 1

and the parity-switching point depends on $E_{\mathrm{YSR}}=\pm \Delta \frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi \rho_0 J,$ 2, unlike the conventional $E_{\mathrm{YSR}}=\pm \Delta \frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi \rho_0 J,$ 3-independent criterion (Lado et al., 2015).

The YSR mechanism has also been generalized beyond superconductivity. In the excitonic insulator Ta $E_{\mathrm{YSR}}=\pm \Delta \frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi \rho_0 J,$ 4Pd $E_{\mathrm{YSR}}=\pm \Delta \frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi \rho_0 J,$ 5Te $E_{\mathrm{YSR}}=\pm \Delta \frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi \rho_0 J,$ 6, STM/STS directly visualized in-gap states bound to Ta-vacancy defects inside the excitonic gap. The theoretical interpretation is a charge-exciton version of the YSR mechanism: a Ta-vacancy-induced charge dipole locally breaks electron-hole pairing in the exciton condensate and creates a pair of impurity-bound states, $E_{\mathrm{YSR}}=\pm \Delta \frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi \rho_0 J,$ 7 and $E_{\mathrm{YSR}}=\pm \Delta \frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi \rho_0 J,$ 8, whose energies track the excitonic gap under local strain and tip-induced carrier injection (Kwon et al., 19 Dec 2025). This establishes a charge analogue of YSR physics in a condensate of excitons rather than Cooper pairs.

Ising superconductors introduce a different kind of unconventional host. In the presence of Ising spin-orbit coupling and an in-plane magnetic field, the rotated pairing contains both a reduced singlet part,

$E_{\mathrm{YSR}}=\pm \Delta \frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi \rho_0 J,$ 9

and a field-induced equal-spin triplet component. The impurity self-energy therefore contains additional spin-dependent normal and anomalous terms, and the YSR spectrum can split into multiple branches, exhibit finite-energy avoided crossings, and depend strongly on impurity orientation and valley structure (Hein et al., 12 May 2026). In this setting, magnetic impurities function as local probes of the spin-valley-locked condensate.

Disorder changes the YSR problem at the level of the spectral singularity itself. In a dirty superconducting film, potential disorder broadens the clean-limit delta-function LDOS peak at the YSR energy into a finite-width resonance. Within the Usadel treatment, the width scales with the square root of the normal-state resistance per square, and mesoscopic fluctuations further reduce the peak height, shift the resonance position, and generate soft tails. A normal-metal STM tip can add a decay channel and mask the YSR feature, whereas a superconducting tip mainly shifts the energy without introducing a decay width (Babkin et al., 2022). This disorder-broadened regime is the appropriate reference for YSR spectroscopy in resistive films and diffusive heterostructures.