Disorder and impurities: pair breaking, bound states and pinning

Scope

Disorder is not one adjustable number. Elastic scattering, magnetic pair breaking, local bound states, localization, vortex pinning and poor grain connectivity affect different observables. Record defect chemistry, irradiation particle and energy, dose, temperature, annealing history and sample geometry. A larger residual resistivity does not by itself specify the pair-breaking rate or prove a change of carrier density. R1 R6 R7

Anderson's time-reversed-state argument explains the robustness of an isotropic, sign-preserving superconducting state to nonmagnetic elastic scattering under restricted assumptions. The participating states must remain extended and the averaged pairing interaction and density of states must not change substantially. It is not immunity to arbitrary chemical substitution, localization, magnetic scattering or spatial granularity. Constant $T_c$ alone does not uniquely determine pairing symmetry. R1 R6

Mathematical assumptions

In the dilute, uncorrelated, weak magnetic-scattering model for an isotropic singlet state, define the pair-breaking energy $\Gamma$ by $\ln(T_{c0}/T_c)=\psi(1/2+\Gamma/(2\pi k_BT_c))-\psi(1/2)$, where $\psi$ is the digamma function. With $g=\Gamma/(k_BT_{c0})$ and $t=T_c/T_{c0}$, the initial slope is $dt/dg=-\pi/4$, and $T_c$ vanishes at $g_c=\pi e^{-\gamma_E}/2\approx0.882$. Rate conventions differ; do not substitute a transport lifetime without deriving its relation to this $\Gamma$. Local numerical checks compare the equation with independently bounded Matsubara sums. The original AG paper remains a bibliographic anchor, not a completed original-text audit. R2 R5 R6

For one classical magnetic scattering channel in a particle–hole-symmetric, isotropic host with a fixed gap, use exchange $J\mathbf S\cdot\mathbf s$, $\mathbf s=\boldsymbol\sigma/2$, and normal density of states per spin $N_0$. Defining $\alpha=\pi N_0JS/2$ and potential strength $\beta=\pi N_0V$, one signed Yu–Shiba–Rusinov branch is $E/\Delta=(1+\beta^2-\alpha^2)/\sqrt{(1+\beta^2-\alpha^2)^2+4\alpha^2}$; its particle–hole partner has opposite energy. At $V=0$, this reduces to $(1-\alpha^2)/(1+\alpha^2)$. A zero crossing requires $\alpha^2=1+\beta^2$ in this model. These conventions and phase-shift checks do not solve quantum Kondo screening, multiple orbital channels or self-consistent gap suppression. R3 R4 R5 R8

Dose is a separate quantity: an electron charge dose of 1 C/cm² corresponds to approximately $6.24\times10^{18}$ incident electrons/cm². The 3.4 C/cm² exposure in Prozorov's protocol is about $2.12\times10^{19}$ electrons/cm². Neither number is the surviving defect concentration; displacement cross sections, recombination and annealing enter that conversion. R6

Experimental criteria

Prozorov studied Ba(Fe1−xRux)2As2 with WDS composition $x=0.24$ using 2.5 MeV electrons. Irradiation and in-situ resistivity measurements took place near 22 K; the initial superconducting transition is reported as 17.8 K in the methods and 17.85 K in the transport figure. Warming to room temperature between runs partially annealed the damage, by up to about 30%. The same contacts were retained. The residual resistivity in Figure 1b is a linear extrapolation from above $T_c$ to zero temperature, not a direct zero-temperature normal-state measurement. Figure 2b includes two other crystals with initial transitions of 15 K and 13.5 K, labeled as Hall-probe measurements; these are separate sample baselines, not successive irradiated transition values of the 17.85 K transport specimen. Their individual Ru fractions and operational transition thresholds are not assigned here. Increased resistivity and suppressed $T_c$ were compatible with the authors' multiband sign-changing scenario, whose interband/intraband scattering ratio matters. An approximately unchanged Hall coefficient is an additional observation, not a complete determination of multiband carrier densities. The 22 K irradiation temperature is not $T_c$, and no unverified onset, midpoint or zero-resistance label is assigned here. R6

Mizukami's BaFe2(As1−xPx)2 crystals had $x=0.33$–0.36 and initial transitions of 28, 29 or 30 K. The 28/29 K batches were tracked by tunnel-diode-oscillator frequency shifts; the 30 K crystal also supplied the resistivity series. Following 2.5 MeV irradiation, low-temperature penetration-depth curves evolved from nodal behavior to an activated regime, then back to gapless behavior with further disorder. In sample #29A after 4.9 C/cm², the penetration depth was flat below approximately $T/T_c=0.05$ within about 1 angstrom, with measurements reaching 80 mK. The absence of a Curie upturn gives a conditional upper limit of approximately $0.2\mu_B$ per Fe defect, using an estimated defect density and penetration depth $\lambda(0)\approx0.2$ micrometres; it does not establish exactly zero magnetic moment. The authors interpreted the irradiation sequence with accidental nodes and sign-changing multiband pairing. Figure 4 shows a two-band model calculation, not a directly measured density-of-states spectrum; the plotted penetration-depth curves are vertically offset for clarity. Here “topological change” concerns the gap's nodal structure; it does not establish a Majorana-carrying topological superconductor. The 80 mK measurement floor and 20–22 K irradiation bath remain separate from transition records. R7

Hatter measured manganese phthalocyanine on Pb(111) at 1.2 K with a superconducting Pb tip. The resolved Shiba triplets have 50–100 microelectronvolt widths and separations up to 400 microelectronvolts. The measured conductance includes the tip density of states: near-zero sample excitation energy appears near the tip-gap bias, not necessarily at zero measured bias. The authors numerically deconvolved the spectra and used relative spectral weights to distinguish impurity ground states and magnetic-anisotropy effects. This is local spectroscopy, not a 1.2 K transition measurement or a standalone Majorana demonstration. R8

Limits and disputes

A suppression curve alone is not a unique symmetry test. Separate changes in pairing interaction, band filling, magnetic moments and defect morphology from elastic scattering; distinguish intra- and interband processes. Gapless superconductivity is also different from destruction of superconducting order. The classical single-impurity formula cannot establish the quantum impurity parity change on its own; spectroscopy, tip calibration, spatial structure and the appropriate many-body model are needed. R3 R5 R6 R7 R8

Defects can help current transport by pinning vortices while harming other properties. BaZrO3-containing YBCO films illustrate an in-field current improvement that must be evaluated at specified temperature, field and orientation, rather than equated with higher $T_c$. Dirty two-gap theory also permits enhanced orbital upper critical fields under particular diffusivity and coupling conditions. Neither result implies monotonic improvement with damage, a universal type-I to type-II conversion, or an enhanced Pauli limit. R9 R10

The present article has selected original-paper and equation checks. The original AG full text and Shiba full derivation have not been read; Yu's original abstract and opening discussion were checked from scans. The detailed AG/YSR equation convention was cross-checked in the Balatsky author manuscript, with final-version comparison still open. Full supplements, raw-data reanalysis and complete scientific acceptance remain unfinished. R2 R3 R4 R5

Discovery implications

Use controlled defect series with repeat measurements before irradiation, after irradiation and after annealing. Compare $T_c$, gap spectra, residual resistivity, magnetic response, $H_{c2}$ and $J_c$ under matched conditions. Preserve sample identifiers and probe-specific transition criteria in machine-readable records; never merge irradiation temperatures, spectroscopy temperatures and superconducting transitions into a single temperature ranking. R6 R7 R8 R9

Related topics

Compare isotropic s-wave pairing, sign-changing s-wave pairing, chemical substitution, magnetic fields and critical current.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.