Dynes-Fulton Analysis in Superconductivity

• Dynes-Fulton Analysis is a method that fits superconducting tunneling data and Josephson interference patterns to extract key parameters like the ideal order parameter and pair-breaking rate.
• It utilizes the Dynes density of states model to parameterize spectral broadening in disordered superconductors and to invert magnetic interference patterns into spatial current profiles.
• Recent advances extend the technique with microscopic disorder interpretations, thermodynamic and electrodynamic consistency, and corrections for nonlinear phase profiles in Josephson junctions.

Dynes-Fulton analysis denotes two closely related but experimentally distinct analysis traditions in superconductivity. In tunneling spectroscopy, the term is used for fitting measured $dI/dV$ curves with the Dynes density of states, often after thermal broadening, in order to extract an ideal order parameter and a broadening scale from dirty or otherwise non-BCS-like superconductors. In Josephson-junction interferometry, the same label refers to the classical inversion of a magnetic interference pattern $I_c(B)$ into a spatial critical current density $J_c(y)$, usually under a linear-phase assumption. Recent work has supplied a microscopic interpretation of the Dynes spectral form, extended it into a thermodynamically consistent theory, generalized it to optical response, and shown that the classical Josephson reconstruction fails when the phase profile is nonlinear (Herman et al., 2016, Herman et al., 2017, Kudriashov et al., 8 Aug 2025).

1. Terminological scope and core objects

The two principal uses of the expression are organized by the measured observable and the inverse problem being solved.

Context	Measured quantity	Reconstructed or fitted quantity
Tunneling spectroscopy	$dI/dV$ or tunneling DOS	$\bar\Delta$, $\Gamma$, and related spectral parameters
Josephson interferometry	$I_c(B)$	Spatial critical current density $J_c(y)$

In the tunneling setting, the central phenomenological object is the Dynes density of states

$$N(\omega)=N_0\,\mathrm{Re}\left[\frac{\omega+i\Gamma}{\sqrt{(\omega+i\Gamma)^2-\bar\Delta^2}}\right],$$

which is widely used because it fits the tunneling density of states of disordered superconductors well. In the Josephson setting, the central object is the interference pattern $I_c(B)$, interpreted through a phase profile across the junction width. The common feature is that both usages treat experimentally accessible data as an encoded representation of a more microscopic quantity, but the mathematical structures are different: spectral broadening in one case, spatial inversion in the other (Herman et al., 2016, Kudriashov et al., 8 Aug 2025).

A recurrent theme across the literature is that neither usage should be treated as a purely formal fitting recipe. In the tunneling case, $I_c(B)$0 may encode pair breaking or environment-assisted tunneling rather than a generic “lifetime broadening.” In the Josephson case, the inversion is exact only if the correct phase profile $I_c(B)$1 is used, and the standard linear-phase version can generate non-physical artifacts when that assumption fails [(Herman et al., 2016); (Pekola et al., 2010); (Kudriashov et al., 8 Aug 2025)].

2. Spectroscopic Dynes-Fulton analysis in tunneling experiments

In practical tunneling spectroscopy, Dynes-Fulton analysis usually means fitting measured $I_c(B)$2 curves with the Dynes DOS, often thermally broadened according to

$I_c(B)$3

Within this usage, $I_c(B)$4 is interpreted as the underlying ideal order parameter and $I_c(B)$5 as the parameter controlling in-gap spectral weight and the suppression of coherence peaks near $I_c(B)$6 (Herman et al., 2016, Lebedeva et al., 2024).

A major conceptual advance was the demonstration that the Dynes formula can be written as the Eliashberg-type expression

$I_c(B)$7

with the causal, frequency-dependent gap function

$I_c(B)$8

This $I_c(B)$9 vanishes at $J_c(y)$0 and tends to $J_c(y)$1 at high frequency. In that sense, the Dynes form corresponds to a gapless but still superconducting state rather than an arbitrary ad hoc broadening rule (Herman et al., 2016).

The phenomenological content of the fit was further developed in analytic studies of the Dynes superconductor model. The gap equation is written as

$J_c(y)$2

so pair breaking enters through the shift

$J_c(y)$3

This formulation distinguishes clean BCS quantities $J_c(y)$4, $J_c(y)$5, and $J_c(y)$6 from Dynes quantities $J_c(y)$7, $J_c(y)$8, and $J_c(y)$9, and treats $dI/dV$0 explicitly as a pair-breaking scattering rate rather than as an unspecified fit width (Lebedeva et al., 2024).

3. Microscopic disorder interpretation and the CPA construction

The microscopic interpretation of the Dynes formula given in "Microscopic interpretation of the Dynes formula for the tunneling density of states" (Herman et al., 2016) is based on a disordered superconducting model with three ingredients: homogeneous pairing $dI/dV$1, pair-conserving potential disorder $dI/dV$2, and magnetic or pair-breaking disorder $dI/dV$3. The local random potential is

$dI/dV$4

with $dI/dV$5 and $dI/dV$6 distributed independently through even probability distributions $dI/dV$7 and $dI/dV$8. The distinction between the two disorder channels is essential: pair-conserving disorder broadens single-particle motion but does not destroy pairing in the Anderson-theorem sense, whereas pair-breaking disorder suppresses superconducting coherence and generates subgap states (Herman et al., 2016, Herman et al., 2017).

The key result is obtained for a Lorentzian distribution of pair-breaking fields,

$dI/dV$9

Within the coherent potential approximation, the disordered system is replaced by an effective translationally invariant medium with self-energy

$\bar\Delta$0

and the CPA condition requires that the average residual scattering vanish: $\bar\Delta$1 For the Lorentzian $\bar\Delta$2, the $\bar\Delta$3-average can be performed analytically, yielding

$\bar\Delta$4

The Dynes DOS then follows exactly. In this construction, $\bar\Delta$5 is the width of the pair-breaking distribution, not merely a phenomenological damping rate, while arbitrary potential disorder does not spoil the Dynes form (Herman et al., 2016).

The same framework also introduces the wave-function renormalization

$\bar\Delta$6

where $\bar\Delta$7 denotes pair-conserving scattering and $\bar\Delta$8 denotes pair breaking. This defines the paper’s “canonical Dynes superconductor”: a system with both pair-conserving and pair-breaking scattering, but with a tunneling DOS governed by the Dynes formula (Herman et al., 2016).

4. Thermodynamic and electrodynamic extensions

The thermodynamic program developed in "Thermodynamic properties of the Dynes superconductors" (Herman et al., 2017) shows that Dynes superconductivity is not only a spectral fit but a thermodynamically consistent theory. The CPA equations are derived from a free-energy functional

$\bar\Delta$9

and the superconducting condensation free-energy difference reduces to

$\Gamma$0

A major consequence is that $\Gamma$1 is independent of the pair-conserving scattering rate $\Gamma$2, which is the Anderson-theorem statement in this setting (Herman et al., 2017).

Because the Dynes state is gapless when $\Gamma$3, its low-temperature asymptotics differ qualitatively from clean BCS behavior. At $\Gamma$4,

$\Gamma$5

so the critical pair-breaking rate is

$\Gamma$6

The low-$\Gamma$7 gap correction is $\Gamma$8 rather than exponential, the specific heat is linear in $\Gamma$9,

$I_c(B)$0

and the specific heat, critical field, gap, and penetration depth all acquire power-law low-$I_c(B)$1 behavior. Near a coupling-constant-controlled superconductor-to-normal-metal transition, the usual dirty-limit Homes relation crosses over from $I_c(B)$2 to the pair-breaking dominated scaling $I_c(B)$3 when $I_c(B)$4 (Herman et al., 2017).

The analytic study in "Detailed Analysis of the Superconducting Gap with Dynes Pair-Breaking Scattering" (Lebedeva et al., 2024) sharpened several of these results. It gave the exact zero-temperature relation

$I_c(B)$5

the critical value

$I_c(B)$6

and the critical ratio

$I_c(B)$7

It also showed that the analytic near-$I_c(B)$8 approximation is accurate to better than $I_c(B)$9 for

$J_c(y)$0

and proposed the global approximation

$J_c(y)$1

with error below about $J_c(y)$2 over the full temperature range and typically below $J_c(y)$3 for practical $J_c(y)$4 (Lebedeva et al., 2024).

The formalism has also been extended to optics. In "Signatures of Dynes superconductivity in the THz response of ALD-grown NbN thin films" (Bolle et al., 16 Feb 2026), terahertz time-domain spectroscopy and frequency-domain spectroscopy were performed on ALD-grown NbN films with thicknesses $J_c(y)$5, $J_c(y)$6, $J_c(y)$7, $J_c(y)$8, and $J_c(y)$9 over the range $$N(\omega)=N_0\,\mathrm{Re}\left[\frac{\omega+i\Gamma}{\sqrt{(\omega+i\Gamma)^2-\bar\Delta^2}}\right],$$0 to $$N(\omega)=N_0\,\mathrm{Re}\left[\frac{\omega+i\Gamma}{\sqrt{(\omega+i\Gamma)^2-\bar\Delta^2}}\right],$$1. Using the optical conductivity model of Herman and Hlubina together with the Dynes DOS, the work identified a step-like onset of absorption at $$N(\omega)=N_0\,\mathrm{Re}\left[\frac{\omega+i\Gamma}{\sqrt{(\omega+i\Gamma)^2-\bar\Delta^2}}\right],$$2, rather than only at the BCS threshold $$N(\omega)=N_0\,\mathrm{Re}\left[\frac{\omega+i\Gamma}{\sqrt{(\omega+i\Gamma)^2-\bar\Delta^2}}\right],$$3. For the $$N(\omega)=N_0\,\mathrm{Re}\left[\frac{\omega+i\Gamma}{\sqrt{(\omega+i\Gamma)^2-\bar\Delta^2}}\right],$$4 film, the fitted pair-breaking rate was essentially temperature independent and equal to about

$$N(\omega)=N_0\,\mathrm{Re}\left[\frac{\omega+i\Gamma}{\sqrt{(\omega+i\Gamma)^2-\bar\Delta^2}}\right],$$5

This extends Dynes-Fulton-style analysis beyond tunneling into bulk-sensitive electrodynamics (Bolle et al., 16 Feb 2026).

5. Dynes-Fulton inversion for Josephson junction interference patterns

In Josephson-junction physics, Dynes-Fulton analysis is the classical method for inferring the spatial critical current density $$N(\omega)=N_0\,\mathrm{Re}\left[\frac{\omega+i\Gamma}{\sqrt{(\omega+i\Gamma)^2-\bar\Delta^2}}\right],$$6 from the magnetic interference pattern $$N(\omega)=N_0\,\mathrm{Re}\left[\frac{\omega+i\Gamma}{\sqrt{(\omega+i\Gamma)^2-\bar\Delta^2}}\right],$$7. The supplementary derivation in "Reconstructing Critical Current Density in Josephson Junctions with Phase Non-linearity" (Kudriashov et al., 8 Aug 2025) writes the supercurrent as

$$N(\omega)=N_0\,\mathrm{Re}\left[\frac{\omega+i\Gamma}{\sqrt{(\omega+i\Gamma)^2-\bar\Delta^2}}\right],$$8

where $$N(\omega)=N_0\,\mathrm{Re}\left[\frac{\omega+i\Gamma}{\sqrt{(\omega+i\Gamma)^2-\bar\Delta^2}}\right],$$9 is the junction width, $I_c(B)$0 is the transverse coordinate, and $I_c(B)$1 is the free Josephson phase. The critical current is the maximum over $I_c(B)$2, and the maximizing phase $I_c(B)$3 satisfies

$I_c(B)$4

The complex representation is

$I_c(B)$5

and the generalized inversion formula is

$I_c(B)$6

When $I_c(B)$7 is linear in $I_c(B)$8, the phase factor becomes a standard Fourier kernel and the reconstruction reduces to the conventional Dynes-Fulton limit. Under that assumption, $I_c(B)$9 behaves like a Fourier transform of $I_c(B)$00, which explains why Fraunhofer-like patterns can be inverted straightforwardly (Kudriashov et al., 8 Aug 2025).

The same work makes explicit that the critical phase is not independent of the current profile: $I_c(B)$01 This is the source of the connection to logarithmic Hilbert-transform structure and phase-retrieval ideas in the classical linear-phase setting. The generalized inversion is mathematically exact when the correct nonlinear phase profile is used, because substituting the forward relation into the inverse produces the delta-function identity and returns $I_c(B)$02 (Kudriashov et al., 8 Aug 2025).

6. Breakdown, ambiguity, and competing interpretations

The most important limitation of the Josephson version of Dynes-Fulton analysis is the linear-phase assumption. In planar Josephson junctions, the local phase $I_c(B)$03 is generally nonlinear because of geometry and field focusing. If one nevertheless applies the conventional inversion as though $I_c(B)$04 were linear, the reconstructed $I_c(B)$05 can exhibit oscillatory ripples, spurious side lobes, incorrect symmetry or asymmetry, and ambiguities when the interference pattern is strongly non-Fraunhofer. For a planar junction with $I_c(B)$06, the supplementary example reports an interference pattern “significantly distorted compared to the conventional Fraunhofer pattern,” with elevated side lobes at the first minima (Kudriashov et al., 8 Aug 2025).

To address this failure mode, the same paper develops an iterative reconstruction algorithm based on the exact forward relation and the generalized inverse formula. The procedure starts from an initial guess, computes $I_c(B)$07, updates $I_c(B)$08, applies the inverse transformation, enforces prior knowledge such as symmetry or expected asymmetry, and iterates to convergence. For symmetric reconstruction, the profile approaches the true distribution after about six iterations; for an asymmetric case initialized with a tilted guess $I_c(B)$09, accurate convergence is achieved by about the 20th iteration. Residual oscillatory artifacts are attributed to finite magnetic-field range and can be reduced by low-pass Fourier filtering (Kudriashov et al., 8 Aug 2025).

In the spectroscopic usage, the main interpretive controversy concerns the meaning of $I_c(B)$10. The CPA-based microscopic theory identifies $I_c(B)$11 with the width of a Lorentzian distribution of pair-breaking fields (Herman et al., 2016). By contrast, "Photon assisted tunneling as an origin of the Dynes density of states" (Pekola et al., 2010) shows that a high-temperature electromagnetic environment can generate an effective Dynes DOS in a normal metal-insulator-superconductor junction through $I_c(B)$12-theory convolution: $I_c(B)$13 For a weak resistive environment with $I_c(B)$14, this becomes a Lorentzian $I_c(B)$15,

$I_c(B)$16

and the effective DOS reduces to the Dynes form with

$I_c(B)$17

This means that a Dynes fit does not by itself distinguish intrinsic pair breaking from environment-assisted tunneling (Pekola et al., 2010).

A related caution arises when comparing tunneling and optics. In ALD-grown NbN, the terahertz response is bulk-sensitive whereas tunneling is surface sensitive, and disorder may be spatially inhomogeneous. The NbN study therefore emphasizes that a Dynes parameter extracted from optics is not directly equivalent to one extracted from tunneling. It also notes that the model is phenomenological even when it fits extremely well, that the microscopic origin of $I_c(B)$18 remains unresolved, and that the suppression of superconductivity in those films appears mainly in the fermionic regime of the disorder-driven SIT rather than in a bosonic or pseudogap regime (Bolle et al., 16 Feb 2026).

Taken together, these developments place Dynes-Fulton analysis on a more explicit footing. In tunneling and optical spectroscopy, it is a controlled way of parameterizing broadened superconducting spectra whose microscopic origin may be pair breaking, environment-assisted tunneling, or related disorder-induced mechanisms. In Josephson interferometry, it is an inverse method whose validity depends decisively on the actual phase profile across the junction. The shared lesson is that successful fitting or inversion does not eliminate the need for a forward model; it instead identifies the forward model that must be tested.