Born-limit scattering and pair-breaking crossover in d-wave superconductivity of (TMTSF)2ClO4

The system under study lends itself to study the effect of (variable density, in situ) disorder on the residual density of states simultaneously with the effect on the superconducting transition temperature.

The manuscript entitled “Born-limit scattering and pair-breaking crossover in d-wave superconductivity of (TMTSF)2ClO4” (Yano, et al.) documents simultaneously the rise of the residual DOS and the depression of the SC critical temperature Tc with increased disorder. The residual DOS and Tc are determined from ac calorimetry measurements. The disorder is indirectly controlled by the rate of cooling through an anion ordering transition that takes place at higher temperature. The authors explain the Fermi surface is indirectly affected as a result since there is a period doubling of the unit cell along the b-direction with slow cooling.
From the variation of the residual DOS with cooling rate, the authors conclude that the defects introduced by increasing the cooling rate are Born-limit scatterers, and consequently very different than defects studied previously in unconventional superconductors. The claim is a strong one that requires explicit defense. My understanding is that neither limit applies generally.
The dependence on the residual DOS decreases with initial cooling rate. This is the most striking result in the article. The authors’ explanation for it is not clearly articulated, at least to this reader. My understanding of the effect of dilute strong (resonant) scatterers (the initial state, slow cooling case) in a d-wave SC is to introduce localized states extending out from the defect location like 4 arms (due to the gap nodal structure).

Is this the right starting point? The authors don’t clearly state it.

Of course, increasing the number of resonant scatterers would only increase the ratio \gamma_0/\gamma_N (defined in the article as the residual DOS relative to normal state DOS). Here, the point is: what about adding anion-orderered domain walls with increasing density? In order for the ratio \gamma_0/\gamma_N to decrease, I must assume the number of localized states also decreases. The authors have discussed the geometry of their model by introducing the Born scatterers as extended domain walls—a configuration not previously addressed to my knowledge. If they made an argument as to why this reduces the number of localized states, then I missed it.

The authors need to address the weaknesses identified above.

Ask for major revision

The manuscript presents a study of the effects of nonmagnetic disorder on the specific heat of unconventional superconductor (TMTSF)₂ClO₄. The authors conclude that the only way to distinguish between Born and unitary scattering limits is to examine the residual density of states, obtained from the specific heat.

There are a number of issues.
1. The manuscript states: “However, while this extension predicts the effect of increased scattering on Tc , it does not provide any information how scattering acts on the superconducting order parameter.” This statement is incorrect. For any superconductor, conventional or unconventional, Tc is always proportional to the magnitude of the order parameter, just with different pre-factors.
2. The discussion of the density of states, N(E,k) ( DOS) in the introduction should be more specific. Heat capacity includes the integral over the Fermi surface containing N(E,k) of DOS, so it includes all energies. Unitary scattering primarily affects the states at energies close to the Fermi level, whereas Born limit influences the states near the gap edge, see V. G. Kogan, R. Prozorov, and V. Mishra, London penetration depth and pair breaking, Phys. Rev. B 88, 224508 (2013). Therefore DOS is affected quite significantly in both regimes. It is true, however, that in the Born limit, a very significant scattering rate is required to suppress Tc and superfluid density, hence affecting any thermodynamic quantity, such as heat capacity. This is the method the manuscript uses to distinguish between two regimes.
3. The authors state that “Almost all unconventional superconductors follow…” unitary limit. This may not quite accurate. Historically, this was the sole theory explaining a rapid change of Tc with non-magnetic disorder in high-Tc. Somehow the Authors do not cite this seminal paper, P. J. Hirschfeld and N. Goldenfeld, Effect of strong scattering on the low-temperature penetration depth of a d-wave superconductor, Phys. Rev. B 48, 4219 (1993).
Over time, it has been recognized that the scattering potential strength is usually intermediate, see for example (and references therein), K. Cho, M. Kończykowski, S. Teknowijoyo, S. Ghimire, M. A. Tanatar, V. Mishra, and R. Prozorov, Intermediate scattering potential strength in electron-irradiated YBa$2$Cu$3$O$ _{7-\delta}$ from London penetration depth measurements, Phys. Rev. B 105, 14514 (2022). This generalized potential is treated withing the well-developed T-matrix approach. It is believed that end-points of the scattering potential (Born and unitary limits) are never realized and the scattering strength is somewhere in between.
4. For a generalized theory of the effects of arbitrary scattering and order parameters, I recommend consulting L. A. Openov's work: Effect of nonmagnetic and magnetic impurities on the specific heat jump in anisotropic superconductors, Phys. Rev. B 69, 224516 (2004).
5. Stylistically, it appears the authors use “resonant scattering” and “unitary scattering” interchangeably, which are distinct regimes. The former produces in-gap bound states, while the latter affects DOS smoothly, as explained in the first referenced paper above.
6. The authors apply an over-simplified background subtraction developed for isotropic metals with isotropic pairing. Phonon modes are sensitive to any anisotropy. More critically, the authors assume there is no electronic contribution above Tc, stating: “after subtracting this background contribution, the phonon contribution was obtained by fitting Cphonon/T = B22T^2+B44*T^4 to the data above Tc.” It is likely they used, but neglected to include, the electronic term (which would be a constant) in the text. This procedure has significant uncertainty due to the fitting region being far from T=0, and because the formula is derived for a simple Fermi-liquid model, which does not apply to many unconventional superconductors.
Usually, phonons are subtracted by suppressing superconductivity using a magnetic field, though magnetoresistance is a problem. (Another way is to use a non-magnetic analog of the studied compound, but this is inapplicable here.) Given the reliance on the subtraction procedure, I do not know how reliable the electronic part is. This important point should be addressed in detail.
7. Another concern is that the manuscript shows only normalized scattering rates, which is obtained from the normalized resistivity Ref.[33], see Fig.2. This is misleading. The authors should derive the absolute values of the scattering rate, \Gamma, and plot it along with the AG-Openov curve for a d-wave superconductor with non-magnetic impurities, which has a specific critical value of \Gamma_c=0.28 when Tc is suppressed to zero. Depending on gap anisotropy, the function Tc(\Gamma) can be anywhere from a scattering-independent constant to the AGO curve.
8. All mentioned theories assume point-like scattering potential. The model suggested in the manuscript involves domain boundaries. Scattering on extended defects is considerably more complex has very different effect on properties .

It is evident that considerable effort was put into this work, and the authors did their best to present and analyze the data. However, the significant issues outlined above need to be addressed before considering it for any journal.

The changes must be substantial, impossible to list here.

Reject