Thermodynamic evidence for a pressure-driven crossover from strong- to weak-coupling superconductivity in Pb

arXiv:2603.22178 · cond-mat.supr-con · Submitted 2026-03-23 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Thermodynamic evidence for a pressure-driven crossover from strong- to weak-coupling superconductivity in Pb".

Mira: The gist: Thermodynamic evidence for a pressure-driven crossover from strong- to weak-coupling superconductivity in Pb indicates that compression drives Lead from the strong-coupling regime toward the weak-coupling limit.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, to wrap up this paper, we have Rustem Khasanov and Mira’s work on "Thermodynamic evidence for a pressure-driven crossover from strong- to weak-coupling superconductivity in Pb." The central finding is that the thermodynamic critical field Bc(zero) shows a pressure dependence that closely follows the superconducting gap ∆(zero) more than the transition temperature Tc <ref:2603.22178#pg1,the thermodynamic critical field Bc>.

Mira: And when they look at the logarithmic pressure derivatives of these parameters, they find them converging at higher pressures, which implies the coupling strength ratio alpha becomes nearly pressure independent in that regime.

Lev: What this means for us is that this thermodynamic evidence suggests a pressure-driven crossover from strong- to weak-coupling superconductivity in lead because it shows how compression affects the fundamental energy scales of the superconducting state.

Kai: So, what does this imply for how we think about these materials? It provides a way to probe the condensation energy directly, which is usually hidden when you only look at Tc.

Mira: Exactly. It gives us a coherent picture of how the superconducting energy scales evolve under compression by combining the data from Bc(T), Bc(zero), Tc, ∆(zero), and alpha over pressure <ref:2603.22178#pg1>.

Lev: If you're working on experimental setups, this suggests that future measurements should focus on tracking those thermodynamic critical field properties rather than just focusing solely on how the transition temperature shifts with pressure.

Conclusion: Kai: So, we’re wrapping up our look at this paper by Khasanov and Mira on lead under pressure. The main idea is that they used thermodynamic measurements to show how lead moves from a strong-coupling superconducting state toward a weak-coupling one when you squeeze it.

Mira: Exactly. They weren't just looking at the temperature shift, which is what most people see, but they looked at the critical field, Bc. That field tells you about the energy scale of superconductivity itself.

Lev: So what does that mean for us on hardware? If we only watch Tc change with pressure, we might miss how the fundamental coupling mechanism is actually changing beneath the surface.

Kai: Right. They found that by looking at how Bc changes, they can map out a clearer path to see this crossover happening in real materials like lead.

Mira: The authors say that when you look at the math—specifically those logarithmic derivatives—the pressure dependence of the gap ratio becomes almost flat at higher pressures. That means the coupling strength isn't changing as much anymore.

Lev: That flatness is important because if it stays flat, it suggests we’re hitting a limit where the material starts behaving more like a standard weak-coupling BCS superconductor, regardless of how much pressure you add.

Kai: So this moves beyond just observing a shift in temperature; they are probing the underlying physics of how the pairing strength itself is evolving under extreme conditions.

Mira: It’s about getting a direct thermodynamic view of that evolution rather than just inferring it from transition temperatures alone.

Lev: If we could replicate those measurements, it would give us a crucial benchmark for understanding pressure effects on pairing in these materials.

Kai: Next time, we’ll take that idea of the coupling ratio changing with pressure and see what kind of experimental signatures that might leave behind.

PSI Center for Neutron and Muon Sciences CNM

cond-mat.supr-con

Submitted: 2026-03-23

Updated: 2026-03-23

Comments: 3 figures, 6 pages

Journal ref: Phys. Rev. Lett. 137, 026002 (2026)

DOI: 10.1103/xqxt-fmh8

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 77/100

The gist: The gist: Thermodynamic evidence for a pressure-driven crossover from strong- to weak-coupling superconductivity in Pb indicates that compression drives Lead from the strong-coupling regime toward

Key concepts

Thermodynamic Critical Field (Bc): This field is measured under pressure and directly relates to the superconducting condensation energy. It offers a more direct measure of the superconducting energy scale compared to the transition temperature alone, which is often influenced by other factors like phonon changes.
Superconducting Condensation Energy Density (U0): This quantity represents the difference in free energy between a normal state and a superconducting state. In weak-coupling BCS theory, it is proportional to the square of the superconducting gap (Delta(0)), linking it directly to the fundamental properties of the superconductor.
Coupling Strength Ratio ($\alpha$): Alpha is defined as the ratio of the zero-temperature energy gap (Delta(0)) to $k_BT_c$. It serves as a measure of how strongly coupled the superconducting electrons are. Values significantly above 1.764 indicate strong-coupling effects, while values near this value suggest weak-coupling behavior.
Pressure-Driven Crossover: This refers to the physical process where applying external pressure changes the fundamental nature of superconductivity in Lead. The study investigates whether compression causes the material to transition from a strong-coupling state (where interactions are significant) to a weak-coupling state (where interactions are less dominant).

Terminology

alpha — 2× in this paper · explained in 1 episode
Benchmark — 1× in this paper · explained in 2 episodes

Summary

The gist: Thermodynamic evidence for a pressure-driven crossover from strong- to weak-coupling superconductivity in Pb indicates that compression drives Lead from the strong-coupling regime toward the weak-coupling limit.

Thermodynamic Probes and Theoretical Framework

The thermodynamic critical field Bc provides direct access to the superconducting condensation energy, which is a more direct view of the superconducting energy scale than the transition temperature alone. The corresponding superconducting condensation-energy density is given by Eq. (1). Within weak-coupling BCS theory, the condensation energy can also be written as Eq. (2). Combining Eqs. (1) and (2) yields the approximate scaling relation Bc(0) ∝ ∆(0)√γe. This relation is expressed in terms of the dimensionless ratio α = ∆(0)/kBTc.

Pressure Dependence Analysis

The relation between the logarithmic pressure derivatives of Bc, Tc, γe, and α is given by d ln Bc(0)dp = d ln Tc/dp + 1/2d ln γe/dp + d ln α/dp. In simple metals, the pressure dependence of the electronic specific-heat coefficient γe is generally weak compared with that of Tc and Bc. This shows that the pressure dependence of the thermodynamic critical field provides direct information on the pressure evolution of the gap ratio α.

Experimental Observations and Interpretation

Muon-spin rotation/relaxation (µSR) is used to determine Bc under hydrostatic pressure up to ≃ 2.3 GPa, providing direct access to the equilibrium thermodynamic critical field. Analysis of the temperature dependence Bc(T) within the α model yields that in the investigated pressure range, the pressure dependence of Bc(0) follows that of the superconducting gap more closely than that of the transition temperature. By combining these results with previously reported high-pressure data, it is found that the pressure derivatives of Bc(0) and Tc approach each other at higher pressures.

Evidence for Crossover

The key result is that the logarithmic pressure derivatives of Tc and Bc(0) become nearly equal above p ∼ 8 GPa, which implies that the strongcoupling corrections to the gap ratio cease to evolve significantly, and Pb approaches a regime in which α is nearly pressure independent. Since the low-pressure µSR data show that α decreases with increasing pressure, this convergence of derivatives at higher pressures is interpreted as thermodynamic evidence that compression drives Pb from the strong-coupling regime toward the weak-coupling limit. This behavior is consistent with the known strong-coupling character of Pb at ambient pressure.

Conclusion

The thermodynamic critical field provides information complementary to that obtained from the transition temperature alone, as it probes the condensation energy and thus the superconducting energy scale. The combined analysis of Bc(T), Bc(0), Tc, ∆(0), and α yields a coherent thermodynamic picture of how the superconducting energy scales of Pb evolve under compression.

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Thermodynamic evidence for a pressure-driven crossover from strong- to weak-coupling superconductivity in Pb Rustem Khasanov1, ∗ 1PSI Center for Neutron and Muon Sciences CNM, 5232 Villigen PSI, Switzerland (Dated: March 24, 2026) The thermodynamic critical field Bc provides direct access to the superconducting condensation energy, yet its pressure dependence has been studied much less extensively than that of the transition temperature. Here, muon-spin-rotation/relaxation measurements of the thermodynamic critical field Bc of elemental Pb under hydrostatic pressure up to ≃ 2.3 GPa are reported. From the magneticfield distribution in the intermediate state, Bc(T) is determined and Bc(0) is extracted at different pressures. In combination with previously reported high-pressure data for Bc and Tc, it is shown that the pressure dependence of Bc(0) follows that of the superconducting gap ∆(0) more closely than that of the transition temperature Tc. At higher pressures, the logarithmic pressure derivatives of Bc(0) and Tc are found to converge, indicating that the coupling strengths ratio α = ∆(0)/kBTc becomes nearly pressure independent. This behavior is interpreted as thermodynamic evidence for a pressure-driven crossover from strong- to weak-coupling superconductivity in Pb. Introduction. Understanding how the characteristic energy scales of a superconductor evolve under external pressure provides important insight into the microscopic mechanisms governing superconductivity [1–4]. In conventional phonon-mediated superconductors, pressure modifies both the electronic structure and the lattice dynamics, typically leading to phonon hardening and a reduction of the electron–phonon coupling strength [3–6]. As a consequence, compression can drive the system gradually from the strong-coupling regime toward the weak-coupling Bardeen–Cooper–Schrieffer (BCS) limit [7]. Experimental signatures of such a crossover are usually inferred from the pressure dependence of the superconducting transition temperature Tc. However, Tc is determined by the linearized gap equation and reflects a delicate balance between competing effects, including phonon hardening and changes in the electron–phonon interaction [5, 6, 8]. Thermodynamic quantities that probe the condensation energy can therefore provide a more direct view of the superconducting energy scale [7]. One such quantity is the thermodynamic critical field Bc, which is directly related to the free-energy difference between the superconducting and normal states [9, 10]. The corresponding superconducting condensation-energy density is given by U0 = B 2c(0) 2µ0, (1) where Bc(0) is the zero-temperature thermodynamic critical field. Within weak-coupling BCS theory, the condensation energy can also be written as U0 = 1/2 N(EF)∆ 2(0) = 3/4πγe∆ 2(0), (2) where N(EF) is the electronic density of states at the Fermi level, ∆(0) is the superconducting energy gap at zero temperature, kB is the Boltzmann constant, and γe = (2π 2/3)k 2BN(EF) is the Sommerfeld coefficient. Combining Eqs. (1) and (2) yields the approximate scaling relation Bc(0) ∝ ∆(0)√γe, (3) showing that the thermodynamic critical field is governed by both the superconducting gap and the electronic density of states. It is convenient to express the gap in terms of the dimensionless ratio α = ∆(0)/kBTc, (4) where Tc is the superconducting transition temperature. The parameter α is commonly used as a measure of coupling strength via comparison with the weak-coupling BCS value αBCS ≡ 1.764. Values significantly above αBCS indicate enhanced strong-coupling effects, whereas values close to αBCS correspond to the weak-coupling limit. Combining Eqs. (3) and (4) gives the relation between the logarithmic pressure derivatives of Bc, Tc, γe, and α: d ln Bc(0)dp = d ln Tc/dp + 1/2d ln γe/dp + d ln α/dp. In simple metals, the pressure dependence of the electronic specific-heat coefficient γe is generally weak compared with that of Tc and Bc [2–4, 13]. Therefore, to a good approximation, the difference between the pressure derivatives of Bc and Tc reflects the pressure evolution of the gap ratio: d ln α/dp ≃ d ln Bc(0)dp − d ln Tc/dp. (5) This relation shows that the pressure dependence of the thermodynamic critical field provides direct information on the pressure evolution of the gap ratio α.

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Bc (mT) T 2 (K 2) (b) D(T/Tc) (c) T/Tc 2.34 GPa 1.25 GPa 0.

Improvements for AI systems

textbfConsultation of Thermodynamic Critical Field Scaling for Crossover Identification: The improved AI system can directly identify thermodynamic evidence for a pressure-driven crossover from strong- to weak-coupling superconductivity by analyzing the logarithmic pressure derivatives of Bc(0) and Tc and their convergence, as stated in the text.
textbfModel Parameterization via the Alpha Model: The system can utilize the α model to extract key physical parameters like the superconducting energy gap ∆(0), p) and the coupling parameter α(p) from temperature dependence data, enabling it to quantify how pressure drives Pb from the strong-coupling regime toward the weak-coupling limit.
textbfMulti-Parameter Correlation Engine: The improved system can correlate the derived parameters to test theoretical relationships, specifically by checking if the logarithmic pressure coefficient of α is nearly equal to the difference between those of Bc(0) and Tc, ensuring consistency with Eq. (5).