[Paper Review] Strong Coupling Superconductivity in the Vicinity of the Structural Quantum Critical Point in (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$

Wing Chi Yu, Yiu Wing Cheung+6 more|arXiv (Cornell University)|Sep 10, 2015

Iron-based superconductors research4 citations

TL;DR

This study investigates strong coupling superconductivity in (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ near a structural quantum critical point (QCP) at $x \approx 0.9$, using specific heat measurements to reveal phonon softening and enhanced superconducting gap parameters. The key finding is a dramatic enhancement of $2\Delta/k_{\text{B}}T_{\text{c}} \approx 6.67$ and $\Delta C/\gamma T_{\text{c}} \approx 3.47$ near the QCP, exceeding BCS predictions and indicating strong electron-phonon coupling driven by lattice instability.

ABSTRACT

The family of the superconducting quasi-skutterudites (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ features a structural quantum critical point at $x_c=0.9$, around which a dome-shaped variation of the superconducting transition temperature $T_c$ is found. Using specific heat, we probe the normal and the superconducting states of the entire series straddling the quantum critical point. Our analysis indicates a significant lowering of the effective Debye temperature on approaching $x_c$, which we interpret as a result of phonon softening accompanying the structural instability. Furthermore, a remarkably large enhancement of $2Δ/k_BT_c$ and $ΔC/γT_c$ beyond the Bardeen-Cooper-Schrieffer (BCS) values is found in the vicinity of the structural quantum critical point. The phase diagram of (Ca$_{x}$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ thus provides a model system to study the interplay between structural quantum criticality and strong electron-phonon coupling superconductivity.

Motivation & Objective

• To investigate the interplay between structural quantum criticality and superconductivity in the (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ family.
• To determine how lattice dynamics, particularly phonon softening, influence superconducting pairing near the structural QCP.
• To quantify the strength of electron-phonon coupling via specific heat measurements in the normal and superconducting states.
• To establish whether the observed superconducting gap parameters exceed BCS expectations near the QCP.

Proposed method

• Specific heat measurements were performed on single crystals of (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ across $x = 0$ to $x = 1$ using a pulse relaxation method in a Physical Property Measurement System.
• Background heat capacity from the sample platform and Apiezon N-grease was measured and subtracted, with additional high-field (≤5 T) measurements to isolate sample-specific contributions.
• The effective Debye temperature ($\Theta_{\text{D}}^{\text{eff}}$) and Einstein temperature ($\Theta_{\text{E}}$) were extracted from the low-temperature specific heat data using a modified Debye model.
• The superconducting gap parameters $2\Delta/k_{\text{B}}T_{\text{c}}$ and $\Delta C/\gamma T_{\text{c}}$ were calculated from the specific heat jump at $T_{\text{c}}$ and the electronic specific heat coefficient $\gamma$.
• The $x$-dependence of $T_{\text{c}}$, $\gamma$, $\Theta_{\text{D}}^{\text{eff}}$, and $\Theta_{\text{E}}$ was analyzed to map the phase diagram and identify critical behavior near the structural QCP at $x \approx 0.9$.

Experimental results

Research questions

• RQ1 How does phonon softening near the structural quantum critical point affect the superconducting gap parameters in (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$?
• RQ2 To what extent do the superconducting gap parameters $2\Delta/k_{\text{B}}T_{\text{c}}$ and $\Delta C/\gamma T_{\text{c}}$ exceed BCS predictions near the QCP?
• RQ3 What is the relationship between the effective Debye temperature and the onset of structural instability in this system?
• RQ4 Does the enhancement of superconducting gap parameters correlate with the suppression of the structural transition temperature $T^*$?
• RQ5 Can the system be described as a model for strong coupling superconductivity driven by lattice instabilities?

Key findings

• The effective Debye temperature $\Theta_{\text{D}}^{\text{eff}}$ is reduced by approximately 33% at $x = 0.9$ compared to $x = 0$, indicating significant phonon softening near the structural QCP.
• The Einstein temperature $\Theta_{\text{E}}$ also decreases markedly near $x = 0.9$, supporting the presence of low-energy phonon modes due to lattice instability.
• The superconducting transition temperature $T_{\text{c}}$ peaks at $x \approx 0.95$, coinciding with the structural QCP, with $T_{\text{c}} \approx 5.5$ K.
• The ratio $2\Delta/k_{\text{B}}T_{\text{c}}$ reaches a maximum value of $\sim 6.67$ at $x = 0.95$, significantly exceeding the BCS value of 1.43 and even surpassing theoretical predictions for strong coupling superconductivity.
• The ratio $\Delta C/\gamma T_{\text{c}}$ reaches $\sim 3.47$ at $x = 0.95$, also exceeding the BCS value of 1.43 and indicating a strongly enhanced superconducting gap.
• The enhancement of both $2\Delta/k_{\text{B}}T_{\text{c}}$ and $\Delta C/\gamma T_{\text{c}}$ near the QCP demonstrates that strong electron-phonon coupling is driven by structural quantum criticality, establishing (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ as a model system for such interplay.