Superconductivity mediated by polar modes in ferroelectric metals

The phenomenon of superconductivity in doped strontium titanate (SrTiO3) at exceptionally low carrier densities has remained a puzzle for decades. This low carrier density suggests an unusually strong pairing interaction, significantly stronger than those observed in conventional superconductors. Understanding the mechanism driving this superconductivity is crucial for advancing our knowledge of unconventional superconductivity and its potential applications. SrTiO3 is an incipient ferroelectric insulator, meaning it is close to exhibiting ferroelectric behavior but requires a 'quantum' tuning parameter, such as chemical substitution (e.g., Nb doping), isotopic substitution, or applied stress, to fully transition into the ferroelectric phase. The phase diagram of SrTiO3 is characterized by a ferroelectric transition temperature (TCurie) that terminates at a quantum critical point (QCP), regions with power law and exponential temperature dependence of the inverse dielectric function, and a high-temperature crossover between classical and quantum behaviors. The introduction of charge carriers, such as via Nb doping, adds another dimension to this phase diagram, allowing for the exploration of the interplay between superconductivity and ferroelectricity. Previous research has explored various mechanisms for superconductivity in SrTiO3, including interactions involving critical transverse optical (TO) modes, plasmons, multi-valley transitions, non-polar phonons, and polaron/bipolaron formation. However, a consensus regarding the dominant pairing mechanism has remained elusive. This study aims to shed light on this fundamental question by investigating the pressure dependence of Tc in Nb-doped SrTiO3, providing critical experimental constraints for theoretical models.

Extensive research has been conducted on the superconductivity of doped SrTiO3, exploring various proposed mechanisms. Early studies focused on the role of soft phonon modes, specifically the transverse optical (TO) phonons associated with the ferroelectric instability. These modes, characterized by their low frequency and strong coupling to the lattice, were suggested to mediate an attractive interaction between electrons, leading to Cooper pair formation. However, this mechanism alone seemed insufficient to explain the observed Tc. Other proposed mechanisms include plasmon-mediated pairing, where electron density fluctuations contribute significantly to superconductivity. The multi-valley nature of the electronic band structure in SrTiO3 was also suggested as a potential factor influencing the pairing. Furthermore, the formation of polarons and bipolarons (localized charge carriers), their interaction, and their potential role in superconductivity have been investigated. Several studies have investigated the connection between the ferroelectric quantum critical point (QCP) and the superconducting dome, suggesting that the proximity to the QCP plays a critical role in enhancing superconductivity. Despite these studies, a complete understanding of the pairing mechanism remained absent, highlighting the need for further investigation.

The researchers used resistivity measurements under varying hydrostatic pressures to investigate the pressure dependence of the superconducting transition temperature (Tc) in Nb-doped SrTiO3 samples. Samples with different Nb doping levels (0.02, 0.2, and 1 at.%) were prepared, resulting in varying carrier concentrations. Ohmic contacts were created using argon-ion plasma etching and gold sputtering. Measurements of the Hall resistance at liquid helium temperatures were used to determine the carrier densities. A sample with 0.2 at.% Nb doping, exhibiting a Tc near the maximum of the superconducting dome, was chosen for detailed high-pressure experiments. Hydrostatic pressure was applied using fluorinert as a pressure-transmitting medium within a piston-cylinder clamp cell. Four-terminal resistivity measurements were conducted down to 50 mK using an adiabatic demagnetization refrigerator. The pressure was calibrated using the superconducting transition temperature of a high-purity tin manometer. Tc was defined as the temperature at which the resistivity dropped by 10% from its normal state value. The theoretical investigation focused on developing a model for superconductivity in SrTiO3. The model incorporated the dynamical screening of the Coulomb interaction between electrons, taking into account contributions from both ionic and electronic degrees of freedom. The interaction was formulated in terms of the dielectric susceptibility, using a simple oscillator model for ionic fluctuations and the Lindhard function for the electrons. The coupled carrier-ion longitudinal modes (hybrid longitudinal modes) were identified as the dominant contributors to electron pairing. The superconducting transition temperature (Tc) was calculated using the Eliashberg equations and the Kirzhnits-Maksimov-Khomskii (KMK) kernel, which accounts for the effects of retardation and carrier dynamics on the pairing interaction. The calculations incorporated realistic material parameters for SrTiO3, allowing for a quantitative comparison with experimental results. A wavevector cut-off was introduced into the gap equation to account for the finite range of the interaction. The model was validated by comparing its predictions to experimental data, encompassing a broad range of carrier densities and pressures.

The experimental measurements revealed a sharp decrease in Tc with increasing pressure, demonstrating that Tc is strongly suppressed as the system moves away from the ferroelectric QCP. The inverse dielectric constant, 1/ε0, which is proportional to the square of the transverse optical phonon frequency, exhibited a similar pressure dependence, implying a strong correlation between the ferroelectric soft mode and superconductivity. In the normal state, the resistivity followed a T² dependence with a coefficient that showed a noticeable pressure dependence. This suggests an influence of pressure on the scattering mechanisms in the normal state. The theoretical analysis proposed a novel mechanism for superconductivity in SrTiO3 mediated by longitudinal hybrid polar modes. These modes arise from the coupled motion of the conduction electrons and the lattice ions. The model showed that the attractive interaction between electrons is facilitated over a range of frequencies, near two hybrid-polar-mode frequencies. It is notable that the interaction strength is substantial due to the high polarizability of the medium. The results of the model calculations showed an increase in Tc with decreasing 1/ε0, indicating that the proximity to the ferroelectric QCP indeed enhances superconductivity. Moreover, the calculated Tc initially rises with carrier density, peaks around the optimal doping level, and then collapses at higher densities, consistent with experimental observations. The model also captured the rapid collapse of Tc with increasing pressure, showcasing the close relationship between the ferroelectric and superconducting phases. The theoretical model demonstrated a remarkable correspondence between the calculated and experimentally observed Tc values, suggesting a dominant role for the longitudinal hybrid polar mode mechanism near optimal doping. The model indicated that a minimum in Tc rather than a maximum is likely at carrier densities significantly lower than optimal doping. The strong interaction stems from polar fluctuations in the highly polarizable medium largely unaffected by electron screening. The retardation effects were incorporated into the description of superconductivity through both the frequency-dependent interaction and the frequency-dependent gap function, significantly aiding superconductivity.

The findings of this study provide compelling evidence for a novel mechanism of superconductivity in SrTiO3 and related materials, mediated by longitudinal hybrid-polar modes. This mechanism relies on the interaction of conduction electrons with coupled ionic and electronic polar modes, demonstrating that the proximity to the ferroelectric QCP significantly enhances the pairing interaction. The strong correlation between Tc and the ferroelectric soft mode is highlighted by the remarkable agreement between the experimental pressure dependence of Tc and the theoretical predictions based on the hybrid-polar-mode model. This work refines earlier models by explicitly incorporating the Coulomb repulsion, screening effects, and the dynamics of the interaction kernel. The model demonstrates the emergence of a superconducting dome in the phase diagram, consistent with observations. The quantitative agreement with experiments, obtained without adjustable parameters, strongly supports the proposed mechanism as the dominant pairing interaction near optimal doping. The observed T² resistivity in the normal state and its pressure dependence also corroborate the model, offering another experimental constraint on the theory. The presented model contrasts with other theories that primarily focus on the coupling to the critical TO mode or neglect essential aspects like Coulomb repulsion and retardation. The concept of 'ion mediation' highlights the critical role of ionic polarizability in driving a strong electron pairing interaction.

This research establishes a quantitative model explaining the doping and pressure dependence of superconductivity in SrTiO3. The study demonstrates that the dominant contribution to Cooper pair formation originates from induced interactions through longitudinal hybrid-polar modes, a mechanism distinct from previously proposed models. The close agreement between theory and experiment, without the use of adjustable parameters, strongly supports this novel mechanism. The findings could provide insights into other unconventional superconductors and stimulate research in exploring materials with similar electron-polar phonon coupling.

The study primarily focuses on the region near optimal doping, and the model's applicability to heavily underdoped or overdoped regimes requires further investigation. The theoretical model employs approximations, such as a simplified oscillator model for ionic fluctuations and the Lindhard function for electrons, which could impact the quantitative accuracy of predictions. Experimental challenges associated with inhomogeneities in samples, especially at lower doping levels, could affect the interpretation of results. Further research could incorporate more sophisticated descriptions of the electronic band structure and electron-phonon interaction.