Doped SrTiO$_3$, one of the most dilute bulk systems to display superconductivity, is perhaps the first example of an unconventional superconductor, as it does not fit into the standard BCS paradigm. More than five decades of research has revealed a rich temperature-carrier concentration phase diagram that showcases a superconducting dome, proximity to a putative quantum critical point, Lifshitz transitions, a multi-gap pairing state and unusual normal-state transport properties. Research has also extended beyond bulk SrTiO$_3$, ushering the new field of SrTiO$_3$-based heterostructures. Because many of these themes are also featured in other quantum materials of contemporary interest, recent years have seen renewed interest in SrTiO$_3$. Here, we review the challenges and recent progress in elucidating the superconducting state of this model system. At the same time that its extreme dilution requires to revisit several of the approximations that constitute the successful Migdal-Eliashberg description of electron-phonon superconductivity, including the suppression of the Coulomb repulsion via the Tolmachev-Anderson-Morel mechanism, it opens interesting routes for alternative pairing mechanisms whose applicability remains under debate. For instance, pairing mechanisms involving longitudinal optical phonons have to overcome the hurdles created by the anti-adiabatic nature of the pairing interaction, whereas mechanisms that rely on the soft transverse optical phonons associated with incipient ferroelectricity face challenges related to the nature of the electron-phonon coupling. Proposals in which pairing is mediated by plasmons or promoted locally by defects are also discussed. We finish by surveying the existing evidence for multi-band superconductivity and outlining promising directions that can potentially shed new light on the rich problem of superconductivity in SrTiO$_3$.

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稀 SrTiO3 中的超导性：综述

掺杂的 SrTiO$_3$ 是显示超导性的最稀的体系统之一，可能是非常规超导体的第一个例子，因为它不符合标准的 BCS 范式。超过五年的研究揭示了丰富的温度 - 载流子浓度相图，展示了超导圆顶、接近假定的量子临界点、Lifshitz 跃迁、多间隙配对状态和不寻常的常态传输特性。研究也超越了块体 SrTiO$_3$，开创了基于 SrTiO$_3$ 的异质结构的新领域。由于这些主题中的许多主题也出现在其他当代感兴趣的量子材料中，因此近年来人们对 SrTiO$_3$ 重新产生了兴趣。这里，我们回顾了在阐明该模型系统的超导状态方面的挑战和最近的进展。同时，它的极端稀释需要重新审视构成成功的 Migdal-Eliashberg 电子-声子超导性描述的几个近似值，包括通过 Tolmachev-Anderson-Morel 机制抑制库仑排斥，它开辟了有趣的路线其适用性仍在争论中的替代配对机制。例如，涉及纵向光学声子的配对机制必须克服配对相互作用的反绝热性质造成的障碍，而依赖于与初始铁电相关的软横向光学声子的机制面临与电子性质相关的挑战 -声子耦合。还讨论了配对由等离子体介导或由缺陷局部促进的建议。最后，我们调查了多波段超导性的现有证据，并概述了可能为 SrTiO$_3$ 中超导性的丰富问题提供新思路的有希望的方向。