A tractable framework for phase transitions in phase-fluctuating disordered 2D superconductors: applications to bilayer MoS$_2$ and disordered InO$_x$ thin films

TL;DR

The paper tackles phase-fluctuation-dominated superconductivity in disordered two-dimensional systems by developing a self-consistent microscopic framework that treats fermionic quasiparticles, Nambu–Goldstone phase fluctuations, BKT vortex fluctuations, long-range Coulomb interactions, and disorder on equal footing. It shows that Coulomb interactions convert NG mode dispersion to a plasmonic form, suppressing thermal NG fluctuations, while BKT physics drives a separation between the gap-closing temperature $T^*$ and the global coherence temperature $T_c$, with disorder and low density enhancing zero-point fluctuations and reducing the zero-temperature gap $| abla|(0)$ and $T^*$; together these yield a density- and disorder-dependent pseudogap regime. The framework quantitatively reproduces experimental trends in gate-tunable bilayer MoS$_2$ and disordered InO$_x$ films, including the suppression of $|"Δ(0)|$, $T_c$, and $T^*$ with density and disorder and the expansion of the pseudogap window. By providing a compact, parameter-efficient approach that links pairing and phase fluctuations, the method offers a practical toolbox for understanding and predicting superconductivity in phase-fluctuation–dominated 2D materials.

Abstract

Starting from the purely microscopic model, we go beyond conventional mean-field theory and develop a self-consistent microscopic thermodynamic framework for disordered 2D superconductors. It incorporates the fermionic Bogoliubov quasiparticles, bosonic Nambu-Goldstone (NG) quantum and thermal phase fluctuations in the presence of long-range Coulomb interactions, and topological Berezinskii-Kosterlitz-Thouless (BKT) vortex-antivortex fluctuations on an equal footing, to self-consistently treat the superconducting gap and superfluid density. This unified phase-fluctuating description naturally recovers the previously known limiting results: the superconducting gap in the 2D limit can remain robust against long-wavelength NG phase fluctuations at $T=0^+$ due to Coulomb-induced regularization, while the gradual proliferation of BKT fluctuations as the system approaches criticality drives a separation between the global superconducting transition temperature $T_c$ and the gap-closing temperature $T^*$. In contrast to mean-field theory, which predicts 2D superconductivity to be independent of carrier density and non-magnetic disorder (Anderson theorem), the incorporation of phase fluctuations generates a density- and disorder-dependent zero-point gap $Δ(0)$ and consequently $T_c$ and $T^*$. Remarkably, applications to bilayer MoS$_2$ [Nat. Nanotechnol. 14, 1123 (2019)] and disordered InO$_x$ thin films [Nat. Phys. 21, 104 (2025)] quantitatively reproduce key experimental observations in excellent agreement. The framework offers a useful theoretical tool for understanding phase-fluctuation-dominated superconductivity.

Figures (4)

• Figure 1: (a) Schematic illustration of the "lower" band in $K$ and $Q$ valleys in bilayer MoS$_2$. (b) Temperature dependence of the superconducting gap $|\Delta(T)|$ and (c) the superfluid density, calculated at a carrier density of $n_{\rm 2D}=4.6\times10^{14}~\mathrm{cm^{-2}}$ and a scattering time $\tau\Delta_{\rm MF}=0.85$, chosen to match the results (compared with experiment in Ref. Zheliuk2019) shown in Fig. \ref{['figyc3']} and its inset.
• Figure 2: Carrier-density dependence of (a) the pair-formation temperature $T^{*}$ and the superconducting transition temperature $T_c$; (b) the zero-point gap $|\Delta(0)|$; and (c) the width of the finite-temperature pseudogap regime, defined as $T^{*}-T_c$, shown for both the clean regime ($\Delta_{\rm MF}\tau=50$) and the dirty regime ($\Delta_{\rm MF}\tau=0.5$). The solid green lines in panels (a) and (b) mark the boundary between the mean-field (MF) regime and the phase-fluctuation(PF)–dominated regime.
• Figure 3: Comparison between theory and experiment. The scattering time $\tau$ is the only fitting parameter, determined by matching the pair-formation temperature $T^{*}$ at $n_{\rm 2D}=4.76\times10^{14}\mathrm{cm^{-2}}$ to the experimental nominal transition temperature $T^*_{\rm e}=6.87~$K in Ref. Zheliuk2019. The resulting density dependence of $T^{*}$ and $T_c$ shows good agreement with experiment without additional fitting parameters. Inset: the calculated temperature dependence of the critical current compared with the experimental data Zheliuk2019. In the calculation of the critical current, a vector potential ${\bf A}$ is introduced through a Doppler shift ${\bf v}_{\bf k}\cdot{\bf A}$ in the quasiparticle spectrum. The critical vector potential ${\bf A}_c(T)$ is determined self-consistently at each temperature, and the critical current is obtained as $J_c(T)\propto n_s(T) A_c(T)$, showing quantitative agreement with the experimental critical current extracted from differential resistance measurements Zheliuk2019.
• Figure 4: (a) Experimentally measured and (b) theoretically calculated disorder dependence of the zero-point gap $|\Delta(0^+)|$ and the superfluid stiffness $\Theta(0^+)$ (in units of K) as well as the critical temperature $T^*$ of superconducting InO$_x$ thin film. Experimental data come from Ref. Charpentier2025. The inset of (a) displays the experimentally measured tunneling spectroscopy for a sample with $R\sim5~\mathrm{k\Omega/sq}$ at $T=T_c=1.7~\mathrm{K}$, showing a pseudogap in the density of states that persists up to $\sim 6~\mathrm{K}$sacepe2011localization. In our simulations, we approximate the $0^+$ limit with $T = 25~\mathrm{mK}$, corresponding to the lowest experimentally accessible temperature Charpentier2025. The inset of (b) shows the disorder dependence of the theoretically calculated temperatures $T^*$, $T_c$, and $T^* - T_c$ (solid curves), while the squares represent the corresponding experimental values from panel (a), showing that the theoretical and experimental $T_c$ curves nearly coincide.