Microwave Signature of the Emerging Abrikosov Lattice Above $H_{c2}$

TL;DR

This work predicts a microwave-scale electromagnetic signature of emergent Abrikosov vortex precursors in the normal state above the upper critical field $H_{c2}$, arising from quantum fluctuations of fluctuating Cooper pairs. By developing a microscopic diagrammatic framework in a magnetic field and exploiting the Lowest Landau Level limit near $H_{c2}$, the authors show that the ac conductivity acquires a pronounced imaginary component at a characteristic frequency $\omega_{QF} \sim \hbar^{-1} \Delta (H-H_{c2})/H_{c2}$, much smaller than the superconducting energy scale. The dominant fluctuation contributions come from the Aslamazov-Larkin and anomalous Maki–Thompson processes, while DOS-related diagrams largely cancel, yielding a measurable inductive signature in the microwave range (e.g., ~0.1–1 GHz for Nb). This provides a practical route to detect vortex-formation precursors via microwave spectroscopy, offering a direct probe of quantum fluctuations in type-II superconductors. The results bridge fluctuation theory with high-frequency experiments, suggesting concrete experimental tests in Nb-based films using modern microwave resonators.

Abstract

The emergence of the Abrikosov lattice in the normal phase of type-II superconducting films when the magnetic field approaches the critical field $H_{c2}$ from above was predicted in Ref.~\cite{GVV2011}. In the quantum fluctuation regime \cite{GL2001} it is characterized by the formation of relatively large (with sizes of order $ξ_{\mathrm{QF}} \sim ξ_{\mathrm{BCS}}\sqrt{H_{c2}/(H-H_{c2})}$) ``long lived'' (lifetime of order $τ_{\mathrm{QF}} \sim \hbar Δ^{-1} H_{c2}/(H-H_{c2})$) clusters of rotating fluctuation Cooper pairs - signatures of developing Abrikosov vortices. We demonstrate that these fluctuation-induced vortex clusters, previously considered unobservable due to their ultrafast dynamics and weak (only logarithmically singular) contribution to the dc-conductivity, can in fact be detected through their distinct electromagnetic signature. By analyzing the high-frequency electromagnetic response of these rotating fluctuation Cooper pairs above the second critical field in superconducting film, we predict a pronounced and measurable enhancement in the imaginary part of the ac-conductivity arising directly from quantum fluctuations. This enhancement is expected to occur at characteristic frequencies $ω_{QF} \sim \hbar^{-1}Δ(H-H_{c2})/H_{c2}$, which are well below the superconducting threshold at $2\hbar^{-1}Δ$, where a similar increase in imaginary conductivity occurs in the superconducting phase. For niobium, a prototypical type II superconductor, $ω_{QF}$ lies in the experimentally accessible microwave range, making the effect directly testable with modern microwave spectroscopy.

Figures (7)

• Figure 1: Schematic phase diagram of type-II superconductors, showing the domains of qualitatively different behavior of fluctuations. Here $H_{c2}(T)$ is the temperature dependent second critical field, while $H^*_{c2}(T)$ is the mirror field, where the magnetic length of the FCP equates to the Ginzburg-Landau length GPV2020.
• Figure 2: (Color online) Feynman diagrams for the leading-order contributions to the electromagnetic response operator. Wavy lines correspond to fluctuation propagators, solid lines with arrows represent impurity-averaged normal state electron Green's functions, crossed circles are electric field vertices, dashed lines with a circle represent additional impurity renormalizations, and triangles and dotted rectangles are impurity ladders accounting for the electron scattering at impurities (Cooperons, see Supplementary Material SM for additional details).
• Figure 3: Imaginary (left) and real (right) parts of total fluctuation conductivity $\sigma$ in units of $e^2/\hbar$ as a function of frequency $\omega/\Delta$ for different magnetic fields $\tilde{h}$ .
• Figure S1: (Color online) The Dyson equation for the fluctuation propagator (wavy line) in the ladder approximation. Solid lines represent one-electron Green's functions, circles represent the electron--electron interaction, and the triangle corresponds to the Cooperon (see Fig. \ref{['fig.cooperon']}).
• Figure S2: (Color online) (a) Dyson equation for Cooperon, i.e. the vertex that accounts for the result of averaging over elastic impurity scattering of electrons in the ladder approximation. Solid lines correspond to bare one-electron Green's functions. The dashed line is associated with an impurity correlator, $\left\langle U^{2}\right\rangle = 1/\left( 2\pi \nu_{0}\tau \right)$. (b) Analogous Dyson equation for the four-leg Cooperon in the ladder approximation.