Microscopic structure of the vortex cores in granular niobium: A coherent quantum puzzle

TL;DR

The study investigates vortex cores in granular Nb films, where grains with size < ξ cause the superconducting gap Δ(r) to drop toward the core via discrete jumps at grain boundaries, deviating from the smooth Caroli–de Gennes–Matricon (CdGM) profile seen in homogeneous superconductors. It combines low-temperature STM/STS at T = 1.1 K with Bogoliubov-de Gennes simulations on a 2D granular landscape derived from topography to link gap maps with local quasiparticle spectra. Bound states are upshifted to higher energies, with a lowest level E0 ≈ 0.8 Δ0 in some cores, and the core gap may remain finite, forming a puzzle-like, grain-based core structure that reflects the granular environment and the presence of 2π phase winding. The findings reveal a two-component mechanism for vortex-core states: the global vortex pair potential with a 2π winding and grain-boundary barriers confining quasiparticles within grains, producing energy bunching correlated with the granular map, and they document ultra-slow vortex dynamics over hours, highlighting new pinning and motion behavior in granular superconductors. These insights bridge CdGM physics in clean systems with disordered, granular environments, with implications for vortex manipulation and electronic device performance in granular Nb films.

Abstract

When macroscopic quantum condensates -- superconductors, superfluids, cold atoms and ions, polaritons etc. -- are put in rotation, a quantum vortex lattice forms inside. In homogeneous type-II superconductors, each vortex has a tiny core where the superconducting gap $Δ(r)$ is known to smoothly vanish towards the core centre on the scale of the coherence length $ξ$. The cores host quantized quasiparticle energy levels known as Caroli-de Gennes-Matricon (CdGM) bound states [Caroli {\it et al.,} Phys. Lett. v. 9, 307 (1964)]. In pure materials, the spectrum of the low-lying CdGM states has the characteristic level spacing $\sim Δ_0^2/E_F$, where $E_F$ is the Fermi energy and $Δ_0$ is the bulk gap. In disordered ones, the CdGM states shift and broaden due to scattering. Here, we show, both experimentally and theoretically, that the situation is completely different in granular Nb films, which are commonly used in superconducting electronics. In these films, in which the grains are smaller than $ξ$, the gap $Δ$ in the quasiparticle spectrum reduces towards the vortex core centres by discrete jumps at the grain boundaries. The bound states adapt to the local environment and appear at unexpectedly high energies. Both $Δ(r)$ and bound states form a puzzle-like spatial structure of the core, elements of which are whole grains. Our discovery shakes up the established understanding of the quantum vortex and encourages a reconsideration of the vortex motion and pinning mechanisms in granular superconductors.

Figures (4)

• Figure 1: Vortex cores in granular niobium.a -- 300 nm $\times$ 300 nm topographic STM image of studied Nb film showing its granular structure, with the grain size $3-10$ nm. The film roughness is about 1.5 nm R.M.S. b -- Color-coded spatial map of the superconducting gap (gap map) of the sample in the region shown in (a) at $T=1.1$ K after field-cooling the sample at 0.25 T. The grain boundaries are outlined by thin black lines. The gaps evolve a little inside each grain, but vary strongly across the grain boundaries. White labels identify eight vortices studied. c -- Left column: detailed gap map of the vortex cores 'i', 'ii' and 'iii'. Right column: respective gap maps calculated within the Bogoliubov-de Gennes framework (see Methods and Section 3 of Supplementary Information). Scale bars in (a-b) correspond to 50 nm, and in (c) -- to 10 nm. d -- Density of quasiparticle excitations $\rho_s(E)$ found in different grains. The colors of curves correspond to those of grains in (c), left column. The dependencies $\rho_s(E)$ are extracted from the fits of the experimental STS data (see Methods and Section 2 in Supplementary Information). In the centre of the vortex core 'iii' the gap does not vanish. The same holds for cores 'iv', 'v' and 'vii'.
• Figure 2: Locating the 2$\pi$-singularities in the vortex cores. Left column: four examples of the gap map calculated for the vortex core 'iii' (see Fig. \ref{['Fig1']}) considering different locations for the 2$\pi$ singularity (shown by white circles). The gap map encountered in red gives the best match with the experimental gap map. Right column: experimental gap map of the core 'iii'. White scale bars correspond to 10 nm.
• Figure 3: Identification of the bound states.a--c -- Histograms depicting the energy distribution of the gap within regions where vortices are localized. a: vortex i; b: vortex ii; c: vortex iii. The lowest-lying states are indicated by $E_0$. Insets compare the experimentally observed gap distribution (left) with that predicted by the puzzle-based model of CdGM states in a granular superconducting film (right). d -- Variation of $E_0$ as a function of size of the central grain containing the 2$\pi$- singularity. Red dashed line: conventional variation of the order parameter in the core of a homogeneous superconductor with $\xi$ = 12 nm and $\Delta_s$ = 1.35 meV.
• Figure 4: Ultra-slow vortex dynamics.a -- Vortex lattice after field-cooling the sample at 0.25 T. In total, 10-11 vortices are visible in this 300 nm × 300 nm area. b -- Vortex lattice in the same region after increasing the applied filed to 0.5 T and a 6 hours delay. The number of vortices increased to 20-22; their locations are completely different from (a). Red circle 'iii' shows the former position of the vortex 'iii'; the circles 'ix' and 'x' are examples of new vortices. c -- The same as in (b) but after the time delay of 12 hours after the field increase. With respect to (b), the number of vortices is the same but their positions and shapes changed significantly. Most of vortices moved by distances $\sim \xi$ (red circles), a few by distances $\ll \xi$ (orange circles); only one vortex, marked by 'x', did not move (white circle). d -- The same as in (c) but after the time delay of 18 hours after the field increase. With respect to (c), the vortex displacement is almost zero; only some vortices moved on a very local scale (orange circles), while the positions of others remained unchanged (white circles).