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

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.

💡 Research Summary

The authors investigate the microscopic structure of vortex cores in granular niobium (Nb) thin films, a material widely used in superconducting electronics. In conventional type‑II superconductors, the superconducting order parameter Δ(r) decays smoothly to zero over the coherence length ξ, and the vortex core hosts Caroli‑de Gennes‑Matricon (CdGM) bound states with a characteristic level spacing Δ₀²/E_F. In contrast, the Nb films studied here consist of grains 3–10 nm in size, i.e., smaller than the bulk Nb coherence length (ξ_Nb≈38 nm). The effective dirty‑limit coherence length is reduced to ξ≈12 nm, comparable to the grain size, creating a network of clean metallic grains linked by weak, atomically thin boundaries.

Using scanning tunneling microscopy and spectroscopy (STM/STS) at 1.1 K, the authors map the local density of states (LDOS) under a perpendicular magnetic field (0.25 T). The gap map Δ(x,y) reveals a disordered Abrikosov lattice, but each vortex core is highly irregular. The key experimental observations are:

1. Δ(r) does not decrease continuously toward the vortex centre; instead it exhibits abrupt jumps at grain boundaries.
2. Within a single grain the gap is essentially constant (or varies only slightly for larger grains).
3. The gap never fully vanishes at the vortex centre; a residual “mini‑gap” of 0.2–0.8 meV persists.
4. The CdGM‑like bound states appear at unexpectedly high energies, close to the bulk gap Δ₀≈1.35 meV, rather than the low‑energy spacing typical of clean materials.

To interpret these findings, the authors perform two‑dimensional Bogoliubov‑de Gennes (BdG) simulations. The model treats each grain as a lattice region with intra‑grain hopping t₀ and inter‑grain hopping t₁ (t₁/t₀≈0.1–0.5). By placing the 2π phase singularity (the vortex centre) at various positions relative to the grain boundaries, the simulations reproduce the experimental gap maps, including the discrete jumps, the residual mini‑gap, and the high‑energy bound states. The best agreement is obtained when the singularity is located near, but not exactly on, a grain boundary, indicating that the vortex can “choose” an energetically optimal path through the granular network.

Theoretical analysis links these phenomena to two mechanisms. First, scattering at grain boundaries introduces kinks in the quasiparticle trajectories. In the one‑dimensional Andreev problem, any trajectory kink that changes the superconducting phase difference shifts the exact zero‑energy CdGM level to a finite energy E₀. Second, the interference of quasiparticle wavefunctions inside a grain is imperfect because of the boundary‑induced phase disorder, preventing the complete destructive interference that would otherwise drive Δ to zero at the singularity. Consequently, a finite mini‑gap survives in the vortex centre.

The authors also study vortex dynamics. After increasing the magnetic field from 0.25 T to 0.5 T, they monitor the same 300 nm × 300 nm area over 18 hours. The number of vortices roughly doubles, and individual vortices relocate by distances of order ξ (≈12 nm) within the first 12 hours, after which motion essentially stops. This ultra‑slow dynamics suggests that the granular landscape provides a broad pinning potential: vortices migrate to locally optimal positions where the 2π singularity aligns with a favorable configuration of grain boundaries, rather than being trapped by isolated point defects.

Overall, the work establishes that when the grain size is comparable to or smaller than the coherence length, the vortex core becomes a “quantum puzzle” composed of whole grains. The superconducting gap and bound‑state spectrum are dictated by the granular geometry, leading to discrete gap jumps, persistent mini‑gaps, and high‑energy CdGM‑like states. These findings challenge the conventional picture of smooth vortex cores, call for revised theories of vortex pinning and motion in granular superconductors, and have practical implications for the design of Nb‑based superconducting devices where vortex dynamics and dissipation are critical.