[Paper Review] Magnetic penetration depth of Aluminum thin films

David López-Núñez, Alba Torras-Coloma+6 more|arXiv (Cornell University)|Nov 23, 2023

Physics of Superconductivity and Magnetism4 citations

TL;DR

This study measures the magnetic penetration depth (λ) in aluminum thin films from 20 nm to 200 nm thickness using two complementary methods: superconducting LC resonator resonance frequency and low-temperature resistance of meander structures. The results show λ decreases from 163.3±0.4 nm in 20 nm films to 53.6±0.4 nm in 200 nm films, indicating a transition from type-II to type-I superconducting behavior near 113–155 nm thickness, with implications for quantum circuit and detector design.

ABSTRACT

We present a study of the superconducting penetration depth $λ$ in aluminum thin films of varying thickness. The range of thicknesses chosen spans from the thin-film regime to the regime approaching bulk behavior. The penetration depths observed range from $λ= 163.3\pm0.4~ m{nm}$ for the thinnest $20~ m{nm}$ samples down to $λ= 53.6\pm0.4~ m{nm}$ for the $200~ m{nm}$-thick ones. In order to accurately determine $λ$, we performed complementary measurements using the frequency of superconducting $LC$ resonators as well as the resistance of normal-state meanders. Both methods yield comparable results, providing a well-characterized set of values of $λ$ in aluminum in the relevant range for applications in fields such as quantum computing and microwave radiation detector technologies.

Motivation & Objective

• To accurately determine the magnetic penetration depth λ in aluminum thin films across a range of thicknesses from 20 nm to 200 nm.
• To compare two independent experimental methods—LC resonator resonance frequency and normal-state resistance of meander structures—for λ extraction to ensure consistency and reliability.
• To estimate the critical thickness range where aluminum transitions from type-II to type-I superconductivity, relevant for vortex-free operation in quantum devices.
• To develop and validate a robust fitting procedure with accurate error estimation for resonator-based λ measurements, improving on sequential fitting methods.
• To provide a well-characterized dataset of λ and kinetic inductance Lk for use in the design of superconducting quantum circuits and kinetic inductance detectors.

Proposed method

• Fabricated superconducting LC resonators on aluminum films of varying thickness (20–200 nm) and measured their resonance frequency as a function of temperature to extract λ.
• Measured the low-temperature normal-state resistance of meander-shaped test structures to determine the sheet resistance and, via the London equation, extract λ.
• Used the iminuit fitting package with initial guesses from sequential fitting to perform simultaneous parameter fitting with accurate error estimation and correlation analysis.
• Applied the London equation, jS = −(μ₀/λ²)AS, to relate the magnetic field penetration to the supercurrent response in thin films.
• Assessed the electron mean-free path l using two models: l = d (film thickness) and l derived from the empirical relation ρ·l = 4×10⁻¹² Ω·cm² for aluminum.
• Estimated the coherence length ξ using the BCS relation ξ = ħvF/(πΔ) and compared ξ with λ to determine the superconducting type transition point via κ = 1/√2.

Figure 1: a) Type-I superconductors, where response to an external perpendicular magnetic field is non-local. Each Cooper pair reacts to the magnetic field at its center of mass, since the average Cooper pair size $\xi$ is much larger than the typical distance at which the magnetic field decays, $\l

Experimental results

Research questions

• RQ1 How does the magnetic penetration depth λ of aluminum thin films vary with film thickness from 20 nm to 200 nm?
• RQ2 What is the critical thickness at which aluminum thin films transition from type-II to type-I superconductivity?
• RQ3 How do two independent measurement techniques—LC resonator frequency and meander resistance—compare in extracting λ, and which provides more reliable error estimation?
• RQ4 To what extent do fabrication variations (e.g., residual resistance ratio RRR) affect the measured λ values across different facilities?
• RQ5 Can the observed thickness dependence of λ be explained by models of electron mean-free path and coherence length, and what does this imply for the superconducting type transition?

Key findings

• The magnetic penetration depth λ decreases from 163.3±0.4 nm in 20 nm thick aluminum films to 53.6±0.4 nm in 200 nm thick films, approaching the bulk value.
• Both the LC resonator and meander resistance methods yielded consistent λ values across all thicknesses, validating the measurement approach.
• The critical thickness range for the transition from type-II to type-I superconductivity in aluminum is estimated between 113 nm and 155 nm, based on ξ = √2λ and two models for electron mean-free path.
• The use of the iminuit fitting package with initial guesses from sequential fitting enabled accurate error estimation and improved parameter correlation analysis compared to sequential fitting alone.
• No significant deviation in λ was observed between samples fabricated at different facilities despite differences in residual resistance ratio (RRR), indicating robustness of the measurement method.
• Thicker films (d > 150 nm) are predicted to enter the type-I superconducting regime, while thinner films (d < 50 nm) behave as local type-II superconductors, with implications for vortex suppression and qubit coherence.

Figure 2: a) $LC$ Resonator equivalent circuit. In the simulations conducted in Sec. IV , no kinetic inductance $L_{k}$ is considered. In the actual experiment, the contribution from $L_{k}$ adds to the geometric inductance $L_{g}$ . b) Simulated and experimental resonances are assumed to only devia