Mattis–Bardeen theory

Introduction

Superconductivity, the phenomenon where a material exhibits zero electrical resistance below a certain critical temperature, has fascinated physicists since its discovery in 1911. Among the many theoretical frameworks developed to understand the intricate behavior of superconductors, the Mattis–Bardeen theory occupies a pivotal place. It provides a quantitative description of how superconductors respond to alternating electromagnetic fields, especially in regimes where conventional, classical electrodynamics fails. By incorporating the microscopic insights of the Bardeen–Cooper–Schrieffer (BCS) theory, Mattis–Bardeen theory explains the anomalous skin effect observed in superconducting materials and has become a cornerstone for optical spectroscopy studies of superconductors.

1. Historical Context

The anomalous skin effect was first recognized in normal metals at very low temperatures and high frequencies. In such conditions, the classical prediction of skin depth – the distance into a conductor at which an alternating electromagnetic field decays to 1/e of its surface value – breaks down. The failure arose because the electron mean free path, the average distance an electron travels between scattering events, becomes comparable to or exceeds the skin depth. This phenomenon was resolved by Robert G. Chambers, who formulated a theory that accounted for the extended mean free path of electrons.

Superconductors, however, present an even more intriguing scenario. Below their critical temperature, the electronic system reorganizes into a condensate of Cooper pairs, and the electrodynamic response is no longer described by the simple Drude model used for normal metals. It became clear that the anomalous skin effect in superconductors required a new theoretical treatment that could simultaneously incorporate the quantum mechanical pairing mechanism and the non-classical response to high-frequency fields. This led to the development of the Mattis–Bardeen theory, which extends the BCS framework to calculate the complex conductivity of superconductors in the presence of electromagnetic radiation.

2. Theoretical Foundations

2.1 Electrodynamic Properties of Superconductivity

Electrodynamic properties refer to how a material responds to time-varying electric and magnetic fields. In a superconductor, this response is dramatically altered by the formation of Cooper pairs and the opening of an energy gap in the electronic density of states. The complex conductivity, σ(ω) = σ₁(ω) + iσ₂(ω), encapsulates both the dissipative (σ₁) and reactive (σ₂) aspects of this response. The Mattis–Bardeen theory provides explicit expressions for σ₁ and σ₂ as functions of frequency, temperature, and material parameters, rooted in the microscopic BCS theory.

2.2 The Bardeen–Cooper–Schrieffer (BCS) Theory

BCS theory, formulated by Bardeen, Cooper, and Schrieffer, explains superconductivity as a macroscopic quantum state formed by electrons pairing through an attractive interaction mediated by lattice vibrations. This pairing leads to a condensate with an energy gap Δ that suppresses scattering processes responsible for electrical resistance. The Mattis–Bardeen theory leverages the BCS description of the superconducting state to compute the material’s response to electromagnetic fields.

3. The Anomalous Skin Effect

3.1 Classical Skin Effect

In ordinary conductors, the skin depth δ is given by the classical formula δ = √(2/μσω), where μ is the magnetic permeability, σ the DC conductivity, and ω the angular frequency of the applied field. This relation assumes that the electron mean free path is short compared to δ, so that electrons scatter frequently and the current distribution is smooth.

3.2 Breakdown at Low Temperatures and High Frequencies

When the temperature drops and the frequency rises, the electron mean free path increases dramatically. In this regime, the classical assumption fails, and the actual skin depth becomes larger than predicted. The field penetrates deeper into the material than expected, and the current distribution deviates from the simple exponential decay. This deviation is what is termed the anomalous skin effect.

3.3 Superconductors and the Anomalous Skin Effect

Superconductors also exhibit an anomalous skin effect, but the underlying physics is more subtle. The superconducting condensate contributes a dissipationless supercurrent that screens magnetic fields, while the residual normal electrons (quasiparticles) still contribute to absorption. The interplay between these two channels, along with the energy gap, leads to a complex frequency-dependent conductivity that cannot be captured by the classical skin depth formula.

4. Derivation of the Mattis–Bardeen Theory

The Mattis–Bardeen theory starts from the Kubo formula for linear response, which relates the complex conductivity to the current–current correlation function. By inserting the BCS quasiparticle spectrum into this formalism, one obtains integrals over the density of states weighted by Fermi–Dirac occupation factors. The resulting expressions for σ₁ and σ₂ naturally incorporate the superconducting energy gap and the temperature dependence of quasiparticle excitations.

A key feature of the Mattis–Bardeen derivation is that it does not assume local electrodynamics. Instead, it retains the full spatial dependence of the current induced by the electromagnetic field, allowing it to capture the nonlocal response characteristic of the anomalous skin effect. In the limit where the mean free path is much shorter than the skin depth, the Mattis–Bardeen formulas reduce to the local, BCS-based results. Conversely, in the extreme anomalous limit, they yield a conductivity that is markedly different from the classical prediction.

5. Applications in Optical Spectroscopy

5.1 Probing Superconducting Gap

Optical spectroscopy, which measures the absorption and reflection of light across a broad frequency range, is a powerful tool for investigating the superconducting energy gap. By measuring the complex conductivity as a function of frequency, researchers can directly test the predictions of the Mattis–Bardeen theory. The theory’s expressions for σ₁ and σ₂ provide a quantitative framework for interpreting the optical spectra of superconductors, especially in the terahertz and far-infrared regimes where the anomalous skin effect is most pronounced.

5.2 Determining Superfluid Density

The imaginary part of the conductivity, σ₂(ω), is directly related to the superfluid density – the density of Cooper pairs that contribute to the dissipationless current. The Mattis–Bardeen theory predicts how σ₂ evolves with temperature, allowing experimentalists to extract the superfluid density from optical measurements.

5.3 Material Characterization

Beyond fundamental physics, the Mattis–Bardeen theory is employed to characterize superconducting materials used in technology, such as superconducting radio-frequency cavities, kinetic inductance detectors, and quantum computing elements. Accurate modeling of the electrodynamic response is essential for optimizing device performance, and the theory’s ability to capture nonlocal effects makes it indispensable for high-frequency applications.

6. Connection to BCS Theory

While the Mattis–Bardeen theory is built upon the BCS framework, it extends it in two crucial ways:

1. Nonlocal Response – Unlike the local conductivity derived from BCS, Mattis–Bardeen incorporates the finite mean free path of quasiparticles, allowing it to describe how the current at one point in the material depends on the electromagnetic field at neighboring points.

1. Finite Frequency – BCS theory typically focuses on the DC or low-frequency limit. The Mattis–Bardeen approach explicitly considers finite frequencies, providing the full frequency dependence of the complex conductivity.

Thus, the Mattis–Bardeen theory can be seen as a bridge between microscopic BCS physics and macroscopic electrodynamic behavior in superconductors.

7. Key Concepts and Definitions

8. Limitations and Extensions

While the Mattis–Bardeen theory has proven remarkably successful, it does have limitations:

• Assumption of Clean Limit – The theory assumes that