[Paper Review] Comment on paper "Extremely Low Frequency Plasmons in Metallic Mesostructures" [J.B. Pendrey, et al., Phys. Rev. Lett. 76, 4773 (1996)]

Andrey K. Sarychev, Vladimir M. Shalaev|arXiv (Cornell University)|Mar 6, 2001

Optical Coatings and Gratings3 citations

TL;DR

This paper challenges the effective electron mass model in Pendry et al.'s theory of extremely low-frequency plasmons in metallic mesostructures, proposing a first-principles solution to Maxwell’s equations that accounts for the skin effect and wire geometry. It derives a dielectric tensor with accurate loss and plasma frequency predictions, showing that Pendry’s formula underestimates losses when skin depth is small, and recommends a revised formula for low-loss applications.

ABSTRACT

A first-principle theory for the effective dielectric permittivity in metallic mesostructures is developed. This theory, in particular, takes into account the skin effect, which is important in practical applications of metal structures.

Motivation & Objective

• To address inconsistencies in Pendry et al.'s model where effective electron mass depends on position, rendering it physically meaningless.
• To develop a first-principles theory for effective dielectric permittivity in metallic mesostructures without relying on mass renormalization.
• To accurately account for skin effect and geometric factors in wire lattices, especially for small wire radii.
• To provide a more accurate prediction of loss factors and plasma frequencies in metallic mesostructures for practical low-frequency plasmon applications.
• To demonstrate that Pendry’s formula fails under strong skin effect conditions, particularly for small r/δ ratios.

Proposed method

• Derives the effective dielectric tensor εₑ for an orthorhombic wire lattice using exact solutions to Maxwell’s equations.
• Incorporates the renormalized metal permittivity ε̃ₘ = εₘF(2πr√εₘ/λ), where F(x) = 2J₁(x)/(xJ₀(x)) involves Bessel functions.
• Introduces a geometric factor ℒ_yz = 2ln(a_y√(1+e²)/(2r)) + (πe)/2 − (e−1/e)arctan e − 3, with e = a_y/a_z.
• Uses the full dielectric tensor expression εₑ,xx = ε_d − (πr²/S_yz) × [ε_d − (1 + π(S_yz/λ²)ε_d)ε̃ₘ] / [1 − (πr/λ)²(ℒ_yz + 1)ε̃ₘ] for accurate permittivity prediction.
• Compares results with Pendry’s Drude-like formula (Eq. 15 of [1]) using Al wire parameters (r = 1 μm, a = 5 mm) and r = 10 μm.
• Analyzes loss factor κ = |εₑ′′/εₑ′| and shows that the new model correctly predicts δ/r dependence, unlike Pendry’s (δ/r)² scaling.

Experimental results

Research questions

• RQ1 Why is the effective electron mass in Pendry et al.’s model physically problematic when it depends on position R?
• RQ2 How does the skin effect influence the accuracy of the Drude-like formula for effective permittivity in metallic mesostructures?
• RQ3 What is the correct analytical form for the effective dielectric tensor in a wire lattice that accounts for geometric and skin effects?
• RQ4 How do the predicted loss factors and plasma frequencies differ between the first-principles model and Pendry’s formula?
• RQ5 Under what conditions does Pendry’s formula break down, and when is the new model necessary for accurate predictions?

Key findings

• The proposed dielectric tensor (Eq. 1) correctly captures δ/r dependence of losses, predicting κ ∝ δ/r, unlike Pendry’s (δ/r)² scaling.
• For r = 1 μm and a = 5 mm, the loss factor difference between the two models is significant, increasing for larger r (e.g., r = 10 μm).
• The plasma frequency is derived as ωₚ² = 4c²π/(a²ℒε_d), differing from Pendry’s by a factor of 2ln(a/r)/(ℒε_d).
• When r ≪ δ (small skin effect), εₑ becomes nearly purely imaginary, making low-frequency plasmons unexcitable due to high losses.
• For practical low-loss applications, the skin effect must be strong (δ/r ≪ 1), validating the use of the new formula (Eq. 3) over Pendry’s.
• The vector potential’s R-dependence in thin wires invalidates the concept of a position-dependent effective mass, undermining Pendry’s model’s physical basis.