[Paper Review] Quantum field theoretical description for the reflectivity of graphene

M. Bordag, G. L. Klimchitskaya+2 more|arXiv (Cornell University)|Jan 30, 2015

Graphene research and applications19 citations

TL;DR

This paper presents a quantum field theoretical description of graphene's reflectivity by deriving a polarization tensor valid across the entire complex frequency plane, enabling accurate analytic continuation to real frequencies. Unlike prior approaches limited to Matsubara frequencies, this method yields consistent reflectivity calculations for both TM and TE polarized light across all frequencies, revealing angular dependence in TE reflectivity previously overlooked and providing asymptotic expressions validated numerically.

ABSTRACT

We derive the polarization tensor of graphene at nonzero temperature in (2+1)-dimensional space-time. The obtained tensor coincides with the previously known one at all Matsubara frequencies, but, in contrast to it, admits analytic continuation to the real frequency axis satisfying all physical requirements. Using the obtained representation for the polarization tensor, we develope quantum field theoretical description for the reflectivity of graphene. The analytic asymptotic expressions for the reflection coefficients and reflectivities at low and high frequencies are derived for both independent polarizations of the electromagnetic field. The dependencies of reflectivities on the frequency and angle of incidence are investigated. Numerical computations using the exact expressions for the polarization tensor are performed and application regions for the analytic asymptotic results are determined.

Motivation & Objective

• To develop a quantum field theoretical framework for graphene reflectivity that is valid at all real frequencies, not just at imaginary Matsubara frequencies.
• To resolve inconsistencies in prior polarization tensor representations that failed to properly allow analytic continuation to the real frequency axis.
• To derive analytic asymptotic expressions for reflectivity at low and high frequencies for both TM and TE polarizations.
• To perform numerical computations and determine the validity ranges of the asymptotic approximations.
• To compare the new results with previous local conductivity models, especially regarding TE reflectivity and angular dependence.

Proposed method

• Derive the polarization tensor of graphene in (2+1)-dimensional space-time using thermal quantum field theory techniques, ensuring validity across the entire complex frequency plane.
• Employ a representation based on Feynman parameters and proper analytic continuation rules, avoiding the pitfalls of earlier methods that failed for real frequencies.
• Use symmetry properties and angular integration in momentum space to simplify the polarization tensor expression, leading to a form amenable to analytic continuation.
• Apply the formula for angular integrals involving cosine terms to reduce the 2D momentum integral to a 1D integral over q⊥.
• Use the square root prescription with consistent sign choice for the imaginary part to ensure correct analytic behavior in the complex plane.
• Numerically evaluate the exact polarization tensor and compare with asymptotic expressions to determine their domain of applicability.

Experimental results

Research questions

• RQ1 How can the polarization tensor of graphene be consistently extended from Matsubara frequencies to the real frequency axis in a way that satisfies physical requirements?
• RQ2 What are the analytic asymptotic expressions for the reflectivity of graphene at low and high frequencies for both TM and TE polarizations?
• RQ3 How does the TE reflectivity of graphene depend on the angle of incidence, and does this contradict previous local conductivity models?
• RQ4 What is the numerical accuracy and domain of validity of the derived asymptotic expressions for reflectivity?
• RQ5 How do the new quantum field theory results compare with previous results based on local conductivity models?

Key findings

• The derived polarization tensor is valid across the entire complex frequency plane, including the real frequency axis, and differs significantly from previous representations except at Matsubara frequencies.
• The TE reflectivity of graphene depends on the angle of incidence, contradicting earlier qualitative conclusions based on local conductivity models.
• Analytic asymptotic expressions for reflectivity are derived for both low and high frequencies, showing good agreement with numerical computations in their respective regimes.
• Numerical results confirm that the asymptotic expressions are accurate for high frequencies and for low frequencies when the temperature is low.
• The TM reflectivity calculated via the new method agrees well with previous local conductivity models at both high and low frequencies.
• The polarization tensor representation ensures correct analytic continuation, resolving inconsistencies in earlier approaches used for Casimir force calculations but not for reflectivity.