Quantum Theory and Unusual Dielectric Functions of Graphene

TL;DR

The paper derives graphene's spatially nonlocal dielectric functions from first principles using the (2+1)-D polarization tensor across all frequencies and temperatures. It shows that $\varepsilon^{\rm L}$ is regular at zero frequency while $\varepsilon^{\rm T}$ contains a notable double pole at $\omega=0$ for $\mathbf{q}\neq 0$, with finite-temperature corrections that keep the imaginary parts positive and consistent with causality. By analyzing both evanescent and propagating regimes, the work connects these nonlocal responses to Casimir physics, offering explanations for experimental puzzles and highlighting the potential for similar nonlocal transverse responses in ordinary metals. The results provide a rigorous, first-principles framework for graphene's electrodynamic response, with implications for Casimir forces, radiative heat transfer, and near-field optics, and motivate extensions to other 2D Dirac materials and metal systems.

Abstract

We address the spatially nonlocal dielectric functions of graphene at any frequency derived starting fromthe first principles of thermal quantum field theory using the formalism of the polarization tensor. After a brief review of this formalism, the longitudinal and transverse dielectric functions are considered at any relationship between the frequency and the wave vector. The analytic properties of their real and imaginary parts are investigated at low and high frequencies. Emphasis is given to the double pole at zero frequency which arises in the transverse dielectric function. The role of this unusual property for solving the problem of disagreement between experiment and theory in the Casimir effect is discussed. We guess that a more complete dielectric response of ordinary metals should also be spatially nonlocal and its transverse part may possess the double pole in the region of evanescent waves.

Figures (4)

• Figure 1: The magnitude of real part of the longitudinal dielectric function of graphene is shown versus frequency for $T=300$ K and $q=100~\hbox{cm}^{-1}$ in the logarithmic scale. The threshold at $\omega=v_Fq$ is marked by the dashed vertical line.
• Figure 2: The imaginary part of the longitudinal dielectric function of graphene is shown versus frequency for $T=300$ K and $q=100~\hbox{cm}^{-1}$ in the logarithmic scale. The threshold at $\omega=v_Fq$ is marked by the dashed vertical line.
• Figure 3: The magnitude of real part of the transverse dielectric function of graphene is shown versus frequency for $T=300$ K and $q=100~\hbox{cm}^{-1}$ in the logarithmic scale. The threshold at $\omega=v_Fq$ is marked by the dashed vertical line.
• Figure 4: The imaginary part of the transverse dielectric function of graphene is shown versus frequency for $T=300$ K and $q=100~\hbox{cm}^{-1}$ in the logarithmic scale. The threshold at $\omega=v_Fq$ is marked by the dashed vertical line.