The optical response of graphene is described by its surface conductivity - a multivariate function of frequency, temperature, chemical potential, and scattering rate. A Kubo formula that accounts for both interband and intraband transitions with two Fermi-Dirac-like integrals is conventionally used to model graphene. The first (intraband) integral can be reduced analytically to a Drude term. The second (intraband) term requires computationally expensive numerical integration over the infinite range of energies, and thus it is usually either neglected or substituted with a simpler approximation (typically valid within a limited range of parameters). Additional challenge is an integral-free time-domain (TD) formulation that would allow efficient coupling of the interband conductivity term to TD electromagnetic solvers.
We propose Kubo-equivalent models of graphene surface conductivity that offer closed-form computationally efficient representations in time and frequency domains. We show that in time domain Kubo’s formula reduces to a combination of rational, trigonometric, hyperbolic, and exponential functions. In frequency domain the integral term is equivalent to an expression with digamma and incomplete gamma functions.
The accuracy and improved performance of our integral-free formulations versus the direct integration of Kubo’s formula is critically analyzed. The result provides efficient broadband multivariate coupling of graphene dispersion to time-domain and frequency-domain solvers. To reinforce theory with practical examples, we use obtained closed-form frequency-domain model to retrieve the optical properties of graphene samples from variable angle spectroscopic ellipsometry (VASE) measurements. . We present ellipsometry fitting cases that are built on an in-the-cloud tool freely available online (https://nanohub.org/resources/photonicvasefit).
Zhaxylyk A. Kudyshev, Ludmila J. Prokopeva, and Alexander V. Kildishev, "Kubo-equivalent closed-form graphene conductivity models
(Conference Presentation)," Proc. SPIE 9920, Active Photonic Materials VIII, 99200Q (Presented at SPIE Nanoscience + Engineering: August 29, 2016; Published: 9 November 2016); https://doi.org/10.1117/12.2237768.5159519623001.
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