Quantum electrodynamic approach to the conductivity of gapped graphene

Abstract

The electrical conductivity of graphene with a nonzero mass-gap parameter is investigated starting from the first principles of quantum electrodynamics in (2+1)-dimensional space-time at any temperature. The formalism of the polarization tensor defined over the entire plane of complex frequency is used. At zero temperature we reproduce the results for both real and imaginary parts of the conductivity, obtained previously in the local approximation, and generalize them taking into account the effects of nonlocality. At nonzero temperature the exact analytic expressions for real and imaginary parts of the longitudinal and transverse conductivities of gapped graphene are derived, as well as their local limits and approximate expressions in several asymptotic regimes. Specifically, a simple local result for the real part of conductivity of gapped graphene valid at any temperature is obtained. According to our results, the real part of the conductivity is not equal to zero for frequencies exceeding the width of the gap and goes to the universal conductivity with increasing frequency. The imaginary part of conductivity of gapped graphene varies from infinity at zero frequency to minus infinity at the frequency defined by the gap parameter and then goes to zero with further increase of frequency. The analytic expressions are accompanied by the results of numerical computations. Possible future generalization of the used formalism is discussed.Comment: 21 pages, 6 figures; to appear in Phys. Rev. B; several typos are correcte

Casimir free energy of dielectric films: classical limit, low-temperature behavior and control

“…Then, we perform similar expansions for the second powers of reflection coefficients (12), where ζ l is replaced with a continuous variable x in accordance to (28). For the TM mode the result is…”

“…In all cases, however, the classical limit is reached for thinner films than for two plates separated by a gap of the same width as the film thickness. We note also that in the case of metallic films described by the plasma model the Casimir free energy does not reach the classical limit at any film thickness and temperature [26,27,28].…”

“…It is shown that, in some analogy to already investigated case of metallic films [26,27,28], the Lifshitz theory leads to considerably different results depending on whether one takes into account the conductivity of film material at nonzero temperature or omits it in calculations. Similar result has been obtained previously in calculations of the measured Casimir forces between two bodies separated by a vacuum gap with one of them made of dielectric material [41,42,43,44,45], but the differences in theoretical predictions were much less than we find here for a dielectric film.…”

“…Large differences in the predictions of both approaches are reached in the recently proposed [38,39,40] differential force measurements, but in the experiments of this kind the measured force signal is also rather low [36]. Unlike the configuration of two metallic plates separated with a gap, the absolute values of the Casimir free energy and pressure of metallic films, as thin as of 50 or 100 nm thickness, can differ by the factors of hundreds and even thousands when the calculation approaches using the Drude and the plasma models are used [26,27,28]. Thus, the Drude-plasma dilemma becomes extremely important not only from the theoretical point of view, but for important technological applications as well.…”

“…Recently it was shown that the Casimir free energy of both nonmagnetic and magnetic metallic films calculated rigorously in the framework of the Lifshitz theory differ considerably depending on whether the Drude or the plasma model is used for extrapolation of the optical data for the complex index of refraction to zero frequency [26,27,28]. It is well known that there is an outstanding problem in the Lifshitz theory [2,3,6].…”