We analyze the Casimir interaction of doped graphene. To this end we derive a simple expression for the finite temperature polarization tensor with a chemical potential. It is found that doping leads to a strong enhancement of the Casimir force reaching almost 60% in quite realistic situations. This result should be important for planning and interpreting the Casimir measurements, especially taking into account that the Casimir interaction of undoped graphene is rather weak.Introduction Graphene, which is a two-dimensional sheet of carbon atoms possesses many unusual properties and attracts a lot of attention. Particular excitement among the theoreticians is caused by the fact that the spectrum of quasi-particles in graphene is described by the quasi-relativistic Dirac model with the effective propagation speed of about 300 times less than the speed of light. This continuous model turned out to be very successful in describing a broad range of effects [1], for instance optical properties of graphene as the absorption of light [2] and the (giant) Faraday effect [3], to mention a few.In the recent years, the Casimir effect for pristine graphene was studied both for zero [4,5] and finite [6][7][8] temperatures. For not too large temperatures (as compared with inverse distance between the interaction sheets) the effect between graphene monolayer and ideal metal is defined by the fine structure constant α 1/137 and is roughly 2.5% of the one between two ideal metal plates. Such small forces are on the limit of sensitivity of modern experimental techniques. For high temperatures (or separations) the effect is hugely reinforced [8], but the measurements under such conditions is a separate quite challenging task, which is not completely solved yet even for metals, see e.g. [9]. It is not surprising therefore that just a single experiment has been performed until now [10]. This experiment revealed [11] a good agreement with the theory [12]. Possibilities, opened by doping, were however not explored there.Previously the Casimir interaction of doped graphene was studied in [13,14]. The reflection coefficients used in that works were expressed though quantities whose explicit dependence on the temperature remained unknown and the results of Refs. [13,14] are mutually contradicting. Therefore, specifically after experimental confirmation of the Dirac approach to Casimir energy of undoped graphene [10,11], it seems to be important to extend the approach to doped graphene.In this letter we consider the Casimir effect at finite temperature, chemical potential and mass, and find a substantial enhancement of the effect in graphenemetal systems which potentially permits to avoid the above mentioned experimental difficulties. Our findings show that for relatively highly (but still feasibly) doped

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Em segundo lugar, investigamos a interação de Casimir entre lmes suspensos de grafeno com um condutor ideal. O efeito é investigado tanto no caso ideal (temperatura nula, amostras ideais) quanto para congurações mais realistas (temperatura não nula e a presença de potencial químico). No caso de temperatura nula, a força de Casimir entre grafeno e condutor ideal é aproximadamente 2.6% da força entre dois condutores ideais. Ao mesmo tempo, no limite de temperatura elevada, o efeito mostrase ser muito forte cerca de 1/2 de efeito entre metais ideais. Os resultados da presente pesquisa foram publicados em [1] [7]. Abstract This research is devoted to investigation of several aspects of the physics of suspended and epitaxial graphene monolayers. The description of graphene is based on the quaserelativistic Dirac model which permits application of the methods of the Quantum Field Theory to investigation of the interaction of graphene with electromagnetic eld. Basing on the path integral formalism we formulate the eective theory for EM eld in presence of graphene monolayers which is governed by the polarization operator of the Dirac quase-particles in graphene. The two main phenomena in the interaction of graphene with electromagnetic eld are studied: the optical properties of graphene (the Faraday rotation in particular), and Casimir interaction between graphene samples and parallel metal. First, we study the propagation of electromagnetic waves in presence of suspended and epitaxial graphene layers. Their dynamics is governed by the modied Maxwell equations obtained from the eective theory for EM eld. We calculate the reection and transmission coecient for linearly polarized light and investigate in detail the quantum Faraday eect in external magnetic eld. In particular we show that the prediction of the Dirac model are in good agreement with recent experimental results on transmission and giant Faraday rotation in cyclotron resonance regime. New regimes are also predicted.Secondly, we investigate Casimir interaction between suspended graphene lms and ideal conductor. The eect is investigated both in the idealistic case (zero temperature, ideal samples) and for realistic congurations (non zero temperature and/or presence of chemical potential).For zero temperature the Casimir force between graphene and a conductor is about 2.6% of that between two ideal conductors. At the same time in the high temperature limit the eect is showed to be greatly enhanced being about 1/2 of that between ideal metals.The results of the present research are published in [1] [7].