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The strong coupling between electromagnetic field and lattice oscillation in piezoelectric materials gives rise to phonon polariton excitations. Such quasiparticles open up new directions in modulating the ubiquitous Casimir force. Here by utilizing the generalized Born-Huang hydrodynamics model, three types of phonons in piezoelectrics are studied: longitudinal optical phonon, transverse optical phonon and phonon polariton. The phonon-electromagnetic coupling results in a complex set of Fresnel reflection matrices which prevents the utilization of the standard Lifshitz approach for calculating Casimir forces in the imaginary frequency domain. Our calculations are based on an approach within real frequency and finite temperatures, through which various regimes of the Casimir interaction are examined. Our study shows that piezoelectrics emerge as a set of materials where this ubiquitous force can be controlled via phonon properties for the first time. The Casimir interaction appears as a suitable means to distinguish between different types of surface phonon polaritons associated with different structural piezoelectric polytypes., Comment: 8 pages, 4 figures, supplementary information with 10 pages, 4 figures; submitted

A new scheme for the computation of dispersive interactions from first principles is presented. This cost-effective approach relies on a Wannier function representation compatible with density function theory descriptions. This is an electronic-based many-body method that captures the full electronic and optical response properties of the materials. It provides the foundation to discern van der Waals and induction energies as well as the role of anisotropy and different stacking patterns when computing dispersive interactions in systems. Calculated results for binding energies in layered materials, such as graphite, hBN, and MoS$_2$ give encouraging comparisons with available experimental data. Strategies for broadened computational descriptions of dispersive interactions are also discussed. Our investigation aims at stimulating new experimental studies to measure van der Waals energies in a wider range of materials, especially in layered systems., Comment: 24 pages, 1 abstract figure, 4 figures, source codes: 10.6084/m9.figshare.26023024.v2 or https://github.com/Diem-TX-Dang/vdW-MBDWan-code/tree/main, submitted

Monolayers of molybdenum disulfide (MoS2) are the most studied two-dimensional (2D) transition-metal dichalcogenides (TMDs), due to its exceptional optical, electronic, and opto-electronic properties. Recent studies have shown the possibility of incorporating a small amount of magnetic transition metals (e.g., Fe, Co, Mn, V) into MoS2 to form a 2D dilute magnetic semiconductor (2D-DMS). However, the origin of the observed ferromagnetism has remained elusive, due to the presence of randomly generated sulfur vacancies during synthesis that can pair with magnetic dopants to form complex dopant-vacancy configurations altering the magnetic order induced by the dopants. By combining high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) imaging with first-principles density functional theory (DFT) calculations and magnetometry data, we demonstrate the critical effects of sulfur vacancies and their pairings with vanadium atoms on the magnetic ordering in V-doped MoS2 (V-MoS2) monolayers. Additionally, we fabricated a series of field effect transistors on these V-MoS2 monolayers and observed the emergence of p-type behavior as the vanadium concentration increased. Our study sheds light on the origin of ferromagnetism in V-MoS2 monolayers and provides a foundation for future research on defect engineering to tune the electronic and magnetic properties of atomically thin TMD-based DMSs.

The Casimir interaction and torque are related phenomena originating from the exchange of electromagnetic excitations between objects. While the Casimir force exists between any types of objects, the materials or geometrical anisotropy drives the emergence of the Casimir torque. Here both phenomena are studied theoretically between dielectric films with immersed parallel single wall carbon nanotubes in the dilute limit with their chirality and collective electronic and optical response properties taken into account. It is found that the Casimir interaction is dominated by thermal fluctuations at sub-micron separations, while the torque is primarily determined by quantum mechanical effects. This peculiar quantum vs. thermal separation is attributed to the strong influence of reduced dimensionality and inherent anisotropy of the materials. Our study suggests that nanostructured anisotropic materials can serve as novel platforms to uncover new functionalities in ubiquitous Casimir phenomena., Comment: 9 pages, 4 figures, submitted to Phys. Rev. B

We study within the framework of the Lifshitz theory the long-range Casimir force for in-plane isotropic and anisotropic free-standing transdimensional material slabs. In the former case, we show that the confinement-induced nonlocality not only weakens the attraction of ultrathin slabs but also changes the distance dependence of the material-dependent correction to the Casimir force to go as $\sim\!1/\!\sqrt{l}$ contrary to the $\sim\!1/l$ dependence of that of the local Lifshitz force. In the latter case, we use closely packed array of parallel aligned single-wall carbon nanotubes in a dielectric layer of finite thickness to demonstrate strong orientational anisotropy and crossover behavior for the inter-slab attractive force in addition to its reduction with decreasing slab thickness. We give physical insight as to why such a pair of ultrathin slabs prefers to stick together in the perpendicularly oriented manner, rather than in the parallel relative orientation as one would customarily expect., Comment: 20 pages, 4 figures, 52 references

Van der Waals interactions are ubiquitous and they play an important role for the stability of materials. Current understanding of this type of coupling is based on linear response theory, while optical nonlinearities are rarely considered in this context. Many materials, however, exhibit strong optical nonlinear response, which prompts further evaluation of dispersive forces beyond linear response. Here we present a $\textit{Discrete Coupled Nonlinear Dipole}$ approach that takes into account linear and nonlinear properties of all dipolar nanoparticles in a given system. This method is based on a Hamiltonian for nonlinear dipoles, which we apply in different systems uncovering a complex interplay of distance, anisotropy, polarizibilities, and hyperpolarizabilities in the vdW energy. This investigation broadens our basic understanding of dispersive interactions, especially in the context of nonlinear materials., Comment: 7 pages, 3 figures; submitted; a part of this work has been presented at APS March Meeting 2023 (N17.00010)

Two-dimensional materials and their heterostructures have opened up new possibilities for magnetism at the nanoscale. In this study, we utilize first-principles simulations to investigate the structural, electronic, and magnetic properties of $\textrm{Fe}/\textrm{WSe}_2/\textrm{Pt}$ systems containing pristine, defective, or doped $\textrm{WSe}_2$ monolayers. The proximity effects of the ferromagnetic Fe layer are studied by considering defective and vanadium-doped $\textrm{WSe}_2$ monolayers. All heterostructures are found to be ferromagnetic, and the insertion of the transition-metal dichalcogenide results in a redistribution of spin orientation and an increased density of magnetic atoms due to the magnetized $\textrm{WSe}_2$. There is an increase in the overall total density of states at the Fermi level due to $\textrm{WSe}_2$; however, the transition-metal dichalcogenide may lose its distinct semiconducting properties due to the stronger than van der Waals coupling. Spin-resolved electronic structure properties are linked to larger spin Seebeck coefficients found in heterostructures with $\textrm{WSe}_2$ monolayers., Comment: 22 pages, 5 figures, 1 table

Twisted bilayered graphenes at magic angles are systems housing long ranged periodicity of Moir\'e pattern together with short ranged periodicity associated with the individual graphenes. Such materials are a fertile ground for novel states largely driven by electronic correlations. Here we find that the ubiquitous Casimir force can serve as a platform for macroscopic manifestations of the quantum effects stemming from the magic angle bilayered graphenes properties and their phases determined by electronic correlations. By utilizing comprehensive calculations for the electronic and optical response, we find that Casimir torque can probe anisotropy from the Drude conductivities in nematic states, while repulsion in the Casimir force can help identify topologically nontrivial phases in magic angle twisted bilayered graphenes., Comment: 9 pages, 6 figures (main), 7 pages, 7 figures (supplementary); provisionally accepted for publication in 2D Materials

The van der Waals (vdW) interaction plays a prominent role between neutral objects at separations where short ranged chemical forces are negligible. This type of dispersive coupling is determined by the interplay between geometry and response properties of the materials making up the objects. Here, we investigate the vdW interaction between 1D, 2D, and 3D standard and Dirac materials within the Random Phase Approximation, which takes into account collective excitations originating from the electronic Coulomb potential. A comprehensive understanding of characteristic functionalities and scaling laws are obtained for systems with parabolic energy dispersion (standard materials) and crossing linear bands (Dirac materials). By comparing the quantum mechanical and thermal limits the onset of thermal fluctuations in the vdW interaction is discussed showing that thermal effects are significantly pronounced at smaller scales in reduced dimensions., Comment: 8 pages (main), 6 pages (supplementary information), 5 figures. Published on Journal of Physics: Materials

The photothermal energy conversion in hanging and floating polyaniline (PANi)-cotton fabrics is investigated using a model based on the heat diffusion equation. Perfect absorption and anti-reflection of wet hanging PANi-cotton fabrics cause quick transfer of total incident light into water confining nearly 100 $\%$ of the sunlight. As a result, a hanging membrane is found to have more attractive properties than a floating above water fabric. We find, however, that the photothermal properties of a floating PANi-cotton membrane can greatly be enhanced by dispersing TiN nanoparticles in the water below the fabric. The calculated temperature gradients for TiN nanoparticle solutions show that the absorbed energy grows with increasing the nanoparticle density and that the photothermal process occurs mostly near the surface. The collective heating effects depend on the size and density of nanoparticles, which can further be used to modulate the photothermal process., Comment: 7 pages, 5 figures, accepted for publication in RSC Advances

The Casimir interaction is induced by electromagnetic fluctuations between objects and it is strongly dependent upon the electronic and optical properties of the materials making up the objects. Here we investigate this ubiquitous interaction between Weyl semimetals, a class of 3D systems with low energy linear dispersion and nontrivial topology due to symmetry conditions and stemming from separated energy cones. A comprehensive examination of all components of the bulk conductivity tensor as well as the surface conductivity due to the Fermi arc states in real and imaginary frequency domains is presented using the Kubo formalism for Weyl semimetals with different degree of tilting of their linear energy cones. The Casimir energy is calculated using a generalized Lifhsitz approach, for which electromagnetic boundary conditions for anisotropic materials were derived and used. We find that the Casimir interaction between Weyl semimetals is metallic-like and its magnitude and characteristic distance dependence can be modified by the degree of tilting and chemical potential. The nontrivial topology plays a secondary role in the Casimir interaction of these 3D materials and thermal fluctuations are expected to have similar effects as in metallic systems., Comment: supplementary information added

We propose a theoretical approach to describe quantitatively the heating process in aqueous solutions of dispersed TiN nanoparticles under solar illumination. The temperature gradients of solution with different concentrations of TiN nanoparticles are calculated when confinement effects of the container on the solar absorption are taken into account. We find that the average penetration of solar radiation into the solution is significantly reduced with increasing the nanoparticle concentration. At high concentrations, our numerical results show that photons are localized near the surface of the solution. Moreover, the heat energy balance equation at the vapor-liquid interface is used to describe the solar steam generation. The theoretical time dependence of temperature rise and vaporization weight losses is consistent with experiments. Our calculations give strong evidence that the substantially localized heating near the vapor-liquid interface is the main reason for the more efficient steam generation process by floating plasmonic membranes when compared to randomly dispersed nanoparticles. The validated theoretical model suggests that our approach can be applied towards new predictions and other experimental data descriptions., Comment: 6 pages, 3 figures, accepted for publication in PCCP

The equations of motion for the position and gauge invariant crystal momentum are considered for multiband wave packets of Bloch electrons. For a localized packet in a subset of bands well-separated from the rest of the band structure of the crystal, one can construct an effective electromagnetic Hamiltonian with respect to the center of the packet. We show that the equations of motion can be obtained via a projection operator procedure, which is derived from the adiabatic approximation within perturbation theory. These relations explicitly contain information from each band captured in the expansion coefficients and energy band structure of the Bloch states as well as non-Abelian features originating from interband Berry phase properties. This general and transparent Hamiltonian-based approach is applied to a wave packet spread over a single band, a set of degenerate bands, and two linear crossing bands. The generalized equations of motion hold promise for novel effects in transport currents and Hall effect phenomena.

We present a compositional and structural investigation of silicene, germanene, and stanene bilayers from first-principles. Due to the staggering of the individual layers, several stacking patterns are possible, most of which are not available to the bilayer graphene. This structural variety, in conjunction with the presence of the spin-orbit coupling, unveil a diversity of the electronic properties, with the appearance of distinct band features, including orbital hybridization and band inversion. We show that for particular cases, the intrinsic spin Hall response exhibits signatures of non-trivial electronic band topology, making these structures promising candidates to probe Dirac-like physics.

Materials exhibiting controllable magnetic phase transitions are currently in demand for many spintronics applications. Here we investigate from first principles the electronic structure and intrinsic anomalous Hall, spin Hall and anomalous Nernst response properties of the FeRh metallic alloy which undergoes a thermally driven antiferromagnetic-to-ferromagnetic phase transition. We show that the energy band structures and underlying Berry curvatures have important signatures in the various Hall effects. Specifically, the suppression of the anomalous Hall and Nernst effects in the AFM state and a sign change in the spin Hall conductivity across the transition are found. It is suggested that the FeRh can be used a spin current detector capable of differentiating the spin Hall effect from other anomalous transverse effects. The implications of this material and its thermally driven phases as a spin current detection scheme are also discussed.

The Casimir and Casimir-Polder interactions are investigated in a stack of equally spaced graphene layers. The optical response of the individual graphene is taken into account using gauge invariant components of the polarization tensor extended to the whole complex frequency plane. The planar symmetry for the electromagnetic boundary conditions is further used to obtain explicit forms for the Casimir energy stored in the stack and the Casimir-Polder energy between an atom above the stack. Our calculations show that these fluctuation induced interactions experience strong thermal effects due to the graphene Dirac-like energy spectrum. The spatial dispersion and temperature dependence in the optical response are also found to be important for enhancing the interactions especially at smaller separations. Analytical expressions for low and high temperature limits and their comparison with corresponding expressions for an infinitely conducting planar stack are further used to expand our understanding of Casimir and Casimir-Polder energies in Dirac materials. Our results may be useful to experimentalists as new ways to probe thermal effects at the nanoscale in such universal interactions., Comment: 8 pages

We investigate the electromagnetic response of staggered two-dimensional materials of the graphene family, including silicene, germanene, and stanene, as they are driven through various topological phase transitions using external fields. Utilizing Kubo formalism, we compute their optical conductivity tensor taking into account the frequency and wave vector of the electromagnetic excitations, and study its behavior over the full electronic phase diagram of the materials. We also consider the plasmon excitations in the graphene family and find that nonlocality in the optical response can affect the plasmon dispersion spectra of the various phases. The expressions for the conductivity components are valid for the entire graphene family and can be readily used by others., Comment: 12 pages, 6 figures

The ability to control electromagnetic fields, heat currents, electric currents, and other physical phenomena by coordinate transformation methods has resulted in novel functionalities, such as cloaking, field rotations, and concentration effects. Transformation optics, as the underlying mathematical tool, has proven to be a versatile approach to achieve such unusual outcomes relying on materials with highly anisotropic and inhomogeneous properties. Most applications and designs thus far have been limited to functionalities within a single physical domain. Here we present transformation optics applied to thermoelectric phenomena, where thermal and electric flows are coupled via the Seebeck coefficient and Joule heating is taken into account. Using laminates, we describe a theoretical thermoelectric cloak capable of hiding objects from thermoelectric flow. Our calculations show that such a cloak does not depend on the particular boundary conditions and can also operate in different single domain regimes. These proof-of-principle results constitute a significant step forward towards finding unexplored ways to control and manipulate coupled transport.

The Casimir force is a universal interaction induced by electromagnetic quantum fluctuations between any types of objects. The expansion of the graphene family by adding silicene, germanene, and stanene, 2D allotropes of Si, Ge, and Sn, lands itself as a platform to probe Dirac-like physics in honeycomb staggered systems in such a ubiquitous interaction. We discover Casimir force phase transitions between these staggered 2D materials induced by the complex interplay between Dirac physics, spin-orbit coupling, and externally applied fields. In particular, we find that the interaction energy experiences different power law distance decays, magnitudes, and dependences on characteristic physical constants. Furthermore, due to the topological properties of these materials, repulsive and quantized Casimir interactions become possible., Comment: 12 pages, 9 Figures (4 pages and 5 figures in the Supporting Information). Discussion about thermal corrections to the Casimir force was included. Updated figures in the manuscript and supplementary information

The Casimir-Polder interaction between an atom and a multilayered system composed of infinitely thin planes is considered using the zeta-function regularization approach with summation of the zero-point energies. As a prototype material, each plane is represented by a graphene sheet whose optical response is described by a constant conductivity or Drude-Lorentz model conductivity. Asymptotic expressions for various separations are derived and compared to numerical calculations. We distinguish between large atom/plane limit, where retardation effects are prominent, and small atom/plane limit, where the typical van der Waals coefficient is found to be dependent on the number of graphenes and characteristic distances. The calculated energies for different atoms and graphene conductivity models brings forward the basic science of the Casimir-Polder effect and suggests ways to manipulate this interaction experimentally., Comment: 9p, Sent to PRA

The Casimir interaction in a stack of equally spaced infinitely thin layers is investigated within the zero-frequency mode summation method. The response properties are considered to be described by a constant conductivity or by a Drude-Lorentz model with a finite set of oscillators consistent with the optical characteristics for graphite. It is found that the asymptotic distance dependence is affected significantly by the specific response. While the energy is $\sim 1/d^3$ for the constant conductivity model, the energy exhibits fractional dependence $\sim 1/d^{5/2}$ for the Drude-Lorentz description. The Casimir force on a plane is also strongly dependent upon the particular plane location in the stack. Furthermore, the calculated Casimir energy within the Drude-Lorentz model yields results in good agreement with measured cohesion energy in graphite., Comment: 9 pages, 4 figures

Charge fluctuations in nano-circuits with capacitor components are shown to give rise to a novel type of long-ranged interaction, which co-exist with the regular Casimir/van der Waals force. The developed theory distinguishes between thermal and quantum mechanical effects, and it is applied to capacitors involving graphene nanostructures. The charge fluctuations mechanism is captured via the capacitance of the system with geometrical and quantum mechanical components. The dependence on the distance separation, temperature, size, and response properties of the system shows that this type of force can have a comparable and even dominant effect to the Casimir interaction. Our results strongly indicate that fluctuations induced interactions due to various thermodynamic quantities can have important thermal and quantum mechanical contributions at the micro- and nanoscale.

We consider the vacuum energy of the electromagnetic field in systems characterized by a constant conductivity using the zeta-regularization approach. The interaction in two cases is investigated: two infinitely thin parallel sheets and an infinitely thin spherical shell. We found that the Casimir energy for the planar system is always attractive and it has the same characteristic distance dependence as the interaction for two perfect semi-infinite metals. The Casimir energy for the spherical shell depends on the inverse radius of the sphere, but it maybe negative or positive depending on the value of the conductivity. If the conductivity is less than a certain critical value, the interaction is attractive, otherwise the Casimir force is repulsive regardless of the spherical shell radius., Comment: 10 pages, 4 figures, Accepted in PRD

The van der Waals interaction between a lipid membrane and a substrate covered by a graphene sheet is investigated using the Lifshitz theory. The reflection coefficients are obtained for a layered planar system submerged in water. The dielectric response properties of the involved materials are also specified and discussed. Our calculations show that a graphene covered substrate can repel the biological membrane in water. This is attributed to the significant changes in the response properties of the system due to the monolayer graphene. It is also found that the van der Waals interaction is mostly dominated by the presence of graphene, while the role of the particular substrate is secondary., Comment: In production Journal of Chemical Physics 2013

The radiative heat transfer between gold nanoparticle layers is presented using the coupled dipole method. Gold nanoparticles are modelled as effective electric and magnetic dipoles interacting via electromagnetic fluctuations. The effect of higher-order multipoles is implemented in the expression of electric polarizability to calculate the interactions at short distances. Our findings show that the near-field radiation reduces as the radius of the nanoparticles is increased. Also, the magnetic dipole contribution to the heat exchange becomes more important for larger particles. When one layer is displayed in parallel with respect to the other layer, the near-field heat transfer exhibits oscillatory-like features due to the influence of the individual nanostructures. Further details about the effect of the nanoparticles size are also discussed., Comment: 5 pages,5 figures; Under review 2013

We present theoretical calculations for the absorption properties of protein-coated gold nanoparticles on graphene and graphite substrates. As the substrate is far away from nanoparticles, numerical results show that the number of protein bovine serum molecules molecules aggregating on gold surfaces can be quantitatively determined for gold nanoparticles with arbitrary size by means of the Mie theory and the absorption spectra. The presence of graphitic substrate near protein-conjugated gold nanoparticles substantially enhances the red shift of the surface plasmon resonances of the nanoparticles. Our findings show that graphene and graphite provide the same absorption band when the distance between the nanoparticles and the substrate is large. However at shorter distances, the resonant wavelength peak of graphene-particle system differs from that of graphite-particle system. Furthermore, the influence of the chemical potential of graphene on the optical spectra is also investigated., Comment: 5 pages, 3 figures; Under review 2013

Graphene nanoribbon folds with single and double closed edges are studied using density functional theory methods. Van der Waals dispersive interactions are included via semi-empirical pairwise optimized potential. The geometrical phases of the single and double folded ribbons are obtained. The electronic structure in terms of energy needed for the folding process, van der Waals contribution, energy band gaps, and bandstructures are also calculated. The results are interpreted in terms of peculiarities of the structures and dispersion interactions. It is shown that significant modifications in the electronic structure can be achieved as a result of folding., Comment: 5 pages, 5 figures

We study the thermodynamic stability at low temperatures of a series of alkali metal/zinc double-cation borohydrides, including LiZn(BH$_4$)$_3$, LiZn$_2$(BH$_4$)$_5$, NaZn(BH$_4$)$_3$, NaZn$_2$(BH$_4$)$_5$, KZn(BH$_4$)$_3$, and KZn$_2$(BH$_4$)$_5$. While LiZn$_2$(BH$_4$)$_5$, NaZn(BH$_4$)$_3$, NaZn$_2$(BH$_4$)$_5$ and KZn(BH$_4$)$_3$ were recently synthesized, LiZn(BH$_4$)$_3$ and KZn$_2$(BH$_4$)$_5$ are hypothetical compounds. Using the minima-hopping method, we discover two new lowest-energy structures for NaZn(BH$_4$)$_3$ and KZn$_2$(BH$_4$)$_5$ which belong to the $C2/c$ and $P2$ space groups, respectively. These structures are predicted to be both thermodynamically stable and dynamically stable, implying that their existence may be possible. On the other hand, the lowest-energy $P1$ structure of LiZn(BH$_4$)$_3$ is predicted to be unstable, suggesting a possible reason elucidating why this compound has not been experimentally identified. In exploring the low-energy structures of these compounds, we find that their energetic ordering is sensitive to the inclusion of the van der Waals interactions. We also find that a proper treatment of these interactions, e.g., as given by a non-local density functional such as vdW-DF2, is necessary to address the stability of the low-energy structures of these compounds., Comment: Final version

The van der Waals interactions between two parallel graphitic nanowiggles (GNWs) are calculated using the coupled dipole method (CDM). The CDM is an efficient and accurate approach to determine such interactions explicitly by taking into account the discrete atomic structure. Our findings show that the van der Waals forces vary from attraction to repulsion as nanoribbons move along their lengths with respect to each other. This feature leads to a number of stable and unstable positions of the system during the movement process. These positions can be tuned by changing the length of GNW. Moreover, the influence of the thermal effect on the van der Waals interactions is also extensively investigated. This work would give good direction for both future theoretical and experimental studies., Comment: 5 pages, 4 figures, submitted 2013

Nanoscale surface forces such as Casimir and the van der Waals forces can have a significant influence on fabrication, handling and assembly processes as well as the performance of micro and nano devices. In this paper, the investigation and the calculation of the Casimir force between a graphene sheet and a ferromagnetic metal substrate in a vacuum are presented. The reflection coefficients of graphene are graphene-conductivity dependent, and the conductivity of graphene is described by the Kubo formalism. There is an effect of magnetic properties of the metal on the Casimir interaction. The magnetic effect plays a significant role at low temperatures or high value of chemical potential. The numerical results also demonstrate that the thickness of a metal slab has a minor influence on the Casimir force. The investigation and findings about the Casimir force in this study would lead to useful information and effective solutions for design and manufacturing of micro and nano devices, especially in the areas of micro and nano machining, fabrication, manipulation, assembly and metrology., Comment: 6 pages, 6 figures, Accepted

Thermal effects contributing to the Casimir interaction between objects are usually small at room temperature and they are difficult to separate from quantum mechanical contributions at higher temperatures. We propose that the thermal Casimir force effect can be observed for a graphene flake suspended in a fluid between substrates at the room temperature regime. The properly chosen materials for the substrates and fluid induce a Casimir repulsion. The balance with the other forces, such as gravity and buoyancy, results in a stable temperature dependent equilibrium separation. The suspended graphene is a promising system due to its potential for observing thermal Casimir effects at room temperature., Comment: 5 pages, 4 figures, in APL production 2012

The Casimir force between two infinitely thin parallel sheets in a setting of $N$ such sheets is found. The finite two-dimensional conductivities, which describe the dispersive and absorptive properties of each sheet, are taken into account, whereupon the theory is applied to interacting graphenes. By exploring similarities with in-plane optical spectra for graphite, the conductivity of graphene is modeled as a combination of Lorentz type oscillators. We find that the graphene transparency and the existence of a universal constant conductivity $e^2/(4\hbar)$ result in graphene/graphene Casimir interaction at large separations to have the same distance dependence as the one for perfect conductors but with much smaller magnitude.