Electrodynamics of graphene heterostructures and electromagnetic applications

Heterostructures combining graphene layers, dielectric slabs and even metallic substrates are interesting systems to look at in view of their potential applications in microwaves and terahertz. A short review of their properties is presented, together with possible applications such as shielding layers, polarizers and plasmonic devices. In the near infrared and visible range, a graphene layer modifies the Brewster angle of the substrate that supports it. This effect leads to a non-destructive technique to count the number of atomic planes the graphene is made of.


I. Introduction
Graphene is a genuine two-dimensional material with remarkable properties in many domain, including electromagnetism. Although infinitely-thin at the macroscopic scale, graphene does interact strongly with electromagnetic radiations. In the infra-red range and a large part of the visible spectrum, a single atomic plane of graphene absorbs 2.3% of the incident radiation at normal incidence. This fraction changes with increasing incidence angle and depends on the polarization [1]. When deposited on a flat dielectric surface, graphene modifies the Brewster angle of the air/substrate interface. This modification can be measured by ellipsometry and used for non-contact optical conductivity characterization [2].
In the THz domain and below, Drude conductivity starts to rule over the electronic inter-band transitions. The electromagnetic absorbance can reach almost 100% at normal incidence on monolayer graphene under special resonant conditions [3]. In the GHz domain, broadband absorption can be realized and adjusted almost at will. A few layers of graphene separated by ultrathin polymer sheets constitute an interesting passive device for GHz applications [4] Shielding layers, polarizers, and absorbing screens are among the few examples that will be described in this short review.

II. Formalism
It will be enough here to focus on the expressions of the reflectance R and absorbance A of graphene -or any conducting layer that is ultra-thin on the scale of its bulk skin depth-placed at the interface between two semi-infinite nonabsorbing media. A plane wave is impinging the conducting layer from medium 1 at an incidence angle θ 1 and is partly transmitted in medium 2 at the refraction angle θ 2 . For the p polarization under consideration here, Fresnel-like formulas write [1] R p = −n 1 sec θ 1 + n 2 sec θ 2 +σ n 1 sec θ 1 + n 2 sec θ 2 +σ 2 (1) From here, n 1 and n 2 are the indices of refraction of the two media andσ is the sheet conductivity of the interfacial layer in units of the intrinsic vacuum admittance ε 0 c, hereafter called the reduced conductivity. Expressions of the dynamical conductivity of graphene have been derived by many authors. The simplest approach relies on the Kubo formalism, complemented by some empirical treatment of the inelastic scattering of the charge carriers described by an average collisional rate γ. The result depends on the temperature T , the absolute value µ of the chemical potential measured from the Dirac point, and the tight-binding π hopping parameter t with α the fine-structure constant and where The first term in eq. 3 accounts for intra-band scattering mechanisms, the next two terms come from ππ * electronic transitions and includes a Fano resonance (factor Fa), with width W and coupling asymmetry parameter q, produced by excitonic corrections to the transition ω = 2t between van Hove singularities at the M point of the first Brillouin zone [5]. The details of the π and π * bands are contained in the expressions of the functions f 1 ( ω t ) and f 2 ( ω t ) [6]. For all the applications that follow, t was taken from the value of the ω = 2t transition predicted at 5.20 eV by GW calculations [5]. The other parameters for CVD graphene are listed in Table I. III. Discussion The reduced conductivity of monolayer graphene computed with the parameters of Table I is plotted in Fig. 1. With this data in hand, interesting phenomena can be derived from the expressions of the transmittance and absorbance of graphene.

A. Absorbing layers
Screening electromagnetic (em) radiations is an important issue in electromagnetic interference and electromagnetic compatibility. Blocking the transmission of em radiations with a reflecting layer is not satisfactory, because it contributes to increase em pollution. The alternative is to absorb as much as possible of the impinging power, like radar absorbing materials do. A conducting layer with thickness much smaller than the skin depth can absorb a substantial fraction of em radiations [10]. This property can readily be caught from eq. 2 in the particular case of normal incidence. The expression of the absorbance then simplifies in irrespective of the polarization. It has a maximum equal to n 1 /(n 1 +n 2 ) whenσ is real and equal to n 1 +n 2 , in other words when the sheet conductivity σ matches the sum of admittance of the surrounding semi-infinite media, n 1 ε 0 c + n 2 ε 0 c. In this respect, graphene is a remarkable material for frequencies below 50 GHz where Reσ is of the order of 0.5 while Imσ Reσ (see Fig. 1). It means that a few graphene planes suffice to realize the admittance matching condition [7]. In free air (n 1 = n 2 = 1), one needsσ = 2, which can be obtained with 6 graphene planes that would absorb 50% of the incoming em power for all frequencies below 50 GHz. If the transmission medium could be a metamaterial with near-zero refractive index (n 2 ≈ 0), for instance an epsilon-near-zero medium, 3 graphene planes would absorb almost 100% [11].
A simple way to mimic a near-zero-parameter medium is to resort to the Salisbury screen. Here, the graphene is supposed being held by a dielectric slab, itself deposited on a metallic plate whose thickness must be larger than its skin depth for the radiations of interest. The graphene layer is exposed to air (n 1 = 1). When the thickness of the slab corresponds to a quarter wavelength, the electric field can simultaneously be zero at the interface with gold and maximum at the graphene layer, maximizing thereby the by Joule power loss. Under the assumption that the metallic plate is an ideal conductor, the absorbance of the system at normal incidence can readily be derived in the form with k 0 = ω/c, and d being the permittivity and thickness of the slab. Maxima of A arise when a = 0 (quarter-wavelength case). Their amplitude is 4Reσ/|1 +σ| 2 , which exactly corresponds to eq. 6 for n 2 = 0. Producing few-layer graphene with a controlled number of atomic planes by CVD is difficult. For that reason, it is more reliable, although time-consuming, to stack individual graphene layers on top of each other. The manipulation of individual graphene planes is usually performed via a thin PMMA film on which graphene sticks. By piling up a few PMMA/graphene units on a substrate with suitable thickness, a heterostructure such as sketched in Fig. 2 is obtained, which can reach about 75% absorbance [8]. Such a high value of the absorbance is possible when the radiations come from the back side of the supporting slab and escape from the graphene side. This geometry is perfect to shield the optical window of an optoelectronic device against microwaves. Indeed, a few layer graphene remains nearly transparent in the NIR and a large fraction of the visible spectrum, while absorbing a great part of microwaves. When the slab thickness corresponds to a quarter wavelength of the microwave to be screen, reflection is minimized. The absorbance can therefore increase to reach the ultimate value /( + 1) where is the permittivity of the slab [12]. In principle, the absorbance and, hence, transmittance of the heterostructure can be tuned by modifying the graphene sheet conductivity σ by electrostatic doping. In the device illustrated in Fig. 2 [13], the application of a static voltage U increases the charge carrier density on one set of graphene planes and decreases it by the same amount on the other set. There is a net change of conductivity σ(U) = σ 0 ( √ |1 + U/U c | + √ |1 − U/U c |)/2. σ 0 is the "intrinsic" conductivity of the graphene planes, U c = N 0 ed/ε 0 , with N 0 the "intrinsic" charge carrier density, d and being the thickness and the static permittivity of the polymer spacers. U c = 7.5 V for 120-nm thick PMMA spacers ( = 3.6), assuming a doping level N 0 = 1.19×10 16 m −2 corresponding to the chemical potential µ of Table I. The induced conductivity σ(U) decreases with increasing |U| in the interval (0, U c ) and then raises to exceed σ 0 whenever |U| > 1.25U c . As a consequence, the transmittance of the multilayer can be modified at will: it increases when |U| varies between 0 and U c and decreases when |U| exceeds U c [13]. The practical problem to overcome here is to avoid leakage current across the PMMA spacers and

B. Microwave polarizer
Self-supported graphene 1 may behave like a polarizer at grazing incidence [14]. This is the case for all frequencies such that the real part of σ remains above a few tenths in units of ε 0 c, typically for ν < 50 GHz, see Fig. 1. In p polarization, the reflectance R p (eq. 1) remains always smaller than |σ/(2 +σ)| 2 whatever the incidence angle θ 1 . As θ 1 → π/2, R p and A p vanish like cos 2 θ 1 and cos θ 1 , respectively, meaning almost perfect transmittance of graphene at grazing incidence. The reflectance for s polarization takes the same expression as eq. 1 after substitution of the existing sec functions with cos functions. Accordingly, when n 1 = n 2 = 1, R s = |σ/(2 cos θ 1 +σ)| 2 . Now R p approaches one at grazing incidence. The opposite behaviors of R p and R s for self-supported graphene makes it easy to separate the s and p components of unpolarized microwaves, see Fig. 3. The degree of polarization φ = (I s −I p )/(I s +I p ) of the reflected and transmitted beams are listed in Table II for different incidence angles (I s and I p being the intensities of the s-and p-polarized waves).
When n 2 2 n 2 1 , this expression reproduces eq. 5 of ref. [2]. Eq. 8 is valid whenσ |n 2 − n 1 |. Then, the modification of the Brewster angle is directly proportional to the real part of the reduced conductivityσ. In the NIR and visible ranges, the angular deviation δθ 1 can be measured by ellipsometry. For graphene on a thin silica plate, it has been found that δθ 1 ≈ 0.3 • at λ = 600 nm, and this angle correction scales linearly with the number of graphene layers [2]. The application of eq. 8 with n 1 = 1, n 2 = 1.5 andσ = απ gives δθ 1 = 0.60 • . The factor of two compared to the measurements comes from the reflection of light at the back side of the silica plate (see supplementary materials of ref [2]). By measuring δθ 1 in the visible range, knowing the optical reduced conductivity of monolayer graphene, one can get the actual number of atomic planes contained in the graphene overlayer.

D. Plasmonics
The surface plasmon of graphene corresponds to an oscillation of charge localized on the atomic plane, with evanescent fields on both sides [15]. Mathematically, this eigenmode can be created by a vanishingly-small external excitation. The TM plasmon is therefore associated with the zero(s) of the denominator of eq. 1. Taking into account the evanescent character of the "reflected" and "transmitted" waves, the secant functions are purely imaginary variables given by sec θ j = −i[(kc/ωn j ) 2 −1] −1/2 , j = 1,2, with k the 2D wave vector of the plasmon. After insertion of this expression in the denominator of eq. 1, the dispersion relation of the TM plasmon is readily derived in the form [16] 1 ω (kc) 2 − 1 ω 2 with j = n 2 j the permittivity of the surrounding media. The existence of a long-lived TM plasmon demands both a vanishing real part of the conductivity σ and a positive imaginary part of σ. Im σ > 0 and Re σ Im σ can be realized near the minimum of Re σ in the frequency domain γ < ω < 2µ/ (see Fig. 1). There are many applications of graphene in plasmonics, most of them require pattering the graphene or depositing a periodic grating on it for exciting the surface plasmon with an external radiation, see ref. [17].