Microwave Absorption in Graphene Films: Theory and Experiment

The interaction of Kα microwave radiation with ultrathin graphene films is studied. Although the thickness of these films is thousands of times smaller than the skin depth, they can absorb a significant fraction of the incident radiation. The possibility of controlling the amount of absorption and reflection of waves incident on graphene is demonstrated. In particular, by choosing the substrate parameters and the angle of incidence, it is possible to increase the absorption in graphene to >50%. For certain angles of incidence it is possible to have the TE-wave reflected, while the TM-wave is transmitted. These effects can be used to create ultrathin (atomic thicknesses) absorbers and polarizers.

where α = k z /(εk 0z ) for the TM-wave and α = k z /k 0z for the TE-wave; k z = k(ε -sin 2 θ) 1/2 ; k 0z = (1 -sin 2 θ) 1/2 ; k = ω/c; ω is the frequency of the electromagnetic radiation; θ is the angle of incidence; and ε and l 1 are the dielectric constant and thickness of the slab. The thickness of the ultrathin slabs considered here is much less than the skin layer, so that the phase shift over the thickness of the slab is very small. In this case, Eqs. (1) and (2) can be simplifi ed by expanding in terms of the small parameter k z l 1 << 1. This expansion for the case of ε' << ε'' = 4πσ/ω yields very simple expressions for refl ection, transmission, and absorption in thin fi lms: T = |1/(1 + l 1 /l σ )| 2 , R = |1/(1 + l σ /l 1 )| 2 , A = 1 -T -R = (2l 1 /l σ )/(1 + l 1 /l σ ) 2 , where l σ = 2cε 0 /(σ cos θ) for the TM-wave and l σ = (2cε 0 cos θ)/σ for the TE-wave. Equations (3) for normal incidence (θ = 0) are given in [5]. An analysis of Eqs. (3) shows that as the thickness l 1 of the slab increases the transmission T decreases while the refl ectivity R increases. This behavior is fully to be expected, but the "unexpected" thing is that the transmission is substantially below unity, while refl ection becomes signifi cant by thicknesses l 1 ∼ l σ . The parameter l σ for metals is several orders of magnitude smaller than the thickness of the skin layer. Even more "unexpected" is the high absorption in a thin metallic slab. Equation (3) implies that the absorption is maximal and reaches 50% at a thickness l 1 = l σ (transmission and refl ection are then each 25%). We note another peculiarity: the different dependence of l σ (and, thereby, of all the coeffi cients T, R, and A (Fig. 1)) on the angle of incidence for different polarizations. Thus, for TM-polarization l σТМ = 2cε 0 /(σ cos θ) increases with increasing angle of incidence, while, on the other hand, for TE-polarization l σТЕ = (2cε 0 cos θ)/σ decreases. This behavior is explained by the fact that, in the case of a TE-wave, the electric fi eld is always in the plane of the slab, so it induces tangential currents, while the "effective" thickness of the slab increases in inverse proportion to cos θ. Thus, for the TE-wave these effects begin to play a role at smaller slab thicknesses compared to the case of normal incidence. In the case of a TM-wave, the angle between the normal and the electric fi eld vector E becomes important: for θ ≠ 0 the tangential component of the vector E is less than for θ = 0, so that the induced tangential current is smaller.
With an appropriate substitution these expressions for T, R, and A are easily generalized to the case where the processes are not taking place in free space but in a waveguide. Thus, for example, it is easy to show that for a rectangular waveguide, expressions for T, R, and A can be obtained with the substitution cos θ → [1 -(πn/ka) 2 -(πm/kb) 2 ] 1/2 [6], where a and b are the linear dimensions of the waveguide and n and m are integers corresponding to the waveguide modes (for a TM-wave both of these indices must be nonzero and for TE-polarization, at least one index must be nonzero). The validity of this substitution is evident since the modes of a rectangular waveguide are combinations of plane waves with an angle of inclination relative to the axis of the waveguide. Similarly to the situation in free space, a reduction in the thickness required for maximum absorption owing to inclined incidence of a TE-wave in a waveguide occurs as the cutoff frequency is approached.
Thus, thin fi lms can be used, on one hand, as concentrators of energy within an extremely small volume (absorption of half the incident radiation for nanometer thicknesses) and, on the other, as effi cient polarizers. In particular, for a thickness and angle of incidence that satisfy the inequality the fi lm refl ects a TE-polarized wave and transmits a TM-wave.
In deriving Eqs. (3) we have used the conditions at both boundaries of the fi lm. The fi lm itself is characterized by a bulk specifi c conductivity σ. Since the thickness of these fi lms is considerably smaller than the thickness of the skin layer, in Eqs. (3) and afterward the bulk conductivity shows up together with the thickness of the slab in the form of the product σl 1 . This is essentially a surface conductivity. In the present limit (k z l 1 << 1) it is simplest to treat the fi lm as infi nitely thin and, instead of using the exact boundary conditions, to use effective boundary conditions, according to which the tangential components of the electric fi elds on different sides of the slab are equal and there is a jump in the tangential magnetic component proportional to the surface current induced in the fi lm (i.e., by the surface conductivity σl 1 ). This approach leads to exactly the same Eq. (3). In addition, the description based on surface currents and surface conductivity also holds in the case where the thickness of the slab is less than the electron mean free path and the concept of a specifi c bulk conductivity loses all meaning [12].
Features of transmission and absorption in metallic fi lms on substrates. As noted above, the maximum absorption in a separately positioned thin metallic fi lm is 50%. The absorption can be increased by combining additional elements with a thin fi lm. One way of increasing the absorption in a fi lm is to optimize the fi lm thickness. Thus, it has been shown [8] that for a system consisting of a thin fi lm on a dielectric substrate, the expressions for the refl ectivity and transmission for a TE-wave are given by TE  TE  TE  TE  TE  TE  TE  TE  TE  TE  TE  TE  TE where α ТЕ = k z /k 0z , B TE = η/(ε 0 c cos θ), e ± = exp (± ik z l), l is the substrate thickness, η is the surface conductivity, given by η = σl 1 , for a thin fi lm with a thickness exceeding the electron mean free path, σ is the conductivity, and l 1 is the fi lm thickness.
It is important to note that Eqs. (5) and (6) have been obtained for a system geometry in which the substrate is facing the incident radiation. It has been shown theoretically and experimentally [6] that in this geometry, absorption in the system can exceed 50%. In a way similar to the above problem, the effect of a thin fi lm with inclined incidence increases for a TE-wave (B TE ∼ 1/cos θ) and decreases for a TM-polarized wave (B TM ∼ cos θ). An analysis of Eqs. (5) and (6) shows that there is a range of thicknesses for which the absorption in the fi lm takes on extreme values. These values occur where the derivative of the absorption with respect to thickness goes to zero. It is easy to show that this condition leads to the simple equality sin (2k z l) = 0, which for the smallest nonzero thickness corresponds to a quarter wave plate. In order for the extremum to correspond to a maximum, the following condition must hold for the conductivity: Under these conditions, the absorption Thus, Eq. (8) implies the existence of an optimum α: Then the maximum absorption in the slab is In Eqs. (7), (9), and (10) the parameters are the dielectric constant, the angle of incidence, etc. These parameters can be chosen for a TE-polarized wave so as to make the absorption >50% (maximum absorption without a substrate), and it is possible to approach complete absorption (an almost black electromagnetic hole for B i >> 1). In the following we use the standard form of the conductivity [12] for graphene. We have made experimental measurements in a waveguide exposed to a TE 10 wave in the K α band. It follows from the general form of the surface conductivity of graphene that in this band, the conductivity is essentially real (the imaginary part is many orders of magnitude smaller).

Results and Discussion.
To confi rm the theoretical results we have made an experimental study of the electromagnetic properties of multilayer (sandwich) graphene/polymer (polymethyl methacrylate PMMA) structures on quartz substrates at microwave frequencies. The synthesis of these structures is discussed in detail elsewhere [8]. We dwell only on the main steps. In the fi rst step, graphene produced by chemical vapor deposition (CVD) at a temperature of 1000 o C in a methane atmosphere on a copper substrate (thickness 25 mm, 99.8% pure) was coated with a ~600 nm-thick PMMA layer by spin coating. The thickness of the polymer layer was monitored with a Veeco Dektak6M profi lometer. The copper substrate was then dissolved in iron chloride (FeCl 3 ). The resulting "freestanding" graphene fi lm coated with the polymer was repeatedly washed in distilled water and transferred to a quartz substrate of specifi ed thickness (7.2 × 3.4 × 0.53 mm substrates were used). Repeating this procedure layer by layer yielded sandwich structures of graphene layers separated by the polymer. Samples with from 1 to 6 graphene/PMMA layers were studied. In order to further monitor the properties of the graphene, a 10 × 10 mm freestanding fi lm was transferred to a separate quartz substrate (radius 55 mm, thickness 0.53 mm) and the surface layer of PMMA was then dissolved in acetone. The quality of the graphene was monitored by scanning microscopy (SEMLEO 1455 Vand) and Raman scattering spectroscopy, and the optical absorption spectra were examined. These data showed that the test fi lms consist predominantly of monolayer graphene.
The electromagnetic response of the samples at frequencies of 26-37.5 GHz was studied using a panoramic standing wave and attenuation measurement system (R2-408 R VSWR and Transmission Loss Meter, Elmika, Vilnius, Lithuania) intended for measuring the moduli of the refl ection and transmission coeffi cients (the ratios of the amplitudes of the incident wave to the amplitude of the refl ected wave (S 11 ) and to the transmitted wave (S 21 )), VSWR, and attenuation of the waveguide structures. In accordance with the measurement techniques (described in detail in [9] and [13]), a sample in the form of a thin fi lm on a quartz substrate was placed in a cross section (7.2 × 3.4 mm) of the waveguide perpendicular to the direction of propagation of the electromagnetic wave (Fig. 2). For testing the theoretical predictions of the effect of substrate thickness on the electromagnetic response of these samples, the thickness of the substrates was varied by placing additional plane-parallel epoxy (EPIKOTETM Resin 828) slabs of thickness 0.7, 0.9, 1.0, 1.2, or 1.5 mm in the waveguide next to the quartz slab (Fig. 2).
There were several reasons for choosing this method of simulating different thicknesses of the substrate. Using additional polymer slabs of different thicknesses makes it easy to change the resultant thickness of the substrate (the thickness of the epoxy + 0.53 mm quartz), which, in turn, makes it much easier to do the experiment by reducing the required number of samples of multilayer structures with specifi ed geometric parameters. At microwave frequencies epoxy, like quartz [8], is a dielectric with low absorption [14]. It is, however, much easier to make these slabs out of polymer materials than quartz. Figure 3 shows the experimental transmission (T = 2 21 S ), refl ectivity (R = 2 11 S ), and absorption (A = 1 -T -R) for samples containing up to six graphene/PMMA layers at a frequency of 31 GHz as functions of the substrate thickness. These curves show that the maximum absorption (~80%) in this frequency range, which corresponds to the refl ection minimum (1-2%), is observed for a substrate thickness of 1.25 mm. These experimental data are in good agreement with the above theoretical predictions (Fig. 4).
The extremely high absorption of microwaves by graphene should be noted. Even a single graphene layer absorbs ~30% of the incident wave. The optical absorption, which is determined by interband transitions, is considerably lower (πα ~ 2.3%) [15]. The absorption in the sandwich structure is exclusively caused by graphene. The imaginary part of the   dielectric constants of quartz and epoxy resin satisfy the relation k ′′ ε i l i << 1 in this frequency range; this in turn determines the low level of absorption in these materials. A PMMA layer with a width of ~600 nm is optically thin in this range and does not contribute to the overall level of absorption.
Conclusions. The feasibility of adjusting the electromagnetic response of thin graphene fi lms deposited on a dielectric substrate over a wide range has been demonstrated experimentally and theoretically. It has been shown that the level of transmission, refl ection, and absorption can be controlled by the substrate thickness, angle of incidence, and polarization of the incident wave. The effects described here can be used to create detectors, sensors, and effi cient polarizers over a wide spectral range. Given the possible absorption of macroscopically large amounts of energy in microscopic volumes, this effect may be useable in a wide variety of applications in power engineering.