Electrodynamics of Graphene/Polymer Multilayers in the GHz Frequency Domain

Abstract

The electromagnetic properties of graphene/PMMA multilayers are calculated by electrodynamics techniques. It is shown that an optimum number of layers exists for which the absorption of GHz radiations by the graphene planes is maximum. Numerical calculations using the rigorous coupled wave analysis method demonstrate that the absorption of GHz radiations by the optimum graphene/PMMA multilayer is robust in the sense that it does not depend on defects of the graphene planes to first order in concentration.

Appendices

Appendix 1: Deviation of Equation (3.1)

In this Appendix, the absorbance A of a conducting slab in normal incidence is derived. The formalism constructed for the s polarization is used, for it readily leads to an approximate, though accurate expression. Starting from $A = 1 - R - T$ and using (3.12), (3.13), (3.9), and (3.5), one obtains

$$\displaystyle{ A = \frac{4n_{I}} {\vert \xi _{s}(0) - in_{I}\vert ^{2}}\,\left (-\mbox{ Im}\,\xi _{s}(0)\, -\, n_{T}\left \vert \frac{b} {a + in_{T}}\right \vert ^{2}\right ) }$$

(3.29)

where, as before, n _I and n _T are the indices of refraction of the incidence and emergence media, respectively (both assumed real), a and b are the coefficients defined by (3.8) for the s polarization. The assumption that will be made is | a | ≫ n _T . If n denotes the complex refractive index of the slab and d the slab thickness, this condition is met when either | n | ≫ 1 or | nk ₀ d | ≪ 1. The limit $d \rightarrow 0$ belongs to the latter case. In both cases, the denominator of the last term in the right-hand side of (3.29) can be approximated to a. Substituting a and b by their expressions (3.8) yields

$$\displaystyle{ A = \frac{4n_{I}} {\vert \xi _{s}(0) - in_{I}\vert ^{2}}\,\left (-\mbox{ Im}\,\xi _{s}(0)\, -\, \frac{n_{T}} {\vert \cos (nk_{0}d)\vert ^{2}}\right )\mbox{.} }$$

(3.30)

It suffices to decompose $n = n_{1} + in_{2}$ in its real and imaginary parts and to use the definition of $\xi _{s}(0)$ in terms of the surface admittance Y _s of the slab to arrive at the final result:

$$\displaystyle{ A = \frac{4n_{I}} {\vert \tilde{Y _{s}} + n_{I}\vert ^{2}}\left (\mbox{ Re}\,\tilde{Y _{s}} - \frac{2n_{T}} {\cos (2n_{1}k_{0}d) +\cosh (2n_{2}k_{0}d)}\right )\mbox{,} }$$

(3.31)

where $\tilde{Y _{s}} = Y _{s}/\epsilon _{0}c$. Equation 3.1 follows from the equation just obtained by setting $n_{I} = n_{T} = 1$. An expression of $\tilde{Y _{s}}$ consistent with the condition | a | ≫ n _T assumed above can be obtained from (3.8) and (3.5):

$$\displaystyle{ \tilde{Y _{s}} = -i\,n\,\tan (nk_{0}d) + \frac{n_{T}} {\cos ^{2}(nk_{0}d)}\,[1 + O(n_{T}/a)]\mbox{.} }$$

(3.32)

Finally, it is worth mentioning that the limit $d \rightarrow 0$ of (3.30) reproduces (3.18) after (3.10) and (3.5) have been used.

Appendix 2: Negative-Imaginary Transformation

In this Appendix, it is demonstrated that the continued-fraction generator (3.7), $\xi (z_{u}) = a - b^{2}/[a +\xi (z_{l})]$, is a complex rational function of $\xi (z_{l})$ whose imaginary part is negative for any complex value $\xi (z_{l})$ such that $\mbox{ Im}\,\xi (z_{l}) < 0$. The demonstration is based on two important inequalities fulfilled by the a and b coefficients (3.8), namely

$$\displaystyle\begin{array}{rcl} \mbox{ Im}\,a \leq 0\qquad (3.33a)\qquad \vert \mbox{ Im}\,b\vert \leq \vert \mbox{ Im}\,a\vert \mbox{.}\qquad (3.33b)& & {}\\ \end{array}$$

The starting point of the proof of (3.33a) and (3.33b) is the following property of the tangent function:

$$\displaystyle\begin{array}{rcl} & & \forall Z \in Q1\qquad \qquad \arg Z \leq \arg \tan Z \leq \pi -\arg Z {}\\ & & \forall Z \in Q2\qquad \qquad \pi -\arg Z \leq \arg \tan Z \leq \arg Z {}\\ \end{array}$$

where Q1 and Q2 denote the first and second quadrants, respectively, of the complex plane, the argument function being defined between −π and +π. Consider two arbitrary complex numbers u and v that both belong to Q1 and any positive real number κ. Let

$$\displaystyle{ C = (u/v)\cot (\kappa uv)\qquad (3.34a)\qquad T = (u/v)\tan (\kappa uv)\mbox{.}\qquad (3.34b) }$$

From the property of the $\tan$ function just underlined, it is easy to show that

$$\displaystyle{ -\pi + 2\theta _{u} \leq \arg C \leq -2\theta _{v}2\theta _{u} \leq \arg T \leq \pi -2\theta _{v} }$$

when uv ∈ Q1, where $\theta _{u} =\arg u$ and $\theta _{v} =\arg v$, and

$$\displaystyle{ -2\theta _{v} \leq \arg C \leq -\pi + 2\theta _{u}\pi - 2\theta _{v} \leq \arg T \leq 2\theta _{u} }$$

when uv ∈ Q2. The conclusion that can immediately be drawn is

$$\displaystyle{ \mbox{ Im}\,C \leq 0\qquad (3.35a)\qquad \mbox{ Im}\,T \geq 0\mbox{.}\qquad (3.35b) }$$

According to (3.8), the a coefficient has the form (3.34a) with $u = \sqrt{f}$, $v = \sqrt{g}$ and κ = k ₀ d. Both f and g have a positive imaginary part, their square roots are located in Q1 or Q3. The first determination can be chosen without loss of generality, because a is an even function of both $\sqrt{f}$ and $\sqrt{g}$. The hypothesis that u and v belong to Q1 is therefore verified. The inequality (3.33a) then follows directly from (3.35a). The expressions (3.8) of the a and b coefficients readily lead to $b + a = \sqrt{f/g}\cot (\sqrt{fg}\,k_{0}d/2)$ and $b - a = \sqrt{f/g}\tan (\sqrt{fg}\,k_{0}d/2)$ which take the forms (3.34a) and (3.34b), respectively, with the consequences Im b ≤ | Im a | and Im b ≥ − | Im a | . Condition (3.33b) is thereby demonstrated.

It becomes now easy to prove the negative imaginary property of the transformation (3.7). Consider first the limiting case Im a = 0 of the condition (3.33a), which implies Im b = 0 according to (3.33b). From (3.7), then, $\mbox{ Im}\,\xi (z_{u}) = (b^{2}/\vert a +\xi (z_{l})\vert ^{2})\,\mbox{ Im}\,\xi (z_{l})$ is indeed negative when $\mbox{ Im}\,\xi (z_{l}) < 0$.

Assume now Im a < 0 and write (3.7) as $\xi (z_{u}) = N/D$ where $D = \vert a +\xi (z_{l})\vert ^{2}$ and $N = [a^{2} - b^{2} + a\xi (z_{l})][a +\xi (z_{l})]^{{\ast}}$. With the hypotheses Im a < 0 and $\mbox{ Im}\,\xi (z_{l}) < 0$, D is strictly positive. All the attention can therefore be put on Im N. Let $a = a_{1} + ia_{2}$, $b = b_{1} + ib_{2}$ and $\xi (z_{l}) =\xi _{1} + i\xi _{2}$. Then $\mbox{ Im}\,N =\alpha \xi _{ 2}^{2} + 2\beta \xi _{2}+\gamma$, where α = a ₂ < 0, $\beta = a_{2}^{2} - b_{2}^{2} + b_{1}^{2} \geq b_{1}^{2} \geq 0$ (see (3.33b)), and $\gamma = a_{2}(a_{1} +\xi _{1})^{2} - 2b_{1}b_{2}(a_{1} +\xi _{1}) + a_{2}(a_{2}^{2} - b_{2}^{2} + b_{1}^{2})$.

The γ coefficient is a negative-definite quadratic function of the variable $a_{1} +\xi _{1}$: the coefficient of its second-degree term is negative (condition 3.33a) and its discriminant $-(a_{2}^{2} - b_{2}^{2})(a_{2}^{2} + b_{1}^{2})$ is also negative (condition 3.33b). As a result, Im N is a quadratic function of $\xi _{2}$ whose value at $\xi _{2} = 0$ is negative, which increases with increasing $\xi _{2}$ and reaches its maximum at the positive value $\xi _{2} = -\beta /\alpha$. Consequently, Im N remains negative for all negative values of $\xi _{2}$, QED.