Fermi energy-dependence of electromagnetic wave absorption in graphene

Graphene is a two-dimensional atomic layer of graphite that has various potential applications because of its unique electronic and optical properties. The linear energy dispersion relation of graphene results in effectively massless Dirac electrons and high carrier mobility.^1–3) The high mobility of graphene has allowed researchers to develop ultrafast optoelectronic devices, such as field effect transistors and photodetectors, from the visible down to the near-infrared regimes.^1,4)

It is well-known that undoped monolayer graphene absorbs 2.3% of visible light perpendicular to its surface, making graphene an almost perfect transparent conductor.²⁾ Transparent graphene is useful for optical devices, such as liquid-crystal displays (LCDs) and light-emitting diodes (LEDs). However, in order to develop other optical devices, such as photodetectors, optical antennas, and solar cells, high optical absorption in graphene is required to generate a sufficiently large photocurrent. In recent years, the enhancement of the optical absorption of graphene has been extensively studied, but most research on the issue has involved the use of complicated techniques, such as using a grating coupler, or shaping the graphene into ribbons or disks.^5–7) Practical optoelectronic applications of graphene thus remain challenging.

In this letter, we propose an alternative method to enhance the optical absorption of graphene for a microwave regime by simply varying the Fermi energy and the angle of incidence of an electromagnetic (EM) wave. The monolayer graphene is placed between two dielectric media to form a total internal reflection (TIR) geometry. Not only does this geometry allow us to enhance the EM wave absorption of monolayer graphene to up to 100%, it also allows us to suppress the absorption to nearly 0% by changing the Fermi energy of graphene.

Our geometrical setup is shown schematically in Fig. 1. Graphene is modeled as a conducting interface between two dielectric media with dielectric constants ε₁ and ε₂. Unless otherwise defined, we assume that ε₁ > ε₂, which allows TIR to occur for the incident EM wave in medium 1. The absorption probability for this geometry can be calculated by utilizing the boundary conditions from Maxwell's equations. If we adopt the p-polarization of the EM wave, as shown in Fig. 1, we can obtain two boundary conditions for electric field E⁽ⁱ⁾ and magnetic field H⁽ⁱ⁾ ($i = 1,2$) as follows:

$$\begin{align} E_{+}^{(1)}\cos\theta + E_{-}^{(1)}\cos\theta & = E_{+}^{(2)}\cos\phi, \end{align} \tag{ 1 }$$

$$\begin{align} H_{+}^{(2)}-(H_{+}^{(1)}-H_{-}^{(1)}) & = -\sigma E_{+}^{(2)}\cos\phi, \end{align} \tag{ 2 }$$

where the + (−) index denotes the right- (left-) going waves according to Fig. 1, θ is the angle of incidence and reflection, ϕ is the angle of refraction, and σ is the optical conductivity of graphene.⁸⁾ The E and H fields are also related to each other in terms of the EM wave impedance in units of Ω for each medium:

$$\begin{equation} Z_{i} = \frac{E_{i}}{H_{i}} = \frac{377}{\varepsilon_{i}}\Omega\quad (i = 1, 2), \end{equation} \tag{ 3 }$$

where the constant 377 Ω is the impedance of vacuum $Z_{0} = \sqrt{\mu _{0}/\varepsilon _{0}}$. The quantities ϕ, θ, and Z_i are related by Snell's law $Z_{2}\sin \theta = Z_{1}\sin \phi$. Solving Eqs. (1)–(3), we obtain the reflectance R, the transmittance T, and the absorption probability A of the EM wave as follows:

$$\begin{align} R & = \left|\frac{E_{1}^{(-)}}{E_{1}^{(+)}}\right|^{2}\\ & = \left|\frac{Z_{2}\cos\phi - Z_{1}\cos\theta - Z_{1}Z_{2}\sigma\cos\theta\cos\phi}{Z_{2}\cos\phi + Z_{1}\cos\theta + Z_{1}Z_{2}\sigma\cos\theta\cos\phi}\right|^{2},\\ T & = \frac{\cos\phi}{\cos\theta}\frac{Z_{1}}{Z_{2}}\left|\frac{E_{2}^{(+)}}{E_{1}^{(+)}}\right|^{2}\\ & = \frac{4Z_{1}Z_{2}\cos\theta\cos\phi}{|Z_{2}\cos\phi + Z_{1}\cos\theta + Z_{1}Z_{2}\sigma\cos\theta\cos\phi|^{2}},\\ A & = 1 - R - T, \end{align} \tag{ 4 }$$

where the values of R, T, and A can be denoted in terms of percentage (0–100%). Note that the factor Z₁/Z₂ in T of Eq. (4) is due to the different velocities of the EM wave in medium 1 and medium 2.

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The EM wave absorption A is determined by the optical conductivity σ, which describes the electron transition due to the optical absorption. Here, the σ for graphene is derived from the dielectric function ε using a random-phase approximation (RPA).^9–11) Since the coupling between the EM wave and graphene occurs only at longer wavelengths, we focus our calculation only on the $q \to 0$ case. At $q \to 0$, the RPA dielectric function of graphene as a function of the wave vector q and the angular frequency ω of the EM wave for a given Fermi energy E_F is expressed by¹⁰⁾

$$\begin{align} &\varepsilon(q\rightarrow 0,\omega) \\ &\quad= 1 - \left(\frac{2\pi e^{2}}{q}\right)\frac{q^{2}}{2\pi\hbar\omega}\biggl[\frac{2E_{\text{F}}}{\hbar\omega} + \frac{1}{2}\ln\left|\frac{2E_{\text{F}} - \hbar\omega}{2E_{\text{F}} + \hbar\omega}\right| \\ &\qquad- i\frac{\pi}{2}\Theta(\hbar\omega-2E_{\text{F}})\biggr], \end{align} \tag{ 5 }$$

where e is the fundamental electron charge and Θ is the Heaviside step function. The relation between σ and ε is given by

$$\begin{equation} \sigma(q,\omega) = \frac{i \omega}{2\pi q}[1-\varepsilon(q,\omega)], \end{equation} \tag{ 6 }$$

and is obtained in $(q,\omega )$ space.¹²⁾ Using Eq. (6), σ can be written as

$$\begin{align} \sigma (\omega)& \equiv\sigma_{\text{D}} + \text{Re}\sigma_{\text{E}} + \text{Im}\sigma_{\text{E}}\\ & = \frac{E_{\text{F}}e^{2}}{\pi\hbar}\frac{i}{\hbar\omega + i\Gamma} + \frac{e^{2}}{4\hbar}\Theta(\hbar\omega - 2E_{\text{F}})\\ &\quad +\frac{ie^{2}}{4\pi\hbar}\ln\left|\frac{2E_{\text{F}} - \hbar\omega}{2E_{\text{F}} + \hbar\omega}\right|, \end{align} \tag{ 7 }$$

in which the q-dependence of σ has now vanished. The first term in Eq. (7) is the intraband conductivity, which is known as the Drude conductivity σ_D. We add spectral width Γ as a phenomenological parameter for the scattering rate. It depends on E_F, as $\Gamma = \hbar ev_{\text{F}}^{2}/\mu E_{\text{F}}$,¹³⁾ where $v_{\text{F}} = 10^{6}$ m/s is the Fermi velocity of graphene and μ = 10⁴ cm² V⁻¹ s⁻¹ is its electron mobility. The second and the third terms in Eq. (7) correspond to the real and imaginary parts, respectively, of interband conductivity σ_E. By inserting Eqs. (7) and (3) into Eq. (4), we get A, R, T as a function of E_F and incidence angle θ. Both σ_D and σ_E affect EM wave absorption, and the contribution of each is discussed below.

Let us see how EM wave absorption in graphene can be modified under certain conditions. In Fig. 2(a), we reproduce the 2.3% optical absorption if graphene is placed in a vacuum,²⁾ i.e., ε₁ = ε₂ = 1 with E_F = 0.64 eV and θ = 0°. The absorption A is associated with the real part of σ, as shown by the identical shape of both curves in Figs. 2(a) and 2(b).¹²⁾ In Fig. 2(b), the conductivity of graphene [Eq. (7)] normalized by $\sigma _{0} = e^{2}/4\hbar$ is shown. The A value of 2.3% is obtained when $\hbar \omega > 2E_{\text{F}}$ [inset of Fig. 2(a)] because at this region, the real part of total conductivity σ is a constant σ₀ [see Fig. 2(b)], which is obtained from $\text{Re} \sigma _{\text{E}}$ (while σ_D is negligible). When $\hbar \omega /E_{\text{F}} \approx 0$ [Fig. 2(a)], it can be seen that the A value becomes large (∼20%) due to Drude conductivity $\text{Re} \sigma _{\text{D}}$, as shown in Fig. 2(b). In this case, $\text{Re} \sigma _{\text{D}}$ plays the major role in σ. Because of the large value of A in this region, we let the parameter $\hbar \omega = 0.1$ meV (equivalent to f = 23.8 GHz, or microwave) and E_F = 0.64 eV such that a large value of A is expected when the incident angle θ is changed.

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If ε₁ > ε₂, the TIR of the EM wave can occur at $\theta \geq \theta _{\text{c}}$, where θ_c is the critical angle defined by $\sin \theta _{\text{c}} = Z_{1}/Z_{2} = \sqrt{\varepsilon _{2}/\varepsilon _{1}}$. When TIR occurs, no EM wave can be transmitted, and thus the EM wave can either be reflected or absorbed by graphene. Here, we set ε₁ = 2.25 and ε₂ = 1.25, which corresponds with θ_c = 48.19°. In Fig. 3(a), we show A, R, and T as functions of θ. As seen in Fig. 3(a), T = 0 if θ ≥ θ_c. Interestingly, A becomes almost unity at an angle of approximately 85°; hence, graphene absorbs all incoming EM waves, where R = 0. The dip in the A spectrum indicates the beginning of TIR, where R reaches its maximum value. In Fig. 3(b), we show the E_F-dependence of A as a function of θ for several E_F values. Furthermore, from each absorption peak obtained in Fig. 3(b), we can plot the maximum value of A and the corresponding peak position θ_max as a function of E_F [see Fig. 3(c)]. The maximum A value rapidly increases with increasing E_F, and is saturated near 100% for E_F ≥ 0.4 eV. For $\hbar \omega /2 < E_{\text{F}} < 0.01$ eV, A is nearly 0, as we can see in Fig. 3(b) and the inset of Fig. 3(c).

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It is expected that when E_F decreases, A will monotonically decrease to zero. However, this is not the case, as we can see in Fig. 3(b) and the inset of Fig. 3(c). Even when we set E_F to zero (or $\hbar \omega \geq 2E_{\text{F}}$), the A value is approximately 2.2%. This is due to the vanishing intraband transition while interband transition dominates with total conductivity σ governed by constant σ₀ for $\hbar \omega \geq 2E_{\text{F}}$. Therefore, although E_F decreases to zero, absorption is not zero. At the same time, for $\hbar \omega < 2E_{\text{F}}$, Drude conductivity dominates, and conductivity is proportional to E_F, as we can realize from Eq. (7). This is why the value of A for E_F = 0.01 eV is smaller than that for E_F = 0 eV. The θ_max value shifts to the lower angle as the Fermi energy decreases, as shown in Fig. 3(c). This angle becomes a constant θ_max = 61.16° at E_F ≤ 0.1 eV. In Fig. 3(d), we define the absorption range (A_R) as the difference between the A value at E_F ≠ 0 and E_F = 0 while keeping θ_max for E_F ≠ 0. We can see that A_R starts to stabilize from E_F = 0.4 eV at approximately 99%. This means that if we change E_F from E_F > 0.4 eV to zero, nearly perfect switching of the reflected EM wave can be observed, and this behavior can be useful for some applications.

The dielectric constant of the surrounding medium can also affect the absorption of graphene, as shown in Fig. 4. In this case, we use $\hbar \omega = 0.1$ meV and E_F = 1 eV. In Fig. 4(a), the ratio n of the dielectric constant of both media is kept constant, i.e., ε₁/ε₂ = n = 2.25/1.25 = 1.8, and we change both values of $(\varepsilon _{1},\varepsilon _{2})$ giving a constant θ_c = 48.19° where the plot starts. The absorption peak for all pairs of $(\varepsilon _{1},\varepsilon _{2})$ yields A values of almost 100%. The peak monotonically decreases from 88 to 84°, with increasing value of ε₁ up to 20.25. When both ε₁ and ε₂ increase while keeping constant ε₁/ε₂ = n = 2.25/1.25 = 1.8, the peak moves to the smaller angle. The A spectrum as a function of θ becomes broader with increasing $(\varepsilon _{1},\varepsilon _{2})$. In Fig. 4(b), we show the case where n is varied but ε₂ = 1.25 is kept constant, i.e., ε₁ = n × ε₂. As the value of n increases, the maximum value of A stays almost 100%, but θ_max is altered. The change in θ_max is not pronounced if n < 100. In both Figs. 4(a) and 4(b), the maximum value of A does not change, even though we tune the surrounding dielectric constant. Hence, tuning the surrounding dielectric constant simply changes θ_max. Figure 4(c) shows the dielectric constant-dependence of θ_max. The relation between θ_max and n can be expressed to contain a sinusoidal function. In Fig. 4(c), $\sin \theta _{\text{max}}$ is plotted as a function of n for ε₂ = 1.25 and ε₂ = 3. It can be seen that $\sin \theta _{\text{max}}$ depends linearly on n. The slope of the line depends on ε₂, and becomes steeper when ε₂ is large.

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To apply all the concepts discussed so far, we show in Fig. 5(a) a possible design for an EM wave-switching device consisting of an EM wave source, a detector, a gate–voltage modulation system, and a graphene layer sandwiched between two dielectric materials $(\varepsilon _{1},\varepsilon _{2})$. The EM wave source is placed at a certain angle where absorption is maximum when E_F ≠ 0. The detector is used to catch the reflected wave. It is necessary to insert thin, multi-layered films attached to medium 1 in front of the EM wave source and the detector in order to suppress unnecessary reflection at the surface of medium 1. The gates in the device are used to change the E_F of the monolayer graphene. We may use electrochemical doping if medium 1 is an electrolyte. The top and bottom dielectric materials $(\varepsilon _{1},\varepsilon _{2})$ can be tuned to obtain large values of A_R for perfect switching. In Fig. 5(b), for example, we used silicon as medium 1 (ε₁ = 11.67 within the microwave frequency range) and vacuum as medium 2 (ε₂ = 1). To obtain the EM wave switch, the EM wave with $\hbar \omega = 0.1$ meV was shot at an incident angle of approximately θ = 62°. If the Fermi energy is changed from E_F = 0.4 to 0 eV, we can detect the reflected EM wave with a reflectance of 0% to be nearly 100%, which corresponds with absorption from 100% to be nearly 0%, respectively.

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In conclusion, we have shown that the absorption of 10–100 GHz electromagnetic waves in graphene can be tuned from nearly 0 to 100% by simply varying the Fermi energy of graphene while keeping the EM wave within TIR geometry. We also designed a possible EM wave-switching device (with $\hbar \omega = 0.1$ meV, or a frequency of approximately 23.8 GHz) where switching behavior can be obtained by changing E_F in the vicinity of 0–0.4 eV. Because of the simplicity of the device structure and its analytical formulation, we expect that experimental verification will be possible in the near future, which will allow us to determine how quickly the device can switch EM waves. We hope that our work also triggers further research on Fermi energy-dependence of EM wave absorption in graphene for frequency ranges beyond the 10–100 GHz regime.

Acknowledgments