1. Introduction

2. Reflectance of Graphene Multilayer on Si Substrate

3. Corrections to Reflectance by Light Emission

3.2. Primary model

Basically, primary model means the Fresnel equation of Equation (1) where ${\epsilon }_g$ is given by Equation (2). ${\epsilon }_g$ has a single unknown constant ${\epsilon }_r$ which can be estimated from the following observations. Ultimately, we conclude that ${\epsilon }_r=4.5$.

First, there must not be a large discrepancy between ${\epsilon }_g$ and the experimentally determined optical constants [19,41]. Experimental values for n and k [19] are shown as • and ∘ in Figure 6a. The lines depict (bare) optical constants plotted using Equation (2) with the refractive index $n_g=\mathrm{Re}\left[\sqrt{{\epsilon }_g}\right]$ and absorption coefficient $k_g=\mathrm{Im}\left[\sqrt{{\epsilon }_g}\right]$ for ${\epsilon }_r=4.5$ and 5.5. When ${\epsilon }_r=4.5$, $k\simeq k_g$ but n has a certain difference from $n_g$. When ${\epsilon }_r=5.5$, $n\simeq n_g$ but k has a certain difference from $k_g$. Because $n_g$ increases with increasing ${\epsilon }_r$ while $k_g$ decreases, there is no ${\epsilon }_r$ value that can reproduce n and k simultaneously. This suggests that there should be such a discrepancy between them which is attributed to the corrections by light emission. Second, ${\epsilon }_g$ has to roughly reproduce the behavior of the reflectance of multilayer graphene. It exhibits a minimum at a certain wavelength, primarily due to destructive interference caused by SiO₂. As shown in Figure 6b, the position is red-shifted by increasing N, indicating that even thin graphite samples significantly impact the light interference effect. When ${\epsilon }_r=1$, the position changes little, and a sizable artificial shift in wavelength is needed to ensure consistency between theory and experiment, which cannot be explained as a correction. When ${\epsilon }_r=4.5$, a small difference between theory and experiment still remains. However, as we show later, the corrections provide better agreement not only for the wavelengths giving minimum reflectivity but also for the minimum reflectivity values, thus accounting for the difference. Third, ${\epsilon }_r=4.5$ roughly reproduces the reflectivity of graphite in the infrared region [42]. Similar ${\epsilon }_r$ values have been used to reproduce the observed reflectivities of graphite and graphene [18,19,41]. We note that the value of ${\epsilon }_r$ is less than the magnitude of the imaginary part of ${\epsilon }_g$, since visible light has a much longer $\lambda$ (400∼800 nm) than d, although $\alpha$ is certainly a small quantity. The optical properties of multilayer graphene are thus characterized by the large imaginary part of ${\epsilon }_g$.

Figure 4. ${\epsilon }_r$ of the primary model (a) Experimental n and k values are represented by dots, taken from Ref. [19]. The lines depict bare optical constants without corrections. (b) Dots indicate measured wavelengths corresponding to the minimum reflectance. A comparison between the measured and calculated results suggests that ${\epsilon }_r=4.5$ is a reasonable value.

Figure 4. ${\epsilon }_r$ of the primary model (a) Experimental n and k values are represented by dots, taken from Ref. [19]. The lines depict bare optical constants without corrections. (b) Dots indicate measured wavelengths corresponding to the minimum reflectance. A comparison between the measured and calculated results suggests that ${\epsilon }_r=4.5$ is a reasonable value.

4. Corrections to Reflectance by Light Emission

4.1. Basic Idea

Figure 5a illustrates our model of reflection, where horizontal lines on the substrate represent N-layer graphene, and the vertical lines depict light rays with arrows indicating the directions of light propagation. The light rays on the left side (black in color) of Figure 5a show the primary processes of reflection (excluding contributions from light emission). In this process, incident light from a light source is transmitted and reflected by graphene, while some energy of light being absorbed by each layer. The reflection coefficient, $r_N$, is calculated from a primary model which is defined in Section 4.2. The light rays on the right side (red in color) correspond to the light emission. Suppose that the jth layer emits light. The emitted light is transmitted and reflected by graphene until the light escapes the system, and it contributes to the reflectance of the system. Thus, there is another “reflection coefficient” when N-layer graphene emits light which is defined in Section 4.3. Let $z_N$ denotes the sum over such amplitudes from all layers. Once we know what $r_N$ and $z_N$ are, then the reflectance is given by $R_N={|r_N+z_N|}^2$.

Figure 5. Model description (a) The primary (left side) and secondary (right side) processes involved in the reflection are physically interconnected through light absorption at each graphene layer. (b) The self-consistent calculation of $Z_j$ is explained in detail in the text.

Figure 5. Model description (a) The primary (left side) and secondary (right side) processes involved in the reflection are physically interconnected through light absorption at each graphene layer. (b) The self-consistent calculation of $Z_j$ is explained in detail in the text.

We use a transfer matrix method to calculate $r_N$ and $z_N$ [21,34]. Transfer matrix method is useful in calculating reflection (up arrow) and transmission (down) coefficients at each layer [$r_N\left(j\right)$ and $t_N\left(j\right)$ in Figure 5a] in addition to the electric field $E_j^N$ ($j=1,\dots ,N$) that determines the absorption of the jth layer as $A_j^N\equiv \pi \alpha {\left|E_j^N\right|}^2$. The total absorption of N-layer graphene is ${\sum }_{j=1}^N{A}_j^N$. [21]

4.2. Primary Model

Basically, primary model means the Fresnel equation of Equation (1) where ${\epsilon }_g$ is given by Equation (2). ${\epsilon }_g$ has a single unknown constant ${\epsilon }_r$ which can be estimated from the following observations. Ultimately, we conclude that ${\epsilon }_r=4.5$.

First, there must not be a large discrepancy between ${\epsilon }_g$ and the experimentally determined optical constants [19,41]. Experimental values for n and k [19] are shown as • and ∘ in Figure 4a. The lines depict (bare) optical constants plotted using Equation (2) with the refractive index $n_g=\mathrm{Re}\left[\sqrt{{\epsilon }_g}\right]$ and absorption coefficient $k_g=\mathrm{Im}\left[\sqrt{{\epsilon }_g}\right]$ for ${\epsilon }_r=4.5$ and 5.5. When ${\epsilon }_r=4.5$, $k\simeq k_g$ but n has a certain difference from $n_g$. When ${\epsilon }_r=5.5$, $n\simeq n_g$ but k has a certain difference from $k_g$. Because $n_g$ increases with increasing ${\epsilon }_r$ while $k_g$ decreases, there is no ${\epsilon }_r$ value that can reproduce n and k simultaneously. This suggests that there should be such a discrepancy between them which is attributed to the corrections by light emission. Second, ${\epsilon }_g$ has to roughly reproduce the behavior of the reflectance of multilayer graphene. It exhibits a minimum at a certain wavelength, primarily due to destructive interference caused by SiO₂. As shown in Figure 4b, the position is red-shifted by increasing N, indicating that even thin graphite samples significantly impact the light interference effect. When ${\epsilon }_r=1$, the position changes little, and a sizable artificial shift in wavelength is needed to ensure consistency between theory and experiment, which cannot be explained as a correction. When ${\epsilon }_r=4.5$, a small difference between theory and experiment still remains. However, as we show later, the corrections provide better agreement not only for the wavelengths giving minimum reflectivity but also for the minimum reflectivity values, thus accounting for the difference. Third, ${\epsilon }_r=4.5$ roughly reproduces the reflectivity of graphite in the infrared region [42]. Similar ${\epsilon }_r$ values have been used to reproduce the observed reflectivities of graphite and graphene [18,19,41]. We note that the value of ${\epsilon }_r$ is less than the magnitude of the imaginary part of ${\epsilon }_g$, since visible light has a much longer $\lambda$ (400∼800 nm) than d, although $\alpha$ is certainly a small quantity. The optical properties of multilayer graphene are thus characterized by the large imaginary part of ${\epsilon }_g$.

Figure 6. ${\epsilon }_r$ of the primary model (a) Experimental n and k values are represented by dots, taken from Ref. [19]. The lines depict bare optical constants without corrections. (b) Dots indicate measured wavelengths corresponding to the minimum reflectance. A comparison between the measured and calculated results suggests that ${\epsilon }_r=4.5$ is a reasonable value.

Figure 6. ${\epsilon }_r$ of the primary model (a) Experimental n and k values are represented by dots, taken from Ref. [19]. The lines depict bare optical constants without corrections. (b) Dots indicate measured wavelengths corresponding to the minimum reflectance. A comparison between the measured and calculated results suggests that ${\epsilon }_r=4.5$ is a reasonable value.

4.3. Model of Light Emission

The corrections to the reflectance are the main subjects of this paper [9,10]. Specifically, we consider corrections where some fraction of the energy absorbed by the jth layer (of N-layer graphene) is transferred to light emitted from that layer [see the right side of Figure 3(a)]. The amplitude of the emitted light is assumed to be the square root of the layer absorption $A_j^N\equiv \pi \alpha {\left|E_j^N\right|}^2$,34] multiplied by the branching ratio, $\mathcal{B}$, i.e., $\sqrt{\left(\mathcal{B}/2\right)A_j^N}$, where $1/\sqrt{2}$ means that the light emission is direction-independent along the c-axis. Note that $A_j^N$ depends not only on j and N but also on $\lambda$ and $d_{{\mathrm{SiO}}_2}$.

To examine how light emitted from the jth layer affects the reflectance, we define two subsystems, as shown in Figure 7b: one is an isolated $(j-1)$-layer graphene in the air; the other is $(N-j)$-layer graphene on SiO₂/Si substrate. Using the transfer matrix method, we can obtain the transmission and reflection coefficients of an isolated $(j-1)$-layer graphene in the air [denoted as $t_{j-1}^g$ and $r_{j-1}^g$] and the reflection coefficient of $(N-j)$-layer graphene on the SiO₂/Si substrate [denoted as $r_{N-j}$] [21,34]. Let the reflection coefficients be $X_j$ and $Y_j$ and transmission coefficient be $Z_j$ for the combined subsystems [see Figure 7b]. These can be obtained by a self-consistent manner as follows. After calculating $X_j^{\left(n\right)}$, we add it to $\sqrt{\left(\mathcal{B}/2\right)A_j^N}$ of the incident light to the $(j-1)$-layer graphene (in the air) as $\sqrt{\left(\mathcal{B}/2\right)A_j^N}+X_j^{\left(n\right)}$ and recalculate $Y_j^{(n+1)}=r_{j-1}^g\left(\sqrt{\left(\mathcal{B}/2\right)A_j^N}+X_j^{\left(n\right)}\right)$ and $Z_j^{(n+1)}=t_{j-1}^g\left(\sqrt{\left(\mathcal{B}/2\right)A_j^N}+X_j^{\left(n\right)}\right)$. Then, we add a new $Y_j^{(n+1)}$ to $\sqrt{\left(\mathcal{B}/2\right)A_j^N}$ of the light incident to the $(N-j)$-layer graphene on the SiO₂/Si substrate as $\sqrt{\left(\mathcal{B}/2\right)A_j^N}+Y_j^{(n+1)}$ and recalculate $X_j^{(n+1)}=r_{N-j}\left(\sqrt{\left(\mathcal{B}/2\right)A_j^N}+Y_j^{(n+1)}\right)$. These computations are repeated until $X_j$ and $Y_j$ converge. In this way, we can obtain analytical expressions for the converged $X_j$, $Y_j$, and $Z_j$ for a given $\mathcal{B}$:

$$\begin{array}{cc}\hfill & X_j\left(\mathcal{B}\right)=r_{N-j}\left\{\frac{1+r_{j-1}^g}{1-r_{N-j}{r}_{j-1}^g}\right\}\sqrt{\left(\frac{\mathcal{B}}{2}\right)A_j^N},\hfill \\ \hfill & Y_j\left(\mathcal{B}\right)=r_{j-1}^g\left\{\frac{1+r_{N-j}}{1-r_{N-j}{r}_{j-1}^g}\right\}\sqrt{\left(\frac{\mathcal{B}}{2}\right)A_j^N},\hfill \\ \hfill & Z_j\left(\mathcal{B}\right)=t_{j-1}^g\left\{\frac{1+r_{N-j}}{1-r_{N-j}{r}_{j-1}^g}\right\}\sqrt{\left(\frac{\mathcal{B}}{2}\right)A_j^N}.\hfill \end{array}$$

(3)

The “corrected” electric fields at an infinitesimal distance above and below the jth layer become $\sqrt{\left(\mathcal{B}/2\right)A_j^N}+X_j^{\left(n\right)}+Y_j^{(n+1)}$ and $\sqrt{\left(\mathcal{B}/2\right)A_j^N}+Y_j^{\left(n\right)}+X_j^{(n+1)}$, respectively. Self-consistency, whereby ${lim}_{n\to \infty }{X}_j^{\left(n\right)}=X_j$ and ${lim}_{n\to \infty }{Y}_j^{\left(n\right)}=Y_j$, is therefore essential to ensuring that the corrected electric field is continuous at the jth layer, which is a requirement of Maxwell equations. The corrected amplitude of the emitted light is written as

$$\begin{array}{c}\hfill \sqrt{\left(\frac{\mathcal{B}}{2}\right)A_j^N}+X_j+Y_j=(1+r_{j-1}^g)\left\{\frac{1+r_{N-j}}{1-r_{N-j}{r}_{j-1}^g}\right\}\sqrt{\left(\frac{\mathcal{B}}{2}\right)A_j^N}.\end{array}$$

(4)

By comparing this with $Z_j\left(\mathcal{B}\right)$, we see that more accurate value of the amplitude of the emitted light is given by multiplying $\{\cdots \}\sqrt{\left(\mathcal{B}/2\right)A_j^N}$ with $1+r_{j-1}^g$ as the renormalization constant, and $(1+r_{j-1}^g)\{\cdots \}\sqrt{\left(\mathcal{B}/2\right)A_j^N}$ is what $b_N\left(j\right)$ in Figure 3(a) represents. Therefore, we redefine $Z_j$ as

$$\begin{array}{cc}\hfill & Z_j\left(\mathcal{B}\right)\equiv t_{j-1}^g{b}_N\left(j\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & b_N\left(j\right)\equiv (1+r_{j-1}^g)\left\{\frac{1+r_{N-j}}{1-r_{N-j}{r}_{j-1}^g}\right\}\sqrt{\left(\frac{\mathcal{B}}{2}\right)A_j^N}.\hfill \end{array}$$

(6)

We can interpret $Z_j\left(\mathcal{B}\right)$ as follows. The transmission coefficient $t_{j-1}^g$ is the direct propagation of the renormalized light emitted from jth layer to the air, and $|t_{j-1}^g{|}^2$ monotonously decreases with increasing j [21]. The effects of scattering and absorption of the emitted light caused by surrounding layers is included by the part in the brace $\{\cdots \}$. It tends to suppress the magnitude of $Z_j$, but sometimes enhance. For example, when $N=1$ (i.e., monolayer on a substrate), the part becomes $1+r_0$ which is larger than unity when $r_0$ is positive.

$Z_j\left(\mathcal{B}\right)$ is the value at zero initial phase, so the transmission coefficient can be given a phase degree of freedom expressing the coherence or incoherence of the light emission from the different layers:

$$\begin{array}{c}\hfill z_N\equiv \sum _{j=1}^N{e}^{i{\theta }_j}{Z}_j\left(\mathcal{B}\right).\end{array}$$

(7)

Accordingly, the corrected reflectance is uniquely determined by $R_N\equiv {\left|r_N+z_N\right|}^2={\left|r_N\right|}^2+2\mathrm{Re}\left[r_N{z}_N^{*}\right]+{\left|z_N\right|}^2$. The value of $z_N$ depends on these phases ${\theta }_j$ [43].

We consider a case in which the phase is given by a coherent phase. The exact derivation of the phase will be shown elsewhere because it is beyond the scope of the present paper. Here, we concisely explain the basic logic leading to the coherent phase in terms of quantum electrodynamics. First, we can define a quantum mechanical state of light ($|{\Psi }_a\rangle$) that the primary model describes (see left side of Figure 5). All the information of light is expressed by the coefficients $r_N\left(j\right)$ and $t_N\left(j\right)$ ($j=1,\dots ,N$). Second, we can also define another quantum state of light ($|{\Psi }_b\rangle$) for the emitted light (see right side of Figure 5). All the information of emitted light is expressed by the coefficients $b_N\left(j\right)$. These two states have an overlap $b_N^{*}\left(j\right)t_N\left(j\right)+b_N^{*}\left(j\right)r_N\left(j\right)$ caused by jth layer graphene. Thus, if we consider a linear superposition of these states as $|{\Psi }_a\rangle +e^{i\theta }|{\Psi }_b\rangle$ to form energy eigenstates, the phase $e^{i{\theta }_j}$ must be chosen so that $e^{-i{\theta }_j}{b}_N^{*}\left(j\right)(t_N\left(j\right)+r_N\left(j\right))$ becomes a real number, namely

$$\begin{array}{c}\hfill e^{i{\theta }_j}=\pm \frac{t_N\left(j\right)+r_N\left(j\right)}{|t_N\left(j\right)+r_N\left(j\right)|}\frac{b_N{\left(j\right)}^{*}}{|b_N\left(j\right)|}.\end{array}$$

(8)

The factor ± is a global phase ($\theta$) in the sense that it is independent of the value of j. Because the scattered light ($r_N$) and the emitted light ($z_N$) form a two-level state, there are two possible linear superpositions of their energy eigenstates, $-1$ ($\theta =\pi$) or $+1$ ($\theta =0$). The minus sign ($e^{i\pi }$) is assigned to the lower energy state. From Equations (5) and (8), we obtain $e^{i{\theta }_j}{Z}_j\left(\mathcal{B}\right)=\pm \frac{t_N\left(j\right)+r_N\left(j\right)}{|t_N\left(j\right)+r_N\left(j\right)|}{t}_{j-1}^g\left|b_N\left(j\right)\right|$.

Including the correction due to coherent light emission leads to

$$\begin{array}{c}\hfill R_N={\left|r_N+\sum _{j=1}^N{e}^{i{\theta }_j}{Z}_j\left({\mathcal{B}}_{coh}\right)\right|}^2,\end{array}$$

(9)

where ${\mathcal{B}}_{coh}$ is the branching ratio of the energy of the emitted coherent photons to that of the absorbed photons. Since coherent photon emission is related to the electron-photon coupling strength of the annihilated photo-excited electron-hole pairs, ${\mathcal{B}}_{coh}$ should be on the order of ${\left(\pi \alpha \right)}^2$ and insensitive to changes in N.

Next, we apply $R_N={\left|r_N\right|}^2+2\mathrm{Re}\left[r_N{z}_N^{*}\right]+{\left|z_N\right|}^2$ to the case that ${\theta }_j$ in $z_N$ is a random variable. Here, the definition of randomness is that if we take the time average regarding ${\theta }_j$, we have $\langle \mathrm{Re}\left[r_N{z}_N^{*}\right]\rangle =0$ and $\langle {\left|z_N\right|}^2\rangle ={\sum }_{j=1}^N{\left|Z_j\left(\mathcal{B}\right)\right|}^2$. We will refer to this case as incoherent corrections, which also include the cases that the global phase takes 0 and $\pi$ if there is a perturbation that can mix the two energy levels. An interference term is now included in ${\left|z_N\right|}^2$ as the last term of

$$\begin{array}{c}\hfill {\left|z_N\right|}^2=\sum _{i=1}^N{\left|Z_i\left(\mathcal{B}\right)\right|}^2+\sum _{i\ne j}{e}^{i({\theta }_i-{\theta }_j)}{Z}_i\left(\mathcal{B}\right)Z_j^{*}\left(\mathcal{B}\right),\end{array}$$

(10)

but it vanishes when taking the time average and only the first term of the incoherent corrections remains. [44] Inelastic scattering of light such as Raman scattering is usually considered to give rise to incoherent photons. Let ${\mathcal{B}}_{inc}$ be the branching ratio of the energy of the emitted incoherent photons to that of absorbed photons. Since $Z_j\left({\mathcal{B}}_{inc}\right)$ is proportional to $\sqrt{{\mathcal{B}}_{inc}{A}_j^N}$ [Equation (3)], the incoherent corrections are proportional to ${\mathcal{B}}_{inc}{A}_j^N$. For Raman scattering, the parameter ${\mathcal{B}}_{inc}$ is fundamentally determined by the electron-photon and electron-phonon coupling strengths, and it should not be so sensitive to the change in N. Indeed, the incoherent corrections with a constant ${\mathcal{B}}_{inc}$ follow the measured N dependence of the G band Raman intensity [Section 5.3]. The G band consists of optical phonons at the $\Gamma$ point, whose lattice vibrations are in-plane.

A generalized reflection formula covering the above two cases (coherent and incoherent corrections) can be written as

$$\begin{array}{c}\hfill R_N(\lambda ,\theta ,{\mathcal{B}}_{coh},{\mathcal{B}}_{inc})\equiv {\left|r_N+e^{i\theta }\sum _{j=1}^N\frac{t_N\left(j\right)+r_N\left(j\right)}{|t_N\left(j\right)+r_N\left(j\right)|}\frac{b_N{\left(j\right)}^{*}}{|b_N\left(j\right)|}{Z}_j\left({\mathcal{B}}_{coh}\right)\right|}^2+\sum _{j=1}^N{\left|Z_j\left({\mathcal{B}}_{inc}\right)\right|}^2.\end{array}$$

(11)

When ${\mathcal{B}}_{coh}={\mathcal{B}}_{inc}=0$, $R_N(\lambda ,\theta ,{\mathcal{B}}_{coh},{\mathcal{B}}_{inc})$ reduces to Equation (1) with Equation (2). Figure 7 shows the N dependence of $R_N(\lambda ,\theta ,0,0)$ (dashed), $R_N(\lambda ,\pi ,{\mathcal{B}}_{coh}=0.0007,0)$ (red), $R_N(\lambda ,0,{\mathcal{B}}_{coh}=0.0007,0)$ (blue), and ${\sum }_{j=1}^N{\left|Z_j({\mathcal{B}}_{inc}=0.1)\right|}^2$ (black), for a fixed $\lambda =540$ nm. Note that the incoherent corrections always increase the reflectance and preclude zero reflections at $N\sim 15$, which is in contrast to the coherent corrections. Moreover, the G band Raman intensity is enhanced when zero reflection occurs [10,45,46,47]. This situation is called interference-enhanced Raman scattering, [9] and it is reasonably reproduced by Equation (11). The experimental fact that zero reflection is observed at $N\sim 15$ [see 14 and 15-layer in Figure 1] shows that ${\mathcal{B}}_{coh}\sim 0.0007$, ${\mathcal{B}}_{inc}$ is much smaller than 0.1, and $\theta =\pi$ (red curve in Figure 7).

Figure 7. Reflectance of N-layer graphene on SiO₂/Si substrate. Dependence of the corrections to $R_N\left(540\mathrm{nm}\right)$ [Equation (11)] on N, where $d_{{\mathrm{SiO}}_2}=268$ nm, ${\mathcal{B}}_{inc}=0.1$, ${\mathcal{B}}_{coh}=0.0007$ and the global phase is − or +. The incoherent components are enhanced (which is called interference-enhanced Raman scattering) at around 20 layers.

Figure 7. Reflectance of N-layer graphene on SiO₂/Si substrate. Dependence of the corrections to $R_N\left(540\mathrm{nm}\right)$ [Equation (11)] on N, where $d_{{\mathrm{SiO}}_2}=268$ nm, ${\mathcal{B}}_{inc}=0.1$, ${\mathcal{B}}_{coh}=0.0007$ and the global phase is − or +. The incoherent components are enhanced (which is called interference-enhanced Raman scattering) at around 20 layers.

5. Applications of Model

6. Discussion

7. Conclusions

References

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