Mode energy of graphene plasmons and its role in determining the local field magnitudes

Linlong Tang; Jinpeng Nong; Wei Wei; Song Zhang; Yuhang Zhu; Zhengguo Shang; Juemin Yi; Wei Wang

doi:10.1364/OE.26.006214

1. Introduction

The intense local fields of surface plasmons can significantly enhance light-matter interactions [1, 2]. Such properties are highly desired in various applications, and hence a great amount of work were devoted to the improvement of the local field magnitudes. For example, surface plasmons of noble metal nanoparticles lie in the heart of surface enhanced Raman spectroscopies (SERS), and a variety of metal nanoparticles with different geometries, morphologies, compositions and inter-particle distances have been produced in order to improve the local field magnitudes [3–5]. These methods, in fact, are the tuning of the field confinement of surface plasmonic modes, because it is widely believed that tight field confinement could enable large local field magnitudes.

In recent years, graphene is demonstrated to be a new material of supporting surface plasmons in the infrared and terahertz region [6–12]. Compared to plasmons of noble metals, graphene plasmons are much more tightly confined, with mode volumes of the orders of 10⁶ times smaller than the excitation wavelengths [6]. This feature is expected to induce large magnitudes of the local fields, which are crucial for applications such as surface enhanced infrared absorption spectroscopies (SEIRAS) [13–16], infrared plasmonic sources [17, 18], and nonlinear optics [19, 20]. However, such expectation is not consistent with current experiment results. For example, the enhancement factors of graphene plasmons based SEIRAS, which are proportional to the square of the local electric fields, are observed to be very low, on the order of only 10 [13, 15]. Therefore, tight field confinement alone is no longer sufficient to enable large local field magnitudes for graphene plasmonic modes. Lately, it is suggested that the quality factor of the plasmonic mode is also an important factor [14, 21], but the quantitative relation between quality factors and local field magnitudes is still unclear.

In this paper, we study the mode energy of graphene plasmons, and demonstrate its fundamental role in determining the local field magnitude of a graphene plasmonic mode. Firstly, it is shown that due to the negligible magnetic field energy of graphene plasmonic modes, a quantitative expression for the mode energy exists, valid for both propagating and localized graphene modes at infrared region. Then, based on the mode energy expression, it is demonstrated that the square of the electric field of a graphene plasmonic mode is proportional to the light absorption rate of the mode and the electron relaxation time of graphene. Finally, it is shown that there are still plenty of rooms for the improvement of the local field magnitudes of graphene plasmonic modes, and the strategies are discussed.

2. Results and discussions

Normally, the total energy stored in a plasmonic mode W_total can be divided into three parts, i.e., the electric field energy W_E, the magnetic field energy W_H and the electron kinetic energy W_K, and the sum of W_H and W_K should be equal to W_E [22]. Hence we have the following relations,

W_{total} = W_{E} + W_{H} + W_{K}, W_{E} = W_{H} + W_{K} .

For a graphene plasmonic mode, its electric field energy and magnetic field energy are stored in the space out of graphene spanned by the fields of the mode, and its electron kinetic energy is stored in the graphene layer. If graphene is surrounded by lossless materials, the electric and the magnetic fields would not loss energy, but the electron kinetic energy should be dissipated constantly due to the collisions of electrons in graphene. Hence, we next calculate the loss power of the electron kinetic energy. Graphene plasmons are usually excited in the middle and far infrared region, in which the conductivity of graphene could be well described by Drude model in the form of [9, 23]

σ (ω) = i e^{2} E_{f} / π ℏ^{2} (ω + i τ^{- 1})

Here, ω is the angular frequency of the incident light, e is the elementary charge, ћ is the reduced Plank constant, E_f is the Fermi energy, and τ is the electron relaxation time. By using Eq. (2) and Ohm’s law J = σ E, the expression for Ohmic loss power of electron kinetic energy, i.e., P_loss = Re {∫_Ω J*•E dr / 2 } [24], can be reformatted into the following concise form:

P_{loss} = 2 W_{K} / τ,

where Ω represents the graphene layer, J is the current due to electron motions, and E denotes the parallel electric field in the graphene layer.

With the assistance of above equations, we can now derive the expression for W_total. As illustrated in Fig. 1, if a graphene plasmonic mode reaches the steady state, its loss power P_loss should be equal to its absorbed power P_abs because the mode energy is invariant. On the other hand, the absorbed power connects with the absorption rate of the mode through P_abs = P_inA with P_in being the incident power of light and A the absorption rate. Therefore, by equalizing P_abs with P_loss, and employing Eq. (1) and (3), we obtain

W_{total} = (1 + W_{H} / W_{K}) P_{in} A τ .

Such equation seems simple, but all the complexity is incorporated in W_H / W_K, a parameter that depends on the details of the mode field distributions, and could only be obtained by numerical calculations in most cases.

Fig. 1 The power balance process of a graphene plasmonic mode. The power of the incident light (P_in) is either absorbed by the plasmonic mode with power of P_abs or reflected / transmitted with power of P_r/t, and P_abs should be equal to the loss power P_loss at the steady state.

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Fortunately, Eq. (4) can be simplified for graphene plasmonic modes. To be specific, the behavior of a graphene plasmonic mode can be well described within electrostatic limit at infrared region because of its ultra-small mode volume, which means its magnetic field energy is much smaller than the electric field energy and electron kinetic energy (i.e., W_H << W_E, W_K). Therefore, W_H / W_K << 1 and we could rewritten Eq. (4) as follows,

W_{total} \approx P_{in} A τ .

The equation shows that in the electrostatic limit, the mode energy no longer depends on W_H / W_K, and it can be directly calculated out by the parameters P_in, A, and τ, all of which are measurable in experiments. More importantly, the derivation of Eq. (5) does not invoke the specific form of the plasmonic mode, hence Eq. (5) should be applicable to all kinds of tightly confined graphene plasmonic modes in lossless dielectric environments, independent on the details of the mode field distributions. It means that no matter the modes are propagating or localized graphene plasmonic modes, their mode energy could be calculated quantitatively by Eq. (5) provided they are in the electrostatic limit.

To demonstrate the validity of Eq. (5), we compare it with numerical results of both propagating and localized graphene plasmonic modes [25–30]. The excitation configurations are schematically shown in Fig. 2, where propagating plasmons of continuous graphene are excited by nano-gratings [Fig. 2(a)], and localized plasmons are excited by pattering graphene into nano-ribbons [Fig. 2(b)]. As a transverse magnetic wave vertically incident on the configurations, graphene plasmons could be excited. The excitation processes are simulated by finite element method (FEM) using Comsol Multiphysics. The parameters in the simulations are set as follows: The period, height, fill factor and refractive index of the grating are 200 nm, 100 nm, 0.5 and 2.4, respectively. The period and width of the nanoribbons are 200 nm and 100 nm. The dielectric layers are semi-infinite thick with refractive index of 1.5. Graphene is modeled as a 0.34 nm layer whose surface conductivity is given by Eq. (2) with E_f = 0.4 eV and τ = 0.2 ps. In the FEM model, periodic boundary conditions is imposed on the left and right sides, and perfect matched layer is applied in the top and bottom domains. Non-uniform meshes are used in the simulation regions, where the maximum element size in the graphene layer is set as 0.1 nm, and the mesh size gradually increases outside graphene. Two dimensional models are employed since the excitation configurations are transitional invariant along the z direction. The results for propagating and localized graphene plasmons are shown in Fig. 2(c) - 2(f). It can be seen that the propagating mode is excited at 10.07 μm [Fig. 2(c)], and the localized mode is excited at 10.25 μm [Fig. 2(e)]. Their mode fields are highly confined, with maximum normalized electric field of 19 for the propagating mode [Fig. 2(d)] and 327 for the localized mode [Fig. 2(f)].

Fig. 2 Schematics of the configurations for the excitations of propagating graphene plasmons (a) and localized graphene plasmons (b). The absorption curves of the propagating plasmonic mode (c) and localized plasmonic mode (e), and the electric field patterns at the resonant centers (d) (f). In (d) and (f), (E) and (E)_in represent the local and incident electric fields, respectively.

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We varied the relaxation time of graphene from 0.05 ps to 0.5 ps to numerically calculate the mode energy and the absorption rates, and the results for propagating and localized modes are plotted in Figs. 3(a) and 3(b), respectively, which show the mode energy and absorption rates both increase with the rise of electron relaxation time. For comparison, the mode energy calculated analytically by Eq. (5) with the absorption rates as inputs are also plotted in the two figures. Clearly, the numerical and analytical mode energy are in good agreement, indicating Eq. (5) is valid for both propagating and localized graphene plasmonic modes. In addition, we numerically calculated the magnetic field energy W_H and the electric field energy W_E of the two types of modes as shown in the two figures, and find that the former is 3 orders smaller than the latter, confirming that the electrostatic limit assumption in derivation of Eq. (5) is reasonable.

Fig. 3 Total mode energy, absorption rates, electric and magnetic field energy versus electron relaxation time of graphene at resonant wavelengths 10 μm for propagating modes (a) and localized modes (b), and versus resonant wavelengths at electron relaxation time of 0.2 ps for propagating modes (c) and localized modes (d). The legend in (a) is also applicable to (b)-(d). The numerical energy are integrated in a period, and the incident power in a period is 1 W.

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At longer wavelengths, the confinements of graphene plasmonic modes become weak, and as a consequence, Eq. (5) may no longer be a good approximation because the plasmonic modes are not in the electrostatic limit. To determine the applicable range of Eq. (5), we numerically calculated the mode energy and the absorption rates of the graphene plasmonic modes at resonant wavelengths from 7 μm to 170 μm by scaling up the grating and graphene ribbon periods of the excitation configurations. The results for propagating and localized graphene plasmonic modes are illustrated in Fig. 3(c) and 3(d), respectively, and the analytical mode energy calculated by Eq. (5) are also plotted in the figures for comparison. We can see that for both type modes, the numerical and analytical mode energy are in good agreement at short wavelengths (7-100 μm), but their difference becomes large at long wavelengths (100-170 μm). The numerical electric and magnetic field energy are also presented in the two figures, and they exhibit consistent trends: the magnetic field energy are 2-3 orders smaller than the electric field energy at short wavelengths, but they are in the same order at long wavelengths, which means the electrostatic limit is no longer appropriate and hence Eq. (5) fails. Therefore, we can conclude that Eq. (5) is applicable in the middle and far infrared region (roughly < 100 μm), but not an excellent approximation at terahertz region. And at terahertz region, Eq. (4) should be employed and then the ratio W_H / W_K need to be calculated numerically in order to obtain the mode energy.

We now investigate how the electric field magnitude of a plasmonic mode relates to its mode energy W_total. The electric field E(r) of a mode can be written as E(r) = a f(r), where a is the mode amplitude and f(r) is the mode pattern function, normalized as ∫_V |f(r)|²dr = 1 with V being the volume spanned by the field of the plasmonic mode. Then, by employing Eq. (1), we can express the mode energy W_total in terms of E(r) as follows,

\begin{array}{l} W_{t o t a l} = 2 W_{E} = 2 \sum_{i = 1}^{N} {\int_{V i} ε_{0} ε_{i} {| Ε (r) |}^{2} d r / 4} \\ = {| a |}^{2} \sum_{i = 1}^{N} ε_{0} ε_{i} β_{i} / 2 \end{array}

Here, ε₀ is the permittivity of vacuum, N is the number of domains of different materials, i denotes the i^th domain with relative permittivity ε_i and volume V_i (e.g., in the configurations shown in Fig. 2(a), N = 3, and 1, 2, 3 denotes the air, grating, dielectric domains, respectively), β_i = ∫_Vi |f(r)|²dr / ∫_V |f(r)|²dr is a constant whose value depends on the mode pattern function f(r), and we call such kind of constants as mode dependent constants.

The mean electric field E_mean and maximum electric field E_max defined as E_mean ≡ ∫_V E(r)dr / V and E_max ≡ max{E(r)}, respectively, are introduced to characterize the electric field magnitude of a graphene plasmonic mode. Based on such definitions, |E|²_mean and |E|²_max of the mode can be cast into the forms as follows,

{| E |}^{2}_{mean} = \int_{V} {| E (r) |}^{2} d r / V = | a |^{2} / V,

{| E |}_{\max}^{2} = \max {{| E (r) |}^{2}} = {| a |}^{2} f_{\max}^{2} = f_{\max}^{2} {| E |}_{mean}^{2} V

In Eq. (8), f_max = max{|f(r)|} is a mode dependent constant. Next, by inserting Eq. (6) into Eq. (7) and (8), respectively, we obtain

{| E |}^{2}_{mean} = C_{1} W_{total} / V, {| E |}^{2}_{max} = C_{2} W_{total},

with C₁ =

2 / \sum_{i = 1}^{N} ε_{0} ε_{i} β_{i}

and C₂ =

2 f_{m a x}^{2} / \sum_{i = 1}^{N} ε_{0} ε_{i} β_{i}

, which are mode depend constants. Finally, by replacing the W_total in Eq. (9) with Eq. (4), we arrive at

{| E |}_{mean}^{2} = C_{3} P_{i n} A τ / V, {| E |}_{\max}^{2} = C_{4} P_{i n} A τ

where C₃ = (1 + W_H / W_K) C₁, C₄ = (1 + W_H / W_K) C₂, and they are also mode dependent constants. Equation (10) is an important result, which shows that |E|²_mean and |E|²_max scale linearly with Aτ, and the scaling factors (i.e., C₃P_in / V for |E|²_mean, C₄P_in for |E|²_max) depend on the details of electric field patterns, so they are generally not identical for different modes. We also compared Eq. (10) with numerical results to demonstrate its validity. As shown in Fig. 4(a) and 4(b), the absorption rates A, |E|²_mean and |E|²_max for propagating and localized graphene plasmonic modes are calculated by varying the electron relaxation time τ of graphene from 0.05 ps to 0.5 ps. Clearly, |E|²_mean /A and |E|²_max /A scale linearly with electron relaxation time τ for both type modes, indicating Eq. (10) is valid.

Fig. 4 The |E|²_mean / A and |E|²_max/ A versus graphene electron relaxation time τ for propagating graphene plasmonic modes (a) and localized graphene plasmonic modes (b). In the numerical calculations, other parameters are kept the same as in calculating Figs. 2(c) and 2(e), except the electron relaxation time varies now.

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Equation (10) signifies that to improve |E|²_mean and |E|²_max of graphene plasmonic modes, we should improve both A and τ to increase the mode energy. The maximum value of A is 1 (i.e. perfect absorption) and τ directly relates to the mobility of graphene μ through τ = μE_f / ev_f² with v_f being the Fermi velocity of graphene [23], which means the ideal case is to use a high mobility graphene to excite a plasmonic mode with perfect absorption. Such ideal case is in fact possible. Because the absorption rate of a graphene plasmonic mode is determined by [30]

A = 4 γ_{0} γ_{1} / [{(ω - ω_{0})}^{2} + {(γ_{0} + γ_{1})}^{2}],

where γ₀ = 1 / (2τ) is the intrinsic loss rate of the plasmonic mode, caused by the loss of electron kinetic energy, γ₁ is the coupling rate between the mode and the incident light, ω and ω₀ are the angular frequency of the incident light and the resonant center of the plasmonic mode, respectively. Equation (11) shows that perfect absorption is achieved at the resonant center (ω = ω₀) when γ₀ = γ₁, this is the so called critical coupling condition. For a high mobility graphene, the γ₀ is small because γ₀ = 1/(2τ) ∝1/μ, hence to maintain perfect absorption, γ₁ should also be small to guarantee γ₀ = γ₁. Such requirement can be met since γ₁ can be arbitrary small by modulating the incident light intensity at the position of graphene using a Fabry-Pérot cavity [30].

Currently, the experimental measured absorption rates of graphene plasmons are typically around 0.05 [13, 15], far from perfect absorption. Also, the mobility of graphene is low, around 1000 cm² / (vs), due to the pattering of graphene into nano structures. If the absorption rates and the mobility can be improved 10 times, respectively, then |E|²_mean and |E|²_max can be improved 100 times. Therefore, there are still plenty of rooms for the improvement of the local field magnitudes.

The quality factor (Q) of a graphene plasmonic mode is given by Q = ω₀ /(1/τ + 2γ₁). Hence, a larger Q usually means a longer τ, but not necessarily mean a higher absorption rate A, indicating the quality factor could partially influence the local field magnitudes.

3. Conclusion

In summary, we studied the mode energy of graphene plasmonic modes, and its role in determining the local field magnitudes. A quantitative expression is derived for the mode energy, which is valid for propagating and localized graphene plasmonic modes at infrared region (roughly <100 μm). It is further demonstrated that the mode energy expression results in the linear dependence of |E|²_mean and |E|²_max on the absorption rate A of the mode and electron relaxation time τ of graphene. Our study shows that |E|²_mean and |E|²_max can still be improved significantly, and further experiments may focus on the improvement of the absorption rate of the plasmons and the mobility of graphene.

Funding

National Natural Science Foundation of China (61675139, 61675037, 11374359, 11574308, 61405021); Chongqing Research Program of Basic Research and Frontier Technology (cstc2017jcyjAX0038, cstc2017jcyjBX0048); the CAS Western Light Program 2016; National High Technology Research and Development Program of China (2015AA034801); and Visiting Scholar Foundation of Key Laboratory of Optoelectronic Technology & Systems (Chongqing University), Ministry of Education.

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Mode energy of graphene plasmons and its role in determining the local field magnitudes

Abstract

1. Introduction

2. Results and discussions

3. Conclusion

Funding

References and links

Cited By

Figures (4)

Equations (11)

Optics Express