Polarization dependent plasmonic modes in elliptical graphene disk arrays

Yuyu Xia; Yunyun Dai; Bo Wang; Ang Chen; Yanbin Zhang; Yiwen Zhang; Fang Guan; Xiaohan Liu; Lei Shi; Jian Zi

doi:10.1364/OE.27.001080

1. Introduction

Graphene nanostructures can support localized plasmonic resonances in the frequency range from terahertz to infrared [1–11]. The behavior of the localized plasmonic resonances depends on the size of the structure [6,12–15], the doping level of graphene [4,8,9,15,16] and the permitivity of the substrate [17,18]. As an example, the plasmon frequency in graphene nanostructures is proportional to the square root of Fermi energy [3,4,6], while inverse proportional to the square root of the corresponding length of graphene nanostructures [4,12,19]; the decay paths of graphene plasmons can be strongly modified by the optical phonons of the substrate [18–20],resulting in the modification of the plasmon lifetime [3]. All these unique properties make it possible in situ control the plasmons in graphene, leading to many applications [1, 2, 5, 21–23], for instance bio-sensor [24–26], THz detector [27, 28], infrared monochromatic light source [29] and modulator [30], with extensive tunabilities.

Up to now, most of the graphene plasmonic nanostructures are based on ribbon [7,15,18,19,31–35] and circular geometries [12,13,16]. As a result, the plasmonic resonant modes of graphene structures are either polarization independent or only available along one single polarization direction. On the other hand, elliptical geometry is inherently anisotropic [36–38]. The optical properties of a graphene ellipse could be expected to be highly polarization dependent. In this paper, theoretical and experimental investigation on the plasmonic resonances of a graphene ellipse array on a substrate are presented. We found that graphene elliptical nanostructures support two plasmonic electrical dipole modes which are orthogonal to each other with dipole moments along major and minor axis of the ellipse respectively, showing polarization-dependent properties. Furthermore, it was experimentally demonstrated that the optical properties of graphene elliptical plasmonic structures can be controlled through polarization of excitation light, structure size, Fermi energy of graphene and permittivity of the substrate material etc. Compared to graphene ribbons and disks, the results obtained provide more degrees of freedom for manipulation of light-matter interactions in the mid-infrared frequency range.

2. Theoretical model

The optical conductivity and its equivalent relative dielectric constant ε_g of graphene are related with the environmental temperature, Fermi level, relaxation time and thickness of the graphene [39,40]. In this work, temperature and thickness were considered as constant 300 K and 1 nm [41,42] respectively. Compared with 0.33 nm or 0.24 nm thickness of graphene used by other researches [43,44], our results are almost similar with the case of 1 nm. The Fermi level E_F, depending on the doping level of graphene, varied from 0.3 to 0.22 eV. The relaxation time τ, depending on the quality of graphene, was taken as 14 fs [6].

To theoretically characterize the polarization dependent plasmonic modes in elliptical graphene disk array, each graphene ellipse was taken as a dipole with polarizability α̃. The peak frequency of the imaginary part of α̃ corresponds to the resonant frequency of the system. And the value of that represents the extinction, which can be directly measured in experiment. In order to analytically obtain the polarizability of graphene ellipse array, each graphene ellipse is assumed as a flat ellipsoid with the length of semi-major axis a and semi-minor axis b far larger than its thickness. It is assumed the volume of such an ellipsoid V is equal to that of the ellipse, i.e. V = 4/3πabc = πabd_g, where c is the length of third semi-axis of the ellipsoid, and d_g is the thickness of the graphene ellipse. Therefore, c = 3/4d_g. Considering the non-local effect of graphene plasmons, c should be modified with an extra term Δc, c = 3/4d_g + Δc, (Δc << d_g).

To calculate the polarizability of graphene ellipsoid array, the local field in the center of each ellipsoid should be obtained. Generally, the local field is the sum of the external incident field E_inc, the field produced by the surrounding ellipsoids E_sur, and the field produced by all the image ellipsoids E_img when considering a substrate, E_loc = E_inc + E_sur + E_img. In mid-infrared frequency range, the local field on each graphene ellipsoid is assumed as a homogeneous quasi-static field.

For a single graphene ellipsoid, the local field is the external incident field, E_loc = E_inc. If the homogeneous quasi-static local field is along axis a(x axis) $E_{loc}^{x}$ or axis b(y axis) $E_{loc}^{y}$ , the induced electric field outside the ellipsoid and parallel to the local field, $E_{ind}^{x}$ or $E_{ind}^{y}$ , can be written as [45]

E_{ind}^{x} (x, y, z) = - E_{loc}^{x} \frac{α_{1}}{V} ({\tilde{L}}_{1} + x \frac{\partial {\tilde{L}}_{1}}{\partial ξ} \frac{\partial ξ}{\partial x})

E_{ind}^{y} (x, y, z) = - E_{loc}^{y} \frac{α_{2}}{V} ({\tilde{L}}_{2} + y \frac{\partial {\tilde{L}}_{2}}{\partial ξ} \frac{\partial ξ}{\partial y})

Where α₁ and α₂ are the polarizability of the ellipsoid along axis a and axis b, respectively; V is the ellipsoid volume;

{\tilde{L}}_{1} = \frac{a b c}{2} \int_{ξ}^{\infty} \frac{d q}{(a^{2} + q) \sqrt{(a^{2} + q) (b^{2} + q) (c^{2} + q)}}

and

{\tilde{L}}_{2} = \frac{a b c}{2} \int_{ξ}^{\infty} \frac{d q}{(b^{2} + q) \sqrt{(a^{2} + q) (b^{2} + q) (c^{2} + q)}}

are the modified geometrical factors along axis a and b; while ξ is one of the ellipsoidal coordinates which is determined by

\frac{x^{2}}{(a^{2} + ξ)} + \frac{y^{2}}{(b^{2} + ξ)} + \frac{z^{2}}{(c^{2} + ξ)} = 1

. [45] For simplicity, let

A_{1} = {\tilde{L}}_{1} + x \frac{\partial {\tilde{L}}_{1}}{ξ} \frac{\partial ξ}{\partial x}

,

A_{2} = {\tilde{L}}_{2} + y \frac{\partial {\tilde{L}}_{2}}{ξ} \frac{\partial ξ}{\partial y}

.

For an ellipsoid placed on an isotropic substrate with dielectric constant ε_s, as shown in Fig. 1, the influence of the substrate can be replaced by the image ellipsoid. And the local field E_loc of the ellipsoid at (0,0,c) is the sum of the incident field and the field produced by its image at (0,0,−c). [46]

E_{loc}^{i} = E_{inc}^{i} + E_{img}^{i}

where i = 1, 2 is an index representing x and y components of the electric field respectively. According to (1),

E_{img}^{i} (0, 0, c) = - E_{{loc}^{'}}^{i} (0, 0, - c) \frac{α_{i}}{V} A_{i} (0, 0, 2 c)

, where

E_{{loc}^{'}}^{i}

is the local field at the image ellipsoid (0,0,−c). The continuity boundary conditions of the electric field and electric displacement give

E_{{loc}^{'}}^{i} = \frac{ε_{0} - ε_{s}}{ε_{0} + ε_{s}} E_{loc}^{i}

, thus [46].

E_{img}^{i} = - E_{loc}^{i} \frac{α_{i}}{V} \frac{ε_{0} - ε_{s}}{ε_{0} + ε_{s}} A_{i} (0, 0, 2 c)

where ε₀ is the vacuum dielectric constant. From (5) and (6),

E_{loc}^{i} = \frac{E_{inc}^{i}}{1 + \frac{α_{i}}{V} \frac{ε_{0} - ε_{s}}{ε_{0} + ε_{s}} A_{i} (0, 0, 2 c)}

The dipole moment p_i of the ellipsoid is

p_{i} = ε_{0} α_{i} E_{loc}^{i} \equiv ε_{0} {\tilde{α}}_{i} E_{inc}^{i}

where the modified polarizability α̃_i is

{\tilde{α}}_{i} = \frac{α_{i}}{1 + \frac{α_{i}}{V} \frac{ε_{0} - ε_{s}}{ε_{0} + ε_{s}} A_{i} (0, 0, 2 c)}

For an ellipsoid array on a substrate, the local field of each ellipsoid is the sum of the field E_inc, E_sur and E_img. [45]

E_{loc}^{i} = E_{inc}^{i} + E_{img}^{i} + E_{sur}^{i}

According to (1), $E_{sur}^{i}$ can be written as

E_{sur}^{i} = - E_{loc}^{i} \frac{α_{i}}{V} \sum_{j \neq 0} A_{i} (x_{j}, y_{j}, 0)

According to (6), $E_{img}^{i}$ can be written as

E_{img}^{i} = - E_{loc}^{i} \frac{α_{i}}{V} \frac{ε_{0} - ε_{s}}{ε_{0} + ε_{s}} \sum A_{i} (x_{j}, y_{j}, 2 c)

where j is an index representing different ellipses. From (10),(11) and (12),

E_{loc}^{i}

can be obtained, therefore the modified polarizability α̃_i can be written as

{\tilde{α}}_{i} = \frac{α_{i}}{1 + \frac{α_{i}}{V} β_{i}} = V \frac{ε_{g} - ε_{0}}{ε_{0} + (L_{i} + β_{i}) (ε_{g} - ε_{0})}

where

L_{1} = \frac{a b c}{2} \int_{0}^{\infty} \frac{d q}{(a^{2} + q) \sqrt{(a^{2} + q) (b^{2} + q) (c^{2} + q)}}

and

L_{2} = \frac{a b c}{2} \int_{0}^{\infty} \frac{d q}{(b^{2} + q) \sqrt{(a^{2} + q) (b^{2} + q) (c^{2} + q)}}

are the geometrical factors along axis a and b respectively.

β_{1} = \sum_{j \neq 0} A_{1} (x_{j}, y_{j}, 0) + \frac{ε_{0} - ε_{s}}{ε_{0} + ε_{s}} \sum_{j} A_{1} (x_{j}, y_{j}, 2 c)

and

β_{2} = \sum_{j \neq 0} A_{2} (x_{j}, y_{j}, 0) + \frac{ε_{0} - ε_{s}}{ε_{0} + ε_{s}} \sum_{j} A_{2} (x_{j}, y_{j}, 2 c)

are the correction terms of L₁ and L₂ respectively.

Fig. 1 The schematic diagram of a single graphene ellipsoid on an isotropic substrate with dielectric constant ε_s. Center coordinates of the ellipsoid and its image are (0,0,c) and (0,0,−c) respectively.

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Then the extinction cross section C_ext can be obtained, C_ext = k × Im[α̃_i] [46], and the extinction rate E_ext is also obtained as E_ext = C_ext/S_period, where k is the wave vector of the incident light, and S_period is the area of a unit cell of the ellipse array. Above results show that for each ellipsoid in the array there exist two independent polarizability α̃₁ and α̃₂ which are related to the permittivity of graphene ε_g and substrate material ε_s, therefore related to frequency, Fermi level, relaxation time τ and so on. Mathematically, by ploting Im[α̃₁] and Im[α̃₂] versus frequency, it can be shown that both of them exist one resonant frequency. In brief, two electrical dipole modes exist in the ellipsoid, whose dipole moments are orthogonal to each other and along the major and minor axis respectively. Many parameters including the length of major and minor axis, Fermi level of graphene and the permittivity of substrate will affect the modified polarizability, therefore the properties of the two modes.

3. Experimental results

To show the two orthogonal graphene plasmonic modes experimentally, four graphene ellipse arrays on BaF₂ with different sizes(a = 77.5 nm, b = 30 nm; a = 72.5 nm, b = 25 nm; a = 67.5 nm, b = 20 nm and a = 62.5 nm, b = 15 nm respectively.) were fabricated. For each array the period along major axis and minor axis is 200 nm and 120 nm, respectively. The fabrication process of graphene ellipse arrays is given in the method section. The schematic diagram and the SEM image of these samples are shown in Fig. 2(a). The extinction spectra 1 − T/T₀ of the four structures were measured and shown in Figs. 2(b) and 2(c), where T and T₀ are the transmittance with and without the graphene array respectively, while the polarization direction of the incident light was set along the major and minor axis respectively. Obviously, extinction peaks are observed, corresponding to the plasmonic resonant modes. Therefore, experimental results demonstrated the existance of two eigen modes for each ellipse array. Such bimodal resonance must be due to the unique asymmetry of the ellipse. To compare the experimental results with the theoretical model, the extinction spectra are calculated, shown in the lower panel of Figs. 2(b) and 2(c). For all different array structures and polarization of the incident light along the major and minor axis of the ellipse, Fermi level E_F and relaxation time τ were properly selected as 0.34 eV and 14 fs respectively. Permittivity of BaF₂ substrate is given in the method section. It can be seen that the theoretical extinction spectra are in good agreement with the experimental spectra both for major axis-polarization mode and minor axis-polarization mode, revealing that the plasmonic modes can be manipulated by adjusting the geometric structures of graphene to meet the specific requirements. As shown in Figs. 2(b) and 2(c), the resonant frequency of minor axis polarization mode is higher than that of major axis polarization mode. For the eigen mode that along the minor axis of the ellipse, the plasmon resonant frequency obviously shows a blueshift, while for the eigen mode that along the major axis, the resonant frequency is almost unchanged. It reveals that the resonant frequency of the mode along minor axis is more sensitive to its length.

Fig. 2 (a) The schematic diagram and the SEM image of periodic graphene ellipse arrays on BaF₂. The length of semi-major axis is a, that of semi-minor axis is b, and half the length of third axis of the effective ellipsoid is c. Scale bar of the SEM image: 140 nm. (b) Experimental (upper panels) and theoretical (lower panels) extinction spectra at normal incidence with the polarization along minor axis (b) and major axis (c), respectively. From yellow line to purple line, the sizes of ellipse are: a = 77.5 nm, b = 30 nm(orange); a = 72.5 nm, b = 25 nm(green); a = 67.5 nm, b = 20 nm(blue); a = 62.5 nm, b = 15 nm(purple), respectively. In lower panel of (b) from yellow line to purple line, the extra terms Δc are: −0.07 nm, 0 nm, 0 nm, 0 nm, respectively. In lower panel of (c) from yellow to purple line, the extra terms Δc are: 0.05 nm, 0.05 nm, 0.07 nm, 0.08 nm, respectively.

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The polarization-dependent extinction spectra were measured, shown in the upper panels of Figs. 3(b) and 3(c). The 0 degree is defined as polarization direction of the incident light along the minor axis of the ellipses and the 90 degree as the polarization direction along the major axis. It is obvious that by varying the polarization direction of normal incident light, the relative strength between the two orthogonal modes can be tuned. When the polarization direction varies from the minor axis (0 degree) to the major axis (90 degree), the high-frequency mode becomes weaker while the low-frequency mode keeps growing. The intensity of the two orthogonal resonant modes for all 360 degree polarization direction is extracted and plotted in the lower panels of Figs. 3(b) and 3(c), showing significant polarization dependence. The maximum intensity along the major axis is larger than that along the minor axis, showing the stronger light-matter interaction along major axis direction. From the polarization-dependent spectra shown in Fig. 3, it can be seen that the larger the size difference between major and minor axis is, the farther apart the two peak positions are. This approach offers good flexibility for light manipulation.

Fig. 3 Polarization-dependent extinction spectra of three different ellipticity structures. The upper panel are extinction spectra of normal incident light with polarizations varied from 0 to 90 degree. The lower panel shows extinction intensities for all polarization direction of the two resonant plasmonic modes. (a) round graphene disk with 45 nm radius. (b) Ellipse with a = 70 nm, b = 38 nm; (c) Ellipse with a = 70 nm, b = 32 nm.

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It is well known that the Fermi energy and then the optical properties of graphene can be tuned by means of applied voltage [7, 31, 32] and chemical doping via UV light [47]. Here, ultraviolet (UV) light was used. The spectra were measured with unpolarized normal incident light. As shown in Fig. 4(a), illumination of UV (355 nm) on the elliptical graphene disk array for 31.8 seconds, resulted in obvious redshift of both the two resonant peaks. This is because UV irradiation can decrease the adsorption of oxygen molecules on graphene, leading to the reduction of carrier (hole) concentration [47]. Therefore, the irradiation of ultraviolet light can continuously change the Fermi energy of graphene, leading to a redshift of the two plasmonic resonant peaks. The peak position shifts by 173 cm⁻¹ for mode along the minor axis and by 85 cm⁻¹ for mode along major axis. The resonant frequency of the two plasmonic modes versus illumination time of ultraviolet light is plotted in Fig. 4(b) with red dots representing high-frequency modes along minor axis and blue dots representing the low-frequency ones along major axis. By fitting the two peak positions, the variation of Fermi energy with irradiation time was extracted, and shown in Fig. 4(b). The Fermi level adjusted by illumination of ultraviolet light, from the pristine 0.3 eV to 0.22 eV. With the change of Fermi energy, the resonant frequencies of two plasmonic modes will shift over a wide range. In addition, the same change of Fermi energy results in a wider tuning range for mode along major axis than that along minor axis. Therefore adjustment of E_F can dynamically tune the two plasmonic modes.

Fig. 4 (a) Redshift of the extinction spectra with illumination time. The green line shows the pristine spectrum without UV light illumination. (b) Black star line shows the variation of Fermi level with illumination time. The red dot line and the light blue dot line represent the peak shift of high-frequency and low-frequency mode, respectively.

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The permittivity of substrate materials will influence the plasmon resonant mode, especially the substrate of polar material with lattice vibration (phonon). Plasmons and phonons represent fundamental collective excitations in solids, and the interaction between them has fundamental implications.

In present experiment, in addition to BaF₂, polar material silicon nitride deposited on silicon (Si₃N₄/Si) was also used as substrate. Upper panels of Figs. 5(a) and 5(b) are spectra measured with polarization of incident light along minor and major axis respectively. Two distinct peaks can be clearly seen for the case along minor axis, while only one peak is observed for the incident lights polarized along the major axis. The increase of extinction with the decrease of wavenumber in low-wavenumber range indicates another peak could be beyond the measurement range. As shown in the lower panel of Figs. 5(a) and 5(b), the theoretical spectra are generally consistent with the experimental results, demonstrating that two modes are generated by the coupling of the plasmonic modes of graphene ellipse array with phonons of the substrate. As a comparison, assuming the imaginary part of the permittivity of the substrate is zero, which means no phonon exists, only one peak at 882 cm⁻¹ and 1163 cm⁻¹ for the two incident polarizations (grey dashed line) exists, obviously different from that with phonon coupling, as shown in the lower panel of Figs. 5(a) and 5(b). The theoretical and experimental results further demonstrate substrate materials will strongly modulate the plasmonic modes of graphene ellipse array.

Fig. 5 Spectra measured on Si₃N₄/Si substrate and comparing the theoretical spectral with and without plsmon-phonon coupling. Upper panels: Red and blue lines are experimental extinction spectra with polarization direction of the incident light along minor and major axis respectively at normal incidence. Lower panels: Theoretical comparison of the extinction spectra with and without plsmon-phonon coupling.

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4. Methods

Sample fabrications: A large-area CVD grown graphene sheet was first transferred onto a BaF₂ substrate. Graphene structures were fabricated with a 100 × 100 μm² area at the center of the graphene layer obtained by standard e-beam lithography followed by oxygen plasma etching. Fabricated samples were then exposed to Br₂ vapor for 30 seconds to increase the doping level.

Spectral measurements: A Bruker Vertex 70 FTIR micro-spectrometer was used for mid-infrared spectral measurements. All measurements were performed at room temperature in air. For UV illumination, a 355 nm solid-state laser was used at power density of 200 mW/cm².

Theoretical parameters: The permittivity of BaF₂ is

ε_{{BaF}_{2}} = 2.15 + \frac{0.2 λ^{2}}{λ^{2} - {29.87}^{2}} + \frac{4.52 λ^{2}}{λ^{2} - {53.82}^{2}}

wave length λ is in micron. The effective permittivity of Si3N4/Si substrate is

ε_{SiN} (k) = ε_{SiN_0} + \frac{Λ \times k_{0}^{2}}{k_{0}^{2} - k^{2} - i \times Γ \times k}

wave number k is in cm⁻¹. ε_{SiN_0} = 3.8, Λ = 1.5, Γ = 100 cm⁻¹, k₀ = 820 cm⁻¹.

5. Conclusions

In summary, the plasmonic modes of graphene ellipse array have been theoretically and experimentally investigated. Firstly, the feasibility of the theory is verified by comparing the theoretical spectra with the experimental ones. Secondly, by changing the polarized direction of normal incident light, the relative strength between the two orthogonal modes can be tuned. Thirdly, the two plasmonic modes show red shift with the increase of the axis length or decrease of the Fermi level of graphene, respectively. Finally, with a polar substrate, the coupling of graphene plasmons and substrate phonons generate two resonant modes in each axis direction of the ellipse. Our findings propose that the unique properties of anisotropic graphene structures may have many potential applications as polarization-dependent detectors, filters and sensors at infrared wavelengths.

Funding

973 and China National Key Basic Research Program (2015CB659400, 2016YFA0301100, 2016YFA0302000); National Natural Science Foundation of China (11774063, 11727811). The research of L.S. was further supported by Science and Technology Commission of Shanghai Municipality (17ZR1442300, 17142200100), State Key Laboratory of Surface Physics (ZA2014-01).

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Polarization dependent plasmonic modes in elliptical graphene disk arrays

Abstract

1. Introduction

2. Theoretical model

3. Experimental results

4. Methods

5. Conclusions

Funding

References

Cited By

Figures (5)

Equations (19)

Optics Express