Localized plasmons induced by spatial conductivity modulation in graphene

Chris Beckerleg; Euan Hendry

doi:10.1364/JOSAB.33.002051

Graphene has been intensely studied since being identified [1]. While initial research was driven by its novel electronic properties [2], the electromagnetic properties of graphene are of increasing interest. One aspect of this is the ability of graphene to support surface plasmons at infrared frequencies [3]. With a broad range of potential infrared applications, the promise of being able to control plasmons in graphene is also of technological interest [4–6]. Moreover, the deeply subwavelength confinement of surface plasmons in graphene and the associated high field enhancement could be of potential benefit for nanophotonic and sensing applications [7–13], while the response is also highly tunable [14].

However, to couple to surface plasmons from incident radiation, one must overcome the wave-vector mismatch between the highly confined plasmons in graphene and the incident light. One approach to overcome this phase matching restriction involves nanostructuring the graphene, either using an external grating [15], patterning the graphene directly by etching ribbons [16] or by corrugating the surface [17]. However, the postfabrication or etching of the graphene and surrounding materials used in these approaches invariably introduces defects, which have a negative impact on the electron mobility and therefore plasmon losses [18]. Peres et al. [19] suggested an alternative approach is to periodically modulate the conductivity of the graphene. This could be achieved using the postapplication of polymer top gates to control the local Fermi level in graphene [20], functionalizing the graphene itself using scanning tunneling microscopy [21], or by supporting the graphene layer with an array [22], removing the need for etching of the graphene. It was demonstrated in the models of Peres et al. that representing a spatial variation in conductivity with a single cosine leads to plasmons being supported by the graphene sheet [23].

In this contribution, we use a full wave modal matching solution to Maxwell’s equations to model a more attainable ribbonlike modulation in the conductivity of a graphene sheet, with areas of high and low conductivity as illustrated in Fig. 1(b). We show that the total response of this periodic system arises from a hybridization between the multipolar resonances supported by two regions in each unit cell. Extending this model to two layers, we show that due to a small interlayer separation, compared to the incident wavelength, this system only supports symmetric resonances in each layer. By carefully tuning the conductivity profile, we show that an increase, approaching 50%, can be achieved in the resonant absorption when both regions are made dipole resonant.

Fig. 1. Schematic of graphene sheets with a periodically modulated conductivity with a period of $L$ and a width of $w$ . (a) Conductivity profile described with a single Fourier term. (b) Conductivity profile with many Fourier terms. Inset: conductivity as a function of the distance along the graphene surface, given by Eq. (4).

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To understand the behavior of a graphene sheet with a modulated conductivity, a solution to Maxwell’s full wave equations is proposed using a modal matching technique. Defining the fields above, (1), and below, (2), a single graphene layer as

B_{(α), y} = A \exp^{i x k_{0}} + \sum_{m} F_{(α), m} \exp^{- i m L z} \exp^{- i q_{(α), m} x},

E_{(α), x} = C \exp^{i x k_{0}} + \sum_{m} G_{(α), m} \exp^{- i m L z} \exp^{- i q_{(α), m} x},

E_{(α), z} = D \exp^{i x k_{0}} + \sum_{m} H_{(α), m} \exp^{- i m L z} \exp^{- i q_{(α), m} x},

with

α

denoting region 1 or 2, as shown in Fig. 1.

A

,

C

,

G

,

D

, and

H

denote the unknown Fourier coefficients and

q_{(α), m} = \sqrt{ω^{2} ϵ_{(α)} / c^{2} - {(m L)}^{2}}

, where

k_{0} = ω^{2} ϵ_{1} / c^{2}

and

L

is the period of the modulated structure. We choose to model a realistic system constructed using ion gel top gates [24]. The relative permittivity,

ϵ_{(α)}

, of a calcium fluoride substrate, region 1, and an ion gel top gate, region 2, are both taken to be 1.96 [25,26]. The spatial modulation of the conductivity is described using the rectangular form

σ (z) = σ_{d} {\frac{w}{L} + \sum_{p} \frac{2}{p π} [\sin \frac{2 p π w}{L} \cos \frac{2 p π z}{L} + \sin \frac{2 p π z}{L} (1 - \cos \frac{2 p π w}{L})]},

represented using a Fourier sum over

p

. This profile creates a periodically alternating array of high- and low-conductivity regions as shown in Fig. 1. Because the range of frequencies considered is below the interband absorption threshold, we can model the conductivity of graphene using the Drude expression [27–29]:

σ_{d} (ω) = \frac{i e^{2} | μ |}{π ℏ (ω + i τ^{- 1})} .

We assume the Drude relaxation time is determined by long-range impurities, so that $τ = μ E_{f} / e V_{f}^{2}$ [30], where $E_{f}$ is the Fermi level and $V_{f}$ is the $Fermi velocity = 1 \times 10^{6} {ms}^{- 1}$ . The mobility, $μ$ , is set to $1000 {cm}^{2} / (Vs)$ , which is typical of chemical vapor deposition graphene [31]. In practice, a modulated conductivity such as in Fig. 1 could be achieved using a patterned gate [32] or functionalization [21], both utilizing the distinctive dependence of the conductivity of graphene on the Fermi level. By imposing the boundary conditions $E_{(1), z} = E_{(2), z}$ and $B_{(1), y} = B_{(2), y}$ on Eqs. (1)–(4), continuity equations for the specular mode,

(\frac{ϵ_{1}}{q_{(1), 0}} + \frac{ϵ_{2}}{q_{(2), 0}}) ω H_{(2), 0} + \frac{1}{ϵ_{1}} \sum_{p} σ_{- p} H_{(2), p} = \frac{2 ω ϵ_{1}}{k_{0}} D,

and for the diffracted orders,

(\frac{ϵ_{1}}{q_{(1), m}} + \frac{ϵ_{2}}{q_{(2), m}}) ω H_{(2), m} + \frac{1}{ϵ_{1}} \sum_{p} σ_{- p} H_{(2), p} = 0,

can be derived. Here,

σ_{- p}

is the

p

th Fourier component from Eq. (4). The summations in Eqs. (4), (6), and (7) are taken over the range of integers in the interval

- p

to

p

resulting in a linear set of simultaneous equations, which are solved numerically for the unknown Fourier coefficients. Using these Fourier coefficients in Eqs. (1)–(3), a full description of the fields surrounding the graphene and the transmission through the structure is calculated.

The conductivity profile presented in Eq. (4) is a sum over a number of Fourier terms. By increasing the number of Fourier components, the edges of the conductivity profile become more sharply defined, as illustrated in Fig. 1(a), with a single term, and Fig. 1(b), with many. In Fig. 2(a), we compare the calculated transmission through a structure with profile described by a single Fourier term to profiles described with an increasing number of Fourier terms. The period is taken to be 200 nm and the width of the high- and low-conductivity regions each taken to be 100 nm. The Fermi level of the low-conductivity region, the light areas in Fig. 1(b), is fixed at a value of 0.1 eV, which is comparable to experimentally determined Fermi levels of ungated graphene [16]. All results presented in Fig. 2 assume the Fermi level of the high-conductivity region to be 0.3 eV. For a conductivity profile with a single Fourier term [blue dashed line of Fig. 2(a)], the transmission displays a single deep minimum, as observed by Peres et al. [19]. This behavior arises due to the singular nature of the Fourier decomposition of the spatially varying response function, i.e., it is similar in origin to singular diffracted order observed for sinusoidal amplitude gratings [33]. Including higher order Fourier terms (red dotted–dashed line) results in the appearance of extra features in transmission. The results converge toward those obtained from a commercial finite-difference time-domain (FDTD) modeling package when we include 10 or more Fourier terms (black line) [34].

Fig. 2. (a) Modeling results for a single Fourier term (blue dashed line), three Fourier terms (red dotted–dashed line), 10 Fourier terms (black solid line), and the FDTD comparison in cyan circles. (b)–(d) Real part of the in-plane electric field ( $E_{z}$ ) normalized to the incident field at 22, 27, and 34 THz as labeled on (a).

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The most striking feature of the data in Fig. 2(a) is the appearance of extra dips in the transmission at 22, 28, and 34 THz as indicated by labels $b$ , $c$ , and $d$ . Through inspecting the electric fields at these resonant frequencies we can comment on the origin of the minima in transmission. The real part of the in-plane electric field is plotted in Figs. 2(b)–2(d), at the frequencies indicated by Fig. 2(a). The regions corresponding to high and low conductivity are indicated by white arrows. Note that, while the out-of-plane electric field components change sign near the tips of a dipole, the in-plane electric field component are expected to be the same sign. The electric field in Figs. 2(b)–2(d) does not change sign across the entire region of high conductivity, indicating a dipolar field. The electric field in the low-conductivity region, on the other hand, displays multiple maxima and minima, indicative of multipolar resonances: one ( $b$ ), three ( $c$ ), and five ( $d$ ) peaks indicating a dipole, quadrupole, and hexapole resonance, respectively. It is somewhat surprising that a modulation of conductivity gives rise to localized surface plasmons. However, it should be noted that if the conductivity in the low- and high-conductivity regions is similar, then the amplitude of the localized plasmons will be small.

It is interesting to compare these complicated transmission spectra, presented in Fig. 2(a), with their multiple resonances, with the transmission expected for an array of isolated graphene ribbons. For a ribbon, the resonant frequency $w_{0}$ of the dipole resonance is determined by

ω_{0} = η \sqrt{\frac{e^{2} E_{f}}{ℏ^{2} ϵ_{avg} ϵ_{0} w}},

where

E_{f}

is the Fermi level of the graphene,

ϵ_{avg}

is the average permittivity of the regions 1 and 2 from Fig. 1,

w

is the width of the graphene ribbon, and

η = 0.62

is a dimensionless constant [16].

In Fig. 3, we compare this prediction for ribbons [Eq. (8), dashed line] with the transmission through our modulated conductivity structure, as we vary the Fermi level of the high-conductivity region. We observe multiple resonances occurring in the vicinity of the predicted dipole resonance given by Eq. (8), when the multipolar modes of the low-conductivity region overlap in frequency with the dipole mode of the high-conductivity region. This indicates that the frequencies of the resonances supported by our structure are broadly determined by the dipole set up in the region of high Fermi level. Rather unexpectedly, these multipolar modes, which clearly correspond to plasmon standing waves excited in the low-conductivity region, also depend on the Fermi level of the high-conductivity region. We find that these multipolar resonances couple more strongly to the incident radiation for frequencies near the expected dipolar resonance of the high-conductivity region. The coupling between the dipole and multipolar modes in the two different regions is a defining feature of the system.

Fig. 3. Transmission as a function of frequency and the difference in Fermi level for the high- and low-conductivity regions. The structure is an array of modulated high- and low-conductivity regions with a period of 200 nm. The high-conductivity region has a width of 100 nm. Dashed line is the position of the dipole resonance of the equivalent high region given by Eq. (8).

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Because the resonance frequencies are determined by both material conductivity and geometry, can the resonance frequencies of the different regions be independently tuned to optimally overlap, thereby increasing the coupling to incident radiation? As we discuss here, this can be achieved by fine-tuning the widths of the high- and low-conductivity regions. To investigate this, we vary the width of the high Fermi level region in a fixed period of 200 nm. The calculations for the transmission through this structure are presented in Fig. 4(a) along with the predictions for the dipole mode for the high- (solid black line) and low- (dashed line) conductivity regions. We again observe several features corresponding to the mixing of the dipole resonance in the high-conductivity region with the multipole resonances supported by the low-conductivity region. However, by far the largest modulation of the transmission occurs where the dipole predictions for the regions of high- and low-conductivity intersect [given by the intersection of the dashed and solid lines in Fig. 4(a)]. The in-plane electric field for the intersection point on Fig. 4(a) is shown in Fig. 4(b), with the regions of high and low conductivity indicated by arrows. As before, the absence of a change in sign of the electric field inside both regions indicates their dipolar nature. This doubly resonant behavior results in a 50% increase in the modulation of the transmission compared to a structure with equally sized high- and low-conductivity regions.

Fig. 4. (a) Transmission through a single layer as a function of frequency and width of the high-conductivity level region. The periodicity of the structure is 200 nm and the high-conductivity region has a Fermi level of 0.3 eV. The position of the dipole frequency for the high-conductivity region (solid line) and the low-conductivity region (dashed line) is shown according to Eq. (8). (b) Real part of the in-plane electric field, normalized to the incident field, at the crossover between the high Fermi level region and low Fermi level region, from (a).

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An additional approach that one can follow to further increase the modulation of the transmission is to create stacked structures. We now extend our model to investigate two equivalent layers of graphene, each with an identical conductivity modulation, separated by a dielectric gap. This requires modification of Eqs. (1)–(3) to describe reflection from the second graphene layer. By applying the boundary conditions to both graphene layers, the modified form of Eqs. (6) and (7) can be solved numerically, as before.

In Fig. 5, we investigate the effect of the separation between the layers on the transmission through the double-layer structure. The conductivity modulation is defined by aligned low- and high-conductivity regions, each 100 nm wide, with Fermi levels of 0.1 and 0.3 eV, respectively. First, we see that the addition of a second layer leads to the expected twofold increase in the modulation of the transmission. However, we observe that the resonant frequencies are dependent on interlayer separation. For large interlayer spacing, greater than the decay length of plasmons, the resonant frequencies approach those of a single layer, as shown by the inset in Fig. 5(a). For interlayer spacing smaller than approximately 70 nm, the transmission minima shift to higher frequencies. This behavior is similar to that observed for symmetrically coupled plasmon resonators [35] and arises from the alignment of dipolar fields of one plasmon resonator with the Coulomb field of the other. Because the separation between our layers is around 2 orders of magnitude smaller than the wavelength of incident light, coupling to the corresponding antisymmetric resonance is expected to be negligible, as confirmed in Fig. 5(b) by the absence of antisymmetric in-plane electric fields.

Fig. 5. (a) Transmission through a double layer of graphene with modulated conductivity regions as a function of the frequency and the interlayer separation. The width of the high-conductivity region is 100 nm in a period of 200 nm and has a Fermi level of 0.3 eV. The minima (1) and maxima (2) in the modulation in the transmission are indicated. Inset: transmission through the equivalent single layer. (b) Real part of the in-plane electric field, normalized to the incident field, for 100 nm conductivity region of 0.3 eV, separated by 30 nm. The regions of high and low conductivity are indicated.

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A rather subtle but surprising feature of the transmission behavior shown in Fig. 5(a) is the observed fluctuation in oscillator strength between the dipole (1) and quadrupole (2) modes as a function of interlayer spacing. The resonant transmission for the low-frequency mode (1) first increases to a maximum at $\sim 20 nm$ , before decreasing again for larger separations. Precisely the opposite behavior is observed for the high-frequency mode (2). We suggest that this effect may arise from the differing spatial decays of dipole and quadrupole fields.

Finally, we end by attempting to tune both the interlayer spacing and widths of the conducting regions, to optimize coupling conditions. In Fig. 6, we investigate varying the width of the high-conductivity region, assuming an interlayer spacing of 30 nm. We observe that the greatest modulation in the transmission occurs when the regions with high and low Fermi levels are both dipole resonant. Repeating these calculations for a range of separations shows that the strength of the modulation in the transmission is relatively insensitive to the separation between the graphene layers. Designing a structure in which both regions in both layers support dipole resonances leads again to a large, 50%, increase in the modulation of the transmission, compared to a double layer of equally sized high- and low-conductivity regions.

Fig. 6. Transmission through a double-layer structure as a function of frequency and the width of the high Fermi level region. The interlayer separation is 30 nm. The dipole and quadrupole frequency for the region of high-conductivity region (solid line) and the dipole, quadrupole and hexapole frequency for the low-conductivity region (dashed line) are shown according to Eq. (8).

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It is interesting to note the dependence of the resonant frequencies on the width of the high-conductivity region ( $y$ axis) as depicted in Fig. 6. We observe certain conditions where the frequencies appear to follow the predictions for the dipole and quadrupole resonances of the high-conductivity region (solid lines), while in other regions, we observe a dependence similar to that predicted for the dipole, quadrupole, and hexapole resonances of the low-conductivity region (dashed lines). This leads to the rather curious S-shaped features in Fig. 6 and is indicative of the anticrossing behavior typical of strongly coupled resonators [36]. This anticrossing behavior is also observed to a lesser extent for a single layer [see Fig. 4(a)].

In conclusion, we have investigated a graphene sheet with a modulation in the conductivity using a modal matching approach. We show that the total response of this periodic system arises from a hybridization between the multipolar resonances supported by two regions in each unit cell. Extending this model to two layers, we show that due to the small interlayer separation compared to the incident wavelength, one can support only symmetric resonances in each layer. By carefully tuning the conductivity profile, we show that an increase, approaching 50%, can be achieved in the resonant absorption, when both regions are made dipole resonant. Given that these structures could in principle be fabricated from a continuous graphene sheet, i.e., without the need for graphene etching, with modulation depths comparable in magnitude to isolated nanostructures [37], such modulated structures may be a promising route to eliminating losses, introduced through fabrication, which currently plague etched samples [18,38].

Funding

European Commission (EC) (FP7-ICT-2013-613024-GRASP); Engineering and Physical Sciences Research Council (EPSRC) (EP/K041215/1).

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Localized plasmons induced by spatial conductivity modulation in graphene

Abstract

Funding

REFERENCES

Cited By

Figures (6)

Equations (8)

Journal of the Optical Society of America B