Enhanced graphene absorption and linewidth ... - OSA Publishing

Enhanced graphene absorption and linewidth sharpening enabled by Fano-like geometric resonance at near-infrared wavelengths ∗

F. Liu,1,3 L. Chen,1 Q. Guo,2 J. Chen,1 X. Zhao,1 and W. Shi1, 1 Department 2 Department

of Physics, Shanghai Normal University, Shanghai 200234, China of Electrical Engineering, Yale University, New Haven, Connecticut 06520, USA 3 [email protected] ∗ [email protected]

Abstract: We theoretically investigate the light-graphene interactions enabled by a single layer of nonlossy nanorods at near-infrared wavelengths. The sustained Fano-like geometric resonance gives rise to enhanced graphene absorption, e.g., 100%, and adjustable absorption linewidth even to be ultra-narrow, e.g., < 1 nm. The conditions for such graphene absorption enhancement and linewidth sharpening are analytically interpreted within the framework of temporal coupled mode theory for the Fano resonance. The geometric resonance enhanced light-graphene interactions are polarization-sensitive and angle-dependent. Our study offers new possibilities towards designing and fabricating novel opto-electronic devices such as graphene-integrated monochromatic photodetectors and ultra-compact modulators. © 2015 Optical Society of America OCIS codes: (230.4555) Coupled resonators; (050.6624) Subwavelength structures; (130.3120) Integrated optics devices; (300.1030) Absorption; (300.3700) Linewidth.

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Received 14 May 2015; revised 7 Jul 2015; accepted 27 Jul 2015; published 4 Aug 2015 10 Aug 2015 | Vol. 23, No. 16 | DOI:10.1364/OE.23.021097 | OPTICS EXPRESS 21097

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1.

Introduction

Owing to its unique electronic and optical properties [1], graphene has received great attentions in fields of plasmonics in mid-infrared to terahertz frequencies [2–12] and opto-electronics at visible to near-infrared wavelengths [13–22]. Although the universal absorption of 2.3% at visible to near-infrared regime is remarkably high considering the atomic thickness of graphene [23], it is still highly desirable to improve the light-graphene interactions in opto-electronic devices such as graphene-integrated photodetectors [14–18], ultrafast lasers [19], and optical modulators [20,21]. In principle, the light-graphene interaction is determined by the interaction strength which can be modified via the electric field or the interaction length. Enabled by the surface plasmon polaritons (SPPs) mode [16–18, 24–26], the guided mode [27–30], the cavity mode [31–33], and waveguide mode [20, 21], the interaction strength is increased effectively, leading to a significant enhancement of graphene absorption. Recently, even total graphene absorption is enabled by SPPs in corrugated metal surface and the guided resonance in the patterned dielectric photonic crystal slab (PCS), providing that the radiative leakage loss rate γR equals to the dissipative loss rate γD [26–29]. On the other hand, the sharpening of the spectral linewidth γ (full width at half maximum, FWHM), in addition to the enhanced absorption, is also crucial to the explorations of novel devices, e.g., the detecting system of high quantum efficiency photodetector used in dense wavelength division multiplexing (DWDM) systems where extremely narrow channel spacing (e.g., less than 1.0 nm) need to be discerned [34]. For graphene coupled structures, the absorption linewidth positively correlates to radiative leakage loss rate γR since only the region near the resonance sustains enhanced field. However, the γR determined by the radiative SPPs in structured metals has an intrinsic lower bound as a result of high dissipative loss of surface plasmons at optical wavelengths (the lifetime as short as femtoseconds in metals) [35, 36], although γR is usually dependent on the metal materials, sizes, and shapes [37]. As a contrast, various dielectric materials are non-dissipative at visible to near-infrared wavelengths, e.g., the wide-gap semiconductors like SiNx , which indicates an unprecedented opportunity to achieve ultra-narrow absorption linewidth for graphene. In this paper, by integrating nonlossy free-standing dielectric nanorod arrays made of SiNx , we theoretically demonstrate enhanced graphene absorption, e.g., up to 100%, and absorption linewidth sharpening, e.g., smaller than 1 nm, enabled by the Fano-like geometric resonance. Within the framework of temporal coupled mode theory (TCMT) for the Fano resonance, the conditions for the geometric resonance enabled graphene absorption enhancement

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and linewidth sharpening are analytically interpreted, in good agreements with the numerical calculations. Furthermore, we show that the light-graphene interactions assisted by such Fano resonance are controllable by tuning the structural parameters, the structural configuration, the incident polarization and angle. (a)

(c)

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Fig. 1. (a) A Sketch of pristine graphene. (b) Schematic illustrations of a single layer of SiNx nanorods. (c) Distribution of normalized |E|2 in the x-z plane at the geometric resonance. It is clear that the electric field is mostly confined within and between the nanorods, rendering enhanced near-field at the surface proximity. The cross section of the nanorods is outlined by bold black lines. (d) The reflection spectra of the nanorods coupled with graphene red-shifts by 0.0027 µm−1 with a lower Q-factor compared with that of the structures without graphene covering. The external radiation loss rate γR of the geometric resonance sustained in the nanorods can be estimated by fitting the Fano-like reflection according to Eq. (6). (e) Calculated absorption of the graphene coupled with the nanorod layer show enhancement than that of pristine graphene at the geometric resonance. The intrinsic dissipative loss rate of γD can be fitted by the Lorentz function of Eq. (8). In the simulations, the periodicity Λ is 800 nm, the nanorod width d is 160 nm, and the layer thickness h is 100 nm.

2.

Models and methods

Graphene is a monolayer consisting of a honeycomb lattice of carbon atoms [Fig. 1(a)], which optical conductivity σ in our simulations are modeled using the result from Kubo formalism within the local random-phase approximation as [38, 39] EF e2 1 1 h¯ ω − 2EF i h¯ ω + 2EF 2ie2 kB T + ( + arctan( )) − ln | | , ln 2 cosh σ (ω) = 2 2kB T 4¯h 2 π 2kB T π h¯ ω − 2EF π h¯ (ω + iτ −1 ) (1) where ω is the angular frequency, e is the charge of an electron, kB is the Boltzmann constant, T is the temperature, h¯ is the reduced Planck constant, τ is the carrier relaxation time, and #240969 © 2015 OSA


EF is the Fermi level. The first term of the equation is responsible for intraband transitions and the second term for interband transitions. If the photon energy h¯ ω > 2EF , the interband transitions dominate, giving rise to a universal limit σ = e2 /4¯h in room temperature. Considering the relation between the in-plane graphene permittivity εk and σ (ω) [39] εk = εg + i

σ (ω) , ε0 ωtG

(2)

where εg = 2.5, ε0 is the vacuum permittivity, and tG = 0.5 nm is the graphene thickness, graphene can be regarded as a lossy thin film in high frequency regime, e.g., a constant absorption of 2.3% throughout the visible wavelengths. In the following calculations, the condition of h¯ ω > 2EF is always fulfilled. In order to calculate the reflectance, transmittance, and absorption, a two-dimensional implementation of the scattering matrix method was used [40]. The central idea of the scattering matrix method is to relate the electric and magnetic fields in different regions by a 2 × 2 matrix. Such matrix can be determined by solving the Maxwells equations and applying proper boundary conditions. A commercial software COMSOL is applied to simulate the electric field distribution via the full-wave simulation package. By meshing the structure with fine enough ˚ for graphene and 1 nm for dielectric nanorods, respecgrids which typical sizes are about 0.5 A tively, the calculated results are well convergent. 3.

Results and discussion

Dielectric nanorod arrays are periodic nanostructures made of tiny rectangle nanorods with a width of d and a height of h (much smaller than the wavelength, h = 100 nm in the following simulations), as sketched in Fig. 1(b). Though individual nanorod is nonresonant in nature, sharp geometric resonance with strong electric field enhancement on the rods and between them occurs in assemblies of the nanorods [Fig.1(c)], which can be understood as a result of coherent multiple scattering of induced ‘dipoles’ by light illumination or Fano resonance arising from the coherent interference between the direct pathway of the broadband Fabry-Pérot reflection with the indirect pathway of the geometric mode [41–45]. Such field enhancement can be dramatically increased by reducing the dimension of nanorod width, following the (λ0 /d)4 law [43], where λ0 is the wavelength of the geometric resonance. The geometric resonance in nanorod array is physically different from the guided mode supported in one-dimensional (1D) subwavelength grating which is featured by much high filling fraction. Unlike that of nanorod, the dielectric scatter in an individual unit of the 1D grating is resonant with the imping light [30]. In the paper, we assume the free-standing nanorod arrays are made of silicon nitride (SiNx ), unless otherwise specified, which can be readily fabricated by etching commercial available SiNx membrane. Importantly, SiNx is a nonlossy material with a constant refraction index n of 2.0 at visible to near-infrared wavelengths, which guarantees perfect optical extinction therefore high-Q at geometric resonance [41]. Under normally incident light with the electric field vector along the axis of the rods (i.e., s-polarized), the reflection spectra are calculated with the array period Λ = 800 nm and d = 160 nm, illustrating a sharp opaque window with a characteristic asymmetric Fano line shape [red line in Fig. 1(d)]. On resonance of 855 nm, the electric field enhancement at the nanorod surface is ∼ 5 at the nanorod surface [Fig. 1(c)]. By resting graphene on surfaces of the arrays, the reflection drops to ∼ 80%, together with slightly linewidth increasing and Q-factor decreasing [blue solid line in Fig. 1(d)]. This is induced by the graphene intrinsic dissipative loss, similar with the results reported in the 1D dielectric gratings [30] and the photonic crystal cavity [32]. The fact that a red-shift of 0.0027 µm−1 (∼ 2.0 nm) occurs is understandable by considering the imping light enters the structure now from graphene layer instead of air. On the other hand, the absorption intensity of graphene is enhanced via the inter#240969 © 2015 OSA


actions between graphene and the enhanced field by about one order of magnitude larger than that of pristine graphene [Fig. 1(e)] on resonance. Finally, we note that the parameters of the nanorod arrays are adopted in order to locate the absorption band at technologically important wavelengths of 850 nm (1.1765 µm−1 ) in optical communications. 1 .0 0

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Fig. 2. Calculated reflection spectra of unloaded structure (a) show that the linewidths of the geometric modes are dependent on the nanorod width d. Smaller d gives rise to sharper linewidths, indicating less radiation loss rate. Such inherited linewidth trend is clearly observed in the calculated absorption spectra for loaded structure (b), which can be understood by the fact that the enhanced light-graphene interaction occurs mainly over the linewidth of geometric mode where enhanced field occurs. The graphene absorption as a function of γR /γD (c) clearly illustrates the coupling strength is determined by the ratio of external radiation loss rate to the dissipative loss rate, which can be understood within the framework of the TCMT for Fano resonance.

Within the framework of TCMT for Fano resonance [46, 47], the spectral characteristics of the unloaded (i.e., nanorod array without graphene covering) and the loaded nanostructures (i.e., nanorod array with graphene resting) can be analytically interpreted. For the loaded structure, the dynamics of the single-mode Fano-like geometric resonance amplitude a is governed by s1+ da , (3) = ( jω0 − γR1 − γR2 − γD )a + d1 d2 s2+ dt d s1− s (4) = C 1+ + 1 a, s2− s2+ d2 where γR1 + γR2 = γR is the total external radiation loss rate, assuming the resonance decays into two ports with the radiation loss rate of γR1 and γR2 , respectively; s1+ and s2+ are the amplitudes of incoming waves from the two ports while s1− and s2− are that for outgoing waves; d1 and d2 are the coupling coefficients between the ports and the resonance. The scattering matrix of C = r jt e jϕ describes the direct coupling between incoming and outgoing waves, where r and jt r #240969 © 2015 OSA


t are the reflection and transmission coefficients of the direct pathway of light with |r2 | + |t 2 | = 1 [48]. According to energy conservation and time-reversal symmetry, the coupling coefficients between the ports and the geometric resonance are constrained by the following equations ∗ d1 d d1 d1∗ = 2γR1 ; d2 d2∗ = 2γR2 ; C =− 1 . (5) d2 d2 For the loaded structure, the radiation loss of the nanorod layer caused by the graphene situation is negligible since the extremely thin graphene layer has little impact on the field distribution pattern of the geometric resonance, as evidenced by the reported cases elsewhere [27–29, 32], manifesting that γR1 = γR2 . As a result, the steady-state solution to the temporal coupled mode equations yields the frequency-dependent reflection R(ω) and transmission T (ω) as R(ω) = |

s1− 2 (rγD )2 + [r(ω − ω0 ) ± tγR ]2 | = , s1+ (ω − ω0 )2 + (γD + γR )2

(6)

T (ω) = |

s2− 2 (tγD )2 + [t(ω − ω0 ) ∓ rγR ]2 | = , s1+ (ω − ω0 )2 + (γD + γR )2

(7)

where ω0 is the central resonant frequency. The absorption A(ω) of the hybrid system is induced by the graphene intrinsic dissipation, which can be derived by calculating 1-R(ω)-T (ω). Accordingly, 2γR γD A(ω) = . (8) (ω − ω0 )2 + (γD + γR )2 Interestingly, the absorption enabled by asymmetric Fano resonance manifests as Lorentzian line shape, similar with the results derived from TCMT for Lorentz resonance [27–29, 32]. The same expressions apply for the unloaded structure just by assuming γD = 0, which reproduces the results for non-dissipative Fano system [46]. The ±/∓ sign is determined by the symmetry of the mode with respect to the mirror plane. In our study, the non-degenerate resonant mode has odd-symmetry, which requires that the ‘minus’ should be taken in R(ω) and ‘plus’ sign adopted in T (ω) expression. From Eq. (8), it is evident that the light-graphene interaction p strength is determined by the ratio of γ /γ . At resonance, the absorption reads A(ω ) = 2/[ γR /γD + R D 0 p γD /γR ]2 , which has a maximum value of 0.5 when γR = γD (i.e., critical coupling). This is consistent with the theoretical predictions from multiple scattering theory for dielectric nanorod arrays conprising lossy materials for s-polarized wave [44] and the discussions based on the coupled mode theory for non-degenerate Lorentz resonance in ‘two-port’ resonator illuminated from one side [49]. On the other hand, the absorption linewidth γ is found to be determined by γR + γD . With the above analytical results in mind, one can retrieve the value of ω0 and γR by fitting the reflection spectra of unloaded structures, and γD from the absorption spectra of loaded structures. For example, ω0 = 1.177 µm−1 , γR = 0.0230 µm−1 [dashed line in Fig. 1(d)], and γD = 0.00335 µm−1 for the nanorod array with d = 160 nm. Although the graphene absorption linewidth for the above nanorod arrays is already comparable with most SPP-assisted enhanced absorption band of graphene, γ can be reduced further, even to be quasimonochromatic, for nonlossy dielectric nanostructures. As γR is mostly proportional to the filling factor f (i.e., d/Λ) of nanorods, we calculate the reflection spectra for unloaded structures and the corresponding absorption for loaded structures by assuming d to scale down from 80 nm to 10 nm with a constant ratio of 2, as shown in Figs. 2(a) and 2(b), respectively. In order to keep ω0 unshifted, the corresponding f varies from 0.0963 (Λ = 831 nm), 0.0473 (Λ = 845 nm), 0.0236 (Λ = 849 nm), to a vanishing value of 0.0118 (Λ = 850 nm). It is found that γR decreases monotonically from 0.00499, 7.13×10−4 , 1.15×10−4 , to 1.08×10−5 #240969 © 2015 OSA


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Frequency (m )

Fig. 3. Linewidths, central wavelengths, and intensity of the graphene absorption are controllable by selecting the constitute materials of the nanorods, the structural parameters, and the structural configurations. Instead of SiO2 nanorods (green open squares), the absorption linewidth is sharpened compared with that of SiNx material (black open circles); With the same material of SiO2 , different Λ is taken in order to tailor the desired wavelengths (red open uptriangles and blue open downtriangles); Finally, the absorption can be enhanced to be 100% (solid circle line) by changing the ‘two-port’ system to ‘one-port’ configuration via adding back mirror under the nanorod layer [the inset].

µm−1 . The results can be understood by realizing that smaller f gives rise to smaller effective refraction index, rendering lower Fabry-Pérot reflectance. The weaker light scattering interferes with the geometric mode channel, manifesting lower γR therefore higher value of Q-factor for the geometric resonance. The value of γD for each f is also evaluated, ranging from 0.00261, 0.00114, 7.09×10−4 to 3.99×10−4 µm−1 , behaving similar with that of γR . This is understandable in that the enhanced light-graphene interaction occurs predominantly in the linewidth of the geometric mode where electric field is enhanced. Interestingly, unlike the photonic crystal slab [27–29], the 1D dielectric grating [30], and the cavity [32], the dissipative loss rate γD of graphene changes rapidly with the f variation. Among the calculations, the lowest γ of 4.10×10−4 µm−1 (0.3 nm) and the highest Q-factor of ∼ 2870 occur for the array of 10nm-width nanorod, however, with a small absorption enhancement. Figure 2(c) illustrates the enhanced graphene absorption as a function of the loss rate ratio γR /γD , in which the numerical and analytical results agrees well with each other. The results clearly show the evolution of the interaction from undercoupling (γR /γD 1) regime. At resonance, the graphene absorption reaches its maximum value of 50% when γR = γD , corresponding to 0.0654 of f and 55 nm of d. The physical scenario of the maximum absorption condition can be understood as: if γR > γD , photons radiate from the nanorod array before being absorbed by the graphene layer; otherwise photons are not efficiently coupled with graphene. Besides the filling factor f , the material dielectric constant (ε = n2 ) of nanorods plays an important role on the linewidths of graphene absorption, thanks to the light scattering dependence on the dielectric contrast between the nanorods and the ambient air as discussed above. In Fig. 3 illustrates the absorption of the graphene coupling with SiO2 (ε = 2.1) nanorods [green open squares]. Compared with the case of SiNx having an identical filling factor, e.g., f = 0.0654, d = 55 nm, and Λ = 841 nm [black open circles], the absorption linewidth is sharpened though accompanying with the detuning of the absorption band, which can be ascribed to the smaller refraction index contrast and the weaker scattering. Meanwhile, the graphene absorption varies

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from ∼ 50% to ∼ 20% as a result of the γR /γD variation. By simply scaling the period Λ of the nanorod arrays, such spectral shift can be compensated and tailored with nearly unchanged γR /γD [red open uptriangles and blue downtriangles] [50]. This is particularly useful in the design of novel graphene-integrated photodectors or modulators with ultranarrow linewidth. Finally, in order to break the upper limit of 50% absorption illuminated by a single source, we modify the ‘two-port’ resonator system to a ‘single-port’ configuration by adding a ‘mirror’ to the system. This ‘mirror’ eliminates the transmission of light, giving rise to t = 0 and r = 1 in the direct coupling process. According to similar procedures discussed in ‘two-port’ system, the R(ω) in ‘one-port’ configuration is derived as R(ω) = |

j(ω − ω0 ) + (γD − γR ) 2 | . j(ω − ω0 ) + (γD + γR )

(9)

As a result, the absorption A(ω) = 1 − R(ω): A(ω) =

4γD γR . (ω − ω0 )2 + (γD + γR )2

(10)

The results reproduce that of ‘one-port’ configuration in the Lorentz resonator in which the direct process is absent [27]. From the analytical results, it is clear that the graphene absorption can reach 100% as the critical coupling condition is fulfilled on resonance. In numerical calculations, the ‘mirror’ can be either a dielectric Bragg multilayer or an ideal metal reflector [the inset of Fig. 3] [27–29]. For simplicity, lossless silver is employed in the loaded nanostructure. As expected, total absorption (i.e., 100%) of graphene is achieved by assuming the Λ = 831 nm and d = 109 nm, respectively [solid circle line].

2 .0 0

0 .0 5 0 0 0

0 .1 0 0 0

0 .1 5 0 0

0 .2 0 0 0

0 .2 5 0 0

(a )

0 .3 0 0 0

0 .3 5 0 0

0 .4 0 0 0

0 .4 5 0 0

0 .5 0 0 0

(b )

s -p o la riz e d F r e q u e n c y ( µm

-1

)

p -p o la riz e d

1 .5

1 .0 0

2 0

4 0

In c id e n t a n g le (d e g re e s )

6 0 0

2 0

4 0

6 0

In c id e n t a n g le (d e g re e s )

Fig. 4. Graphene absorption spectra of the hybrid nanostructures as a function of frequency and angle of incidence for s-polarized (a) and p-polarized (b) illumination. It is clearly that enhanced light-graphene interactions occur over broad range of incident angles for both sand p-wave. However, the absorption linewidth is much narrower for p-polarized light as a result of smaller external radiation loss rate of the geometric resonance. In the simulations, the period Λ is assuming to be 841 nm and the rod width d is 55 nm.

Considering the dependence of the geometric resonance on the incident angle and the light polarization, the light-graphene interaction is expected to show strong correlations with these -1 -1 two factors. In Figs.F 4(a) F r e q u e n c y of ( µm the) absorption specr e q u e and n c y 4(b) ( µm are ) plotted the angle dependence tra on light frequency and polarization for d = 55 nm of the loaded nanorod array. Explicitly, there are broad angular ranges in which enhanced light-graphene interaction occurs for both #240969 © 2015 OSA


s- and p-polarization (i.e., the electric field vector perpendicular to the axis of the rods). For s-polarization, the enhanced lowest-order graphene absorption at 1.176 µm−1 under normal incidence splits as incident angle increases, however, with similar linewidths preserved. For ppolarization, there is no graphene absorption enhancement as a result of the absence of coupling between the induced ‘dipoles’ pointing in the x direction at normal incidence. The enhanced graphene absorption is enabled due to the geometric resonance excitations at oblique incidence with a much narrower spectral linewidth compared with that of s-polarization. The difference can be ascribed to the way of coupling between the induced ‘dipoles’ with different polarizability tensor [43]. For p-polarized illumination, like the guided resonance supported in photonic slab [51], γR is much smaller than that of s-wave. 4.

Conclusion

To conclude, we have theoretically investigated the enhanced interactions between light and graphene coupled with a single layer of nonlossy dielectric nanorods at near-infrared wavelengths. By TCMT for Fano resonance and numerical simulations, the study make it evident that the magnitude of the graphene absorption is tunable with the adjustment of the ratio γR /γD . Such graphene absorption and linewidth are readily tailored by scaling the dimension of the nanostructure, modifying the structural configuration, selecting different f , or using different dielectric materials. Total absorption or ultranarrow absorption linewidth is achieved. Finally, the enhanced light-graphene interactions are strongly dependent on the incident angle and light polarization. We believe the results could directly inspire a wide range of high-performance optoelectronic devices, e.g., ranging from graphene-integrated quasimonochromatic photodetectors, ultra-compact modulators, to other two-dimensional material incorporated applications. Acknowledgments The research was supported by the Natural Science Foundation of China (Grant Nos. 11104187 and 61376010).

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