Supporting Information Launching and control of graphene plasmon by nanoridge structures Sanpon Vantasin, Yoshito Tanaka,* Tsutomu Shimura 1. Launching and stationary modes of single nanoridge structure Figure S1 S4 shows launching and stationary modes of plasmon coupling by nanoridge with 100 250 nm width. These figures represent the exactly same system as in Figure 2 of the main manuscript. The λ ex symbols denote excitation wavelength. Figure S1. Launching and stationary modes of 100 nm ridge. Every image in this figure shares a same color scale. Figure S2. Launching and stationary modes of 150 nm ridge. Every image in this figure shares a same color scale.
Figure S3. Launching and stationary modes of 200 nm ridge. Every image in this figure shares a same color scale. Figure S4. Launching and stationary modes of 250 nm ridge. Every image in this figure shares a same color scale. 2. Ridge curve length calculation As mentioned in the Method section of the main manuscript, the nanoridges are modeled as circular curves, both on main ridge body and at the corner, presented in Figure S5 as the curves with radius r 1 and r 2, respectively. For every numerical simulation in the main manuscript, r 2 was defined as r 1 /10. In the figure, h and w are reported width and height of the ridge, respectively.
Figure S5. Model of nanoridge. (Only right half of the ridge is shown.) From the diagram in Figure S5, it is obvious that and curve length 2 2.2 arcsin /. The ridge curve length in the case of 30 nm ridge height is shown in Figure S6. Figure S6. Ridge curve length plot against ridge width, in the case of 30 nm height. The length under nanoridge is 2.2 sin. The difference between ridge curve length and length under the ridge is important, because it indicate the extra distance when SPP wave follows ridge curve, compared to SPP wave propagating on flat graphene. This difference is shown in Figure S7. As discussed in the main manuscript, smaller ridge gives SPP wave more extra distance compared to larger ridge, resulting in more phase delay.
Figure S7. The difference between ridge curve length and under ridge length, plot against ridge width, in the case of ridge with 30 nm height. 3. Relationship of excitation wavelength and plasmon wavelength Forati et. al. 1 explained (in the Supplemental Materials) that graphene plasmon wavelength and excitation wavelength have a relationship: 1 2., where λ 0, λ spp, η 0, and σ are plasmon wavelength, excitation wavelength, impedance of free space, and conductivity of graphene, respectively. Noted that the conductivity is a complex value and the imaginary part must be included in the calculation. The result plasmon wavelength, however, is considered only for its real part. For the numerical simulation, plasmon wavelength can be acquired from damped harmonic fitting as explained in the main manuscript. The relationship between plasmon wavelength and excitation wavelength, both for analytic from the equation above, and for the numerical simulation with several ridge sizes, is presented in Figure S8. Figure S8. Graphene plasmon wavelength plotted against excitation wavelength.
As discussed in the main manuscript, there is no difference in plasmon wavelength between the plasmon launched by nanoridges of different sizes because we consider plasmon that launched onto flat graphene (not on the ridge). In this case, the plasmon wavelength does not dependent on the coupler (ridges, gold antennae, tips, etc.), but the permittivity of graphene and air. The close similarity between numerical result and analytic values confirms the validity of the simulation. 4. Substrate effect The effect of refractive index of substrate (media under graphene) is shown in Figure S9 and Figure S10. The discussion about this effect is in the main manuscript. Figure S9. Amplitude of SPP wave launched from a nanoridge with width of 150 nm and height of 30 nm, on the substrate with permittivity of 1.0, 1.2, 1.4, and 1.6. Figure S10. Plasmon wavelength of SPP wave launched from a nanoridge with width of 150 nm and height of 30 nm, on the substrate with permittivity of 1.0, 1.2, 1.4, and 1.6.
5. Symmetric plasmon launching of symmetric double ridge structure In the analytic model of symmetric double ridges, plasmon launched to left and right side has exactly same parameter, therefore there is no different in the amplitude between each side. Figure S11 shows that the amplitude in both side are same in the numerical simulation as well. The small variation in the left/right amplitude ratio is due to the calculation error (mainly meshing). However, this variation never exceeds 1% for any data point. The standard deviation of this ratio is less than 0.0025 for each data set. This emphasize the importance of the different phase delay for each side in asymmetric double ridge system, which allows asymmetric plasmon launching. Figure S11. Left/right plasmon amplitude ratio for symmetric double nanoridge system. 6. Separation-independent and separation-dependent stationary modes of symmetric double nanoridge systems Figure S12 and Figure S13 depict separation-independent and separation-dependent stationary mode of double 100 nm ridges and double 150 nm ridges, respectively. Each separation-independent stationary mode has a corresponding single ridge stationary mode in Figure S1 and Figure S2, this indicate that the separation-independent stationary modes are intra-ridge phenomenon, and can be explained by the mechanism of single ridge plasmon coupling. The separation-dependent modes are obviously inter-ridge phenomenon. Figure S12. Separation independent and separation dependent stationary modes of double 100 nm ridges.
Figure S13. Separation independent and separation dependent stationary modes of double 150 nm ridges. 7. Plasmon amplitude of asymmetric double ridge system Asymmetric double ridge structure launch plasmon wave with different amplitude to the left and right side. Using the analytic model in the manuscript, the amplitude values for various condition can be calculated. The amplitude values are shown in Figure S14. The log 10 of ratio between the two datasets resulting in Figure 6A of the main manuscript. Noted that, in the case of amplitude calculation of symmetric double ridge (Figure 5), the initial phase of the SPP wave launch from each ridge is always identical, thus the phase difference is ridge separation divide by plasmon wavelength, plus phase delay when SPP wave climbs over another ridge. In the case of asymmetric double ridge, the initial phase of each SPP wave (Figure 2B of the main manuscript) have to be accounted, in addition to the phase difference from ridge separation and phase delay. Figure S14. Amplitude of plasmon launched onto left and right side of symmetric double nanoridge structure. 8. Effects of Fermi energy Figure S15 and Figure S16 represent the effect of graphene Fermi energy in the plasmon launching on single nanoridge system. Fermi energy plays a big role in the permittivity of graphene, this is reflected on the change in the position of stationary and launching modes. The decay rate presented in the Figure S16 is the decay coefficient C in the damped harmonic fitting function e sin. The smaller decay rate for higher Fermi energy can be simply explain as more free electrons in the conduction band, resulting in lower loss.
Figure S15. Effect of Fermi energy of graphene on the launched amplitude of the 150 nm wide, 30 nm high single nanoridge system. Figure S16. Effect of Fermi energy of graphene on the plasmon decay rate of the 150-nm wide, 30-nm high single nanoridge system. Since the plasmon amplitude (and phase) at certain excitation wavelength is affected by Fermi energy, the adjustment of Fermi energy (by doping or gate tuning) can be applied to control the direction of unidirectional plasmon launching in the double ridge system. In Figure S17, unidirectional plasmon launching of double asymmetric ridges with 100 and 150 nm width, 30 nm height, and 275 nm separation is presented. At the 3.58 µm excitation wavelength and 0.5 ev Fermi energy, which is the same condition as in the main manuscript, SPP wave is launched to the right side. With the exactly same structure and illumination, but with 0.56 ev Fermi energy, the SPP wave is launched to the left instead. Noted that this
is just an example of many setups that provide such phenomenon, rather than the optimized condition for highest directivity (left/right amplitude ratio). Nevertheless, it demonstrates the active control of unidirectional launching, as an extension of the passive control unidirectional launching in the main manuscript. Figure S17. Unidirectional plasmon launching of asymmetric double ridge system with 275 nm separation and 3.58 µm excitation wavelength, at 0.50 and 0.56 ev Fermi energy. 9. Effect of radius of curvature ratio between main body of ridge and ridge-flat junction Figure S5 present the double circular curve of structure of nanoridge model in this study. The radius of curvature of the main ridge body, r 1, is governed by the width and height of nanoridge. The radius of curvature of the ridge-flat junction, r 2, is arbitrarily set as / 10 for all simulation presented in the main manuscript. Here, we show that this ratio does not significantly affect the result if the ridge curve length is preserved. (Although r 2 should be kept quite smaller than r 1 otherwise the ridge shape would be very unnatural.) Figure S18 presents the amplitude of SPP wave launched from a single nanoridge system with ridge curve length of 182 nm (in the case of / 10, it is the same condition as the ridge with 150 nm width and 30 nm height in the main manuscript). Figure S18. Amplitude of SPP wave launched from a single nanoridge with a curve length of 182 nm, with the / ratio of 6, 8, and 10. 10. Controlling the wavefront of launched SPP wave
Figure S19 present plasmon launching by using nanobump, which is an analog of nanoridge, but with curved in both X and Y axes instead of just X axis. Note that this simulation was done in three dimensions. Also called as graphene nanobubble, this structure can be naturally occurred in graphene on boron nitride, 4 and can also be induced by substrate sublimation. 5 The launched SPP wave in Figure S19 clearly shows circular wavefront, following the shape of the launching structure. This, together with the planar wavefront of SPP wave from nanoridge, demonstrated that the wavefront of launched SPP wave can be controlled by the morphology of the launching structure. The wavefront control by the shape of gold antenna was previously developed, 6 but the wavefront control by graphene ridge and bump has an advantage that no foreign entity is added to the system. Figure S19. SPP wave (represented by electric field perpendicular to the grapehen plane) launched from a bump strcutrue with 10 nm height and 100 nm width. The excitation wavelength is 3.5 µm. 11. Effect of nanoridge height Figure S20 presents the launched amplitude from a single nanoridge with 200 nm width and 10 50 nm height. As discussed in the main manuscript, the launched amplitude increase following the ridge height. The plasmon wavelength and decay rate are unaffected by ridge height due to the same reason that they are not affected by ridge width, as already discussed. Figure S20. Launched plasmon amplitude of the single ridge system with 200 nm width and 10 50 nm height.
12. SPP launching efficiency of nanoridge Figure 21 presents a comparison between graphene SPP launching by graphene nanoridge and by a gold nanoparticle. The nanoridge is 100 nm in width and 30 nm in height. The gold nanoparticle has 100 nm diameter. The gold nanoparticle is floating 2 nm above flat graphene. The excitation wavelength is chosen to provide a launching mode from nanoridge, maximizing launch amplitude. This demonstrates that, for normal illumination, size of gold nanoparticle comparable to the 100-nm nanoridge, and excitation wavelength for the launching mode, nanoridge is can launch SPP wave more efficiently than a gold nanoparticle. Figure S21. E y image plot presenting a launching of graphene SPP by A) 100-nm wide, 30-nm high graphene nanoridge and B) 100-nm diameter gold nanoparticle. The two images share a color scale. 13. Multiple nanoridges Figure S22 demonstrate a use of multiple nanoridge to amplify launch SPP amplitude. The ridges have 100 nm width, 30 nm height, and 250 nm center-to-center separation. The mechanism is exactly same as the case of double ridge in the main manuscript, as ridge separation and phase delay resulting in the constructive/destructive interference for some plasmon wavelength. As the ridge number increase, the structure become more similar to grating and the allowed band of excitation frequency become narrower.
Figure S22. SPP amplitude launched from 1 to 5 ridge(s) with 100 nm width, 30 nm height, and 250 nm center-to-center separation between each ridge. References 1 E. Forati, G. W. Hanson, A. B. Yakovlev and A. Alù, Planar hyperlens based on a modulated graphene monolayer, Phys. Rev. B, 2014, 89, 081410. 2 Z.-Y. Ong and M. V. Fischetti, Theory of interfacial plasmon-phonon scattering in supported graphene, Phys. Rev. B, 2012, 86, 165422. 3 S. Dai, Q. Ma, M. K. Liu, T. Andersen, Z. Fei, M. D. Goldflam, M. Wagner, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, G. C. a. M. Janssen, S.-E. Zhu, P. Jarillo-Herrero, M. M. Fogler and D. N. Basov, Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial, Nat. Nanotechnol., 2015, 10, 682 686. 4 E. Khestanova, F. Guinea, L. Fumagalli, A. K. Geim and I. V. Grigorieva, Universal shape and pressure inside bubbles appearing in van der Waals heterostructures, Nat. Commun., 2016, 7, ncomms12587. 5 J. H. Lee, J. Y. Tan, C.-T. Toh, S. P. Koenig, V. E. Fedorov, A. H. Castro Neto and B. Özyilmaz, Nanometer Thick Elastic Graphene Engine, Nano Lett., 2014, 14, 2677 2680. 6 P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso and R. Hillenbrand, Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns, Science, 2014, 344, 1369 1373.