Hexagonal Boron Nitride Self-Launches Hyperbolic. Phonon Polaritons

Transcription

1 Hexagonal Boron Nitride Self-Launches Hyperbolic Phonon Polaritons Leonid Gilburd, Kris S. Kim, Kevin Ho, Daniel Trajanoski, Aniket Maiti,, Duncan Halverson, Sissi de Beer,, and Gilbert C. Walker, * Department of Chemistry, University of Toronto, 80 St. George Street Toronto, Ontario M5S 3H6, Canada Department of Physics, Indian Institute of Technology, Kanpur, , India Materials Science and Technology of Polymers, MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, the Netherlands * Corresponding author: gwalker@chem.utoronto.ca 1

2 SUPPLEMENTARY INFORMATION Dielectric functions of hbn. As described by S. Dai and co-authors 1, the dielectric functions of hbn can be approximated by a single-lorentzian function. LO TO 2 2 i TO 2 2 SI Figure 1 shows the relevant range of the calculated in-plane and out-of-plane complex dielectric functions based on the LRB and URB, which are found at cm -1 and cm -1, respectively. We use the same broadening values -1 4cm and 5cm 1. The macroscopic dielectric constants for both in- and out-of-plane components are found in this reference 2. SI Figure 1. The calculated in- and out-of-plane hbn dielectric functions. Blue and orange lines represent the real and the imaginary parts of the (b) in- and (a) out-of-plane dielectric functions. 2

3 The thickness of the SiO2 layer. The thickness of the silicon dioxide (SiO2) layer is calculated from XPS spectra using a standard technique 3 5. The thickness (t SiO2 ) is given by: exp t SiO2 = λ SiO2 sin(θ) ln[( 1 β ) (I SiO 2 exp ) + 1] where λ SiO2 is the attenuation length of the Si 2p photoelectrons in SiO2, θ is the angle between the sample surface plane and the detector, β = I SiO2 I (where I SiO2 and I Si are the Si 2p electron Si intensity from infinitely thick SiO2 and Si, respectively), and I exp SiO2 I Si and I exp Si are the experimentally measured Si 2p electron intensities from SiO2 and Si, respectively (data not shown). λ SiO2 = 2.7 nm θ = 20 o β = 0.83 t SiO2 = 0.6 nm Because the measured SiO2 layer is thin, it is neglected in our calculations; i.e. in the dispersion relation calculation we assume an infinite Si layer under hbn crystal with the dielectric constant of e Si =

4 Calculation of the dispersion relation of hbn. We follow the phenomenological model proposed by S. Dai and co-authors 1, which is derived from the Fresnel equations for a three-layer (air-hbn-si) structure. A more detailed derivation is found in the Supplementary Information of Ref. 1 The black dashed lines in Figure 2 of the main text show the polariton branches, given by the large momentum approximation from the same reference 1. As mentioned above, the main difference in our calculation compared with that in Ref. 1 is that the silicon oxide layer is not included. 4

5 The topography of the fold from the main text. To characterize the topography at the fold (from Figure 1 in the main text) and to avoid scan artifacts due to convolution with the AFM probe, scans at different angles relative to the fold were collected and are shown in SI Figure 2. These data show that the fold is asymmetric, which, as mentioned in the main text, causes a small optical phase shift in the IR profiles in the immediate vicinity of the fold feature. SI Figure 2. Asymmetry of the fold. AFM topography images (a-c) of the fold at different probesample orientations collected after physically rotating the underlying sample relative to the AFM probe. (d) The corresponding height profiles show different material slopes on either side of the fold. 5

6 Near-field IR microscope. The microscope (Inspire, Bruker) employs the homodyne s-snom detection technique 7 9. The microscope utilizes a quantum cascade IR laser (MirCat, Daylight Solutions) as a light source which is focused at the apex of a metallic AFM tip. The back-scattered IR field is interferometrically homodyned with a reference field. The detected field is demodulated at harmonics of the tip-oscillation frequency and the second, third and fourth harmonics are recorded. The Inspire software utilizes Bruker s interleave mode in which the AFM tip repeats the same line scan twice, allowing the interferometer to move the reflector distance equivalent to /4 of the applied excitation wavenumber. This allows one to semi-simultaneously obtain the back-scattered in- and out-of-phase IR signals from the tip, which carry the sample s reflection and absorption information, respectively. The Inspire microscope is equipped with a variety of different modes, allowing to simultaneously characterize the sample and obtain information about the nanoscale electrical, mechanical and chemical properties. In this work, only the Tapping IR mode was used. 6

7 Signatures of the asymmetry of the fold in the field phase. For the sake of the derivation, we define the obtained in- and out-of-phase demodulated (at either second or third harmonics of the AFM tapping frequency) signals, E in and E out, respectively. These are projections of the imaginary field vector on the real and imaginary (orthogonal) axes. To normalize the field, we use the following relationship: E i norm = E i E 2 2 in + E out i = in, out The phase of the normalized field is then given by: φ(x) = atan ( E out norm (x) E norm in (x) ) For the case in which the waves are self-launched at the fold we investigate the effect of the fold s asymmetry on the HPhPs propagating into the flake. By taking the phase difference of the field in each direction with respect to the topographical center of the fold we observe that the phase asymmetry does not extend beyond ~0.2 µm on either side of the fold (see SI Figure 3). The phase difference is calculated using the following equation: φ(x) = φ(x) φ( x) where x = 0 at the topographical center of the fold. The results show that the tip detects an additional field in the vicinity of the slopes of the fold. The localization of the field, whose decay length is smaller than the surface wavelength of HPhPs, suggests that a possible origin of this additional field is the near-field decay of the HPhPs at the slope. Since the metallic AFM tip is 7

8 sensitive to p-polarized fields, the amplitudes of the p-polarized components of the evanescent fields perpendicular to the sloped regions contribute to the detected signal near the fold. SI Figure 3. Difference in phases of hbn self-launched HPhPs. The difference in the phase of the fields ( (φ)) of the hbn self-launched HPhPs propagating away from the fold (topography shown in black) in either direction is shown at 1490 cm -1 (blue), 1510 cm -1 (orange), 1541 cm -1 (yellow) and 1581 cm -1 (purple). The asymmetry of the fold gives rise to a local asymmetry in the phase of the fields. The asymmetry disappears as the tip moves more than ~0.2 µm beyond the fold. This suggests that the local asymmetry of the phase of the fields arises from the asymmetry of the fold and provides additional contribution to the total field sensed by the tip in the vicinity of the fold. The IR responses are shifted up for clarity. 8

9 The simulations of the tip-launched and hbn self-launched HPhPs. The simulations are based on the proposed algorithm by S. Dai et al. 1, which assume that the tip is launching waves in hbn and the sum of the intensities reflected back to the tip is detected. In addition to the reflections, here we add another field, which represents the fold-launched wave. To reproduce the experimental in Figure 1b in the main text we use a 2D simulation (see Figure 1c). To reproduce the cross-sections shown in Figure 3 of the main text we use a simplified 1D simulation, which calculates the IR signal in cross-sections perpendicular to the fold. In the 1D simulation we neglect the contribution of the reflected wave from the upper edge of the crystal. Both, the tip- and the fold-launched waves have the same loss factor, which is found to be in the range of at 1510cm -1. To match the standing wave at the edge (of the upper flat crystal surface), the reflection coefficient is found to be r i. No reflection coefficient has been implemented at the fold because no reflection is detected there. The fold is modelled as a narrow strip, wave launching source.. The amplitude of the fold-launched waves in the simulation is Afold 0.25, which is one quarter of the amplitude of the tip-launched waves. The simulated IR profile is shown in SI Figure 4. Note, due to the comparison to the out-of-phase experimental results, the imaginary part of the simulated fields are shown. 9

10 SI Figure 4. Simulation of the hbn HPhPs. A line profile of the simulated result (orange) and the corresponding experimental result (blue) obtained at 1510 cm -1 are compared. A topography profile is shown in black. 10

11 How hbn self-launches HPhPs. To further illustrate the experimentally observed fringes around the fold, we introduce another simulation. In this simulation we included neither the tip-launched HPhPs nor the reflections from the edges. This simulation shows the effect of the material self-launching HPhPs at all points across the surface. Due to the slopes of the fold, the overall fold has more IR-active material per lateral displacement than do the adjacent, flat regions of the sample. For this reason, waves launched at the fold are injected into the crystal with an extra phase relative to any other point on the crystal. This is supported by the experimental results (see Figure 3 in the main text), which show an increased out-of-phase signal on top of the fold that can be caused by a larger volume of the absorptive material at the fold. In addition, the sub-wavelength dimensions of the fold scatter the incoming field differently than any other point on the crystal. A simulation was conducted in which energy injection from all points along the surface of the crystal was considered. The energy in the thin crystal propagates in a 3D cone-like shape. A 2D depiction of these propagating waves on the surface is illustrated in the inset of SI Figure 5a. Thus, the injection of the waves from the fold into the crystal is simulated with a different amplitude and phase. The sum of the injected fields are then calculated at any point across the surface. The calculated field has a generic form of: r 2 r E Aexp cos where r is the distance between any two points, is the corresponding wavelength of the HPhP, and A and are the relative injection amplitude and phase. In the representative example shown in SI Figure 5a any point at the fold injects 10% less field and has 0.04π radian shift compared to 11

12 any other point on the crystal. The average amplitude of the field in the simulation was normalized to match the experimental DC background. SI Figure 5b shows a comparison of the line profiles of the simulation with the experimental data. SI Figure 5. Simulation of the self-launched HPhPs. (a) A 2D simulation reproducing oscillations similar to the experimentally observed pattern. In the simulation, 2D waves similar to those shown in the inset are launched from each point in space. (b) A line profile of the simulated result (orange) and the corresponding experimental result (blue) obtained at 1510 cm -1 are compared. A topography profile is shown in black. 12

13 Fragments of hbn on flat hbn crystals. Examples of hbn fragments on a flat surface of an hbn crystal are shown in SI Figure 6. There is discontinuity between the layered structure of the crystal and the disordered structure of hbn fragments. These hbn fragments are highly IR active in the range of excitation and thus can act as a scattering center for the HPhPs inside the crystal. Auger Electron Spectroscopy (AES) results confirm that the chemical composition of the fragments is similar to the underlying crystal (see SI Table 1). The simulated damping coefficients (SI Figure 6a) show that the highest sensitivity is expected around 1500 cm -1. The IR out-of-phase responses, shown in the middle section of SI Figure 6c-e, were collected at 1500 cm -1. As seen, the detected fringe spacing is of the standing wave. For visual confirmation, the standing wave pattern occurring at the edge of the underlying crystal is shown in the middle section of SI Figure 6e. Tip-launched waves can scatter from these fragments and travel within the crystal in a cone-like pattern, which are then detected by the tip. Not all fragments detectably scatter the HPhPs, as seen from the comparison of fragments in SI Figure 6c-e, leading us to infer that closeness between the lattices of the fragment and underlying hbn is relevant. 13

14 SI Figure 6. Fragments of hbn on a flat hbn crystal. (a) Simulated damping coefficient of HPhPs, calculated from the dispersion relation. b, An AFM topography image of a flat hbn crystal with fragments of hbn on top (indicated by yellow, pink and blue squares). (c-e) The corresponding AFM topography, IR out-of-phase signal and SEM side profiles of the color-coded hbn fragments in (b). Scale bar in (b) is 5 µm and in (c-e) are 500 nm. 14

15 SI Table 1. Elemental composition of hbn fragments. The relative elemental composition (on / off the fragments in %) as measured by AES reveals similar BN composition on- and off- the fragments, indicative of the same material. The colors in the table correspond to the color-labels of the fragments in SI Figure 6. A uniform presence of carbon is also observed. Carbon functionalities are not IR active in the range of URB of hbn and thus do not have a significant effect on the visibility of HPhPs. Boron Nitride Carbon Yellow / / / Pink 9.07 / / / Blue / / /

16 Other hbn folds and their IR response. We have studied many different hbn crystals that exhibit folds. In addition to the one in the main text, here we show more crystals and the corresponding IR responses (see SI Figure 7). As one can see, on each of those crystals there are oscillations parallel to the folds (self-launched waves), as well as double-frequency oscillations (standing waves) close to edges. SI Figure 7. Different hbn crystals with folds and their IR responses. Topography maps are on the left (a,c,e,g) and their corresponding IR responses are on the right (b,d,f,h). Excitation wavenumbers are noted. 16

17 Closer investigation of another hbn fold SEM image. SI Figure 8 shows the fold indicated in SI Fig 7d, accompanied by an SEM image. SI Figure 8. The fold of hbn that generates the signal in SI Figure 7d, above. (a) AFM topography image and (b) SEM side view of the fold. 17

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