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Nanomechanical electro-optical modulator based on atomic heterostructures

Two-dimensional atomic heterostructures combined with metallic nanostructures allow one to realize strong light–matter interactions. Metallic nanostructures possess plasmonic resonances that can be modulated by graphene gating. In particular, spectrally narrow plasmon resonances potentially allow for very high graphene-enabled modulation depth. However, the modulation depths achieved with this approach have so far been low and the modulation wavelength range limited. Here we demonstrate a device in which a graphene/hexagonal boron nitride heterostructure is suspended over a gold nanostripe array. A gate voltage across these devices alters the location of the two-dimensional crystals, creating strong optical modulation of its reflection spectra at multiple wavelengths: in ultraviolet Fabry–Perot resonances, in visible and near-infrared diffraction-coupled plasmonic resonances and in the mid-infrared range of hexagonal boron nitride's upper Reststrahlen band. Devices can be extremely subwavelength in thickness and exhibit compact and truly broadband modulation of optical signals using heterostructures of two-dimensional materials.

(1) An equivalent expression can be found for a . We can then use this value to calculate the Maxwell stress experienced by each dielectric 1 , where 0 = 8.854 × 10 −12 F/m is the permittivity of free space. This yields a Maxwell pressure of around 20 atm (atmospheres) at the air-hBN interface at gating voltage of 100 V. The Grüneisen parameter is close to zero for most phonon modes in hBN, including the transverse optic phonon mode shown in figure 4, see Ref. 2 , meaning that the modulation of the absorption feature at ~7.35 µm can only be attributed to the movement of the hBN flake.

Supplementary Note 2 | Optical Pauli blocking and the Fermi energy
In the modelling we have used values for the Fermi energy calculated with the help of a simple capacitance model. In order to check these calculations we have measured gated optical spectra of our devices at low gate voltages such that the onset of motion in the graphene/hBN stack was not yet reached due to mechanical hysteresis. As a consequence, the optical reflection of the device only changes due to the effect of optical Pauli blocking.
Supplementary Figure 5 shows the spectral dependence of relative reflection of our device in the wavelength range 2-7 m (the relative reflection was measured as R(V g )/R(0)). The black and red curves show the relative spectra at V g = 20 V and V g = -20 V. Since the initial doping of our exfoliated graphene was high, these spectral curves are close to unity, which is characteristic of the forbidden interband transition (and negligible intraband contribution). However, at V g = -50 V we see change in the reflection centred around twice the Fermi energy position (at a wavelength of 3.8 m) which is in a good agreement with a simple capacitance model for the Pauli Blocking (provided an initial doping is taken into account).

Supplementary Note 3 | Modulation frequency and amplitude
To estimate the maximum modulation frequency of our device we approximate our device as a slab of hBN vibrating in one dimension with both ends fixed, see Supplementary Figure 6.
In our case ≈ 50 μm is the length of unfixed hBN that is free to vibrate and ≈ 110 nm is the thickness of the hBN. We shall take the hBN width (going into the page) to be approximately equal to . Euler-Bernoulli beam theory gives the fundamental frequency of the simply supported hBN plate as 3 where ≈ 70 GPa is the Young's modulus of the hBN and = 2.

Supplementary Note 4 | Frequency dependence of modulation
As mentioned in the main text, high speed experimental characterization of the device was beyond the scope of this work, particularly as the device was not optimized for high frequency operation (e.g. we used graphene as one of the contacts, the area size was not optimized, etc.). However, we did measure the optical response of our modulator in the frequency range of 100 Hz -100 kHz available in our setup. The fit is quite good except in the high frequency range approaching ~100 kHz where we see a shallow peak of the experimental response (which is most probably connected with the mechanical resonance). In order to describe the combined electro-mechanical properties of the device we have fitted the measured data as the product of a harmonic oscillator response and the overdamped electrical response (green line in Supplementary Figure 8b), which fits the experimental data extremely well. This fit provides a mechanical resonance frequency of ~120 kHz, which is in good agreement with estimates described in Supplementary Note 3.