Graphene mechanical pixels for Interferometric Modulator Displays

Electro-optic modulators based on micro-electromechanical systems have found success as elements for optical projectors, for simplified optical spectrometers, and as reflective-type screens that make use of light interference (Interferometric Modulator Display technology). The latter concept offers an exciting avenue for graphene nanomechanical structures to replace classical micro-electromechanical devices and bring about enhancement in performance, especially switching speed and voltage. In this work we study the optical response of electrically actuated graphene drumheads by means of spectrometric and stroboscopic experiments. The color reproducibility and speed of these membranes in producing the desired electro-optic modulation makes them suitable as pixels for high refresh rate displays. As a proof of concept, we demonstrate a Graphene Interferometric Modulator Display prototype with 5 μm-in-diameter pixels that compose a high resolution image (2500 pixels per inch)—equivalent to a 5″ display of 12K—whose color can be changed at frame rates of at least 400 Hz.


Supplementary Figures
Supplementary Figure 1: Optical microscope image showing the light collection area of the spectrometer. Here the spectrometer was replaced by a light source to project a spot on the sample.

Supplementary Note 1: Area measured by spectrometer
For the spectrometer measurements, we used a focused fiber to collect the reflected light from the graphene drumheads. In order to align the fiber so it points and covers the desired device, we first connect the fiber to a light source so we can observe a bright spot on top of the sample.
Supplementary Figure 1 shows the light spot of 27.5 µm in diameter located away from the devices, covering an area of graphene on SiO 2 .

Supplementary Note 2: Capacitance of the deflected circular membrane
We apply a 1 degree of freedom graphene membrane model that neglects bending rigidity, and describes the drums' deflection by an axisymmetric parabolic profile given by where δ c is the center deflection, a is the radius of the drum, and r is the radial distance away from the drum's center. The capacitance of the circular membrane is now given by: We defineδ = δc g 0 , Y = 1 − r 2 a 2 , and C 0 = 0 πa 2 g 0 , and re-express Supplementary Equation 1 as: Supplementary Note 3: Electrostatic force on a circular graphene membrane We obtain the electrostatic force under the effect of an applied voltage V by taking the spatial derivative of the electrostatic potential energy, thus:

Supplementary Note 4: Force equilibrium equation
The deflection of the graphene membrane, if the perfect hermeticity assumption is to be maintained, requires that the force balance equation accounts for the hydrostatic pressure that develops from the compression of the gas trapped within the cavity and cannot escape it. To calculate this last effect, we assume that the parabolic profile of membrane deflection applies and that the compression is isothermal, thus applying the perfect gas law the gas pressure within the cavity reads: (Supplementary Equation 4) where A is the area of the circular membrane, P 0 is the initial (ambient) pressure, and the factor 1 2 in the denominator on the rightmost hand side term comes from integrating the area under the parabolic profile 1 . Thus the hydrostatic force acting on the membrane due to the change in the cavity pressure can be written as: Note that Supplementary Equation 5 applies in case of both positive and negative deflection, i.e. positive and negative differential pressure.
Combining all the terms we obtain the following force equilibrium equation: Supplementary Equation 6 can be rewritten in the following non-dimensional form: where the non-dimensional parameters are given ask 3 = k 3 g 2 0 /k 1 , β = AP 0 /k 1 g 0 , andV 2 = C 0 V 2 /k 1 g 0 .
In Supplementary Figure 2  is negligibly small compared to the travel path term. By integrating the reflectance over the drum's area, we obtain: where Y = 1 − r 2 a 2 . Dividing Supplementary Equation 9 by πa 2 to obtain the drum's average reflectivity, and integrating we arrive at: In order to plot these values as trajectories on an RGB color triangle, the following coordinate transformation is applied 3 : x = Blue Red + Green + Blue y = Green Red + Green + Blue In order to find out what would be an optimal thickness, i.e. number of graphene layers, for a GIMOD pixel we resort to more detailed simulations. These simulations are based on a full optical model that accounts for the multiple reflectance and absorbance of the successive optical layers, i.e. air-graphene-air-silicon, as described in 5 . According to this optical model, the reflectance of a 14 graphene layer suspended on top of a Silicon cavity is given as:

(Supplementary Equation 14)
where r 1 and r 2 are the Fresnel reflection coefficients of air-graphene and air-silicon interfaces, respectively. φ 1 and φ 2 are the phase changes induced by the optical path through the graphene and the cavity respectively.
Next we maintain the assumed parabolic deflection profile of the graphene film, this is reasonable since literature suggests that 2D materials continue to act as membranes for thicknesses up to 50 layers 6 . In order to obtain the drum averaged reflection of the GIMOD, we perform the following integral: where L * is the "Lightness" obtained from the CIE standard from: Whereas the color difference (∆E) between two points in color space is defined as: We repeat the previous simulation for a range of graphene layers going from 1 to 50, the resulting contrast ratio and color difference obtained are shown in the two dimensional plots of Supplementary Figure 11. From these simulations we conclude that the number of graphene layers that provide the highest contrast ratio and richest color gamut is around 29 layers, with a corresponding optimal cavity gap of around 390 nm. Note that color difference values have a maximum value around gap of 740 nm, however that cavity depth corresponds to a low contrast ratio. A gap on the order of 400 nm thus offers the best values for both color difference and contrast ratio.