### A Graphene Drum

September 22, 2014

The cantilever, as shown in the figure, is a simple mechanical object whose properties are defined by its dimensions and the properties of the material from which it is made. Cantilevers can be used as the basis of a variety of devices, as I've written in some previous articles (Weighing Attograms, January 27, 2014; Energy Harvesting Cantilevers, January 14, 2013; Combinatorial Research, December 12, 2011; Cantilever Energy Harvesting, August 16, 2011; and Paper Accelerometer, March 10, 2011.

The mechanical properties of a cantilever of uniform cross-section are defined by its length L, width w, and thickness t, along with the material properties of Young's modulus E, and density ρ.

(Illustration by the author, rendered using Inkscape.)

A cantilever will "ring" at a characteristic resonant frequency, fr, when you pluck it,

where Kn is a constant that specifies the resonant mode (3.516 for the first mode), E is Young's modulus, I is the moment of inertia, A is the cross-sectional area (w⋅t), ρ is the density, and L is the length. The moment of inertia of a cantilever is easy to calculate, but it's even easier to just list,

Combining the equations gives the following result,

What's interesting about this is that the resonance frequency is independent of the width of the cantilever beam. More importantly, the density term means that the frequency will change with mass loading. It will, of course, depend on where the mass is added, but the change will be in proportion to m-1/2.

The mass response of the cantilever resonant frequency became important with the advent of microelectromechanical systems (MEMS), since it was then possible to make silicon cantilever resonators with attogram (10-18 gram) mass sensitivity. This is about the weight of individual large protein molecules, so cantilever immunoassay sensors were possible.

Since mass detection depends on the initial mass of a resonator, just how small can you make a resonator? In 2012, scientists from the US National Institute of Standards and Technology (NIST, Gaithersburg, Maryland), and the University of Maryland (College Park, Maryland) made drumhead resonators of graphene by suspending it over shallow holes in a silicon wafer.[1-2] Although their objective was the modification of graphene's electrical properties, they found that they could use the van der Waals forces from the tip of a scanning tunneling microscope to move the graphene membrane.[1-2]

Now, scientists from the Kavli Institute of NanoScience at the Delft University of Technology (Delft, The Netherlands) have created graphene drumheads over superconducting cavities and used microwave photons to both tap these drumheads and "listen" to their response.[3-7] Says Vibhor Singh, first author of the paper describing this research and a scientist at the Kavli Institute of NanoScience,
"In this experiment, we shot microwave photons at a tiny graphene drum. The drum acts as a mirror: by looking at the interference of the microwave photons bouncing off of the drum, we are able to sense minute changes in the position of the graphene sheet of only 17 femtometers, nearly 1/10000th of the diameter of an atom."[4]

Artist's impression of the graphene drumhead.

(Kavli Institute of Nanoscience, still image from a YouTube video.)

The tapping force is from the very small momentum of the microwave photons. I discussed photon momentum in a previous article (The Pioneer Anomaly, December 19, 2012). Although the photon momentum is small, the mass of the graphene sheet is likewise small, so displacement of the graphene is detectable.[4,6] The graphene resonators had quality factors up to 220,000 and displacement sensitivity of 17 fmHz−1/2.[3] The Kavli researchers were able to achieve microwave amplification with these devices. Seventeen dB of microwave amplification was achieved.[3] A YouTube video shows the action of these drumheads.[7]

Since graphene is an atomic layer material, its properties verge on the quantum mechanical realm, and the Kavli research team thinks that similar devices may be used as quantum computing elements. Kavli scientist, Gary Steele, describes the quantum mechanical aspects, as follows:
"If you hit a classical drum with a stick, the drumhead will start oscillating, shaking up and down. With a quantum drum, however, you cannot only make the drumhead move up and then down, but also make it into a 'quantum superposition', in which the drum head is both moving up and moving down at the same time... This 'strange' quantum motion is not only of scientific relevance, but also could have very practical applications in a quantum computer as a quantum 'memory chip'."[5]

### References:

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