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Impermeable Atomic Membranes from Graphene Sheets

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1Supplementary InformationImpermeable Atomic Membranes from Graphene SheetsJ. Scott Bunch, Scott S. Verbridge, Jonathan S. Alden, Arend M. van der Zande, Jeevak M. Parpia, Harold G. Craighead, Paul L. McEuenCornell Center for Materials Research, Cornell University, Ithaca NY 14853Experimental MethodsGraphene drumheads are fabricated by a combination of standard photolithography and mechanical exfoliation of graphene sheets. First, a series of squares with areas of 1 to 100 µm2 are defined by photolithography on an oxidized silicon wafer with a silicon oxide thickness of 285 nm or 440 nm. Reactive ion etching is then used to etch the squares to a depth of 250 nm to 3 µm leaving a series of wells on the wafer. Mechanical exfoliation of Kish graphite using Scotch tape is then used to deposit suspended graphene sheets over the wells.To determine the elastic constants of graphene using equation (3), we extrapolate the deflection in Fig. 1e (inset) to z = 181 nm to account for a 40-minute sample-load time, assume an initial pressure difference across the membrane, p = 93 kPa, and a negligible initial tension. The latter two assumptions are verified using resonance measurements. The actual deflection used in equation (3) is obtained by subtracting the extrapolated deflection z = 181 nm from the initial deflection z0 = 23 ± 3 nm at p = 0. This initial deflection is determined from the AFM image in Fig. 4a and AFM force-distance curves Fig. S1.2Slack and Self Tensioning at p = 0Since the cantilever-surface interaction is expected to be different for AFM measurements over the relatively-pliable suspended and the rigid SiO2-supported graphene, the depth of the membrane z0, at p = 0 must be determined via force and amplitude calibrations of the cantilever over each surface(Whittaker, Minot et al. 2006). A representative calibration measurement is shown in Fig. S.1. Both the amplitude (upper) and deflection (lower) of the AFM tip is measured while approaching the surface.Over the SiO2-supported surface, the difference between the actual surface position and the position given by the image in Fig. 4a can be determined by subtracting the height at which the AFM tip begins to bend due to unbroken contact with the surface (A) from the height at which the amplitude setpoint intersects with the amplitude response curve (B) (Fig. S.1). The surface is determined to be 30 nm below the amplitude setpoint position.Since suspended graphene is more pliable than supported graphene, the onset of the AFM cantilever’s deflection of Fig. S.1 is more gradual, and thus cannot be readily used to determine the equilibrium height of the suspended graphene. Instead, we note that when in unbroken-contact with the graphene surface, any deviations of the AFM tip from the equilibrium (lowest-strain) depth of the membrane will result in an increase in the membrane tension as the tip either pulls up or pushes down on the membrane. This increase in tension on either side of the equilibrium position will cause a decrease in cantilever response amplitude, resulting in a peak in the cantilever-amplitude response at the equilibrium position, similar to what has been observed for suspended carbon nanotubes(Whittaker, Minot et al. 2006). This occurs at ~100 nm, or 34 nm below the amplitude setpoint position (C).Comparing these setpoint-to-surface depths for suspended and supported graphene, we find that the equilibrium depth of the suspended membrane is 17 + (34 – 30) = 21 nm below the SiO2-supported surface where 17 nm is the distance measured in Fig. 4a. Repeating these measurements across the3center of the membrane yields an average equilibrium membrane-depth depth z0 = 17 ± 1 nm + (6 ± 2 nm) = 23 ± 3 nm.Measuring the Gas Leak RatesThe gas leak rate is measured by monitoring the internal pressure, pint, vs. time. For the case of the leak rate of air, the microchamber begins with pint ~ 100 kPa Air. This is verified by a scan of frequency vs. pext, as in Fig. 3. A similar scan is performed once every few hours to monitor pint while the device is left at pext ~ 0.1 mPa between each measurement (Fig. 3a and 3b). The leak rate of argon is measured in a similar manner except the microchamber begins with a pint ~ 0 kPa argon and ~ 10 kPa air. The microchamber is left in pext ~ 100 kPa argon between measurements to allow argon to diffuse into the microchamber. This diffusion is monitored by finding the minimum pressure in a scan of frequency vs. pext. To measure the helium leak rate we apply a p ~ 40 – 50 kPa He and monitor the resonance frequency as helium diffuses into the microchamber. It will diffuse until the partial pressure of helium is the same inside and outside the microchamber (Fig. S2). From the slope of the line we extract a heliumleak rate for the devices using equation (1). Leak rates from square membranes with sides varying from 2.5 to 4.8 µm were measured with no noticeable dependence of the leak rate on area.Transmission ProbabilityUsing this measured leak rate, we estimate an upper bound for the average transmission probability of a He atom impinging on a graphene surface as:<NvddtdN 2 10-11(S.1)where dN/dt is the measured leak rate, d is the depth of the microchamber, and v is the velocity of He atoms. The number of He atoms/second impinging on the graphene seal is given by ~Nv/2d, since each atom takes a time ~2d/v to make a round trip in the chamber. Dividing the measured rate by this value,4gives us the upper bound estimate given above in (S.1).Tunneling of He Atoms across a Graphene SheetIn the WKB approximation of tunneling probability, the probability of a particle with a mass m, tunneling across a finite potential barrier with a height V, and distance x, is given by:()hEVmxep=22` (S.2)To estimate tunneling through a perfect graphene sheet with E ~ 25 meV, we assume a barrier has V ~ 8.7 eV and a thickness x ~ 0.3 nm. This gives a tunneling probability p ~ 1 x 10-335. For the case of tunneling through a “window” mechanism whereby temporary bond breaking lowers the barrier height to ~ 3.5 eV, the tunneling probability at room temperature is p ~ 1 x 10-212 which is still many orders of magnitude smaller than we observe (Hrusak, Bohme et al. 1992; Saunders, Jimenez-Vazquez et al. 1993; Murry and


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