We study the transport properties of graphene layers placed over ∼200 nm triangular holes via attached electrodes under applied pressure. We find that the injected current division between counter electrodes depends on pressure and can be used to realize a nanoscale pressure sensor. Estimating various potential contributions to the resistivity change of the deflected graphene membrane including piezoresistivity, changing gate capacitance, and the valley Hall effect due to the pressure-induced synthetic magnetic field, we find that the valley Hall effect yields the largest expected contribution to the longitudinal resistivity modulation for accessible device parameters. Such devices in the ballistic transport regime may enable the realization of tunable valley polarized electron sources.

Graphene is a single atomic layer material with exceptional electronic and mechanical properties,1 with a band structure that consists of two inequivalent Dirac point valleys, each with dispersion corresponding to Dirac fermions.1 Graphene has formed the basis of nanoelectromechanical and strain sensing devices,2–12 including on elastomeric substrates5,7 and supported by thin insulating silicon nitride membranes.6 However, the ultimate limit of miniaturization of such sensors has not yet been ascertained, although predictions have been made towards the operation of nanoscaled pressure sensors.13,14 Here, we report measurements on nanoscale pressure sensors realized from graphene sheets suspended over 200 nm triangular holes with a ground plane at the bottom acting as an electrostatic gate electrode. Electrical transport measurements are performed while pressurizing the graphene, which acts as an impermeable membrane,15 with an inert gas. The pressure deflects the graphene sheet into the hole, inducing lattice strain. Based on the signal to noise ratio found in our devices, we find a typical pressure sensitivity of ∼20 kPa/√Hz. We consider a number of pressure transduction mechanisms for realistic device parameters including piezoresistivity6,8,10 and deflection of the device towards the gate, and find that at these small length scales and device geometry, the largest mechanism contributing to the resistivity change for typical devices is expected to arise from a valley Hall effect,16 in which a pseudomagnetic field generated from the nonuniform strain increases the resistivity of the membrane. This may enable the fabrication of very small pressure sensors and tunable valley splitter devices.

Figure 1 shows the fabrication process. Our devices were fabricated on a Si/SiO2 substrate with 300 nm of oxide (Silicon Quest International). Using electron beam (e-beam) lithography an array of 200 nm-side equilateral triangles is defined in a polymethylmethacrylate (PMMA) resist layer and subsequently 150 nm of the oxide is etched using reactive ion etching (Surface Technology Systems) with CHF3 (Fig. 1(a)). The PMMA is removed by dipping in acetone overnight and high temperature oxygen annealing. Graphene is exfoliated from kish graphite (Covalent Materials Corporation) and transferred mechanically over the triangular hole (Fig. 1(b)). A resist bilayer of copolymer poly(methylmethacrylate-methacrylic acid) and PMMA, [P(MMA-MAA)/PMMA] is spun on and windows are defined using e-beam lithography (Fig. 1(c)). 10 nm of Ti and 65 nm of Au are then deposited with an e-beam evaporator (Temescal) at a deposition rate of 1 Å/s and 2 Å/s, respectively. The P(MMA-MAA)/PMMA bilayer is then removed by soaking the sample in acetone overnight (Fig. 1(d)). A colorized scanning electron microscope image of a completed device showing the graphene with attached electrodes is shown in Fig. 1(e). After fabrication, devices are probed at room temperature by applying a voltage to one contact and collecting the current at two other ones. An example of the measurement geometry is shown in the inset of Fig. 2. A current is injected into the green contact and the currents at the two red contacts I1 and I2 are measured. The two-terminal conductance versus gate voltage for typical samples indicates a field effect mobility of ∼1000–3000 cm2/(Vs), and thus, the electronic mean free path is expected to be tens of nanometers or less even at large doping. (This behavior may result from resist residues, which could potentially be removed in future experiments using, e.g., low temperature current annealing.17 Because our experiment is performed at room temperature, such an annealing step was not taken.) Therefore, electron transport in the triangular region can be considered to be diffusive despite its relatively small size.

FIG. 1.

(a) Triangular holes are created in the Si/SiO2 wafer using e-beam lithography and etched with CHF3 using a reactive ion etch. (b) Graphene is exfoliated over the etched triangles. (c) Electrodes are created using a P(MMA-MAA)/PMMA resist bilayer using e-beam lithography. (d) 10 nm of Ti and 65 nm of Au are deposited with an e-beam evaporator. (e) Colorized electron microscope image of completed device. Dark purple, graphene; purple, substrate; and white, attached electrodes. Scale bar: 300 nm.

FIG. 1.

(a) Triangular holes are created in the Si/SiO2 wafer using e-beam lithography and etched with CHF3 using a reactive ion etch. (b) Graphene is exfoliated over the etched triangles. (c) Electrodes are created using a P(MMA-MAA)/PMMA resist bilayer using e-beam lithography. (d) 10 nm of Ti and 65 nm of Au are deposited with an e-beam evaporator. (e) Colorized electron microscope image of completed device. Dark purple, graphene; purple, substrate; and white, attached electrodes. Scale bar: 300 nm.

Close modal
FIG. 2.

Main panel: Plot of the fractional current difference between contacts 1 and 2 versus time. The graphene membrane is pressurized with He gas and the He pump-out initiated at the points in time indicated by the arrows. Inset: Schematic diagram of pressure sensor device.

FIG. 2.

Main panel: Plot of the fractional current difference between contacts 1 and 2 versus time. The graphene membrane is pressurized with He gas and the He pump-out initiated at the points in time indicated by the arrows. Inset: Schematic diagram of pressure sensor device.

Close modal

Previously, a number of experiments have investigated the response of graphene to an applied hydrostatic pressure.4,6,10 However, in these experiments, the graphene devices were much larger, and in some cases also supported by substrates.6 In the main panel of Fig. 2, the fractional current difference

ΔI=I1I2I1+I2
(1)

for a nanoscaled device D1 (∼200 nm triangular region edge length) with zero gate voltage applied is monitored versus time. After maintaining vacuum for ∼1–3 days to enable any trapped gas in the triangular hole to diffuse out,15 and avoid outward pressure18 at time t ∼ 120 s, ∼30 kPa of He gas is let into the sample chamber to pressurize it. The fractional current difference decreases and reaches a plateau. Slightly before t ∼ 300 s, He pump-out is initiated and ΔI returns to the previous level as the pressure drops. The resistance change acts together with the asymmetric placement of the triangular hole (or other present asymmetry) with respect to the leads to produce a differential signal dependent on the resistivity change within the strained region of the membrane. The deviation of ΔI under the pressure of 30 kPa is ∼5 × 10−3, as shown by the dashed lines and double arrow in Fig. 2. Similar behavior has been observed on four additional samples.

Here, we consider a number of possible mechanisms for resistance change of the device. The first is piezoresistivity, in which a strain causes a change in resistivity due to the lattice strain. In a number of experiments,2,5,6,8 the gauge factor giving the ratio of the fractional change in resistivity to the strain was found to be2,5,6,8

η=ΔRεR2,
(2)

with one experiment yielding a somewhat larger value of η = 6.7 The characteristic strain in a triangular membrane under pressure has been previously estimated as16 

ε=(PLμ)2/3,
(3)

where P is the pressure, L is the length of the triangular hole side, and μ is the graphene shear modulus, which we take to be μ = 150 N/m. This gives for this mechanism

ΔRR=η(PLμ)2/3.
(4)

The resistivity change is positive.

Another potential mechanism is the change in sheet resistance due to the charge density changing when the graphene membrane approaches the gate due to the applied pressure. This leads to an increase in its capacitance to the gate. Considering this mechanism and approximating the triangular hole as circular gives an estimated19 

ΔRRR4GVgVghd,
(5)

where G is the conductance, R = 1/G is the resistance, Vg is the gate voltage, h = 0.15 L(PL/μ)1/3 is the defection at the center towards the gate due to the pressure, and d is the initial distance of the membrane from the back gate. Note that an additional factor of ¼ occurs compared to the parallel plate result due to the fixed boundary conditions, which results in the resistance change being concentrated near the center.

Finally, when the membrane is deflected to produce a nano-bubble shape, it has been predicted16 and demonstrated experimentally20 that a pseudomagnetic field develops of magnitude

B=0.3βhcaeL(PLμ)2/3,
(6)

where a is the C-C bond length, β = ∂(log t)/∂(log a) = 2–3 with t the nearest neighbor electron hopping parameter, c is a factor of order unity, e is the electric charge, and h is Plank's constant.16,21 Here, we take c = 1 and β = 2.5. This results from the displacement in momentum space of the Dirac points by the local strain, which acts as an effective vector potential. A nonuniform strain then produces an effective magnetic field. However, because the strain does not break time-reversal symmetry, the pseudomagnetic field has the opposite sign for the carriers with momenta near each of the two inequivalent Dirac points, which maintains overall time-reversal symmetry in the crystal.16 As a result, in the diffusive transport regime, the electric field and current remain parallel. However, the longitudinal resistivity is nevertheless altered by the deflection of the carriers between scattering events. Using a Drude model,22 we find

ΔRR=(μmB)2,
(7)

where μm is the carrier mobility. The off-diagonal elements of the resistivity tensor are zero because of the opposite pseudomagnetic field experienced by the carriers in the two valleys.

The contributions of these various potential resistivity modulations are summarized in Table I, for a particular set of device parameters for a model device denoted D2 that has similar parameters to the one studied experimentally but with somewhat higher mobility and at nonzero Vg. The device parameters are: L = 200 nm, P = 30 kPa, d = 300 nm, Vg = 20 V, μm = 5000 cm2/(Vs), dG/dVg = 15 μS/V, and a = 0.14 nm is the graphene nearest neighbor bond length, a typical value of the gauge factor η ≈ 2,2,5,6,8R = 3.4 kΩ. The largest contribution for this device is expected to arise from the valley Hall effect. It is interesting to study how the various contributions depend on the device size. The results for ΔR/R for a device identical to D2 except with different triangle edge lengths are plotted in Fig. 3. The crossover to the regime where the valley Hall effect dominates is L ∼ 250 nm, and the valley Hall effect becomes yet more dominant as the device size is reduced. This indicates that for optimizing the pressure sensitivity of submicron pressure sensors, L ∼ 250 nm leads to the least predicted sensitivity for these device parameters. The valley Hall effect also has the strongest dependence on pressure (∝P4/3), so at larger pressures, the valley Hall effect would be expected to be even more dominant over the other mechanisms. For experimental sample D1, the change in ΔI is approximately 0.5%, which suggests that the gauge factor is ∼4 and is somewhat larger than that typically found in larger membranes.2–8,10 The mobility for this device was found to be somewhat lower than for D2, and the valley Hall contribution to ΔR/R is expected to be smaller but a similar order of magnitude to that expected from piezoresistivity. The gate charging mechanism is negligible since the data were taken with Vg = 0. Assuming that the valley Hall effect is dominant, based on the root mean square noise level, measured to be 5.7 × 10−4 for a bandwidth of 0.1 Hz, and the value of the step in ΔI determined from Fig. 2, we estimate the pressure sensitivity for our devices under these experimental conditions to be ∼20 kPa/√Hz by determining the threshold for the signal to exceed the noise floor using the results in Table I. Assuming the piezoresistivity mechanism is dominant yields a similar order of magnitude result. More experiments will be necessary to determine the origin of the resistance changes in these devices. Nevertheless, this demonstrates that nanoscale-area graphene membranes are capable of pressure sensitivity with a large gauge factor. Further work will explore the low doping and ballistic transport regimes. In this regime, the cyclotron orbit radius for a given pseudomagnetic field becomes large. Because of time-reversal symmetry, the effective pseudomagnetic field has the opposite sign for electrons in the two valley states of graphene.16,21 Thus, if a ballistic electron beam were to cross a region with a nonzero pseudomagnetic field, electrons in the two valley states would follow diverging trajectories. This may enable the realization of devices that are able to split a charge current by valley index to produce a tunable source of valley polarized electrons towards realizing valleytronics.21,23–32

TABLE I.

General equations and specific values for a model device for the three considered resistivity modulation mechanisms of membrane approach to gate, piezoresistivity, and the valley Hall effect.

MechanismΔR/RΔR/R for device D2 (×10−3)
Approach of membrane to gate 0.0375RGVgVgdL(PLμ)1/3 −0.87 
Piezoresistivity η(PLμ)2/3 2.3 
Valley Hall effect 0.56μm2h2(aeL)2(PLμ)4/3 4.2 
MechanismΔR/RΔR/R for device D2 (×10−3)
Approach of membrane to gate 0.0375RGVgVgdL(PLμ)1/3 −0.87 
Piezoresistivity η(PLμ)2/3 2.3 
Valley Hall effect 0.56μm2h2(aeL)2(PLμ)4/3 4.2 
FIG. 3.

Plot of ΔR/R for three different indicated mechanisms of resistivity modulation in suspended graphene devices with pressure versus the side length of the triangular suspended region, for device D2. The sign of the gate approach mechanism has been inverted to facilitate comparison of its magnitude with the other contributions. The valley Hall effect dominates for sufficiently small devices.

FIG. 3.

Plot of ΔR/R for three different indicated mechanisms of resistivity modulation in suspended graphene devices with pressure versus the side length of the triangular suspended region, for device D2. The sign of the gate approach mechanism has been inverted to facilitate comparison of its magnitude with the other contributions. The valley Hall effect dominates for sufficiently small devices.

Close modal

J.A.-S. was supported by UCMEXUS-CONACYT. J.A.-S. and T.M. were supported by DOE ER 46940-DE-SC0010597. Additional support for this work was from the UCR CONSEPT Center.

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