pubs.acs.org/NanoLett
Bistability and Oscillatory Motion of Natural Nanomembranes Appearing within Monolayer Graphene on Silicon Dioxide T. Mashoff,†,‡ M. Pratzer,*,†,‡ V. Geringer,†,‡ T. J. Echtermeyer,§ M. C. Lemme,§,| M. Liebmann,†,‡ and M. Morgenstern†,‡ †
2nd Institute of Physics B, RWTH Aachen University, 52074 Aachen, Germany ‡ Ju¨lich-Aachen Research Alliance: Fundamentals of Future Information Technology (JARA-FIT) § Advanced Microelectronic Center Aachen (AMICA), AMO GmbH, Otto-Blumenthal-Strasse 25, 52074 Aachen, Germany ABSTRACT The truly two-dimensional material graphene is an ideal candidate for nanoelectromechanics due to its large strength and mobility. Here we show that graphene flakes provide natural nanomembranes of diameter down to 3 nm within its intrinsic rippling. The membranes can be lifted either reversibly or hysteretically by the tip of a scanning tunneling microscope. The clampedmembrane model including van-der-Waals and dielectric forces explains the results quantitatively. AC-fields oscillate the membranes, which might lead to a completely novel approach to controlled quantized oscillations or single atom mass detection. KEYWORDS Graphene, nanoelectromechanics, scanning tunneling microscopy
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anoelectromechanical systems (NEMS) are promising, for example, as ultralow mass detectors, where one measures the shift in resonance frequency of a micro- or nano-object when a particle adsorbs onto it. Mostly, silicon nanobeams with sensitivities down to attograms are used so far.1-4 For further improvement of mass sensitivity, the most direct approach would be to use lighter, for example, thinner oscillators. Graphene with its thickness of one atomic layer and its extreme mechanical strength1,4-6 could be ideal. Indeed, for this purpose resonators using double layer graphene have already been demonstrated implying a mass sensitivity down to 4.5 × 10-22 kg.1 The highest sensitivity so far is found for a double wall carbon nanotube, which exhibits resonance frequency shifts compatible with the mass of one gold atom.5 Here, we study a graphene monolayer on SiO27 by scanning tunneling microscopy (STM). We find movable areas within the valleys of the intrinsic rippling8,9 exhibiting an extremely small size of 6-21 nm2 (230-800 carbon atoms). These natural nanomembranes show either a hysteretic or a reversible deflection in response to the tip bias or to a change in tip-graphene distance. The areas can be forced to oscillatory motion by an ac-field. Both, the bistability and the oscillatory motion are reproduced using the clamped membrane model,10-12 which includes electrostatic and van-der-Waals (vdW) forces of tip and substrate. We deduce resonance frequencies of the nanomembrane
up to 400 GHz corresponding to a vibronic energy of 1.7 meV. Such a large value might provide easy cooling to the ground state leading to a novel access to quantum-mechanical manipulation of vibrons13,14 or in combination with the low membrane mass to ultrasensitive mass detection, which likely goes down to a single hydrogen atom. The preparation of the graphene sample is done by mechanical exfoliation on a SiO2 substrate as described elsewhere.7,15 A graphene flake containing a monolayer region is identified, first, by an optical microscope and, in addition, by Raman spectroscopy.8 A gold contact surrounding the graphene is produced using e-beam lithography. To remove the residual resist and other adsorbates, the sample is heated to 170 °C for a few hours, first in air, and then, directly before the measurement, in ultrahigh vacuum (p ) 2 × 10-10 mbar). The monolayer region of (18 × 26) µm2 is positioned below the tip of the STM by a piezo motor using the control by an optical long distance microscope.8 The measurements are performed at T ) 4.8 K with a high resolution STM.16 As probe tips we used an electrochemically edged tungsten wire. The initial tip has been checked by scanning electron microscopy and exhibits a radius of about 90 nm. The effective tip radius concerning the tunneling current, caused by a local asperity, is much smaller (0.7 nm, see Supporting Information) and has been tuned by further tip preparation prior to measurements using field emission and voltage pulses. Therefore we applied a bias voltage of 100 V for 12 h (feedback switched on) on the gold contact, followed by millisecond voltage pulses up to 10 V. All STM images are measured in constant-current mode with the voltage U applied to the sample.
* To whom correspondence should be addressed. E-mail: pratzer@ physik.rwth-aachen.de. | Present address: Department of Physics, Harvard University, 11 Oxford Street, Cambridge, MA 02138. Received for review: 09/21/2009 Published on Web: 01/08/2010
© 2010 American Chemical Society
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DOI: 10.1021/nl903133w | Nano Lett. 2010, 10, 461-465
FIGURE 1. Reversible lifting of the graphene nanomembrane. (a) Three-dimensional representations of an atomically resolved monolayer graphene at different tip-sample distance as indicated on the right. At a large distance (U ) 1 V, I ) 0.1 nA) carbon hexagons are visible. Decreasing the distance, a hill appears in the center of the valley and the hexagons are transformed to bumps at every second atom position. (b) High-pass filtered STM image (U ) 1 V and I ) 0.1 nA). Circles denote the hill and valley areas marked in panel a. The apparent lattice constants show a difference of ∆a ) 0.03 nm due to the tilting of the π-orbitals. (c) Model explaining the mechanical behavior of the nanomembrane. Because of the locally applied force between tip and graphene, the valley is continuously lifted while scanning the tip and the STM image shows a hill in the center of the valley (dotted line).
The morphology of graphene has been studied previously by STM.8,17-19 By comparing with the morphology of SiO2, we revealed that graphene often exhibits an intrinsic rippling not induced by the substrate. We argued that the rippling appears, if graphene is freely suspended between hills of the SiO2.8 The rippling (amplitude: 0.3-0.5 nm) exhibits a preferential wavelength of 15 nm at 300 K.8 The wavelength is slightly increased at 4.8 K to 20 nm. At smaller length scales, we observe additional corrugation of (70 pm with a wavelength of only 5 nm. Figure 1a shows this smaller corrugation at several tip-sample distances. At large distance (top), the corrugated surface and the typical carbon hexagons are visible. Interestingly, the apparent lattice constant differs by 14% between oppositely curved areas (Figure 1b). This can be explained quantitatively by the tilting of the pz-orbitals within the curved surface assuming an effective pz-length of 0.26 ( 0.05 nm in accordance with theory.20 Decreasing the tip-graphene distance (middle, Figure 1a), a small bump appears within the valley, which at even smaller distance (bottom) increases in height and diameter. With respect to the neighboring hill, the valley is lifted by 32 pm, that is, about half of the initial height difference between hill and valley. Within the lifted area, the atomic structure changes from hexagons to bumps appearing at every second atom position, which has been checked © 2010 American Chemical Society
carefully by following up lines of atomic corrugation (see Supporting Information). The symmetry between A and B lattice is broken, most probably due to in-plane compressive stress. An explanation of these results is sketched in Figure 1c. The electrostatic and vdW force of the tip (Fel, FvdW(tg)) lift the graphene valley until compensated by the restoring elastic force Fmem of the membrane and the vdW force FvdW(sg) of the substrate. Since the two tip forces change with lateral tip position, a dynamic image of the lifting process results as indicated by white dots in Figure 1c. While a hill appears within the STM image (dotted line), the membrane still maintains its valley-like shape but with reduced curvature, as indicated by the gray lines. The resulting compressive stress within the lifted area can be reduced during lifting by a vertical zigzag atomic arrangement (see Supporting Information) straightforwardly explaining the observed symmetry breaking between A and B lattice. While the membrane’s lifting in Figure 1 is reversible, other valleys exhibit hysteresis. Figure 2b shows the I(z) curve of a hysteretic area at a sample voltage of U ) 1 V. While approaching or retracting the tip, a jump in current I by 3 orders of magnitude is observed with a hysteresis of 50 pm. Such hysteretic membranes can be identified directly within constant-current images, if I is chosen within the bistable range. This is demonstrated in Figure 2a, where a valley (hill) 462
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FIGURE 2. Hysteretic nanomembrane. (a) Constant-current images of a graphene area taken at different tip-surface distances. At intermediate distance (I ) 3 nA, 15 nA), a part of the valley exhibits unstable behavior visible as spikes. Higher current leads to stable imaging again, but the valley has turned into a hill. (b) I(z)-spectrum revealing hysteretic behavior. (c) Calculated potential energy Φsum as a function of substrategraphene distance zsg at fixed tip-substrate distance. Φsum consists of electrostatic potential Φel, van-der-Waals potentials between tip and graphene ΦvdW(tg) and SiO2 and graphene ΦvdW(sg), and the elastic membrane potential Φmem. The two local minima in Φsum indicate two metastable positions. The definition of distances is sketched at the top. (d) Using an initial tip-substrate distance ztg + zsg ) 1.46 nm, (top) the hysteretic switching in constant-current mode can be reproduced by first applying a voltage moving the membrane toward the tip (middle) and then retracting the tip to reduce current again, thereby switching the membrane back (bottom).
potentials.21 The elastic potential Φmem is modeled by a 4 with A taken clamped membrane implying Φmem ) A·z+ 10 from experiment and z+ being the membrane deflection. We assume two minima (two relaxed membrane positions) of two z4+-curves with origins separated by 0.12 nm to describe the fact that the membrane is first laterally compressed by lifting with respect to the hills but is relaxed again after being higher than the surrounding hills. We do not include a pretension term,10 since pretension is given explicitly by the vdW forces. The electrostatic energy Φel ) 2 is calculated using Cm as the tip-membrane 1/2·CmUeff capacitance modeled as a sphere in front of a plane and Ueff as the voltage drop between tip and graphene reduced with respect to U due to graphene’s finite carrier density. The calculated potentials are plotted in Figure 2c as a function of vertical graphene position for a fixed tip-SiO2 distance (1.63 nm). The summed up potential Φsum exhibits two local minima representing the observed bistability. Figure 2d shows two nearly degenerate minima at U ) 0 V and tip-substrate distance 1.46 nm (top). The lower minimum transforms into a saddle at U ) 1.3 V (middle) forcing the membrane to the upper minimum, that is, 0.34 nm closer to the tip leading to 3 orders of magnitude increase in current
showing atomic hexagons is visible at low (high) I, while a noisy area appears at intermediate currents. The noisy behavior is caused by the feedback loop. The tip approaches, which increases the tip forces and induces a snapping of the membrane toward the tip; this snapping increases I by 3 orders of magnitude and the tip retracts leading to reduced tip forces and a snapping back of the membrane. We observed bistability only in valleys and never on hills. Although analyzing several tens of valleys, we could not find any correlation between hysteretic or reversible behavior and depth, width, or curvature of the valleys. Thus, we conclude that graphene-substrate interactions not visible by STM are crucial. To model the observed behavior, we analyze the involved interaction potentials. Besides the electrostatic potential Φel between tip and graphene, the vdW potentials ΦvdW for tip/ graphene and graphene/SiO2,21 and the elastic potential of the membrane Φmem are considered. Calculations of the interaction potentials acting on the graphene membrane are described in detail in the supplement. In short, we use a dielectric plane for SiO2, a metallic circle of cosine shape in vertical direction for the membrane and a metallic W tip of parabolic shape (central radius: 0.7 nm) to calculate the vdW © 2010 American Chemical Society
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FIGURE 3. Oscillatory deflection of the graphene nanomembrane. (a) I(U)-curve (Ustab ) -0.3 V, Istab ) 100 pA) recorded on a valley (red) and on a hill position (black) as marked in the lower inset. Upper inset: logarithmic plot of the ratio of the two I(U) curves being proportional to the deflection z+(U). The minimum marks the contact potential UC; blue line: parabolic fit. (b) Calculated potential Φsum for an initial substrate-graphene distance zsg ) 0.63 nm and tip-graphene distance ztg ) 0.56 nm. Even at U ) 1.85 V the system cannot move to the right minimum but exhibits only a reversible shift of 30 pm. (c) Current response (black, hill; red, valley) to an oscillating tip voltage Umod (blue) measured at reference (hill) and on nanomembrane (valley). (d) Upper inset: in-phase lock-in signal recorded on reference (black) and nanomembrane (red). Main: ratio of the two lock-in signals rLI from the upper inset (symbols) in comparison with a fit using the model of a prestrained clamped membrane (blue line).10 Right and upper scale show the deduced deflection amplitude ∆z (right axis) and the applied electric field amplitude E0 (upper axis). Lower inset: STM image with the two measurement positions (white circles) and area of the membrane (dashed circle) marked.
in excellent agreement with the experiment. Increasing now the tip-graphene distance ztg by 0.34 nm, to reduce I again, shakes the potential back and the membrane flips back to its original minimum (bottom). Note that ztg = 0.56 nm used in this simulation is estimated by extrapolation of I(ztg) toward the contact conduction G0 ) 2e2/h.22 This leaves only the initial substrate-graphene distance zsg ) 0.9 nm as a fit parameter. By decreasing the initial zsg to 0.63 nm, we still obtain two minima, but we are not able to calculate a switch of the membrane to the upper position by reasonable U. This explains the reversible behavior observed in Figure 1 as due to strong vdW forces of the substrate. To explore the electric response of the membrane, we varied U with the feedback switched off at stabilization voltage Ustab ) -0.3 V and stabilization current Istab ) 100 pA. First, we measured I(U) on a hysteretic valley in comparison with I0(U) on a hill (Figure 3a). Assuming the same electronic structure at both positions, the ratio of the two curves is given by I(U)/I0(U) ) exp(-2κ(U)·z+(U)), where κ(U) is the electron’s decay constant determined within the supplement. Thus, the logarithmic plot of I(U)/I(U0) directly displays the deflection z+ of the valley with respect to the © 2010 American Chemical Society
hill (upper inset, Figure 3a). The minimum of z+ (U) marks the absence of electrostatic force at the contact potential UC ) -0.45 V implying a tip work function of 5.11 eV (graphene: 4.66 eV23). Next, we apply an ac-voltage U(t) ) Udc + U0 cos(2πνt) with varying amplitude U0 at a dc-offset Udc ) -0.3 V = UC to compensate the contact potential and fixed tip-substrate distance. The tip is placed above the nanomembrane marked by the dashed circle in the inset of Figure 3d. We choose Istab ) 0.2 nA low enough to avoid snapping of the membrane. Figure 3c shows the resulting I(t) and the applied Umod(t) ) U(t) - Udc for the membrane (bottom) and on a reference (hill) position (top). At reference, I(t) is dominated by the capacitive crosstalk (phase shift of 90°). At the membrane, an additional in-phase signal, nonlinear with respect to Umod(t), indicates the reversible membrane movement. Figure 3b displays the corresponding Φsum(zsg) at the two extrema of electrostatic force (ztg ) 0.56 nm). The membrane does not switch between the two minima but oscillates reversibly within one valley in agreement with experiment. 464
DOI: 10.1021/nl903133w | Nano Lett. 2010, 10, 461-465
oscillators at adequate anisotropy28 or Landau-level Lasers using graphene in B-field29 are possible candidates.
Finally, we use lock-in technique to measure the deflection amplitude ∆z with respect to U0. The inset of Figure 3d shows the lock-in output measured on the nanomembrane (red) and on a reference position (black). To get rid of unknown parameters we use the ratio of the two lock-in signals
∫t ∫t t+T I0(t)cos(ωt)dt t+T
rLI )
Acknowledgment. We appreciate financial support of the German Science foundation via Mo 858/8-1 and Mo 858/11-1. Supporting Information Available. Determination of the position of bumps in stressed graphene areas, calculation of the involved interaction potentials acting on the nanomembrane, and determination of the deflection of the nanomembrane. This material is available free of charge via the Internet at http://pubs.acs.org.
I(t)cos(ωt)dt
(1)
From rLI we deduce ∆z numerically (see Supporting Information) as displayed on the right scale of Figure 3d. The displayed ∆z and the corresponding electric field amplitude E0 deduced from U0 and ∆z (see Supporting Information) give an impression of the amplitude-field relation achievable by low-frequency external excitation. The basic resonance frequency of the nanomembrane ν0 can be estimated by clamped membrane theory using the known values E2D ) 340 N/m,10 membrane radius r ) 2.58 nm, and mass m0 ) 1.59 × 10-23 kg and assuming HOPG values for thickness h ) 0.335 nm, and Poisson-ratio s ) 0.1624
REFERENCES AND NOTES (1) (2) (3) (4) (5) (6) (7) (8)
ν0 )
h 4r
3 2D
πE = 430 GHz 3m0(1 - s2)
(2)
(9) (10) (11)
More adequate molecular dynamics simulations (MDS) for monolayer graphene reveal ν0 ) 400 GHz for a smaller area (4.3 nm2) exhibiting 75% deviation from eq 2.25 The high resonance frequency corresponding to 1.7 meV might be favorable for quantum electromechanics.13,14 Moreover, the nanomembranes consisting of only 200-800 atoms are ideal resonators for ultrasensitive mass detection. Adsorbing one hydrogen atom of mass ma would lead to a relative frequency shift of
∆ν ) ν0
[(
1+
ma m0
)
-0.5
]
- 1 = 10-4
(12) (13) (14) (15) (16) (17) (18) (19)
(3)
(20) (21)
MDS finds quality factors Q ) 2500 (273 000) for unsupported graphene monolayers at 300 (3) K.26 Experiments reveal lower values for multilayer graphene probably due to interlayer friction,1,4,6,27 but Q ) 4000 has been achieved using oxidized multilayer graphene at 300 K.24 This would be sufficient to detect frequency shifts of 10-4 easily. Measuring the membrane oscillations at 400 GHz would be eased by the nonlinearity of I(z) requiring only dc detection. The excitation in the range of 400 GHz remains a technical challenge, but approaches using, for example, spin torque © 2010 American Chemical Society
(22) (23) (24) (25) (26) (27) (28) (29)
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