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At present, the most sensitive displacement detectors for nanoscale mechanical resonators rely on the mechanical modulation of current flow through a single electron transistor or atomic point contact2,3,4,5, achieving resolutions around 10−15 m Hz−1/2, which in the case of the work by LaHaye et al. is only several times the quantum limit. These devices, however, feature a resonator and sensor integrated into a single unit, limiting their versatility for some force-sensing applications. Although high-finesse interferometers also achieve nearly quantum-limited displacement resolution—down to an astounding 10−20 m Hz−1/2 (refs 10, 11)—their application to micro- and nanomechanical oscillators is challenging, especially as oscillator size is reduced. A fundamental obstacle is the optical diffraction limit, which sets a rough lower bound on the size of the measured oscillator. In addition, many requirements of a high-finesse cavity (for example, thick substrates and stiff multilayer mirror stacks for maximum reflection) run counter to the requirements of the most sensitive microelectromechanical and nanoelectromechanical systems (for example, low spring constants and thin membranes for sensitive force detection). Optical interferometers encounter some limitations at the low temperatures often required in ultrasensitive force microscopy. For temperatures below 1 K, the absorption of light from a typical interferometer laser—even for incident powers less than 100 nW—has been observed to heat Si cantilevers through absorption12,13. As a result, the resolution of typical optical interferometry of micromechanical force sensors hovers above 10−13 m Hz−1/2 (refs 13, 14).

Here, we show how simply bringing a micromechanical oscillator in close proximity to an off-board quantum point contact (QPC) enables sensitive displacement measurements. In demonstrating this principle, our QPC detector achieves a resolution of 10−12 m Hz−1/2, which is comparable to that achieved by optical interferometry on resonators of similar size. Our QPC transducer, however, has the fundamental advantage that it can be applied to measurements of oscillators with dimensions smaller than the optical diffraction limit. In addition, by virtue of its off-board design, the QPC can be used in conjunction with sensitive cantilevers in a variety of force-sensing applications including magnetic resonance force microscopy15. Although the resolution of our QPC is limited by device imperfections, QPC transducers of this type should have the properties required to reach the quantum limit on continuous position detection8,9.

In the years since the discovery of quantized conductance through semiconductor QPCs (refs 16, 17), these devices have been used as sensitive charge detectors in a variety of applications. The dependence of a QPC’s source–drain conductance on small changes in electrostatic fields makes it useful as a detector of single electrons in gate-defined quantum dots18 or of charge motion through electron interferometers19,20. This extreme sensitivity to charge has also been applied in the detection of mechanical motion on minute scales. In 2002, Cleland et al. demonstrated a displacement detector with a resolution of 3×10−12 m Hz−1/2 at 1.5 MHz using the piezoelectric effect in a GaAs micromechanical resonator to modulate current through an integrated QPC (ref. 21). As this method requires the QPC to be built into a piezoelectric resonator, device processing can degrade both the mobility of the two-dimensional electron gas (2DEG) forming the QPC and the quality factor of the resonator. Furthermore, the stiff doubly clamped geometry of the resonator and the requirement that it be made from a single-crystal GaAs heterostructure limit its application as a sensitive force detector. Here, we use a different scheme for displacement detection, wherein a cantilever is brought close to an off-board QPC causing the lever’s motion to modulate the QPC conductance. In principle, the motion of an arbitrary resonator, without any integrated devices, can be detected in this way.

The displacement measurement, carried out in vacuum (pressure<1×10−6 torr) at T=4.2 K, is made by positioning the tip of a metal-coated Si cantilever about 100 nm above a QPC, as shown schematically in Fig. 1a. Owing to the tip’s proximity to the QPC itself—the narrow channel of electron conduction directly between the gates—the lever tip and the QPC are capacitively coupled. The tip acts as a movable third gate above the device surface. Changes in the cantilever potential Vl affect the potential landscape of the QPC channel and thereby alter its conductance G. A voltage Vg applied to the two gates patterned on the surface modifies G in the same manner.

Figure 1: Components and geometry of the experiment.
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a, Scaled schematic diagram of the experimental set-up. A close-up view of the QPC—with the cantilever in close proximity—is shown in the inset. The laser beam is part of the low-power interferometer used to calibrate displacement measurements made by the QPC. b, Scanning electron micrograph of the cantilever and its Au-coated tip (inset). c, Scanning electron micrograph of the QPC with a high-resolution view of the active region (inset). The red dot indicates the position of the cantilever tip during the experiment.

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Figure 2 shows the dependence of G on both Vg and Vl with the cantilever positioned near the QPC. The tip is located z=70 nm above the surface of the QPC device and x=660 nm directly in front of the point contact, as indicated in the inset of Fig. 1c. As Vg and Vl are made more negative, both act to decrease G in steps of the conductance quantum 2e2/h until the conductance through the point contact pinches off. From Fig. 2a,b, we determine that G is about 14 times more sensitive to changes in Vg than to changes in Vl. This factor corresponds to the ratio between the gate–QPC capacitance and the tip–QPC capacitance, Cg/Cl.

Figure 2: Pinching off the QPC using both gate voltage and lever voltage.
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a, Conductance plotted as a function of gate voltage and lever voltage with the cantilever tip positioned as shown in the inset of Fig. 1c and at z=70 nm above the QPC. b,c, Linecuts of a for constant Vl (b) and Vg (c).

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The tip–QPC capacitive coupling depends strongly on their relative separation; only when the tip is positioned near the QPC does Vl affect G strongly. By moving the tip over the device surface at fixed distance z and with a voltage Vl applied, we can make an image of its effect on G. Because the cantilever oscillates in the x direction, we are particularly interested in ∂ G/∂ x, shown in Fig. 3. The ring pattern observed in this spatial map results from the quantized steps in conductance caused by changes in the tip–QPC separation. In regions near the point contact, where changes in the lever position most strongly affect G, we find a conductance response of up to 0.005 (2e2/h) nm−1 of cantilever motion along x as shown in Fig. 3. The x y position of the cantilever tip during the following experiments, also indicated in Fig. 3, corresponds to a region where G responds most sensitively to changes in lever position. This response similarly depends on the z position of the tip, dropping off roughly as z−1.

Figure 3: ∂ G/∂ x plotted as a function of cantilever x and y over the QPC device.
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The blue dotted lines show the position of the QPC gates, whereas the red dot and red dotted lines indicate the extent of the cantilever tip and its position during the other measurements. z=70 nm, Vl=−3.0 V and Vg=−1.75 V.

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With the cantilever so positioned, we study the QPC’s effectiveness as a transducer of tip motion. Figure 4 shows the displacement resolution of the QPC as compared with a low-power laser interferometer simultaneously detecting the cantilever’s average thermal motion. At T=4.2 K, the cantilever tip has a thermal motion amplitude of xth=1.6 År.m.s.. In Fig. 4a, the spectral density of the cantilever displacement Sx driven by thermal force noise as measured by the optical interferometer is shown in red. The spectral density Si of the current driven through the point contact with a d.c. source–drain voltage Vsd=2.0 mV, Vg=−1.74 V and Vl=−3.0 V is shown in black. The current response of the QPC matches the cantilever thermal noise in both frequency and quality factor. Once we normalize the peak QPC response to the peak amplitude of cantilever motion as measured by the interferometer, we calculate a conductance sensitivity of 0.004 (2e2/h) nm−1 of cantilever motion. Furthermore, this normalization enables us to plot the spectral density of the QPC response in Fig. 4, both in terms of current on the right axis and in terms of cantilever motion on the left axis. The level of the noise floor sets the resolution of the QPC displacement transducer at 10−12 m Hz−1/2, which is over an order of magnitude better than the low-power optical interferometer used here. In our experiment, the noise on the QPC current and thus the displacement resolution of the transducer is limited by charging noise (charges fluctuating near the QPC) with a 1/f-like dependence (see Supplementary Information, Fig. S1)22. Within the half-power bandwidth of the cantilever, this displacement noise is 4×10−12 m, which is over 100 times the zero-point motion of the cantilever.

Figure 4: Cantilever thermal noise spectrum observed using a QPC transducer.
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a, QPC with d.c. source–drain bias. The thermal motion of the cantilever tip, measured by an optical interferometer, is plotted in red in Å2 Hz−1. The tip is z=70 nm above the QPC. The response to this motion by the QPC is plotted in black for Vsd=2.0 mV d.c., Vg=−1.74 V and Vl=−3.0 V in both A2 Hz−1 and Å2 Hz−1. b, QPC with a.c. source–drain bias. The same description applies as in a except that we apply a 2.0 mVr.m.s. sinusoid at 291 Hz to Vsd with Vg=−1.45 V and Vl=−3.0 V.

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To verify that the QPC response at the cantilever resonant frequency νc is not produced by electrical feedthrough (for example, due to stray capacitive coupling between measurement lines), we drive the QPC with an a.c. source–drain voltage Vsd=2.0 mVr.m.s. at 291 Hz. In Fig. 4b, the spectrum of the source–drain current through the QPC reveals a response centred on νc and split between two peaks spaced by twice the source–drain drive frequency. These sidebands are the signature of a mixer and they confirm that the response of the QPC results from the thermal motion of the nearby cantilever. The cantilever tip indeed acts as an oscillating gate that modulates the QPC conductance. Further evidence comes from exciting cantilever oscillations using a mechanically coupled piezoelectric element: the sideband amplitude increases as a function of increasing excitation amplitude.

To study the QPC’s performance as a displacement transducer and to determine optimal operating conditions, we vary several parameters. Figure 5a shows the QPC transduction factor η—defined as the ratio of the cantilever’s thermal motion xth to the QPC’s corresponding current response ith (see the Methods section)—as a function of Vg for several values of Vsd. η is proportional to the derivative of the conductance G with respect to Vg and hence shows an oscillatory behaviour as a function of Vg. As a function of Vg, the maxima of η are aligned with the steps in G. The steps—and indeed these oscillations in η—are manifestations of conductance quantization in QPCs. η has a similar oscillatory dependence as a function of Vl. In addition, η increases as a function of increasing Vsd amplitude. As shown for a fixed Vg in Fig. 5b,c, ith increases with increasing QPC current, whereas xth and Q remain unaffected, even for source–drain amplitudes up to 2.0 mVr.m.s..

Figure 5: The response of the QPC to the thermal motion of the cantilever tip.
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The cantilever tip is z=70 nm above the QPC with Vl=−3.0 V. a, The transduction efficiency η plotted at different Vsd as a function of Vg. b–d, The current response ith (b), the cantilever thermal motion xth (c) and the cantilever Q (d) plotted as functions of Vsd for Vl=−3.0 V and Vg=−1.8 V.

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Whereas the cantilever Q is not affected by the QPC source–drain current, it is affected by the voltages applied both to the lever and to the QPC gates. As has been observed before23,24, non-contact friction between a cantilever and a surface degrades the cantilever’s Q with decreasing tip–sample spacing. This degradation is exacerbated by the application of voltages to either the cantilever or the surface gates. The observed dissipation is a result of tip–sample electric fields, which can be present even when the lever and surface gates are grounded23. Under typical operating conditions (Vg=−1.75 V, Vl=−3.0 V and z=70 nm), the cantilever Q is around 2,500, much lower than the intrinsic Q0=22,500 measured far from the QPC surface. To minimize this external cantilever dissipation, future QPC transducers could be designed without surface gates25 and without the need to apply a voltage to the cantilever. Our experiments show that the QPC has a substantial response even to a grounded cantilever, possibly due to trapped charges or surface potential differences. In this case (Vl=0 V) with Vsd=1 mVr.m.s., Vg=−1.98 V and z=70 nm, we find a reasonably large η (0.16 A m−1) and much larger Q (7,500) than with a voltage applied to the lever.

The prospects for improving the resolution of future QPC transducers on the basis of this proof-of-principle are bright. By reducing the temperature below 4.2 K, the conductance steps will become sharper, thus increasing the response of the QPC transducer to a given displacement. We have also observed that lower operating temperatures decrease the 1/f charging noise in some QPCs. This noise currently limits our displacement resolution. Furthermore, the impact of 1/f noise will diminish for smaller cantilevers with higher resonant frequencies, potentially enabling us to reach the QPC’s shot-noise limit. The bandwidth of the QPC is limited to around 104 or 105 Hz owing to its resistance coupled with unavoidable stray cabling capacitance. Careful design of both device and material parameters to maximize the QPC transduction factor and minimize noise may also improve future results.

Methods

In this experiment, the cantilever and QPC are mounted in a vacuum chamber at the bottom of a 4He cryostat, which is isolated from environmental vibrations. A three-dimensional positioning stage with nanometre precision and stability moves the QPC relative to the cantilever. No feedback was used to stabilize the position of the cantilever tip with respect to the QPC.

The QPC device is made from a heterostructure grown by molecular-beam epitaxy on a GaAs substrate by Sumitomo Electric Industries (which no longer makes molecular-beam epitaxy samples). The structure consists of an 800 nm GaAs layer grown on top of the substrate, followed by a 15 nm Al0.265Ga0.735As layer, a 40 nm Si-doped Al0.265Ga0.735As layer and finally a 5 nm GaAs cap. The 2DEG lies 60 nm below the surface with mobility μ=1.0×106 cm2 V−1 s−1 and a carrier density n=4.5×1011 cm−2 at T=4.2 K, corresponding to a mean free path λ=11 μm. Ti/Au (10/20 nm) gates patterned by electron-beam lithography and shown in Fig. 1c are used to define the QPC within the 2DEG. The application of a negative potential Vg between the gates and the 2DEG forms a variable-width channel through which electrons flow. The conductance G of the QPC is measured between two ohmic contacts to the 2DEG on either side of the channel. A source–drain voltage Vsd is applied across these ohmic contacts to drive the conductance.

We use a metallized cantilever made from single-crystal Si that is 350 μm long, 3 μm wide and 1 μm thick24. At T=4.2 K, the cantilever has a resonant frequency νc=5.2 kHz and an intrinsic quality factor Q0=22,500. The oscillator’s spring constant is determined to be k=2.1 mN m−1 through measurements of its thermal noise spectrum at several different base temperatures. The motion of the cantilever can be detected using laser light focused onto a 25-μm-wide paddle 100 μm from its tip and reflected back into an optical fibre interferometer 13. When the interferometer is in use, 20 nW of light is incident on the lever from a temperature-tuned 1,550 nm distributed-feedback laser diode26. As shown in Fig. 1b, the cantilever includes a 5-μm-wide section beyond the paddle, which ends with a fine 5-μm-long, 1-μm-wide tip. A thin metallic film of Cr/Au (10/30 nm), with Cr as an adhesion layer, is evaporated onto the end of the cantilever. 22 nm of Pt sputtered on the lever’s entire surface provides a conductive path from the cantilever base to its tip. A voltage Vl is applied to the cantilever through a pressed-indium contact at the base of the cantilever chip.

To extract parameters such as the cantilever quality factor Q and average thermal motion xth, we fit the spectral density of the cantilever displacement as measured by the optical interferometer to the expected (single-sided) spectrum from a simple harmonic oscillator, Sx(ω)=(4ωc3xth2/Q)(1/(ωc2−ω2)2+(ωcω/Q)2)+Sxn, where ω is an angular frequency, ωc=2πνc and Sxn is the white spectral density of the interferometer measurement noise. Similarly, we can quantify the current response ith of the QPC displacement transducer by fitting the spectral density of the current through the point contact to Si(ω)=(4ωc3ith2/Q)(1/(ωc2−ω2)2+(ωcω/Q)2)+Sin, where Sin is the baseline spectral density of the current measurement noise. We therefore define the QPC transduction factor as η=ith/xth.

η is determined from the QPC’s response to an a.c. source–drain drive voltage at 291 Hz, as shown in Fig. 4b. We fit to this doubly peaked response rather than to the singly peaked response to a d.c. source–drain voltage to avoid spurious signals at νc. Such signals can appear because of small fluctuations (<100 μV) in the virtual ground of our current amplifier. Note that the baseline noise of the displacement measurement in Fig. 4b is worse than the noise in Fig. 4a because a smaller Vg is used. The measurement shown in Fig. 4a is made with all voltages chosen to optimize the QPC transducer resolution.

The dependence of Q on Vsd shown in Fig. 5d is extracted from spectra of the cantilever’s thermal motion. As these spectra are averaged over several minutes, they are susceptible to slow drift in the QPC conductance due to charges rearranging in the device. The plotted Q is therefore lower than the actual Q, which is more accurately measured to be around 2,500 using a ‘ring-down’ method.