Cavity optomechanics deals with the detection and manipulation of massive “test objects” at the fundamental limits imposed by quantum physics . By detecting the motion of the test object, we can sense an external force, for example, gravitational waves acting on a 40 kg mirror in LIGO , or atomic-scale tip–surface forces acting on a 40 pg cantilever in an atomic force microscope (AFM). For AFM cantilevers operating at room temperature close to their fundamental resonant frequency in the kilohertz-to-megahertz range, optical interferometric and beam-deflection detectors of motion are sufficient to resolve the thermal noise force determined by the damping of the cantilever eigenmode in thermal equilibrium with its environment. Operation in high vacuum and at cryogenic temperatures reduces this force noise, improving sensitivity to the point where motion detection becomes the limiting source of noise. In this context, the principles of cavity optomechanics may improve the sensitivity of AFM force sensors. Cryogenic AFM further enables the use of superconducting microwave resonators in a cavity optomechanical detection scheme . We recently introduced such a sensor based on the electromechanical coupling between surface strain and kinetic inductance of a superconducting nanowire . In this paper, we describe in detail the fabrication and characterization methods of these kinetic inductive mechano-electric coupling (KIMEC) sensors.

A force sensor designed specifically for scanning probe microscopy must have a sharp tip that is readily positioned and scanned over a surface. We operate the sensor near a mechanical resonance with a high quality factor Q for enhanced responsivity to force. The mechanical resonator is a transducer, converting the tip–surface force to changes in its sinusoidal motion as the tip oscillates above the surface with amplitude A ≃ 1 nm, the typical range of tip–surface forces. Furthermore, we desire that the stiffness of the mechanical mode is k ≃ 100 N/m, the typical change in tip–surface force gradient during oscillation. Although many types of resonators could fulfill these requirements in principle, it is hard to beat the microcantilever for ease of fabrication.

To sense force on the tip we need to measure the motion of the mechanical resonator, detecting its deflection from mechanical equilibrium. With AFM, this is typically done by optoelectronic means, using either interferometry or beam deflection . These optical methods often require delicate in situ alignment of the detector to the mechanical force transducer. An integrated detector requiring no alignment is highly desirable. Furthermore, we would like the integrated sensor package, that is, transducer and detector, to be easily exchangeable, as AFM tips are frequently damaged when scanning over unknown surface features.

Dynamic AFM is typically operated in two alternative modes of scanning feedback, namely amplitude modulation AFM (AM-AFM) and frequency modulation AFM (FM-AFM). Both modes, and their many variants, rely on lock-in detection of the motion signal, and in most cases, this signal is at the same frequency as the excitation. Cavity optomechanical detection principles can be used for both AM-AFM and FM-AFM, as well as additional driving and read-out schemes. In contrast to optical cavities, superconducting lumped-element microwave resonators easily reach the resolved-sideband regime, where the cavity linewidth is smaller than the mechanical resonant frequency . Typically, lumped-element microwave resonators are coupled to mechanical motion through a change in capacitance. Here, we detect motion through a change in kinetic inductance.

Kinetic inductance is an electromechanical phenomenon resulting from Cooper pair mass and the kinetic energy of a supercurrent. It can be orders of magnitude larger than the geometric (electromagnetic) inductance in thin films and nanowires made of amorphous superconductors . It is, therefore, useful in applications that require compact microwave resonators with low loss , including microwave filters and resonant radiation detectors . Large kinetic inductance also comes with intrinsic nonlinearity, or current dependence of the inductance, which enables low-noise microwave parametric amplification . Different materials studied in the literature include niobium nitride (Nb-N) , titanium nitride (Ti-N) , niobium titanium nitride (Nb-Ti-N) , or granular aluminum (grAl) , wolfram (W) , and silicon doped with boron (Si:B) .

Figure 1: (a) Scanning electron microscope (SEM) image of a fabricated sensor seen from an angled topside view. The cantilever is formed from a Si-N plate protruding from and supported by a Si substrate. A thin film of Nb-Ti-N is deposited on top of the Si-N and patterned to form the microwave resonator. A short nanowire in parallel forms a shunt inductance Ls, affecting the coupling between the resonator and the transmission line. The signal line of a coplanar waveguide is connected to the circuit. (b) Equivalent circuit diagram of the device, where the microwave mode is modeled as a series RLkC circuit in parallel with a shunt inductance Ls, directly connected to a transmission line and measured in reflection. The resistance R models the internal losses of the microwave resonator. (c) Topside view of the meandering nanowire inductor at the base of the cantilever with kinetic inductance Lk. The inductor is placed transversely to the base of the released cantilever. (d–f) SEM images of nanowires from three different devices, showing three different nominal nanowire widths: 200, 100, and 75 nm. (g) SEM image of an underside view of the clamping line of a released cantilever using an isotropic silicon etch. The etch produces an uneven clamping line, affecting the mechanical frequency of the cantilever.

We view the sensors as composed of the cantilever, which transduces force to displacement (transducer), and the microwave resonator, which detects the displacement (detector). For this reason, we separately discuss the mechanical resonator, the microwave resonator, and their electromechanical coupling. The coupling of the microwave resonator to a transmission line is also an important design consideration.

Our primary goal is high sensitivity to tip–surface forces, given the constraints of the AFM application. If the detector is limited by thermal noise, the sensitivity is given by the power spectral density of fluctuations in force (force noise) SFF(ω), which sets a minimum detectable force (signal-to-noise ratio equals one) in a given measurement bandwidth, that is, signal integration time. The fluctuation–dissipation theorem applied to the harmonic oscillator gives

where Tm is the temperature of the mechanical mode, and the damping coefficient is expressed as the product of the effective mass meff and the linewidth γm of the mechanical resonance. Decreasing temperature and damping improves force sensitivity. It is important to understand the sources of damping for cryogenic AFM, where cantilevers oscillate in vacuum. Force sensitivity will improve with smaller meff only if mechanisms of damping, such as clamping loss and surface effects, do not increase disproportionately.

The detection scheme imprints the mechanical motion into the field of the microwave resonator, leading to motional sidebands in the measured output microwave field SVV(ω). The thermal noise force is detected at these sidebands ,

where is the added noise of the detector, nc is the number of circulating intra-cavity photons in the microwave resonator, g0 is the single-photon electromechanical coupling rate, and α is a proportionality factor, which depends on the excitation power, the resonator parameters, and the measurement line. If the motional sideband is larger than the noise floor of the detector, then force sensitivity is set by the properties of the mechanical resonator.

First in the hierarchy of design constraints comes the spring constant k, which should match the maximum tip–surface force gradient. For a given k, the mechanical resonant frequency is set by meff,

A larger mechanical resonant frequency gives a larger integration bandwidth for a given mechanical quality factor Qm = ωm/γm. However, increasing ωm also requires decreasing meff to maintain k. Practically, the limits of ωm and meff are set by material choices, fabrication possibilities, and cantilever dimensions. These factors also affect the surface strain at the base of the cantilever where the nanowire is located.

Second in the hierarchy are the resonant frequency ωc and the linewidth κ of the microwave resonator used to detect cantilever motion. A practical consideration is that our multifrequency measurement apparatus works in the frequency band of 4–8 GHz. This frequency range constrains the possible values of the circuit’s inductance L and capacitance C. The relatively small size available from the cantilever dimensions motivates the use of kinetic inductance to achieve a compact lumped-element inductor with negligible stray capacitance. Mattis–Bardeen theory relates the kinetic inductance of a film with thickness t much less than the London penetration depth to the normal-state sheet resistance R□ and the superconducting energy gap Δ0. For a conductor of length ℓ and width w, the kinetic inductance is

characterizing the microwave frequency shift G per zero-point motion (in the z direction) of the mechanical mode zzpf. The shift will depend as

where the last term requires a microscopic theory of the effect of strain on the kinetic inductance of the nanowire. In our design, a sufficient requirement on g0 is that the motional sidebands due to the thermal noise force can be resolved above the noise floor of the detector. We prioritize a larger critical current of the nanowire to compensate a potentially smaller g0.

For excitation and read-out, the microwave resonator is coupled to the transmission line with impedance Z0 = 50 Ω, giving a measured (total) quality factor Qtot,

where Qint and Qext are the internal and external quality factors of the microwave resonator. We introduce a shunt inductor with inductance Ls in order to tune Qext and, therefore, the coupling parameter η of the circuit to the transmission line. The coupling parameter is given by the ratio of external losses, that is, useful signal to total cavity losses, or equivalently,

where R models the losses of the microwave resonator. In the undercoupled case, η < 0.5, the internal losses of the resonator dominate, while in the overcoupled case, η > 0.5, the losses in the transmission line dominate. Large Qtot and η are desirable to maximize the output signal. A smaller shunt inductance increases Qext. However, the total quality factor is bounded by the internal losses. To decrease η while remaining overcoupled, the shunt inductor Ls should satisfy 50 Ω > iωcLs> R.

As a final consideration, the temperature when operating low-temperature AFMs is typically around 1 K. For temperatures closer to Tc, the internal losses of the microwave circuit increases because of increasing quasiparticle population, leading us to select Nb-Ti-N with its high (bulk) critical temperature Tc ≈ 14 K.

Several considerations determine the design of the Si-N cantilever in the scanning force sensor. The Si-N plate thickness is fixed when fabricating a wafer of sensor chips. We design the cantilever’s plane-view dimensions to achieve ωm/2π in the range of 0.5–10 MHz, corresponding to mechanical spring constant values k in the range of 2–160 N/m for typical device parameters. The wide frequency range allows us to fabricate devices working in either the sideband-resolved or sideband-unresolved regime. For the given thickness, we simulate the eigenfrequencies of the cantilever using the finite-element method (FEM) implemented in COMSOL Multiphysics , with the boundary condition of a perfectly rigid clamp along the line where the plate meets the Si substrate.

Figure 2: (a) Distribution of longitudinal strain at the surface for displacement in the z direction, εxx(x, y), a dimensionless quantity (ε = Δl/l). The strain is simulated for a 1 nm displacement at the apex of the triangular Si-N plate in the z direction, assuming a perfect clamp along the base of the plate. A lighter color indicates a larger strain. The displacement of the cantilever is exaggerated for clarity. (b) Maximum strain at the point indicated by the dark dot [see panel (a)] as a function of cantilever width b and length l. (c) Resonant frequency of the cantilever ωm as a function of its width b and length l.

Figure 3: (a) Microwave simulations of the normalized current density of the first electromagnetic eigenfrequency ω of the meandering nanowire, for widths w = 75, 100, and 200 nm and using Lk,□ = 36 pH/□, with the current node at the center of the nanowire. The frequencies land in the range of 18–28 GHz. (b) Simulation of the first resonant mode of the entire structure, using a nanowire width w = 200 nm. On resonance, the current density is uniformly distributed in the meandering nanowire, behaving as a lumped-element inductor.

As discussed previously, we can further increase the total quality factor by increasing the total inductance L, either geometrically, that is, making the nanowire longer, or by increasing the kinetic inductance per square Lk,□, that is, making the film thinner or the nanowire narrower. If we arbitrarily increase the total length ℓ of the nanowire, the parasitic capacitance of the meandering structure will eventually become significant enough to decrease the first eigenmode frequency to the point that it can no longer be treated as a lumped-element inductor. Additionally, to maintain the resonant frequency in the band of 4–8 GHz, an increasing L must be matched by a decreasing capacitance C, which cannot be made too small in relation to the parasitic capacitance. The measured parameters of our devices given below represent an adequate trade-off between these design considerations.

The narrative above attempts to convey the complex interplay between the mechanical and electrical considerations when designing the force sensor. Some important constraints are the cavity resonant frequency ωc/2π ∼ 5 GHz, which must be inside the band of 4–8 GHz of our digital multifrequency lock-in measurement system. Higher frequencies are possible, but the cost of such equipment increases steeply with frequency. Our designs also started with a single thickness of the Si-N plate that forms the cantilever, which is natural as all chips must be fabricated on the same Si-N layer. By increasing the thickness of the Si-N plate, we increase the surface strain for the same curvature of the cantilever, giving larger values of

which also increases the stiffness of the bending mode. We can compensate for the latter by increasing the length and reducing the width of the triangular plate. Eventually, we run out of space to accommodate the kinetic inductor. However, the inductor’s footprint can be reduced by increasing its sheet kinetic inductance using a thinner superconducting film. Another option is to work with a smaller inductance, which would require increasing the capacitance of the resonator to keep the resonant frequency constant. However, as previously mentioned, L or C cannot be arbitrarily large or small without considering parasitic inductance or capacitance.

We do not claim to have achieved an optimal design, and clearly there is plenty of room to explore variations. We have discussed the importance of G as a figure of merit. This parameter provides the calibration of cantilever tip motion, which is an important part in design optimization. A good design must also be fabricated at the wafer scale with a reasonable number of process steps and at a reasonable cost and uniformity.

Figure 5: (a–h) Illustration of the main fabrication steps for one device (not to scale). Details of each step are given in the main text. (i) Photograph of a 100 mm wafer after sensor release, frame removal, and removal of broken chips, with remaining chips attached to the wafer tape. The inset is an optical image of one chip. The cavity and the cantilever, too small to be seen in this image, are located at the left-side end of the chip.

We tested an alternative method to release the cantilever using a wet-etch in potassium hydroxide (KOH). The wet-etch has a high selectivity between silicon and silicon nitride, and KOH etches silicon with different rates in the ⟨100⟩ and ⟨111⟩ crystalline directions. With proper orientation of the cantilever mask to the crystalline axes of the wafer, one can etch under the triangular silicon nitride plate and form a very straight clamping line to the Si substrate. However, the KOH etch is slow compared to the isotropic RIE process and attacks the Nb-Ti-N superconducting film. An additional lithography step was needed to protect the superconducting circuit with a mask consisting of a 190 nm thick layer of Cr and PMMA. After the release, we strip the PMMA and Cr layers, while taking care to not break the cantilevers. The difficulties associated with using KOH lead us to prefer the dry-etch described above in step (h).

Figure 6: (a) An SEM image of a top-tilted view of an electron-beam deposited platinum tip on the released Si-N cantilever of one chip. The tip has total height in the range of 1–2 μm. (b) An SEM image of the reverse cone structure, deposited using multiple layers of platinum. Note the larger connection to the Nb-Ti-N thin film to the left of the tip, ensuring an ohmic contact with the ground plane of the chip. (c) An SEM image of a deposited tip, showing a radius of curvature smaller than 10 nm.

Figure 7: Mechanical resonant frequency ωm of a long cantilever as a function of the radial distance D from the center of the wafer (illustrated as an inset). The resonant frequency decreases with the radial distance and with increased spread. Finite-element method simulations give an expected resonant frequency of 700 kHz.

Figure 8: (a) Schematic of the measurement setup. The excitation signal is band-pass filtered at room temperature, attenuated at low temperature and passed to the sample. A directional coupler separates incoming and reflected signal. The reflected signal is amplified with a cryogenic low-noise amplifier and additionally amplified at room temperature. A separate port on the microwave platform generates the low-frequency drive to a piezoelectric shaker. (b) Photograph of the sample, the custom-made launcher, and the directional coupler integrated at the bottom of the microwave inset. The piezoelectric shaker used to inertially actuate the cantilever of the sample (held by three prongs at the front end of the launcher) is visible as the white disk beneath.

Figure 9: Temperature-dependent (a) magnitude and (b) phase response of the microwave resonator as functions of the probe frequency with an on-chip power of −127 dBm, with a fit to theory (dashed lines). The cavity is overcoupled at low temperatures T ≈ 1.7 K, apparent from the 2π phase flip in the reflection measurement. As the temperature increases over the range 1.7–5.0 K, the cavity becomes critically coupled at T = 2.8 K, above which it becomes undercoupled. (c) The extracted internal and external quality factors Qint and Qext as functions of the temperature. (d) The coupling parameter η as a function of temperature.

Table 1: Summary of the mean kinetic inductance per square , mean resonant frequency , and the range of measured values, grouped by the nominal nanowire width w.

We also studied the temperature dependence of the microwave response in the range 1.7–5.0 K, where we find a shift of the resonant frequency ωc by many linewidths, and a change of the coupling parameter η.