Design, fabrication, and characterization of kinetic-inductive force sensors for scanning probe applications

Introduction

Cavity optomechanics [1] deals with the detection and manipulation of massive “test objects” at the fundamental limits imposed by quantum physics [2]. 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 [3], 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 [4-7]. We recently introduced such a sensor based on the electromechanical coupling between surface strain and kinetic inductance of a superconducting nanowire [8]. 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 [9-11] or beam deflection [12-14]. 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 [1,15]. 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 [16]. It is, therefore, useful in applications that require compact microwave resonators with low loss [17], including microwave filters [18] and resonant radiation detectors [19]. Large kinetic inductance also comes with intrinsic nonlinearity, or current dependence of the inductance, which enables low-noise microwave parametric amplification [20]. Different materials studied in the literature include niobium nitride (Nb-N) [21,22], titanium nitride (Ti-N) [23,24], niobium titanium nitride (Nb-Ti-N) [20,25], or granular aluminum (grAl) [26,27], wolfram (W) [28], and silicon doped with boron (Si:B) [29].

Results and Discussion

Figure 1 gives an overview of our fabricated sensor, showing the main components. The cantilever is a 600 nm thick Si-N triangular plate released from a much thicker silicon (Si) support, as shown in Figure 1a and Figure 1g. Figure 1a shows the microwave resonant circuit with its interdigital capacitor in series with the nanowire inductor. The meandering nanowire is placed at the base of the cantilever as shown in Figure 1c, in the area of largest strain. Details of three different nanowire widths are shown in Figure 1d–f. The circuit is measured in reflection, as illustrated in the device schematic in Figure 1b, through a coaxial transmission line that launches to the coplanar waveguide on the sensor chip (not shown). We now consider the sensor design in more detail.

Sensor design

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) S_FF(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 T_m is the temperature of the mechanical mode, and the damping coefficient is expressed as the product of the effective mass m_eff and the linewidth γ_m 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 m_eff 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 S_VV(ω). The thermal noise force is detected at these sidebands [8],

where is the added noise of the detector, n_c 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 m_eff,

A larger mechanical resonant frequency gives a larger integration bandwidth for a given mechanical quality factor Q_m = ω_m/γ_m. However, increasing ω_m also requires decreasing m_eff to maintain k. Practically, the limits of ω_m and m_eff 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 ω_c 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 [30] 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

where L_k,□ is the kinetic inductance per square. From Equation 4, we require a superconducting film with large normal-state resistance in geometry with many squares, that is, a long and narrow strip. For a given total inductance L, the meandering structure allows for a physically compact shape. A larger L also leads to a larger maximum number of intra-cavity photons n_c for a given current I,

where the total current through the inductor is limited by the critical current I_c, also set by material choice and physical dimensions. From Equation 5, we can increase n_c for a given material by, again, making the nanowire longer or narrower, or by increasing L_k,□ through a thinner film. However, longer and narrower nanowires decrease the participation ratio, that is, the fraction of the nanowire that experiences strain, and thus g₀. The nanowire can instead be made more compact by decreasing the width and adjusting the length so that the same number of squares experience strain. Conversely, reducing the cross-sectional area of the nanowire reduces I_c and decreases the maximum n_c possible before undesirable nonlinear effects become significant. A nonlinear microwave mode is not part of the standard electromechanical formulation, complicating the analysis in, for example, the phase-sensitive detection scheme.

Third in the hierarchy is the value of the single-photon electromechanical coupling strength,

characterizing the microwave frequency shift G per zero-point motion (in the z direction) of the mechanical mode z_zpf. 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 Q_tot,

where Q_int and Q_ext are the internal and external quality factors of the microwave resonator. We introduce a shunt inductor with inductance L_s in order to tune Q_ext 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 Q_tot and η are desirable to maximize the output signal. A smaller shunt inductance increases Q_ext. However, the total quality factor is bounded by the internal losses. To decrease η while remaining overcoupled, the shunt inductor L_s should satisfy 50 Ω > iω_cL_s > R.

As a final consideration, the temperature when operating low-temperature AFMs is typically around 1 K. For temperatures closer to T_c, 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 T_c ≈ 14 K.

Mechanical simulations

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 ω_m/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 [31], with the boundary condition of a perfectly rigid clamp along the line where the plate meets the Si substrate.

The FEM model gives the distribution of strain at the surface. Figure 2a shows the distribution of longitudinal strain ε_xx(x, y) for the fundamental bending mode of interest. The strain is normalized to its maximum value at the center of the clamping line. Figure 2b displays this maximum value of the surface strain as a function of the length l and the width b of the triangular plate calculated for a z displacement of 1 nm at the apex of the triangle, a typical tip displacement for measuring surface forces in AFM.

We place the nanowire inductor in this region of maximum longitudinal surface strain, with the long segments of the nanowire oriented parallel to the x axis. The nanowire will thus experience compression or tension as the tip deflects in positive or negative z direction, respectively. We can then vary b and l of the triangular cantilever to increase the strain for a given deflection and to achieve the desired mechanical resonant frequency ω_m, as shown in Figure 2c. We see that the length of the cantilever is the main factor affecting the strain and the resonant frequency. The width b must also be adjusted to accommodate a meandering nanowire with a total length ℓ large enough to realize the desired kinetic inductance. To understand this constraint, we turn to electromagnetic simulations.

Electromagnetic simulations

In this work, we explore thin-film nanowires of width w = 75, 100, and 200 nm, which we can fabricate with a high degree of uniformity using electron-beam lithography and reactive-ion etching. We simulate the electromagnetic response of the meandering nanowire inductors using Sonnet, a quasi-3D electromagnetic simulator [32], which has the feature of including sheet kinetic inductance L_k,□. We begin by simulating the meandering inductor itself to find the lowest-frequency eigenmode. We require that this eigenfrequency falls well above the target frequency of our resonator ∼5 GHz so that we may, to a good approximation, treat the meandering nanowire as a lumped-element inductor. Figure 3a shows simulations of the current distribution of a typical inductor for all three nanowire widths at their lowest eigenfrequency in the range of 18–28 GHz, where we see the current node located in the center of the meander. Figure 3b shows the microwave simulation of the entire circuit, including the shunt inductance formed from a short nanowire. At the lower resonant frequencies of the inductor and the series capacitor, the current is uniformly distributed inside the meandering nanowire, confirming that it behaves as a lumped-element inductor. We also see that, on resonance, the current in the impedance-transforming shunt inductor reaches a magnitude similar to that in the meandering 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 L_k,□, 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 shunting inductance forms a part of the circuit design. Figure 4a shows the simulated total quality factor Q_tot, and Figure 4b shows the coupling parameters as a function of the shunt inductance, using typical circuit parameters for our devices. Figure 4c and Figure 4d display the measured magnitude and phase response of two nominally identical devices, both with nanowire width w = 200 nm, where one device has the shunt inductance and the other does not. For a shunt with inductance L_s = 195 pH, we increase Q_ext by a factor of roughly twenty at the cost of a slight reduction in η while remaining overcoupled.

Design summary

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 ω_c/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.

Fabrication

The main fabrication steps are illustrated in Figure 5. The fabrication produces around 400 chips per wafer with a yield of about 80%. We start with a double-side polished Si wafer, 100 mm in diameter and 525 μm thick, coated on both sides with 600 nm low-stress (<100 MPa) Si-N films. The sensor chips are 1.6 mm by 3.4 mm, about the size of a standard AFM cantilever chip. The steps are as follows:

(a) Superconducting film. We first deposit a 15 nm thick thin film of superconducting Nb₆₀Ti₄₀N by reactive co-sputtering from separate niobium and titanium targets [33] in an ATC2200 from AJA International Inc., with a deposition rate of roughly 3 nm/min.

(b) Pads and markers. A lift-off process defines the gold contact pads and alignment marks. We spin a 400 nm thick photoresist (maN1407), bake on a hotplate at 100 °C for 60 s and expose with a dose of 450 mJ/cm² using an MLA150 from Heidelberg Instruments. We develop the pattern in maD533s for roughly 45 s and then deposit 10 nm chromium and 40 nm of gold by electron-beam evaporation in an Auto306 from Edwards Vacuum. Lift-off in mrREM700 removes the resist mask, and the patterned wafer is ultrasonically cleaned and rinsed with IPA.

(c) Back-side mask. Before fabricating the chromium etch mask on the back side, we first protect the wafer’s front side with a thin PMMA layer. We then define the lift-off mask on the wafer back side by spinning a 400 nm thick photoresist (maN1407), baking at 100 °C on a hotplate for 60 s. We expose the pattern with a dose of 450 mJ/cm², aligning to the markers on the front, and we develop in maD533s for roughly 45 s. A subsequent short soft-ashing step in a Plasmalab 80 ICP65 from Oxford Instruments removes residual resist and improves the adhesion of the following 150 nm deposition of chromium with electron-beam evaporation in the Auto306 from Edwards. After the lift-off in mrREM700, we also strip the protective PMMA layer on the front side with AR600–71, and we clean the wafer in isopropanol (IPA).

(d) Coarse circuit pattern. A layer of photolithography defines the coarse circuit features in the superconducting film, such as coplanar waveguide, ground planes, and signal line. We use the same recipe as in steps (b) and (c) to define a resist etch mask; after development, we transfer the pattern into the superconducting film with a CF₄/O₂ reactive-ion etch (RIE) process in a Plasmapro 100 ICP300 from Oxford Instruments, with an etch rate of roughly 8 nm/min.

(e) Fine circuit pattern. Electron-beam lithography defines the finer structures, such as the meandering nanowire inductor, the shunt inductor, and the interdigital gap of the capacitor. We first spin a thin layer of an adhesion promoter (AR 300–80) before spinning a roughly 170 nm thick layer of the electron-beam resist ARP–6200–09 (CSAR 09), baking at 150 °C for 1 min. We expose with a dose of 110 μC/cm² in a Voyager EBL system from Raith Nanofabrication and etch the Nb-Ti-N film using the same CF₄/O₂ RIE process as in step (d). In our design, we vary the widths (w = 75, 100, and 200 nm) of the nanowires across the wafer, adjusting the total number of squares (total inductance) and the capacitor to obtain a resonant frequency ω_c/2π ∼ 4.5 GHz, see Figure 1d–f.

(f) Cantilever pattern. Photolithography defines the chip and cantilever. We spin a 1.7 μm thick photoresist maP1225, bake at 105 °C for 2 min. We then expose with a dose of 300 mJ/cm² in the MLA150, and develop in maD331 for 45 s. We etch through the Si-N layer using a CHF₃/SF₆ process with an etch rate of roughly 100 nm/min in the Plasmapro 100 ICP300.

(g) Back-side through-etch. Before etching through the back side of the wafer, we first spin a protective positive resist on the front side and pattern an opening, or a “trench”, around the chip, which we will use to complete the etch once a larger portion of the wafer has been etched through from the back. We design the trench so that all cantilevers on the wafer are released at the same time, regardless of their length. To this end, we spin a roughly 2.2 μm thick layer of photoresist (maP1225), bake it at 105 °C for 3 min, expose with a dose of 550 μC/cm² and develop it in maD331 for 60 s. With the front side of the wafer protected, we flip over the wafer and etch through the Si-N using the same CHF₃/SF₆ process as in step (f) and the etch mask defined in step (c). We then use a Bosch process to etch through most of the Si substrate (approximately 450 μm deep) with an etch rate of roughly 6 μm/min. This results in all the samples on the wafer being supported by a thin layer of silicon close to the top side.

(h) Release and cantilever under-etch. A simple and fast method of release uses an isotropic dry-etch that completes the chip’s release from the wafer and removes the unwanted silicon support underneath the silicon nitride cantilever. We etch the silicon through the “trench” defined in step (g), from the top side with a short Bosch etch, followed by an isotropic etch that undercuts the cantilever. We use an SF₆/O₂ RIE process in the Plasmapro 100 ICP300 with lateral etch rate of 10 μm/min. The isotropic etch results in an uneven clamping line, as shown in Figure 1g, leading to variations in the mechanical resonant frequency.

Before the final etch and release step, we apply a Nitto Semiconductor Wafer Tape to the back side of the wafer, holding it together during release. After release, the wafer is cleaned in mrREM7000 and IPA, and the individual sensor chips are separated from the wafer in a single step by lifting away the outer frame, with the chips remaining on the wafer tape, as shown in Figure 5i.

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).

Tip deposition

In scanning probe microscopy (SPM), the tip plays a fundamental role in the achievable lateral resolution of the image. The focused electron-beam induced deposition (FEBID) [34] technique has been adapted to fabricate tips for SPM, for example, to enhance commercial platinum–iridium alloy (Pt-Ir)-coated conductive tips [35], or to realize laterally grown high-aspect ratio nanopillars [36]. We realize sharp, vertically grown conductive tips at the apex of the Si-N cantilever using FEBID with a Pt precursor gas. Figure 6 shows the resulting structure. We obtain the conical shape by stacking multiple depositions with different radii to achieve a total tip height in the range of 1–2 μm. This conical structure gives added rigidity to lateral forces while scanning. We form a sharp tip at the apex of the cone by exposing a circular area with a diameter of 10 nm, which is smaller than the nominal electron-beam spot size, and by setting the deposition height to 10 μm. Defocusing of the electron spot during vertical growth naturally forms a narrowing conical structure. At the apex of this cone, we routinely achieve a curvature radius of less than 10 nm, as verified by the SEM image in Figure 6c. Finally, we deposit a thin strip to connect the base of the cone to the Nb-Ti-N film, which is the ground plane of the microwave circuit. This feature enables the measurement of the tunneling current between the tip (grounded) when a DC bias is applied to a conductive sample surface. Scanning tunneling microscopy (STM) operation was verified both at room temperature and in a cryogenic environment. Thus, the deposited material is suitably conductive for STM and various electrostatic AFM techniques that require applying a low-frequency voltage to the tip.

Mechanical mode

Our chips were the same size as an AFM cantilever chip, making it easy to load them into a commercial AFM for mechanical characterization of the cantilever. For the design with nominal cantilever width 40 μm, length 50 μm, and thickness 600 nm, the optical lever detector in our AFM had sufficient bandwidth to detect the fundamental bending mode. We measured 60 chips from one wafer, detecting the thermal fluctuations at room temperature under ambient conditions and fitting them to a Lorentzian lineshape. We found that ω_m decreases with the increasing radial distance D from the center of the wafer, as shown in Figure 7, probably because of non-uniform etching conditions across the wafer. To some extent, one could change the mask design and adjust the dimensions of the cantilever to compensate for this effect. Using the mean value of 641 ± 42 kHz and adjusting the Young’s modulus of our Si-N plate to 208 GPa, we find good agreement between mechanical simulation and experiment.

Electrical mode

From the measured normal-state resistance of our nanowires and the measured thickness and width, we find a sheet resistance R_□ = 243 Ω/□, corresponding to a resistivity of ρ_n = 365 μΩ cm. We monitor the microwave response during cool-down and estimate a critical temperature T_c = 9.6 K, from which we estimate the superconducting energy gap with the BCS relation Δ₀ = 1.76k_BT_c = 1.46 meV. Using Equation 4, we find a kinetic inductance per square L_k,□ = 35 pH/□ for the 200 nm wide nanowires, corresponding to a kinetic inductance per unit length L_k/ℓ = 175 pH/μm. We compare this to the estimated geometric inductance per unit length, using the thin-ribbon formula [27,37] L_g ≈ (μ₀/2π)ℓ ln(2ℓ/w), from which we obtain L_g/ℓ = 17 pH/μm for our 200 nm wide nanowires. The ratio of kinetic inductance to total inductance is α = L_k/(L_k + L_g) ≃ 1, meaning that we can safely neglect the geometric contribution to the total inductance of our nanowires. This approximation is also valid for other samples with smaller nanowire widths, which are expected to have a higher value of L_k/ℓ.

We measured the microwave cavity resonant frequency ω_c on 26 chips. In some cases, we studied the temperature dependence of ω_c and verified electromechanical coupling between the cavity mode and the cantilever mode. These measurements were performed in a dry cryostat (DynaCool Physical Properties Measurement System from Quantum Design) with a base temperature of 1.7 K. We modified a measurement stick by adding high-frequency coaxial cabling for microwave signals to probe the cavity response and twisted pairs for lower-frequency signals, such as the voltage applied to the piezo disk that inertially actuates the cantilever. The stick is equipped with a cold attenuator, a directional coupler, and a cryogenic amplifier for low-noise microwave reflection measurement, as shown in Figure 8. Low- and high-frequency signals are synchronously synthesized and measured with a digital multifrequency microwave measurement device (Vivace from Intermodulation Products AB) to measure phase-sensitive electromechanical transduction [8].

We sweep the microwave frequency and measure the reflected signal (amplitude and phase) to locate the resonance. A sharp dip in reflection is easily observed in the more slowly varying background. We zoom in on the dip to capture the resonance lineshape at low power, that is, in the linear response regime. Using standard methods [38], we analyze the frequency dependence of the reflected amplitude and phase to determine the cavity resonant frequency ω_c, the internal quality factor Q_int, and the external quality factor Q_ext. From the measured ω_c and the simulated capacitance, we extract a kinetic inductance per square L_k,□ for the meandering structures of different nanowire widths. For the sample shown in Figure 9, we find ω_c/2π = 4.637 GHz, Q_ext = 4747, and Q_int = 17690 at T = 1.7 K with an on-chip power of −127 dBm. However, most of the tested samples were strongly overcoupled, making it difficult to extract a reliable value for Q_int using the standard fitting methods. Nevertheless, we can reliably determine the resonant frequency for all measured samples and calculate the kinetic inductance using the simulated capacitance and nominal number of squares in the meander. Table 1 summarizes the kinetic inductance per square thus determined for different nanowire widths w. The values of L_k,□, in the range of 32–60 pH/□ across all nanowire widths, are in approximate agreement with nanowires of similar materials and dimensions [21,37,39,40].

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 ω_c by many linewidths, and a change of the coupling parameter η. Figure 9a and Figure 9b show the amplitude and phase of the reflected signal for one of the chips as functions of the temperature. At each temperature, we fit to extract ω_c, Q_ext, and Q_int. As shown in Figure 9c, the external losses are roughly independent of the temperature, while the internal quality factor degrades with temperature. The change in ω_c results from a temperature dependence of L_k, which, together with the change in Q_int, results in a transition from overcoupled to undercoupled at T ≈ 2.8 K, as shown in Figure 9d.

Electromechanical coupling

Electromechanical coupling allows us to detect forces on the tip by measuring the bending of the cantilever. Measurements of the harmonic motion of a nanomechanical resonator coupled to a microwave electrical mode using multiple microwave tones have demonstrated very high force sensitivities [41]. We use a multifrequency drive and read-out scheme, facilitated by the device being in the sideband-resolved regime [8]. Figure 10a illustrates the working principle, where two microwave tones are applied at ω_c ± ω_m, while simultaneously actuating the cantilever through a coherent drive. Sidebands generated by the mechanical motion interfere with each other at the cavity’s resonant frequency ω_c, either constructively or destructively, depending on the phase of the mechanical motion ϕ_m, which is a controllable parameter. Figure 10b shows this phase-sensitive detection of the signal as a function of the mechanical phase ϕ_m relative to the microwave drive tones. This measurement was made in a dilution refrigerator on a sample with ω_m/2π = 5.828 MHz.

Conclusion

We described our approach to designing cantilever force sensors with integrated microwave cavity electromechanical sensing of flexural motion, based on the strain-dependent kinetic inductance of a superconducting nanowire. This type of force sensor is potentially interesting for low-temperature AFM as the superconducting cavity read-out scheme is intrinsically low noise. Compared to other low-temperature AFM probes, such as the qPlus sensor [42], our versatile design allows for achieving a wide variety of the parameters of the mechanical mode (force transducer) with a much lower effective mass and, consequently, an improved force sensitivity. The integrated resonator used for motion detection also provides a potentially quantum-limited approach to motion detection. It is difficult to make a complete and detailed comparison of AFM force sensors using a microwave resonator with those using an optical cavity to detect motion [10,43] because of numerous differences in the entire measurement apparatus. However, in comparison to optical resonators, the ability of microwave resonators to easily operate in the sideband-resolved regime allows for additional and potentially more efficient schemes of sensing force with minimal back-action from motion detection.

Our design covers a vast parameter space, balancing different considerations for both the electrical mode, such as critical current I_c, critical temperature T_c, and kinetic inductance per square L_k,□, and the mechanical mode, such as resonance frequency ω_m, quality factor Q_m, and spring constant k of the cantilever. These trade-offs affect the transduction efficiency and force sensitivity. Although a large single-photon coupling rate g₀ is desirable, we prioritize a cavity that we can pump to large intra-cavity photon numbers while maintaining linearity. In this regard, variations on the design presented here need to be fabricated and tested. The sensors described here represent the first generation of devices, where there is room for improvement. These devices served to establish the fabrication process that we described in detail herein and to verify that kinetic inductive mechano-electric coupling (KIMEC) is a useful, albeit not entirely understood, physical effect. Further investigation of alternative designs will help to shed light on the underlying physical mechanism behind KIMEC. We hope that this study motivates future work in this direction.