Direct-write nanoscale printing of nanogranular tunnelling strain sensors for sub-micrometre cantilevers

The sensitivity and detection speed of cantilever-based mechanical sensors increases drastically through size reduction. The need for such increased performance for high-speed nanocharacterization and bio-sensing, drives their sub-micrometre miniaturization in a variety of research fields. However, existing detection methods of the cantilever motion do not scale down easily, prohibiting further increase in the sensitivity and detection speed. Here we report a nanomechanical sensor readout based on electron co-tunnelling through a nanogranular metal. The sensors can be deposited with lateral dimensions down to tens of nm, allowing the readout of nanoscale cantilevers without constraints on their size, geometry or material. By modifying the inter-granular tunnel-coupling strength, the sensors' conductivity can be tuned by up to four orders of magnitude, to optimize their performance. We show that the nanoscale printed sensors are functional on 500 nm wide cantilevers and that their sensitivity is suited even for demanding applications such as atomic force microscopy.

The region marked in green indicates expected geometries where optical sensing outperforms nanogranular tunneling resistor (NTR) sensing, if optical detection were practical to implement for these cantilever geometries. The region marked in yellow indicates where NTR sensing is expected to outperform optical sensing. Unallowed geometries are such that the cantilever thickness would exceed the width. Figure 8 | Estimation of gauge factor for small sized nanogranular tunneling resistor. a, Strain distribution in the structure simulated in COMSOL for 50 nm cantilever deflection. Cantilever size: 1.5×0.5×0.1 μm 3 , active nanogranular tunneling resistor (NTR) size: 150×100×20 nm. b, estimated gauge factors from simulated strain in the structure and measured change of the relative resistance. c+d, possible scenarios for the reduction of the apparent NTR gauge factor: current flow directly from one electrode to another through parasitic resistance % & (c) and misalignment of the gold contact electrodes, for which only part of the NTR acting as strain sensor (d). Figure 9 | Scanning electron microscopy image of the nanogranular tunneling resistors deposited on a cantilever using the semi-automated deposition process.

Supplementary Note 1: influence of parasitic resistances on the signal-to-noise ratio (SNR)
One suggested method for the deflection readout of small sized cantilevers is resistive strain sensing using thin gold films 4 . However, as the size of the cantilever is decreased, the resistance of the thin gold film resistor decreases as well. In the measurement setup there will always exist some parasitic resistances of the connection lines. When these resistances become on par with the resistance of the strain sensing resistor, the SNR of the readout will start to decrease significantly. One benefit of the NTRs is that they can be deposited with sub-micron dimensions while their resistance can be tuned by post irradiation, hence always keeping the resistance in an optimal range.
In order to compare the SNR loss due to parasitic resistances in thin gold films and NTR deflection readout, we performed a case study. We assume that the change of the resistance is measured using a Wheatstone bridge, with one active resistor positioned on the cantilever and three passive resistors. We will denote the total parasitic resistance of each trace connecting the bridge as 2 3 * (see Supplementary Figure 1a). Parasitic resistances of the connection lines in the bridge itself are neglected as they can be designed to be fairly small.
For the ease of writing, we define 2 3 = 22 3 * . We would like to maximize the input bridge voltage 6 78 in order to get a higher output signal. The maximum input voltage that we can apply is defined by the maximum allowed current through the resistor 9 /01 as 6 78,/01 = 2 2 + 2 3 9 /01 , where 2 is resistance of the bridge resistors. In this case, for a certain cantilever deflection <, the maximum achievable output voltage of the circuit is (1) where ' and + are the cantilever length and thickness; ' ( and + ( represent the length and thickness of the sensing resistor and , is sensor gauge factor. In order to estimate the SNR of the readout, we are considering the intrinsic noises of the circuit resistors. For simplicity, we will not take into account the low frequency 1 E noise as we are only considering the case where resistors are used for strain sensing at higher frequencies, with band pass filtering around the working frequency. In this case the overall circuit noise comes from the circuit Johnson noise: The SNR of the readout for a given cantilever deflection < can be calculated as

Case study
The cantilever used in our calculations is a silicon nitride cantilever (N = 250 RST, -= 3100 HU/W C ), having dimensions of 3×1×0.1 ZW C . The measurement bandwidth is set to 500 H[\ and the point load induced deflection of the cantilever is set to < = 1 ]W. Calculations were made for three values of the connection lines parasitic resistance 2 3 * = 5 Ω, 25 Ω and 50 Ω.
Two types of resistive strain sensors were compared: a U-shaped thin gold film resistor and a cuboid shaped NTR resistor. The assumed properties of each resistor are given in Supplementary Results obtained by the calculation are plotted in Supplementary Figure 2. SNR losses due to parasitic resistances are much larger for the case of the thin gold film resistor. This is because the maximum achievable resistance for the gold film resistor is around 50 Ω (when the gold trace goes along the whole cantilever length). By varying post deposition irradiation dose and resistor length, for chosen thickness and width of the sensor (Supplementary Table 1), the resistance of the NTR can be tuned from several 100s Ω to several 10s of kΩ.

Supplementary Note 2: explanation of the signal-to-noise ratio expression
In the paper equation (1), the signal-to-noise ratio is defined as where b /01 represents the peak of the cantilever thermomechanical noise spectral density at cantilever resonance frequency, ] F is the NTR noise spectral density at cantilever resonance frequency and cL is the deflection sensitivity of the measurement system.
b /01 is given by where H I and J are Boltzmann constant and temperature; d , H and E " are cantilever quality factor, spring constant and resonance frequency. The parameter 4/3 represents a correction factor due to a non-perfect harmonic oscillator behaviour of the cantilever and the fact that induced strain, rather than deflection is measured with strain-sensing element [5][6][7] .
The NTR noise spectral density ] F 8 and the deflection sensitivity cL 9 are given by where 2 is resistance of the NTR, 6 g is supply voltage to the Wheatstone bridge, , is gauge factor, h is NTR sensor length, and ' and + are cantilever length and thickness. For simplicity, in the paper, the numerical constant 0.173 in the SNR expression is calculated from all other numerical constants.

Supplementary Note 3: I(V)-characteristics nonlinearity due to local heating effects
A local heating predominately results in changes of the conductivity of the NTR. We have derived the following analysis of the contribution of the heating and matched the theoretical explanation to the to the measured data.
We start with the geometry of the Pt(C) deposits with different gap sizes between Au electrodes, as shown below, in Supplementary Figure 3. We begin our analysis with the entropy continuity equation, as follows where the entropy rate of change iL/i+ in the deposit is given by the entropy production rate j due to the current flow 9 and the entropy current 9 g leaving the deposit towards the Au electrodes and substrate. We have furthermore j = 6 a 2 1 J = 6 a k J *ℎ h " 1 J where 6, 2, k J and J denote the applied voltage, the sample resistance, the sample conductivity and temperature, respectively. For the entropy current we note J " is the bath temperature referring to the Au electrodes and substrate. We set J " = 295 p. k g is the entropy conductivity which we assume to be approximately temperature independent, for simplicity. For the effective area b mnn and length h mnn we set in reasonable approximation b mnn = * h " + 2∆h and h mnn = q a .
Under steady-state conditions iL/i+ = 0 we obtain immediately from Eq. (8) Solving for J we finally have with J r being physically relevant for calculating the steady-state temperature of the deposit due to energy dissipation during current flow.
The temperature-dependent conductivity of the deposits we obtain from a wealth of independent measurements in the temperature range from 295 K down to about 2 K. In the present case, the deposits were irradiated with 74 ]`/ZW a which we take into account. Also, from our data, presented in paper Figure 3e, we can directly obtain the conductance at J " by linear extrapolation of the I(V )-characteristics.
The entropy conductivity k g or, alternatively, heat conductivity t = k g J is not readily available. From recent theoretical work on nano-granular metals in the strong-coupling limit 10 we estimate k g ≈ 2×10 va w/W p a (14) For the sample with h " = 500 ]W gap size and adapt it accordingly for the other samples depending on their respective electrical conductivity, as k g ∝ k for granular metals in the strong-coupling regime 10 . The results of our calculation providing an estimate of the samples' temperature for different gap lengths, as well as for the I(V)-characteristics are shown in Supplementary Figure 4.
Considering the rather crude model used for the calculation, we can state a rather good correspondence between the simulation and the measured data. We conclude that the dominating factor for the non-linear behavior of the I(V)characteristics at small gap sizes is a heating effect.

Supplementary Note 4: Influence of the FEBID sensor deposition on the cantilever dynamics
We investigated the effect of the sensor mass and stiffening on the cantilever resonance frequency and spring constants using finite element analysis (FEA) The obtained results are shown in Supplementary Figure 5. Even when going to extreme sensor thicknesses of 30% of the cantilever thickness, the effect is only a ~10% shift, in both the resonance frequency and spring constant. We chose finite element analysis rather than actual experiments since the spread in mechanical properties of the base cantilevers due to the typical fabrication tolerances is much larger than the effect of the added mass.

Supplementary Note 5: AFM measurement setup and procedure
AFM imaging with NTR cantilevers was performed using a custom-made electronics setup and custom-made AFM head (see Supplementary Figure 6a-c). Details regarding the design of the AFM head are previously published in reference 11 . The custom-made AFM head allowed us to compare the self-sensing performance of the small 20×8 µm 2 sized NTR cantilevers to optical beam deflection (OBD) readout. A custom-made fluid-sealed cantilever holder allowed for imaging in both air and liquid environments (see Supplementary Figure 6d). All connection lines on the cantilever and the PCB chip were fluid isolated, except for the NTRs themselves, which were in direct contact with fluid.
The electrical readout of the NTR cantilevers consisted of a full Wheatstone bridge of NTR resistors on the cantilever and subsequent amplification stages (Supplementary Figure 6a). The first amplification stage, a low noise instrumentation amplifier (AD8250), was positioned in the AFM head close to the cantilever sensor to reduce noise and stray capacitances from the Wheatstone bridge electrical connections. A high precision voltage reference (ADR130) biased the bridge. The subsequent amplification stages were located in an electronics module below the standard Bruker MultiMode AFM base and designed to intersect between the Bruker Nanoscope controller and the Bruker MultiMode AFM base (see Supplementary Figure 6b-c).
For AFM measurements the DC offset of the Wheatstone bridge was first zeroed. The electrical and optical deflection signals were verified by a frequency sweep of the tapping excitation piezo; the appropriate working frequency was further confirmed via a thermal tune using optical deflection detection. Finally, appropriate cantilever free and set point oscillation amplitudes, gains and other imaging parameters were chosen.

Supplementary Note 6: Estimate of cantilever length scale where NTRs outperform optical detection
Here we consider as the signal to noise ratio (SNR) the total thermomechanical peak height ptm versus the baseline noise floor. Both NTR Johnson noise and optical beam deflection photodetector shot noise may be considered as white noise sources with noise floor n and ripple δ. The total peak height of the thermomechanical peak is the square root of the sum of the squared thermomechanical noise ttm and the white noise The SNR is then the height of the peak above the baseline, divided by the ripple in the noise floor where VG is the Wheatstone bridge voltage, \ is the cantilever end deflection, , the NTR gauge factor, Vs the Wheatstone bridge supply voltage, lg the length between contacts of the NTR sensor, l the cantilever length, and t the cantilever thickness. The Johnson noise of the NTR element is then where kB is the Boltzmann constant, T the temperature of the sensor (assumed 298 K), and RNTR the resistance of the NTR element. The noise floor of the optical beam deflection system is given by 12 where a is the size of the laser spot on the photodetector, s the distance from the cantilever to the photodetector, e the electron charge, S the photosensitivity of the photodetector, and P the laser power at the photodetector. For practical geometrical parameters (see Supplementary Table 2), this noise floor is about 50 fm/√Hz at 0.5 mW of laser power at the photodetector. The thermal peak for the optical deflection measurement is given by and for the NTR deflection by The difference is due to the optical system measuring an angle change as opposed to a displacement 5 . The cantilever resonance frequency f0, spring constant k and quality factor Q are calculated from where E is the cantilever Young's modulus,the cantilever density, w the cantilever width, Ñ the viscosity of the surrounding medium (air), M the molecular mass of air, R the universal gas constant and p the pressure of the surrounding medium (assumed 1 atm).
In order to simplify the parameter space, we assume a cantilever planar shape such that ' = 3*. Supplementary Table 2 lists values of the relevant parameters. Supplementary Figure 7 illustrates the regions where SNR âäã > SNR çãé and where SNR çãé > SNR âäã . Starting below 3 µm length, there exists a broad range of cantilever geometries where the expected NTR signal to noise ratio is larger than the optical signal to noise ratio. This NTR performance advantage becomes especially prominent at submicron cantilever lengths, in addition to the extreme practical limitations of optically detecting these cantilevers.

Supplementary Note 7: Apparent gauge factor of the smallest NTR sensors
In order to estimate gauge factors of the NTR structure deposited on the 1.5×0.5×0.1 μm silicon nitride cantilever, a finite element analysis simulation was performed in COMSOL Multiphysics (see Supplementary Figure 8a). To calculate the strains in the structure, we estimated that the elastic modulus of the NTR is close to the Young's modulus of a diamond-like structure of amorphous carbon 13,14 . Therefore, in simulations we assumed the Young's modulus for NTR of 760 GPa.
For irradiation doses in the range from 0 -500 nC/μm 2 , estimated gauge factor values were in the range of 2 -5 (see Supplementary Figure 8b). This range of values is lower than gauge factors measured for larger cantilevers and NTR structures. We suspect that this is due to several reasons: 1. Assumed values for the NTR Young's modulus might be off and we might have stronger stiffening effects of the cantilever. 2. If the NTR is too broad, a fraction of the current can directly flow from the left to the right electrode and not over the bending edge (see Supplementary Figure 8c). For this part the resistance does not change during bending of the cantilever. 3. If the gold contacts are not located at the bending edge but are slightly offset back on the chip body, the effective change of resistance ∆2 is reduced to è ' ( • ∆2 (see Supplementary Figure 8d). This effect was incorporated in the COMSOL simulation. However it was difficult to estimate the exact position of the cantilever bending edge from the SEM cantilever image.