Comparison of force sensors for atomic force microscopy based on quartz tuning forks and length-extensional resonators

The force sensor is key to the performance of atomic force microscopy (AFM). Nowadays, most atomic force microscopes use micromachined force sensors made from silicon, but piezoelectric quartz sensors are being applied at an increasing rate, mainly in vacuum. These self-sensing force sensors allow a relatively easy upgrade of a scanning tunneling microscope to a combined scanning tunneling/atomic force microscope. Two fundamentally different types of quartz sensors have achieved atomic resolution: the “needle sensor,” which is based on a length-extensional resonator, and the “qPlus sensor,” which is based on a tuning fork. Here, we calculate and measure the noise characteristics of these sensors. We find four noise sources: deflection detector noise, thermal noise, oscillator noise, and thermal drift noise. We calculate the effect of these noise sources as a factor of sensor stiffness, bandwidth, and oscillation amplitude. We find that for self-sensing quartz sensors, the deflection detector noise is independent of sensor stiffness, while the remaining three noise sources increase strongly with sensor stiffness. Deflection detector noise increases with bandwidth to the power of 1.5, while thermal noise and oscillator noise are proportional to the square root of the bandwidth. Thermal drift noise, however, is inversely proportional to bandwidth. The first three noise sources are inversely proportional to amplitude while thermal drift noise is independent of the amplitude. Thus, we show that the earlier finding that quoted an optimal signal-to-noise ratio for oscillation amplitudes similar to the range of the forces is still correct when considering all four frequency noise contributions. Finally, we suggest how the signal-to-noise ratio of the sensors can be improved further, we briefly discuss the challenges of mounting tips, and we compare the noise performance of self-sensing quartz sensors and optically detected Si cantilevers.

Figure 1

Parameter fields of cantilever spring constants $k$ and oscillation amplitudes $A$ for classic Si cantilevers, qPlus sensors, and needle sensors. The $(k,A)$ data points for Si cantilevers and qPlus sensors are adapted from Table  in Ref. 5, and those from shortened qPlus sensors are taken from Ref. 7. To enable stable oscillation of the cantilever at the optimal amplitudes around 100 pm, it was necessary to increase the spring constants of cantilevers (“classic” FM-AFM) from about 10 N/m by more than two orders of magnitude (qPlus sensors). The needle sensor has a stiffness that is almost three orders of magnitude larger than that of the qPlus sensor. The question of whether this further increase is beneficial is addressed in this paper.

Figure 2

(a) Needle sensor. (b) qPlus sensor. The scale bar is valid for both sensors.

Figure 3

Geometry of sensors based on quartz tuning forks (a)–(e) and length-extensional resonators (f)–(j). A qPlus sensor (a) is created by attaching one of the prongs of the tuning fork to a substrate and attaching a tip to the other prong. For clarity, only the electrodes on the free prong are shown. The prong without displayed electrodes is fixed to a massive substrate (not shown here; see Fig. 2b). A needle sensor (f) is built by attaching a light tip to one prong of the length-extensional resonator. Parts (a), (b), (f), and (g) illustrate the geometrical dimensions as listed in Table 1; parts (c) and (h) show a schematic view of the electrostatic field in the cross sections; and parts (d) and (i) show the mechanical stress profile along a cross section. Parts (e) and (j) show the idealized field distribution within the quartz crystals. The qPlus sensor uses a bending mode, thus the mechanical stress is maximal where the charge-collecting electrodes are located (d), while the length-extensional resonator develops a uniform stress profile (i). The idealized field distribution (e,j) is much closer to the actual field distribution (c,h) for the needle sensor [(j) vs (h)] than for the qPlus sensor [(e) vs (c)].

Figure 4

(a) Mechanical analog of a single oscillator-type force sensor [standard cantilever or qPlus sensor as in Fig. 2b], consisting of a single oscillating beam. The single oscillator has only one degree of freedom, its deflection $q$. (b) Mechanical analog of a coupled oscillator used as a force sensor [tuning fork or length-extensional resonator as in Fig. 2a]. The coupled oscillator has three degrees of freedom: the deflection of the central mount $q_c$ and the deflections of the two coupled oscillators $q_{1,2}$.

Figure 5

Signal-to-noise ratio (SNR) as a function of the product between decay constant $\kappa$ and amplitude $A$, where the decay constant $\kappa$ is inverse to the interaction length $\lambda$, thus $\kappa =1/\lambda$. Optimal SNR is obtained for $\kappa A=A/\lambda =1.545.$

Figure 6

Force gradient (red) and calculated frequency shift for the interaction of a silicon tip with an adatom on the Si(111)-(7$\times$7) surface modeled by a Morse potential with $E_{\text{bond}}=2.273$ eV, $\sigma =235.7$ pm, and $\kappa =12.70$ nm${}^{-1}$ for a qPlus sensor with $k=1800$ N/m and $f_0=30$ kHz and various amplitudes (see legend). If a standard needle sensor was used here, the frequency shift values denoted on the right vertical axis would have to be multiplied by 1/20, because the frequency shift is proportional to $f_0/k$. For the qPlus sensor, a minimal frequency shift of $-70$ Hz results at the optimal amplitude $A=122$ pm, while the needle sensor only yields a minimal frequency shift of $-3.5$ Hz.

Figure 7

Schematic of a quartz sensor, cable, and current-to-voltage converter that is often used for amplifying deflection data from quartz sensors. The gain of the amplifier is given by $V_{\text{out}}=-RI/(1+if/f_{c1})$ with its first corner frequency $f_{c1}$ given by $f_{c1}=1/\left(2\pi RC\right)$. The capacity of the cable should be as low as possible—cable capacity increases noise in the amplifier output. If the amplifier is vacuum-compatible, it can be placed close to the sensor, thus reducing cable capacity and noise. The sensor can be excited electrically, as shown in this figure, or mechanically—the drive signal is grounded in this case.

Figure 8

Current gain vs frequency for a current-to-voltage converter built from an ideal operational amplifier and a 100 M$\Omega$ feedback resistor with a parasitic capacitance of 0.2 pF (red line), yielding a first corner frequency (here, $f_{c1}=8$ kHz). For frequencies higher than $f_{c1}$, the gain drops proportional to $1/f$. Typically, these simple amplifiers develop a second corner frequency (here $f_{c2}=80$ kHz);[40] for frequencies higher than $f_{c2}$, their gain drops proportional to $1/f^2$. The black line displays the gain of a commercial charge amplifier[41] with a constant gain of ${10}^{13}$ V/C (black line) for a remarkably large frequency range from 250 Hz to 15 MHz.

Figure 9

Effect of temperature changes on the measured tip-sample force gradient. Both the needle sensor and the qPlus sensor change their frequency as a function of temperature. Although the relative frequency shift is much smaller than for silicon cantilevers, the effect on the measured force gradient scales with stiffness $k$. This thermal frequency drift noise is almost three orders of magnitude smaller for the qPlus sensor than for the needle sensor.

Figure 10

Effect of temperature drift on frequency drift, frequency noise at the PLL output, and force gradient noise. (a) A temperature drift of 125 $\mu$K/s is assumed, yielding a temperature increase of 75 mK over 10 min. (b) Frequency drift at at temperature 10 K above or below the turnover temperature $T_p$; see Eq. (47). For the needle sensor, the absolute frequency change over 10 min is 78 mHz, while for the qPlus sensor it is 2.5 mHz. (c) Power spectral density of the frequency drift noise for the needle and the qPlus sensors. A linear frequency drift with time causes a $1/f$ power spectrum. (d) Power spectral density of the tip-sample force gradient noise due to drift. This noise contribution is linear with the force constant of the sensor, i.e., it is 600 times larger for the needle sensor than for the qPlus sensor.

Figure 11

Thermal spectrum of a needle sensor with standard dimensions at room temperature and ambient pressure. A commercial preamplifier[41] was used. The sensitivity of the sensor is calculated to 45.4 $\mu$C/m, the $Q$ factor is 18 500, and the deflection detector noise density is 1.89 fm$/\sqrt{\text{Hz}}$. The $Q$-factor is determined by fitting the resonance curve of a damped harmonic oscillator to the thermal spectrum.

Figure 12

Thermal spectrum of a qPlus sensor with standard dimensions at room temperature and ambient pressure. A home-built preamplifier was used. The sensitivity of the sensor is calculated to 1.44 $\mu$C/m, the $Q$ factor is 2900, and the deflection detector noise density is 62 fm$/\sqrt{\text{Hz}}$.

Figure 13

Total experimental and calculated force-gradient noise densities as a function of modulation frequency for the needle sensor (red line) and the qPlus sensor (blue line) at room temperature. The calculated force-gradient noise densities are derived with the experimental values for $S,k,n_{\text{amp}},Q$, and $f_0$ at an amplitude of $A=100$ pm. The $1/f$ component for small $f_{\text{mod}}$ is due to thermal frequency drift noise [see Eq. (55)].

Figure 14

Calculated force-gradient noise densities $n_{k_{\text{t}s}}$ as a function of modulation frequency for the standard needle sensor (red line), qPlus sensor (blue line), and a modified qPlus sensor with $f_0=92.8$ kHz, $k=3500$ N/m, $Q=1650$, and $n_q=28$ fm$/\sqrt{\text{Hz}}$ (green line). The calculated values for $n_{k_{\text{t}s}}$ are based on measured values of $n_{\text{amp}},S,k,f_0$, and $Q$.