Thermal noise limit for ultra-high vacuum noncontact atomic force microscopy

Summary The noise of the frequency-shift signal Δf in noncontact atomic force microscopy (NC-AFM) consists of cantilever thermal noise, tip–surface-interaction noise and instrumental noise from the detection and signal processing systems. We investigate how the displacement-noise spectral density d z at the input of the frequency demodulator propagates to the frequency-shift-noise spectral density d Δ f at the demodulator output in dependence of cantilever properties and settings of the signal processing electronics in the limit of a negligible tip–surface interaction and a measurement under ultrahigh-vacuum conditions. For a quantification of the noise figures, we calibrate the cantilever displacement signal and determine the transfer function of the signal-processing electronics. From the transfer function and the measured d z, we predict d Δ f for specific filter settings, a given level of detection-system noise spectral density d z ds and the cantilever-thermal-noise spectral density d z th. We find an excellent agreement between the calculated and measured values for d Δ f. Furthermore, we demonstrate that thermal noise in d Δ f, defining the ultimate limit in NC-AFM signal detection, can be kept low by a proper choice of the cantilever whereby its Q-factor should be given most attention. A system with a low-noise signal detection and a suitable cantilever, operated with appropriate filter and feedback-loop settings allows room temperature NC-AFM measurements at a low thermal-noise limit with a significant bandwidth.


Amplitude calibration
In the optical beam-deflection detection scheme, the cantilever oscillation yields a nearly harmonically oscillating voltage V z at the output of the photodetector preamplifier as illustrated in Figure S1. For the calibration, we relate the voltage amplitude V A of the oscillating voltage V z = V A sin(2π f t + φ ) to the amplitude A of the mechanical cantilever oscillation with frequency f using a procedure where the tip-surface interaction is kept constant at a certain value while A is varied [1]. As the cantilever is inclined at an angle θ towards the sample surface, we distinguish between A, the oscillation amplitude of the cantilever and the component A z of the oscillation amplitude perpendicular to the sample surface. These two quantities are related by A deflection of the cantilever by an angle ∆θ changes the angle of the reflected laser beam by 2∆θ , resulting in a displacement of the laser spot on the PSD. The displacement on the PSD results in a difference ∆I z of the photocurrent from the PSD quadrants, which is in turn converted to a voltage signal V z by the preamplifier. The calibration establishes the relation between the amplitude A of the oscillating cantilever and the voltage amplitude V A A = S ×V A (13) where S is the sensitivity or calibration factor. Here, we obtain S in a noncontact measurement without any knowledge of the details of the detection system except the tilt angle θ . To accomplish this, the normalised frequency shift derived from the measured frequency shift ∆ f [2] γ(z ts , A) is kept constant during the entire calibration procedure to maintain a certain level of tip-surface interaction independent of the cantilever oscillation amplitude (see main manuscript text and Figure S1 S1 for a description of the quantities involved). Note that this concept is valid only for amplitudes larger than a critical value, which is typically 1 nm [3]. Figure S1: Schematic representation of the signal path in an NC-AFM system based on the optical beam-deflection scheme. The relation between the amplitude A of the oscillating cantilever and its component A z perpendicular to the surface is illustrated as well as the periodic cantilever deflection ∆θ , the PSD output current difference ∆I z and the preamplifier output voltage V z = V A sin(2π f t + φ ). The quantity z ts represents the tip-sample distance in the lower turning point of the cantilever oscillation.
In the calibration experiment, A is stepwise increased or decreased by a variation of the amplitude feedback setpoint defining V A , while the normalised frequency shift γ is kept constant by the choice of an appropriate frequency-shift setpoint ∆ f set . This causes the topography feedback to readjust the z-piezo position z p , which is recorded as a function of the varied oscillation amplitude V A as shown in Figure S1. A MATLAB script (The MathWorks, Inc., Natick, MA, USA) is used to control the setpoints of V A and ∆ f and to determine the corresponding z p value. Initially, the tip is approached to the surface and stabilised at a distance in the long-range interaction regime by the choice of an appropriate frequency shift setpoint ∆ f set . After waiting for a while to reduce piezo creep effects, the piezo position z p is recorded for each oscillation voltage amplitude V A , while V

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A ∆ f is kept constant by adjusting the ∆ f setpoint accordingly. Typical parameters and results are shown in Table S1 and Figure S2, respectively.

S4
The calibration factor S is obtained from the slope in the plot z p versus V A and considering the correction for the tilt angle θ : To compensate for thermal drift as well as piezo creep, this procedure is first performed with increasing oscillation amplitude and then repeated with decreasing oscillation amplitude. Respective measurements are denoted as "up" and "down" in the plots of Figure S2. While Figure S2a shows a measurement with a significant drift in the z direction, for the small drift measurement in Figure S2b, the "up" and "down" curves almost coincide with each other. To reduce the systematic error due to drift, the mean slope is used as the final result. The calibration measurement is repeated several times and results are averaged to reduce the statistical error. The standard deviation for a series of calibration runs is typically 1%.

Thermally excited cantilever oscillation
Thermally excited fluctuations of the cantilever are of special interest as they determine the ultimate noise limit in NC-AFM measurements. Based on an approach originally proposed to describe thermal noise in a resistor [4], we derive the power spectral density for the displacement D z th of a cantilever in thermal equilibrium with a thermal bath at temperature T . According to the equipartition theorem [5], we assume that the mean potential energy of the one-dimensional cantilever oscillator in contact with the thermal bath is equal to k B T /2 with k being the static stiffness of the cantilever and k B the Boltzmann constant. The cantilever is assumed to be excited by random thermal fluctuations in the thermal bath. According to the Wiener-Khinchin theorem [5], the cantilever mean-square displacement of this random stationary process S5 can be related to the power spectral density of the cantilever displacement D z th by The total mean-square displacement can be decomposed into the sum of displacements originating from the cantilever eigenmodes [6] and is related to constants of modal stiffness k n for the nth eigenmode. Using Equation 16 we find: where z n is the displacement corresponding to the nth eigenmode. Equation 18 demonstrates that the contribution of the nth eigenmode to the total energy of the thermally excited cantilever is It has been shown that the modal stiffness k n , specifically for the higher harmonics, strongly depends on the mass ratio µ = m tip /m beam [7,8], where m tip and m beam are the masses of tip and cantilever beam, respectively. Calculated relations between the modal stiffness and the static stiffness depending on the mass ratio are given in Table S2 for the first four eigenmodes. The Table shows that 97% of the total oscillation energy is in the fundamental mode for a beam without a tip and this fraction increases as the tip mass increases.
The ratios between the higher harmonic eigenfrequencies f n to the fundamental eigenfrequency f 0 in dependence of the mass ratio µ are given in Table S3.
Similar to the mean-square cantilever displacement, the total thermal displacement power spectral density is represented as the sum of contributions from the cantilever eigenmodes:  To determine the displacement power spectral density for each mode n, we assume a constant energy spectral density Ψ n from the thermal bath exciting the cantilever following the arguments given by Nyquist for the explanation of a thermally excited electrical current in a resistor [4].
The response of the cantilever to this thermal excitation is determined by its amplitude response G(ω), which is the amplitude response of a damped harmonic oscillator [10]. For each eigenmode we assume where ω n = 2π f n and Q n are the angular eigenfrequencies and quality factors of the nth mode, respectively. Therefore, we represent the energy spectral density of the nth mode, being (1/2)kD z th,n , Considering the result of Equation 18, stating that a fraction k/k n of the total energy k B T /2 goes to the nth eigenmode, and integrating over D z th,n we find for each mode: dω .
We substitute ζ n = ω/ω n and dω = ω n dζ n and find

Solving the integral using
with a = (1/Q 2 n − 2) and b = 1 yields the excitation energy spectral density for the nth mode: Inserting this result in Equation 21 yields the displacement power spectral density of the thermally excited cantilever: for the nth oscillation mode. This agrees well with results reported by other authors [11][12][13]. A S8 calculated example of the displacement thermal noise spectral density d z th = D z th for a typical cantilever at room temperature for the first four oscillation modes is given in Figure S3.   The bandwidth B −3dB is defined as the frequency f m at which the signal is attenuated by 3 dB: For the 300 Hz filter, we obtain B −3dB = 392 Hz. For noise considerations, often the equivalent noise bandwidth B ENBW defined as [14] is used instead. Here, we yield B ENBW = 450 Hz for the 300 Hz filter.
For the easyPLL plus system, we investigate two different filter settings with nominal frequencies of 120 Hz and 400 Hz and obtain the amplitude response curves shown in Figure <S6 with  For the PLLpro2 system, the transfer function is generally more sophisticated as this system offers a large variety of filter settings. Here, the adjusTable low-pass filter is not located at the ∆ f output but inside the feedback loop of the PLL. Therefore, the frequency response not only depends on the low-pass filter, but also on the proportional-integral (PI) controller settings and the system has to be described for closed-loop operation [15].
To describe this system in detail, we first recall that the transfer function of the open loop with S12 the low-pass characteristics H LP and an amplification factor K of the voltage controlled oscillator can be written as [15] H OL (s) = K × H LP (s) s .
The amplification K is split into a proportional part K P and an integral part K I . The PI controller determines the response of the feedback system.
The transfer function of the PLLpro2 can hardly be modelled in total due to the complexity of the system. Thus, the amplitude response is measured using the procedure described above. Several curves ∆ f m versus f m are recorded to investigate the influence of the loop filter H LP of order o and cutoff frequency f c as well as the settings of the PI controller on the frequency response. To relate the settings P and I in the PLL software to the parameters K P and K I of our model, a fit of the model to the measured curve is performed for various filter settings and yields prefactors of 369 deg for the P gain and 3.2 for the I gain. These values have to be multiplied to the values of P and I gain, set in the PLL software to obtain a quantitative description of the frequency response. To demonstrate the precision of the predictions, we plot two amplitude response curves in Figure S7. These curves show the amplitude response of the PLL system with f c = 1000 Hz, n = 3 and a P gain setting of −2 Hz/deg as a comparison between model and experiment. While we obtain a low-pass behaviour S13 with varying steepness for an I gain setting of 1 Hz (see Figure S7a), the I gain setting of 200 Hz yields a slight gain peaking at 100 Hz (see Figure S7b).   Figure S8) yielding a rather flat amplitude response over the desired bandwidth and a steep slope S14 above f c . If the P and I gain settings are too large, gain peaking occurs. In contrast, a significant attenuation is observed if P is too small. The phase response is also affected by P and I gain settings.
Here, the optimum settings yield a smooth response curve with the least pronounced changes in the gradient. to the product of the transfer function H CL (s) and the unit step function 1/s [17]. This is performed numerically [18] using a MATLAB script (The MathWorks, Inc., Natick, MA, USA). The respective graphs are shown in Figure S8c, where the curves demonstrate the result of wrong P and I settings for the step response in terms of overshoot or slow response. As a check for consistency, we directly measure the step response as shown in Figure S8d and find excellent agreement with data from Figure S8c. Note that the optimum for the amplitude and phase response corresponds to the optimum step response (P = −2.2 Hz/deg, I = 1 Hz).
Obviously, such simulations are not only useful for the noise analysis, but can also be used to find optimum values for the PI controller settings of the PLL system. In Table S4, the ideal P gain values for several combinations of filters and cutoff frequencies are shown to provide a proven set of filter parameters for the PLLpro2 system useful in practice. As a rule of thumb, we find that the higher the bandwidth, the more the P gain has to be reduced. We find that adjusting the P gain is critical, as a wrong setting easily results in gain peaking in the vicinity of f c . Adjusting the I gain is less critical. High values yield a moderate gain enhancement in a frequency region below f c . Therefore, we chose a setting of I = 1 Hz for most of our measurements and also for the simulations presented here. A typical setting for our scan environment for highresolution measurements is f c = 500 Hz, o = 3, P = −2.0 Hz/deg and I = 1 Hz.

RMS noise
The RMS-noise δ f tot in the frequency-shift signal ∆ f can be obtained from the integration of the ∆ f noise power spectral density over the entire frequency range For the analysis of our experiments, we perform a numerical integration of D ∆ f tot ( f m ) from 0 to 10 kHz as the noise power spectral density is always negligible above 10 kHz.
For the case where the spectral function D ∆ f tot ( f m ) is not known, the following approximative expression has been suggested [19] to estimate the RMS noise: This expression results from integrating the unfiltered frequency-shift noise D ∆ f tot over the bandwidth B. To reveal the discrepancy between the precise result from Equation 26 and the approximation S17 from Equation 27, we represent δ f 2 tot as: reveals that these expressions are identical for the thermal noise but not for the detection system noise. Comparing the approximative result of Equation 27 to the precise result of Equation 26 for realistic PLL filter settings, we find that the approximation underestimates the total noise typically by 20% to 30%. A comparison for various filter settings is shown in Table S5.