Noise Calibration¶

A quantitative measurement of tip-surface force requires an accurate calibration of the cantilever. Noise calibration uses the enhanced force sensitivity of a high Q resonance to measure the thermal equilibrium fluctuations of cantilever deflection, and thereby extract the calibration constants of both the cantilever and the detector. The method models the cantilever’s free dynamics as a simple harmonic oscillator; an accurate model for one cantilever eigenmode, in a frequency band near resonance.

ImAFM™ also takes advantage of this enhanced sensitivity and high accuracy, using only signals near resonance to reconstruct the force between the tip and the surface. Furthermore, ImAFM™ determines tip-surface force as a function of the cantilever deflection at fixed probe height. Thus, both axes of the reconstructed force vs. deflection curve (or force-quadrature vs. amplitude curve) rely on one and the same calibration. Quantitative measurement with ImAFM™ is therefore completely independent of the scanner calibration [Platz-2012a] , [Forchheimer-2012]. Noise calibration is performed for each cantilever at the start of the ImAFM™ session and it should be redone if any adjustments are made to the detector. Calibration is easily done at any time during a session so it can be frequently redone to check for consistency. Since the accuracy of quantitative ImAFM™ is traceable to this one measurement, it is important to understand the physical ideas behind noise calibration.

Calibration in the field of AFM is traditionally discussed in terms of finding the spring constant $k_s$ associated with the static (independent of time) deflection of the cantilever. Finding this one constant is sufficient to determine a static force $F$ , which gives rise to the static deflection $d$ ,

$d = \frac{1}{k_s} F$

ImAFM™ is a dynamic method, where force is determined from cantilever motion $d(t)$ . A linear relation between force and motion also exists for dynamic AFM if deflection is small in comparison to the cantilever length. However, due to inertia and damping, the constant of proportionality is frequency dependent. This linear relation between dynamic force and dynamic deflection is most easily expressed in the frequency domain.

$\hat{d}(\omega) = \frac{1}{k} \hat{G}(\omega) \hat{F}(\omega)$

The frequency dependence is contained in the dimensionless factor $\hat{G}$ , called the transfer gain, and the hat denotes a complex quantity, with the real part being the Fourier cosine component, and the imaginary part the sine component at the angular frequency $\omega$ . The complex, frequency-dependent quantity which relates each frequency component of the force to the component of the motion at the same frequency, is called the linear response function, and it is usually denoted $\hat{\chi}(\omega)$ . For the simple harmonic motion of each eigenmode of the cantilever,

$\hat{\chi}(\omega) = \frac{1}{k} \hat{G}(\omega;\omega_0,Q) = \frac{1}{k} [ 1 + i \frac{\omega}{\omega_0 Q} - \frac{\omega^2}{\omega_0^2} ]^{-1}$

Three parameters are needed to calibrate each eigenmode: The mode stiffness $k$ , effective mass $m$ and viscous damping coefficient $\eta$ , or alternatively, the stiffness $k$ , resonant frequency $\omega_0 = \sqrt{k/m}$ and the quality factor $Q=\sqrt{m k} / \eta$ . A complete calibration requires a fourth constant that converts the detector signal as measured in the digital counting units used by the MLA, or Analog-to-Digital Units ( $\text{ADU}$ ), to the actual cantilever deflection in nanometers. This fourth constant is called the detector responsivity $\alpha \text{ [ADU/m]}$ . Note that here we could express the responsivity in SI units $\text{Volts/m}$ , which would further require a calibration of the MLA against a voltage standard.

All four constants can be determined from one thermal noise measurement as described in [Higgins-2006]. Here we give a somewhat different derivation of the method which is based on the idea that the cantilever is in thermal equilibrium with the damping medium at temperature $T$ , and that the damping can be calculated using hydrodynamic theory. Thermal equilibrium and linear response allow us to use the fluctuation-dissipation theorem to relate the power spectral density of fluctuations of cantilever deflection, to the imaginary part (the dissipative or damping part) of the linear response function.

$S_{dd}(\omega) = \frac{2 k_B T}{\omega} \textbf{Im}[\hat{\chi}(\omega)] = 2 k_B T \eta \frac{1}{k^2}|\hat{G}(\omega)|^2 \text{ [m}^2/\text{Hz}]$

The simple harmonic motion gives us the functional form of $|\hat{G}(\omega)|^2$ . We fit this function to the measured noise peak at resonance, thus determining $\omega_0$ and $Q$ . To get the magnitude of $S_{dd}$ we require knowledge of the damping coefficient $\eta$ . Hydrodynamic theory is used to calculate the damping due to the surrounding fluid (air, water, etc.) [Sader-1998]. The result can be written as:

$\eta = L \mu \text{ Re } \Lambda(\text{Re})$

where $L$ is the length of the beam and $\Lambda(\text{Re})$ is a dimensionless hydrodynamic function of the Reynolds number,

$\text{Re} = \frac{\rho b^2 \omega_0}{4 \mu }$

The Reynolds number is a dimensionless quantity, which physically corresponds to the ratio of inertial forces to viscous forces in the hydrodynamic flow around the beam. It is formed from the resonant frequency, the fluid density $\rho$ and viscosity $\mu$ , and the width of the cantilever $b$ which is the relevant length scale for desribing the flow.

The hydrodynamic function $\Lambda(\text{Re})$ has been calculated for beams of arbitrary cross-section with length much larger than width (either transverse dimension), $L \gg b$ [Sader-1998]. However, a generalization of the theory shows that the basic idea of desribing the damping in terms of a hydrodynamic function of a Reynolds number, applies to an any elastic body of arbitrary shape which is damped by motion in a fliud [Sader-2005]. The beauty of this approach is that once the relevant hydrodynamic function $\Lambda(\text{Re})$ has been determined for a cantilever (or plate) of a particular shape as described by it’s plan-view dimensions (effective length and width), we can use this knowledge to calculate the fluctuation force on any cantilever of that shape simply by measuring the resonant frequency in the fluid [Sader-2012].

We measure the total noise which consists of two contributions: the cantilever fluctuations $S_{dd} \text{[nm}^2/\text{Hz]}$ , and the detector noise $S_{DD} \text{[ADU}^2/\text{Hz]}$ . Since the cantilever noise is statistically independent from the detector noise, we can simply add their noise powers to get the total noise power spectrum at the output of the detector,

$S_\text{tot}(\omega) \text{[ADU}^2/\text{Hz]} = S_{DD} + \alpha^2 S_{dd}(\omega)$

We fit this expression to the measured data, with $\alpha$ , $\omega_0$ , $Q$ and $S_{DD}$ as fitting parameters. Using the value of $\omega_0$ determined from the fit, we can calculate the Reynolds number $\text{Re}$ and thereby the hydrodynamic function to determine the damping coefficient $\eta$ . This coefficient, together with the value of $\omega_0$ and $Q$ determined from the fit, give the mode stiffness $k$ . Thus, thermal equilibrium linear response theory and the hydrodynamic damping theory together give us all quantities needed to determine the linear response function of the cantilever and the detector.

It is important to point out that the hydrodynamic damping function, and thereby the theoretical value of the damping coefficient $\eta$ , depends on the mode shape and is therefore specific to the particular eigenmode. Calculations show that the mode shape of the fundamental eigenmode of a long rectangular beam is particularly insensitive to the Reynolds number. Higher eigenmode shapes are sensitive to $\text{Re}$ [Sader-1998] and the mass of the tip placed at the end of the cantilever. This calibration method is therefore applicable to the fundamental eigenmode and we can expect it to be valid in different fluids, but its applicability to higher eigenmodes is questionable.

We should also stress that reconstructing force from the measured deflection over a wide frequency band, for example from many harmonics of a single drive frequency, requires determination of the detector response function $\hat{\alpha}(\omega)$ . An ideal detector should have a flat (frequency-independent) response function up to it’s cut-off frequency where it it will start to roll-off, or decrease with frequency to some power. However, real detectors do not always behave in this ideal way, and the detector electronics can introduce frequency-dependent amplitude changes and phase shifts over a wide frequency band. Since ImAFM™ works in a narrow band, we escape this difficulty and we can expect accurate calibration and therefore accurate force reconstruction with the assumption of a frequency-independent detector response function.

Note on sensitivity and accuracy¶

In the above text we emphasized that accuracy is higher when we base force reconstruction on the callibration of high Q resonance. This notion of accuracy is rooted in physical understanding: We are better able to model the force transducer and deflection detector in a narrow frequency band near a single eigenmode resonance, and thus make a good calibration in this narrow band. We also argued that sensitivity is higher near resonance. Our notion of sensitivity is independent of that regarding accuracy and it does not originate from theoretical understanding of the physics of cantilever beams or opto-electronic detection. Sensitivity has to do with the signal-to-noise ratio ( $SNR$ ) of measurement.

The field of AFM has unfortunately adopted the term ‘inverse optical lever sensitivity’ (or invOLS) to describe the constant $\alpha^{-1}$ which converts the measured detector signal (e.g. Volts) to nm of cantilever deflection. We use the term responsivity for $\alpha$ because it is actually the magnitude of a linear response function, and it does not tell us anything about the $SNR$ of the measurement. The sensitivity of the measurement is described by two quantities which are displayed in the Calibration Result panel.

Thermal noise force is the random force due to collisions of the molecules in the damping fluid with the cantilever. This random force has zero average value but the average of the force squared is not zero. The thermal noise force is the square root of the power spectral density of force fluctuations $\sqrt{S_{FF}(\omega)} [\text{fN} / \sqrt{\text{Hz}}]$ . The force noise power spectrum is independent of frequency, $S_{FF} = 2 k_B T \eta$ and this type of ‘white noise’ arises whenever the damping force is simply proportional to velocity. With this thermal noise force we can calculate a minimum detectable force $F_{\mathrm{min}}$ , or the force measured with $SNR=1$ in a given measurement time $T$ (measurement bandwidth $B=1/T$ ).

For example: The scan speed and image resolution dictate a pixel time of $T=\text{1 ms}$ , and the thermal noise force is $\sqrt{S_{FF}} = \text{22 fN}/ \sqrt{\text{Hz}}$ . The minimum detectable force at each pixel is therefore $F_{\mathrm{min}} = \sqrt{S_{FF} B} = \text{696 fN}$ . This minimum force is the ‘signal’ which just equals the noise. A good measurement is typically characterized by a signal which is many times the noise level, or $SNR \gg 1$ . Note that you must scan 4 times more slowly to decrease $F_{\mathrm{min}}$ by a factor of 2.

The detector Noise floor is the second quantitative measure of sensitivity. If you measure at frequencies away from the high-Q resonance, $SNR$ is typically not determined by the actual force noise of the cantilever in its damping medium, but rather by the electronic noise of the detector, $S_{DD}$ . We express this detector noise as an equivalent deflection noise by dividing it with the detector responsivity, $\sqrt{S_{dd}^{\mathrm{equiv}}} = \alpha^{-1} \sqrt{S_{DD}}$ . Here again we assume that the detector noise is independent of frequency in the band of interest. If the pixel time is $T=\text{1 ms}$ and $\sqrt{S_{dd}^{\mathrm{equiv}}} = \text{150 fm}/ \sqrt{\text{Hz}}$ , the minimum detectable cantilever deflection at each pixel would be, $d_{\mathrm{min}} = \sqrt{S_{dd}^{\mathrm{equiv}} B} = \text{4.7 pm}$ .

It is also interesting to express the detector noise as an Equivalent force noise $\sqrt{S_{FF}^{\mathrm{equiv}}} = k \sqrt{S_{dd}^{\mathrm{equiv}}}$ . For the example given above and $k = \text{40 N/m}$ , we find $\sqrt{S_{FF}^{\mathrm{equiv}}} = \text{6 pN/}\sqrt{\text{Hz}}$ , which, in a measurement time $T=\text{1 ms}$ gives an equivalent minimum detectable force of $F_{\mathrm{min}}^{\mathrm{equiv}} = k d_{\mathrm{min}} = \text{0.19 nN}$ . This would be the minimum detectable force with the cantilever if the measurement were dominated by detector noise, as is often the case for quasi-static force measurements.

Thus, sensitivity of the measurement is characterized by two quantities: The noise force of the cantilever, which represents a fundamental limit, and the noise of the detector. One would like the detector noise to be as low as possible and a very useful way to compare different detectors is to use the ImAFM™ system to perform thermal noise calibration on the same cantilever mounted in two different host AFMs. Since the cantilever force noise is identical in each case, we can compare the ratio of the thermal noise peak, to the detector background noise, or the Peak to flat ratio. The larger this ratio, the better your detection. It is also interesting to track this number as you measure. If this ratio becomes smaller, it may mean that the detector is poorly adjusted, or there may be some additional source of detector noise.

Table of Contents

Previous topic

Next topic

This Page

Noise Calibration¶

Note on sensitivity and accuracy¶