AFM cantilever calibration

The cantilever-associated tip is the “finger” of an AFM. Usually the tip sharpness and cantilever stiffness are reflected in the image quality of the sample. Optimizing both parameters does not necessarily mean choosing a sharper tip and a stiffer cantilever, they must be suitable to the corresponding samples.

Meanwhile, the physical properties of the cantilever and the tip need to be well characterized so that the artifacts can be minimized.

In most cases, the physical properties of the AFM cantilever need to be accurately obtained. To obtain accurate results each cantilever should be individually calibrated before an experiment, rather than merely use the parameters given by the manufacturer. Of all the properties, the cantilever resonance frequency and spring constant are the most important for tapping mode imaging and mechanical measurements.

If imaging is performed in tapping mode, the cantilever needs to be vibrated at a constant frequency and amplitude. If vibrated at resonance frequency, the cantilever achieves maximum amplitude at minimal excitation. To obtain the resonance frequency, the cantilever is vibrated by a piezoelectric shaker with a continuously increasing frequency, meanwhile the vibration amplitude is obtained via the photodetector signal from the lock-in amplifier. This process is often referred to as the tuning of the cantilever. Plotting the amplitude against frequency, several peaks are visible (Fig 1.3.2), which consist of the real resonance frequencies of the cantilever but also those of other parts in the AFM head (132).

The peak which is closest to the first peak of the thermal noise power spectral density (Fig 1.3.5), or the highest peak will usually be chosen. The value will be used by the AFM software, so that when imaging in tapping mode the cantilever will be vibrated at this frequency. It is worth mentioning that, if small temperature changes occur, the peaks are likely to shift, therefore it is recommended to retune the cantilever on an hourly basis and each time after changing the sample.

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Fig 1.3.2 Tuning graph of a cantilever. A piezoelectric shaker which is associated to the cantilever vibrates it with frequencies increased from 2 to 12 kHz. The amplitude (recorded via the photodetector signal) is plotted against the frequency. (Black) Multiple major peaks are visible on the plot, known as the forest of peaks. (Green) The phase shift between the drive and detector signal is shown. (The image is obtained by a MFP-3D AFM with an Olympus bio-lever BL-RC150VB.)

The characterization of cantilever spring constant is essential for mechanical measurements. The cantilever and sample can be seen as two springs in series Fig 1.3.3:

Fig 1.3.3 Schematic drawing shows the relation of cantilever, sample and measured spring constants.

35 The directly measured stiffness (kmeas.), the real stiffness of the sample (kreal) and the spring constant of the cantilever (kcanti.) follow the equation:

1

𝑘_{𝑚𝑒𝑎𝑠 .} = 1

𝑘_{𝑐𝑎𝑛𝑡𝑖 .}+ 1 𝑘_{𝑟𝑒𝑎𝑙}

(1.3.1) To obtain kreal, one needs to subtract kcanti. from kmeas.. There are several methods to obtain kcanti. (133), only the thermal noise fitting method will be introduced here, since it is well accepted and used by us.

Intuitively, the elasticity of a rectangular cantilever shall be dependent on its geometry and material.

Indeed, the spring constant can be written as:

𝑘 =𝐸𝑤𝑡³ 4𝐿³

(1.3.2) where E is the Young’s modulus of the cantilever material; w, t and L are the width, thickness and length of the cantilever.

The equation (1.3.2) is called cantilever beam theory. Although this method is simple and straightforward, it is usually difficult to measure the geometric parameters precisely. The thickness of the cantilever is especially difficult to determine, since the cantilevers are coated with a reflective gold layer.

A more precise and time-saving way to determine kcanti. is by fitting the thermal noise curve of the cantilever. When the cantilever is kept in equilibrium in the imaging media, the media particles (liquid or air molecules) bounce onto the cantilever because of the Brownian motion. The resulting bending fluctuations of the cantilever are recorded by the photodetector in Volts over time. The signal is then Fourier-transformed to obtain the power spectral density (PSD, unit V²/Hz). The PSD recorded on the detector can be translated to the PSD of the cantilever (m²/Hz) by:

𝑃_𝐶𝐿 = 𝑃_𝑉∗ 1 𝑆²∗ 1

𝑐𝑜𝑠²𝛼∗ 𝜒²

(1.3.3) where PCL is the PSD of the cantilever; PV is the PSD recorded via the detector; S is the sensitivity of the cantilever bending detection (nm/V), which can be measured by performing a force distance curve on empty hard surface (Fig 1.3.4); α is the angle between the cantilever and the surface (usually 10-15 degree); χ is a factor which is related to the size and position of the laser spot that is reflected on the back of the cantilever (usually taken as constant 1.09, according to (134)).

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Fig 1.3.4 Typical force-distance curve obtained by performing a single indentation of the cantilever on the empty hard surface. (A) The red curve represents for the trace, which is obtained by pushing the cantilever towards the surface; the blue curve represents for the retrace, which is obtained by pulling the cantilever away from the surface. (B) Deflection of the cantilever (recorded in Volts) vs. relative distance of the cantilever tip to the surface. Similarly, before contacting the surface, the deflection remains zero. When the cantilever gets in contact with the surface, the deflection rapidly increases. The slope of the curve (V/m) gives the sensitivity of the detected system.

According to equipartition theorem, the energy of a thermally equilibrated object is equally distributed over all its possible forms. In the specific case of an AFM cantilever (which can be simplified as a spring) in thermal equilibrium, its potential energy equals to its kinetic energy. Together, both forms of the energy compose of the thermal energy:

𝐻_{𝑝𝑜𝑡 .} =𝑘 ∙< 𝑥² >

2 , 𝐻_{𝑘𝑖𝑛 .} = 𝑚𝑣²

2 , 𝐻_{𝑝𝑜𝑡 .}= 𝐻_{𝑘𝑖𝑛 .} = 𝑘_𝐵 ∙ 𝑇 2

(1.3.4) where kB is the Boltzmann-constant, T is the temperature of the thermal equilibrium in Kelvin, k is the spring constant of the spring and x is the average fluctuation of the oscillator, m is the mass of the oscillator, and v is the speed of the oscillator.

To be able to obtain <x²>, the cantilever is theoretically simulated as a simple harmonic oscillator (SHO).

The spring constant of the cantilever can be written as a function of x (dependent on frequency) (133):

𝑘 =2

𝜋∗ 𝑘_𝐵𝑇𝑄 𝑥² 𝜈_𝑟 ∗∆𝜈

𝜈_𝑟

(1.3.5) where ν is the frequency, νr is the resonance frequency and Q is the quality factor.

The resulting function to fit to equation (1.3.3) is:

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< 𝑥² 𝜈 >=𝐴

𝜈+ 𝐵 +< 𝑥² 𝜈_𝑟 >

𝑄² ∗ 1

(1 − (_𝜈^𝜈

𝑟)²)² + (_𝜈^𝜈

𝑟∗𝑄)²

(1.3.6) where ν is the frequency; νr is the resonance frequency; <x²(ν)> is the frequency dependent coordinate of the oscillator, which is fitted to the power spectral of the cantilever PCL; Q is the quality factor; A and B describe the pink noise (frequency dependent) and white noise (frequency independent).

Fitting equation (1.3.6) to equation (1.3.3) by free parameters A, B, Q, νr and <x²(νr)>, one is able to calculate the spring constant of the cantilever.

An example of SHO fitting to the PSD of the thermal noise is shown in Fig 1.3.5:

Fig 1.3.5 (Black) Thermal noise PSD of an Olympus bio-lever BL-RC150VB. (Blue) The PSD is fitted by equation (1.3.6). Both x and y axis have a logarithmic scale. The other peaks are the second, third…harmonic frequencies of the cantilever.