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Cantilever array-based sensor devices widely utilise the laser-based optical deflection method for measuring static cantilever deflections mostly with home-built devices with individual geometries. In contrast to scanning probe microscopes, cantilever array devices have no additional positioning device like a piezo stage. As the cantilevers are used in more and more sensitive measurements, it is important to have a simple, rapid, and reliable calibration relating the deflection of the cantilever to the change in position measured by the position-sensitive detector. We present here a simple method for calibrating such systems utilising commercially available AFM cantilevers and the equipartition theorem.

Cantilever-based sensor devices have extensively developed from the atomic force microscope (AFM) operating in the static mode [

The laser beam-based deflection system [

The displacement of the laser spot on the PSD

Schematic representation of the geometry of the laser deflection setup. The bending of the cantilever represented by

The equipartition theorem relates the thermal energy of a system to its temperature in classical thermodynamics. Thermal noise of a cantilever can be quantified using this theorem [

Combining (

It is important to note that only the area under the first resonance peak is considered in further measurements, neglecting the higher modes since their contribution was seen to be minor (modelled as a simple harmonic oscillator with one degree of freedom). The spring constant of the calibration cantilevers hence needs to be measured as well. There are several methods available to perform such calibration to obtain spring constants [

Different sets of commercially available AFM cantilevers were used namely Mikromasch CSC38/AIBS “B” (Mikromasch, Estonia) and NTMDT CSCS12 “E” (NT-MDT, Russia) cantilevers for the measurement of the thermal noise spectrum and final calibration. The cantilevers were calibrated using the Asylum MFP-3D AFM to get individual values for their spring constants

Manufacturer specifications of the cantilevers used for calibration factor measurement.

Specifications | Mikromasch CSC38/AIBS “B” | NTMDT CSCS12 “E” | ||||

Min | Typical | Max | Min | Typical | Max | |

Length ( | 350 | 350 | ||||

Width ( | 35 | 35 | ||||

Thickness, | 0.7 | 1.0 | 1.3 | 0.9 | 1.0 | 1.1 |

Resonant frequency (kHz) | 7 | 10 | 14 | 8 | 10 | 12 |

Force constant (N/m) | 0.01 | 0.03 | 0.08 | 0.02 | 0.03 | 0.04 |

The power spectrum of the thermal noise was obtained using a 150 kHz band pass position-sensitive detector (SiTek, Sweden). This detector is a modified version of the low-pass 5 Hz sensor which is used for performing static mode biological experiments. A Labview program was used to obtain the averaged power spectrum from the differential and sum signals from the PSD. The parameters for obtaining the power spectrum had to be chosen so as to eliminate effects like aliasing which leads to truncated or artificially small resonance peaks and also electronic noise. Also it was necessary to choose the number of samples and the sampling frequency such that it avoided overloading the system and the data acquisition card (DAQ, National instruments). Keeping in mind all these details and following the Nyquist theorem (signal must be sampled at a rate at least greater than twice the highest frequency component of the signal) the parameters which were chosen for the power spectral analysis were as follows: sampling frequency: 100 kHz, number of samples: 10,000, and number of averages: 5000. The area under the first resonance peak was obtained using a Lorentzian fit in origin graphical software (OriginLab Corporation, USA). The area hence calculated along with the spring constant values was then used to determine the value of

For our present instrumental scheme, the geometrical calculation for both the setups is the same as derived below. For:

Equation (

The spring constants for the calibration cantilevers were determined as an average of three trials during which the cantilevers were removed and replaced in the AFM setup in order to average out errors. The averaged values of the cantilevers are summarized in Table

Spring constants

Spring constant | Micromasch B cantilevers | NTMDT cantilevers E | ||

B1 | B2 | E1 | E3 | |

69.66 | 166.74 | 32.64 | 53.13 |

Calibration factor,

Thermal noise power spectrum of NTMDT cantilever E1 on trial 2 for calibration of Setup 2. The area obtained under the peak after a Lorentzian fit (uniform broadening and best fitting parameters) is later used for determining the calibration factor.

Table

Calibration factors for the cantilever deflection Setups.

Area under curve | Average | |||

Deflection factor | ||||

Cant B1 | Trial 1 | 2128 | 2077.5 | |

Trial 2 | 2284 | |||

Trial 3 | 2175 | |||

Cant B2 | Trial 1 | 1986 | ||

Trial 2 | 1982 | |||

Trial 3 | 1910 | |||

Deflection factor | ||||

Cant E1 | Trial 1 | 2517 | 2679.5 | |

Trial 2 | 3062 | |||

Trial 3 | 2891 | |||

Cant E3 | Trial 1 | 2807 | ||

Trial 2 | 2368 | |||

Trial 3 | 2432 |

From the above set of values of the

The importance of having sensitive measurements especially in systems involving a differential analysis is of foremost significance for ensuring the reliability of cantilever sensor systems. Establishing the occurrence of an event of interest on the cantilever surface using

We demonstrate here a simple and reliable method for rapid calibration of laser-based deflection systems. Using commercially available AFM cantilevers, we can show that the relationship between the spot movement on the PSD and the actual cantilever deflection can be determined although within the accuracy of the assumptions and the thermal calibration method (~5–10%) [

The authors would like to thank the Science Foundation Ireland and F. Hoffman La Roche for their support through research Grants (SFI 00/PI.1/C02, 09IN.1B2623, and Roche 5AAF11).