Atomic force microscopy: loading position dependence of cantilever spring constants and detector sensitivity

A simple and accurate experimental method is described for determining the effective cantilever spring constant and the detector sensitivity of atomic force microscopy cantilevers on which a colloidal particle is attached. By attaching large (approximately 85 microm diameter) latex particles at various positions along the V-shaped cantilevers, we demonstrate how the normal and lateral spring constants as well as the sensitivity vary with loading position. Comparison with an explicit point-load theoretical model has also been used to verify the accuracy of the method.     

Interlaboratory round robin on cantilever calibration for AFM force spectroscopy

Single-molecule force spectroscopy studies performed by Atomic Force Microscopes (AFMs) strongly rely on accurately determined cantilever spring constants. Hence, to calibrate cantilevers, a reliable calibration protocol is essential. Although the thermal noise method and the direct Sader method are frequently used for cantilever calibration, there is no consensus on the optimal calibration of soft and V-shaped cantilevers, especially those used in force spectroscopy. Therefore, in this study we aimed at establishing a commonly accepted approach to accurately calibrate compliant and V-shaped cantilevers. In a round robin experiment involving eight different laboratories we compared the thermal noise and the Sader method on ten commercial and custom-built AFMs. We found that spring constants of both rectangular and V-shaped cantilevers can accurately be determined with both methods, although the Sader method proved to be superior. Furthermore, we observed that simultaneous application of both methods on an AFM proved an accurate consistency check of the instrument and thus provides optimal and highly reproducible calibration. To illustrate the importance of optimal calibration, we show that for biological force spectroscopy studies, an erroneously calibrated cantilever can significantly affect the derived (bio)physical parameters. Taken together, our findings demonstrated that with the pre-established protocol described reliable spring constants can be obtained for different types of cantilevers.     

Quantitative comparison of two independent lateral force calibration techniques for the atomic force microscope

Two independent lateral-force calibration methods for the atomic force microscope (AFM)--the hammerhead (HH) technique and the diamagnetic lateral force calibrator (D-LFC)--are systematically compared and found to agree to within 5?% or less, but with precision limited to about 15?%, using four different tee-shaped HH reference probes. The limitations of each method, both of which offer independent yet feasible paths toward traceable accuracy, are discussed and investigated. We find that stiff cantilevers may produce inconsistent D-LFC values through the application of excessively high normal loads. In addition, D-LFC results vary when the method is implemented using different modes of AFM feedback control, constant height and constant force modes, where the latter is more consistent with the HH method and closer to typical experimental conditions. Specifically, for the D-LFC apparatus used here, calibration in constant height mode introduced errors up to 14?%. In constant force mode using a relatively stiff cantilever, we observed an ≈?4?% systematic error per μN of applied load for loads ≤?1 μN. The issue of excessive load typically emerges for cantilevers whose flexural spring constant is large compared with the normal spring constant of the D-LFC setup (such that relatively small cantilever flexural displacements produce relatively large loads). Overall, the HH method carries a larger uncertainty, which is dominated by uncertainty in measurement of the flexural spring constant of the HH cantilever as well as in the effective length dimension of the cantilever probe. The D-LFC method relies on fewer parameters and thus has fewer uncertainties associated with it. We thus show that it is the preferred method of the two, as long as care is taken to perform the calibration in constant force mode with low applied loads.     

Precise atomic force microscope cantilever spring constant calibration using a reference cantilever array

A method for calibrating the stiffness of atomic force microscope (AFM) cantilevers is demonstrated using an array of uniform microfabricated reference cantilevers. A series of force-displacement curves was obtained using a commercial AFM test cantilever on the reference cantilever array, and the data were analyzed using an implied Euler-Bernoulli model to extract the test cantilever spring constant from linear regression fitting. The method offers a factor of 5 improvement over the precision of the usual reference cantilever calibration method and, when combined with the Systeme International traceability potential of the cantilever array, can provide very accurate spring constant calibrations.     

Calibration of AFM cantilever spring constants

In this paper we present two simple, reliable and readily applicable methods for calibrating cantilevers and measuring the thickness of thin gold films. The spring constant calibration requires knowledge of the Young's modulus, density of the cantilever and resonant frequency. The thickness of thin gold layers was determined by measuring changes in the resonant frequency and Q-factor of beam shaped AFM cantilevers before and after coating.The techniques for measuring the spring constant and thin film thickness provide accuracy on the order of 10-15%.     

Study of thermal and acoustic noise interferences in low stiffness atomic force microscope cantilevers and characterization of their dynamic properties

The atomic force microscope (AFM) is a powerful tool for the measurement of forces at the micro/nano scale when calibrated cantilevers are used. Besides many existing calibration techniques, the thermal calibration is one of the simplest and fastest methods for the dynamic characterization of an AFM cantilever. This method is efficient provided that the Brownian motion (thermal noise) is the most important source of excitation during the calibration process. Otherwise, the value of spring constant is underestimated. This paper investigates noise interference ranges in low stiffness AFM cantilevers taking into account thermal fluctuations and acoustic pressures as two main sources of noise. As a result, a preliminary knowledge about the conditions in which thermal fluctuations and acoustic pressures have closely the same effect on the AFM cantilever (noise interference) is provided with both theoretical and experimental arguments. Consequently, beyond the noise interference range, commercial low stiffness AFM cantilevers are calibrated in two ways: using the thermal noise (in a wide temperature range) and acoustic pressures generated by a loudspeaker. We then demonstrate that acoustic noises can also be used for an efficient characterization and calibration of low stiffness AFM cantilevers. The accuracy of the acoustic characterization is evaluated by comparison with results from the thermal calibration.     

Calibration of the normal spring constant of microcantilevers in a parallel fluid flow

We demonstrate a novel approach to determine the normal spring constant of microcantilevers. The cantilevers are placed parallel to a fluid flow thus establishing one of the walls of the flow channel. Resonance frequencies are recorded depending on the velocity of the fluid. The pressure gradient resulting from the flow causes the resonance frequency to change. This change can be exploited to deduce the cantilever spring constant with high precision. The method we present can be performed in situ and does not involve any contact of the cantilever with a surface thus having great potential for the calibration of modified probes and for being incorporated in microfluidic systems. In case the spring constant is known, the setup can also be employed to determine the velocity of fluid flows and the flow rate with high precision and up to high speeds.     

Calibration of tapping AFM cantilevers and uncertainty estimation: Comparison between different methods

We present a comparative study between three different methods for the spring constant calibration of silicon beam-shaped Atomic Force Microscope (AFM) cantilevers, used in tapping AFM mode in air. The geometries of these levers can be quite different from the standard rectangular cross section. We examine a method that combines the knowledge of cantilever dimensions and eigenfrequencies (Cleveland formula), the Sader method and we build cantilever models based on Finite Element Analysis (FEA). We demonstrate that with accurate measurement of dimensions, resonance frequency and quality factor, the Cleveland formula yields a combined cantilever stiffness uncertainty of approximately ±7% and the Sader method an uncertainty of ±5%. We also use FEA to show that when trying to approximate a realistic trapezoidal 3D tipped geometry, there exists a systematic overestimation in cantilever stiffness of ±2%, compared to when considering a simple rectangular cross section. Our constructed FE models are able to account for inhomogeneities in material properties as well as the influence of the added reflective coating in the cantilever stiffness estimation.     

Dynamic thermomechanical response of bimaterial microcantilevers to periodic heating by infrared radiation

This paper investigates the dynamic thermomechanical response of bimaterial microcantilevers to periodic heating by an infrared laser operating at a wavelenegth of 10.35 μm. A model relates incident radiation, heat transfer, temperature distribution in the cantilever, and thermal expansion mismatch to find the cantilever displacement. Experiments were conducted on two custom-fabricated bimaterial cantilevers and two commercially available bimaterial microcantilevers. The cantilever response was measured as a function of the modulation frequency of the laser over the range of 0.01-30 kHz. The model and the method of cantilever displacement calibration can be applied for bimaterial cantilever with thick coating layer. The sensitivity and signal-to-noise of bimaterial cantilevers were evaluated in terms of either total incident power or incident flux. The custom-fabricated bimaterial cantilevers showed 9X or 190X sensitivity improvement compared to commercial cantilevers. The detection limit on incident flux is as small as 0.10 pW μm(-2) Hz(-1/2).     

Influence of Poisson's ratio variation on lateral spring constant of atomic force microscopy cantilevers

Atomic force microscopy (AFM) can be used to measure the surface morphologies and the mechanical properties of nanostructures. The force acting on the AFM cantilever can be obtained by multiplying the spring constant of AFM cantilever and the corresponding deformation. To improve the accuracy of force experiments, the spring constant of AFM cantilever must be calibrated carefully. Many methods, such as theoretical equations, the finite element method, and the use of reference cantilever, were reported to obtain the spring constant of AFM cantilevers. For the cantilever made of single crystal, the Poisson's ratio varies with different cantilever-crystal angles. In this paper, the influences of Poisson's ratio variation on the lateral spring constant and axial spring constant of rectangular and V-shaped AFM cantilevers, with different tilt angles and normal forces, were investigated by the finite element analysis. When the cantilever's tilt angle is 20 degrees and the Poisson's ratio varies from 0.02 to 0.4, the finite element results show that the lateral spring constants decrease 11.75% for the rectangular cantilever with 1muN landing force and decrease 18.60% for the V-shaped cantilever with 50nN landing force, respectively. The influence of Poisson's ratio variation on axial spring constant is less than 3% for both rectangular and V-shaped cantilevers. As the tilt angle increases, the axial spring constants for rectangular and V-shaped cantilevers decrease substantially. The results obtained can be used to improve the accuracy of the lateral force measurement when using atomic force microscopy.     

Prototype cantilevers for quantitative lateral force microscopy

Prototype cantilevers are presented that enable quantitative surface force measurements using contact-mode atomic force microscopy (AFM). The "hammerhead" cantilevers facilitate precise optical lever system calibrations for cantilever flexure and torsion, enabling quantifiable adhesion measurements and friction measurements by lateral force microscopy (LFM). Critically, a single hammerhead cantilever of known flexural stiffness and probe length dimension can be used to perform both a system calibration as well as surface force measurements in situ, which greatly increases force measurement precision and accuracy. During LFM calibration mode, a hammerhead cantilever allows an optical lever "torque sensitivity" to be generated for the quantification of LFM friction forces. Precise calibrations were performed on two different AFM instruments, in which torque sensitivity values were specified with sub-percent relative uncertainty. To examine the potential for accurate lateral force measurements using the prototype cantilevers, finite element analysis predicted measurement errors of a few percent or less, which could be reduced via refinement of calibration methodology or cantilever design. The cantilevers are compatible with commercial AFM instrumentation and can be used for other AFM techniques such as contact imaging and dynamic mode measurements.     

Spring constant calibration of atomic force microscopy cantilevers with a piezosensor transfer standard

We describe a method to calibrate the spring constants of cantilevers for atomic force microscopy (AFM). The method makes use of a "piezosensor" composed of a piezoresistive cantilever and accompanying electronics. The piezosensor was calibrated before use with an absolute force standard, the NIST electrostatic force balance (EFB). In this way, the piezosensor acts as a force transfer standard traceable to the International System of Units. Seven single-crystal silicon cantilevers with rectangular geometries and nominal spring constants from 0.2 to 40 Nm were measured with the piezosensor method. The values obtained for the spring constant were compared to measurements by four other techniques: the thermal noise method, the Sader method, force loading by a calibrated nanoindentation load cell, and direct calibration by force loading with the EFB. Results from different methods for the same cantilever were generally in agreement, but differed by up to 300% from nominal values. When used properly, the piezosensor approach provides spring-constant values that are accurate to +/-10% or better. Methods such as this will improve the ability to extract quantitative information from AFM methods.     

Dynamic spring constants for higher flexural modes of cantilever plates with applications to atomic force microscopy

In atomic force microscopy (AFM) a sharp tip fixed close to the free end of a cantilever beam interacts with a surface. The interaction can be described by a point-mass model of an equivalent oscillator with a single spring located at the position of the tip. However, other spring constants have to be used to describe the oscillation behavior correctly if forces are acting on the cantilever over an extended lateral range. A point-mass model is then no longer valid. In the present study we derive expressions for the spring constants of cantilevers that can interact with any part of their plan view area along the beam and for all flexural modes. The equations describe the oscillation behavior in the corresponding mass model and are based on the eigenfrequencies and modal shapes of the free cantilever. The results are of high practical relevance, for example if an AFM is operated in a higher flexural mode, if the tip is not located at the free end of the cantilever beam, or if the external conservative forces affecting cantilever movement are not restricted to a single point. The limitations of the approach are discussed.     

Expanded beam deflection method for simultaneous measurement of displacement and vibrations of multiple microcantilevers

Here we present an extension of optical beam deflection (OBD) method for measuring displacement and vibrations of an array of microcantilevers. Instead of focusing on the cantilever, the optical beam is either focused above or below the cantilever array, or focused only in the axis parallel to the cantilevers length, allowing a wide optical line to span multiple cantilevers in the array. Each cantilever reflects a part of the incident beam, which is then directed onto a photodiode array detector in a manner allowing distinguishing between individual beams. Each part of reflected beam behaves like a single beam of roughly the same divergence angle in the bending sensing axis as the incident beam. Since sensitivity of the OBD method depends on the divergence angle of deflected beam, high sensitivity is preserved in proposed expanded beam deflection (EBD) method. At the detector, each spot's position is measured at the same time, without time multiplexing of light sources. This provides real simultaneous readout of entire array, unavailable in most of competitive methods, and thus increases time resolution of the measurement. Expanded beam can also span another line of cantilevers allowing monitoring of specially designed two-dimensional arrays. In this paper, we present first results of application of EBD method to cantilever sensors. We show how thermal noise resolution can be easily achieved and combined with thermal noise based resonance frequency measurement.     

Calibration and examination of piezoresistive Wheatstone bridge cantilevers for scanning probe microscopy

This paper describes the method of determining the force constant and displacement sensitivity of piezoresistive Wheatstone bridge cantilevers applied in scanning probe microscopy (SPM). In the procedure presented here, the force constant for beams with various geometry is determined based on resonance frequency measurement. The displacement sensitivity is measured by the deflection of the cantilever with the calibrated piezoactuator stage. Preliminary results show that our method is capable of measuring the force constant of Wheatstone bridge cantilevers with an accuracy of better than 5% and this is used as feedback for improvement of sensor micromachining process.     

Direct measurement of cantilever spring constants and correction for cantilever irregularities using an instrumented indenter

A method is presented that allows direct measurement of a wide range of spring constants of cantilevers using an indentation instrument with an integrated optical microscope. An uncertainty of less than 10% can be achieved for spring constants from 0.1 to 10(2) Nm. The technique makes it possible to measure the spring constant at any desired location on a cantilever of any shape, particularly at the tip location of an atomic force microscopy cantilever. The article also demonstrates a technique to detect and correct apparent length anomalies of cantilevers by analyzing spring constants at multiple positions.     

Calibration of AFM cantilever stiffness: a microfabricated array of reflective springs

Calibration of the spring constant of atomic force microscope (AFM) cantilevers is necessary for the measurement of nanonewton and piconewton forces, which are critical to analytical applications of AFM in the analysis of polymer surfaces, biological structures and organic molecules.

We have developed a compact and easy-to-use reference standard for this calibration. The new artifact consists of an array of 12 dual spiral-cantilever springs, each supporting a mirrored polycrystalline silicon disc of 160 μm in diameter. These devices were fabricated by a three-layer polysilicon surface micromachining method, including a reflective layer of gold on chromium. We call such an array a Microfabricated Array of Reference Springs (MARS). These devices have a number of advantages. Cantilever calibration using this device is straightforward and rapid. The devices have very small inertia, and are therefore resistant to shock and vibration. This means they need no careful treatment except reasonably clean laboratory conditions.

The array spans the range of spring constant from around 0.16 to 11 N/m important in AFM, allowing almost all contact-mode AFM cantilevers to be calibrated easily and rapidly. Each device incorporates its own discrete gold mirror to improve reflectivity. The incorporation of a gold mirror both simplifies calibration of the devices themselves (via Doppler velocimetry) and allows interferometric calibration of the AFM z-axis using the apparent periodicity in the force–distance curve before contact. Therefore, from a single force–distance curve, taking about one second to acquire, one can calibrate the cantilever spring constant and, optionally, the z-axis scale. These are all the data one needs to make accurate and reliable force measurements.     

Kinetostatic model of spring constant ratios for an AFM cantilever with end extended mass

In previous work we showed that the kinetostatic method is very effective in computing the increase in value of the spring constants of an AFM free (with or without added mass) and supported rectangular cantilever for higher mode oscillations relative to their values for natural vibration. We have considered in all previous cases that added mass is a concentrated one. However, the additional mass may be an extended one particularly in the case of a V-shaped cantilever. In this article we consider the influence of the constituent beam’s (leg’s) mutual skew and the altered position of the nodal points in the case when the attached extended triangular (trapezoid) mass of the V-shaped cantilever has a significant moment of rotational inertia and a center of this mass gravity located beyond the constituent beam end. We show that considering these effects in using the kinetostatic model yields results for the ratios of the spring constants at higher modes of oscillation and their values at the first frequency natural vibration for a V-shaped cantilever which are in good agreement with the thermomechanical noise amplitudes obtained by other researchers. This should prove helpful for the proper calibration of V-shaped cantilevers whose application with higher modes oscillation provides increased measurement sensitivity.     

Chemical vapor detection using nanomechanical platform

For high sensitive and multiplexed chemical analysis, an opto-mechanical detection platform has been built. To check the performance of the platform, we performed water vapor response measurements for the cantilevers coated with alkane thiols having different functional end groups. Furthermore, for the exposure of 50 ppb toluene vapor to carboxylic benzene thiol coating layer, nanoscale static deflection of the cantilever sensors has been measured simultaneously. The nanomechanical platform using cantilever sensors can be miniaturized to be used for high sensitive and selective environmental monitoring for indoor ar outdoor air pollutants, mold, heavy metals, and other health hazard materials.     

High-resolution constant-height imaging with apertured silicon cantilever probes

We present high-resolution aperture probes based on non-contact silicon atomic force microscopy (AFM) cantilevers for simultaneous AFM and near-infrared scanning near-field optical microscopy (SNOM). For use in near-field optical microscopy, conventional AFM cantilevers are modified by covering their tip side with an opaque aluminium layer. To fabricate an aperture, this metal layer is opened at the end of the polyhedral probe using focused ion beams (FIB). Here we show that apertures of less than 50 nm can be obtained using this technique, which actually yield a resolution of about 50 nm, corresponding to λ/20 at the wavelength used. To exclude artefacts induced by distance control, we work in constant-height mode. Our attention is particularly focused on the distance dependence of resolution and to the influence of slight cantilever bending on the optical images when scanning at such low scan heights, where first small attractive forces exerted on the cantilever become detectable.