Accurate analytical measurements in the atomic force microscope: a microfabricated spring constant standard potentially traceable to the SI

Characterisation of the Material and Mechanical Properties of Atomic Force Microscope Cantilevers with a Plan-View Trapezoidal Geometry

Cantilever devices have found applications in numerous scientific fields and instruments, including the atomic force microscope (AFM), and as sensors to detect a wide range of chemical and biological species. The mechanical properties, in particular, the spring constant of these devices is crucial when quantifying adhesive forces, material properties of surfaces, and in determining deposited mass for sensing applications. A key component in the spring constant of a cantilever is the plan-view shape. In recent years, the trapezoidal plan-view shape has become available since it offers certain advantages to fast-scanning AFM and can improve sensor performance in fluid environments. Euler beam equations relating cantilever stiffness to the cantilever dimensions and Young’s modulus have been proven useful and are used extensively to model cantilever mechanical behaviour and calibrate the spring constant. In this work, we derive a simple correction factor to the Euler beam equation for a beam-shaped cantilever that is applicable to any cantilever with a trapezoidal plan-view shape. This correction factor is based upon previous analytical work and simplifies the application of the previous researchers formula. A correction factor to the spring constant of an AFM cantilever is also required to calculate the torque produced by the tip when it contacts the sample surface, which is also dependent on the plan-view shape. In this work, we also derive a simple expression for the torque for triangular plan-view shaped cantilevers and show that for the current generation of trapezoidal plan-view shaped AFM cantilevers, this will be a good approximation. We shall apply both these correction factors to determine Young’s modulus for a range of trapezoidal-shaped AFM cantilevers, which are specially designed for fast-scanning. These types of AFM probes are much smaller in size when compared to standard AFM probes. In the process of analysing the mechanical properties of these cantilevers, important insights are also gained into their spring constant calibration and dimensional factors that contribute to the variability in their spring constant.

A novel method and system for calibrating the spring constant of atomic force microscope cantilever based on electromagnetic actuation

It is crucial to calibrate atomic force microscope (AFM) cantilevers for the development and further applications of AFM in precision engineering such as nanonewton force measurement. This paper presents a novel approach to calibrate the spring constant of an AFM cantilever based on electromagnetic actuation and null position measurement. According to the method, a calibration system was designed. In order to optimize the static and dynamic characteristics of the calibration system, the analytical models for the electromagnetic force and the suspension mechanism stiffness have been developed. Finite element analysis has been utilized to further investigate the precision of analytical modeling. The null position measurement method was utilized to monitor the deformation of the flexible beam, and then the deformation was compensated by the electromagnetic force. Experiments were carried out based on the developed prototype, and the results show that the electromagnetic force conversion rate is 40.08 µN/mA. Finally, a typical AFM cantilever was calibrated and the spring constant is (30.83 ± 0.24) N/m. The uncertainty of the proposed null position measurement method is better than 0.78%, which verifies the effectiveness and feasibility of the calibration method and system.

Experimental confirmation of the atomic force microscope cantilever stiffness tilt correction

The tilt angle (angle of repose) of an AFM cantilever relative to the surface it is interrogating affects the effective stiffness of the cantilever as it analyzes the surface. For typical AFMs and cantilevers that incline from 10° to 15° tilt, this is thought to be a 3%-7% stiffness increase correction. While the theoretical geometric analysis of this effect may have reached a consensus that it varies with cos^-2 θ, there is very little experimental evidence to confirm this using AFM cantilevers. Recently, the laser Doppler vibrometry thermal calibration method utilized at NIST has demonstrated sufficient stiffness calibration accuracy, and precision to allow a definitive experimental confirmation of the particular trigonometric form of this tilt effect using a commercial microfabricated AFM cantilever specially modified to allow strongly tilted (up to 15°) effective cantilever stiffness measurements.

An analytic model for accurate spring constant calibration of rectangular atomic force microscope cantilevers

Spring constant calibration of the atomic force microscope (AFM) cantilever is of fundamental importance for quantifying the force between the AFM cantilever tip and the sample. The calibration within the framework of thin plate theory undoubtedly has a higher accuracy and broader scope than that within the well-established beam theory. However, thin plate theory-based accurate analytic determination of the constant has been perceived as an extremely difficult issue. In this paper, we implement the thin plate theory-based analytic modeling for the static behavior of rectangular AFM cantilevers, which reveals that the three-dimensional effect and Poisson effect play important roles in accurate determination of the spring constants. A quantitative scaling law is found that the normalized spring constant depends only on the Poisson's ratio, normalized dimension and normalized load coordinate. Both the literature and our refined finite element model validate the present results. The developed model is expected to serve as the benchmark for accurate calibration of rectangular AFM cantilevers.

A compact torsional reference device for easy, accurate and traceable AFM piconewton calibration

The invention of the atomic force microscope led directly to the possibility of carrying out nanomechanical tests with forces below the nanonewton and the ability to test nanomaterials and single molecules. As a result there is a pressing need for accurate and traceable force calibration of AFM measurements that is not satisfactorily met by existing calibration methods. Here we present a force reference device that makes it possible to calibrate the normal stiffness of typical AFM microcantilevers down to 90 pN nm(-1) with very high accuracy and repeatability and describe how it can be calibrated traceably to the International System of Units via the ampere and the metre, avoiding in that way the difficulties associated with traceability to the SI kilogram. We estimate the total uncertainty associated with cantilever calibration including traceability to be better than 3.5%, thus still offering room for future improvement.

Calibration of atomic force microscope cantilevers using standard and inverted static methods assisted by FIB-milled spatial markers

Static methods to determine the spring constant of AFM cantilevers have been widely used in the scientific community since the importance of such calibration techniques was established nearly 20 years ago. The most commonly used static techniques involve loading a trial cantilever with a known force by pressing it against a pre-calibrated standard or reference cantilever. These reference cantilever methods have a number of sources of uncertainty, which include the uncertainty in the measured spring constant of the standard cantilever, the exact position of the loading point on the reference cantilever and how closely the spring constant of the trial and reference cantilever match. We present a technique that enables users to minimize these uncertainties by creating spatial markers on reference cantilevers using a focused ion beam (FIB). We demonstrate that by combining FIB spatial markers with an inverted reference cantilever method, AFM cantilevers can be accurately calibrated without the tip of the test cantilever contacting a surface. This work also demonstrates that for V-shaped cantilevers it is possible to determine the precise loading position by AFM imaging the section of the cantilever where the two arms join. Removing tip-to-surface contact in both the reference cantilever method and sensitivity calibration is a significant improvement, since this is an important consideration for AFM users that require the imaging tip to remain in pristine condition before commencing measurements. Uncertainties of between 5 and 10% are routinely achievable with these methods.

Atomic force microscope cantilever calibration using a focused ion beam

A calibration method is presented for determining the spring constant of atomic force microscope (AFM) cantilevers, which is a modification of the established Cleveland added mass technique. A focused ion beam (FIB) is used to remove a well-defined volume from a cantilever with known density, substantially reducing the uncertainty usually present in the added mass method. The technique can be applied to any type of AFM cantilever; but for the lowest uncertainty it is best applied to silicon cantilevers with spring constants above 0.7 N m(-1), where uncertainty is demonstrated to be typically between 7 and 10%. Despite the removal of mass from the cantilever, the calibration method presented does not impair the probes' ability to acquire data. The technique has been extensively tested in order to verify the underlying assumptions in the method. This method was compared to a number of other calibration methods and practical improvements to some of these techniques were developed, as well as important insights into the behavior of FIB modified cantilevers. These results will prove useful to research groups concerned with the application of microcantilevers to nanoscience, in particular for cases where maintaining pristine AFM tip condition is critical.

A Piezoresistive Cantilever Force Sensor for Direct AFM Force Calibration

We describe the design, fabrication, and calibration testing of a new piezoresistive cantilever force sensor suitable for the force calibration of atomic force microscopes in a range between tens of nanonewtons to hundreds of micronewtons. The sensor is calibrated using the NIST Electrostatic Force Balance (EFB) and functions either as a force reference or stiffness artifact that is traceable to the International System of Units. The cantilever has evenly spaced fiducial marks along its length. We report stiffnesses that vary quadratically with location, from a high of 12.1 N/m at the first fiducial to a low of 0.394 N/m at the last; with force sensitivities that vary linearly, ranging from 18.1 Ù/mN to 106 Ù/mN. We also test the device to transfer the unit of force to an atomic force microscope, finding that force and stiffness based approaches yield independent estimates of the contact force consistent within 2 % of each other.

Accurate noncontact calibration of colloidal probe sensitivities in atomic force microscopy

The absolute force sensitivities of colloidal probes comprised of atomic force microscope, or AFM, cantilevers with microspheres attached to their distal ends are measured. The force sensitivities are calibrated through reference to accurate electrostatic forces, the realizations of which are described in detail. Furthermore, the absolute accuracy of a common AFM force calibration scheme, known as the thermal noise method, is evaluated. It is demonstrated that the thermal noise method can be applied with great success to colloidal probe calibration in air and in liquid to yield force measurements with relative standard uncertainties below 5%. Techniques to combine the electrostatics-based determination of the AFM force sensitivity with measurements of the colloidal probe's thermal noise spectrum to compute noncontact estimates of the displacement sensitivity and spring constant are also developed.

Effects of anisotropic material property on the spring constant and the resonant frequency of atomic force microscope cantilever

Atomic force microscope (AFM) is a powerful tool for force measurement in nanoscale. Many methods have been developed to obtain the precise cantilever's spring constant for improving the accuracy of force measurement. AFM cantilevers are usually made by single crystal silicon of which the anisotropic material property seriously affects the spring constant of cantilevers and has not considered before. In this paper, the density function theory was used to calculate the anisotropic stiffness matrix of crystal silicon, which was used in the finite element analysis to calculate lateral, axial, bending spring constants, and resonant frequencies of rectangular AFM cantilevers. These results were compared with those derived from other theoretical methods and with those provided by the manufacturers. The results showed that the anisotropic material property significantly affected the spring constants and the resonant frequencies of the AFM cantilever. The assumption of equivalent isotropic property of the rectangular AFM cantilever would cause an error up to 29.72%. Furthermore, two equations were proposed to obtain the spring constants and the resonant frequencies of crystal silicon AFM cantilever with the axis located at different cantilever-crystal angles.

SI traceable calibration of an instrumented indentation sensor spring constant using electrostatic force

We present a measurement scheme for creating reference electrostatic forces that are traceable to the International System of Units. This scheme yields reference forces suitable for calibrating the force sensitivity of instrumented indentation machines and atomic force microscopes. Forces between 10 and 200 muN were created and expressed in terms of the voltage, length, and capacitance between a pair of interacting electrodes. The electrodes comprised an electrically conductive sphere mounted as a tip on an instrumented indentation sensor, and a planar counterelectrode fixed to a sample stage in close proximity to the sphere. For comparison, we applied mechanical forces of similar magnitudes, first using deadweights and then using a reference force sensor. The deflection of the sensor due to the various applied forces was measured using an interferometer. A spring constant for the sensor was computed from the observed records of force versus displacement. Each procedure yielded a relative standard uncertainty of approximately 1%; however, the electrostatic technique is scalable and could provide traceable reference forces as small as a few hundred piconewtons, a range far below anything yet achieved using deadweights.

The investigation of Protein A and Salmonella antibody adsorption onto biosensor surfaces by atomic force microscopy

The investigation of Protein A and antibody adsorption on surfaces in a biological environment is an important and fundamental step for increasing biosensor sensitivity and specificity. The atomic force microscope (AFM) is a powerful tool that is frequently used to characterize surfaces coated with a variety of molecules. We used AFM in conjunction with scanning electron microscopy to characterize the attachment of protein A and its subsequent binding to the antibody and Salmonella bacteria using a gold quartz crystal. The rms roughness of the base gold surface was determined to be approximately 1.30 nm. The average step height change between the solid gold and protein A layer was approximately 3.0 +/- 1.0 nm, while the average step height of the protein A with attached antibody was approximately 6.0 +/- 1.0 nm. We found that the antibodies did not completely cover the protein A layer, instead the attachment follows an island model. Salt crystals and water trapped under the protein A layer were also observed. The uneven adsorption of antibodies onto the biosensor surface might have led to a decrease in the sensitivity of the biosensor. The presence of salt crystals and water under the protein A layer may deteriorate the sensor specificity. In this report, we have discussed the application and characterization of protein A bound to antibodies which can be used to detect bacterial and viral pathogens.

The role of few-asperity contacts in adhesion

The surface roughness of a few asperities and their influence on the work of adhesion is of scientific interest. Macroscale and nanoscale adhesion data have seemingly given inconsistent results. Despite the importance of bridging the gap between the two regimes, little experimental work has been done, presumably due to the difficulty of the experiment needed to determine how small amounts of surface roughness might influence adhesion data lying in between the two scales. To investigate the role of few-asperity contacts in adhesion, the pull-off force was measured between different sized atomic-force microscope (AFM) tips (with different roughnesses) and sample surfaces that had well-controlled material properties. There were seventeen tips of four different types, with radii from 200 nm to 60 microm. The samples were unpatterned single crystal silicon with a chemical silicon dioxide surface resulting from a standard silicon wafer clean. Some of the samples were treated with a few angstroms of vapor deposited diphenylsiloxane. We observed that the uncorrected (for surface roughness) pull-off force was independent of the radius of the AFM tip, which was contrary to all continuum-mechanics model predictions. To explain this behavior, we assumed that the interactions between the AFM tip and sample were additive, material properties were constant, and that the AFM tip, asperities, and sample surfaces were of uniform density. Based on these assumptions, we calculated a simple correction due to the measured root mean square (RMS) surface roughness of the AFM tips. The simple correction for the RMS surface roughness resulted in the expected dependence of the pull-off force on radius, but the magnitudes were higher than expected. Commercial and heat-treated AFM tips have minimal surface roughness and result in magnitudes that are more reliable. The relative uncertainty for the pull-off force was estimated to be 10%. In this paper, we derive how the cantilever and tip parameters contribute to the measured pull-off force and show how the corrected results compare with theory. Although much work is still needed, the work presented here should advance the understanding of adhesion between the macroscale and nanoscale regimes.

Calibration of silicon atomic force microscope cantilevers

We present a comparison of three different methods to calibrate the spring constant of two different types of silicon beam shaped atomic force microscope (AFM) cantilevers to determine each method's accuracy, ease of use and potential destructiveness. The majority of research in calibrating AFM cantilevers has been concerned with contact mode levers. The two types of levers we have studied are used in force modulation and tapping mode in air. Not only can these types of cantilevers have spring constants an order of magnitude greater than contact mode levers, but also their geometries can be quite different from the standard V-shape contact lever. In this work we experimentally determine the correction factors for two of the calibration methods when applied to the tapping mode cantilevers and also demonstrate that the force modulation levers can be calibrated easily and accurately using these same techniques.

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 microm 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.