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Reconstructing force from harmonic motion
KTH, School of Engineering Sciences (SCI), Applied Physics, Nanostructure Physics.ORCID iD: 0000-0002-5923-0279
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

High-quality factor oscillators are often used in measurements of verysmall force since they exhibit an enhanced sensitivity in the narrow frequencyband around resonance. Forces containing frequencies outside this frequencyband are often not detectable and the total force acting on the oscillatorremains unknown. In this thesis we present methods to eciently use theavailable bandwidth around resonance to reconstruct the force from partialspectral information.We apply the methods to dynamic atomic force microscopy (AFM) wherea tip at the end of a small micro-cantilever oscillates close to a sample surface.By reconstructing the force between the tip and the surface we can deducedierent properties of the surface. In contrast, in conventional AFM only oneof the many frequency components of the time-dependent tip-surface forceallowing for only qualitative conclusions about the tip-surface force.To increase the number of measurable frequency components we developed Intermodulation AFM (ImAFM). ImAFM utilizes frequency mixing ofa multifrequency drive scheme which generates many frequencies in the response to the nonlinear character of the tip-surface interaction. ImAFM,amplitude-modulated AFM and frequency-modulated AFM can be considered as special cases of narrow-band AFM, where the tip motion can bedescribed by a rapidly oscillating part and a slowly-varying envelope function. Using the concept of force quadratures, each rapid oscillation cycle canbe analyzed individually and ImAFM measurements can be interpreted as arapid measurement of the dependence of the force quadratures on the oscillation amplitude or frequency. To explore the limits of the force quadraturesdescription we introduce the force disk which is a complete description of thetip-surface force in narrow-band AFM at xed static probe height.We present a polynomial force reconstruction method for multifrequencyAFM data. The polynomial force reconstruction is a linear approximativeforce reconstruction method which is based on nding the parameters of amodel force which best approximates the tip-surface force. Another classof reconstruction methods are integral techniques which aim to invert theintegral relation between the tip-surface force and the measured spectraldata. We present an integral method, amplitude-dependence force spectroscopy (ADFS), which reconstructs the conservative tip-surface force fromthe amplitude-dependence of the force quadratures. Together with ImAFMwe use ADFS to combine high-resolution AFM imaging at high speeds withhighly accurate force measurements in each point of an image. For the measurement of dissipative forces we discuss how methods from tomography canbe used to reconstruct forces that are a function of both tip position andvelocity.The methods developed in this thesis are not limited to dynamic AFM andwe describe them in the general context of a harmonic oscillator subject to anexternal force. We hope that theses methods contribute to the transformationof AFM from a qualitative imaging modality into quantitative microscopy andwe hope that they nd application in other measurements which exploit theenhanced sensitivity of a high-quality factor oscillator.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. , v, 91 p.
Series
Trita-FYS, ISSN 0280-316X ; 2013:21
Keyword [en]
oscillator, force spectroscopy, atomic force microscopy, intermodulation, multifrequency, inverse problem, high quality factor
National Category
Nano Technology
Identifiers
URN: urn:nbn:se:kth:diva-122583ISBN: 978-91-7501-792-1 (print)OAI: oai:DiVA.org:kth-122583DiVA: diva2:622877
Public defence
2013-06-14, FA31, Albanova University Center, Roslagstullsbacken 21, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20130527

Available from: 2013-05-27 Created: 2013-05-23 Last updated: 2013-05-27Bibliographically approved
List of papers
1. Intermodulation atomic force microscopy
Open this publication in new window or tab >>Intermodulation atomic force microscopy
2008 (English)In: Applied Physics Letters, ISSN 0003-6951, E-ISSN 1077-3118, Vol. 92, 153106- p.Article in journal (Refereed) Published
Abstract [en]

A mode of atomic force microscopy ͑AFM͒ is demonstrated where an oscillating AFM cantileverhaving linear response is driven with two frequencies in the vicinity of a resonance. Newfrequencies in the response, known as intermodulation products, are generated when the linearity ofthe cantilever response is perturbed by the nonlinear tip-surface interaction. A rich structure of theintermodulation products is observed as a function of the probe-surface separation, indicating thatit is possible to extract much more detailed information about the tip-surface interaction than ispossible with the standard amplitude and phase imaging methods.

Keyword
Atomic force microscopy, Nonlinear oscillations, Cantilevers; Surfaces; Tip
National Category
Condensed Matter Physics
Identifiers
urn:nbn:se:kth:diva-11551 (URN)10.1063/1.2909569 (DOI)000255117100081 ()2-s2.0-42349116023 (Scopus ID)
Note

QC 20100812

Available from: 2009-11-20 Created: 2009-11-19 Last updated: 2013-05-27Bibliographically approved
2. Phase imaging with intermodulation atomic force microscopy
Open this publication in new window or tab >>Phase imaging with intermodulation atomic force microscopy
Show others...
2010 (English)In: Ultramicroscopy, ISSN 0304-3991, E-ISSN 1879-2723, Vol. 110, no 6, 573-577 p.Article in journal (Refereed) Published
Abstract [en]

Intermodulation atomic force microscopy (IMAFM) is a dynamic mode of atomic force microscopy (AFM) with two-tone excitation. The oscillating AFM cantilever in close proximity to a surface experiences the nonlinear tip-sample force which mixes the drive tones and generates new frequency components in the cantilever response known as intermodulation products (IMPs). We present a procedure for extracting the phase at each IMP and demonstrate phase images made by recording this phase while scanning. Amplitude and phase images at intermodulation frequencies exhibit enhanced topographic and material contrast.

Keyword
Atomic force microscopy, Nonlinear oscillations
National Category
Condensed Matter Physics
Identifiers
urn:nbn:se:kth:diva-11557 (URN)10.1016/j.ultramic.2010.02.012 (DOI)000279482900002 ()2-s2.0-77953541578 (Scopus ID)
Note

Uppdaterad från manuskript till artikel: 20100812 QC 20100812

Available from: 2009-11-20 Created: 2009-11-20 Last updated: 2013-05-27Bibliographically approved
3. Reconstructing nonlinearities with intermodulation spectroscopy
Open this publication in new window or tab >>Reconstructing nonlinearities with intermodulation spectroscopy
Show others...
2010 (English)In: Physical Review Letters, ISSN 0031-9007, Vol. 104, no 5, 050801- p.Article in journal (Refereed) Published
Abstract [en]

We describe a method of analysis which allows for reconstructing the nonlinear disturbance of a high Q harmonic oscillator. When the oscillator is driven with two or more frequencies, the nonlinearity causes intermodulation of the drives, resulting in a complicated spectral response. Analysis of this spectrum allows one to approximate the nonlinearity. The method, which is generally applicable to measurements based on resonant detection, increases the information content of the measurement without requiring large detection bandwidth, and optimally uses the enhanced sensitivity near resonance to extract information and minimize error due to detector noise.

Keyword
Nonlinear oscillations, Atomic force microscopy
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-11558 (URN)10.1103/PhysRevLett.104.050801 (DOI)000274336800009 ()2-s2.0-76249130755 (Scopus ID)
Funder
Swedish Research Council
Note

Uppdaterad från manuskript till artikel: 20100812 QC 20110126

Available from: 2009-11-20 Created: 2009-11-20 Last updated: 2013-05-27Bibliographically approved
4. Note: The intermodulation lockin analyzer
Open this publication in new window or tab >>Note: The intermodulation lockin analyzer
Show others...
2011 (English)In: Review of Scientific Instruments, ISSN 0034-6748, E-ISSN 1089-7623, Vol. 82, no 2, 026109- p.Article in journal (Refereed) Published
Abstract [en]

Nonlinear systems can be probed by driving them with two or more pure tones while measuring the intermodulation products of the drive tones in the response. We describe a digital lockin analyzer which is designed explicitly for this purpose. The analyzer is implemented on a field-programmable gate array, providing speed in analysis, real-time feedback, and stability in operation. The use of the analyzer is demonstrated for intermodulation atomic force microscopy. A generalization of the intermodulation spectral technique to arbitrary drive waveforms is discussed.

National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-31596 (URN)10.1063/1.3541791 (DOI)000287813400082 ()2-s2.0-79952143423 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20110325

Available from: 2011-03-25 Created: 2011-03-21 Last updated: 2017-12-11Bibliographically approved
5. The role of nonlinear dynamics in quantitative atomic force microscopy
Open this publication in new window or tab >>The role of nonlinear dynamics in quantitative atomic force microscopy
2012 (English)In: Nanotechnology, ISSN 0957-4484, E-ISSN 1361-6528, Vol. 23, no 26, 265705- p.Article in journal (Refereed) Published
Abstract [en]

Various methods of force measurement with the atomic force microscope are compared for their ability to accurately determine the tip-surface force from analysis of the nonlinear cantilever motion. It is explained how intermodulation, or the frequency mixing of multiple drive tones by the nonlinear tip-surface force, can be used to concentrate the nonlinear motion in a narrow band of frequency near the cantilever's fundamental resonance, where accuracy and sensitivity of force measurement are greatest. Two different methods for reconstructing tip-surface forces from intermodulation spectra are explained. The reconstruction of both conservative and dissipative tip-surface interactions from intermodulation spectra are demonstrated on simulated data.

National Category
Nano Technology
Identifiers
urn:nbn:se:kth:diva-99062 (URN)10.1088/0957-4484/23/26/265705 (DOI)000305411400017 ()2-s2.0-84862638379 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20120719

Available from: 2012-07-19 Created: 2012-07-13 Last updated: 2017-12-07Bibliographically approved
6. Model-based extraction of material properties in multifrequency atomic force microscopy
Open this publication in new window or tab >>Model-based extraction of material properties in multifrequency atomic force microscopy
2012 (English)In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 85, no 19, 195449- p.Article in journal (Refereed) Published
Abstract [en]

We present a method to reconstruct the nonlinear tip-surface force and extract material properties from a multifrequency atomic force microscopy (AFM) measurement with a high-quality-factor cantilever resonance. In a measurement time of similar to 2 ms, we are able to accurately reconstruct the tip-surface force-displacement curve, allowing simultaneous high-resolution imaging of both topography and material properties at typical AFM scan rates. We verify the method using numerical simulations, apply it to experimental data, and use it to image mechanical properties of a polymer blend. We further discuss the limitations of the method and identify suitable operating conditions for AFM experiments.

National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-98006 (URN)10.1103/PhysRevB.85.195449 (DOI)000304395300007 ()2-s2.0-84861707952 (Scopus ID)
Funder
Swedish Research Council
Note
QC 20120618Available from: 2012-06-18 Created: 2012-06-18 Last updated: 2017-12-07Bibliographically approved
7. Interaction imaging with amplitude-dependence force spectroscopy
Open this publication in new window or tab >>Interaction imaging with amplitude-dependence force spectroscopy
2013 (English)In: Nature Communications, ISSN 2041-1723, E-ISSN 2041-1723, Vol. 4, 1360- p.Article in journal (Refereed) Published
Abstract [en]

Knowledge of surface forces is the key to understanding a large number of processes in fields ranging from physics to material science and biology. The most common method to study surfaces is dynamic atomic force microscopy (AFM). Dynamic AFM has been enormously successful in imaging surface topography, even to atomic resolution, but the force between the AFM tip and the surface remains unknown during imaging. Here we present a new approach that combines high-accuracy force measurements and high-resolution scanning. The method, called amplitude-dependence force spectroscopy (ADFS), is based on the amplitude dependence of the cantilever's response near resonance and allows for separate determination of both conservative and dissipative tip-surface interactions. We use ADFS to quantitatively study and map the nano-mechanical interaction between the AFM tip and heterogeneous polymer surfaces. ADFS is compatible with commercial atomic force microscopes and we anticipate its widespread use in taking AFM toward quantitative microscopy.

Keyword
Atomic-Resolution, Microscopy, Probe, Cantilevers, Nanoscale, Contact, Surface, Chaos, AFM
National Category
Engineering and Technology
Identifiers
urn:nbn:se:kth:diva-121505 (URN)10.1038/ncomms2365 (DOI)000316614600030 ()2-s2.0-84876120115 (Scopus ID)
Funder
Swedish Research CouncilVinnova
Note

QC 20130506

Available from: 2013-05-06 Created: 2013-04-29 Last updated: 2017-12-06Bibliographically approved
8. Interpreting motion and force for narrow-band intermodulation atomic force microscopy
Open this publication in new window or tab >>Interpreting motion and force for narrow-band intermodulation atomic force microscopy
2013 (English)In: Beilstein Journal of Nanotechnology, ISSN 2190-4286, Vol. 4, 45-56 p.Article in journal (Refereed) Published
Abstract [en]

Intermodulation atomic force microscopy (ImAFM) is a mode of dynamic atomic force microscopy that probes the nonlinear tip-surface force by measurement of the mixing of multiple modes in a frequency comb. A high-quality factor cantilever resonance and a suitable drive comb will result in tip motion described by a narrow-band frequency comb. We show, by a separation of time scales, that such motion is equivalent to rapid oscillations at the cantilever resonance with a slow amplitude and phase or frequency modulation. With this time-domain perspective, we analyze single oscillation cycles in ImAFM to extract the Fourier components of the tip-surface force that are in-phase with the tip motion (F-I) and quadrature to the motion (F-Q). Traditionally, these force components have been considered as a function of the static-probe height only. Here we show that F-I and F-Q actually depend on both static-probe height and oscillation amplitude. We demonstrate on simulated data how to reconstruct the amplitude dependence of F-I and F-Q from a single ImAFM measurement. Furthermore, we introduce ImAFM approach measurements with which we reconstruct the full amplitude and probe-height dependence of the force components F-I and F-Q, providing deeper insight into the tip-surface interaction. We demonstrate the capabilities of ImAFM approach measurements on a polystyrene polymer surface.

Keyword
atomic force microscopy, AFM, frequency combs, force spectroscopy, high-quality-factor resonators, intermodulation, multifrequency
National Category
Nano Technology
Identifiers
urn:nbn:se:kth:diva-118243 (URN)10.3762/bjnano.4.5 (DOI)000313722200001 ()2-s2.0-84876109153 (Scopus ID)
Note

QC 20130214

Available from: 2013-02-14 Created: 2013-02-14 Last updated: 2015-02-09Bibliographically approved
9. Polynomial force approximations and multifrequency atomic force microscopy
Open this publication in new window or tab >>Polynomial force approximations and multifrequency atomic force microscopy
2013 (English)In: Beilstein Journal of Nanotechnology, ISSN 2190-4286, Vol. 4, no 1, 352-360 p.Article in journal (Refereed) Published
Abstract [en]

We present polynomial force reconstruction from experimental intermodulation atomic force microscopy (ImAFM) data. We study the tip-surface force during a slow surface approach and compare the results with amplitude-dependence force spectroscopy (ADFS). Based on polynomial force reconstruction we generate high-resolution surface-property maps of polymer blend samples. The polynomial method is described as a special example of a more general approximative force reconstruction, where the aim is to determine model parameters that best approximate the measured force spectrum. This approximative approach is not limited to spectral data, and we demonstrate how it can be adapted to a force quadrature picture.

Keyword
AFM, Atomic force microscopy, Force spectroscopy, Intermodulation, Multifrequency, Polynomial
National Category
Nano Technology
Identifiers
urn:nbn:se:kth:diva-122572 (URN)10.3762/bjnano.4.41 (DOI)000320048600001 ()2-s2.0-84880255850 (Scopus ID)
Funder
Swedish Research CouncilVinnovaKnut and Alice Wallenberg Foundation
Note

QC 20131204. Updated from accepted to published.

Available from: 2013-05-23 Created: 2013-05-23 Last updated: 2017-12-06Bibliographically approved
10. Tip-surface interactions in dynamic atomic forcemicroscopy
Open this publication in new window or tab >>Tip-surface interactions in dynamic atomic forcemicroscopy
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In atomic force microscopy (AFM) tip-surface interactions are usuallyconsidered as functions of the tip position only, so-called force curves. However,tip-surface interactions often depend on the tip velocity and the past tip trajectory.Here, we introduce a compact and general description of these interactionsappropriate to dynamic AFM where the measurement of force is restricted toa narrow frequency band. We represent the tip-surface interaction in termsof a force disk in the phase space of position and velocity. Determination ofthe amplitude dependence of tip-surface forces at a fixed static probe heightallows for a comprehensive treatment of conservative and dissipative interactions.We illuminate the fundamental limitations of force reconstruction with narrowband dynamic AFM and we show how the amplitude dependence of the F ouriercomponent of the force at the tip oscillation frequency, gives qualitative insight intothe detailed nature of the tip-surface interaction. With minimal assumptions thisamplitude dependence force spectroscopy allows for a quantitative reconstruction ofthe effective conservative tip-surface force as well as a position-dependent dampingfactor. We demonstrate this reconstruction on simulated intermodulation AFMdata.

Keyword
atomic force microscopy, measurement of force, mechanical resonators, MEMS/NEMS, dissipation, intermodulation
National Category
Nano Technology
Identifiers
urn:nbn:se:kth:diva-122581 (URN)
Funder
Swedish Research CouncilVinnova
Note

QS 2013

Available from: 2013-05-23 Created: 2013-05-23 Last updated: 2013-05-27Bibliographically approved

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