Bimodal atomic force microscopy

Excitation and detection scheme in bimodal AFM. The cantilever is excited at its first two eigenmodes with frequencies ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$. Upon interaction with the sample, the components of the tip's response are processed. The topography is obtained by keeping ${\displaystyle A_{1}}$= constant. In a bimodal AM-FM configuration, two feedback loops act on the 2nd mode. One keeps ${\displaystyle A_{2}}$ fixed while a phase-lock loop keeps ${\displaystyle \phi _{2}=90^{o}}$.

Bimodal Atomic Force Microscopy (bimodal AFM) is an advanced atomic force microscopy technique characterized by generating high-spatial resolution maps of material properties. Topography, deformation, elastic modulus, viscosity coefficient or magnetic field maps might be generated. Bimodal AFM is based on the simultaneous excitation and detection of two eigenmodes (resonances) of a force microscope microcantilever.

History

Numerical and theoretical considerations ^{[1] [2]} prompted the development of bimodal AFM. The method was initially thought to enhance topographic contrast in air environments. ^{[3] [4]} Three subsequent advances such as the capability to detect non-topography properties such electrostatic ^[5] and magnetic ^[6] interactions; imaging in liquid ^[7] and ultra-high vacuum ^[8] and its genuine quantitative features ^{[9] [10]} set the stage for further developments and applications.

Principles of Bimodal AFM

Nanomechanical maps of a polymer blend generated by bimodal AFM. Young's modulus (top) and viscosity coefficient (bottom) maps of a PS-b-PMMA block co-polymer. The stiffer domains (PMMA) show a lower viscosity coefficient and a faster response time.

Angstrom-scale resolution. Topography (top) and elastic modulus (middle) maps of metal-organic-framework (MOF). The MOF structure on the basal plane is being overlaid. Angstrom-scale variations of the elastic modulus are associated with the chemical structure of the MOF. Distribution of the elastic modulus values. Data obtained from the middle map (bottom). The curve can be decomposed in four individual Gaussian curves, centered respectively, at 25.7, 27.5, 29.3 and 32.3 GPa.

The interaction of the tip with the sample modifies the amplitudes, phase shifts and frequency resonances of the excited modes. Those changes are detected and processed by the feedback of the instrument. Several features make bimodal AFM a very powerful surface characterization method at the nanoscale. (i) Resolution. Atomic, molecular or nanoscale spatial resolution was demonstrated. (ii) Simultaneity. Maps of different properties are generated at the same time. (iii) Efficiency. A maximum number of four data points per pixel are needed to generate material property maps. (iv) Speed. Analytical solutions link observables with material properties.

Configurations

In AFM, feedback loops control the operation of the microscope by keeping a fixed value a parameter of the tip's oscillation. ^[11] If the main feedback loop operates with the amplitude, the AFM mode is called amplitude modulation (AM). If it operates with the frequency shift, the AFM mode is called frequency modulation (FM). Bimodal AFM might be operated with several feedback loops. This gives rise to a variety of bimodal configurations. ^[12] The configurations are termed AM-open loop, AM-FM, FM-FM. ^[13] For example, bimodal AM-FM means that the first mode is operated with an amplitude modulation loop while the 2nd mode is operated with a frequency modulation loop. The configurations might not be equivalent in terms of sensitivity, signal-to-noise ratio or complexity.

Let's consider the AM-FM configuration. The first mode is excited to reach free amplitude (no interaction) and the changes of its amplitude and phase shift are tracked by a lock-in amplifier. The main feedback loop keeps constant the amplitude, at a certain set-point ${\displaystyle A_{1}}$ by modifying the tip vertical position (AM). In a nanomechanical mapping experiment, ${\displaystyle \phi _{1}}$ must be kept below 90°, i.e., the AFM is operated in the repulsive regime. At the same time, an FM loop acts on the second eigenmode. A phase-lock-loop regulates the excitation frequency ${\displaystyle f_{2}}$ by keeping the phase shift of the second mode at 90°. An additional feedback loop might be used to maintain the amplitude ${\displaystyle A_{2}}$ constant.

Theory

The theory of bimodal AFM operation encompasses several aspects. Among them, the approximations to express the Euler-Bernoulli equation of a continuous cantilever beam in terms of the equations of the excited modes, ^{[2] [8] [10] [14]} the type of interaction forces acting on the tip, ^{[15] [16] [17]} the theory of demodulation methods ^[18] or the introduction of finite-size effects. ^[19]

In a nutshell, the tip displacement in AFM is approximated by a point-mass model, ^[13]

${\displaystyle {\frac {k_{i}}{4\pi ^{2}f_{i}^{2}}}{\ddot {z_{i}}}+{\frac {k_{i}}{2\pi f_{0i}Q}}{\dot {z_{i}}}+k_{i}z_{i}=F_{i}\cos(2\pi f_{i}t)+F_{ts}(t)\,}$

where ${\displaystyle f_{i}}$, ${\displaystyle f_{0}i}$, ${\displaystyle Q_{i}}$, ${\displaystyle k_{i}}$, ${\displaystyle F_{i}}$, and ${\displaystyle F_{ts}}$ are, respectively, the driving frequency, the free resonant frequency, the quality factor, the stiffness, the driving force of the i-th mode, and the tip–sample interaction force. In bimodal AFM, the vertical motion of the tip (deflection) has two components, one for each mode,

${\displaystyle z(t)=z_{0}+z_{1}(t)+z_{2}(t)\approx A_{1}cos\left(2\pi f_{1}t-\phi _{1}\right)+A_{2}cos\left(2\pi f_{2}t-{\frac {\pi }{2}}\right)\,}$

with ${\displaystyle z_{0}}$, ${\displaystyle z_{1}}$, ${\displaystyle z_{2}}$, as the static, the first, and the second mode deflections; ${\displaystyle A_{i}}$, ${\displaystyle f_{i}}$ and ${\displaystyle \phi _{i}}$ are, respectively, the amplitude, frequency and phase shift of mode i.

The theory that transforms bimodal AFM observables into material properties is based on applying the virial ${\displaystyle V_{i}}$ and energy dissipation ${\displaystyle E_{diss}}$ theorems to the equations of motion of the excited modes. The following equations were derived ^[20]

${\displaystyle V_{1}={\frac {1}{T}}\int _{0}^{T}F_{ts}(t)z_{1}(t)dt=-{\frac {k_{1}A_{1}A_{01}}{2Q_{1}}}\cos {\phi _{1}}\,}$

${\displaystyle V_{2}={\frac {1}{T}}\int _{0}^{T}F_{ts}(t)z_{2}(t)dt\approx -{\frac {k_{2}A_{2}^{2}\Delta f_{2}}{f_{02}}}\,}$

${\displaystyle E_{diss1}=\int _{0}^{T}F_{ts}(t){\dot {z}}_{1}(t)dt=-{\frac {\pi k_{1}A_{1}}{Q_{1}}}(A_{1}-A_{01}\sin(\phi _{1}))\,}$

where ${\displaystyle T=T_{1}T_{2}}$ is a time where the oscillation of both modes are periodic; ${\displaystyle Q_{i}}$ the quality factor of mode i. Bimodal AFM operation might be involve any pair of eigenmodes. However, experiments are commonly performed by exciting the first two eigenmodes.

The theory of bimodal AFM provides analytical expressions to link material properties with microscope observables. For example, for a paraboloid probe (radius ${\displaystyle R}$) and a tip-sample force given by the linear viscoelastic Kelvin-Voigt model, the effective elastic modulus ${\displaystyle E_{eff}}$ of the sample, viscous coefficient of compressibility ${\displaystyle \eta _{com}}$, loss tangent ${\displaystyle \tan \rho }$ or retardation time ${\displaystyle \tau }$ are expressed by ^[21]

${\displaystyle E_{eff}=4{\sqrt {2}}{\frac {Q_{1}}{\sqrt {R}}}{\frac {k_{2}^{2}}{k_{1}}}{\frac {\Delta f_{2}^{2}}{f_{02}^{2}}}{\frac {A_{1}^{3/2}}{A_{01}^{2}-A_{1}^{2}}}\,}$

${\displaystyle \eta _{com}={\frac {E_{eff}}{\omega _{1}}}\left[{\frac {A_{01}\sin {\phi _{1}}-A_{1}}{A_{01}\cos {\phi _{1}}}}\right]\,}$

${\displaystyle \tan \rho =2\pi \omega _{1}{\frac {\eta _{com}}{E_{eff}}}=2\pi \omega _{1}\tau \,}$

For an elastic material, the second term of equation to calculate ${\displaystyle \eta }$ disappears because ${\displaystyle A_{1}=A_{01}\sin {\phi _{1}}}$ which gives ${\displaystyle \eta =0}$. The elastic modulus is obtained from the equation above. Other analytical expressions were proposed for the determination of the Hamaker constant ^[15] and the magnetic parameters of a ferromagnetic sample. ^[16]

Applications

Bimodal AFM is applied to characterize a large variety of surfaces and interfaces. Some applications exploit the sensitivity of bimodal observables to enhance spatial resolution. However, the full capabilities of bimodal AFM are shown in the generation of quantitative maps of material properties. The section is divided in terms of the achieved spatial resolution, atomic-scale or nanoscale.

Atomic and molecular-scale resolution

Atomic-scale imaging of graphene, ^[22] semiconductor surfaces ^[23] and adsorbed organic molecules ^[24] were obtained in ultra high-vacuum. Angstrom-resolution images of hydration layers formed on proteins and Young's modulus map of a metal-organic frame work, purple membrane and a lipid bilayer were reported in aqueous solutions. ^{[20] [25] [26]}

Material property applications

Bimodal AFM is widely used to provide high-spatial resolution maps of material properties, in particular, mechanical properties. Elastic and/or viscoelastic property maps of polymers, ^{[27] [28] [29] [30] [31]} DNA, ^[32] proteins, protein fibers, ^[33] lipids ^{[34] [19]} or 2D materials ^{[35] [36]} were generated. Non-mechanical properties and interactions including crystal magnetic garnets, electrostatic strain, superparamagnetic particles and high-density disks were also mapped. ^{[37] [38]} Quantitative property mapping requires the calibration of the force constants of the excited modes. ^[39]

References

1. Rodrı́guez, Tomás R.; Garcı́a, Ricardo (2004). "Compositional mapping of surfaces in atomic force microscopy by excitation of the second normal mode of the microcantilever". Applied Physics Letters. 84 (3): 449–451. Bibcode:2004ApPhL..84..449R. doi:10.1063/1.1642273. hdl: 10261/23475 . ISSN   0003-6951.
2. Lozano, Jose R.; Garcia, Ricardo (2008). "Theory of Multifrequency Atomic Force Microscopy". Physical Review Letters. 100 (7): 076102. Bibcode:2008PhRvL.100g6102L. doi:10.1103/PhysRevLett.100.076102. hdl: 10261/11867 . ISSN   0031-9007. PMID   18352572.
3. Proksch, Roger (2006). "Multifrequency, repulsive-mode amplitude-modulated atomic force microscopy". Applied Physics Letters. 89 (11): 113121. Bibcode:2006ApPhL..89k3121P. doi:10.1063/1.2345593. ISSN   0003-6951.
4. Martinez, N. F.; Patil, S.; Lozano, J. R.; Garcia, R. (2006). "Enhanced compositional sensitivity in atomic force microscopy by the excitation of the first two flexural modes". Applied Physics Letters. 89 (15): 153115. Bibcode:2006ApPhL..89o3115M. doi:10.1063/1.2360894. hdl: 10261/17931 . ISSN   0003-6951.
5. Stark, Robert W; Naujoks, Nicola; Stemmer, Andreas (2007). "Multifrequency electrostatic force microscopy in the repulsive regime". Nanotechnology. 18 (6): 065502. Bibcode:2007Nanot..18f5502S. doi:10.1088/0957-4484/18/6/065502. ISSN   0957-4484. S2CID   135615561.
6. Li, Jason W.; Cleveland, Jason P.; Proksch, Roger (2009). "Bimodal magnetic force microscopy: Separation of short and long range forces". Applied Physics Letters. 94 (16): 163118. Bibcode:2009ApPhL..94p3118L. doi:10.1063/1.3126521. ISSN   0003-6951.
7. Martínez, N F; Lozano, J R; Herruzo, E T; Garcia, F; Richter, C; Sulzbach, T; Garcia, R (2008). "Bimodal atomic force microscopy imaging of isolated antibodies in air and liquids". Nanotechnology. 19 (38): 384011. Bibcode:2008Nanot..19L4011M. doi:10.1088/0957-4484/19/38/384011. hdl: 10261/14884 . ISSN   0957-4484. PMID   21832570. S2CID   16928752.
8. Kawai, Shigeki; Glatzel, Thilo; Koch, Sascha; Such, Bartosz; Baratoff, Alexis; Meyer, Ernst (2009). "Systematic Achievement of Improved Atomic-Scale Contrast via Bimodal Dynamic Force Microscopy". Physical Review Letters. 103 (22): 220801. Bibcode:2009PhRvL.103v0801K. doi:10.1103/PhysRevLett.103.220801. ISSN   0031-9007. PMID   20366084.
9. Martinez-Martin, D.; Herruzo, E. T.; Dietz, C.; Gomez-Herrero, J.; Garcia, R. (2011). "Noninvasive Protein Structural Flexibility Mapping by Bimodal Dynamic Force Microscopy". Physical Review Letters. 106 (19): 198101. Bibcode:2011PhRvL.106s8101M. doi:10.1103/PhysRevLett.106.198101. hdl: 10486/668698 . ISSN   0031-9007. PMID   21668203.
10. Herruzo, Elena T.; Perrino, Alma P.; Garcia, Ricardo (2014). "Fast nanomechanical spectroscopy of soft matter". Nature Communications. 5 (1): 3126. Bibcode:2014NatCo...5.3126H. doi: 10.1038/ncomms4126 . hdl: 10261/169779 . ISSN   2041-1723. PMID   24445593.
11. García, R (2002). "Dynamic atomic force microscopy methods". Surface Science Reports. 47 (6–8): 197–301. Bibcode:2002SurSR..47..197G. doi:10.1016/S0167-5729(02)00077-8. ISSN   0167-5729.
12. Solares, Santiago D.; Chawla, Gaurav (2010). "Triple-frequency intermittent contact atomic force microscopy characterization: Simultaneous topographical, phase, and frequency shift contrast in ambient air". Journal of Applied Physics. 108 (5): 054901–054901–5. Bibcode:2010JAP...108e4901S. doi:10.1063/1.3475644. ISSN   0021-8979.
13. Garcia, Ricardo; Proksch, Roger (2013). "Nanomechanical mapping of soft matter by bimodal force microscopy". European Polymer Journal. 49 (8): 1897–1906. doi: 10.1016/j.eurpolymj.2013.03.037 . ISSN   0014-3057.
14. Labuda, Aleksander; Kocuń, Marta; Meinhold, Waiman; Walters, Deron; Proksch, Roger (2016). "Generalized Hertz model for bimodal nanomechanical mapping". Beilstein Journal of Nanotechnology. 7: 970–982. doi:10.3762/bjnano.7.89. ISSN   2190-4286. PMC   4979904 . PMID   27547614.
15. Lai, Chia-Yun; Santos, Sergio; Chiesa, Matteo (2016). "Systematic Multidimensional Quantification of Nanoscale Systems From Bimodal Atomic Force Microscopy Data". ACS Nano. 10 (6): 6265–6272. doi:10.1021/acsnano.6b02455. ISSN   1936-0851. PMID   27172380.
16. Gisbert, Victor G.; Amo, Carlos A.; Jaafar, Miriam; Asenjo, Agustina; Garcia, Ricardo (2021). "Quantitative mapping of magnetic properties at the nanoscale with bimodal AFM". Nanoscale. 13 (3): 2026–2033. doi: 10.1039/D0NR08662B . hdl: 10261/251842 . ISSN   2040-3364. PMID   33449980. S2CID   231622985.
17. Rajabifar, Bahram; Bajaj, Anil; Reifenberger, Ronald; Proksch, Roger; Raman, Arvind (2021). "Discrimination of adhesion and viscoelasticity from nanoscale maps of polymer surfaces using bimodal atomic force microscopy". Nanoscale. 13 (41): 17428–17441. doi:10.1039/D1NR03437E. ISSN   2040-3364. PMID   34647552. S2CID   238858285.
18. Harcombe, David M; Ruppert, Michael G; Fleming, Andrew J (2020). "A review of demodulation techniques for multifrequency atomic force microscopy". Beilstein Journal of Nanotechnology. 11: 76–91. doi:10.3762/bjnano.11.8. ISSN   2190-4286. PMC   6964647 . PMID   31976199.
19. Gisbert, Victor G.; Garcia, Ricardo (2021). "Accurate Wide-Modulus-Range Nanomechanical Mapping of Ultrathin Interfaces with Bimodal Atomic Force Microscopy". ACS Nano. 15 (12): 20574–20581. doi:10.1021/acsnano.1c09178. hdl: 10261/255640 . ISSN   1936-0851. PMID   34851086. S2CID   244835977.
20. Amo, Carlos A.; Perrino, Alma P.; Payam, Amir F.; Garcia, Ricardo (2017). "Mapping Elastic Properties of Heterogeneous Materials in Liquid with Angstrom-Scale Resolution". ACS Nano. 11 (9): 8650–8659. doi: 10.1021/acsnano.7b04381 . hdl: 1983/c802e856-cf8e-496a-bf06-36667ec8d992 . ISSN   1936-0851. PMID   28770996.
21. Benaglia, Simone; Amo, Carlos A.; Garcia, Ricardo (2019). "Fast, quantitative and high resolution mapping of viscoelastic properties with bimodal AFM". Nanoscale. 11 (32): 15289–15297. doi: 10.1039/C9NR04396A . hdl: 10261/189448 . ISSN   2040-3364. PMID   31386741. S2CID   206145305.
22. Kawai, Shigeki; Eren, Baran; Marot, Laurent; Meyer, Ernst (2014). "Graphene Synthesis via Thermal Polymerization of Aromatic Quinone Molecules". ACS Nano. 8 (6): 5932–5938. doi:10.1021/nn501047v. ISSN   1936-0851. PMID   24873393.
23. Naitoh, Yoshitaka; Turanský, Robert; Brndiar, Ján; Li, Yan Jun; Štich, Ivan; Sugawara, Yasuhiro (2017). "Subatomic-scale force vector mapping above a Ge(001) dimer using bimodal atomic force microscopy". Nature Physics. 13 (7): 663–667. Bibcode:2017NatPh..13..663N. doi:10.1038/nphys4083. ISSN   1745-2473. S2CID   125626854.
24. Martin-Jimenez, Daniel; Ruppert, Michael G.; Ihle, Alexander; Ahles, Sebastian; Wegner, Hermann A.; Schirmeisen, André; Ebeling, Daniel (2022). "Chemical bond imaging using torsional and flexural higher eigenmodes of qPlus sensors". Nanoscale. 14 (14): 5329–5339. doi:10.1039/D2NR01062C. ISSN   2040-3364. PMID   35348167. S2CID   247776212.
25. Garcia, Ricardo; Herruzo, Elena T. (2012). "The emergence of multifrequency force microscopy". Nature Nanotechnology. 7 (4): 217–226. Bibcode:2012NatNa...7..217G. doi:10.1038/nnano.2012.38. hdl: 10261/49721 . ISSN   1748-3387. PMID   22466857.
26. Trewby, William; Faraudo, Jordi; Voïtchovsky, Kislon (2019). "Long-lived ionic nano-domains can modulate the stiffness of soft interfaces". Nanoscale. 11 (10): 4376–4384. doi: 10.1039/C8NR06339G . hdl: 10261/203212 . ISSN   2040-3364. PMID   30801089. S2CID   73461360.
27. Kocun, Marta; Labuda, Aleksander; Meinhold, Waiman; Revenko, Irène; Proksch, Roger (2017). "Fast, High Resolution, and Wide Modulus Range Nanomechanical Mapping with Bimodal Tapping Mode". ACS Nano. 11 (10): 10097–10105. arXiv: 1702.06842 . doi:10.1021/acsnano.7b04530. ISSN   1936-0851. PMID   28953363. S2CID   22705354.
28. Ebeling, Daniel; Eslami, Babak; Solares, Santiago De Jesus (2013). "Visualizing the Subsurface of Soft Matter: Simultaneous Topographical Imaging, Depth Modulation, and Compositional Mapping with Triple Frequency Atomic Force Microscopy". ACS Nano. 7 (11): 10387–10396. doi:10.1021/nn404845q. ISSN   1936-0851. PMID   24131492.
29. Nguyen, Hung K.; Ito, Makiko; Nakajima, Ken (2016). "Elastic and viscoelastic characterization of inhomogeneous polymers by bimodal atomic force microscopy". Japanese Journal of Applied Physics. 55 (8S1): 08–06. Bibcode:2016JaJAP..55hNB06N. doi:10.7567/JJAP.55.08NB06. ISSN   0021-4922. S2CID   124918476.
30. Dietz, Christian (2018). "Sensing in-plane nanomechanical surface and sub-surface properties of polymers: local shear stress as function of the indentation depth". Nanoscale. 10 (1): 460–468. doi:10.1039/C7NR07147G. ISSN   2040-3364. PMID   29227502.
31. Nguyen, Hung K.; Aoki, Mika; Liang, Xiaobin; Yamamoto, Satoru; Tanaka, Keiji; Nakajima, Ken (2021). "Local Mechanical Properties of Heterogeneous Nanostructures Developed in a Cured Epoxy Network: Implications for Innovative Adhesion Technology". ACS Applied Nano Materials. 4 (11): 12188–12196. doi:10.1021/acsanm.1c02692. ISSN   2574-0970. S2CID   240137740.
32. Bose, Krishnashish; Lech, Christopher J.; Heddi, Brahim; Phan, Anh Tuân (2018). "High-resolution AFM structure of DNA G-wires in aqueous solution". Nature Communications. 9 (1): 1959. Bibcode:2018NatCo...9.1959B. doi:10.1038/s41467-018-04016-y. ISSN   2041-1723. PMC   5958085 . PMID   29773796.
33. Gisbert, Victor G.; Benaglia, Simone; Uhlig, Manuel R.; Proksch, Roger; Garcia, Ricardo (2021). "High-Speed Nanomechanical Mapping of the Early Stages of Collagen Growth by Bimodal Force Microscopy". ACS Nano. 15 (1): 1850–1857. doi:10.1021/acsnano.0c10159. ISSN   1936-0851. PMC   8477367 . PMID   33412008.
34. Al-Rekabi, Zeinab; Contera, Sonia (2018). "Multifrequency AFM reveals lipid membrane mechanical properties and the effect of cholesterol in modulating viscoelasticity". Proceedings of the National Academy of Sciences. 115 (11): 2658–2663. Bibcode:2018PNAS..115.2658A. doi: 10.1073/pnas.1719065115 . ISSN   0027-8424. PMC   5856542 . PMID   29483271.
35. Li, Yuhao; Yu, Chuanbin; Gan, Yingye; Jiang, Peng; Yu, Junxi; Ou, Yun; Zou, Dai-Feng; Huang, Cheng; Wang, Jiahong; Jia, Tingting; Luo, Qian; Yu, Xue-Feng; Zhao, Huijuan; Gao, Cun-Fa; Li, Jiangyu (2018). "Mapping the elastic properties of two-dimensional MoS2 via bimodal atomic force microscopy and finite element simulation". npj Computational Materials. 4 (1): 49. Bibcode:2018npjCM...4...49L. doi: 10.1038/s41524-018-0105-8 . ISSN   2057-3960. S2CID   139573874.
36. Lai, Chia-Yun; Santos, Sergio; Chiesa, Matteo (2019). "Machine learning assisted quantification of graphitic surfaces exposure to defined environments". Applied Physics Letters. 114 (24): 241601. Bibcode:2019ApPhL.114x1601L. doi:10.1063/1.5095704. hdl: 10037/17650 . ISSN   0003-6951. S2CID   197510684.
37. Dietz, Christian; Herruzo, Elena T; Lozano, Jose R; Garcia, Ricardo (2011). "Nanomechanical coupling enables detection and imaging of 5 nm superparamagnetic particles in liquid". Nanotechnology. 22 (12): 125708. Bibcode:2011Nanot..22l5708D. doi:10.1088/0957-4484/22/12/125708. hdl: 10261/47513 . ISSN   0957-4484. PMID   21325711. S2CID   27542582.
38. Giridharagopal, R.; Flagg, L. Q.; Harrison, J. S.; Ziffer, M. E.; Onorato, J.; Luscombe, C. K.; Ginger, D.S. (2017). "Electrochemical strain microscopy probes morphology-induced variations in ion uptake and performance in organic electrochemical transistors". Nature Materials. 16 (7): 737–742. Bibcode:2017NatMa..16..737G. doi:10.1038/nmat4918. ISSN   1476-1122. PMID   28628123.
39. Labuda, Aleksander; Kocun, Marta; Lysy, Martin; Walsh, Tim; Meinhold, Jieh; Proksch, Tania; Meinhold, Waiman; Anderson, Caleb; Proksch, Roger (2016). "Calibration of higher eigenmodes of cantilevers". Review of Scientific Instruments. 87 (7): 073705. arXiv: 1604.00601 . Bibcode:2016RScI...87g3705L. doi:10.1063/1.4955122. ISSN   0034-6748. PMID   27475563. S2CID   9353030.