No Access Submitted: 22 June 2018 Accepted: 17 August 2018 Published Online: 04 September 2018
Review of Scientific Instruments 89, 093701 (2018); https://doi.org/10.1063/1.5045679
more...View AffiliationsView Contributors
  • Aleksander Labuda
  • Changhong Cao
  • Tim Walsh
  • Jieh Meinhold
  • Roger Proksch
  • Yu Sun
  • Tobin Filleter
A method for calibrating the dynamic torsional spring constant of cantilevers by directly measuring the thermally driven motion of the cantilever with an interferometer is presented. Random errors in calibration were made negligible (<1%) by averaging over multiple measurements. The errors in accuracy of ±5% or ±10% for both of the cantilevers calibrated in this study were limited only by the accuracy of the laser Doppler vibrometer (LDV) used to measure thermal fluctuations. This is a significant improvement over commonly used methods that result in large and untraceable errors resulting from assumptions made about the cantilever geometry, material properties, and/or hydrodynamic physics of the surroundings. Subsequently, the static torsional spring constant is determined from its dynamic counterpart after careful LDV measurements of the torsional mode shape, backed by finite element analysis simulations. A meticulously calibrated cantilever is used in a friction force microscopy experiment that measures the friction difference and interfacial shear strength (ISS) between graphene and a silicon dioxide AFM probe. Accurate calibration can resolve discrepancies between different experimental methods, which have contributed to a large scatter in the reported friction and ISS values in the literature to date.
We would like to thank Mark Reitsma, Ted Limpoco, John Sader, Deron Walters, Ryan Wagner, and Donna Hurley for valuable discussions.
  1. 1. S. Maier, Y. Sang, T. Filleter, M. Grant, R. Bennewitz, E. Gnecco, and E. Meyer, Phys. Rev. B 72, 245418 (2005). https://doi.org/10.1103/physrevb.72.245418, Google ScholarCrossref
  2. 2. I. Barel, M. Urbakh, L. Jansen, and A. Schirmeisen, Phys. Rev. B 84, 115417 (2011). https://doi.org/10.1103/physrevb.84.115417, Google ScholarCrossref
  3. 3. A. Labuda, F. Hausen, N. N. Gosvami, P. H. Grütter, R. B. Lennox, and R. Bennewitz, Langmuir 27, 2561 (2011). https://doi.org/10.1021/la104497t, Google ScholarCrossref
  4. 4. C. Loppacher, R. Bennewitz, O. Pfeiffer, M. Guggisberg, M. Bammerlin, S. Schär, V. Barwich, A. Baratoff, E. Meyer, and H.-J. Güntherodt, Phys. Rev. B 62, 13674 (2000). https://doi.org/10.1103/physrevb.62.13674, Google ScholarCrossref
  5. 5. Z. Liu, S. M. Zhang, J. R. Yang, J. Z. YangLiu, Y. L. Yang, and Q. S. Zheng, Acta Mech. Sin. 28, 978 (2012). https://doi.org/10.1007/s10409-012-0137-0, Google ScholarCrossref
  6. 6. A. Labuda, M. Lysy, W. Paul, Y. Miyahara, P. Grütter, R. Bennewitz, and M. Sutton, Phys. Rev. E 86, 031104 (2012). https://doi.org/10.1103/physreve.86.031104, Google ScholarCrossref
  7. 7. Y. Dong, H. Gao, A. Martini, and P. Egberts, Phys. Rev. E 90, 12125 (2014). https://doi.org/10.1103/physreve.90.012125, Google ScholarCrossref
  8. 8. X.-Z. Liu, Z. Ye, Y. Dong, P. Egberts, R. W. Carpick, and A. Martini, Phys. Rev. Lett. 114, 146102 (2015). https://doi.org/10.1103/physrevlett.114.146102, Google ScholarCrossref
  9. 9. Y. Dong, Q. Li, and A. Martini, J. Vac. Sci. Technol., A 31, 030801 (2013). https://doi.org/10.1116/1.4794357, Google ScholarCrossref, ISI
  10. 10. M. Müser, Phys. Rev. B 84, 125419 (2011). https://doi.org/10.1103/physrevb.84.125419, Google ScholarCrossref
  11. 11. E. Gnecco, R. Bennewitz, T. Gyalog, C. Loppacher, M. Bammerlin, E. Meyer, and H. Guntherodt, Phys. Rev. Lett. 84, 1172 (2000). https://doi.org/10.1103/physrevlett.84.1172, Google ScholarCrossref
  12. 12. D. Maugis, J. Colloid Interface Sci. 150, 243 (1992). https://doi.org/10.1016/0021-9797(92)90285-t, Google ScholarCrossref
  13. 13. R. W. Carpick, D. F. Ogletree, and M. Salmeron, J. Colloid Interface Sci. 211, 395 (1999). https://doi.org/10.1006/jcis.1998.6027, Google ScholarCrossref
  14. 14. E. Liu, B. Blanpain, and J. P. Celis, Wear 192, 141 (1996). https://doi.org/10.1016/0043-1648(95)06784-1, Google ScholarCrossref, ISI
  15. 15. R. G. Cain, M. G. Reitsma, S. Biggs, and N. W. Page, Rev. Sci. Instrum. 72, 3304 (2001). https://doi.org/10.1063/1.1386631, Google ScholarScitation, ISI
  16. 16. J. E. Sader and C. P. Green, Rev. Sci. Instrum. 75, 878 (2004). https://doi.org/10.1063/1.1667252, Google ScholarScitation, ISI
  17. 17. M. A. Lantz, S. J. O’Shea, A. C. F. Hoole, and M. E. Welland, Appl. Phys. Lett. 70, 970 (1997). https://doi.org/10.1063/1.118476, Google ScholarScitation, ISI
  18. 18. R. W. Carpick, D. F. Ogletree, and M. Salmeron, Appl. Phys. Lett. 70, 1548 (1997). https://doi.org/10.1063/1.118639, Google ScholarScitation, ISI
  19. 19. D. B. Asay and S. H. Kim, Rev. Sci. Instrum. 77, 043903 (2006). https://doi.org/10.1063/1.2190210, Google ScholarScitation, ISI
  20. 20. R. J. Cannara, M. Eglin, and R. W. Carpick, Rev. Sci. Instrum. 77, 053701 (2006). https://doi.org/10.1063/1.2198768, Google ScholarScitation, ISI
  21. 21. Q. Li, K.-S. Kim, and A. Rydberg, Rev. Sci. Instrum. 77, 065105 (2006). https://doi.org/10.1063/1.2209953, Google ScholarScitation, ISI
  22. 22. E. Tocha, H. Schönherr, and G. J. Vancso, Langmuir 22, 2340 (2006). https://doi.org/10.1021/la052969c, Google ScholarCrossref
  23. 23. M. Munz, J. Phys. D: Appl. Phys. 43, 063001 (2010). https://doi.org/10.1088/0022-3727/43/6/063001, Google ScholarCrossref, ISI
  24. 24. N. Mullin and J. K. Hobbs, Rev. Sci. Instrum. 77, 113703 (2014). https://doi.org/10.1063/1.4901221, Google ScholarScitation
  25. 25. J. E. Sader, Rev. Sci. Instrum. 74, 2438 (2003). https://doi.org/10.1063/1.1544421, Google ScholarScitation, ISI
  26. 26. N. A. Burnham, X. Chen, C. S. Hodges, G. A. Matei, E. J. Thoreson, C. J. Roberts, M. C. Davies, and S. J. B. Tendler, Nanotechnology 14, 1 (2003). https://doi.org/10.1088/0957-4484/14/1/301, Google ScholarCrossref, ISI
  27. 27. C. P. Green, H. Lioe, J. P. Cleveland, R. Proksch, P. Mulvaney, and J. E. Sader, Rev. Sci. Instrum. 75, 1988 (2004). https://doi.org/10.1063/1.1753100, Google ScholarScitation, ISI
  28. 28. D. F. Ogletree, R. W. Carpick, and M. Salmeron, Rev. Sci. Instrum. 67, 3298 (1996). https://doi.org/10.1063/1.1147411, Google ScholarScitation, ISI
  29. 29. A. Labuda, J. Cleveland, N. Geisse, M. Kocun, B. Ohler, R. Proksch, M. Viani, and D. Walters, Microsc. Anal. 28, 23 (2014). Google Scholar
  30. 30. A. Labuda and R. Proksch, Appl. Phys. Lett. 106, 253103 (2015). https://doi.org/10.1063/1.4922210, Google ScholarScitation, ISI
  31. 31. J. E. Sader, J. A. Sanelli, B. D. Adamson, J. P. Monty, X. Wei, S. A. Crawford, J. R. Friend, I. Marusic, P. Mulvaney, and E. J. Bieske, Rev. Sci. Instrum. 83, 103705 (2012). https://doi.org/10.1063/1.4757398, Google ScholarScitation, ISI
  32. 32. A. Labuda, M. Kocun, M. Lysy, T. Walsh, J. Meinhold, T. Proksch, W. Meinhold, C. Anderson, and R. Proksch, Rev. Sci. Instrum. 87, 073705 (2016). https://doi.org/10.1063/1.4955122, Google ScholarScitation, ISI
  33. 33. A. Labuda, Rev. Sci. Instrum. 87, 033704 (2016). https://doi.org/10.1063/1.4943292, Google ScholarScitation, ISI
  34. 34. R. Gates, W. Osborn, and G. Shaw, Nanotechnology 26, 235704 (2015). https://doi.org/10.1088/0957-4484/26/23/235704, Google ScholarCrossref
  35. 35. R. S. Gates, W. A. Osborn, and J. R. Pratt, Nanotechnology 24, 255706 (2013). https://doi.org/10.1088/0957-4484/24/25/255706, Google ScholarCrossref
  36. 36. W. C. Young and R. G. Budynas, Roark’s Formulas for Stress and Strain, 7th ed. (McGraw-Hill, New York, 1975). Google Scholar
  37. 37. M. J. Higgins, R. Proksch, J. E. Sader, M. Polcik, S. Mc Endoo, J. P. Cleveland, and S. P. Jarvis, Rev. Sci. Instrum. 77, 013701 (2006). https://doi.org/10.1063/1.2162455, Google ScholarScitation, ISI
  38. 38. K. Wagner, P. Cheng, and D. Vezenov, Langmuir 27, 4635 (2011). https://doi.org/10.1021/la1046172, Google ScholarCrossref
  39. 39. A. Labuda, W. Paul, B. Pietrobon, R. B. Lennox, P. H. Gruätter, and R. Bennewitz, Rev. Sci. Instrum. 81, 083701 (2010). https://doi.org/10.1063/1.3470107, Google ScholarScitation, ISI
  40. 40. A. Yurtsever, A. M. Gigler, E. Macias, and R. W. Stark, Appl. Phys. Lett. 91, 253120 (2007). https://doi.org/10.1063/1.2826285, Google ScholarScitation, ISI
  41. 41. R. Meyer, E. Hug, and H. Bennewitz, Scanning Probe Microscopy: The Lab on a Tip (Springer, 2004), Vol. 45. Google ScholarCrossref
  42. 42. A. Khan, J. Philip, and P. Hess, J. Appl. Phys. 95, 1667 (2004). https://doi.org/10.1063/1.1638886, Google ScholarScitation, ISI
  43. 43. M. J. Lachut and J. E. Sader, Phys. Rev. Lett. 99, 206102 (2007). https://doi.org/10.1103/physrevlett.99.206102, Google ScholarCrossref
  44. 44. J. E. Sader, J. Appl. Phys. 84, 64 (1998). https://doi.org/10.1063/1.368002, Google ScholarScitation, ISI
  45. 45. A. Labuda and P. H. Grütter, Rev. Sci. Instrum. 82, 013704 (2011). https://doi.org/10.1063/1.3503220, Google ScholarScitation, ISI
  46. 46. J. Tamayo, V. Pini, P. Kosaka, N. F. Martinez, O. Ahumada, and M. Calleja, Nanotechnology 23, 315501 (2012). https://doi.org/10.1088/0957-4484/23/31/315501, Google ScholarCrossref
  47. 47. R. Proksch, J. Appl. Phys. 118, 072011 (2015). https://doi.org/10.1063/1.4927809, Google ScholarScitation, ISI
  48. 48. H. Chen and T. Filleter, Nanotechnology 26, 135702 (2015). https://doi.org/10.1088/0957-4484/26/13/135702, Google ScholarCrossref
  49. 49. M. Daly, C. Cao, H. Sun, Y. Sun, T. Filleter, and C. V. Singh, ACS Nano 10, 1939 (2016). https://doi.org/10.1021/acsnano.5b05771, Google ScholarCrossref
  50. 50.This equation was derived by equating the strain energy integrals of both mode shapes to the respective stiffnesses by the equipartition theorem and then taking the ratio of both equations. Combining both equations into a single one is valid because both mode shapes share the same coordinate system.
  51. 51.At small amplitudes, the digital-to-analog digitization of the cantilever deflection signal by the Polytec LDV causes a bias in the estimation of the overall thermal fluctuations measured by the Cypher AFM electronics.
  1. © 2018 Author(s). Published by AIP Publishing.