1. Atomic force microscopy Jiří Boldyš

2. Outline Motivation Minisurvey of scanning probe microscopies Imaging principles Ideas about application of moment invariants Alternative reconstruction approach Image artifacts

3. Classification of blur kernel symmetries n-fold circular symmetry (Cn symmetry) Dihedral symmetry (Dn) Radial symmetry

4. Common blurs Atmospheric – radial symmetry Out-of-focus – radial, cyclic or dihedral symmetry Motion – central symmetry

5. What invariants we have Model: g = f * h I ( f ) = I ( g ) Invariance x discriminability Invariants to kernels with Cn and Dn symmetry Potentially we can, we have not done that - arbitrary symmetry, arbitrary dimension

6. Other potential applications Electron microscopy? – N-fold symmetrical correction elements Atomic force microscopy (AFM)? … But – Is there a convolution???

7. Scanning probe microscopy - classification Scanning tunneling microscopy - STM Atomic force microscopy - AFM Electric force microscopy - EFM Magnetic force microscopy - MFM Scanning near-field optical microscopy - SNOM...

8. Scanning tunneling microscopy Mironov: Fundamentals of scanning probe microscopy, 2004

9. Scanning tunneling microscopy 1981 – Swiss scientists Gerd Binnig and Heinrich Rohrer Atomic resolution, simple 1986 – Nobel prize Chen: Introduction to scanning tunneling microscopy, 1993

10. Scanning tunneling microscopy The first demonstration of the atomic-resolution capability of STM – Si(111)-7x7, Binnig, Rohrer, Gerber, Weibel, 1983 Chen: Introduction to scanning tunneling microscopy, 1993

11. Scanning tunneling microscopy Chen: Introduction to scanning tunneling microscopy, 1993

12. Atomic force microscopy Mironov: Fundamentals of scanning probe microscopy, 2004 1986, Binnig, Quate, Gerber 1989 – the first commercially available AFM

13. Magnetic force microscopy Mironov: Fundamentals of scanning probe microscopy, 2004 Local magnetic properties AFM + tip covered by a layer of ferromagnetic material with specific magnetization

14. Atomic force microscopy in detail Mironov: Fundamentals of scanning probe microscopy, 2004 Forces can be explained by e.g. van der Waals forces – approximated by Lennard-Jones potential

15. Tip – sample force Mironov: Fundamentals of scanning probe microscopy, 2004 Energy of interaction Force – normal + lateral component Corresponds to deflections of an elastic cantilever

16. STM vs. AFM Giessibl, Quate: Physics Today, 2006 STM Tunneling current drops off exponentially  spatially confined to the frontmost atom of the tip and surface Distance dependence is monotonic  simple feedback scheme Modest experimental means, excellent SNR AFM Force – short range + long range – less tractable as a feedback signal Not monotonic with distance

17. Beam-bounce technique Mironov: Fundamentals of scanning probe microscopy, 2004

18. Feedback system Mironov: Fundamentals of scanning probe microscopy, 2004

19. Examples of cantilevers Images: Mironov, Fundamentals of scanning probe microscopy, 2004 Si3N4, Si Different spring constants and resonant frequencies

20. Methods used to acquire images Mironov: Fundamentals of scanning probe microscopy, 2004 Contact vs. non-contact modes Contact modes attractive or repulsive Balance between atomic and elastic forces Small stiffness – high sensitivity, gentle to the sample Tip breakage, surface damages Not suitable for soft samples (biological) Constant force Constant average distance

21. AFM image acquisition at constant force Mironov: Fundamentals of scanning probe microscopy, 2004

22. AFM image acquisition at average distance Mironov: Fundamentals of scanning probe microscopy, 2004

23. Force-distance curves – elastic interaction Mironov: Fundamentals of scanning probe microscopy, 2004

24. Force-distance curves – plastic interaction Mironov: Fundamentals of scanning probe microscopy, 2004

25. Forced oscillations of a cantilever Mironov: Fundamentals of scanning probe microscopy, 2004 Better for soft samples Reduce mechanical influence of the tip on the surface Possible to investigate more surface properties Piezo-vibrator Motion equation

26. Forced oscillations of a cantilever Mironov: Fundamentals of scanning probe microscopy, 2004

27. Contactless mode of AFM cantilever oscillations Mironov: Fundamentals of scanning probe microscopy, 2004 Small forced oscillations amplitude – 1nm Close to surface – additional force Small oscillation around z0 Presence of a gradient in the tip-surface interaction force  Additional shift of the amplitude and phase response curves Additional phase shift  phase contrast AFM image

28. Contactless mode of AFM cantilever oscillations Mironov: Fundamentals of scanning probe microscopy, 2004

29. Semi-contact mode of AFM cantilever oscillations Mironov: Fundamentals of scanning probe microscopy, 2004 Before – high sensitivity and stability feedback required In practice often semi-contact mode Excited near resonance frequency, amplitude 10-100nm Working point:

30. Semi-contact mode of AFM cantilever oscillations Mironov: Fundamentals of scanning probe microscopy, 2004

31. Frequency modulation AFM Si(111) Reactive surface →  Dynamic mode  Ultrahigh vacuum  FM-AFM – frequency modulation – introduced in 1991  First with commercial cantilevers with a limited range of spring constants  Strong bonding energy Si-Si  large amplitude of vibrations 34nm  no atomic resolution   small amplitudes  stiff cantilevers  dramatic improvement in spatial resolution Giessibl, Quate: Physics Today, 2006

32. From static to dynamic mode Static approach still in use  Materials in liquids  Tip subject to wear  Large lateral forces  Absolute force measurements are noisy Amplitude modulation AFM  Driven near fundamental resonance frequency  Less noise  Sensing variations in amplitude  Lateral forces minimized – broken contacts Giessibl, Quate: Physics Today, 2006

33. From static to dynamic mode Frequency modulation AFM  Even less noisy  Fixed amplitude  Frequency as a feedback signal  Lateral forces minimized – broken contacts  Average tip-sample force gradient  Frequency shift  Further improvement – exploiting signal proportional to higher-order derivative – better spatial resolution  And next – reconstruction using the frequency shift and higher-harmonic components of the cantilever vibrations  Higher harm. can be viewed as a convolution of the nth-order derivative of the force with some weight function Giessibl, Quate: Physics Today, 2006

34. Revealing angular symmetry of chemical bonds Combined STM and FM AFM Angular dependence of chemical bonding forces between CO on copper surface Cu(111) and the terminal atom of metallic tip Forces depend also on angles between atoms Other opinions: feedback artifact or multiple-atom tips 3D force spectroscopy used Welker, Giessibl, Science, 2012

35. Revealing angular symmetry … Welker, Giessibl, Science, 2012

36. Silicon (111)-(7x7) surface Giessibl, Hembacher, Bielefeldt, Mannhart, Science, 2000

37. Non-expert ideas in the field of AFM Two ways of imaging – Tip imaging – Surface imaging Surface symmetry Tip symmetry Do we need to register (align) two blurred images, or one sharp and one blurred? Images with different class of blur – generates new mathematical task for us

38. Registration of images blurred by kernels with different symmetry Example: – Tip imaging by surface with 4-fold (C4) symmetry … – Followed by tip imaging (the same one) by surface with 3-fold symmetry – Both kill different frequencies – together we might reconstruct them more easily and precisely Another example: – Scanning the same surface with 4-fold symmetrical and 3-fold symmetrical tip (due to crystallic structure)

39. Is there any convolution? We are not sensing surface height (z-coordinate) - we are sensing force / potential energy Can potential energy be calculated as a 3D convolution? What we measure is a 2D surface in the 3D potential map Probe influences atom distribution We are sensing force through approx. linear dependence F(z) What if we do not measure pure force but rather frequency shift or higher order force derivatives?

40. Mathematical morphology based reconstruction Intuitively degradation corresponds to the dilation operation in mathematical morphology Why not if the region where the forces are significant is << tip size Villarrubia, J. Res. Natl. Inst. Stand. Technol., 1997

41. Critical dimension AFM Higher throughput during quality control Not so ambitious resolution Dahlen et al., Veeco

42. AFM imaging artifacts West, Starostina, Pacific Nanotechnology, Inc.

43. AFM imaging artifacts West, Starostina, Pacific Nanotechnology, Inc.

44. AFM imaging artifacts West, Starostina, Pacific Nanotechnology, Inc.

45. AFM imaging artifacts West, Starostina, Pacific Nanotechnology, Inc.

46. AFM imaging artifacts Mates, Summer school of SPM microscopy, 2007

47. Thank you! Recommended reading: Giessibl, Quate: Physics Today, Exploring the nanoworld with atomic force microscopy, 2006