Point mass approximation. Rigid beam mechanics. spring constant k N effective mass m e. Simple Harmonic Motion.. m e z = - k N z

Transcription

1 Free end Rigid beam mechanics Fixed end think of cantilever as a mass on a spring Point mass approximation z F Hooke s law k N = F / z This is beam mechanics, standard in engineering textbooks. For a rectangular cross section we find k N = E w t 3 / 4 l 3 w t l spring constant k N effective mass m e Simple Harmonic Motion.. m e z = - k N z Implies resonant frequency of oscillation 0 = (k N / m e ) 1/2 m e Effective mass ~ 1/3 actual mass need to solve beam equation to work it out. k N z 1 2

2 Spring constant Cantilever constraints - I Interatomic bond - spring constant Hydrogen bond - spring constant Force between two electrons? Conclusion spring constant k N ~ nn nm -1 Resonant frequency ~ 10 times higher than scan rate * no. of points (bandwidth), ~ 10 times higher than building vibrations i.e. > khz Given a density of Silicon of = 2330 kg m -3 this gives a volume of V ~. m 3 So microfabrication required: standard dimensions are t ~ 2 m, w ~ 30 m, l ~ 100 m Cantilever constraints - II So for the spring constants k N ~ nn nm-1 effective mass should be m e < to keep the resonance frequency high enough Thickness of the cantilever is the critical parameter: hardest to control and measure, but has a big effect k N = E w t 3 / 4 l 3 0 = (k N / 0.24 m c ) 1/2 / 2 3 4

3 Driven oscillator In reality we have driving force in TM, and damping Steady state response z This is easily solved (1 st year physics problem) Experimental Results transient term steady state term 5 6

4 Transient response Low and high Q Typical Q in water < 10 air ~ 100 vacuum > Low High Amplitude responds on a time scale of 1 / For a cantilever with frequency 100 khz and Q of 300 this is a time of ~ 1ms. Bandwidth of measurement is thus ~ 1 khz. Scan speed is thus limited to about 1 Hz Phase and frequency respond on a time scale of 1/ for low Q dynamics of oscillation poorly defined, sensitivity poor and interaction forces high for Q > 50 the amplitude of oscillation on resonance is given by A cant = A drive Q for high Q bandwidth is too low for resonance frequency 100 khz, and Q 30000, transient decay time is 0.1 s, bandwidth 10 Hz implies scan rate of < 0.01 Hz or image time ~ 10 hrs (512 by 512) 7 8

5 Effect of F ts If long range force is just a perturbation it effectively alters the spring constant Tip-surface forces Strength, range and direction of forces important Linear superposition so all forces important AFM detects total force (and force gradient) Force gradient leads to a shift in resonance frequency Shift in resonance frequency alters phase (at set ) force separation as in MHB 9 10

6 F ts - short range chemical forces attractive (bonding) and repulsive (ion cores) Decay length in the angstroms Forces ~ nn per interacting atom Model interactions such as Lennard Jones Morse F ts - van der Waals short range and long range (retarded) for AFM can approximate by a sphere approaching a semi-infinite body giving A tip of radius 30 nm, 0.5 nm from the surface the force in vacuum is F ts ~ 2nN Very dependent on medium, for example greatly reduced in water 11 12

7 F ts - electrostatic / magnetic F ts - capillary Both long range forces Trapped charges, work function differences, applied potentials, surface charges all sources of potential differences between tip and surface. Typical values ~ N Magnetic force requires magnetic tip Typical values ~ N Sharp point acts as condensation nucleus Tip to surface water meniscus formed Force depends on partial pressure of water vapour and tip and surface contact angles Typical forces in ambient ~ nn 13 14

8 Contact areas Contact mechanics F ts nonconservative For conservative forces no energy dissipated in the sample Nonconservative forces result in hysteresis in F ts (z) Examples include.. Contact pressures Deformation of the tip Hertz model, DMT, JKR If motion is sinusoidal, oscillation is stable, and the amplitude is fixed the phase response is determined by energy dissipation sample stiffness adhesion topography See Garcia 2002 for more details 15 16

9 Oscillation constraints Forces and tm amplitudes jump to contact occurs if avoided by high enough oscillation amplitudes if energy is dissipated (hysteresis in F ts (z)) PROBLEM DUE TO NONMONOTONIC DISTANCE DEPENDENCE OF FORCE Average F ts over oscillation Phase -> ~ average force gradient Large A => more likely to have a larger repulsive interaction - more stable in this regime Small A can be more stable in attractive regime typical oscillation amplitudes ~ 5 50 nm, k ~ nn nm -1 Veeco SPM guide 17 18

10 tm force curve Phase bistability what it means how can you see it effect inaccurate topography what to do if you see it? Garcia and San Paolo, PRB 61, R13381 (2000) 19 20

11 Thermal noise Harmonic oscillator =>Thermal noise Equipartition theory Sensitivity / Resolution Ultimately will be limited by thermal oscillations in the cantilever Contact mode force sensitivity thermal noise at best, position ~ k -1/2 nm Dynamic mode position as above, force gradient sensitivity 21 22

12 Beyond point mass Calibrating the spring constant only first resonance Beam mechanics gives higher order resonances (1, 6.25, 17.5, ) Analytical methods continuous approximation FEMLAB Thermal noise Added mass Sader method Reference cantilever NEMS based Standardised cantilevers

13 Thermal noise Sader Method Sources of error Accuracy Caution on using the in-built thermal tune John Sader University of Melbourne Principle frequency and spring constant depend on thickness, measure both Method measure length, width, frequency and Q alibration.html Advantages Accuracy 25 26

14 To be aware of References and links spring constant force at tip or end higher modes relation between deflection measured and deflection in theory temperature effects tip movement in force curves force direction boundary conditions tilt of cantilever. SPM The lab on a tip SPM and Spectroscopy, Wiesendanger Giessibl 2003 Garcia 2002 Dietler 1999 John Sader s website Veeco application note on spring constant calibration AN