1. Dissipation in Nanomechanical Resonators Peter Kirton

2. Overview Part I: Theory Part I: Theory Introduce the Euler-Bernoulli theory of beam vibrationsIntroduce the Euler-Bernoulli theory of beam vibrations Thermoelastic DampingThermoelastic Damping Zener’s model Zener’s model Lifshitz and Roukes’ solution Lifshitz and Roukes’ solution Regimes where this is not applicable Regimes where this is not applicable Part II: Experimental Results Part II: Experimental Results How does size affect the achievable quality factor?How does size affect the achievable quality factor? Review of some recent experimental resultsReview of some recent experimental results

3. Specification of the system Beam fixed at both ends Length L Cross section a × b Relaxed value of Young’s Modulus, E R Density, ρ Heat Capacity, C P Coefficient of linear expansion,α 2 nd moment of inertia I y Use the Euler-Bernoulli approximation: a,b<<L Allows us to neglect the effect of shear, etc.

4. Equation of motion Newton’s Laws give the equation of motion for the displacement of the beam Assume displacement is harmonic in time   Equation of motion reduces to 4 th order ODE with general solution 4 th order ODE with general solution

5. Solutions Boundary conditions for a beam fixed at both ends Boundary conditions for a beam fixed at both ends So the solution to the equation of motion becomes So the solution to the equation of motion becomes And β n satisfies We can simply find the frequency of the n th mode from the known properties of the beam We can simply find the frequency of the n th mode from the known properties of the beam e.g.

6. Damping Clamping Losses Clamping Losses Beam is fixed to a supportBeam is fixed to a support Lattice Defects Lattice Defects Impure crystalsImpure crystals Phonon Losses Phonon Losses High temperature phonon interactions Thermoelastic Thermoelastic Internal friction

7. Quality Factors Quantify the amount of damping a process creates by its associated quality factor - Q Quantify the amount of damping a process creates by its associated quality factor - Q Can then sum the losses due to many different sources to find the total Q Can then sum the losses due to many different sources to find the total Q

8. The process of thermoelastic damping One side of the beam compressed - heated One side of the beam compressed - heated Other side stretched - cooled Other side stretched - cooled Creates a temperature gradient across the beam Creates a temperature gradient across the beam Energy loss - damping Energy loss - damping

9. Zener’s Model Consider the beam to made from an anelastic solid Consider the beam to made from an anelastic solid Assume stress and strain to be harmonic in time Assume stress and strain to be harmonic in time C. Zener, Phys. Rev. 52, 230 (1937), C. Zener, Phys. Rev. 53, 90 (1938). Replaced by: Modify Hooke’s Law to take account of stress and strain being out of phase

10. Quality factor from Zener’s model Quality Factor can be defined as Quality Factor can be defined as Which when substituted into Zener’s model gives the Lorentzian Which when substituted into Zener’s model gives the Lorentzian Where: All known quantities so we can calculate and test this All known quantities so we can calculate and test this

11. Lifshitz and Roukes’ solutions Introduce full, coupled equations of motion for the stress and temperature fields of the beam Introduce full, coupled equations of motion for the stress and temperature fields of the beam They neglect temperature gradients along the rod (z- direction) and so find the exact solution when They neglect temperature gradients along the rod (z- direction) and so find the exact solution when Again we can measure all these quantities and so can predict the thermoelastic limit of the quality factor. Again we can measure all these quantities and so can predict the thermoelastic limit of the quality factor. R. Lifshitz and M. L. Roukes, Phys. Rev. B 61, 5600 (2000)

12. Comparison to simulation results

13. Physical Interpretation Low frequencies: large temperature gradients can’t form, beam is Isothermal Low frequencies: large temperature gradients can’t form, beam is Isothermal High frequencies: thermal diffusion doesn’t have time to take place, beam is adiabatic High frequencies: thermal diffusion doesn’t have time to take place, beam is adiabatic Intermediate frequencies: thermal and mechanical timescales are similar: thermoelastic damping becomes important Intermediate frequencies: thermal and mechanical timescales are similar: thermoelastic damping becomes important isothermal adiabatic

14. Problems with the Theory Make the beam too small and the simulation results start to diverge Make the beam too small and the simulation results start to diverge Can bring the results back together by reducing the diffusivity, χ Can bring the results back together by reducing the diffusivity, χ This means that for very small beams conduction across the rod becomes important This means that for very small beams conduction across the rod becomes important

15. More Difficulties Lifshitz and Roukes’ ignored diffusion along the length of the rod Lifshitz and Roukes’ ignored diffusion along the length of the rod Solution only works if the ends are perfectly insulating Solution only works if the ends are perfectly insulating If we attach heat baths at the ends of the rod: If we attach heat baths at the ends of the rod:

16. How to approach solving these problems Add in the diffusion term for conduction along the length of the rod Add in the diffusion term for conduction along the length of the rod Solve the new coupled equations of motion Solve the new coupled equations of motion More difficult than it sounds! More difficult than it sounds! Work still ongoing…. Work still ongoing….

17. Part II: Experimental Results A recent review paper by Ekinki and Roukes compiled quality factor data A recent review paper by Ekinki and Roukes compiled quality factor data Found that quality factor generally decreases with ‘size’ of the resonator Found that quality factor generally decreases with ‘size’ of the resonatorBUT Results taken from many different sources using different types of resonator Results taken from many different sources using different types of resonator Is volume really a good quantity to use? Is volume really a good quantity to use? K. L. Ekinci and M. L. Roukes, Review of Scientific Instruments, 76, 061101 (2005)

18. Kleinman et al. Torsional oscillators, length 1.91cm Torsional oscillators, length 1.91cm Quality factors at low temperatures Quality factors at low temperatures Q dependence on resonance mode Q dependence on resonance mode Due to defects in silicon wafers? Due to defects in silicon wafers? R. N. Kleiman, G. Agnolet, and D. J. Bishop, Phys. Rev. Lett. 59, 2079 (1987).

19. Klitsner and Pohl 2cm long torsional oscillators 2cm long torsional oscillators Temperature dependence of Q over a larger range Temperature dependence of Q over a larger range Fundamental mode only Fundamental mode only Increase in Q when heated? Increase in Q when heated? T. Klitsner and R. O. Pohl, Phys. Rev. B 36, 6551 (1987).

20. Greywall et Al. Beams of length 550μm Beams of length 550μm Q measured at very low temperatures Q measured at very low temperatures Oscillatory behaviour Oscillatory behaviour Effect reduced by magnetic field Effect reduced by magnetic field D. S. Greywall, B. Yurke, P. A. Busch, and S. C. Arney, Europhys. Lett. 34, 37 (1996).

21. Mihailovich and Parpia Torsional oscillators, 200μm thick Torsional oscillators, 200μm thick Various levels of Boron doping were used Various levels of Boron doping were used Q recorded at low temperatures for different doping levels. Q recorded at low temperatures for different doping levels. Doping effect reduced at higher temperatures Doping effect reduced at higher temperatures Increased doping R. E. Mihailovich and J. M. Parpia, Phys. Rev. Lett. 68, 3052 (1992).

22. Carr et Al. Beams length 2-8μm long Beams length 2-8μm long Strong linear dependence of Q on surface area to volume ratio Strong linear dependence of Q on surface area to volume ratio Indicates that surface effects can considerably reduce Q Indicates that surface effects can considerably reduce Q D. W. Carr, S. Evoy, L. Sekaric, H. G. Craighead, and J. M. Parpia, Applied Physics Letters 75, 920 (1999)

23. Conclusions from these results Many different types of behaviour measured with many variables Many different types of behaviour measured with many variables The volume of a resonator isn’t a good a measure of it’s dissipative qualities The volume of a resonator isn’t a good a measure of it’s dissipative qualities Thermoelastic, clamping losses and other forms of dissipation are more sensitive to the thickness Thermoelastic, clamping losses and other forms of dissipation are more sensitive to the thickness

24. Putting all these (and more) Together Forbidden region? Anomalous point: S.S. Verbridge et Al., J. App. Phys, 99, 124304, (2006)

25. Conclusions Euler-Bernoulli Theory allows us to predict the frequency of beams, ignoring thermal effects Euler-Bernoulli Theory allows us to predict the frequency of beams, ignoring thermal effects Lifshitz and Roukes’ solution allows accurate prediction of thermoelastic damping in most circumstances. Lifshitz and Roukes’ solution allows accurate prediction of thermoelastic damping in most circumstances. But this is still not a fully general theory… But this is still not a fully general theory… Can’t include conduction at the ends of the beamCan’t include conduction at the ends of the beam Breaks down if the beam is made too smallBreaks down if the beam is made too small Recent measurements are inconclusive about Q behaviour of small resonators, with some contradictory results Recent measurements are inconclusive about Q behaviour of small resonators, with some contradictory results Compilation of many sets of results shows a region where no Q values have been measured Compilation of many sets of results shows a region where no Q values have been measured Still lots of work needed to decide exactly what factors are important to energy loss in these nanomechanical resonators Still lots of work needed to decide exactly what factors are important to energy loss in these nanomechanical resonators