Dissipation in Nanomechanical Resonators

1. Dissipation in Nanomechanical Resonators Peter Kirton

2. Overview • Part I: Theory • Introduce the Euler-Bernoulli theory of beam vibrations • Thermoelastic Damping • Zener’s model • Lifshitz and Roukes’ solution • Regimes where this is not applicable • Part II: Experimental Results • How does size affect the achievable quality factor? • Review 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, ER • Density, ρ • Heat Capacity, CP • Coefficient of linear expansion,α • 2nd moment of inertia Iy • 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 4th order ODE with general solution • Equation of motion reduces to

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

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

7. Quality Factors • 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

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

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

10. Quality factor from Zener’s model Quality Factor can be defined as Which when substituted into Zener’s model gives the Lorentzian Where: 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 • 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. 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 • 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 isothermal adiabatic

14. Problems with the Theory • Make the beam too small and the simulation results start to diverge • Can bring the results back together by reducing the diffusivity, χ • 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 • Solution only works if the ends are perfectly insulating • 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 • Solve the new coupled equations of motion • More difficult than it sounds! • Work still ongoing….

17. Part II: Experimental Results • A recent review paper by Ekinki and Roukes compiled quality factor data • Found that quality factor generally decreases with ‘size’ of the resonator BUT • Results taken from many different sources using different types of resonator • 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 • Quality factors at low temperatures • Q dependence on resonance mode • 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 • Temperature dependence of Q over a larger range • Fundamental mode only • 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 • Q measured at very low temperatures • Oscillatory behaviour • 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 • Various levels of Boron doping were used • Q recorded at low temperatures for different doping levels. • 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 • Strong linear dependence of Q on surface area to volume ratio • 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 • 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

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 • Lifshitz and Roukes’ solution allows accurate prediction of thermoelastic damping in most circumstances. • But this is still not a fully general theory… • Can’t include conduction at the ends of the beam • Breaks down if the beam is made too small • 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 • Still lots of work needed to decide exactly what factors are important to energy loss in these nanomechanical resonators