Dissipation in Dissipation in Nanomechanical Nanomechanical

ResonatorsResonators

Peter KirtonPeter Kirton

OverviewOverview Part I: TheoryPart I: Theory

• Introduce the Euler-Bernoulli theory of beam Introduce the Euler-Bernoulli theory of beam vibrationsvibrations

• Thermoelastic DampingThermoelastic Damping Zener’s modelZener’s model Lifshitz and Roukes’ solutionLifshitz and Roukes’ solution Regimes where this is not applicableRegimes where this is not applicable

Part II: Experimental ResultsPart II: Experimental Results• How does size affect the achievable quality How does size affect the achievable quality

factor?factor?• Review of some recent experimental resultsReview of some recent experimental results

Specification of the systemSpecification 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.

Equation of motionEquation of motion

2

2

4

4

t

Uab

z

UEI y

tin

nezUtzU )(),(

ny

nn U

EI

ab

dz

Ud

24

4

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

44thth order ODE with general order ODE with general solutionsolution

zdzczbzaU nnnnnnnnn sinhcoshsincos

2

14

1

ny

n EI

ab

SolutionsSolutions Boundary conditions Boundary conditions

for a beam fixed at for a beam fixed at both endsboth ends

So the solution to the So the solution to the equation of motion equation of motion becomesbecomes

0,0

Lz

n

dz

dU0

,0

LznU

zzbzzaU nnnnnnn sinhsincoshcos

And βn satisfies ...00.11,85.7,73.41coshcos LLL nnn

We can simply find the frequency We can simply find the frequency of the nof the nth th mode from the known mode from the known properties of the beamproperties of the beam

2

1

2

2

1

73.4

ab

EI

Ly

e.g.e.g.

DampingDamping

Clamping LossesClamping Losses• Beam is fixed to a Beam is fixed to a

supportsupport

Lattice DefectsLattice Defects• Impure crystalsImpure crystals

Phonon LossesPhonon Losses• High temperature High temperature

phonon interactionsphonon interactions

ThermoelasticThermoelastic• Internal frictionInternal friction

Quality FactorsQuality Factors

Quantify the Quantify the amount of damping amount of damping a process creates a process creates by its associated by its associated quality factorquality factor - Q - Q

Can then sum the Can then sum the losses due to many losses due to many different sources to different sources to find the total Qfind the total Q

W

WQ

02

0

1

2 W

WQ i

i

tot

The process of thermoelastic The process of thermoelastic dampingdamping

One side of the beam One side of the beam compressed - compressed - heatedheated

Other side stretched - Other side stretched - cooledcooled

Creates a temperature Creates a temperature gradient across the gradient across the beambeam

Energy loss - Energy loss - dampingdamping

Zener’s ModelZener’s Model Consider the beam to Consider the beam to

made from an made from an anelasticanelastic solidsolid

Assume stress and Assume stress and strain to be harmonic strain to be harmonic in timein time

tiet 0)( tiet 0)( )( RE

C. Zener, Phys. Rev. 52, 230 (1937), C. Zener, Phys. Rev. 53, 90 (1938).

Replaced byReplaced by::

Modify Hooke’s Law to take account of stress and strain being out of phase

Quality factor from Zener’s modelQuality factor from Zener’s model

)Re(

)Im(1

R

R

E

EQ

21

)(1

EQ

P

R

R

RUE C

TE

E

EE

02

Quality FactorQuality Factor can be can be defined asdefined as

Which when Which when substituted into substituted into Zener’s model gives Zener’s model gives the the LorentzianLorentzian

2

2aWhere:Where:

All known quantities All known quantities so we can calculate so we can calculate and test thisand test this

Lifshitz and Roukes’ solutionsLifshitz and Roukes’ solutions Introduce full, coupled Introduce full, coupled

equations of motion for the equations of motion for the stressstress and and temperaturetemperature fields of the beamfields of the beam

They neglect temperature They neglect temperature gradients along the rod (z-gradients along the rod (z-direction) and so find the direction) and so find the exact solution whenexact solution when

Again we can measure all Again we can measure all these quantities and so these quantities and so can predict the can predict the thermoelastic limit of the thermoelastic limit of the quality factor.quality factor.

2

2

2

2

4

4

t

Uab

z

IE

z

UEI T

y

2

22

z

U

tx

tE

2

22

x

coscosh

sinsinh6632

1EQ

2

a

R. Lifshitz and M. L. Roukes, Phys. Rev. B 61, 5600 (2000)

Comparison to simulation resultsComparison to simulation results

Physical InterpretationPhysical Interpretation Low frequencies: large Low frequencies: large

temperature gradients temperature gradients can’t form, beam is can’t form, beam is IsothermalIsothermal

High frequencies: thermal High frequencies: thermal diffusion doesn’t have time diffusion doesn’t have time to take place, beam is to take place, beam is adiabaticadiabatic

Intermediate frequencies: Intermediate frequencies: thermal and mechanical thermal and mechanical timescales are similar: timescales are similar: thermoelastic damping thermoelastic damping becomes importantbecomes important

isothermalisothermal

adiabaticadiabatic

1

Problems with the TheoryProblems with the Theory Make the beam too Make the beam too

small and the small and the simulation results start simulation results start to divergeto diverge

Can bring the results Can bring the results back together by back together by reducing the reducing the diffusivity, diffusivity, χχ

This means that for This means that for very small beams very small beams conduction across the conduction across the rod becomes rod becomes important important

More DifficultiesMore Difficulties Lifshitz and Roukes’ Lifshitz and Roukes’

ignored diffusion along ignored diffusion along the length of the rodthe length of the rod

Solution only works if Solution only works if the ends are perfectly the ends are perfectly insulatinginsulating

If we attach heat If we attach heat baths at the ends of baths at the ends of the rod:the rod:

How to approach solving these How to approach solving these problemsproblems

Add in the diffusion term for conduction Add in the diffusion term for conduction along the length of the rodalong the length of the rod

Solve the new coupled equations of motionSolve the new coupled equations of motion

More difficult than it sounds!More difficult than it sounds!

Work still ongoing….Work still ongoing….

Part II: Experimental ResultsPart II: Experimental Results A recent review paper by A recent review paper by

Ekinki and Roukes compiled Ekinki and Roukes compiled quality factor data quality factor data

Found that quality factor Found that quality factor generally decreases with generally decreases with ‘size’ of the resonator‘size’ of the resonator

BUTBUT

Results taken from many Results taken from many different sources using different sources using different types of resonatordifferent types of resonator

Is volume really a good Is volume really a good quantity to use?quantity to use?

K. L. Ekinci and M. L. Roukes, Review of Scientific Instruments, 76, 061101 (2005)

Kleinman et al.Kleinman et al. Torsional oscillators, Torsional oscillators,

length 1.91cmlength 1.91cm

Quality factors at low Quality factors at low temperaturestemperatures

Q dependence on Q dependence on resonance moderesonance mode

Due to defects in Due to defects in silicon wafers?silicon wafers?

R. N. Kleiman, G. Agnolet, and D. J. Bishop, Phys. Rev. Lett. 59, 2079 (1987).

Klitsner and PohlKlitsner and Pohl 2cm long torsional 2cm long torsional

oscillators oscillators

Temperature Temperature dependence of Q over dependence of Q over a larger rangea larger range

Fundamental mode Fundamental mode onlyonly

Increase in Q when Increase in Q when heated?heated?

T. Klitsner and R. O. Pohl, Phys. Rev. B 36, 6551 (1987).

Greywall et Al.Greywall et Al.

Beams of length Beams of length 550550μμmm

Q measured at very Q measured at very low temperatureslow temperatures

Oscillatory behaviourOscillatory behaviour

Effect reduced by Effect reduced by magnetic fieldmagnetic field

D. S. Greywall, B. Yurke, P. A. Busch, and S. C. Arney, Europhys. Lett. 34, 37 (1996).

Mihailovich and ParpiaMihailovich and Parpia Torsional oscillators, Torsional oscillators,

200200μμm thickm thick

Various levels of Boron Various levels of Boron doping were useddoping were used

Q recorded at low Q recorded at low temperatures for different temperatures for different doping levels.doping levels.

Doping effect reduced at Doping effect reduced at higher temperatureshigher temperatures

Increased dopingIncreased doping

R. E. Mihailovich and J. M. Parpia, Phys. Rev. Lett. 68, 3052 (1992).

Carr et Al.Carr et Al. Beams length 2-8Beams length 2-8μμm m

longlong

Strong linear Strong linear dependence of Q on dependence of Q on surface area to volume surface area to volume ratioratio

Indicates that surface Indicates that surface effects can considerably effects can considerably reduce Qreduce Q

D. W. Carr, S. Evoy, L. Sekaric, H. G. Craighead, and J. M. Parpia, Applied Physics Letters 75, 920 (1999)

Conclusions from these resultsConclusions from these results

Many different types of behaviour Many different types of behaviour measured with many variablesmeasured with many variables

The volume of a resonator isn’t a good a The volume of a resonator isn’t a good a measure of it’s dissipative qualitiesmeasure of it’s dissipative qualities

Thermoelastic, clamping losses and other Thermoelastic, clamping losses and other forms of dissipation are more sensitive to forms of dissipation are more sensitive to the thicknessthe thickness

Putting all these (and more) Putting all these (and more) TogetherTogether

Forbidden Forbidden region?region?

Anomalous point:Anomalous point:

S.S. Verbridge et Al., J. S.S. Verbridge et Al., J. App. Phys, 99, App. Phys, 99, 124304, (2006)124304, (2006)

ConclusionsConclusions Euler-Bernoulli Theory allows us to predict the frequency of Euler-Bernoulli Theory allows us to predict the frequency of

beams, ignoring thermal effectsbeams, ignoring thermal effects

Lifshitz and Roukes’ solution allows accurate prediction of Lifshitz and Roukes’ solution allows accurate prediction of thermoelastic damping in most circumstances.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 Recent measurements are inconclusive about Q behaviour of small resonators, with some contradictory resultssmall resonators, with some contradictory results

Compilation of many sets of results shows a region where no Q Compilation of many sets of results shows a region where no Q values have been measuredvalues have been measured

Still lots of work needed to decide exactly what factors are Still lots of work needed to decide exactly what factors are important to energy loss in these nanomechanical resonatorsimportant to energy loss in these nanomechanical resonators