Nanomechanics II Andreas Isacsson

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1 Nanomechanics II Andreas Isacsson Condensed Matter Theory Department of Applied Physics Chalmers University of Technology A. Isacsson, MCC026,

2 Outline A brief overview Mechanics, micromechanics, nanomechanics and molecular mechanics. Commonly studied systems and research directions Sensitivity to force, mass and thermal fluctuations Quantum nanomechanical systems Current trends Into the details Reduced models Problem 1. Transduction. Problem 2. Equivalent circuit modeling. Problem 3. Quantum nanomechanics. Problem 4. A. Isacsson, MCC026,

Graphene as 2D-membrane >> Material parameters >> Model example Continuum

Varying on a scale large (long wavelengths) compared to the scale of material

Large scale defined in comparison to unit cell.

(10-100 microns) Example3: Concrete. Large scale compared to granules and pores.

3 Graphene as 2D-membrane >> Material parameters >> Model example Continuum mechanics Continuum mechanics deals with deformation of material systems Varying on a scale large (long wavelengths) compared to the scale of material inhomogenieties. Example1: Single crystal of Si. Large scale defined in comparison to unit cell. (tens of nanometers) Example2: Steel. Large scale compared to grain size. ( microns) Example3: Concrete. Large scale compared to granules and pores. (1 mm 1 m) Size scale varies, but as long as we know the applicable scale, structure of theory is the same. Only parameters differ. A. Isacsson, MCC026,

4 Scaling it down km m mm micromechanics nanomechanics molecular mechanics mm nm Å Why study this size range when mechanical equations mostly the same? Nanomechanics: Small, but continuum theory mostly valid A. Isacsson, MCC026,

5 NEM-resonators ω 0 1 L T ρ 1 MHz 1 GHz F~ ω 2 x ~ 1 nn ħω 0 1 mk 100 mk E ~ mω 0 2 x 2 TL 0.1 mev 100 mev v~ ωx ~ 1 m/s a~ ω 2 x ~ 10 8 m/s 2 Nanomechanics: Mechanics of objects with sub ev scale vibrational energy. Mechanics still obey continuum mechanics. Interesting physics happens when systems with similar energy scales interact. Photons (optical, microwave), quantum dots, qbits, superconductors, single-electron transistors, thermal fluctuations A. Isacsson, MCC026,

6 Common systems Metallic or SiN beams Ring resonators Weig, Kotthaus (LMU) Kippenberg (MPD) Harris (Yale) Carbon nanotubes Graphene Steele, Zant (TU-DELFT) A. Isacsson, SO-Nano, Hone (Columbia)

7 Active research areas Applications Sensitive detectors Mass-sensors, Force sensors RF-electronics filters Nanoscience Fluctation phenomena Dissipative processes Material properties A. Isacsson, SO-Nano, Quantum nanomechanics Observing quantum coherent phenomena in systems described by continuum mechanics

B E m ~ 6 μev 180 mev ~ 10 5 Rugar et al.

8 Sensitivity to force Atomic force microscopy Typical freq. khz Small changes in frequency can be detected, Gives large sensitivity, δf f 0 E B E m ~ 6 μev 180 mev ~ 10 5 Rugar et al., Nature 430, 329 (2004) Large sensitivity due to small energy scale. Well defined due to top down fabrication. A. Isacsson, MCC026,

9 Sensitivity to mass A resonance frequency shift occurs each time an analyte is adsorbed on the nanoresonator. M. L. Roukes, et. al., Nature Nanotechn. 4, (2009) m Rx ( ) m With CNTs, sub-zeptogram (1zg = 1E-21g) sensitivity is possible! A. Isacsson, MCC026,

10 Sensitivity to mass Recently (Roukes, 2012) actual spectrum could be taken. Using two vibrational modes, mass of each individual particle could be determined A. Isacsson, MCC026,

Thermal fluctuations, noise and transduction Transduction = conversion of energy in one form to another. For instance mechanical to electric. A harder problem than one might think.

11 Thermal fluctuations, noise and transduction Transduction = conversion of energy in one form to another. For instance mechanical to electric. A harder problem than one might think. - Small energies - High frequencies - Noisy environment - Other resonances may lie at nearby frequencies AFM readout, Photothermal actuation, Optical detection, Piezo: A. Isacsson, MCC026,

12 Dissipation Definition of Q-factor: Q-factor x x 2 0 x f0 t cos( ) Q 0 / For most applications the higher the Q the better. Gas-damping: Collision with molecules in a gas is detrimental. Clamping losses: Radiation of phonons into the supports Electrical losses: Displacement currents flowing in and out of the resonator. Two level fluctuators. Defect position fluctuations and charge traps. Mysterious losses: The bandwidth (apparent Q-factor) increases dramatically with temperature. Exact source is still unknonwn. Others yet unknown... A. Isacsson, MCC026,

13 Quantum fluctuations Quantum mechanics Governed by Schrödinger Eq. Linear, allows superposition of solutions Quantum Harmonic oscillator k m x(t)

14 A. Isacsson, MCC026, Let s get down to it.

A. Isacsson, MCC026, 2012-11-13 A reduced model = Despite small size, still many atoms. Can use continuum mechanics to study deformations.

15 A. Isacsson, MCC026, A reduced model = Despite small size, still many atoms. Can use continuum mechanics to study deformations. Euler-Bernoulli beamequation (see wikipedia) E ~ Youngs Modulus, I ~ Moment of inerta w~ deflection m ~ mass density/unit length q ~ force/unit length Usually we are only interested in a single eigenmode. 2 q q 0 q f ( t )

Problem 1. A carbon nanotube resonator is clamped at both ends and is under no tension. What are the frequencies of the three lowest flexural eigenmodes if the tube can be described by beam theory.

18 Problem 1. A carbon nanotube resonator is clamped at both ends and is under no tension. What are the frequencies of the three lowest flexural eigenmodes if the tube can be described by beam theory. m 2 t w( x, t) EI 4 x w( x, t) 0 Assume the suspended part of the tube to be 1 mm long and have a diameter of 10 nm. For simplicity assume the graphene the tube is made up of to have a thickness of 0.34 nm and a Young modulus of E=1TPa.

19 Transduction Magnetomotive - Metallic structures - Metal coated insulating structures (SiN) Actuation: Lorentz-force Readout: Motionally induced EMF Requires to local B-field. Not so good for integration.

20 Electromechanical mixing Semiconducting resonators (eg. CNT, graphene) change conductance as they resonate. Conductance is modulated with frequency Bias is modulated with frequency D Current through device will have slow comonent with frequency D I ~ cos(t) cos(t+d t) ~ cos(2t+dt)+ cos(dt)

Dielectric and other methods Other methods AFM

heating) Optical detection: Works amazingly well but,

21 Dielectric and other methods Other methods AFM readout: Directly visualize the mode shapes. Really hard Photothermal actuation: Local timedependent heating with laser (Causes unwanted heating) Optical detection: Works amazingly well but, again heating problems. Piezo: Works rather well but needs good material compatibility.

22 Transduction Capacitive/direct Small capactiances => Impedance matching problems. Need to Tailor the circuitry to specific frequency.

Problem 2: Consider the nanotube studied in problem 1. Assume it suspended 300 nm above a metallic backgate. Estimate the capacitance between the tube and the backgate.

24 Problem 2: Consider the nanotube studied in problem 1. Assume it suspended 300 nm above a metallic backgate. Estimate the capacitance between the tube and the backgate. Determine the frequency tuning of the fundamental mode due to electrostatic spring softening as function of backgate-voltage (CNT is grounded). In these kind of structures one often finds that the frequency tunes upwards when the backgate voltage is increased. Why is this?

Equivalent circuit modelling With capacitive transduction the resonator can be replaced by equivalent circuit having motional inductance L m, motional

25 Equivalent circuit modelling With capacitive transduction the resonator can be replaced by equivalent circuit having motional inductance L m, motional capacitance C m and motional resistance R m. = R m L m C m 1 L m 2 L m V M C( 0 ) 2 0 x C g C( x 0 ) Problem 3. Derive these relations for C m, L m and R m.

27 The first to succeed (A. Cleland)

oscillator quantum. Then tune out of resonance and measuring state of qubit.

28 What was measured? Qubit level spacing can be tuned Using flux bias. When it is close to resonant frequency it can be excited by absorbing an oscillator quantum. Then tune out of resonance and measuring state of qubit. If excited, the excitaion likely from oscillator quantum. Qubit was with 95% prob. in ground state! => <n> = 0.07

Changing capacitance changes resonant frequency of LC-circuit.

29 Dispersive Sideband readout cooling and strong coupling Resonator is part of capacitor. Changing capacitance changes resonant frequency of LC-circuit. For strong RF-drives the interaction can be linearized. Result is two coupled harmonic oscillators

32 The strenght of the drive field determines the effective coupling g The two interacting oscillators (mechanical and electric) hybridize and a frequency split is observed. Split becomes larger as the drive (coupling) is increased

energy exceeds the rate to scatter to lower energy Thus, the net

33 Experiments Sideband cooling When the circuit is excited with a detuned microwave drive the rate to scatter photons to higher energy exceeds the rate to scatter to lower energy Thus, the net scattering rate (blue solid arrow) provides a cooling mechanism for the membrane.

34 Problem 4: In the experiment by Teufel et al, the quantum zero point fluctuations of the fundamental mode is estimated to be x 0 =4.1 fm. One may now put forth the argument that it is nonsense to speak of displacements of a large object like a membrane on this scale. Indeed x 0 is roughly the size of a single nucleus and hence much smaller than the individual atoms the membrane consists of. Nevertheless it seems that the measurment firmly confirms that the fundamental mode is in its ground state. Consequently, the measurment does correspond to resolving displacements of this size. But does this make sense?

35 Current trends Sensing: Quantum: Nonlinear: Robust larger scale sensors. Individual mass detection Active cooling. Readout schemes. Nonclassical states, entangled resonators. Quantum optomechanical systems. Mode coupling Fluctuations Dissipation and spectral broadening A. Isacsson, SO-Nano,

36 IF TIME PERMITS, there s some more...

37 Fluctuations Particles inside or adsorbed on resonator can diffuse (thermal motion). Resonance frequency Depends on particle distribution. Leads to effectively nonlinear behavior with stochastic switching. A. Isacsson, SO-Nano,

38 Dissipation in graphene A systematic study Observed behavior with frequency, size and temperature not explained. resonators A. Isacsson, SO-Nano,

39 resonance frequency (MHz) Q Nonlinear damping - A. Eichler, J. Moser, J. Chaste, M. Zdrojek, I. Wilson-Rae, A. Bachtold Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene Nature Nanotechnology 6, 339 (2011) Nonlinear damping also seen in Graphene resonators from AALTO (P. Hakonen, Barcelona, June 1, 2012) f0 Q annealing step A. Isacsson, SO-Nano,

40 Nonlinearity and modecoupling Amplitude > thickness gives nonlinear response For graphene resonators, corresponding equivalent thickness is 40 pm. A. Isacsson, SO-Nano,

Initial coherent state Non-classical states

n H t i i t i i e e e e t 4 / / 4 2 1 2 /,,

41 Initial coherent state Non-classical states A. Isacsson, SO-Nano, In graphene drums, nonlinearity strong enough to induce evolution into nonclassical states 2 0 ˆ ˆ n n H t i i t i i e e e e t 4 / / /,, 0 t

42 Thank you for listening..... and to the people I have and have had the pleasure to work with Prof. Jari Kinaret, Prof. Mark Dykman (MSU), Prof. Eva Weig (LMU), Prof. Eleanor Campbell (UE), Prof. H. Park (BU). Dr. Alexander Croy, Chalmers, Dr. Juan Atalaya, Dr. Niklas Lindahl (GU). MSc. Aurora Voje, MSc. Daniel Midtvedt, MSc. Johannes Rieger (LMU), MSc. Christin Edblom, MSc. Zenan Ki (BU). Bsc. Martin Eriksson. A. Isacsson, SO-Nano,

Nanomechanics II. Why vibrating beams become interesting at the nanoscale. Andreas Isacsson Department of Physics Chalmers University of Technology

Nanomechanics II. Why vibrating beams become interesting at the nanoscale. Andreas Isacsson Department of Physics Chalmers University of Technology Nanomechanics II Why vibrating beams become interesting at the nanoscale Andreas Isacsson Department of Physics Chalmers University of Technology Continuum mechanics Continuum mechanics deals with deformation