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

Transcription

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

2 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.

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

4 Vibrating beams are not new Nonlinear response Electromagnetic environment Defects in the material Noise Dissipation ff = (1.875)2 2ππLL 2 EEEE ρρρρ

5 Scales change for nano-beams M. Roukes. ωω 0 1 LL TT ρρ vv~ ωωωω ~ 1 m/s 1 MHz 1 GHz aa~ ωω 2 xx ~ 10 8 m/s 2 FF~ mmmm 2 xx ~ 1 nn ħωω 0 1 mk 100 mk EE ~ mmmm 0 2 xx 2 TTTT 0.1 mev 100 mev Nanomechanics: Mechanics of objects with sub ev scale vibrational energy. Mechanics still obey continuum mechanics.

6 Scales allow coupling to... Optics/quantum optics Quantum transport Sensing (force, charge, mass, magnetic) Nonlinear dynamics Strongly correlated systems Surface physics Materials science Quantum mechanics Thermal fluctuations

7 Nano-resonators are sensitive Record Nanomechanical Force sensor: 12 zn Hz 1/2 Rugar et al., Nature 430, 329 (2004) δδδδ ΔEE ~ 6 µev ff 0 EE m 180 mev ~ 10 5 (12 zn=gravitational force between to humans separated a distance 5000 km) Next challenge: Single nuclear spin, isotope detection (proteomics)

8 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) δω ω = δ m Rx ( ) m With CNTs, sub-zeptogram (1 zg = 1E-21 g) sensitivity is possible! A. Isacsson, MCC026,

9 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,

10 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,

11 Dissipation Q-factor Definition of Q-factor: + cos( ω ) 2 x γ x + ω0 x = f0 t For most applications the higher the Q the better. Q = ω 0 / γ 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...

12 They are very nonlinear R. B. Karabalin, et al., PRB 79, (2009). Next challenge: Can we tame the nonlinear response to become useful?

13 A. Isacsson, Chalmers Noise becomes important c qq + γγqq + ωω 0 2 qq = ff D + DDξξ(tt) Very easy Very hard interesting qq + γγqq + ωω 0 2 qq + VV qq = ff D + DDξξ(tt) R. Landauer (1998): The noise is the signal Mark Dykman (2012) Next challenge(s): Tame the noise in nonlinear systems. What can we learn from observing the noise?

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

15 c Let s get down to it.

16 Reduced models c = Despite small size, still many atoms. Can use continuum mechanics to study deformations. Euler-Bernoulli beam-equation (see wikipedia) E - Youngs Modulus, I - Moment of inerta w - Deflection µ - Mass/unit length q - Force/unit length Usually we are only interested in a single eigenmode. 2 q + γq + ω0 q = f ( t )

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18 c

19 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. µ 2 t w( x, t) + c EI 4 x w( x, t) = 0 Assume the suspended part of the tube to be 1 µm long and have a diameter of 10 nm. For simplicity assume the graphene the tube is made have a thickness of 0.34 nm and a Young modulus of E=1 TPa.

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

21 Transduction (mixing) c Semiconducting resonators (eg. CNT, graphene) change conductance as they resonate Conductance is modulated with frequency ω Bias is modulated with frequency ω+ ω Current through device will have slow component with frequency ω I cos(ωt) cos(ωt+ ω t) ~ cos(2ωt+ ωt)+ cos( ωt)

22 Transduction (other methods) c 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.

23 Transduction (capacitive/direct) c Capacitive/direct Small capacitances => Impedance matching problems. Need to Tailor the circuitry to specific frequency.

24 Lots of noise, hard to readout, high frequencies, all sort of resonances in circuit. c How do we know we have a mechanical one?

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26 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? c

27 Equivalent circuit modeling c 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 = γ m L m C m 1 = L Ω m 2 L m = V 2 0 M C ( x 0 ) C = C( x 0 ) g Problem 3. Derive these relations for C m, L m and R m.

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29 The first to succeed (A. Cleland) c

30 What was measured? c 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 excitation likely from oscillator quantum. Qubit was with 95% prob. in ground state! => <n> = 0.07

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

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34 c 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

35 c 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.

36 c 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 measurement 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?

37 A. Isacsson, Chalmers Strongly correlated systems? Bachtold, Steele, Science (2009) Coulomb Blockade + nanoresonators Quantum Hall Effect Desmukh, Hone, Nat. Phys. (2016) Superfluid helium FQHE, Wigner crystal in graphene? Probing excitations in superfluids (supersolids?) Next challenge: Show that using nanoresonators we can learn something new. (Majorana Fermions+Resonators=True?)

38 Current trends c Sensing: Robust larger scale sensors Individual mass detection Quantum: Active cooling Readout schemes Nonclassical states Entangled resonators Quantum optomechanical systems Nonlinear: Mode coupling Fluctuations Dissipation and spectral broadening

39 They couple to photons Teufel/Lehnert Nature, (2010,2011) Next challenge: RF OPTICAL Quantum transducers

40 THANK YOU FOR YOUR ATTENTION