nanophysics: main branches
TRANSCRIPT
Nan
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ech
anic
s
Nano-opto-electronics
Nanophysics:Main branches
1. What are main branches of Nanophysics? What is
the place and role of Nanomechanics there?
Nanomechanics is a branch of nanoscience studying fundamental mechanical (elastic, thermal and kinetic) properties of physical systems at the nanometer scale.
Nanomechanics has emerged on the crossroads of classical mechanics, solid-state physics, statistical mechanics, materials science, quantum chemistry and biology. As an area of nanoscience, nanomechanics provides a scientific foundation of nanotechnology.
IBMQSystemOne
https://newsroom.ibm.com/2019-01-08-IBM-Unveils-Worlds-First-Integrated-Quantum-Computing-System-for-Commercial-Use
https://www.nextbigfuture.com/2019/03/ibm-quantum-computer-roadmap.html
Moor’s law -II
Quantum brain
https://www.epfl.ch/campus/spiritual-care/wp-content/uploads/2018/10/quantum-mind.jpg
By Ziko van Dijk, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=33747872
Odhner Arithmometer
Charles Babbage’s The Difference Engine No. 2
https://en.wikipedia.org/wiki/Difference_engine
https://www.youtube.com/watch?v=qctHEGKr9Zs
Charles Babbage (1791-1871), computer pioneer, designedthe first automatic computing engines. He inventedcomputers but failed to build them. The first completeBabbage Engine was completed in London in 2002, 153years after it was designed.
Consists of 8,000parts, weighs fivetons, and measures11 feet long.
Mechanical computer
http://www.computerhistory.org/babbage/
Smaller, cheaper, faster, lower power consumption
“Phones of the future”. Nano-electromechanics (NEM)-devices are in the right frequency range (1-5 GHz) to replace elements in cell phones
Better frequency selectivity (higher Q), lower power consumption
New sensor applications
Motivation
Device applications
Requirements: High Q, high frequency
… and interesting “cutting edge” physics.
https://eda360insider.wordpress.com/
2. Give some examples of Nanomechanics devices.
What are their functions? Why are they needed? What
are the promises of Nanomechanics devices?
Introduction 9
2018 highlights
• Engineers produce smallest 3-D transistor yet: 2.5 nm
• Light triggers gold in unexpected way
• Borophene advances as 2-D materials platform
• Insights into magnetic bacteria may guide research into medical nanorobots
• Nanoscale tweezers can perform single-molecule 'biopsies' on individual cells
• Holey graphene as Holy Grail alternative to silicon chips
Nanotweezers extracting a mitochondrion from a cell. Credit: Imperial College London
Next-generation optical components. Credit: Link Research Group/Rice University
Transistors that measure only 3 nm wide. Credit: Massachusetts Institute of Technology
Nano machinesTechnology updateFeb 19, 2018Frustrated liquid crystal film makes continuously rotating nanomachineLight-operated molecular machines can be used to rotate objects muchbigger than themselves but they stop rotating once they reach the so-calledphotostationary state. By using a photo-sensitive chiral liquid crystal, it isnow possible to generate continuously rotating supramolecular structuresfuelled by light. Such machines might be used in applications like syntheticmuscles, nano- and micro-robots and advanced mechanical motors.
http://nanotechweb.org/cws/article/tech/71138
Motor-doped liquid crystals
winding under irradiation,
and associated chiral
structures.
Scale bar is 20 m
11
2019 highlights
• Use a microscope as a shovel? Researchers dig it.
• Light connects two worlds on a single chip
• Researchers develop direct-write quantum calligraphy in monolayer semiconductors allowing single-photon emission.
• When cold atoms meet nano. Using arrays of cold
cesium atoms around a nanofiber, researchers
reported the first wired entangled state of atoms
and single photons.
AFM indents produce single photonemitter 'ornaments' on a monolayer WSe2
Credit: US Naval Research LaboratoryArtist impression of the optocoupler. Credit:University of Twente
Credit: Proceedings of the National Academy of Sciences (2019). DOI: 0.1073/pnas.1806074116
Credit: Kastler Brossel Laboratory.
12
2020 highlight
• A glass sphere about150 nm in diameter hasbeen cooled to itsmotional quantumground state usingoptical tweezers.
• It was achieved byconfining the oscillatingobject using light in anoptical cavity (opticaltweezers) and thenreducing its vibrationsby laser cooling.
• The apparatus doesnot need to be cooled.
https://physicsworld.com/a/glass-sphere-is-cooled-to-its-motional-quantum-ground-state/?utm_medium=email&utm_source=iop&utm_term=&utm_campaign=17973-45194&utm_content=QUANTUM%20MECHANICS%20%7C%20RESEARCH%20UPDATE&Campaign+Owner=
Caught in a trap: a levitated nanoparticle has been cooled to its quantum ground state.(Courtesy: Lorenzo Magrini and Yuriy Coroli/University of Vienna)
Glass sphere is cooled to its motional quantum ground state03 Feb 2020
13
2021 highlight
• First scanning force microscope is reported, in which the tip is at rest whilethe substrate with the samples on it vibrates.
• The new method promises to push the sensitivity of force microscopy to itsfundamental limit, beyond what can be expected from further improvementsof the conventional "finger tapping" approach.
• A perforated membrane made of silicon nitride, a mere 41 nm in thickness.The low-mass membrane is outstanding nanomechanical resonator withextreme quality factor.
https://phys.org/news/2021-02-microscopy-concept.html
A perforated silicon nitride membrane servesas force sensor. Two coupled 'islands'undergo out-of-plane vibrations. On one ofthem the samples are loaded and the otheris used to measure the vibrations with a laserinterferometer. A metallic scanning tipinteracts with the samples and modifies thevibrations. Credit: Alexander Eichler, ETHZurich
New microscopy concept enters into force FEBRUARY 8, 2021
by ETH Zurich
Hälg D et al. Membrane-based scanning force microscopy. Phys. Rev. Appl. 15, L021001 (2021).
2018
https://www.nobelprize.org/prizes/physics/2018/summary/
The Nobel Prize in Physics 2018 was awarded "for groundbreaking inventions in the
field of laser physics" with one half to Arthur Ashkin "for the optical tweezers and their
application to biological systems", the other half jointly to Gérard Mourou and Donna
Strickland "for their method of generating high-intensity, ultra-short optical pulses."
2018
https://www.nobelprize.org/prizes/physics/2018/summary/
Press Release: The Nobel Prize in Chemistry 20183 October 2018The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Chemistry 2018 toFrances H. Arnold "for the directed evolution of enzymes.“George P. Smith "for the phage display of peptides and antibodies." Sir Gregory P. Winter "for the phage display of peptides and antibodies."
Frances H. Arnold George P. Smith
2018
https://www.nobelprize.org/nobel_prizes/chemistry/
Sir Gregory P. Winter
Ill. Niklas Elmehed. © Nobel Media
2018
https://www.nobelprize.org/nobel_prizes/chemistry/
Introduction 18
Chemistry Nobel Prize 2016 for molecular nanomachines
Jean-Pierre Sauvage, Sir J Fraser Stoddart and Bernard L Feringa
Membrane molecular motors
https://www.google.com/search?q=membrane+nano+motors&source=lnms&tbm=isch&sa=X&ved=0ahUKEwivgdqbz-3gAhWDs4sKHTNVAHsQ_AUIDigB&biw=1280&bih=603#imgrc=Jzx6z5PGeay2mM:
DOI: 10.1021/acsnano.7b04747
Nano machines
https://www.youtube.com/watch?v=YdjERhTczAsNano robots inside you
https://www.youtube.com/watch?v=sjxTfaWhVs8Evolution
https://www.youtube.com/watch?v=s1NkvH98yEENano motor
https://www.youtube.com/watch?v=AJ09dLUWBxENano motor inside cell
https://www.youtube.com/watch?v=1aOrq5E8rbMOptofluidics tweezers
Microtubules in living cells
https://www.youtube.com/watch?v=sjxTfaWhVs8
Microtubules in living cells
https://www.youtube.com/watch?v=sjxTfaWhVs8
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Transport measurements of brain tissue
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0
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Current-voltage characteristic of the brain
slice shown in three-probe configuration with
current leads 2 and 4, and potential leads 2
and 3.
Current-voltage characteristic of the brain tissue
measured in the four-probe configuration. Small
black arrows show direction of the record.http://jonlieffmd.com/blog/human-brain/could-the-brain-and-mind-be-a-
quantum-computer-quantum-effects-in-brain-and-mind
https://medicalxpress.com/news/2014-05-neural-smallest-scales-traumatic-brain.html
https://www.pinterest.com/pin/171910910752672164/
P. Mikheenko, Graphene-assisted Transport Measurements of Biological Samples. IEEE Xplore Digital Library
7757272 (2016). DOI: 10.1109/NAP.2016.7757272. http://ieeexplore.ieee.org/document/7757272/ .
Possible superconductivity in the brainhttps://link.springer.com/article/10.1007/s10948-018-4965-4
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Josephson generation in living organisms
Typical voltages in a human body vary between 20 and 200 mV with the average membrane potential
of about 70 mV. These voltages correspond to 4.8, 48.4 and 16.9 THz if generated by normal
conductor and to 9.6, 96.8 and 33.8 THz if generated by Josephson junctions. The corresponding
wavelengths are in the range from 3.1 to 31 m in superconducting and 6.2 to 62 m in normal case.
The Josephson membrane-potential voltage could produce radiation with wavelength 8.8 m, which
is in the transparency window for surrounding atmosphere. There is possibility of
electromechanical resonance in cell microtubules.
Atmospheric absorption across the electromagnetic
spectrum. Image Credit: NASA (public domain).
Mikheenko, Pavlo (2020). Nano superconductivity and quantum processing of information in living organisms, In 2020
IEEE 10th International Conference on “Nanomaterials: Applications & Properties” (NAP – 2020). IEEE. ISBN 978-1-
7281-8506-4. Article. s 02SNS02-1 - 02SNS02-4
https://www.youtube.com/watch?v=Q3M4S7_ISs0
Nano robots
https://www.youtube.com/watch?v=Q8tAj8A4pc0
https://www.youtube.com/watch?v=4UK4tilHjD8
https://www.youtube.com/watch?v=F7REp0Y9edA
Organic computing
Flexural Resonators
Series of four AlN beams, with the undercut lengths ranging from 3.9 to 5.6 μm.
The beams are 0.17 μm thick, and the widths are 0.2 μm, increasing to widths of 2.4 μm at either end.
Flexural resonators, either in the form ofcantilevers or doubly-clamped beams, form a basicelement in nano- mechanical structures.
A beam fabricated from Si with thickness t = 0.2 μm and a length of 100 μm has a resonance frequency of about 170 kHz; shortening the length to 10 μm increases the frequency to 17 MHz, and a further reduction to 1 μm brings the frequency up to 1.7 GHz. Quality factors for the fundamental resonance have been measured in excess of 20,000. As stable local oscillators, such resonators are good candidates for time-base and frequency standards applications .
A.N. Cleland’s group, Santa Barbara
3. Describe Flexural Resonators. What are their main types? What are their typical frequencies and quality factors? How is
possible to control their parameters? What materials are good for their preparation and what techniques are needed for that? How
are flexural resonators coupled to external electronic circuitry?
Q = f/f,f - FWHM
AlN was used because of its low density, high elastic modulus, and relatively high fracture strength.
This material can be grown as single crystal films on <111>-orientation Si wafers using metal-organic chemical vapor deposition.
The films can subsequently be patterned using a combination of electron beam lithography and thin metal film deposition.
The AlN can be anisotropically etched using Cl2-based reactive ion etching, andthe Si substrate selectively removed using an isotropic Si etch (ammonium fluoride, nitric acid and water).
These structures are coupled to external electronic circuitry using a number of techniques based on magnetomotive transduction, electrostatic displacement sensing and piezoelectric effect.The magnetomotive technique involves placing the mechanically active element in a transverse magnetic field, and passing a current through a metal film integrated with the mechanical structure. A Lorentz force develops as a result, generating a displacement; the displacement then generates an electromotive voltage, which can be sensed using external electronics.
Movie 1
Preparation
A doubly-clamped resonator with thickness t and length L has a fundamental flexural resonance
frequency given by
Observed resonance of a 3.9 μm long beam, measured at 4.2 K in a transverse magnetic field of 8 T. Inset: Measured resonance while varying the magnetic
field from 1 T (smallest peak) to 8 T (largest peak).
Capacitive coupled resonators:Single-crystal silicon flexural resonators coupled by capacitive transducers, allowing both displacement actuation and sensing.The horizontal scale bar is 1 μm. The flexural resonance frequency of these structures is measured to be 68 MHz.
Young modulus
Doubly-clamped resonator
4. How does the frequency of doubly-clamped resonator depend on its mass, length,
thickness and Young’s modulus? How can it be actuated and sensed?
Young modulus
Young modulus
Young's modulus is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression. Sometimes referred to as the modulus of elasticity, Young's modulus is equal to the longitudinal stress divided by the strain
Resonant Mass Sensors (mass sensing on the level of single molecules)
= 0 M
New Functionality and Applications NEM sensing (sensing of mass, displacements and forces on an atomic scale)
Mechanical control of quantum point contacts and transportation of single electrons
mmin ≈ M/Qlow M , high 0 , high Q
See review in Nature Nanotech. 4, 445 (2009)[Roukes’ group (Caltech)] Sensitivity: ~200 DaNature Nanotech. 3, 533 (2008);Nano Lett. 8, 4342 (2008)
1 Dalton=1.66 10-24 g – mass of proton, 1ag=10-18 g
Q = f/f,f - FWHM
5. What are Resonant Mass Sensors? Describe principle of their work. How does minimum detection mass depend on the mass of
resonator, its frequency and quality factor? What is the sensitivity of resonant nano mass sensors? Can it be compared with 1 Dalton? What
is the principle of biomolecular recognition?
The long and short scales are two of several large-number naming systems for integer powers of ten that use the same words with different meanings. The long scale is based on powers of one million, whereas the short scale is based on powers of one thousand.
https://en.wikipedia.org/wiki/Zepto-
THz Flexural Resonator
A novel device consisting of a subwavelength terahertz meta-atom resonator, which integrates a nanomechanicalelement and allows energy exchange between the mechanical motion and the electromagnetic degrees of freedomis proposed. An incident terahertz wave thus produces a nanomechanical signal that can be read out optically withhigh precision.
3. Describe Flexural Resonators. What are their main types? What are their typical frequencies and quality factors? How is
possible to control their parameters? What materials are good for their preparation and what techniques are needed for that? How
are flexural resonators coupled to external electronic circuitry?
Belacel, C., Todorov, Y., Barbieri, S. et al. Optomechanical terahertz detection with single meta-atom resonator. Nat Commun 8, 1578 (2017). https://doi.org/10.1038/s41467-017-01840-6
a Side view of the split-ring resonator (SRR), the scale bar is 1 μm. b Far-infrared transmission spectrum of an arrayof identical SRRs. c Distribution of the electric energy density in the SRR. d Illustration of the normalized THz eddycurrents. The scale bar indicated in c, d is 2 μm.
Quantum nanomechanical devices
Cold, macroscopic mechanical systems are expected tobehave contrary to our usual classical understanding; themost striking and counterintuitive predictions involve theexistence of states in which the mechanical system islocated in two places simultaneously. This requiresmechanical states that are close to the lowest energyeigenstate, the mechanical ground state. It can be achievedby the cooling of the motion of a radiofrequencynanomechanical resonator by parametric coupling to adriven, microwave-frequency superconducting resonator.
Nb, 170 nm x 140 nm x 30 μm, ωm = 7.5 GHz - quantum length scale is 3 10–15 m, comparable with size of proton (0.84–0.87 fm).
Quantum nanomechanical devices
Quantum length scale rq = (ћ/2mω)1/2.
The energy of mechanical resonator should be quantized: En = ћωm(n+1/2), where n is an integer and ωm is the resonant frequency. Ћ = h/2π, h = 6.6 10-34
m2kg/s, for a particular device ωm is 7.5 109 Hz. In thermal equilibrium, an average occupation factor is expected to follow the Bose–Einstein distribution: nT
m~(eћωm
/kB
T - 1)-1, where T and kB = 1.38 10-23 m2kg/s2K are the temperature and Boltzmann’s constant, respectively.
ћωm/kBT = 2 10-4/T, typical nTm are large, starting with 480 in this work and
being reduced by parametric cooling to 3.8.
The occupation factor of 3.8 corresponds to probability P0 = 1/(nTm + 1) of
0.21 at which the mechanical resonator is in the quantum ground state of motion.
Quantum nanomechanics
Preparation and detection of a mechanical resonator near the ground state of motion
T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk & K. C. Schwab, Nature 2010
Nb–Al–SiN sample: the nanomechanical resonator is 30 μmlong, 170nm wide and 140nm thick, is formed of 60nm of stoichiometric, high-stress, low-pressure chemical-vapour-deposition SiN29 and 80nm of Al, and is located 75nm from the gate electrode connected to the Superconducting Microwave Resonator (SMR). The SMR is made from a 345-nm-thick Nb film.
A radiofrequency nano-resonator (NR) was parametrically coupled to a driven SMR. Starting from a thermal occupation of 480 quanta, the authors have observed occupation factors as low as 3.8±1.3 and expect the mechanical resonator to be found with probability 0.21 in the quantum ground state of motion.
9. Describe possible quantum behaviour of mechanical resonator. Can resonator operate near the ground state of motion? What is the
benefit of connecting it to the Superconducting Microwave Resonator? What is the principle of parametric cooling? Can you describe it in
terms of parametric amplification? What are occupation factors already achieved by cooling? What is resonator motion sensitivity
comparable with quantum length-scale of the device? What is the typical value of the latter?
Parametric amplifier
For example, a well known parametric oscillator is a child on a swing where periodically changing the child's center of gravity causes the swing to oscillate. The varying of the parameters drives the system. Examples of parameters that may be varied are its resonance
frequency ω and damping β.
A parametric oscillator is a harmonic oscillator whose parameters oscillate in time.
Mathematical background
Variable transform:
with
Small variation:
Therefore, we have to solve:
Assume:
Search solution as:
and are slowly varying functions of time
Substitute the above form to the initial equation. Then multiply by and integrate
over t between and . This will give an equation connecting and .While integrating we take into account time dependences of these quantities only in time derivatives. After that we repeat the procedure replacing . In this way we get the second equation.
Gain is maximal at zero detuning
Substitute the above form to the initial equation. Then multiply by and integrate
over t between and . This will give an equation connecting and .While integrating we take into account time dependences of these quantities only in time derivatives. After that we repeat the procedure replacing . In this way we get the second equation.
Parametric amplification
https://en.wikipedia.org/wiki/Parametric_oscillator
At sufficiently small detuning . Then we can denote . The solution close to is:
The occupation number was measured by registering thermal noise
The level of cooling is expected to make possible a series of fundamental quantummechanical observations including direct measurement of the Heisenberguncertainty principle and quantum entanglement of qubits.
Cooling of the NR is similar to parametric amplification. SMR is driven witheigenfrequency, which depends on the position of NR.
When pumping the SMR at ωp = ωSMR - ωm, harmonic motion of the nanomechanicalresonator preferentially up-converts microwave photons to frequency ωSMR ,extracting one radio-frequency nanomechanical-resonator quantum for each up-converted microwave SMR photon, a process that both damps and cools thenanomechanical resonator’s motion. This cooling process is analogous to Ramanscattering and the process used to cool an atomic ion to the quantum ground stateof motion.
Quantum cooling
Optically driven ultra-stable nanomechanical rotorStefan Kuhn, Benjamin A. Stickler, Alon Kosloff, Fernando Patolsky, Klaus Hornberger,
Markus Arndt & James Millen, NATURE COMMUNICATIONS November 2017
Frequency locking. a A silicon nanorod is optically
trapped in a standing wave formed by counter
propagating focused laser beams at λ = 1550 nm.
The polarization of the light is controlled using a
fiber-EOM driven by a signal generator. We detect
the motion of the nanorod via the scattered light
collected in a multimode optical fiber. The signal is
mixed down with a local oscillator fLO to record the
spectrum and the phase of the rotation with
respect to the drive frequency fd. Both frequencies
fd and fLO are synced to a common clock. b Power
spectral density (gray points) of the frequency-
locked rotation at 1.11 MHz, taken over 4
continuous days, fit with a Lorentzian (red curve).
The upper bound on the FWHM is 1.3 μHz. c
Comparison of driven rotation when frequency
locked (“locked rotation”, red, right axis) and un-locked (“threshold rotation”, blue, left axis)
‘We levitate a nanofabricated silicon nanorod in
the standing light wave formed by two
counterpropagating laser beams, and track its
motion by monitoring the scattered light.’
Optically driven ultra-stable nanomechanical rotorStefan Kuhn, Benjamin A. Stickler, Alon Kosloff, Fernando Patolsky, Klaus Hornberger,
Markus Arndt & James Millen, NATURE COMMUNICATIONS November 2017
‘Nano- and micromechanical systems are of great technological
interest, due to their low mass and extreme sensitivity to external
forces4–6, torques7, 8, acceleration9, displacement10, 11, charge12, and
added mass13, 14. Many of these systems are themselves realizations
of harmonic oscillators, with frequency stabilities reaching f/Δf = 108
(refs. 15, 16), which can be further improved through mechanical
engineering17, injection locking18, 19, electronic feedback20, or
parametric driving21. The contact-free mechanical motion of particles
suspended by external fields in vacuum22–29 can reach a frequency
stability that is only limited by laser power fluctuations and collisionaldamping by residual gas particles30.’
Optically driven ultra-stable nanomechanical rotorStefan Kuhn, Benjamin A. Stickler, Alon Kosloff, Fernando Patolsky, Klaus Hornberger,
Markus Arndt & James Millen, NATURE COMMUNICATIONS November 2017
‘In this work, we transduce clock stability into the rotation of an
optically trapped silicon nanorod in vacuum. By periodically driving the
rotation with circularly polarized light, we create a nanomechanical
rotor whose rotation frequency fr, and frequency noise, is determined
by the periodic drive alone. This driven rotor is sensitive to non-
conservative forces, and the operating frequency can be tuned by
almost 1012 times its linewidth. Through our method, the frequency
stability is independent of material stress, laser noise, and collisional
damping. The driven nanorotor operates at room temperature, and
across a wide pressure range from low vacuum to medium vacuum,
achieving a pressure resolution of three parts-per-thousand, and inprinciple allowing a torque sensitivity below the zepto-Nm level.’
The long and short scales are two of several large-number naming systems for integer powers of ten that use the same words with different meanings. The long scale is based on powers of one million, whereas the short scale is based on powers of one thousand.
https://en.wikipedia.org/wiki/Zepto-
Surface stress changes the nanomechanical response of cantilevers. Bending of cantilevers is detected by an optical deflection technique.
Biomolecular Recognition
J. Fritz et al., Science 288, 316 (2000)
Scanning electron micrograph of a section of amicrofabricated silicon cantilever array (eightcantilevers, each 1 μm thick, 500 μm long, and 100 μmwide, with a pitch of 250 μm, spring constant 0.02 Nm−1; Micro- and Nanomechanics Group, IBM ZurichResearch Laboratory, Switzerland)
Typical deflections –few nm.
Focus on spatial displacements of bodies and their parts
m( )F r
Motion of a point-like mass Rotational displacement +center-of-mass motion
F F
F
Elastic deformations
Displacements: Classical or Quantum
The discrete nature of solids can be ignored on the nanometer length scale.
2
2( )
d rm F r
dtPoint-like mass – Newton’s equations:
2( ) / 2,U x kxHarmonic oscillator: ( )dU
F x kxdx
2
20
d x dxm kx
dt dt
Dynamics of nanostructures – sketch of theory
Consider an elastic wave in a long bar of cross-sectional area A and mass density ρ = M /V .
Longitudinal wave of compression/expansion
Newton’s law for the shown segment:
Assuming that the wave propagates along the direction [100] of a cubic crystal we can write the Hooke’s law as:
Solution of the above wave equation is
vl is the longitudinal sound velocity.
Elastic waves in a bar
stress
P(x)
U(x)
2
2
2 2
2 2
( , )( ) ( )
( ) ( , )
el
el
U x tA P x P x
t
P x EI U x tx x
E – Young’s modulus – represents rigidity of the materialI – Second moment of crossection – represent influence of the crossectional geometry
Easy to bend Dificult to bend
Why there is sensitivity to geometry of the beam crossection?
Euler-Bernoulli Equation
P is load
https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory
https://en.wikipedia.org/wiki/Bending
Transverse wave
Sketch of derivation
Longitudinal and Flexural Vibrations
Longitudinal deformation: Compression across the whole crossection
Flexural deformation. Compression and streching occur at different parts of the crossection
Dispersion law for plane waves:
2 2
2 2
2 4
2 4
( , )
( , )
U x t UA k
t x
U x t UA k
t x
Londitudinal vibrations
Flexural vibrations
12. Describe longitudinal and flexural vibrations of a strained nano-beam. What is the
difference between longitudinal and flexural deformations? How do resonanse
frequency depend on the length of the beam in these cases?
2 4 2
2 4 2
u u uA EI T
t x x
APL 78 (2001) 162
Flexural Vibrations of a Strained Beam
Quantum ”bell”
A. Erbe et al., PRL 87, 96106 (2001);
D. Scheible et al. NJP 4, 86.1 (2002)
Here: Nanoelectromechanics caused by or associated with single-charge tunneling effects
H. Park et al., Nature 407, 57 (2000)
Single C60 Transistor
Shuttles
13. Describe principle of electron shuttle. How can it be realised experimentally in a quantum ‘bell’ or C60 transistor
configurations? Is tunnelling involved in this process?
Nanoelectromechanics caused by or associated with single-chargetunneling effects
Recent implementation:D. Scheible and R. Blick , NJP 2010
Single-charge tunneling effects
(a) Tuning fork resonator (TFR) with two integrated metallic islands— electron shuttles—placed on the tips of the resonator’s arms on the left and right.
(b) Schematics of the TFR mechanism: leaving the clamping points at rest minimizes dissipation in the resonator.
(c) Sketch of the resonator indicating the wiggled resonator arms: the asymmetry of the resonator arms leads to a slight imbalance in the mechanical restoring forces if electric fields are applied to contact A or B.
(d) Magnified scanning electron microscopy picture of the electron shuttle and the circuit diagram. The resonator arms or clapper are made of crystalline silicon (blue) and a thin layer of gold on the top (yellow). The clapper can be set into motion by ac/dc voltages on electrodes A and B, so that the electron island (I) shuttles electrons from the source (S) to the drain (D). The measurements are performed at room temperature in a vacuum chamber.
D. Scheible and R. Blick, 2010
Tuning fork resonator for phase-coherent frequency conversion
14. What is principle of work of tuning fork resonator? How can
it be set in motion? What can it be used for?
2 / 2 ,2
L Rn n n
C CVE Q C Q
C
Electrostatic energy of the charged grain
nQ neRL CCC
2
( )1, , , ' 1
( ) 1 exp { ( )}
s n n sn n
s n n s
E E eV n nG s L R n n
e R x E E eV n n
Electron tunneling rates:
Grain
Define the probability, pn, of finding nelectrons on the grain. Then
( ) n
n
Q x e np
})()({
2,11'
'''
snn
n
s
nnn
s
nn
n pxGpxGt
p
22
2
( )d x dx eQ x Ex
dt dt M
The device is governed by the set of equations:
Master equation:
Newton equation:
Multiply by en and
then sum over n to get
at equation for Q(x)
Description of shuttling
1 1 R L
L R L R
C CdQ Q V
dt C R R C R R
22
2
( )d x dx eQ x Ex
dt dt M
Result:
, 0
1 1 ( )1
R L
x t
R R
0( ) exp{ }x t x i t t Search solution as:Assume weak electromechanical coupling:
12 22
M
CVE
thr
1{ },
2
2
thr 2 2
0
1;
2
RR
R R C
Shuttling instability at !thr
Description of shuttling (continued)
‘The result is an electromechanical instability in the sense that even if the grain is initially
at rest a sufficiently strong electrostatic field will cause the grain to start vibrating.’
L. Y. Gorelik, A. Isacsson, M. V. Voinova, B. Kasemo, R. I. Shekhter, and M. Jonson 1998 Phys. Rev. Lett. 80 4526.
A tunable carbon nanotube electromechanical oscillator
V. Sazonova, Y. Yaish, H. Üstünel, D. Roundy, T. Arias & P.L. McEuen, Nature 2004
Device geometry and diagram of experimental set-up.
a, A false-color SEM image of a suspended device (top) and a schematic of device geometry (bottom). Scale bar, 300 nm. Metal electrodes (Au/Cr) are shown in yellow, and the silicon oxide surface in grey. The sides of the trench, typically 1.2–1.5 µm wide and 500 nm deep, are marked with dashed lines. A suspended nanotube can be seen bridging the trench.
b, A diagram of the experimental set-up. A local oscillator (LO) voltage (usually around 7 mV) is applied to the source (S) electrode at a frequency offset from the high frequency (HF) gate voltage signal by an intermediate frequency of 10 kHz.
6. Describe the geometry and the principle of work of tunable carbon nanotube electromechanical oscillators. What are their resonance
frequencies? How do actuation and detection of nanotube motion achieved? What are different regimes of oscillations and their
oscillation modes? How could motion of nanotube be detected?
Carbon nanotube electromechanical oscillator
V. Sazonova, Y. Yaish, H. Üstünel, D. Roundy, T. Arias & P.L. McEuen, Nature 2004
• Resonance frequencies vary from 3 to 200MHzfor different samples and gate voltages.• A motion of 0.5nm was detected with forcesensitivity of 10-15 NHz-1/2
• Actuation and detection ofnanotube motion was achievedusing the electrostatic interactionwith the gate electrode underneaththe tube.• The attraction between thecharge q and its opposite charge -qon the gate causes an electrostaticforce downward on the nanotube.• The AC voltage produces aperiodic electric force, which setsthe nanotube into motion. As thedriving frequency approaches theresonance frequency of the tube,the displacement becomes large.• Three different regimes ofoscillations and several oscillationmodes was observed andexplained.
High-frequency nanotube resonators
(a) AFM image of the device. The scale bar is 300 nm. The white arrows point to the nanotube. (b) Schematic of the actuation/detection setup. A frequency modulated voltage VFM is applied to the device. The motion is detected by measuring the mixing current Imix.
J. Chaste et al. Cond. Mat. ArXiv 1207 4874
Xe atoms added
Resonance frequency can be as high as 4.2 GHz for the fundamental eigenmode and 11 GHz for higher order eigenmodes.
Detection technique
• The devices show resonances in the megahertz range, and the strong dependence of resonant frequency on applied gate voltage can be fitted to a membrane model to yield the mass density and built-in strain of the graphene.
• Following the removal and addition of mass, changes in both density and strain are observed, indicating that adsorbents impart tension to the graphene.
• On cooling, the frequency increases, and the shift rate can be used to measure the unusual negative thermal expansion coefficient of graphene.
• The quality factor increases with decreasing temperature, reaching 104 at 5 K.
• By establishing many of the basic attributes of monolayer graphene resonators, the groundwork for applications of these devices, including high-sensitivity mass detectors, is put in place.
Graphene nanomechanical resonators
7. Can monolayer graphene be used in nanomechanical resonators? What could be advantage of graphene comparable with other
materials? Describe a model that could be used to describe these resonators. How do their frequency and quality factor change with
cooling? What could graphene resonators be used for?
Monolayer graphene nanomechanical resonators
Columbia group, Nature Nanotechnology 2009
The enormous stiffness and low density of graphene make it an ideal material for nano-electromechanical applications.
Schematic of suspended graphene.
The SiO2 under the entire graphene flake is etched evenly,
including the contact region
SEM image of several resonators
SEM image of suspended graphene nanoribbon
Diagram of electronic circuit
Electrothermally Tunable Graphene Resonators Operating at Very High Temperature up to 1200KFan Ye, Jaesung Lee, and Philip X.-L. FengNano Lett., 31 Jan 2018
Graphene nanomechanical resonators
‘The unique negative thermal expansion coefficient and remarkable thermal stabilityof graphene make it an ideal candidate for nanoelectromechanical systems (NEMS)with electrothermal tuning... negative thermal expansion coefficient of graphene andits excellent tolerance to very high temperature can be exploited for engineeringhighly tunable and robust graphene transducers for harsh and extremeenvironments.’
A potentially quantum-limited displacement sensing
Electron micrograph of a doubly-clamped beam coupled to a single-
electron transistor
The resonating beam includes an
inter-digitated capacitor Cc(x)coupled to the gate of the single-electron transistor; the
displacement x of the beam center point changes the coupled charge, and modulates the current through the transistor.
The sensitivity is limited by the signal-to noise ratio. For the device under consideration the noise-limited sensitivity is estimated as 4 × 10−14 m (the radius of the proton is about 0.84–0.87 10-15 m).
SET-based displacement sensing
8. How can displacement be sensed with the help of single electron transistor? Describe the geometry and principle of work of a
relevant device. What is its noise-limited sensitivity? Is it comparable with the radius of proton?
Nanometer-scale mechanical electrometer
The inset schematically depicts itsprincipal components: torsion mechanical resonator, detection electrode, and gate electrode used to couple charge to the mechanical element. An external, parallel magnetic field is employed for readout.
The resonator was fabricated from a single-crystal Si-on-insulator substrate, with a 0.2-µm-thick Si layer on a 0.4-µm insulating layer.
The Au electrodes and resonator structure are patterned using electron beam lithography.
Perspective: to reach the charge sensitivity comparable with the sensitivity of cryogenic SETs
A. N. Cleland & M. L. Roukes, Nature 1998
10. Give an example of nanometer-scale mechanical electrometer. What is the principle of its readout? What could be its advantage
comparable with single electron transistor?
A bolometer consists of a volume whose temperature is changed by absorbing infrared light, and a thermistor element that detects the changes in temperature.
A nanometer-scale bolometer, based on a mechanically suspended volume of semi-insulating GaAs, which is patterned using EBL to define a volume, 2 × 3 × 0.2 μm3, suspended by thin, 4 × 0.2 × 0.2 μm3 legs from the bulk substrate.
The thermistor element is a Al-AlO-Cu-AlO-Al tunnel (SINIS) junction, fabricated on the surface of the GaAs. The tunnel characteristics of the junction are sensitive to the temperature of the electrons in this normal metal.
Suspended structures provide low specific heat and thermal conductance.
This is due to a specific vibration spectrum of suspended membranes and beams.
A. N. Cleland’s group, USA
Nanoscale bolometer
11. Describe a nanoscale bolometer. What is the advantage of
its small volume and principle of suspension? What kind of
thermistor element may it use?
The standard setup of optomechanics
An optical cavity driven by a laser impinging on the cavity through a fixed mirror.
The other mirror of the cavity is movable. For example, it may be attached to a micro-cantilever as used in atomic force spectroscopy.
In such a setup the mechanical effects of light are enhanced, as the light field is resonantly increased in the cavity and each photon will transfer momentum to the mirror in each of the numerous reflections it undergoes, until finally leaving the cavity.
The coupling can be described through dependence of the cavity’s resonance frequency on the displacement of the cantilever.
Optomechanical devices
15. Describe an optomechanical device. What is the principle of the enhancement of the mechanical effects of light? How is cavity’s
resonance frequency influenced by the displacement of cantilever?
Near-field cavity optomechanics
‘Cavity-enhanced radiation-pressure coupling between optical andmechanical degrees of freedom allows quantum-limited positionmeasurements and gives rise to dynamical backaction, enablingamplification and cooling of mechanical motion. Here, we demonstratepurely dispersive coupling of high-Q nanomechanical oscillators to anultrahigh-finesse optical microresonator via its evanescent field, extendingcavity optomechanics to nanomechanical oscillators. Dynamical backactionmediated by the optical dipole force is observed, leading to laser-likecoherent nanomechanical oscillations solely due to radiation pressure.’
Near-field cavity optomechanicswith nanomechanical oscillators,
G. Anetsberger et al., Germany, Switzerland
16. Comment on near-field cavity optomechanics devices linked with nanomechanical oscillators. Do they allow quantum-limited position
measurements or enable amplification and cooling of mechanical motion? Can they be used as readout techniques for quantum
nanomechanical motion? Are there any chances of using them at room temperature?
Near-field cavity optomechanics
‘Moreover, sub-fm Hz1/2 displacement sensitivity is achieved, with ameasurement imprecision equal to the standard quantum limit (SQL), whichcoincides with the nanomechanical oscillator’s zero-point fluctuations. Theachievement of an imprecision at the SQL and radiation-pressure dynamicalbackaction for nanomechanical oscillators may have implications not only fordetecting quantum phenomena in mechanical systems, but also for a variety ofother precision experiments. Owing to the flexibility of the near-field couplingplatform, it can be readily extended to a diverse set of nanomechanicaloscillators. In addition, the approach provides a route to experiments whereradiation-pressure quantum backaction dominates at room temperature,enabling ponderomotive squeezing or quantum non-demolitionmeasurements.’
Near-field cavity optomechanicswith nanomechanical oscillators,
G. Anetsberger et al., Germany, Switzerland
In physics, a ponderomotive force is a nonlinear force that a charged particle experiences in an inhomogeneous oscillating electromagnetic field.
www.photonics.com
Optical microresonators
http://phys.org/news180363327.html
Schematic of the experiment, showing a tapered-fiber-interfaced optical microresonator dispersively coupled to an array of nanomechanical oscillators
Scanning electron micrograph of an array of doubly clamped SiN nanostring oscillators with dimensions 110nm x ( 300–500)nm x (15–40) mkm.
Scanning electron micrograph of a toroid silica microcavityacting as an optomechanicalnear-field sensor.
Optomechanical devices offer sensitive readout techniques for quantum nanomechanicalmotion. They can act as parametric motion transducers based on behavior of photons in acavity.Example: evanescently coupled to the tightly confined optical field of an ultrahigh-finessetoroidal silica microresonator, high-Q nanomechanical oscillators .
Movie 2
Readout techniques for quantum nanomechanical motion
The finesse of an optical resonator (cavity) is fullydetermined by the resonator losses and is independent ofthe resonator length. If a fraction ρ of the circulating power isleft after one round-trip (i.e., a fraction 1 − ρ of the power islost) when there is no incident field from outside theresonator, the finesse is
where the approximation holds for low round-trip losses (e.g., <10%), i.e., only for high finesse values.
The finesse of an optical resonator
F = 230,000
DOI: 10.1038/NPHYS1425
Development of NEMS resonators coupled to a nanoSQUID, T B Patel et al., National PhysicalLaboratory, UK, University College London, UK, Physikalisch-Technische Bundesanstalt, Germany
The growing interest in nanoelectromechanical systems (NEMS) devices is due to vast potentialapplications in both classical and quantum measurements.
At low temperatures, nanoscale mechanical resonators exhibit quantum behavior. NanoscaleSuperconducting Quantum Interference Devices (nanoSQUIDs) are capable of detecting thisquantum behavior which is subsequently converted into an electrical readout.
The eventual aim of this project is to use a graphene membrane as the resonator in a ‘drum’configuration. The mechanical motion of this conducting membrane will be sensed by a closelycoupled nanoSQUID.
For effective direct coupling, the size of the SQUID coil must match the shape and size of themoving parts of the NEMS resonator. Thus the SQUID loop must be sub-micron in size. Byfabricating weak link ‘nanobridge’ junctions made by focused ion beam milling (FIB) we are ableto produce junctions approximately 65nm in length and width and SQUID loop dimensionsdown to 1μm by 100nm.
• Micro- and nanomechanics deals with modeling, design, fabrication and applications of three-dimensional structures with dimensions of micrometers and below.
• These systems incorporate a number of unusual features. The classical fabrication methods of micromachining are extended by those developed in the semiconductor industry during the passed decades. Different quantities scale differently when moving from large to small structures demanding new models to describe physics on small scale.
• Devices used to perform certain function on macroscopic scale are replaced by others exploiting various physical effects of microscopic world.
• The limits of classical continuum mechanics have to be explored and extended. New methods need to be developed in order to quantify bonding between different layers, residual stresses caused by the manufacturing processes as well as elastic constants.
Summary