nanomechanicsnanomechanics andand mesoscopicmesoscopic phononsphonons

Courtesy CT-Cnyugen (Berkeley)

Smaller, cheaper, faster, lower power consumption

“Phones of the future”: NEM-devices are in the right frequency range (1-5 GHz) to replace elements in cell phones

Better frequency selectivity (higher Q)

Bulk passive components replaced by smaller size MEMS/NEMS components

New sensor applications

Why NEMS? Device applicationsWhy NEMS? Device applications

needed: high Q; high frequency

Nanotube RAM (including movie): http://www.nantero.com/index.html

mass sensing on the level of single moleculesmass sensing on the level of single molecules

Doubly-clamped beamDoubly-clamped beam

m

M

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

ω / ω0

Am

plitu

de ∂ω ω0 012min

~Q

mass loading

QmM /min ≈∂f0 L × w × t (µm) Meff Q ∂ mmin (d)

7.7 MHz 10 x 0.2 x 0.1 230 fg 10,000 1.4×108 " " " 100,000 1.4×107

380 MHz 1 x 0.05 x 0.05 3 fg 10,000 1.8×106 " " " 100,000 1.8×105

7.7 GHz 0.1 x 0.01 x 0.01 23 ag 10,000 1400 " " " 100,000 140

low M (high f); high Q

carboncarbon--nanotube nanotube oscillators towards oscillators towards zeptogram zeptogram detectiondetection

M. Nishio et al. APL 86 (2005) 133111

nanorelaysnanorelays: : lowlow--powerpower mechanical switchesmechanical switches

S.N. Cha et al. APL 86 (2005) 083105 S.W. Lee et al. Nanoletters (2004)

nanotubenanotube--nanomechanicsnanomechanics: rotator: rotator

rotating mirror

low-friction internal motion of the shells of a multi-wall nanotube makes motion possible

Fennimore et al, Nature 286 (1999) 2148

biobio--nanomechanics nanomechanics IIII

bio-motors: ATP fuelled biomotor with a fluorescent filament of a few micron length attached (time between pictures 133 ms)

H. Nori et al. Nature 386 (1997) 299

nanomechanicsnanomechanics of breaking of breaking an atomic gold wirean atomic gold wire

estimate force:

nN2

N/m10

eV 3 nm2.0

0

2

2

0

==

≈∂

∂≈

==

kxFx

V(x)k

Ex

b

G. Rubio et al., PRL 67 (1996) 2302

vibrationalvibrational modes in Ptmodes in Pt--HH22--Pt junctionsPt junctions

lithographic break junctionInelastic tunneling spectroscopy (IETS)

Djukic et al. cond/mat 0409640

-0,7 -0,6 -0,5 -0,4 -0,3 -0,2 -0,1 0,0

45

50

85

90PtD2 stretching

Ener

gy (m

eV)

δX (A)

stretching the contact: longitudinal or stretching the contact: longitudinal or transversal modes?transversal modes?

-100 -80 -60 -40 -20 0 20 40 60 80 100

-0,1

0,0

0,1

0,981,001,021,04

same as stretched by 0.05 nm

d2I/d

V2 (a

.u)

dI/d

V (G

0)

Energy (meV)

mk

=ω

CasimirCasimir force: a force of nothingforce: a force of nothing

L∫

∑∑∞

∞

=

=

==

0

1

d :cavity theoutside

:cavity theinside

nnL

cE

nL

cEnk

h

hh

π

πω

2

0 0

1 0

12force

12

0)0('',121)0(',0)0(

......)0("720

1)0('121)0(

21d)()(

:Maclauren-Euler

d

Lc

LcE

fff nf(n)

fffnnfnf

nnnL

cE

n

n

hh

h

ππ

π

=⇒−=∆

=−==⇒=

−+−=−

−=∆

∑ ∫

∑ ∫

∞

=

∞

∞

=

∞

can be generalized to other geometries (3D, plate-sphere), temperature effects and non-ideal conductors

vacuum fluctuations: vacuum fluctuations: Casimir Casimir effecteffectIn absence of electrostatic forces, attractive force between plate and sphere: F ∝ z-3

z-3

z-2

MEMS technology(MicroElectroMechanical Systems)

H.B. Chan et al., Science 291 (2001) 1941

mesoscopicmesoscopic phononsphonons

Tkhv

B

λνmax max

= ≈ =c hc

k Tnm

TB3112

Inte

nsity

0 2 4 6 8 10 12 14

3max ≈Tk

hv

B

Planck Distribution

λν

max

max

=

=

11625

nmGHz

λ µν

max

max

=

=

11625

mMHz

at 10K,

at 10 mK,

By nanofabrication techniques, we can enter domain where particle wavelength is comparable to channel dimensions.

Courtesy Keith Schwab

the thermal conductance “quantum”

[ ]

Thk

TJG

TJ

kkv

ekvkkJ

vnJ

BTOTth

TTOT

Tk B

3

d

dd)(;

11)()()()(

2d

220

/

π

ηηωηηω

ωωηωηωπ

ω

η

ω

=∆

=

∆=−−=

=−

==

=

∂∂−+

∞−+∫

∫

h

h

h

h

the thermal conductance “quantum”derivation analogous to that of the electrical conductance quantum but

look at the energy flux (instead of electron flux) and use Bose-Einstein statistics(consider two reservoirs with a T difference ∆T coupled by a 1D channel)

Wiedemann-Franz law:Gth = LGT, L= Lorentz constantWith G0 = e2/h one finds Gth = π2kB2T/3h

the existence of a thermal conductance quantum also indicates that it is difficult to cool nano-objects to their ground state !!

quantum phonon transportquantum phonon transport

g0 = π2kB2 T/3h

Schwab et al, Nature 404 (2000) 974

eigenfrequencieseigenfrequencies cantilevers and beamscantilevers and beams

classical elasticity, Euler-Bernoulli theory:

solve and use appropriate boundary conditions

top-down fabrication: surface nanomachining

1.starting material:heterostructurewith a singlestructural layer

4.selective etch:removes thesacrificial layer;final structuresare suspended

3.pattern transfer:anisotropic etch definesstructurevertically

2.mask definition:by optical ande-beam lithographyand thin film deposition

• high resolution e-beam lithography (x,y) • mono-crystalline epitaxial layers (z)

• GaAs-based systems• Si-based systems (SOI)• Silicon Carbide

eigenfrequencieseigenfrequencies cantilevercantilever

0)(''')(''0)0(')0(

====LL ξξ

ξξ

0)cos()cosh(1 =+ kLkL)12(

2......,,6941.4,8751.1; 21 −==== nLk nnn

παααα

ρπα

12212 223 E

LHfWHI nn =⇒=

boundary conditions:

for the first modes: not harmonicsound velocity: v = (E/ρ)1/2

eigenmodeseigenmodes of cantilever of cantilever nanotube nanotube

TEM work: Gao et al, PRL 85 (2000) 622 P. Poncheral et al., Science 283 (1999) 1513

eigenfrequencieseigenfrequencies double clamped beamsdouble clamped beams

0)(')(0)0(')0(

====

LL ξξξξ

0)cos()cosh(1 =− LkLk nn)12(

2......,,8532.7,7300.4; 21 +==== nLk nnn

πββββ

ρπβ

12212 223 ELHfWHI nn =⇒=

boundary conditions:

silicon:L=1 µm, H=W=0.1 µm, f0= 1GHz

nanotube:L= 100 nm, d = 1.4 nm, f0= 5 GHz

double clamped strings/beamsdouble clamped strings/beams

022

4

4

=−∂∂

−∂∂ K

zuT

zuEI

APL 78 (2001) 162

frequency ∝ L-1frequency ∝ L-2

magnetomotive method: conducting beam (50 Ω)

V

Cleland and Roukes, Sensors and Actuators 72, 256 (1999)

example of a measurementexample of a measurement

Rm ∝ B2

Q decreases with B

For Q > 10: response of the damped, driven the same as that of a harmonic oscillator.Lorentzian peak shape; the width defines the Q-factor.

increasing drive: nonlinear response increasing drive: nonlinear response ((DuffingDuffing oscillator)oscillator)

“Jimmy Hendrix regime”

Husain et al, APL 83 (2003) 1240

interesting for applications (parametric amplification) and for new quantum effects (Thorwart et al. (2005))

quantumquantum--limited detection of motionlimited detection of motion

detection of gravitational waves

The MiniGRAIL detector is a cryogenic 68 cm diameter spherical gravitational wave antenna made of CuAl(6%) alloy with a mass of 1400 Kg, a resonance frequency of 2.9 kHz and a bandwidth around 230 Hz, possibly higher. The quantum-limited strain sensitivity dL/L would be ~4x10-21. The antenna will operate at a temperature of 20 mK. An other similar detector is being built in São Paulo, which will strongly increase the chances of detection by looking at coincidences. The sources we are aiming at are for instance, non-axisymmetricinstabilities in rotating single and binary neutron stars, small black-hole or neutron-star mergers etc.

MiniGRAIL: Frossati (Leiden University)

quantum effects in a harmonic oscillatorquantum effects in a harmonic oscillator

mpkxH

22

21

21

+=

+=⇒

21nEn ωh

-2 0 2

E=7hv/2

E=5hv/2

E=3hv/2

n=0

n=3

n=2

n=1

E=hv/2

Ene

rgy

position

|0>

|1>

|2>

zero-point motion:

00 ωm

x h=

energy stored in the oscillator:1e2 / −

+= TkBE ωωω

h

hh

quantum regime if occupation number (Nth) is of order one: 1 GHz ⇒ Tsample < 50 mK

zero point motionzero point motion

expected rms displacement noise of a 4.4 GHz nanotube resonator as a function of temperature

needed: high Q; high frequency, low mass

MEMS devices as electrometersMEMS devices as electrometers

•measured sensitivity (300 K): 0.1 eHz-1/2

•ultimate sens. (300 K): 2 x 10 -5 eHz-1/2

Cleland and Roukes, Nature 392 (1998) 160

towardstowards zerozero--point motion detection using point motion detection using a SET and a mixing techniquea SET and a mixing technique

00 ωm

x h=

x0 a factor 100 above the quantum limit; Nth = 10

Knobel and Cleland, Nature 424 (2003) 291

almostalmost quantum detection limit with RFquantum detection limit with RF--SETSET

22

21

21

RMSB xmTk ω=

x0 a factor 4.3 above the quantum limit; Nth = 58

LaHaye et al., Science 304 (2004) 74

Coulomb blockade in SET that can moveCoulomb blockade in SET that can move

• usual CB theory does not apply since the gate capacitance is distance dependent

• mechanical degrees of freedom via classical theory of elasticity

• nanotube modeled as an elastic rod

• applicable to other suspended structures

• nanotube: hope for high Q-factors (smooth on a nanometer level)

S. Sapmaz et al., PRB 67 (2003) 235414

bottombottom--up fabrication: suspended up fabrication: suspended nanotubesnanotubes

AFM markers

catalyst

ISWNTs

electrodes catalyst + nanotubes

electrodes ISWNT lateral Gate

bottombottom--up fabrication: suspended up fabrication: suspended nanotubesnanotubes

Aunanotube

Crbefore etchingSiO2

doped-Si (backgate)

after etching in acid (BHF)

Nygård and Cobden, APL 79 (2001) 4216

electrostatic term in Coulomb Blockade is electrostatic term in Coulomb Blockade is displacement dependentdisplacement dependent

Σ

+≠

CQneW

2)( 2

Electrostatic force (z

the the nanotube nanotube moves in discrete steps every time moves in discrete steps every time an additional electron tunnels onto it an additional electron tunnels onto it

0 10

1

2

3

~VG

2/3

~VG

2

z max

(nm

)V

G (V)

S. Sapmaz et al., PRB 67 (2003) 235414

r = 0.65

r = 0.65 nm L = 500 nmR = 100 nmT0= 0; CR=CL=0

weak bending:zmax < r, zmax ∝ VG2

strong bending:zmax > r, zmax ∝ VG2/3

As soon as zmax>d, the strain due to deformation has to be taken into account

eV

new equilibrium position

l

electronelectron--vibron coupling in a movable dotvibron coupling in a movable dot

xxmm

pH λω ++= 2202

21

2)

λ: e-ph coupling

ωλ

mg hl

l

l=

== 0

2

0212

21

nmPRL,2

beforeafterRL,RL, | Γ=ΨΨΓ⇒Γ

in practice: steps visible if g > 0.1

Frank - Condon factors determine step heights in IV

!)(|)(

20 n

gexxPng

n

−

=Ψ−Ψ= lFrank-Condon factors: overlap

between the harmonic oscillator wave functions in the

initial ground state and the displaced final electronic state Braig and Flensberg, PRB 68 (2003) 205323

FrankFrank--Condon factors reproduce step heights in Condon factors reproduce step heights in the experiment reasonably well

g = 0.5g = 0.95g = 0.9

the experiment reasonably well

e-ph coupling constant is about 1 and approximately length and gate independent: displacement ~1 pm !

a nanotube of 1 micron length acts as a single quantum harmonic oscillator (ћω >> kBT) !!!

A better fit may be obtained if the influence of a gate voltage and asymmetric coupling is considered (Braig and Flensberg, 2003) or if non-equilibrium phonons are present (Mitra, Aleiner and Milles, 2004); Negative Differential Resistance not understood

some some vibrationalvibrational modes modes nanotubesnanotubes

squashing modef~0.54-1.0 THz, E~2.2-4.2 meV

bending mode

f~2.5 GHz-25 MHz(100 nm-1 µm)E~10-0.1 µeV

Length indep.Length dep.

radial breathing modef~4.5-6.2 THzE~18-25 meV

f~1.3-0.13 THz(100 nm-1 µm)E~600-60 µeV

stretching (longitudinal) mode

data fit longitudinal (stretching) modes the data fit longitudinal (stretching) modes the bestbest

stretching

RBM

bendingd ≈ 1.5 nm E = 0.1 µV L[µm]-2

E = 110 µV L[µm]-1

E = 18.7 meV

E = 1.67 meV L[µm]-1elec

0.0 0.5 1.0

10-4

10-3

10-2

10-1

100

101

E (m

eV)

L (µm)

Sapmaz et al. PRL 96 (2006) 26801