Capri School, April 2016Yaroslav M. Blanter

Nanomechanics

Yaroslav M. Blanter

Kavli Institute of Nanoscience, Delft University of Technology

Detection Shuttling Doubly-clamped beams Graphene Strain engineering

Capri School, April 2016Yaroslav M. Blanter

Persistent currents

22

22

20

22

20

ˆ ˆ2

ˆ ;2

( )

2

iN

eH p A

m c

H imR

e

E NmR

f

f

f

æ ö= - Þç ÷è ø

æ ö¶ F= - -ç ÷¶ Fè ø

Y µ Þ

æ öF= -ç ÷Fè ø

r rh

h

h

Interference is affected by Aharomov-Bohm flux

Energy levels vs. flux

RA

0

0-2

2 -2

22

2/

2E

mR

h

Capri School, April 2016Yaroslav M. Blanter

Measurements of persistent currents

A. C. Bleszynski-Jayich et al, Science 326, 272 (2009)

i

filledlevels

EEI

¶¶= =¶F ¶Få

Amplitude clean, single ring:2

e

mR

h

Amplitude disordered:

20

2 4sin ,l l

l

l eI I I

p dp

F= = -

Få h

Difficult to measure!

Capri School, April 2016Yaroslav M. Blanter

A. C. Bleszynski-Jayich et al, Science 326, 272 (2009)

Measurements of persistent currents

Currents produce tork and shiftthe cantilever frequency

Good agreement with the theorypredictions

Bt m= ´

Capri School, April 2016Yaroslav M. Blanter

Mass detection

Y. T. Yang et al (Roukes group, Caltech) Nano Letters 6, 583 (2006)

Resonant frequency: 133 MHzSize: 2300 x 150 x 70 nmMass sensitivity: 100 zg

Capri School, April 2016Yaroslav M. Blanter

M. S. Hanay et al, Nature nanotech. 7, 602 (2012)

Single-molecule detection

Capri School, April 2016Yaroslav M. Blanter

Shuttling

Couples phonons to charge due tothe position dependence of tunnelrates and of Coulomb-induced force

D. R. König and E. M. Weig APL 101, 213111 (2012)

L. Y. Gorelik et al PRL 80, 4526 (1998)

S D

Capri School, April 2016Yaroslav M. Blanter

Double-clamped beam

Couples phonons to charge due tothe Coulomb-induced force

G. Steele et al Science 325, 1103 (2009)

Size: 500 nmFrequency: 140 MHzQ-factor: over

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

L R

G

Lz

[ ]C z

510

Capri School, April 2016Yaroslav M. Blanter

Backaction in a double-clamped beam

H. B. Meerwaldt, G. Labadze, B. H. Schneider, A. Taspinar, YMB, H. S. J. van der Zant, and G. A. Steele, PRB 86, 115454 (2012)

200

2 3

cos [ ]

[ ]

RF

MMx x M x F t F x

Q

F x kx x x

ww w

b a

+ + = +

= -D - -

&& &

( )21

2g

g CNT

dCF V V

dx= -

Renormalization of frequency and introducing of Duffing parameter

Capri School, April 2016Yaroslav M. Blanter

Frequency stiffening by Coulomb effects

0 ~ 1 tot

g g g

NC e

C C Vw

¶D µ - -

¶G. Steele et al Science 325, 1103 (2009)

Capri School, April 2016Yaroslav M. Blanter

Coulomb blockade

LV

0RV =

1nS +

nS

Conditions that current is not flowing:

1n LS eV+ >(a)

1 0nS + >(c)

n LS eV<(b)

0nS <(d)GV

V

0n =1n =

Capri School, April 2016Yaroslav M. Blanter

Frequency softening by Coulomb effects

0 ~ 1 tot

g g g

NC e

C C Vw

¶D µ - -

¶

H. B. Meerwaldt et al, PRB 86, 115454 (2012)

Capri School, April 2016Yaroslav M. Blanter

Coulomb-induced nonlinearity

2 3[ ]F x kx x xb a= -D - -

Capri School, April 2016Yaroslav M. Blanter

Coulomb-induced damping

0 0

0

1stoch g g

g tot g

F V dC N

Q Q mC dx V

w w ¶= +

G ¶

O. Usmani, YMB, and Yu. V.Nazarov, PRB 75, 195312 (2007)

Capri School, April 2016Yaroslav M. Blanter

Backaction in SET coupled to a resonator

O. Usmani, YMB, and Yu. V. Nazarov, ‘04, '07

( , , )nP x v t

[ ]nn

P Fv P St P

t x v M

¶ ¶ ¶æ ö+ + =ç ÷¶ ¶ ¶è øAdiabaticity: reduce to Fokker-Planck equation

- obeys master equation x – positionv - velocity

2( , )F x v M x M v Fnw g= - - +

22

2( ) ( ) ( )

P P P Pv x x vP D x

t x v Q v v

ww é ù¶ ¶ ¶ ¶ ¶+ - - + Y =ê ú¶ ¶ ¶ ¶ ¶ë û

Built-in dissipationDissipation due to tunneling Diffusion in

velocity space2

2 3

( ) ( ) ( ) ( )( ) x x

t

x x x xFx

M

- + + -G ¶ G -G ¶ GY =

G

Capri School, April 2016Yaroslav M. Blanter

Coulomb-induced instabilities

0 0

0

1stoch g g

g tot g

F V dC N

Q Q mC dx V

w w ¶= +

G ¶

O. Usmani, Ya. M. Blanter, and Yu. V.Nazarov, PRB 75, 195312 (2007)

, ,( ),

,

2

( ( ))

L R L Ra E FxL R

F L R

e

f E Fx

D +±G =

´ ± D +

Capri School, April 2016Yaroslav M. Blanter

Strain in graphene

Suzuura, Ando ‘02Guinea, Katsnelson, Vozmediano '08

Deformation of a graphene sheet acts at electrons as pseudomagnetic field in the Dirac equation

( ); 2 ; 3x xx yy y xyA t u u A t ub b b= - = - »

Deformation caused by uniform load:

22 204

( )h

h r R rR

Uniform load Local load (center)

( ) ( ) ( )Fv p A r E rs + Y = Yrr r r rDirac equation: with added gauge fields

Deformation caused by local load:

( )2 2 202

1( ) ln

2

h Rh r R r r

R ræ ö= - -ç ÷è ø

R0h

K.-J.Kim, YMB, K.-H.AhnPhys. Rev. B 84, 081401 (2011)

Capri School, April 2016Yaroslav M. Blanter

Piezoconductivity in graphene

Deformation: Creates strain: pseudomagnetic gauge fields Creates density redistribution; the profile needs inprinciple to be calculated self-consistently

Fogler, Guinea, and KatsnelsonPhys. Rev. Lett. 101, 226804 (2008)

Local shift of the Dirac cones:Predicted metal-insulator transition at certaindeformation

Capri School, April 2016Yaroslav M. Blanter

Piezoconductivity in graphene

M. V. Medvedyeva and YMBPhys. Rev. B 83, 045426 (2011)

Capri School, April 2016Yaroslav M. Blanter

Strain engineering in MoS2

A. Castellanos-Gomez et alNano Lett. 13, 5361 (2013)

Wrinkles: large strain difference

Capri School, April 2016Yaroslav M. Blanter

Strain engineering in MoS2

A. Castellanos-Gomez et alNano Lett. 13, 5361 (2013)

Funneling of excitons