Yaroslav M. Blanterindico.ictp.it/event/7991/session/0/contribution/2/material/slides/0.pdf · Yaroslav M. Blanter ICTP, September 2017 Piezoconductivity in graphene Deformation:

ICTP, September 2017Yaroslav M. Blanter

Nanomechanics

Yaroslav M. Blanter

Kavli Institute of Nanoscience, Delft University of Technology

Coupling Detection Doubly-clamped beams Graphene and 2D materials


Mechanical oscillator

200 cos

Fx x x t

Q M

ww w+ + =&& &

A

w0w

( )cosx A tw q= +A peak of the amplitude and a jump of the phase

Driven harmonic oscillator:


Mechanical oscillator

200

( , )( )

( )

F tx x x

xf

Q Mx

x

ww+ + + =&& &

( )cosx A tw q¹ +

More advanced oscillator:

Nano- or optomechanical system:

200

( , , )( )

( )

F tx x x

xx

x

yf

Q M

ww+ + + =&& &

and another equation for evolution of y – the degree of freedom to which the mechanical oscillator is coupled


Coupling

Mechanical resonators can be coupled to:

– Charge: capacitive coupling– EM radiation: radiation pressure coupling– Spin– Magnetic flux: inductive coupling– Other mechanical resonators


Capacitive coupling

Couples phonons to charge due tothe Coulomb-induced force

L R

G

Lz

[ ]C z

Electrostatic energy:

In the simplest form

2 2[ ]

2 [ ] 2

Q C z V

C z=

(Other mechanisms of coupling to the charge: e.g. piezoelectric coupling)


Capacitive coupling

T. Miao, S. Yeom, P. Wang,B. Standley, M. Bockrath, Nano Lett. 14, 2982 (2014)

Graphene resonatoron a hole over a backgate


Radiation pressure coupling

2

( )2m

cav

M xH x n

ww= +h

Cavity Mechanical resonator

( ) cavcav cavx x

x

ww w

¶» +

¶

Kippenberg's group website

Movable mirror Static mirror

Radiation pressure: cav xnx

w¶¶h

Other mechanisms of interaction withradiation: e.g optical phononsin bulk solids


Radiation pressure coupling

S. Hong, R. Riedinger, I. Marinkovic, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, S. Gröblacher, arXiv:1706.03777


Coupling to spin

D. Rugar, R. Budakian, H. J. Mamin,B. W. Chui, Nature 430, 329 (2004)

Two magnets exert mechanicalforce on each other, dependent on their orientation

A magnet on a cantilever cansense a spin

Magnetic resonance forcemicroscopy


Coupling to spin

O. Arcizet, V. Jacques, A. Siria, P. Poncharal, P. Vincent, and S. Seidelin Nature Physics 7, 879 (2011)


Coherent spin driving

A. Barfuss, J. Teissier, E. Neu, A. Nunnenkamp, P. Maletinsky Nature Physics 11, 820 (2015)

Spin state depends on the strain;piezoelectric substrate driven


Inductive coupling

See Lecture 3

Inductance: can depend on the position of a mechanical resonator

LI V LIF = Þ -F = = && 2( )

2

L x IE =


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 Aharonov-Bohm flux

Energy levels vs. flux

RA

0

0-2

2 -2

2

2

2/

2E

mR

h


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!


A. C. Bleszynski-Jayich, W. E. Shanks,B. Peaudecerf, E. Ginossar, F. von Oppen, L. Glazman, J. G. E. HarrisScience 326, 272 (2009)

Measurements of persistent currents

Currents produce tork and shiftthe cantilever frequency

Good agreement with the theorypredictions

Bt m= ´


Coherent two-mode manipulation

T. Faust, J. Rieger, M. J. Seitner, P. Krenn, J. P. Kotthaus, E. M. Weig, PRL 109, 037205 (2012)

T. Faust, J. Rieger, M. J. Seitner, J. P. Kotthaus, E. M. Weig, Nature Physics 9, 485 (2013)


Mass detection

Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, M. L. Roukes,Nano Letters 6, 583 (2006)

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


M. S. Hanay, S. Kelber, A. K. Naik, D. Chi, S. Hentz, E. C. Bullard,E. Colinet, L. Duraffourg, M. L. Roukes, Nature Nanotech. 7, 602 (2012)

Single-molecule detection

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, A. BachtoldNature Nanotech. 7, 301 (2012) – 1 yg resolution



Double-clamped beam

Couples phonons to charge due tothe Coulomb-induced force

G. Steele, A. K. Hüttel, B. Witkamp, M. Poot, H. B. Meerwaldt, L. P. Kouwenhoven, H. S. J. van der Zant Science 325, 1103 (2009)

Size: 500 nmFrequency: 140 MHz

S. Sapmaz . YMB. L. Gurevich, H. S. J. van der Zant, PRB 67, 235414 (2003)

L R

G

Lz

[ ]C z


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

[ ]

[ ]

MMx x M x F x

Q

F x kx x x

ww

b a

+ + =

= -D - -

&& &( )21

( ) ( )2 g g CNT

dF C x V V x

dx= -

The beam is stretched by the gate voltage, and this shifts the frequency (optical spring effect)


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 =


Frequency softening by Coulomb effects

0 1 tot

g g g

NC e

C C Vw

¶D µ - -

¶

N

CgVg/e

-4

-2

0

2

4

-4 -3 -2 -1 0 1 2 3 4

(zero bias)



Frequency softening by Coulomb effects

0 1 tot

g g g

NC e

C C Vw

¶D µ - -

¶



Coulomb-induced nonlinearity

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


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)F. Pistolesi, YMB, and I. Martin, PRB 78, 085127 (2008)


Backaction in SET coupled to a resonator

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

( , , )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


Strain in graphene

H. Suzuura, T. Ando Phys. Rev. B 65, 235412F. Guinea, M. I. Katsnelson, M. A. H. Vozmediano Phys. Rev. B 77, 075422 (2008)

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

( );

2 ; 3

x xx yy

y xy

A t u u

A t u

b

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)


Piezoconductivity in graphene

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

M. Fogler, F. Guinea, M. I. KatsnelsonPhys. Rev. Lett. 101, 226804 (2008)

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


Piezoconductivity in graphene

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


Strain engineering in MoS2

A. Castellanos-Gomez, R. Roldán, E. Cappelluti, M. Buschema, F. Guinea, H. S. J. van der Zant, G. A. Steele, Nano Lett. 13, 5361 (2013)

Wrinkles: large strain difference


Strain engineering in MoS2

Funneling of excitonsA. Castellanos-Gomez, R. Roldán, E. Cappelluti, M. Buschema, F. Guinea, H. S. J. van der Zant, G. A. Steele, Nano Lett. 13, 5361 (2013)

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Yaroslav M. Blanterindico.ictp.it/event/7991/session/0/contribution/2/material/slides/0.pdf · Yaroslav M. Blanter ICTP, September 2017 Piezoconductivity in graphene Deformation: