Casimir Friction

Aleksandr VolokitinResearch Center Jülich and Samara State Technical University

Casimir Forces Near-Field Radiative Heat Transfer

Casimir Friction

Non-Contact Friction

Frictional Drag

Fluctuational Electrodynamics

– p. 1

Fluctuation-Dissipation Theorem

mẍ+mω20x+ Γẋ = F (t)

Γ = (kBT )−1Re

∫

∞

0 dt〈

F̂(t)F̂(0)〉

The detection of single spins by MRFM for: (a) 3Datomic imaging (b) quantum computation

Measurements of gravitation force at short length scale

Measurements of Casimir forces

– p. 2

Rytov’s Theory

∇× E = iω

cB

∇×H = −iω

cD+

4π

cjf

〈

jfi (r)jf∗k (r

′)〉

ω=

~

(2π)2

(

1

2+ n(ω)

)

ω2Imεik(r, r′, ω)

n(ω) = [e~ω/kBT − 1]−1

– p. 3

Origin of Non-Contact Friction

A.I. Volokitin and B.N.J. Persson, Rev.Mod.Phys., 79, 1291(2007)

– p. 4

Non-contact Friction

Experiment: Stipe B.C. et.al PRL, 87, 096801 (2001)

Ffriction = Γv, Γ ∼ 10−13 − 10−12kg/s at d ∼ 1− 100 nm.

Γ ≈ d−n with n = 1.3± 0.3 and n = 0.5± 0.3, Γ ∼ V 2

Theory: A.I. Volokitin and B.N.J. Persson, PRB, 73, 165423(2006)

– p. 5

Electronic versus Phononic Friction

Experiment: M. Kisiel et.al Nat. Materials, 10, 119 (2011)

– p. 6

Electronic versus Phononic Friction

Theory: A.I. Volokitin and B.N.J. Persson, PRB, 73, 165423(2006)

– p. 7

Casimir Friction

v

ω+q vx ω-q vx

Pendry J.B. JPCM 9, 10301 (1997)A.I. Volokitin and B.N.J. Persson, JPCM, 11, 345 (1999)

– p. 8

Frictional Stress

σxz =

∫

∞

0

dω

2π

∑

i=(s,p)

∫

d2q

(2π)2~qxsgn(ω

′)(

n2(ω′)− n1(ω)

)

Γi(ω,q),

Γi(ω,q) =1

2kz | 1− e2ikzdR1i(ω)R2i(ω′) |2

×[

(kz + k∗

z)(1− | R1i(ω) |2)(1− | R2i(ω

′ |2)

+4(kz − k∗

z)ImR1i(ω)ImR2i(ω′)e−2dImkz

]

ω′ = ω−qxv, ni(ω) = [exp(~ω/kBTi)−1]−1, kz =

√

(ω/c)2 − q2

A.I. Volokitin and B.N.J. Persson, JPCM, 11, 345 (1999)

– p. 9

Small Velocities

v ≪ vT = kBTd/~, d ≪ λT = c~/kBT,

σxz = γv,

γ =~2

8π3kBT

∫

∞

0

dω

sinh(~ω/kBT )

∑

i=p,s

∫

d2qq2xe−2qd

×ImR1i(ω)ImR2i(ω)

| 1− e−2qdR1i(ω)R2i(ω) |2,

Γ =

∫

dSγ(z(x, y))

– p. 10

Adsorbate enhancement of Casimir friction

log d (Angstrom)

log

Γ (k

g/s)

Cs/Cu(100)

A.I. Volokitin and B.N.J. Persson, PRB, 73, 165423 (2006)

– p. 11

Radiative Heat Transfer.

0

1

2

3

4

5

6

7

8

9

10

0.5 1 1.5 2 2.5 3 3.5 4 4.50

5

10

1 2 3 4

log

(S /

Jm

s

)-2

-1

with adsorbates

clean surfaces

log (d / A)o

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

0.5 1 1.5 2 2.5 3

1 2 34

6

8

log

(S /

1Jm

s

)

-2-1

log (d / 1A)o

K/Cu(001) SiC

T1 = 273 K, T2 = 0 K

Volokitin A.I. and Persson B.N.J. PRB, 69, 045417 (2004)

– p. 12

Two ways to study Casimir friction

v

U

2

1

1

J = n ev22

+++

---

..

..

.

.

.

. ...

.......

.

.. .

..

.. .

..

.

. . .

..

.. .

.

.

.

. .

..

.

. .

. ..

. ..

..

.

..

Left:Upper block is sliding relative to block at the bottomRight: The current is induced in the upper block .

– p. 13

Frictional Drag in 2D-systems

V

Layer 2

Layer 1 J 1

E 2 (J =0)2

d

Theory - Coulomb Drag. M. B. PogrebenskiiSov.Phys.Second.,11 (1977) 372, P. J. Price PhysicaB+C,117 (1983) 750Experiment - Quantum wells T. J. Gramila et.al PRL,66(1991) 1216, U. Sivan et.al PRL,68 (1992) 1196Experiment - Graphene Sheets S. Kim et.al PRB,83 (2011)161401, R.V. Gorbachev et.al Nat.Phys.,8 (2012) 896Theory - Casimir Friction. A.I. Volokitin and B.N.J. Persson,J.Phys.:Condens.Matter, 13, 859 (2001); ibid, EPL 10324002 (2013)

– p. 14

Quantum Friction

qxv = ω1 + ω2Pendry J.B. JPCM 9, 10301 (1997)Volokitin A.I. and Persson B.N.J. JPCM, 11, 345 (1999)Volokitin A.I. and Persson B.N.J. PRB, 78, 155437 (2008)

– p. 15

Anomalous Doppler Effect

v

ω+q vx ω-q vx

Normal Doppler effect ω′ = ω − qxv > 0Anomalous Doppler effect ω′ = ω − qxv vT = kBTd/~

– p. 16

Classical Vavilov-Cherenkov Radiation

Cherenkov P.A. Dokl. Akad. Nauk SSSR, 2, 451 (1934)Resonant condition: qxv = cq/n > v0 = cqx/nThreshold velocity: v > c/n

– p. 17

Quantum Vavilov-Cherenkov Radiation

Frank M.I. J.Phys.USSR, 7, 49 (1943)Doppler effect (a): ωph = ω0 − gxv > 0

Doppler effect (b): ω0 − gxv < 0;ωph = qxv − ω0

– p. 18

The Landau criterion for the critical

velocity of a superfluid

p

ε(p)

E = Mv2

2 + ε(p)− pv

ε(p)− pv < 0

v > vc = min(

ε(p)p

)

Landau L.D., Zh. Eksp. Teor. Fiz. 11, 592 (1941)

– p. 19

Radiation at shearing two transparent plates

Pendry J.B. JMO 45, 2389 (1998).Magreby F.M., Golestian R., and Kardar M., PRA 88,042509 (2013).Volokitin A.I. and Persson B.N.J. PRB 93, 035407 (2016).

– p. 20

Non-Relativistic Theory

0 10 20 30 40 50 60 70 80 90 100

v/vo

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fs 1x [

norm

aliz

ed]

Resonant condition: ωph = qxv − cq/n = cq/n, v > v0 = 2c/n

F s1x =~v0π3d4F

s1x[normalized]

Rs =

√

(ω/c)2 − q2 −√

(nω/c)2 − q2√

(ω/c)2 − q2 +√

(nω/c)2 − q2

Magreby F.M., Golestian R., and Kardar M., PRA 88,042509 (2013).Volokitin A.I. and Persson B.N.J. PRB 93, 035407 (2016).

– p. 21

Relativistic Theory

0.8 0.84 0.88 0.92 0.960

1

2

3

4

v/c

F1x

[nor

mal

ized

]

0.96 0.97 0.98 0.99 1.00

20

40

60

80

v/cF

1x

[norm

aliz

ed]

ωph = qxv − cq′/γn = cq/n, v > v0 = 2cn/(n

2 + 1)

Volokitin A.I. and Persson B.N.J. PRB 78, 155437 (2008)Volokitin A.I. and Persson B.N.J. PRB 93, 035407 (2016).

– p. 22

Radiation From Moving Neutral Particle

2

1 1 1

v v

a) b) c)

v

ωph = qxv − ω0/γ = cq/n, v > v0 = c/n

Pieplow G. and Henkel C., JPCM 27, 035407 (2015).Volokitin A.I. and Persson B.N.J. arxiv.org/abs/1512.04366(2016).

– p. 23

Quantum Friction for Particle

0.5 0.6 0.7 0.8 0.9 1.0

-22

-20

-18

-16

-14

-12

v/c

log

10f fr

ictio

n(N

)

1.6log10γ

0.4 0.6 0.8 1.0 1.2 1.4 1.8-12.2

-12

-11.8

-11.6

-11.4

-11.2

-11

-10.8

(a) (b)

– p. 24

Polar Dielectric SiO2

0 1 2 3 4 5 6v, 105 m/s

0

1

2

3

4

5

6

Ffr

ictio

n, 1

05 N

/m2

Resonant condition: ωph = qxv − ω0 = ω0Threshold velocity: v > 2ω0/qxmax ∼ 2ω0d ∼ 2 · 10

5 m/s

– p. 25

Graphene on SiO2

grapheneelectrode

SiO2

Threshold velocity: v > vF + ω0/qxmax ≈ vF ∼ 106m/s

Resonant condition: ωph = qxv − vF q = ω0Experiment:Freitag M.,Steiner M., Martin Y., Perebeinos V.,Chen Z., Tsang J.C., and Avouris P., Nano Lett. 9, 1883(2009).Theory:Volokitin A.I. and Persson B.N.J. PRL 106 094502(2011)

– p. 26

Current density-electric field dependence

in graphene on SiO2

J (m

A/µ

m)

0 0.2 0.4 0.6 0.8E (V/µm)

0

0.4

0.8

1.2

1.6

T=0 KT = 150 K

450 K

0 1 2 43

E (V/µm)

0

0.4

0.8

1.2

1.6

J (m

A/µ

m)

300 K

a)

0

3

6

9

5 7 9 11

T=300 K

0 K

v, 105 m/s

Fx,

103

N/m

2

b)

vsat ∼ vF ∼ 106m/s

Jsat = ensvsat ∼ 1mA/µm

– p. 27

Frictional Drag between Graphene Sheets

0 4 8 12 16 200

400

800

1200

v, 105 m/s

Fx,

N/m

2

T=600 K

300 K

0 K

a)

0 4 8 12 16 200

0.4

0.8

1.2

1.6

T=300 K

0 K100 K

Fx,

N/m

2

v, 105 m/s

b)

d=1 nm d=10 nmAt l v ≪ vF induced electric field E = ρDJ = µ

−1v.

ρD =Γ

(ne)2=

h

e2πζ(3)

32

(

kBT

ǫF

)21

(kFd)21

(kTFd)2,

Fx0 =~v

d415ζ(5)

128π2

(

v

vF

)21

(kTFd)2.

Volokitin A.I. and Persson B.N.J. EPL 103 24002 (2013)

– p. 28

Acoustic Phonon Emission

z

d + u - ueq 0 1

solid 1

solid 0solid 0solid 0

σ(x, t) = K[u0(x− vt, t)− u1(x, t)],

Persson B.N.J., Volokitin A.I. and Ueba H., JPCM 23045009 (2011)

– p. 29

Acoustic Phonon Emission

3000 3400 3800 42004600 5000

v, m/s

10-1

100

101

102

103

σ, N

/m2

– p. 30

Summary

Casimir friction can be studied using modernexperimental setups for observation of frictional drag ingraphene systems..

The challenging problem is to study frictional drag forgraphene for strong electric field (large drift velocity)when Casimir friction is dominated by quantumfluctuations (quantum friction).

The challenging problem for experimentalists is todevelop setup for mechanical detection of Casimirfriction using AFM.

Quantum friction is strongly enhanced above thethreshold velocity.

The threshold velocity is determined by a type ofexcitations which are responsible for quantum friction.

– p. 31

mbox {hskip 1mm}Blue {Fluctuation-Dissipation Theorem}mbox {hskip 8mm}Rytov's Theorycenterline {Origin of Non-Contact Friction}centerline {Non-contact Friction}centerline {Electronic versus Phononic Friction}centerline {Electronic versus Phononic Friction}centerline {Casimir Friction}centerline {Frictional Stress}centerline {Small Velocities}centerline {Adsorbate enhancement of Casimir friction}mbox {hskip 3mm} Radiative Heat Transfer. Blue {Two ways to study Casimir friction}mbox {hskip 2cm}Blue {Frictional Drag in 2D-systems }centerline {Quantum Friction}mbox {hskip 2cm}Anomalous Doppler Effect mbox {hskip 2cm}Classical Vavilov-Cherenkov Radiationmbox {hskip 1cm}mbox {hskip 1mm}Blue {Large {�f The Landau criterion for the critical}}mbox {hskip 1mm}Radiation at shearing two transparent platesmbox {hskip 2cm}Non-Relativistic Theorymbox {hskip 2cm}Relativistic Theorymbox {hskip 1cm}Radiation From Moving Neutral Particlembox {hskip 1cm}Quantum Friction for Particlembox {hskip 3cm}Polar Dielectric SiO$_2$mbox {hskip 4cm} Blue {Large {�f Graphene on SiO$_2$}}mbox {hskip 1mm}Blue {Large {�f Current density-electric field dependence}}Blue {Frictional Drag between Graphene Sheets}centerline {Blue {Acoustic Phonon Emission}}centerline {Blue {Acoustic Phonon Emission}}centerline {Blue {Summary}}