Casimir Friction

Aleksandr VolokitinResearch Center Julich 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

mx+mω20x+ Γx = F (t)

Γ = (kBT )−1Re

∫

∞

0 dt⟨

F(t)F(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 J1

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 + ω2

Pendry 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<0Thermal fluctuations dominate at v < vT = kBTd/~Quantum fluctuations dominate at v > 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 [n

orm

aliz

ed]

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

F s1x = ~v0

π3d4Fs1x[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/(n2 + 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/m

2

Resonant condition: ωph = qxv − ω0 = ω0

Threshold velocity: v > 2ω0/qxmax ∼ 2ω0d ∼ 2 · 105 m/s

– p. 25

Graphene on SiO2

grapheneelectrode

SiO2

Threshold velocity: v > vF + ω0/qxmax ≈ vF ∼ 106m/sResonant condition: ωph = qxv − vF q = ω0

Experiment: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

/m2

b)

vsat ∼ vF ∼ 106m/sJsat = 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