casimir friction aleksandr volokitinaleksandr volokitin research center ju¨lich and samara state...
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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