Viscosity in electron liquids
Davide Forcella - Université de Bruxelles
Jan Zaanen – Universiteit Leiden
Davide Valentinis - Université de Genève
Dirk van der Marel - Université de Genève
Specific Heat
Landau-Fermi liquids
Example: CeAl3
Heavy fermions:
m*/m≈300
1
t
»l
2
T2
e F*
Resistivity (μΩcm)
T (mK)
Andres,Gräbner&Ott,
PRL 35,1779 (1975)
T2 (K2)
Not-So-Fermi liquids
C.M. Varma, Z. Nussinov, Wim van Saarloos, Physics Reports 361, 267 (2002)
J. Zaanen,
Nature 430, 512 (2004)
J. Moreno and P. Coleman,
PRB 53, R2995 (1996).
Hole doping
Resistivity (μΩcm)
Resistivity (μΩcm)
Magnetic field
H. von Löhneysen et al.,
PRL 72, 3262 (1994)
Temperature (K)
Y. Ando et al,
PRL 93, 267001 (2004)
A brief reminder of what viscosity is
A brief reminder of what viscosity is
Navier-Stokes for current flow with transverse (shear) stress :
éë¶t - n Ñ2 ùûu = f
Velocity field
External forces, friction,…
Accelleration
Dynamic Shear Viscosity
Viscosity According to Fermi Liquid theory
A. A. Abrikosov and I. M. Khalatnikov, Zh. Eksperim. Teor. Fiz. 33, 110 (1957)
What is more viscous ?
Strongly interacting
Weakly interacting
Fermi liquids
Fermi liquids
(3He,Sr2RuO4,UPt3,UBe13,CeCu
6..)
(Aluminium, silver…)
Some Viscosity Numbers
Quantum critical matter
Fermi Liquids
Abrikosov & Khalatnikov,
Zh. Eksperim. Teor. Fiz. 33 (1957):
Kovtun, Son & Starinets,
JHEP 10 (2003) 064:
ß
Material
uF / c
TF
Al, C
0.007
10 5 K
Sr2 RuO 4
0.003
10 4 K
1.6 ×10 -6
2K
3
He
n min
3
ìï10 -6 n FL (T = 300 K)
æT ö
» ç ÷ n FL » í -12
ïî10 n FL (T = 3 K)
è TF ø
Weakly interacting
Fermi liquid
Strongly interacting
Fermi liquid
Whatisis more
more viscous
? ?
What
viscous
Weakly interacting
Fermions
(Aluminium, silver…)
Canonical Neutral Fermi liquid:3He
D. Forcella, J. Zaanen, DvdM (2013)
Viscosity
Viscosity of 3He
Black, Hall & Thompson, J Phys C 4, 129 (1971)
Propagation of sound in a neutral viscous liquid
Viscosity According to Fermi Liquid theory
A. A. Abrikosov and I. M. Khalatnikov, Zh. Eksperim. Teor. Fiz. 33, 110 (1957)
Consequence for the transverse accoustic impedance:
Transverse
sound velocity
=
2100 cm/s
Transverse
sound
Hydrodynamic
regime
Pat R. Roach & J. B. Ketterson, Phys. Rev. Lett. 36 (1976)
Consequence for ω >> 1/τ:
Transverse Sound
Transverse
sound
Transverse
m*/m
sound= 3
Hydrodynamic
regime
And now: Electron Liquids
THE ANOMALOUS SKIN EFFECT AND THE REFLECTIVITY OF METALS
C. W. Benthem and R. Kronig
Physica 20, 293 (1954)
ON THE POSSIBLE INFLUENCE OF ELECTRON INTERACTION ON THE
REFLECTIVITY OF METALS
C. W. Benthem
Appl. Sci. Res. B7, 275 (1958)
Hydrodynamic Effects in Solids at Low Temperature
R. N. Gurzhi,
Usp. Fiz. Nauk 94, 689 [Sov. Phys. Usp. 11, 255 (1968)]
Two-fluid hydrodynamic description of ordered systems
Charles P. Enz
Rev. Mod. Phys. 46, 705 (1974)
Electronfluid model for dc size effect
R. Jaggi
Helvetica Physica Acta, 52, 258 (1980)
Helvetica Physica Acta 62, 752 (1989)
Journal of Applied Physics 69, 816 (1991)
Dotted: classical (fluid) behaviour
Solid: viscous flow
Electronfluid model for dc size effect
R. Jaggi
Helvetica Physica Acta, 52, 258 (1980)
Helvetica Physica Acta 62, 752 (1989)
Journal of Applied Physics 69, 816 (1991)
Hydrodynamic electron flow in high-mobility wires
M. J. M. de Jong, L. W. Molenkamp
Phys. Rev. B 51, 13389-13402 (1985)
Dependence of the thermovoltage Vtrans= V6-V3 and of the
difference between the electron and the lattice temperature Tc-T
on the heating current I measured for wire 1 at T = 1.5 K. Point
contact AB is adjusted for maximum, CD for zero thermopower.
The inset shows the schematical layout of the gates (hatched
areas) used to define a wire with point-contact voltage probes.
The wire width W is typically 4 μm, the length L varies between 20
and 120 μm. The crossed boxes denote Ohmic contacts. The
coordinates used for the theory are indicated.
Differential resistance dV/dI versus current. The top
curve is the experimental result at T = 1.8 K. The
other curves are theoretical results for various
boundary-scattering parameters. The dotted lines
are calculated with a constant specularity
coefficient p = 0.845, 0.87, 0.895 (top to bottom).
The solid lines are calculated for angle-dependent
boundary scattering, with α = 0.8, 0.7, 0.6 (top to
bottom). Best agreement with experiment is found
for α = 0.7 (thick curve).
Perfect fluids: Some recent investigations
Quark gluon plasma
Graphene: A Nearly
Perfect Fluid
Nearly Perfect
Fluidity in a High
Temperature
Superconductor
M. Müller et al, PRL
103, 025301 (2009)
Rameau,Reber,Yang,
Akhanjee,Gu,
Johnson& Campbell
arXiv:1409.5820
E. Shuryak, Prog. Part. Nucl.
Phys. 53, 273 (2004).
A. Rebhan and D. Steineder,
Phys. Rev. Lett. 108, 021601
(2012).
D. Forcella, J. Zaanen, DvdM (2013)
Evidence for hydrodynamic electron flow in PdCoO2
P. J. W. Moll, P. Kushwaha, N. Nandi, B. Schmidt, and A. P. Mackenzie
arXiv:1509.05691
Hydrodynamic effect on transport.
The measured resistivity of PdCoO2 channels
normalised to that of the widest channel (ρ0),
plotted against the inverse channel width 1/W
multiplied by the bulk momentum- relaxing
mean free path.
Electron viscosity, current vortices and negative nonlocal resistance in graphene
L Levitov and G Falkovich
arXiv:1508.00836
Negative local resistance due to viscous electron backflow in graphene
D. A. Bandurin, I. Torre, R. Krishna Kumar, M. Ben Shalom, A. Tomadin, A. Principi, G. H. Auton, E.
Khestanova, K. S. Novoselov, I. V. Grigorieva, L. A. Ponomarenko, A. K. Geim, M. Polini
arXiv:1509.04165
Measured viscosity of the Fermi liquids in
graphene. (a,b) ν as a function of n for
different T for SLG and BLG, respectively. The
grey‐shaded areas indicate regions around the
CNP where our hydrodynamic model is not
applicable. Dashed curves: Calculations based
on many‐body diagrammatic perturbation
theory (no fitting parameters).
Propagation of coupled transverse sound and
EM field in viscous charged liquids
Maxwell equations (c =1) Þ
é¶2z -¶t2 ù E = 4p¶t J
ë
û
2
ne
Navier-Stokes+damping Þ éët -1 + ¶t - n¶2z ùû J =
E
m
Forcella, Zaanen, Valentinis, vdMarel, PRB 90, 035143 (2014)
ransverse sound in a charged Fermi liquid @ 300 Kelvi
Mode -1
m*/m = 3
Mode 1
Negative Index of Refraction in the Quark Gluon
Plasma
A. Amariti, D. Forcella, A. Mariotti and G. Policastro, JHEP 1104, 036 (2011).
A. Amariti, D. Forcella and A. Mariotti, JHEP 1301, 105 (2013).
A. Amariti, D. Forcella and A. Mariotti, arXiv:1010.1297.
?
Reflection and transmission of EM waves at the
boundary
from vacuum to a viscous charged fluid
e
iw ( z-t )
re
-iw ( z+t )
Constituant relations at the interface:
(
)
2 na -1
1- ilw na
(1 From Maxwell) E = continuous ü ì
ï ïJa =
na +1 na + nb 1+ ilw 1- na - nb
ï ï
ï ï
(2 From Maxwell) H = continuous ý Þ í
ï ï
ï ï
(3 From N&S)
u 0 = lu ' 0 ïþ ïr = J + J -1
1
2
î
(
()
l º "slip length"
()
)(
)
(
)
Surface Impedance
The electromagnetic response of a metal, whether
normal or superconducting, is described by a complex
surface impedance:
Surface Impedance: Z s = Rs + iX s =
E(0)
¥
ò J(z)dz
0
Z 0 = Vacuum impedance = 377 W
Re Zs = Rs = surface resistance
Im Zs = Xs = surface reactance
Classical skin effect : Z s =
2
ipwr DC
c
Brian Pippard, Proc. R. Soc. Lond. 191, 385 (1947)
Brian Pippard
The theory of the anomalous skin effect in metals
G. E. H. Reuter and E. H. Sondheimer,
Proc. R. Soc. A 195, 336 (1948)
mu F
l = mean free path = 2
ne r DC
c
d = skin depth =
2pw
r DC
What happens when l >> d ?
Anomalouus regime
Franz Sondheimer
The theory of the anomalous skin effect in metals
G. E. H. Reuter and E. H. Sondheimer,
Proc. R. Soc. A 195, 336 (1948)
Franz Sondheimer
"Under the combined action of the applied electromagnetic field
and the collisions of the electrons with the lattice, a steady state is
set up, and the distribution function in the steady state is
determined by the Boltzmann equation"
Experimental surface impedance
Pippard
Theoretical surface impedance
Reuter&Sondheimer
How to calculate the Surface Impedance
in the Anomalous Skin Regime
Reuter-Sondheimer equations for Zs
æ 1 1 ö -s u
0) Kernel: K ( u) º ò ç - 3 ÷ e ds
è s ø
1 s
¥
¥
ü
d 2 E ( z)
æz- yö
3 ì ¥ æz- yö
1) Solve:
=
i
p
K
E
y
dy
+
1p
K
E
y
dy
í
ò ç ÷ ( ) ( ) ò çè l ÷ø ( ) ýþ
dz 2
2d 2l î -¥ è l ø
0
2) Compute: Z s = Z0
iw E ( 0)
c [ dE / dz ] z=0
Reuter&Sondheimer, Proc. R. Soc. A 195, 336 (1948)
FZVM equations for Zs
2
1
t -1 - iw é t -1 - iw ù i4w p
1) n± =
1± ê1+
ú +
w 2n
w 2n û w 3n
2
ë
2
n+ + n- + ilw (1- n+2 - n-2 - n+n- )
2) Z s = Z 0
1+ n+ n- - ilw n+ n- (n+ + n- )
Forcella, Zaanen, Valentinis, vdMarel, PRB 90, 035143 (2014)
Comparison of hydrodynamical approach (present work) and
Reuter-Sondheimer model of anomalous skin effect (Proc.R.Soc.
A195 (1948))
Forcella, Zaanen, Valentinis, vdMarel, PRB 90, 035143 (2014)
Surface Impedance of correlated electrons
The anomalous skin effect and the reflectivity of
metals
Hydronamical approach
C. W. Benthem and R. Kronig, Physica 20. 293 (1954)
"internal friction" from the formula for the stopping power
On the possible influence of electron interaction on
the reflectivity of metals
C. W. Benthum, Appl. Sci. Research B 1, 275 (1959)
"It seems, therefore, that the effect …
…. will be too small to measure"
Ralph de Laer Kronig
Holography and hydrodynamics: diffusion on stretched horizons
P. Kovtun, D.T. Son and A.O. Starinets, JHEP 10 (2003) 064
Kovtun
Son
Starinets
Reflection of EM waves at the boundary
from vacuum to a viscous charged fluid
Smooth surface
High temperature
Drude
Rough
surface
Low temperature
Viscous
Forcella, Zaanen, Valentinis, vdMarel, PRB 90, 035143 (2014)
Sr2RuO4: the `Helium 3’ of
transition-metal oxides !
Beautiful review articles:
•
A.Mackenzie & Y.Maeno, RMP 75, 657 (2003)
•
C. Bergemann, Adv. Phys. 52, 639 (2003)
Resistivity anisotropy
D. Stricker et al, unpublished
Reflectivity
E
r(ω)
ε(ω)=(1-r)2/(1+r)2
E
r(ω)
ε(ω)=(1-r)2/(1+r)2
Reflectivity
Experimental test on Sr2RuO4
The overlap of the spectra with and without viscosity implies an upper limit for ν / τ
n
-7 2
< 10 c
t
Reflection and transmission of EM waves at the
boundary
from vacuum to a viscous charged fluid
e
iw ( z-t )
re
-iw ( z+t )
(
E ( z) / E ( 0) = J + + J - + 2 Re J +J -e
2
2
2
iw ( n+ +n- ) z
)
Interference between the two modes inside the
material:
m*/m = 3
Transmission through a film
E
τ(ω)
E ( z ) / E ( 0) = eiw z + e-iw z
(z < 0)
= J +eiwn+z + y+e-iwn+z + J -eiwn-z + y-e-iwn-z
(0 < z < d)
= J eiw z
(z > d)
Material: aluminum
Film thickness = 100 nm
Transmitted Intensity
Drude
Perfectly smooth
interface
Wavenumber (cm-1)
Transmitted Intensity
Transmitted Intensity
Wavenumber (cm-1)
Wavenumber (cm-1)
Maximally rough
interface
Summary
Viscosity is a manifestation of non-local correlations
Viscosity renders the response to a force non-local
In electron liquids two coupled charge-matter modes exist for given ω
One of these two modes has a negative index of refraction
In the optical properties interference between the two modes occurs
New challenges: non-local (q-dispersive) optics
Application to strongly interacting matter
Jackpot: disappearance of viscosity in quantum critical matter
Forcella, Zaanen, Valentinis, vdMarel, PRB 90, 035143 (2014)
Reflection and transmission of EM waves at the
boundary
from vacuum to a non-viscous charged fluid
e
iw ( z-t )
re
-iw ( z+t )
Two solutions for the index of refraction  Two “modes” for each frequency
m*/m = 3