Table Of ContentA Corbino disk viscometer for 2D quantum electron liquids
Andrea Tomadin,1,∗ Giovanni Vignale,2 and Marco Polini1
1NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy
2Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
The shear viscosity of a variety of strongly interacting quantum fluids, ranging from ultracold
atomic Fermi gases to quark-gluon plasmas, can be accurately measured. On the contrary, no
experimental data exist, to the best of our knowledge, on the shear viscosity of two-dimensional
quantum electron liquids hosted in a solid-state matrix. In this Letter we propose a Corbino disk
4 device, which allows a determination of the viscosity of a quantum electron liquid from the dc
1
potential difference that arises between the inner and the outer edge of the disk in response to an
0
oscillating magnetic flux.
2
n PACSnumbers: 66.20.-d,47.80.-v,73.23.-b,71.10.-w
a
J
5
Introduction.—Theshearviscosityη,whichdescribesthe
] diffusion of the average momentum density orthogonally
l
l to its direction, is one of the cornerstones of the hydro-
a
h dynamic theory of fluids [1, 2]. The first estimate of
- the shear viscosity of a dilute gas as a function of its
s
e density and temperature was given by Maxwell in his
m celebrated article on the “Dynamical Theory of Gases”.
He found that the shear viscosity of a dilute gas is inde-
.
t
a pendent of its density, a counterintuitive result that he
m felt needed immediate experimental testing [3]. Recent
- yearshavewitnessedasurgeofinterestintheviscousflow FIG.1. (Coloronline)Aviscometerfor2Dquantumelectron
d of strongly interacting quantum fluids, for which hydro- liquids. The light gray surface represents the 2D electron
n
dynamics provides a powerful non-perturbative descrip- system, which is shaped into a Corbino disk geometry. The
o
tion [4]. Experimentally, the shear viscosity of quantum intermediategreenregionrepresentsadielectriclayerofthick-
c
[ liquids like 3He and 4He can be measured by a variety ness d, which separates the 2D electron system from a back
gate(darkgreyregion). TheinternalholeoftheCorbinodisk
of tools including capillary, rotation, and vibration vis-
1 isthreadedbyanoscillatingmagneticfluxΦ(t)=Φ cos(Ωt),
0
v cometers [5]. The shear viscosity of cold atomic gases whichinducesanazimuthalelectricfieldE oscillatingatthe
8 can be inferred from measurements of collective modes same frequency Ω. The magnitude of theθˆazimuthal electric
3 or by looking at the expansion of the gas in a deformed field decreases as one goes from the inner to the outer rim.
9 trap after the trapping potential is turned off [6–8]. The The inner rim of the disk is grounded, while the outer rim
0 shear viscosity of quark-gluon plasmas can be extracted is free to adjust its voltage. A dc electrical potential energy
.
1 from elliptic flow measurements at relativistic heavy-ion difference∆U appearsbetweentheinnerandtheouterrimin
0 response to the oscillating magnetic flux. The quantity ∆U
colliders [9]. To the best of our knowledge, however,
4 sensitively depends on the shear viscosity of the 2D electron
no protocols exist for measuring the shear viscosity of
1 fluid and the frequency Ω of the magnetic flux.
: two-dimensional (2D) quantum electron liquids (QELs)
v
in solid-state matrices [10, 11]. This gap is truly sur-
i
X prising in view of the large body of theoretical work [12]
this is ensured by the following chains of inequalities:
r that has been carried out in connection with the shear (cid:96) L (cid:96) , where (cid:96) is the mean free path be-
a viscosity of these systems. In this paper we try and fill ee (cid:28) (cid:28) p ee
tween quasiparticle collisions [13] and the length scale
the gap by proposing a method to measure the viscosity
overwhichlocalthermodynamicequilibriumisachieved,
of electrons in a realistic experimental setup.
while (cid:96) is the length scale over which electron-impurity
p
The concept of hydrodynamic viscosity η [1, 2, 4] be- and electron-phonon scattering break momentum con-
comes relevant in a regime of parameters in which the servation. It is well known [14–16] that the above in-
electron liquid is well described by a quasi-equilibrium equalities are satisfied in highly-pure 2DQELs e.g. in
distribution function characterized by slowly time- modulation-doped GaAs/AlGaAs semiconductor hetero-
dependent density and drift velocity—the local counter- junctions [10] for typical electron densities, in the tem-
parts of globally conserved particle number and momen- perature range 5 K (cid:46) T (cid:46) 35 K, and for devices with
tum. In a solid-state device with linear dimension L linear size 10 µm(cid:46)L(cid:46)50 µm. In this range of param-
2
1.0
0.6
eters momentum-non-conserving collisions can be safely 0.5
0.8
0.4
neglected, while electron-electron interactions establish n 0.3
local equilibrium with a slowly-varying density n(r,t) )/vi 0.6 00..12
and drift velocity v(r,t). Hydrodynamic electron flow 0 0.0
has indeed been experimentally generated and hydrody- (r, 0.4 1.0 1.5 2.0 2.5 3.0 3.5 4.0
)
1
namic effects have been measured [17]. We now describe (θ
v
our method for determining the viscosity. 0.2
Electrical measurement of the viscosity.—We consider a η
2DQEL shaped in a Corbino disk (CD) geometry—see 0.0
1 2 3 4 5 6 7 8 9 10
Fig. 1. The CD lies in the z = 0 plane and has an
r/r
inner radius r and an outer radius r . It is sepa- in
in out
rated from a back gate by a dielectric layer of thickness
d and dielectric constant (cid:15). An oscillating magnetic flux FIG. 2. Radial profile of the azimuthal component of the
Φ(t) = Φ0cos(Ωt) oriented along the zˆ axis threads the velocity—in units of vin as defined in Eq. (4)—at t = 0,
obtained by solving Eq. (15) with boundary conditions (12)
inner hole of the CD and induces, by Faraday’s law, an
(main panel) and (13) (inset). The results are obtained with
azimuthal electric field, which is given by
r /r = 10.0, and for several values of the dimensionless
out in
parameter ξ defined in Eq. (16): ξ = 0.2 (dashed line), 0.5
1
E (r,t)= ∂ Φ(t)θˆ, (1) (dotted line), and 1.0 (dash-dotted line). The value of the
θˆ −2πcr t
viscosity in the different solutions increases as shown by the
arrow. Thesolidlinecorrespondstotheanalyticalresult(4),
where c is the speed of light and θˆ is the unit vector in
which holds at ξ=0.
the azimuthal direction. Fluctuations of the circularly-
symmetric electron density n(r,t) on the surface of the
CD generate a radial electric field of the form
(η =0),itiseasilyshown(seediscussionbelow)thatthe
e
E (r,t) ∂ n(r,t)rˆ, (2) flow is given by
rˆ r
(cid:39) C
(cid:12) e Φ r
where e is the absolute value of the electron charge, C = v(1)(r)(cid:12) = 0 v in , (4)
(cid:15)/(4πd) is the geometric capacitance per unit area, and θ (cid:12)η=0 mc2πr ≡ in r
rˆ is the unit vector in the radial direction. This “local whichisirrotational(i.e.curl-free)intheregionr <r <
in
capacitance approximation” is valid as long as d is much r . Putting this in Eq. (3) and assuming r r we
out out in
smaller than any lateral length scale [14]. The electron obtain (cid:29)
velocity v(r,t), with radial and azimuthal components
e2 1 1
vr(r,t) and vθ(r,t), respectively, fluctuates in response ∆U = Φ2 . (5)
to the total electric field E(r,t)=Eθˆ(r,t)+Erˆ(r,t). |η=0 mc2(2πrin)24 0
We now apply the Navier-Stokes equations of hydro-
Notice that ∆U is independent of the frequency Ω
dynamics [1, 2] to the calculation of the time-evolution |η=0
of the driving flux. To estimate the magnitude of the dc
of n(r,t) and v(r,t). At first order in Φ , the elec-
0 response, we use the following parameters [18]:
tron density fluctuations n(1)(r,t) and the radial veloc-
ity fluctuations v(1)(r,t) vanish, while the azimuthal ve- r =2.0 µm, r =20.0 µm,
r in out
locity oscillates at the frequency of the magnetic flux, Φ /(πr2)=10 mT, m=0.067 m , (6)
v(1)(r,t) = e[v(1)(r)e−iΩt]. At second order in Φ , the 0 in e
θ (cid:60) θ 0 where m is the bare electron mass in vacuum and the
circularmotionoftheelectronsintheCDorbitingaround e
the zˆaxis generates a dc potential energy difference ∆U value of m given above is appropriate for electrons in
GaAs[10]. FromEq.(5),wefind∆U =63µeV. The
between the inner and outer edges of the CD, given by η=0
|
analytical result (4) is plotted as a solid line in Fig. 2.
m(cid:90) rout 1 Including the shear viscosity has three main effects on
∆U = dr v(1)(r)2 , (3)
2 r| θ | which we further elaborate below: (i) the spatial vari-
rin
ation of the velocity field is considerably reduced (see
wheremistheeffectiveelectronmass. Thequantity∆U Fig. 2), (ii) the flow acquires a non-curl-free dependence
in Eq. (3)—independent of time—can be readily recog- on the radial position r, i.e., a non-zero vorticity [1, 2]
nized as the work of the centripetal force acting on an ω = v appears near the inner and outer edges
r
∇ ×
electron which drifts from the inner to the outer rim of (see Fig. 4), and (iii) the dc potential drop ∆U becomes
the CD and is subject to a constant acceleration in the strongly frequency-dependent (see Fig. 3). Indeed, ∆U
radialdirectiongivenby v(1)(r)2/(2r). Foranidealfluid decreases by a factor 20 as the frequency decreases from
| θ |
3
100 where the electric potential energy U(r,t) is obtained by
solving the Poisson equation in the CD geometry with a
50 constantboundaryconditionatthegatepositionz = d.
−
If the typical wavelength of density fluctuations is larger
V]
20 than d, it is easy to see that
e
µ
[
U 10 U(r,t) e2n(r,t)/C , (10)
∆ (cid:39)−
5 whichimmediatelyleadstoEq.(2). Finally,thepressure
gradient in Eq. (8) can be neglected when ∂P(n)/∂n
(cid:28)
e2n¯/C, i.e. when d a(cid:63)/4. This inequality is always
21 5 10 50 100 500 well satisfied since a(cid:63)(cid:29) (cid:15)B(cid:126)2/(me2) is the material Bohr
B ≡
f [MHz] radius, which is 10 nm for GaAs.
∼
The Navier-Stokes and continuity equations (8)-(9)
must be complemented by suitable boundary conditions
FIG.3. Thedcpotentialenergydifference∆U betweenouter
(BCs) expressed in terms of the flux density tensor [1, 2]
and inner rims of the Corbino disk as a function of the fre-
quencyf =Ω/(2π)oftheoscillatingmagneticfluxΦ(t). Re- Πi,k(r,t)=P(r,t)δi,k+ρvi(r,t)vk(r,t)−σi(cid:48),k(r,t), where
sultsinthisplothavebeenobtainedwiththeparametersasin σ(cid:48) (r,t) is the viscous stress tensor. We require the ra-
i,k
Eq.(6). Differentcurvescorrespondtodifferentvaluesofthe dial diffusion of azimuthal momentum, which is propor-
kinematic viscosity: ν = 1.0 cm2/s (dashed line), 5.0 cm2/s tionaltotheviscosityη, tovanishattheouterandinner
(dotted line), and 15.0 cm2/s (dash-dotted line). The solid
rims of the CD, i.e.
line represents the viscous-free analytical result in Eq. (5).
σ(cid:48) (r,t)(cid:12)(cid:12) = σ(cid:48) (r,t)(cid:12)(cid:12) 0 . (11)
r,θ r=rin r,θ r=rout ≡
The off-diagonal component of the viscous stress tensor
500 MHz (where the effect the viscosity is practically reads σ(cid:48) = η(∂ v +∂ v /r v /r) [1, 2]. Because of
r,θ r θ θ r − θ
negligible) to 1 MHz. The profile of ∆U depends only circular symmetry, the BCs (11) reduce to
on the ratio of the viscosity to the average mass density
(cid:12)
mn¯, i.e. the kinematic viscosity [1, 2] ∂rvθ(r,t)|r=ri = vθ(rr,t)(cid:12)(cid:12)(cid:12) , (12)
η r=ri
ν = . (7)
mn¯ whereri =rin,rout. Moreover,forthesetupinFig.1,two
furtherBCsshouldbeimposed. First, theradialcompo-
Bymeasuringthefrequencydependenceof∆U,subtract- nent of the current j(r,t)=n(r,t)v(r,t) must vanish at
ing the frequency-independent background (5), and fit-
theouterrim,wheretheCDisisolated. Second,theelec-
ting the theoretical curve to the experimental result one
tricpotentialattheinnerrimisfixed,hencetheelectron
candetermine ν. Intheremainder ofthisLetterwesup-
density n(r ,t) at the inner rim of the CD must coin-
in
ply the main steps of the calculation of ∆U.
cide with the average value n¯ =eV /C fixed by the gate
G
Hydrodynamic equations and their solution.—In the hy-
voltage V .
G
drodynamic regime, the response of the 2DQEL is gov-
We notice that the BCs (12) differ from the standard
erned by the Navier-Stokes equation [1, 2]
“no-slip” BCs
ρ(r,t){∂tv(r,t)+[v(r,t)·∇r]v(r,t)}= vθ(r,t)|r=rin =vθ(r,t)|r=rout ≡0 , (13)
eE(r,t)n(r,t) P(r,t)+η 2v(r,t) (8)
− −∇r ∇r whicharecommonlyemployed[1,2]todescribefluidad-
+ζ r[ r v(r,t)] , hesion to the walls of a container. While the use of the
∇ ∇ ·
BCs in Eq. (13) is not immediately justified in our case,
combined with the continuity equation
we have checked that the results in Fig. 3 do not change
qualitatively if the BCs in Eq. (13) are used instead of
∂ n(r,t)+ [n(r,t)v(r,t)]=0 . (9)
t ∇r· thoseinEq.(12). Theagreementbetweentheresultsob-
tained with two different sets of BCs gives us confidence
Here, ρ(r,t)=mn(r,t) is the mass density, P(r,t) is the
in the robustness of the effect illustrated in Fig. 3.
pressure, and ζ is the bulk viscosity, which we can ne-
We solve Eqs. (8)-(9) by expanding the hydrodynamic
glect here since it vanishes at long wavelengths [1, 13].
variables in powers of the amplitude Φ of the magnetic
In Eq. (8) we have also included the contribution of the 0
flux [19]:
electricfieldE(r,t). Theazimuthalsymmetryofthesys-
temimpliesthatallquantitiesdependontheradialcoor- v(r,t)=v(0)(r,t)+v(1)(r,t)+[δv(r)+v(2)(r,t)]+...
dinater onlyandthatthederivativeswithrespecttothe
n(r,t)=n(0)(r,t)+n(1)(r,t)+[δn(r)+n(2)(r,t)]+...,
azimuthalangleθ vanish. TheradialcomponentE (r,t)
rˆ
(14)
of the electric field is given by E (r,t) = rˆ∂ U(r,t)/e
rˆ r
−
4
1.0
1.0
0.8 n 0.8
htwhyhederreoexdpvyan(kna)sm(iroi,nct),(en1q4(uk)a)t(wiro,ent)sin∼ocflu(dmΦe0o)tckioocnnosstaa(rkneΩttnc)oo.nntlSirniinbecauert,iothnines r/vinin 0.6 L/rmaxi 000...246
0) 0.4 0.0
δsivg(nra)l,δant(rfr)e∼quOen(cΦy20)Ωa[r1is9i]n.gAfrtomfirsttheorsdelefr-m, wixeinfignodf tthhee 1)(r, 0.2 η 0.0 0.2 0L.4η/r0i.n6 0.8 1.0
(z
followingdifferentialequationfortheradialprofileofthe ω
0.0
azimuthal component of the velocity:
(cid:20) (cid:21) 0.2
iΩv(1)(r)=ν ∂2v(1)(r)+ 1∂ v(1)(r) 1 v(1)(r) − 1 2 3 4 5 6 7 8 9 10
− θ r θ r r θ − r2 θ r/r
in
e Φ
0
iΩ . (15)
− mc2πr FIG.4. SameasinFig.2butfortheradialprofileofthevor-
ticity ω(1)(r,t) (in units of v /r ) at time t = 0. The inset
In the general ν = 0 case, Eq. (15) must be solved nu- z in in
merically [20]. It(cid:54) is convenient to introduce dimension- shows the position Lmax of the maximum of vθ(1)(r,t=0), as
displayedinFig.2,forseveralvaluesofthevorticitypenetra-
lessvariablesbyrescalingr withr andv(1)(r)withv ,
in θ in tion depth Lη (in units of rin). The dashed line corresponds
which has been defined in Eq. (4). Then, Eq. (15) de- to the expected relation L =L .
max η
pends only on the dimensionless parameter
(cid:18) (cid:19)2
Lη obtained by looking at the radial profile of the vorticity
ξ , (16)
≡ rin ωz(r,t) = ∂r[rvθ(r,t)]/r, whose first-order contribution
ω(1)(r,t)inpowersofΦ isshowninFig.4. Intheregions
(cid:112) z 0
whereL = ν/Ωisthevorticitypenetrationdepth[1,2]
η near the rims of the CD the non-curl-free dependence of
during a time 1/Ω. The solution of Eq. (15) is shown in the velocity flow v(1)(r,t) on r leads to large values of
Fig. 2 for several values of ξ and both sets of BCs. We θ
the vorticity. Moving away from the rims, the vorticity
note that with increasing viscosity the amplitude of the
decreases to zero on a length scale L , which confirms
velocity flow diminishes. Since according to Eq. (3) ∆U η
the interpretation of this quantity as the vorticity pen-
dependsontheintegralofthesquareofthevelocitypro-
etration depth [1, 2, 21]. On the same length scale the
file, increasing the viscosity suppresses the dc response
velocity flow crosses over to the curl-free profile (4)—see
∆U, thereby explaining the results in Fig. 3.
inset in Fig. 4.
In the limit ξ 1 (low viscosity or high frequency)
(cid:28) Beforeconcluding,wewouldliketoquantifythepower
the solution of Eq. (15) is given by the curl-free profile
W dissipated in the fluid by viscosity [1, 2] in a period
in Eq. (4). In the opposite ξ 1 limit (high viscosity
(cid:29) of the oscillating flux. This is given by the following
orlowfrequency)thesolutionisfoundbysettingtozero
expression:
the term in square brackets in Eq. (15). In this case, it
is easy to demonstrate that a non-curl-free linear profile (cid:90) rout 1 (cid:12) (cid:12)2
vθ(1T)(or)ob∝tarinsotlhveesdtcherepsrpoobnlseem∆, sUatiisnfyEinqg. t(h3)e BweCse.xpand W =−πη rin dr r (cid:12)(cid:12)vθ(1)(r)−r∂rvθ(1)(r)(cid:12)(cid:12) . (18)
Eq. (9) and the radial component of Eq. (8) to second To leading order in η for η 0 we obtain W
→ →
orderandweaveragetheresultingexpressionsoverape- 8π(η/m)∆U . As far as order-of-magnitudes are
η=0
− |
riod T =2π/Ω of the oscillating flux Φ(t). The solution concerned, we find that W is of the order of microwatts
of the time-averaged equations yields δvr(r) 0 and for ν 10 cm2/s and n¯ 1011 cm−2. Away from
≡ ∼ ∼
the perturbative regime, we have verified that η
mC (cid:90) r 1 ∼
δn(r)= dr(cid:48) v(1)(r,t)2 , (17) amW/(8π∆U)wherea 0.5isanumericalcoefficient.
e2 rin r(cid:48)(cid:104) θ (cid:105)t −This relation is valid in th∼e range 0.25(cid:46)ξ (cid:46)1.0. Thus,
an independent measure of W and ∆U at a single fre-
where(cid:104)g(t)(cid:105)t ≡T−1(cid:82)0T dt(cid:48)g(t(cid:48))denotesthetime-average quency Ωoftheoscillatingmagneticfluxyieldsthevalue
of a function g(t) over one period of the oscillating of η. This provides a possible alternative to the method
magnetic flux. Finally, Eq. (3) can be easily obtained describedafterEq.(7), which requiresameasurementof
by setting r = rout in Eq. (17) and making use of ∆U as a function of frequency.
v(1)(r,t)2 = v(1)(r)2/2 and of the local capacitance In summary, we have demonstrated that the hydro-
(cid:104) θ (cid:105)t | θ |
formula, Eq. (10). dynamic shear viscosity of a two-dimensional quantum
Vorticity and dissipation.—Furtherinsightsonthephys- electron liquid can be obtained by studying the response
ical properties of the solution shown in Fig. 2 can be ofthesystemtoanoscillatingmagneticfluxinaCorbino
5
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