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Normal Fluid Behavior and
Superfluid Density of 3He in axially
compressed aerogel
Cornell
Robert Bennett, Nik Zhelev and Andrew Fefferman
Royal Holloway: Priya Sharma
Northwestern (Aerogel): Pollanen, Halperin
Materials World Network NSF & Thanks to Dynasty Foundation!
Jeevak Parpia, Cornell University, Ithaca NY USA
15 сентября 2010
Introduction
0. Background
1. Description of experimental arrangement
2. Normal Fluid measurements and assumptions
a. Results
b. Discussion
3. Expected effect of uniaxial compression on superfluid
4. Past measurements of ρs/ρ in aerogel
5. Superfluid density and dissipation in compressed
aerogel
6. Future work and Conclusions
2
0. Study heff , rs of impurity-limited 3He
Usual measurements on TO employ
a right circular cylinder. Unsuited
for normal state measurements. In
this geometry, the normal
component is not well-clamped
leads to large dissipation (Q-1).
Will use the Q-1 and frequency shift
to establish the effective viscosity of
the normal fluid. Equivalent to
resistivity of normal state in a
superconductor. Such a
measurement could also be used to
yield the impurity limited viscosity in
the superfluid phases and other
useful parameters.
R. Nomura, G. Gervais, T.M. Haard, Y. Lee, N. Mulders, and W.P. Halperin,
Phys. Rev. Lett., 85 4325 (2000).
The attenuation of 14.6 MHz
sound for 3He in aerogel.
Dashed line: calculated by
the viscoelastic model. The
dot-dashed line includes a
decoupling effect. The
attenuation in bulk 3He (solid
line) is shown for
comparison.
3
I: Aerogel Sample
Aerogel grown at Northwestern
inside a stainless steel cavity, id
6.95 mm, target height 400mm.
After drying, the shims were
carefully removed and a nominal
force applied to the faces of the
cavity to achieve 10% compression.
The aerogel was optically
characterized through small holes
in the cavity wall before (after)
compression to verify the lack
(presence) of an optical anisotropy
axis as described in Pollanen et al.
The cavity was embedded in an
epoxy head on a double pendulum
torsional oscillator.
Stainless cell for aerogel. Radial holes
allow for optical characterization of the
aerogel.
J. Pollanen, J.P. Davis, B. Reddy,
K.R. Shirer, H. Choi, and W.P.
Halperin, J. of Phys. Conf. Ser. 150
032084, (2009).
4
1- Background (dissipation and period)
The background period and Q-1
were measured at const.
amplitude. The cell was filled at
>5 K to ~ 5 bar, cooled to 0.1K,
and pressure lowered to 0.14
bar, 3 bar etc. The period and
Q were measured to 0.5 mK.
The Q-1 and period for empty
(Pe , Qe) and 0.14 bar 3He filled
cell. The background is fitted
(solid line) and subtracted from
P(T) to obtain period shift and
the dissipation due to the “dirty”
Fermi fluid together with any
(small) bulk contribution.
6
2a- Normal state behavior
Q-1 after background
subtraction
Qf-1= Qmeas-1- Qe-1
Fraction of inertia decoupled =
1-[P(T)-Pe(T)]/[P(T=0)-Pe(T=0)]
Red Lines show expected behavior
for 3He filling a 400 µm cavity (without
aerogel)
With aerogel present,
Fluid velocity
profile shows
velocity
profile shows
viscous penetration
“Drude-like”
character: depth
no
<< channel
width
variation
in fluid
velocity
across channel width. As
Viscous(resistivity)
penetration depth
viscosity
=0.5 channel
width
increases,
the velocity
profile shows a smaller
Viscous
penetration
relative
velcity
to walls.depth
Thus
>>fluid
channel
width solidthe
approaches
body rotation.
V=0
V=0
V=0
7
2a- Normal state behavior
Q-1 after background subtraction
Qf-1= Qmeas-1- Qe-1
Fraction of inertia decoupled =
1-[P(T)-Pe(T)]/[P(T=0)-Pe(T=0)]
Red Lines show expected
behavior for 3He filling a 400 µm
cavity (without aerogel)
Dashed red line shows behavior
for a scaled bulk cavity of height
340µm with an inertia of 0.75% of
fluid
Subtract scaled behavior from
data to obtain contribution to
dissipation and fraction decoupled
from aerogel
8
2a- Aerogel acts as an impurity for Fermi Liquid
Landau – Fermi liquid theory shows
viscosity η and mean free path, in ~
T-2, due to decreasing excitation
density as T decreases.
The quasiparticle-quasiparticle mean
free path (inelastic scattering)
increases as the temperature is
lowered and is ~ 100 µm at 1 mK.
Aerogel introduces an additional
elastic scattering mechanism, e. 3500 Å
Since the scattering rates add,
τ -1=τe-1 + τin-1,  is limited by elastic
scattering at low temperatures. Thus
η and  are bounded from above.
9
2b-Analysis of Normal State Results
Calculate the effective
viscosity from the measured
dissipation of the aerogel-Fermi
liquid mixture. The inferred
viscosity is very large.
We have to find some method
by which the stiffness of the
aerogel matrix is taken into
account.
One way would be to introduce
a collection of point scatterers
quasi-one dimensional rods
embedded in the liquid.
10
2b-Analysis of Normal State Results
For simplicity we replicate the
stiffness of the aerogel by a
collection of space-filling
slabs, each spaced a distance
“tslab ” from each other.
Such a collection of slabs
would lock the fluid to the
torsional oscillator, the same
way as the 3He is locked to
the aerogel structure. The
viscosity of the 3He would still
be limited by the elastic
scattering of quasiparticles
from the aerogel.
tslab
11
2b-Analysis of Normal State Results
If the slabs are filled with
“pure” bulk 3He, we expect that
the dissipation will vary as the
slab separation is altered.
Here we show the result with
a slab separation of 4.15 µm.
When we introduce a mean
free path limited viscosity, we
see that there is now much
better agreement with the
background and bulk subtracted
data.
Fit parameters are:
t bulk = 340µm, fraction bulk 0.0075, tslab = 4.15 µm, =0.15 µm
12
2b-Analysis of Normal State Results
We use exactly the same
procedure and parameters to
fit data obtained in the normal
state for 3 bar.
Dissipation fit for impurity
limited viscosity is still
excellent, the mass
decoupling is not well
replicated.
This is understandable since
the slab model would give a
smaller mass decoupling than
a rod model or point
scatterers.
Fit parameters are:
t bulk = 340µm, fraction bulk 0.0075, tslab = 4.15 µm, =0.15 µm
13
2- Summary of normal state
• We can accurately assay the dissipation
and mass decoupling in the normal state
• These reveal that there is a crossover to
impurity dominated transport where the mean
free path is set by elastic scattering from the
aerogel.
• We find a long mean free path =0.15 µm
• Correlation of impurity scattering length
may be used to predict Tc suppression (eg via
Abrikosov-Gorkov model).
• Tc in this sample should be much higher
than in usual 98% open aerogel.
14
2- Summary of normal state – A-G model
 Tc 0 
1 r 
1
ln 
          ,   digamma function
2 2
2
 Tca 
r  /(2 k BTca ),   vF
 Tca 
From G-L theory one expects rs / r     

 Tco 
2
2
2
0
• Such an analysis should allow us to link Tc
suppression to mean free path (normal state
properties) and ρs once phase diagram is
mapped out.
15
3-“Dirty” superfluid phases & stability
TA*→B
Tcaerogel (P)
A
B
Nazaretski, Mulders, Parpia JETP Lett. 79, 470 (2004)
In ordinary “isotropic” 98%
open aerogel, it is common
to observe a large region of
A – like (A*) phase when
cooling.
Contrast to the extent of A
phase in the bulk.
Question why does the A
phase occur so reliably and
over such a broad extent in
aerogel when cooling? Is it
nucleated from some
precursor state in the so
called normal fluid?
16
3 - Larkin-Imry-Ma state
Could the A* phase be the Larkin-Imry-Ma state? In such a state, l should be
pinned by fluctuations of random orientations of impurities. This should be
modified by compression which would restore the A like phase and yield (ρs/ρ) .
┴
See Volovik Journal of Low Temperature Physics, 150, p. 453-463 (2008).
17
3 - Bulk ρs/ρ Results
(ρs/ρ)
┴
(ρs/ρ)
║
In the bulk, A phase
superfluid density can be
greater or less than that of
the B phase, (depends on
texture).
(ρs/ρ)// ~ 0.5 (ρs/ρ)B
Without a field, textural
alignment is unlikely.
(ρs/ρ)// <(ρs/ρ)B
(ρs/ρ) > (ρs/ρ)B
┴
Is the A phase correctly
identified?
See Fomin, JLTP, 79, 134
(2004).
Berthold, Giannetta, Smith, Reppy, PRL 37, 1138 (1976).
18
4 - 3He A, B phase Superfluid density in Aerogel
No sign of A phase on warming?
Finite width of TA→B. (ρs/ρ)A< (ρs/ρ)B
Nazaretski, Mulders, Parpia JETP Lett. 79, 470 (2004)
19
4- Making mixtures of A and B in aerogel
Main figure:
Mixture made
by cooling into
TA→B.
Inset: Mixture
made by
warming into
TB→N or TB→A
band.
Nazaretski, Mulders, Parpia JETP Lett. 79, 470 (2004)
20
4- Superfluid density ratio
Superfluid density
in A phase is ~½
that of B phase,
suggests (ρs/ρ)//
Results at 3
different pressures
at right.
Nazaretski, Mulders, Parpia JETP Lett. 79, 470 (2004)
(rAs/r)/(rBs/r)0.5
21
4- Superfluid density in magnetic fields
Ratio of rs/r A* and B
phase, hysteresis does
not change with field.
Range of expected
B→A* transition
No B → A* transition
seen on warming?
Evidence for
superheating? Is the A*
phase the stable phase
between normal and
the B phase?
Nazaretski, Lee, Parpia Phys Rev. B
22
4- B→A* transition only seen in higher field
In experiments at Northwestern using sound, A→B and B→A
transitions were seen in higher magnetic fields but not in low
fields.
Gervais et al Phys. Rev B.
Nazaretski, Lee, Parpia Phys Rev B
23
4 - Polycritical point
Polycritical point (bulk)
1  TAB / Tc  g ( )(B / B0 )
8 2 kBTca
B0 
1  F0a
7 (3) 
(
2
)
For 3He in usual 98% open
aerogel, the polycritical point
is eliminated
Gervais, Yawata, Mulders and Halperin, PRB 66 054528 (2002)
24
5- Superfluid density in compressed aerogel
Proposed Experiment
Measure the dissipation and period shift of 3He.
Expect to see:
1. Broader region of A phase
2. rsA > rsB since l should be perpendicular to
flow, yielding (ρs/ρ)
┴
3. A phase to reappear on warming.
Need high precision to resolve superfluidity in
impurity limited 3He near Tc from that of small
amount of bulk fluid.
25
5-Measurements at 25.7 bar in superfluid state
We measure period shift and
dissipation in the superfluid state
while warming and cooling.
The region below Tc has
numerous 2nd sound
resonances. These resonances
occur at the same values of rs/r
while cooling and warming.
rs/r in the A phase is less than
that of the B phase (contrary to
expectation).
26
5-Measurements at 29.1 bar in superfluid state
(rs/r) A cooling < (rs/r) B warming
(contrary to expectation).
Same effect is seen at 25.7 bar
Width of A
B transition
essentially unchanged in
compressed aerogel.
28 bar
27
5-The A to B Transition
Dissipation and rs/r near the
A*-B transition. The A*
phase converts gradually to
the B phase with at least
one jump in dissipation. The
dissipation decreases while
rs/r increases as the A*
transforms to the B phase.
On warming there is no
evidence for a B- A*
transition in this region.
28
5-The A to B Transition- (dissipation)
Dissipation and rs/r near the
A*-B transition. We see a
two step transition in the
dissipation as the 3He
converts from A phase to B
phase. Note the width of the
transition is not rate
dependent and shows the
effects of pinning. In contrast
the bulk transition occurs
over a negligible time
(temperature) interval. We
see the bulk transition at the
same temperature as that in
the fork
T/Tc(bulk)
29
5-Region near Tc at 29.1 bar
rs/r near the onset of
29.1bar
bar
29.2
Tc bulk
superfluidity shows some
unexpected behavior.
We see a clear bulk fluid
transition that coincides with the
transition seen in a vibrating
quartz fork.
At a lower temperature we see
a kink in the resonant period
marked as Tc aerogel.
At a slightly lower temperature
we see a divergence of warming
and cooling data that we believe is
evidence for a zero field TB→A
transition.
30
5-Compare regions near Tc at 25.7 and 29.1 bar
Focus on the B→A transition on
rs/r near the onset of superfluidity.
29.2 bar
At this point in time (two
pressures examined) there is
some evidence for a B→A
transition on warming.
Turn around experiments also
show that the state (A phase or B
phase) is altered depending on
which side of the point marked
TB→A we warm the cell to.
31
5-Compare rs/r to previous experiments
rs/r (A phase) shows
continuous change with
temperature (ie power law is
different from B phase result).
Quite different from results in
uncompressed aerogel. But not
similar to bulk behavior or what
was expected from original
experiment.
(rAs/r)/(rBs/r)0.
5
32
5-Compare rs/r to previous experiments
rs/r is only slightly enhanced in the present experiment
compared to previous results. Inconsistent with G-L model.
33
5- “Dirty” superfluid phases & stability
From Pollanen et al.
This work
Pollanen (uncompressed)
Isotropic compression of the
aerogel should result in a
preferred axis. The expectation
is that the A phase should be
stabilized by uniaxial
compression. We find the A
phase extent is broadened by
compression.
We do (tentatively) see a
B→A transition in zero magnetic
field in compressed aerogel.
There should also be a
textural orientation effect. Flow
measured perp. to compression
should sample an enhanced rsA.
Such an enhancement was not
observed.
34
5-“Dirty” superfluid phases & stability
TA*→B
Tcaerogel (P)
A
B
In ordinary “isotropic” 98%
open aerogel, it is common
to observe a large region of
A – like (A*) phase when
cooling.
Contrast to the extent of A
phase in the bulk.
We see a much enhanced
Tc (correlated to long mean
free path in normal state?).
We also see enhanced
stability region for the A
phase.
Also see some evidence for
recurrence of the A phase
on warming.
35
5- Hysteretic dissipation signature
Dissipation is seen to decrease abruptly at low temperature
(deep in B phase). Recovers on warming with differing
amounts of hysteresis.
Possibly related to vortex-core transition?
Will investigate by cooling through Tc without oscillation,
examine at other pressures.
J.P.Pekola et al, Phys. Rev. Lett., 53 584 (1984).
36
5- Dissipative transition?
TA*→B
Tcaerogel (P)
A
B
Thuneberg .
37
5- Phase diagram
Tc Suppression
Simple model for Tc
suppression
35
Aerogel Tc
calculated
Pressure (Bar)
30
Bulk Tc
25
Aerogel Tc
measured
20
15
10
5
0
0
0.5
1
1.5
2
2.5
Temperature (mK)
 (T , P)T a  constant for a particular aerogel = L o  50.2nm
c
Use L0 
0
(1  Tca / Tco )1/ 2
to generate phase diagram as a function of pressure
38
6 - Future
Normal state:
• Measure/analyze results at higher pressures.
• Develop better model for hydrodynamics in hybrid system
• Compare results for consistency across all pressures
Superfluid
• Map phase diagram (Tc, TA→B, TB→A) as a function of pressure
• Correlate phase diagram to  in normal state
• Pursue dissipative transition
• Re-examine low pressure region for quantum phase transition.
39
6 - Conclusions
Conclusions
Normal state:
• Single step aerogel is different: has longer scattering length,
less suppression of Tc.
Superfluid
• Compressed aerogel does increase metastable region of A
phase, but no alignment of rs/r.
• Possibly see A phase reappear on warming – connection to
bulk phase diagram. Viscosity is different in A and B phases.
• See dissipation signature at low temp in B phase? Is it
connected to Vortex transition? Check by rotation?
• Relate Tc suppression to mean free path (normal state
transport).
40
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