Supersolidity and disorder

1
Supersolidity and disorder
S. Sasaki, R. Ishiguro , F. Caupin, E.
Rolley, H.J. Maris and S. Balibar Laboratoire
de Physique Statistique (ENS-Paris) now at
Osaka University Brown University, Providence
(RI, USA)
a reference Science 313, 1098 (25 Aug.
2006) more to be published in 2007
U. Mass. Amherst, May 9th, 2007
2
outline
1 - Introduction to supersolidity The
experimental discovery by Kim and Chan
(2004) early theoretical ideas recent
theoretical ideas more experimental results
(Cornell, Yokohama...)
2 - The flow experiment at ENS -Paris
(2006) supersolidity requires disorder
3 - Recent results on grain boundaries
4 - some discussion of possible scenarios for
supersolidity
3
Could a solid flow like a superfluid ?
E. Kim and M. Chan (Penn. State U. 2004)
a torsional oscillator (1 kHz) a change in the
period of oscillation below 200 mK 1 of the
solid mass decouples from the oscillating walls ?
4
1 superfluid density in solid 4He ?
FNCRI (non classical rotational inertia
fraction) 1 at 51 bar
no effect in pure 3He
5
early theoretical ideas
Penrose and Onsager 1956 generalize BEC to
condensed matter systems Off diagonal long range
order in the density matrix BEC is impossible in
a solid (but they used non-symmetrized wave
fonctions) Andreev and Lifshitz 1969
delocalized defects (vacancies) could exist at
T0 ( the crystal would be  incommensurate )
BEC gt superplasticity at low velocity or long
times Reatto, Chester and Leggett 1969-70 NCRI
is possible if atoms are delocalized (if there
are free vacancies ?)
6
recent theoretical ideas
Prokofev and Svistunov 2005 no BEC in crystals
without free vacancies (commensurate crystal,
vacancy-interstitial pairs) 4He crystals are
commensurate BEC possible in a 4He glass
(Boninsegni et al. PRL 2006)
Galli and Reatto 2006 superfluidity in
simulations with trial functions ( SWF )
which reproduce the properties of solid 4He
Clark and Ceperley (2006) superfluidity
depends on the trial functions not found in
quantum Monte Carlo simulations the crystal is
commensurate, no vacancies at T 0

• Anderson Brinkman and Huse 2005 a new analysis
  of the T variation of the
• lattice spacing (old experiments by Simmons) T4
  instead of exp(-Evac/T) ?
• and the specific heat Cv(T) AT3 BT7
• a low density of zero-point vacancies (lt 10-3
  ?) TBEC a few mK ?s ?

PG de Gennes (CR-Physique 2006) superplasticity
due to dislocations being mobile at low T. It
should depend on frequency. Contradiction with
Kojima 2007 ...
7
puzzling experimental results
8
annealing the crystals, adding 3He
Rittner and Reppy (Cornell, 2006-7) annealing
destroys supersolid behavior quenched cooled
crystals show a very large ?s up to 20 at least
In 2006, Kim and Chan (Penn State) claimed that
annealing the samples increased the supersolid
fraction ! In 2007 they partially agree
Shirahama et al. (Yokohama, 2006-7) some
effect of annealing but the supersolid density ?s
0.1, not 1 ...
Kim and Chan (Penn State, 2006) 3He impurities
increase Tc but decrease ?s but ultrapure 4He
shows very small ?s J. Beamish (Edmonton,
Canada) 3He impurities must be bound to 1D or 2D
defects which control supersolidity
9
ENS 2006 experimental setup
liquid helium
window
in a glass tube (1 cm ??) grow a crystal at 1.3
K lower T down to 50 mK melt the outside gt
height difference follow the level inside
solid helium
any change in the level inside requires a mass
flow through the solid since ?C 1.1 ?L
if critical velocity vc 10 ?m/s and superfluid
density ?s 10-2 ?C gt melting velocity V 3
mm/h
10
filling the tube with solid 4He makes defects
liquid
liquid
liquid
the inside crystallizes only if a substantial
stress is applied. For example if the outside is
warmed up to 1.4K for a few seconds while the
inside is at 1.3K
solid
Pm( 1.4 K) - Pm( 1.3 K) 0.3 bar fast growth
under inhomogeneous stress creates defects
11
cusps and grain boundaries
mechanical equilibrium of surface tensions at
the liquid-solid interface ?GB ????LS
cos? each cusp signals the existence of an
emerging grain boundary
most cusps move away in a few hours
(melting-crystallization pinning) some GBs stay
pinned on walls
12
no flow in good quality crystals
for 10 crystals with no or very few cusps the
tube, we could see no flow no mass leak along the
glass wall either if supersolidity were due to a
1 superfluid density in the bulk with a
critical velocity vc 10 ?m/s the interface
should relax at V ?s/(?C - ?L)vc 1 ?m/s,
that is 3.6 mm in 1 hour Instead, we see no flow
within 50 ?m in 4 hours, meaning at least 300
times less

• supersolidity is not due to the superfluidity of
  a
• 1 equilibrium density of vacancies moving at
  10?m/s

13
mass flow in crystals with enough grain boudaries
for 3 crystals with some cusps inside the tube we
observed a mass flow crystal 1 when the cusp
disappears, the mass flow stops
superflow of mass through solid 4He is associated
with the existence of grain boundaries
14
crystal 1 relaxed 1 mm down and stopped
15
crystal 2 had many defects
Many grain boundaries more in the lower
part faster flow down to equilibrium at h 0
16
crystal 2 relaxed down to eq. (h 0)
time x 250 5 s 20 min
17
crystal 2relaxation at 50 mK
relaxation is not exponential but linear with two
successive regimes, constant velocity 6 ?m/s
for 0 lt t lt 500 s 11 ?m/s for 500 lt t lt 1000
s more defects in the lower part of crystal
2 typical of superfluid flow at its critical
velocity
18
crystal 1 a single grain boundary
the relaxation at V 0.6 ?m/s stops when the
cusp disappears (the grain boundary moves away,
unpinning from the wall somewhere)
Assume 1 grain boundary (thickness e a 0.3 nm
, width w D 1cm) the critical velocity
inside is vcGB (?D2/4ew?s)(?C-?L)V 1.5
(a/e)(D/w)(?C /?s) m/s comparable to 2 m/s
measured by Telschow et al. (1974) on free liquid
films
19
Numerical simulation of grain boundaries
Nature 21 octobre 2006
20
Pollet et al. PRL 98, 135301, 2007
Grain boundaries are 3 atoms thick and
superfluid except in special directions. Tc
0.5 K vc ?
21
have we seen the same effect as Kim and Chan and
other groups using torsional oscillators ?
the effect of annealing (Cornell , Keio Univ.,
Rutgers, Kharkov...) and the large scatter in the
data gt evidence for the importance of quenched
disorder most natural defect grain boundaries
no vacancies in single crystals at T 0 Fraass
and Simmons (1989) Blackburn et al. (2007)
condmat-0702537 agreement with Prokofev et al.
05 and with Ceperley and Clark 05 not with
Anderson et al. 05
increase of ?s (P) more grain boundaries ?
decrease of at large P superfluidity disappears
at high density
22
E. Blackburn et al. (ISIS , UK)cond-mat/0702537
(23 Feb. 2007)

• a neutron scattering experiment
• no T - dependence
• lattice parameters a and c
• fluctuations lt?ugt2 are purely quantum with
• no change at the supersolid transition
• no vacancies at low T,
• no BEC of vacancies

23
H.J Maris and S. Balibar (J. Low Temp. Phys. 147,
539, 2007) further critics on the model by
Anderson, Brinkman and Huse (Science 2005)
the specific heat of solid helium is well
described by Cv A T3 B T7 ABW no
exp(-Evac/T) term gt vacancies are not thermally
activated. A simple model of zero point vacancies
leads to a T7 term MB Evac is not very
precisely known ABW the T7 term cannot be due to
the dispersion of phonons because B (???D)4
with ?D 25 K MB the coefficient ? is large,
a good fit of the T7 term could be found
24
1 superfluid density is large ! (Rittner and
Reppy 2007 20 in thin quenched cooled samples !)
We used to grow crystals at constant P from the
superfluid, but in torsional oscillator
experiments, crystal growth at constant V from
the normal liquid, i.e. at variable T and P gt
defects gt polycrystals with grain boundaries
every 100 à 200 a , about 50nm ??
1 vacancies would be very large too (6 to 20
...)
25
crystal growth from the normal liquid is dendritic
a wet snowball of helium grown at 1.9K strong
light scattering by a high density of defects
26
a high pressure cell to grow He crystals at
constant V
two cubic cells 11 x 11 x 10 mm3 or 11
x 11 x 3 mm3 thermal contact via 10 mm thick
copper walls 2 glass windows (4 mm
thick) indium rings stands 65 bar at
300K Straty-Adams pressure gauge (0 to 37 bar)
connection through a 3 cm long CuNi capillary
(0.6 mm ID)
27
crystal growth at constant V
starting pressure 61 bar
25 jan 07 164022 T 2.98 K
28
crystal growth at constant V
25 jan 07 170231 T 2.42 K
growth from the walls in a radial T
gradient start crystallizing at 2.42 K
29
crystal growth at constant V
25 jan 07 170331 T 2.405 K
further growth as we cool down
30
crystal growth at constant V
25 jan 07 170431 T 2.395 K
31
crystal growth at constant V
25 jan 07 170818 T 2.33 K
32
crystal growth at constant V
25 jan 07 170922 T 2.30 K
a thin solid layer appears on the windows
33
crystal growth at constant V
25 jan 07 171222 T 2.25 K
growth from the walls in a radial T gradient the
hcp crystal is transparent
34
crystal growth at constant V
25 jan 07 172724 T 2.08 K
further growth from the walls in the T gradient
35
crystal growth at constant V
25 jan 07 174344 T 2.06 K
the hcp crystal is transparent slow growth in a
T-gradient is not dendritic the solid film on
the windows shows boundaries between grains
36
crystal growth at constant V
25 jan 07 180340 T 1.88 K
further growth
37
crystal growth at constant V
25 jan 07 181520 T 1.81 K
growth nearly completed the hcp crystal is still
transparent no evidence for macroscopic liquid
regions inside
38
crystal growth at constant V
end of crystallization after 100 min at
1.78K where Pm 31 bar the pressure gauge
indicates P gt 37 bar (no relaxation of the
pressure through the 0.6 mm capillary) no
hydrostatic equilibrium in the solid
25 jan 07 181750 T 1.78 K
cooling further the last liquid drop disappears
39
going through the bcc phase
29 jan 07 105328 T 2.38 K all liquid
crystallization starts at 2.26 K and ends at
1.52 K
40
going through the bcc phase
29 jan 07 110330 T 2.25 K
crystallization starts at 2.26 K and ends at
1.52 K
41
going through the bcc phase
29 jan 07 111328 T 2.09 K
crystallization starts at 2.26 K and ends at
1.52 K
42
going through the bcc phase
29 jan 07 112328 T 1.88 K
crystallization starts at 2.26 K and ends at
1.52 K
43
going through the bcc phase
29 jan 07 113329 T 1.79 K
crystallization starts at 2.26 K and ends at
1.52 K
44
going through the bcc phase
29 jan 07 114329 T 1.69 K
hcp
bcc
liquid
close to the triple point 3 phases
coexist there is a T gradient in the cell the
bcc solid is a poly-crystal with macroscopic
grains
45
going through the bcc phase
29 jan 07 105328 T 2.38 K
hcp
29 jan 07 115329 T 1.69 K
bcc
crystallization starts at 2.26 K and ends at
1.52 K
46
going through the bcc phase
29 jan 07 120726 T 1.52 K all hcp
crystallization starts at 2.26 K and ends at
1.52 K the hcp solid is transparent
47
cooling down
29 jan 07 120726 T 1.52 K all hcp
at low T (40 mK), the hcp crystal is still
transparent no evidence for macroscopic liquid
regions
30 jan 07 182820 T 0.040 K
48
melting at low T
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P
-gt Pm ?
30 jan 07 182830 T 0.040 K
49
melting at low T
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P -gt
Pm ?
30 jan 07 182840 T 0.040 K
50
melting at low T
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P -gt
Pm ?
30 jan 07 182851 T 0.040 K
51
melting at low T
29 jan 07 182830 T 0.040 K
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P -gt
Pm ?
30 jan 07 182901 T 0.040 K
52
melting at low T
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P -gt
Pm ?
30 jan 07 182911 T 0.040 K
53
melting at low T
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P -gt
Pm ?
30 jan 07 182916 T 0.040 K
54
melting at low T
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P -gt
Pm ?
30 jan 07 182952 T 0.040 K
55
melting at low T
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P -gt
Pm ?
30 jan 07 183032 T 0.040 K
56
melting at low T
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P -gt
Pm ?
30 jan 07 183112 T 0.040 K
57
melting at low T
liquid
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P -gt
Pm ?
30 jan 07 183152 T 0.040 K
58
melting at low T
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P -gt
Pm ?
30 jan 07 183202 T 0.040 K
59
melting at low T
evidence for large number of defects but crystal
grains have macroscopic sizes coarsening as P -gt
Pm ?
30 jan 07 183212 T 0.040 K
60
melting at low T
liquid
grain boundaries touch windows as visible
lines macroscopic liquid channels ?
30 jan 07 183522 T 0.041 K
61
dendritic growth
T 1.87 K
fast mass injection through the fill line in the
normal liquid (here at 1.87 K) leads to dendritic
growth but not slow growth at constant V in a
T-gradient
62
very fast growth from the superfluid
63
melting a crystal after fast growth
11 mm

• a the fast grown solid
• is transparent
• b to f in 11 seconds
• some bulk liquid
• appears in f
• small size crystal grains
• ripening of the solid foam in a few seconds at
  the melting pressure

64
further melting often produces bi-crystals
a 10 mm thick sample
a 3 mm thick sample
angle 2?
from the measurement of the cusp angle 2? 25
/- 10 degrees , we find the gain boundary
energy ?GB 1.85 /- 0.08 ?LS (preliminary
result) the thickness of grain boundaries is
microscopic (? gt 0, i.e. partial wetting)
complete wetting would imply ?GB ?????LS (2
liq-sol interfaces with bulk liquid in between)
65
growth oscillations its a crystal !
real time, 60 mK
66
wetting of grain boundaries near a wall

• consequences on the interpretation of our flow
  experiment (Sasaki et al. Science 2006) the flow
  could be
• either along the GBs (then vc 1 m/s)
• or at the GB-wall contact (then vc 1 cm/s)
• to be checked by changing the aspect ratio of
  the cell

67
the contact line on the window is a liquid
channel
a fit of the asymptotic variation of the width w
with the depth z leads to a cusp angle ? 25
if the contact angle ?c 45 consistent with
the direct measurement but ?c is hysteretic
68
hysteresis of the contact angle
growing
melting
more hysteresis on copper rough walls than on
smooth glass walls, as expected from E. Rolley
and C. Guthmann (ENS-Paris) PRL 98, 166105 (2007)
69
the (partial) wetting of grain boundaries should
depend on orientation
J.P. Franck et al. (Edmonton, Alberta, Canada)
PRL 50, 1463 (1983) and JLTP 58, 153 (1985)
early study of solid helium films (50
microns) adsorbed on a sapphire window (which
favors the presence of the liquid phase) at high
pressure (0.5 to 9 kbar) wetting of the
boundaries between fcc grains in the adsorbed
film for the fcc phase, partial ? ? between 0 and
30 degrees no wetting for hcp grains at 1
kbar but only for stacking faults ? the wetting
should depend on orientation (to be studied) the
superfluidity of grains also (see Pollet et al.
PRL 2007 and Reatto 2007)
70
temporary conclusions
No superflow through solid He without defects
When grown at constant V, solid 4He is usually
polycrystalline with macroscopic grains
?S 1 looks impossible to achieve with grain
boundaries only (a fortiori 20 !)
Annealing produces a drop in P (Grigorev et al.
07, Rittner and Reppy 07 gt existence of low
density (liquid or glassy ?) regions in the
disordered solid. Even at high P, they can be
stabilized by stress gradients in the solid
No macroscopic droplets can be seen in our
experiment Submicron size droplets ? to be
checked by light scattering Cv measurements

• Are these hypothetical droplets connected by
  grain boundaries or dislocation cores?
• this would provide
• phase coherence
• a low Tc associated with the superfluid
  transition inside the GB (to be determined)
• an apparently small critical velocity (10 ?m/s
  is not a local value)
• but 1 to 20 supersolid density is not obvious
  to obtain gt glassy regions ?

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the pressure is far from homogeneous in the solid
a Straty-Adams pressure gauge is connected to
the cell via a capillary (0.6 mm ID, 3 cm long)
even at 40 mK, P is not homogeneous in the thin
space of the gauge the 0.6 mm capillary is
usually blocked with P 26 to 31 bar in the
cell, the gauge indicates P gt 37 bar (where C
8 )
One run showed partial relaxation down to 30 bar
although P should be 26 bar everywhere
The gauge pressure usually relaxes when P is
lowered down to Pm and liquid appears in the
cell two examples
74
Pressure relaxation (crystal 4)
even at 1.6 K, on the melting curve in the
cell, the pressure does not relax to Pm 27.4
bar in the gauge volume
75
Pressure relaxation (crystal 6)
at 40 mK, Pm 25.3 bar in the cell, crystal 6
showed a relaxation down to 26.5 bar in the
gauge volume pressure relaxations are not
reproducible at low temperature gt quenched
disorder
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