Supersolidity and disorder

1
Supersolidity and disorder
S. Sasaki, R. Ishiguro, F. Caupin, H.J. Maris
and S. Balibar Laboratoire de Physique
Statistique (ENS-Paris) now at NWU, USA now at
Osaka University Brown University, Providence
(RI, USA)
- S. Sasaki et al. , Science 313, 1098 (2006) -
H.J. Maris and S. Balibar, J. Low Temp. Phys.
147, 539 (2007) - S. Sasaki , F. Caupin and S.
Balibar, Phys. Rev. Lett. 99, 205302 (2007) -
S. Balibar and F. Caupin  topical review  J.
Phys. Cond. Mat. 20, 173201 (2008)
UFL, April 2008
2
a  supersolid  is a solid which is superfluid
a paradoxical idea
a solid transverse elasticity non zero shear
modulus a consequence of atom localization cryst
als - glasses
a superfluid a quantum fluid with zero
viscosity a consequence of   Bose-Einstein
condensation  atoms are undistinguishable and
delocalized
3
Could a solid flow like a superfluid ?
E. Kim and M. Chan (Penn. State U. 2004)
Be-Cu Torsion Rod
a torsional oscillator (1 kHz) a change in the
period of oscillation below 100 mK 1 of the
solid mass decouples from the oscillating walls ?
Torsion Bob containing helium
Drive
Detection
4
early theoretical ideas
Penrose and Onsager 1956 generalize Bose
Einstein Condensation (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)
Reatto and Chester 1969 some models of Q-solids
show BEC symmetry of the wave function ? overlap
is necessary (Imry and Schwartz 1975) a large
class of Q-solids do NOT show BEC Leggett 1970
non-classical rotation inertia (NCRI) if atoms
are delocalized (if there are free vacancies ?)
5
recent theoretical ideas (a selection...)
Prokofev , Svistunov, Boninsegni, et al. 2005-6
(MC calc.) no BEC in crystals without free
vacancies the probability that a real Q-crystal
is commensurate and superfluid is zero 4He
crystals are commensurate (Evac 13K) BEC is
possible in a 4He glass (Boninsegni et al. PRL
2006) also in grain boundaries (2D He glassy
systems, see Pollet et al. 2007)
Galli and Reatto 2006 (variational method)
superfluidity is found with trial functions
( SWF ) which reproduce the properties of
solid 4He superfluidty in a commensurate crystal
!
Clark and Ceperley (2006) superfluidity
depends on the trial functions not found in
quantum Path Integral Monte Carlo simulations
which are  exact  a perfect 4He crystal is
commensurate, with no vacancies at T 0 and no
superfluidity
Cazorla and Boronat PRB 2006 a lattice-gas
model with the right  Lindemann ratio  shows
BEC in the commensurate case. What is the
difference between lattice gas models and real
crystals ?
6
more theoretical ideas

• Anderson Brinkman and Huse (Science 2005)
• a new analysis of the lattice parameter ?a/a (T)
  and specific heat Cv(T)
• a low density of zero-point vacancies (lt 10-3
  ?) TBEC a few mK ?s ?
• not confirmed by new neutron scattering
  measurements (Blackburn et al. PRB 2007)
• criticized by H.J. Maris and S. Balibar (J. Low
  Temp. Phys. 147, 539, 2007)
• Anderson has now turned to the hypothesis of a
   vortex liquid , in analogy with high Tc
  superconductors (Nature Physics 2007) but where
  are the mobile particles ?

could supersolidity be due to disorder , i.e.
the presence of defects in the crystals ? Dash
and Wetlaufer 2005 the He-wall interface can be
superfluid PG de Gennes (Comptes Rendus -
Physique 7, 561, 2006) the change in the
rotational inertia could be due to the change in
the mobility of dislocations. It should depend on
frequency (see Aoki, Kojima PRL 2007) Pollet et
al. 2007 grain boundaries are
superfluid Shevchenko1987, Toner 2006, Boninsegni
et al. 2007 the core of screw dislocations is
superfluid Biroli and Bouchaud 2008 transverse
fluctuations of dislocations
7
the role of disorderannealing , grain
boundaries
Rittner and Reppy (Cornell, 2006-7) annealing
destroys supersolid behavior quenched cooled
crystals show a very large ?s , up to 20 at
least
Sasaki, Ishiguro, Caupin, Maris and Balibar
(Science 2006) superfluid mass transport
through solid He in the presence of grain
boundaries
8
S. Sasaki et al. ENS 2006 experimental setup
liquid helium
window
in a glass tube (1 cm ??) grow a crystal from
the superfluid 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
9
filling the tube with solid 4He makes defects
liquid
liquid
liquid
solid
the inside crystallizes only if a substantial
stress is applied.
solid

• fast mass injection at 1.3K
• gt defects
• grain boundaries
• grooves

10
grooves 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
at Pm , most cusps move away in a few hours
(melting-crystallization pinning) some GBs stay
pinned on walls
11
no flow in good quality crystals
with no or very few cusps the tube, no flow no
observable mass leak along the glass wall
either If supersolidity was due to a 1
superfluid density in the bulk with a critical
velocity vc 10 ?m/s the interface should relax
at V 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 an intrinsic property of
  He4 crystals
• it is not due to the superfluidity of a 1 (even
  0.03) equilibrium density of vacancies moving
  in the bulk at 10?m/s.

12
mass flow in crystals with enough grain boudaries
crystal 1 when the cusp disappears, the mass
flow stops
superflow of mass through solid 4He requires the
existence of grain boundaries
13
crystal 1 relaxed 1 mm down and stopped
14
crystal 2 had many defects
Many grain boundaries more in the lower
part faster flow down to equilibrium at h 0
15
crystal 2 relaxed down to eq. (h 0)
time x 250 5 s 20 min
16
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
17
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 of atomic thickness
18
Numerical simulation of grain boundaries
Nature 21 octobre 2006
19
Pollet et al. PRL 98, 135301, 2007
Grain boundaries are 3 atoms thick and
superfluid except in special directions. Tc
0.2 to 1 K depending on orientation critical
velocity ?
20
1 superfluid density is large ! (Rittner and
Reppy 2007 20 in thin quenched cooled samples !)

• 1 matter with grain boundaries
• 1 atomic layer of superfluid matter
• grain size 100 nm
• 3 ?m for 0.03
• Is this possible ? may be
• We used to grow single crystals at constant P
  from the superfluid,
• but
• crystals grown at constant V from the normal
  liquid are usually polycrystals with much more
  disorder

21
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)
22
at T lt 100 mK from the superfluid, fast growth
and melting of single crystals
real time, 60 mK
23
at T gt 1.8 K fast growth from the normal
liquiddendrites
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
strong light scattering as also observed by Ford
et al. 2007 Maekawa et al. 2002
24
more helium snowflakes
facetted ?? the roughening transition would be
re-entrant , due to a change in the surface
tension ?
25
slow growth at constant volume from the normal
liquid
slow growth (3 hours) in a temperature gradient
(Twalls lt Tcenter)
hcp solid
liquid
liquid 2.56K
the solid is transparent but polycrystalline
hcp solid 1.95 K
26
melting a crystal grown at constant volume
solid at 40 mK
liquid channels appear at the contact line of
each grain boundary with the windows grain size
?m ripening
27
melting a crystal after fast growth from the
superfluid

• a the fast grown solid
• is transparent
• but polycrystalline
• 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

28
further melting gt 2 crystals 1 grain boundary

• the cusp angle is non-zero gt the gain boundary
  energy ?GB is strictly lt 2 ?LS
• gt partial wetting of the GB by the liquid, the
  thickness of grain boundaries is microscopic as
  calculated by Pollet et al. (2007).
• complete wetting would imply ?GB ?????LS (2
  liq-sol interfaces with bulk liquid in between)

the contact of the GB with each window is a
liquid channel
29
angle measurements gt the grain boundary energy

• here, the GB is parallel to the optical axis
• a fit with Laplace equation near the cusp leads
  to
• ? 14.5 4
• ??GB (1.93 0.04) ?LS
• other crystals
• ? 11 3
• 16 3
• for more details see Sasaki, Caupin and Balibar,
  to be submitted to
• J. Low Temp. Phys. (April 2008)

30
wetting of grain boundaries near a wall
GB
grain 2
grain 1
liquid
grain 1
grain 2
wall
wall
S. Sasaki, F. Caupin, and S. Balibar, PRL 99,
205302 (2007)
If ?GB?is large enough, more precisely if???
?c lt ?/2 the liquid wets the contact of the GB
with the wall. wetting and premelting of grain
boundaries an important problem in materials
science (see for ex. JG Dash Rep. Prog. Phys. 58,
115, 1995) this effect is responsible for the
apparent wetting of GBs observed with fcc
crystals by Franck et al. (Edmonton, 1983-5)
31
the contact line on the window is a liquid
channel whose width w (P-Pm)-1
the width w of the triangular liquid channel
decreases as 1/ z (the inverse of the departure
from the equilibrium melting pressure
Pm) consistent with the direct measurement but ?c
is hysteretic the channel should disappear
around Pm 10 bar (where 2w 1 nm)
32
hysteresis of the contact angle
advancing angle 22 6 (copper) 26 7
(glass) receding angle 55 6 (copper) 51
5 (glass) 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)
33
the (partial) wetting of grain boundaries
depends on orientation
no liquid channel along the wall if the GB has a
low energy (large ?)
the superfluidity of GBs should also depend on
orientation (see Pollet et al. PRL 2007)
34
two possible interpretations of Sasaki et al.
(Science 2006)
liquid
liquid

• the flow could be
• either along the GBs (then vc 1 m/s)
• or at the GB-wall contact (then vc 1 cm/s).
  This would explain why we saw mass flow up to
  1.1K at least.
• to be checked by changing the shape of the cell
• or by squeezing the side channels with an
  electric field

solid
solid
35
supersolidity inside grain boundaries no
experimental evidence yet
Pollet et al. (PRL 98, 135301, 2007) Tc 0.5 K
with 3 layers thickness but less than 1
superfluid layer Tc is 3 times less than the
bulk T? at the solid density (1.5K), even for a
liquid film 3 atomic layers thick ? a
Kosterlitz-Thouless transition ? possible
measurement a cell with two parts, one fixed
grain boundary, squeeze the liquid channels with
an electric field make a height
difference measure the mass flow as a function of
T another experiment in progress at U. Mass
Amherst (Hallock et al. 2008)
36
single crystals Clark et al. 2007
crystals grown at constant T1.3K from the
superfluid gt single crystalline no grain
boundaries gt dislocations ? the superfluid
density is 0.4 but 0.04 in another cell gt
dislocations ??
37
superfluidity of screw dislocation cores
phase coherence along the core of screw
dislocations on a distance 1/T a true 1D-
supersolid a network of dislocations with density
ns Tc 1/l ns1/2 if dislocations are immobile,
a very large dislocation density is needed to
build a superfluid density of order 0.1 but
dislocations are probably highly mobile and
fluctuate through the crystal G. Biroli and JP
Bouchaud (Saclay, France) preprint oct. 2007
a Monte Carlo calculation
38
not a single mechanism ?

• the origin of rotation anomalies is not
  necessarily the same in all samples
• dislocations in single crystals
• grain boundaries in polycrystals grown at
  constant V
• where ?S 1 ?
• liquid or glassy regions connected by grain
  boundaries in quenched samples
• where ?S up to 20 (Rittner 06-07)) ?

39
no superfluidity ?
no evidence for phase coherence no permanent
nor dc- mass currents no critical velocity ?
large hysteresis found by Aoki et al.
2007-08 dissipation found below 1 quantum of
circulation by Clark et al.
a change in elastic properties ? from PG de
Gennes to Day and Beamish 2007
40
Day and Beamish (Nature 2007) a measurement of
the shear modulus of crystals grown at constant V
similar anomalies
the shear modulus increases by 15 below 100
mK dislocation pinning by 3He impurity adsorption
? relation to torsional oscillator experiments
? less inertia for a stiffer crystal ?? a new
theoretical challenge
41
Clark and Chan an increase in stiffness ?
Clark et al. preprint 2007 to be published in PRB
2008 could the change in period be due to an
increase in the shear modulus ? check the
frequency dependence see Aoki, Graves and
Kojima 2007-2008 (expts) Dorsey et al. 2008 to be
published (theory)
42
Chan et al. 2007 evidence for a true phase
transition from a peak in the specific heat
Dorsey, Goldbart and Toner PRL 96, 055301, 2006
a ?-peak in the solid Chans
experiment substract cell AT3 (phonons ) BT
(3He) ... a peak around 75 mK consistent with ?s
0.06 a true phase transition ? other
contributions from dislocations or from grain
boundaries or glassy regions ??
43
conclusion
the superfluidity of solid helium 4 is not yet
established see Balibar and Caupin, topical
review, J. Phys. Cond. Mat. 20, 173201, 2008
future work measure properties of samples with
known disorder
44
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45
hysteresis Aoki, Graves and Kojima PRL 2007
is there really a critical velocity ? vc from
less than 1 to more than 600 ?m/s 1 ?m/s (see
Clark Maynard and Chan PRB 2008) is less than 1
quantum of circulation !
46
Lattice-gas modelscold atoms in optical lattices
cold atoms in an optical lattice on site
repulsion U, tunneling energy J transition from
 Mott insulator  to  superfluid  when U/J
5.8 z and n Nat / Ns 1 (Greiner et al.
Nature 2002) BUT, compared to real crystals -
the symmetry is not spontaneously broken - double
occupancy of sites - non-homogeneous potential
- Nat and Ns are independent
H. Stoof, Nature 415, 25, 2002
47
Bose-Hubbard model with density nthe
commensurate case n1 is singular
n 1 commensurate crystal, insulator-superfluid
transition at U/J 5 z n 1 - ? or 1 ?
superfluid transition of adatoms or interstitials
below Tc
I. Bloch, J. Dalibard and W. Zwerger to appear
soon in Rev. Mod. Phys. 2008
F. Gerbier, PRL 99, 120405 (2007)
in solid 4He , U 10K , J lt 10 - 3K gt J/U lt 10
- 4 ?
48
have we seen the same effect as in torsional
oscillators experiments on samples grown at
constant V ?
the effect of annealing (Cornell , Penn State,
Keio Univ., Rutgers, Kharkov...) and the large
scatter in the data gt evidence for the
importance of quenched disorder
increase of ?s (P) from 25 to 60 bar more
grain boundaries ? decrease of at large P
superfluidity disappears at high density
Kim and Chan 04
49
crystal growth from the normal liquid is
dendritic if fast enough
a wet snowball of helium grown at 1.9K strong
light scattering by a high density of defects
50
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 large
macroscopic grains
51
The T7 term in the specific heatH.J Maris and
S. Balibar (J. Low Temp. Phys. 147, 539, 2007)
the specific heat of solid helium is well
described by Cv A T3 B T7 Anderson Brinkman
and Huse (2006) no exp(-Evac/T) term gt
vacancies are not thermally activated a simple
model of zero point vacancies leads to a T7
term Maris and Balibar (2007) Evac is not very
precisely known ABH the T7 term cannot be due to
the dispersion of phonons because it would lead
to B A ??D4 with ?D 25 K MB B is
typically 100 times larger, a good fit of the T7
term could be found, depending on the dispersion
of phonons
52
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

53
Pressure relaxation at 1.6 K
grow a crystal at constant volume from liquid at
60 bar
the pressure gauge works only below 37 bar

• at time t 0,
• open the fill line
• liquid appears in the cell
• where P Pm (1.6) 27.4 bar
• smooth variation of P in about 10 hours
• evidence for thermally activated vacancies ?
• the pressure does not relax to
• the equilibrium Pm 27.4 bar

54
Pressure relaxation at 40 mK
at 40 mK, most crystals show no relaxation at
all, even with Pm 25.3 bar in the cell
crystal 6 showed a relaxation down to 26.5 bar
in the gauge volume are discontinuous jumps due
to unpinning of grain boundaries ?
55
light scattering from crystals quenched at low T
laser beam
liquid
4He cell
quenched solid close to Pm liquid drops with
submicron size ?
no observable light scattering in the case of
solid grown slowly at constant V liquid droplets
too small ?
56
which kind of disorder ?
If supersolidity is not due to an equilibrium
density of vacancies, what else ? dislocations ?
(edge or screw ?) grain boundaries ? liquid or
glassy regions ? (how large ? how many ?) 3He
? everything ? no clear answers yet not
necessarily the same defects in samples prepared
in different ways we have focused on grain
boundaries