Could a quantum solid flow like a superfluid ?

Could a quantum solid flow Could a quantum solid flow like a superfluid ? like a superfluid ?

S. Sasaki, R. Ishiguro , F. Caupin, H.J. Maris* S. Sasaki, R. Ishiguro , F. Caupin, H.J. Maris*

and and S. BalibarS. Balibar

Laboratoire de Physique Statistique (ENS-Paris)Laboratoire de Physique Statistique (ENS-Paris)

* Brown University, Providence (RI, USA)* Brown University, Providence (RI, USA)

Oxford, 25 jan 2007Oxford, 25 jan 2007

A reference: Science 313, 1098 (25 aug. 2006) A reference: Science 313, 1098 (25 aug. 2006)

Evangelista Torricelli (1608-1647)

Galileos friendGalileos friendinvented the first barometerinvented the first barometer

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liquid Hgliquid Hg

1 atm = 760 mmHg 1 atm = 760 mmHg

vacuumvacuum

two communicating vessels (inside and outside the tube)two communicating vessels (inside and outside the tube)hydrostatic equilibrium hydrostatic equilibrium the weight of the liquid column is compensated by the atmospheric pressurethe weight of the liquid column is compensated by the atmospheric pressure

under vacuum: same level

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when Torricelli pumped through E:when Torricelli pumped through E:liquid-gas equilibrium in A and Bliquid-gas equilibrium in A and B

same temperaturesame temperaturesame vapor pressuresame vapor pressure

same levelssame levelsbecause a liquid allows the mass flow because a liquid allows the mass flow

which is necessary to achieve hydrostatic equilibriumwhich is necessary to achieve hydrostatic equilibrium

we did the same experiment we did the same experiment with solid with solid 44He in eq. with liquid He in eq. with liquid 44HeHe

E. Torricelli, Florence 1644E. Torricelli, Florence 1644

Motivation : is solid 4He « supersolid »?

E. Kim and M. Chan E. Kim and M. Chan (Penn. State U. 2004):(Penn. State U. 2004):

a torsional oscillator (a torsional oscillator (~1 kHz)~1 kHz)

a change in the period of a change in the period of oscillationoscillation below 200 mKbelow 200 mK

1 % of the solid mass decouples 1 % of the solid mass decouples from the oscillating walls ?from the oscillating walls ?

Be-Cu Torsion Rod

Torsion Bobcontaining helium

Drive

Detection

K

Io πτ 2=

1% superfluid density in solid 4He ?

NCRI NCRI (non classical rotational inertia) (non classical rotational inertia) ~1% at 51 bar~1% at 51 bar

no effect in no effect in 33HeHe

the effect is strongly reduced the effect is strongly reduced with a barrier in the rotating annuluswith a barrier in the rotating annulus

early theoretical ideas

Penrose and Onsager 1956: Penrose and Onsager 1956: BEC is impossible in a solid BEC is impossible in a solid (but they used non-symetrized wave fonctions)(but they used non-symetrized wave fonctions)

Andreev and Lifshitz 1969: Andreev and Lifshitz 1969: delocalized defects (vacancies) could exist at T=0 delocalized defects (vacancies) could exist at T=0 ( the crystal would be « incommensurate ») ( the crystal would be « incommensurate ») BEC => superplasticity at low velocity or long timesBEC => superplasticity at low velocity or long times

Reatto, Chester and Leggett 1969-70: Reatto, Chester and Leggett 1969-70: NCRI is possible if atoms are delocalizedNCRI is possible if atoms are delocalized(if there are free vacancies ?)(if there are free vacancies ?)

Imry and Schwartz (1975): Imry and Schwartz (1975): no supersolidity in a true crystal without free vacancies no supersolidity in a true crystal without free vacancies (a lattice gas is different)(a lattice gas is different)......

EE00

recent theoretical ideasProkofev and Svistunov 2005: no BEC in crystals without free vacanciesProkofev and Svistunov 2005: no BEC in crystals without free vacancies(commensurate crystal, vacancy-interstitial pairs); BEC in a (commensurate crystal, vacancy-interstitial pairs); BEC in a 44He glass He glass (Boninsegni et al. PRL 2006)(Boninsegni et al. PRL 2006)

Galli and Reatto 2006: superfluidity in simulations with trial functions Galli and Reatto 2006: superfluidity in simulations with trial functions (« SWF ») which reproduce the properties of solid 4He(« SWF ») which reproduce the properties of solid 4He

Clark and Ceperley (2006) : superfluidity depends on the trial functionsClark and Ceperley (2006) : superfluidity depends on the trial functionsnot found in quantum Monte Carlo simulations; not found in quantum Monte Carlo simulations; the crystal is commensurate, no vacancies at T =0the crystal is commensurate, no vacancies at T =0

Anderson Brinkman and Huse 2005: a new analysis of the T variation of the Anderson Brinkman and Huse 2005: a new analysis of the T variation of the lattice spacing (old experiments by Simmons)lattice spacing (old experiments by Simmons)and the specific heat Cand the specific heat Cvv(T) = AT(T) = AT33 + BT + BT77

a low density of zero-point vacancies (< 10a low density of zero-point vacancies (< 10-3-3 ?); T ?); TBECBEC ~ a few mK ; ~ a few mK ; ss ? ?

PG de Gennes (CR-Physique 2006): quantum dislocations are mobile at low TPG de Gennes (CR-Physique 2006): quantum dislocations are mobile at low T......

puzzling experimental results

Kim and Chan: the critical velocityKim and Chan: the critical velocity is 10 is 10 m/s, independent of P m/s, independent of P The critical temperature isThe critical temperature is also independent of Palso independent of P

the superfluid fraction increasesthe superfluid fraction increasesbefore decreasing as a fct of Pbefore decreasing as a fct of Palthough atoms should be less mobilealthough atoms should be less mobileand vacancies should disappearand vacancies should disappearas P increasesas P increases

annealing the crystals, adding 3He

Rittner and Reppy (Cornell, 2006): annealing destroys supersolid behavior Rittner and Reppy (Cornell, 2006): annealing destroys supersolid behavior

Kim and Chan (Penn State, 2006): annealing enhances supersolid behavior ! Kim and Chan (Penn State, 2006): annealing enhances supersolid behavior !

Shirahama et al. (Tokyo, 2006): Shirahama et al. (Tokyo, 2006): no effect of annealing but the supersolid density no effect of annealing but the supersolid density ss = 0.1%, not 1% ... = 0.1%, not 1% ...

Kim and Chan (Penn State, 2006): Kim and Chan (Penn State, 2006): 33He impurities increase Tc but decrease He impurities increase Tc but decrease ss

but ultrapure 4He shows very small but ultrapure 4He shows very small ss

thermodynamic quantities :thermodynamic quantities : very small change in the specific heat (Kim and Chan)very small change in the specific heat (Kim and Chan)no singularity in the melting curve (Todoshchenko et al. Helsinki 2006)no singularity in the melting curve (Todoshchenko et al. Helsinki 2006)

two previous experiments on superflow

.... ..Day, Herman and Beamish (PRL 2005):Day, Herman and Beamish (PRL 2005):no flow in Vycor glassno flow in Vycor glass

the lattice is probably pinned at low T, the lattice is probably pinned at low T, mass flow requires motion of the latticemass flow requires motion of the latticeBut probably not in the new expt through But probably not in the new expt through capillaries (PRL 2006)capillaries (PRL 2006)

crystalcrystal

liquidliquid

Bonfait, Godfrin and Castaing (J. Physique 1989) Bonfait, Godfrin and Castaing (J. Physique 1989) growth inside a thin capacitor at T < 20 mK growth inside a thin capacitor at T < 20 mK

blocked by a facet at the entrance ?blocked by a facet at the entrance ?

ENS 2006: experimental setup

Fill a test tube (1 cm Fill a test tube (1 cm ) at 1.3 K) at 1.3 K

lower T down to 50 mKlower T down to 50 mK

melt the outsidemelt the outside

follow the level insidefollow the level inside

any change in the level inside any change in the level inside requires a mass flow through the requires a mass flow through the

solid (solid (CC = 1.1 = 1.1 LL))

melting velocity V = 3 mm/hmelting velocity V = 3 mm/hif critical velocity 10 if critical velocity 10 m/s and m/s and superfluid density superfluid density ss / / CC = 10 = 10-2-2

VV

liquidliquid

solidsolid

Ishiguro’s tube

the ENS fridge the ENS fridge with optical accesswith optical access

large optical accesslarge optical access through sets of windows through sets of windows

down to 30 mKdown to 30 mK

filling the tube filling the tube with solid 4He with solid 4He makes defectsmakes defects

liquidliquid

solidsolid

the inside crystallizes the inside crystallizes only if a substantial only if a substantial stress is applied.stress is applied.For example if the For example if the outside is warmed up outside is warmed up to 1.4K for a few to 1.4K for a few seconds while the seconds while the inside is at 1.3Kinside is at 1.3K

PPmm( 1.4 K) - P( 1.4 K) - Pmm( 1.3 K) = 0.3 bar( 1.3 K) = 0.3 bar

fast growth under inhomogeneous stress creates defectsfast growth under inhomogeneous stress creates defects

liquidliquid liquidliquid

cusps and grain boundaries

crystal 1crystal 1crystal 2crystal 2

grain boundarygrain boundary

liquidliquid

mechanical equilibrium mechanical equilibrium of surface tensionsof surface tensionsat the liquid-solid interface:at the liquid-solid interface:

each cusp signals the existenceeach cusp signals the existenceof an emerging grain boundary (GB)of an emerging grain boundary (GB)

most cusps move away in most cusps move away in a few hoursa few hours(melting-crystallization + pinning)(melting-crystallization + pinning)some GBs stay pinnedsome GBs stay pinned

no flow in good quality crystals

for 10 crystals with no for 10 crystals with no or very few cusps the tubeor very few cusps the tubewe could see no flowwe could see no flowno mass leak along the glass wallno mass leak along the glass wallif supersolidity were due to a if supersolidity were due to a 1% superfluid density in the bulk1% superfluid density in the bulkwith a critical velocity vwith a critical velocity vcc = 10 = 10 m/sm/s

the interface should relax at the interface should relax at V = [V = [ss/(/(CC - - LL)]v)]vcc = 1 = 1 m/sm/s

that is 3.6 mm in 1 hourthat is 3.6 mm in 1 hourInstead, we see no flow within Instead, we see no flow within 50 50 m in 4 hours,m in 4 hours,meaning 300 times lessmeaning 300 times less

=> supersolidity is not due to the superfluidity of => supersolidity is not due to the superfluidity of a 1% equilibrium density of vacanciesa 1% equilibrium density of vacancies



mass flow in crystals with enough grain boudaries

for 3 crystals with some cusps inside the tube we observed a mass flowfor 3 crystals with some cusps inside the tube we observed a mass flowIf the cusps disappear, the mass flow stops (see crystal #1)If the cusps disappear, the mass flow stops (see crystal #1)Mass flows along grain boudariesMass flows along grain boudaries

Solids with grain boudaries may be supersolidSolids with grain boudaries may be supersolid(polycrystals) but not single crystals(polycrystals) but not single crystals

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crystal 1 relaxed 1 mm down and stopped



crystal 1



crystal 2 had many defects

Many grain boundariesMany grain boundariesmore in the lower partmore in the lower partfasterfaster flow down to flow down to equilibrium at h = 0equilibrium at h = 0



crystal 2 relaxed down to eq. (h = 0)



time time xx 250 2505 s = 20 min5 s = 20 min

crystal 2:relaxation at 50 mK

0.0

2.0

4.0

6.0

8.0

0 500 1000 1500 2000

crystal #2

time t (seconds)

relaxation is not exponential but linearrelaxation is not exponential but linear with two successive regimes, with two successive regimes, constant velocity : 6 constant velocity : 6 m/s for 0 < t < 500 sm/s for 0 < t < 500 s

11 11 m/s for 500 < t < 1000 sm/s for 500 < t < 1000 smore defects in the lower part of crystal 2more defects in the lower part of crystal 2

typical of superfluid flow at a critical velocitytypical of superfluid flow at a critical velocity

crystal 1 : a single grain boundary

7.6

7.8

8.0

8.2

8.4

0 500 1000 1500 2000

crystal #1

time (seconds)

The relaxation at V = 0.6 The relaxation at V = 0.6 m/s stops when the cusp disappears m/s stops when the cusp disappears (the grain boundary moves away, unpinning from the wall (the grain boundary moves away, unpinning from the wall somewhere)somewhere)

grain boundaries at Pm are comparable to liquid films with atomic thickness

If we assume the existence of a single grain boundary withIf we assume the existence of a single grain boundary withthickness e , width w , thickness e , width w , the critical velocity inside isthe critical velocity inside is

vvccGB GB = (= (ππDD22/4ew/4ewss)()(CC--LL)V = 1.5 (a/e)(D/w)()V = 1.5 (a/e)(D/w)(CC / /ss) ) m/sm/s

comparable to 2 comparable to 2 m/sm/s measured by Telschow et al. (1974) measured by Telschow et al. (1974) on free adsorbed films of liquid Heon free adsorbed films of liquid He

agreement with the prediction by Burovski, Prokof’ev andagreement with the prediction by Burovski, Prokof’ev and Svistunov (PRL 2005)Svistunov (PRL 2005)in a general model.in a general model.simulations of GBs in solid helium 4 are in progress simulations of GBs in solid helium 4 are in progress in their group (U. Mass. Amherst) and at Urbana (Ceperley and Clark)in their group (U. Mass. Amherst) and at Urbana (Ceperley and Clark)

Numerical simulation of grain boundaries

Nature 21 octobre 2006Nature 21 octobre 2006

crystal 4 at 1.13 K

0.0

1.0

2.0

3.0

1500 2000 2500 3000 3500 4000 4500

time t (seconds)

crystal #4

a highly distorted crystal ; final relaxation at 0.9 a highly distorted crystal ; final relaxation at 0.9 m/sm/s

grain boundaries are superfluid up to 1.13 K at leastgrain boundaries are superfluid up to 1.13 K at leastconsistent with e consistent with e ~ 2~ 2a and a and ss ~~CC at P = P at P = Pmm

have we seen the same effect as Kim and Chan ?

the effect of annealing: the effect of annealing: Rittner and Reppy (2006) vs Kim and Chan (2004)Rittner and Reppy (2006) vs Kim and Chan (2004)large scatter of datalarge scatter of dataevidence for the importance of quenched disorderevidence for the importance of quenched disordernot an intrinsic property of He crystalsnot an intrinsic property of He crystalsmost natural defect: grain boundariesmost natural defect: grain boundaries

increase of increase of ss (P) : more (P) : more

grain boundaries ?grain boundaries ? decrease of at large P: decrease of at large P: superfluidity disappears superfluidity disappears at high densityat high density

Tc and vc are different

at P = Pat P = Pmm, equilibrium with the liquid: , equilibrium with the liquid:

Partial wetting of grain boundaries by the liquid phasePartial wetting of grain boundaries by the liquid phase (long range van der Waals forces)(long range van der Waals forces)The thickness is microscopic (a few times a)The thickness is microscopic (a few times a)

Out of equilibrium at high P:Out of equilibrium at high P:prewetting near Pprewetting near Pm m , ,

e(P) should decrease, e(P) should decrease, T Tcc et v et vc c as well below one layeras well below one layer

PP

ee( or( or T Tc c ))

( or( or v vc c ) )

PPmm

1% superfluid density is large

In torsional oscillator In torsional oscillator experiments, crystallization at experiments, crystallization at constant Vconstant Vfrom the normal liquidfrom the normal liquidAt variable T and PAt variable T and P

=> polycrystals=> polycrystalsgrain boundaries grain boundaries every 100 à 200 a every 100 à 200 a , , about 50nm ??about 50nm ??

1% vacancies would be very large too1% vacancies would be very large too

crystals grown from the normal liquid at 1.9 K

dendritic growthdendritic growthstrong light scattering by a high density of defectsstrong light scattering by a high density of defects

work in progress

The research is now focusing on the effect of disorder, The research is now focusing on the effect of disorder, especially grain boundaires (GB):especially grain boundaires (GB):

calculate the thickness e and superfluid transition temperature Tcalculate the thickness e and superfluid transition temperature Tcc of GBs of GBs

measure the Tmeasure the Tcc of GBs with variable misorientation of GBs with variable misorientation

measure vmeasure vcc in fixed GBs, find a model for it in fixed GBs, find a model for it

measure GBs at P > Pmeasure GBs at P > Pmm : thinner ? lower T : thinner ? lower Tcc ? lower v ? lower vcc ? ?

measure the adsorption of 3He on GBsmeasure the adsorption of 3He on GBscharacterize the density of GBs in crystals grown at cst V : X rays, light scatteringcharacterize the density of GBs in crystals grown at cst V : X rays, light scatteringstudy the pinning of GBs on different wallsstudy the pinning of GBs on different wallstorsional oscillator experiments in good quality crystals grown at cst T and Ptorsional oscillator experiments in good quality crystals grown at cst T and Psupersolidity under rotationsupersolidity under rotationreproduce the measurement of the vacaqncy density vs Treproduce the measurement of the vacaqncy density vs Tchange the frequency of torsional oscillator measurementschange the frequency of torsional oscillator measurements......

1% superfluid density is large

In torsional oscillator experiments, all crystals In torsional oscillator experiments, all crystals have been grown at constant Vhave been grown at constant Vfrom the normal liquid phasefrom the normal liquid phase

variable T and Pvariable T and P => polycrystals=> polycrystalsgrain boundaries every 100 to 200 a grain boundaries every 100 to 200 a ~ 50 nm ?~ 50 nm ?a very high densitya very high density

facets block the growth

no growth if the crystal no growth if the crystal level is raised again outsidelevel is raised again outside

except if a large except if a large P is P is applied: applied:

facets are easily pinned to facets are easily pinned to wall defectswall defects

facets disappear during facets disappear during melting ( a geometrical melting ( a geometrical effect) => no pinningeffect) => no pinning

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Could a quantum solid flow like a superfluid ?