Supersolidity, disorder and grain boundaries S. Sasaki, R. Ishiguro, F. Caupin, H.J. Maris and S. Balibar Laboratoire de Physique Statistique (ENS-Paris) now at North Western Univ. (USA) now at North Western Univ. (USA) now at Tokyo University (Japan) Brown University, Providence (RI, USA)  de la Physique, Saclay, 23 oct S. Sasaki et al., Science 313, 1098 (2006) - S. Sasaki, F. Caupin and S. Balibar, Phys. Rev. Lett. 99, (2007) - S. Balibar and F. Caupin « topical review » J. Phys. Cond. Mat. 20, (2008) - S. Sasaki, F. Caupin and S. Balibar, to appear in J. Low Temp. Phys. (Nov. 2008)

a « supersolid » is a solid which is also superfluid solid : transverse elasticity, i.e. non-zero shear modulus a consequence of the localization of atoms crystals - glasses superfluid : a quantum fluid with zero viscosity a property of interacting Bose particles, which are indistinguishable and delocalized a paradoxical idea

could solid helium 4 flow like a superfluid ? E. Kim and M. Chan (Penn. State U. 2004): a « torsional oscillator » (~1 kHz) a change in the resonance period below ~100 mK 1 % of the solid mass decouples from the oscillating walls ? no effect in helium 3 (fermions) rigid axis ( Be-Cu) solid He in a box excitation detection Temperature (K) superfluid fraction (NCRIF)

early theoretical ideas Andreev and Lifshitz 1969: delocalized vacancies could exist at T = 0 ( the crystal would be « incommensurate ») BEC => superplasticity at low velocity and long times coexistence of non-zero shear modulus and mass superflow ! E0E0E0E0 zh zh Among others, off diagonal order (Penrose et Onsager 1956): no supersolidity symmetrization and overlap of wave functions (Reatto 1969, Chester 1969, Leggett 1970, Imry et Schwartz 1975), the rotation of a quantum solid (Leggett 1970)...

more recent ideas (a selection...) Prokofev, Svistunov, Boninsegni, Pollet, Troyer (MC calc.): no supersolidity without free vacancies ; the probability for a real quantum crystal to be commensurate and supersolid is zero 4 He crystals are commensurate (E vac = 13K) => not supersolid BUT BEC is possible in a 4 He glass (Boninsegni et al. PRL 2006, see also AF Andreev JETP Lett. 2007) Galli and Reatto 2006 : variational calculation with particular trial wave functions (« SWF ») which describe the properties of solid 4 He => supersolidity in a commensurate crystal ! Clark and Ceperley (2006) : supersolidity depends on trial wave functions no supersolidity in « exact » calculations ( quantum path Integral Monte Carlo); crystals are commensurate, no vacancies at T = 0, no supersolidity in perfect crystals other models of quantum crystals : Cazorla and Boronat (PRB 2006) Josserand, Rica and Pomeau (PRL 2007) supersolidity but are these models realistic ?

other theoretical ideas Anderson Brinkman and Huse (Science 2005): new analysis of the lattice parameter  a/a (T) and the specific heat C v (T): a small density of zero-point vacancies (< ?); T BEC ~ a few mK ;  s ? not confirmed by neutron scattering (Blackburn et al. PRB 2007) critics by H.J. Maris and S. Balibar (J. Low Temp. Phys. 147, 539, 2007) two transitions ? a vortex liquid ? could supersolidity be due to the presence of defects in crystals ? Dash and Wetlaufer 2005: the He-wall interface could be superfluid PG de Gennes (Comptes Rendus - Physique 7, 561, 2006): the mobility of dislocations could depend on T; frequency dependence ? (see Kojima et al. 2007) Pollet et al. (Monte Carlo): grain boundaries should be supersolid Boninsegni et al. 2007: dislocation cores should be supersolid Biroli and Bouchaud (2008): dislocations fluctuate and favor atom exchange

the role of disorder : annealing, quench-cooling, grain boundaries Rittner and Reppy (Cornell, ): supersolidity disappears after annealing  s up to 20 % if the samples are grown by quench-cooling the liquid from its normal state Sasaki, Ishiguro, Caupin, Maris et Balibar (ENS, Science 2006) : a dc-flow experiment with solid helium 4 superfluid mass transport in the presence of grain boundaries

the role of disorder S. Sasaki et al. Science 313, 1098, 2006 a glass tube (1 cm  ) cristallization from superfluid liquid at 1.3 K cool down to 50 mK a height difference between inside and outside any relaxation ? hélium solide hélium liquide fenêtre any variation of the inside level requires a mass current through the solid because  C = 1.1  L

the physics of grain boundaries solid liquid liquid solid liquid crystal 1 crystal 2 grain boundary liquid phase  GB   LS mechanical equilibrium of the liquid-solid interface :  GB  LS cos  each groove signals the existence of an emerging grain boundary dynamics of grain boundaries : large mobility + pinning on walls

without grain boundaries, no flow with grain boundaries, superfluid flow time accelerated x 250 supersolidity is not an intrinsic property of the crystalline state of helium 4, a property associated with its defects annealing - quench (Rittner and Reppy )

cristal 2: relaxation at50 mK relaxation is linear, not exponential a superfluid flow at its critical velocity 2 successive regimes: 6  m/s for 0 < t < 500 s 11  m/s for 500 < t < 1000 s more defects in the lower part of crystal 2

crystal 1 : only one grain boundary relaxation at relaxation at V = 0.6  m/s stops when the grain boundary unpins and suddenly disappears If 1 grain boundary only with a superfluid thickness e ~ 0.3 nm, width w ~ 1cm the critical velocity inside should be: v c GB = (  D 2 /4ew  s )(  C -  L )V = 1.5 (a/e)(D/w)(  C /  s ) m/s comparable with 2 m/s measured by Telschow et al. (1974) for liquid films of atomic thickness

Pollet et al. PRL 98, , 2007 grain boundaries : ~ 3 atomic layers superfluid except in special directions T c ~ 0.2 à 1 K depending on orientation critical velocity ?

a 1% superfluid density is large! (Rittner and Reppy 2007: 20% in quenched samples !)  s = 1% => grain size ~ 100 nm  s = 0.03% => 3  m is it possible ? may be how does the grain size depend on the growth method ? constant P or contant V from the normal liquid from the superfluid slow or fast cell geometry (T gradients)

the ENS high pressure cell 2 cells : 11 x 11 x 10 mm 3 or 11 x 11 x 3 mm 3 thermal contact via copper walls thickness 10 mm 2 glass windows (4 mm thickness) indium rings => leak tight stands 65 bar at 300K pressure gauge (0 to 37 bar)

at T < 100 mK from the superfluid, fast growth and melting of a high quality single crystal real time, 60 mK 11 mm

fast growth from the normal liquid at T > 1.8 K dendrites fast pressurization of the normal liquid => dendritic growth similar to Rittner et Reppy’s quenched samples ? T = 1.87 K 11 mm

helium snow flakes T = 2.58 K is the roughening transition reentrant ? facets at T < 1.3 K and at T > 2 K ? Balibar, Alles and Parshin, Rev. Mod. Phys remember Burton, Cabrera and Frank

slow growth from the normal liquid at constant V growth in ~ 3 hours a temperature gradient : T walls < T center the solid is transparent but polycrystalline

T = 0.04 K melting a crystal after growth at constant V T = 0.04 K liquid channels appear where each grain boundary meets the glass windows grains < 10  m ripening

T < 0.1 K ripening at the liquid-solid equilibrium (another example) T < 0.1 K minimization of surface energy mass transport through the superfluid latent heat L = 0

2 crystals + 1 grain boundary the groove angle is non-zero => the grain boundary energy  GB  is strictly the grain boundary energy  GB  is strictly < 2  LS => microscopic thickness, in agreement with Pollet et al. (2007). a complete wetting would imply  GB  LS (2 liq-sol interfaces with liquid in between) thickness 10 mm thickness 3 mm angle 2  the contact line of grain boundaries with each window is a liquid channel

angle measurement => grain boundary energy align the optical axis fit with Laplace’ equation near the cusp  = 14.5 ± 4 °  = 14.5 ± 4 °  GB  = (1.93 ± 0.04)  LS other crystals :  = 11 ± 3 °  = 11 ± 3 °  = 16 ± 3 ° angle 2 

near a wall : wetting of grain boundaries The wall is favorable to the liquid. If  GB  is large enough, more precisely if  +  c <  /2 the liquid phase wets the contact line. an important problem in materials science (see JG Dash Rep. Prog. Phys. 58, 115, 1995) prediction: liquid channels also where grain boundaries meet each other => between 25 and 35 bars a polycrystal contains many liquid channels grain 1 grain 2 liquid wall S. Sasaki, F. Caupin, and S. Balibar, PRL 99, (2007) wall grain 1 grain 2 GB largeur w ; épaisseur e invers t prop. à la profondeur z (w ~ 20  m à z = 1 cm) l c : longueur capillaire the width w and the thickness e are inversely proportional to the depth z l c : capillary length

between 3 grains stable channels if Miller and Chadwick Acta metall. ’67 Raj Acta metall. mater. ’90

grain boundaries in classical crystals Besold and Mouritsen ’94 Monte-Carlo simulation colloidal crystals Alsayed et al. Science ’05 T m =28.3  C Alsayed et al. Science ’05 T m =28.3  C 5 µm wetting of boundaries by the liquid phase ?

short and long range L S S t Schick and Shih PRB ’87 short range forces: complete wetting the thickness t diverges at the liquid-solid equilibrium long range forces : incomplete wetting, microscopic thickness consequences : impurity diffusion (ice), mechanical rigidity

the channel width w ~ (P-P m ) -1 the channel width w decreases as 1/ z (the inverse of the departure from the liquid-solid equilibrium pressure P m ) agreement with new measurements of the contact angle  c liquid channels should disappear around P m + 10 bar (where 2w ~1 nm)

hysteresis of the contact angle advancing angle : 22 ± 6 ° (copper) 26 ± 7 ° (glass) 37 ± 6 ° (graphite) receeding angle : 55 ± 6 ° (copper) 51 ± 5 ° (glass) 53 ± 9° (graphite) perhaps more hysteresis on copper rough walls than on glass or graphite walls whose roughness is smaller. as expected from E. Rolley and C. Guthmann (ENS-Paris) PRL 98, (2007) meltinggrowing copper copper

a stacking fault : low energy, no liquid channel growth shape between 2 crystals with same orientation  GB smaller   larger   +  c >  /2  no liquid channel equilibrium shape see also H. Junes et al. (Helsinki) JLTP 2008 : 2  = 155 ± 5°

2 possible interpretations of the experiment by Sasaki et al. (Science 2006) mass transport either - along the grain boundaries (then v c ~ 1 m/s) - or along the liquid channels (then v c ~ 3 mm/s). This would explain why flow was observed also at 1.13 K - future experiments : change the cell geometry, reduce the width of liquid channels with an electric field study h(t) with more accuracy with 1 fixed grain boundary liquid liquid solid solid

are grain boundaries (together with their liquid channels) responsible for everything ? Chan et al. (sept. 2007): anomalies in single crystals grown at constant T and P NO!

a superfluid network of dislocations ? along one dislocation, a coherence length l ~ a (T*/T) percolation of quantum coherence in a 3D network ? May be, but T c << 100 mK... furthermore, helium 3 impurities increase T c ! Boninsegni et al. PRL 2007

shear modulus oscillator period Day and Beamish (Nature 2007): the shear modulus increases ! the shear modulus increases by ~ 15 % below 100 mK (depending on He3 content!) pinning of dislocations by 3He impurities ? and grain boundaries (samples grown at constant V) ? another explanation for torsional oscialltor measurements : K increases ? similar anomalies

my opinion in June experiments: 2 main interpretations are possible 1 - superfluidity of some fraction of the mass, associated with defects in a mysterious way  (I/K) 1/2 decreases because the inertia I decreases  (I/K) 1/2 decreases because the inertia I decreases 2 - no superfluidity a change in the dynamics of defects (dislocations ? grain boundaries ?) the elastic constant K increases theory: a quasi-general consensus no supersolidity in a perfect crystal (without defects) dislocations, grain boundaries and glassy helium (if it exists...) should be superfluid at low T

new experiments by Rittner and Reppy 2008 no period change in a blocked annulus there is a macroscopic mass current in a free annulus

Kim and Chan 2008: influence of 3 He impurities the dependence on 3He concentration is consistent with a simple model of adsorption on dislocations

new experiments by Beamish et al He3 hcp and He4 hcp : same elastic anomalies He3 bcc: no elastic anomalies He4 bcc: supersolidity ? supersolidity is clearly linked to quantum statistics

... and my opinion at the end of August 2008 Supersolidity appears when 3He adsorbs on defects ! WHY ?? in liquid 4 He, 3 He acts against superfluidity the existence of supersolidity in solid helium 4 is confirmed it is associated with the existence of defects but also to the presence of 3He impurities it is not a purely elastic effect An idea by B. Svistunov (Trieste, 22 August 08): by condensing at the nodes of a dislocation network, the 3He could provide a connection between the dislocations of the 4He crystal... provide a connection between the dislocations of the 4He crystal...

perspectives Measure the rotational inertia and the shear modulus in samples with well characterized disorder : single crystals (oriented ? measure the dislocation density ?), polycrystals, glassy samples... optics, X - rays, neutrons, thermal conductivity... search for superfluidity inside grain boundaries : flow experiments in a cell with 1 stable grain boundary and reduce the effect of liquid channels (electrostriction) measure the quantum dynamics of defects theory of quantum defects (quantum metallurgy)