The surface of helium crystals:

review and open questions

Sébastien Balibar Laboratoire de Physique Statistique de l ’ENS (Paris, France)

for references and files, including video sequences, go to http://www.lps.ens.fr/~balibar/

CC2004, Wroclaw, sept. 2004

to appear in Rev. Mod. Phys. (jan. 05) download from: http://www.lps.ens.fr/~balibar/ 4 He and 3 He crystals: model crystals with both universal and exotic quantum properties static and dynamic properties: roughening and growth mechanisms open problems

hcp-helium 4 crystals

helium 4 crystals growing from superfluid helium 4 photographs by S.Balibar, C. Guthmann and E. Rolley, ENS, 1994 hexagonal close packed structure just like any other crystal, more facets at low T : successive "roughening transitions" 0.1 K 1.4 K 1.1 K 0.5 K

crystal shapes: lead crystallites

T > 120 °C T > 120 °C

growth shapes the growth reveals facetted directions more facets at low T electron microscope photographs by JJ Metois and JC Heyraud (CRMC2 - Marseille, France)

50 °C < T < 120 °C T < 50 °C

indium

more facets at low T photographs by JJ Metois and JC Heyraud CRMC2 Marseille

40 < T < 100 °C 10 < T < 20 °C T > 100 °C 20 < T < 40 °C T < 10 °C

video sequence

crystallization waves

eq. shape at 320 mK;

g

= 0.060 erg.cm

-2

bcc - helium 3 crystals

1 mm helium 3 atoms are lighter larger quantum fluctuations in the solid larger zero point energy smaller surface tension facetting at lower T

E. Rolley, S. Balibar, F. Gallet, F. Graner and C. Guthmann, Europhys. Lett. 8, 523 (1989)

(110) facets at 80 mK

E. Rolley, S. Balibar and F. Gallet, Europhys. Lett. 2, 247 (1986)

coalescence of

3

He crystals at 320 mK

R. Ishiguro and S. Balibar, submitted to PRL (2004) the neck radius varies as t 1/3 after contact instead of t ln(t) or t 1/2 for viscous liquid drops

facet sizes are enlarged by a slow growth

facets grow and melt much more slowly than rough corners

up to 11 different facets on helium 3 crystals 0.55 mK 2.2 mK (110) (100) (100) (110) (110) Wagner et al., Leiden 1996 : (100) and (211) facets Alles et al. , Helsinki 2001 : up to 11 different facets

the roughening transition

at T = 0 atoms minimize their potential energy the surface is localized near a lattice plane, i.e. "smooth" Landau 1949: crystal surfaces are smooth in all rational directions (n,p,q) at T=0 at T > 0 , fluctuations: adatoms, vacancies, steps with kinks, terraces...

the surfaces are "rough" a roughening temperature T the crystal surface is free from the influence of the lattice above R numerical simulations by Leamy and Gilmer 1975 solid on solid model, bond energy J per atom T R = 0.63 J

roughening and facetting: coupling of the surface to the lattice vs thermal fluctuations d weak coupling: wide steps with a small energy

b

(

b

<<

g

d

g

: surface tension) ex: liq-sol interface helium 4, liquid crystals strong coupling: narrow steps with a large energy

b

(

g b

~

g

d : surface tension) ex: metal-vacuum interface helium 3 : weak for 60 < T< 100 mK ? strong below 1 mK ?



the roughening transition of soft crystals shear modulus << surface tension : steps penetrate as edge dislocations below the crystal surface -> the step energy steps are very broad but

d b

~ (

g

and a)

d

2 / surfaces is

m

l 2

b

is large ~

m

a 2 /4

p

compensate each other

m

a << is very small their interaction the roughening temperature for (1,n,0)

g

T Rn

 2 p g  g //

a n

2  2 p 6 bd

a

2

a n

2  g

a

2

n

2

in the end, many facets because the unit cell a ~ 50 Angström is large for (1,1,2) surfaces T R ~ 27000 K !

for (9,8,15) surfaces T R ~ 360 K experiments: Pieranski et al. PRL 84, 2409 (2000); Eur. Phys. J. E5, 317 (2001) theory: P. Nozières, F. Pistolesi and S. Balibar Eur. Phys. J. B24, 387 (2001)

First estimates of the step energy on (110) 3He facets: Rolley et al., Paris, 1986 eq. shape at 320 mK;

g

= 0.060 erg.cm

-2 Measurement of the surface tension from the equilibrium shape of large crystals:

g

= 0.060 +/- 0.011 erg/cm 2 1 mm The roughening temperature of (110) facets should be T R = (2/

p) g

d 2 = 260 mK (110) facets at 80 mK Why no visible facets above 100 mK ?

dynamic roughening

dynamic roughening

the critical radius r the nucleation of terraces: r c =

b/ Dm  m

d L

r

c

Dm

where c for

- m

C : chemical potential difference the correlation length

x

= 2

g

d 2 / (

p

2

b) x

r c the surface is dynamically rough is r c i.e. if

b 2

< 2

r

c

Dm g

d 3 /

p

2 in 3 He (Rolley et al. 1986), if

b

< 10 -11 <

x

, erg/cm above 100 mK

dynamic roughening in helium 4

liquid H v helium crystal

grow a crystal through a hole watch the relaxation of the surface to its equilibrium height (Wolf, Gallet, Balibar et al. (1983-87)

50

from linear iin to non-linear growth 4 He

40 30 20 1.205K

1.218K

1.234K

1.252K

1.285K

10

closer to the roughening temperature r c <

x

0 0 200 400 600 height difference (

m

m) 800

T < T T > T R : non-linear growth (v is quadratic or exponential in the applied force) ( spiral growth due to step motion around dislocations or nucleation of terraces) R : linear growth v = k

Dm

(sticking of atoms one by one)

1000

critical behaviour of the growth rate

Nozières's RG calculation also describes the evolution of the growth rate (i.e. the surface mobility) fits with the same values of the parameters as for the step energy (T dynamic roughening R = 1.30 K ; t c = 0.58 ; L 0 : facets are destroyed by a fast growth ( a"finite size effect" in the renormalization calculation) = 4 a )

comparison with experiments in helium: the step free energy the step free energy is calculated from the relation

b

=(4a/

p

) [ where L max

g

(L max )/V(L max ) stopped ("truncated") ] 1/2 is the max scale at which the renormalization is it vanishes exponentially as:

b

~ exp [ -

p

/2(tt c ) 1/2 ] where t = 1 - T R /T is the reduced temperature and t T R c measures the strength of the coupling to the lattice a measurement in helium (ENS group 1983-92) : = 1.30 K t c = 0.58 (weak coupling)

1.5

1.0

0.5

0.0

1.1

1.15

1.2

1.25

Temperature (K) 1.3

1.35

the universal relation

k B T R = (2/

p) g

(T R ) a 2 the surface stiffness tends to

g

(T R ) =

p

k B T R / 2 a 2

g

= 0.315 erg.cm

-2 at zero tilt angle if T R = 1.30 K and t c = 0.58

agreement with the curvature measurements by Wolf et al. (ENS-Paris) and by Babkin et al. (Moscow) universal : no dependence on microscopic quantities (lattice potential ...) Nozières's theory also predicts the angular variation of

g

, as another finite size effect

Nozières’RG-theory of roughening



The sine - Gordon model an effective hamiltonian for a surface deformation z(r):

H

 

d

2

r

  1 2 g  ) 2 

V

cos 2 p

z d

  a ' a a g

=

a

+ d 2

a

/d

f

2 : surface stiffness

a :

surface tension V : lattice potential near T R , assumptions : small height z weak coupling to the lattice

f

we use the renormalization calculation by Nozières who revisited this problem in 1985-94, using several previous works, in particular Knops and den Ouden Physica A103, 579, 1980) => the renormalization trajectories

[g

L) , V(L)]

a '

the coupling strength in Nozières’s theory

H

 

d

2

r

  1 2 g  ) 2 

V

cos 2 p

z d

 

assume that

g

a coarse graining at variable scale L (L) and V(L) depend on scale L start at the microscopic scale inject fluctuations at larger and larger scale, calculate the free energy of the surface for each coarse graining deduce the L dependence of

g g

(L 0 ) = and V

g

0 ; V (L 0 ) = V 0 the « microscopic scale » : where the surface starts feeling thermal fluctuations the parameter t c ~ V 0 /

g

0 measures the coupling strength

the T-variation of the step energy

A. Hazareesing and J.P. Bouchaud Eur. Phys. J. B 14, 713 (2000) : functional renormalization calculation of the step energy the coupling strength : Nozieres' parameter t c ≈ 13 V 0 /

g

0 helium 4 : t c = 0.58 medium strength at microscopic scale helium 3 : dynamic roughening at 100 mK ~ 0.4 T R implies t c << 1

t c ≈ 0.01

weak coupling t c ≈ 1 strong coupling

helium 3 : weak coupling at high T

Todoshchenko et al.

(Helsinki, aug. 2004) step energy from v ( in the range 60 -110 mK weak coupling compatible with upper bound by Rolley et al. and universal relation T R

d

p) (spiral growth) = 260 mK

V. Tsepelin et al. (Helsinki + Leiden): strong coupling at 0.55 mK at 0.55 mK the step energy

b

is comparable with the surface energy

g b

~ 0.3

g

d strong coupling ?

d:

a possible explanation : quantum fluctuations (Todoshchenko et al. , preprint aug. 2004)

due to quantum fluctuations, the solid - liquid interface is thick compared to the lattice spacing this implies weak coupling of the surface to the lattice Todoshchenko et al. : in 3 He , quantum fluctuations are damped at low T, not at high T according to Puech et al. 1983 , the growth rate k = v/

Dm

is proportional to the sticking probability

a

~ (S C - S L )/S L ~ 1/T at low T where S C = k ln2 and S L ~T << S

a

C of 3 He atoms : but above the superfluid transition at T c =2mK and the antiferromagnetic transition at T N = 1 mK

Todoshchenko et al.

extend Nozières’ renormalization theory

In Nozières’ theory, the effect of quantum fluctuations is included in the value of the lattice potential V 0 at the atomic scale L 0 no problem in 4 He, the quantum fluctuations are always there and make the liquid-solid interface rather thick at the scale L 0 Todoshchenko et al. start the renormalization procedure at the atomic scale d but include quantum effects in the renormalization treatment of surface fluctuations This allows them to caculate the case of 3 He where the amplitude of quantum fluctuations strongly depends on T

new fit of the step energy by Todoshchenko’s RG-theory Todoshchenko’s theory Nozières’ theory good agreeement but: 1- the theory is valid only for weak coupling 2- only for 2 < T < 100 mK where S L ~T <

b

and

g

accross T N and T also as a function of magnetic field c

two-dimensional nucleation of terraces

10 -1

interferometric measurement of the relaxation of a crystal surface to its equilibrium height experimental evidence : velocity:

v =k

Dm

exp[-

b

2 /(3a

r

C

difference in chemical potential:

Dm

= H (

r

C -

r

L )/

r

C

r

L

slope -> step energy

b Dm

k B T)] 10 -2 10 -3 10 -4 10 -5 0.2

0.4

0.6

0.8

1 1/H (mm -1 ) 1.2

1.4

1.13K

1.145K

1.155K

1.173K

1.178K

1.19K

1.23K

1.6

1.8

some results of the renormalization calculation

as first predicted by several groups in the late 70's , the roughening transition is a "Kosterlitz - Thouless transition" like the superfluid transition in 2D, the 2D-crystallization, XY model...

(H. van Beijeren, H.J.F. Knops, S.T. Chui and J.D. Weeks...) infinite order : the step free energy vanishes exponentially the surface stiffness shows a " universal jump " and a square root cusp : T < T R : infinite surface stiffness (the facet is flat) T = T R :

g

(T R ) =

p

T R / 2a 2 T > T R :

g

(T) =

g

(T R ) where t = T/T R [ 1 - (tt c ) 1/2 ] - 1 is the reduced temperature

the remarkable growth dynamics of helium crystals

25 solid superfluid normal liquid gas 0 1 2 temperature (K)

helium 4 crystals grow from a superfluid (no viscosity, large thermal conductivity) the latent heat is very small (see phase diagram) the crystals are very pure wih a high thermal conductivity -> no bulk resistance to the growth, the growth velocity is limited by surface effects smooth surfaces : step motion rough surfaces : collisisions with phonons (cf. electron mobility in metals) v = k

Dm

with k ~ T -4 : the growth rate is very large at low T helium crystals can grow and melt so fast that crystallization waves propagate at their surfaces as if they were liquids.

the dispersion relation of crystallization waves

superfluid crystal 2 restoring forces - surface stiffness :

g

(at high frequency or short wavelength)

-

gravity g ( at low frequency or large wavelength) inertia : mass flow in the liquid (

r

C >

r

L )

 2   r

C

r

L

r

L

) 2 [ g

q

3   r

C

r

L

)

gq



-> experimental measurement of the stiffness

g 

surface stiffness measurements

Rolley et al. (ENS - Paris) PRL 72, 872 (1994) J. Low Temp. Phys. 99, 851 (1995)

the anisotropy of stepped surfaces

a

f  

for a stepped surface: small tilt angle

f

with respect to a facet two stiffness components

g  

a

b f b

: step energy

g //  6 d f

a

3 d

: interaction between steps wide steps : crossover to rough at

f

≈ a/6L 0 ≈ 1/24 rad

step-step interactions

entropic interaction: steps do not cross (no overhangs) steps are confined by their neighbours entropy reduction entropic repulsion

d

S l

2  p 2 6 (

k B T

) 2 b

l

2

elastic interaction:

d

el /l 2 ~

g

2



2 (E : Young modulus) elastic repulsion

d

el l

2  g 2

E l

2

l l



elastic + entropic interactions

solid line: prediction for thin steps but, in helium, the steps are very wide (weak coupling to the lattice) the measurement needs to be done at very small tilt angle or calculate a correction due to the finite step width

terrace width distributions on Si surfaces Schartzentruber et al. PRL 65, 1913 (1990) E.D. Williams and N.C. Bartelt, Science 251, 393 (1991)

the step energy in helium 3

the T variation of the step energy theory and very weak coupling (t but

b b

agrees with RG c ≈ 0.01), (T=0) ≈ 0.3

g

d is much too large (Tsepelin et al. Helsinki 2002) a change in coupling strength between 0.55 mK and 100 mK ?

- Fermi liquid - superfluid transition - magnetic ordering in the solid

the truncation of the renormalization

Our analysis was done by integrating the RG trajectories up to a max scale such that the lattice potential U = VL However, in his 1992 lectures at Beg Rohu, Nozieres explains that the criterion for weak coupling is U < k 2 max B T/4 ≈ k

p

B T Should one stop using the theory where it fails ?

the values of the fitting parameters depend on this One would like to do an independant measurement of both

x

= (a/2

p) g

/V) 1/2 and

b

= (4a/

p) g

V) 1/2

a possible measurement of the correlation length

X ray scattering on a solid 4He film grown by epitaxy on a Si(111) substrate ? hcp 4He crystals grow by epitaxy on graphite, why not on Si(111) ?

study the continuous evolution of critical layering transitions towards the roughening transition as a function of film thickness T c (n) ≈ T 1984) R [1 - c/ln 2 (n)] (Huse 1984, Nightingale, Saam and Schick

Ramesh and Maynard, PRL 1982-84

color strioscopy

white light glass prism helium crystal imaging lens mask