The surface of helium crystals: review and open questions

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The surface of helium crystalsreview 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
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to appear in Rev. Mod. Phys. (jan. 05)download
fromhttp//www.lps.ens.fr/balibar/
4He and 3He crystals model crystals with
both universal and exotic quantum
properties static and dynamic properties
roughening and growth mechanisms open problems
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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"
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crystal shapes lead crystallites
T gt 120 C
T gt 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 lt T lt 120 C
T lt 50 C
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indium
T gt 100 C
20 lt T lt 40 C
more facets at low T photographs by JJ Metois
and JC Heyraud CRMC2 Marseille
40 lt T lt 100 C
T lt 10 C
10 lt T lt 20 C
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video sequence
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crystallization waves
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bcc - helium 3crystals
E. Rolley, S. Balibar, F. Gallet, F. Graner and
C. Guthmann, Europhys. Lett. 8, 523 (1989)
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 and F. Gallet, Europhys.
Lett. 2, 247 (1986)
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coalescence of 3He crystals at 320 mKR. 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
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facet sizes are enlarged by a slow growth
facets grow and melt much more slowly than rough
corners
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up to 11 different facets on helium 3 crystals
0.55 mK
2.2 mK
(100)
(110)
(110)
(110)
(100)
Wagner et al., Leiden 1996 (100) and (211)
facets
Alles et al. , Helsinki 2001 up to 11 different
facets
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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 T0 at T gt 0 , fluctuations adatoms,
vacancies, steps with kinks, terraces... the
surfaces are "rough" above a roughening
temperature TR the crystal surface is free from
the influence of the lattice
numerical simulations by Leamy and Gilmer
1975 solid on solid model, bond energy J per
atom TR 0.63 J
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roughening and facetting coupling of the
surface to the lattice vs thermal fluctuations

• weak coupling
• wide steps
• with a small energy b
• b ltlt g d
• (g surface tension)
• ex liq-sol interface
• helium 4, liquid crystals

• strong coupling
• narrow steps
• with a large energy b
• b g d
• (g surface tension)
• ex metal-vacuum interface

helium 3 weak for 60 lt Tlt 100 mK ? strong below
1 mK ?
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the roughening transition of soft crystals
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First estimates of the step energy on (110) 3He
facets Rolley et al., Paris, 1986
Measurement of the surface tension from the
equilibrium shape of large crystals g 0.060
/- 0.011 erg/cm2
The roughening temperature of (110) facets should
be TR (2/p) g d2 260 mK
Why no visible facets above 100 mK ?
dynamic roughening
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dynamic roughening
the critical radius rc for the nucleation of
terraces rc b/drcDm where Dm mL - mC
chemical potential difference
x
rc
the correlation length x 2g d2 / (p 2b)
the surface is dynamically rough is rc lt x ,
i.e. if b2 lt 2rc Dm g d3 / p2 in 3He (Rolley et
al. 1986), if b lt  10-11 erg/cm above 100 mK
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dynamic roughening in helium 4
grow a crystal through a hole watch the
relaxation of the surface to its equilibrium
height (Wolf, Gallet, Balibar et al. (1983-87)
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from linear to non-lineargrowthiin 4He
closer to the roughening temperature rc lt x
T lt TR non-linear growth (v is quadratic or
exponential in the applied force) ( spiral
growth due to step motion around dislocations or
nucleation of terraces) T gt TR linear growth v
k Dm (sticking of atoms one by one)
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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 (TR 1.30 K
tc 0.58 L0 4 a ) dynamic roughening
facets are destroyed by a fast growth ( a"finite
size effect" in the renormalization calculation)
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comparison with experiments in heliumthe step
free energy

• the step free energy is calculated from the
  relation
• b (4a/p) g (Lmax)/V(Lmax)1/2
• where Lmax is the max scale at which the
  renormalization is stopped ("truncated")
• it vanishes exponentially as
• b exp -p /2(ttc)1/2
• where t 1 - TR/T is the reduced temperature
• and tc measures the strength of the coupling to
  the lattice
• a measurement in helium
• (ENS group 1983-92)
• TR 1.30 K
• tc 0.58 (weak coupling)

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the universal relation kBTR (2/p) g (TR) a2

• the surface stiffness tends to
• (TR) p kBTR / 2 a2
• 0.315 erg.cm-2
• at zero tilt angle
• if TR 1.30 K and tc 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
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NozièresRG-theory of roughening
The sine - Gordon model an effective
hamiltonian for a surface deformation z(r)
g a d2a /df2 surface stiffness a surface
tension V lattice potential near TR ,
assumptions small height z weak coupling to
the lattice
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) gt
the renormalization trajectories g(L) , V(L)
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the coupling strength in Nozièress theory
principle of the calculation a coarse graining
at variable scale L assume that g(L) and V(L)
depend on scale L start at the microscopic scale
g (L0) g0 V (L0) V0 inject fluctuations at
larger and larger scale, calculate the free
energy of the surface for each coarse
graining deduce the L dependence of g and V
the  microscopic scale  where the surface
starts feeling thermal fluctuations
the parameter tc V0/g0 measures the coupling
strength
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the T-variationof 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 tc 13 V0
/g0 helium 4 tc 0.58 medium strength at
microscopic scale helium 3 dynamic roughening
at 100 mK 0.4 TR implies tc ltlt 1
tc 1 strong coupling
tc 0.01 weak coupling
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helium 3 weak coupling at high T
Todoshchenko et al. (Helsinki, aug. 2004) step
energy from v (d p) (spiral growth) in the range
60 -110 mK weak coupling compatible with upper
bound by Rolley et al. and universal relation TR
260 mK
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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 d b 0.3 g d strong
coupling ?
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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 3He , 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 of 3He atoms a (SC - SL)/SL
1/T at low T where SC k ln2 and SLT ltlt
SC but above the superfluid transition at
Tc2mK and the antiferromagnetic transition at TN
1 mK
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Todoshchenko et al.extend Nozières
renormalization theory
In Nozières theory, the effect of quantum
fluctuations is included in the value of the
lattice potential V0 at the atomic scale L0 no
problem in 4He, the quantum fluctuations are
always there and make the liquid-solid interface
rather thick at the scale L0
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 3He where the amplitude
of quantum fluctuations strongly depends on T
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new fit of the step energy by Todoshchenkos
RG-theory
good agreeement but 1- the theory is valid
only for weak coupling 2- only for 2 lt T lt 100
mK where SL T ltltSC
needed measurements of b and g accross TN and
Tc also as a function of magnetic field
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(No Transcript)
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two-dimensional nucleation of terraces
experimental evidence velocity v k Dm exp-b
2/(3arC Dm kBT) difference in chemical
potential Dm H (rC-rL)/rCrL slope -gt step
energy b
interferometric measurement of the relaxation of
a crystal surface to its equilibrium height
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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 lt TR infinite surface
stiffness (the facet is flat) T TR g (TR)
p TR / 2a2 T gt TR g (T) g (TR) 1 -
(ttc)1/2 where t T/TR - 1 is the reduced
temperature
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the remarkable growth dynamics of helium crystals
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 -gt 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.
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the dispersion relation of crystallization waves
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surface stiffness measurements
Rolley et al. (ENS - Paris) PRL 72, 872
(1994) J. Low Temp. Phys. 99, 851 (1995)
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the anisotropy of stepped surfaces
a
f
for a stepped surface small tilt angle f with
respect to a facet two stiffness
components b step energy d
interaction between steps
wide steps crossover to rough at f a/6L0
1/24 rad
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step-step interactions
entropic interaction steps do not cross (no
overhangs) steps are confined by their
neighbours entropy reduction entropic repulsion
elastic interaction overlap of strain fields
del/l2 g2/El2 (E Young modulus) elastic
repulsion
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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
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terrace width distributionsonSi surfaces
E.D. Williams and N.C. Bartelt, Science 251, 393
(1991)
Schartzentruber et al. PRL 65, 1913 (1990)
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the step energy in helium 3
the T variation of the step energy b agrees with
RG-theory and very weak coupling (tc 0.01), but
b (T0) 0.3 gd 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
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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 VL2max kBT However, in
his 1992 lectures at Beg Rohu, Nozieres explains
that the criterion for weak coupling is U lt
kBT/4p 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/2p)
(g/V)1/2 and b (4a/p) (gV)1/2
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a possible measurementof 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 Tc(n) TR 1 - c/ln2(n) (Huse 1984,
Nightingale, Saam and Schick 1984)
Ramesh and Maynard, PRL 1982-84
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color strioscopy
white light
mask
helium crystal
glass prism
imaging lens