Helium crystals as a probe in materials science

Pergamon

Solid State Communications, Vol. 92, Nos 1-2, pp. 19-29, 1994 Elsevier Science Ltd Printed in Great Britain 0038-1098(94)00535-4 0038-1098/94 $7.00+.00 HELIUM CRYSTALS AS A PROBE IN MA'I'ERJAI~ SCIENCE S. Balibar* and P. Nozi/~res** * Laboratoire de Physique Statistique de l'Ecole Normale Sup~rieure, 24 rue Lhomond, 75231 Paris Cedex 05, France ** Institut Laue Langevin, 156X, 38042 Grenoble Cedex, France

We review here a number of properties of helium crystals which are paradoxically

classical, not quantum. We believe that the study of these properties are of general interest in materials science. Among them are the roughening transition and other properties of steps and terraces on hep helium 4 crystals. We also deserihe the stress induced instability of helium crystal surfaces. We eventually mention other general problems whose study is interesting in helium because its extreme purity and quasi-infinite thermal conductivity offer unique opportunities for an accurate control of the relevant physical parameters. Dans cet article, nous pr6sentons une revue de certaines propri6t~s classiques des cristaux d'h61ium a basse teml~rature, dont rinterl~ratation, paradoxalement, ne fait pas appel/t la m6canique quant/que. Ces propri6t6s nous semblent presenter un int6ret g~n&al en science des mat~riaux. Parmi elles figurent la transition rugueuse et les autres propri6t6s des marches et des terrasses/t la surface des cristaux hexagonaux d'h61ium 4. Nous d6erivons aussi l'instabilit6 m6canique de la surface des cristaux sons contrainte. Enfin, nuns mentionnons quelques autres probl/~mes dont l'6tude est int6ressante darts l'h61ium parce que son extreme puret~ et sa conduction thermique quasi-inf'mie offrent d'uniques possibilit6s de contrOle des param6tres physiques pertinents. Keywords:D.surfaces and interfaces,phase transitions,mechanical properties

At low temperature, the condensed phases of helium are well known for quantum properties such as superfluidityl. However, many other properties can be understood by only considering that liquid and solid helium are extremely pure and homogeneous classical systems. This purity is estonishing: by filtering superfluid helium 4 through a porous glass 1, one can eliminate all chemical impurities and reduce the concentration in isotopic impurities (helium 3) below 10-12! Moreover, there is no resistance to mass transport in the liquid which has no viscosity. No resistance either to heat transport at low temperature: both phases have quasi-infinite thermal conductivities. As a result, thermodynamic equilibrium is reached very fast in helium, and departures from it can be controlled with a rare accuracy (10. 8 or less). Interracial properties become accessible, while they are hidden by diffusion problems in usual materials. Another unusual feature of helium is its phase diagram. It is well known that the van der Waals attraction is weak between helium atoms and that, for such light atoms, a large quantum kinetic energy prevents solidification at low pressure, even down to absolute zero. 4He solidifies at 25 bars and the melting curve is nearly flat at low temperature. As a result one can easily scan the temperature along the melting curve, a most unusual situation at a liquid-solid interface. Phase transitions of that interface are easily accessible. In this review we first summarize the results recently obtained on the roughening transition of helium crystals. We then describe how one later succeeded with these crystals in measuring the fundamental interactions between steps. We finally present the mechanical instability which was discovered at the surface of helium crystals under

stress, and which was predicted by Asaro and Tiller as a precursor of fracture. Our conclusion contains a few suggestions for other such studies in helium, whose results might be of interest for a better understanding of the general properties of classical materials. 1- The Roughening Transition. At zero temperature a crystal surface is a smooth flat face 2. As the temperature increases thermal fluctuations create terraces. At some critical temperature TR, these terraces percolate through the interface : the height fluctuations become much larger and the surface is rough. Because terraces can pile up on each other, such a roughening transition is not a real percolation process; it belongs to a different universality class, that of "KosterfitzThouless" uansitions (together with the two dimensional xy model) 3-4-5. Physically the transition corresponds to a vanishing step free energy : above TR the steps proliferate. A sharp roughening transition occurs only for a facet in equilibrium. The transition is blurred for a tilted interface : roughening sets up progressively when the distance between steps is comparable to their width. It is equally blurred for a growing facet : one goes continuously from homogeneous 2d nucleation to free growth when the critical nucleation radius is comparable to the step width 5. Note that the whole effect is a consequence of the crystal periodicity. The "dynamic roughening" just described should not be confused with instabilities due to non linearities of the growth rate. From the point of view of experiments 6-14, the roughening transition leads to profound changes in crystal shape and growth. The equilibrium shape is facetted below 19

20

Vol.92, Nos 1-2

HELIUM CRYSTALS

the roughening temperature TR, and rounded above. Each surface crystal plane has a roughening temperature which strongly depends on orientation, and for each direction the shape progressively changes from rounded to facetted as the temperature is lowered. The change is equally spectacular for growth, which proceeds by sticking of individual atoms for rough surfaces (above TR), while it involves the lateral displacement of steps for smooth surfaces (below TR). Consequently, the growth dynamics changes from fast and linear above TR to slow and nonlinear below TR10, II. Both the experimental observation and the theoretical interpretation of critical roughening are delicate problems. Usually equilibrium is not reached and one does not really control the dynamics. Helium is the exception for which a quantitative comparison of theory and experiment is p o s s i b l e , b l e n d i n g e q u i l i b r i u m and g r o w t h properties 10,11,12. Hexagonal helium crystals can be observed at 25 bars and down to T = 0, in equilibrium with a superfluid liquid and through the windows of an optical cryostat (fig. 1). Three different types of facets have been seeng: the (0001) or "c" facets below TRI = 1.30 K, the (10T0) facets below TR2 --- IK and the (10T 1) facets below TR3 ~ OAK. Here, we are mainly interested in the c facets, the only ones whose roughening transition was studied in full quantitative details 9-14. Let us start with one of the static properties, namely the capillary shift of the equilibrium pressure at TR, ~p

-

Pc - PL

2~, (PL -P0) = ~ - -

(1)

PL where PC,L is the crystal (resp. liquid) density, PL is the liquid pressure and P0 the equilibrium pressure of a fiat interface. Since the c axis has cylindrical symmetry near TR, only one curvature radius R appears in eq. 1. The quantity T is the surface stiffness of the crystal, related to the surface tension or free energy per unit area ct by the relation ~' = 0t + ~2(X/~2

(2)

where 0 is the angle between the normal to the surface and the c axis. The most famous theoretical prediction concerns the crystal curvature. According to mean field arguments 15 one could expect this curvature to vanish linearly when the temperature T tends to TR. However, rigorous theories predict a universal jump as for other "Kosteflitz-Thouless" transitions3,4,5: here, the curvature should jump from zero below TR (the facet) to a universal value 16,17 above. A universal relation is predicted to exist for the surface stiffness TR at TR: 2 kBTR = ~ "~Ra2

(3)

where a is the step height. Eq.(3) can be verified experimentally - but it turns out that a rather detailed theoretical picture is needed. For a weakly pinned liquidsolid interface a convenient description is the continuous "sine-Gordon" model 17. The influence of the lattice is treated as a periodic term in the effective surface hamiltonian H=

21th (Vh) 2 + V c o s T

(4)

where h is the local height of the surface whose average orientation is supposed horizontal. In order to calculate the effect of thermal fluctuations, a renormalization procedure is used in which the potential V is treated as a small perturbation. Fluctuations with a scale smaller than L renormalize y and the pinning energy U = VL 2. Above TR, U goes to zero as L increases: the height-height correlation function G(r) -- <(h(r) - h(0))2> is found to diverge with the distance r as a21og(r/a). This divergence shows that the surface is thermodynamically rough like that of a liquid. On the contrary, below TR, G(r) is predicted to saturate within the correlation length ~ to a value of order a21og(~a). In order to check such a property, one should measure ~. That can be done directly using diffraction methods, as done on copper surfaces by the group of Lapujoulade 8- Alternately one can rely on the step free energy 13which behaves as

[3~ = kBT R

(5)

The sine-Gordon model predicts [3 = (4a//t)(yV) 1/2 and [3 should vanish as [3 = exp {-rd(2(ttc) It2) }

(6)

where t = I-T/TR is the reduced temperature and tc = 1.25 7t2 U0 y0a2

(7)

is a parameter describing the strength of the coupling of the surface to the underlying lattice. Due to such exponential critical behaviours, Kosterlitz-Thouless transitions are considered as "infinite order" phase transitions. The above theory contains three adjustable parameters which are necessary for any comparison with experiments, namely the temperature TR, the coupling parameter tc and the cutoff wavevector AO = l/L0. In order to check eq.6, some authors tried to measure the step energy from facet sizes. However, facets hardly reach thermodynamic equilibrium as soon as the step energy is not extremely small, so that their size depends on the growth history and usually differs from the equilibrium size l = 2[~/(a5P). A much better determination was made by Wolf, Gallet et al. 10,11 who studied the growth dynamics below TR. They measured the relaxation of the height h of a helium crystal surface to its equifibrium value h0. The force driving the growth was the difference in chemical potential between liquid and solid, which writes ~Y~tt=(1 - pL/Pc)g(h-ho)

(8)

for an isothermal liquid under hydrostatic equilibrium. In the temperature interval 1.13K - 1.23 K, i.e. slightly below TR, they found one of the best present evidences of a growth proceeding from the 2D-nucleation of terraces. Indeed, if this growth mode dominates all others, the growth velocity v is expected to vary as

~h v = ~ - = kAltt exp

_ n~2 3aPc~ttkBT

(9)

where k is a surface mobility and the factor 3 comes from the hypothesis that successive layers are completed from the nucleation and coalescence of a large number of terraces. As shown by fig.2, a semilog plot of v/h as a function of lha shows very good straight lines over several

Fig.l.An ultrapure crystal of helium 4 growing at 0.1 K from superfluid helium, as seen by Rolley et a1.13 through the windows of their cryostat (typical crystal size: lcm). Colors are obtained by illuminating the crystal with white light dispersed by a glass prism. Each color corresponds to a couple of facets forming a small helium prism. The "c" facet is visible as a yellow-blue-orange hexagon on top.

Vol.92, Nos 1-2

23

HELIUM CRYSTALS 6

lffI

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I

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Experiment

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¢D

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Fig.2. If a crystal grows by the two-dimensional nucleation of terraces, its growth velocity v should vary exponentially with the driving force, here a difference inchemical potential which is proportional to a crystal height h (see text). This is precisely what was shown by Wolf et al. 10 with this semilog plot of v/h vs 1/h where straight lines are obtained over several decades. The successive slopes are proportional 132, the square of the step free energy which vanishes when the temperature goes up to the roughening temperature TR = 1.30K.

1.3

1.35

the roughening transition"• Theoretically, it is understood

as a finite size effect• A finite driving force introduces a finite time x = a/v, which is the time necessary for the crystal to grow by one period of the lattice potential The diffusion length of surface fluctuations during that time x is L* = (Ta/pcSp.) I/2. On a scale L* the diffusion time is comparable to the sweeping time of crystal planes by the interface : scaling stops! The growth rate is not renormalized to zero•

decades in growth velocity. It is this good agreement which allowed the precise determination of the step free energy 1~ by Wolf, Gallet et al.• As shown on fig.3, the vanishing of 13 is well described by theory 17. The dotted line on fig.3 corresponds to a numerical integration of the renormalization equations in ref.17, since eq.6 is only an approximation very close to TR. A best fit was obtained 12 with the following set of parameters:

A further comment is needed in order to distinguish the above type of dynamic roughening (called "type I" by 50

li

0u

o

40

(10)

While negligible at low temperature, the 2d-nucleation of terraces is dominant close to TR because the step energy vanishes (Note that the step energy is ~/a = 0.014 erg/cm 2 at low temperature, a much bigger value). Close to TR, but not too much, however! Gallet et al.lt (as later Balibar et al.12) noticed that eq.9 can no longer be used above 1.23K, when the critical radius for the nucleation of terraces becomes smaller than the correlation length ~. This is the regime of "dynamic roughening" where a different approach is needed 17, based on a Langevin equation which describes the motion of the crystal surface in the presence of an applied force, a capillary pressure, a laUice potential and thermal noise. Using the same renormalization method, a non linear growth law is found : the mobility depends on the growth velocity. As shown on rigA, the growth preogressively evolves from non-linear below TR to linear at and above TR. Good fits of these data were obtained using the same set of parameters as for the step energy (eqs. 10). In the limit of zero driving force, the growth rate would jump from a finite value at TR to zero below. As shown by fig.4, this jump is broadened by a non-zero 6~t since the growth rate is non-zero, even below TR. This effect was called "dynamic roughening" or "dynamic broadening of

1.2 1.25 Temperature (K)

Fig•3. The temperature variation of the step fm energy, as obtained by Gallet et ak I 1 from graphs such as shown on fig.2. The exponential vanishing of 13 is characteristic of Kosterlitz-Thouless transitions. The solid line represents a best fit obtained by Balibar et al. t2 from the numerical integration of the renormalization theory by Nozi6res5,17.

(ram-1 )

TR = 1.30 K ; tc = 0.58 ; AO = 0.25/a

1.15

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30

A

T-I•205K

*

T - ] • 21BK

0

T - I . 234K

m

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o

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0 °°= Oo

0[3

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"'oo

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÷÷

A

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~bo

6bo

ebo

1000

HEIGHT (MICRON)

Fig.4. It is possible to see a change from non-linear to linear growth very close to the roughening temperature TR = 1.30K. This was shown by Gallet et at.l! by applying a very small departure from equilibrium, here a difference in height corresponding to a difference in chemical potential of order 10.6 to 10.8 kT per atom. These data were taken in the critical region of the roughening transition where a a renormalization theory of the growth rate is necessary. Here again, good agreement was obtained with Nozi~res'theory and the same adjustment of parameters as for the step energy (fig.3).

24

HELIUM CRYSTALS

Balibar and Bouchaud 18) from the dynamic roughening ("type II" in ref. 18) introduced by Kardar Parisi and Zhang (KPZ) 19. By including a term (~2)(Vh) 2 in the Langevin equation, KPZ found that at large enough scale, a growing surface should show algebraic roughening, i.e. beyond another length L** the height correlation function G(r) should diverge as rrl. Hwa et al.20 proposed to include such a term in the interpretation of the experiment by Gallet et all 1. However, in such experiments the departure from equilibrium is very small (apcSll/T is typically 10-7). The crossover from logarithmic (type I) to algebraic (type II) • 18 , since • "L * * ~- a roughening occurs at unrealistic scales exp(T/aPc~lX)2 . Let us finally come to the universal curvature of crystals, more precisely to the experimental check of eq.3. In the experiment by Wolf et al.lO small crystals are obtained in equilibrium with liquid helium and their shape measured. The pressure difference (PL - Po) is accurately known, but the curvature, inferred from the second derivative of the profile equation, cannot be obtained in the very small angular region of interest around the c direction (~ = 0). As a result a detailed theory of the angular dependence of Y is needed. This is achieved once more via an "interrupted scaling" method (when the length scale is comparable to the step distance). The resulting angular variation of "/has a logarithmic cuspS,lOA 7 in a narrow region around #~ = 0 where the maximum value TR is reached. For the precise facet direction (~ = 0), the temperature variation above T R is described by the well known "square root cusp" ~,=~

(1 - (ttc)l/2 + ...)

(ll)

On fig.5, the dotted line corresponds to a numerical integration with the 3 parameters already adjusted as explained above 12. Of course there is some scatter in the data, and this indicates how delicate such a measurement is, but the general agreement with theory is rather good. The curve has an inflexion point around a crossover angle ~ 0 = 2.5 °. Above 0c0, no temperature variation is found At large angle, the variation is non-singular, due to the general anisotropy of the hep crystal and well described by ~/=TO(I - 12q~2)

(ll)

with To = 0.245 erg/cm 2. To is in turn related to YR according to TR = TO(1 + tc/2)

(12)

The unrenormalized value TO was measured by Wolf et al. and the values of to and TR had already been adjusted on the respective variations of the step energy and of the growth rate. The width ~c0 is related to the microscopic cutoff L0. Collecting numerical factors we find that the crystal surface cannot feel the influence of the lattice if the terrace width is not at least three times the minimum width of steps 12. We shall come back to this point in the next section, when considering the measurement of step energies and step interactions. Let us conclude this section with a few numbers which illustrate the validity of the weak coupling hypothesis for helium crystals. From the value of to one can infer that the unrenormalized value of the potential is U0 = 0.036 kBTR. As explained in ref.5, the coupling is weak if U is less than

Vol.92, Nos I-2 0.4!

.

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,

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.=

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theory (1.2K)

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Andreevaet al. (0.4K) Babkinet al. (I.2K) Babkinet al. (1.2K)

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Tilt angle ~ (degrees) Fig.5. As a function of angle, and close to the roughening temperature, the surface stiffness has a critical variation in an angular region which depends on the strength of the coupling of the surface to the crystal lattice. In the case of helium crystals, this coupling is weak, and the critical variation only occurs below a small angle of order 2.5 ° where the dotted line has an inflexion point. Despite some scatter in the data obtained by Babkin et al. 14, agreement is found with Nozi~res'theory whose parameters had already been adjusted on the variation of the step energy and the growth rate.

kaT/41t, so that the weak coupling hypothesis seems to be justified even down to the microscopic scale which can be identified with the unrenormalized situation at T = 0. More physically, a weak coupling of the surface to the lattice means that steps should be smooth defects with a large width, typically 8a here 12. 2- Stepped Crystal Surfaces During the last decade, the measurement of step-step interactions progressively appeared as an important challenge in the physics of crystal surfaces. Indeed these interactions determine not only the equilibrium shape of crystals near facet edges 21 but also the magnitude of step fluctuations and the distribution of terrace widths on vicinal surfaces 22, eventually the roughening transition of these vicinal surfaces 8. Most existing theories predict that neighbouring steps should repell each other with a positive interaction energy proportional to lid 2, the inverse square of the average distance d between steps. The physical origin for such a l/d 2 repulsion can be elastic (the overlap of strain fields around neighbouring steps), entropic (steps do not cross each other since overhangs are unlikely), or dipolar (on metallic surfaces). However, disorder could change this law 23 to lid and other possible mechanisms have also been proposed 24. Moreover various experiments led to contradictory results. Some authors tried to measure the equilibrium shape of Pb, In or He crystals. A lid 2 repulsion implies that, near the facet edge, the profile equation should be z = x 3/2, while a lid repulsion would lead to z = x 2. However, the exact location of the facet edge is very difficult, and it is hard to distinguish between the two possible shapes which were both apparently observed 25-26. Wang et al.27 and Alfonso et al. 28 later measured the width a of the distribution of terrace sizes on Si surfaces; unfortunately, o depends only weakly 22 on the nature of interactions (6

HELIUM CRYSTALS

Voi.92, Nos I-2

d for I/d2 and 0 ,* d 3/4 for I/d interactions) so that a definite conclusion was hard to draw 29. Finally, the case of C u (I In) surfaces also seems unclear, since I/d2 repulsions had been observed from the study of roughening transitions8, but attractive interactions were found from

STM studies 3°. With helium 4 crystals, step energies and step interactions can be measured from a rather different type of experiment. As mentionned above, these crystals are able to grow or melt so easily, i.e. with such a small dissipation, that it is possible to propagate melting-freezing waves 3z waves on their surface. Sometimes also called "crystallization raves", these waves result from the alternative growth and melting of the crystals. It is a macroscopic phenomenon which is visible to the naked eye, and it is analogous to the waves one can see at the free surface of a liquid. Indeed, the restoring forces to a flat horizontal surface are the same (gravity and surface tension) and the inertia is similar (growth and melting implie a flow of mass in the liquid since the two densities PC and PL arc different). A simplified dispersion relation for melting-freezing waves is: to2 =

PL

yq3

(13)

(Pc - PL) 2

where to and q are the respective pulsation and wave vector of the waves. For simplicity, we have neglected dissipation in cq. 13, i.e. imaginary terms involving the growth rate. Wc also neglected the effect of gravity which is relevant at low frequencies (below a few hundred Hertz) and large wavelength (above about 1 mm, the capillary length). As a remarkable consequence, accurate measurements of the surface tension are possible from the study of meltingfreezing waves on the surface of helium 4 crystals, as for liquid surfaces. Of course, a major difference with liquid surfaces is that the surface of helium crystals is anisotropic. This is precisely why the quantity in eq.13 is not exactly the surface tension ~ but the surface stiffness y (as defined by eq.2). Consider for instance a vicinal surface with a small tilt angle 0: the surface displays well separated steps. The wave vector q of the capillary waves may lie either perpendicular or parallel to the steps. In the former case, the wave induces a local modulation of the step density (the steps remain straight), y is controlled by the compressibility of the step system5, and equal to

~2~

In order to measure step energies and step interactions, one can thus study the propagation of melting-freezing waves whose velocity should be very different along or across steps. This was done recently by Rolley et al.13. In their experiment, a crystal is first grown with a c axis along the vertical direction Oz in a box which can rotate around two perpendicular axes Ox and Oy thank~ to two cryogenic micromotors. The box has a few cm size in all directions, so that gravity forces the equilibrium crystal shape to keep a horizontal liquid-solid interface. The waves are excited along Oy and they measure y//(0) or y±(0) depending on whether the crystal is rotated around Ox or Oy. A unique crystal can thus be used to study the whole angular variation of the surface stiffness; no cut or surface preparation is needed, they just use the effect of gravity on the equilibrium shape of crystals which grow and melt very easily. As shown on fig.6, Rolley et at.13 succeeded in this measurement. They first found that the anisotropy of the surface stiffness only appears below a crossover angle of about 2.5 °. This is the same crossover angle Oc0 as mentionned above in the problem of the universal relation of roughening. Its physical meaning is simple: if 0 > 0c0, the distance between steps is not large enough compared to the step width, terraces are too narrow to be well defined, ,

,

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r~

0.5

66

Yi/= Ct + ~ b2 - a3 ~b

(14)

where w e have supposed thai the magnitude of step interactions is & d 2 (the notation y//refers to the projection of the c axis being parallel to the wave vector q). Note that such longitudinal waves become softer and softer as the tilt angle ~b goes to zero (the distance d increases and the step interactionsvanish, itcosts a very small energy to m o v e the steps with respect to one another).

On the contrary, if q is perpendicular to the c axis, the wave bends the steps without changing their distance and the restoring force for such a transverse wave is the line tension, i.e. the step free energy 13. More and more bending of steps is needed as ¢~ tends to zero, because the step density also vanishes. The corresponding surface stiffness is 5 7.1.= 0~ 4- - -

25

tanO &~

aO

(15)

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0

.

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.

1

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tilt angle (degrees) Fig.6. If tilted by a sufficiently small angle 0 with respect to some high symmetry direction, a vicinal crystal surface is a stepped surface. Such a surface is very anisotropic, as shown by the large difference between the two components of its surface stiffness ts. Above the same crossover angle 0c "~ 2.5 ° as on fig.5, the helium surface becomes truely rough and neatly isotropic because terraces are not well defined between steps which are too close to each other. From the analysis of these data, Rolley et al.t3 obtained the value of the step energy and the magnitude of" the interactions between steps.

26

Voi.92, Nos 1-2

HELIUM CRYSTALS

the surface is truely rough because it is a set of entangled steps. On the contrary, for ~ < ~c0, the steps are well separated and the stepped character of the surface shows up as a large anisotropy. For strongly coupled surfaces such as a metal-vacuum interface in some high symmetry direction, the crossover from stepped to rough behaviour would take place at large angles of order one radian since the step width is only about one lattice spacing. In the case of helium crystals, the small value found for this crossover further confirms the large width of steps, i.e. the weak coupling already found in the analysis of the roughening transition (see the end of section I). Rolley et al. 13 also showed that the angular variation of the two surface stiffness components is well described by eqs.14-15 in the limit of zero angle. They deduced a precise value of the step energy and found step-step interactions in good agreement with the prediction of a lid 2 repulsion. Indeed, a lid repulsion would need to have a very small amplitude to be compatible with their tesults. On the contrary, it was found very likely that the interaction has a double origin: an elastic interaction 8el/d 2 and an entropic interaction 8s/d 2. The elastic interaction originates in the overlap of strain fields around each step 5. The entropic interaction corresponds to a reduction in the entropy of steps when their fluctuations are limited by the presence of neighbours (crossings would generate overhangs which cost too much in energy) 5. In the case of insulators, these two interactions are usually thought as dominant, but they don't add linearly. As explained in ref.22 where a previous analogy with one dimensional fermions33 is used, the total interaction is given by 2

4L V

6

I~

When a crystal is subject to a non hydrostatic stress, a planar solid-liquid interface may become unstable : through melting and freezing the surface corrugates spontaneously. (The same effect may occur at a solid-vacuum interface through surface diffusion). Here the primary role is not played by surface properties, in contrast to the preceding two sections : the surface deformation acts to reduce the bulk elastic energy. Such an effect was described long ago by Asaro and Tiller 35 as a possible mechanism for corrosion under stress : the instability may proceed as far as cracking! It was rediscovered independently by Grinfeld36, who considerably extended its range of application. The physical origin of that instability is simple5,37. Let us consider an horizontal liquid-solid interface z = 0, and also call oij is the stress tensor and uij the strain tensor. Mechanical equilibrium implies Ozz = - PL , Ozx = azy = 0

(I 8)

Assume that the solid is subject to an extra horizontal stress, for instance (19)

Oxx=Ozz+OO , Uyy=O

The solid stores more energy : it is less competitive as compared to the liquid and it thus melts. For our planar interface melting equilibrium implies

2

3~,J

(16)

fc - Ozz -

-

eft -

~c

= ~l'L

(20)

Pc

Rolley et al. 13 were able to measure both 6el and 6s by measuring the temperature variation of the total interaction 5. Indeed the elastic term is predicted to be the temperature independent quantity 6el=(2/g)f2(l-02)/E where E is Young's modulus, o = l / 3 is Poisson's ratio and f is the amplitude of the force doublet producing the strain field around each step. On the contrary, it was predicted 34 that the entropic interaction rapidly increases with temperature according to n 2 (kl)T)2 8s = - -

3- Stress induced Surface Instability

(17)

Rolley et al. indeed found that the temperature variation of the limiting slope of y//is well described by eqs.14 to 17. They found a reasonable elastic interaction leading to surfaces forces f with the same order of magnitude as the surface tension ~. Since they extracted the value of 13from the measurement of transverse waves, they also found that the numerical factor in eq.17 is indeed close to g2/6, as predicted by Akutsu et al.34. To our knowledge, this is the first experimental check of the magnitude of the entropic interaction between steps. Eq.17 is very general since it describes the interaction between any lines fluctuating in a 2-d space with a no-crossing condition. The experiment by Rolley et al. is an important complement to the prevnous study of the roughening transition, since it is a study of the behaviour of surfaces with well defined steps near T = 0, while the roughening transition study concerns the critical vanishing of crystal steps at high temperature.

where fc is the free energy per unit volume of the solid and lttL the liquid chemical potential per unit mass. The extra uniaxial stress o0 shifts the equilibrium pressure by 5PL. Melting equilibrium is maintained if ~teff

(1-o2)o2 + ~iaL -

- -

2pcE

Pc

8PL = laL -

(21)

PL

(as above, o and E are respectively Poisson's coefficient and Young's modulus). In a gravity field g the interface goes down by an amount

h -

2 (1-02) GO

(22)

2Eg(pc - PL) in order to increase the equilibrium pressure. Assume now that the interface corrugates : the extra stress oo decreases in the hills (which feel the substrate less), while it increases in the troughs. As a result the hills will melt less and the troughs will melt more - the corrugation grows. Clearly the instability is countered by gravity and by capillarity - as usual it is a matter of wavelength. In order to make that argument quantitative, we displace the interface by an amount h(x) via growth (not to be confused with the elastic displacement). Due to the interface rotation Oh//)x the extra stress o0 breaks mechanical equilibrium at the surface unless we add a first order

HELIUM CRYSTALS

Vol.92, Nos 1-2

correction to the bulk Strain and stress fields uij and oij. The corresponding change in bulk elastic energy is 1 S "3

Eel [ h ( x ) ] = - ~

a

(1)..(1) r uij uij

(23)

(the minus sign comes from the work of the zeroth order stress). For a periodic displacement h = 11 cosqx, and up to second order in 11, the resulting bulk energy is

(l-a2)

Eel=- ~

ff0q112

(24)

(the change is monitored by the slope qh and it penetrates on a wavelength 2x/q). Combining that contribution with those due to gravity and to capillarity, we obtain the net energy cost for the deformation h(x) : 112

F-,tot= -]- ~.(q)

(25)

with ~(q)

=

(Pc- PL)g + 7q2- 2 ~

0~0q

(26)

The situation is similar to that of ferrofluids under magnetic field or of charged interfaces under electric field: the instability appears at the capillary wavevector q* past a critical stress excess o0=o*, with q, = ~ ( P c

O'* = ~ /

~L)g '

-7-F-'q---~* 1 - 02

(27)

Well beyond threshold gravity becomes negligible : all Fourier components are unstable up to a maximum wave vector

2o~ qm

--~ - -

27

the strain is measured by an interferometric technique from the displacement of the pistons, and we focus on its results. As expected the interface goes down by an amount h when a tangential strain is applied. The result shown on Fig.7 is in fair agreement with the above theory. Note that when one pu//s on the crystal the interface lowering is sometimes divided by a factor 4 : one of the transducers does not stick to the sample and the resulting o0 is twice smaller! When the applied strain exceeds a threshold of order 7.5 10.5 corrugations appear (fig.8). The threshold o* agrees with quantitative predictions37. The threshold q* - 1.3 cmI as well. The orientation of the ~x>oves is found to depend on the anisotropy of both the applied stress and the surface stiffness, an expected phenomenon which would need further theoretical and experimental analysis. Finally, the transition is found to display hysteresis, which means a subcritical bifurcation. S.harl~grooves develop downward (on the solid side): all of that is in agreement with the above amplitude expansion. (As a byproduct, the distortion disappears in 10 s e c o n d s at 1.2K, and plastic relaxation seems fast above this temperature). We thus have a neat experimental demonstration of the instability. As they stand the experiments are still somewhat crude - it is certainly possible to improve their accuracy. Such a stress driven instability has actually many experimental implications. We already mentioned the possible appearance of cracks that was the original motivation of Asaro and Tiller35. J. Berrehar et al.41 also found evidence for this instability in a very different context, at the surface of diacetylene crystals after some surface polymerization. Another possible application is heteroepitaxy42, when there is a misfit in lattice spacing. In the standard Frank-van der Merwe picture, epitaxial dislocations appear past a critical thickness d* in order to restore the equilibrium spacing. Another possibility is the so called "Stranski-Krastanow" regime, in which the layer develops islands past a minimum thickness. These islands are subsequently amplified by the growth and they are clearly visible in electron microscopy. The epitaxial layer is indeed highly strained, and our instability should occur. The geometry is that of a finite layer instead of a semiinfinite material - but that does not matter if the two materials have similar elastic coefficients. We expect

(28) E ~J

Such a linear stability argument cannot tell what happens afterwards. A hint is obtained pushing the amplitude expansion to fourth order in 11. After some messy algebra35 it is found that (i) the bifurcation is first order Csubcritical") (ii) the second harmonic of the deformation h(x) has a sign which corresponds to sharp grooves on the back side of the interface (towards the solid). So much for theory. Very likely such an instability has been observed in a number of situations, perhaps as a spontaneous corrugation of solid layers deposited on substrates when the temperature is quenched: differential thermal expansion creates a tangential stress - hence the instability. But the first quantitative verification was achieved on the solid-superfluid interface of 4He, for which the kinetics is practically instantaneous. In these experiments a controlled stress 00 is applied to the crystal. The resulting lowering of the interface is measured, and the change of morphology is observed directly. In the experiment of Thiel et.ai.39 the stress is created either by a temperature quench or by a piezoelectric bimorph, while Torii and Balibar40 use pistons driven by piezoelectric cylinders. The latter geometry is better controlled because

10

Q

8 ~O e,

6

e.,

4

.E

2 0 -2

-1 0 1 uniaxial strain ux~ 00 "5)

2

Fig.7. The melting of a helium crystal induced by small non-hydrostatic strains 40. The strain is obtained by displacing two symmetric pistons. When only one piston acts on the crystal, the real strain is divided by 2 and the resulting amount of melting by 4. Good agreement is found with theory (eq. 22).

28

HELIUM CRYSTALS

Fig.8. The upper photograph shows an unstrained helium crystal, as seen from above with an interferometric technique 40. The lower one shows the same crystal after a 9.10 -5 strain was applied in the horizontal direction. This is 25% more than the instability threshold (eq.27) and three grooves are visible as deformations of the fringe pattern.

spontaneous corrugations up to the maximum wave vector qm (limited by diffusion kinetics). In practice many questions are unanswered (Why is there a minimum thickness? What is the role of van der Waals interactions?). The issue is open but the stress induced instability is certainly an important feature: understanding its mechanism is a prerequisite. 4- Conclusion With these few examples, we tried to convince the reader that 4He is an interesting system for materials science exotic as it may look for a metallurgist! Among its many virtues, one is crucial : the rapidity of kinetics. Typical relaxation times are very short : they allow to reach

Voi.92, Nos I-2

thermodynamic equilibrium, a very unusual situation in crystal physics. Equilibrium shapes are within reach, allowing a detailed application of the Wulff construction. One can also study dynamical properties - for instance the melting-freezing waves that give direct and detailed information on crystal steps. Equally useful is the possibility to vary the temperature along the melting curve: impressive examples are the demonstration of the homogeneous nucleation of terraces or that of the T 2 dependence of the entropic repulsion between steps. Finally the extreme purity of superfluid 4He avoids all uncertainties due to uncontrolled surface adsorption. Indeed one may even turn the problem around and study for their own sake the effect of 3He impurities on surface properties. That was done for the propagation of capillary waves on vicinal surfaces 43. We find that an impurity concentration of 0. lppm is enough to change markedly the step energy by 30% - a lesson that should he kept in mind. The examples we gave are in no way limitative : helium has something to say on many other material properties! The plasticity of crystals is one such example. It has been known for a long time that bcc He crystals are unusually soft. They flow easily through an orifice, under small stresses and in a small time. We also mentioned that the stress applied in the elastic instability experiments could relax in a few seconds. Such easy plastic flow might be due to a large concentration of vacancies, for which there is some evidence44; but it may also involve dislocations. Until now results are mostly qualitative : they deserve a more systematic analysis, as a function of temperature and stress. Another fascinating problem in which helium may provide a real breakthrough is that of cavitation. In ordinary liquids which are impure, cavitation results from the heterogeneous nucleation of gas bubbles in regions where the flow decreases the local pressure. It is heterogeneous because it takes place on solid impurities or on bubbles of foreign gases already present in the liquid. A small depression is therefore sufficient to trigger cavitation. On the contrary, it was shown recently that the extreme purity of superfluid helium 4 offers a unique opportunity to study cavitation under controlled conditions. In particular, Pettersen et al. 45 obtained evidence that cavitation is a thermally activated process in helium 4 around I K, also that helium 4 can be kept as a metastable liquid down to negative pressures of several bars, eventually that vortices play an important role in this problem. It thus seems that even superfluid helium can be considered as an interesting classical liquid, since such experiments should provide very useful information on the stability limit of simple liquids, i.e. on their intrinsic tensile strength.

REFERENCES 1- As a first approach to the Physics of Helium, the famous book by J.Wilks (The Properties of Liquid and Solid Helium, Clarendon Press, Oxford, 1967) is still very useful. For the purification techniques, see P.V.E. McClintock, Cryogenics 18, 201 (1978) and references therein. 2- W.K.Burton, N.Cabrera and F.C.Frank, Phil. Trans. Roy. Soc. London 243A, 299 (195 ! ). 3- J.D.Weeks and G.H.Gilmer, Adv. in Chem. Phys. 40, 157, 1979. 4- H. van Beijeren and I.Nolden, in "Structure and dynamics of surfaces II", ed. by W.Schommers and P.van Blanckenhagen, Springer 1987. 5- P.Nozi~,res, in "Solids Far From Equilibrium", Lectures at the Beg-Rohu summer school, C.Godr~:che ed. (Cambridge University Press, 1991 ).

6- To our knowledge, the roughening transition was discovereded by Kaischew on urotropine crystals 7. Since then, this phase transition was observed on organic crystals (C2H 6, NH4Br, CCI 4 ...), metals (In, Pb, Cu, Ni, Ag...), ionic crystals (Ag2S), liquid crystals, hcp helium 4 and bec helium 3, etc. However, quantitative measurement have only been done with Copper 8 and with Helium crystals yet. Moreover, only the c facet of helium 4 crystals has allowed a full quantitative experimental study of the nature of the roughening transition. 7- R.Kaischew, Godishnik Sofiiskiya Univ. Fiz. Mat. Fak, 43, 99 (1946). 8- J.Lapujoulade, J.Perrau and A.Kora, Surf.Sci. 129, 59 (1983); J.Villain, D.R.Grempel and J.Lapujoulade. J. Phys. F

Voi.92, Nos 1-2

HELIUM CRYSTALS

15, 804 (1985); J. Lapujoulade, in "Interaction of atoms and molecules with solid surfaces", ed. by V. Bortolani et al. p. 381 (Plenum, 1990). 9- J.E.Avron, L.S. Balfour, C.G. Kuper, J. Landau, S.G. Lipson and L.S. Schulman, Phys. Rev. Lett. 45, 814, 1980. S. Balibar and B. Castaing, J. Physique Lett. 41,329, 1980. K.O. Keshishev, A.Ya. Parshin and A.V. Babkin, Zh. Eksp. Teor. Fiz. 9180, 716, 1981 (Soy. Phys. JETP 53, 362, 1981). P.E. Wolf, S. Balibar and F. Gallet, Phys. Rev. Lett. 51, 1366, 1983 O.A. Andreeva and K.OK Keshishev, Pis'ma Zh. Eksp. Teor. Fiz. 52, 799, 1990 (Soy. Phys. JETP Lett. 52, 164, 1990). 10- P.E. Wolf, F. Gallet, S. Balibar, E. Rolley and P. Nozi~res, J. Physique 46, 1987, 1985. 11- F. Gallet, S. Balibar and E. Rolley, J. Phys. (France) 48, 369, 1987. 12- S.Balibar, C.Guthmann and E.Rolley, J. Phys. I France 3, 1475, 1993. 13- E.Rolley, E.Chevalier, C.Guthmann and S.Balibar, Phys. Rev. Lett. 72, 872, 1994. 14- A.V. Babkin, D.B. Kopeliovitch and A.Ya. Parshin, Zh. Eksp. Teor. Fiz. 89, 2288, 1985 (Sov. Phys. JETP 62, 1322, 1985). 15- A.F. Andreev, Zh. Eksp. Teor. Fiz 80, 2042, 1981 (Sov. Phys. JETP 53, 1063, 1981). 16- C. Jayaprakash, W.F. Saarn and S. Teitel, Phys. Rev. Lett. 50, 2017, 1983. D.S. Fisher and J.D. Weeks, Phys. Rev. Lett. 50, 1077, 1983. 17- P. Nozi~res and F. Gallet, J. Phys. (France) 48, 353, 1987. 18- S.Balibar and J.P.Bouchaud, Phys. Rev. Lett. 69, 862, 1992 S. Balibar, J.P. Bouchaud, F. Gallet, C. Guthmann and E. Rolley, Proc. of "Surface Disordering, Growth, Roughening and Phase Transitions", R. Jullien, J. Kertesz, P. Meakin and D. Wolf ed., p.63, Nova Science Publishers (New York, 1992). 19- M. Kardar, G. Parisi and Y.C. Zhang, Phys. Rev. Lett. 56, 889, 1986. 20- T. Hwa, M. Kardar and M. Paczuski, Phys. Rev. Lett. 66, 441, 1991. 21- C.Jayaprakash, W.F.Saam and S.Teitel, Phys. Rev. Lett. 50, 2017 (1983) C. Rottman and M. Wortis, Phys. Rev. B29, 328 (1984). 22- N.C.Bartelt, T.L.Einstein and E.Williams, Surf. Sci. Lett. 240, 591 (1990); E.Williams and N.C.Bartelt, Science 251,393 (1991). 23- M.Kardar and D.R.Nelson, Phys.Rev.Lett. 55, 1157 (1985). 24- M.Uwaha, J.Phys.(France) 51, 2743 (1990); H. van Beijeren, private comunication. 25- C.Rottman, M.Wortis, J.C.Heyraud and J.J.M~tois, Phys.Rev.Lett. 52, 1009, 1984 J.J.Saenz and N. Garcia, Surf.Sci. 155, 24, 1985 J.J. M~tois and J.C. Heyrand, Surf.Sci. 180, 647, 1987 F.Gallet, Ph.D. thesis, unpublished, Paris, 1986 Y.Carmi, S.G. Lipson and E. Polturak, Phys. Rev. B36, 1894, 1987 New results have very recently been obtained on helium 4 crystals by A.V.Babkin (private communication) with

29

an improved interferometric technique. They show a precise x3/2 behaviour and should soon be published. 26- O.A. Andreeva and K.O. Keshishev, Pis'ma Zh. Eksp. Teor. Fiz. 46, 160, 1987 (Soy. Phys. JETP Lett. 46, 200, 1987). O.A. Andreeva, K.O. Keshishev and S. Yu. Osip'yan, Pis'ma Zh.Eksp.Teor.Fiz. 49, 661, 1989 (Sov. Phys. JETP Lett. 49, 759, 1989). 27- X.S.Wang, J.L.Goldberg, N.C.Bartelt, T.L.Einstein and E.Williams, Phys.Rev.Lett. 65, 2430 (1990); 28- C.Alfonso, LM.Bermond, J.C.Heyraud and J.J. Metois, Surf.Sci. 262, 371 (1992). 29- S.Balibar, C.Guthmann and E.Rolley, Surf.Sci. 283, 290 (1993). 30- J.Frohn, M.Giesen, M.Poengsen, J.F. Wolf and H. Ibach, Phys.Rev.Lett. 67, 3543 (1991) 31- K.O.Keshishev, A.Ya.Parshin and A.V.Babkin, Pis'ma Zh.Eksp.Teor.Fiz. 30, 63 (1979) (Sov.Phys.JETP Lctt. 30, 56, 1979)" 32- Strictly speaking, a stepped surface of orientation is rough if its height-height correlation function diverges, i.e. above its own roughening temperature TRn. Below TRn, steps are registered with respect to the lattice, that is the phenomenon which is considered in ref.8. However, if the step density is small, TRn is very small. Here we only consider temperatures between TRn and TR0, the roughening temperature of the main terraces. For simplicity we call "stepped" the surfaces whose large anisotropy is due to the existence of well dfined steps and terraces, and we restrict the use of "rough" to qualify surfaces at larger til angle, whose anisotropy is small because terraces ate too narrow to be well defined. 33- C.Jayaprakash, C.Rottman and W.F.Saam, Phys. Rev. B30, 6549, 1984. 34- Y.Akutsu, N.Akutsu and T.Yamamoto, Phys. Rev. Lett. 61,424 (1988). Their prediction for the entropic interaction in terms of the step energy can also be obtained from the work of Bartelt, Williams et al.22 who used the work of C.Jayaprakash et al.33. However, the hopping matrix element in ref.33 should be doubled (W.F.Saam, private communication). 35- I. Asaro and W.A. Tiller, Metall. Trans. 3, 1789 (1972). 36- M.Ya. Grinfeld, Soy. Phys; Dokl. 31,831 (1986). A recent review is published in J. Non Linear Sci. 3, 35 (1993). See also D.J. Srolowitz, Acta Metall. 37, 621 (1989). 37- S.Balibar, D.O.Edwards and W.F. Saam, J. Low Temp. Phys; 82, 119 (1991) 38- P.Nozi~res, J. Physique I (France) 3, 681 (1993) 39- M.Thiel, A.Willibald, P.Evers, A. Levchenko, P. Leiderer and S. Balibar, Europhys. Lett. 26, 707 (1992). 40- R. Torii and S. Balibar, J. Low Temp. Phys. 89, 391 (1992). 41- J. Berr~har, C. Caroli, C. Lapersonne-Meyer and M. Schott, Phys. Rev. B46, 13487 (1992). 42- M. Grinfeld, Europhys. Lett. 22, 723 (1993). 43- E.Rol|ey, S.Balibar, C.Guthmann and P.Nozi/~res, to appear in Physica (1994). 44- C.A. Bums and J.M. Goodkind, J. Low Temp. Phys. 95, 695 (1994). 45- M.S. Pettersen, S. Balibar and H.L Marls, Phys. Rev. B49, 12062 (I 994) and references therein.