Chapter 3: The Surface of Helium Crystals

CHAPTER 3

THE SURFACE OF KELIUM CRYSTALS BY

S.G. LIPSON and E. POLTURAK Physics Department, Technion - Israel Institute of Technology, Haga, Israel

Progress in Low Temperuture Physics, Volume X I Edited by D.F. Brewer 0Elsevier Science Publishers B. V.. 1987

Contents ........... ........ ........................ 1.1. The phases of sol ...... 1.2. Surface physics with solid helium . . . . . . . . . . . . . . . . . . . . . . 2. The morphology of helium crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Crystal shapes in the absence of gravity: the Wulff construction . . . . . . 2.2. Surface stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Crystal shapes in a gravitational field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... 2.4. Analogy between crystal shapes and ferromagnetism . . . . . 2.5. Classical calculations of the evolution of equilibrium crystal 2.6. Are equilibrium morphologies obtainable? . . . . . . . . . . . . . . . . . . . . . . . 2.7. The roughening transition: elementary theory . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Experimental results on crystal shapes and roughening transitions . . . . . . . . . . 3. Recent theoretical advances regarding roughening . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Roughening temperatures . . . . . . . . . . . . . . . . . . . . . . . . ........ 3.2. Critical behaviour of facet sizes and surface stiffness . . . . . ........ 4. The growth of helium crystals . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Rough surfaces: theory of the growth resistance . . . . . . . . . 4.2. Crystallization waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Transmission of sound through the interface . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Experiments using charged helium surfaces . . . . . . . . . . . . . . . . . . . 4.5. Theory of the Kapitza resistance of helium surfaces . . . . . . . . . . . . . 4.6. Experiments on the Kapitza resistance . . ................. 4.7. Growth of faceted surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8. Experimental work on facet growth . . .............. 4.9. Growth of crystals from dilute 3He-4He mixtures . . . . . . . . . . . . . . . . . . . . . ..................... 5. Substrate-induced phenomena . 5.1. Solid multilayer growth on matching crystalline substrates . . . . . . . . . . . . . . . 5.2. The wetting transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Note added in proof . . . .......... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction . . . . . .

128

129 129 131 132 139 140 149 151 156 156 161 166

174 176 177

1. Introduction

Were it not for the discovery of roughening transitions on the surface of helium crystals in 1980, it would be unlikely that the helium hcp-superfluid interface would have been recognized as one of the most nearly ideal interfaces available to the physicist. The helium surface is unique in being the only solid-liquid interface that exists over a wide continuous range of temperatures, so that the physical mechanisms of crystal growth and equilibrium can be investigated as a function of temperature; in any other material the temperature range over which this is possible is limited by the triple point on one side and the steepness of the melting curve on the other. Other significant factors contributing to the importance of the 4He surface are in particular the high inherent purity of the system and the fact that, at temperatures below 1.8 K, the fluid phase is superfluid. In this article we shall review the various experiments that have been performed on the helium surface and indicate their significance in terms of simple theoretical models. We shall not attempt to review the theories themselves in depth. Balibar and Castaing (1985) have also recently reviewed this field. 1.1.T H E

PHASES OF SOLID HELIUM

Since we shall need to make several references to their phase diagrams, we shall briefly discuss what is known at present about the bulk equilibrium of the helium isotopes (fig. 1).The absence of a gas-liquid-solid triple point was explained by London (1954) [see Wilks (1967)l. Much of the high-pressure region is still “terra incognita”; it has been explored only along the melting-curve coastline, and reminds one of nineteenth century maps of Africa. The highest temperature regions have been investigated using helium bubbles formed in aluminium and molybdenum by alpha-particle irradiation (Rife et al. 1981, Evans et al. 198l), and the actual structure of the phase labelled “bcc?’ is still in doubt (Levesque et al. 1983). As far as the low-temperature regions go, the major 4He phase is hcp, with the exception of a thin sliver of bcc between 1.4 and 1.8 K (Vignos and Fairbanks 1961). In contrast, in 3He the bcc phase predominates, the hcp only occurring at higher pressure. Then, both isotopes transform to fcc at still higher pressure. The obviously delicate considerations leading to the 129

S.G. LIPSON AND E. POLTURAK

I30

b.c.c. ?

I

TEMP. ( K )

I0 '

Y

n

Id 10 .I

I

100

10 TEMP.

(K)

Fig. 1. Bulk phase diagrams of (a) 4He and (b) 'He.

choice of structure at absolute zero have been discussed by Niebel and Venables (1974). They attribute the choice of hcp to the spherical symmetry of the helium atom; the other inert gases, which differ from helium in this respect, have fcc structures at T = 0. The hexagonal phase of 4He is noteworthy in that it is very close indeed to the ideal close-packed structure; it has a c / a ratio of 1.632, to be compared to the ratio 1.633 for perfect close-packing (Vos et al. 1967). This

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phase of helium is the only one to exhibit birefringence, which was used by Vos et al. to create contrast in their film of crystallization processes in helium. The 4He surface comes into its element at temperatures below the intersection of the lambda line with the melting curve. When the temperature is low enough, both the solid and the liquid are highly ordered, the former in real space and the latter in momentum space. As a result, neither bulk phase has appreciable entropy and the surface entropy becomes prominent. This is the situation from zero up to about 0.9 K, in which region the latent heat L = T(S,- S,) 0 and the melting curve is almost a horizontal line (Clausius-Clapeyron), although surface quantum effects continue to be important up to 1.4 K, the hcp-bcc transition temperature (Keshishev et al. 1981). The mechanisms involved in atomic movements in solid helium have been reviewed by Andreev (1982), who postulated but did not prove the existence of zero-point vacancies at T = 0. The dominant process of atomic motion appears to be quantum tunnelling into vacancies. This process is temperature independent and so the time constants for structural changes in helium do not become asymptotically infinite as the temperature falls. Relatively little is known about dislocations in helium crystals (Paalanen et al. 1981).

-

1.2. SURFACE PHYSICS WITH

SOLID HELIUM

At this point we can summarize the main reasons for the suitability of helium for surface studies: 1. The equilibrium melting curve can be followed in a single phase (hcp) in contact with superfluid from 0 to 1.4 K, and through most of this region the surface properties dominate. 2. The rate at which morphological changes take place does not generally become infinitely slow as absolute zero is approached. 3. The absence of latent heat of melting at low temperatures together with the superfluid nature of liquid helium means that crystal growth and equilibrium are independent of heat-flow considerations. 4. The 4He system can be produced with very high purity (< lo-”) (McClintock 1978) and therefore ideally clean surfaces can easily be investigated. There is also scope for investigating pure 3He and 3He-4He crystal surfaces, although little has yet been done in this field. Helium crystals have so far provided important information in several fields, which will be discussed in greater detail in later sections: 1. Equilibrium crystal shapes and their relationship to the anisotropy of surface tension. It is not generally appreciated that all natural crystals have

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shapes determined by their growth, and not by thermodynamic equilibrium. 2. The roughening transition of a crystal surface: at the transition temperature the surface structure changes from smooth to rough on an atomic scale. Macroscopically, a smooth surface is a crystal facet. 3. The mechanisms of crystal growth of rough and faceted surfaces. 4. The wetting of one solid by another. 2. The morphology of helium crystals

Until 1980 (Landau et al. 1980) helium crystals were thought to be essentially spherical and devoid of any morphological interest (Keshshev et al. 1979, Vos et al. 1967, Fraas et al. 1977). This observation was explained in a paper by Andreev and Parshin (1978) [see also Andreev (1982)], who attributed the generally rounded shape to the existence of zero-point atomic excitations on the surface of a quantum crystal. They argued that the atomic excitations, being non-localized, would fall into energy bands, and they estimated these bands to be wide enough to include the zero of energy. Thus zero-point excitations, in the form of kinks and ledges, would destroy any tendency of the crystal to show facets, even at absolute zero. We shall discuss t h s paper in greater detail in section 3. Landau et al. (1980) reported optical observations of the dynamics of equilibration of helium crystals and noticed that below about 1 K the facets, whch had developed on the surface during growth, persisted on the equilibrium form. These observations were the first hint of the existence of a roughening transition, a subject which will be developed in sections 2.7 and 3. On the assumption that what is observed as the final shape of a helium crystal is in fact the true thermodynamic equilibrium shape, the whole relationship between the anisotropic surface tension and the crystal shape then becomes experimentally accessible in a macroscopic system. 2.1. CRYSTAL SHAPES IN

THE ABSENCE OF GRAVITY: THE

WULFFCONSTRUC-

TION

Wulff (1901) first formulated the relationship between the equilibrium shape of a crystal and its surface free energy (or surface tension). T h s topic is extensively discussed in many texts dealing with crystal growth [e.g. Woodruff (1973) and Landau and Lifshitz (1959)l and we present it here only for the reader unfamiliar with the subject. Imagine two crystals having the same volume and number of atoms and differing only in their external shapes. The difference between the free energies of the two is due solely to the atoms on their surfaces, these atoms having a lesser number of satisfied bonds than the atoms in the bulk. At thermodynamic equilibrium, all other

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parameters being held constant, the crystal would like to have the shape with the minimum free energy. Because the unit cell is not isotropic, it costs less energy to break bonds in certain directions, and the crystal with more surface in these directions will be preferred. Thus the crystal shape is not simply a sphere, which has the minimum surface area, but is more complicated. Wulff's construction allows the shape of the equilibrium crystal to be deduced geometrically from knowledge of the directional dependence of the surface free energy, y ( i ) , where i is the unit vector normal to the crystal surface. We first plot y ( h ) in polar coordinates; this is known as the " Wulff diagram" or "y-plot". The construction is carried out by drawing a plane normal to 8 through each point on the y-plot. The volume enclosed by the tangent planes has the shape of the equilibrium crystal (fig. 2). Since some planes will be sufficiently far from the origin that they do not contribute to the resultant shape, it follows that some portions of the y-plot may not be represented on the equilibrium crystal surface. The most elegant proof of the Wulff construction is given by Herring (1952). We shall give an alternative simple proof here (Wortis 1985). I

y-plot

Fig. 2. Wulffs construction.Through C, which is a point on y(A), draw the plane ABC normal to ri (OC). The crystal has the shape of the region enclosed by all the planes such as ACB. Note that the point P on the surface has the surface energy corresponding to point C on the y-plot.

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tangent plane \

A

' 0 Fig.3. Geometry of the proof of Wulffs construction.

Consider, as in fig. 3, an element ii d S of the surface of the equilibrium crystal: the element has normal vector ii and its vector distance from the origin is r. The total surface energy of the crystal is then

E = / y ( S ) dS(ii). The volume of the crystal V=

+ Ir - i i d S ( h ) .

To find the surface S which has minimum E under the constraint that V = constant we write d ( E - XV)= 0, dS(R)

y ( h ) = )Ar*ii.

(3)

Thus the equilibrium crystal surface is bounded by tangent planes at distance r * h , proportional to y ( h ) , from the origin. The parameter A, which determines the overall size of the crystal, has the dimension of pressure, and is in fact the pressure difference 6 p across the surface [see eq.

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135

(7)]. Frank (1963) has described an alternative method of deriving the

equilibrium surface from y ( R ) . The characteristics of the crystal surfaces generated in this way have been discussed in detail by Herring (1951, 1952). Since these characteristics are not widely known, we shall discuss them briefly. First, the properties of the y-plot in general should be recognized. A spherical y-plot corresponds to an isotropic medium. If the medium is crystalline it will generally be anisotropic, and a surface lying normal to a high-symmetry direction will usually show a local minimum in y, since creation of such a surface from the bulk crystal means breaking fewer nearest-neighbour bonds than for any neighbouring orientation. A surface at a small angle 8 to a high-symmetry direction with atomic plane separation a requires the creation of 18 I / a atomic ledges per unit distance. If the energy required to create a step of unit length is a, we have

Y ( 0 ) = Yo + fJ I 8 I/a,

(4)

which defines a singularity in y ( R ) . This is generally called a cusp. At the singularity dy/d8 changes discontinuously by 2a/a. The argument has been extended by Landau (1965) to include further neighbours and he has shown that y ( R ) has a cusp in every rational orientation (a “raspberry” shape - Frank’s (1963) description). In fact, thermal vibrations should wipe out all the cusps except those in the highest-symmetry directions (Andreev 1981). Several important characteristics of crystal shapes can be summarized from Herring’s (1951) paper: 1. The shape of a free crystal is convex at all points. This does not apply to crystals with boundaries determined by external constraints: think, for example, of a pendant drop of water waiting to drip! (section 5). 2. Facets correspond to the cusps in y ( R ) , although not every cusp gives rise to a facet on the equilibrium crystal. The size of a facet L relative to the crystal size Ro can be related to the discontinuity in y at the cusp in cases where both facets and rounded surfaces coexist. From fig. 4a one can easily see in two dimensions that

unless the facet is intersected at a non-zero angle by another facet or a rounded surface (fig. 4b). The difference between the cases in figs. 4a and b seems quite prominent, but may be quite subtle in practice: compare the examples of Pb and Au in the work of Rottman et al. (1984) and Heyraud and MCtois (1980a, b).

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a

tangent circle at cusp

- --_

b

0 Fig. 4. Details of Wulffs construction around a cusp in y ( 0 ) in two dimensions. (a) When y ( 0 ) lies outside the circle which is tangent at the cusp and passes through 0, the facet of length 2 L through G terminates at B. When dB is small, similarity of the triangles OGX and BCX proves eq. (5). (b) When y ( 0 ) lies inside the tangent circle the normal to OC’ through C’ curtails the facet at B’. resulting in a discontinuity in surface slope at B’. In this case eq. ( 5 ) does not hold.

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a

b Fig. 5. Wulffs construction for faceted cubic crystals, (a) with nearest-neighbour interactions only, (b) with next-nearest-neighbour interactions as well.

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3. If the anisotropy of the surface tension is large the crystal may be completely faceted. D e t d s of the behaviour of y ( i ) around the cusps then have no influence on the equilibrium shape, since ledges do not exist on facets and therefore the step energy is meaningless. An example of such behaviour can be found in the ideal nearest-neighbour-bonded crystal, which is often considered as a simple model for calculations. By counting broken bonds, each of energy E, one can easily show that the energy of the surface having direction cosines (I, m , n) is ( E / a ) ( I I I + I m I + I n I) and the polar plot of this in two dimensions, for example, is y ( 6 ) = ( E / u ) ( lcos

e l + /sin 61).

(6)

This defines arcs of four circles passing through the origin. The corresponding crystal shape has four facets only (fig. 5a). The addition of attractive further-neighbour bonds leads to cusps in other directions, which then lead to additional facets (fig. 5b). 2.2. SURFACE STIFFNESS

The shape of the equilibrium crystal can be expressed in a different way using the Young-Laplace equation (Landau and Lifshitz 1959). Balancing the excess chemical potential inside the crystal against the surface tension forces gives a pressure difference between the two phases:

In ths equation C , and C, are the two principal curvatures of the surface and

where x, is the coordinate in the plane of C,. The second term of 7 is analogous to the lateral force required to bend a bar, and consequently 7 is called the “surface stiffness”. In the case of zero gravity the pressure and chemical potential within each phase are constants, and so 6 p is a constant and eq. (7) can be integrated if y ( i ) and hence ?(I;)are known. For example, if y ( $ ) is given by eq. (6), f(R) contains four delta functions in the principal directions and the facets follow. The surface stiffness is most real in dynamic situations, such as surface waves, but it does have a very important general aspect: it is the only true measure of y, and is the quantity actually measured in any experiment to

THE SURFACE OF HELIUM CRYSTALS ,T-

draws y - plot

139

square

A 1 1 h

profile

of card

Fig. 6 . A simple analogue computer to construct y ( S ) from the crystal shape. The T-square rotates around the pin while maintaining contact with the crystal profile at a point or along a facet.

determine surface tension. For example, a given crystal shape may be related by the inverse Wulff construction to a family of y-plots, depending on where the origin is taken. It is instructive to make a simple analogue computer to illustrate this point (fig. 6). If the crystal is centro-symmetrical, it can be assumed that the origin is at the centre of symmetry, but this need not be the case. The family is related of course by an arbitrary shift of ongin ( A , 71,

y ( e ) = yo(^) + A cos(e + 7). Clearly ?( e) = yo( 8 ) ; the surface stiffness is invariant. 2.3. CRYSTAL SHAPES IN

(9)

A GRAVITATIONAL FIELD

Although the relationship between y ( i ) and the crystal shape in zero gravity is a nice closed problem with an elegant solution, the same is nowhere near the truth in a non-zero field. Consequently, the latter problem has received scanty attention; gravity and capillarity present competing forces whose relative importance depends on the size of the crystal. The corresponding isotropic problem, the drop of liquid in a field, has been studied extensively, and has been reviewed by Boucher (1980). Only in 1980 was the uniqueness of the solution proved (Finn 1980)!

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The parameter which expresses the relative importance of the two fields is the capillary length I = (2y/Sp g)1/2, where 6p = p, - pt is the difference in density between the two phases. Gravitational effects can be neglected if the crystal size is less than I. In the case of solid-liquid helium, 1 has the value of about 1.4 mm. Amongst the few attempts to get some feeling for the effects of gravity is a paper by Avron et al. (1983), which considers the problem of “ a crystal on a table”, written with special reference to the shapes of helium crystals. It refers to the problem of a sessile drop of an anisotropic medium on a horizontal substrate. While the authors are not able to solve any three-dimensional problem in detail, they enumerate a number of theorems relating the shapes of crystals V in a gravitational field to those of their counterparts W in zero field (W for Wulff). The important physical results are: (1) V can have facets in any direction in which W has, and in addition it may have a horizontal “gravity-induced” facet, which W does not have. If the orientation of this horizontal surface is one that does not appear in W, then it will spontaneously break up into a microscopic pattern of “hills and valleys” (microfacets), which result in a lower free energy. (2) Apart from the horizontal facet, no other new facets can be induced by gravity. (3) The contact angle is unchanged by gravity (see section 5). In two dimensions Avron et al. (1983) have given a general solution, and Andreev and Grischchyuk (1985) have given solutions in particular cases. But it does not seem impossible to formulate the complete three-dimensional problem in a way which might lead to a solution. The addition of a gravitational term to eq. (1) leads to the analogy of eqs. (3):

where 8 is the angle between the vertical and 2, and q!I is the angle between the verticai and the centre of the facet or surface element defined by the tangent plane to P (fig. 2). Solution of this would require iteration with the gravity being “turned on” slowly, but it seems that the results might be interesting.

2.4. ANALOGY BETWEEN

CRYSTAL SHAPES AND FERROMAGNETISM

Andreev (1982) and, independently, Garcia et al. (1984) have pointed out the analogy between the Wulff construction for a crystal with anisotropic surface tension and the relationship between magnetization M and applied field H in a ferromagnetic material below its Curie temperature. The clearest treatment is that of Wortis (1985). In t h s analogy the surface

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normal ri is replaced by M , and the radius vector r by H. The equivalent of eq. (3) is then an equation for the bulk energy:

which is physically quite reasonable. The interpretation is that, given G ( M ) , we can use the Wulff construction to find a “crystal shape”, which in fact allows us to determine the relationship between H and M. Clearly various situations can follow: as H is rotated M can follow smoothly (if the shape is continuously rounded) or it can jump discontinuously from easy magnetization to easy magnetization. Shape effects in the ferromagnet are equivalent to the “gravity-induced facets” of Avron et al. (1983) and if the anisotropy is strong enough a domain structure is formed, which of course is analogous to the “hill and valley” behaviour. 2.5 CLASSICAL

CALCULATIONS OF THE EVOLUTION OF EQUILIBRIUM CRYSTAL

SHAPES

The evolution of crystal shapes as a function of temperature has recently been studied by two types of approach: a mean field theory (Rottman and Wortis 1984a, b) and exactly soluble microscopic models (Jayaprakash et al. 1983, Jayaprakash and Saam 1984). The studies have been applied to different crystal structures: simple cubic, bcc and fcc, respectively. Although the hcp structure of helium has not been considered explicitly, there is little doubt what the results would be. These theories are completely classical; we shall discuss quantum corrections in section 3. The models assume attractive nearest-neighbour interactions J1 and next-nearest-neighbour interactions J2 of variable size and sign. The form of y ( r i ) is first calculated, and then the crystal shape by using the Wulff construction. This may not be possible in some directions, but physically intuitive interpolation is used to complete the picture. The result in all cases is a steady evolution from a completely faceted crystal at T = 0 to an isotropic spherical crystal at a critical temperature T,. A roughening transition appears naturally only in the microscopic models, since the mean field theory does not recognize fluctuations. The “lattice gas” model, which is formulated for this problem, puts all the atoms at sites on the lattice in question, labelling those in the solid phase by M = +1, and those in the liquid phase by M = -1. After specifying the orientation of the interface with respect to the lattice, y ( h ) is computed by summing the energies of interaction between the atoms. The crystal shape follows from the Wulff construction.

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The system has Hamiltonian

H = -J1

M;M,-J2 n.n.

M;M,. n.n.n.

Boundary conditions are imposed on the system so as to force an interface in the right average direction. In the mean field approach M, is replaced by a suitable field, which is varied to find the minimum value for the free energy. Recalling the Heisenberg ferromagnet [see, for example, Kittel (1971)l one can appreciate that this results in an equation of the form

M, = t a n h { ( J , ~ M , + J , ~ M , ) / k , T ) . By starting with a reasonable trial function for Mi, the minimum free energy is found in terms of (12) by iteration for each orientation. From the ferromagnet analogy, it follows that eq. (13) has the solution M = 0 if T is above the critical temperature

in which z1 and z2 are the numbers of nearest and next-nearest neighbours, 6 and 12, respectively, in the simple cubic structure. This critical temperature is an artifact of both the liquid and solid phases being restricted to a lattice: the solid-liquid phase diagram really has no critical point. Jayaprakash et al. (1983) and Jayaprakash and Saam (1984) addressed the question of crystal shapes using the exactly soluble “solid-on-solid” (SOS) model for bcc and fcc structures, respectively. In this model, the free energy can be minimized exactly, using mapping of the problem onto a previously known exactly soluble system (van Beijeren 1977). The essence of this model is that any column of atoms on the lattice starts in one phase and changes at one point only to the other phase before the far boundary is reached. Thus overhangs, voids and evaporated islands are not allowed. The avoidance of overhangs in particular restricts use of the model to orientations close to high-symmetry directions (fig. 7). For the fcc lattice, Jayaprakash and Saam find that the (100) facet undergoes a roughening transition, while the (111) orientation is not exactly soluble but appears to remain rough. The exact results, including details of the predicted behaviour around the roughening transition, will be discussed further in section 3. Although the mean field theory does not predict it, a roughening transition was introduced empirically into the results in order to complete

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143

(b) Fig. 7. An interface permitted by the SOS model is illustrated by (a), where + represents solid and - liquid. The broken line indicates the mean surface. (b) Features not allowed in the model (voids, overhangs and solid nuclei in the liquid) are encircled. (c) Overhangs are necessary to represent a noticeably tilted surface in an unbiased manner.

the picture. The results of Wortis (1985) are shown in fig. 8. The situation for J, > 0 fits helium qualitatively. It is not obvious that the crystal of a quantum solid must be completely faceted at absolute zero because of zero-point motion. Wulff plot

Y (G, T I

Crystal Shape

(equaforial

(equatorial plane)

plane)

( perspective)

Fig. 8. y-plots and crystal shapes for a cubic crystal in the mean field model for J2 > 0 (Wortis 1985). The dots on the crystal outlines represent second-order junctions between facets and rounded surfaces.

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Another critical aspect which appears in the SOS theory refers to the way in which a facet joins on to a rounded surface. As we showed in fig. 4b, this junction can be continuous in slope or discontinuous (second or first order). Jayaprakash and Saam gwe a definite prediction that the junction must be second order, and that the transition at ( x o , z o ) from the plane of the facet z = zo at x < xo should give a crystal profile

zo - z

- (x - x o

)3’2

+ ... ,

(15)

provided that one is not too close to the roughening transition. This form has been confirmed for Pb crystals by Rottman et al. (1984), but not for Au crystals (Heyraud and Metois 1980a, b). Saenz and Garcia (1985) have shown in a mean field approach that both first- and second-order behaviour can be accommodated by taking into account interaction between steps on the crystal surface. Experimentally, the confirmation of the above results in helium depends on whether true equilibrium crystals can be obtained. It is important to do one’s best since the theoretical work has held great hope for exact confirmation in the helium system. Crystals during slow growth do seem to bear out the theory for J2 > 0 quite well in a qualitative manner (after all, the

Fig. 9 A crystal in slow growth at 0.35 K, showing all three types of facet: (OOOl), (1070) and ( l O i l , (Wolf et al. 1983a. b). Photograph taken from the side.

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145

Fig. 10. Crystal profiles during slow growth in the horizontal plane, around T,(lOiO): (a) crystal axes, (b) T < 7, (c) T 5 T,. (d) T > T, (Carmi et a]. 1984).

theoretical results are quite qualitative also!). Wolf et al. (1983a) (fig. 9) show the lowest-temperature form of their crystals as having (OOOl), (lOi0) and (1011) facets, which are equivalent to the fcc results; the fact that the last facet becomes too small to be seen under equilibrium conditions may correspond to J2 being very small, as one might expect for a van der Waals solid. Carmi et al. (1984) show crystal profiles at temperatures around for the (lOi0) facet (fig. 10). Results in helium illustrating the junction between a facet and a rough surface are shown in fig. 11 and confirm the 3/2 power law of eq. (15) (Carmi, to be published). 2.6. ARE EQUILIBRIUM MORPHOLOGIES OBTAINABLE? Since anything more than the qualitative results presented so far requires exact equilibrium of the system, we must ask how this is to be ensured. This

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0.5mm t

I

A Fig. 11. Junction between an (OOO1) facet and a rounded surface for a crystal in the horizontal plane at 0.80 K. The crosses represent a digitization of the crystal profile, and the curve is a fit to eq. (15) with exponent 1.52. The curve was fitted to the data between the points A (the junction between the facet and rounded part. determined by a best fit) and B. The bump beyond B is caused by a spurious interference fringe which crosses the photograph (Carmi. to be published).

is much more difficult than might be expected at first sight. The reason is that the growth rate of crystal surfaces is strongly anisotropic (see sections 4.1 and 4.7). Facets grow slowly and the more rapidly growing rounded parts of the crystal near the corners gain on them until they catch up and become part of the facets themselves. Then they continue to grow with the facets, which therefore become larger and larger until they dominate the crystal shape. During melting the reverse happens, and facets appear very small, if at all. Thus growth and melting shapes of crystal are considerably different from the equilibrium shapes (fig. 12). The earliest work on equilibrium crystals was done on microscopic samples, a few microns in size, by Sundqvist (1964) and Winterbottom and Gjostein (1966a, b), who allowed various metal samples to equilibrate in contact with their vapours. Calculations based on surface diffusion and evaporation or condensation showed that equilibrium shapes should be attained after several days of annealing at temperatures close to melting. Helium crystals have been considered as ideal for equilibrium studies because the first experiments in wtuch their morphologies were observed (Landau et al. 1980, Keshishev et a]. 1981) showed the speed with which they respond to changes in environment. This speed, which apparently resulted in a final crystal shape being achieved in a minute or so, was attributed to the superfluid characteristics. the absence of latent heat at the temperatures of interest and the process of quantum tunnelling in the solid (Andreev 1982). Recently Wolf et al, (1985) cast doubt on whether the crystals observed

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147

Fig. 12. Comparison of (a) growth and (b) melting profiles of a crystal; ( c ) an equilibrium shape after cooling slowly through the roughening transition.

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S.G. LIPSON AND E. POLTURAK

really have equilibrium shapes, and we shall now consider briefly the possible equilibration mechanisms: surface diffusion and crystal growth. Surface diffusion probably plays little part in the equilibrium of the crystals. The time constant to involved can easily be shown to depend on the third power of the crystal size r (Drechsler 1983). Despite the relatively fast diffusion in helium, and the increased density of vacancies at the surface, it is still unlikely that to will be smaller than 1014r3 s cmP3,which limits equilibration by this means to microscopic crystals. Crystal growth mechanisms will be discussed in detail in section 4, and here we shall quote only the quantities necessary to estimate the time constants for equilibration. Atomically rough surfaces (sections 4.1-4.4) grow very rapidly. The time constant 1, scales as r 2 , and is proportional to the growth resistance * ( K m 4 ) - ' , values of which can be found in fig. 20:

Substitution shows that for a 1 mm crystal at 1 K, t o is of the order of 1 s. Facets (sections 4.7, 4.8) grow differently by mechanisms which are dislocation dominated unless very close to a roughening transition. The growth rate is not linearly related to the chemical potential difference, and so a time constant cannot be defined, but typical times can be seen from fig. 24 to be greater than lo5 s under common conditions. For all practical purposes, facet growth should be considered as hysteretic, i.e. as showing a growth threshold. We can summarize this discussion by saying that, if we observe a helium crystal of millimeter size for, say, 100 s, the rounded parts will come into complete equilibrium, whereas the facet sizes will still be determined by their history, although the crystal morphology will be correct. The facets may grow so slowly that within any reasonable time no shape changes will be observed and an illusion of equilibrium is achieved. Close to roughening, the facets may also equilibrate, but will be too small to see. This means that the complete y-plot cannot be determined from the crystal shape, as was assumed by Carmi et al. (1984); nor can step energies be deduced from facet sizes, as was done by Balibar and Castaing (1980) and Keshishev et al. (1981). But the surface stiffness, 7, which is related to the crystal curvature only, can be deduced unambiguously; and this in fact contains most of the physical information. * The growth coefficient K is defined in eq. (26). The useful form is ( Krii4)-*, which recurs frequently in this field. m i 4 is the helium atomic mass.

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149

Just the same, it may be possible to create equilibrium shapes by slowly cooling a crystal through a roughening temperature, under conditions in which it neither grows nor melts. All the shape change necessary in order to maintain the equilibrium shape can then be achieved by changes in the rounded parts. This will be easiest in a temperature region where there is negligible thermal expansion of either phase, i.e. below about 1 K. A comparison between crystals grown in this way and by increasing the amount of solid at constant temperature is shown in figs. 12c and a, respectively. 2.7. THEROUGHENING TRANSITION: ELEMENTARY THEORY We have already mentioned the existence of roughening transitions as the most important reason for the prominence of helium surface studies, and we have seen above how the phenomenon fits into the scheme of the morphology of crystals. At this stage we shall give a simple description of the physics of the transition, based on the original discussion by Burton et al. (1951). Today we know that this is a very much over-simplified modeI, but it has the advantage of being easy to grasp; recent theoretical advances have added greater detail to our understanding of the phenomenon, but their reliance on the exact solutions of certain mathematical models have made the physics behind the results rather obscure to the average experimentalist. These models and their experimental investigation will be discussed in section 3. The model of Burton et al., in fact, refers to a solid-vapour interface, but can readily be reworded to apply to the liquid-solid case. Consider an ideal faceted high-symmetry crystal surface whose atoms lie in a single plane. Allow the surface to be covered with half (say) of a monolayer of additional atoms. The additional atoms are bound to the facet by a number of interatomic bonds which is generally less than the number for an atom which is completely surrounded by other ad-atoms. The total free energy of this half-layer is determined by statistical mechanics. At T = 0, the lowest free energy P = U - TS is achieved by grouping the atoms together in a single island covering half the surface: at higher temperatures a more random structure will be preferred because of the entropy term in F, and eventually complete disorder will reign. To show that there is a phase transition from an ordered to a disordered state, we label, similarly to section 2.5, the occupied lattice positions by M = + 1, and the unoccupied ones by M = - 1. If we assume nearest-neighbour forces between ad-atoms, the Hamiltonian is then, as in eq. (12), given by H = -E

C Mi-Mj. n.n.

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150

This is the two-dimensional nearest-neighbour Ising lattice problem, which has a phase transition many of whose properties can be calculated exactly [Onsager (1944); see also McCoy and Wu (1973)]. (Notice that even here we are relying on the well-known exact solution of a mathematical model!) E is the bond energy between neighbours. In the early work on this subject E was deduced from the latent heat of evaporation [Jackson (1958); see also Woodruff (1973)l; in helium this is not possible (see section 1.1) and we use the surface tension, which is the surface free energy per unit area, and is directIy related to the bond energy. The model shows a phase transition at temperature

where c = 0.57 for a square lattice, 0.91 for a triangular one. The interpretation is that below T, the surface is covered by a small number of island domains of ad-atoms separated by large vacant regions; it is atomically flat and has relatively few steps at which new ad-atoms can attach themselves easily. As the transition is approached the islands become smaller and above T, random ad-atoms are the favoured structure; ths is the atomically rough situation, which provides innumerable attachment sites for new atoms. It is easy to estimate the value of from eq. (18). For metals the estimates give temperatures around, usually above, the melting temperature [see Woodruff (1973)]. But for some organic materials with low latent heats the transition was expected at lower temperatures. The first evidence of a faceting transition was published by Pavlovska and Nenow (1971, 1972) for diphenyl and naphthalene. Since then other organic materials have been shown to have transitions, and Nenow (1984) has recently reviewed that area. In helium, substitution of values taken from the surface tension gives for the basal plane (Oool) or c-facet:

<

q = 0.9ld’y k,

=

1.3 K ,

where d is the lattice spacing. T h s was the incentive for the search for roughening on 4He crystals and has turned out to be substantially correct. Balibar and Castaing (1985) have suggested that 3He should show a roughening transition in the range 90-200 mK, on the basis of a crude estimate of the surface tension. (See note added in proof.)

THE SURFACE OF HELIUM CRYSTALS

2.8. EXPERIMENTAL RESULTS ON CRYSTAL SHAPES AND ROUGHENING

151 TRANSI-

TIONS

Much of the information we now have about helium crystal surfaces has come from the study of equilibrium crystal morphology and it will be appropriate at this stage to discuss the experiments which have been performed. Other studies have of course supplemented information, notably sound and heat transmission, dynamic studies of crystal growth, crystallization waves and surface electrification, which will all be mentioned in due course. Because of the size of the crystals involved (millimeters) it has been natural to use optical techniques. At the temperature involved there is no particular problem regarding the heat load associated with optical observation, since helium has no absorption bands in the visible or in any of the adjacent spectral regions. The earliest observations of the shapes of helium crystals (Keshishev et al. 1973, Vos et al. 1967) did not remark on any interesting morphological properties, although the ease of growing single crystals of the hcp phase from the superfluid was demonstrated both by Vos et al. and by Fraas et al. (1977), who used X-rays to study the perfection and growth characteristics of helium amongst other quantum crystals. Vos et al. employed the birefringence of the hcp 4He to get contrast between various crystallites and the fluid. Most of their work related to higher pressures and temperatures, up to the fcc-hcp transition, where the crystal is of course in contact with normal fluid and crystal growth is dendritic unless extremely slow. In the region of contact with superfluid they reported the crystals to be exactly circular. The first experiments in which aspects of an equilibrium shape were investigated were those of Balibar et al. (1979), who measured, as a function of temperature, the pressure difference necessary to extrude solid helium through a circular hole about 1 mm in diameter. They also measured the capillary rise of the solid helium between the plates of a cylinder capacitor (fig. 13). These two measurements give, respectively, y and 7 cos a, where a is the contact angle between the solid helium and the capacitor plates (section 5). These authors also reported an abrupt change in both 7 and (Y around 1 K, and time constants for the attainment of equilibrium of the order of 1 min in most cases. Following up on these observations Landau et al. (1980) used an optical cryostat to observe the morphology of the crystals directly. They used a 3He refrigerator to cool a disk shaped optical cell of volume 0.5 cm3 and 10 mm diameter with its axis horizontal (Pipman et al. 1978), in which crystals of 4He with random orientation could easily be grown by compressing the

S.G. LXPSON AND E. POLTURAK

152

liquid

liquid

solid

Fig. 13. Schematic diagram of apparatus used by Balibar et al. (1979) to measure (a) capillary depression of the interface and (b) bubble stability of solid 4He.

erated shutter

Image lens

\

Focal plane sto Electronic clack

3 mW He - Nc Laser Hologrophic plate

Fig. 14. Diagram of the optical system used by Landau et al. (1980) to obtain holographic interferograms of helium crystals.

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153

Fig. 15. One of the first holographic interferograms of a helium crystal showing facets (regions with straight parallel interference fringes). The crystal occupies the lower third of the cell (Landau et al. 1980).

superfluid to the melting pressure. The technique of holographic interferometry (fig. 14) allowed them to visualize details of the crystal morphology by the formation of interference fringes. Their observations confirmed the contact angle measured by Balibar et al. above 1 K, and analysis of the crystal shape allowed y to be measured. No temperature dependence of either parameter was seen, nor any great anisotropy in 7. They reported values of 7 of 0.16 erg cm-2 in the hcp phase and 0.84 erg cmP2 in the bcc phase and contact angles of about 140". But the most significant observation in the experiments of Landau et al. was the persistence of facets on the crystal profile even after long periods of equilibrium at temperatures below about 1 K (fig. 15). This observation was interpreted by Avron et al. (1980) and by Balibar and Castaing (1980) as pointing to the existence of a roughening transition. Avron et al. supported their thesis by demonstrating two faceting transitions in optical experiments on a small crystal. The transitions were observed at 1.08 K on the (0001) or c-facet, and at about 0.85 K on the a-facet (1010) [then guessed incorrectly as (1120)]. The observations used interferometry in order to identify facets on the crystal, and gave a rough indication of the manner in which the facet

S.G. LIPSON AND E. POLTURAK

154

/

He - Ne laser beorn

bifilor copocitor t o excite waves

dif f roct ion

\

def lexion observed crystal

Fig. 16. Cell used by Keshishev et al. (1981) to excite and study crystallization waves, and to measure the size of facets in the region of ~(OOOI).

size went to zero as the upper transition was approached. T h s transition was observed to be reversible. The lower transition was inferred from the disappearance of the a-facet between two observations. Their paper also inchded a theoretical discussion wtuch established a hierarchy for the transition temperatures of different facets (section 3.1). Keshishev et al. (1981) identified the upper transition in several crystals at 1.15 -t 0.03 K and also made measurements of the way in which the facet disappeared. They measured the equilibrium sizes of the facets as a function of temperature by using the diffraction pattern of light reflected at glancing angle from them (fig. 16). These observations were made on crystals which were apparently at equilibrium. Wolf et al. (1983a) reported observing a third transition on the s-facet (1011) at 0.365 K, but only on slowly growing crystals (fig. 9); the facet did not appear to remain on equilibration. No further transitions have been observed at lower temperatures on 4He, but it is possible that 3He shows a transition at about 90 mK (Rolley et al. 1986). In order to study the morphology in greater detail it is necessary to contend with gravity. Two ways of doing this have been considered. The first (Wolf et al. 1985) is to use very small crystals so that gravity can be neglected. The second is to work in the horizontal plane (Carmi e: al. 1984, 1985).

THE SURFACE OF HELIUM CRYSTALS

5

155

to camera

light beam -

c

( f r o m laser or other source )

&--beam

sphtter

N

m

Fig. 17. System used by Carmi et al. to observe crystal equilibrium and growth in the horizontal plane.

In their work on very small crystals, Wolf et al. grow their crystals by forcing them through a small hole as in their earlier work (Balibar et al. 1979). They photograph the crystal profiles from the side using incoherent light, which is well known to produce clearer images than laser light, and digitally analyse the highly magnified images obtained. Their work has concentrated on the immediate vicinity of the roughening transition of the (0001) facet (see fig. 23). Carmi et al. use a cylindrical cell 10 mm diameter, which has its axis vertical. The lower surface of this cell is a polished beryllium-copper mirror, gold plated, on which the crystals sit. The cell is illuminated and observed from above with either incoherent or coherent light using a symmetrical 1 : l imaging lens (fig. 17). In order to achieve equilibrium conditions, the experimental cell can be isolated by a cryogenic valve at its entrance. Some results from this work have been shown in figs. 10-12. It is important to remember that whereas the rounded parts of the crystals are in equilibrium, the facet sizes may be determined by their growth. If equilibrium facets could be observed, it would be possible to measure the step

156

S.G. LIPSON AND E. POLTURAK

energy u [eq. ( 5 ) ] from their sizes; the fact is that various attempts to do this (Balibar and Castaing 1980, Keshtshev et al. 1981, Babkin et al. 1984) have given widely ranging results because of the uncertainty of equilibrium, while the only independent measurements of the step energy (see section 3.2) are limited to the close vicinity of the roughening transition. More detailed experiments of behaviour around the transitions has been published by Wolf et al. (1984, 1985), by Babkin et al. (1984) and by Carmi et al. (1984) and will be discussed in section 3.2. The latest values for the transition temperatures are those appearing in table 1 in section 3.2. Although the origmal measurements of by Landau et al. (1980) did not show any noticeable temperature dependence or anisotropy, more recent experiments by Gallet et al. (1984) have shown that a small temperature dependence exists. Regarding the anisotropy of 7, various experiments [Keshishev et al. (1981), see section 4.2; Maris and Huber (1982), see section 4.6; Wolf et al. (1985), see section 4.81 show that it is not negligible. This can, in fact, be seen by comparing radii of curvature in the principal directions in fig. 12c, which shows a profile in the horizontal plane and to which eq. (7) can be applied directly.

3. Recent theoretical advances regarding roughening The mode1 of Burton et a]. (1951) has the drawback that it is difficult to extend it to more than a single layer of ad-atoms in a consistent manner, as was shown in the original paper, and it therefore arbitrarily limits the possible fluctuations in surface height. More recent approaches allow the surface to fluctuate at will whtle retaining a fixed mean orientation. For example, if we tie at both ends a one-dimensional string of N atoms, and stipulate that steps of height no greater than 5 1 may occur, the RMS fluctuation is well known (random walk) to be N i l 2 . In two dimensions, a similar structure of N by N atoms with steps limited to k 1 in any direction has RMS fluctuation of log N. (In higher dimensions, the RMS fluctuation remains finite even as N -+ 00.) When the introduction of a step is made to cost energy, it is found that in the two-dimensional case there is a roughening transition; below T, the RMS fluctuation is finite as N -+ co, while above T, it becomes infinite. (The transition only occurs on a two-dimensional surface; a one-dimensional interface is always rough, three and more are never rough. We are lucky to live in such an interesting worid!) 3.1. ROUGHENING TEMPERATURES As we have pointed out in section 2.4 there exist three-dimensional models

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157

which allow the surface to fluctuate and are exactly soluble under some conditions. The solid-on-solid model (van Beijeren 1977) has been used to make definite predictions about the transition and its critical behaviour by Jayaprakash et al. (1983). Moreover, the problem is amenable to computer simulation using Monte-Carlo methods (Muller-Krumbhaar 1977). It is interesting to see how the experimental results compare with these theories. But we must remember that these theories are classical, and helium is quantal. The classical discussion we have used so far predicts that at absolute zero all the crystal surfaces will be faceted. Andreev and Parshin (1978), recalling the early observations that helium crystals are invariably rounded in equilibrium (a fact afterwards shown not to be true), introduced quantum considerations in order to explain this observation. They considered a surface close to a high-symmetry direction which has a structure of atomic ledges. Dents or bumps (“kinks” in general) in the ledges were treated as delocalized elementary excitations moving in the periodic atomic potential. The energies of such kinks then lie in a band which would have a width equal to the amplitude of the periodic potential and centred about the energy of an isolated free kink. They estimated these parameters as about 1 K and 0.1 K, respectively, although the basis of these estimates is not clear. Thus the lowest energies in the band are below zero, implying that the excitation of kinks is advantageous even at absolute zero. Therefore, it seems that the crystal would always be rough. Andreev (1982) has discussed this theory in greater detail, in the light of discoveries made after its publication. The ideas seem to be partially true because helium crystals still contain rounded parts at the lowest temperatures investigated but it does not apply generally. The conclusions of Andreev and Parshin were contested by Fisher and Weeks (1983), who described the fluctuations of a crystal surface by a superposition of surface wave excitations obeying Bose-Einstein statistics. They calculate the mean square fluctuation amplitude as a function of temperature for a surface of infinite size and observe that the integral so obtained converges to a finite value at T = 0; thus the surface is atomically smooth at that temperature. The result is not particularly dependent on the form of the dispersion relation a( k ) for surface waves, provided there is no energy gap around k = 0. Crystallization waves (section 4.2) are safe from this feature. Specifically, the surface fluctuation f = ( I h ( r ) I ’) is related to the amplitudes of the crystallization wave modes a ( k ) by

S.G. LIPSON AND E.POLTURAK

158

For the two-dimensional infinite system this becomes an integral,

The integral is dominated by small w as T -+ 0 and converges to a finite value at T = 0, suggesting a smooth surface at that temperature. Fradhn (1983) has obtained a similar result to Fisher and Weeks in a different way. He shows that, while in a classical system the spatial and momentum coordinates can be separated, and roughening discussed in terms of the former only, t h s is not possible in a quantum system because of the uncertainty principle. This makes the surface behave as if it were three dimensional at T = 0 and it therefore does not roughen. Lower bounds for the roughening temperature as a function of crystal parameters have been given by Avron et al. (1980), by Fisher and Weeks (1983) and by Jayaprakash et al. (1983). The first of the above obtain their results by considering certain 2-D Ising models which represent sections of the 3-D structure and therefore have different lattices and nearest-neighbour interactions. The latter two give formulae involving the surface stiffness and spacing between lattice planes, d , for an isotropic interface:

T, = 2d27/akB.

(22)

This result arises from a mapping of the surface fluctuation problem onto the 2-D X-Y model as opposed to the Ising model. Table 1 Comparison of experimental and theoretical roughening temperatures facet

index

plane spacing d

(Oool). f (1010). a (1011). s

(A)

2.1 1.5 2.4

observed T, ( K )

T, / T,( f ) (experimental)

1.2550.03 b.c 0.95 + 0.05 b.d 0.36 5 0.03

1 0.75 0.29

Theoretical relationships [d/d(c)12 a 1 0.29 0.79

T,/T,(C) 1 0.20 0.16

According to Fisher and Weeks (1983) [cf. eq. (22)] this factor multiplied by ( y.yyt,)''2 is proportional to T,. But y is not isotropic (section 2.8) and insufficient data are available at present to compare the theory with experiment. The plane spacings d are between parallel, but not necessarily crystallographically equivalent, atomic planes. * Wolf et al. (1985). Babkin et al. (1984). Avron et al. (1980). Wolf et al. (1983a).

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159

Jayaprakash and Saam (1984) also introduce next-nearest-neighbour interactions and are able to predict roughening temperatures for a larger range of orientations in the fcc system. For example, when J2 = 0, T,(lOO) = 1.44 J l / k B ,but T,(111) = 0; introduction of J2 = 0.1 J1 adds T,(110) = 0.3 J l / k B . It is difficult to compare these figures with experiments on helium, and one awaits calculations for the hcp structure. Table 1 compares the observed values with estimates based on nearestneighbour interactions only, 3.2. CRITICAL BEHAVIOUR

OF FACET SIZES AND SURFACE STIFFNESS

Next we can look at the critical behaviour of facet sizes near T,. These have been measured roughly by Avron et al. (1980), and more precisely by Keshishev et al. (1981), who showed that the facet size falls continuously to zero as the roughening transition is approached from below. Following Edwards (1983) we shall call the temperature at which facets disappear the “faceting transition” temperature, q. The distinction between TI and T, arises from the important role of the coherence length ,$ near a phase transition when applied to this problem (Jayaprakash and Saam 1984). Using the continuum theory of fluctuations [see Landau and Lifshitz (1959), Ch. 121, making appropriate modifications to apply it to the surface problem and ignoring gravitational effects, we can show that ,$ = y / a ( T ) and the width of the power spectrum of fluctuations is 2n/& As o ( T ) + 0 at T,, 5 diverges as expected at a phase transition. It remains infinite at T > T,. The interpretation is that fluctuations of the surface about its mean position are correlated only at pairs of points separated by less than 6. This means that in order to determine the mean orientation of the surface it is necessary to take a sample having dimensions much larger than 5; in particular, a facet with equilibrium dimension L ,< ,$ cannot be resolved. As T - , T, from below, L shrinks [eq. (5)] and ,$ diverges, and the two intersect when they are both equal to (R,a)’/*; this condition defines the temperature q. The exact relationship ,$(T) and hence T, are model dependent; for example, in the SOS model (case (c) below), on a crystal of radius R , = 0.1 cm, a facet will disappear when its cm, at t = (T, - T ) / T , between 0.02 and 0.08. These size L = 5 X considerations are important when we are interpreting experiments on crystal morphologies carried out close to T,. The structure of the surface on a microscopic scale ( Q 6) is not described by the continuum theory. Here we recall the remarks at the beginning of section 3 about microscopic roughness, from which it can be seen that the RMS surface fluctuation fl/’ within the correlated area (,$/a X Z/a atoms) is of order a ln(,$/a).

160

S.G. LIPSON AND E. POLTURAK

The form of the critical behaviour can be deduced from the various models in terms of t : (a) The mean field calculations of Andreev (1981) are based on a Landau-Ginsburg expansion of the free energy using the facet size as an order parameter (cf. the analogy between this and magnetization, section 2.4) and give a facet size t"*. (b) The original model of Burton et al. (1951) follows the Ising model and leads to a facet size t In(t) (McCoy and Wu 1973). (c) The microscopic models whch derive from the X-Y model all give , which C is of the order of 1-2. critical behaviour exp( - C / t * / * ) in (d) Computer simulations based on the discrete Gaussian model (Miiller-Krumbhaar 1977) also show the above behaviour. It seems that the facet size falls to zero in two stages. The lower stage would correspond to the faceting transition while the upper one exhibits critical behaviour similar to the SOS model (Muller-Krumbhaar, private communication). Babkin et al. (1984, 1985) have carried out experiments in which they analyze the facet size and the way in which the facets join on to the rounded part of large crystals. They show that the junction is of second order (section 2.5), but claim that none of the above forms of critical behaviour fit their measurements. However, we shall see (section 4.8) that studies of crystal growth mechanisms favour (c) above. Another prediction of the SOS and other microscopic models is that the surface stiffness J has no singular behaviour on going through the roughening transition, and as a result there is a jump in curvature of the crystal at the transition, from the Young-Laplace value [eq. (7)] above T, to the value zero below it. In contrast, Andreev's (1981) mean field theory predicts that J should diverge as the transition is approached from above, so that the curvature goes continuously to zero. The jump in curvature at T, has been demonstrated experimentally by both Babkin et al. (1984) and Wolf et al. (1985). It is interesting to compare the predicted forms for y ( h ) around the cusp as a function of T in the region of T,(fig. 18). When interactions between kinks are taken into account, Cabrera and Garcia (1982) and Cabrera et al. (1984) write the angular dependence y(S) as

-

-

in whch /3, = o/a. It follows that if P2 > 0 (repulsion between neighbouring steps) the joint between a facet and a rounded surface is second order, whereas if p2 < 0 the joint is first order (see section 2.5). Following through the Wulff construction we find that the curvature of the surface goes

THE SURFACE OF HELIUM CRYSTALS

161

I

*

I

b

I

I

Fig. 18. Evolution of y ( R ) around a high-symmetry direction, as T rises through q.In (a) the cusp angle increases with T , and becomes 180° at T,, at which point the minimum in y has disappeared. In (b) the minimum remains at all T; it is a cusp below T,, and rounded above it. WulfPs construction from (a) results in a jump in curvature at T,; (b) results in a continuous change in curvature on going through T,.

continuously to zero at T, in the repulsive case, and discontinuously to zero in the attractive one. These authors have made a careful analysis of the data published by Landau et al. (1980) for one particular temperature and have come to the conclusion that & < O for helium, but we feel that their analysis places unjustifiable reliance on one set of data. 4. The growth of helium crystals

We have seen that, depending on temperature and orientation, the surface of helium crystals can be either atomically rough or smooth. The two types of surface grow in completely different manners. However, as a facet approaches its roughening temperature it becomes progressively rougher, while still remaining macroscopically smooth, and its growth behaviour changes dramatically as a result. Experiments on the growth of rough surfaces have been reported by Leiderer (1984), Bodensohn and Leiderer (1984), Keshishev et al. (1981) and Castaing et al. (1980), while Wolf et al. (1984,1985) have investigated facet growth. 4.1. ROUGHSURFACES: THEORY

OF THE GROWTH RESISTANCE

Suppose that the solid is crystallizing from the fluid, causing the interface to move at velocity u, and that a heat flux JE is flowing through the interface. Castaing and Nozibres (1980) write down a phenomenological relationship between the matter and energy fluxes, J = ups and JE, and the temperature and chemical potential differences, 6T and 6p, between the

S.G. LIPSON AND E. POLTURAK

162

two phases. They assume linear dependence and obtain a pair of Onsager relations:

~ T / = T b~~ + CJ,. We can define the growth coefficient K

(25)

= u / m 4 6p when

6T = 0 and get:

In addition the Kapitza resistance (sections 4.5, 4.6) R , , which relates heat flow to 6T when the surface is stationary ( J = 0), is given by RK'

(27)

=cTi,

and the cross coefficient b tells us how the latent heat of melting is divided between the two sides of the interface (Edwards 1983). Andreev and Parshin (1978) [see also Andreev (1982)] suggested the growth coefficient to be limited by the interaction between surface steps and elementary excitations of the fluid, phonons and rotons, giving ( K m 4 ) - ' = AT4

+ B exp(-A/k,T),

(28)

where A is the roton excitation energy, and A and B are constants. Thus the growth rate becomes infinitely large at absolute zero. These ideas were quantified by Andreev and Knizhnik (1982) and by Bowley and Edwards (1983), who analyzed in some detail the growth of a rough surface. They calculated the transfer of energy and momentum between the excitations in the liquid phase, longitudinal phonons 2nd rotons, and those in the solid, phonons of three different polarizations. In the ballistic regime (very long mean free path) Bowley and Edwards consider a frame of reference moving with the interface at velocity u . The excitation spectra, determined by equilibrium conditions in the lab frame remote from the interface, are not in equilibrium in the moving frame, and so reflection and transmission of phonons at the interface result in transfer of energy and momentum. Since transverse phonons do not propagate in a liquid, only longitudinal phonons can be transferred across the surface; these contribute mainly to R K. On the other hand, the reflected excitations, mainly transverse phonons and rotons, determine the growth coefficient. The suggestions of Andreev and Parshin (1978) are justified, even if the specific role of the kinks is not identified.

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163

Assuming the crystal to be isotropic (anisotropy is shown to make some 10% difference) Bowley and Edwards calculate at low T

where cL is the sound velocity in the liquid, cg that for longitudinal waves in the solid and c, for transverse waves. Since cJcL = 1.33 and c J c L = 0.67 in the isotropic model, transverse sound clearly dominates the growth resistance in the phonon regime. The roton term, dominant at high T, is:

where qo is the roton wavevector and S is the ratio of conversion of normal (group and phase velocity parallel) into anomalous (antiparallel) rotons, thought to be about 50%. These results agree reasonably with experiments carried out not only in the ballistic regime, but also at temperatures where the phonons should propagate hydrodynamically. The reason for this is not clear. If 3He atoms are present, they introduce an additional growth resistance

depending on whether T >< Tf,-, usual in Fermi liquid theory.

where the symbols have the meanings

4.2. CRYSTALLIZATION WAVES

The study of crystallization waves and their damping (Keshishev et al. 1979, 1981) was the first method used to determine the growth coefficient of the rough surfaces of helium crystals. This work has recently been reviewed by Parshin (1982) and Andreev (1982). Surface waves on a solid are generally of a type determined by the elastic restoring force of the solid (Rayleigh waves), whereas those on a liquid are either short-wavelength capillary waves, dominated by surface tension, or long-wavelength gravity waves. The only solid surface on which capillary waves seem to be possible is the atomically rough surface of hcp helium crystals in contact with the superfluid. In essence these waves are also really waves on the surface of the fluid, in that no motion occurs in the solid at all; it simply grows and melts in harmony with the changes in chemical

164

S.G. LIPSON A N D E. POLTURAK

Fig. 19. An example of a standing crystallizationwave on the rounded surface of a 4He crystal at 0.5 K (a single frame showing one complete cycle from a cine-film of the wave). Notice that the facet on the left does not participate in the wave motion.

potential dictated by surface tension and the curvature of the interface. The actual atomic motion takes place in a layer of liquid one wavelength thick in contact with the crystal and compensates for the difference Zip between the densities of the liquid and solid. The superfluid offers no viscous resistance to the wave motion; damping of the waves occurs because the growth rate of the crystal is not infinite, and therefore the surface motion lags behind the changes in chemical potential. An example of crystallization waves is shown in fig. 19; it looks as if the crystal is vibrating like a jelly, but in fact the rough surface is melting and freezing at about 200 Hz. Facets adjacent to the rough surface take no part in the vibrations. Crystallization waves were predicted by Andreev and Parshin (1978), who derived the dispersion relation and discussed the relationship between the wave damping and the growth coefficient K. (The waves had in fact been suggested earlier by Kuper (1976), who unfortunately only published h s work in the form of a conference abstract!) The waves were discovered experimentally by Keshishev et al. (1979) the following year. An extension of the dispersion relation to high frequencies by the inclusion of coupling to Rayleigh waves in the solid has been given by Uwaha and Baym (1982).

THE SURFACE OF HELIUM CRYSTALS

165

At low frequencies the dispersion relation o ( k ) was shown to be

This equation shows wave propagation of the normal capillary wave type when K is large (rough surfaces):

whereas in the limit K D = yKm4/ps:

+

0 the motion is diffusive with diffusion constant

In general, the first term on the right in eq. (32) represents damping of the waves. This has made it possible to measure both 7 and K for rough surfaces by measuring the velocity and attenuation of the waves.

Fig. 20. Data on the growth resistance ( K m , ) - ' for rough helium surfaces. The full line represents a fit to eq. (28) with A = 7.8 K. The trend of numerous data of Keshishev et al. (1981) is represented by X , that of Bodensohn and Leiderer (1984) by circles. The triangles and -k show data of Castaing et al. (1980) and Keshishev et al. after the phonon contribution has been subtracted (Castaing 1984). The broken line shows the roton contribution expected for A = 7.21 K, which is the roton energy at the melting curve (Bowley and Edwards 1983).

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S.G. LIPSON AND E. POLTURAK

The experiments of Keshishev et al. (1979, 1981) confirm the theory. They used a transparent cell (fig. 16), in which the waves were excited on a horizontal surface by applying a periodic electric field. A laser beam scanning the surface at glancing angle allowed the wavelength and amplitude to be measured and hence k and the imaginary part of a. The dispersion relation was confirmed and the growth coefficient measured from the decay rate. The range of temperature over which this could usefully be done was limited by a minimum observable damping at low temperatures and critical damping at high temperatures. Values of 7 were also measured, and a variation of more than a factor of two reported between various sets of data. This was attributed to anisotropy of the surface free energy (section 2.8), although no systematic variation with orientation was seen. No temperature variation of 7 was seen in the region 0.36 K to 1.0 K, but the variation in growth coefficient confirmed the prediction of Andreev and Parshm [Eq. (28) and fig. 201.

4.3. TRANSMISSION OF SOUND THROUGH

THE INTERFACE

The remarkable speed with which the rough surface can grow and melt also markedly affects the transmission of sound through the interface (Castaing and Nozi&es 1980). Density variations associated with the sound wave in the liquid can all but disappear at the interface if it can adjust its position fast enough to cancel any excess density by melting or growing solid. In addition, latent heat, if any, is continuously absorbed and released, creating temperature fluctuations which are propagated away as second sound. The conversion to second sound decreases the sound transmission even further. If the acoustic impedances of the solid and fluid are Z, and Z,, the transmission coefficient T is given by 7 =

2 z, z,+ z,+252,z;

(35)

Here $. is the effective surface mobility which is proportional to K. The transmission of 1 MHz sound through the interface between 0.4 and 1 K was measured by Castaing et al. (1980) using a pair of transducers mounted vertically above one another. Ths was done in order to ensure parallelism of the horizontal interface with the transducers and therefore to avoid spurious energy losses caused by refraction of the sound waves. The crystal direction was determined by measuring the anisotropic velocity of the sound when the cell was filled with solid. When part of the solid had been melted, and the interface was between the transducers, the position of

THE SURFACE OF HELIUM CRYSTALS

167

the interface was found by measuring the time of flight. A similar method was used by Dyumin et al. (1984) to measure the conversion from first to second sound in the region between 1.0 and 1.7 K, where latent heat is important: they used porous membrane transducers to generate both types of sound. The temperature dependence of the growth coefficient measured in these experiments agrees well with the theories. In the region investigated the dominant excitations are rotons, and the exponential dependence on A / k , T is shown in fig. 20.

4.4. EXPERIMENTS USING

CHARGED HELIUM SURFACES

The use of electrostatic deformation of a surface also allows measurement of the growth coefficient (Bodensohn and Leiderer 1984). In this method, a layer of charge from a field emission tip is trapped on the crystal surface (fig. 21). The force resulting from the application of an electric field then melts the surface, and when the field is turned off the crystal regrows. If the wavelength of the distortion is long, the regrowth is a response to a gravitational tip; if it is short, surface tension provides the restoring force. The rate of return to equilibrium, measured interferometrically, leads to the growth coefficient in the former case (fig. 20). It is interesting that, if the charge density on the surface is large enough, the effective surface tension becomes negative and the surface becomes unstable, spontaneously developing an irregular deformation pattern which has a rather well-defined spatial frequency spectrum (Leiderer 1984). This effect has been analyzed in some detail because of its application to electro-optic recording devices (Budd 1965) and Leiderer has used it to determine 7 for helium surfaces.

1

S

LU

I* IU ,-!a1

I

I

Fig. 21. Schematic diagram of the cell used by Bodensohn and Leiderer (1984) to investigate charges on the surface of helium crystals. S is the field-emission tip used to charge the surface.

S.G. LIPSON AND E. POLTURAK

168

4.5. THEORY OF THE

KAPITZA RESISTANCE OF HELIUM SURFACES

The acoustic mismatch between the two media gives rise to the well-known Kapitza resistance to heat transfer across the surface [see, e.g., Challis (1974)J which is usually proportional to T P 3 .The Kapitza resistance between solid and superfluid helium is modified by the extraordinary mobility of the rough interface. Apparently the melting-freezing process of the rough interface can take place efficiently on a time scale comparable to that of the period of thermal phonons, i.e. in about lo-” s at 1 K! Therefore one would expect a drastic decrease in the transmission of phonons through the interface, just as in the case of sound transmission. However, Marchenko and Parshin (1980) have suggested that when the surface is perturbed by very-short-wavelength phonons an additional 6 p due to surface stiffness will become important. Thermal phonon wavelengths may be in the range 10 to 100 A. Such phonons, obliquely incident on the surface, wrinkle it with a spatial period A comparable to their wavelength. A surface perturbation of amplitude 6 yields a radius of curvature of the order A2/6, which then gives rise to a pressure difference according to the Laplace-Young relation [eq. (7)]. It follows that higher-frequency phonons will then generate larger pressure amplitudes and have a higher transmission probability. dependence Marchenko and Parshin (1980) showed that this leads to a T 5 for the Kapitza resistance. Maris and Huber (1982) subsequently made a detailed calculation of this effect, involving averaging over all angles of incidence to obtain the frequency dependence of the transmission probability, and then over a thermal phonon distribution. The results are expected to be valid only at low temperatures, where phonons dominate. 4.6. EXPERIMENTS ON

THE KAPITZA RESISTANCE

Experiments on the Kapitza resistance of rough surfaces have been carried out by Huber and Maris (1982), by Puech et al. (1982) and by Wolf et al. (1983b). The last of these experiments was carried out in a similar type of apparatus to that used to measure the growth coefficient (see section 4.8), but in which a temperature difference across the surface was balanced against a chemical potential difference in the presence of a heat current. Huber and Mans used a pair of fast bolometers made from thin films of Aquadag painted onto insulating boards, one in each phase. Following a heat pulse in one of the phases, the temperature evolution of each bolometer was followed. From the time constant for recovery of equilibrium R , can be deduced. Below 0.2 K the results were in agreement with the predicted T - 5 dependence. An interesting “greenhouse effect” was also observed at temperatures below 0.15 K. When the liquid phonons are

THE SURFACE OF HELIUM CRYSTALS

169

ballistic, a pulse of heat emitted in the liquid does not thermalize, and the high-frequency phonons all enter the solid. After thermalizing in the solid the phonons, now having much lower frequencies, are trapped in the solid because of the frequency dependence of the transmission probability. So the solid ends up with a higher temperature than the liquid, where the pulse originated. Puech et al. (1982) measured the temperature difference between two thermometers located one above the other in their cell, in the presence of a heat flux. As the solid slowly grows from below the lower thermometer to above the upper one the temperature difference reflects the change in geometry; while the interface is between the two an additional temperature difference appears because of the Kapitza resistance. The experiment was also carried out as the solid melts. The difference between the resistance of the smooth surface during growth and the rough one during melting could then be seen. In addition, the crystal direction could be deduced from the anisotropy in heat conduction. This tells the whole story. R , for the smooth interface varies as T P 3 ,while the rough interface shows a T-’ dependence. The coefficient of T-5was found to be orientation dependent, and this variation was attributed by Mans and Huber (1982) to the dependence of the phonon transmission on the anisotropic surface stiffness (see section 2.3). The data are summarized in fig. 22.

,

t

*-O

0

-I

I

T

T - ~K( - ~ I Fig. 22. Orientation and temperature dependence of the Kapitza resistance of 4He surfaces (Puech et al. 1982). Closed triangles: faceted surface; open triangles: surface at 4 5 O to (0001); circles: 80° to (0001); squares: 30° to (0001). The line “M.P.” indicates the prediction of Marchenko and Parshin (1980).

170

S.G. LIPSON AND E. POLTURAK

The Kapitza resistance between solid 4He and liquid 3He-4He solutions has also been measured by Graf et al. (1984, 1985). This brings into play another channel of energy transfer, from phonons in the solid to the 3He quasiparticles in the liquid, and gives rise to another term depending on T - 7 / 2 ,which was observed in the experiment. The magnitude and temperature dependence of the total resistance indicated that the surface was rough, in contradiction to the visual observations of Landau et al. (1980), who observed stable facets on crystals grown from solutions. 4.7. GROWTH OF

FACETED SURFACES

The way that faceted surfaces grow is quite different from rough ones, and has been studied in great detail in many systems. The basic principles were first laid down in the classic paper of Burton et al. (1951) and are discussed in many texts (e.g. Woodruff (1973), who concentrates particularly on the liquid-solid interface). In general a facet can grow by two types of mechanism: (a) by two-dimensional nucleation, i.e., by growth at steps around ad-atoms and islands created statistically on the surface; (b) by means of Frank-Read sources, which are ledges joining pairs of screw dislocations of opposite sign penetrating the facet, or occasionally by means of a ledge radiating from a screw dislocation in the middle of a facet. In (a) the island must have a minimum size in order to grow, otherwise it prefers to m e k The minimum size depends on 6y and its free energy is given by (Weeks and Gilmer 1979):

The rate of growth resulting from such two-dimensional islands is found by summing their contributions and is found to depend on 8p as

This relationship is strongly non-linear. When growth is in this regime, measurements of K ( 8 p ) allow u to be determined independently of facet sizes (section 4.8). In (b) the growth results from the presence of dislocations and Burton et al. (1951) showed that the rate at which a facet advances is proportional to 6y2, but the proportionality constant depends on the ledge mobility. This brings us back to the concepts introduced by Andreev and Parshin (1978),

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171

which have been discussed in section 3: we expect the growth rate to become rapidly larger as T falls [eq. (28)j. 4.8. EXPERIMENTAL WORK ON FACET GROWTH Facet growth has been investigated in several crystals [see Woodruff (1973)l and gallium in particular has proved an ideal medium for studying dislocation-controlled growth. But the recent work of Wolf et al. (1985) on helium has shed quantitative light on all the aspects in a single reproducible medium, and in particular has illuminated the behaviour around a roughening transition, a region which has never been studied in any other material. In the system used by these workers (fig. 23), the chemical potential difference is applied gravitationally, by using the height difference between the surfaces of a large crystal acting as reservoir and of a small crystal under investigation, produced by extrusion as in fig. 13b. The surface being investigated is horizontal, an the rate at which it is growing is measured optically. The highlights of this exciting and important work are as follows. For c-facets, the rate of growth was found to obey the exponential law [eq. (37)], which allowed u to be deduced. This was done for temperatures above 1.14 K; below this, the rate of growth was too small to be measured. The approach to the roughening temperature was heralded by a decrease in u as predicted by the X- Y model, but apparently in disagreement with the other models (see section 3.2). Fitting to this model allowed the thermodynamic roughening temperature to be determined as 1.28 f 0.02 K for the facet. In fig. 24 we show the exponential growth law and in fig. 25 the way in which u approached zero in comparison with the X-Y model. However, we growing crystal \

transparent box

.window

Fig. 23. Schematic diagram of the experimental cell used by Wolf et al. (1985) to study facet growth. The height difference H gives 6p. After Wolf (1985).

S.G. LIPSON AND E. POLTURAK

172

Fig. 24. Growth rates for (OOO1) facets as a function of H . The growth coefficient K seen to be exponentially dependent on 8 p - l [eq. (37)] (Wolf et al. 1985).

= ii/6p

is

5 4 N

5 3

t

0

-

I

0

1.10

1.15

I .20 TEMPERATURE

1.25

1.30

(K )

Fig. 25. The critical dependence of the step energy a ( T ) near Tc, for (OOO1) facets. The lines show the fit to the SOS model for two values of T, (Wolf et al. 1985).

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173

should point out that the value of T, so deduced is very dependent on the applicability of this particular model, although the present authors have indeed observed faceted growth up to about 1.26 K. In addition Wolf et al. were able to confirm the discontinuity in surface curvature (section 3.2) at the transition; the fact that it was necessary to measure a finite portion of the crystal surface, subtending between 0.3 and 0.5 rad, in order to determine the curvature meant that the jump only became apparent when the equilibrium facet size was large enough. This occurs at about 1.12 K. It is conceivable that this curvature jump could be the result of the rapid disappearance of anisotropy within a small temperature range above T,; however, at face value it is very convincing confirmation of the applicability of the X-Y model to the helium surface. Observations of the crystal shape just below T, allow y to be measured. It was confirmed that it does not go to zero a; T, and that its value, 0.242 A 0.015 erg cm-’, is consistent with the lower bound to T, [eq.(22)], which from 7 is calculated to be 0.85 of the observed value. Notice that this value of 7 is rather different from the “average” values given in section 2.8, suggesting a fairly sizeable anisotropy in y. For a-facets both growth regimes were observed, two-dimensional nucleation being dominant from 0.5 K upwards and dislocation-dominated growth below this temperature. The roughening temperature for this facet can be estimated at between 0.9 and 1.0 K from the growth data (fig. 26). (See note added in proof.) A (

A

IOTO)Orientation

0

t

I %

c

->

0

W 0

2 . 1 4

a I I

I

0

I

0.2

I

I

0.4 0.6

I

0.8

I

1.0

I

1.2

T (K) Fig. 26. Growth rates for (lOi0)facets (Wolf et al. 1985).

174

S.G. LIPSON AND E. POLTURAK

4.9. GROWTH OF CRYSTALS FROM

DILUTE

3HE-4HE MIXTURES

In experiments using dilute mixtures Landau et al. (1980) and Balfour and Lipson (1984) have observed quite considerable differences between the forms of the crystals obtained and those of pure 4He grown under similar conditions. Hcp crystals near 0.5 K were observed to have large well-defined facets and did not show a tendency to rounding at the edges (fig. 27); the bcc crystals at higher temperatures were observed to grow dendritically (fig. 28). These phenomena have not been studied in any detail, mainly because much complementary information about the equilibrium properties of mixtures is not available. For example, their phase diagram is complicated (Tedrow and Lee 1969, Vvedenskii 1976) and only very recently have accurate details about dilute mixtures been published (Lopatik 1984); no measurements have yet been made of the surface tension of the interface between solid and fluid mixtures. This field still offers a large number of interesting unanswered questions. The observations of persistent facets on mixture crystals were first interpreted by Avron et al. (1980) as a result of the 3He in the superfluid damping the surface fluctuations of the crystal, thereby raising its roughening temperature. It is significant that the concentration of 3He at which mixture crystals started to differ from pure 4He crystals, about 10 ppm, corresponds roughly to the amount needed to create a monolayer of 3He on the crystal surface. But subsequent measurements of the Kapitza resistance of the interface [Graf et al. (1984, 1985); see section 4.61 suggested the interface to be rough at lower temperatures. From another point of view, in

Fig. 27. Crystals of 0.1%3He-4He mixtures showing stable facets near 0.6 K (Landau and Lipson. unpublished).

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175

Fig. 28. Dendrites of 0.4%bcc 'He-4He mixtures grown at 1.04 K. The crystals grow radially inwards from the cell wall as a pressure pulse is applied (Balfour and Lipson 1984).

their theory of the thermodynamics of the liquid-solid interface of mixtures, Castaing et al. (1982) analyzed an experiment on the heat capacity of dilute solid mixtures at high pressures by Hebral et al. (1981). This was intended to study a conjectural phase separation upon solidification into solid 4He containing bubbles of pure 3He (Greenberg et al. 1972). Castaing et al. suggested also here an interpretation in which a monolayer of 3He atoms be adsorbed on the solid surface, similar to that which has been shown to accumulate at the liquid-vapour interface of mixtures (Andreev 1966, Edwards and Saam 1978). Castaing et al. point out that at first sight this seems improbable, because the difference in density between the liquid and solid phases is too small to produce a potential well with even one bound state for a 3He atom. However, they claim that this argument does not take into account the structural difference between the two phases and the potential well may be stronger than anticipated, and they analyse the results of Landau et al. (1980) as a kind of type-I1 adsorption phenomenon (Dash 1977). Clearly there is much scope for new experiments in this field. Another interesting experiment (Lipson 1982) that has not yet been explained is the apparent formation of a surface shell on crystals grown from 0.7% 3He mixtures. The shell is the last part of the crystal to melt when the pressure is reduced (fig. 29). This cannot be explained by any phase diagram effect, since the last part to solidify will always be the closest

176

S.G. LIPSON AND E. POLTURAK

Fig. 29. Hcp crystals grown from a 0.7% 3He-4He mixture at 0.5 K. (a) Shell remaining as the last stage of melting; (b) the same crystal before melting began.

to the final melting pressure and therefore the first to melt. A possible explanation requires diffusion of 3He into a surface region of the solid to take place on a time scale similar to that of the crystal growth. In the bcc phase the formation of dendrites is prominent at high enough rates of growth [fig. 28; Balfour and Lipson (1984)], but their type and scale is very dependent on the temperature and concentration of the mixture. Dendritic growth has become a very active subject in recent years [see, for example, Langer (1980)] and today it is thought that the physics of the process is understood, including the factors controlling the choice of the pattern and symmetry of the dendrites, even if the exact solution of the resultant non-linear heat and mass flow equations is complicated. Essentially the crystal surface tries to maximize its area during growth in order to facilitate the rejection of the latent heat into the fluid phase, consistent with local satisfaction of growth conditions and surface equilibrium. In 3He- 4He fluid mixtures the thermal and mass diffusivities are strong functions of the temperature and He concentration, and the growth coefficient, surface tension and Kapitza resistance are all anisotropic, so it is not surprising that the dendrites obtained are quite varied. However, it seems that the basic problems in this field have now been solved and probably the helium system is too complicated to contribute much new information.

5. Substrate-induced phenomena The amount of influence that a substrate has upon a crystal in contact with it can conveniently be classified according to the degree of wetting of the

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177

substrate by the crystal. The practical importance of the wetting process is recognized by all low-temperature workers who wish their solder joints to be vacuum tight! Non-wetting implies that the extra energy associated with contact with the substrate is so high that the crystal avoids it completely; in that case the substrate has no effect on the crystal shape. In the complete wetting situation the substrate has a dramatic effect on the crystal shape, since the crystal will try to maximize its area of contact with it. An intermediate situation of partial wetting also occurs, in which the minimum free energy situation demands a contact angle (Y defined by [e.g. Woodruff (1973)l cos a = YfS - Yss , YCS

where the subscripts S, l and s refer to substrate, liquid and solid, respectively. A wetting transition may also occur. This is a situation in which the solid suddenly (first order) or continuously (second order) wets the substrate as the temperature is raised, and (Y changes in consequence. At the microscopic level, wetting properties can be used to understand the mode of growth of multilayer films, and the way in which a transition is made from an adsorbed surface solid to bulk material (Dash 1977). In the following we shall describe studies at both the microscopic and macroscopic scales; in the microscopic regime we shall restrict ourselves to multilayers since they possess many of the characteristics of the bulk solid, including the fact that their interface is with the liquid phase. Helium is particularly interesting in this respect because of the possible influence of the substrate on the roughening transition. When the surface of the adsorbed crystal is rough and undergoes macroscopic fluctuations, then the concept of distinct atomic layers becomes meaningless. On the other hand, one would expect the substrate somehow to damp fluctuations in a crystal surface only a few atomic spacings away from it, and therefore to raise T, in such films. Useful reviews of the possible behaviour of multilayers on a substrate are given by Pandit et al. (1982) and Wortis (1985), although, as in the crystal morphology studies (section 2.5), these are mean field theories and roughening is put in “by hand”. 5.1. SOLIDMULTILAYER GROWTH ON MATCHING CRYSTALLINE SUBSTRATES Solid 4He multilayers were grown on the hexagonal surface of graphite (Landau and Saam 1977, Wiechert et al. 1982), while solid ’He layers on cubic MgO have been investigated by Eckstein et al. (1980). Both of these systems exhibited uniform layer-by-layer growth, which corresponds to complete wetting of the substrate (Dash 1977). In the first of the above

178

S.G. LIPSON AND E. POLTURAK

experiments a fixed amount ( N atoms) of helium was sealed off in a cell containing clean exfoliated graphite (“Grafoil”). Plotting p ( T ) at constant N (isopycnal), Landau and Saam observed deviations from bulk behaviour just before reaching the melting curve, which they interpreted as solidification of the helium on the graphite. The thickness of the solid was calculated by using particle number conservation. Eckstein et al. carried out similar experiments with 3He on both MgO smoke and Grafoil, and found complete wetting on MgO but no wetting of Grafoil. The combined conclusion of these experiments was that wetting depends crucially on matching of the crystal symmetry of the substrate to that of the helium layer; hcp 4He wets the hexagonal face of graphite, and bcc 3He wets the cubic face of MgO, but not vice versa. Complementary information regarding the 4He-graphite system was obtained by Wiechert et al. (1982) in neutron scattering experiments. The influence of the graphite symmetry turned out to be so strong that the solid hcp structure grew even in the region of the phase diagram where the bcc structure is stable. The thickness of the solid layer was about 100 A when the melting curve in this region was reached, and only then did the bcc phase appear. Thls is indeed a long range for the Van der Waals potential, which is thought to have the value of 50 K/n3 at a distance of n atomic K at 100 A. Small as it layers (Vidali et al. 1979). This gives 2 X seems, this number is of the same order of magnitude as the difference between the ground state energies of the cubic (albeit fcc) and hexagonal structures of helium as estimated by Niebel and Venables (1974). However, we should note that, contrary to expectation, no growth of the hcp phase of ’He has been observed on graphite (Tiby et al. 1981) and so it appears that the last word on this subject has yet to be said. The finer details of the uniform mode of growth of 4He on Grafoil were investigated recently by Maynard et al. (1981), Gridin et al. (1984) and Polturak et al. (1985). A layer-by-layer mode of growth was detected by Maynard et al. in fourth-sound experiments in which Grafoil was used as the superleak. As the pressure rose from 5 bar towards the melting curve, the sound velocity showed an oscillatory behaviour (fig. 30) instead of the monotonic increase expected under isothermal conditions. Ramesh and Maynard (1982) and Adler et al. (1983) suggested that the sound pressure induces crystallization waves in the solid 4He layer on the surface of the Grafoil, and the strength of the coupling between the two waves depends on the degree of filling of the last atomic layer. The strongest coupling will occur when the layer is half full and the surface is roughest, and the weakest when it is complete. The type of roughening envisaged is more akin to the Burton et al. (1951) model in this case. As layer after layer of solid is deposited, the variable mode coupling causes the fourth-sound frequency to

THE SURFACE OF HELIUM CRYSTALS

179

go through successive minima. Additional, more direct information was obtained when adsorption experiments were carried out simultaneously with the fourth-sound measurements (Gridin et al. 1984). To measure the adsorption a novel differential technique was used in order to separate out of the total amount of helium added to the cell the very small part that solidifies on the graphite. The continuously rising adsorption isotherm can be seen (fig. 31) to be punctuated by periodic dips. A correlation between the fourth-sound data and the adsorption isotherms carried out by Polturak et al. (1985) showed that the oscillations of the sound velocity occur simultaneously with the dips. These dips were interpreted as phase transitions of the solid layer. Two types of transition take place. One occurs on completion of successive monolayers (de Oliveira and Griffiths 1978). Transitions of this type only appear in the data at 0.1 K, since at 1.3 K, above T, for the hexagonal face, the monolayers are indistinct. The second type occurs near the melting curve when the last unfilled layer is about 60% full. At this coverage Bretz et al. (1973) found the first 4He layer on graphite to order in an epitaxial 6 X J?; structure. Epitaxial regions like these in the phase diagram were predicted by Ebner et al. (1983), but it is surprising that such features occur after the completion of several atomic layers. Once again, the main open question regards the long-range effect of the substrate symmetry.

- 800 N

I

0 >-

I

1

I

f

v

z W

2

W 0 LL LL W 0

/s1^ I

:

0

.

,P*, t

;; V

.*-,;*

700

*.' .*

z a z

2W U

600-

I

I

I

I

I

I

Fig. 30. Resonance frequency near 1 K in a fourth-sound cell with a Grafoil superleak as a function of pressure, after Ramesh (1985). The solid thickness is estimated using a Franchetti relation (Ramesh and Maynard 1982).

180

8

1

S.G. LIPSON AND E.POLTURAK

-

7t

-I

31

e0

0

0

n

0

5

10 15 20 PRESSURE (bar)

25

30

Fig. 31. Adsorption isotherms for multilayers of 4He on Grafoil at 0.1 K (top) and 1.3 K (bottom). The curves are offset by 1 layer (Gridin et al. 1984).

Another substrate of fundamental importance is solid 4He itself. This situation arises at grain boundaries and stacking faults, where a contact angle can be observed between adjoining crystals. Franck et al. (1985) have shown in crystals grown at 15 K and 1.13 kbar that fluid 4He wets the crystal completely at grain boundaries between fcc crystals, whereas in the hcp phase at slightly lower pressure the fluid does not wet the crystal completely (fig. 32). In terms of surface energies 2Y(,C-,, < Y(fcc-fcc)~ 'Y(hcp-f)

< Y(hcp-hcp).

(39) (40)

In the case of superfluid in contact with a stacking fault, fig. 33, taken at 0.8 K (Lipson, unpublished), allows the contact angle [eq. (38)] to be estimated as 117", which can be compared with the value of 140" determined on other substrates (section 2.8).

THE SURFACE OF HELIUM CRYSTALS

181

Fig. 32. Crystals of fcc 4He at 2.8 kbar, showing complete wetting of grain boundaries by the fluid (Franck et al. 1983).

The reluctance of solid 4He to wet another 4He surface that is strained leads to a dramatic modification of the phase diagram for helium confined inside porous materials, such as compressed powders or Vycor glass. The internal surfaces of the pores, the matrix being amorphous, are covered with a highly strained monolayer of solid helium, which inhibits further

Fig. 33. A stacking fault separates the upper and lower crystals; (OOO1) vertical. The contact angle, marked by the arrow, is 117'.

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growth of solid at the interface with the liquid, On further compression, solid nuclei can be formed either attached to the disordered substrate (Dash 1982) or freely floating in the liquid (Lie-zhao et al. 1986). The pore size, which limits the diameter of the crystallites that can be formed, dictates the minimum overpressure A p necessary for solid to form. Nuclei of solid smaller than the critical radius rc = 2 7 K / A p ( V t - V,) are unstable and will remelt; thus rc must be smaller than the pore size. For pore radius 20 A, typical of Vycor, A p is 20 bar, which is added to the melting pressure. The solidification of helium in porous materials has been investigated in DC flow experiments and torsional oscillator (AC flow) experiments, reviewed by Lie-zhao et al. (1986). Beamish et al. (1983) have used sound propagation, Shmoda et al. (1984) specific heat and NMR, and Adams et al. (1984) isochore measurements to investigate the same effects. All these experiments show a shift of the solidification curves of both 3He and 4He to higher pressures. Superfluidity in 4He in Vycor has also been observed up to 50 bar. 5.2. THEWETTING TRANSITION

Here we shall discuss the wetting process of macroscopic 4He crystals. The shape of a crystal on a substrate can be determined from the Wulff y - plat

Fig. 34. Wulffs construction of a crystal grown on a substrate oriented as shown (cf. fig. 2 for the free crystal). The effective surface tension in the direction of the substrate is y, = yss - yfs.

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construction if yss - yrs = ,y is known (Winterbottom 1967). It gives, in general, a truncated crystal (fig. 34) and the crystal shape will obviously change if ,y changes. From the figure it should be clear that for a faceted crystal the contact angle between facet and substrate does not characterize the degree of wetting as it does in a liquid. Measurements of the contact angle between solid 4He and polycrystalline substrates originally showed non-wetting behaviour (Balibar et al. 1979, Landau et al. 1980) in a vertical geometry. This was, in a way, rather surprising, since solid surfaces are generally considered to be covered with a few monolayers of solid helium at all pressures, and this situation would seem to provide ideal conditions for wetting. But Dash (1982) has suggested that crystalline mismatch energy can explain the reluctance of the few monolayers to grow further, and

Fig. 35. Stages during reversible continuous wetting of the copper cell peephery by a 4He crystal as T increases. In (a) the (1010) facets are visible; (b) is around T(1010) and (c) above T,. The (OOO1) axis is normal to the paper (Carmi, unpublished).

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therefore the type-I1 behaviour. Recently, a first observation of a transition from a non-wetting to a partially wetting behaviour of hcp 4He on copper was reported by Carmi et al. (1985) in a horizontal geometry. The experimental cell used for this experiment has already been shown in fig. 17. It seems that pressure fluctuations played a part in overcoming pinning by the substrate; the inhibition of the spreading of liquid drops by irregularities of the substrate has been discussed by de Gennes (1985). Experiments carried out on crystals with their c-axis normal to the substrate (fig. 35) show behaviour characteristic of continuous (secondorder) wetting, whereas those with the c- and a-axis in the plane of the substrate show first-order wetting. The interpretation of these results is at the moment very sketchy, since the theories (Pandit et al. 1982) do not treat the case of an anisotropic medium. Nightingale et al. (1983) have claimed that a system governed by Van der Waals forces cannot undergo a continuous transition, which does not agree with the observations; however, Dietrich and Schick (1985) have recently admitted the possibility of such a transition for a propitious combination of substrate and wetting medium. In conclusion, it seems that the wetting of copper by hcp 4He crystals first demonstrated the dynamic macroscopic wetting of one solid by another, and the system allows the possibility of studying anisotropy effects. However, it is very clear that “innocent-bystander”-type walls are getting hard to find,

6. Conclusion In this review we have presented a range of topics on the growth of crystals and the equilibrium properties of their surfaces which have been investigated in the helium system. Many of the investigations parallel studies made in other systems, but in most cases helium has provided the experimenter with distinct advantages. In particular, questions about the existence and nature of the roughening transition, and its influence on crystal properties, have recently been answered clearly in 4He. However, there are still many remaining puzzles; for example, what should the true equilibrium shape of a quantum crystal be at absolute zero? This has neither an experimental nor a theoretical answer yet. In addition, the surfaces of 3He and isotopic mixtures have received as yet little attention, and open questions still remain regarding the interaction between a quantum crystal and a crystalline substrate. These subjects will almost certainly provide much amusement and some surprises in the future.

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Note added in proof (Jan. 1987) Since the completion of this article, we have learned of several new developments which deserve mention. First, Rolley et al. (1986) have observed a change in the growth shapes of bcc 3He crystals from rounded to faceted on the (110) surface at 80 mK, and have interpreted it as indicating a roughening transition on this surface. The transition temperature is consistent with the prediction of eq. (22) using an estimate of the interfacial surface tension recalculated from the results of Eckstein et al. (1980). In another experiment, Gallet et al. (1986) reexamined the growth of the c-facet of 4He in more detail in the critical region near T, using an improvement of the technique used by Wolf et al. (1985). They showed that growth of the crystal causes a dynamic broadening of the transition, supporting the picture that the roughening transition is of infinite order. Finally a very useful review of the theoretical models pertinent to the roughening transition has been published by van Beijeren (1986).

Acknowledgements We should like to acknowledge stimulating discussions with J.E. Avron, S. Fishman and P. Nozihres during preparation of this review. We also thank J.E. Avron, C.G. Kuper, Y. Eckstein and S. Balibar for constructive criticism of the manuscript. We are grateful for support of this research by the US-Israel Binational Science Founcation, and the Fund for Basic Research administered by the Israel National Academy of Sciences and Humanities.

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