Faceting on ³He crystals

Abstract

It has been predicted by Landau that, ideally at low temperatures, crystals should show many different types of facets, i.e., flat smooth faces on their surface, but this so-called “devil's staircase” phenomenon has been difficult to observe experimentally. In this paper we describe our recent experiments, in which altogether 11 different types of facets have been identified on growing ³He crystals at the temperature of 0.55 mK by using a unique low-temperature Fabry–Pérot interferometer. Previously only 3 types of facets had been seen in this system. We have also measured the growth velocities of different facets, and our interpretation of the obtained results yields the conclusion that ³He has much stronger coupling of the liquid–solid interface to the crystal lattice than has been expected. After an introduction we present a short theoretical background about the equilibrium crystal shape and the roughening transitions, which is followed by the description of our experimental results and discussion.

The crystals that we find in nature have usually a polyhedral shape. Their surface is covered with smooth faces or facets that correspond to high-symmetry crystallographic orientations (see Fig. 1). These facets have developed on the crystal surfaces during growth when the crystals were formed. It must be pointed out that it is difficult to study the growth dynamics as well as the equilibrium shape of ordinary crystals. The difficulties are caused by big differences in entropy and density between the solid and the adjacent melt or vapor phases and finite thermal conductivities of these bulk phases. As a result, the relaxation processes are highly dissipative and the corresponding time scales extremely long. The equilibrium shapes have therefore been obtained with microscopic metallic crystals which after annealing of several days have showed facets surrounded by rough (rounded) areas on their surface (1).

Figure 1.

Computer-generated shape of a body-centered cubic (bcc) crystal is presented together with an elementary patch of a crystal habit where the facets are labeled with Miller indices. The depicted facets are of the types observed in our experiments with ³He crystals. On the elementary patch, the facets that were expected to be seen, as explained in the text, are marked with empty circles. The diameters of the circles are proportional to the interplanar distances of corresponding orientations.

It was predicted by Landau in the middle of the last century that, in the limit of low temperatures, any ideal crystal should be completely faceted, with an infinite number of facets on its surface (2). This is the so-called “devil's staircase” phenomenon (3, 4). The most complicated crystals, however, have shown at most six types of facets. Only very recently Pieranski and his coworkers (5) were able to grow lyotropic monocrystals that revealed more than 60 types of facets on their surface, confirming experimentally the devil's staircase phenomenon. They interpreted their results as the coincidence of a quite large surface tension and interplanar distance with an exceptionally low elastic modulus. But with liquid lyotropic crystals, faceting can be studied in a very limited temperature range and it is hard to measure their equilibrium shape.

The experimental situation is more promising in helium crystals, which exist at low temperatures and high pressures. The dynamics of helium crystals, which are surrounded by a superfluid with high thermal conductivity, can be extremely fast. For instance, ⁴He crystals have been observed to relax to their equilibrium shape within a fraction of second at temperatures below 1.5 K. In ⁴He also, a phenomenon not observable in ordinary systems, the crystallization waves, has been found (6). These waves are periodic melting–freezing waves on the superfluid–solid interface that are characterized by a very small dissipation.

In addition to fast dynamics, the helium crystals allow studies in a wide temperature range along their melting curve down to absolute zero. Helium crystals have also an advantage in that they are a very clean system because all impurities have frozen out at low temperatures; this means that purely interfacial phenomena, such as faceting, can be relatively easily studied.

Together with their successful prediction of crystallization waves in helium, it was proposed by Andreev and Parshin (7) in 1978 that when the temperature goes down to absolute zero, the helium crystals would be only partly faceted or even completely rounded because of quantum delocalization of steps and kinks. Soon after that, c facets [the (0001) orientation in the hexagonal crystal lattice] were observed on ⁴He crystals at slightly above 1 K by Landau et al. (8). Afterward, Fisher and Weeks (9) argued that the equilibrium shape of all crystals (classical or quantum) is faceted at sufficiently low temperatures (see also ref. 10). Later, two more types of facets were found on ⁴He crystals at T = 0.9 K and at 0.36 K (for a review see ref. 11). Despite the general belief that more and more different types of facets will show up when cooling the helium crystals to lower and lower temperatures, no more facets were seen on ⁴He crystals down to 2 mK (12).

Although the shape of ⁴He crystals has been an object of visual observations for almost 40 years, the crystals of ³He have been optically much less studied. This lack of study is, first of all, because of the fact that the superfluid transition of ³He takes place at about three orders of magnitude lower temperatures than the λ-point of ⁴He (T_λ = 1.76 K at the melting pressure), see the phase diagram of ³He in Fig. 2. In addition, solid ³He has a quite large latent heat of crystallization even at millikelvin temperatures. The result is that ³He crystals behave as ordinary crystals in this temperature range. However, below the Néel temperature, T_N = 0.93 mK (13), the nuclear spins of solid ³He order into an antiferromagnetic state, and the latent heat decreases as T⁴. Thus, well below 1 mK, the dynamics of the superfluid–solid interface of ³He is expected to become similar to that of ⁴He at much higher temperatures.

Figure 2.

Phase diagram of ³He. (One bar = 100 kPa.)

Before our experiments on ³He crystals, only three different types of facets had been observed, as in ⁴He. First, facets of the (110) type were detected at about 80 mK by Rolley et al. (14), and 10 years later two more types of facets, (100) and (211), were observed on body-centered cubic (bcc) ³He crystals by Wagner et al. (15) with a new type of optical cryostat that had a cooled charge-coupled device chip in addition to the light source (a red light-emitting diode) installed inside the cryostat.

In this paper, the interferometric experiments in which we have been able to identify 11 different types of facets on growing ³He crystals are described (16). We also measured the growth velocities of almost all observed facets (17). Before our measurements, only an average growth rate of ³He crystals had been reported (18–20).

Equilibrium Crystal Shape and Roughening Transitions

The equilibrium shape of a crystal is determined by the free energy per unit area of the interface between the solid and the liquid or vapor phase, i.e., the surface tension (21). If the surface tension would be isotropic, then the interface would be a perfect sphere in the absence of gravity. However, crystals have a periodic lattice structure that results in an anisotropy in the surface tension. It was first shown by Landau (2) that the surface tension of a crystal has cusps for the main crystallographic orientations and because of that smooth plane areas (facets) are present on the crystal surface. The planes that are most likely to appear as external faces are the ones with the largest step height, which is directly proportional to the interplanar spacing d of a given type of plane. In the bcc lattice d_hkl = [1/2]a(h² + k² + l²)^−1/2, where a is the lattice constant and 〈hkl〉 is the reciprocal lattice vector.

Faceting can be observed only if the temperature is low enough; thermal fluctuations do not wash out then the cusps in the surface tension. Each type of facet has its own roughening transition temperature, T_R. According to Fisher and Weeks (9), the lower limit for T_R is given by

1

where k_B is the Boltzmann constant and γ is the surface stiffness for a given surface (the surface stiffness is the sum of the surface tension and its second derivative with respect to the surface orientation). Above the roughening transition temperature, the surface of a crystal is rough. The actual temperatures for roughening transitions have been measured in ⁴He crystals, and the detailed study, which is reviewed in ref. 22, shows good agreement and strong support to Eq. 1 for the c facet.

In ³He, the lower limit for roughening transition has been found to be about 100 mK for the (110) facets, which have the largest interplanar distance in the bcc lattice (23). It has to be noted from Eq. 1 that in ³He the roughening transitions of all observed facets fall into the range well above the superfluid transition temperature, where the dynamics of the liquid–solid interface is very slow, as in ordinary crystals. The roughening transitions of the higher-order facets are expected to be at lower temperatures, but at the same time the lower the transition temperature the smaller the equilibrium size of the facets (see Discussion). Thus the observation of a roughening transition is a really challenging task in ³He. Facets much larger than the equilibrium size, on the other hand, can be observed during growth, when crystals typically get completely faceted, because the growth is strongly anisotropic. The rough surfaces of crystals grow easily because of the abundance of sticking sites for extra atoms, but the smooth facets grow much more slowly, and thus rough surfaces between them will be “eaten away” by facets.

Results

In our experiments, single ³He crystals were grown in a compressional cell mounted inside a nuclear demagnetization cryostat. The cell has an optical part that consists of a cylindrical (16 mm diameter, 12 mm height) copper volume sealed off with two optical windows. Below and above the optical part of the cell there are two semitransparent mirrors forming a multiple-beam Fabry–Pérot interferometer (see Fig. 3), which operates at a temperature of about 10 mK and is probably the coldest multiple-beam interferometer ever made (24). The He–Ne laser light (λ = 632.8 nm) is guided into the cryostat through an optical fiber and is expanded to a beam 8 mm in diameter. Twenty-millisecond light pulses were used for imaging and an interference pattern (see interferograms in Fig. 4) is focused to a slow scan charge-coupled device sensor which is located inside the 4-K vacuum can. The vertical resolution of our interferometer is a few micrometers, whereas the horizontal resolution of about 15 μm is limited by the pixel size of the sensor. For further details of our optical setup, see ref. 25.

Figure 3.

Principal scheme of our Fabry–Pérot interferometer. M50 and M70 are mirrors with 50% and 70% reflectivities, respectively. The figure has been modified from ref. 24.

Figure 4.

The sequence of interferograms of a growing ³He crystal at T = 0.55 mK. The dashed white lines outline the identified facets marked with Miller indices.

First, we cooled liquid ³He well below 1 mK and then the pressure of ³He was slowly increased above the melting curve value by applying a small steady flow of ⁴He from room temperature to the compressional cell. At the same time a high voltage was applied to a nucleator (a sharp tungsten tip just outside the visible area of the cell) to nucleate the crystal in the field of view. After nucleation the crystals were grown to a suitable size, typically several millimeters.

In Fig. 4, a sequence of interferograms is shown which has been taken after a ³He crystal was first melted (to get a rounded shape) before a slow and steady compression of ³He was started. During growth every 4 s an interferogram was taken. In Fig. 4a, which was taken soon after starting the growth, one can see that the interference fringes are already not nicely rounded, and there are facets evolving on the crystal surface that show up later as areas with equidistant parallel straight lines. At the very initial stage the types of facets cannot be specified because we need at least three fringes to identify a facet.

Later on, as in Fig. 4b, several facets can be identified. During the growth process slower facets increase in size at the expense of faster ones, which finally disappear from the crystal surface. This exchange can be seen from Fig. 4 b–f where the (110) facet on the right side of the crystal continuously increases throughout the whole sequence whereas, for instance, the (510) facet on the left is at the same time shrinking in size and is not visible in the last interferogram anymore.

The identification of facets, i.e., the comparison of the measured angles between the facets with the theoretically possible ones for the perfect bcc structure, was started from the last interferogram of the sequence in which there are only a few large facets present (see Fig. 4f). When all facets were identified, the preceding interferogram was taken and so on backward toward the beginning of the growth sequence.

In our interferograms the fringes correspond to multiples of λ/2Δn in crystal thickness, where λ is the laser wavelength and Δn is the difference in the refractive indices of the solid and liquid ³He. At temperatures below 1 mK, this corresponding thickness difference is about 190 μm. Determining the positions of fringes by the intensity-based analysis methods (26, 27) allows us to reconstruct the crystal profile that is equivalent to the thickness plot when the lower part of the crystal is a single flat surface.

In our experiments, we have identified altogether 11 different types of facets. Fig. 1 shows a computer-generated picture of a crystal (the upper part) with all observed facet types together with an elementary patch, in which also the facets are marked that were not observed but have a higher (or equal) interplanar distance than the highest order facet we did observe, the (311) facet. The sequence of observed facets from the (100) to (110), namely the (210), (310), and (510), forms the precursor to the devil's staircase.

During the sequence, partly presented in Fig. 4, we have also been able to measure the growth velocities of individual facets. The whole growth series took about 25 min, during which the rate of the volume change of our compressional cell was increased several times to obtain data for different growth rates. The results are presented in Fig. 5, where the velocities of different facets v are plotted versus driving overpressure δp, which is the pressure difference between the actual pressure and the equilibrium melting pressure. As we observed two different (210) facets to grow with somewhat different rates, there are two data sets for this type of facet.

Figure 5.

Anisotropy of ³He crystal growth at T = 0.55 mK. The dashed lines are the linear fits to the data for different facets marked with Miller indices. The figure has been adapted from ref. 17.

Discussion

From Fig. 5 one can see that the measured growth anisotropy is rather strong in ³He crystals. The velocities of the (110) and (510) facets, for instance, differ by about 1 order of magnitude. The obtained growth velocities have linear dependence on the applied overpressure, which suggests that the mechanism is spiral growth in the regime of the so-called suppressed step mobility (1, 12):

2

where β is the step energy, d is the height of an elementary step on a facet, ρ_s and ρ_l are the densities of the solid and liquid phases, respectively, and K is the number of steps produced by one dislocation. The critical velocity v_c in Eq. 2 is the velocity above which the step mobility suddenly drops down. The lowest critical velocities in ³He are the magnon velocity in the solid and the pair-breaking velocity in superfluid ³He, which, at low magnetic fields, are both about 7 cm/s (28, 29).

We used Eq. 2 and linear fits to the data in Fig. 5 to calculate the step energies β of different facets, and these values are plotted in Fig. 6 versus the interplanar distance d. We have assumed that one dislocation produces a single step, i.e., K = 1 in Eq. 2 because in our experiments the most stable facets have been observed, which have the lowest growth velocities, hence also the minimum K value. The linear fit to the data [except of the (411) facet] in the log–log coordinates gives approximately a fourth power dependence of β versus d. It can be shown that this kind of dependence is a result of elastic interaction between steps which has r⁻² behavior (30). It is, therefore, surprising that the observed quartic power-law dependence extends not only to higher-order facets like (510), for which these calculations are supposed to be valid, but to the more closely packed lower-order facets as well.

Figure 6.

Step energies β of different facets on a ³He crystal at T = 0.55 mK. The linear fit has a slope of 3.95 ± 0.25. The figure has been adapted from ref. 17.

The energy of an elementary step on the (110) facet, β₁₁₀ = 6.6 ⋅ 10⁻¹⁰ erg/cm, is of the same order of magnitude as the value measured for the c facet on ⁴He crystals (31). Because the measured surface tension of ³He crystals, γ = 0.06 erg/cm² at T = (0.15 … 0.33) K (23), is about 4 times smaller than on ⁴He crystals (31), this yields the conclusion that the step width in ³He is also 4 times smaller than in ⁴He because the step width should be proportional to the surface stiffness according to refs. 32 and 33. Basically, it means that the coupling of the liquid–solid interface to the crystal lattice is stronger in ³He compared with ⁴He. So far it has been assumed that because of a larger zero-point motion, this coupling is weaker in ³He.

We want to note that we were unable to measure the equilibrium shape of the crystals (and the equilibrium size of the facets) in our experiments because we could not stabilize the pressure of ³He well enough. The equilibrium size of a certain facet should be on the order of βR/(dγ), where R is a characteristic size of the crystal (2). Using the surface stiffness value measured at high temperatures, the equilibrium size of (110) facets should be about 0.3 R on ³He crystals and the higher-order facets are expected to be accordingly smaller than that.

Conclusions

We have identified in our experiments altogether 11 different types of facets on growing ³He crystals compared with only 3 types observed earlier in this system. As the higher-order facets grow fast and have small equilibrium sizes, a very delicate choice of the growth regime is needed to make their observation possible. If the crystals would be grown so slowly that they have close to their equilibrium shape, the higher-order facets are probably too small to be observed with our present optical setup. This difficulty might be actually the reason why on ⁴He crystals only 3 types of facets have been seen so far.

We have measured also the growth anisotropy of ³He crystals and extracted the values for step energies. Our results lead therefore to the conclusion that the coupling of the liquid–solid interface to the crystal lattice is much stronger in ³He than expected.

Abbreviation

bcc

body-centered cubic