The surface of helium crystals

Helium crystals exhibit faceting as do ordinary crystals, but no other crystals can grow and melt sufficiently fast to make the propagation of crystallization waves possible at their surfaces. After nearly two decades of controversy, it is now generally accepted that helium crystals are model systems for the general study of crystal surfaces, but also exceptional in having unique quantum properties. This review, which summarizes 25 years of research on the surface of helium crystals, treats both what is general and what is particular to helium. A central issue among the general properties is the “roughening transition,” the phase transition from a smooth faceted state of the crystal surface at low temperature to a rough and fluctuating state at high temperature. This review describes the series of experiments that have significantly improved our understanding of this transition and its related critical phenomena. Some attention is paid to the experimental techniques, which are rather unusual in many cases. The Nozières renormalization theory of roughening is also described in some details and compared with experiment. Other general properties of crystal surfaces have also been studied in helium, such as the energy of steps on the facets and their mutual interactions, and several instabilities. The relevant experiments are presented together with their theoretical interpretation. The quantum mechanisms that control the growth dynamics of helium crystals are also reviewed. Here, too, theory is compared with experiment, not only in the matter of crystallization waves, but more generally on the mass and heat flows in nonequilibrium situations, including the two stable helium isotopes ${}^4\mathrm{He}$ and ${}^3\mathrm{He}$, which behave in quite different ways. Finally, a list of open questions is presented for future research.

Figure 1

Crystallization waves in ${}^4\mathrm{He}$. Keshishev et al. discovered crystallization waves in 1979 by shaking their cryostat: the interface between a ${}^4\mathrm{He}$ crystal and superfluid ${}^4\mathrm{He}$ moves so easily by growth and melting that it looks like a free liquid surface. From 124.

Figure 2

(Color) Faceting of ${}^4\mathrm{He}$ crystals. As temperature goes down, more and more facets appear at the surface of ${}^4\mathrm{He}$ crystals. From top to bottom, the temperature is successively 1.4, 1, 0.4, and 0.1 K. The size of the facets is larger than on equilibrium shapes, due to the slow growing from the surrounding superfluid (see Fig. 20). The colors are real, obtained by Balibar, Guthmann, and Rolley (1994) with a prism, a lens, and a small mask (see Sec. 2B1).

Figure 3

(Color in online edition) Phase diagram of ${}^4\mathrm{He}$. There is no triple point where the liquid, solid, and gas phases would meet. On the melting curve at low temperature, solid ${}^4\mathrm{He}$ has a hcp structure; between 1.46 and 1.76 K, it is bcc.

Figure 4

(Color in online edition) Phase diagram of ${}^3\mathrm{He}$. As in ${}^4\mathrm{He}$, there is no liquid-solid-gas triple point. Two distinct superfluid phases exist below $T_c=2.5\mathrm{mK}$, and the bcc crystal is antiferromagnetic below $T_N=0.93\mathrm{mK}$.

Figure 5

Experimental cell used in Moscow. From 125.

Figure 6

Optical setup of the Paris group. From 187.

Figure 7

(Color in online edition) Leiden optical setup for studies of ${}^3\mathrm{He}$ crystals in a high magnetic field. From 225.

Figure 8

(Color) Principle of the color imaging technique used by Balibar, Guthmann, and Rolley in Paris (not to scale). In practice, the mask was a small black cylinder with a diameter of 2 mm. By moving this mask in the focal plane of the imaging lens, one could change the color of the background. Each pair of facets forms a helium prism whose angle determines the refraction and thus the color.

Figure 9

Interferometric setup of the Helsinki group for the studies of ${}^4\mathrm{He}$ crystals built inside the 4-K vacuum can of the nuclear demagnetization cryostat. From 190.

Figure 10

(Color in online edition) Compressional cell and a Fabry-Pérot type of interferometer for the studies of ${}^3\mathrm{He}$ crystals. From 211.

Figure 11

Sequence of photographs showing a ${}^4\mathrm{He}$ crystal falling from the top, where it nucleates, to the bottom of the cell. If its shape is a flat prism, the crystal flies like a sheet of paper and lands horizontal. From 33.

Figure 12

Rotating box in the Paris setup. Two micromotors (M1 and M2) allow rotation by $\pm 6$° around the $X$ and $Y$ axes. The motors are coupled to the box (B) and to the plate (C) thanks to flexible metallic wires (W) attached to screws (S). From 100.

Figure 13

(Color in online edition) Contact angle of the liquid-solid interface of ${}^4\mathrm{He}$. As shown by this photograph, taken by Balibar, Guthmann, and Rolley (1994), the contact angle of the liquid-solid interface of ${}^4\mathrm{He}$ is about 135°; the walls are preferably wet by the liquid phase (39). ${}^3\mathrm{He}$ crystals show a similar property (see Fig. 34).

Figure 14

(Color in online edition) Standing crystallization waves in ${}^4\mathrm{He}$. The photograph is from a video sequence taken in 1994 by Balibar, Guthmann, and Rolley and shows the experimental cell with the oscillating interface between the crystal in the lower part and the superfluid above it. The wave was excited with an interdigital capacitor on the tilted plate on the right and it resonated across the width of the cell. The excitation frequency was about 30 Hz, so that the wavelength was close to $2∕3$ of the cell width (24 mm).

Figure 15

Profile of a crystallization wave propagating at the surface of a ${}^4\mathrm{He}$ crystal, as measured by 186. In this particular case, $T=280\mathrm{mK}$, the surface was oriented 3° away from the (0001) plane, and the frequency was 1946 Hz, so that the wavelength was 0.660 mm. The recorded quantity is the local tilt angle of the crystal surface with respect to the horizontal plane. From 186.

Figure 16

Interference pattern of a charged interface between liquid and solid ${}^4\mathrm{He}$. Here the electrons are concentrated in the central region to enhance the visibility of the deformation. The parallel fringe pattern outside the center results from a small angle between the two interferometer plates. The field of vision is about 2 cm in diameter. From 194.

Figure 17

Charge-induced instability of a superfluid-hcp ${}^4\mathrm{He}$ interface, as observed by 193 above the critical electric field. The instability develops in the form of a lattice of dimples, where electrons accumulate (dark spots on the image). The diameter of this pattern is about 1.5 cm. From 193.

Figure 18

Horizontal $c$ facet at the surface of a ${}^4\mathrm{He}$ crystal as seen by Keshishev et al. From 125.

Figure 19

(Color in online edition) Unit cell of the hcp ${}^4\mathrm{He}$ crystals. The plane containing three atoms in the center of the unit cell has the same energy as the bottom and top (0001) planes, so that the step height is half the lattice period in the [0001] direction.

Figure 20

Growth and melting shapes of a crystal. Facets grow and melt more slowly than rough corners; as a consequence, facets are larger on growth shapes than on melting shapes.

Figure 21

(Color in online edition) Numerical simulations by 138 illustrating the basic physics of a roughening transition. The crystal has a simple cubic lattice and each atom is represented by a cube. At low temperature, there are very few defects such as adatoms, surface vacancies, steps, and terraces. As temperature increases, steps proliferate and the crystal surface loses reference to the lattice. The temperature is expressed as a function of the bond energy $J$. The roughening transition occurs at $T_R=0.632J$. From 138.

Figure 22

Experimental measurement by 241 of the growth characteristics of the $c$ facet in ${}^4\mathrm{He}$: semilog plot of the growth velocity $v$ divided by a departure of the height $H$ from the equilibrium position $(H=0)$, as a function of $1∕H$. The difference in chemical potential across the interface is $\delta \mu =(\delta \rho ∕{\rho }_C)gH$. Good agreement was found with Eq. (24), from which values of the step free energy were obtained. From 241.

Figure 23

Temperature variation of the step free energy $\beta$ for $c$ facets in ${}^4\mathrm{He}$. The symbols correspond to the successive experiments of 241 and 86. The solid line is the last fit to the RG theory of 157 that was done by 42 using the parameters of Eq. (25). From 42.

Figure 24

Growth characteristics of ${}^4\mathrm{He}$ crystals in the [0001] direction. As the temperature approaches the roughening temperature $T_{R1}=1.3\mathrm{K}$, the growth characteristics evolve smoothly from nonlinear to linear behavior. These measurements were made by 86 and were analyzed using the RG theory of dynamic roughening (see Fig. 25). From 86.

Figure 25

Measurements by 86 of the growth velocity of $c$ facets: 엯, height difference of 0.7 mm; +, height difference of 0.07 mm. The “reduced velocity” is the ratio of the velocity $v$ of the (0001) surface to the velocity $v_r$ of a typical rough surface at the same temperature (taken in another crystalline direction). The two solid lines are theoretical fits by 42, who used the theory of dynamic roughening by 157 and the parameters from Eq. (25). From 42.

Figure 26

Angular variation of the surface stiffness of hcp ${}^4\mathrm{He}$ crystals close to the [0001] direction: 엯, +, measurements by 33 at 1.2 K; ▵, measurements by 26 at 0.4 K; dashed line, measurements by 241 in the interval from 1.18 to 1.41 K [Eq. (23)]. Given the scatter in the data, the agreement is good with the result from the RG theory at 1.2 K [solid line, calculated by 42]. It confirms that the critical angular domain is small, about 2° or 3°, the only region where the surface feels the formation of the facet and where the stiffness has some temperature dependence. 42.

Figure 27

Set of measurements of the surface stiffness $\gamma$ of hcp ${}^4\mathrm{He}$ crystals in the $\left(10\overline{1}0\right)$ plane. The angle $\varphi =0$ corresponds to the (0001) plane and $\varphi =90°$ to the $\left(11\overline{2}0\right)$ plane. From 25.

Figure 28

General variation of the surface stiffness $\gamma$ of hcp ${}^4\mathrm{He}$ crystals, as measured from the dispersion relation of crystallization waves by 26 and 123. From 15.

Figure 29

Bending of a vicinal surface. It is easy to bend a vicinal surface in a plane parallel to the $c$ axis, but difficult in the plane perpendicular to it. For waves propagating with $q$ parallel to the projection of $c$, the relevant stiffness component ${\gamma }_{\Vert }$ is proportional to the interaction between steps and vanishes as the tilt angle $\varphi$ tends to zero. In contrast, the component ${\gamma }_{\perp }$ is proportional to the step energy $\beta$ and diverges as $1∕\varphi$.

Figure 30

No-crossing condition for steps. A crossing between steps (left) would induce a local overhang at the surface (cross section below). This is rather unlikely; thus the steps do not cross (right). The no-crossing condition for steps implies a reduction of the step entropy, hence a repulsive interaction.

Figure 31

Surface profile of a ${}^4\mathrm{He}$ crystal at 0.05 K (a) in a wide angular region $0⩽\Theta ⩽4.5\mathrm{mrad}$; the dashed line corresponds to the gravitational horizon and the arrow indicates the slope discontinuity, and (b) at small $\Theta$ (magnified view); the dashed line is an exponential fit (see text). From 31.

Figure 32

Adsorption isotherms of solid ${}^4\mathrm{He}$ on graphite. At low temperature, the coverage shows jumps, that is, first-order transitions, between successive layers indexed with the integer number $j$ by 180. The first-order transitions have critical end points at successive temperatures $T_c$ which depend on the number of atomic layers. From 180.

Figure 33

Extrapolation of the critical layering transition temperatures $T_c\left(n\right)$ in ${}^4\mathrm{He}$ as a function of the number $n$ of layers, more precisely ${\left(\ln n\right)}^{-2}$. According to 109 and 153, $T_c$ tends to the roughening temperature $T_R$ at infinite thickness. From 180.

Figure 34

${}^3\mathrm{He}$ surface: (a) ${}^3\mathrm{He}$ crystal in equilibrium at 320 mK; (b) a fit of the surface profile with the Laplace equation, which yielded the value of the surface tension $\alpha =0.06\mathrm{erg}∕{\mathrm{cm}}^2$ for ${}^3\mathrm{He}$ crystals. From 183.

Figure 35

${}^3\mathrm{He}$ crystals growing at 70 mK. The facets are the (110) surfaces. From 182.

Figure 36

Growing ${}^3\mathrm{He}$ crystal at 2.2 mK. Three types of facets, [(110), (100), and (211)], were identified from a comparison of this image with computer simulations. The (211) facets are barely visible in between some of the two neighboring (110) facets. From 230.

Figure 37

Interferogram of a growing ${}^3\mathrm{He}$ crystal at 0.55 mK, revealing seven different types of facets which are labeled with Miller indices. From 6.

Figure 38

Measurements by 125 of the dispersion of crystallization waves in ${}^4\mathrm{He}$. At high enough frequencies, a $3∕2$ power law was found, in a very good agreement with Eq. (38). Below about $20{\mathrm{cm}}^{-1}$, the slight deviation from the $3∕2$ power law indicates a crossover to gravitational waves. From 125.

Figure 39

Stiffness of the stepped surfaces of ${}^4\mathrm{He}$ crystals. As measured by 184 and 187, the stepped surfaces show a large anisotropy as their tilt angle $\varphi$ with respect to a high-symmetry plane tends to zero. One component of the surface stiffness tensor tends to zero while the other diverges, as predicted by Eqs. (28, 29). From 187.

Figure 40

Surface stiffness coefficient ${\gamma }_{\perp }$ as a function of the tilt angle $\varphi$. With this measurement, 185 verified that ${\gamma }_{\perp }$ is inversely proportional to $\varphi$, as predicted by Eq. (29); from the initial slope, they measured the step free energy $\beta ∕d=0.014\mathrm{erg}∕{\mathrm{cm}}^2$ for ultrapure ${}^4\mathrm{He}$ ($0.011\mathrm{erg}∕{\mathrm{cm}}^2$ with 130 ppb of ${}^3\mathrm{He}$ impurities). From 185.

Figure 41

Surface stiffness coefficient ${\gamma }_{\Vert }$ as a function of the tilt angle $\varphi$. 186 and 187 verified that ${\gamma }_{\Vert }$ vanishes as $\varphi$ tends to zero. They could not measure it at an angle small enough to verify the linear asymptotic behavior predicted by Eq. (28), but their measurements are consistent with Eqs. (30, 31) (solid lines). From 187.

Figure 42

Diagram of entropy conservation at the liquid-solid interface during growth, as represented by 40 following 159. From 40.

Figure 43

Contribution of rotons to the growth resistance $k^{-1}$ of solid ${}^4\mathrm{He}$ labeled ${\left(m_{4}K\right)}^{-1}$, as calculated by 46: +, experiments of 125; ×, experiments of 60; ●, ∎, ◆, experiments of 46; solid line, best fit with a simple exponential function [Eq. (59)]. The phonon contribution has been subtracted in the case of Keshishev’s measurements. From 46.

Figure 44

Transmission coefficient of thermal phonons through the liquid-solid interface of ${}^4\mathrm{He}$: filled symbols, measurements of 240; open symbols, measurements of 176; dashed curve, calculation of 75 using a temperature-independent surface inertia $m_I^0=2.1\times {10}^{-10}\mathrm{g}∕{\mathrm{cm}}^3$. A better fit (solid curve) was obtained with a temperature-dependent inertia $m_I=\gamma m_I^0$, where $\gamma =m_k\left(T\right)∕m_k^0$ is the kink mass ratio shown in the inset. From 75.

Figure 45

Transmission coefficient of acoustic waves in ${}^4\mathrm{He}$ as a function of the inverse temperature $1∕T$, at three different frequencies. These measurements by 10 show that, at low enough temperature, the mobility of rough crystal surfaces is limited by the surface inertia, so that the transmission is larger at higher frequency. From 10.

Figure 46

Surface inertia of ${}^4\mathrm{He}$ crystals as deduced by 10 from their sound transmission measurements. This graph shows the angular dependence of the inertia, which was found to be in a good agreement with the theory of 75 (solid line). The dashed line represents the theory of 133 in the limit of well-separated steps, that is, for a small tilt angle $\varphi$. The horizontal dotted line indicates the surface inertia of rough surfaces at large $\varphi$. From 10.

Figure 47

Setup used by 240 to measure the Onsager cross coefficient $B$ and the Kapitza resistance $R_K$ at the liquid-solid interface of ${}^4\mathrm{He}$. A heat current is applied at the bottom of a transparent box. It flows through the left side, which is open; the right side is closed by a superleak, which ensures thermodynamic equilibrium but does not allow any heat flow. The double horizontal gray line across the whole box is the liquid-solid interface outside the box. In this case ($J_E=0.503\mathrm{mW}∕{\mathrm{cm}}^2$ and $T=0.27\mathrm{K}$) it was about 2 mm below the levels inside the box. From the level changes on each side of the box, the pressure and the temperature of the crystal were measured, and $R_K$ and $B$ deduced. From 239.

Figure 48

Growth resistance of the vicinal surfaces of ${}^4\mathrm{He}$ crystals as a function of temperature and orientation, as measured by 187; the angles measure a tilt with respect to the $c$ facet. From 187.

Figure 49

Mobility of the vicinal surfaces of ${}^4\mathrm{He}$ crystals. Once the mobility of the vicinal surfaces is normalized by that of rough surfaces and plotted against $n_s∕q_{\mathrm{ph}}$, the step density divided by the phonon wave vector, all measurements collapse on a single curve. From 187.

Figure 50

Temperature dependence of the growth resistance of ${}^4\mathrm{He}$ crystals, for a vicinal surface tilted by 2° with respect to the $c$ facet. In the crossover, which corresponds to the phonon wavelength’s being comparable to the interstep distance, the growth resistance increases faster than $T^4$ (coherent scattering for rough surfaces or stepped surfaces at low temperature) and $T^3$ (incoherent scattering for stepped surfaces at higher temperature). From 187.

Figure 51

${}^3\mathrm{He}$ crystal relaxing towards its equilibrium shape. The time delay between the two photographs is 10 s. The temperature is close to $T_{\min }=320\mathrm{mK}$. From 95.

Figure 52

Inverse mobility of the liquid-solid interface of ${}^3\mathrm{He}$ near the melting curve minimum at 320 mK. The solid line is a fit with Eq. (73). From 93.

Figure 53

Measurements of the growth rates of ${}^3\mathrm{He}$ crystals below $T_N=0.93\mathrm{mK}$, the antiferromagnetic transition temperature of bulk solid ${}^3\mathrm{He}$. The solid line shows the $T^7$ dependence.

Figure 54

Spiral growth of a crystal induced by screw dislocations. Growth of (a) a single dislocation under a small excess pressure and (b) a Frank-Read source. At higher growth rates the dislocation winds around itself to produce a spiral as shown in (c) for a screw dislocation and (d) for a Frank-Read source. From 190.

Figure 55

Velocity of the $c$ facet on ${}^4\mathrm{He}$ crystals as a function of the driving pressure $\Delta p:엯,T=2\mathrm{mK}$; ●, 20 mK; ▿, 50 mK; ◆, 100 mK; ◇, 150 mK; ▵, 200 mK. Solid lines are guides to the eye. From 189.

Figure 56

Step mobility on ${}^4\mathrm{He}$ crystals as a function of temperature: 엯, results of 190, which follow the $T^{-3}$ law predicted by 159; ◻, results of 187, who measured a slightly larger mobility, probably because the phonon scattering was intermediate between coherent and incoherent in their experiment. From 190.

Figure 57

Step mobilities $\mu$ on the $c$ facets of ${}^4\mathrm{He}$ crystals calculated from measurements with different ${}^3\mathrm{He}$ concentrations: ◻, 40 ppm; ●, 110 ppm; ◇, 220 ppm; dashed line, data taken with regular ${}^4\mathrm{He}$ (0.1 ppm); solid lines, theoretical curves. From 213.

Figure 58

Step free energies of different facets at 0.55 mK, plotted versus corresponding step heights. From 212.

Figure 59

Pressure trace indicating a burstlike growth of ${}^4\mathrm{He}$ crystals. From 189.

Figure 60

Mechanical instability due to the appearance of stresses in a ${}^4\mathrm{He}$ crystal after a rapid quench in temperature around 1 K. The distance between parallel grooves which appear at the crystal surface is approximately $2\pi l_c\approx 6.3\mathrm{mm}$, the length of the scale bar on the image ($l_c$ is the capillary length). From 46.

Figure 61

Amount of melting produced when a uniaxial stress is applied to a ${}^4\mathrm{He}$ crystal in equilibrium with its superfluid phase; the experimental results (various symbols) are close to the prediction of Eq. (105) (solid line). An instability occurs at the threshold strain $u_c$ indicated by arrows. From 208.

Figure 62

${}^3\mathrm{He}$ dendrite growing at 100 mK with a tip velocity of $30\mu \mathrm{m}∕\mathrm{s}$. The cell width is 4 mm. Side branches appear at a distance of about 50 times the tip radius from this tip. From 184.

Figure 63

Magnetization at the liquid-solid interface: (a) Sketch of the magnetization profile of a planar liquid-solid interface melting with a constant speed, as analyzed by 175. On the solid side, the magnetization profile falls off exponentially and there is no gradient on the liquid side. (b) Sketch of the buildup of magnetization profiles as concluded by 1 at time $t=0$, when the melting is started (black line) and at three successive moments during the initial melting phase. While the boundary layer with increased magnetization is building up in the solid, a stabilizing gradient is also building up in the liquid. The magnetization of the bulk solid (at right) decreases due to the increase in temperature during the rapid melting experiment. Both pictures are drawn in a frame moving with the interface. From 1.

Figure 64

Sequence of images during rapid melting of solid ${}^3\mathrm{He}$ in an 8.9-T magnetic field. The instability occurs about 45 s after cell decompression was started: cellular dendrites with a size of $50–100\mu \mathrm{m}$ appear in the vertical direction. From 147.