Charge Motion in Solid Helium

Abstract

We overview and discuss several advances in the understanding of the motion of ions through solid helium. Positive and negative ions in solid helium are microscopic complexes with internal structures embedded into the host crystal lattice. Their low-field mobilities normally decrease with cooling in both bcc and hcp crystals of either isotope of helium ($^3$He and $^4$He). Depending on the density of solid helium (as well as, in the case of positive ions in hcp $^4$He, on the crystal orientation), the corresponding activation energies for the mobility of the two species of ions may or may not coincide, and they are found to be typically either equal to or about twice the vacancy creation energy. In strong electric fields, the field dependence of the drift velocity of ions is often nonlinear but monotonic. However, for positive ions in hcp $^4$He, non-monotonic anomalies in both temperature and field dependences of the drift velocity were observed. We discuss these features within the framework of the theory of ion motion via inelastic scattering of low-energy vacancions put forward by Alexander Andreev and co-workers.

1 Introduction

Quantum diffusion is a transport of particles, in which quantum-mechanical tunnelling is essential [1,2,3,4,5,6]. Its spectacular manifestation is the motion of $^3$He impurities through the matrix of solid $^4$He [7,8,9,10,11]. Macroscopic objects in solid helium, such as dislocations, are also believed to be moving by means of quantum tunnelling: dislocations in hcp $^4$He can glide in the basal plane with a virtually absent Peierls barrier [12] and climb with the help of delocalized vacancies (and even, perhaps, superclimb with the help of the quantum transport of atoms along dislocation cores [13,14,15]).

Ions in solid helium (see a review by Dahm [16] for the state of the problem in 1986, as well as his paper of 1998 [17]) are complexes of several crystal lattice parameters across. Unlike atomic impurities, they have an internal structure. The positive ion—a snowball [18]—consists of a molecular-ion core surrounded by a highly compressed, due to the induced-dipole attraction to the molecular ion, shell of helium atoms of radius $\sim 1$ nm. The negative ion—an electron bubble [19]—is an excess electron that has repelled surrounding helium atoms, thus localizing in a void of radius $\sim 1$ nm which decreases with increasing pressure.

Both types of ions played a vital role in the studies of excitations in superfluid $^3$He and $^4$He. In liquid helium, the atomic arrangements of either the snowball’s or electron bubble’s surface are largely irrelevant because the surrounding fluid is essentially structureless on the relevant length scale. This is different in solid helium, where details of the atomic arrangement of the ion in a particular crystal lattice (either bcc or hcp depending on the isotope and pressure) could strongly affect the mechanism of the ion’s motion. This consideration is borne out by the fact that ion mobilities were found to be very different in solid helium of different crystalline structures: for instance, in solid $^4$He of molar volume $V_\mathrm{{m}} \sim 21$ cm$^3$mol$^{-1}$, snowballs move much faster than electron bubbles when in hcp crystals, while in bcc crystals it is the other way around [20, 21].

Fig. 1

Temperature dependence of the diffusion coefficient in hcp $^4$He of molar volume $V_\mathrm{{m}} \approx 20.7$ cm$^3$ mol$^{-1}$ for: $^3$He impurities of concentration $x_3=0.75\%$ (open green diamonds) and $x_3=2.17\%$ (solid green diamonds) [8, 9]; positive ions (red circles) measured in diodes with gap $d=0.3$ mm ($\bullet$) [22,23,24] and $d=0.2$ mm ($\circ$) [25]; vacancies (blue asterisks) calculated from the rate of recovery of dislocations after cold working [26, 27]. Dashed line $D = D_0 \exp (-\varepsilon _\mathrm{{d}}/k_\mathrm{{B}}T)$ with $D_0=1.6\times 10^{-6}$ cm$^2$s$^{-1}$ and $\varepsilon _\mathrm{{d}} = 11.1$ K guides the eye

In Fig. 1, the temperature dependence of the diffusion coefficient is shown for three types of objects in hcp $^4$He: $^3$He impurities, positive ions, and vacancies responsible for the recovery of dislocations introduced by cold working. The Arrhenius-type temperature dependence¹$D = D_0 \exp (-\varepsilon _\mathrm{{d}}/k_\mathrm{{B}}T)$ is usually attributed to the vacancy-assisted mechanism of diffusion. For $^3$He impurities, the cross-over, at $T \sim 1$ K, to the temperature-independent diffusion coefficient $D \propto x_3^{-1}$ at lower temperatures is due to them behaving as a gas of delocalized quasiparticles. A discussion of the reasons—why D(T) for positive ions deviates, around $T \sim 0.7$ K, from the Arrhenius-type dependence—is one of the focuses of this paper.

2 Theories of Ion Motion

Early theories considered ions as macroscopic objects moving through solid helium under applied force.² Each event of absorption and emission of a vacancy by the ion results in its displacement by the distance $b \approx \frac{\Omega }{\pi R^2} \sim \frac{a^3}{\pi R^2} \sim 0.2$ Å, where $R\sim 10$ Å is the radius of the ion, $a \sim 4$ Å is the typical inter-atomic distance in solid helium and $\Omega \sim a^3$ is the vacancy volume [29,30,31,32,33,34,35,36,37,38,39,40,41,42]. Also, a possibility of the diffusion of adatoms along the inner surface of the electron bubble was suggested [43]. These models would predict a monotonic Arrhenius-type temperature dependence of the ion mobility as well as, in stronger fields E such that $eEb > k_\mathrm{{B}}T$, a monotonic super-linear field dependence of the drift velocity similar to $v(E) \propto \exp (eEb/k_\mathrm{{B}}T)$. Within the vacancy-assisted mechanism, the activation energy $\varepsilon$ for the ion mobility would not necessarily coincide with the energy of vacancy creation $\varepsilon _\mathrm{{v}}$: the induced-dipole repulsion of a vacancy from the ion results in an additional potential barrier while the effective attraction of a vacancy to the compressed region of solid helium in front of the ion contributes to the reduction of $\varepsilon$. While these simple types of dependence of the ion drift velocity on temperature and electric field have indeed been observed, anomalous non-monotonic deviations have been reported as well. Also, while only a smooth monotonic dependence of ion mobility on molar volume would be predicted by the above macroscopic models of ions, abrupt deviations from a smooth dependence on $V_\mathrm{{m}}$ have been observed. One hence needs to take into account the microscopic structures of both the ion and host lattice as well as the quantum nature of solid helium—for instance, when the existence of finite energy bands for the vacancion (a delocalized vacancy) and phonon excitations might lead to an anomalous behaviour.

Indeed, a vacancy in solid helium is expected to behave as a delocalized quasiparticle (vacancion) with energy within the band of $\Delta _\mathrm{{v}} \sim$ 5–10 K [44, 45]. At temperatures $T\sim$ 1–2 K the de Broglie wavelength of thermal vacancions would exceed the diameter of an ion; this was noticed by Andreev, Meyerovich and Savishchev who came up with the theory of ion motion via inelastic scattering of vacancions off ions within the general framework of scattering of slow particles off point objects [46,47,48,49,50,51,52,53,54].

Since an ion is an object with its own atomic structure, the elementary hop between two stable configurations, whether classical thermally activated or quantum, involves effective displacements of dozens of atoms. Such a complex would have a set of configurations of identical energies separated by vectors $\textbf{b}_i$ not necessarily coinciding with the set of elementary translations of the host lattice. A transition between these configurations would be facilitated by the inelastic scattering of one or more vacancions, with or without the involvement of phonons [46,47,48,49,50,51,52,53]. With either the one-vacancion or two-vacancion processes dominating, the effective activation energy $\varepsilon$ of ion motion would hence be equal to either $\varepsilon _\mathrm{{v}}$ or $2\varepsilon _\mathrm{{v}}$, respectively [54]. In strong electric field $\textbf{E}$, when the energy $e\textbf{E}\cdot \textbf{b}_i$ gained upon the hop in the direction of the field, exceeds the width of the vacancy energy band $\Delta _\mathrm{{v}}$, the mechanism would cease to work resulting in a sub-linear type $v \propto E^{1/2}$ when approaching the localizing field $E_\mathrm{{c}} \sim \Delta _\mathrm{{v}}/(eb_i)$ from below. The spontaneous emission of single phonons [46,47,48,49,50,51] could help lift this localization and was predicted to yield a super-linear contribution of form $v(E) \propto E^3$, but this would also work only in a limited range of field E due to the finite width of the phonon energy band $\sim k_\mathrm{{B}}\Theta _\mathrm{{D}}$, where the Debye temperature is $\Theta _\mathrm{{D}} \sim 30$ K. Thus, the theory of the vacancion scattering off an ion, developed by Andreev and co-workers, could explain a non-monotonic field dependence of the ion’s drift velocity v(E) as well as its anisotropy with respect to the crystal orientation.

Since an electron bubble can be thought of as consisting of an integer number of vacancies $N_-$, its competing configurations with different numbers $N_-$ but similar energies could exist at certain values of molar volume $V_\mathrm{{m}}$ [55, 56]—thus allowing for an additional mechanism for ion transport. This might manifest itself as a peculiar modulation of the ion mobility as a function of $V_\mathrm{{m}}$.

In an attempt to explain the non-monotonic temperature dependence of the mobility of positive ions in hcp $^4$He, Levchenko et al. [57] discussed the potential implications of the case of a narrow vacancy band $\Delta _\mathrm{{v}} \ll k_\mathrm{{B}}T$. Whether such a scenario is applicable to solid helium at molar volumes $V_\mathrm{{m}} \sim 20$ cm$^3$mol$^{-1}$ is yet unclear, because, according to modern calculations [44, 45], the vacancy energy bandwidth $\Delta _\mathrm{{v}}$ should be of order 5–10 K.

3 Experimental Techniques

The first observations of the motion of ions through solid helium were made by Shal’nikov [58,59,60,61]. These were followed by several groups using a range of techniques for generating ions and measuring their drift velocity [23,24,25, 43, 62,63,64,65,66,67,68,69,70,71,72,73].

In all our experiments [23,24,25, 74, 75], samples of solid helium were grown at constant pressure P between two flat electrodes, a distance d (in the range of 0.2–1 mm) apart. One electrode (source) was covered with tritium, whose $\beta$-electrons continuously generated both species of ion within a narrow layer of solid helium $\sim 10$ $\upmu$m thick. The space-charge-limited current to the opposite electrode (collector) was monitored as a function of time. A step-like application of a voltage U between the electrodes resulted in the transport of the chosen species of ions (depending on the sign of U) all the way to the collector. Before settling at its steady value, the collector current goes through a peak corresponding to the arrival at the collector of the leading front of ions that have been travelling from the source in the field $\gtrsim U/d$. Calculations show that the timing of the peak, $\tau (U)$, is related to the drift velocity of ions, v(E), in the field $E=U/d$ as $\tau = \zeta \frac{d}{v(U/d)}$, where $\zeta \sim 1$ ($\zeta = 0.8$ for the linear case $v \propto E$, and varies in the range 0.6–1.6 depending on the type of nonlinearity in v(E)) [17, 23, 24, 56, 74, 75]. In what follows, by the experimental drift velocity v(E) of ions in the diode in voltage U we mean $v_* \equiv d/\tau (U)$ in the field $E_* \equiv U/d$—with the understanding that it is systematically off the true value of v(E) by not more than $\sim 50 \%$ (usually better). The “box averaging” nature of the field $\gtrsim U/d$, in which the ion front travels, causes the apparent $v_*(E_*)$ to iron out sharp features, if any, of the true v(E) (and to locate them at slightly reduced values of $E_*$ as compared to true E).

In experiments by Dahm’s group [17, 43, 66], a diode with a radioactive source of ionization was also used. The majority of their experimental ion drift velocities have been obtained from the analysis of the dependences of the steady space-charge-limited collector current on the applied voltage.

In experiments by Keshishev’s group [72, 73], the injection of ions into solid helium was controlled by a grid above a radioactive source of ionization. This allowed to limit the density of injected ions (thus avoiding space-charge effects) and hence to determine their drift velocity to the collector in a uniform electric field $E=U/d$. So far only this group succeeded in measuring the ion drift velocity in single crystals of hcp $^4$He of known orientations of their $C_6$ axis with respect to the direction of the applied electric field $\textbf{E}$.

Similar to the case of superfluid helium, where both ion species can be trapped by quantum vortices, ions in solid helium are attracted to strained regions near dislocation cores. Like in many other insulators, the trapping of ions by crystal defects results in the static space charge that causes a reduction of the ion current for the given external field. In fact, this phenomenon can be used to characterize the density of defects in helium samples—as was done by Efimov and Mezhov-Deglin [67, 76, 77], and Gudenko and Tsymbalenko [68, 69]. Also, ions can be used as means of exerting force on crystal defects: Lau et al. [66, 78], and later Golov and Mezhov-Deglin [79, 80] observed periodic discharges of up to $\sim 10^7$ ions in strained samples of $^3$He  and $^4$He. This was interpreted as forcing, by an applied electric field, of a grain boundary (which was being charged continuously by the current of injected ions) to periodically move towards the collector electrode, where it would discharge and then elastically bounce back—thus completing a cycle of oscillations of the collector current.

4 Simple Cases

4.1 Weak Electric Fields, High Temperatures

In weak fields, typically $E<10^4$ V cm$^{-1}$, the ion drift velocity is linear in E: $\textbf{v} = \mu \textbf{E}$. From the melting temperature down, the Arrhenius dependence for the mobility

$$\begin{aligned} \mu (T) = \mu _0\exp (-\varepsilon /k_\mathrm{{B}}T) \end{aligned}$$

(1)

was observed for both ion species in both bcc and hcp lattices. It allowed characterization of the high-temperature mechanism of ion motion by the activation energy $\varepsilon$ and prefactor $\mu _0$.

The equilibrium concentration of thermal vacancies is $x_\mathrm{{v}} \propto \exp (-\varepsilon _v/k_\mathrm{{B}}T)$, with the vacancy creation energy in the tight-binding approximation following [1, 2],

$$\begin{aligned} \varepsilon _v \approx \varepsilon _{v0} - \frac{1}{2}\Delta _\mathrm{{v}}, \end{aligned}$$

(2)

where $\varepsilon _{v0}$ is the classical energy of Schottky vacancies and $\Delta _\mathrm{{v}}$ is the bandwidth of delocalized vacancies. Both terms $\varepsilon _{v0}(V_\mathrm{{m}})$ and $-\frac{1}{2}\Delta _\mathrm{{v}}(V_\mathrm{{m}})$ in the RHS increase with decreasing $V_\mathrm{{m}}$, hence the dependence $\varepsilon _\mathrm{{v}}(V_\mathrm{{m}})$ is monotonic. $\varepsilon _\mathrm{{v}}(V_\mathrm{{m}})$ was measured by various techniques [81,82,83,84] and also calculated theoretically [44, 85, 86]. In hcp $^4$He at the largest $V_\mathrm{{m}}=21.0$ cm$^3$ mol$^{-1}$, the bandwidth $\Delta _\mathrm{{v}}$ was calculated [44, 45] to be of order 5–10 K. Several experimental properties, relying on matter transport through solid helium, reveal the existence of an activation energy, monotonically increasing with decreasing $V_\mathrm{{m}}$: self-diffusion of $^3$He in bcc $^3$He [87, 88], diffusion of $^3$He impurities in hcp $^4$He [89,90,91,92], recovery of macroscopic strain via dislocation climb after cold working in $^4$He [26, 27, 93], etc.; these are usually associated with vacancies and can hence be used to estimate $\varepsilon _\mathrm{{v}}$.

Most experiments on ion motion in solid helium have been carried out with single crystals of unknown orientation. It turned out that the values of mobility, measured in samples of the same density, were reproducible for bcc crystals but revealed a range of values for positive ions in the hcp lattice. The latter has been traditionally attributed to the anisotropy of hcp crystals.

Fig. 2

Activation energies for various processes in BCC $^3$He. The black solid line guides the eye approximately through the values of vacancy creation energy $\varepsilon _\mathrm{{v}}(V_\mathrm{{m}})$ obtained via X-ray, heat capacity and self-diffusion measurements. The black dashed line is a factor of two higher, $2\varepsilon _\mathrm{{v}}(V_\mathrm{{m}})$. References to the legend: Efimov [74, 75], Marty [65], Simmons [81], Sample [84], Sullivan [88], Reich [87]

In Fig. 2, we show the available values of $\varepsilon$ for the mobility³ of positive and negative ions in bcc $^3$He along with the vacancy creation energy $\varepsilon _v$. It seems positive ions have $\varepsilon _+ \approx 2\varepsilon _\mathrm{{v}}$ in the whole explored range of $V_\mathrm{{m}}$ between 19.9 and 22.9 cm$^3$ mol$^{-1}$, indicating that two-vacancion scattering is likely the dominant mechanism of their transport here. The negative ions, on the other hand, only move via the two-vacancion mechanism ($\varepsilon _+ \approx 2\varepsilon _\mathrm{{v}}$) at molar volumes $V_\mathrm{{m}} \lesssim 21.3$ cm$^3$ mol$^{-1}$, while at larger $V_\mathrm{{m}}$ the one-vacancion scattering ($\varepsilon _+ \approx \varepsilon _\mathrm{{v}}$) seems dominant.

Fig. 3

Temperature dependence of the mobility of positive and negative ions in samples of hcp $^4$He of different orientations. Our data [23, 24] (measured at voltage $U=300$ V in a diode with $d=0.3$ mm): positive ions (red open circles) and negative ions (black open diamonds)—$\phi \approx 0$ at pressure 28.8 atm ($V_\mathrm{{m}}=20.75$ cm$^3$ mol$^{-1}$) and positive ions (green solid circles) and negative ions (black solid diamonds)—$\phi \approx 72^{\circ }$ (according to Eq. 4) at pressure 28.5 atm ($V_\mathrm{{m}}=20.73$ cm$^3$ mol$^{-1}$). For comparison, mobilities of positive ions in oriented samples of $V_\mathrm{{m}}=21.0$ cm$^3$ mol$^{-1}$ from Andreeva et al. [72, 73] are shown; red open triangles—$\phi = 0$, cyan solid triangles—$\phi = 90^{\circ }$, straight lines through points correspond to activation energies of $\varepsilon _{\parallel } = 5.3$ K ($\phi =0$) and $\varepsilon _{\perp } = 11$ K ($\phi = 90^{\circ }$). Also, examples of positive ion mobilities measured by Lau and Dahm [17] in samples of $V_\mathrm{{m}}=20.54$ cm$^3$ mol$^{-1}$ with apparent orientations $\phi = 0$ (red open squares) and $73^{\circ }$ (green solid squares), as well as their expected asymptote for $\theta = 90^{\circ }$ (cyan dashed line) are shown

Fig. 4

The same data and legend as in Fig. 3 but with temperatures for Andreeva et al. [72, 73] and Lau and Dahm [17] re-scaled by constant factors to make the values of the positive ion mobility in the $C_6$ direction ($\phi =0$, i. e. in the samples with the highest mobility of positive ions) coincide with that from Golov et al. [23, 24] at $T^{-1}=0.8$ K$^{-1}$. This is done to simplify the comparison of ion mobilities measured at different molar volumes (see text)

The case of positive ions in hcp $^4$He is more complicated due to the strong anisotropy of their mobility. In Fig. 3, we show examples of ion mobilities in samples of hcp $^4$He with different angles $\phi$ between the direction of electric field $\textbf{E}$ and the hexagonal axis $C_6$, measured by three different groups: from Andreeva et al. [72, 73] for $\phi =0$ and $\phi =90^{\circ }$ (measured directly) at $V_\mathrm{{m}}=21.0$ cm$^3$, from Lau and Dahm [17] for $\phi =0$ and $\phi =73^{\circ }$ (inferred after fitting $\mu (T)$ to Eq. 3 for a range of orientations $\phi$ at $V_\mathrm{{m}}=20.54$ cm$^3$), and from Golov et al. [23, 24], for $\phi \approx 0$ and $\phi \approx 73^{\circ }$ (inferred after using Eq. 4, see below). Here we concentrate on the high-temperature part, $T \gtrsim 0.9$ K (i. e. $T^{-1} \lesssim 1.1$ K$^{-1}$), where the low-temperature anomaly (see next Section) is absent.

Unfortunately, measurements from these three groups were made at different molar volumes, which complicates comparisons due to the dependence of vacancy creation energy on $V_\mathrm{{m}}$. In order to help the discussion, in Fig. 4 we show the same data but with temperatures for Andreeva et al. [72, 73] and Lau and Dahm [17] re-scaled by constant factors to make the values of the positive ion mobility in the $C_6$ direction ($\phi =0$) coincide with that in the samples with the highest mobility of positive ions from Golov et al. [23, 24] at $T^{-1}=0.8$ K$^{-1}$. Here, we can see that the positive ion mobilities $\mu _+(T)$ split into two distinctive branches: those for $\phi =0$ are the largest and with the shallowest slope (smallest $\varepsilon$), while those for large $\phi$ have smaller values and steeper slope. Those for an intermediate angle $\phi$ of $72-73^{\circ }$ actually do not look like a straight line with a single activation energy $\varepsilon$ but more like a cross-over between two asymptotic straight lines with quite different values of $\varepsilon$.

Lau and Dahm [17] and Andreeva et al. [72, 73, 94, 95] described $\mu _+$ as an anisotropic property,

$$\begin{aligned} \mu _+(\phi ,T) = \mu _{0\parallel }\cos ^2(\phi )\exp (-\varepsilon _{\parallel }/k_\mathrm{{B}}T) + \mu _{0\perp }\sin ^2(\phi )\exp (-\varepsilon _{\perp }/k_\mathrm{{B}}T). \end{aligned}$$

(3)

The experimental values for $\varepsilon _{\parallel }$ and $\varepsilon _{\perp }$ from Lau and Dahm [17] are $\varepsilon _{\parallel }=11.5$ K and $\varepsilon _{\perp }=23.5$ K for $V_\mathrm{{m}}\approx 20.54$ cm$^3$ mol$^{-1}$, and from Andreeva et al. [72, 73, 94] are $\varepsilon _{\parallel }=5.3$ K and $\varepsilon _{\perp }=11$ K for $V_\mathrm{{m}}\approx 21.0$ cm$^3$ mol$^{-1}$ (it is plausible that the values from Andreeva et al. [72, 73, 94] are underestimated due to the procedure used$^{3}$). Quite a large difference between the activation energies $\varepsilon _{\perp }$ and $\varepsilon _{\parallel }$ explains why previous analyses of $\mu (T)$, measured in samples with intermediate $\phi$ between 0 and $90^{\circ }$ and in a limited temperature range, often resulted in apparent values of $\varepsilon _+$ in between these two extremes $\varepsilon _{\parallel }$ and $\varepsilon _{\perp }$. One should hence take the multitude of values of $\varepsilon _+$ for hcp $^4$He, previously reported by several authors, with caution.

Eq. 3 explains why, provided angle $\phi$ is not too close to $90^{\circ }$, at sufficiently low temperatures of order $T \sim 1$ K (where the first term in the RHS of Eq. 3 dominates), the observed slope of $\mu _+(T)$ usually coincides with $\varepsilon _{\parallel }$. And for samples with a substantial angle $\phi$, its value can be determined from the ratio (at T in the range 0.9–1.1 K, see Fig. 3)⁴ of the mobility $\mu _+(T,\phi )$ to that for $\phi \approx 0$, $\mu _+(T,0)$:

$$\begin{aligned} \phi = \arccos \left[ \left( \frac{\mu _+(T,\phi )}{\mu _+(T,0)}\right) ^{1/2} \right] . \end{aligned}$$

(4)

As we see, the experimental activation energy for the diffusion in the basal-plane direction $\varepsilon _{\perp }$ is, to a good accuracy, twice the value $\varepsilon _{\parallel }$ along the $C_6$ axis. Within the approach of Andreev and Savishchev [54], these two values, $\varepsilon _{\parallel } = \varepsilon _\mathrm{{v}}$ and $\varepsilon _{\perp } = 2\varepsilon _\mathrm{{v}}$, are associated with the inelastic scattering of either one or two vacancions off the ion.

At the same time, the experimental values of activation energies for negative ions $\varepsilon _-$ showed little anisotropy (in Fig. 3 they are identical for two samples with very different mobilities of positive ions—i. e. with apparent angles $\phi \approx 0$ and $\phi \approx 72^{\circ }$), and were approximately equal to $\varepsilon _{\perp }$ for positive ions, i. e. $\varepsilon _- \approx 2\varepsilon _\mathrm{{v}}$ (and even the absolute values of $\mu _-(T)$ were close to $\mu _{\perp }(T)$ for positive ions in samples with $\phi = 90^{\circ}$). It is then plausible that, at molar volumes 20.0–21.0 cm$^3$ mol$^{-1}$, negative ions move in all directions by scattering pairs of vacancions at once.

Fig. 5

Activation energies for various processes in HCP $^4$He. For the ion mobility, the angle $\phi$ was only known for the data from Andreeva et al. [72, 73] and Lau & Dahm [17]. The black solid lines guide the eye approximately through the values of vacancy creation energy $\varepsilon _\mathrm{{v}}(V_\mathrm{{m}})$ obtained via measurements of X-ray and diffusion of $^3$He impurities. The black dashed lines are a factor of two higher, $2\varepsilon _\mathrm{{v}}(V_\mathrm{{m}})$. References to the legend: Shalnikov [60, 61], Keshishev [70, 71], Marty [65], Dahm [43], Golov [23, 24], Andreeva [72, 73], Lau [17], Simmons [83], Mikheev [90, 91], Mizusaki [92]

To survey the situation for a broader range of molar volumes of hcp $^4$He, in Fig. 5 we plot activation energies $\varepsilon$ for the mobility² of positive and negative ions along with $\varepsilon _\mathrm{{v}}$ for vacancies as a function of $V_\mathrm{{m}}$. A plausible trend for $\varepsilon _\mathrm{{v}}(V_\mathrm{{m}})$ is outlined by the black solid line; the change of its slope near $V_\mathrm{{m}} = 20$ cm$^3$ could be related (see Eq. 2) to the cross-over from the limit $\Delta _\mathrm{{v}} \ll \varepsilon _{v0}$ at $V_\mathrm{{m}} < 20$ cm$^3$ to $\Delta _\mathrm{{v}} \sim \varepsilon _{v0}$ at $V_\mathrm{{m}} > 20$ cm$^3$. Unfortunately, there is no reliable data for $\varepsilon _\mathrm{{v}}$ at $V_\mathrm{{m}} < 19$ cm$^3$; we can only speculate that the shallower slope of $\varepsilon _\mathrm{{v}}(V_\mathrm{{m}})$ continues down to $V_\mathrm{{m}} =17$ cm$^3$ (as extrapolated by the solid line). Our main conclusion here would be that, depending on the sample’s orientation and density, there are cases for the activation energy for the mobility of both positive and negative ions to be equal to either $\varepsilon _\mathrm{{v}}$ (one-vacancion scattering) or $2\varepsilon _\mathrm{{v}}$ (two-vacancion scattering).

4.2 Strong Fields, Monotonic Field Dependence of Drift Velocity

Fig. 6

Field dependence of the drift velocity of negative ions in a hcp $^4$He sample grown at 31.0 atm (same as in Fig. 7) shown at various temperatures from Golov et al. [23, 24]. Dashed lines indicate linear $v\propto E$ dependence at small E

Fig. 7

Field dependence of the drift velocity of negative ions in a sample of hcp $^4$He grown at 31.0 atm (same as in Fig. 6) shown at various temperatures from Golov et al. [23, 24]. The same data as in Fig. 6 but plotted in semi-log scale; lines indicate $v \propto \exp (\gamma E)$ dependence at large E

Negative ions in bcc $^3$He and hcp $^4$He revealed super-linear $v_-(E)$ similar to [23, 24, 74, 75] $v_- \propto \sinh \gamma E$ up to $E\sim 10^5$ V cm$^{-1}$. For bcc $^3$He, the data are consistent with the model $\gamma = eb/k_\mathrm{{B}}T$ with $b \approx 0.5$ Å, which implies a thermal activation over an energy barrier, modified by $\pm eEb$ in the presence of driving field E [74, 75]. Positive ions in bcc $^3$He [74, 75] revealed a similar trend but with a somehow poorer collapse on a single exponential dependence $v \propto \exp (\gamma E)$ in strong fields (and with possibly temperature-independent parameter $\gamma$). For hcp $^4$He, an example of a cross-over, around $E \sim 2\times 10^4$ V cm$^{-1}$, from a linear to exponential v(E) is shown in Figs. 6 and 7. For different samples, the values of $\gamma$ varied (likely due to their different orientation). Equating $\gamma = eb/k_\mathrm{{B}}T$ at temperatures $T \sim 1$ K results in b within the range 0.35–0.8 Å for different samples.

5 Anomalous Behaviour of Positive Ions in hcp $^4$He

It is the positive ions in hcp crystals which brought up a surprise at low temperatures $T \lesssim 1$ K. As one can see from Figs. 1 and 8, the temperature dependence of their low-field mobility $\mu _+(T)$ reveals an upward deviation [22,23,24] from the Arrhenius form at $T<0.9$ K. This anomaly generally becomes stronger with increasing pressure in the investigated range of 25–40 bar, see Fig. 9 featuring samples of high mobility (i.e. with the orientation of $\textbf{E}$ closer to $C_6$, $\phi \approx 0$).

Fig. 8

Temperature dependences of the drift velocity of positive ions in hcp $^4$He, measured in a diode with $d=0.3$ mm at voltage U (shown in legend) in a sample of pressure 28.8 atm ($V_\mathrm{{m}}=20.75$ cm$^3$ mol$^{-1}$) from Golov et al. [23, 24]

Fig. 9

Temperature dependences of the mobility of positive ions measured in a diode with $d=0.3$ mm and voltage $U=300$ V in hcp $^4$He at several pressures (shown in legend) from Golov et al. [23, 24]. Lines extrapolate the high-temperature dependence given by Eq. 1

Furthermore, with increasing field E above some $\sim 10^4$ V cm$^{-1}$, new types of nonlinearity in v(E) appear [20, 21, 23, 24], see Fig. 10. In the samples with the highest mobilities (i. e. with $\phi \approx 0$), v(E) displays the familiar super-linear growth up to $E\sim 5\times 10^4$ V cm$^{-1}$—which is then followed by a rather abrupt drop until $E\sim 10^5$ V cm$^{-1}$—above which the growth of v(E) resumes. In samples with the lowest mobilities (corresponding to the field $\textbf{E}$ perpendicular to the $C_6$ axis, $\phi \approx 90^{\circ }$), only a weakly sub-linear [23, 24, 70, 71] growth of v(E) was seen in fields above $10^4$ V cm$^{-1}$, which is then followed by a strong super-linear exponential trend at $E\sim 10^5$ V cm$^{-1}$ (see examples in Fig. 11).

Fig. 10

Field dependences of the drift velocity of positive ions in hcp $^4$He from Golov et al. [23, 24] at various temperatures in a sample of pressure $P = 31.0$ atm. Lines guide the eye through data at the same temperature. Dashed lines indicate linear regime $v_+ \propto E$

Fig. 11

Field dependences of the drift velocity of positive ions in hcp $^4$He from Golov et al. [23, 24], shown at the temperature of 0.72 K in several samples of different pressures. Lines guide the eye through data at the same temperature. The blue solid line gives an example of $v_+ \propto \sinh (eEb/k_\mathrm{{B}}T)$ with $b=0.25$ Å

A selection of field dependences of the drift velocity for samples grown at pressures in the range 26–34 atm is shown in Fig. 11 [23, 24]. While there is a general trend for a greater amplitude of the anomaly at higher pressures, substantial differences are also observed between different samples of the same density but different orientations with respect to the electric field (as an example in Fig. 11, there are three samples of the same pressure 29.1 atm). In the strongest fields of order $E\sim 10^5$ V cm$^{-1}$, above those of the anomaly, the Arrhenius temperature dependence seems resumed (see Fig. 8) with activation energies similar to those for weak fields (see Fig. 5). It seems there is a common underlying field dependence (indicated by the blue line in Fig. 11) of type $v \propto \sinh (\gamma E)$ at the highest studied fields E, on top of which an anomalous non-monotonic contribution (of magnitude strongly sensitive to the crystal orientation) at intermediate fields E is superimposed.

Fig. 12

Anomalous contribution to the mobility of positive ions in hcp $^4$He over the high-temperature Arrhenius extrapolation. Same data as in Fig. 9

Fig. 13

Anomalous contribution to the drift velocity of positive ions in hcp $^4$He at $T=0.72$ K over the “classical” monotonic $v \propto \sinh (\gamma E)$ shown by the blue line in Fig. 11

Taking a phenomenological approach, one may assume that both anomalies (the excess contribution to the low-field mobility and excess contribution to the drift velocity in the strong field) originate from an additional mechanism that allows positive ions to move faster but is only limited to temperatures below some $T_\mathrm{{c}} \sim 0.9$ K and fields below some $E_\mathrm{{c}} \sim 4\times 10^4$ V cm$^{-1}$. The existence of a limiting field $E_\mathrm{{c}}$ supports the idea that only a limited range of energy can be removed at each elementary hop of the ion—as would be if the hop is accompanied by either emission or inelastic scattering of either a vacancion or a phonon. Similarly, the existence of a limiting temperature $T_\mathrm{{c}}$ might imply that this energy balance is fragile due to a certain limitation on the available energy band $\sim k_\mathrm{{B}}T_\mathrm{{c}}$ (this line of thought was followed [57] but seems controversial due to the contradictory theoretical estimate for the vacancion band [44, 45] $\Delta _\mathrm{{v}} \sim$ 5–10 K).

To isolate the anomalous contributions, in Figs. 12 and 13 we subtracted the underlying “classical” dependences of form $v \propto \exp {-\frac{\varepsilon }{k_\mathrm{{B}}T}}\sinh \left( \frac{eEb}{k_\mathrm{{B}}T}\right)$ from both the low-field and high-field experimental results from Figs. 9 and 11. As was expected by design, both residual contributions reveal bell-shaped anomalies as a function of either T (Fig. 12) or E (Fig. 13). In fact, a similar bell-shaped temperature dependence of the diffusion coefficient has been observed [10, 11] and explained [96] for concentrated $^3$He impurities in hcp $^4$He, where the energy of elastic interaction of neighbouring $^3$He atoms $U_3$ causes their self-localization at low T due to the narrowness of the energy band of $^3$He atom’s quasi-free states $\Delta _3 < U_3$. Increasing temperature towards $k_\mathrm{{B}}T \sim U_3$ would initially help lift this localization but eventually breaks the coherent tunnelling when $k_\mathrm{{B}}T \gg U_3$. In the spirit of the theory of the motion of ions subject to inelastic scattering of delocalized vacancies by Andreev and co-workers [52,53,54], the incipient localization of positive ions, in strong fields, could be naturally explained by the limitations due to the finite width of the vacancion energy band $\Delta _\mathrm{{v}}$. Unfortunately, neither the typical length of vectors $\textbf{b}_i$ nor the width of the vacancion energy band are currently known. By assuming $b\sim 1$ Å, and using the observed $E_\mathrm{{c}} \sim 4 \times 10^4$ V cm$^{-1}$, from the condition $eEb = \Delta _\mathrm{{v}}$ for a one-vacancion process we would arrive at $\Delta _\mathrm{{v}} \sim 4$ K—which is close to the numerical values [44, 45] of 5–10 K.

6 Strong-field Anomalies in bcc Lattice

Fig. 14

Field dependence of the drift velocity of positive ions in bcc $^4$He with 5% $^3$He impurities (closed circles) and in a low-mobility sample of hcp $^4$He (open diamonds)—at similar densities ($V_m \approx 21$ cm$^3$mol$^{-1}$) and temperatures ($T\approx 1.3$ K), measured in the same diode with gap $d=0.3$ mm—from Golov [56]. Dashed lines indicate linear dependence $v \propto E$

As was mentioned above, no significant deviations from the Arrhenius temperature dependence of ion velocities in other systems (apart from the positive ions in hcp $^4$He) have so far been observed—quite possibly due to the experimental limitations on the range of measurable drift velocities at low temperatures. It would be of great interest to compare the observed anomalous behaviour, at $T\lesssim 1$ K of positive ions in hcp $^4$He at molar volumes 20–21 cm$^3$mol$^{-1}$and pressures 25–40 bar, with those in bcc crystals at similar densities and pressures. With $^3$He, an accurate comparison cannot be made due to much higher pressures for the same values of $V_\mathrm{{m}}$, while pure $^4$He does not exist in the bcc phase below 1.4 K. However, it is possible to extend the bcc phase of solid $^4$He down to $T=0$ by adding several per cent of $^3$He [97]. Our preliminary experiments with samples of bcc $^4$He containing 5% $^3$He revealed no substantial deviation from the Arrhenius-type temperature dependence for the mobility of positive ions down to 1 K and for the negative ions down to 0.6 K but hinted at a possible existence of high-field anomalies. In Fig. 14, we present an example [56] of a sub-linear field dependence $v_+(E)$ of the velocity of positive ions in such a bcc $^3$He–$^4$He mixture of $V_\mathrm{{m}}=21.3$ cm$^3$ mol$^{-1}$ ($P\approx 27$ bar) at $T=1.25$ K, alongside a strikingly similar $v_+(E)$ in hcp $^4$He of $V_\mathrm{{m}}=20.75$ cm$^3$ mol$^{-1}$ $P\approx 28$ bar) at $T=1.32$ K. Note that in both samples the dominant mechanism of ion transport in low fields is likely two-vacancion inelastic scattering (as seen from the activation energies $\varepsilon _+ \approx 2 \varepsilon _\mathrm{{v}}$ in both bcc lattice, Fig. 2, and hcp samples with $\phi \approx 90^{\circ }$ orientation, Fig. 5). It is also worth mentioning that sub-linear $v_-(E)$ was also reported for negative ions in hcp $^4$He [17].

7 Summary

Let us summarize several advances and outstanding questions:

1. The direct verification and calibration [72, 73, 94] of the anisotropy in the low-field mobility of positive ions in hcp $^4$He can now be used to determine the orientation of the $C_6$ axis of a single crystal with respect to the applied electric field—see data assembled in Fig. 3 and Eqs. 3–4.

2. Our new analysis (see solid and dashed lines in Figs. 2 and 5) presents a strong case in favour of the existence of both one- and two-vacancion mechanisms [54] of motion of both positive and negative ions in bcc and hcp solid helium. It seems positive ions move by the two-vacancion mechanism at all available densities in bcc $^3$He and $^4$He, while in hcp $^4$He they only do this within the basal plane. The negative ions, on the other hand, in all helium lattices and isotopes have windows of density within which they predominantly move by either one-vacancion or two-vacancion mechanism.

3. Positive ions in hcp $^4$He reveal a bell-shaped anomalous enhancement of $v_+(E)$ (see our analysis in Figs. 11–13) of large magnitude when moving parallel to $C_6$ (one-vacancion scattering). It is terminated above $E_\mathrm{{c}} \sim 4 \times 10^4$ V cm$^{-1}$; with $eE_\mathrm{{c}}b=\Delta _v$ and $\Delta \sim 4$ K this gives $b \sim 1$ Å, which seems reasonable—this would be consistent with the phonon-less one-vacancion scattering. Obviously, a better understanding of the microscopic structure of the positive ion is eagerly desired. We showed new evidence (Fig. 14) that a similar strong-field anomaly in $v_+(E)$ might exist in bcc helium too.

4. The strong-field anomaly in $v_+(E)$ for positive ions in hcp $^4$He seems to be present only below $\sim 1$ K—precisely where a bell-shaped anomalous contribution to the low-field mobility $\mu _+(T)$ was observed [22] (see Fig. 8). Whether these two anomalies are related and what is their underlying mechanisms are still open questions.

5. A related question arises: why, in hcp $^4$He, do negative ions not show these anomalies? So far, negative ions in hcp $^4$He were mainly studied at low densities ($V_m$ in the range 20.5–21.0 cm$^3$ mol$^{-1}$), where they seem to hop via two-vacancion processes in all directions. Perhaps one should explore the range 19–20 cm$^3$ mol$^{-1}$), where the one-vacancion process seems dominant.