Anomalous Positron Lifetime in Single Crystal of Weyl Semimetal CoSi

Abstract

The positron annihilation lifetimes were measured using a ${}^{48}$V positron source in noncentrosymmetric cubic single crystals of CoSi, FeSi and MnSi. The following lifetimes were determined from the positron annihilation time spectra: 168(1) ps for CoSi, 114(1) ps for FeSi and 111(1) ps for MnSi. For single-crystal CoSi, the positron annihilation lifetime was also determined with a ${}^{22}$Na positron source. For CoSi, the lifetimes obtained from different positron sources are consistent. The differences in the positron annihilation lifetimes in MnSi and FeSi, on the one hand, and in the Weyl semimetal CoSi, on the other hand, are possibly caused by the formation of a positron + electron bound state (positronium).

1. Introduction

Positronium (Ps) is a purely lepton bound state consisting of an electron and a positron [1,2]. As the simplest electromagnetically bound state, it is an ideal system for studying in quantum electrodynamics. Ps can exist either in a spin-singlet (para(p)–Ps) state or in a spin-triplet (ortho(o)–Ps) state. In vacuum, these states have an average lifetime of 0.125 ns and 142 ns, respectively [3,4]. The Ps lifetimes in solids differ from their values in vacuum due to the many-particle Coulomb interaction. For example, in the $\alpha$-SiO${}_2$ single crystal, the lifetime of p-Ps amounts to ${\tau }_{\mathrm{p}-\mathrm{Ps}}=156\left(4\right)$ ps [5]. The formation of Ps both on the surface of a solid [6] and in the bulk [7] depends on the electronic structure near the Fermi level of the sample under study. In studying the formation of Ps in multilayer graphene, grown on a polycrystalline copper substrate, it has been found that Ps in graphene and Ps on the surface of copper are different [8]. On the other hand, it is well known that the electronic structure of graphene, characterized by the presence of quasi-two-dimensional Dirac fermions, gives rise to a discovery of Weyl semimetals that contain massless Dirac–Weyl fermions in three dimensions [9,10,11,12,13,14].

This suggests that the formation of Ps can also be expected in single crystals of Weyl semimetals, including, in particular, CoSi, RhSi, RhSn, and PtAl [15,16,17,18,19], which are crystallized in the noncentrosymmetric structure of the FeSi-type (the $P{2}_{1}3$ space group). In addition, in these quantum materials, such unusual objects as chiral fermions have been found. For CoSi and RhSi, the existence of two types of chiral fermions with nonzero Chern numbers was confirmed: spin-1 and charge-2. Ab initio calculations [20] showed that their band structure near the Fermi energy ($E_{\mathrm{F}}$) has a threefold degenerate point at the $\Gamma$ point and a fourfold degenerate point at the R point of the Brilloin zone. It is also worth mentioning that in CoSi and RhSi the topological superconductivity was theoretically predicted in Ref. [21]. New spin-related transport properties of these unconventional chiral fermionic semimetals make them very promising for future spintronic and spin caloritronics applications [22], while the quantum combination of topological and optical properties can be used for new information technologies [23].

The electronic properties of CoSi are often compared with those of MnSi and FeSi. Notice that, since Mn, Fe and Co are consecutive elements of the same row in the periodic table, in going from Mn to Fe and then to Co, the $3d$ electron band acquires more electrons and the Fermi energy ($E_{\mathrm{F}}$) is increased. As a result, in FeSi, the added electron populates the unoccupied states below the energy gap, while in CoSi the extra electron is accommodated by filling electron states above the gap, with $E_{\mathrm{F}}$ reaching a three-fold degenerate state at the $\Gamma$ point and a four-fold degenerate level at the R point [24]. Correspondingly, one observes a change in properties from the helicoid magnet MnSi [25] through the formation of the Kondo state in FeSi [26] to the chiral diamagnetic Weyl semimetal ground state in CoSi. This change, caused by differences in the electronic band structure near the Fermi level, can be probed by positron annihilation, and, in this paper, we present our results of the positron lifetime annihilation spectroscopy (PALS) study performed in CoSi, FeSi, and MnSi single crystals.

Usually, the positron annihilation rate (inversely proportional to the annihilation lifetime $\tau$) is determined by the overlap of the positron ${\rho }_{+}\left(r\right)={\left|{\psi }_{+}\left(r\right)\right|}^2$ and electron ${\rho }_{-}\left(r\right)$ densities in the localization region: $\lambda =\frac{1}{\tau }=\pi r_0^{2}c\int {\left|{\psi }_{+}\left(r\right)\right|}^2{\rho }_{-}\left(r\right)\gamma dr$, where $r_0$ is the classical electron radius, whereas $\gamma =\gamma \left[{\rho }_{-}\left(r\right)\right]=1+\frac{\Delta {\rho }_{-}}{{\rho }_{-}}$ describes the increase in the electron density as a result of the mutual Coulomb attraction of the electron and the positron [27]. For performing PALS experiments in laboratories, the preferred source of positrons is usually ${}^{22}$NaCl on thin metal (Ni, Al) or polymer foil (Kapton, Mylar). However, the monosilicides MnSi, FeSi and CoSi have already been studied using this source or accelerator beams [28,29,30,31]. In this work, as a source of positrons, we have decided to use the ${}^{48}$V isotope in titanium foil with a half-life $T_{1⁄2}\approx$ 16 days, which makes PALS measurements more convenient and environmentally friendly. First, titanium has a high melting point (1668 °C), which makes it possible to use the obtained source in various experiments at high temperatures (for example, in liquid metals or boiling solutions). On the other hand, at low temperatures, this source is also convenient in experiments due to its low thermal conductivity (13 times less than the thermal conductivity of aluminum and 4 times less than that of iron), making it possible to carry out positron measurements with high accuracy in a thermostat, and also without the possibility of dissolving the source itself in the process of its operation. A chemically aggressive environment (for example, molten salts) is also not a problem for such a source due to the high corrosion resistance of titanium. Due to its convenient shape (thin metal foil), ${}^{48}$V can also be used in high pressure experiments. Secondly, after titanium foil irradiation at the cyclotron, active nuclides ${}^{48}$V are distributed in the titanium node, and the foil becomes a closed source of positrons, which is already ready for experiments without requiring radiochemical preparation. This allows you not to worry about additional personal protective equipment in the future due to the fact that the source will be lost or smeared on the surfaces of other materials. Moreover, the relatively short half-life makes it possible to store the source without excessive radiation and biological protection almost immediately after its use in a nuclear physics experiment. Based on this, excessive environmental pollution does not occur, and the process of “recharging” the source can be easily controlled by having a cyclotron with a beam of low-energy protons. Compared to ${}^{22}$Na, the positron spectrum from ${}^{48}$V has higher energy, with a maximum of about 700 keV (for ${}^{22}$Na, the maximum energy is 545 keV). A higher positron energy will make it possible to study slightly thicker layers of the studied substance (and hence the volume) and is commensurate with the energies of the ${}^{22}$Na source; thus, it does not require additional serious changes in the reconfiguration of the experimental technique and changes in the design of the detectors (replacement of crystals and the electronic part of the equipment).

2. Materials and Methods

The samples of MnSi ($a=4.5598\left(2\right)$ Å) and CoSi ($a=4.444\left(1\right)$ Å) single crystals studied in this work were grown by the Bridgman method [32,33] (Ames Laboratory), and their crystal structure was well defined and they have been characterized by various macroscopic measurements (see [32,33,34]). The sample of FeSi ($a=4.486\left(2\right)$ Å) single crystal was grown by the Czochralski method. The lattice parameters of the crystals, determined by X-ray diffraction, correspond well to literature data [33,35,36]. As has been mentioned earlier, as a source of positrons, we used the ${}^{48}$V isotope obtained by the reaction ${}^{48}$Ti(p,n)${}^{48}$V by irradiating titanium foil (50 $\mathsf{\mu }$m) with protons with an energy of 7.8 MeV (in the cyclotron of the Institute of Nuclear Physics, Moscow State University). The scattering cross section for protons was about 65 mbarn [37,38]. The use of relatively low proton energy proved to be sufficient to obtain the required ${}^{48}$V activity. Measurements of the energy spectrum of the irradiated foil showed the absence of "spurious" lines from another isotope (see Figure 1).

Annihilation time spectra with the ${}^{48}$V positron source were measured using the “VUKAP” four-detector compact digital spectrometer equipped with two LaBr${}_3$:Ce detectors (BrilLanCe${}^{\mathrm{TM}}$ 380) [39]. The time resolution of the spectrometer (FWHM at ${}^{60}$Co) was 380 ps. Two detectors were installed at an angle of 90$°$ to each other at a short distance. The irradiated titanium foil was sandwiched between two samples of the same single crystal. Such a “sandwich” was installed at an angle of 45$°$ to the detectors. All measurements were carried out at room temperature using the same positron source. The source size was 5 × 5 mm${}^2$, which made it possible to completely cover the source with samples. The initial activity of the source was about 150 kBq. The 1312 keV $\gamma$-ray photon from the ${}^{48}$V decay was used for the “START” and the 511 keV annihilation gamma-ray photon for the “STOP”. Each PALS spectrum contained more than 6 million counts.

3. Results and Discussion

The time spectra of bulk positron annihilation and their characteristics, described with the two-exponential model in the LT10 program [40], are shown in Figure 2 and in

Table 1. Measured positron annihilation lifetimes (${\tau }_{\exp }$) in single crystals of Si, CoSi, MnSi, and FeSi (this work), compared with ${\tau }_{\mathrm{b}}$ in polycristalline samples (CoSi, MnSi, FeSi) and the Si single crystal with ${}^{22}$Na as a source of positrons.

Material	${\mathit{\tau }}_{\mathbf{exp}}$, ps (in This Work)	${\mathit{\tau }}_{\mathbf{b}}$, ps (According to Literature)
Si	218(1) (with ${}^{48}$V)	218(1) (single crystal) [43]
CoSi	168(1) (with ${}^{48}$V), 166(2) (with ${}^{22}$Na)	115(2) (polycrystal) [28]
FeSi	114(1) (with ${}^{48}$V)	130(3) (polycrystal) [31]
MnSi	111(1) (with ${}^{48}$V)	119(3) (single crystal) [29]

. Our results for MnSi and FeSi agree with previous measurements performed with other experimental setups with ${}^{22}$Na as a positron source [29,31]. It has been found that the value of $\tau$ for the second component varies from 1700 ps to 4000 ps depending on the sample under study. Therefore, we attribute the second component to the medium between the samples and the source. Its partial contribution to the intensity varies from 0.5% to 3%. The intensity contribution from the annihilation in the source (the 50 $\mathsf{\mu }$m width Ti foil) should be within 15–20% [41]. As pointed out in [42], the main component (about 90%) from a proton-irradiated unannealed Ti-foil source is described by $\tau \approx 37$ ps. We were unable to resolve such a low lifetime due to the finite time resolution of the spectrometer. As a result, the contribution of the positron source to the spectra could not be reliably determined. We then measured the PALS spectrum for the CoSi single crystal using another spectrometer with a time resolution of about 200 ps and the ${}^{22}$NaCl positron source (Joint Institute for Nuclear Research, Dubna). In this case, the positron lifetime is 166(2) ps (see Figure 3 and Table 1), which coincides with the result obtained with a ${}^{48}$V positron source. We also measured the PALS spectrum for the Si single crystal with a ${}^{48}$V positron source. The positron lifetime is 218(1) ps (see Figure 3 and Table 1), which coincides with the result obtained with a ${}^{48}$V positron source [43]. The lifetimes for MnSi and FeSi are found to be approximately the same, despite the different densities of electron states at $E_{\mathrm{F}}$. While MnSi is a metal, FeSi has a small energy gap of about 75 $\mathsf{\mu }$eV above the filled electron band. On the other hand, our high-quality single crystal positron annihilation lifetime in CoSi has turned out to be 1.5 times larger than the found values of $\tau$ in MnSi, FeSi and polycrystal CoSi (see Figure 2 and Table 1). This result contradicts the previous value of the positron lifetime, obtained in polycrystalline samples of CoSi [28]. According to Ref. [28], the lifetime in polycrystalline CoSi amounts to 115 ps, which is close to the value of $\tau$ both in the MnSi single crystal (Table 1) and in FeSi and apparently can be explained by positron annihilation on conducting electrons, despite the fact that defects may exist in a polycrystalline sample of CoSi. In our case, we consider that the positron annihilation in the single-crystal CoSi can be associated with its electronic features [17,44]. In particular, since the chiral Weyl fermions near the Fermi energy are absent in MnSi and FeSi, their presence can probably account for the formation and subsequent decay of the Ps, which results in a longer positron annihilation lifetime observed experimentally. The peculiarities of CoSi in this B20-compounds row have been recently observed on the temperature dependence of elastic constants; the values of c${}_{11}$ and c${}_{12}$ are highest for CoSi and then follow FeSi and MnSi [33,45]. In contrast to FeSi, inelastic neutron scattering measurements have shown that CoSi exhibits normal phonon behavior, which is clear from its different electronic structure [46]. Our experimental results certainly indicate the need for further research to elucidate the mechanism of positron annihilation in topological semimetal single crystals with Weyl singular points.

4. Conclusions

Our measurements of the PALS spectra in the single crystals of MnSi, FeSi and CoSi indicate that the positron annihilation lifetime in CoSi is approximately 1.5 times longer than $\tau$ in MnSi and FeSi, and by the same amount exceeds the value of $\tau$, obtained in polycrystalline CoSi. The longer lifetime implies a possible formation of the Ps. The peculiarities of CoSi in this B20-compounds row also have been recently observed on the temperature dependence of elastic constants, the small value of shear modulo and phonon density of states.

In the future, we plan to continue our PALS study of other topological Weyl semimetals at various temperatures. In addition, we prepare the PALS experiment at high pressure thanks to the good possibilities of the ${}^{48}$V positron source.