Single crystal growth and magnetoresistivity study of topological semimetal CoSi

TL;DR

This study reports the successful growth of CoSi single crystals and investigates their magnetoresistivity, revealing quantum oscillations consistent with topological semimetal properties and providing insights into their electronic structure.

Contribution

It presents the first detailed analysis of quantum oscillations in CoSi crystals, linking experimental results with band structure calculations for topological semimetals.

Findings

Observation of Shubnikov-de Haas oscillations in CoSi

Identification of two isotropic bulk electron Fermi surfaces

Extraction of effective mass and scattering rates

Abstract

We report single crystal growth of CoSi, which has recently been recognized as a new type of topological semimetal hosting fourfold and sixfold degenerate nodes. The Shubnikov-de Haas quantum oscillation (QO) is observed on our crystals. There are two frequencies originating from almost isotropic bulk electron Fermi surfaces, in accordance with band structure calculations. The effective mass, scattering rate, and QO phase difference of the two frequencies are extracted and discussed.

Equations

$$\rho (T)={\rho }_0+C(\frac{T}{{\Theta }_R}{)}^n{\int }_0^{\frac{{\Theta }_R}{T}}\frac{t^n}{(e^t-1)(1-e^{-t})}dt$$

$$\begin{array}{rl}& {\text{MR}}_{OSC}\sim \\ & \sum _i\sum _p\frac{5}{2}\sqrt{\frac{B}{2p{F}_i}}{R}_T^p{R}_D^p{R}_s^p\cos \{2\pi p[\frac{F_i}{B}+\frac{1}{2}]+{\delta }_i+{\varphi }_i\},\end{array}$$

Full text

CoSi is a long known material with FeSi-type cubic structure (B20, space group P213), of which the thermoelectric property and application have been the focus of studySakai et al. (2007); Longhin et al. (2017), it is generally regarded as a semimetal. With the development of characterizing materials by the topology of their electronic band structure, it has recently been suggested that in crystalline systems, in addition to Dirac, Weyl, and nodal line semimetals, band crossing points (nodes) with three-, six-, or eight-fold degeneracies can be stablizedBradlyn et al. (2016), the low energy fermionic excitations close to the nodes are called ”new fermions”. For CoSi and isostructural transition metal silicides RhSi, RhGe, and CoGe, the crystal structure is asymmorphic with threefold rotation symmetry and twofold screw symmetry. Theoretical studiesTang et al. (2017); Chang et al. (2017) found that with inclusion of spin-orbit-coupling (SOC), close to the Fermi level there only exist bands containing a fourfold degenerate node at the Brillouin zone (BZ) center $\Gamma$, and a sixfold degenerate node at BZ corner R, both are chiral, thus could serve as a model system in the study of unconventional chiral fermions. Besides, the phonon spectra of these monosilicides host double Weyl pointsZhang et al. (2018). Subsequent angle resolved photoemission spectrescopy (ARPES) measurementsTakane et al. (2019); Rao et al. (2019); Sanchez et al. (2019) have confirmed the overall bulk band structure and the existence of surface Fermi arcs. However, it has been noted that for CoSi preparation of a flat surface for ARPES is at least challenging. Therefore other probes of its bulk electronic structure are needed. At the same time, in the context of searching for feature characteristic of chiral fermions, the physical properties of CoSi are also worth revisiting on high quality single crystalline samples.

In this work, we report single crystal growth by the flux method, different from the Czochralski, Bridgman, or chemical vapor transport techniques previously used. The resulting crystals are of decent quality. To our knowledge, this is the first time SdH is observed in CoSi. The angular dependence of the QO frequencies reveals that they are from the bulk electron Fermi surface centered at R in BZ, agreeing with band calculation and ARPES data. Thus, it is a good starting point for further investigations.

The single crystal growth starts with Co powder (99.999%, Alfa), Si powder (99.999%, Alfa), and Sn grains (99.99%), which were mixed with a molar ratio of 1:1:20, loaded into an alumina crucible, then sealed with a partial Ar pressure inside a silica capsule. The reactant was quickly heated up to 1273 K and maintained there for 40 hours, followed by cooling down in 3 days to 1073 K, at which temperature the flux was separated by centrifuge. The harvested crystals with well-defined surfaces were further washed by dilute HCl to remove possible tin coating. There were mainly two types of shapes, one is thin, prismatic rod, the other is polyhedron close to a cube. To check for chemical composition, we employed energy dispersive X-ray (EDX) spectroscopy equipped on a Hitachi S-4800 scanning electron microscope (SEM). The rod type crystals showed good consistency among different growth batches with the Co:Si atomic ratio close to unit, the Co percentage was ranging from 49.5% to 50.7%. However, the polyhedron type crystals showed a large variation in composition, with the Co% expanding from 39% to 61%, indicating formation of other Co-Si binary compoundsIshida et al. (1991). The quality of the rod type crystals were then checked by single crystal x-ray diffraction at room temperature in a Bruker D8 Venture diffractometer using Mo K radiation, $\lambda$ = 0.71073 Å. Structure refinement was performed by the program SHELXL-2014/7SHE embedded in the program suite Apex3. Employing the known crystal structure of CoSi, the best refinement had R-indices of R = 0.0334(74) and wR2 = 0.0767(74), respectively. The lattice constant determined is a=4.4376(6) Å, in accordance with reported valuesStishov et al. (2012). In addition, the orientation along the rod is determined as the $\langle 011\rangle$ direction of the crystal, an optical image of a typical crystal in shown as inset of Fig.1.

Based on initial composition and crystallographic screening, all the electrical resistivity measurements were performed on rod-type single crystals by standard four-probe method with electrodes along the rod made by silver paste (DuPont 4922N) and platinum wires. Environment of cryogenic temperature and magnetic field was provided by either a 14 Tesla Quantum Design physical property measurement system(PPMS-14), or an Oxford 14 Tesla magnet with top-loading He-3 insert, both equipped with single-axis rotator. An external AC current source and lock-in amplifier were utilized. We note here that there are only a few inconsistent reportsAmamou et al. (1972); Narozhnyi and Krasnorussky (2013) on the temperature and magnetic field dependent magnetization of CoSi . Due to our current crystal size and limitation of instrumental resolution, it is planned in our future work.

The temperature dependence of resistivity $\rho(T)$ with $I\parallel\langle 011\rangle$ for several samples is shown in Fig.1. It is overall metallic while there is no up-turn at low temperature, against that of sample having nostoichiometric cobaltBurkov et al. (2017). The residual resistivity $\rho$(2 K) for 3 out of 4 samples is around 20 $\mu\Omega$ cm, at the low side of reported dataStishov et al. (2012); Petrova et al. (2010); Ohta et al. (1998); Ou-Yang et al. (2017); van der Marel et al. (1998). The largest residual resistivity ratio [RRR = $\rho$(300 K)/$\rho$(2 K)] appears in Sample S4, RRR$\sim$7.6, at the high side of reported values. Since the residual resistivity and the RRR are commonly used as indicators of crystal quality of metallic systems, we judge that our flux grown crystals have at least reached the same level of quality as crystals grown by other methods. Compared to other known Dirac and Weyl semimetals, e.g., Cd3As2, for which residual resistivity $\sim$ 20 $n\Omega$ cm, and RRR$\sim$4000 has been reportedLiang et al. (2015), the corresponding values for CoSi are moderate. Simply considering that the residual resistivity is proportional to the product of carrier density and carrier mobility, while the carrier density of Cd3As2 is lower than that of CoSi($\sim$1019 vs. $\sim$1020 cm*-3*), the carrier mobility of CoSi should be lower by orders of magnitude. Whether it is intrinsic property of CoSi or that samples at current stage still have high level of impurity scattering requires further investigation. A simulation of $\rho(T)$ is also tested by the Bloch-Gr$\ddot{\textrm{u}}$neisen (BG) formulaCvijović (2011):

[TABLE]

in which $\Theta_{R}$ is a characteristic temperature that is usually equal to the Debye temperature $\Theta_{D}$, and n is an integer depending on the scattering mechanism, n = 5 for electron-phonon scattering. As shown in Fig.1, for sample S4 BG simulation with n fixed at 5 could only closely follow the temperature dependence below 170 K, while the derived $\Theta_{R}$ is 300 K, only about 60% of $\Theta_{D}$ obtained by heat capacity measurementAmamou et al. (1972). Therefore, it is clear that an one channel electron-phonon scattering mechanism could not explain the $\rho(T)$ behavior of CoSi. Other causes, including an non-constant $\Theta_{D}$, multiple conducting channels, and electron-electron scattering, may also need to be considered.

With the aim of probing the bulk Fermi surface properties of CoSi, angular and temperature dependent magnetoresistivity (MR) measurements were carried out on sample S4. Thanks to the thin rod shape of the crystal, the mixing of MR and Hall signals is minimal. Fig.2 (a) shows the isothermal MR curves covering over 90*∘, the measurement configuration is illustrated as the inset, both B and I are parallel to the $\langle 011\rangle$ direction at $\theta$ = 0∘. The MR reaches 120% at 14 T, 350 mK, and $\theta\sim$90∘, about 2 times larger than reported values for CoSiOhta et al. (1998), although it is at least one order of magnitude smaller than the huge transverse MR reported for other Dirac and Weyl semimetalsXu and Jia (2016). The MR versus tilt angle at selected magnetic fields is plotted in Fig.2 (b), which is symmetric with respect to 0∘* and 90*∘* as expected. The data could be tracked by a sin2($\theta$) function, suggesting that the MR is proportional to a simple power law of H, with the power close to 2, and is mainly orbital. Besides, the mobility of associated charge carriers should not have a significant anisotropy. The SdH is already apparent as small wiggles on those MR curves in Fig.2 (a). Since the amplitude of the oscillation is less than 2% of the total resistivity, it is feasible to keep the analysis of SdH on MR only. Fig.2 (c) shows the oscillating part of MR (MROSC) versus the reciprocal of magnetic field after a smooth background substraction by a forth order polynomial. A beating pattern is evident for most of the curves, while the position of nodes of the envelop line has a weak evolution with angle. By fast Fouier transform (FFT) from 7 to 14 T, two prominent fundamental frequencies are identified on the spectra for all the angles stacked in Fig.2 (d), their 1st harmonics are also visible when the oscillation amplitude is strong around $\theta$ = 90*∘. By Gaussian peak fit, F1 = 563.9 T and F2 = 664.6T at $\theta\sim$ 87.7∘, and F1 = 564.5 T and F2 = 664.1T at $\theta\sim$ 3.1∘*, respectively. The standard errors of fitting are all less than 1 T, while the full width at half maximum (FWHM) is less than 30 T. If the FWHM was taken as the upper limit of frequency shift, then for both F1 and F2 the changes are less than 5% when B rotates from parallel to perpendicular to I.

Since the oscillation frequency F is directly proportional to the extremal cross-sectional area of a Fermi surface ($A_{F}$) by the Onsager relationShoenberg (1984) $F=(\hbar/2\pi e)A_{F}$, here the non-dispersive angular dependence of oscillation frequency F1 and F2 with respect to magnetic field direction indicates that the underlying bulk Fermi surfaces are close to spherical. From recent band structure calculationsTang et al. (2017); Pshenay-Severin et al. (2018), it is shown that those bands crossing the Fermi level close to the BZ center $\Gamma$ point have their node above the Fermi energy (EF), while the Fermi wavevector (kF) is different along $\Gamma$-X, $\Gamma$-M, and $\Gamma$-R, which are directions from $\Gamma$ to the face center, the edge center, and the corner of BZ, respectively. At the same time, for those bands crossing close to the R point, their node is $\sim$ 200 meV below EF. Without inclusion of SOC, there are two adjacent bands crossing the Fermi level along R-$\Gamma$, one along R-M, and one along R-X, the kF is similar along the three aforementioned directions. Thus there exist two electron Fermi Surfaces (FSs) close in size centered at R, from which we infer the observed two oscillation frequencies are originated. For F1 = 564 T and F2 = 665 T, the corresponding $A_{F}$1 = 5.39$\times$10*-2Å-2* and $A_{F}$2 = 6.34$\times$10*-2Å-2*, respectively. The averaged value is compatible with that obtained from EF intensity mapping at R point in recent ARPES experimentsTakane et al. (2019); Rao et al. (2019). Here, quantum oscillation shows the advantage of resolving two frequencies. However, we have not yet detected any oscillation from the hole FSs at $\Gamma$.

For the observed oscillations, more information about charge carriers at the FS could be extracted utilizing the Lifshitz-Kosevitch (LK) formulaShoenberg (1984); Murakawa et al. (2013), considering summations over different oscillation frequencies and their harmonics,

[TABLE]

in which $R_{T}^{p}=\alpha pm^{*}(T/B)/\sinh[\alpha pm^{*}(T/B)]$, $R_{D}^{p}=\exp[-\alpha pm^{*}(T_{D}/B)]$, and $R_{s}^{p}=\cos(p\pi gm^{*}/2)$ are the thermal damping factor, the Dingle damping factor, and the spin factor, respectively. $\alpha=2\pi^{2}k_{B}m_{e}/e\hbar$ is a prefactor constructed by the Boltzmann constant $k_{B}$, the bare electron mass $m_{e}$, the electron charge e, and the reduced Plank constant $\hbar$. For the additional phase shift of the cosine function, $\delta$ is the dimensional correction, and $\phi$ is often denoted as the Berry phase. The effective mass $m^{*}$ at a given field direction can be obtained by the temperature dependence of oscillation amplitude through RT. Fig.3 shows the analysis of MR data at $\theta\sim$ 90*∘* up to $\sim$ 9 K. The effective masses for previously determined F1 and F2 are m${}_{1}^{*}$ = 0.82 and m${}_{2}^{*}$ = 0.88, respectively. While in early experimental work, the effective mass of electrons was estimated to be 2Asanabe et al. (1964). By assuming a circular cross-section of FS, the Fermi wave vectors are estimated as $k_{F1}$ = $1.31\times 10^{-1}$ $\AA^{-1}$, and $k_{F2}$ = $1.42\times 10^{-1}$ $\AA^{-1}$, and the Fermi velocities ($v_{F}=\hbar k_{F}/m^{*}$) are $v_{F1}=1.85\times 10^{5}m\cdot s^{-1}$ and $v_{F2}=1.87\times 10^{5}m\cdot s^{-1}$, respectively. As a comparison, $v_{F}$ value of Cd3As2 from quantum oscillation experiments is around $1\times 10^{6}m\cdot s^{-1}$ Liang et al. (2015); He et al. (2014).

Since there are two nearby oscillation frequencies, the extraction of Dingle temperature TD is performed by a direct fitting of $\textit{MR}_{OSC}$ to the LK formula with known F and m*∗. In addition, the Landé g-factor in $R_{s}$ takes simply the free electron value of 2.0023, and the dimensional phase factor $\delta$ for both F1 and F2 takes the value of (-1/4)$\pi$ for three dimensional electron FS with maximum cross section. By only including fundamental harmonics, a good fit is achieved, as shown in Fig.4 (a) for $\theta$ = 83∘*, while the residual mainly contains 2nd harmonics. Within fitting error the obtained $T_{D}$ for both F1 and F2 is 3.2 K. The quantum scattering time $\tau_{q}=\hbar/(2\pi k_{B}T_{D})$ for electrons is then computed to be $\sim 3.8\times 10^{-13}s$. By taking the values of electron mobility and effective mass from Ref.Asanabe et al. (1964), $\mu_{e}=$ 800 $cm^{2}V^{-1}s^{-1}$ and $m^{*}=2$, the transport scattering time $\tau_{t}=\mu m^{*}/e$ = $9.1\times 10^{-13}s$. The small difference between $\tau_{t}$ and $\tau_{q}$ could be one reason for the relatively small MR in CoSi when compared to other topological semimetals. As in Cd3As2, $\tau_{t}$ is 4 orders larger than $\tau_{q}$ because of the suppression of backscattering, which is lifted in magnetic field and leads to huge positive MRLiang et al. (2015).

The importance of phase factor in quantum oscillation associated with the symmetry of FS has recently been further stressedAlexandradinata et al. (2018). In the above LK fitting, the Berry phase is free parameter and non-zero values within [0,2$\pi$] for both F1 and F2 are obtained. However, we treat them as nominal value and make our interpretation in another route without Landau level index plot. Since the two observed frequencies are close with similar effective masses and Dingle temperatures, a clear beating pattern is present. Then without the field dependent damping factors, the envelop line is given by a ’slow’ cosine function $\cos[\pi(F1-F2)/B+(\delta_{1}-\delta_{2})/2+(\phi_{1}-\phi_{2})/2]$. Given that the dimensional correction terms $\delta_{i}$ are the same for F1 and F2, the position of nodes of the envelop line is dependent on $(\phi_{1}-\phi_{2})/2$. It is illustrated in Fig.4 (b) that a finite value of phase shift in $\cos[\pi(F1-F2)/B]$ is needed to correctly track the beating nodes, thus the difference of Berry phase terms for F1 and F2 is non-zero at this field direction, here the absolute value $\sim$0.32$\pi$. As a deduction, at least one of the terms should have a non-zero value, possibly due to chirality of the associated Fermions. For our current angular dependent $MR_{OSC}$ of CoSi, the node positions of the envelop line with respect to field direction have a weak but observable evolution, meanwhile the oscillation frequencies are almost constant, then following the above analysis, without invoking other trivial phase factors, the difference of the two Berry phase terms will show an angular dependence, this could serve as a constraint when analyzing how the individual Berry phase term associated with a FS changes with respect to different k-space directions.

In summary, decent quality CoSi single crystals have been grown with the aid of tin flux. QO originated from the bulk electron FSs at the R point of BZ are observed in magnetoresistivity, and the additional phase in QO arising from electron orbit geometry varies with field direction with non-zero values present. However, a full characterization of the bulk electronic structure of CoSi is still at the beginning stage. Further studies are needed to probe the electronic structure at $\Gamma$, due to the heavier effective mass, we planned to carry out transport and magnetization measurements at lower temperature and up to higher magnetic field.

Note added During preparation of this manuscript, we become aware a preprint by Xu et. al.Xu et al. (2019). The authors used Te flux for crystal growth and reached a better RRR for CoSi. QO was observed and yields similar frequencies as ours.

We thank H. M. Weng, Y J. Sun and T. Qian for helpful discussions. This work is supported by the Natural Science Foundation of China with grant No. 11874399, and the National Key Research Program of China with Grant No. 2016YFA0300602.