In ultrapure single HgSe crystal, a large linear magnetoresistance with a record magnitude of 15 000% in mercury chalcogenides was discovered in a quantum limit at the magnetic field of 12 T in the temperature range of 20–40 K. The effect was described in terms of Abrikosov's theory of quantum linear magnetoresistance, thus providing its first experimental confirmation. In the temperature range of 0.4–4.2 K, a magnetic field-induced metal–insulator transition appeared in the ultrapure HgSe. Estimates show that the critical magnetic field of this transition of 6.8 T satisfies the criterion for stabilization of a condensed Wigner phase in the quantum limit.

## I. INTRODUCTION

In recent years, the study of topological semimetals representing a 3D counterpart of graphene has been of particular interest in condensed matter physics. These include Dirac^{1} and Weyl^{2,3} semimetals (DSMs and WSMs). In the former, the band structure contains Dirac points protected against gap formation by crystalline symmetry, in the vicinity of which energy dispersion is linear. In the latter, energy dispersion is also linear near the band torching points (referred to as Weyl nodes), which always come in pairs of opposite chirality. These Weyl nodes can be viewed as magnetic monopoles in *k*-space, generating a Berry curvature.^{4} Additionally, in WSMs, charge carriers are compiled of low-energy quasiparticle excitations known as Weyl fermions. In the present report, we will limit our focus to WSMs. Currently, there are major WSM research areas that include the 3D quantum Hall effect,^{5–9} chiral magnetic effect,^{10–14} planar Hall effect,^{15–20} and valleytronics.^{21,22} All of the above provide both fundamental interest and potential for practical applications in spintronics and optoelectronics.^{23} Therefore, the quest to search and engineer new members of the WSM class of materials is crucial. That said, to isolate WSM-associated physics, it is preferable to deal with a minimum amount of Weyl node types and hence fewer Fermi surface pockets. Mercury selenide crystallizing in a zinc-blend structure may be attributed to such WSMs. The zinc-blende structure consists of two mutually interpenetrating face-centered cubic lattices with a tetrahedral coordination of atoms. HgSe belongs to the space symmetry group $F 4 \xaf3 m$_{,} with the lattice constant *а* = 6.074 Å and coordination number Z = 4. According to our previous studies,^{24–28} there are reasons to consider HgSe as a non-centrosymmetric WSM family candidate.

In the present study, a unique “ultrapure” single HgSe crystal sample was chosen to be investigated. The term “ultrapure” is used to outline both the lowest electron concentration of *n _{e}* = 5.5 × 10

^{15}cm

^{−3}and the highest Hall mobility of

*μ*= 3.4 × 10

_{H}^{5}cm

^{2}V

^{−1}s

^{−1}in HgSe at

*T*= 4.2 K, resulting from long-term passive annealing of this sample.

^{28}An estimate shows that at such a low $ n e$, the share by volume of a Fermi sphere in a whole Brillouin zone is only ∼10

^{−6}, which is an order of magnitude less than that of, for example, high-purity elemental bismuth.

^{29}The remainder of the electronic and geometric sample parameters are given in Ref. 27. The sample had the shape of a rectangular parallelepiped and was not specially oriented. The procedure of sample etching and electrical probe preparation is described in Ref. 24. In previous studies of this sample, the chiral magnetic effect

^{27}and the giant planar Hall effect

^{28}were discovered. Simultaneous observation of these effects in isotropic non-magnetic materials, such as HgSe, provides a strong argument for their occurrence due to chiral anomaly. The aim of the present study was to reveal features of transverse magnetoresistance (MR) and the Hall effect in ultrapure HgSe, which would confirm realization of the topological semimetal phase in mercury selenide. Magnetotransport properties of HgSe were previously studied in Refs. 30 and 31. However, these studies focused mainly on the analysis of Shubnikov–de Haas oscillations in the samples with

*n*≥ 3.5 × 10

_{e}^{16}cm

^{−3}in a magnetic field up to 6 T. In the relevant present discussion case of low electron density, the magnetotransport of the HgSe samples with

*n*= 8.9 × 10

_{e}^{15}and 2.5 × 10

^{16}cm

^{−3}was studied in the magnetic field up to 12 T in Refs. 24 and 26, respectively. The comparison of these results with those obtained for the ultrapure HgSe sample will be discussed briefly in Sec. III.

## II. EXPERIMENTAL DETAILS

Transverse MR and Hall effect measurements were accomplished using a standard four-probe technique involving the equipment of the “Testing Center of Nanotechnology and Advanced Materials” Collaborative Access Center of the Institute of Metal Physics of the Ural Branch of the Russian Academy of Sciences. Voltage signals $ V x x$ and $ V x y$ providing transverse magnetoresistance $ \rho x x$ and Hall resistance $ \rho x y$ were registered using two pairs of electrical probes each [Fig. 1(a)]. During the investigation, it was revealed that the difference in signals registered from these two pairs of probes is insignificant. For this reason, the experimental data on $ \rho x y$ and $ \rho x x$ will, hereinafter, represent values averaged over two pairs of probes. Magnetotransport properties were studied in magnetic field $ B \u2192$ up to 12 T, aligned perpendicular to the electrical current $ J \u2192$ ( $ B \u2192\u22a5 J \u2192$, so-called “transverse configuration”). Measurements were performed either in a variable temperature insert system, with a temperature variable between 2 and 40 K, or in a helium-3 system with the base temperature down to 0.4 K. To eliminate nonsymmetrical effects of the electrical probes, magnetotransport measurements were performed in two opposite directions of the magnetic field.

## III. RESULTS AND DISCUSSIONS

Figure 1(b) shows the dependence of reduced transverse magnetoresistance $MR= \rho x x ( B ) \u2212 \rho 0 \rho 0\xd7100%$ on magnetic field *B* for the ultrapure sample at various temperatures (given $ \rho 0$ is zero field resistance). Figure 2(a) shows $ \rho x y(B)$ dependence for the ultrapure HgSe sample at the same temperatures as for the $MR(B)$ dependence shown in Fig. 1(b). It is convenient to discuss the specifics of magnetotransport at relatively high temperatures (20–40) K first. At these temperatures, $MR$ demonstrates an increase with *B*, reaching ≈15 000% at *B* = 12 T, without a saturation tendency [Fig. 1(b)]. Previously, we found peak values of the transverse $MR$ increase in HgSe to be ≈7000% for the sample with *n _{e}* = 2.5 × 10

^{16}cm

^{−3}(Ref. 24) and 5500% for the sample with

*n*= 8.9 × 10

_{e}^{15}cm

^{−3}.

^{26}Therefore, Fig. 1(b) shows the largest $MR$ observed so far in mercury selenide. Overall, the large positive transverse MR seems to be a common feature upon topological semimetals possessing high mobility of charge carriers,

^{32,33}in which, generally, the transport relaxation time $ \tau t r= \mu H m \u2217 e$ (given $ m \u2217$ is the electron’s effective mass and

*e*is the electron charge) significantly exceeds the quantum scattering time $ \tau Q= \u210f 2 \pi k B T D$ (given $ k B$ is the Boltzmann constant, $ T D$ is the Dingle temperature, and ℏ is the reduced Planck constant).

The condition $ \tau t r\u226b \tau Q$ suggests the dominance of small-angle scattering, which, in turn, implies protection against large-angle scattering (or “backscattering”) in a zero magnetic field.^{32,33} Such a protection mechanism was developed on the basis of the quantum mechanical theory of chiral wave-packet scattering by a single impurity.^{34} According to the theory, in DSMs and WSMs, electrons gain a chirality-protected shift to side jump due to the definite chirality. The lifting of such protection against backscattering by the applied magnetic field leads to a large positive MR. It is important to note that the main applicability condition for the theory^{34 }—that is, well-defined chirality—holds in ultrapure HgSe. This condition implies that the Fermi energy is close to Weyl nodes, or in other terms, $ k F\u226a\Delta k W$, given *k _{F} *= 5.18 × 10

^{−3}Å

^{−1}is the Fermi wave vector

^{27}and Δ

*k*≈ 8 × 10

_{W}^{−2}Å

^{−1}is an estimated distance between Weyl nodes in HgSe.

^{25}Additionally, the condition $ \tau t r\u226b \tau Q$ also holds in HgSe with sufficiently low $ n e$.

^{24}Another possible source of the large positive MR in topological semimetals is a pronounced charge compensation, as in WTe

_{2}.

^{35}However, this mechanism is irrelevant in HgSe since its Fermi surface lacks hole pockets.

^{36}

Summarizing all of the above, there is a single general approach for interpreting large positive MR without a saturation trend in topological semimetals,^{34} which could have means to be applied to HgSe. However, in the ultrapure HgSe sample, the dependence of $ \rho x x$ vs *B* possesses one nontrivial feature at *T* = 20 and 40 K, namely, its distinct linearity at $B\u22733 T$ [Fig. 3(a)], which is not provided by Ref. 34. The discovered linear MR in ultrapure HgSe requires detailed consideration. To begin with, this effect emerges at unusually low $ n e$, which suggests the reaching of an extreme quantum limit (i.e., when only the zeroth Landau level remains occupied) at a moderate magnetic field $ B Q L$. Moreover, the 3D electronic system transforms into a quasi-one-dimensional metal in the quantum limit. Such a system with low electronic density exhibits pronounced Fermi surface instabilities, driven by interaction effects, toward new electronic phases, such as a spin density wave, a charge density wave, and a Wigner crystal.^{37} For ultrapure HgSe, the experimentally obtained field accounts for *B _{QL} *≈ 2 T,

^{27}which is marked by an arrow in Fig. 3(a). It is common knowledge that the magnetic length $ l B= ( \u210f e B ) 1 2$ in the quantum limit becomes less than the Fermi wavelength $ \lambda F= 2 \pi k F$. At the magnetic field $B= B Q L$, an equality $ \lambda F=2\pi l B Q L$ becomes valid. This gives the estimate of $ B Q L= \u210f k F 2 e=1.8 T$, which is in good agreement with the experiment. Therefore, the linear MR in ultrapure HgSe does emerge in the extreme quantum limit.

^{38}

*n*are Landau indexes, $ k z$ is a wave vector in the direction of the magnetic field, $ v F$ is Fermi velocity, and “ $\xb1$” corresponds to different Weyl nodes with opposite chirality. Therefore, at $ k z=0$, for $n=1$, the energy is given by $ \epsilon 1=\xb1 v F 2 \u210f e B$. For ultrapure HgSe,

*v*= 1.85 × 10

_{F}^{5}m s

^{−1}and Fermi energy

*ɛ*= 6.3 meV.

_{F}^{27}By substituting the $ v F$ value, the result is $ \epsilon 1=6.5 B meV$ (given

*B*is in tesla). This implies that in HgSe, within the linear MR domain, the Fermi energy is less than the distance between the zeroth and the first Landau levels (i.e., $ \epsilon F< \epsilon 1$). This is depicted in Fig. 3(b), which also demonstrates the fact that the WSM energy spectrum in the quantum limit is relativistic. Thus, the observed linear MR in ultrapure HgSe is distinctive since it is observed in the quantum limit involving charge carriers with the linear dispersion law. Therefore, this unusual effect is optimal for treatment using the quantum linear MR theory developed by Abrikosov for semimetals having very small closed Fermi surfaces such as bismuth.

^{39–41}According to Ref. 41, quantum linear MR can be written as follows:

^{42}From a physical standpoint, in HgSe, $\gamma $ should be in the order of 1. In Fig. 3(a), solid lines depict linear MR fitting by means of Eq. (2), with $\gamma $ acting as a fitting parameter. Fitting results in values of $\gamma =0.6 and 0.7$ at

*T*= 20 and 40 K, respectively. A slight excess of $ n e$ over $ N i$ can be associated with intrinsic charge carriers in HgSe at

*T*= 20 and 40 K. As can be seen from Fig. 3(c), a relative deviation $\Delta = \rho x x e x p \u2212 \rho x x f i t t \rho x x e x p\xd7100%$ of experimental MR from the fitting does not exceed 3%, which is within the experimental accuracy. It should be noted that the linear MR changes rather slightly with an increase in

*T*from 20 to 40 K [Fig. 3(a)]. This is in qualitative agreement with Abrikosov's theory, since according to Eq. (2), $ \rho x x$ does not explicitly depend on temperature (see Ref. 39 for details). Another important aspect is that the $ \rho x y(B)$ dependence appears to be essentially linear in the quantum limit at

*T*= 20 and 40 K [Fig. 2(a)]. This satisfies Abrikosov's theory outcomes, which in quantum limit for relativistic electron spectrum predicts the formula $ \rho x y= B e n e$, reminiscent of the ordinary Hall effect in a weak magnetic field limit.

^{39}Based on this formula, an estimate of

*n*= 1.6 × 10

_{e}^{16}cm

^{−3}was obtained (provided that temperature effects on concentration are omitted) using linear fitting of $ \rho x y(B)$ at

*T*= 20 K [dashed line in Fig. 2(a)], which is consistent with

*n*= 5.5 × 10

_{e}^{15}cm

^{−3}evaluated from Shubnikov–de Haas oscillations.

Thus, the linear MR in ultrapure HgSe is quantitatively described by Abrikosov's theory of quantum linear MR. Previously, linear transverse MR in the quantum limit was observed in *n*-InSb with low electron density.^{43} However, in this case, Abrikosov's theory would be inappropriate to apply since this classical narrow-gap semiconductor does not possess the linear electron spectrum essential for Abrikosov's model. Thereby, the possibility of implementing Abrikosov's quantum linear MR in topological materials in the quantum limit was demonstrated for the first time by the example of HgSe. It is important to note that the quantum limit becomes a basic element for the emergence of the linear MR in HgSe. Unlike the ultrapure sample, in HgSe possessing *n _{e} *= 2.5 × 10

^{17}cm

^{−3}, linear MR was not observed,

^{44}which correlates with inaccessibility of the quantum limit for this sample in the magnetic field up to 12 T.

Now, let us discuss the specifics of magnetotransport at relatively low temperatures, (0.4–4.2) K. As can be seen from Fig. 1(b), in this temperature range, $MR(B)$ decreases in magnitude down to (6–8) × 10^{3}% in *B* = 12 T, while losing linearity. The cause of this effect is in the threefold decrease in $ \rho 0$ from ≈4 × 10^{−3} Ohm cm at *T* = (0.4–4.2) K to ≈1.4 × 10^{−3} Ohm cm at *T* = (20–40) K. In turn, such a decrease in $ \rho 0$ with temperature stems from an increase in the contribution of intrinsic carriers to electrical conductivity. For the same reason, in aforementioned HgSe samples with *n _{e}* = 8.9 × 10

^{15}and 2.5 × 10

^{16}cm

^{−3}, $ \rho 0$ also decreases with

*T*. Therefore, the data from Refs. 24 and 26 on $MR(B)$ at different temperatures correlate with those depicted in Fig. 1(b). That said, another feature emerges in magnetoresistance in ultrapure HgSe at low temperatures. As can be seen from Fig. 1(c), all $ \rho x x(B)$ curves intersect in a single point at

*B*≈ 6.8 T. Above $ B c$, the magnitude of $ \rho x x(B)$ decreases with the increase in

_{c}*T*as expected from an insulator, whereas below $ B c$, the magnitude of $ \rho x x(B)$ increases with temperature, behaving like in a metal. This feature clearly signals a critical point for metal–insulator (MI) transition in the quantum limit. Recently, a similar electronic phase transition was found in bulk topological semimetals ZrTe

_{5}

^{37}and HfTe

_{5}

^{45}in the quantum limit at

*B*≈ 7 T. It should be noted that the linear dependences of $ \rho x x(B)$ in Fig. 3(a) also intersect in a single point, although at ≈10 T. However, since Abrikosov's theory operates with a free electron gas, the emergence of such an intersection being random in character should not be regarded as the magnetic-field-induced MI transition.

_{c}According to a scaling theory for the magnetic-field-induced 3D MI transition in the quantum limit,^{46} this can be a consequence of the formation of either a charge density wave driven by a strong electron–electron interaction in addition to an electron–phonon interaction or the Wigner crystal, in case, the electron-phonon interaction can be omitted. In the former, Peierls transition leads to a gap formation at the Fermi level along the *z*-direction with a period corresponding to a nesting wave vector close to $2 k F , z$. Such a charge density wave can manifest itself as a quasi-plateau in the magnetic field dependence of the Hall resistance $ \rho x y(B)$ in the quantum limit.^{37,45} As can be seen in Fig. 2(a), neither a pronounced plateau nor any other features are observed in $ \rho x y(B)$ at $B> B c $ up to 12 T. On the other hand, the Hall effect responds to MI transition with a minimum in derivative $ d | \rho x y | d B$ in a magnetic field close to $ B c$ [Fig. 2(b)]. With this in mind, the origin of MI transition in the quantum limit might be related to a spatial electronic density modulation as in Wigner crystallization. It is noteworthy that for the ultrapure HgSe crystal, the condition $V>K$ for the emergence of a Wigner crystal is satisfied even in a zero magnetic field (given $V= e 2 \epsilon r 0 $ is an average energy of the Coulomb interaction, $K= \u210f 2 2 m \u2217 r 0 2$ is an average kinetic energy, and $ r 0= ( 3 4 \pi n e ) 1 3$ is an average inter-electron spacing). This condition is equivalent to the more descriptive one $ n e\u226a n 0$,^{47} where the parameter $ n 0= ( m \u2217 e 2 \u210f 2 \epsilon ) 3\u22487\xd7 10 16 c m \u2212 3$ for HgSe (given the effective mass $ m \u2217=0.032 m 0$).^{25,27} Quantizing magnetic field increases $ V K$ by lowering $ \epsilon F\u223c n e B 2$,^{48} which facilitates electron ordering and strengthens the electronic lattice. The hypothesis of the emergence of the Wigner crystal in ultrapure HgSe is supported by the fulfillment of the criterion of free electron gas crystallization $ l B \u2272 r 0$ in the magnetic field $B= B c$.^{49} Indeed, for the ultrapure HgSe sample, the estimated values of spacing $ r 0$ and magnetic length $ l B c$, which is inherently a radius of electron wavefunction in the plane perpendicular to the magnetic field direction, are ≈30 and ≈10 nm, respectively. With this in mind, fading of a minimum in $ d | \rho x y | d B$ at $T=4.2 K$ relative to *T* = 0.4 and 1 K suggests proximity to the melting point of the Wigner crystal.

## IV. CONCLUSIONS

In summary, a large positive transverse magnetoresistance without saturation tendency was observed in the ultrapure single HgSe crystal sample. The magnitude of the effect reaches 15 000% in the magnetic field of 12 T, which is a record value for mercury chalcogenides. Moreover, in the quantum limit, the magnetoresistance becomes linear at relatively high temperatures. We argue that a topological semimetal having low electron density is an ideal system to study Abrikosov's quantum linear magnetoresistance. It was demonstrated that Abrikosov's theory quantitatively describes the observed phenomenon. Such an extraordinary response to a magnetic field as a large linear magnetoresistance makes pure HgSe a promising candidate for magnetic sensors,^{50} magnetic random access memory,^{51} and other spintronic devices.^{52} From the fundamental perspective, the revealed Abrikosov's quantum linear magnetoresistance can be considered as the substantive confirmation of the nontrivial electronic topology of HgSe since this effect requires the electron spectrum in the quantum limit to be purely relativistic. Additionally, metal–insulator transition was revealed in ultrapure HgSe in the quantum limit at low temperatures. It was assumed that the origin of this transition involves Wigner crystallization of free electron gas in the magnetic field.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**A. T. Lonchakov:** Conceptualization (lead); Data curation (equal); Investigation (equal); Writing – original draft (lead). **S. B. Bobin:** Data curation (equal); Investigation (equal); Visualization (lead); Writing – review & editing (equal).

### Author Contributions

**A. T. Lonchakov:** Conceptualization (lead); Data curation (equal); Investigation (equal); Writing – original draft (lead). **S. B. Bobin:** Data curation (equal); Investigation (equal); Visualization (lead); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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