For the first time, magnetoresistive properties of the single crystal of HgSe with a low electron concentration were studied in wide range of temperature and magnetic field. Some fundamental parameters of spectrum and scattering of electrons were experimentally determined.Two important features of magnetic transport were foundstrong transverse magnetoresistance (МR) and negative longitudinal MR, which can indicate the existence of the topological phase of the Weyl semimetal (WSM) in HgSe. Taking this hypothesis into account we suggest a modified band diagram of the mercury selenide at low electron energies. The obtained results are essential for the deeper understanding of both physics of gapless semiconductors and WSMspromising materials for various applications in electronics, spintronics, computer and laser technologies. Weyl and Dirac semimetals have recently attracted great attention as materials that possess strong spin-orbit coupling (SOC) and relativistic electron spectrum. Such materials can be considered to be three-dimensional analogues of graphene. For an WSM phase to exist there should be breaking of either inversion symmetry, as in monophosphides and monoarsenides of tantalum and niobium TaP [1,2], TaAs [3], NbP [4], NbAs [5], or time reversal symmetry, as in the ferromagnetic spinel HgCr2Se4 [6]. The breaking of inversion symmetry also occurs in mercury selenidegapless semiconductor that crystallizes into a zinc-blende structure and has inverted electron spectrum in the center of the Brillouin zone. 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 F43m with the lattice constant а = 6.074Å and coordination number Z = 4. The band order is inverted because of relativistic effects including SOC [7]. Mercury selenide with low electron concentration can become relevant material in the topological condensed matter physics. In this regard, it should be noted that in HgSe under no annealing condition could the electron concentration be reduced below ~ 10 16 cm -3 at 4.2K [7]. Smallness of the electron concentration is essential to reveal the Weyl nodesfeatures of an energy spectrum of topological nature. The Weyl nodes (or magnetic monopoles) are band touching points in the momentum space, which always come in pairs of opposite chirality [8]. Near

In this paper, the authors report the results of an experimental study of effective mass, electron mobility and phase shift of Shubnikov-de Haas oscillations of transverse magnetoresistance in an extended electron concentration region from 8.8×10 15 cm -3 to 4.3×10 18 cm -3 in single crystals of mercury selenide. The revealed features indicate that Weyl semimetal phase may exist in HgSe at low electron density. The most significant result is the discovery of an abrupt change of Berry phase ≈ at electron concentration ≈ 2×10 18 cm -3 , which we explain in terms of a manifestation of topological Lifshitz transition in HgSe that occurs by tuning Fermi energy via doping.

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The magnetoresistance (MR) and Hall effect of a single HgSe crystal with an extremely low electron concentration of 8.8 × 10 15 cm −3 were studied in a quantising magnetic field applied both along and across the direction of the electric current. As the result, a broad plateau was discovered in the ordinary (transverse) Hall resistance in the quantum limit. Within a framework of quantum spin Hall effect for an inversion breaking Weyl semimetal, we associate this plateau with a contribution to Hall conductivity from Chern insulator edge states when only a zero Landau level is occupied. In addition to the plateau in the quantum limit, we also detected a well-developed plateau-like behaviour in a phenomenologicallyintroduced 'longitudinal' Hall resistivity. In the 'longitudinal' Hall conductivity, a step-like behaviour was revealed, which we identify with the discovery of half-integer quantum spin Hall effect in HgSe. This effect, being purely topological in origin, supplements the non-trivial Weyl semimetal physics and may serve as a promising magnetotransport method for the detection of Weyl nodes in a studied material.