Introduction

Weyl fermions were originally introduced as elementary particles in high energy physics more than 80 years ago1. While evidences of Weyl fermions as elementary particles remain elusive, realizing them as low-energy quasiparticles in solids has recently attracted increasing interest2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18, especially after the discovery of topological insulators19,20. In solids, Weyl nodes arise as discrete band-crossing points in crystal momentum space near the Fermi level, and Weyl fermions, as quasiparticles with linear dispersions across the Weyl nodes, can be described by the massless Dirac equation21. Weyl nodes carry a left- or right-handed chirality and they always appear in pairs of opposite chiralities according to the no-go theorem21. Intriguingly, it has been shown that the chiral anomaly in Weyl semimetals induces many manifestations in transport properties21,22,23,24,25,26,27,28,29,30 such as negative magnetoresistance, anomalous Hall effect and chiral magnetic effects. Another important hallmark of Weyl semimetals is its topologically protected unusual surface states—surface Fermi arcs2.

To realize Weyl semimetals in solids, either the time-reversal or lattice inversion symmetry has to be broken2,3,4,5,6,7,8,9,10,11,12,13. Recently, Weyl nodes were experimentally observed in noncentrosymmetric TaAs-class materials14,15,16,17,18. It was reported14,15,16 that these compounds host 24 Weyl nodes which are close to but not exactly at the Fermi level due to the fact that not all Weyl nodes are symmetry related. Moreover, there are also trivial Fermi pockets at the Fermi level in addition to the Weyl nodes. It is thus desirable to search for ideal Weyl semimetals with all Weyl nodes exactly residing at the Fermi level at stoichiometry composition (as shown in Fig. 1a) considering that ideal Weyl semimetals could maximize the potential for transport phenomena induced by the chiral anomaly and that supersymmetry may emerge at the pair-density-wave quantum criticality in an ideal Weyl semimetal31.

Figure 1: Phase diagram of strained HgTe.
figure 1

(a) The Weyl nodes in ideal Weyl semimetals are exactly at the Fermi level without coexisting trivial Fermi pockets. (b) The schematic representation of HgTe lattice with a typical zinc-blende structure. (c) For a certain range of strain in the xy plane, HgTe has three phases: ideal Weyl semimetals, type-II Weyl semimetals and topological insulators. (d) When the compressive strain is large enough , HgTe turns into the ideal Weyl semimetal phase. At stoichiometry, all Weyl nodes are exactly at the Fermi level. (e) When the strain is small enough , HgTe is in the type-II Weyl semimetal phase. (f) When the tensile strain is large enough , HgTe turns into the topological insulator phase.

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In this work, we show that the HgTe-class materials with both band inversion and lattice noncentrosymmetry, including mercury chalcogenide HgX (X=Te, Se)32 and the multifunctional half-Heusler compounds33,34,35, can realize symmetry-protected ideal Weyl semimetals with four pairs of Weyl nodes under a broad range of in-plane biaxial compressive strain. Note that strain engineering has been successfully employed in condensed matter systems to realize many physics36,37, which makes the experimental realization of strained HgX and half-Heusler compounds highly feasible. For simplicity, we focus on HgTe to discuss concrete results and to illustrate the general guiding principle to realize ideal Weyl semimetals in such class of materials.

Results

The effective model of HgTe

HgTe has a typical zinc-blende structure with the space group (Fig. 1b) and it is known to have an inverted band structure32 such that it is a semimetal and its Γ8 bands are half-filled. The Γ8 bands at the Γ point are fourfold degenerate as J=3/2 multiplet. Another important feature of HgTe is its bulk inversion asymmetry (BIA), which plays an essential role in realizing ideal Weyl fermions in strained HgTe as we shall show below. The Γ8 bands around the Γ point can be effectively described by the following Luttinger Hamiltonian38 plus perturbations due to the BIA:

where Ji are spin-3/2 matrices and αi are constants characterizing the band structure. The main perturbations induced by BIA are given by , where {} represents anti-commutator, c.p. means cyclic permutations, and α,β are constants characterizing the strength of BIA39,40. By fitting the first-principle band structures around the Γ point, parameters above can be determined: for HgTe, α0≈109.8 Å2 eV, α1≈−45.87 Å2 eV, α2≈−19.73 Å2 eV and α≈0.208 Å2 eV.

Because of the BIA, dispersions around the Γ point are not purely quadratic and the Fermi level at stoichiometry is slightly above the fourfold degeneracy energy of the Γ8 bands41. Consequently, there exist tiny electron and hole pockets at the Fermi level. The existence of small Fermi pockets is further indicated by the line crossings between two intermediate bands protected by the crystalline symmetries of the cubic HgTe (Supplementary Fig. 1; Supplementary Note 1). Breaking the crystalline symmetry from Td to D2d by an in-plane strain can remove the line crossings and render the realization of ideal Weyl semimetals generically inevitable as we show below.

Strain in the xy plane generates a perturbation , where g depends on the strength of strain and the lattice constant in the xy plane changes to a=(1+δ)a0. As explained in Supplementary Note 1, g>0 (g<0) for tensile strain δ>0 (compressive strain δ<0). The in-plane strain reduces the original cubic symmetry Td to the tetragonal symmetry D2d in which only two mirror planes (the kx=±ky planes) survive and the other four are broken. As a result, line crossings originally protected by the broken mirror symmetries split except at discrete points in the kx=0 or ky=0 plane. The crossing points in the kx=0 or ky=0 plane are robust because these planes respect a special symmetry C2T=C2·T, where C2 is the two-fold rotation along the x or y axis and T is the time-reversal transformation (see Supplementary Note 1 for details). In total, there are eight Weyl nodes with four in the kx=0 or ky=0 plane, respectively.

The phase diagram of strained HgTe

Under a sufficiently small strain, these Weyl nodes are type-II (ref. 13; see also Supplementary Fig. 2); they are close to but not exactly at the Fermi level and there are also other coexisting trivial Fermi pockets. Note that tensile and compressive strains have qualitatively different effect in moving these type-II Weyl nodes. Under a large enough tensile strain, the eight Weyl points annihilate with each other in the kx=±ky planes, leading to a strong topological insulator, as shown in Fig. 1f, which agrees with previous theoretical calculations40 and experimental observations42. Intriguingly, under an increasing compressive strain, these type-II Weyl nodes quickly evolve to type-I Weyl nodes, shown in Fig. 1d, all of which lie exactly at the Fermi level leading to an ideal Weyl semimetal at stoichiometry, similar to graphene! Based on the k·p Hamiltonian , we obtain the whole phase diagram, schematically shown in Fig. 1c, where the strained HgTe is a strong topological insulator when , and is an ideal Weyl semimetal when . In the ideal Weyl semimetal phase, HgTe has eight Weyl nodes at and , as schematically shown in Fig. 2.

Figure 2: Schematic of Weyl points in Brillouin zone.
figure 2

The bulk Brillouin zone (BZ), and (001) and (010) surface BZs of HgTe. In the ideal Weyl semimetal phase of the strained HgTe, there are four pairs of ideal Weyl nodes in the bulk BZ, schematically shown as red (chirality +1) and blue (chirality −1) circles. The pink (yellow) plane is for the ky=0 (kx=0) plane. The projections of bulk Weyl nodes onto the (001) and (010) surface BZs are shown; there are four gapless points in the (001) surface BZ but six gapless points in the (010) surface BZ.

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The analysis of topological properties

Now, we use the k·p theory to analyse topological properties of ideal Weyl nodes. We first consider the effect of the strain and treat the BIA as a perturbation. It is straightforward to show that hosts two Dirac points in the kz axis. The BIA perturbation splits each Dirac point into four Weyl nodes. The effective Hamiltonian around one of the Weyl points is described by the Weyl equation:

where vi are Fermi velocities given in Supplementary Note 3. For HgTe, we find that the Weyl node located at is right-handed, from which the chirality of other Weyl nodes can be derived since all of them are related by symmetry. The locations and chiralities of the eight ideal Weyl nodes are shown in Fig. 2.

One hallmark of Weyl semimetal is the existing of topologically protected surface Fermi arcs. In the ideal Weyl semimetal phase, the electronic states in the kz=0, kx=ky and kx=−ky planes are all gapped so that the Z2 topological invariant in any of these two-dimensional time-reversal-invariant subsystems is well-defined; these Z2 invariants are all nontrivial because of the band inversion at the Γ point. Consequently, these subsystems carry gapless helical edge modes, which have important implications to possible Fermi arc patterns. Combined with the known chirality of different Weyl nodes, it is expected that the Fermi arcs form a closed circle in (001) surface Brillouin zone (BZ) and that there are open Fermi arcs on the (100) or (010) surface BZ.

The first-principles calculations

To confirm the results predicted in the effective k·p theory above, we carry out first-principles calculations on HgTe. For the compressive in-plane strain with a=0.99a0 and c=1.02a0, while the fourfold degeneracy of the Γ8 band at the Γ point is lifted to open a gap, the two intermediate bands touch at eight discrete points: and with Å−1 and Å−1, schematically shown in Fig. 2. These eight gapless points are exactly the Weyl nodes predicted by the effective k·p model above. Importantly, these Weyl nodes all exactly locate at the Fermi level without coexisting with any trivial bands. It precisely realizes an ideal Weyl semimetal! On the contrary, with the tensile strain with a=1.01a0 and c=0.98a0, a full band gap opens in the entire BZ, which results in a strong topological insulator because of the band inversion between the Γ6 and Γ8 bands. Qualitative difference between tensile and compressive in-plane strains in HgTe is due to the different response of px,y and pz orbitals to the strain.

A key feature of the Weyl semimetal is its unusual surface Fermi arcs, which start from one Weyl node and end at another with opposite chirality2. Figure 3a shows the projected bands along in the (001) surface BZ and surface states can be seen clearly. As the gapless point in the line, projected from the two Weyl nodes , has monopole charge −2, two Fermi arcs have to connect to this point. Similar consideration can be applied to other gapless points and in the (001) surface BZ. The Fermi arcs form a closed loop, as shown in Fig. 3b, which is consistent with the nontrivial Z2 topological invariant defined in the two-dimensional planes of kx=±ky. Along the bulk-gapped line of , there is a single edge mode across the Fermi level, consistent with the topological properties of Fermi arcs forming a loop on the (001) surface BZ.

Figure 3: Surface states and Fermi arcs.
figure 3

Electronic structure of HgTe with surfaces in the ideal Weyl semimetal phase. (a) Band structure projected onto the (001) surface BZ. The red lines denote the surface states. (b) Surface Fermi arcs in the (001) surface BZ form a closed loop, because each gapless point has the monopole charge of +2 or −2. (c) Band structure projected onto the (010) surface BZ. (d) Surface Fermi arcs in the (010) surface BZ. (e,f) To see Fermi arcs clearly, the Fermi surfaces around the Weyl nodes are zoomed in.

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The Fermi arcs on the (010) surface is even more interesting. From the projected bands shown in Fig. 3c, we can see that a single chiral edge mode comes out of the gapless points and two chiral edge modes come out of the gapless point , which are consistent with their monopole charges +1 and −2, respectively. The corresponding Fermi surface is shown in Fig. 3d, and the marked regions are zoomed-in in Fig. 3e,f, where discontinuous Fermi arcs are clearly seen. The Fermi arc originating from the point and terminating at the point spans across a significant portion of the BZ (about 9% of reciprocal lattice constant), which is of great advantage for experimental detection such as angle resolved photoemission spectroscopy (ARPES). The Fermi arc pattern may change upon varying chemical environment on the surface, but the parity of number of Fermi arcs is topologically stable.

The analysis of realizing ideal Weyl fermions in strained HgTe leads to a general guiding principle to search for ideal Weyl semimetals in noncentrosymmetric materials with nontrivial band inversion. Noncentrosymmetric half-Heusler compounds with band inversion are another family of such materials which could also realize ideal Weyl semimetals under a compressive in-plane strain. For illustration, we perform first-principles calculations on LaPtBi (Fig. 4a) with a compressive in-plane strain of a=0.99a0 and c=1.02a0. Indeed, it also shows eight ideal Weyl nodes at and with Å−1 and Å−1. Similar to HgTe, the surface Fermi arcs also forms a closed loop in the (001) surface BZ, as shown in Fig. 4b,c. The Fermi arcs in the (010) surface are shown in Fig. 4d, whose marked regions are zoomed-in in Fig. 4e,f.

Figure 4: Surface states and Fermi arcs of strained LaPtBi.
figure 4

Electronic structure of LaPtBi with surfaces in the ideal Weyl semimetal phase. (a) The stuffed zinc-blende structure of LaPtBi. (b) Surface Fermi arcs in the (001) surface BZ also forms a closed loop, though the pattern is different from that of HgTe. (c) The surface states around are zoomed in. (d,e,f) Fermi surfaces in the (010) surface BZ exhibit Fermi arcs in d, and the marked regions are zoomed-in in e and f, respectively.

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Discussion

We have shown that applying in-plane strain in HgTe-class materials can give rise to ideal Weyl semimetals with only four pairs of Weyl nodes exactly locating at the Fermi level. Experimentally, such strain in the xy plane may be effectively obtained by growing samples on substrates with a smaller lattice constant. For example, materials with zinc-blende structure such as GaSb, InAs, CdSe, ZnTe and HgSe are promising substrates for growing HgTe to achieve the desired in-plane compressive strain. The locations of Weyl nodes for various strength of strains in HgTe and LaPtBi are calculated and summarized in Supplementary Table 1 (Supplementary Note 2). The predicted Weyl nodes and surface Fermi arcs can be directly verified by ARPES measurements. Moreover, transport experiments can be used to detect unusual bulk magneto-transport properties induced by the chiral anomaly of ideal Weyl fermions27.

Ideal Weyl semimetals predicted in the strained HgTe-class materials are interesting on their own. It further provides a perfect platform to study the interplay between ideal Weyl fermions and other exotic phenomena, especially in half-Heusler compounds, including superconductivity in LaPtBi (ref. 43), magnetism in GdPtBi, and heavy fermion behaviour in YbPtBi (ref. 44). Interestingly, space-time supersymmetry may also emerge at the superconducting quantum critical point in ideal Weyl fermions31. This class of ideal Weyl semimetals thus opens a broad avenue for fundamental research of emergent physics as well as potential applications of low-power quantum devices.

Methods

The first-principles calculations

The first-principle calculations are carried out in the framework of the Perdew–Burke–Ernzerhof-type generalized gradient approximation of the density functional theory through employing the BSTATE package45 with the plane-wave pseudo-potential method. The kinetic energy cutoff is fixed to 340 eV, and the k-point mesh is taken as 16 × 16 × 16 for the bulk calculations. The spin-orbit coupling is self-consistently included. The experimental lattice constants are used with a0=6.46 Å for HgTe and 6.829 Å for LaPtBi. In order to simulate the uniaxial strain along the [001] direction, we fix the experimental volume but change the ration a/c, where a is the lattice constant in the x–y plane and c is the lattice constant along z axis (the [001] direction). Without loss of generality, we take the parameters a=0.99a0 and c=1.02a0 for the tensile strain and a=1.01a0 and c=0.98a0 for the compressive strain. To exhibit unique surface states and Fermi arcs on the surface, we employ maximally localized Wannier functions46,47 to first obtain the ab initio tight-binding model of the bulk HgTe and LaPtBi and then pick these bulk hopping parameters to construct the tight-binding model of the semi-infinite system with the (001) or (010) surface as the boundary. The surface Green’s function of the semi-infinite system, whose imaginary part is the local density of states to obtain the dispersion of the surface states, can be calculated through an iterative method48.

Additional information

How to cite this article: Ruan, J. et al. Symmetry-protected ideal Weyl semimetal in HgTe-class materials. Nat. Commun. 7:11136 doi: 10.1038/ncomms11136 (2016).