Pressure-induced topological phases of KNa2Bi

We report an ab initio study of the effect of hydrostatic pressure and uniaxial strain on electronic properties of KNa2Bi, a cubic bialkali bismuthide. It is found that this zero-gap semimetal with an inverted band structure at the Brillouin zone center can be driven into various topological phases under proper external pressure. We show that upon hydrostatic compression KNa2Bi turns into a trivial semiconductor with a conical Dirac-type dispersion of electronic bands at the point of the topological transition while the breaking of cubic symmetry by applying a uniaxial strain converts the compound into a topological insulator or into a three-dimensional Dirac semimetal with nontrivial surface Fermi arcs depending on the sign of strain. The calculated phonon dispersions show that KNa2Bi is dynamically stable both in the cubic structure (at any considered pressures) and in the tetragonal phase (under uniaxial strain).

We report an ab initio study of the effect of hydrostatic pressure and uniaxial strain on electronic properties of KNa 2 Bi, a cubic bialkali bismuthide. It is found that this zero-gap semimetal with an inverted band structure at the Brillouin zone center can be driven into various topological phases under proper external pressure. We show that upon hydrostatic compression KNa 2 Bi turns into a trivial semiconductor with a conical Dirac-type dispersion of electronic bands at the point of the topological transition while the breaking of cubic symmetry by applying a uniaxial strain converts the compound into a topological insulator or into a three-dimensional Dirac semimetal with nontrivial surface Fermi arcs depending on the sign of strain. The calculated phonon dispersions show that KNa 2 Bi is dynamically stable both in the cubic structure (at any considered pressures) and in the tetragonal phase (under uniaxial strain).
Alkali metal bismuthides form a group of compounds which attract much attention as materials with a variety of interesting physical properties. The Bi-based binary compounds, A 3 Bi (A = Na, K, Rb), belong to a special class of topological electronic states, three-dimensional (3D) Dirac semimetals [1][2][3] , where the valence and conduction bands touch at discrete points at the Fermi level. Such materials with a conical 3D dispersion of electronic bands at one or more crossing points and with nontrivial surface states (so-called Fermi arcs) are being given much attention [1][2][3][4][5][6][7][8] .
The 3D conical dispersion relations can also be achieved at the point of semimetal (metal) -to -semiconductor topological transition by tuning a system parameter such as alloying composition or chemical doping [9][10][11] . For example, in zinc-blende crystals of HgCdTe the conical dispersion with some peculiar properties similar to those in Dirac and Weyl semimetals is achieved by changing cadmium concentration 11 . By varying a chemical composition and thereby tuning both the strength of spin-orbit coupling and the lattice parameters, one can also convert a system into a topological insulator 9 . Another possibility of tuning the topological order is to apply an external strain what was theoretically demonstrated for a number of narrow gap cubic semiconductors like α-Sn (grey tin), HgTe, and InSb [12][13][14][15][16][17] or to combine the strain with adjusting the alloy composition 18 .
Up to now there is a lack of data concerning the electronic properties of ternary alkali bismuthides, such as KNa 2 Bi. Unlike the well known K 3 Bi and Na 3 Bi which have a hexagonal low temperature phase 19,20 the crystal structure of KNa 2 Bi is cubic, Fm m 3 21 . In this paper we theoretically demonstrate that KNa 2 Bi, a zero-gap semimetal with an inverted band structure at the Brillouin zone center, can be driven into different topological phases by applying pressure. We show that under hydrostatic compression the compound undergoes a transition to a conventional semiconductor with a conical Dirac-like dispersion of electronic bands at the point of the topological transition whereas upon uniaxial strain the compound can be converted into a topological insulator or into a three-dimensional Dirac semimetal with nontrivial Fermi arcs on the surfaces.

Results and Discussion
KNa 2 Bi crystallizes in the cubic Fm m 3 structure 21 composed of four face-centered sublattices mutually shifted by ( ) parameter, a = 7.97 Å, which is used in the calculation, was obtained by the total energy minimization and is slightly larger than the experimental value, a exp = 7.896 Å. Positions of atoms are shown in Table 1.
Equilibrium electronic structure. Figure 1(a,b) displays the effect of spin-orbit coupling on electronic bands of KNa 2 Bi. We also calculated the phonon spectrum to show the dynamical stability of KNa 2 Bi in the cubic structure ( Fig. 1(e)). It is obvious that low-energy electronic properties of KNa 2 Bi are determined by the bands located around the Γ point since away from the Brillouin zone center the valence and conduction bands are well separated. The arrangement of bands around the Fermi level near the BZ center is shown in Fig. 1(c,d).
In the absence of spin-orbit coupling (SOC), three bands which are linear combinations of p-like states are degenerate in the Γ point at the Fermi level, and the s-type band is lower in energy ( Fig. 1(c)). The SOC changes the structure of the valence bands ( Fig. 1(d)): three degenerate bands split into two Γ 8 bands degenerate at Γ and a split-off state, Γ 7 . The s-type band, Γ 6 , is now between the Γ 8 and Γ 7 bands. Thus, with and without taking into account the spin-orbit interaction KNa 2 Bi is a zero-gap semimetal in which the conduction and valence bands are degenerate at the Γ point. The account of SOC cannot lift the degeneracy of the Γ 8 bands, which is a consequence of the cubic symmetry.
Another feature is the inversion of band structure around the Γ point. According to the normal band-filling order, the valence band edge should be formed of Bi p orbitals while the conduction band edge should be composed of s-type states. However in KNa 2 Bi the Γ 6 band appears below the Γ 8 bands composed of Bi p states. The energy gap (ε S (Γ 6 ) − ε P (Γ 8 )) is negative: about − 0.09 eV in the calculation without spin-orbit coupling and increases up to − 0.65 eV with the account of SOC.
Electronic structure under hydrostatic lattice compression. First we analyzed the electronic structure of KNa 2 Bi under hydrostatic lattice compression. Figure 2 shows the evolution of electronic bands under pressure in the vicinity of the Γ point. Upon the compression, the cubic phase remains dynamically stable. At low pressures, the compound is still a zero-gap semimetal with an inverted band order. The compression leads to a decrease of the gap between bands Γ 6 and Γ 8 at the Γ point. The variation of the energy gap with pressure is shown in Fig. 2(d). At V/V 0 ≈ 0.81, the gap vanishes and all three bands become degenerate at the Fermi level ( Fig. 2(b)). At the crossing point, two of the bands have a conical dispersion, which is crossed in the middle by the third band. A subsequent increase of pressure opens up an energy gap between the valence and conduction bands. Simultaneously the inverted band character around the Γ point disappears, i.e. the sequence of bands restores to normal: the s-type band, Γ 6 , is now above the Γ 8 bands composed of Bi p-type states ( Fig. 2(c)).
A similar semiconductor-to-semimetal topological transition in the crystals of HgCdTe was reported 11 . Upon tuning of the alloying composition the semiconductor band structure becomes gapless at the critical value of cadmium concentration and the conical band dispersion with peculiar properties appears at the BZ center.
Electronic structure under uniaxial strain. Topological properties of KNa 2 Bi can be also changed by applying an external strain and thereby breaking the cubic symmetry. A strain was applied in the [001] direction (c axis). Such a strain lowers the crystal symmetry to a tetragonal one ( Fig. 3(a)). The tetragonal structure of KNa 2 Bi is dynamically stable at all considered uniaxial strains. It turned out that the system responds differently to the compression and expansion: upon expansion the material becomes a 3D topological Dirac semimetal while the uniaxial compression converts the compound into a topological insulator. In both cases, the inverted band order in the vicinity of the Γ point remains that is an important sign of nontrivial topology. Figure 3(c,d) show the electronic band structure of KNa 2 Bi calculated with c = a 0 ± 10% a 0 at constant volume. The lattice optimization does not introduce any qualitative changes. To facilitate comparison we also used the primitive tetragonal unit cell to calculate the electronic structure of the unstrained compound ( Fig. 3(b)).
A topological insulator. When a uniaxial compression breaks the cubic symmetry the degeneracy of the Γ 8 bands is lifted. The splitting creates a direct energy gap at the Γ point between the valence p x,y -type Bi states and the conduction band composed mainly of p z Bi orbitals (Fig. 3(d)). And an indirect energy gap appears near the Γ point because of the band inversion between Bi p x,y and p z states in the (k x , k y ) plane. Since the inverted band order at the Γ point is preserved, the phase remains topologically nontrivial.
The topologically nontrivial character of the phase should be manifested in the existence of topological surface states. The calculated density of electronic states for the semi-infinite Bi&K-and Na-terminated (001) surfaces and for the (100) surface of KNa 2 Bi is shown in Fig. 4 together with the corresponding Fermi surfaces (in the inserts). All the data are presented as a color intensity plot. In the unstrained case ( Fig. 4(b)), the surface states with a Dirac-type crossing are inside the projected valence bands due to the mixing with bulk electronic states. These topological surface states are clearly seen on the (001) surface ( Fig. 4(b, left, middle)). The crossing (Dirac) point is located about 0.03 eV (0.45 eV) below the Fermi level depending on the surface termination. Upon a uniaxial lattice compression, the surface Dirac point on the Bi&K-terminated (001) surface moves up towards the energy gap. With increasing strain, the surface states are split off from the gap edges and form a Dirac cone inside the gap (Fig. 4(a, left)). Such gapless topological states are also observed on surface (100) (Fig. 4(a,  right)). On the (001) surface the spin texture in the upper part of the Dirac cone has a clockwise helical structure whereas in the case of the (100) surface it is strongly warped because of the lack of C 4 symmetry in the (k z , k y ) plane. Additionally, each layer along the [100] direction is composed of four crystallographically non equivalent atoms. In the case of Na termination, the topological states with a Dirac-type crossing remain inside the bulk valence bands ( Fig. 4(a, middle)). However there is another surface state, it split off from the edges of the conduction bands, but when merging with bulk valence bands the state loses its surface character.
A Dirac semimetal. When a uniaxial expansion removes the band degeneracy at the Γ point a fundamental band gap does not appear (Fig. 3(c)). The valence and conduction bands cross at two points exactly at the Fermi level. As a result, a pair of 3D Dirac nodes shows up on the rotational k z axis (the Γ Z direction) at ± k z ≈ 0.2π/c (the position depends on applied strain). One of the crossing bands is composed of Bi 6p x,y states and the other mostly of Bi 6p z states. Since both bands are doubly degenerate each crossing point has a four-fold degeneracy.
The 3D Dirac cones and the inverted band order at the Γ point suggest the existence of nontrivial surface states (Fermi arcs). In the case of (001) surface, the projection of bulk electronic bands is superimposed on the nontrivial surface states arising owing to the band inversion at the Γ point and the Fermi surface represents just a point. The surface states show up along the edges of projected valence and conduction bands with the crossing at the Γ point (Fig. 4(c, left)).
On the (100) surface, however, the projected bulk Dirac nodes and the nontrivial surface states are separated in the momentum space and can be visible (Fig. 4(c, right)). The Fermi surface composed of two half-circle Fermi arcs is closed with two singular points corresponding to the projection of bulk Dirac nodes. The spin texture of Fermi arcs has a helical structure, but the value of spin is not defined at the singular points.

Discussion
To describe the change in the band dispersion at the point of extremum (the Γ point) under pressure, we have constructed a low-energy effective k · p-model Hamiltonian 22 by considering the rotational (C 4 , along the k z axis), time-reversal and inversion symmetries. Since the inverted band order in the vicinity of Γ is a characteristic feature of the compound and is conserved under any uniaxial strain we do no consider s-type states. Only low-energy electronic states around the Γ point which are mostly consist of Bi 6p x,y,z orbitals are used as basis states:   basis states with the same parity (p-orbital hybridization) is also accounted for. So the parity operator is τ 0 , a zero Pauly matrix. In such a basis set, the Hamiltonian can be written as  To describe possible spin textures of surface states let us consider the Hamiltonian consisting of the spin-orbit contribution, which includes both Bychkov-Rashba (BR) and Dresselhaus terms, and an additional "kinetic"-like term 23 : x y x y 2 1 2 1 2 3 Here k x , k y are in-plane components of momentum, = + k k k x y 2 2 2 . α and β determine the strength of the RB and Dresselhaus spin-orbit interactions and γ defines the "kinetic"-like term. σ 1 , σ 2 and σ 3 are Pauly matrices. The BR and "kinetic"-like terms are isotropic, the Dresselhaus term has C 2 symmetry and so can be considered as a warping.
In Fig. 6 three spin configurations corresponding to those shown in Fig. 4 are presented. Upon uniaxial compression, on the (001) surface of KNa 2 Bi ( Fig. 4(a, left)) the spin texture in the upper part of the Dirac cone has a strong clockwise helicity without any warping. To obtain a similar spin texture we set α = 1, β = 0, γ = 10 ( Fig. 6(a)). In this case, only the BR interaction contributes to the in-plane components of spin while the out-of-plane component is determined by the "kinetic"-like term. The existence of the non-zero out-of-plane spin component is proved by our TB calculations. The "kinetic" term also reduces the radius of isoenergetic contour and leads to a square dispersion of surface states with increasing k-vector.
The in-plane spin texture in the Dirac cone on the (100) surface ( Fig. 4(a, left)) is fully described by the Dresselhaus term which has a C 2 symmetry (we set α = 0, β = 1, γ = 10, Fig. 6(b)). In this case the spin texture is strongly warped because of the lack of C 4 rotational symmetry on the (100) and (010) surfaces due to the strain along the [001] direction. Such a warping was also observed in the spin texture of bulk states in the tensile-strained HgTe 24 . In both considered cases there is no warping in the energy spectrum: the dispersions along k x and k y directions are the same.
The most interesting is the case of surface states on the (100) surface of KNa 2 Bi under uniaxial expansion (Fig. 4(c, right)). If in the previous cases the spin texture can be described using either the BR or Dresselhaus contributions, in this case, you need to consider all three terms in the Hamiltonian. The difference between α and β leads to spin helicity, so if α = β, the spin will be directed strictly along the positive (negative) k y axis, depending on whether k x < 0 (k x > 0). Finally, the "kinetic"-like term results in a more isotropic dispersion. Without this term (γ = 0) in the α = β limit the isoenergetic contour would consist of two parallel lines with the opposite spin direction. If α ≠ β the dispersion along the k x and k y directions is different (Fig. 6(c)). And when k x = 0 the in-plane components of spin disappear, which corresponds to the bulk Dirac points. Thus, using a simple k · p-model, we can describe all possible spin texture configurations on the surfaces of KNa 2 Bi. This model can also be used for related compounds with cubic symmetry.
Similar strain-induced transitions were predicted theoretically for gray tin (α-Sn) and HgTe 12,13,16,17,25 . The existence of topological surface states in the strained HgTe was also demonstrated experimentally by ARPES measurements 14 . At zero pressure, like KNa 2 Bi, α-Sn and HgTe are gapless semimetals with the inverted bulk band structure in the vicinity of Γ and surface states located on the edge of the conduction bands and inside the valence bands, indicating a nontrivial topology 25 . These semimetals are turned into 3D topological insulators by applying a uniaxial compression 12 . The strain induces a band gap in the bulk whereas the topological surface states acquire a gapless Dirac-cone dispersion, thereby giving a metallic character to the 3D topological insulators 12,13,25 . Like in the case of KNa 2 Bi the gapless surface states in HgTe is found to be sensitive to the surface termination 14 . Under uniaxial expansion of HgTe the degeneracy at Γ is also lifted but a fundamental band gap does not appear. Like in KNa 2 Bi the valence and conduction bands form a pair of Dirac points in the Z-Γ -Z line 18 .

Conclusion
We have analyzed theoretically the effect of external strain on the electronic properties of KNa 2 Bi, a cubic bialkali bismuthide. Under hydrostatic compression KNa 2 Bi, a zero-gap semimetal with the inverted band order around the BZ center, undergoes a transition to a conventional gapped semiconductor with a conical Dirac-like dispersion of electronic bands at the point of the topological transition. Simultaneously the sequence of different symmetry bands restores to normal. We find that topological Dirac-like surface states exist even in the unstrained compound as long as the band structure is inverted. It is also shown that KNa 2 Bi can be driven into distinct topologically nontrivial phases by breaking the cubic symmetry. Under uniaxial lattice expansion the compound is converted into a three-dimensional Dirac semimetal with nontrivial Fermi arcs on the surface while a nontrivial topological insulating phase can be realized by applying a uniaxial lattice compression.

Methods
The electronic structure calculations were performed in the mixed-basis pseudopotential approach 26-28 with the exchange and correlation energy functional evaluated within the generalized gradient approximation 29 . Spin-orbit coupling (SOC) was incorporated into the pseudopotential scheme via Kleinman's formulation 30 and was treated fully self-consistently 31 . Phonon dispersions were calculated using the linear response technique 32 in combination with the mixed-basis pseudopotential method 28 . Integrations over the Brillouin zone (BZ) were performed by sampling a 8 × 8 × 8 mesh corresponding to 29 and 75 k points in the irreducible part of the cubic and tetragonal BZ, respectively, combined with a Gaussian broadening with a smearing parameter of 0.01 eV. The projected density of surface states was calculated using ab initio based tight-binding formalism 33,34 .