Abstract
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 zerogap 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 Diractype 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 threedimensional 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).
Introduction
Alkali metal bismuthides form a group of compounds which attract much attention as materials with a variety of interesting physical properties. The Bibased binary compounds, A_{3}Bi (A = Na, K, Rb), belong to a special class of topological electronic states, threedimensional (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 (socalled 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 zincblende 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 spinorbit 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 αSn (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, ^{21}. In this paper we theoretically demonstrate that KNa_{2}Bi, a zerogap 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 Diraclike 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 threedimensional Dirac semimetal with nontrivial Fermi arcs on the surfaces.
Results and Discussion
KNa_{2}Bi crystallizes in the cubic structure^{21} composed of four facecentered sublattices mutually shifted by along the body diagonal. The sublattices occupied by Na atoms are symmetry equivalent. The lattice 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 spinorbit 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 lowenergy 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 spinorbit coupling (SOC), three bands which are linear combinations of plike states are degenerate in the Γ point at the Fermi level, and the stype 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 splitoff state, Γ_{7}. The stype band, Γ_{6}, is now between the Γ_{8} and Γ_{7} bands. Thus, with and without taking into account the spinorbit interaction KNa_{2}Bi is a zerogap 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 bandfilling order, the valence band edge should be formed of Bi p orbitals while the conduction band edge should be composed of stype 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 spinorbit 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 zerogap 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 stype band, Γ_{6}, is now above the Γ_{8} bands composed of Bi ptype states (Fig. 2(c)).
A similar semiconductortosemimetal 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 semiinfinite Bi&K and Naterminated (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 Diractype 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&Kterminated (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 Diractype 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 ΓZ 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 fourfold 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 halfcircle 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 lowenergy effective k · pmodel Hamiltonian^{22} by considering the rotational (C_{4}, along the k_{z} axis), timereversal 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 stype states. Only lowenergy electronic states around the Γ point which are mostly consist of Bi 6p_{x,y,z} orbitals are used as basis states: , , and . The superscript indicates the parity, and the arrow — the direction of spin moment. Due to the inplane isotropy, components can be easily replaced by . The coupling of the basis states with the same parity (porbital 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
Here , , k_{±} = k_{x} ± ik_{y}. So the parameters of the effective model are M_{0,1,2}, A and β. In the Hamiltonian the topology of band structure is solely determined by parameter M_{0} which corresponds to the energy difference between basis orbitals. Other parameters are fixed in our calculation so as to reproduce both topological transitions: M_{1} = 1.0 eVÅ^{2}, M_{2} = −1.0 eVÅ^{2}, A = 1.0 eVÅ^{2}, and β = 1.0 eVÅ). If M_{0}M_{1} < 0 the system is a topological insulator, otherwise (M_{0}M_{1} > 0) it is a Dirac semimetal. The spectra of Hamiltonian, , are presented in Fig. 5.
Let us consider three possible situation: M_{0} = 0, M_{0} > 0 and M_{0} < 0. If M_{0} = 0 (Fig. 5(a)) the compound is a zerogap semimetal. Two double degenerate bands with square dispersion along all the directions of the BZ are touching at the Γ point and the crossing point has a fourfold degeneracy. Under hydrostatic compression, the stability of the system in the topological semimetal phase (M = 0) is provided by the cubic symmetry when all the basis orbitals are equivalent and have the same energy. To make M_{0} nonzero the band degeneracy at the Γ point should be lifted. Upon positive (negative) uniaxial strain, E(P_{z}, ↑(↓)〉) is greater (less) than E(P_{x(y)}, ↑(↓)〉) and, so, M_{0} < 0 (M_{0} > 0). In the case of M_{0} > 0 (Fig. 5(b)), the band gap is not global. The system posses a semimetal band structure with two Dirac points at . Along the k_{x} direction the gapped bands have a square dispersion. The case of M_{0} < 0 (Fig. 5(c)) corresponds to a topological insulator with hybridized gap. All three presented situations are similar to the tightbinding results.
An analysis of the parity of occupied electronic states also reveals two phases in KNa_{2}Bi from the viewpoint of index classification. The index was obtained from the parity of occupied electronic states at the timereversal invariant points of the bulk Brillouin zone^{12} and also using Z2Pack software package. In the case of topological insulator (M_{0} < 0), the invariants are . It coincides with the results of surface band structure calculations: topological surface states exist on all possible surfaces. In the case of a Dirac semimetal (M_{0} > 0), the situation is different: that corresponds to a weak topological insulating phase. This fact is also supported by the existence of surface states only on the (100) and (010) surfaces.
To describe possible spin textures of surface states let us consider the Hamiltonian consisting of the spinorbit contribution, which includes both BychkovRashba (BR) and Dresselhaus terms, and an additional “kinetic”like term^{23}:
Here k_{x}, k_{y} are inplane components of momentum, . α and β determine the strength of the RB and Dresselhaus spinorbit 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 inplane components of spin while the outofplane component is determined by the “kinetic”like term. The existence of the nonzero outofplane 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 kvector.
The inplane 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 tensilestrained 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 inplane components of spin disappear, which corresponds to the bulk Dirac points. Thus, using a simple k · pmodel, 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 straininduced transitions were predicted theoretically for gray tin (αSn) 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, αSn 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 Diraccone 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 zerogap semimetal with the inverted band order around the BZ center, undergoes a transition to a conventional gapped semiconductor with a conical Diraclike 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 Diraclike 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 threedimensional 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 mixedbasis pseudopotential approach^{26,27,28} with the exchange and correlation energy functional evaluated within the generalized gradient approximation^{29}. Spinorbit coupling (SOC) was incorporated into the pseudopotential scheme via Kleinman’s formulation^{30} and was treated fully selfconsistently^{31}. Phonon dispersions were calculated using the linear response technique^{32} in combination with the mixedbasis 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 tightbinding formalism^{33,34}.
Additional Information
How to cite this article: Sklyadneva, I. Y. et al. Pressureinduced topological phases of KNa_{2}Bi. Sci. Rep. 6, 24137; doi: 10.1038/srep24137 (2016).
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Acknowledgements
We acknowledge financial support of the University of the Basque Country UPV/EHU (grant No. GIC13IT75613), the Departamento de Educación del Gobierno Vasco, the Spanish Ministerio de Ciencia e Innovación (Grant No. FIS201019609C0201), the Spanish Ministry of Economy and Competitiveness MINECO Project FIS201348286C21P, and the Saint Petersburg State University (project No. 15.61.202.2015).
Author information
Affiliations
Donostia International Physics Center (DIPC), 20018 San Sebastián/Donostia, Basque Country, Spain
 I. Yu. Sklyadneva
 , P. M. Echenique
 & E. V. Chulkov
Karlsruher Institut für Technologie, Institut für Festkörperphysik, D76021 Karlsruhe, Germany
 I. Yu. Sklyadneva
 , R. Heid
 & K.P. Bohnen
Institute of Strength Physics and Materials Science, pr. Academicheskii 2/1, 634021, Tomsk, Russian Federation
 I. Yu. Sklyadneva
Tomsk State University, 634050, Tomsk, Russian Federation
 I. Yu. Sklyadneva
 & I. P. Rusinov
St. Petersburg State University, 199034, St. Petersburg, Russian Federation
 I. P. Rusinov
Departamento de Física de Materiales, Facultad de Ciencias Químicas, UPV/EHU, Apdo. 1072, 20080 San Sebastián/Donostia, Basque Country, Spain
 P. M. Echenique
 & E. V. Chulkov
Centro de Física de Materiales CFM  Materials Physics Center MPC, Centro Mixto CSICUPV/EHU, 20018 San Sebastian/Donostia, Spain
 P. M. Echenique
 & E. V. Chulkov
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Contributions
The calculations were performed mainly by I.Y.S. and I.P.R. with contributions by R.H. and K.P.B. The idea of the study was proposed by E.V.C. and P.M.E.; I.Y.S. and I.P.R. wrote the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to I. P. Rusinov.
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