Introduction

Topological insulator (TI) is a new kind of material which has gapped bulk state and gapless surface state with the latter protected by the topological character of TI1,2,3,4,5,6,7,8,9,10. For TIs with conserved spin along quantized axis, the topological order parameter is spin Chern number and TI under time reversal symmetry is characterized by Z2 quantum number6. The unique features of its surface state make TI have potential applications in spintronics and quantum information devices. TI is also the breeding ground for a good number of interesting quantum phenomena such as quantum anomalous Hall effect11,12,13,14, Majorana fermions15,16 and topological magnetoelectric effect4. TIs usually appear in those materials containing elements with strong spin-orbit coupling, for example, the bismuth element in Bi2Se35,17, BiTeI18,19 and ScPtBi3. Moreover, pressure and strain has been demonstrated as an effective way to modulate the topological property of materials. For instance, CdSnAs2 under a 7% decrease in the lattice constant will become topological insulator20 while a 6% change in the length of c-axis will drive Bi2Se3 from topological non-trivial phase into topological trivial phase21. However, more interesting phenomena only can be induced by strong electronic interaction, such as the transition in correlated Dirac fermions23 and interaction induced topological Fermi liquids24. Consequently, those TIs beyond p-band inversion arouse intensive research interest25,26,27,28. Here, we predict a new d-p band-inversion topological insulator in bismuth-based skutterudites in which the bands involved in the topological band-inversion process are d- and p-orbitals. Due to the present of large d-electronic states, the electronic interaction in this topological insulator is much stronger than that in other conventional topological insulators27,28,29,30,31,32.

Skutterudites, such as RhAs3, IrAs3, IrSb3 and IrP3, crystallize in a cage-like crystal structure in which each transition metal atom octahedrally coordinates to six pnictide atoms33,34 (see Fig. 1 (a) where IrBi3 is illustrated). They have large Seebeck coefficients and therefore can behave as excellent thermoelectric materials44. The discovery of heavy fermion superconductivity45, exciton-mediated superconducting state46 and Weyl fermions47 in this system makes skutterudites a hot spot in condensed matter physics. Besides those skutterudites naturally exist, a number of new members in skutterudites have been experimentally synthesized, such as NiSb348 in 2002 and RuSb334 in 2004. However, those materials are composed of elements with relatively weak spin-orbit coupling (SOC). Knowing that topological insulators are usually those materials containing elements with strong spin-orbit coupling strength, such as the bismuth element in topological insulator Bi2Se35,17, BiTeI18,19 and LaPtBi22, It is reasonable to ask whether or not skutterudites composed of elements with strong spin-orbit coupling strength, i.e. bismuth, can exist stably and whether they can be topologically non-trivial? This new topological insulator in bismuth-based skutterudites, is exactly such kind of skutterudite material which is able to exist stably, contains elements with strong SOC and has controllable topological phase transition.

Figure 1
figure 1

Crystal structure and Brillouin zone.

(a) unit cell of IrBi3, including 8 Ir atoms (green) and 24 Bi atoms (pink). Each Ir atom is surrounded by 6 Bi atoms and each Bi atom has 2 Ir nearest neighbors. (b) The equivalent primitive cell of IrBi3, containing 4 Ir (green) atoms and 12 Bi atoms (pink). (c) The corresponding Brillouin zone and high symmetric points with Γ (0,0,0), H (0,1/2,0), N (1/4,1/4,0), P (1/4,1/4,1/4). (d) Free energy as a function of lattice constant.

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In this work, we predict a new d-p band inversion topological insulator in bismuth-based skutterudites, which is distinctive from usual topological insulators, for instance in Bi2Se3 and BiTeI the bands involved in the topological band-inversion process are only p-orbitals. Due to the present of large d-electronic states, the electronic interaction in this topological insulator is much stronger than that in other conventional topological insulators. The stability of the new material is verified by binding energy calculation, phonon modes analysis and the finite temperature molecular dynamics (FTMD) simulations. We demonstrate that external strains are able to induce a topological phase transition in this system via band structure calculations. We confirm its topological non-trivial property by Z2 quantum number calculation.

Results

Crystal structure and optimized lattice parameter

The bismuth-based skutterudite IrBi3 investigated here has space group and its crystal structure is shown in Fig. 1. There are 8 Ir atoms and 24 Bi atoms in a unit cell. Each Ir atom is surrounded by 6 Bi atoms and each Bi atom has 2 Ir nearest neighbors (see Fig. 1 (a)). The structure has space inversion symmetry with the inversion center (1/2,1/2,1/2). The structure belongs to the body-centered lattice type and its primitive cell (Fig. 1 (b)) has a half volume of the unit cell. Fig. 1 (c) shows the Brillouin zone and high symmetric points with Γ (0,0,0), H (0,1/2,0), N (1/4,1/4,0), P (1/4,1/4,1/4).

We first optimize the lattice parameter and ionic positions. The calculated total free energy (solid line) as a function of lattice parameter is shown in Fig. 1 (d). It can be clearly seen that the optimized lattice parameter (corresponding to the position of free energy minimum) of the primitive cell is 8.493Å. This value is 6% larger than that of IrSb352, which can be explained that Bi atom has a larger atomic radius than Sb atom.

Binding energy calculation, phonon modes analysis and the finite temperature molecular dynamics simulations

In order to verify the stability of the new material, the authors perform the binding energy calculation, phonon modes analysis and the finite temperature molecular dynamics (FTMD) simulations. The binding energy is calculated by

where denotes the free energy of IrBi3 per primitive cell, EIr and EBi the free energy of crystalline Ir and Bi per atom, nIr and nBi the number of Ir and Bi atoms in IrBi3 primitive cell. By simple calculation [There are nIr = 4 Ir atoms and nBi = 12 Bi atoms in an IrBi3 primitive cell. At GGA level, EIr = −8.69 eV for crystalline Ir with space group and EBi = −3.70 eV for crystalline Bi with space group . From Fig. 1(d) we read . Substituting the above values in Eq.(1), we arrived at the binding energy Eb = −3.65 eV.], Eb is found to be equal to −3.65 eV per primitive cell. The negative value of binding energy infers a stable state of IrBi3.

Fig. 2 shows the phonon dispersion and phonon density of states (DOS) for IrBi3 at zero strain. In the phonon DOS subfigure, the black solid line represents the total phonon density of states, while the green and red shaded areas represent the states coming from Ir and Bi atoms, respectively. Phonon states in the low energy range are mostly composed of states from Bi atoms, indicating that Bi atoms in IrBi3 are much easier to vibrate than the Ir atoms. The phonon dispersion and phonon DOS show no imaginary frequency, indicating that IrBi3 is stable.

Figure 2
figure 2

Phonon dispersion and phonon density of states for IrBi3.

Orange dotted lines in all subfigures denotes the zero frequency. Calculations are performed at zero strain. (a) phonon dispersion curves for IrBi3, in which the inset shows the dispersion near the zero energy. (b) phonon density of states for IrBi3, in which black solid line represents the total phonon density of states, while the green and red shaded areas represent the states coming from Ir and Bi atoms, respectively. Phonon states in the low energy range are mostly composed of states of Bi atoms, indicating that Bi atoms in IrBi3 are much easier to vibrate than the Ir atoms. The phonon dispersion and phonon density of states shows no imaginary frequency, indicating that IrBi3 is stable.

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In addition, the dynamical stability of the material is further checked by finite temperature molecular dynamics simulations at temperature 300 K for room temperature and 30 K for low temperature. During the simulations, a 2 × 2 × 2 supercell containing 256 atoms is used. The length of time-step is chosen as 5 fs and simulations with 1000 steps are executed. It is observed that, the atoms shake around the equilibrium positions back and forth while the extent of such motion under 300 K is larger than under 30 K (the evolution of atomic positions can be found in movies in supplementary information). However, no structural collapse happens throughout the simulations, which can also be seen from the free energies curves as the functions of time-step shown in Fig. 3. It is also observed that, the crystal structure always remains nearly the same as the initial crystal structure. Actually, as is shown in the inset of Fig. 3, the crystal structure corresponding to the last free energy maximum in T = 300 K case (right), still shows no significant structural differences as compared with the initial crystal structure (left). The lattice relaxation, binding energy calculation, phonon modes analysis together with FTMD simulations mentioned above provide an authentic test for the stability of bismuth-based skutterudite IrBi3.

Figure 3
figure 3

Finite temperature molecular dynamics.

Free energies as functions of time-step at temperature T = 30 K (blue curve) and T = 300 K (red curve). The slight shift of the free energy curves corresponds to the oscillations of each atom around their equilibrium position. The absence of sharp changes in such curves indicates that no structural phase-transition happens throughout the whole simulation process. The initial crystal structure (denoted by the orange circle on the free energy curve) is plotted in inset (a). The crystal structure corresponding to the last free energy maximum (denoted by the green circle on the free energy curve) is shown in inset (b) as a comparison. It can be seen that, the latter still shows no significant structural differences as compared with the initial crystal structure.

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Strain-induced d-p band-inversion topological insulator

The calculated band structures are listed in Fig. 4, where the black and blue lines represent the GGA and GGA+U band structures, respectively. As is shown in Fig. 4 (a), before exerting pressure, IrBi3 resides in the normal metal state with its bands crossing the Fermi level several times. Subfigure (b) to (d) represent the band structures at isotropic strain 3%, 6%, 9% respectively. With the increase of isotropic strain ((a) to (d)), the valence band crossing the EF along H-N moves downwards and the density of states (DOS) at Fermi level decreases gradually. Under a 9% isotropic strain, the bands go across the Fermi level at Γ point but not at other points in Brillouin Zone (BZ) (see Fig. 4 (d)) and the conduction band minimum and valence band maximum degenerate so that the material behaves as a semi-metal which have a zero energy gap, just like Graphene and CeOs4As128. This degeneracy at Γ is protected by the cubic symmetry of crystal, which, as is tested by us, cannot be eliminated by small changes of the lattice constant. In order to shift the degeneracy at Γ, one needs to break that symmetry. An unsophisticated way is to add an anisotropy just like what was done on CdSnAs220. Here, we simply further impose a 2% suppression on the c-axis of the primitive cell while remaining the length of a- and b-axis unchanged, which imposes anisotropy on the system. While the anisotropy does not change the parities of each band, it opens a gap at the Fermi level, dragging the system in the insulating state (see Fig. 4 (e)). Fig. 4 (f) shows the Ir-d projected band structure near the Fermi level and near Γ point, in which the radii of red circles correspond to the proportion of Ir-d electrons. It can be seen that, those localized bands above the Fermi level are mainly contributed by d-orbitals of Ir atoms. The highly dispersive band below the Fermi level is mainly contributed by p-orbitals of Bi atoms and it has little weight of Ir atoms in those k-points far away from Γ point. However, in the vicinity of Γ point, the weight of Ir atoms in that band increases rapidly and becomes dominating orbital component, showing an apparent band inversion. Such band-inversion character is further checked by the modified Becke-Johnson (mBJ) potential (see supplementary information), which is proved to be able to predict an accurate band gap and band order53,54,55. In order to further confirm the topological property in such condition, we calculate the Z2 topological quantum number of the system by the Fu-Kane method6. The index for strong topological insulators v0 is expressed as in which represents the product of the parities of the occupied band at 8 time-reversal invariant momenta Γi. The calculated parities of top-most isolated valence bands (here refers to the isolated block of states between −8.0 eV to 0 eV in Fig. 5) at eight time-reversal invariant momenta are listed in Table 1, where the deeper states (those states lower than −9.5 eV in Fig. 5) separated far from top-most isolated valence bands are ignored because they don't change system's band topology. As is shown, the product of parities of occupied bands contributes a −1 at Γ while +1 at the seven other time-reversal invariant momenta. As a result, Z2 quantum number is ν0 = 1, ν1 = ν2 = ν3 = 0, which corresponds to a strong topological insulator.

Table 1 Parities of top-most isolated valence bands at eight time-reversal invariant momenta. Positive parity is denoted by + while negative denoted by −. Products of the occupied bands at each time-reversal invariant momentum are listed in the right-most column. As is shown, the product of parities of occupied bands contributes a −1 at (0,0,0) while +1 at the seven other time-reversal invariant momenta, resulting in ν0 = 1, ν1 = ν2 = ν3 = 0
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Figure 4
figure 4

Band structures of IrBi3.

The black and blue lines in all subfigures represent the GGA band structures and GGA+U band structures respectively. (a) band structure without exerting pressure, the system is in normal metal state with its bands go across the Fermi level several times. (b) to (d) represent the band structures at isotropic strain 3%, 6%, 9% respectively. With the increase of isotropic strain ((a) to (d)), the valence band crossing the EF along H-N moves downwards gradually. In the band structure under 9% uniform strain (d), a zero gap metal state is obtained. (e) further impose a 2% suppression on the length of c-axis of the primitive cell, a gap appeared at the Fermi level due to the breaking of the cubic symmetry. The inset of (e) is the zoom-in of the band structure close to the Fermi level. (f) Ir-d projected band structure near Fermi level, the radii of red circles are proportional to the weight of Ir-d states, showing a significant band inversion.

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Figure 5
figure 5

The atomic- and orbital-resolved density of states.

The black solid lines in all subfigures represent the total density of states (DOS). (a) atomic-resolved DOS, in which the green curve represents the states of Ir and the red curve represents the states of Bi. It's clear that both type of atom made a significant contribution to the total DOS, different from MoS2 where states near Fermi level are dominated by Mo. (b) and (c) are orbital-resolved DOS of Ir and Bi atom respectively. Green, blue and red curves represent s-, p- and d-orbitals.

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Partial-density of states and the d-p orbitals dominating property near the Fermi level

Fig. 5 depicts the atomic- and orbital-resolved density of states (DOS). The black solid lines in subfigure (a)(b)(c) represent the total DOS. Fig. 5 (a) is atomic-resolved DOS, in which the green curve represents the states of Ir while the red curve represents the states of Bi. It's clear that the DOS of both types of atoms is in quite large values, indicating that both types of atoms make a significant contribution to the total DOS. This is different from MoS2 where states near Fermi level are dominated by only one kind of atom (Mo)56. Fig. 5 (b) and (c) are orbital-resolved DOS of Ir and Bi atoms, respectively. Green, blue and red curves represent s-, p- and d-orbitals. One character of the material introduced here is a large proportion of d-states near EF.

Further, we calculate the d-orbitals projected band structures (see Fig. 6) in the local coordinate of the Bi octahedral. The orange, violet, red, green and blue colors in Fig. 6 represent the , , dxy, dyz and dxz orbitals respectively. The radii of circles are proportional to the weights of corresponding orbitals. It can be seen that, the t2g orbitals (including the dxy, dyz and dxz orbitals) reside far below the Fermi level and are fully occupied. While, the lowest three conduction bands are mainly contributed by the eg orbitals (including the and orbitals). More specifically, the orbital makes an even larger contribution than the the orbital for the lowest conduction band. The large proportion of d-states near EF is distinctively different from usual TI materials, for example states near EF mainly containing s-p electrons in HgTe and p-electrons in Bi2Se3. The large proportion of d-states near EF indicates that electrons in such material process strong electronic correlations and are more localized than other TI materials. The strong electronic correlations make the material a good platform for investigating the effect of correlations on the topology, as well as a candidate for realizing the quantum information device based on correlations. The localization will enhance the effective mass of bulk electrons and hence reduce the bulk contribution to the local current at finite temperature, making the spin-binding property more apparent, which is helpful in fabricating spintronics devices with higher stability.

Figure 6
figure 6

Orbital-projected band structures.

The orange, violet, red, green and blue colors in subfigures represent the , and dxy, dyz and dxz orbitals respectively. The radii of circles are proportional to the weights of corresponding orbitals. The Fermi level is set to be zero energy. It can be seen that, the t2g orbitals (including the dxy, dyz and dxz orbitals) reside far below the Fermi level and are fully occupied. While, the lowest three conduction bands are mainly contributed by the eg orbitals (including the and orbitals). More specifically, the orbital makes an even larger contribution than the the orbital for the lowest conduction band.

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Discussion

Experimentally, the new strain-induced topological insulator IrBi3 could be grown using the Bridgman method, by which the CoP333 and the RuSb334 crystals have been successfully synthesized. The crystal growth should be conducted in a sealed quartz ampoule. The iridium and bismuth should be coated by graphite and then introduced into the quartz ampoule. A temperature gradient of about 50°C/cm should be maintained at the growth interface, just like in the case of RhSb335. To remove the excess bismuth in the as-grown crystal, post-annealing should be performed36. After the synthesis of the new material, its crystal structure could be characterized by the X-ray diffraction using the monochromatic Cu Kα radiation37. Then, the strains could be generated by a pair of diamond anvils38, which was used to generate strong pressure even above 200 GPa39. Moreover, the real-time pressure strength could be detected by ruby fluorescence method38,40. In order to verify the topological property of the material, it is suggested to perform the transport measurements41. Similar to Bi2Se3, the observation of the spin-Hall current42 and the non-equally spaced Landau levels43 in IrBi3 will be signatures of the Dirac fermions in surface of the topological insulator1.

In this work, we predict a d-p band inversion topological insulator bismuth-based skutterudite IrBi3 and verify its stability. Our results indicate that this material is zero gap semi-metal after imposing uniform strain and it can become topological insulator if an anisotropy is further applied to break the cubic symmetry. Furthermore, near the Fermi level there is a large proportion of d-electronic states which is distinctive from usual topological insulators, for instance in Bi2Se3 and BiTeI the bands involved in the topological band-inversion process are only p-orbitals. Consequently, the electronic interaction in this topological insulator is much stronger than that in other conventional topological insulators. This provides realistic material for investigating the effect of correlations on the topology, fabricating quantum information devices and spintronics devices with higher stability.

Methods

Our first principle calculations are in the framework of the generalized gradient approximation (GGA) of the density functional theory. The VASP package49,50 has been employed and the projector-augmented-wave pseudo-potentials51 are used. Plane waves with a kinetic energy cut-off Ec of 400 eV are used as basis sets and k-point grids in Brillouin zone is chosen as 6 × 6 × 6 according to the Monkhorst-Pack scheme. The relaxations are carefully made so that the forces on atoms are smaller than 0.0003 eV/Å, in which the conjugate gradient algorithm is utilised. In the finite temperature molecular dynamics simulations, a 2 × 2 × 2 supercell containing 256 atoms is used and the length of time-step is chosen as 5 fs. The phonon dispersion curve and phonon density of states are obtained using the force-constant method by phonopy code57. The effect of spin-orbit coupling (SOC) is included in the calculations after the structural relaxations. GGA+U calculations are based on the Dudarev's approach implemented in VASP, with the effective on site Coulomb interaction parameter U = 3.0 eV and the effective on site exchange interaction parameter J = 0.5 eV for d-orbitals of Ir atoms58. The GGA band structures are checked by the full-potential DFT code WIEN2k59 in the supplementary information. We also use the modified Becke-Johnson (mBJ) semilocal exchange-correlation potential53,54 to further check the band order and the magnitude of energy gap and in this process the GGA wave function is used to initialize the mBJ calculation.