, a layered three-dimensional (3D) material, exhibits topological insulating properties due to the presence of surface states and a bandgap of 0.3 eV in the bulk. We study the effect of hydrostatic pressure and doping with rare earth elements on the topological aspect of this material in bulk from a first principles perspective. Our study shows that under a moderate pressure of , the bulk electronic properties show a transition from an insulating to a Weyl semi-metal state due to band inversion. This electronic topological transition may be correlated to a structural change from a layered van der Waals material to a 3D system observed at . At large , the density of states have a significant value at the Fermi energy. Intercalating Gd with a small doping fraction between layers drives the system to a metallic anti-ferromagnetic state, with Weyl nodes below the Fermi energy. At the Weyl nodes, time reversal symmetry is broken due to the finite local field induced by large magnetic moments on Gd atoms. However, substituting Bi with Gd induces anti-ferromagnetic order with an increased direct bandgap. Our study provides novel approaches to tune topological transitions, particularly in capturing the elusive Weyl semimetal states, in 3D topological materials.
Topological insulators (TIs) have potential future application in quantum computers1–3 and spintronics4–6 owing to the existence of symmetry protected edge or surface states and also provide a fundamental bridge between high-energy and condensed-matter physics due to the presence of exotic physical states in the system. These materials exhibit insulating bulk and metallic surface states and these properties lead to extensive theoretical and experimental studies.2,7–11 These systems show non-trivial topological order by conserving the particle number and time reversal symmetry.10,12
The primary feature of the TI state is the inverted band structure, which results from the crossing of the valence and conduction bands of different parity symmetry,13,14 and family of materials are a prime example of three-dimensional (3D) materials with invariant symmetry. , and are 3D material topological layered materials. forms two-dimensional (2D) layered structures effectively, and the theoretical studies predict that has a topologically non-trivial bulk energy gap of 0.3 eV.14–19 The topological surface states are described by a single gapless Dirac cone at the point.15,18,20,21 It is interesting to understand the mechanism that may close bulk energy gap in .
and its family of materials have been extensively explored, by applying pressure,22–28 doping Bi with rare earth (RE) atoms like Gd and Sm,29–34 and also with transition metal (TM) atoms like Cr and Fe.35–38 Incorporating impurities with large magnetic moment may break time reversal symmetry and lead to many exotic phenomena like quantum anomalous Hall effect (QAHE), which supports dissipationless charge transport. Mn-doped shows spin glass like behavior,35 whereas Fe and Cr doping leads to a dominant ferromagnetic and dominant anti-ferromagnetic interactions, respectively.36 A small (less than 0.1%) doping of Cr in this material is reported as ferromagnetic, while Fe-doped tends to be a weakly anti-ferromagnetic system. Cr and Fe-doped are insulating, but the bandgaps are substantially reduced due to the strong hybridization between the orbitals of the dopants and the orbitals of the neighboring Se atoms.39
Doping of with TM atoms can lead to many exotic phases; e.g., doping with Cu with doping fractions of 0.12 and 0.15 shows intermixing of both Cu-intercalation between Se–Se layers and Cu-substitution in Bi-layer sites.40 In Cu-intercalated , superconducting transitions are observed experimentally with a of 3.5 K and 3.6 K for doping fractions of 0.12 and 0.15, respectively. However, at low temperature, Cu-doped crystals behave like a paramagnetic metal.40 Intercalation of Cu in this material also holds the promise of topological superconductivity.41
RE atoms with electrons are expected to be better candidates for introducing magnetic order in TIs compared to TM atoms,34,42 because of their larger number of possible unpaired electrons, larger ionic radii compared to TM atoms, as well as the fact that RE radii are comparable to Bi atoms of . Therefore, a smaller structural distortion is expected due to small atomic radii mismatch, and no disorder is envisioned.34 RE atoms are better dopants in avoiding impurity aggregation which is seen in Cr (TM) doped and these aggregations are responsible for magnetic disorder.42 In addition, orbitals can have more unpaired spins compared to orbitals; therefore, RE can induce larger magnetic moments compared to TM atoms. RE electrons are more localized with a maximum of seven unpaired electrons compared to a maximum of five unpaired electrons in TMs. In some cases, a bulk paramagnetic behavior with a large magnetic moment from substituted ion is reported in ;29 whereas a doping induced paramagnetic to anti-ferromagnetic phase transition is observed experimentally on substitutional doping of the system with Gd.32 Intercalating Rb atoms between the quintuple layer structure of can form a quantum-confined two-dimensional electron gas (2DEG) state with a strong Rashba-type spin–orbit splitting.43
Application of pressure is an important tool to enhance the hybridization between the orbitals, and a recent study shows the emergence of an unconventional superconducting phase in topological at a critical pressure of 11 GPa on application of pressure via diamond anvil cell (DAC).44 Pressure induced structural phase transitions have been experimentally observed at very high pressures.45 The experiments indicate that a progressive structural evolution occurs from an ambient rhombohedral phase (Space group (SG): ) to monoclinic phase (SG: C2/m) at 36 GPa and eventually to a high pressure body-centered tetragonal phase (SG: I4/mmm) at 81 GPa on application of pressure via DAC. A pressure induced transition to a topological phase has been found in at a pressure of 5.3 GPa exerted via DAC.46
In some of the topological materials like TaAs single crystals, the valence band (VB) and the conduction band (CB) intersects at two points at near the Fermi energy. If band dispersion near the crossing point is linear due to the relativistic nature of fermions and the system preserves the time reversal and inversion symmetry, then the system may be characterized as a Weyl semimetal (WSM).47,48 In WSMs, non-orthogonal magnetic and electric fields result in a novel observation of chiral anomaly. This results in the chiral-magnetic effect, which is the observation of an unconventional negative longitudinal magnetoresistance.49
In this article, we investigate the effects of hydrostatic pressure (HP) as well as doping with a rare earth element on the bulk phase of , a probable 3D TI, from the perspective of ab initio density functional theory (DFT) based calculations. The material shows an electronic topological transition from a small bulk bandgap insulator at low HP, to a gapless dirac state at a critical pressure of 7.9 GPa, and to a WSM beyond the critical pressure. To the best of our knowledge, this is the first prediction of a WSM state arising due to the application of pressure in family of materials. We note that undergoes a transition from a layered quasi-2D van der Waals crystal to 3D topological material on applying HP . We also study the effect of doping of with rare earth elements. While intercalating Gd between the quintuple layers (QLs) shows a broad bandwidth metallic ground state, with a time reversal symmetry broken Weyl like feature in the band structure below the Fermi energy, substituting Bi with Gd shows an increase in bandgap and an insulating state. Thus, we propose that a tunable topological transition to Weyl states may be driven in bulk by both application of hydrostatic pressure and intercalating with rare earth elements to attain the exciting and elusive Weyl like states.
The paper is divided into five sections, we describe the computational method in Sec. II. The results are divided into two major sections. In Sec. III, we describe the effect of pressure on structural and electronic properties. The effect of doping is studied in Sec. IV. The summary and conclusion drawn from the results are given in Sec. V.
II. COMPUTATIONAL DETAILS
Our first-principles calculations were carried out on the plane wave basis as implemented in the Vienna Ab-initio Simulation Package (VASP)50 with projector-augmented wave (PAW) potential.51 The exchange-correlation functional used in the calculations is the generalized gradient approximation (GGA) implemented following the Perdew–Burke–Ernzerhof52 prescription. Local correlations are taken into account wherever necessary with the energy correction within the framework of formalism primarily for dopant Gd atoms, with values of , . For ionic relaxations, internal positions of the atoms are allowed to relax until the forces became less than . The energy cutoff used for calculations is 500 eV, and Monkhorst-Pack -points mesh provide a good convergence of the total energy in self-consistent field calculations. The spin–orbit coupling (SOC) in Bi atoms is treated as a perturbative non-self consistent correction, which is better suited for topological materials.53 In order to study the effect of hydrostatic pressure, calculations are done by first changing the volume of the unit cell isotropically and then relaxing the ionic positions for each of the modified volume. The van der Waals corrections (á la DFT-D3 method of Grimme with Becke–Johnson damping) were also included in our VASP simulations. The phonon spectrum was calculated based on the density functional perturbation theory (DFPT) as implemented in the VASP package. A supercell and a centered Monkhorst-Pack mesh were used. The phonon frequencies were calculated using the Phonopy code.54
III. EFFECT OF HYDROSTATIC PRESSURE P
In this section, we discuss the effects of the application of hydrostatic pressure (HP) on the bulk band structure of . The HP induced structural transition and changes in lattice parameters are analyzed in Subsection III A, and a systematic study of the effect of structural transition on the band structure properties of are provided in Subsection III B. Generally, pressure pushes the atoms closer to each other and leads to enhancement in effective hybridization of orbitals, which result in the reduction of bandgap. In the large limit, structural transition is also a possibility. In this section, we study the structural behavior of first and then analyze the effect of pressure induced structural transitions on the electronic properties.
A. Change in the crystal structure
We first discuss the basic structural details of the material at , and thereafter the structural changes are analyzed at various pressures. has a hexagonal symmetry with space group and with lattice parameters , and lattice angles , . forms quintuple layers (QLs) within the hexagonal unit cell as seen from Fig. 1(a). The crystal structure along the -axis direction consists of QLs of two Bi layers sandwiched between three Se layers, Se-Bi-Se-Bi-Se, where the subscript indicates that the two Se atoms are in-equivalent by symmetry (as shown in Fig. 1). The atoms within each QL are chemically bonded, but the QLs are weakly bonded through van der Waals interaction. We show in Fig. 2(b) the electron localization function (ELF) at 0 Gpa. It is seen that within the QLs there is covalent bonding by electron sharing between Bi and Se. No such electron overlap or sharing is seen between the layers, which are loosely connected by van der Waals interaction, and shows a clear gap in the ELF between the QLs. slabs consist of an integer number of QLs.
To understand the structural changes with pressure , ratio and the distance between Se–Se atoms sitting at two nearest QL are studied, and we note that the ratio decreases with up to , and then it increases on increment of as shown in Fig. 2(a). Variation of lattice parameters a, b, and c and c/a with are provided in Table I in the Appendix. The distance between the two atoms from two nearest QL continuously decreases with , and at the Se–Se distance decreases to below 2.17 Å, which is less than the bond length of diatomic .55 Therefore, for , behaves like three-dimensional (3D) structure rather than a layered structure, as demonstrated in Fig. 2(c). The bond distances from Bi to Se1, , and to Se2, , decrease on application of pressure. The bond angles and also decrease with an increase of ; however, due to bending of the bonds, increases with an increase of as shown in Table II in the Appendix.
|P (GPa) .||0 .||5.5 .||10 .||16.4 .||24.6 .||33.4 .|
|P (GPa) .||0 .||5.5 .||10 .||16.4 .||24.6 .||33.4 .|
|P (GPa) .||0 .||5.5 .||10 .||16.4 .||24.6 .||33.4 .|
|P (GPa) .||0 .||5.5 .||10 .||16.4 .||24.6 .||33.4 .|
The structural change from a quasi-2D layered van der Waals crystal to a 3D crystal can be seen in the ELF plotted in Fig. 2(c), unlike the case, there is an electron sharing or overlap of orbitals between the QLs. Therefore, we can confirm that the structural change leads to the electronic topological transition. We also note that our calculation shows no significant effect of the vdW correction (á la DFT-D3 method of Grimme with Becke–Johnson damping) implemented in VASP.
We studied the dynamical stability of the by performing phonon calculations and analyzing the phonon spectrum. In the absence of pressure, has small negative frequencies , which is consistent with the earlier studies.56 The negative frequencies are related to the negative stress tensor of the optimized system. The lattice parameters being optimized using GGA to reduce the internal force overestimate the lattice constant. As a result, the optimized structure has a negative stress tensor that leads to negative frequencies. The phonon modes along the z-direction are sensitive to stress because of the weak vdW force along that direction. External pressure favors positive frequencies by decreasing the lattice constant. This is shown in Fig. 8 in the Appendix which depicts that the contribution from the negative frequencies to the phonon density of states decreases with increasing pressure and, at , the negative frequencies are negligibly small. Thus, higher pressure leads to greater stability in the system, which has been predicted in the existing literature.56
B. Change in the electronic structure
We study the change in the electronic structure of with application of hydrostatic pressure as shown in Fig. 3. The band structure at zero applied pressure has a bandgap of 0.3 eV at the point and both Bi and Se have partially filled -orbitals which participate to form energy bands as seen from the projected DOS in Fig. 5. The valence bands are primarily Se- bands, whereas the conduction bands are Bi- bands as shown in Fig. 3. Based on the symmetry analysis, we find that the levels on Bi and Se1 are split by the crystal symmetry into and () at the point. The bandgap is formed between the bonding and anti-bonding states resulting from the hybridization of orbitals on the Bi and Se1 sites. Considering surface calculations, the band structure of shows the presence of surface states at the point, which is a typical signature of topological insulators at [Not shown here. This has already been shown in several papers (Fig. 4 of Ref. 15, Fig. 1 of Ref. 57)].
We apply HP systematically on the system and notice that the bandgap decreases with and it is 0.05 eV and 0.009 eV at and 7.3 GPa, respectively. The bandgap vanishes completely at , and this gapless state can be correlated to the structural phase transition at . Surprisingly, the CB and VB move toward the fermi energy with increasing as shown in Fig. 3 before the critical pressure at which VB and CB meet each other and the bandgap vanishes. Whereas, for , these two bands cross at two points points around the point at the fermi energy. There is no spin-splitting at these crossing points and these crossing points are possible signatures of WSM as shown in Fig. 4.
In a Weyl semimetal other than CB and VB coinciding within some energy window, the degeneracy is expected to be robust to small parametric perturbation. The double degeneracy may arise in the presence of time reversal and inversion symmetry or their combined PT presence in the system,47 i.e., for inversion symmetry , for time reversal symmetry , and in combined PT symmetry conditions are satisfied. These conditions are easily fulfilled in the case of band inversion, i.e., the two branches of a band undergo an accidental band crossing and give rise to Weyl points, and this is applicable in our case. However, the degeneracy of crossing point is preserved only in the case of special symmetry in the lattice, and the symmetry prevents the repulsion of degenerate points to keep the fourfold degeneracy intact. has crystal symmetry and under pressure the distance between the QLs reduces and gives rise to a three-dimensional structure. For , the Bi-p band and the Se-p band cross at and at as shown in Fig. 4(a). The crossing point shown in Fig. 4 is along the -C line, but a similar crossing can be found along other -K momentum axis.
In the absence of any external perturbation, the band dispersion near the Weyl point varies as with momentum , where and are mass and velocity parameter. In our system, the CB and the VB are formed from two different orbitals, which have different chemical potential. Therefore, the dispersion relation for VB and CB near the Weyl point can be fitted with
as shown in Fig. 4(a). We notice that the crossing point is symmetric about the point and spin up and down channel of the band is degenerated in the absence of the SOC. Therefore, this system preserves both time reversal and inversion symmetry. The crossing points act as a source or a sink of the Berry curvature. The Barry curvature calculated using Vaspberry58 shown in the inset of Fig. 4(b). The Berry curvature has the highest value near one crossing point, whereas it has the lowest value near the second crossing points; the direction of curvature is shown with arrow. In Fig. 5, projected density of states (PDOS) are shown for four different pressures, and the Bi and Se bands are marked separately. The contribution of the Bi-p band is higher near the in the CB than the Se-p band, whereas in VB the Se-p band has higher contribution. At low pressure , there is no density of states at , and PDOS at increases with , large PDOS can be seen at .
IV. EFFECT OF DOPING
In this section, we discuss the effect of doping with a rare earth element Gd. The system can be doped either by substituting Bi with Gd or intercalating Gd between the QLs of . Intercalating one Gd per unit cell gives rise to a doping fraction of 16.67% whereas substituting one Bi with Gd per unit cell gives rise to a doping fraction of 20%. To our surprise, the two different methods of doping, with similar doping fractions, resulted in two completely different electronic ground states. The highest valence and the lowest conduction bands are shown in Fig. 6(a) for pure, substituted, and intercalated Gd in the system. On substituting Bi with Gd, there is an increase in the split between the valence band maxima and conduction band minima, and hence the direct bandgap increases to 0.5 eV. The intercalation of Gd shifts the entire band structure to a lower energy in such a way that we now have a partially occupied conduction band, and a fully occupied VB as shown in Fig. 6(a), and therefore it is a wideband metal. The intercalated Gd induces rearrangement of energy bands, which leads to band inversion below ; however, it is different from the pressure induced inversion. The band inversion below is shown in the circle in Fig. 6(a). In this case, the up- and down-spin band split due to the internal magnetic field induced by Gd atoms. Therefore, the resulting time reversal symmetry is not preserved. The upper band is contributed from the Bi band, whereas the lower band contributed by Se atoms. This is shown in the orbital resolved band structure for the Gd intercalated system in Fig. 6(b). Thus, we observe a time reversal symmetry broken Weyl state whose existence has been discussed in the literature.59
We note here that on carrying out intercalation of Gd at a lower doping fraction of 8.33%, which is 1 Gd atom intercalated in a supercell of undoped , we find the qualitatively same electronic structure, with very similar DOS and a similar band inversion below the Fermi energy.
The total DOS for three different cases pure, substituted, and intercalated are shown in Fig. 6(b), and we note that intercalation of Gd can lead to finite density of states at , which leads to an insulator to metal transition. The large DOS comes from Gd orbitals, which are half filled and have high spin splitting. The rest of the DOS has usual contributions from Bi and Se orbitals with some mixing with filled Gd orbitals. The contribution to DOS at the Fermi in the case of Gd intercalated is from the Bi bands, which is the partially filled conduction band. It is also to be noted that although the direct bandgap changes in the case of substitutional doping from pure , the integrated bandgap does not undergo any substantial change as observed from the total DOS.
The magnetic properties of the system under the influence of the dopant atom change significantly. The substitutional doping gives rise to a AFM ordering consistent with the earlier results.32 We find a similar AFM ordering between intercalated Gd atoms, as shown in Fig. 7. The large magnetic moment of on Gd, which is consistent with the literature.32 The large magnetic moment on Gd induces a small moment of on Se. The large magnetic moment may be because of larger exchange splitting compared to the crystal field splitting, in this case, the orbitals likely to be occupied by electrons with parallel spin. The large local magnetic moments induce Zeeman splitting in the Bi and Se bands in both doping methods, although intercalation of Gd has a much stronger splitting effect on the Bi and Se bands than substitutional doping as seen from Fig. 7. Thus, intercalation is more effective in inducing magnetism than previously reported methods of substitution.
To estimate the spin exchange coupling in the solid state system, generally one considers at least two magnetic atoms per supercell. We consider a supercell for this purpose. We consider the moments to interact via a simple nearest neighbor Ising model , where . We find a magnetic exchange meV between the Gd atoms, which is mediated by the Se orbitals interacting anti-feromagnetically, as seen in from the plot of magnetization density which is shown in Fig. 7. However, the orbital structures for the two different doping are quite different, which might be related to the different conductivity behavior. Although in both cases majority of magnetization comes from the orbitals [in both cases a combination of orbitals], there is a rather significant contribution to the total moment of as well from orbitals. In the case of intercalation, has a major contribution while in the case of substitution has a major contribution. This shows up in the magnetization density plotted in Fig. 7.
Thus, our study on doping not only shows a tunable topological transition to Weyl like states depending on the method of doping but also shows induced magnetism in the system.
V. SUMMARY AND CONCLUSION
In our ab initio DFT study, we note that bulk shows a tunable topological transition with the application of hydrostatic pressure and by doping with rare earth elements. Bulk , upon application of hydrostatic pressure, shows an electronic topological transition from a surface state driven topological insulator to a Weyl semimetal for , and the transition can be associated with structural changes from a layered quasi-2D vdW material to a 3D material. For , Se-p band and Bi-p band show band inversion and these two bands cross at two points, points or Weyl points around the point at the Fermi energy. There is no spin-splitting at these crossing points, and . We also notice that crossing points shift to higher momentum for larger , whereas the DOS at the Fermi energy have a finite value at large .
Our DFT calculations also suggest that a topological transition may also be achieved by doping, albeit depending on the type of doping. Intercalating a rare earth atom, Gd, between the QL leads to a metallic state with a large bandwidth and shows the presence of Weyl points below the , while substitution leads to a larger direct bandgap compared to undoped . However, both types of doping induce anti-ferromagnetic ordering in the system. We also note an induced magnetism in the system owing to the large magnetic moment on the rare earth dopant Gd atom.
The anti-ferromagnetic metallic state is particularly important for spintronics based application, which may be driven in this material. This anti-ferromagnetic metallic state arises in the case of intercalation of Gd between the QLs and could be of immense importance in spintronics based applications.60 Anti-ferromagnetic metals have primarily been seen as exotic electronic structure states. Our study opens up new application possibilities in this 3D topological material and most importantly shows the emergence of the Weyl semimetal state in the family of materials by application of pressure, which may be easily verified experimentally. Similar experiments may also be carried out with other materials in this class like , , and .
We hope our study motivates further theoretical and particularly experimental studies to explore the tunable topological transitions, particularly in the search for exotic Weyl semimetals, with associated magnetism in TIs both from the perspective of understanding of novel states and device applications as well.
The authors thank DST India for the funding and also computational facilities at S.N. Bose National Centre for Basic Sciences, Kolkata, provided by the Thematic Unit of Excellence on Computational Materials Science. M.K. thanks DST India for a Ramanujan Fellowship SR/S2/RJN-69/2012. H.B. thanks the Austrian Science Fund (FWF) for funding through START project Y746 and DST India for funding support during execution of the project. S.K.S. thanks DST-INSPIRE for financial support. H.B. acknowledges useful discussions with Dr. Sudipta Kanungo and Dr. Oindrila Deb. M.K. thanks Dr. Sandip Chatterjee and Dr. T. Setti for useful discussion.
The data that support the findings of this study are available from the corresponding author upon reasonable request.