Topological transitions to Weyl states in bulk Bi$_2$Se$_3$: Effect of hydrostatic pressure and doping

Bi$_2$Se$_3$, a layered three dimensional (3D) material, exhibits topological insulating properties due to presence of surface states and a band gap of 0.3 eV in the bulk. We study the effect hydrostatic pressure $P$ 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 P$>$7.9 GPa, the bulk electronic properties show a transition from an insulating to a Weyl semi-metal state due to band inversion. This transition may be correlated to a structural transition from a layered material to a 3D system observed at $P$=7.9 GPa. At large $P$ density of states have significant value at the Fermi-energy. Intercalating Gd with a small doping fraction between Bi$_2$Se$_3$ 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 finite local field induced by large magnetic moments on Gd atoms. However, substituting Bi with Gd induces anti-ferromagnetic order with an increased direct band gap. Our studies provides novel approaches to tune topological transitions, particularly in capturing the elusive Weyl semimetal states, in 3D topological materials.


I. INTRODUCTION
Topological insulators (TI) have potential future application in quantum computers 1-3 and spintronics 4-6 owing to existence of symmetry protected edge or surface states, and also provide a fundamental bridge between highenergy 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 Bi x Sb 1−x family of materials are a prime example of three dimensional (3D) materials with Z 2 invariant symmetry. Sb 2 Te 3 , Bi 2 Te 3 and Bi 2 Se 3 are 3D material topological layered materials. Bi 2 Se 3 forms effectively two dimensional (2D) layered structures and theoretical studies predict that Bi 2 Se 3 has a topologically non-trivial bulk energy gap of 0.3 eV [14][15][16][17][18][19] . The topological surface states are described by a single gapless Dirac cone at the Γ point 15,18,20,21 . It is interesting to understand the mechanism which may close bulk energy gap in Bi 2 Se 3 . Bi 2 Se 3 and its family of materials have been extensively explored, by applying pressure [22][23][24][25][26][27][28] , doping Bi with rare earth (RE) atoms like Gd, Sm etc. [29][30][31][32][33][34] and also with transition metal (TM) atoms like Cr, Fe, etc. [35][36][37][38] . Incorporating impurities with large magnetic moment may break time reversal symmetry and lead to many exotic phenomena one of which is quantum anomalous hall ef-fect (QAHE). The QAHE in material supports dissipationless charge transport which is very highly desirable experimentally. Mn-doped Bi 2 Se 3 shows spin glass like behavior 35 , and Fe-doped Bi 2 Se 3 exhibits dominance of ferromagnetic interactions, whereas Cr-doped Bi 2 Se 3 favors anti-ferromagnetic interactions 36 . However, small ( less than 0.1 %) doping of Cr doped materials are reported as ferromagnetic, while Fe-doped Bi 2 Se 3 tends to be weakly anti-ferromagnetic. Cr and Fe-doped Bi 2 Se 3 are insulating, but the band gaps are substantially reduced due to the strong hybridization between the d orbital of the dopants and the p orbitals of the neighboring Se atoms 39 . There are some reports of intercalating and doping topological materials which exhibit exotic behaviour like Bi 2 Se 3 doped with Cu at doping fractions of 0.12 and 0.15 show intermixing of both Cuintercalation between Se-Se layers and Cu-substitution in Bi-layer sites 40 . Superconducting transitions in Cuintercalated Bi 2 Se 3 is observed experimentally with a T C of 3.5 K and 3.6 K for doping fraction 0.12 and 0.15 respectively. Metallic behavior in the Bi 2 Se 3 crystals and paramagnetic features are observed in the low temperature region of the Cu-doped samples 40 . Intercalating of Cu in this material also holds the promise of topological superconductivity 41 .
Rare earth (RE) atoms with 4f electrons are expected to be better candidates to introduce FM order in TIs compared to 3d TMs 34,42 . Ionic radii of RE atoms are larger than TM atoms and RE elements match Bi 2 Se 3 lattice better than TM. Hence a smaller structural distortion is expected due to small lattice mismatch, and no disorder is observed 34 . RE atoms are better dopants in avoiding impurity aggregation which is seen in Cr (TM) doped (Bi,Sb) 2 Te 3 which is responsible for mag-arXiv:2011.07574v1 [cond-mat.mtrl-sci] 15 Nov 2020 netic disorder 42 . In addition f orbitals have 2 extra sub-orbitals compared to d orbitals and hence they may introduce larger magnetic moments in the system. In case of substitutional doping 3d TMs are divalent substitution for trivalent Bi, and hence generally have a reduced average moment. 4f RE electrons are more localized with a maximum of 7 unpaired electrons compared to 5 unpaired electrons in 3d TMs. A bulk paramagnetic behavior with a large magnetic moment from substituted Gd 3+ ion is reported in Bi 2 Se 3 29 . However, a weak anti-ferromagnetic coupling between the Gd 3+ ions with an exchange strength of -0.5K is shown in Bi 2 Te 3 31 . In an experimental study of substitutional doping of Gd in Bi 2 Se 3 , doping induced transition from a paramagnetic to anti-ferromagnetic phase is observed 32 . Intercalating Rb atoms, between the quintuple layer structure of Bi 2 Se 3 can form a quantum-confined two-dimensional electron gas state (2DEG) with a strong Rashba-type spin-orbit splitting 43 .
Substitutional Gd doping effects have been studied in Bi 2 Te 3 44 , and these systems have shown paramagnetic (PM) to anti-ferromagnetic (AFM) phase transition. For example, Gd doped Bi 2 Te 3 shows an effective AFM coupling between dopants mediated by Te atoms 45 .
Angle-resolved photoemission spectroscopy shows that the topological surface state remains intact up to large doping concentration of Gd, in Bi 2 Te 3 44 . Magnetoresistance measurements show a weak anti-localization, indicating strong spin orbit interaction [46][47][48][49] , and magnetometry reveals that these films are paramagnetic with a magnetic moment of 6.93 µ B per Gd 3+ ion 33 . However to the best of our knowledge no theoretical explanation of this phenomena exists till date.
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 Bi 2 Se 3 at a critical pressure of 11GPa on application of pressure via diamond anvil cell (DAC) 50 . Pressure induced structural phase transitions have been experimentally observed at very high pressures 51 . The experiments indicate that a progressive structural evolution occurs from an ambient rhombohedral phase (Space group (SG): R3m) to monoclinic phase (SG: C2/m) at 36GPa and eventually to a high pressure body-centered tetragonal phase (SG: I4/mmm) at 81GPa on application of pressure via DAC. A pressure induced transition to a topological phase has been found in Bi 2 S 3 at a pressure of 5.3GPa exerted via DAC 52 .
In some of topological materials like TaAs single crystals the valance band (VB) and the conduction band (CB) intersects at two points at ± k near the Fermi energy. If band dispersion near the crossing point is linear due to relativistic nature of fermions and the system preserves the time-reversal and inversion symmetry, then the system may be characterised as a Weyl semimetal (WSM) 53,54 . In WSMs non-orthogonal magnetic and electric fields results in a novel observation of chiral anomaly. This results in the chiral-magnetic effect which is the observation of an unconventional negative longitudinal magnetoresistance 55 .
In this article we investigate the effect of hydrostatic pressure (HP), and doping with a rare earth element on bulk phase of Bi 2 Se 3 , a probable 3D TI, from the perspective of ab-initio density functional theory (DFT) based calculations. The material shows a transition from a small bulk band gap insulator at low HP, to a gapless dirac state at a critical pressure 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 Bi 2 Se 3 family of materials. We also note that Bi 2 Se 3 undergoes a transition from a layered material to 3D topological material on applying 7.9 GPa HP.
We also study the effect of intercalation of Gd atoms between the Bi 2 Se 3 quintuple layers (QL) as well as partially substituting Bi with Gd in bulk Bi 2 Se 3 . While intercalating Gd between the 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 band gap and an insulating state. This results from differential bonding introduced due to the different ways of doping the material. Thus we propose that a tunable topological transition to Weyl states may be driven in 3D bulk Bi 2 Se 3 by the effect of both application of hydrostatic pressure and intercalating with rare earth elements, to attain the exciting and elusive Weyl like states. Our study is expected to open up further experimental investigations in the tunability of topological transitions in bulk phases of 3D topological materials particularly in context of experimental realisation of such Weyl semimetal states.
The paper is divided into five sections, we describe the computational method in section II. The results are divided into two major sections. In section III we describe the effect of pressure on structural and electronic properties. The effect of doping is studied in section IV. The summary and comparison of results are given in section V.

II. COMPUTATIONAL DETAILS
Our first-principles calculations were carried out in the plane wave basis as implemented in the Vienna Ab-initio Simulation Package (VASP) 56 with projector-augmented wave (PAW) potential 57 . The exchange-correlation functional used in the calculations is the generalized gradient approximation (GGA) implemented following the Perdew-Burke-Ernzerhof 58 prescription. Local correlations are taken into account wherever necessary with the energy correction within the framework of GGA+U formalism primarily for dopant Gd atoms, with values of U = 6eV, J = 1eV . For ionic relaxations, internal positions of the atoms are allowed to relax until the forces became less than 0.005 eV/A 0 . Energy cutoff used for calculations is 500 eV, and 6 × 6 × 4 Monkhorst-Pack kpoints 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 59 . 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.

III. EFFECT OF HYDROSTATIC PRESSURE P
In this section we discuss the effects of the application of hydrostatic pressure (HP) P on bulk band structure of Bi 2 Se 3 . The HP induced structural transition, and changes in lattice parameters are analysed in first subsection, and a systematic study of the effect of structural transition on the band structure properties of Bi 2 Se 3 are provided in the next subsection. Generally, pressure pushes the atoms closer to each other, and leads to enhancement in effective hybridization of orbitals which result in reduction of band gap. In large P limit structural transition is also a possibility. In this section we study the structural behaviour of Bi 2 Se 3 first, and then analyse the effect of pressure induced structural transitions on the electronic properties.

A. Change in crystal structure
We first discuss the basic structural details of the material at P = 0GPa, and thereafter, the structural changes are analysed at various pressures. Bi 2 Se 3 has a hexagonal symmetry with space group R3m, and with lattice parameters a=b=4.142Å, c=28.637Å and lattice angles α = β = 90 • , γ = 120 • . Bi 2 Se 3 forms quintuple layers (QL) within the hexagonal unit cell as seen from Fig 1 a. The crystal structure along the c-axis direction consists of QLs of two Bi layers sandwiched between three Se layers, Se 1 -Bi-Se 2 -Bi-Se 1 , where the subscript indicates that the two Se atoms are in-equivalent by symmetry (cf Fig.1). The atoms within each QL are chemically bonded, but the QLs are weakly bonded through van der Waals interaction. Bi 2 Se 3 slabs consist of an integer number of QLs.
To understand the structural phase transition with pressure P , c/a ratio and distance between Se-Se atoms sitting at two nearest QL d Se−Se are studied, and we note that c/a ratio decreases with P up to P = P c = 7.9 GPa, and then it increases on increment of P as shown in Fig. 2 a. Variation of lattice parameter a, b and c and c/a with P are provided in Table I in the Appendix. The distance between the two Se atoms from two nearest QL d Se−Se continuously decreases with P , and at P = P c GPa the Se-Se distance decreases to below 2.17Å which is less than bond length of diatomic Se 2 60 , therefore at P c GPa For P > P c Bi 2 Se 3 behaves like three dimensional (3D) structure rather a layered structure, as demonstrated in Fig 2 c, which is also confirmed from the charge density which are not shown here for the sake of brevity. The bond distances from Bi to Se1, d Bi−Se1 , and to Se2, d Bi−Se2 , decrease on application of pressure. The bond angles ∠Se1-Bi-Se1 and ∠Se2-Bi-Se2 also decrease with increase of P , however due to bending of the bonds, ∠Se1-Bi-Se2 increases with increase of P as shown in Table II in the appendix.

B. Change in electronic structure
We study the change in electronic structure of Bi 2 Se 3 with application of hydrostatic pressure as shown in Fig.  3. The band structure at zero applied pressure has a band gap of 0.3 eV at the Γ point and both Bi and Se have partially filled p-orbitals which participate to form energy bands as seen from the projected DOS in between the bonding and anti-bonding states resulting from the hybridization of p z orbitals on the Bi and Se1 sites. Considering surface calculations the band structure of Bi 2 Se 3 shows the presence of surface states at the Γ point which is a typical signature of topological insulators at P = 0 (not shown here).
We apply HP P systematically on the system and notice that the band gap decrease with P and it is 0.05 eV and 0.009eV at P = 5.5 and 7.3 GPa respectively. The band gap vanishes completely at P c = 7.9 GPa, and this gapless state can be correlated to the structural phase transition at P > 7.9GPa. Surprisingly, the CB and VB moves towards the fermi-energy with increasing P as shown in Fig.3 before the critical pressure P c at which VB and CB meet each other and the band gap vanishes. Whereas, for P > P c these two bands crosses at two points ± k points around the Γ point at the fermienergy. There is no spin-splitting at these crossing points and these crossing points are possible signatures of WSM 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 presence of time reversal T and inversion symmetry P or their combined PT presence in the system 53 i.e. for inversion symmetry E nσ (k) = E nσ (−k) , for time reversal symmetry E n↑ (k) = E n↓ (k) and in combined PT symmetry E n↑ (k) = E n↓ (−k) conditions are satisfied. These conditions are easily fulfilled in 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 are preserved only in case special symmetry in the lattice, and the symmetry prevents the repulsion of degenerate points to keep the four fold degeneracy intact. Bi 2 Se 3 has R3m crystal symmetry and under pressure the distance between the QLs reduces and gives rise to three dimensional structure. For P > 7.3 GPa, the Bi-p band and Se-p band crosses at K x = ±0.045 and E f = 0 at P = 24.6 GPa as shown in Fig. 4 a. The crossing point shown in the Fig. 4 is along the Γ-C line, but similar crossing can be found along other Γ-K momentum axis.
In absence of any external perturbation, the band dispersion near the Weyl point varies as E(k) = √ m 2 + v 2 k 2 with momentum k where m and v 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 (2) as shown in Fig. 4 a. We notice that crossing point is symmetric about the Γ point and spin up and down channel of the band is degenerate in absence of the SOC. Therefore, this system preserves the both time reversal and inversion symmetry. The crossing points acts as source or sink of the Berry curvature. The Barry curvature calculated using Vaspberry 61 shown in inset Fig.  4 b. The Berry curvature have highest value near one crossing point whereas it 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 p bands are marked separately. The contribution of Bi-p band is higher near the E f in the CB than the Se-p band, whereas in VB Se-p band has higher contribution. At low pressure P < 7.3 GPa there is no density of states at E f , and PDOS at E f increases with P , large PDOS can be seen at P = 33.4 GPa.

IV. EFFECT OF DOPING
In this section we discuss the effect of doping Bi 2 Se 3 with a rare earth element Gd. The system can be doped either by the substituting Bi with Gd, or intercalating Gd between the QLs of Bi 2 Se 3 . 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 Bi 2 Se 3 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 band gap increases to 0.5eV. 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 wide band metal. The intercalated Gd induces rearrangement of energy bands which leads to band inversion below E f , however, it is different from the pressure induced inversion. The band inversion below E f is shown in the circle in Fig. 6 (a). In this case the up and down spin band split due to 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. Thus we observe a time reversal symmetry broken Weyl state whose existence has been discussed in literature. 62 . The total DOS for three different cases pure, substituted and intercalated Bi 2 Se 3 are shown in Fig 6(b), and we note that intercalation of Gd can lead to finite density of states at E f which leads to an insulator to metal transition. The large DOS comes from Gd f orbitals which are half filled and have high spin splitting. The rest of the DOS has usual contributions from Bi and Se p orbitals with some mixing with filled Gd d orbitals. The contribution to DOS at the Fermi in case of Gd intercalated Bi 2 Se 3 is from the Bi p bands which is the partially filled conduction band. It is also to be noted that although the direct band gap changes in case of substitutional doping from pure Bi 2 Se 3 , the integrated band gap 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 changes significantly. The substitutional doping gives rise to a AFM ordering consistent with earlier results 32 . We find a similar AFM ordering between intercalated Gd atoms, as shown in Fig.  7. The large magnetic moment of ∼ 7µ B on Gd which is consistent with literature 32 . The large magnetic moment on Gd induces a small moment of ∼ 0.2µ B on Se. The large magnetic moment may be because of larger exchange splitting compared to the crystal field splitting, in this case, the f -orbitals likely to be occupied by elec- trons with parallel spin. The large local magnetic moments induces Zeeman splitting in the Bi and Se p bands in both doping method, although intercalation of Gd has a much stronger splitting effect on the Bi and Se p bands than substitutional doping as seen from Fig7. 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 2×1×1 supercell for this purpose. We consider the moments to interact via a simple nearest neighbour Ising model E = J ×S i S j , where S=7/2. We find a magnetic exchange J=9.6meV between the Gd atoms, which is mediated by the Se p orbitals interacting anti-feromagnetically, as seen in from the plot of magnetisation 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 behaviour. Although in both cases majority of magnetisation comes from the f orbitals (in both cases a combination of f (3x 2 − y 2 ), f (xyz), f (yz 2 ) orbitals, there is a rather significant contribution to the total moment of 7µ B as well from d orbitals. In case of intercalation d(x 2 −y 2 ) has a major contribution while in case of substitution d(xz) has a major contribution. This shows up in the magnetisation 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 Bi 2 Se 3 shows a tunable topological transition with the application of hydrostatic pressure and doping by rare earth elements. Upon application of hydrostatic pressure bulk Bi 2 Se 3 shows a topological transition from a surface states driven topological insulator to a Weyl semimetal in bulk for P > P c , and the topological transition may be correlated with structural phase transition from a layered to 3D material. For P > P c Se-p band and Bi-p band shows band inversion and these two bands cross at two points, ± k points or Weyl points around the Γ point at the Fermi-energy. There is no spin-splitting at these crossing points, and E n↑ (k) = E n↓ (−k). We notice that crossing points shifts to higher momentum k for larger P , and the DOS at the Fermi-energy have finite value at large P .
We have also shown that a topological transition may also be achieved by doping. Our DFT calculations show that intercalating a rare earth element, Gd, between the QL has very different effects from the conventional methods of substitutional doping. While substitution leads to a larger direct band gap, intercalation leads to a metallic state with a large band width, and presence of Weyl points below the E f , however, both cases of doping induces 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 case of intercalation of Gd between the QLs, and could be of immense importance in spintronics based applications 63 . 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 Weyl semimetal state in the Bi 2 Se 3 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 Sb 2 Te 3 , Bi 2 Te 3 and Bi 2 Se 3 .
We hope our study motivates further theoretical and particularly experimental studies to explore the tunable topological transitions, particularly in the search for Weyl semimetals, with associated magnetism in TIs both from the perspective of understanding of exotic states and device applications as well.
Science. MK thanks DST India for a Ramanujan Fellowship SR/S2/RJN-69/2012. HB thanks the Austrian Science Fund (FWF), for funding through START project Y746, and DST India for funding support during execution of the project. SKS thanks DST-INSPIRE for financial support. HB acknowledges useful discussions with Dr. Sudipta Kanungo and Dr. Oindrila Deb.