XFe4Ge2 (X = Y, Lu) and Mn3Pt: Filling-enforced magnetic topological metals
Di Wang, Feng Tang, Hoi Chun Po, Ashvin Vishwanath, Xiangang Wan
XXFe Ge (X = Y, Lu) and Mn Pt: Filling-enforced magnetic topological metals
Di Wang,
1, 2
Feng Tang,
1, 2
Hoi Chun Po, Ashvin Vishwanath, and Xiangang Wan
1, 2 National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA Department of Physics, Harvard University, Cambridge, MA, USA (Dated: October 30, 2019)Magnetism, coupled with nontrivial band topology, can bring about many interesting and exoticphenomena, so that magnetic topological materials have attracted persistent research interest. How-ever, compared with non-magnetic topological materials (TMs), the magnetic TMs are less studied,since their magnetic structures and topological phase transitions are usually complex and the first-principles predictions are usually sensitive on the effect of Coulomb interaction. In this work, wepresent a comprehensive investigation of XFe Ge (X = Y, Lu) and Mn Pt, and find these mate-rials to be filling-enforced magnetic topological metals. Our first-principles calculations show thatXFe Ge (X = Y, Lu) host Dirac points near the Fermi level at high symmetry point S . These Diracpoints are protected by P T symmetry ( P and T are inversion and time-reversal transformations,respectively) and a 2-fold screw rotation symmetry. Moreover, through breaking P T symmetry, theDirac points would split into Weyl nodes. Mn Pt is found to host 4-fold degenerate band crossingsin the whole high symmetry path of A - Z . We also utilize the GGA + U scheme to take into accountthe effect of Coulomb repulsion and find that the filling-enforced topological properties are naturallyinsensitive on U . I. INTRODUCTION
The topological nature of electronic bands has at-tracted tremendous attention in condensed matterphysics since the birth of topological insulator [1, 2].During the past decade, a variety of topological materi-als have been discovered, including topological insulators[1–5], Dirac semimetals [6–10], Weyl semimetals [11–13],node-line semimetals [14–18], topological crystalline in-sulators [19, 20], and various other topological phases[21–27]. It is well-known that the nontrivial band topol-ogy is usually protected by time-reversal ( T ) symmetryor other spatial symmetries such as mirror symmetry,glide symmetry, etc [26]. Furthermore, symmetry andband topology are intertwined with each other, and as aresult symmetry information can be used to diagnose theband topology in a highly efficient manner. Nowadaysthe time-reversal-invariant topological materials (TMs)have been extensively studied in both theories and exper-iments. Recently, symmetry-based methods for the effi-cient discovery of topological materials were developed[28–31], and thousands of nonmagnetic TM candidateshave been proposed [32–34].Compared to the time-reversal-invariant topologicalmaterials, magnetic topological materials are also ex-pected to show rich exotic phenomena [35], such as ax-ion insulators [36–43], antiferromagnetic topological in-sulator [40–47], magnetic Dirac semimetal [48–50], andmagnetic Weyl semimetals [11, 51]. However, the pre-dictions on magnetic TMs are relatively rare and veryfew of them have been realized in experiments up to now[35]. This limitation is originated from the fact that thetopological properties are usually accompanied by sig-nificant spin-orbit coupling (SOC), while SOC typicallyleads to complex magnetic structures which is difficult to characterize experimentally and theoretically. More-over, unlike non-magnetic systems, Coulomb interactionis of substantial importance in most magnetic systems,and the Coulomb repulsion is usually incorporated by theparameter U in first principles calculations. Therefore,the first-principles predictions for magnetic topologicalmaterials usually depends on the value of U [11, 38, 51].Recently, the filling constraint for band insulatorhas been established to discover topological semimetals[52, 53]. This method enables the efficient search forfilling-enforced topological materials solely on their spacegroup (SG) and the filling electron number. The centrallogic is that there is a tight bound for fillings of bandinsulator in a SG [52]. Once a material crystallizing inthis SG own the number of valence electrons per unit cellout of the tight bound, it cannot be a insulator. Basedon the filling-constraint method [52, 53], one can readilycalculate the filling constraint ν BS for a material accord-ing to its SG, where ν represents the filling number ofoccupied electrons per unit cell. If ν / ∈ ν BS · Z (here Z represents any integer), this material has an electron fill-ing incompatible with any band insulator, and it musthave symmetry-protected gaplessness near the Fermi en-ergy (unless a further symmetry-breaking or correlatedphase is realized). This type of material is referred toas filling-enforced (semi)metals [52, 53]. Moreover, thefilling-constraint method has been extended to magneticmaterials based on their magnetic space group (MSG)symmetries [54]. Note that the filling-constraint methodis based on the interplay between electron filling and(magnetic) space group symmetries, and therefore it isinsensitive to the precise value of U so long as the rele-vant symmetries are preserved.In this work, by applying the filling-constraint method[52–54], we find serveral magnetic topological metals: a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t FIG. 1. Crystal structure of YFe Ge . The green, blue, andpurple balls represent the Y, Fe, and Ge ions, respectively.The arrows denote the ground magnetic order measured byRef. [55]. XFe Ge (X = Y, Lu) and Mn Pt. We display detailedanalysis for the topological features of XFe Ge , whereDirac points are located at S point near the Fermi en-ergy. Moreover, the calculations show that the Diracpoints would split into Weyl nodes by a small perturba-tion. We also perform the first-principles calculations forthe high-temperature phase of Mn Pt, where the bandsalong the A - Z path are four fold degenerate. The resultsshow that the essential properties and our conclusions donot depend on the value of U as we expected. II. METHOD
The calculations of electronic band structure and den-sity of states have been carried out as implemented inthe Vienna ab-initio simulation package (VASP) [56–58].The Perdew–Burke–Ernzerhof (PBE) of generalized gra-dient approximation (GGA) is chosen as the exchange-correlation functional [59]. 6 × ×
12 and 16 × × Ge (X = Y, Lu) and Mn Pt system, respec-tively. The self-consistent calculations are considered tobe converged when the difference in the total energy ofthe crystal does not exceed 0.01 mRy. The effect of spin-orbit coupling (SOC) [60] is considered self-consistentlyin all the calculations. We also utilize the GGA + U scheme [61] to take into account the effect of Coulombrepulsion in 3 d orbital and the value of parameter U isvaried between 0 and 4 eV. III. RESULTS
YFe Ge was previously reported to crystallize inthe ZrFe Ge type of structure with the space group P /mnm at room temperature [62]. In this tetragonalstructure, YFe Ge has two formula units in the primi-tive unit cell [62]. In 2001, Schobinger-Papamantellos etal. [55] measured the neutron diffraction and magneticproperties of YFe Ge , and found a magnetostructural(ferroelastic and antiferromagnetic) transition, where the FIG. 2. Partial density of states (PDOS) of YFe Ge (left) and LuFe Ge (right) from GGA calculation with non-collinear antiferromagnetic configuration. The Fermi energyis set to zero. magnetic transition at T N = 43.5 K is accompaniedby a first-order phase transition from tetragonal struc-ture ( P /mnm ) to orthorhombic structure ( P nnm ).The magnetic structure below T N is non-collinear an-tiferromagnetic with the type-III magnetic space group P n (cid:48) n (cid:48) m (cid:48) (58.399 in the Belov-Neronova-Smirnova (BNS)settings [63]), as shown in Fig. 1. Note that the mag-netic moments on two sites related by the inversion sym-metry ( P ) point in opposite directions, thus YFe Ge is invariant under P T symmetry ( T is the time-reversaltransformation). The magnetic moments of Fe ions attwo sites are measured to be 0.63 µ B per Fe ion equallyat 1.5 K. Similar to YFe Ge , LuFe Ge has the first-order magnetoelastic transition at T N = 32 K from non-magnetic tetragonal structure to antiferromagnetic or-thorhombic structure, while the Fe moment value is 0.45 µ B [64]. These materials are suggestted to be filling-enforced topological materials [54], as we show in thefollowing.Based on the non-collinear antiferromagnetic configu-ration suggested by neutron diffraction experiment [55]as shown in Fig. 1, we perform the first-principles cal-culations for YFe Ge . The density of states and theband structures are shown in Fig. 2 (a) and Fig. 3 (a)-(c), respectively. It should be noted that, due to P T symmetry, the electronic bands in the whole BZ are dou-bly degenerate, as shown in Fig. 3. The bands in theenergy range from − − s states, while Y bands appear mainlyabove 3.0 eV. The 3 d states of Fe ions are mainly locatedfrom − p states appear mainlybetween − − FIG. 3. Band structures of XFe Ge (X = Y, Lu) with experimental magnetic configuration [55]. (a)-(c) Band structures ofYFe Ge from GGA, GGA+ U ( U = 2 eV) and GGA+ U ( U = 4 eV) calculations, respectively. (d)-(f) Band structures ofLuFe Ge from GGA, GGA+ U ( U = 2 eV) and GGA+ U ( U = 4 eV) calculations, respectively. The Fermi energy is set tozero. The Dirac points near the Fermi level are marked with red circle. has a small calculated magnetic moment ( ∼ µ B ),but the major magnetic moment is still located at theFe site. Our calculated magnetic moments of Fe ionsincorporates all the symmetry restrictions, and the abso-lute values of magnetic moments at two Fe sites are 1.86 µ B and 1.70 µ B , which is larger than the experimentalvalue. Similar discrepancy has also been reported in thecalculations for other Fe-based intermetallic compounds[65]. For YFe Ge system, the filling constraint ν BS forits MSG 58.399 is 4 [52–54]. Meanwhile, the number ofelectrons per unit cell for YFe Ge is to ν = 414, thus ν / ∈ ν BS · Z , indicating that YFe Ge must be a filling-enforced material.However, the filling-constraint method does not pro-vide the detailed topological properties. As shown in thesymmetry analysis at S point in the next section, only a4-dimensional irreducible representation is allowed, thusall the states at the S point must be grouped into Diracpoints. While this conclusion holds for all materials inthe same magnetic space group, the filling of YFe Ge implies these Dirac points are naturally close to the Fermienergy. As shown in Fig. 3 (a), there is a Dirac point atonly 56 meV above the Fermi level at S point, while theDirac point below the Fermi level is relatively far away (atabout −
120 meV). We also take into account the effectof Coulomb repulsion in Fe-3 d orbital by performing theGGA + U calculations. The value of U around 2 eV iscommonly used in the Fe-based intermetallic compound[66, 67]. We have varied the value of U from 0 to 4.0eV ( U = 0 eV represents GGA calculation without U ), and the calculations show that the position of the Diracpoint is kept at the S point with slightly varying energynear the Fermi level, as shown in Fig. 3 (a)-(c). As men-tioned above, the filling-constraint method depends onlyon electron filling and magnetic space group symmetries,thus the filling-enforced topology is not sensitive to thecalculation details.We also perform the first-principles calculations ofLuFe Ge whose the band structures and the density ofstates are shown in Fig. 3 (d)-(f) and Fig. 2 (b), respec-tively. Except Lu-4 f states which are located around − Ge are very sim-ilar to YFe Ge , as shown in Fig. 2 (b) and Fig. 2 (a).The filling number of electrons per unit cell is found to be ν = 478, thus ν / ∈ ν BS · Z , also identifying LuFe Ge asfeaturing Dirac points pinned at S point near the Fermilevel.As mentioned above, YFe Ge exhibits an antiferro-magnetic order with opposite spins related by inversion,and so P T symmetry is present and the electronic bandsare doubly degenerate everywhere. Upon breaking the P T symmetry, the Dirac cone may split into Weyl nodes[68]. Note that, with the P T symmetry and the two-fold rotation { | } symmetry coexisting in this sys-tem, the z-direction component of Fe magnetic moment m z should be zero, and the magnetic moments are lyingin the xy-plane. By a small perturbation such as exter-nal field, the magnetic configuration may have nonzeroz-direction component with P T symmetry broken while { | } preserved, and the Dirac cone may split into FIG. 4. Band structures of YFe Ge from GGA calculationwith the magnetic configuration that is slightly deviated fromthe ground state. The Fermi energy is set to zero. The insetis the detailed structure around S point. The red and blueline represent the eigenstates of { | / , / , / } with theeigenvalues − ie − iπk y and + ie − iπk y , respectively. The Weylpoint along X - S line are marked with black circle. Weyl nodes. Accordingly, we perform the GGA calcu-lations with the magnetic state where the magnetic mo-ments have the deflection angle about 2 ◦ from xy-plane.As shown in Fig. 4, at S point, all the Dirac points in-deed split into Weyl points. In addition, there is also asymmetry-protected band crossing in the path X - S , asshown in Fig. 4. As discussed in the symmetry analysisbelow, only a two-fold screw rotation { | / , / , / } is preserved for the k point (1/2, k y , 0) in the path X - S .The first-principles results show that the red and blueline represent the eigenstates of { | / , / , / } withthe eigenvalues − ie − iπk y and + ie − iπk y respectively, asshown in Fig. 4. Therefore the hybridization betweenthese two bands is forbidden and there is a unavoidablecrossing point located at X - S line. Similarly, the first-principles results show that there is also a unavoidablecrossing point along the path Y - S .We also find the high-temperature phase of a cubicantiferromagnetic intermetallic compound Mn Pt as an-other filling-enforced topological material. Experimentreveals that Mn Pt crystallizes in a cubic crystal struc-ture (space group
P m -3 m ) at room temperature and hasa long-range antiferromagnetic order with T N = 475 K[69–71]. Neutron diffraction experiments show a first-order magnetic transition in Mn Pt system at about 365K, between a low-temperature non-collinear antiferro-magnetic state and a high-temperature collinear antifer-romagnetic state [69–71]. The high-temperature phaseof Mn Pt is collinear antiferromagnetic with the mag-netic space group P c /mcm (132.456), where the Mnatoms in xy-plane couple antiferromagnetically, and theMn atoms along z-direction also have opposite spin ori-entations. Very recently, Liu et al. [72] report the obser- FIG. 5. Band structures of Mn Pt with high-temperaturecollinear antiferromagnetic configuration from GGA calcula-tion. The Fermi energy is set to zero. vation of the anomalous Hall effect in thin films of thelow-temperature phase for Mn Pt. They also show thatthe anomalous Hall effect can be turned on and off byapplying a small electric field at a temperature around360 K and the Mn Pt is close to the phase transition [72].Therefore exploring the possible exotic properties of thehigh-temperature phase for Mn Pt is also an interestingproblem.Similarly, we perform the first-principles calculationsbased on the high-temperature phase of Mn Pt andthe band structures are shown in Fig. 5. The high-temperature phase of Mn Pt also has P T symmetrylike YFe Ge , thus the electronic band structures aresymmetry-protected doubly degenerate in whole BZ. Thecalculated magnetic moment at the Mn site is 2.9 µ B perMn ion, which is in reasonable agreement with the exper-iment value 3.4 µ B [69–71]. The Pt-5 d states are mainlylocated from − − d states of Mn ionsappear mainly from − Pt, the fillingnumber of electrons per unit cell is 306 while the ν BS forits MSG (132.456) is also 4, thus ν / ∈ ν BS · Z , indicatingthat there is a half-occupied four-fold energy level nearFermi energy. As shown in Fig. 5, the bands are four folddegenerate in the path of A - Z , which is protected by thesymmetry operations of magnetic stucture, as shown inthe detailed symmetry analysis in the next section. Wealso vary the value of U from 0 to 4.0 eV and find thatthe four-fold energy level always exists. IV. SYMMETRY ANALYSIS
In this section, we show the detailed symmetryanalysis for the Dirac band crossings. We will firstfocus on the S point in XFe Ge (X = Y, Lu), followedby a corresponding discussion for Mn Pt. For the S point (1/2, 1/2, 0), eight symmetry operations aregenerated by three symmetries: the P T symmetry {− (cid:48) | } , a two-fold screw rotation { | / , / , / } and a two-fold rotation { | } , where the left partrepresents the rotation and the right part means thelattice translation. Note that − T here. Since { | / , / , / } = −{ | , , } , where the minus signoriginates from the spin rotation, the momentum phasefactor equals to − S point. Thus,the eigenvalues for { | / , / , / } is ±
1. We canthen choose the eigenstates ψ ± nS of { | / , / , / } at S point, where the superscript denotes the eigen-value of { | / , / , / } and n is the band index.Because [ {− (cid:48) | } , { | / , / , / } ] = 0 when actingon the Bloch states at S, operation of {− (cid:48) | } willpreserve the eigenvalue of { | / , / , / } andresult in the other state in the Kramers doublet:i.e. {− (cid:48) | } ψ + nS is orthogonal to ψ + nS but with thesame eigenvalues of { | / , / , / } . Besides, itshould be noted that { | / , / , / }{ | } = −{ | }{ | / , / , / } , thus { | } ψ ± nS reverses the eigenvalue of { | / , / , / } . So the four orthogonal states ψ + nS , {− (cid:48) | } ψ + nS , { | } ψ + nS , { | }{− (cid:48) | } ψ + nS aredegenerate, constituting the basis of the 4-dimensionalirreducible representation. Therefore, in XFe Ge system, the S point (1/2, 1/2, 0) only allow for4-dimensional irreducible representation.Based on the k · p method, we build theeffective Hamiltonian by using the four rele-vant states as basis vectors, in the order of ψ + nS , {− (cid:48) | } ψ + nS , { | } ψ + nS , { | }{− (cid:48) | } ψ + nS .To the lowest order in q , the Hamiltonian H ( q ) can bewritten as q x C − i q z C − q y C √ q z ( iC − C ) √ q x C q z ( iC + C ) √ i q z C − q y C √ i q z C − q y C √ q z ( − iC + C ) √ − q x C q z ( − iC − C ) √ − i q z C − q y C √ − q x C , (1)where q = k − S , and C i ( i = 1, 2, ..., 5) are parameters.The effective Hamiltonian suggests a linear dispersion inneighbourhood of S . It is worth mentioning that there isalso only one 4-dimensional irreducible representation in Z point (0, 0, 1/2), and the dispersion around Z point isalso linear.When P T symmetry is broken, there are only foursymmetry operations for the S point (1/2, 1/2, 0) gener-ated by { | / , / , / } and { | } . As mentionedbefore, for the eigenstates ψ ± nS of { | / , / , / } at S point, { | } ψ ± nS reverses the eigenvalue of { | / , / , / } . So ψ + nS and { | } ψ + nS own thesame energy and they are orthogonal with each other,constituting the basis of the 2-dimensional irreduciblerepresentation. Therefore all the states at the S pointmust be grouped pairwise. Meanwhile, for the k point (1/2, k y , 0) in the path X (1/2, 0, 0) − S (1/2, 1/2, 0),only a two-fold screw rotation { | / , / , / } is pre-served. Note that { | / , / , / } = −{ | , , } ,for k point (1/2, k y , 0), the eigenvalues should be ± ie − iπk y . As shown in Fig. 4, the first-principles re-sults show that two bands along this line belong to theeigenstates of { | / , / , / } with different eigenval-ues. Thus the hybridization between these two bandsis forbidden and the band crossing is symmetry pro-tected, as shown in Fig. 4. Similarly, for the k point( k x , 1/2, 0) in the path Y (0, 1/2, 0) − S (1/2, 1/2, 0), { | / , / , / } is preserved and the first-principlesresults show that there is also a symmetry protected bandcrossing along the Y - S path.In Mn Pt, similar to the discussion above, westudy the symmetry operations of this magnetic struc-ture and find that there is only one 4-dimensionalirreducible representation along A - Z path in theBZ: For the path of A (1 / , / , / Z (0 , , / P T symmetry {− (cid:48) | } , a two-foldscrew rotation { | , , / } and a mirror operation { m − | } . Since { | , , / } = −{ | , , } (theminus sign is coming from the electron spin), theeigenvalues for { | , , / } is ± i . We can thenchoose the eigenstates ψ ± nA − Z of { | , , / } in A - Z path, where the superscript denotes the eigen-value of ± i and n is the band index. Note that {− (cid:48) | }{ | , , / } = −{ | , , / }{− (cid:48) | } ,indicating that {− (cid:48) | } ψ + nA − Z is orthogonalto ψ + nA − Z but with the same eigenvalues of { | , , / } . Besides, it should be noted that { | , , / }{ m − | } = −{ m − | }{ | , , / } ,thus { m − | } ψ ± nA − Z reverses the eigenvalueof { | , , / } . So ψ + nA − Z , {− (cid:48) | } ψ + nA − Z , { m − | } ψ + nA − Z , { m − | }{− (cid:48) | } ψ + nA − Z are againorthogonal and degenerate, constituting the basis ofthe 4-dimensional irreducible representation. Therefore,in Mn Pt system, only a 4-dimensional irreduciblerepresentation is allowed along the A (1 / , / , / Z (0 , , /
2) path.
V. CONCLUSION
In conclusion, by applying the filling constraints,we discover several magnetic topological semimetals:XFe Ge (X = Y, Lu) and Mn Pt. The first-principlescalculations show that YFe Ge is a metal with a Diraccone located at S point near the Fermi level, which isprotected by the symmetry operations of magnetic stuc-ture. We have varied the value of U from 0 to 4.0 eV,and the results show that Dirac point always exists, sincethe topological property is filling-enforced and indepen-dent on U . When the magnetic moments have a smallnonzero z-direction component, the Dirac point wouldsplit into Weyl nodes around the S point. We also per-form the first-principles calculations based on the high-temperature collinear antiferromagnetic configuration ofMn Pt. The calculation results and symmetry analysisshow that it is also a topological material.
VI. ACKNOWLEDGEMENTS
DW, FT and XW were supported by the NSFC(No.11525417, 11834006, 51721001 and 11790311),National Key R&D Program of China (No.2018YFA0305704 and 2017YFA0303203) and theexcellent programme in Nanjing University. AV wassupported by a Simons Investigator Award and by theCenter for Advancement of Topological Semimetals, anEnergy Frontier Research Center funded by the U.S.Department of Energy Office of Science, Office of BasicEnergy Sciences, through the Ames Laboratory underits Contract No. DE-AC02-07CH11358. HCP wassupported by a Pappalardo Fellowship at MIT and aCroucher Foundation Fellowship. [1] M. Z. Hasan and C. L. Kane, “Colloquium: topologicalinsulators,” Rev. Mod. Phys. , 3045 (2010).[2] X.-L. Qi and S.-C. Zhang, “Topological insulators andsuperconductors,” Rev. Mod. Phys. , 1057 (2011).[3] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, “Topological insulators in Bi2Se3, Bi2Te3 andSb2Te3 with a single Dirac cone on the surface,” Nat.phys. , 438 (2009).[4] Y. 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