Two-dimensional topological semimetal states in monolayers Cu 2 Ge, Fe 2 Ge, and Fe 2 Sn
Liangliang Liu, Chongze Wang, Jiangxu Li, Xing-Qiu Chen, Yu Jia, Jun-Hyung Cho
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Two-dimensional topological semimetal states in monolayers Cu Ge, Fe Ge, and Fe Sn Liangliang Liu , , Chongze Wang , Jiangxu Li , Xing-Qiu Chen , Yu Jia , and Jun-Hyung Cho ∗ Key Laboratory for Special Functional Materials of Ministry of Education,Henan University, Kaifeng 475004, People’s Republic of China Department of Physics, Research Institute for Natural Science,and HYU-HPSTAR-CIS High Pressure Research Center, Hanyang University,222 Wangsimni-ro, Seongdong-Ku, Seoul 04763, Republic of Korea Shenyang National Laboratory for Materials Science, Institute of Metal Research,Chinese Academy of Sciences, Shenyang 110016, China (Dated: November 18, 2019)Recent experimental realizations of the topological semimetal states in several monolayer systems are veryattractive because of their exotic quantum phenomena and technological applications. Based on first-principlesdensity-functional theory calculations including spin-orbit coupling, we here explore the drastically differenttwo-dimensional (2D) topological semimetal states in three monolayers Cu Ge, Fe Ge, and Fe Sn, which areisostructural with a combination of the honeycomb Cu or Fe lattice and the triangular Ge or Sn lattice. Wefind that (i) the nonmagnetic (NM) Cu Ge monolayer having a planar geometry exhibits the massive Diracnodal lines, (ii) the ferromagentic (FM) Fe Ge monolayer having a buckled geometry exhibits the massive Weylpoints, and (iii) the FM Fe Sn monolayer having a planar geometry and an out-of-plane magnetic easy axisexhibits the massless Weyl nodal lines. It is therefore revealed that mirror symmetry cannot protect the four-fold degenerate Dirac nodal lines in the NM Cu Ge monolayer, but preserves the doubly degenerate Weyl nodallines in the FM Fe Sn monolayer. Our findings demonstrate that the interplay of crystal symmetry, magneticeasy axis, and band topology is of importance for tailoring various 2D topological states in Cu Ge, Fe Ge, andFe Sn monlayers.
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I. INTRODUCTION
In the past decade, topological insulators and topologicalsemimetals have attracted considerable attention because oftheir promising prospects in both fundamental research andtechnological applications [1–5]. Specifically, topologicalsemimetals are characterized by the nontrivial topology ofgapless bulk bands near the Fermi energy E F and its associ-ated robust surface states [6–8]. There are several types oftopological semimetals such as Dirac semimetal (DSM), Weylsemimetal (WSM), and nodal-line semimetal (NLS) [9–12].The DSM (WSM) states have four-fold (two-fold) degenerateband crossings at discrete k points in momentum space, whilethe NLS states have band crossings along the closed or openlines within the Brillouin zone [13–17]. Interestingly, the NLSsystems have drumhead-like surface states with narrow banddispersions, thereby giving rise to a high density of states near E F . As a result, such topologically nontrivial surface statesare very vulnerable to various exotic phenomenona such asflatband ferromagnetism, Mott physics, high- T c superconduc-tivity, and other electronic instabilities [18–20].Most of the NLS states have so far been experimentally ob-served in three-dimensional (3D) materials such as PtSn [21],ZrSiS [22], and PbTaSe [23]. However, recent theoreticaland experimental studies of such NLS states have been ex-tended to 2D monolayers [24–27] whose electronic propertiescan be easily tuned by mechanical strains [28, 29]. Basedon the combined angle-resolved photoemission spectroscopymeasurements and density-functional theory (DFT) calcula-tions, Feng et al . [24] reported the presence of Dirac nodal lines (DNLs) in Cu Si monolayer which is composed of ahoneycomb Cu lattice and a triangular Si lattice. Here, Cu Simonolayer has a planar geometry, which is identical to thatof Cu Ge monolayer [see Fig. 1(a)]. Subsequently, Feng et al . [25] also synthesized another isostructural monolayerof Ag Gd to observe Weyl nodal lines (WNLs) in the fer-romagnetic (FM) phase. It is, however, noticeable that theDNLs of Cu Si monolayer and the WNLs of Ag Gd mono-layer were predicted to lift their four-fold and two-fold de-generacies with including spin-orbit coupling (SOC) [24, 25],respectively. Therefore, Cu Si and Ag Gd monolayers havenon-zero masses in DNLs and WNLs, respectively. We notethat the nonmagnetic (NM) phase of Cu Si monolayer pre-serves the mirror symmetry, while the FM phase of Ag Gdmonolayer having the in-plane magnetic easy axis breaks themirror symmetry [30, 31]. However, if the magnetizationdirection in the latter monolayer were reoriented along theout-of-plane direction via external perturbations, e.g., spin-orbit torque, the mirror symmetry would be respected to pro-tect massless WNLs against SOC, as demonstrated below inFe Sn monolayer. Thus, the symmetry protection of WNLs inFM monolayers can be manipulated by the spin reorientationeffect [32–34].In this paper, we systematically investigate the different 2Dtopological states in three monolayers Cu Ge, Fe Ge, andFe Sn using DFT calculations with the inclusion of SOC.We find that these monolayers have different ground statesdepending on the presence/absence of magnetism and mirrorsymmetry: i.e., massive DNLs for the NM Cu Ge monolayerhaving the mirror symmetry with respect to the x - y plane [seeFig. 1(a)], massive Weyl points for the FM Fe Ge mono-Typeset by REVTEX (a) (c)(b) h xy Fe Ge Fe SnCu Ge a a a a a a FIG. 1: (Color online) Optimized structures of (a) Cu Ge, (b) Fe Ge, and (c) Fe Sn monolayers. The top and side views are given in the upperand lower panels, respectively. The arrows represent the primitive lattice vectors a and a in each unit cell (indicated by the dashed lines).The structures of Cu Ge and Fe Sn monolayers are planar, while that of Fe Ge monolayer is buckled with a height difference h betweenneighboring Fe atoms. layer breaking the mirror symmetry [Fig. 1(b)], and masslessWNLs for the FM Fe Sn monolayer having the mirror sym-metry with an out-of-plane magnetic easy axis [Fig. 1(c)].It is thus revealed that for the NM Cu Ge monolayer, themirror symmetry cannot protect the four-fold degeneracy ofDNLs, whereas for the FM Fe Sn monolayer, it protects thetwo-fold degeneracy of WNLs. Interestingly, unlike Fe Snmonolayer, the geometry of Fe Ge monolayer is found to bebuckled due to an increased magnetic stress arising from itsrelatively smaller lattice constants. The resulting broken mir-ror symmetry in Fe Ge monolayer induces a transformationfrom the WNLs to Weyl points. Therefore, our comprehen-sive investigation of different 2D topological quantum statesin Cu Ge, Fe Ge, and Fe Sn monolayers demonstrates thatthe versatile topological behaviors can be entangled with mir-ror symmetry and magnetism.
II. Calculational methods
The present DFT calculations were performed using theVienna ab initio simulation package with the projector-augmented wave method [35–37]. For the exchange-correlation energy, we employed the generalized-gradientapproximation functional of Perdew-Burke-Ernzerhof(PBE) [38]. The present monoalyer systems were modeled bya periodic slab geometry with ∼
30 ˚A of vacuum in betweenthe slabs. A plane-wave basis was employed with a kineticenergy cutoff of 500 eV, and the k -space integration was donewith the 21 ×
21 meshes in the 2D Brillouin zone. All atomswere allowed to relax along the calculated forces until allthe residual force components were less than 0.005 eV/ ˚A.To investigate the topological properties of Cu Ge, Fe Geand Fe Sn monolayers, we constructed Wannier functionsby projecting the Bloch electronic states obtained from DFTcalculations onto a set of Cu (Fe) s , Cu (Fe) d , and Ge (Sn) p orbitals. Based on the tight-binding Hamiltonian with a basisof maximally localized Wannier functions [39], we not onlyidentified the existence of nodal lines but also calculated theBerry curvature around the band crossing points by using theWANNIERTOOLS package [40]. III. Results
We begin by optimizing the atomic structures of Cu Ge,Fe Ge, and Fe Sn monolayers, where Cu or Fe (Ge or Sn)atoms form a honeycomb (triangular) lattice. Their optimizedstructures are displayed in Figs. 1(a), 1(b), and 1(c), respec-tively. For Cu Ge monolayer, we obtain the lattice constants a = a = 4.218 ˚A with a planar geometry, in good agreementwith those ( a = a = 4.214 ˚A) of a previous DFT calcula-tion [41]. Therefore, Cu Ge monolayer has the point group of D with the mirror symmetry M z about the x - y plane. Mean-while, Fe Ge monolayer is found to be buckled with a = a = 4.147 ˚A and a height difference of 0.403 ˚A between neigh-boring Fe atoms [see Fig. 1(b)]. This buckling of Fe atomsis induced by the emergence of FM order, as discussed be-low. Therefore, Fe Ge monolayer has the broken M z mirrorsymmetry, leading to a reduced crystalline point group D .Meanwhile, the FM Fe Sn monolayer has a planar geometrywith a = a = 4.453 ˚A, preserving the crystalline point groupof D with the M z mirror symmetry. We note that the equi-librium structures of Cu Ge, Fe Ge, and Fe Sn monolayersdo not exhibit any imaginary phonon mode in their calculatedphonon dispersions, indicating that they are thermodynami-cally stable (see Fig. S1 in the Supplemental Material [42]).Figure 2(a) shows the electronic band structure of Cu Gemonolayer in the absence of SOC. We find that the two hole-like bands (labeled as α and β ) and one electron-like band( γ ) overlap with each other near the Fermi level E F . As aresult, there are four nodal points [designated as A, A ′ , B,and B ′ in Fig. 2(a)] along the Γ − M and Γ − K lines. Usingthe tight-binding Hamiltonian with a basis of maximally lo-calized Wannier functions [39, 40], we reveal the existenceof two DNLs around the Γ point [see Fig. 2(b)]. As shownin Fig. S2 in the Supplemental Material [42], the Wannierbands near E F are in good agreement with the DFT bands. Itis noticeable that the two DNLs are protected by the M z mir-ror symmetry with two different one-dimensional irreduciblesymmetry representations: i.e., if the two crossing bands havethe opposite eigenvalues of M z , they cannot hybridize witheach other [43, 44]. The band projections onto the Cu 3 d andGe 4 p orbitals show that the α and β bands are mainly com-posed of the Cu d xy / d x − y and Ge p x / p y orbitals, while the γ band arises from the Cu d xz and Ge p z orbitals (see Fig. S3in the Supplemental Material [42]). Therefore, the α and β bands have the even parity eigenvalue of M z , which is oppo-site to the odd parity eigenvalue for the γ band [see Fig. 2(a)].We note that, when M z is broken by the buckling of two Cuatoms in the primitive unit cell, the DNL containing A andA ′ (B and B ′ ) is transformed into three Dirac points along thethree nonequivalent Γ − K ( Γ − M) lines (see Fig. S4 in theSupplemental Material [42]).In order to examine how SOC influences the four-fold de-generacy of DNLs in Cu Ge monolayer, we perform the DFTcalculations with including SOC. Figure 2(c) displays the cal-culated band structures along the Γ − M and Γ − K lines aroundthe DNLs. We find the band-gap openings of about 45 − ± i of M z . It is noted that along the DNLs with thefour-fold degeneracy, the two bands having the same parityeigenvalue can hybridize with each other, thereby openingband gaps. Therefore, M z in Cu Ge monolayer can not protectthe DNLs against SOC, leading to the formation of massiveDNLs.To realize the symmetry-protected nodal lines in 2D mono-layers, it is prerequisite to split two-fold degenerate bandsvia the emergence of ferromagnetism that breaks time re-verse symmetry. In the present study, we consider the twoFM Fe Ge and Fe Sn monolayers. For Fe Ge and Fe Snmonolayers, the FM phase is found to be more stable than theNM (antiferromagnetic) one by 0.758 (0.404) eV and 1.181(0.413) eV per unit cell, respectively. The calculated mag-netic moments for the FM Fe Ge and Fe Sn monolayers are m = 2.08 and 2.38 µ B per Fe atom, respectively. Figures 1(b)and 1(c) display the optimized structures of the FM Fe Geand Fe Sn monolayers, respectively. Interestingly, for Fe Gemonolayer, the FM structure is buckled, while the NM one isplanar. Here, the lattice constants ( a = a = 4.147 ˚A) of theFM structure are larger than those ( a = a = 4.063 ˚A) of theNM one (see Table S1 in the Supplemental Materials [42]).Such buckling of Fe atoms and larger lattice constants in theFM phase can be attributed to the magnetic stress generatedby the exchange interactions [45] of spin-polarized electrons.By contrast, the FM Fe Sn monolayer having the relativelylarger lattice constants ( a = a = 4.453 ˚A) exhibits a planar -4-202 M (cid:1) K M E ( e V ) (a) A' B'AB (cid:2) ( ) + (cid:3) ( ) + (cid:4) ( ) - (c) E ( e V ) (cid:1) - -- -- -- - (63 meV)(47.2) M (45.1)(59.5) - -- -- -- - K (cid:1) A B A' B' (b) -0.4 -0.8
E (eV) -0.2-0.4-0.6-0.3-0.5-0.7(eV) (cid:1) M K
A A' B B' FIG. 2: (Color online) (a) Calculated band structure of Cu Ge mono-layer in the absence of SOC. The four crossing points of three bands(labeled as α , β , and γ ) along the Γ − M and Γ − K lines are designatedas A, A ′ , B, and B ′ . The Γ − M direction is parallel to the x axis. Forthe α , β , and γ bands, the parity of mirror symmetry is labeled plus orminus sign in parentheses. (b) Energy dispersions of the two DNLspassing through the A and A ′ (B and B ′ ) points, together with theirprojections onto the Brillouin zone using the color scale. (c) Zoom-in band structures around the A and B (A ′ and B ′ ) points, obtainedwith including SOC. The numbers represent the gaps (in meV) at theA, A ′ , B, and B ′ points. geometry [see Fig. 1(c)].Next, we examine why the FM instability exists in Fe Geand Fe Sn monolayers. For this, we calculate the band struc- M (cid:1) K M -224 E ( e V ) d xz d xy / d x - y d yz / d z (b) DOS (states/eV)DOS (states/eV) d xz d xy / d x - y d yz / d z FIG. 3: (Color online) Calculated band structures and DOS of theNM (a) Fe Ge and (b) Fe Sn monolayers. The band projections ontothe Fe 3 d orbitals are displayed with circles whose radii are propor-tional to the weights of the d xy / d x − y , d xz , and d z / d yz orbitals. ture and density of states (DOS) for their NM phases. Thecalculated NM band structure of Fe Ge monolayer [see Fig.3(a)] is similar to that [Fig. 3(b)] of Fe Sn monolayer, show-ing that the electronic states around E F are mostly composedof the Fe 3 d orbitals (see also Fig. S5 in the SupplementalMaterial [42]). We find that above E F , the two hole-like andone electron-like parabolic bands arise from the Fe d xy , d x − y ,and d xz orbitals with effectively high neighbor hoppings, giv-ing rise to large energy dispersions [see Figs. 3(a) and 3(b)].Meanwhile, close to E F , there exist the flatbands arising fromthe Fe d z and d yz orbitals, which are relatively more localizedthan the d xy , d x − y , and d xz components. Here, unlike the d xz orbital, the d yz orbital is deviated away from the Fe − Fe bondsdirecting parallel to the x axis. It is noted that the 3 d s va-lence electrons of Fe atom are less than those (3 d s ) ofCu atom. Therefore, the positions of E F in Fe Ge and Fe Snmonolayers [see Figs. 3(a) and 3(b)] shift downward relativeto that in Cu Ge monolayer [Fig. 2(a)]. As a consequenceof such Fermi level shifts, Fe Ge and Fe Sn monolayers havehigh DOS at E F , which in turn induces a FM order via theStoner criterion D ( E F ) I > D ( E F ) is the total DOS at E F and the Stoner parameter I can be estimated with dividingthe exchange splitting of spin-up and spin-down bands by thecorresponding magnetic moment. Figure 4(a) shows the band structure of the FM phase ofFe Ge monolayer, computed without including SOC. We findthat near E F , there are three spinful Weyl points W + , W + ,and W + along the Γ − M, Γ − K, and K − M lines, respectively[see Fig. 4(b) and Fig. S7 in the Supplemental Material [42]].Here, we consider the crossings of the same spin-polarizedbands because the absence of SOC decouples two differentspin channels. It is noted that the crystalline point group D d of buckled Fe Ge monolayer has three generators includingthreefold rotational symmetry C z about the z axis, inversionsymmetry P , and mirror symmetry M y about the x - z plane.Therefore, we have not only nine nonequivalent Weyl pointsof W + , W + , and W + in the whole Brillouin zone, but alsotheir paired Weyl points W − , W − , and W − of opposite chi-rality at inversion symmetric k -points [see Fig. 4(c)]. It hasbeen known that specific crystalline symmetries are neededto guarantee 2D massless Weyl points [48, 49]. For Fe Gemonolayer, the twofold degeneracy of Weyl points in the Γ − Kand K − M lines is mandated by C (equivalent to the combi-nation of P and M y ), because the two crossing bands have theopposite parity eigenvalues ± C . Meanwhile, the Weylpoints in the Γ − M line are protected by M y (see Fig. S8 in theSupplemental Material [42]). Using the WANNIERTOOLSpackage [40], we demonstrate that each pair of Weyl pointshave the positive and negative Berry curvature distributions[see Fig. 4(c)]. However, the inclusion of SOC lifts the two-fold degeneracy of all Weyl points, leading to massive Weylpoints. Figure 4(d) displays the gap openings of W + , W + , andW + . It is noteworthy that SOC aligns the spontaneous mag-netization direction parallel to the z axis, as discussed below.Such a magnetic anisotropy breaks C and M y symmetries,thereby giving rise to the SOC-induced gap opening at eachWeyl point.Contrasting with the buckled geometry of Fe Ge mono-layer, Fe Sn monolayer has a planar geometry which involvesthe same crystalline point group of D as Cu Ge monolayer.The resulting preservation of mirror symmetry M z in Fe Snmonolayer will be demonstrated to allow the protection ofWNLs in the presence of SOC. Figure 5(a) shows the cal-culated band structure of the FM Fe Sn monolayer withoutSOC. We find that the spin-up bands exhibit the overlaps oftwo hole-like (labeled as α and β ) and one electron-like ( γ )bands around E F , giving rise to four crossing points [desig-nated as A, A ′ , B, and B ′ in Fig. 5(b)] along the Γ − M and Γ − K lines. These nodal points evolve into the two WNLs inthe whole Brillouin zone: one passes through the A and A ′ points and the other passes through the B and B ′ points (seeFig. S9 in the Supplemental Material). The band projectionsonto the Fe 3 d orbitals show that the α ( β ) band is mainlycomposed of the Fe d x − y ( d xy ) orbital with the even parity of M z , while the γ band arises from the Fe d xz orbital with the oddparity of M z (see Fig. S10 in the Supplemental Material [42]).Therefore, the two-fold degeneracy of the two WNLs is re-spected by M z . When SOC is included, the spontaneous mag-netization direction is also parallel to the z axis, as discussedbelow. Therefore, the point group becomes C h containing W +0(cid:0)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) -(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23) W (cid:24) (b) E(cid:25)(cid:26)(cid:27) ) (a) M K (cid:1) (cid:28) -2 (cid:29) ) u"d W $ W % --- W & x ’ z ()*6 ,./12 35789:; (cid:1) < y W - W = W > W - W - W ? @ x A y - BCDFGHIJLNOPQR - STUVWXYZ[\]^_‘ab ) W cefghij W klmno pqr ) - stvw - x . yz - {|}~ W (cid:127)(cid:128)(cid:129)(cid:130)(cid:131) - (cid:132)(cid:133)(cid:134)(cid:135) - (cid:136) . (cid:137)(cid:138) - (cid:139)(cid:140)(cid:141)(cid:142) FIG. 4: (Color online) (a) Calculated band structure of the FM Fe Ge monolayer in the absence of SOC and (b) zoom-in band structuresaround the three Wely points W + , W + , and W + . (c) Distribution of all the Weyl points in the whole Brillouin zone, together with the Berrycurvature component Ω z around W − . (d) Zoom-in band structures around the W + , W + , and W + points in (b), obtained with including SOC.The numbers represent the gaps (in meV) at W + , W + , and W + . M z . Figure 5(c) displays the SOC-included band structure ofFe Sn monolayer along the Γ -M and Γ -K lines around thecrossing points. Since spin is not a good quantum numberin the presence of SOC, the spin-up bands could hybridizewith the spin-down ones. As shown in Fig. 5(c), the originalcrossing points (A, A ′ , B, and B ′ ) are still reserved, but someadditional crossing points (C, C ′ , D, and D ′ ) appear becausea spin-down band (labeled as χ ) overlaps with the spin-up α , β and γ bands. Such nodal points evolve into the four WNLspassing through the A − A ′ , C − B ′ , B − C ′ , and D − D ′ points,respectively (see Fig. S11 in the Supplemental Material [42]).Here, the three WNLs passing through the C − B ′ , B − C ′ , andD − D ′ points are newly formed by the hybridization of spin-up and spin-down bands through SOC. Figure 5(d) shows theenergy dispersion of the WNL passing through the A and A ′ points, which has a bandwidth of ∼
100 meV. The other WNLshave relatively larger bandwidths (see Fig. S11 in the Supple-mental Material [42]). It is noted that each WNL is composedof two bands of the opposite M z parity eigenvalues ± i (seeFig. S10 in the Supplemental Material [42]). Therefore, wecan say that the predicted four WNLs in Fe Sn monolayer are robust against breaking the M z symmetry. TABLE I: Calculated MAE values (in µ eV per Fe atom) of Fe Geand Fe Sn monolayers with respect to the magnetic easy axis in theout-of-plane direction. ∆ E [100] ∆ E [110] ∆ E [3.732,1,0] ∆ E [111]Fe Ge 605 605 605 403Fe Sn 890 890 890 593
Finally, we discuss magnetocrystalline anisotropic energy(MAE) which determines the orientation of magnetization aswell as topological property. Table I shows the relative energydifferences of Fe Ge and Fe Sn monolayers, depending ondifferent magnetization directions. We find that the magneticeasy axes of both Fe Ge and Fe Sn monolayers are parallelto the out-of-plane direction. The calculated MAE values ofFe Ge (Fe Sn) monolayer are 605 (890) and 403 (593) µ eVper Fe atom along the [100] and [111] directions, respectively.It is noted that the MAE values along other in-plane directions (a) (cid:143)(cid:144)(cid:145)(cid:146)(cid:147) (cid:1) (cid:148) (cid:149) I II -2024 E ( e V ) (cid:2) (cid:3) (cid:4) (cid:5) (b) (c) C'D'A' B' E ( e V ) (cid:1) K I (cid:1) B A (cid:5) (cid:2)(cid:6) (+) (cid:3) (+) (cid:4) (-) II A DB I (cid:2)(cid:6) (-i) (cid:3) (-i) (cid:4) (+i) (cid:5) (+i) (cid:1) E ( e V ) C (cid:4) (+i) (cid:5) (+i) (cid:2)(cid:6) (-i) B'A' (cid:3) (-i) (cid:3) (+) (cid:2)(cid:6) (+) (cid:4) (-) (cid:5) (cid:1) K II A' A FIG. 5: (Color online) (a) Calculated spin-polarized band structure of Fe Sn monolayer in the absence of SOC. The three spin-up and onespin-down bands around E F are labeled as α , β , γ , and χ (thick red line), respectively. Zoom-in band structures in the dashed square are shownin (b), where the crossing points of the α , β , and γ bands are designated as A, B, A ′ and B ′ . The parity of mirror symmetry is labeled plus orminus sign in the α , β , and γ bands. The corresponding zoom-in band structures in the presence of SOC are given in (c), where ± i representsthe parity of mirror symmetry. (d) Energy dispersion of the WNL passing through the A and A ′ points in (c). such as the [110] and [3.732,1,0] directions are the same asthat along the [100] direction (see Table I). This invariance ofMAE in the x - y plane is likely to be due to the fact that theFM instability is induced by the highly localized Fe d z and d yz orbitals near E F [see Figs. 3(a) and 3(b)]. Interestingly,the magnitudes of MAE in Fe Ge and Fe Sn monolayers aremuch larger than those of the typical FM crystals such as Fe( ∼ µ eV), Co ( ∼ µ eV), and Ni ( ∼ µ eV) [50, 51], as wellas ∼ µ eV of the previously predicted 2D nodal-line ma-terials InC [27] and MnN [52]. Therefore, Fe Ge and Fe Snmonolayers can be classified as hard magnetic 2D materials,which barely change the spin orientations via externally ap-plied magnetic field. It is noteworthy that Cu Si monolayerwas synthesized by the deposition of Si atoms on the Cu(111)surface [24] and Ag Gd monolayer was also synthesized bythe deposition of Gd atoms on the Ag (111) surface [25]. Inthis sense, we anticipate that Cu Ge, Fe Ge, and Fe Sn mono-layers could be synthesized by using the atomic layer deposi-tion technique [53] in future experiments.
IV. Summary
We have performed first-principles calculations for Cu Ge,Fe Ge and Fe Sn monolayers to investigate their different 2Dtopological states. By a systematic study of the electronicstructures of Cu Ge, Fe Ge, and Fe Sn monolayers, we re-vealed the existence of massive DNLs, massive Weyl points,and massless WNLs, respectively. Such different topologi-cal states were identified to be formed depending on the crys-talline symmetries and NM/FM orders in the three monolay-ers, i.e., the planar NM Cu Ge monolayer with the mirrorsymmetry of M z , the buckled FM Fe Ge monolayer break-ing M z , and the planar FM Fe Sn monolayer preserving M z .Therefore, for Cu Ge monolayer, the mirror symmetry can-not protect the four-fold degeneracy of DNLs, but for Fe Snmonolayer, it protects the two-fold degeneracy of WNLs.Specifically, Fe Ge and Fe Sn monolayers have the sizable MAE values of one or two orders larger than those of Fe, Co,and Ni crystals. The resulting topological and magnetic prop-erties of Fe Ge and Fe Sn monolayers are anticipated to bevery promising for the utilization of future spintronics appli-cations. Our findings demonstrated that the versatile topo-logical properties can be entangled with mirror symmetry andtime reversal symmetry in atomically thin monolayer systems.This work was supported by the National ResearchFoundation of Korea (NRF) Grant funded by the Ko-rean Government (Grants No. 2019R1A2C1002975, No.2016K1A4A3914691, and No. 2015M3D1A1070609), theNSFC (Grant No. 11774078), and the Innovation Scien-tists and Technicians Troop Construction Projects of HenanProvince (Grant No. 10094100510025). 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Supplemental Material for Two-dimensional topological semimetalstates in monolayers Cu Ge, Fe Ge, and Fe Sn
1. Calculated phonon spectra of Cu Ge, Fe Ge and Fe Sn monolayers.
M K M (cid:150)(cid:151)(cid:152)(cid:153)(cid:154) en cy ( T H z ) M K M F r equen cy ( T H z ) (a) (b) F r equen cy ( T H z ) M K M (c)
FIG. S1: Calculated phonon spectra of the ground-state structures of (a) Cu Ge, (b) Fe Ge, and (c) Fe Sn monolayers using the Phonopycode [1]. The three equilibrium structures are found to be stable without any imaginary phonon mode.
2. Electronic band structure of Cu Ge monolayer. -4-2024M K M (cid:155)(cid:156)(cid:157)(cid:158)(cid:159)(cid:160)¡¢£⁄ E ( e V ) FIG. S2: Band structure of Cu Ge monolayer, obtained using the tight-binding Hamiltonian with maximally localized Wannier functions(MLWF) [2]. The Wannier bands near the Fermi energy fit well with the DFT bands.
3. Band projections onto Cu and Ge orbitals.
M K M-4-2024 E ( e V ) M K M-4-2024 E ( e V ) p ¥ƒ p § p z FIG. S3: Calculated projected bands of Cu Ge monolayer onto the Cu 3 d and Ge 4 p orbitals. Here, the radii of circles are proportional to theweights of the corresponding orbitals.
4. Electronic structure of buckled Cu Ge monolayer. -4-2024 E ( e V ) M K M ¤' “«‹› fifl(cid:176) –† ‡·(cid:181)
FIG. S4: Calculated band structure of buckled Cu Ge monolayer where neighbouring Cu atoms are buckled with a height difference of 0.1 ˚A.We find that the DNL containing A and A ′ (B and B ′ ) is transformed into three Dirac points (DPs) along the three nonequivalent Γ − K ( Γ − M)lines. These DPs are protected by three nonequivalent M σ or C symmetries, similar to the case of Cu Si monolayer [3].The sizes of gapsalong the Γ − M and Γ − K lines are given.
5. Projected bands onto Fe and Ge/Sn orbitals. Fe s Fe p Fe d Ge s Ge p (a) Fe s Fe p Fe d Sn s Sn p (b) FIG. S5: Calculated projected bands of the NM (a) Fe Ge and (b) Fe Sn monolayers onto the Fe 4 s , 4 p , 3 d orbitals, Ge 4 s , 4 p orbitals, andSn 5 s , 5 p orbitals. Here, the radii of circles are proportional to the weights of the corresponding orbitals. The Fe 3 d orbitals are more dominantcomponents of the electronic states around E F , compared to other orbitals.
6. Stoner criteria for ferromagnetism in Fe Ge and Fe Sn monolayers. (b) up down D O S ( s t a t e s / e V ) E (eV) a FIG. S6: Calculated spin-polarized density of states (DOS) for the FM (a) Fe Ge and (b) Fe Sn monolayers. The Stoner parameter I can beestimated with dividing the exchange splitting ∆ E of spin-up and spin-down state density by the corresponding magnetic moment m . Here, weobtain ∆ E = 1.152 eV for Fe Ge monolayer and ∆ E = 1.503 eV for Fe Sn monolayer by calculating the average difference of the Kohn-Shameigenvalues of spin-up and spin-down bands below Fermi level [4, 5]. Using the relation ∆ E = Im , the Stoner parameter I = ∆ E / m is calculatedto be 0.553 eV (0.632 eV) for Fe Ge (Fe Sn) monolayer. Meanwhile, from Fig. 3(a) [3(b)], the DOS of the NM Fe Ge (Fe Sn) monolayeris 4.32 state/eV (5.53 states/eV) per spin at the Fermi level. It is thus demonstrated that the Stoners criterion D( E F ) I > Ge (Fe Sn) monolayer.
7. Band structure of the FM Fe Ge monolayer.
M K M-2024 E ( e V ) up ¶• K M-0.4-0.3-0.2-0.1-0.15-0.05M K
II II ‚„” »…‰ (cid:190)¿(cid:192)` ´ˆ˜
FIG. S7: Calculated band structure of the FM Fe Ge monolayer in the absence of SOC and zoom-in band structures below E F . The numbersrepresent the gaps.
8. Symmetry-protected 2D Weyl points in Fe Ge monolayer.
M M M M up -2024 E ( e V ) M K M d x ¯˘ d ˙¨ d (cid:201)˚ d ¸ d x (cid:204) down ˝ ˛ ˇ d — (cid:209)(cid:210) d (cid:211)(cid:212) d (cid:213)(cid:214) d (cid:215) d x (cid:216)(cid:217)(cid:218) (cid:219) (cid:220) (cid:221) (cid:222) -2024 E ( e V ) (cid:223) (cid:224)Æ (cid:226) ª (cid:228) FIG. S8: Projected (a) spin-up and (b) spin-down bands of Fe Ge monolayer onto the Fe 3 d orbitals, obtained using DFT calculations withoutSOC. Here, the radii of circles are proportional to the weights of the corresponding orbitals. In (a), we find that the two crossing bands at W + arise from the Fe d xy and d xz orbitals, whereas those at W + arise from the Fe d yz and d x − y orbitals. Note that, along the Γ − K and M − Kdirections, the little group has three nonequivalent C rotation symmetries, as shown in (c). The two crossing bands at W + or W + located in the Γ − K and M − K lines have the opposite parity of C rotation symmetry, thereby remaining gapless. In (b), we find that the two crossing bandsat W + arise from the Fe d xz and d yz orbitals. Note that, along the Γ − M direction, the little group has three out-of-plane mirror symmetries, M σ , M σ and M σ , as shown (d). The crossing bands at W + located in the Γ − M line have the opposite parity of M σ , thereby remaininggapless.
9. WNLs of Fe Sn monolayer in the absence of SOC. A A' BB' (cid:229)(cid:230)(cid:231) ŁØŒ
MK MK
FIG. S9: Momentum distribution of WNLs passing through the (a) A-A ′ and (b) B-B ′ points in Fe Sn monolayer, obtained using DFTcalculations without SOC. The energy eigenvalues along the WNLs are drawn in the color scale.
10. Projected bands of FM Fe Sn monolayer onto the Fe 3 d orbitals. -2024 E ( e V ) M K M M K M-2024 E ( e V ) º(cid:236)(cid:237) (cid:238)(cid:239)(cid:240) up down d æ (cid:242)(cid:243) d (cid:244)ı d (cid:246)(cid:247) d ł d x øœß(cid:252) (cid:253)(cid:254)(cid:255) W(cid:0)(cid:1) (cid:2)(cid:3)(cid:4) d x -(cid:5) d (cid:6)(cid:7) d y(cid:8) d z d x (cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17) E ( e V ) M K((cid:20)(cid:21)
FIG. S10: Projected (a) spin-up and (b) spin-down bands of Fe Sn monolayer onto the Fe 3 d orbitals, obtained using DFT calculationswithout SOC. Here, the radii of circles are proportional to the weights of the corresponding orbitals. In (a), we find that the α ( β ) band mainlyarises from the Fe d x − y ( d xy ) orbital with the even parity of M z , while the γ band is composed of the Fe d xz orbital with the odd parity of M z .The projected bands with including SOC are given in (c). Here, the χ band arising from the Fe d z orbital hybridizes with the γ band becausethe two bands have same parity (+ i ) of M z . Meanwhile, the χ band does not hybridize with the α and β bands having the opposite parity (- i ) of M z , forming WNLs.
11. WNLs in Fe Sn monolayer in the presence of SOC. A A' DD' CB' (cid:22)(cid:23)(cid:24) (cid:25)(cid:26)(cid:27)
MK MKBC' (cid:28)(cid:29)(cid:30) KM FIG. S11: Momentum distribution of WNLs passing through the (a) A − A ′ , (b) C − B ′ and D − D ′ , and (c) B − C ′ points in Fe Sn monolayer,obtained using DFT calculations with SOC. The energy eigenvalues along the WNLs are drawn in the color scale.
Table S1: Calculated lattice constants for various monolayers. For Fe Ge and Fe Sn, the values for the NM and FMphases are given.
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