Multiple surface states,nontrivial band topology,and antiferromagnetism in GdAuAl4Ge2

Chengcheng Zhang(张成成), Yuan Wang(王渊), Fayuan Zhang(张发远), Hongtao Rong(戎洪涛),Yongqing Cai(蔡永青), Le Wang(王乐), Xiao-Ming Ma(马小明), Shu Guo(郭抒), Zhongjia Chen(陈仲佳),Yanan Wang(王亚南), Zhicheng Jiang(江志诚), Yichen Yang(杨逸尘), Zhengtai Liu(刘正太),Mao Ye(叶茂), Junhao Lin(林君浩), Jiawei Mei(梅佳伟), Zhanyang Hao(郝占阳),†,Zijuan Xie(谢子娟), and Chaoyu Chen(陈朝宇),§

1Shenzhen Institute for Quantum Science and Engineering(SIQSE)and Department of Physics,Southern University of Science and Technology(SUSTech),Shenzhen 518055,China

2Songshan Lake Materials Laboratory,Dongguan 523000,China

3Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,Chinese Academy of Sciences,Beijing 100190,China

4State Key Laboratory of Functional Materials for Informatics and Center for Excellence in Superconducting Electronics,Shanghai Institute of Microsystem and Information Technology,Chinese Academy of Sciences,Shanghai 200050,China

5International School of Microelectronics,Dongguan University of Technology,Dongguan 523000,China

Keywords: magnetic topological material,surface state,ARPES,topological invariant

1.Introduction

The integration of band topology and magnetism in quantum materials gives rise to novel states of matter such as Chern insulator, axion insulator, topological M¨obius insulator, Weyl semimetal,etc.[1–5]Such states usually host an enhanced Berry curvature and surface/edge/hinge/corner states due to the bulk–boundary correspondence, manifesting extreme responses to external stimuli such as electric/magntic field,temperature gradient and optical excitation,e.g.,(quantum)anomalous Hall effect,anomalous Nernst effect,topological magneto–optical effect, and negative magnetoresistance as signature of chiral anomaly.Functionalization of these effects may boost industrial development in dissipationless spintronics, information storage and quantum computation.[6–18]As a current example, (MnBi2Te4)·(Bi2Te3)n(n=0, 1, 2, 3)family of materials has intrigued the community as the first realization of intrinsic magnetic topological insulator.[3,4,19–24]Its lattice structure, magnetic configuration, and electronic structure possess fertile tunability,[25]which enables the realization of novel phenomena such as magnetic gap opening of topological Dirac surface state,[26]layer Hall effect,[27]Chern insulator, and axion insulator phases.[28–31]The second example is theT3SnX(T=Fe, Mn;X=1, 2, 3) compounds in which magnetic transition metal Fe or Mn atoms form kagome layers.The band topology inherited from the kagome layer and the multiple types of magnetic orders leads to gapped/gapless Dirac cones,[32,33]Weyl points,[34]anomalous Hall effect,[33,35]anomalous Nernst effect,[36]etc.These topological phenomena have also been observed in other magnetic topological materials such as Co3Sn2S2,[37,38]Co2MnGa,[39,40]andRMn6Sn6(R=rare earth metal).[41–49]In fact,exploring new topological states and novel phenomena based on 3d magnetic transition metal atoms(Mn,Fe,Co,Ni)represents a main trend.By comparison, magnetic topological states of matter based on magnetic rare earth atoms remain largely unexplored.

In this paper, we systematically study the magnetism,electronic structure and band topology of a newly discovered metallic antiferromagnet GdAuAl4Ge2, in which rare earth magnetic Gd3+ions form a layered triangular lattice.Magnetic property measurements have revealed multiple antiferromagnetic transitions belowTN=17.8 K.Angle resolved photoemission spectroscopy(ARPES)measurements and density functional theory (DFT) calculations have revealed multiple bulk bands crossing the Fermi level,which possess nontrivialZ2topology according to our parity and Wannier charge center analyses, establishing a strong topological insulator state in its paramagnetic phase.Furthermore,multiple pairs of low energy surface states residing in the bulk gap have been observed by ARPES.Our termination dependent surface spectra calculations exhibit acceptable agreement with the ARPES spectra, providing insight into their atomic layer origin.Our results establish rare earth antiferromagnet GdAuAl4Ge2as a new material platform and call for further magnetic,electronic and transport studies to investigate the interplay between band topology and magnetism.

2.Methods

Single crystals of GdAuAl4Ge2were grown by the selfflux method.[50–52]High purity Gd (block), Au (powder), Al(rod),and Ge(lump)were mixed with a ratio of 1:1:10:5 in the glovebox and placed into an alumina crucible.The crucible was then sealed into an evacuated quartz tube under vacuum to avoid oxidization during the reaction.In the furnace, the quartz tube was heated up to 1000◦C and kept at this temperature for 12 hours.Then the temperature was slowly decreased down to 700◦C in 100 hours.After the heating procedure,the quartz tube was taken into the centrifuge quickly to remove excess flux.A large number of millimeter-size single crystals were obtained.

The structure of the crystals was determined by x-ray diffraction with CuKαradiation at room temperature using a Rigaku MiniFlex diffractometer.Temperature and magnetic field-dependent magnetization measurements were performed for temperatureT=2 K–300 K and magnetic fieldµ0H=0 T–7 T applied inabplane and alongcaxis using magnetic property measurement system(MPMS3,Quantum Design).

ARPES measurements were performed at the BL03U beamline of the Shanghai Synchrotron Radiation Facility with a DA30 electron analyser.The energy and angular resolution were set to be better than 10 meV and less than 0.05◦,respectively.Samples were cleavedin situwith pressure better than 5×10−11mbar(1 bar=105Pa)and temperatures below 20 K.

First-principles calculations are carried out within the framework of density functional theory (DFT)[53,54]by using the Viennaab-initioSimulation Package(VASP).[55]The planewave-basis cutoff energy of 500 eV is used.The first Brillouin zone(BZ)is sampled by a 5×5×5k-mesh-grid.[56]We employ GGA+Umethod[57]by choosing the on-site Coulomb repulsion ofU=4 eV for the Au element.The felectrons are treated as core states for the Gd element.The spin–orbit coupling effect is taken into consideration.The maximally localized Wannier functions (MLWFs) of Gd-p/d,Au-s/p/d, Al-s/p/d, and Ge-s/p ortitals are selected to construst the tight-binding model as implemented in the Wannier90 package.[58,59]

3.Results

Single crystals of GdAuAl4Ge2adapt a rhombohedral space groupR¯3m(No.166) with lattice parametersa=4.2123(6) ˚A andc=30.994(6) ˚A.The crystal structure of GdAuAl4Ge2is shown in Fig.1(a) where Gd atoms form a triangular lattice inab-plane.The unit cell is generally composed of staggered layers of Gd–Ge octahedra and Au-Al polyhedral alongcaxis.Powder x-ray diffraction is used to determine the crystal structure, as shown in Fig.1(b),the main phase is consistent with the simulation results of GdAuAl4Ge2, and peaks of impurities introduced by the flux are marked with asterisks.[50,51]

Temperature-dependent magnetic susceptibility of GdAuAl4Ge2is shown in Fig.1(c).Forµ0H= 0.5 T, the magnetic susceptibility curves for the field applied in and out of theab-plane coincide at the high-temperature region.No splitting is observed with the decreasing of temperature for both zero-field-cooling (ZFC) and field-cooling (FC) measurements.At low temperature, the magnetic susceptibility behaves differently forH ‖abandH ‖c, indicating the existence of magnetic anisotropy.As highlighted in Fig.1(d),forH‖ab,there are three phase transition anomalies occurring atTN1=17.8 K,TN2=15.6 K,andTN3=13.8 K.Whereas onlyTN1andTN3can be observed forH ‖c.ForH ‖ab,the magnitude ofχbegins to decline belowTN1atµ0H=0.5 T.With the increase of magnetic field, the magnetic phase transitions are gradually suppressed and tend to saturate forµ0H>1.5 T as shown in Fig.1(e).Similar magnetic behaviors can also be seen forH ‖c(Fig.1(f)).The field dependence of magnetization curves at various temperatures forH ‖abare presented in Fig.1(g).AtT=1.8 K, magnetization starts out as a linear increase and subsequently undergoes an apparent upturn atHC1∼1.9 T, which is considered as a spin flop transition.Then magnetization continues to increase linearly without sign of saturation up to 7 T.With increasing temperature, the transitions moves to lower fields and disappears above 15 K.ForH‖c,magnetization exhibits linear behavior and no phase transition is observed.The magnetic behaviors of GdAuAl4Ge2agree with recent work which reported it as a reservoir for complex magnetism and electronic behaviors such as the topological Hall effect.[51]Therefore,here we are prompted to focus on the electronic structure to directly unveil its topological property,leaving it unclear what is the detailed magnetic configuration.

Fig.1.Crystal structure and characterization of single-crystalline GdAuAl4Ge2.(a) Crystal structure of GdAuAl4Ge2.The local atomic arrangement of Gd–Ge and Au–Al and the triangular configuration of the Gd atomic plane are present in the right bottom.(b)Powder x-ray diffraction data for GdAuAl4Ge2.Simulated pattern is plotted in light blue for comparison and the peaks of impurities are marked with asterisks.(c)Temperature-dependent magnetic susceptibility measurements with ZFC and FC modes atµ0H=0.5 T for H‖ab and H‖c respectively.(d)Expanded view of panel(c).(e)–(f)Temperature-dependent magnetic susceptibility at different fields for H‖ab and H‖c respectively.(g)–(h)Field-dependent magnetization at different temperatures with fields applied in ab-plane and along c axis,respectively.The unit 1 Oe=79.5775 A·m−1.

Fig.2.Electronic structure of GdAuAl4Ge2.(a) The three-dimensional Brillouin zone (BZ) of bulk GdAuAl4Ge2 and the projected BZ for the ARPES measured (00l) surface with high symmetry points specified.(b) Fermi surface mapping on the (00l) surface using incident photons of 120 eV.The twodimensional(2D)BZ is represented by the red lines.(c)Photon energy-dependent ARPES from 60 eV to 80 eV(kz mapping)at E −EF=−2.1 eV(left)and−0.35 eV(right)respectively,with inner potential V0=14 eV.The 2D BZ boundary is traced in red.(d)Spectra taken along high-symmetry path–––at variable temperatures.(e)–(f)ARPES measured spectra and DFT calculated bulk bands along high symmetry directions.

To uncover the topological properties of GdAuAl4Ge2,we combine ARPES measurement and DFT calculations to systematically explore its electronic structure.ARPES measurements were carried out on the cleaved (00l) surface (abplane).Figure 2(b) shows the Fermi surface mapped using photon energy of 120 eV.A hexagram-like Fermi surface can be clearly identified.The hexagonal pockets surrounding thepoint and the elliptical pockets centered at theMpoints can be clearly distinguished.Photon energy dependent ARPES from 60 eV to 80 eV are utilized to clarify itskzdispersion,as shown in Fig.2(c).One can slightly identify thekzdependent shadow features aroundkx=0 in the left panel.It is noted that it has a period of 2π/(c/3) rather than 2π/c,which is related to the smallest repeating unit of crystal structure alongcaxis.Except for this, other energy bands, especially these within 2 eV below the Fermi level,have no observablekzdispersion, in line with the layered lattice.Referring to the magnetic phase transitions in GdAuAl4Ge2, ARPES measurements were carried out at four different temperatures.As shown in Fig.2(d), ARPES spectra show no noticeable temperature dependence, similar to some reported magnetic topological materials not sensitive to magnetic order, such asRMn6Sn6(R=rare earth metal).[41–49]Therefore,unless otherwise emphasized,all ARPES spectra in this work are based on data measured at 14.7 K and the energy bands calculations are based on the nonmagnetic phase.

We employ termination-dependent band structure projections to illustrate the origin of these bands which are absent in the bulk calculation.Figures 3(a) and 3(b) show ARPES spectra alongtaken with various photon energies.Potential surface state pairs at thepoint are labeled as SS1–SS4 (red arrows).For different photon energies, while the intensity of these band pairs shows some variation, their energy location and band shape remain constant, suggesting a two dimensional behavior.For comparison, energy distribution curves(EDCs)atKare plotted to clarify the positions of the four sets of surface states at different photon energies.To shed light on the surface origin of these band pairs, DFT calculated band structures are projected onto (00l) surface with different terminating layers.As shown in Figs.3(c)and 3(d), we treat all the atomic layers forming the repeating octuple layer as possible terminating layers (T1–T8) and present their corresponding projected surface and bulk spectra.In the energy window shown here, the surface band pairs show obvious termination dependence concerning the number of pairs and their location and dispersion.Basis similarity can be found between the ARPES measured surface spectra and the one calculated based on Au atomic layer termination(T1).All four sets of surface states are reproduced by the theoretical calculations.It is noted that some bands such as the bands with strong spectral weight right below SS4 and the deeper bands at aboutE −EF=−2.3 eV in T1 panel are not observed experimentally,which may be due to the matrix element effect.

Fig.3.Multiple surface states in GdAuAl4Ge2.(a)ARPES spectrum using incident photon energy of 120 eV alongΓ–K–M.The red arrows point to four sets of surface states at .(b)Same ARPES spectra but taken with various photon energies and corresponding EDCs at K.(c)Crystal structure to illustrate the different cleavage planes for panel(d).The atomic layer crossed by each dashed line indicates the topmost atoms after cleavage.(d)Calculated surface states and bulk states projection onto(00l)surface with different terminating layers as shown in panel(c).

For band topology analysis, we perform parity calculations at the eight time-reversal invariant momenta (TRIM) to obtain theZ2topological invariant.As shown in Fig.4(a),the bulk band structures are plotted along the high-symmetryk-paths.All the bands are two-fold degenerate due to the combined operation of time-reversal and inversion symmetries.The four bands highlighted by distinct colors near the fermi level are highly associated with the ARPES spectrums discussed above.For the band indexesn=56, 58, 60, 62,parity calculations run over all thenoccupied bands, respectively.It is worth to note that the global band gap exists for all the selected occupied band numbers,making it possible to define the topological phases just as insulator systems.By taking the inversion symmetry into consideration,theZ2topological invariant is given by

wherekTRIMare eight TRIM,ξnis the parity eigenvalue of then-th band andλis the strongZ2topological invariant.The results show that GdAuAl4Ge2is a strong topological insulator with nontrivial surface states for the 56, 58, and 60 occupied bands.As for the 62 occupied band, GdAuAl4Ge2is a weak topological insulator.To further examine the nontrivial topology, we also perform Wannier charge center calculations for the 60 occupied bands in the six key planes.As shown in Fig.4(b),Z2= 0 is obtained in the three planeskx=0/ky=0/kz=0 andZ2=1 is obtained in the three planeskx=π/ky=π/kz=π,which are in good agreement with the results of parity calculations.To further elucidate the origin of topological band structures, we conduct an orbital analysis around the TRIM and obtain the projected band structures.The results demonstrate that band inversion occurs at multiplek-points,collectively leading to a nontrivial topology(see the detailed discussion in the supplementary material).In Figs.4(c)–4(f), we plot the projected band structures for orbitals with dominant contributions near theΓandFpoints as our examples.Band inversion features are clearly observed at bothΓandFpoints.

Fig.4.Nontrivial Z2 topology in GdAuAl4Ge2.(a)Band structures of GdAuAl4Ge2 and Z2 topological invariants of specific band index.λ;(v1v2v3)includes one strong topological invariant λ and three weak topological invariants vi (i=1,2,3).Bold solid lines with distinct colors represent the n-th band(n=56,58,60,62).(b)The Wilson loop of GdAuAl4Ge2 for the band index 60,showing Wannier charge centers along specific k-path in the high-symmetry planes.(c)–(f) Projected band structures of specific orbitals near the time-reversal invariant momenta.The size of the red dots indicates the relative contribution weight of the associated orbitals.The coordinates of points A,B,C,and D are kΓ −0.5k1,kΓ +0.5k1,kF −0.5k3,and kF+0.5k3,respectively,where k1,k2,and k3 are the basis vectors of the reciprocal space.

4.Discussion

Combining experimental and calculational analyses, we have proved that there exists a strong topological insulator state in rare earth antiferromagnet GdAuAl4Ge2with nontrivial surface states in its nonmagnetic phase.We have directly revealed multiple pairs of surface states whose topological nature remains to be clarified.In addition, while our first principles, parity, and Wilson loop calculations are all performed based on nonmagnetic phase,incorporating the antiferromagnetic order into calculation is highly desired to specify the magnetic topological phase.The magnetic fine structure of GdAuAl4Ge2deserves further x-ray scattering study.Furthermore, Hall and Nernst measurements are needed to demonstrate its transport response steming from the interplay between magenetism and band topology.

Acknowledgments

Project supported by the National Key Research and Development Program of China(Grant No.2022YFA1403700),the National Natural Science Foundation of China(Grant No.12074163), the Basic and Applied Basic Research Foundation of Guangdong Province, China(Grants Nos.2022B1515020046, 2022B1515130005, and 2021B1515130007), the Innovative and Entrepreneurial Research Team Program of Guangdong Province, China (Grant Nos.2019ZT08C044), and Shenzhen Science and Technology Program (Grant No.KQTD20190929173815000).C.C.acknowledges the assistance of SUSTech Core Research Facilities.