Topological edge states in a high-temperature superconductor FeSe/SrTiO 3 (001) film Z. F. Wang 1,2,3+, Huimin Zhang 2,4+, Defa Liu 5, Chong Liu 2, Chenjia Tang 2, Canli Song 2, Yong Zhong 2, Junping Peng 2,4, Fangsen Li 2, Caina Nie 2,4, Lili Wang 2,6, X. J. Zhou 5,6*, Xucun Ma 2,4,6*, Q. K. Xue 2,6* and Feng Liu 2,3,6* 1 Hefei National Laboratory for Physical Sciences at the Microscale, Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 2 State Key Lab of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China 3 Department of Materials Science and Engineering, University of Utah, UT 84112, USA 4 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 5 National Lab for Superconductivity, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 6 Collaborative Innovation Center of Quantum Matter, Beijing 100084, China + These authors contributed equally to this work. *Correspondence to: XJZhou@aphy.iphy.ac.cn; xucunma@mail.tsinghua.edu.cn; qkxue@mail.tsinghua.edu.cn; fliu@eng.utah.edu NATURE MATERIALS www.nature.com/naturematerials 1
FeSe/STO interface structure. FeSe/STO interface structure. Experimentally, we observe the half-unit-cell shift across the domain boundaries, as shown in Fig. S6, which is consistent with our previous results and indicates a complex interface structure between FeSe and STO. Therefore, we also did calculations by laterally shifting the FeSe half-unit-cell relative to the STO substrate, as shown in Fig. S7. All the key features, the FeSe band shape and the SOC gap at the M point, remain the same. There is only a small difference that the bands around the SOC gap becomes slightly anisotropic along MX and MX directions due to lower symmetry. Moreover, the FeSe band is totally separated from the valence band of STO in energy space around Fermi-level. In real samples, the interface structures will be more complex involving different configurations, so the effect of STO substrate on FeSe electronic structure would be an average effect, which cannot be fully represented by any single interface-configuration calculation. But we believe the predicted topological edge state is rather robust against variations in FeSe/STO interface structures, because the main effect of STO substrate is to introduce electron doping in FeSe, as shown before and also confirmed in our first-principles calculations. Tight-binding model Hamiltonian of FeSe/STO. A tight-binding (TB) model Hamiltonian is constructed to further explore the SOC mechanism in FeSe/STO. Since spin-up and spin-down electrons are decoupled, the Hamiltonians can be written separately for different spin components. Zheng et al. (Ref. 15) have written a 4 4 Hamiltonian to describe the band structures of FeSe/STO around 2 NATURE MATERIALS www.nature.com/naturematerials
SUPPLEMENTARY INFORMATION M point, however, their model failed to describe the band structures in the whole BZ. Therefore, we extended their model and wrote an 11 11 Hamiltonian, including five d orbitals of Fe atom and six p orbitals of two Se atoms. The corresponding TB parameters are obtained from Wannier fitting of collinear spin-polarized first-principle calculations for free-standing FeSe. Figures S14a and S14b are the band structures of freestanding FeSe without and with SOC, respectively. We can see that the SOC opens a gap at M point. We have tested different SOC terms in the Hamiltonian, only the on-site coupling between dxz and dyz orbital can open the gap at M point and at the same time conserve the spin. This is consistent with previous orbital-components analysis, since dxz and dyz orbital of Fe atoms have the largest contribution around M point. The fitted SOC coupling strength is soc=17 mev. Next, including the STO substrate induced electric field effect, the band structures of FeSe without and with SOC are shown in Figs. S14d and S14e, respectively. The charge transfer between STO and FeSe induces an internal electric field and splits the bands around M point, which confirms the previous work. The electric field effect is modeled by tuning the on-site energy of pz orbital. The fitting parameters are ESe1=1 ev and ESe2= 1 ev. The spin Berry curvatures for the bands shown in Figs. S14b and S14e are shown in Figs. S14c and S14f, respectively. The reciprocal-space distribution of spin Berry curvature agrees with the first-principles results shown in Fig. 1e, and the integration value is also Cs= 1. Thus, this TB model captures the main physics found in the first-principles calculations. NATURE MATERIALS www.nature.com/naturematerials 3
Figure S1 Non-magnetic band structure of free-standing FeSe. a, Unit cell of nonmagnetic FeSe. b, Unit-cell Brillouin zone with high-symmetry k points. c, Band structure. 4 NATURE MATERIALS www.nature.com/naturematerials
SUPPLEMENTARY INFORMATION Figure S2 Collinear AFM band structure of free-standing FeSe. a, Supercell of FeSe in collinear AFM spin configuration. The dashed line denotes the unit cell. The blue arrows denote spin-up and spin-down along z-direction. b, Supercell Brillouin zone with high-symmetry k points. c, The corresponding unit-cell Brillouin zone with high-symmetry k points. d, Folded band Structure in the supercell Brillouin zone. e, Unfolded band structure in the unit-cell Brillouin zone. NATURE MATERIALS www.nature.com/naturematerials 5
Figure S3 Bicollinear AFM band structure of free-standing FeSe. a, Supercell of FeSe in bicollinear AFM spin configuration. The dashed line denotes the unit cell. The blue arrows denote spin-up and spin-down along z-direction. b, Supercell Brillouin zone with high-symmetry k points. c, The corresponding unit-cell Brillouin zone with high-symmetry k points. d, Folded band Structure in the supercell Brillouin zone. e, Unfolded band structure in the unit-cell Brillouin zone. 6 NATURE MATERIALS www.nature.com/naturematerials
SUPPLEMENTARY INFORMATION Figure S4 Pair-checkerboard AFM band structure of free-standing FeSe. a, Supercell of FeSe in pair-checkerboard AFM spin configuration. The dashed line denotes the unit cell. The blue arrows denote spin-up and spin-down along z-direction. b, Supercell Brillouin zone with high-symmetry k points. c, The corresponding unitcell Brillouin zone with high-symmetry k points. d, Folded band Structure in the supercell Brillouin zone. e, Unfolded band structure in the unit-cell Brillouin zone. NATURE MATERIALS www.nature.com/naturematerials 7
Figure S5 Checkerboard AFM band structures of FeSe/STO and FeSe with SOC for different spin orientations. The band structure for z spin direction clearly agrees better with the ARPES data, as highlighted by the blue circle around M point. a-c, Band structures of FeSe/STO (the bottom Se atom in FeSe is directly above the top Ti atom in STO, as shown in Fig. 1a) with SOC for spin along z, xy and x direction, respectively. d-f, Band structures of freestanding FeSe with SOC for spin along z, xy and x direction, respectively. U=0.4 ev in a-f. g-i, Definition of spin directions along z, xy and x, respectively. z is the vertical direction normal to FeSe plane. xy is along nearestneighbor Fe-Fe bond direction. x is along next-nearest-neighbor Fe-Fe bond direction. 8 NATURE MATERIALS www.nature.com/naturematerials
SUPPLEMENTARY INFORMATION Figure S6 Boundary structure of FeSe/STO showing half-unit-cell shift. The STM topography (4 4 nm 2, 0.021 na, 4.0 V) showing atom arrangement near the boundary. The gray balls represent the Se atoms on the top surface and the dotted line is drawn to show the relative position of atoms across the boundary. NATURE MATERIALS www.nature.com/naturematerials 9
Figure S7 Atomic structure and band structure of FeSe/STO with half-unit-cell shift. a, Atomic structure of FeSe/STO with half-unit-cell shift. The bottom Se atom is directly above the top O atom in STO substrate. b, Band Structure of FeSe/STO with SOC (U=0.4 ev). 10 NATURE MATERIALS www.nature.com/naturematerials
SUPPLEMENTARY INFORMATION Figure S8 Checkerboard AFM band structure of FeSe/STO without and with SOC. a, The spin-polarized band structures of FeSe/STO with U=0.4eV. The red and blue colors are spin-up and spin-down bands, respectively. b, The band structures of FeSe/STO with SOC. NATURE MATERIALS www.nature.com/naturematerials 11
Figure S9 U effect on the checkerboard AFM band structure of free standing FeSe without SOC. a, U=0 ev. b, U=0.2 ev. c, U=0.4 ev. d, U=0.6 ev. e, U=0.8 ev. f, U=1.0 ev. g, U=1.1 ev. h, U=1.2 ev. i, U=1.4 ev. j, U=2.0 ev. k, U=3.0 ev. Note that for large U, the band degeneracy at M point is broken and a trivial gap is opened. However, the valence band continues to move down and the overall band shows significant deviation from the ARPES spectra with the increasing U value. The same trend is also observed with SOC in Fig. S10 below. 12 NATURE MATERIALS www.nature.com/naturematerials
SUPPLEMENTARY INFORMATION Figure S10 U effect on the checkerboard AFM band structure of free standing FeSe with SOC. a, U=0 ev. b, U=0.2 ev. c, U=0.4 ev. d, U=0.6 ev. e, U=0.8 ev. f, U=1.0 ev. g, U=1.1 ev. h, U=1.2 ev. i, U=1.4 ev. j, U=2.0 ev. k, U=3.0 ev. NATURE MATERIALS www.nature.com/naturematerials 13
Figure S11 Comparison between theoretical band structure of free standing FeSe with U=1.1 ev and second derivative ARPES spectra. a,b, Theoretical band structures overlaid with ARPES around and M points, respectively. c-e, Theoretical band structures overlapped with ARPES along different high symmetry directions. Although there happens a trivial gap of ~40 mev at M point, one clearly sees the dramatic difference between theoretical and experimental band structures, and the agreement between theory and experiment is much worse than Fig. 2. Therefore, a trivial band gap at M point using larger U is very unlikely to explain our experimental data. 14 NATURE MATERIALS www.nature.com/naturematerials
SUPPLEMENTARY INFORMATION Figure S12 Raw ARPES spectra. a,b, ARPES spectra around (cut #1 in Fig. 2a) and M (cut #2 in Fig. 2a) points, respectively. c-e, ARPES spectra along different high symmetry directions. NATURE MATERIALS www.nature.com/naturematerials 15
Figure S13 ARPES spectra. a,b, Raw and second derivative ARPES spectra around M3 along the perpendicular direction of cut #2 in Fig. 2a, respectively. The ARPES spectral intensity for the Fermi surface is not uniform around the M point, which may originate from the photoemission-matrix-element effect. 16 NATURE MATERIALS www.nature.com/naturematerials
SUPPLEMENTARY INFORMATION Figure S14 TB band structure and spin Berry curvature of FeSe. a,b, TB band structures of FeSe without and with SOC, respectively. c, Reciprocal-space distribution of the spin Berry curvature for the band in b. d-e, TB band structures of FeSe under electric field without and with SOC, respectively. f, Reciprocal-space distribution of the spin Berry curvature for the band in e. NATURE MATERIALS www.nature.com/naturematerials 17
Figure S15 STS measurement of the topological edge states, showing the reproducibility of Fig. 4c. a-c, STM topographies of the FeSe/STO with size of 500 500 nm 2 (0.02 na, 5.0 V), 200 200 nm 2 (0.01 na, 4.0 V) and 50 50 nm 2 (0.01 na, -1.0 V) respectively. d,h, STM topographies of two FM edges of FeSe/STO at (0.1 na, -300 mv) and (0.01 na, 4.0 V). e,i, Atomic-resolution STM topographies at the bulk position of d (0.1 na, 220 mv) and h (0.1 na, 600 mv), showing the topmost Se atom arrangement (the crystal orientations are labeled) and the orientation of FM edges. f,j, STS line scan at the marked positions along the blue arrow direction in d (0.1 na, 100 mv) and h (0.1 na, 100 mv). g,k, STS spectra of FM edge and bulk states extracted from f and j, showing the same feature as that in Fig. 4i (left panel). 18 NATURE MATERIALS www.nature.com/naturematerials
SUPPLEMENTARY INFORMATION Figure S16 2D STS mapping of the edge state of FeSe/STO. a, Topographic image of FM edge of FeSe/STO (0.1 na, -100 mv). b-h, 2D STS mapping of the edge state at different bias voltages (0.1 na, 100 mv).. NATURE MATERIALS www.nature.com/naturematerials 19
Figure S17 Superconducting gap at the FeSe/STO boundary. a, The STM topography (0.21 na, 4.0 V) of FeSe film. b, A series of STS spectra taken along the green line in a near the domain boundary (0.1 na, 50 mv). The spectra are shifted along the y axis to guide the eyes. 20 NATURE MATERIALS www.nature.com/naturematerials
SUPPLEMENTARY INFORMATION Figure S18 Theoretical LDOS for edge and bulk states. a,b, Theoretical LDOS for edge and bulk states of FM and AFM edges with single-unit-cell projection (bottom curves) and three-unit-cell projection (top curves) with a broadening parameter of 20 mev. The curves for different number of unit-cell projection are vertically moved to distinguish with each other. Three-unit-cell projection resembles the experimental STS spectra in Fig. 4i much better than single-unit-cell projection. NATURE MATERIALS www.nature.com/naturematerials 21
Figure S19 The spin-polarized band structures of FeSe/STO with different thickness of STO substrate. a, 10 layers STO. b, 14 layers STO. U=0.4eV. The red and blue colors are spin-up and spin-down bands, respectively. 22 NATURE MATERIALS www.nature.com/naturematerials