Abstract
Kagomenets, appearing in electronic, photonic and coldatom systems, host frustrated fermionic and bosonic excitations. However, it is rare to find a system to study their fermion–boson manybody interplay. Here we use stateoftheart scanning tunneling microscopy/spectroscopy to discover unusual electronic coupling to flatband phonons in a layered kagome paramagnet, CoSn. We image the kagome structure with unprecedented atomic resolution and observe the striking bosonic mode interacting with dispersive kagome electrons near the Fermi surface. At this mode energy, the fermionic quasiparticle dispersion exhibits a pronounced renormalization, signaling a giant coupling to bosons. Through the selfenergy analysis, firstprinciples calculation, and a lattice vibration model, we present evidence that this mode arises from the geometrically frustrated phonon flatband, which is the lattice bosonic analog of the kagome electron flatband. Our findings provide the first example of kagome bosonic mode (flatband phonon) in electronic excitations and its strong interaction with fermionic degrees of freedom in kagomenet materials.
Introduction
The kagomenet, a pattern of cornersharing triangular plaquettes, has been a fundamental model platform for exotic states of matter, including quantum spin liquids and topological band structures^{1,2,3}. Recently, the transition metalbased kagome metals^{4,5,6,7,8,9,10,11,12,13} are emerging as a new class of topological quantum materials to explore the interplay between frustrated lattice geometry, nontrivial band topology, symmetrybreaking order, and manybody interaction. A kagome lattice tightbinding model generically features a Dirac crossing and a flatband, which are the fundamental sources of nontrivial topology and strong correlation. Such topological fermionic structures arising from the correlated 3d electrons in the kagome lattice have been widely reported in several quantum materials^{4,5,6,7,8,9,10,11,12,13}, including Mn_{3}Sn, Fe_{3}Sn_{2}, Co_{3}Sn_{2}S_{2}, TbMn_{6}Sn_{6}, FeSn, and CoSn. In parallel, the band dispersion of bosonic excitations on a kagome lattice also features Dirac crossings and flatbands, as demonstrated in photonic crystals^{14,15}. A question naturally arising when studying kagome lattice electrons is the possibility of a nontrivial manybody interplay between the bosonic kagome lattice phonons and the fermionic quasiparticles.
Such fermion–boson interactions often manifest as a perturbation of the bare band structures at very lowenergy scales. However, most kagome lattice materials exhibit complicated multibands near the Fermi energy^{4,5,6,7,8,9,10,11,12}, severely challenging the clear identification of the manybody effect by spectroscopic methods. Among all known kagome lattice materials, the paramagnetic CoSn is recently highlighted as an outstanding kagome topological metal with much cleaner bands and simpler Fermi surface^{13}, making it an ideal platform to search for the geometrical frustrated fermion–boson interaction. Here we report the discovery of fermion–boson manybody interplay in kagome lattice of CoSn, which arises from the coupling of the phonon flatband with the kagome electrons, utilizing the lowtemperature (T = 4.2 K), high energyresolution (ΔE < 0.3 meV), atomic layerresolved scanning tunneling microscopy.
Results
Atomicscale visualization of kagome lattice
CoSn has a hexagonal structure (space group P6/mmm) with lattice constants^{13,16}a = 5.3 Å and c = 4.4 Å. It consists of a Co_{3}Sn kagome layer and an Sn_{2} honeycomb layer (Fig. 1a) with alternating stacking. The sideplane atomically resolved map of the crystal measured by transmission electron microscopy (Fig. 1b) directly demonstrates this atomic stacking sequence along the c axis. Upon cryogenic cleaving, the surface yields either the Co_{3}Sn or Sn_{2}terminated atomic layer. The lower panel in Fig. 1c shows a highly rare topographic image of the cleaving surface that contains both terminations. It consists of a Co_{3}Sn surface and islands of Sn_{2} layer sitting on top. The simultaneously obtained differential conductance map directly reveals their difference in the electronic structure shown in the upper panel of Fig. 1c, where the Co_{3}Sn surface has a higher density of states at the bias voltage of 100 mV. From this map, we also find that the Co_{3}Sn surface has detectable impurities induced quasiparticle interferences, which is the basis for our further spacemomentum investigation. Scanning the Sn_{2} (Fig. 1d) and Co_{3}Sn (Fig. 1e) surfaces with higher magnification, we directly reveal their honeycomb and kagome lattice symmetry, respectively. Remarkably, our topographic image was able to resolve the fine cornersharing triangle structure of the Co kagome lattice and the Sn atom in the kagome center (Fig. 1e). Such ultrahigh atomic resolution has not been achieved in the previous scanning tunneling studies of kagome lattice materials.
Bosonic mode coupling from kagome electrons
With the lattice structure of CoSn visualized at the atomic scale, we now study their electronic structure by measuring the differential conductance as shown in Fig. 2a. According to the firstprinciple calculations and photoemission measurement, the Fermi surface is dominated by a fairly simple electronlike band^{13}. Strikingly, we find pronounced lowenergy modulations for the spectra taken on the Co_{3}Sn layer, while this feature is absent on the Sn_{2} layer as shown in Fig. 2b. The observed peakdiphump modulation can indicate the strong bosonic mode coupling with the coherent state at the Fermi level, as similar spectroscopic features have been found in many strong coupling superconductors, including lead^{17}, cuprates^{18}, and ironpnictides^{19}. Such a bosonic mode often arises from phonons or spin resonances. As this material is nonmagnetic and has no detectable magnetic field dependence of tunneling spectra up to 8T (Fig. 2c), the bosonic mode is more likely to arise from the coupling to phonons. Moreover, the electronic coupling to the bosonic mode can be described within the Eliashberg theory^{20,21} with α^{2}F(ω) where α is the coupling matrix element and F(ω) is the bosonic density of states, and this can be studied in the tunneling experiments. α^{2}F(ω) is intimately related to the second differentiation of the tunneling spectra^{17,18,19,20,21,22}. Analyzing the measured tunneling spectra (Fig. 2d, e), we find a Gaussianlike state in the second derivative centered at E_{M} = 15 meV with a full width at half maximum of 9 meV (Fig. 2e), and identify it as a candidate signal related to the Eliashberg function (Fig. 2e, blue curves). The other peak features at lower energies of the second differential spectra can be expected from a coherent state at the Fermi energy as shown by the simulation curve in Fig. 2d, e.
Manybody kink in the kagome dispersion
To gain deeper insight into the bosonic mode coupling on the kagome lattice, we perform systematic spectroscopic imaging on a large Co_{3}Sn area with only a few Sn_{2} adatoms (Fig. 3a). By taking the Fourier transform of the differential conductance map (Fig. 3b), we obtain the quasiparticle interference (QPI) data. The QPI data at 0 meV (Fig. 3c) shows a clear ringlike signal, consistent with the intraband scattering of the dominant electronlike Fermi surface^{13} centered at Γ. Thus, the lowenergy QPI dispersion reflects the behavior of the electronlike band crossing the Fermi surface (Q = 2k). Analyzing the QPI dispersion along ΓM direction in Fig. 3d, we observe a pronounced double kink feature, different from its bare band dispersion calculated by the firstprinciples (dashed line). The spectroscopic kink feature has been identified as a fingerprint of the bosonic mode coupling^{23,24,25,26,27,28,29} and indicates a giant mode coupling strength. The energy of the QPI kink is around ±15 meV, well consistent with the mode energy E_{M} in the second differential conductance signal. The coupling strength can be estimated from Fermi velocity renormalization λ = v_{f0}/v_{f} − 1 = 1.8 with s.d. error bar of 0.3, where v_{f0} and v_{f} are the Fermi velocities of the bare QPI band and renormalized QPI band, respectively. We also explored the QPI on the Sn_{2} honeycomb lattice but did not find any clear kink. Hence the unique kagome lattice resolving capability combined with low temperature and high energyresolution of our advanced spectroscopic technique can be the key for the kink discovery in this material.
Discussion
The pronounced kink signal from the kagome lattice allows us to analyze the electron manybody selfenergy Σ(ω) and further quantify the Eliashberg function α^{2}F(ω). Σ(ω) is intimately related to α^{2}F(ω), the Fermi–Dirac distribution f(E) and the Bose–Einstein distribution n(ω) (see Eq. (4) in “Methods” section). The real part of the selfenergy Re(Σ) is related to the energy difference between the observed kink dispersion and the bare QPI dispersion, while the imaginary part of the selfenergy Im(Σ) is related to the electron band broadening, which is inversely proportional to the QPI intensity. Re(Σ) and Im(Σ) are tied to one another through the Kramers–Kronig relation, and we can more accurately acquire Re(Σ) from the QPI data in Fig. 3e. We take the shape of α^{2}F(ω) in reference to the second differential conductance (Fig. 2), and tune the coupling strength \(\lambda = {\int} {\alpha ^2} F\left( \omega \right){\mathrm{/}}\omega \,{\mathrm{d}}\omega\) and calculate the real part of the selfenergy Re(Σ). We find that with λ = 1.9 with s.d. error bar of 0.3, the calculated Re(Σ) can account for that determined by the experiment (Fig. 3e), which agrees with the estimated λ from Fermi velocity renormalization. We further simulate the QPI signal with this α^{2}F(ω) under the Green function formalism in Fig. 3f, which also shows reasonable consistency with the experimental data both in dispersion and intensity evolution, providing key support to our manybody analysis of the kagome fermion–boson interaction.
Having characterized the manybody fermion–boson interaction in the kagome lattice, we perform firstprinciples calculations of the phonon band to understand the nature of the bosonic mode. Firstly, the calculated phonon density of states exhibits a pronounced peak at 15 meV (Fig. 4a), coincide with the mode energy in experiments. Secondly, this phonon mode mainly arises from the Co_{3}Sn kagome layer, consistent with the experiments. Thirdly, the calculation provides momentum space insight into the origin of this peak, in that it arises from a flatband in phonon momentum space as shown in Fig. 4b. Lastly, through the atomic displacement resolved calculation, we identify that the flatband phonon is mainly associated with the Co kagome lattice vibrations confined to the line connecting the centers of two neighboring triangles (Fig. 4b, inset).
In light of the firstprinciple calculation, we build a kagome lattice vibration model to elucidate the striking physics. The essential momentum features of the flatband can be well captured by this model (Fig. 4c), with the flatband toughing a parabolic band bottom. This model is highly analytical and offers a heuristic understanding of the nonpropagating nature of the kagome flatband phonon mode. We find that the collective lattice displacement shown in the inset of Fig. 4c, a deformation from a hexagonal ring (inner six atoms) rotating clockwise or anticlockwise, would not exert any net force on the outer atoms. Hence, such geometrically frustrated vibrations can be localized forming the phonon flatband. It is also clear now that this phonon flatband is the lattice analog of kagome electron flatband^{8,13}, whose quadratic band touching feature distinguishes them from the isolated flatbands in heavyfermion systems^{30} and the Dirac cone touching flatbands in Moire lattices^{31}. The coupling to the kagome phonon flatband is not predicted by existing research papers but is highly anticipated to explain the giant fermion–boson interaction observed here. Our findings suggest that the flat phonon dispersion can be probed by future momentumresolved phononsensitive scattering experiments including inelastic Xray scattering and neutron scattering.
The correspondence between the kagome lattice, tunneling conductance, magnetic field response, double kink feature, selfenergy analysis, firstprinciples, and lattice vibration model provides strong evidence and conceptual framework for the fermion–boson interaction in a geometrically frustrated topological quantum material. The nontrivial kagome band structures have been widely observed in both fermionic and bosonic systems^{15,32}, but their manybody interactions were rarely experimentally observed previously. We expect the latter to be quite general in many topological quantum materials with flatbands. Such fermion–boson interactions can be the driving force for future discovery of incipient density waves and superconductivity in kagome lattice materials through pressure tuning or chemical engineering. Although our research addresses the coupling of the dispersive electrons and the flatband phonon, it would be interesting to explore in the future the intriguing possibility of coupling of the kagome flat electron band and flat phonon band when they are tuned to the similar energies.
Methods
Sample preparation
Highquality single crystals of CoSn were synthesized by the Sn flux method. The starting elements of Co (99.99%), Sn (99.99%) were put into an alumina crucible, with a molar ratio of Co:Sn = 3:20. The mixture was sealed in a quartz ampoule under a partial argon atmosphere and heated up to 1173 K, then cooled down to 873 K with 2 K/h. The CoSn single crystals were separated from the Sn flux by using a centrifuge.
Scanning tunneling microscopy characterization
Single crystals with size up to 2 mm × 2 mm × 1 mm were cleaved mechanically in situ at 77 K in ultrahigh vacuum conditions, and then immediately inserted into the STM head, already at He4 base temperature (4.2 K). The magnetic field was applied with zerofield cooling. Tunneling conductance spectra were obtained with an Ir/Pt tip using standard lockin amplifier techniques with a root mean square oscillation voltage of 0.3 meV and a lockin frequency of 977 Hz. Topographic images were taken with tunneling junction set up: V = −100 mV, I = 2–0.05 nA. The conductance maps are taken with tunneling junction set up: V = −50 mV, I = 0.5 nA.
Transmission electron microscopy characterization
Thin lamellae were prepared by focused ion beam cutting using a ThermoFisher Helios NanoLab G3 UC DualBeam. All samples for experiments were polished by 2 kV Ga ion beam to minimize the surface damage caused by the high energy ion beam. Transmission electron microscopy imaging, atomic resolution highangle annular darkfield scanning transmission electron microscopy imaging and atomiclevel energydispersive Xray spectroscopy mapping were performed on a double Cs corrected ThermoFisher Titan Cubed Themis 300 scanning/transmission electron microscope equipped with an extreme field emission gun (XFEG) source operated at 300 kV and superX energydispersive spectrometry system.
Manybody theory for bosonic mode coupling
Due to the scattering by bosonic modes, the electrons do not have a definite energy but rather a finite lifetime and a broadened spectral function. In manybody theory, this phenomenon is characterized by the electron selfenergy Σ(ω), which can be regarded as a correction to the freeelectron Green function. The electron Green function is
where \({\it{\epsilon }}_{{k}}^0\) is the electron bare energy dispersion based on firstprinciple calculation. And the electron spectral function is
The QPI dispersion is further described by
With the Eliashberg function, the electron selfenergy can be described by
where α^{2}F(ω) is Eliashberg coupling function describing the electronbosonic mode interaction, f(E) is the Fermi–Dirac distribution, and n(ω) is Bose–Einstein distribution at temperature T. Since in most cases, the bosonic mode energy is far less than the Fermi energy ε_{F}, we can make the approximation that the initial and final energies of scattered electrons are close to Fermi energy. In this way, we obtain an Eliashberg coupling function solely determined by the bosonic energy distribution. An effective coupling constant can be defined as
In our experiment, we found a kink in the electron dispersion at E_{M} = ±15 meV, which implies a dominant bosonic mode with ω_{E} = 15 meV. In the calculation, we use the Einstein model and take α^{2}F(ω) as a gaussian peak centered at ω_{E}. The calculated Re Σ(k, ω) is given in Fig. 3e, with comparison to experimental data. With the electron Green function, we can simulate the QPI dispersion Q(q, ω) by Eq. (3) convoluted with the experimental resolution, which is given in Fig. 3f.
Firstprinciples calculation
We perform electronic structure calculations within the framework of the density functional theory using normconserving pseudopotentials^{33} as implemented in the Quantum Espresso simulation package^{34}. The exchange–correlation effects are treated within the local density approximation with the Perdew–Zunger parametrization^{35}. The electronic calculations used a planewave energy cutoff of 80 Ry and a 12 × 12 × 12 Γcentered kmesh to sample the Brillouin zone. Total energies were converged to 10^{−7} Ry in combination with Methfessel–Paxton type broadening of 0.01 Ry. The phonon calculation is done by using a 2 × 2 × 2 qmesh grid centered at Γ for the sampling of phonon momenta. Starting with the experimental structure, we have fully optimized both the ionic positions and lattice parameters until the Hellmann–Feynman force on each atom is <10^{−3} Ry/a.u. (10^{−4} Ry) and zerostress tensors are obtained. We find that the flatband phonon is mainly associated with the vibrations of Co atoms in Co_{3}Sn lattice confined to the line connecting the centers of two neighboring triangles as shown by the dotted line in Fig. 5a. Figure 5b shows the atom displacement resolved phonon band structure.
Kagome lattice vibration model
We consider a kagome lattice vibration model, assuming the motion of the atoms is highly anisotropic and confined to the dotted line in Fig. 5a. The analysis reproduces the kagome band structure with a flat band quadratically touching another band.
We choose the vectors in real space connecting nearest neighbor kagome atoms as
The three atoms α = 1, 2, 3 in each kagome unit cell can move along the directions
All atoms are then coupled with the same spring constant β, except for the sign. The atoms in one unit cell have the potential energy
In contrast, the interunit cell coupling reads
After Fourier transformation, the potential energy reads
with
The spectrum of v_{k} is given by
where k_{1}, k_{2} stand for the inner product of k vector with a_{1}, a_{2}. We can observe the characteristic flatband and two quadratic bands.
Data availability
All relevant data are available from the corresponding authors upon reasonable request.
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Acknowledgements
We acknowledge insightful discussions with Biao Lian and Zhida Song, and technical assistance from Limin Liu and Gaihua Ye. Experimental and theoretical work at Princeton University was supported by the Gordon and Betty Moore Foundation (Grant No. GBMF4547 and GBMF9461/Hasan). Sample characterization was supported by the United States Department of Energy (US DOE) under the Basic Energy Sciences programme (Grant No. DOE/BES DEFG0205ER46200). M.Z.H. acknowledges support from Lawrence Berkeley National Laboratory and the Miller Institute of Basic Research in Science at the University of California, Berkeley in the form of a Visiting Miller Professorship. This work benefited from partial lab infrastructure support under NSFDMR1507585. M.Z.H. also acknowledges visiting scientist support from IQIMat the California Institute of Technology. The work at Renmin University was supported by the National Key R&D Program of China (Grants Nos. 2016YFA0300504 and 2018YFE0202600), the National Natural Science Foundation of China (No. 11774423,11822412), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (RUC) (18XNLG14, 19XNLG17). The authors acknowledge the use of Princeton’s Imaging and Analysis Center, which is partially supported by the Princeton Center for Complex Materials, a National Science Foundation (NSF)MRSEC program (DMR1420541). T.R.C. was supported from the Young Scholar Fellowship Program by the Ministry of Science and Technology (MOST) in Taiwan, under MOST Grant for the Columbus Program No. MOST1092636M006002, National Cheng Kung University, Taiwan, and National Center for Theoretical Sciences (NCTS), Taiwan. This work was supported partially by the MOST, Taiwan, grant MOST1072627E006001. This research was supported in part by Higher Education Sprout Project, Ministry of Education to the Headquarters of University. Z.W. and K.J. acknowledge US DOE Grant No. DEFG0299ER45747. T.N. acknowledges support from the European Union’s Horizon 2020 research and innovation programme (ERCStGNeupert757867PARATOP). The work performed at the Texas Center for Superconductivity at the University of Houston is supported by the U.S. Air Force Office of Scientific Research Grant FA95501510236, the T.L.L. Temple Foundation, the John J. and Rebecca Moores Endowment, and the State of Texas through the Texas Center for Superconductivity at the University of Houston. R.H. acknowledges support by NSF CAREER Grant No. DMR1760668 and NSF MRI Grant No. DMR1337207.
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J.X.Y., N.S., S.S.Z., and Y.J. conducted tunneling experiments in consultation with M.Z.H; Q.W., D.W., and H.L. synthesized and characterized the sequence of samples; S.M., H.J.T., G.Chang, K.J., Z.W., T.N., A.A., and T.R.C. carried out theoretical analysis in consultation with J.X.Y. and M.Z.H.; D.M., G.Cheng, N.Y., S.W., L.D., Z.Y., R.H., Z.L., and C.W.C. contributed to sample characterization. J.X.Y., N.S., and M.Z.H. performed the data analysis and figure development, and wrote the paper with contributions from all authors; M.Z.H. supervised the project. All authors discussed the results, interpretation, and conclusion.
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Yin, JX., Shumiya, N., Mardanya, S. et al. Fermion–boson manybody interplay in a frustrated kagome paramagnet. Nat Commun 11, 4003 (2020). https://doi.org/10.1038/s41467020174642
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DOI: https://doi.org/10.1038/s41467020174642
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