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Electronic nematicity without charge density waves in titanium-based kagome metal

Nature Physics volume 19, pages 1591–1598 (2023)Cite this article

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Abstract

Layered crystalline materials that consist of transition metal atoms on a kagome network have emerged as a versatile platform for the study of unusual electronic phenomena. For example, in the vanadium-based kagome superconductors AV3Sb5 (where A can stand for K, Cs or Rb), there is a parent charge density wave phase that appears to simultaneously break both the translational and rotational symmetries of the lattice. Here we show a contrasting situation, where electronic nematic order—the breaking of rotational symmetry without the breaking of translational symmetry—can occur without a corresponding charge density wave. We use spectroscopic-imaging scanning tunnelling microscopy to study the kagome metal CsTi3Bi5 that is isostructural to AV3Sb5 but with a titanium atom kagome network. CsTi3Bi5 does not exhibit any detectable charge density wave state, but a comparison to density functional theory calculations reveals substantial electronic correlation effects at low energies. In comparing the amplitudes of scattering wave vectors along different directions, we discover an electronic anisotropy that breaks the sixfold symmetry of the lattice, arising from both in-plane and out-of-plane titanium-derived d orbitals. Our work uncovers the role of electronic orbitals in CsTi3Bi5, suggestive of a hexagonal analogue of the nematic bond order in Fe-based superconductors.

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Fig. 1: Crystalline structure and topographic characterization of CsTi3Bi5.
Fig. 2: Large-scale electronic band structure and the absence of CDW in CsTi3Bi5.
Fig. 3: Twofold symmetric electronic signature.
Fig. 4: Mapping the origin of quasiparticle scattering and interference.

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Data availability

The data supporting the findings of this study are available from Zenodo at https://doi.org/10.5281/zenodo.8092076 and also upon request from the corresponding author. Source data are provided with this paper.

Code availability

The computer code used for data analysis is available upon request from the corresponding author.

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Acknowledgements

I.Z. gratefully acknowledges the support from the National Science Foundation (NSF), Division of Materials Research 2216080. S.D.W. and B.R.O. acknowledge financial support from the US Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Grant No. DE-SC0020305. This work used facilities supported via the University of California, Santa Barbara, NSF Quantum Foundry funded via the Quantum Materials Science, Engineering and Information program under award DMR-1906325. Z.W. acknowledges the support of the US Department of Energy, Basic Energy Sciences Grant No. DE-FG02-99ER45747 and the Cottrell Singular Exceptional Endeavors of Discovery Award No. 27856 from Research Corporation for Science Advancement. D.W. and D.J. acknowledge the support from the Bavaria California Technology Center Grant 7 [2021-2]. B.Y. acknowledges the financial support by the European Research Council (ERC Consolidator Grant ‘NonlinearTopo’, No. 815869) and the ISF - Personal Research Grant (No. 2932/21).

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Authors and Affiliations

  1. Department of Physics, Boston College, Chestnut Hill, MA, USA

    Hong Li, Siyu Cheng, Keyu Zeng, Ziqiang Wang & Ilija Zeljkovic

  2. Materials Department, University of California, Santa Barbara, Santa Barbara, CA, USA

    Brenden R. Ortiz & Stephen D. Wilson

  3. Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel

    Hengxin Tan & Binghai Yan

  4. Department of Chemistry, Ludwig-Maximilians-Universität München, München, Germany

    Dominik Werhahn & Dirk Johrendt

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Contributions

STM experiments and data analysis were performed by H.L., with the help from S.C. and B.R.O. D.W. synthesized and characterized the samples under the supervision of S.D.W. and D.J. Z.W. and K.Z. provided theoretical input on data interpretation. H.T. and B.Y. performed band structure calculations. H.L., S.D.W., Z.W. and I.Z. wrote the paper, with input from all authors. I.Z. supervised the project.

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Correspondence to Ilija Zeljkovic.

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Extended data

Extended Data Fig. 1 Low-temperature differential conductance spectra.

Average dI/dV spectra on the Bi surface acquired at approximately 350 mK (left) and 2 K (right) temperature showing the absence of a clear superconducting gap. STM setup conditions: (left) Iset = 500 pA, Vsample = 5 mV, Vexc = 0.1 mV, 0 T; (right) Iset = 1 nA, Vsample = 5 mV, Vexc = 0.1 mV, 0 T.

Extended Data Fig. 2 Comparison of QPI wave vectors and theoretical band structure.

(a) Fourier transform (FT) of a Bi surface dI/dV (r, V = 20 mV). Three pink arrows denote q1, q2 and q3 scattering wave vectors. (b) FT linecut taken along the blue dashed arrow in (a). The three dispersive QPI wave vectors are clearly seen, which are denoted as q1, q2 and q3. (c) The summary of energy dependent vector position in k space. Data in (b,c) was converted from momentum-transfer q-space to k-space by ½ multiplication factor. Each point in (c) is extracted by fitting a Gaussian peak function to the linecuts in (b). The error bars represent standard errors of peak positions. The comparison of the linearly fitted band velocities with electronic band velocities in (d) points to the origin of different q-vectors as discussed in the main text: q1 comes from intrascattering within the Bi orbital derived Г pocket, q3 arises from the intrascattering within the Ti orbital derived, and q2 is likely due to scattering between the Bi and the Ti orbital derived bands. (d) The DFT calculated band structure along Г-K direction. STM setup conditions: Iset = 600 pA, Vsample = 100 mV, Vexc = 4 mV (a).

Extended Data Fig. 3 Robustness of observed anisotropy to using different tips.

(a) Fourier transforms (FTs) of dI/dV maps taken on the same Bi surface region with different tips (left and right panel). Blue and red arrows denote q2 and q3 wave vectors. (b) Angle-dependent polar plots of q2 and q3 amplitudes before and after the intentional tip change. Red line corresponds to the q3 amplitude and the blue line corresponds to the q2 amplitude. (c,d) Equivalent to (a,b) but for a different region of the sample. It is evident that amplitudes exhibits a strong C2-symmetric signal on different regions with different tips, with the dominant axis pointing along a Γ-K direction in each dataset. Note that the scan frame was slightly rotated between region A and B, which is why the lattice also appears rotated between panels (a) and (c). These experiments provide further evidence that C2-symmetric electron scattering is not an artifact caused by the tip shape. STM setup condition: Iset = 200 pA, Vsample = -10 mV, Vexc = 2 mV (region A); Iset = 500 pA, Vsample = 10 mV, Vexc = 5 mV (region B).

Extended Data Fig. 4 Robustness of observed anisotropy to scan direction.

(a) A 35 nm square STM topograph taken over the region A of sample 2. (b, c) Fourier transforms (FTs) of dI/dV maps taken over the region in (a) scanned along (b) vertical and (c) horizontal direction at -7 mV bias. Atomic Bragg peaks are circled in black. Blue and red arrows denote q2 and q3. (d,e) The amplitudes of q2 and q3 plotted as a function of angle in polar axis corresponding to data in (b,c). It is clear that both patterns are C2-symmetric, and the symmetry axis does not change due to a different scanning direction. STM setup condition: Iset = 200 pA, Vsample = -10 mV, Vexc = 2 mV (a-c).

Extended Data Fig. 5 Theoretically calculated constant energy contours.

Constant energy contours at -50 meV (left), 0 mV (middle) and +50 meV (right) calculated by DFT for kz = 0 (red lines) and kz = 0.5 (blue lines).

Extended Data Fig. 6 Additional scattering wave vector dispersions.

(a) Fourier transform of a normalized differential conductance L(r,V = -14 mV) map taken on the Bi surface (also shown in Fig. 3a of the main text). Atomic Bragg peaks are circled in black. (b) Momentum-transfer space positions of the wave vectors q1, q2 and q3 along the three Γ-K directions. Error bars in (b) represent the standard errors obtained by Lorentzian fits to the data. The anisotropy of q1 is much smaller compared to the other two wave vectors.

Extended Data Fig. 7 Calculated phonon dispersions.

Calculated phonon dispersion in (a) CsTi3Bi5 and (b) CsV3Sb5 using the same procedure as described in Tan et al, PRL 127, 046401 (2021). Imaginary (negative) phonon frequencies in (b) marked by red arrows corresponds to the breathing modes of the kagome lattice, which likely contributes to the CDW distortions in CsV3Sb5. Corresponding instability in CsTi3Bi5 is however absent.

Extended Data Fig. 8 STM topograph spanning both surface terminations.

(a) STM topograph spanning both the Cs and the Bi surface termination. (b) Zoom-in on a smaller region showing the atomic structure of both surfaces across a step edge. The lattice of small white circles shows the positions of individual Cs atoms based on the known atomic structure of the Bi termination. It can be seen that Cs atoms on the Cs termination reside on top of bright features in the topograph. STM setup conditions: Iset = 200 pA, Vsample = 100 mV.

Extended Data Fig. 9 Relating QPI anisotropy and spectral weight anisotropy.

(a) Form factor function f(ϴ,N = 1.5) = 1 + (cos(2ϴ)-1)/(2 N) used to simulate the spectral weight anisotropy. (b) Autocorrelation of the constant energy contour (CEC) at zero energy, which is multiplied by f(ϴ, N = 1.5) prior to the autocorrelation procedure. This qualitatively simulates the C2-symmetric QPI by adding an angle-dependent function to the weight of the Fermi surfaces to break their C6 symmetry. ϴ is angle calculated with respect to the Γ-K direction. As seen in (c), to approximately get the ratio of q3 wave vectors along the two directions to be 2, N factor is about 1.5. In this scenario, the spectral weight varies by about a factor of 2 as seen in panel (a), consistent with QPI anisotropy.

Extended Data Fig. 10 Energy-dependence of the signal at atomic Bragg peak wave vectors.

(a) Fourier transform of the L-map from the main text Fig. 3 showing the three atomic Bragg peak positions used for the analysis. The peak labeled A is along a mirror symmetry axis. (b) Amplitude of the atomic Bragg peaks as a function of energy from dI/dV maps. (c,d) Kagome lattice simulation. We generate a Kagome lattice real space density of states (DOS) distribution by using a simple trigonometric function: \({\left|{{\psi }}({\bf{r}})\right|}^{2}=\sum _{{\rm{j}}={\rm{A}},{\rm{B}},{\rm{C}}}{\left|{{\psi }}({\rm{j}})\right|}^{2}\prod _{{\rm{i}}=0,1,2}(1+{{\cos }}[\frac{4{{\pi }}}{\sqrt{3}{\rm{a}}}{{\bf{e}}}_{{\rm{x}}}\bullet \left({\bf{R}}\left[\frac{\left(2{\rm{i}}-1\right){{\pi }}}{6}\right]\bullet \left({\bf{r}}{\boldsymbol{-}}{{\bf{r}}}_{{\bf{j}}}\right)\right)])\), where \({{\bf{r}}}_{{\bf{A}}}{\boldsymbol{=}}(0,0){\rm{;}}{{\bf{r}}}_{{\bf{B}}}{\boldsymbol{=}}\left(1,0\right){\rm{a}}/2{\rm{;}}{{\bf{r}}}_{{\bf{C}}}{\boldsymbol{=}}\left(1,\sqrt{3}\right){\rm{a}}/4\). \({{\bf{e}}}_{{\rm{x}}}=\left(1,0\right),\) \({\bf{R}}\left[{{\theta }}\right]\) is the rotation matrix, and \({\rm{a}}\) is the lattice constant. When \({\left|{{\psi }}({\rm{A}})\right|}^{2}={\left|{{\psi }}({\rm{B}})\right|}^{2}={\left|{{\psi }}({\rm{C}})\right|}^{2}\), the DOS at the three sub-lattices are all the same. However, when (c) \({\left|{{\psi }}({\rm{A}})\right|}^{2} > {\left|{{\psi }}({\rm{B}})\right|}^{2}={\left|{{\psi }}({\rm{C}})\right|}^{2}\) (A site has a higher DOS than B and C), or (d) \({\left|{{\psi }}({\rm{A}})\right|}^{2} < {\left|{{\psi }}({\rm{B}})\right|}^{2}={\left|{{\psi }}({\rm{C}})\right|}^{2}\) (A site has a smaller DOS compared to B and C), the simulated lattices break the C6 symmetry. By tuning \({\left|{{\psi }}({\rm{B}}={\rm{C}})\right|}^{2}/{\left|{{\psi }}({\rm{A}})\right|}^{2}\) from 1 to 0, sites B and C gradually fade away, and the lattice becomes triangular, as shown in (e). By tuning \({\left|{{\psi }}({\rm{A}})\right|}^{2}/{\left|{{\psi }}({\rm{B}}={\rm{C}})\right|}^{2}\) from 1 to 0, the A site gradually disappears as shown in (f), which ultimately leads to a rectangular lattice. Fourier transforms (FTs) of the simulated topographs are shown in the second row of (e,f). In both cases, the lattice symmetry varies due to the variation of A site DOS relative to that of B and C sites. (g,h) Atomic Bragg peak QBragg intensities depending on the DOS ratios between the sub-lattices.

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Li, H., Cheng, S., Ortiz, B.R. et al. Electronic nematicity without charge density waves in titanium-based kagome metal. Nat. Phys. 19, 1591–1598 (2023). https://doi.org/10.1038/s41567-023-02176-3

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