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Charge density wave order in the kagome metal AV3Sb5 (A=Cs,Rb,K)

Shangfei Wu, Brenden R. Ortiz, Hengxin Tan, Stephen D. Wilson, Binghai Yan, Turan Birol, and Girsh Blumberg
Phys. Rev. B 105, 155106 – Published 4 April 2022

Abstract

We employ polarization-resolved electronic Raman spectroscopy and density functional theory to study the primary and secondary order parameters, as well as their interplay, in the charge density wave (CDW) state of the kagome metal AV3Sb5. Previous x-ray diffraction data at 15 K established that the CDW order in CsV3Sb5 comprises of a 2×2×4 structure: one layer of inverse-star-of-David and three consecutive layers of star-of-David pattern. We analyze the lattice distortions based on the 2×2×4 structure at 15 K, and find that the U1 lattice distortion is the primarylike (leading) order parameter while M1+ and L2 distortions are secondarylike order parameters for vanadium displacements. This conclusion is confirmed by the calculation of bare susceptibility χ0(q) that shows a broad peak at around qz=0.25 along the hexagonal Brillouin zone face central line (U line). We also identify several phonon modes emerging in the CDW state, which are lattice vibration modes related to V and Sb atoms as well as alkali-metal atoms. The detailed temperature evolution of these modes' frequencies, half-width at half-maximums, and integrated intensities support a phase diagram with two successive structural phase transitions in CsV3Sb5: the first one with a primarylike order parameter appearing at TS=94K and the second isostructural one appearing at around T*=70K. Furthermore, the T dependence of the integrated intensity for these modes shows two types of behavior below TS: the low-energy modes show a plateaulike behavior below T* while the high-energy modes monotonically increase below TS. These two behaviors are captured by the Landau free-energy model incorporating the interplay between the primarylike and the secondarylike order parameters via trilinear coupling. Especially, the sign of the trilinear term that couples order parameters with different wave vectors determines whether the primarylike and secondarylike order parameters cooperate or compete with each other, thus determining the shape of the T dependence of the intensities of Bragg peak in x-ray data and the amplitude modes in Raman data below TS. These results provide an accurate basis for studying the interplay between multiple CDW order parameters in kagome metal systems.

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  • Received 13 January 2022
  • Accepted 14 March 2022

DOI:https://doi.org/10.1103/PhysRevB.105.155106

©2022 American Physical Society

Physics Subject Headings (PhySH)

  1. Research Areas
  1. Physical Systems
Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Shangfei Wu1,*, Brenden R. Ortiz2, Hengxin Tan3, Stephen D. Wilson2, Binghai Yan3, Turan Birol4, and Girsh Blumberg1,5,†

  • 1Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
  • 2Materials Department and California Nanosystems Institute, University of California Santa Barbara, Santa Barbara, California 93106, USA
  • 3Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel
  • 4Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA
  • 5National Institute of Chemical Physics and Biophysics, 12618 Tallinn, Estonia

  • *sw666@physics.rutgers.edu
  • girsh@physics.rutgers.edu

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Vol. 105, Iss. 15 — 15 April 2022

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  • Figure 1
    Figure 1

    (a) Crystal structure of CsV3Sb5 in the high-temperature phase (space group: P6/mmm, No. 191; point group: D6h). (b) Top view of the high-temperature crystal structure along the c axis and definitions of X and Y directions. (c) The inverse star of David (iSoD) 2×2×1 CDW phase. The black, blue, and green arrows represent the three V-V displacement directions. (d) The star of David (SoD) 2×2×1 CDW phase, which is obtained by merely reversing the direction of black arrows in (c). (e) The shorter V-V bond pattern for iSoD structure. (f) Same as (f) but for SoD structure.

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  • Figure 2
    Figure 2

    (a) The color plot of the T dependence of Raman spectra in RR (A1g+A2g) scattering geometry for CsV3Sb5. (b) The corresponding Raman spectra of (a). (c), (d) Same as (a) and (b) but for RL (E2g) scattering geometry. The dashed lines in (a) and (c) represent the structure phase transition temperature TS. The dashed lines in (b) and (d) represent the level of zero for the vertically shifted spectra.

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  • Figure 3
    Figure 3

    The 3D hexagonal Brillouin zone (BZ) corresponds to the space group P6/mmm with M, L, and U points highlighted, as well as the U line connecting M and L shown in red. (a) The three vectors in the stars of M points, leading to 2×2×1 CDW ordering shown as bond patterns in (b). (c) The three vectors in the stars of L points, contributing to 2×2×2 CDW ordering shown as bond patterns in (d). (e) The six vectors in the stars of U points, which give rise to 2×2×4 CDW ordering shown as bond patterns in (f). For these vectors shown in (a), (c), and (e), the solid arrows are pointing to the front faces while the dashed arrows are pointing to the back faces. (g) The normalized in-plane amplitude of V displacements δ/δmax,V (at 6j sites in the first kagome layer, 12n sites in the second kagome layer, and 6k sites in the third kagome layer) as a function of the normalized coordinate (z/c0) for the 2×2×4 structure of CsV3Sb5 at 15 K. c0 is the c-axis lattice constant for the four-layer structure. The solid red, blue, and green lines represent the total fitted curve, the c0-cosinusoidal modulation component, and the c0/2-cosinusoidal modulation component, respectively. (h) Illustration of the 2×2×4 structure for vanadium lattice [36]. In (b), (d), (f), and (h), the Cs and Sb atoms are omitted for simplification.

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  • Figure 4
    Figure 4

    Illustration of the D6h point group symmetries of the high-temperature, SoD, and iSoD phases of CsV3Sb5. The black lines represent the mirror planes. The C6 rotational symmetry is preserved in the above three phases. The Sb and Cs atoms are omitted for simplification.

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  • Figure 5
    Figure 5

    (a) The bare susceptibility χ0(qx,qx,qz) and limω0χ(qx,qy,qy,ω)/ω for CsV3Sb5 in the high-temperature phase for qz=0, 0.125, 0.25, 0.375, and 0.5, respectively. (b) Three-dimensional Fermi surfaces for CsV3Sb5 adapted from Ref. [21]. (c) Two-dimensional Fermi-surface cut for CsV3Sb5 at kz=0. (d) Same as (c) but for kz=π. (e) χ0(12,0,qz) along the LUMUL line as a function of qz. The solid red line is the guide line.

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  • Figure 6
    Figure 6

    Wyckoff-site-dependent displacement pattern for CsV3Sb5 in the 2×2×4 CDW phase at 15 K [36]. The Wyckoff sites are indexed according to the space group P6/mmm based on the unit cell of the 2×2×4 CDW phase. The arrows represent the direction of displacement for specific atoms at 15 K. The length of arrows for different atoms are scaled for visualization purposes.

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  • Figure 7
    Figure 7

    Illustration of the Landau free-energy model Eq. (6). The parameters used in the model are αu0=1, αl0=1, γ=0.2, β=0.1, λu=1.3, λl=4, TS=94, and T*=70. (a) The free energy at 100 K. (b) The free energy at 20 K. (c) The free energy at 20 K in the case of γ=0.2 while the other parameters remain unchanged. In (a)–(c), the global minima are represented by a black dot at (u0,l0). (d) T dependence of the A1g amplitude mode frequencies for modes 1 and 2. (e) T dependence of (u0+l0)2. The black solid circles represent the T dependence of the integrated intensity for the A1g 105 cm1 mode in CsV3Sb5. The inset of (e) shows the T dependence of u0(T) and l0(T) order parameters. (f) T dependence of (u0+l0)2 in the case of γ=0.2 while the other parameters remain unchanged. The inset of (f) shows the T dependence of u0(T) and l0(T) order parameters.

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  • Figure 8
    Figure 8

    Symmetry-resolved spectra of CsV3Sb5 above and below TS. (a) Raman spectra of CsV3Sb5 on a cleaved ab surface for the XX, XY, RR, and RL scattering geometries at 110 K. (b) Symmetry decompositions into separate irreducible representations according to the point group D6h using the algebra shown in Table 2. (c), (d) Same as (a) and (b) but at 20 K.

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  • Figure 9
    Figure 9

    (a) Comparison of the Raman spectra taken in RR (A1g+A2g) scattering geometry for the three compounds CsV3Sb5, RbV3Sb5, KV3Sb5. (b) Same as (a) but in RL (E2g) scattering geometry. The green arrows in (a) and (b) locate the weak low-energy phonons for KV3Sb5.

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  • Figure 10
    Figure 10

    An example of fitting for the Raman response in RR (a) and RL (b) scattering geometries at 20 K for CsV3Sb5. The red, blue, and green lines represent the total fitted response, the individual Lorentzian components, and a smooth background, respectively.

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  • Figure 11
    Figure 11

    T dependence of the peak position (a), HWHM (half-width at half-maximum) (b), and integrated intensity (c) for the main E2g phonon at 119 cm1 for CsV3Sb5. (d)–(f) Same as (a)–(c) but for the main A1g phonon at 137 cm1 . The error bars represent one standard deviation. The solid black and blue lines represent the fitting of the phononic self-energy T dependence above TS and below TS by anharmonic decay model, respectively [Eqs. (F1) and (F2)]. The solid vertical lines represent TS while the dashed black lines represent T*. The insets of (a) and (d) show the lattice vibration patterns for the E2g phonon and A1g phonon, respectively.

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  • Figure 12
    Figure 12

    T dependence of the peak position, HWHM, and integrated intensity for the new A1g phonon mode below TS for CsV3Sb5. (a)–(c) For the mode at 45 cm1 . (d)–(f) For the mode at 105 cm1 . (g)–(i) For the mode at 197 cm1 . (j)–(l) For the mode at 202 cm1 . (m)–(o) For the mode at 241 cm1 . The error bars represent one standard deviation. For fitting of the phononic self-energy T dependence for the mode at 45 and 241 cm1 , Eqs. (F1) and Eq. (F2) are used. For the modes at 105, 197, and 202 cm1 , Eqs. (F3) and (F4) are used.

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  • Figure 13
    Figure 13

    T dependence of the peak position, HWHM, and integrated intensity for the new E2g phonon modes below TS for CsV3Sb5. (a)–(c) For the mode at 45 cm1 . (d)–(f) For the mode at 61 cm1 . (g)–(i) For the mode at 102 cm1 . (j)–(l) For the mode at 181 cm1 . (m)–(o) For the mode at 207 cm1 . (p)–(r) For the mode at 223 cm1 . (s)–(u) For the mode at 237 cm1 . The error bars represent one standard deviation. For fitting of the phononic self-energy T dependence for the modes at 45, 61, and 102 cm1 , Eqs. (F1) and (F2) are used. For the modes at 181, 207, and 223 cm1 , Eqs. (F1) and (F2) are used.

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  • Figure 14
    Figure 14

    Comparison of raw data and polarization leakage removed spectra, taken in RR (top panel) and RL (bottom panel) polarization scattering geometry from the ab surface of CsV3Sb5 at 60 K with 647-nm laser excitation.

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  • Figure 15
    Figure 15

    DFT calculation of phonon dispersion in the whole Brillouin zone for the high-temperature phase of CsV3Sb5.

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