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Hierarchical excitations from correlated spin tetrahedra on the breathing pyrochlore lattice

Shang Gao, Andrew F. May, Mao-Hua Du, Joseph A. M. Paddison, Hasitha Suriya Arachchige, Ganesh Pokharel, Clarina dela Cruz, Qiang Zhang, Georg Ehlers, David S. Parker, David G. Mandrus, Matthew B. Stone, and Andrew D. Christianson
Phys. Rev. B 103, 214418 – Published 9 June 2021
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  • INTRODUCTION
  • RESULTS AND DISCUSSIONS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The hierarchy of the coupling strengths in a physical system often engenders an effective model at low energies where the decoupled high-energy modes are integrated out. Here, using neutron scattering, we show that the spin excitations in the breathing pyrochlore lattice compound CuInCr4S8 are hierarchical and can be approximated by an effective model of correlated tetrahedra at low energies. At higher energies, intratetrahedron excitations together with strong magnon-phonon couplings are observed, which suggests the possible role of the lattice degree of freedom in stabilizing the spin tetrahedra. Our work illustrates the spin dynamics in CuInCr4S8 and demonstrates a general effective-cluster approach to understand the dynamics on the breathing-type lattices.

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    • Received 23 January 2021
    • Revised 26 April 2021
    • Accepted 17 May 2021

    DOI:https://doi.org/10.1103/PhysRevB.103.214418

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Frustrated magnetismSpin dynamics
    1. Physical Systems
    Pyrochlores
    1. Techniques
    Density functional calculationsNeutron scattering
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Shang Gao1,2, Andrew F. May1, Mao-Hua Du1, Joseph A. M. Paddison1, Hasitha Suriya Arachchige3,1, Ganesh Pokharel3,1, Clarina dela Cruz2, Qiang Zhang2, Georg Ehlers4, David S. Parker1, David G. Mandrus3,1,5, Matthew B. Stone2, and Andrew D. Christianson1

    • 1Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
    • 2Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
    • 3Department of Physics & Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
    • 4Neutron Technologies Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
    • 5Department of Material Science & Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA

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    Issue

    Vol. 103, Iss. 21 — 1 June 2021

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    Images

    • Figure 1
      Figure 1

      (a) The breathing pyrochlore lattice formed by the Cr3+ spins (shown as gray spheres) in CuInCr4S8. The complete crystal structure can be found in Appendix pp2. The bonds of the smaller and larger tetrahedra are shown in blue and red, respectively. Further-neighbor couplings including the second-neighbor coupling J2 and two third-neighbor couplings J3a and J3b are indicated. (b) Collinear magnetic structure viewed along the c axis. φ denotes the tilting angle from the b axis. (c) Coplanar magnetic structure that is composed of two AF spin pairs over the smaller tetrahedra. ψ denotes the angle between the two spin pairs. (d) Refinement result of the powder neutron diffraction data measured on HB-2A at T=2K assuming a collinear structure with φ=25∘. Data were collected with a constant incident neutron wavelength of 2.41 Å. The impurity peaks marked by stars are excluded in the refinements. Data points are shown as red circles. The calculated pattern is shown as the black solid line. The vertical bars indicate the positions of the structural (upper) and magnetic (lower) Bragg peaks for CuInCr4S8. The blue line at the bottom shows the difference of measured and calculated intensities. A similarly good refinement is obtained for nonzero ψ, with the refined moment magnitude staying constant at 2.54(2) μB. (e) Map of the goodness-of-fit χ2 as a function of ψ and φ.

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    • Figure 2
      Figure 2

      (a) Scattering intensity of CuInCr4S8 measured with Ei=100meV at T=5K. (b) Scattering intensity as a function of E integrated at Q=1.8 Å−1 (red circles) and 5.8 Å−1 (blue squares) with an integration width of 0.2 Å−1 as indicated by the white markers in (a). Solid line is the phonon scattering intensity at Q=5.8 Å−1 obtained from the DFT calculations after convolution with the instrumental energy resolution. (c) Phonon scattering intensity from the DFT calculations. The calculated data are convoluted with the tabulated instrumental energy resolution. (d) Total and projected density of states (PDOS) of phonons in CuInCr4S8 from the DFT calculations.

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    • Figure 3
      Figure 3

      (a) Scattering intensity of CuInCr4S8 measured with Ei=25meV at T=5K. (b) Effective fcc cluster model where the J1′-bonded tetrahedra are treated as rigid FM clusters. The nearest- and second-neighbor couplings among the tetrahedra are indicated as Jt1 and Jt2, respectively. (c) Zero-energy mode along the (1q0) direction in the fcc cluster model with the intertetrahedron couplings limited to the nearest neighbors, i.e., Jt2=0. (d) Intensity calculated by linear spin wave theory using the fcc cluster model with fitted coupling strengths of Jt1=0.099(9)meV and Jt2=−0.022(2)meV. The coplanar structure with ψ=90∘ as shown in Fig. 1 is assumed as the ground state. The calculated data are convoluted with a Gaussian function with a fitted full-width at half-maximum (FWHM) of 1.6(1) meV. (e) The corresponding fits for the constant-Q scans that are integrated at Q=0.5, 0.7, 0.9, and 1.8 Å−1 with an integration width of 0.1 Å−1 as indicated by the white markers in (a). Data at 0.7, 0.9, and 1.8 Å−1 are shifted horizontally by 1.5, 3.0, and 3.5 units for clarity of the figure, respectively. The reduced-χ2 factor for the fit is 7.4. (f) and (g) Similar to (d) and (e) but assuming a collinear ground state shown in Fig. 1. The fitted coupling strengths are Jt1=0.074(7)meV and Jt2=−0.035(3)meV. The reduced-χ2 factor for the fit is 13.2.

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    • Figure 4
      Figure 4

      (a) Scattering intensity of CuInCr4S8 measured with Ei=50meV at T=5K. (b) Intensity map calculated by the linear spin wave theory using the minimal J1−J1′−J4 model on the breathing pyrochlore lattice. The strength of J1′ is fixed at −2.6meV to reproduce the central positions of the high-E modes. For the coplanar ground state with ψ=90∘, the coupling strengths obtained by fitting the low-energy spectra are J1=1.8(1)meV, J4=−0.11(1)meV, close to the expected values of J1=16J1t and J4=4Jt2 of the fcc cluster model. The calculated data are convoluted by a Gaussian function with a FWHM that is two times broader than the tabulated instrumental energy resolution. Similar results can be obtained for the ground states with different ψ.

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    • Figure 5
      Figure 5

      (a) Refinement result of the powder neutron diffraction data measured on HB-2A at T=160K. The vertical bars indicate the positions of the structural Bragg peaks for CuInCr4S8. Data points and calculated pattern are presented similarly to Fig. 1 of the main text. (b) Crystal structure of CuInCr4S8 with the Cr-S bonds shown explicitly. (c) Refinement result of the powder neutron diffraction data measured on HB-2A at T=2K using the single basis vector collinear structure (ψ=0). Data points and calculated pattern are presented similarly to Fig. 1 of the main text. (d) The collinear magnetic structure with ψ=0.

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    • Figure 6
      Figure 6

      The Q dependence of the exclusive structure factor for a FM tetrahedron with d=3.73Å (red dashed lines) with S(Q)=1+3j0(Qd) in (a) for the static correlation and S(Q)=1−j0(Qd) in (b) for the first excitation. The blue solid lines are the structure factor multiplied by the squared Cr3+ magnetic form factor. Gray circles are intensities in arb. units integrated in the energy range of [2, 6] meV for the Ei=50meV spectra (a) and [15, 25] meV for the Ei=100meV spectra (b). Error bars representing the standard deviations are smaller than the marker size. The deviation of the integrated intensities from the free tetrahedron predictions at low Q is due to kinematic constraints of the neutron resulting in a partial region of energy transfer and wave-vector transfer sampled for fixed incident energy neutrons at small wave vectors.

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    • Figure 7
      Figure 7

      J1−J2 spin chain model with two spins at r1 and r2 in one unit cell. a defines the vector along the chain direction with a length equal to the lattice constant.

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    • Figure 8
      Figure 8

      INS spectra calculated using the effective fcc model with coupling strengths Jt1=0.099meV, Jt2=−0.022meV in (a) and Jt1=0.099meV, Jt2=0 in (b). (c) The four exchange paths of J4 (blue dashed lines) that contribute to the effective Jt2 interactions.

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