Ab initio determination of spin Hamiltonians with anisotropic exchange interactions: The case of the pyrochlore ferromagnet ${\mathrm{Lu}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{7}$

Ab initio determination of spin Hamiltonians with anisotropic exchange interactions: The case of the pyrochlore ferromagnet ${Lu}_{2} V_{2} O_{7}$

Kira Riedl, Daniel Guterding, Harald O. Jeschke, Michel J. P. Gingras, and Roser Valentí

Phys. Rev. B 94, 014410 – Published 8 July 2016

Abstract

We present a general framework for deriving effective spin Hamiltonians of correlated magnetic systems based on a combination of relativistic ab initio density functional theory calculations, exact diagonalization of a generalized Hubbard Hamiltonian on finite clusters, and spin projections onto the low-energy subspace. A key motivation is to determine anisotropic bilinear exchange couplings in materials of interest. As an example, we apply this method to the pyrochlore ${Lu}_{2} V_{2} O_{7}$ where the vanadium ions form a lattice of corner-sharing spin-1/2 tetrahedra. In this compound, anisotropic Dzyaloshinskii-Moriya interactions (DMIs) play an essential role in inducing a magnon Hall effect. We obtain quantitative estimates of the nearest-neighbor Heisenberg exchange, the DMI, and the symmetric part of the anisotropic exchange tensor. Finally, we compare our results with experimental ones on the ${Lu}_{2} V_{2} O_{7}$ compound.

Received 12 April 2016
Revised 7 June 2016

DOI:https://doi.org/10.1103/PhysRevB.94.014410

Physics Subject Headings (PhySH)

Ferromagnets Pyrochlores

Density functional theory Heisenberg model Hubbard model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Kira Riedl^1,*, Daniel Guterding¹, Harald O. Jeschke¹, Michel J. P. Gingras^2,3,4, and Roser Valentí¹

¹Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany
²Department of Physics and Astronomy, University of Waterloo, Ontario, N2L 3G1, Canada
³Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada
⁴Canadian Institute for Advanced Research, 180 Dundas Street West, Suite 1400, Toronto, Ontario, M5G 1Z8, Canada

^*riedl@itp.uni-frankfurt.de

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Vol. 94, Iss. 1 — 1 July 2016

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Figure 1
(a) Network of corner-sharing vanadium tetrahedra in the pyrochlore ${Lu}_{2} V_{2} O_{7}$ . (b) Oxygen environment around a vanadium tetrahedron.
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Figure 2
(a) Nonrelativistic density of states (DOS) of ${Lu}_{2} V_{2} O_{7}$ in the energy range $[- 7 eV, 4 eV]$ obtained in the GGA approximation. Shown are the total, as well as partial DOS corresponding to V, O, and Lu. (b) Vanadium orbital-resolved DOS around the Fermi level. (c) Illustration of the local reference frame in one tetrahedron of vanadium atoms, and orbital energy hierarchy in ${Lu}_{2} V_{2} O_{7}$ .
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Figure 3
Reference frames of spin and orbital degrees of freedom on two neighboring sites. The spins $S_{1}, S_{2}$ are given in the global reference frame and the orbitals are $d_{z^{2}}$ orbitals in the local reference frame at each site. Due to the site-dependent local coordinate frame, the spin operators have to be rotated in each local reference frame.
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Figure 4
(a) Chosen high-symmetry $k$ path in the Brillouin zone of pyrochlore. (b) Relativistic band structure of ${Lu}_{2} V_{2} O_{7}$ on the high-symmetry path. (c) Relativistic band structure between the high symmetry points L and W in a smaller energy window around the Fermi energy [gray shaded region in (b)]. The dark blue ( $DFT + SOC$ ) curve is the result of the fully relativistic band structure calculation. The purple curve (TB) represents the tight-binding band structure from the nonrelativistic calculation. The red ( $TB + SOC$ ) curve is the result of the tight-binding band structure taking the spin-orbit coupling term into account. The band splitting caused by relativistic effects is well reproduced, as can be seen by comparing the blue ( $DFT + SOC$ ) and red ( $TB + SOC$ ) curves.
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Figure 5
(a) Vanadium tetrahedron showing the two mutually perpendicular mirror planes relevant to the A-B bond. One plane includes the bond A-B, while the other bisects this bond at C. The presence of such mirror planes constrains the form of the local DM vectors and symmetric tensors. (b) Direction of the Dzyaloshinskii-Moriya vectors in a pyrochlore system in the global coordinate system considered in the calculations.
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Figure 6
Dependence of the calculated exchange parameters on the interaction parameters $U_{0}$ and $J_{avg}$ . The results for a certain choice of interaction parameters are given in Eq. (10). The gray area far left is the forbidden region where, for $U_{0} - 3 J_{α β} < 0$ , the projection on singly occupied states is no longer justified and the white lines correspond to the theoretical results given in Eq. (10). (a) Heisenberg exchange $J_{12}$ . For comparison, experiment [25] estimates $| J | = 8.22 meV$ . (b) $D_{12}^{x}$ , given in the global coordinate system as in Fig. 5. The results are below the experimental estimates, which correspond to $D_{12}^{x} = - 0.8 meV$ [18] and $D_{12}^{x} = - 1.05 meV$ [25]. Note, that we are reporting $D_{12}^{x}$ and not $| {\vec{D}}_{12} | = \sqrt{2} | D_{12}^{x} |$ . (c), (d) Independent contributions to the symmetric anisotropic exchange $\hat{K}$ . The calculated values are for none of the interaction parameters $U_{0}, J_{avg}$ small enough to neglect them.
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Physical Review B

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Ab initio determination of spin Hamiltonians with anisotropic exchange interactions: The case of the pyrochlore ferromagnet ${Lu}_{2} V_{2} O_{7}$

Kira Riedl, Daniel Guterding, Harald O. Jeschke, Michel J. P. Gingras, and Roser Valentí

Phys. Rev. B 94, 014410 – Published 8 July 2016

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Ab initio determination of spin Hamiltonians with anisotropic exchange interactions: The case of the pyrochlore ferromagnet Lu2V2O7

Kira Riedl, Daniel Guterding, Harald O. Jeschke, Michel J. P. Gingras, and Roser Valentí

Phys. Rev. B 94, 014410 – Published 8 July 2016

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Ab initio determination of spin Hamiltonians with anisotropic exchange interactions: The case of the pyrochlore ferromagnet ${Lu}_{2} V_{2} O_{7}$