Terahertz magneto-optical investigation of quadrupolar spin-lattice effects in magnetically frustrated ${\mathrm{Tb}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$

Terahertz magneto-optical investigation of quadrupolar spin-lattice effects in magnetically frustrated ${Tb}_{2} {Ti}_{2} O_{7}$

K. Amelin, Y. Alexanian, U. Nagel, T. Rõõm, J. Robert, J. Debray, V. Simonet, C. Decorse, Z. Wang, R. Ballou, E. Constable, and S. de Brion

Phys. Rev. B 102, 134428 – Published 20 October 2020

Abstract

Condensed matter magneto-optical investigations can be a powerful probe of a material's microscopic magnetoelectric properties. This is because subtle interactions between electric and magnetic multipoles on a crystal lattice show up in predictable and testable ways in a material's optical response tensor, which dictates the polarization state and absorption spectrum of propagating electromagnetic waves. Magneto-optical techniques are therefore strong complements to probes such as neutron scattering, particularly when spin-lattice coupling effects are present. Here we perform a magneto-optical investigation of vibronic spin-lattice coupling in the magnetically frustrated pyrochlore ${Tb}_{2} {Ti}_{2} O_{7}$ . Coupling of this nature involving quadrupolar mixing between the ${Tb}^{3 +}$ electronic levels and phonons in ${Tb}_{2} {Ti}_{2} O_{7}$ has been a topic of debate for some time. This is particularly due to its implication for describing the exotic spin-liquid phase diagram of this highly debated system. A manifestation of this vibronic effect is observed as splitting of the ground and first excited crystal field doublets of the ${Tb}^{3 +}$ electronic levels, providing a fine structure to the absorption spectra in the terahertz (THz) frequency range. In this investigation, we apply a static magnetic field along the cubic [111] direction while probing with linearly polarized THz radiation. Through the Zeeman effect, the magnetic field enhances the splitting within the low-energy crystal field transitions revealing new details in our THz spectra. Complementary magneto-optical quantum calculations including quadrupolar terms show that indeed vibronic effects are required to describe our observations at 3 K. A further prediction of our theoretical model is the presence of a novel magneto-optical birefringence as a result of this vibronic process. Essentially, spin-lattice coupling within ${Tb}_{2} {Ti}_{2} O_{7}$ may break the optical isotropy of the cubic system, supporting two different electromagnetic wave propagations within the crystal. Together our results reveal the significance of considering quadrupolar spin-lattice effects when describing the spin-liquid ground state of ${Tb}_{2} {Ti}_{2} O_{7}$ . They also highlight the potential for future magneto-optical investigations to probe complex materials where spin-lattice coupling is present and reveal new magneto-optical activity in the THz range.

Received 20 July 2020
Accepted 29 September 2020

DOI:https://doi.org/10.1103/PhysRevB.102.134428

Physics Subject Headings (PhySH)

Exotic phases of matter Extraordinary optical transmission Magnetic coupling Magnetism Phonons Quantum phase transitions Spin dynamics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

K. Amelin ¹, Y. Alexanian ², U. Nagel ¹, T. Rõõm ¹, J. Robert², J. Debray², V. Simonet ², C. Decorse ³, Z. Wang^4,5,†, R. Ballou ², E. Constable ⁶, and S. de Brion ^2,*

¹National Institute of Chemical Physics and Biophysics, Tallinn 12618, Estonia
²Université Grenoble Alpes, CNRS, Institut Néel, 38000 Grenoble, France
³ICMMO, Université Paris-Saclay, CNRS, 91400 Orsay, France
⁴Institute of Radiation Physics, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
⁵Institute of Physics II, University of Cologne, 50937 Cologne, Germany
⁶Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria

^*sophie.debrion@neel.cnrs.fr
^†Present address: Fakultät Physik, Technische Universität Dortmund, 44221 Dortmund, Germany

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Vol. 102, Iss. 13 — 1 October 2020

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Figure 1
${Tb}_{2} {Ti}_{2} O_{7}$ cubic structure with the ${Tb}^{3 +}$ network. Upper part: Cross section viewed along the [111] direction showing the connection between tetrahedra. Lower part: one single tetrahedron. Two orthogonal linear polarizations of the incident light beam with $k ∥ [111]$ are shown in the upper corners, where the electric field vector E $^{ω}$ and magnetic field vector H $^{ω}$ oscillate parallel to $[\bar{1} \bar{1} 2]$ or $[\bar{1} 10]$ . The static magnetic field H, shown in black, is applied perpendicular to the THz polarization plane along one of the diagonals of the cube, the [111] direction.
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Figure 2
$H ∥ [111]$ magnetic-field dependence of the differential absorption $Δ α (H)$ in ${Tb}_{2} {Ti}_{2} O_{7}$ measured at (a) 3 K and (b) 60 K, with the reference absorption measured in zero field for two different THz polarizations (blue: H $^{ω}$ along $[\bar{1} \bar{1} 2]$ , red: H $^{ω}$ along $[\bar{1} 10]$ ). The spectra are offset vertically in proportion to $H$ . Shaded areas below the curves are included as a guide to the eye to highlight the different absorption bands.
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Figure 3
Experimental and calculated THz absorption as a function of magnetic field applied along the [111] direction of ${Tb}_{2} {Ti}_{2} O_{7}$ . The panels show the experimental results for 3 K (a) and 60 K (b), and the theoretical calculations for 3 K (c) and 60 K (d). The four ${Tb}^{3 +}$ sites on the tetrahedron (one with the field along its threefold axis shown in black and the three remaining sites shown in red) and the corresponding field dependence of the calculated energy of their absorption branches at 3 K are presented in the middle panels (e). The different observed branches are labeled $ν_{1}$ to $ν_{6}$ .
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Figure 4
THz wave propagation in an anisotropic medium. The wave, linearly polarized along $B^{ω}$ (blue), is decomposed into the two—orthogonal and linearly polarized for simplicity of the picture—normal modes polarized along $B_{1}^{ω}$ and $B_{2}^{ω}$ (green). Inside the crystal the two waves are normal modes, and the rays propagate independently in the direction of their Poynting vector $S_{1}$ and $S_{2}$ (red) with wave vector $k_{1}$ and $k_{2}$ , respectively, which have the same direction (orange). At the output face, in vacuum, the Poynting vectors are collinear again.
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Figure 5
Calculated absorption as a function of magnetic field applied along [111] at 3 K and 60 K for two THz polarizations with no vibron [(a),(b)], with a vibron associated with $O_{2}^{1}$ [(c)–(f)], and a vibron associated with $O_{2}^{2}$ [(g)–(j)] quadrupolar operators.
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Physical Review B

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Terahertz magneto-optical investigation of quadrupolar spin-lattice effects in magnetically frustrated ${Tb}_{2} {Ti}_{2} O_{7}$

K. Amelin, Y. Alexanian, U. Nagel, T. Rõõm, J. Robert, J. Debray, V. Simonet, C. Decorse, Z. Wang, R. Ballou, E. Constable, and S. de Brion

Phys. Rev. B 102, 134428 – Published 20 October 2020

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Physical Review B

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Terahertz magneto-optical investigation of quadrupolar spin-lattice effects in magnetically frustrated Tb2Ti2O7

K. Amelin, Y. Alexanian, U. Nagel, T. Rõõm, J. Robert, J. Debray, V. Simonet, C. Decorse, Z. Wang, R. Ballou, E. Constable, and S. de Brion

Phys. Rev. B 102, 134428 – Published 20 October 2020

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Terahertz magneto-optical investigation of quadrupolar spin-lattice effects in magnetically frustrated ${Tb}_{2} {Ti}_{2} O_{7}$