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Low-energy magneto-optics of Tb 2 Ti 2 O 7 in a [111] magnetic field

The pyrochlore magnet Tb 2 Ti 2 O 7 shows a lack of magnetic order to low temperatures and is considered to be a quantum spin liquid candidate. Here we perform time-domain THz spectroscopy on high-quality Tb 2 Ti 2 O 7 crystals and study the low-energy excitations as a function of [111] magnetic field with high energy resolution. The low-energy crystal-field excitations change their energies anomalously under magnetic field. Despite several sharp field-dependent changes, we show that the material's spectrum can be described not by phase transitions but by field-dependent hybridization between the low-energy crystal-field levels. We highlight the strong coupling between spin and lattice degrees of freedom in Tb 2 Ti 2 O 7 as evidenced by the magnetic-field tunable crystal-field environment. Calculations based on single ion physics with field-induced symmetry reduction of the crystal-field environment can reproduce our data.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Materials Data on Tb(Ni2B)6 by Materials Project

Tb(Ni2B)6 crystallizes in the orthorhombic Cmc2_1 space group. The structure is three-dimensional. Tb3+ is bonded in a distorted hexagonal planar geometry to six B3- atoms. There are a spread of Tb–B bond distances ranging from 2.99–3.29 Å. There are seven inequivalent Ni+1.25+ sites. In the first Ni+1.25+ site, Ni+1.25+ is bonded to four B3- atoms to form a mixture of distorted edge and corner-sharing NiB4 tetrahedra. There are a spread of Ni–B bond distances ranging from 2.02–2.10 Å. In the second Ni+1.25+ site, Ni+1.25+ is bonded to four B3- atoms to form a mixture of edge and corner-sharing NiB4 tetrahedra. There are a spread of Ni–B bond distances ranging from 2.02–2.09 Å. In the third Ni+1.25+ site, Ni+1.25+ is bonded in a distorted T-shaped geometry to three B3- atoms. There are two shorter (2.10 Å) and one longer (2.17 Å) Ni–B bond lengths. In the fourth Ni+1.25+ site, Ni+1.25+ is bonded in a distorted T-shaped geometry to three B3- atoms. There are a spread of Ni–B bond distances ranging from 2.06–2.14 Å. In the fifth Ni+1.25+ site, Ni+1.25+ is bonded in a T-shaped geometry to three B3- atoms. There are two shorter (1.99 Å) and one longer (2.08 Å) Ni–B bond lengths. In the sixth Ni+1.25+ site, Ni+1.25+ is bonded in a T-shaped geometry to three B3- atoms. There are a spread of Ni–B bond distances ranging from 2.03–2.09 Å. In the seventh Ni+1.25+ site, Ni+1.25+ is bonded to four B3- atoms to form a mixture of edge and corner-sharing NiB4 tetrahedra. There are a spread of Ni–B bond distances ranging from 1.98–2.09 Å. There are four inequivalent B3- sites. In the first B3- site, B3- is bonded in a 7-coordinate geometry to one Tb3+ and seven Ni+1.25+ atoms. In the second B3- site, B3- is bonded in a 7-coordinate geometry to one Tb3+ and seven Ni+1.25+ atoms. In the third B3- site, B3- is bonded in a 7-coordinate geometry to one Tb3+ and seven Ni+1.25+ atoms. In the fourth B3- site, B3- is bonded in a 7-coordinate geometry to one Tb3+ and seven Ni+1.25+ atoms.

36 MATERIALS SCIENCE↗

Materials Data on Tb(BO2)3 by Materials Project

Tb(BO2)3 crystallizes in the monoclinic C2/c space group. The structure is three-dimensional. Tb3+ is bonded in a 8-coordinate geometry to eight O2- atoms. There are a spread of Tb–O bond distances ranging from 2.40–2.43 Å. There are two inequivalent B3+ sites. In the first B3+ site, B3+ is bonded in a tetrahedral geometry to four O2- atoms. All B–O bond lengths are 1.47 Å. In the second B3+ site, B3+ is bonded in a trigonal planar geometry to three O2- atoms. There is one shorter (1.35 Å) and two longer (1.40 Å) B–O bond length. There are three inequivalent O2- sites. In the first O2- site, O2- is bonded in a 2-coordinate geometry to one Tb3+ and two B3+ atoms. In the second O2- site, O2- is bonded in a distorted single-bond geometry to two equivalent Tb3+ and one B3+ atom. In the third O2- site, O2- is bonded in a 2-coordinate geometry to one Tb3+ and two B3+ atoms.

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Materials Data on Tb(HO)3 by Materials Project

Tb(OH)3 crystallizes in the hexagonal P6_3/m space group. The structure is three-dimensional. Tb3+ is bonded in a 9-coordinate geometry to nine equivalent O2- atoms. There are six shorter (2.44 Å) and three longer (2.47 Å) Tb–O bond lengths. H1+ is bonded in a single-bond geometry to one O2- atom. The H–O bond length is 0.97 Å. O2- is bonded in a single-bond geometry to three equivalent Tb3+ and one H1+ atom.

36 MATERIALS SCIENCE↗

Materials Data on Tb(NiB)2 by Materials Project

Tb(NiB)2 crystallizes in the monoclinic C2/c space group. The structure is three-dimensional. Tb3+ is bonded in a 6-coordinate geometry to six equivalent B3- atoms. There are a spread of Tb–B bond distances ranging from 2.71–2.88 Å. Ni+1.50+ is bonded in a 4-coordinate geometry to four equivalent B3- atoms. There are a spread of Ni–B bond distances ranging from 2.03–2.07 Å. B3- is bonded in a 8-coordinate geometry to three equivalent Tb3+, four equivalent Ni+1.50+, and one B3- atom. The B–B bond length is 1.75 Å.

36 MATERIALS SCIENCE↗

Materials Data on Tb(IO3)3 by Materials Project

Tb(O3I)3 crystallizes in the monoclinic P2_1/c space group. The structure is three-dimensional. Tb3+ is bonded in a 8-coordinate geometry to eight O2- atoms. There are a spread of Tb–O bond distances ranging from 2.28–2.60 Å. There are nine inequivalent O2- sites. In the first O2- site, O2- is bonded in a distorted bent 150 degrees geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.84 Å. In the second O2- site, O2- is bonded in a bent 150 degrees geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.81 Å. In the third O2- site, O2- is bonded in a distorted bent 120 degrees geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.83 Å. In the fourth O2- site, O2- is bonded in a 2-coordinate geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.82 Å. In the fifth O2- site, O2- is bonded in a 2-coordinate geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.83 Å. In the sixth O2- site, O2- is bonded in a distorted bent 150 degrees geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.83 Å. In the seventh O2- site, O2- is bonded in a distorted single-bond geometry to one I5+ atom. The O–I bond length is 1.83 Å. In the eighth O2- site, O2- is bonded in a bent 150 degrees geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.83 Å. In the ninth O2- site, O2- is bonded in a distorted bent 120 degrees geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.85 Å. There are three inequivalent I5+ sites. In the first I5+ site, I5+ is bonded in a distorted trigonal non-coplanar geometry to three O2- atoms. In the second I5+ site, I5+ is bonded in a 3-coordinate geometry to three O2- atoms. In the third I5+ site, I5+ is bonded in a 3-coordinate geometry to three O2- atoms.

36 MATERIALS SCIENCE↗

Materials Data on Tb(IO3)3 by Materials Project

Tb(O3I)3 crystallizes in the monoclinic P2_1/c space group. The structure is three-dimensional. Tb3+ is bonded in a 7-coordinate geometry to eight O2- atoms. There are a spread of Tb–O bond distances ranging from 2.32–2.79 Å. There are nine inequivalent O2- sites. In the first O2- site, O2- is bonded in a distorted bent 120 degrees geometry to one Tb3+ and two I5+ atoms. There are one shorter (1.85 Å) and one longer (2.92 Å) O–I bond lengths. In the second O2- site, O2- is bonded in a distorted bent 120 degrees geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.84 Å. In the third O2- site, O2- is bonded in a distorted single-bond geometry to two I5+ atoms. There are one shorter (1.84 Å) and one longer (2.74 Å) O–I bond lengths. In the fourth O2- site, O2- is bonded in a bent 150 degrees geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.82 Å. In the fifth O2- site, O2- is bonded in a distorted water-like geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.85 Å. In the sixth O2- site, O2- is bonded in a 2-coordinate geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.85 Å. In the seventh O2- site, O2- is bonded in a bent 120 degrees geometry to one Tb3+ and one I5+ atom. The O–I bond length is 1.84 Å. In the eighth O2- site, O2- is bonded in a 1-coordinate geometry to two equivalent I5+ atoms. There are one shorter (1.87 Å) and one longer (2.40 Å) O–I bond lengths. In the ninth O2- site, O2- is bonded in a distorted bent 150 degrees geometry to two equivalent Tb3+ and one I5+ atom. The O–I bond length is 1.86 Å. There are three inequivalent I5+ sites. In the first I5+ site, I5+ is bonded in a 6-coordinate geometry to six O2- atoms. In the second I5+ site, I5+ is bonded in a 5-coordinate geometry to three O2- atoms. In the third I5+ site, I5+ is bonded in a 3-coordinate geometry to three O2- atoms.

36 MATERIALS SCIENCE↗

Materials Data on Tb(BO2)3 by Materials Project

Tb(BO2)3 crystallizes in the monoclinic P2_1/c space group. The structure is three-dimensional. Tb3+ is bonded in a 8-coordinate geometry to eight O2- atoms. There are a spread of Tb–O bond distances ranging from 2.37–2.44 Å. There are three inequivalent B3+ sites. In the first B3+ site, B3+ is bonded in a trigonal planar geometry to three O2- atoms. There are a spread of B–O bond distances ranging from 1.34–1.41 Å. In the second B3+ site, B3+ is bonded in a tetrahedral geometry to four O2- atoms. There is two shorter (1.47 Å) and two longer (1.48 Å) B–O bond length. In the third B3+ site, B3+ is bonded in a trigonal planar geometry to three O2- atoms. There are a spread of B–O bond distances ranging from 1.34–1.41 Å. There are six inequivalent O2- sites. In the first O2- site, O2- is bonded in a distorted single-bond geometry to two equivalent Tb3+ and one B3+ atom. In the second O2- site, O2- is bonded in a 2-coordinate geometry to one Tb3+ and two B3+ atoms. In the third O2- site, O2- is bonded in a 2-coordinate geometry to one Tb3+ and two B3+ atoms. In the fourth O2- site, O2- is bonded in a distorted bent 120 degrees geometry to one Tb3+ and two B3+ atoms. In the fifth O2- site, O2- is bonded in a distorted bent 120 degrees geometry to one Tb3+ and two B3+ atoms. In the sixth O2- site, O2- is bonded in a distorted single-bond geometry to two equivalent Tb3+ and one B3+ atom.

36 MATERIALS SCIENCE↗

Materials Data on Tb(AlC)3 by Materials Project

Tb(AlC)3 crystallizes in the hexagonal P6_3/mmc space group. The structure is three-dimensional. Tb3+ is bonded to six equivalent C4- atoms to form TbC6 octahedra that share corners with six equivalent AlC4 tetrahedra, edges with six equivalent TbC6 octahedra, and edges with six equivalent AlC4 tetrahedra. All Tb–C bond lengths are 2.57 Å. There are two inequivalent Al3+ sites. In the first Al3+ site, Al3+ is bonded to four C4- atoms to form AlC4 tetrahedra that share corners with three equivalent TbC6 octahedra, corners with seven equivalent AlC4 tetrahedra, and edges with three equivalent TbC6 octahedra. The corner-sharing octahedral tilt angles are 20°. There are one shorter (2.02 Å) and three longer (2.11 Å) Al–C bond lengths. In the second Al3+ site, Al3+ is bonded in a trigonal planar geometry to three equivalent C4- atoms. All Al–C bond lengths are 1.99 Å. There are two inequivalent C4- sites. In the first C4- site, C4- is bonded in a 6-coordinate geometry to three equivalent Tb3+ and three equivalent Al3+ atoms. In the second C4- site, C4- is bonded to five Al3+ atoms to form corner-sharing CAl5 trigonal bipyramids.

36 MATERIALS SCIENCE↗

Materials Data on Tb(SiPt)2 by Materials Project

Tb(PtSi)2 crystallizes in the tetragonal I4/mmm space group. The structure is three-dimensional. Tb4+ is bonded in a 8-coordinate geometry to eight equivalent Pt2- atoms. All Tb–Pt bond lengths are 3.24 Å. Pt2- is bonded in a 4-coordinate geometry to four equivalent Tb4+ and four equivalent Si atoms. All Pt–Si bond lengths are 2.47 Å. Si is bonded in a 5-coordinate geometry to four equivalent Pt2- and one Si atom. The Si–Si bond length is 2.33 Å.

36 MATERIALS SCIENCE↗

Materials Data on Tb(HO)3 by Materials Project

Tb(OH)3 crystallizes in the monoclinic P2_1/m space group. The structure is three-dimensional. Tb3+ is bonded in a 9-coordinate geometry to nine O2- atoms. There are a spread of Tb–O bond distances ranging from 2.41–2.61 Å. There are three inequivalent H1+ sites. In the first H1+ site, H1+ is bonded in a single-bond geometry to one O2- atom. The H–O bond length is 0.97 Å. In the second H1+ site, H1+ is bonded in a single-bond geometry to one O2- atom. The H–O bond length is 0.98 Å. In the third H1+ site, H1+ is bonded in a single-bond geometry to one O2- atom. The H–O bond length is 0.97 Å. There are three inequivalent O2- sites. In the first O2- site, O2- is bonded in a single-bond geometry to three equivalent Tb3+ and one H1+ atom. In the second O2- site, O2- is bonded in a single-bond geometry to three equivalent Tb3+ and one H1+ atom. In the third O2- site, O2- is bonded in a single-bond geometry to three equivalent Tb3+ and one H1+ atom.

36 MATERIALS SCIENCE↗

Materials Data on Tb(BiO2)3 by Materials Project

Tb(BiO2)3 is Ilmenite-like structured and crystallizes in the cubic I2_13 space group. The structure is three-dimensional. Tb3+ is bonded to six O2- atoms to form TbO6 octahedra that share corners with six BiO6 octahedra and edges with six BiO6 octahedra. The corner-sharing octahedra tilt angles range from 53–54°. There are three shorter (2.32 Å) and three longer (2.34 Å) Tb–O bond lengths. There are two inequivalent Bi3+ sites. In the first Bi3+ site, Bi3+ is bonded to six O2- atoms to form distorted BiO6 octahedra that share corners with two equivalent TbO6 octahedra, corners with four equivalent BiO6 octahedra, edges with two equivalent TbO6 octahedra, and edges with four equivalent BiO6 octahedra. The corner-sharing octahedra tilt angles range from 54–55°. There are a spread of Bi–O bond distances ranging from 2.35–2.48 Å. In the second Bi3+ site, Bi3+ is bonded to six O2- atoms to form distorted BiO6 octahedra that share corners with two equivalent TbO6 octahedra, corners with four equivalent BiO6 octahedra, edges with two equivalent TbO6 octahedra, and edges with four equivalent BiO6 octahedra. The corner-sharing octahedra tilt angles range from 53–57°. There are a spread of Bi–O bond distances ranging from 2.36–2.47 Å. There are two inequivalent O2- sites. In the first O2- site, O2- is bonded to one Tb3+ and three Bi3+ atoms to form a mixture of distorted edge and corner-sharing OTbBi3 trigonal pyramids. In the second O2- site, O2- is bonded to one Tb3+ and three Bi3+ atoms to form a mixture of distorted edge and corner-sharing OTbBi3 trigonal pyramids.

36 MATERIALS SCIENCE↗

Materials Data on Tb(ZnP)3 by Materials Project

Tb(ZnP)3 crystallizes in the hexagonal P6_3/mmc space group. The structure is three-dimensional. Tb3+ is bonded to six equivalent P3- atoms to form TbP6 octahedra that share corners with six equivalent ZnP4 tetrahedra, edges with six equivalent TbP6 octahedra, and edges with six equivalent ZnP4 tetrahedra. All Tb–P bond lengths are 2.86 Å. There are two inequivalent Zn2+ sites. In the first Zn2+ site, Zn2+ is bonded in a trigonal planar geometry to three equivalent P3- atoms. All Zn–P bond lengths are 2.31 Å. In the second Zn2+ site, Zn2+ is bonded to four P3- atoms to form ZnP4 tetrahedra that share corners with three equivalent TbP6 octahedra, corners with seven equivalent ZnP4 tetrahedra, and edges with three equivalent TbP6 octahedra. The corner-sharing octahedral tilt angles are 15°. There are one shorter (2.42 Å) and three longer (2.48 Å) Zn–P bond lengths. There are two inequivalent P3- sites. In the first P3- site, P3- is bonded to three equivalent Tb3+ and three equivalent Zn2+ atoms to form PTb3Zn3 octahedra that share corners with three equivalent PTb3Zn3 octahedra, corners with three equivalent PZn5 trigonal bipyramids, and edges with nine equivalent PTb3Zn3 octahedra. The corner-sharing octahedral tilt angles are 0°. In the second P3- site, P3- is bonded to five Zn2+ atoms to form PZn5 trigonal bipyramids that share corners with six equivalent PTb3Zn3 octahedra and corners with six equivalent PZn5 trigonal bipyramids. The corner-sharing octahedral tilt angles are 69°.

36 MATERIALS SCIENCE↗

Materials Data on Tb(BRu)4 by Materials Project

Tb(RuB)4 crystallizes in the tetragonal I4_1/acd space group. The structure is three-dimensional. Tb3+ is bonded in a 12-coordinate geometry to twelve equivalent B3- atoms. There are a spread of Tb–B bond distances ranging from 2.95–3.23 Å. Ru+2.25+ is bonded to five equivalent B3- atoms to form a mixture of distorted corner and edge-sharing RuB5 trigonal bipyramids. There are a spread of Ru–B bond distances ranging from 2.15–2.30 Å. B3- is bonded in a 6-coordinate geometry to three equivalent Tb3+, five equivalent Ru+2.25+, and one B3- atom. The B–B bond length is 1.79 Å.

36 MATERIALS SCIENCE↗

Materials Data on Tb(Mo3Se4)2 by Materials Project

Tb(Mo3Se4)2 crystallizes in the trigonal R-3 space group. The structure is three-dimensional. Tb3+ is bonded in a body-centered cubic geometry to eight Se2- atoms. There are two shorter (2.83 Å) and six longer (3.11 Å) Tb–Se bond lengths. Mo+2.17+ is bonded to five Se2- atoms to form a mixture of corner and edge-sharing MoSe5 square pyramids. There are a spread of Mo–Se bond distances ranging from 2.54–2.75 Å. There are two inequivalent Se2- sites. In the first Se2- site, Se2- is bonded in a 1-coordinate geometry to one Tb3+ and three equivalent Mo+2.17+ atoms. In the second Se2- site, Se2- is bonded in a 5-coordinate geometry to one Tb3+ and four equivalent Mo+2.17+ atoms.

36 MATERIALS SCIENCE↗

Materials Data on Tb(NiSb)2 by Materials Project

Tb(NiSb)2 crystallizes in the tetragonal P4/nmm space group. The structure is three-dimensional. Tb3+ is bonded in a 8-coordinate geometry to eight Sb3- atoms. There are four shorter (3.33 Å) and four longer (3.42 Å) Tb–Sb bond lengths. There are two inequivalent Ni+1.50+ sites. In the first Ni+1.50+ site, Ni+1.50+ is bonded to four equivalent Sb3- atoms to form a mixture of edge and corner-sharing NiSb4 tetrahedra. All Ni–Sb bond lengths are 2.56 Å. In the second Ni+1.50+ site, Ni+1.50+ is bonded in a 5-coordinate geometry to five Sb3- atoms. There are one shorter (2.47 Å) and four longer (2.54 Å) Ni–Sb bond lengths. There are two inequivalent Sb3- sites. In the first Sb3- site, Sb3- is bonded in a 4-coordinate geometry to four equivalent Tb3+ and four equivalent Ni+1.50+ atoms. In the second Sb3- site, Sb3- is bonded in a 9-coordinate geometry to four equivalent Tb3+ and five Ni+1.50+ atoms.

36 MATERIALS SCIENCE↗

Materials Data on Tb(CuTe)3 by Materials Project

Tb(CuTe)3 crystallizes in the trigonal R-3 space group. The structure is three-dimensional. Tb3+ is bonded to six equivalent Te2- atoms to form TbTe6 octahedra that share corners with twelve equivalent CuTe4 tetrahedra, edges with three equivalent TbTe6 octahedra, and edges with six equivalent CuTe4 tetrahedra. All Tb–Te bond lengths are 3.08 Å. Cu1+ is bonded to four equivalent Te2- atoms to form CuTe4 tetrahedra that share corners with four equivalent TbTe6 octahedra, corners with six equivalent CuTe4 tetrahedra, edges with two equivalent TbTe6 octahedra, and edges with three equivalent CuTe4 tetrahedra. The corner-sharing octahedra tilt angles range from 17–59°. There are a spread of Cu–Te bond distances ranging from 2.58–2.68 Å. Te2- is bonded in a 6-coordinate geometry to two equivalent Tb3+ and four equivalent Cu1+ atoms.

36 MATERIALS SCIENCE↗

Materials Data on Tb(CuSe)3 by Materials Project

Tb(CuSe)3 crystallizes in the trigonal R-3 space group. The structure is three-dimensional. Tb3+ is bonded to six equivalent Se2- atoms to form TbSe6 octahedra that share corners with twelve equivalent CuSe4 tetrahedra, edges with three equivalent TbSe6 octahedra, and edges with six equivalent CuSe4 tetrahedra. All Tb–Se bond lengths are 2.87 Å. Cu1+ is bonded to four equivalent Se2- atoms to form CuSe4 tetrahedra that share corners with four equivalent TbSe6 octahedra, corners with six equivalent CuSe4 tetrahedra, edges with two equivalent TbSe6 octahedra, and edges with three equivalent CuSe4 tetrahedra. The corner-sharing octahedra tilt angles range from 16–57°. There are a spread of Cu–Se bond distances ranging from 2.43–2.50 Å. Se2- is bonded in a 6-coordinate geometry to two equivalent Tb3+ and four equivalent Cu1+ atoms.

36 MATERIALS SCIENCE↗