Chapter 6 Six-fold clock-anisotropy in honeycomb titanates ${\text{CoTiO}}_3$and ${\text{FeTiO}}_3$

${\text{CoTiO}}_3$has an Ilmenite crystal structure comprised of stacked layers of honeycombs with an $R\overline{3}$ space group. Figure 6.1a shows that the layers alternate between containing honeycombs of edge sharing ${\text{CoO}}_6$ octahedra and edge sharing ${\text{TiO}}_6$ octahedra. The honeycomb layers are not quite in the $𝒂$, $𝒃$ plane, but the two sites of the honeycomb are displaced relative to one another along the $𝒄$ axis, leading to some buckling of the planes.

Figure 6.1b shows an alternative setting for the structure, the rhombohedral setting. This setting is the primitive structural unit cell and contains two ${\text{Co}}^{2+}$sites, two ${\text{Ti}}^{4+}$ sites, and six ${\text{O}}^{2-}$ sites. The hexagonal basis vectors $(𝒂,𝒃,𝒄)$ can be written it terms of the primitive vectors $(𝑨,𝑩,𝑪)$ as

(6.1)

Previous studies have reported the crystal structure to a high quality standard [52, 75]. Table 6.1 shows a summary of the structural parameters and atomic fractional coordinates of the material’s ions, with data taken from [52].

In addition, previous reports [26, 27] pointed out commonly occurring twins in single crystals. These are the A domain, with parameters consistent with [52], shown in Table 6.1, and the B domain, which is rotated $180\mathrm{°}$ around the $(110)$ axis in the hexagonal setting. This rotation leaves the positions of the ${\text{Co}}^{2+}$and ${\text{Ti}}^{4+}$ ions unchanged, but does change the ${\text{O}}^{2-}$ ion positions. Because of this change in oxygen positions, one can determine which domains are present in a sample through X-ray diffraction. This is possible by finding Bragg peaks where the structure factor changes significantly in magnitude between the two domains. In this case, because the transition metal ion positions are invariant between the domains, this change in structure factor will be large if the contribution from the ${\text{O}}^{2-}$ ions changes sign between the two domains.

Figure 6.2 shows results of calculations of Bragg peak intensities in the $(hk1)$ crystallographic plane for domains A and B. On these are highlighted a trio of peaks whose intensity varies strongly between the domains. X-ray diffraction was performed at room temperature with an Oxford Diffraction Supernova diffractometer on all ${\text{CoTiO}}_3$crystals used in this chapter to verify sample quality. By comparing measured Bragg peak intensities to the calculated values, the domain populations can be estimated. Each crystal used for the torque measurements is an untwinned A-domain sample.

The magnetic ions in ${\text{CoTiO}}_3$are $3d^7$ ${\text{Co}}^{2+}$ions in an octahedron of oxygen ions, this octahedral crystal environment is trigonally distorted along the $𝒄$ direction. The Hund’s rules ground state for this electron configuration is $L=3$, $S=\frac{3}{2}$. The crystal field Hamiltonian can be written

$${\widehat{\mathscr{H}}}_{\text{CEF}}=\frac{2}{3}{B}_4\left(O_4^0+20\sqrt{2}{O}_4^3\right)+B_2^0{O}_2^0+B_4^0{O}_4^0$$

(6.2)

using the Stevens operators $O_n^k$ in the Cartesian basis with $x,y,z$ along the $a^{\ast },b,c$ axes respectively. $B_4>0$ parametrises the octahedral cubic crystal field environment, and the additional terms characterise a trigonal distortion along the $𝒄$ axis. With this, spin-orbit coupling is included to give a single-ion Hamiltonian

$$\widehat{\mathscr{H}}=-\lambda \widehat{𝑳}\cdot \widehat{𝑺}+{\widehat{\mathscr{H}}}_{\text{CEF}}$$

(6.3)

where $\lambda >0$ because the shell is more than half filled.

This can be approached perturbatively by assuming that the octahedral cubic crystal field (OCF)

$${\widehat{\mathscr{H}}}_{\text{OCF}}=\frac{2}{3}{B}_4\left(O_4^0+20\sqrt{2}{O}_4^3\right)$$

(6.4)

is the largest energy scale.

In the $|L_z⟩$ basis, the OCF gives a ground state $T_{1u}$ triplet at energy $-480B_4$, an excited $T_{2u}$ triplet at $+120B_4$, and a further excited excited $A_{2u}$ singlet at $+600B_4$ [1]. The ground state has basis vectors

	$|\tilde{0}⟩$	$=-\sqrt{{\displaystyle \frac{4}{9}}}|0⟩+\sqrt{{\displaystyle \frac{5}{9}}}{\displaystyle \frac{1}{\sqrt{2}}}\left[|+3⟩-|-3⟩\right]$		(6.5a)
	$|\pm \tilde{1}⟩$	$=\pm \sqrt{{\displaystyle \frac{5}{6}}}|\mp 2⟩+\sqrt{{\displaystyle \frac{1}{6}}}|\pm 1⟩$		(6.5b)

where the states are labelled by the effective angular momentum operator $|l_z⟩$ with the equivalence relation $\widehat{𝑳}\equiv -\frac{3}{2}\widehat{𝒍}$, valid on the projected subspace of this triplet of states. The spin-orbit and trigonal distortion terms can then be treated as a perturbation on this ground state manifold.

The spin-orbit coupling operator ${\widehat{\mathscr{H}}}_{\text{SO}}=-\lambda \widehat{𝑳}\cdot \widehat{𝑺}$ can be rewritten in terms of the effective orbital angular momentum operator ${\widehat{\mathscr{H}}}_{\text{SO}}=\frac{3}{2}\lambda \widehat{𝒍}\cdot \widehat{𝑺}$. By itself, this couples the spin and orbital parts of the angular momentum into a total effective angular momentum $J_{\text{eff}}$, with values $J_{\text{eff}}=1/2,3/2,5/2$.

The trigonal distortion operator can be written in terms of effective orbital angular momentum operator as ${\widehat{\mathscr{H}}}_{\text{trig}}=\delta \left({\widehat{l}}_z^2-\frac{2}{3}\right)=\frac{1}{3}\delta O_2^0$. Acting on the octahedral crystal field $l=1,S=3/2$ ground state, this splits the 12-fold degenerate manifold into an a 8-fold degenerate $l_z=\pm 1$ manifold of states and a 4-fold degenerate $l_z=0$ manifold. Acting as on the $J_{\text{eff}}$ states from spin-orbit coupling, it mixes the states with same $J_{\text{eff}}^z$, as seen in the level splitting diagram in Figure 6.3. The result is a ground state Kramers doublet with $J_{\text{eff}}^z=\pm 1/2$, and the next excited level a $J_{\text{eff}}^z=\pm 3/2$ doublet at $\approx 15\mathrm{meV}$. These levels are determined by diagonalising the perturbation Hamiltonian, treating the spin-orbit coupling and trigonal distortion at the same level. The effects of each of these terms on the energy levels can be seen in Figure 6.3.

The values of $\lambda$ and $\delta$ used were determined by previous reports by fitting the energy levels to an inelastic neutron scattering energy scan [26, 27].

${\text{CoTiO}}_3$has been shown previously to order at $38\mathrm{K}$ into an antiferromagnetic arrangement [38, 26, 27, 94, 6]. The structure consists of spins ferromagnetically aligned within each honeycomb layer, the layers alternate so that each layer has opposite spin alignment to the previous. The primitive magnetic unit cell has lattice vectors ${𝑴}_a,{𝑴}_b,{𝑴}_c$ which can be written in terms of the structural lattice vectors as

(6.6)

The primitive magnetic unit cell is a doubling of the primitive (rhombohedral) structural unit cell. The magnetic cell unit vectors $({𝑴}_a,{𝑴}_b,{𝑴}_c)$ are rotated relative to $(𝑨,𝑩,𝑪)$ by $\frac{\pi }{3}$ around the rhombohedral $(111)$ axis. The cell is also stretched along the same axis, doubling the unit cell volume relative to the primitive structural unit cell. In total there are four ${\text{Co}}^{2+}$sites in the magnetic unit cell.

Alternatively, Figure 6.4a) shows how the magnetic unit cell can be constructed in the hexagonal setting. In this setting, the magnetic unit cell is a simple doubling of the hexagonal structural unit cell along the $𝒄$ direction. This doubling occurs because there are three layers per structural unit cell in this setting; the layers alternate in magnetisation so six layers, or two structural cells, are needed for the magnetic structure to repeat.

No single-ion anisotropy is allowed in the ground state, because the ground state doublet is described by an effective $S=1/2$ and all even polynomials of $S_x^2,S_y^2,S_z^2$ are constants. The anisotropy effect, an energy preference of spins to point in particular directions over another, must be caused by exchange anisotropy. An inelastic neutron scattering study fit a minimal XXZ exchange model including nearest-neighbour intralayer exchange $J_1$ and next-nearest-neighbour interlayer Heisenberg exchanges $J_4$, $J_6$ [111]. These parameters can be seen in Table 6.2. By using these parameters in a classical model, which is solved as discussed in Chapter 3, the magnetic ground state described above is stabilised up to a spontaneous orientation of spins in the $𝒂𝒃$ plane.

The XXZ Hamiltonian as described has no dependence on the orientation of spins in the $𝒂𝒃$ plane. This is because it has a continuous $U(1)$ symmetry, higher than the discrete $\overline{3}$ point group symmetry of the material. If a $U(1)$ symmetry was realised, one would expect an associated gapless Goldstone mode in the excitation spectrum [30], whereas measurements observe a clear spin-gap [26]. The model developed by Elliot et.al [26] lifted this in-plane degeneracy by including a bond-dependent anisotropic exchange term on the nearest-neighbour honeycomb bonds through a parameter $\eta$ defined such that

$$\eta =J^y-J^x$$

(6.7)

in the the local coordinates of the each nearest-neighbour bond. In these coordinates, $𝒛$ is the $𝒄$ axis, $𝒙$ is the projection of the bond direction into the $𝒂𝒃$ plane, and $𝒚=𝒛\times 𝒙$. As such the bond’s Hamiltonian is written as

$${\widehat{\mathscr{H}}}_{ij}=J^x{S}_i^x{S}_j^x+(J^x+\eta )S_i^y{S}_j^y+J^z{S}_i^z{S}_j^z$$

(6.8)

when $\eta$ is non-zero, it introduced a dependence of the spin-wave spectrum on the azimuthal angle $\varphi$, defined as the angle between the ordered spin direction and the ${𝒂}^{\mathbf{\ast }}$ axis, as seen in Fig.6.4. As a consequence, the zero-point contribution to the ground state energy acquires a dependence on $\varphi$ and discrete spin orientations in-plane are selected, with a $60\mathrm{°}$ periodicity.

The classical, mean field calculation developed in this thesis cannot capture this effect. This can be demonstrated by explicitly computing the effective interaction between sublattices discussed in Chapter 3.

The most general bilinear exchange matrix can be written as

(6.9)

where $D$ terms parametrise a Dzyaloshinskii–Moriya (DM) interaction, and $\mathrm{\Gamma }$ terms parametrise symmetric off-diagonal exchange. In this coordinate system, $𝒛$ along the local 3-fold axis, $𝒙$ is the ${𝒂}^{\mathbf{\ast }}$ direction, and $𝒚=𝒛\times 𝒙$. For each bond not parallel to the $𝒛$ axis, there is another symmetry equivalent bond obtained by a $3_z^{+}$ rotation with exchange matrix

$${𝒥}^{\prime }={ℛ}_z\left(\frac{2\pi }{3}\right)𝒥{ℛ}_z\left(\frac{-2\pi }{3}\right)$$

(6.10)

where ${ℛ}_z\left(\pm \frac{2\pi }{3}\right)$ are matrix representations of the the 3-fold rotation about the $𝒛$ axis for spins in these coordinates. This has components given in Table 6.3.

In the magnetic structure of ${\text{CoTiO}}_3$, the spins in the honeycomb plane are all parallel, so the total exchange energy between sites ${𝑺}_A$ and ${𝑺}_B$ for these three bonds can be written as the sum

				(6.11a)
		$={𝑺}_A^T{𝒥}_T{𝑺}_B$		(6.11b)

where ${𝒥}_T$ can be written in terms of the single bond exchange

(6.12)

The off-diagonal terms show a DM contribution which also cancels. This is because all the bonds with multiplicity of three, such as nearest-neighbour bonds, have an inversion centre at their midpoint so $D_z=0$ by symmetry. Other bonds with a multiplicity of six, such as next-nearest neighbour, in these the other trio of bonds must have opposite $D_z$ such that the total is zero. The diagonal terms show no dependence on orientation in the $𝒂𝒃$ plane so at the mean field level, no bilinear exchange term can introduce an energy dependence on the alignment of spins in the $xy$ plane.

Calculations of the zero-point contribution to the ground state energy find a clear energy dependence on the spin orientation in the $𝒂𝒃$ plane of the form

$$E(\varphi )=\mathrm{\Lambda }\cos 6\varphi$$

(6.13)

where $\mathrm{\Lambda }\sim {\eta }^3$ to leading order [26]. This 6-fold periodicity is a requirement of the $\overline{3}$ point group symmetry of the crystal combined with time reversal. The energy of the system cannot change if the system is transformed under a symmetry operation of the Hamiltonian, for example if all spin operators are reversed.

In the mean field approximation, spin operators are replaced with their expectation values, which in shorthand can be written as . In order to preserve the $3$ point group site symmetry of the ${\text{Co}}^{2+}$ion, the basis functions must be unchanged under the symmetry operations of that group. Additionally, time reversal maps $𝑺\to -𝑺$, so the basis functions must also be unchanged under inversion. Up to 6^thorder polynomials of $𝗑,𝗒,𝗓$, this energy is

(6.14)

These functions are plotted over the unit sphere in Figure 6.5, labelled by their coefficient, which shows the negative energies (easy axes) in blue and positive energies (hard axes) in red. For spins in the $𝒂𝒃$ plane $𝗓=0$, so only the $E$ and $E^{\prime }$ functions are non-zero, and the anisotropy energy can be written

	$E_A$	$={\mathrm{\Lambda }}_0{\sin }^6\theta \cos 6(\varphi -{\varphi }_6)$		(6.15a)
		$=\mathrm{\Lambda }\cos 6(\varphi -{\varphi }_6)$		(6.15b)

with $\varphi$ the azimuthal angle measured from the ${𝒂}^{\mathbf{\ast }}$ axis, $\tan 6{\varphi }_6={\displaystyle \frac{E^{\prime }}{E}}$, and $|\mathrm{\Lambda }|=\sqrt{E^2+E^{\prime 2}}$. When ${\varphi }_6=0$, $\mathrm{\Lambda }>0$ the easy axes are at $\varphi =\frac{\pi }{2}+n\frac{\pi }{3}$ and change to $\varphi =n\frac{\pi }{3}$ for . This behaviour matches that of the quantum-fluctuation induced anisotropy energy in the presence of bond-dependent anisotropy $\eta$ [26].

It is worth highlighting that the $E$ and $E^{\prime }$ terms cannot come from single-ion anisotropy in ${\text{CoTiO}}_3$. This is because the quantum form of the single-ion anisotropy terms involves Stevens operators. In order to get the $E$ and $E^{\prime }$ terms, the $O_6^{\pm 6}$ operators are needed which have the form

	$O_6^{+6}$	$={\displaystyle \frac{1}{2}}\left[{\widehat{L}}_{+}^6+{\widehat{L}}_{-}^6\right]$		(6.16a)
		$={\widehat{L}}_x^6-15{\widehat{L}}_x^4{\widehat{L}}_y^2+15{\widehat{L}}_x^2{\widehat{L}}_y^4-{\widehat{L}}_y^6$		(6.16b)
	$O_6^{-6}$	$={\displaystyle \frac{1}{2i}}\left[{\widehat{L}}_{+}^6-{\widehat{L}}_{-}^6\right]$		(6.16c)
		$=6{\widehat{L}}_x^5{\widehat{L}}_y-20{\widehat{L}}_x^3{\widehat{L}}_y^3+6{\widehat{L}}_x{\widehat{L}}_y^5$		(6.16d)

These operators give no energy change for ${\text{Co}}^{2+}$ions. This is because the $L=3$ quantum state in ${\text{Co}}^{2+}$is made up of $d$ orbital electrons, so the highest order Stevens operator with a non-zero eigenvalue is $O_4^{\pm n}$, where $n$ can be 0, 2, or 4. The anisotropy being modelled originates from interactions; it is not a single-ion effect.

In order to determine the high temperature magnetic properties of ${\text{CoTiO}}_3$, the magnetic susceptibility of single crystals was measured using vibrating sample magnetometry. Figure 6.6a) shows the evolution of the magnetic susceptibility between $2\mathrm{K}$ and $300\mathrm{K}$, measured at $0.1\mathrm{T}$ for orthogonal field orientations relative to the crystallographic axes. The curves in green and blue show the susceptibility along the ${𝒂}^{\mathbf{\ast }}$ and $𝒃$ directions respectively, while the curve in red shows the susceptibility along the $𝒄$ direction. There is a remarkably strong anisotropy between the directions parallel and normal to the $𝒄$ axis, showing an overall easy-plane arrangement. There is a strong suppression of susceptibility for temperatures below  $38\mathrm{K}$ in the $𝒂𝒃$ plane, while the suppression along the normal direction is much smaller. This confirms that ${\text{CoTiO}}_3$orders into an antiferromagnet with a Néel temperature of $T_N=38\mathrm{K}$. There is no significant difference in susceptibilities for fields along the two directions in the $𝒂𝒃$ plane across the entire temperature range, suggesting that ${\text{CoTiO}}_3$behaves very close to uniaxially in its magnetic behaviour, i.e. the anisotropy in the $𝒂𝒃$ plane is very small.

Temperature dependent susceptibility data (Fig. 6.6) can be fitted to a Curie-Weiss relationship

$$\chi ={\chi }_0+\frac{C}{T+{\theta }_{\text{CW}}}$$

(6.17)

over the temperature range $50\mathrm{K}$-$200\mathrm{K}$.

The fitted parameters can be seen in Table 6.4. These data in some areas agree well with previously published data [38, 94, 103]: the low-temperature shapes of the curves and absolute scale of the susceptibilities, and the strong anisotropy of susceptibilities that is seen at all temperatures. The high-temperature data does not agree well with the previously published results, here a larger value of ${\chi }_{⟂}$ is observed than previously reported. This discrepancy is likely caused by a magnetic background from the brass sample holder, which has not been subtracted. Where susceptibility data is used for calibration in later sections, the reference data [38] is used. While this high temperature discrepancy may also account for disagreement between the fitted and published Curie-Weiss parameters [103], if the fitting process is repeated on more recently published data in [38], the fitted moment ${\mu }_{\text{eff}}$ agrees relatively well. This data gives ${\mu }_{\text{eff}}=$ $2.5{\mu }_B$, $7.3{\mu }_B$ along the $𝒄$ and $𝒂$ directions respectively. This is because a background term ${\chi }_0$ is captured in the fit. The poor fit of Curie-Weiss behaviour at high temperature is expected from the relatively small energy gap of $\approx 15\mathrm{meV}\approx 175\mathrm{K}$ to the next excited doublet. For temperatures greater than this value, the excited doublet states are populated and the CW treatment fails. A more thorough treatment including all of the energy levels predicted from the level splitting would capture this effect, but as the crystal field parameters and $\lambda$ have already been established, this is not necessary.

Figure 6.7a) shows the evolution of magnetisation with applied magnetic field up to $4\mathrm{T}$ for temperatures between $2\mathrm{K}$ and $40\mathrm{K}$. Above $T_N$, the relationship is linear. As the temperature falls below the Néel temperature, the linear relationship no longer holds. For fields below $B^{\ast }\approx 2.5\mathrm{T}$ and temperatures below $T_N$, the differential susceptibility is suppressed significantly, as can be seen in Fig 6.7b). This effect was reported previously [38], where the peak in $\frac{\partial M}{\partial H}$ at approximately $2.5\mathrm{T}$ was labelled $B^{\ast }$. This can be attributed to the in-plane anisotropy proposed previously, as magnetic domains exist in the material at low field, with ordered moments in the $𝒂𝒃$ plane. This suggests that a single domain is selected for fields ${\mu }_{0}H\gg B^{\ast }$.

As has been discussed in Chapter 2.2, torque magnetometry is a powerful tool for exploring the anisotropy of a material. In this section I discuss how this technique was used to study the in-plane anisotropy of ${\text{CoTiO}}_3$.

While I was able to cut large crystals of ${\text{CoTiO}}_3$into arbitrary shapes to align the field along the different crystallographic axes for the VSM experiments described above, the size of the cantilevers requires very small single crystals of order $\sim 100\mathrm{\mu }\mathrm{m}$ to be able to measure torque. To prepare these crystals I crushed small amounts of the larger ${\text{CoTiO}}_3$crystals in grease. This allowed small crystallites to be collected from the resulting material. As ${\text{CoTiO}}_3$is a layered material, it preferentially broke between layers, leading to large numbers of plate-like single crystals with a normal vector parallel to $𝒄$. An example of such a crystal can be seen in Figure 6.8a), which shows a plate-like single crystal of ${\text{CoTiO}}_3$with a diameter of $\sim 100\mathrm{\mu }\mathrm{m}$, mounted ready for torque measurements in the ${𝒂}^{\mathbf{\ast }}𝒄$ plane. This plate-like arrangement made mounting the crystal for torque in the $𝒂𝒃$ plane much more challenging, as the plane normal $𝒄$ had to be perpendicular to the lever surface normal. In order to measure the crystal in such an orientation, I mounted it on the side of the lever, as seen in Figure 6.8b), supported by a small quantity grease on the surface. This made precise alignment very difficult. In this image, the crystal is seen ‘edge-on’. As the in-plane susceptibility is much larger than the out-of-plane susceptibility ${\chi }_{⟂}\gg {\chi }_{\parallel }$ even at high temperature, subjecting the crystal to a large magnetic field causes a torque on the sample which would tend to improve the alignment of the crystal. That is, the energy is minimised if the field is in the $𝒂𝒃$ plane. This effect was used to improve the crystal alignment in the $𝒂𝒃$ plane by rotating the crystals in a $16\mathrm{T}$ field at $300\mathrm{K}$, such that the grease was a unfrozen. The crystal was then cooled in field to base temperature.

Unlike in the other systems studies in this thesis, I have been able to normalise the torque signal in ${\text{CoTiO}}_3$in absolute units. This is because of the availability of high quality published data of anisotropic susceptibilities along different crystal axes in absolute units, as discussed above. Above the Néel temperature, ${\text{CoTiO}}_3$behaves as a uniaxial paramagnet, so the torque at high temperatures can be calculated from the susceptibility data (Eqn. 2.36) using the susceptibility measured parallel to the $𝒄$ axis ${\chi }_{\parallel }$, and measured in the $𝒂𝒃$ plane ${\chi }_{⟂}$:

	$\tau (\theta ,T)$	$={\displaystyle \frac{1}{2}}{B}^2\left({\chi }_{⟂}(T)-{\chi }_{\parallel }(T)\right)\sin 2\theta$		(6.18a)
		$=-{\tau }_2(T)\sin 2\theta$		(6.18b)

where $\theta$ is the angle between the field and the $𝒄$ axis. As the anisotropic susceptibilities are known in absolute units, torque in the ${𝒂}^{\mathbf{\ast }}𝒄$ plane can be measured for a large temperature range above the $T_N$ and the results compared to the predicted values to determine the conversion factor between the measured signal and the physical torque, which will capture both the sensitivity of the lever and the mass of the sample.

Figure 6.9 shows measured magnetic torque for rotations in the ${𝒂}^{\mathbf{\ast }}𝒄$ plane at $1\mathrm{T}$ and temperatures between $10\mathrm{K}$ and $200\mathrm{K}$. These show an overall $\sin 2\theta$ dependence, with a positive gradient when fields are parallel to the $𝒄$ axis and a negative gradient when fields are parallel to the ${𝒂}^{\mathbf{\ast }}$ axis at all temperatures, indicating a smaller susceptibility in-plane than out of plane. This is in agreement with easy-plane type antiferromagnet model determined from the VSM data reported previously. These piezocantilever voltages, in units of $\mathrm{mV}$ were fitted to the form

$$V=-V_0\sin 2(\theta -{\theta }_0)$$

(6.19)

and the values $V_0$ compared to the predicted values

$${\tau }_2(T)=-\frac{1}{2}{B}^2\left({\chi }_{⟂}(T)-{\chi }_{\parallel }(T)\right)$$

(6.20)

for temperatures $T>50\mathrm{K}$. Figure 6.10 shows the plot of $V_0$ vs ${\tau }_2$ for all temperatures measured, the green line shows a linear fit to these data. The dashed red line shows a line with the same gradient and zero intercept. This calibrated scaling factor comes to

$$\tau =V\cdot 5.938(10)\times {10}^{-3}\frac{{\mathrm{meVrad}}^{-1}/{\text{Co}}^{2+}}{\mathrm{mV}}$$

(6.21)

which can be used to calibrate all other measurements on the same sample-lever pair. It should be noted that these samples are too small for their mass to be measured using standard equipment, so this calibration factor is not able to calibrate other samples on the same lever. For this reason, all torque data reported in this chapter has been measured using the same calibrated sample shown in Figure 6.8. Figure 6.10 shows how this calibration was verified by comparing the temperature dependence of the experimentally observed torque amplitudes, scaled by this calibration factor, against the predicted temperature dependence for the paramagnetic phase from (6.20). These show good agreement over the high temperature region where (6.20) is valid, indicating that this calibration procedure was effective.

Figure 6.11a) shows the evolution of magnetic torque with field in the $𝒂𝒃$ plane, measured at the experimental base temperature of $2\mathrm{K}$, far below the ordering temperature for this material. These torque profiles show complex behaviour at all fields, unlike the simple $\sin 2\theta$ behaviour seen in the ${𝒂}^{\mathbf{\ast }}𝒄$ orientations. Despite the complexity, six peaks and six troughs can be clearly seen in all fields in the $\varphi =0\mathrm{°}\to 360\mathrm{°}$ angle range. This suggests a significant contribution from a six-fold energy term, however the signal also contains other components. Some of these will be contributed by the lever background, as the resistance of the bridge will vary slightly with applied magnetic field. Others will be caused by a small projection of the $𝒄$ axis onto the rotation plane, as the anisotropy between the $𝒂𝒃$ plane and the $𝒄$ axis is much stronger than any anisotropy within the $𝒂𝒃$ plane, only a small projection of this axis is needed for the signal to be significant compared to the very small contribution from in-plane anisotropy.

In order to better show the six-fold contribution to the torque signal, I have performed a simple background subtraction by removing low order Fourier components. To do this I fit the signal to the form

	$\tau$	$=C_0+C_1\varphi +{\displaystyle \sum _{n=1}^m}{A}_n\sin n\varphi +B_n\cos n\varphi$		(6.22a)
		$=C_0+C_1\varphi +{\displaystyle \sum _{n=1}^m}{\tau }_n\sin n(\varphi -{\varphi }_n)$		(6.22b)

where all $A_n,B_n,C_n$ are free fitting parameters, and $m=12$ is chosen to be sufficiently large that increasing it does not significantly improve the fit. Here ${\tau }_n^2=A_n^2+B_n^2$, and $\tan n{\varphi }_n=\frac{A_n}{B_n}$. This form is motivated from the experimental design: $C_0$ is non-zero if the electronic Wheatstone bridge balancing is not perfect. $C_1$ can account for a linear drift in the lever balancing over the course of the experiment, which occurs as the temperature stability is not ideal. The choice of the Fourier series is because the rotation is a physical rotation of the levers in the magnetic field. Physically, the torque applied to the lever must be periodic in $\varphi$, unless it is hysteretic. I then take the fitted curve with parameters as the ‘background’ to be removed, which is subtracted from the raw signal. Note that Fourier components $n=1,3,4$ are very small, but above the noise level, likely coming from non-linearities and background in the lever assembly itself.

Figure 6.11b) shows the same signal measured at $2\mathrm{K}$ with the background subtracted as described above. This shows an overall pattern of a six-fold torque signal, which increases in magnitude as the strength of the applied magnetic field increases from $1\mathrm{T}$ to $16\mathrm{T}$. Additionally the phase and amplitude of the signal is modulated with the magnetic field strength. In order to explore these effects, Figure 6.12 shows the evolution of the fitted parameters in Eqn. (6.22a) as a function of applied magnetic field. This shows that the six-fold components are comparable in magnitude to the two-fold components. Additionally it shows the modulation of the $A_6$ and $B_6$ components, which increase in magnitude with applied magnetic field and change sign at approximately equal magnetic fields. This effect can be more clearly seen in Figure 6.13, which shows select torque traces, with background subtracted, by themselves. These clearly show the change in sign of the signal as the applied magnetic field increases.

I have drawn a vertical line at $0\mathrm{°}$ on Figure 6.11b) to indicate where the magnetic field is approximately parallel to the ${𝒂}^{\mathbf{\ast }}$ axis. This is close to the zeros of the six-fold torque signal. In low fields, below $\approx 4\mathrm{T}$, the gradient through this point is positive, indicating an unstable equilibrium. At large fields (above $\approx 14\mathrm{T}$), the gradient is negative indicating a stable equilibrium. At intermediate fields the gradient changes three times, at $4\mathrm{T}$, $10\mathrm{T}$, and $14\mathrm{T}$. The gradient is negative between $\approx 4\mathrm{T}$ and $\approx 10\mathrm{T}$ and positive between $\approx 10\mathrm{T}$ and $\approx 14\mathrm{T}$.

These changes can be naturally explained by considering the changes in the anisotropy energy as the material becomes magnetised. At the largest field of $16\mathrm{T}$ in-plane the magnetisation is almost saturated; the spins are approximately parallel with the magnetic field. An overall stable equilibrium with the field along ${𝒂}^{\mathbf{\ast }}$ suggests that the local anisotropy for the spins has a stable equilibrium at the same position. In small magnetic fields below $\approx 4\mathrm{T}$ the spins are approximately antiparallel, and orthogonal to the field direction in order to minimise energy. With $𝑩\parallel {𝒂}^{\mathbf{\ast }}$, this constrains the spins to be collinear with $𝒃$ ($⟂{𝒂}^{\mathbf{\ast }}$). The overall system is in an unstable equilibrium, indicating that spins along this orientation is an energy maximum, determined by the positive torque gradient $\frac{\partial \tau }{\partial \varphi }>0$ when the $𝑩\parallel {𝒂}^{\mathbf{\ast }}$ (see Fig. 6.13). The intermediate field behaviour can be understood as caused by the modulations in local energy of the spins as they rotate towards the field. I illustrate this is Figure 6.14, which shows the evolution in the anisotropy energy as field is increased along the ${𝒂}^{\mathbf{\ast }}$ direction. With the magnetic field below $\sin 15\mathrm{°}\cdot B_{sat}$, the system is in an unstable equilibrium. For the system is stable and so on. The numbers on the x-axis indicate the fields when these switches will occur for a saturation field of $16\mathrm{T}$. These fields are overlaid as vertical lines on Figure 6.12, which show good agreement with the minima in the fitted amplitudes of the ${\tau }_6$ components.

From this model and the data in Figure 6.11, I can determine the anisotropy parameters of interest. In particular, at $16\mathrm{T}$, the angular dependence of torque can be fit to the form

$$\tau (\varphi )={\tau }_6\sin 6(\varphi -{\varphi }_6)$$

(6.23)

with ${\tau }_6=-8.5(2)\times {10}^{-4}{\mathrm{meVrad}}^{-1}/C{o}^{2+}$ and ${\varphi }_6\approx +2\mathrm{°}$. Note that the absence of mirror planes containing the $𝒄$-axis means that ${\varphi }_6$ is not symmetry constrained and could in principle take any value. As the material should be (approximately) fully polarised at this magnetic field strength, this torque should be coming from the full scale of the anisotropy energy

	$E_A$	$=\mathrm{\Lambda }\cos 6(\varphi -{\varphi }_6)$		(6.24)
	$⟹{\tau }_6$	$=6\mathrm{\Lambda }\sin 6(\varphi -{\varphi }_6)$		(6.25)

so the local anisotropy energy can be read off simply as $\mathrm{\Lambda }=-1.4(1)\times {10}^{-4}\mathrm{meV}/{\text{Co}}^{2+}$.

The model for a quantum-fluctuation induced six-fold clock anisotropy [26] suggests that at leading order

$$\mathrm{\Lambda }=\frac{1}{8}a{\eta }^3$$

(6.26)

where $a$ is a constant estimated at $a=6.13(7)\times {10}^{-5}{\mathrm{meV}}^{-2}$ and $\eta$, defined in Eqn. (6.7), $\approx -1.7\mathrm{meV}$, suggesting a value of $\mathrm{\Lambda }\approxeq -3.8\times {10}^{-5}\mathrm{meV}$. This is of the same order of magnitude as the value estimated from torque.

Practical considerations with the mounting mean that precisely aligning crystallographic axes within the plane is not possible, so the phase ${\varphi }_6$ should not be taken to be a precise offset. However, it is clear from the data that the stable equilibrium is close to the ${𝒂}^{\mathbf{\ast }}$ axis and the unstable equilibrium close to the $𝒃$ axis, tentatively placed within $5\mathrm{°}$ of these. In the quantum fluctuations model, the negative favoured spins parallel to the bond direction, which is parallel to ${𝒂}^{\mathbf{\ast }}$. Together with the order of magnitude quantitative agreement of the size of the magnetic anisotropy this agreement on the locations of the in-plane easy axes provides strong support for the interpretation that quantum fluctuations are the origin of the anisotropy.

Figure 6.15 shows a numerical calculation for torque in the $𝒂𝒃$ plane for fields between $2\mathrm{T}$ and $16\mathrm{T}$ with the anisotropy energy taken as above. This calculation captures the modulations in torque strength and phase change with magnetic field strength. The calculation predicts a larger magnitude of torque at small fields than is measured, this is likely a combination of two main factors. It is clear from the suppression of susceptibility in the $𝒂𝒃$ plane that ${\text{CoTiO}}_3$has magnetic domains which remain in small magnetic fields, the pinning of these domains is not modelled by the calculation and their presence is likely to suppress the energy modulation as the field is rotated. If the anisotropy amplitude $\mathrm{\Lambda }$ originates from zero-point quantum fluctuations, one would expect a variation with field strength. Most straightforwardly the quantum fluctuations would be expected to be suppressed at high field, inconsistent with what is seen experimentally. It should be noted however that the energy scale here is $g{\mu }_{B}B\ll J^x$, so quantum fluctuations should still be significant.

The magnetic ions in ${\text{FeTiO}}_3$are $3d^6$ ${\text{Fe}}^{2+}$ions in an octahedral crystal environment with trigonal distortion. The Hund’s rules ground state for this electronic configuration is $L=2$, $S=2$. The octahedral crystal field Hamiltonian can be written

$${\widehat{\mathscr{H}}}_{\text{CEF}}=-\frac{2}{3}{B}_4\left(O_4^0+20\sqrt{2}{O}_4^3\right)+B_2^0{O}_2^0+B_4^0{O}_4^0$$

(6.27)

using the Stevens operators $O_n^k$ in the Cartesian basis with $x,y,z$ along the ${𝒂}^{\mathbf{\ast }},𝒃,𝒄$ axes respectively [1]. $B_4>0$ parametrises the octahedral cubic crystal field environment, and the additional terms characterise a trigonal distortion along the $𝒄$ axis. With this, spin-orbit coupling is included to give a single-ion Hamiltonian

$$\widehat{\mathscr{H}}=-\lambda \widehat{𝑳}\cdot \widehat{𝑺}+{\widehat{\mathscr{H}}}_{\text{CEF}}$$

(6.28)

where $\lambda >0$ because the shell is more than half filled.

This can be approached perturbatively by assuming that the cubic octahedral crystal field ${\widehat{\mathscr{H}}}_{\text{OCF}}$ is the largest energy scale in $\widehat{\mathscr{H}}$.

In the $|L_z⟩$ basis, the octahedral field gives a ground state $T_{2g}$ triplet at energy $-48B_4$, and an excited $E_g$ doublet at $+72B_4$ [1]. The ground state has basis vectors

	$|\tilde{0}⟩$	$=|0⟩$		(6.29a)
	$|\pm \tilde{1}⟩$	$=\pm \sqrt{{\displaystyle \frac{2}{3}}}|\mp 2⟩-\sqrt{{\displaystyle \frac{1}{3}}}|\pm 1⟩$		(6.29b)

where the states are labelled by the effective angular momentum operator $|l_z⟩$ ($z$ along the trigonal $𝒄$ axis) with the equivalence relation $\widehat{𝑳}\equiv -\widehat{𝒍}$, valid on the projected subspace of this triplet of states. The spin-orbit and trigonal distortion terms can then be treated as a perturbation on this ground state manifold.

The spin-orbit coupling operator ${\widehat{\mathscr{H}}}_{\text{SO}}=-\lambda \widehat{𝑳}\cdot \widehat{𝑺}$ can be rewritten in terms of the effective orbital angular momentum operator ${\widehat{\mathscr{H}}}_{\text{SO}}=\lambda \widehat{𝒍}\cdot \widehat{𝑺}$. By itself, this couples the spin and orbital parts of the angular momentum into a total effective angular momentum $J_{\text{eff}}$, with values $J_{\text{eff}}=1,2,3$.

The trigonal distortion operator can be written in terms of the effective orbital angular momentum operator as ${\widehat{\mathscr{H}}}_{\text{trig}}=\delta \left({\widehat{l}}_z^2-\frac{2}{3}\right)=\frac{1}{3}\delta O_2^0$. Acting on the 15-fold degenerate octahedral crystal field $\widehat{l}=1,S=2$ ground state manifold, this splits the state into an a 10-fold degenerate ${\widehat{l}}_z=\pm 1,S=2$ manifold and a 5-fold degenerate $l_z=0,S=2$ manifold. Acting on the $J_{\text{eff}}$ states from spin-orbit coupling, it mixes the states with same $J_{\text{eff}}^z$, as seen in the level splitting diagram in Figure 6.17. The result is a ground state doublet with $J_{\text{eff}}^z=\pm 1$, and the next excited level a $J_{\text{eff}}^z=0$ singlet. These levels are determined by diagonalising the perturbation Hamiltonian, treating the spin-orbit coupling and trigonal distortion together. The effects of each of these terms on the energy levels can be seen in Figure 6.17.

In the presence of this trigonal distortion, as appears to be the experimental case [27], the ground state is the easy-axis situation $J_{\text{eff}}^z=\pm 1$, with next excited $J_{\text{eff}}^z=0$ singlet at $3\mathrm{meV}$. By applying a small field to split the ground state doublet, the moment along the $z$ axis can be determined as ${\mu }_z=4.43{\mu }_B$. In qafm, the classical spin is taken as $J_{\text{eff}}=1$ with a single-ion anisotropy term $A=-D{(J_{\text{eff}}^z)}^2$, where $D>0$ because the system is easy-axis. The energy $D$ can be read off as as the gap between the ground state doublet and excited singlet as $D\approx 3\mathrm{meV}$, if the values from [27] are used. This easy-axis situation is confirmed by the metamagnetic spin-flip transition seen at $8\mathrm{T}$ for fields along the $𝒄$ axis. These transitions are only present for large $-D{(J_{\text{eff}}^z)}^2$-type anisotropy, while XXZ easy-axis antiferromagnetism will always favour a spin-flop transition.

Strictly, the above single-ion anisotropy of a octahedral cubic environment with trigonal distortion assumes a $\overline{3}m$ point group for the ${\text{Fe}}^{2+}$ion. The full symmetry allowed anisotropy for the $3$ point group is however [48, 74]

$$B_2^0{O}_2^0+B_4^0{O}_4^0+B_4^{-3}{O}_4^{-3}+B_4^3{O}_4^3$$

(6.30)

Piezoresistive cantilever torque magnetometry was performed on single crystals of ${\text{FeTiO}}_3$in order to explore the anisotropy of the material. A photo of the sample reported throughout this section can be seen in Figure 6.18.

Figure 6.19 shows how the torque profiles in the ${𝒂}^{\mathbf{\ast }}𝒄$ plane change with temperature through the magnetic phase transition. At all temperatures the profile is a good fit to

$$\tau (\theta )={\tau }_2\sin 2(\theta -{\theta }_0)$$

(6.31)

where $\theta$ is measured from the $𝒄$ axis towards the $a^{\ast }$ axis and ${\theta }_0$ is interpreted as an offset between the nominal $𝒄$ orientation and the true orientation. In the high-temperature paramagnetic region, the gradient through the $𝒄$ axis is negative, indicating that it is a stable equilibrium. This is expected as the susceptibility measured along the $𝒄$ axis ${\chi }_{\parallel }$ is larger than the in-plane susceptibility ${\chi }_{⟂}$.

Figure 6.20 shows the plot of fitted piezocantilever voltage amplitude $V_2(T)$ against predicted torque $\frac{1}{2}{B}^2\left({\chi }_{\parallel }(T)-{\chi }_{⟂}(T)\right)$ for temperature range , the green line shows a linear fit to these data. The dashed red line shows a line with the same gradient and zero intercept, this fit was used to calibrate the absolute value of the torque as described in the previous section. At the phase transition the difference ${\chi }_{\parallel }-{\chi }_{⟂}$ is maximised, which is seen as a peak in the amplitude in the torque at $58\mathrm{K}$ in Figure 6.20b). As the temperature cools further below the transition the torque amplitude drops as the out-of-plane susceptibility is suppressed by the antiferromagnetic order. At approximately $39\mathrm{K}$ the torque in minimised at ${\chi }_{⟂}\approx {\chi }_{\parallel }$. This temperature is lower than the $46\mathrm{K}$ expected from the susceptibility data. The most likely cause of this disagreement is local sample heating; as domain reorientation while the sample is rotated in field could cause the sample to be heated above the bath temperature. At lower temperatures than this the stability of the axes changes, such that the system in unstable for fields close to $𝒄$, and stable for fields in the plane. This is because the out-of-plane susceptibility ${\chi }_{\parallel }$ has been suppressed below the in-plane value ${\chi }_{⟂}$. The evolution of this torque in the paramagnetic phase $T>T_N$ was used to calibrate the magnitude of the torque for these crystals by comparing the measured torque to the predicted value from $\frac{1}{2}{B}^2({\chi }_{⟂}-{\chi }_{\parallel })\sin 2\theta$ as was described in the previous section, all further results in this chapter are on the same lever sample pair, although the results were verified on a second sample.

In order to explore the existence of in-plane anisotropy in ${\text{FeTiO}}_3$, angular dependence of torque was measured for rotations in the $𝒂𝒃$ plane. Figure 6.21 shows how the magnetic torque measured in the $𝒂𝒃$ plane varies with applied magnetic field at $2\mathrm{K}$. A dashed vertical line indicates where field is parallel to the ${𝒂}^{\mathbf{\ast }}$ direction. These show a very clear 6-fold torque signal such that the system is in a stable equilibrium for fields close to the ${𝒂}^{\mathbf{\ast }}$ direction and an unstable equilibrium for fields close to $𝒃$. This suggests that the local energy environment for the spins in minimised for spins parallel to ${𝒂}^{\mathbf{\ast }}$, that is parallel to the bond directions. This matches the case for ${\text{CoTiO}}_3$. Unlike in the case for ${\text{CoTiO}}_3$, this six-fold torque signal shows no change in the phase of the signal as field increases, and the magnitude increases monotonically. This is because in ${\text{FeTiO}}_3$, the spins are canting from the $𝒄$ axis, so their projection in the $𝒂𝒃$ plane is directly towards the magnetic field, it is not dependent on field magnitude.

The largest field I was able to test with this setup was $6\mathrm{T}$, this is because the growing torques became a danger of breaking the magnetometer levers. At this field, the amplitude of the 6-fold component of the torque is ${\tau }_6=4.2\times {10}^{-3}{\mathrm{meVrad}}^{-1}/F{e}^{2+}$, approximately 5 times larger than the largest value measured in ${\text{CoTiO}}_3$. Although the moments do not lie in-plane, so the full scale of the anisotropy is not exerting a torque on the sample, the observed size of the ${\tau }_6$ component of the torque allows a lower bound to be placed that $|{\mathrm{\Lambda }}_0|>7\times {10}^{-4}\mathrm{meV}/F{e}^{2+}$. However, it should be noted that the value for the energy scale in ${\text{CoTiO}}_3$was determined at $16\mathrm{T}$, when the magnetisation was nearly saturated. At $6\mathrm{T}$, far below the saturation field of the material, only a small part of the anisotropy energy will be seen, suggesting the six-fold clock anisotropy energy in ${\text{FeTiO}}_3$is much larger than in ${\text{CoTiO}}_3$.

The field scaling of this torque parameter was determined by plotting the amplitude of the six-fold torque ${\tau }_6$ against the magnetic field, as can be seen in Figure 6.22. These amplitudes were fitted to the form

$${\tau }_n\sim {({\mu }_{0}H)}^{{\alpha }_n}$$

(6.32)

with the curves for the fitted parameters overlaid on the figure. The exponent for the six-fold anisotropy ${\alpha }_6\approx 3.5$. If the origin of the anisotropy was single-ion, then it should have the form from Eqn. (6.15)

	$E_A$	$={\mathrm{\Lambda }}_0{\sin }^6\theta \cos 6(\varphi -{\varphi }_6)$		(6.33a)
		$={\mathrm{\Lambda }}_0{\left({\displaystyle \frac{M_{xy}}{M_{sat}}}\right)}^6\cos 6(\varphi -{\varphi }_6)$		(6.33b)

so a value of ${\alpha }_6=6$ would be expected as $M_{xy}={\chi }_{⟂}{\mu }_0{H}_{xy}$. This is larger than the value found experimentally, providing further support that the anisotropy does not originate in single-ion effects.

Figure 6.21b) shows the results of a calculation, performed as in the case of ${\text{CoTiO}}_3$, with exchange parameters taken from [27]. To this I added an anisotropy energy of the form in Equation 6.14, with , such that individual spins minimise their energy by pointing along nearest-neighbour bond directions. These show qualitative agreement with the data, with a six-fold contribution to the torque of the same sign and phase as is seen in the data. The calculations results also predict a torque that increases with the magnetic field faster than seen experimentally with exponent ${\alpha }_n=6$, larger than the fitted value of 3.5.

Figure 6.23 shows the temperature evolution of the magnetic torque measured in the $𝒂𝒃$ plane measured at $6\mathrm{T}$, between $100\mathrm{K}$ and $2\mathrm{K}$. In the high temperature regime, a 2-fold torque signal can be seen. This 2-fold signal is symmetry disallowed for the $𝒂𝒃$ plane because of the rhombohedral symmetry of the crystal. For this reason I attribute this signal primarily to a small misalignment, such that the rotation plane is skewed from the $𝒂𝒃$ plane. Fits to the 2-fold component of the signal seen in Figure 6.23b) show a peak, associated with the onset of magnetic order. This can be understood as a projection of this much larger easy-axis anisotropy into the measurement plane. As the temperature cools below the transition, a large 6-fold component of torque can be seen, increasing monotonically as temperature decreases. Figure 6.23b) shows the evolution with temperature of the fitted components of the torque signal. As can be seen, the six-fold component can be detected only below the magnetic phase transition.

The lowest order single-ion anisotropy term which could induce a $\cos 6\varphi$ energy profile is the $O_6^{\pm 6}$ Stevens operator. The $d$-orbital valence electrons of ${\text{Fe}}^{2+}$do not permit any high order $O_n^q,n>4$ terms in the CEF Hamiltonian. Because of this, the 6-fold torque profile seen at low temperatures in ${\text{FeTiO}}_3$cannot have a single-ion anisotropy origin. This is further evidenced by the temperature dependence of the 6-fold component of the torque, which has onset alongside magnetic order, as can be seen in Figure 6.23. From this evidence I suggest that the anisotropy seen must originate from magnetic interactions and set in alongside the onset of the magnetic order, through quantum fluctuations inducing an angular dependence of the anisotropy energy or a spin-orbital exchange mechanism as proposed in ${\text{CoTiO}}_3$[26].