Magnetic anisotropy in the frustrated spin chain compound $\beta$-TeVO₄

All calculations were performed for the experimental crystal structure of $\beta$-TeVO₄ determined at 10 K (Table 1). Band structure calculated on the LDA level is shown in Fig. 12. Its apparent metallicity is related to the fact that LDA does not capture effects of strong electronic correlations pertinent to the $3d$ shell of V^4+ 111Note that in DFT+$U$ we reproduce the insulating state with a band gap of about 3.0 eV.. We find predominantly V $3d$ states at the Fermi level. Crystal-field effects split the V $3d$ states into two narrow band complexes that can be ascribed ²²2Here, we used the coordinate frame with the $z$-axis directed along the short V–O bond in the axial position of the VO₅ square pyramids. The $x$ and $y$ axes are directed roughly toward the oxygen atoms in the basal plane of the pyramid, subject to a condition that $x$ and $y$ are perpendicular to $z$. to the $d_{xy}$ and $d_{yz}+d_{xz}$ orbitals around 0 eV and 1 eV, respectively. Above 1.2 eV, the spectrum is dominated by the $d_{x^{2}-y^{2}}+d_{3z^{2}-r^{2}}$ orbitals that strongly hybridize with Te $5p$-orbitals above 2.5 eV. The $d_{xy}$-states have the lowest energy, in agreement with the crystal-field splitting expected for the 5-fold oxygen coordination in the square pyramid.

The LDA band structure was analyzed using a 12-band tight-binding model, where we included all $d_{xy}$, $d_{yz}$, and $d_{xz}$ bands lying between $-0.3$ eV and $1.0$ eV. The hopping parameters $t$ are then used to calculate individual exchange integrals according to the Kugel-Khomskii model Mazurenko et al. (2006); Tsirlin et al. (2011a):

where $\varepsilon_{\alpha}$ are orbital energies, $\Delta_{\alpha}=\varepsilon_{\alpha}-\varepsilon_{xy}$ are crystal-field splittings, and $U_{\rm eff}=4$ eV and $J_{\rm eff}=1$ eV are the effective Coulomb repulsion and Hund’s exchange in the V $3d$ shell Tsirlin and Rosner (2011); Tsirlin et al. (2011b). The higher-lying $d$-orbitals ($\alpha=3z^{2}-r^{2},x^{2}-y^{2}$) were excluded because of their strong mixing with the Te $5p$-states that renders the fitting procedure somewhat ambiguous. We checked, however, that different fits, where the higher-lying orbitals were included, produce similar results and do not influence any of the conclusions drawn below.

In Table 2, we list both FM and AFM contributions to the exchange corresponding to the second and first terms of Eq. (2), respectively. Alternatively, we evaluated exchange couplings by calculating total energies of several collinear spin configurations using DFT+$U$. Here, we used the generalized gradient approximation (GGA) Perdew et al. (1996), the on-site Coulomb repulsion $U_{d}=4$ eV, the on-site Hund’s exchange $J_{d}=1$ eV, and the atomic-limit version of the double-counting correction term. We have also checked that a different double-counting correction scheme or different values of $U_{d}$ lead to marginal changes in the exchange couplings without altering the resulting magnetic model.

Our results for the isotropic exchange couplings are in line with those from Ref. Saúl and Radtke, 2014. The two leading interactions are FM $J_{1}$ and AFM $J_{2}$ forming frustrated spin chains along the $c$ direction. The strongest interchain coupling $J_{b2}$ connects the chains in the $bc$ plane, within the structural layers. The coupling between the layers is facilitated by FM $J_{a}$, $J_{a1}$, and $J_{a2}$. Other couplings along the $a$ direction are below 0.2 K (by absolute value), and the couplings in the $bc$ plane beyond $J_{b2}$ are 2.0 K or less. Therefore, the magnetic model with $J_{1}$ and $J_{2}$ as intrachain couplings and $J_{b2}$, $J_{a}$, $J_{a1}$, and $J_{a2}$ as interchain couplings provides an exhaustive microscopic description of $\beta$-TeVO₄ on the isotropic (Heisenberg) level.

The microscopic origin of these couplings can be understood as follows. The coupling $J_{1}$ involves only one oxygen atom and corresponds to the V–O–V pathway with the bridging angle of $133.6^{\circ}$. While Goodenough-Kanamori-Anderson rules prescribe that such a coupling should be AFM, its AFM part is in fact negligible, and the FM part dominates. This is in line with the results of Ref. Saúl and Radtke, 2014 and may be related to the enhanced hybridization between V $3d$ and Te $5p$ states. Te $5p$ orbitals contribute 6.5 % of states in the vicinity of the Fermi level, which is comparable to 9.7 % contributed by O $2p$ (see also Fig. 12). A somewhat similar microscopic scenario has been reported for CdVO₃ Tsirlin et al. (2011a), where $5s$ orbitals of Cd admix to the V $3d$ states and trigger ferromagnetic exchange couplings that give rise to the overall ferromagnetic long-range order, a very rare case among V⁴⁺ oxides.

The couplings beyond $J_{1}$ are long-range. Their mechanism is usually understood as V–O$\ldots$O–V superexchange controlled by the V–O$\ldots$O angles defining the linearity of the superexchange pathway, and by the O$\ldots$O distance. Indeed, the larger values of $J_{2}$ and $J_{b2}$ can be ascribed to the shortest O$\ldots$O distances of $2.82-2.84$ Å and 2.87 Å, respectively. In contrast, the longer O$\ldots$O distance of 3.31 Å disfavors the interchain coupling $J_{b1}$, even though its V–V distance is 0.55 Å shorter than those of $J_{2}$ and $J_{b2}$. Finally, the weakly FM nature of $J_{a}$, $J_{a1}$, and $J_{a2}$ is typical for interactions in the direction perpendicular to basal planes of VO₅ pyramids, as in Pb₂V₃O₉ Tsirlin and Rosner (2011) and Zn₂VO(PO${}_{4})_{2}$ Yusuf et al. (2010). Here, no suitable V–O$\ldots$O–V pathway for an efficient $xy\rightarrow xy$ hopping can be formed, hence the AFM contribution is very small. On the other hand, the V–V distance is short enough to induce non-zero $xy\rightarrow yz$ and $xy\rightarrow xz$ hoppings resulting in weakly FM superexchange.

In $\beta$-TeVO₄, frustrated interactions $J_{1}$ and $J_{2}$ along the spin chains manifest themselves already in thermodynamic properties. In Fig. 13, we compare both temperature dependence of the magnetic susceptibility and field dependence of the magnetization with two models: i) uniform Heisenberg chain (HC) proposed in Ref. Savina et al., 2011; and ii) $J_{1}-J_{2}$ frustrated Heisenberg chain (FHC) supported by our calculations. Both models provide good fits of the magnetic susceptibility resulting in $J=20.2$ K, $g=2.09$ for the HC and $J_{2}=-J_{1}=26.4$ K, $g=2.00$ for the FHC. However, a closer examination of the difference between the simulated and experimental curves $\left(\Delta\chi=(\chi_{\rm calc}-\chi_{\rm exp})^{2}/\chi_{\rm exp}^{2}\right)$ reveals better agreement for the FHC model (Fig. 13, top inset). This model also provides a better description of the high-field magnetization curve, although the saturation field is slightly overestimated. Finally, the fitted $g$-value for the FHC model is in good agreement with $g=2.01$ measured by electron spin resonance Pregelj et al. (2015), while the fitted $g$-value for the HC model would be too high.

We can now rationalize the incommensurate magnetic order that has been observed in $\beta$-TeVO₄ experimentally Pregelj et al. (2015). The propagation vector $\mathbf{k}=(-0.208,0,0.423)$ below $T_{\rm N3}$ implies the pitch angle $\alpha=76.1^{\circ}$ for the helix propagating along the $c$ direction. This is in very good agreement with $J_{2}/J_{1}=-1$ that yields the classical pitch angle $\alpha_{\rm cl}=\arccos(-J_{1}/4J_{2})=75.5^{\circ}$, whereas quantum corrections to the pitch angle are small in this range of $J_{2}/J_{1}$ Zinke et al. (2009).

The leading interchain coupling along the $b$ direction, $J_{b2}$, is compatible with the helical order and leads to the same spin arrangement within every second chain, so that, e.g., both atoms 3 in Fig. 1 (middle) feature parallel spins resulting in $k_{y}=0$. On the other hand, the FM interchain couplings are incompatible with the helical order (Fig. 14, left). The couplings $J_{1}$, $J_{a}$, and $J_{a1}$ ($J_{a2}$) build triangles, where the non-collinear order of the two spins coupled by $J_{1}$ cannot be combined with the FM order imposed by $J_{a}$ and $J_{a1}$ ($J_{a2}$). This frustration is alleviated by introducing a non-collinear spin arrangement on $J_{a}$ and $J_{a1}$ ($J_{a2}$) too. By a classical energy minimization for the magnetic model defined by the exchange couplings from Table 2 ³³3Here, we fixed the angle between spins 1 and 2 to the pitch angle of the helix $\alpha$. We checked that the simultaneous variation of $\varphi$ and $\alpha$ leads to a nearly indistinguishable result, because the value of $\alpha$ is fixed by a much larger energy scale of $J_{2}\simeq-J_{1}\simeq 25$ K., we find that the spins on atoms 1 and $1^{*}$ should be turned by an angle

corresponding to $k_{x}=\varphi/360=-0.150$, which is somewhat lower than $k_{x}=-0.208$ in the experiment, but shows the correct sign and explains the opposite directions of rotation along $a$ and $c$.

We find that the incommensurability along the $a$ direction is controlled by a rather subtle difference between the diagonal interactions $J_{a1}$ and $J_{a2}$ that correspond to the bond vectors $(1,\Delta y,\frac{1}{2})$ and $(1,\Delta y,-\frac{1}{2})$, respectively, where $\Delta y$ is the difference between the $y$-coordinates of two neighboring V atoms along the chain. The interaction $J_{a1}$ is more FM than $J_{a2}$. Therefore, the angle $\varphi$ has to counteract the rotation introduced by $\alpha$ in order to bring spins 1 and 2* connected by $J_{a1}$ closer to the parallel configuration.

In the following, we extend our microscopic analysis to the anisotropy parameters entering Eq. (1). Technically, these parameters could be evaluated in the same spirit of perturbation theory as in Eq. (2), following Refs. Mazurenko et al., 2008; Vasiliev et al., 2013. However, the evaluation of magnetic anisotropy requires that hoppings to all four unoccupied $d$-orbitals, including $d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$, are defined, which is not the case in $\beta$-TeVO₄, where the strong hybridization between Te $5p$ and V $3d$ orbitals prevents unambiguous modeling of the higher-lying $d$-bands. Therefore, we restrict ourselves to a qualitative analysis that provides useful insight into the role of magnetic anisotropy in $\beta$-TeVO₄.

The DM terms are allowed by symmetry for both intrachain couplings. Symmetry relations between different components of the $\mathbf{D}_{1}$ and $\mathbf{D}_{2}$ on individual lattice bonds are summarized in Table 3. Helical ground state implies that same rotations occur between every two contiguous spins in a chain (Fig. 14). Since all three components of $\mathbf{D}_{1}$ change sign within the chain, the nearest-neighbor DM couplings do not gain any energy from the helical order. The same argument shows that $d_{2y}$ cannot stabilize the helical order, because it changes sign between $\mathbf{D}_{2}^{(11)}$ and $\mathbf{D}_{2}^{(22)}$. On the other hand, $d_{2x}$ and $d_{2z}$ gain energy from the helical order and should thus stabilize the helix in the plane perpendicular to $(d_{2x},0,d_{2z})$ within the $1-2$ chain. However, the relevant DM components in the neighboring $3-4$ chains are $(-d_{2x},0,-d_{2z})$ implying that the helix with the opposite sense of rotation will be stabilized. On the other hand, the AFM interchain coupling $J_{b2}$ imposes same sense of rotation in the helices $1-2$ and $3-4$ (Fig. 14, right).

The sign change of $\mathbf{D}_{2}$ between the two contiguous chains implies that the helix plane in $\beta$-TeVO₄ cannot be uniquely defined when the DM anisotropy is considered. This striking observation should be parallel to the following. First, the helical order is stabilized only below $T_{3}$ upon a first-order phase transition that entails an abrupt shrinkage of the unit cell along the $b$- and $c$-directions implying a structural effect that probably alleviates this frustration. Second, neutron-scattering data indicate two different helix planes for the chains $1-2$ and $3-4$, respectively Zaharko et al. (2014). This observation is hard to reconcile with the monoclinic crystal structure of $\beta$-TeVO₄, where these two chains are crystallographically equivalent (related by an inversion symmetry), and thus a structural distortion must be involved.

The incompatibility of the helical order with the DM anisotropy in $\beta$-TeVO₄ is also a plausible reason behind the delayed formation of the helical phase upon cooling in zero field. Other $J_{1}-J_{2}$ frustrated-chain compounds reported so far Schäpers et al. (2013) undergo a direct (and second-order) transition from the paramagnetic phase to the helically-ordered phase in zero field. In $\beta$-TeVO₄, however, the paramagnetic and helically-ordered phases are separated by the SDW phase (between $T_{\rm N1}$ and $T_{\rm N2}$) and by the long-period stripe order (between $T_{\rm N2}$ and $T_{\rm N3}$), see Fig. 8. We can also conclude that the helical order is largely destabilized in the magnetic field applied along the $c$-direction (compare the two panels of Fig. 8), which should then be the common direction of the two helices, i.e., the direction that is most vulnerable to the application of the magnetic field.