Striped Magnetic Ground State of the Kagome Lattice in Fe4Si2Sn7O16

TL;DR

This study uncovers a unique magnetic ground state in Fe4Si2Sn7O16, where two-thirds of the spins form canted antiferromagnetic chains while one-third remains dynamically frustrated, revealing complex partial order in kagome lattices.

Contribution

It experimentally demonstrates a new partial magnetic order in a kagome lattice, including a frustrated liquid-like state, and relates it to theoretical models of symmetry breaking ground states.

Findings

Two-thirds of spins form canted antiferromagnetic chains

One-third of spins remain in a liquid-like frustrated state

No lattice distortion observed despite symmetry-breaking propagation vector

Abstract

We have experimentally identified a new magnetic ground state for the kagome lattice, in the perfectly hexagonal Fe2+ (3d6, S = 2) compound Fe4Si2Sn7O16. Representational symmetry analysis of neutron diffraction data shows that below T_N = 3.5 K, the spins on 2/3 of the magnetic ions order into canted antiferromagnetic chains, separated by the remaining 1/3 which are geometrically frustrated and show no long-range order down to at least T = 0.1 K. Moessbauer spectroscopy confirms that there is no static order on the latter 1/3 of the magnetic ions - i.e., they are in a liquid-like rather than a frozen state - down to at least 1.65 K. A heavily Mn-doped sample Fe1.45Mn2.55Si2Sn7O16 has the same magnetic structure. Although the propagation vector q = (0, 1/2 , 1/2 ) breaks hexagonal symmetry, we see no evidence for magnetostriction in the form of a lattice distortion within the resolution of our data. We discuss the relationship to partially frustrated magnetic order on the pyrochlore lattice of Gd2Ti2O7, and to theoretical models that predict symmetry breaking ground states for perfect kagome lattices.

Figures

Full text

PACS numbers

75.10.Jm, 75.25.-j, 75.47.Lx, 75.50.Ee, 76.80.+y

††preprint: APS/123-QED

Ever since the first published consideration of the ground state of a triangular lattice of Ising spins, Anderson (1950) the pursuit of materials with geometrically frustrated magnetic (GFM) lattices has been an important driver in experimental condensed matter physics. Balents (2010) Perfect GFM lattices are proving grounds for a number of predicted exotic states of matter. The most famous of these is the quantum spin liquid (QSL), in which there is effectively no energy barrier between macroscopically degenerate ground states for $S=\frac{1}{2}$ spins, which can therefore continue to fluctuate down to $T=0$ K. Anderson (1973)

The simplest GFM case is a triangular lattice, followed by the expanded triangular network known as the kagomé lattice. Undistorted (perfectly hexagonal) magnetic kagomé lattices are rare – the most studied examples are naturally occurring minerals or synthetic versions thereof, notably the jarosites $AB_{3}$\ce(SO4)2(OH)6 where $B^{3+}$ can be Fe3+ ($S=\frac{5}{2}$), Wills and Harrison (1996) Cr3+ ($S=\frac{3}{2}$), Inami et al. (2001) or V3+ ($S=1$), Papoutsakis et al. (2002) which generally still undergo Néel ordering due to Dzyaloshinkii-Moriya anisotropy. A series of quinternary oxalates studies by Lhotel et al., Lhotel et al. (2011, 2013) which contain Fe2+ ($S=2$) lattices equivalent to kagomés (for 1st neighbor interactions only), all freeze into a $q=0$ Néel state below $\sim 3.2$ K. More recently, the Cu2+ ($S=\frac{1}{2}$) undistorted kagomé compound herbertsmithite \ceZnCu3(OH)6Cl2 Shores et al. (2005) has attracted a great deal of attention as arguably the most promising QSL candidate so far discovered (for a recent review see ref. Mendels and Bert (2016)).

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Fe4Si2Sn7O16 Söhnel et al. (1998) is a synthetic compound that incorporates an undistorted kagomé lattice of high-spin (HS) Fe2+ (3d6, $S=2$) magnetic ions on the 3f Wyckoff sites of its hexagonal (trigonal space group $P\bar{3}m1$, $\#164$) structure. The kagomé lattice is located in layers of edge-sharing \ceFeO6 and \ceSnO6 octahedra (hereafter called the oxide layer), which alternate with layers of oxygen-linked \ceFeSn6 octahedra (the stannide layer). The layers are separated by \ceSiO4 tetrahedra (see Fig. 1). A triangular lattice of low-spin (LS) Fe2+ (3d6, $S=0$) on the 1a Wyckoff site in the stannide layer is magnetically inactive.

We recently reported a long-range antiferromagnetic (AFM) Néel ordering transition in \ceFe4Si2Sn7O16 at $T_{N}=3.0$ K and its Mn-doped (in the oxide layer) analogue Fe1.45Mn2.55Si2Sn7O16 at $T_{N}=2.5$ K. Allison et al. (2016) There was no evidence of spin-glass behavior or a ferromagnetic (FM) component to the ground state. Given their perfectly hexagonal lattices above $T_{N}$, the ordered magnetic ground state was expected to be either the conventional $q=0$ or $(\sqrt{3}\times\sqrt{3})$ solution, which preserve hexagonal symmetry. Harris et al. (1992) In the work reported in this Letter, we set out to test this by collecting low-temperature neutron powder diffraction (NPD) and Mößbauer spectroscopy data above and below $T_{N}$. Contrary to expectation, we found that the ground state is a striped AFM structure in which $\frac{2}{3}$ of the magnetic sites are completely ordered, while the other $\frac{1}{3}$ are frustrated and remain completely disordered down to at least 0.1 K. To the best of our knowledge, this state has no precedent experimentally and has not been explicitly predicted theoretically.

New magnetometry data collected for the present study were identical to those in ref. Allison et al. (2016) apart from slightly revised values for \ceFe4Si2Sn7O16 of $T_{N}=3.5$ K, $\mu_{\textrm{eff}}=5.45~{}\mu_{B}$ per HS Fe2+ and $\theta=-12.7$ K, which corresponds to a modest frustration index Ramirez (1996) $f=|\theta/T_{N}|=3.6$. 111See Figs. 1–3 of Supplemental Material at [URL will be inserted by publisher] Note that the orbital angular momentum $L$ is not completely quenched ($\mu_{\textrm{eff spin-only}}=4.90~{}\mu_{B}$), as is commonly observed in Fe2+ oxides.

High-resolution NPD data were collected on the instrument Echidna Liss et al. (2006) at the OPAL reactor, Lucas Heights, Australia. Samples were placed in 6 mm diameter vanadium cans using neutrons of wavelength $\lambda=2.4395$ Å, over the range $2.75-162^{\circ}2\theta$ with a step size of $0.125^{\circ}2\theta$. Low-temperature data were collected to 1.6 K in a cryostat and 0.1 K in a dilution fridge for \ceFe4Si2Sn7O16, and to 1.8 K for Fe1.45Mn2.55Si2Sn7O16. Rietveld-refinements of the nuclear structure above $T_{N}$ were consistent with our previous work using single-crystal X-ray diffraction Söhnel et al. (1998) and a combination of synchrotron X-ray and neutron powder diffraction.Allison et al. (2016)

New low-angle Bragg peaks emerge for \ceFe4Si2Sn7O16 at 1.6 K, below $T_{N}$, indicative of 3D long-range ordered magnetism (Fig. 2). The same peaks are observed at 0.1 K, i.e., the magnetic structure shows no further change down to at least this temperature; and at 1.8 K for Fe1.45Mn2.55Si2Sn7O16, i.e., the magnetic structure is not fundamentally changed by almost complete substitution of Mn2+ for Fe2+. 222See Fig. 2 of Supplemental Material at [URL will be inserted by publisher] All peaks could be indexed to a propagation vector $q=(0,\frac{1}{2},\frac{1}{2})$, which breaks hexagonal symmetry. However, we do not observe any peak splitting or broadening within the resolution of our NPD data, indicating that magnetostriction is very small.

We solved the magnetic structure by representational symmetry analysis using the BasIreps routine in the program Fullprof. Rodriguez-Carvajal (1993) The propagation vector $q=(0,\frac{1}{2},\frac{1}{2})$ acting on the space group $P\bar{3}m1$ splits the magnetic HS Fe2+ on the 3f site in the oxide layer into two orbits, Fe-3f(1) and Fe-3f(2), in a 1:2 ratio. The irreducible representations (irreps) for each orbit decompose in terms of two 1D irreps for Fe-3f(1), $\Gamma_{\textrm{mag}}=2\Gamma_{2}+\Gamma_{4}$, and another two for Fe-3f(2), $\Gamma_{\textrm{mag}}=3\Gamma_{1}+3\Gamma_{3}$. The basis vectors are given in Table 1 of Supplementary Information.

We first tested these four possible representations independently (i.e., with only the Fe-3f(1) or Fe-3f(2) site magnetically active) by Rietveld refinement against 1.6 K data using Fullprof. We found that the $\Gamma_{1}$ representation for Fe-3f(2) gave by far the best fit. We tried adding the $\Gamma_{2}$ and $\Gamma_{4}$ representations for Fe-3f(1) to the refinement, but in neither case was the fit improved, and the moments on Fe-3f(1) refined to zero within error. The component of the moment on Fe-3f(2) along the $z$ axis also refined to zero within error. The final refinement, for which the fit is shown in Fig. 2, was therefore carried out using only the $x$ and $y$ axis components of the $\Gamma_{1}$ representation for Fe-3f(2). This structure is equivalent to the Shubnikov magnetic space group (Opechowski-Guccione setting, $C_{2c}2/m$ #12.6.71).

Fig. 3 shows the final refined magnetic structure at 1.6 K, with an ordered moment of 2.52(6) $\mu_{B}$ ($\mu_{x}=0.44(6)$, $\mu_{y}=2.71(5)$ $\mu_{B}$). The refinement at 0.1 K yielded increased moment of 3.2(2) $\mu_{B}$ ($\mu_{x}=0.3(2)$, $\mu_{y}=3.3(2)$ $\mu_{B}$). The slight reduction compared to the total spin-only moment of 4 $\mu_{B}$ for HS Fe2+ may be an effect of crystal field and spin-orbit coupling. The most striking feature is the absence of long-range magnetic order on the Fe-3f(1) sites, which sit on the kagomé legs between magnetically ordered rows of Fe-3f(2) sites along the $x$ direction, despite all Fe-3f sites being HS Fe2+ (d6, $S=2$) at all temperatures. Note that although the spin orientation in the $x-y$ plane refined robustly, NPD cannot distinguish between the model shown in Fig. 3 and an alternative version in which the ordered rows of spins are shifted by $\frac{1}{2}a$, to point approximately towards/away from the non-magnetically ordered Fe-3f(1) site rather than towards/away from the center of the hexagon.

It is clear from Fig. 3 that the Fe-3f(1) site is geometrically frustrated by its four nearest-neighbor Fe-3f(2) sites, regardless of the sign of magnetic exchange with those sites. This is consistent with the zero refined moment for Fe-3f(1), and the fact that because the propagation vector splits the Fe sites into two independent orbits, the molecular field created by one sublattice on the other one is zero, i.e., the 3f(1) magnetic moments can only couple to each other through $J_{d}$ interactions. However, NPD cannot determine whether Fe-3f(1) is locally ordered but long-range disordered, or completely locally disordered. More importantly, it cannot distinguish between the model shown in Fig. 3 and a multi-$q$ structure with three arms of the propagation vector star $q_{1}=(0,\frac{1}{2},\frac{1}{2}$), $q_{2}=(\frac{1}{2},0,\frac{1}{2}$), $q_{3}=(\frac{1}{2},\frac{1}{2},\frac{1}{2}$). Such multi-$q$ structures could preserve trigonal symmetry without requiring stripes of disordered Fe2+ ions. The anisotropy of Fe2+ could be an important parameter in the Hamiltonian to stabilize such a structure. We therefore tested all 1-, 2-, and 3-$q$ symmetry allowed models, with two of the 3-$q$ models (Shubnikov groups $P_{C}2/m$, $\#$10.8.56 and $P_{2c}\bar{3}m1$, $\#$164.6.1320 in Opechowski-Guccione settings) giving comparable fits to the striped 1-$q$ model discussed above. To resolve the single-$q$ vs. multi-$q$ question, we conducted a Mößbauer spectroscopy experiment.

The Mößbauer spectra above and below $T_{N}$ were obtained on a conventional spectrometer operated in sine-mode with both the sample and 57CoRh source cooled by flowing He gas. The system was calibrated using $\alpha-$Fe metal at room temperature and the spectra were fitted to a sum of Lorentzians with positions and intensities derived from a full solution to the nuclear Hamiltonian. Voyer and Ryan (2006) The spectrum above $T_{N}$ at 5 K shown in Fig. 4, which is effectively identical to the data published in ref. Söhnel et al. (1998), shows that the subspectra from the Fe in the 1a and 3f sites are clearly resolved, being distinct in both spectral area (1:3 as expected from site multiplicities) and hyperfine parameters (as expected from the different spin configurations). The LS Fe2+ on the 1a site gives an isomer shift $\delta=0.33(1)$ mm/s with a quadrupole splitting $\Delta=0.48(1)$ mm/s, while the HS Fe2+ on the 3f site gives $\delta=1.19(1)$ mm/s and $\Delta=2.41(1)$ mm/s. Cooling through $T_{N}$ to 1.8 K leads to remarkably limited changes. The subspectrum from the Fe-1a site is completely unchanged. There is no evidence for magnetic splitting at this site and therefore no magnetic order, consistent with its LS d6 ($S=0$) configuration. The well-split doublet from the Fe-3f site also persists unchanged, but its intensity is greatly reduced to equal that of the 1a subspectrum. This non-magnetically ordered subspectrum therefore now reflects only $\frac{1}{3}$ of HS Fe2+ on the 3f site, consistent with the Fe-3f(1) site in our single-$q$ striped model. The remaining contribution to the 1.8 K spectrum in Fig. 4 comes from the $\frac{2}{3}$ of HS Fe2+ 3f sites that do order, consistent with Fe-3f(2) in our striped model. The Fe-3f(2) subspectrum can be fitted assuming the same values for $\delta$ and $\Delta$ as for Fe-3f(1), but with a small additional hyperfine magnetic field ($B_{\textrm{hf}}$). The magnetic component could not be fitted as a single subspectrum, and required further splitting into two equal (within error) area sub-components. This is very weak effect (the fields are only 7.5 T and 4.3 T) that does not change the main result. It may indicate that the moments on the two Fe-3f sub-sites do not make exactly the same angle with the electric field gradient axes in the kagomé plane, as a result of which they are not truly equivalent, producing some small difference in orbital contribution.

Crucially, the splitting of the Mößbauer spectra is not consistent with any of the multi-$q$ models, including the two mentioned above that fit our NPD fits as well as the striped model does. Finally, the observed hyperfine fields are remarkably small (7.5(3) T and 4.3(3) T), and further cooling to 1.65 K did not lead to a significant increase and so these values appear to be close to their $T=0$ limits. The simplest explanation for such small fields is highly dynamic spins, but the dynamic component would have to be very fast ($>100$ MHz) because we not see any line broadening due to slower dynamics.

We note here that while the Mößbauer analysis fully and independently confirms the striped model from NPD in which the Fe-1a site does not order and only $\frac{2}{3}$ of the Fe-3f site orders, it allows us to go further. For moments on a crystallographic site to contribute to a (Bragg) diffraction peak they must be long-range-ordered. However, for a non-zero hyperfine field to be observed in a Mößbauer spectrum, the moments need only be static on a time scale of $\sim$0.1 $\mu$s – they just need to have a non-zero time average over a relatively short time. Thus, our observation that $B_{\textrm{hf}}$ is zero for $\frac{1}{3}$ of the Fe-3f sites means that we can rule out any “frozen” random spin configuration that does not give magnetic Bragg peaks in NPD, as well as the multi-$q$ state. There is no static order at $\frac{1}{3}$ of these Fe-3f sites, at least down to 1.65 K. Furthermore, since there are no significant changes in either $\delta$ or $\Delta$, we can also rule out changes in the electronic configuration of some or all of the Fe ions (e.g., a HS $\rightarrow$ LS transition making them non-magnetic).

A further striking aspect of the striped state of \ceFe4Si2Sn7O16 is that it breaks hexagonal symmetry. Zorko et al. Zorko et al. (2017) recently presented experimental evidence for symmetry breaking in herbertsmithite, but this appears to be related to the presence of significant ($5-8\%$) disorder on its otherwise “perfect” kagomé lattice, which our diffraction data rule out in the present case. A number of theoretical models predict symmetry breaking on $S=\frac{1}{2}$ kagomé lattices, notably the valence bond crystal (VBC) state Singh and Huse (2007) (with the help of magnetoelastic coupling) and the striped spin-liquid crystal state. Clark et al. (2013) However, these models are based on the resonance valence bond (RVB) picture of paired-up $S=\frac{1}{2}$ spins, so their relevance to the present $S=2$ case is unclear. Similarly, a type of striped order was predicted by Ballou Ballou (1998) in itinerant electron kagomé systems (the disordered sites being really non-magnetic in this case), but the mechanism should be different to the present insulating case. The magnetic structure of \ceFe4Si2Sn7O16 – and the fact that it is not altered by substituting Mn2+ (HS d5, $S=\frac{5}{2}$) for Fe2+ (HS d6, $S=2$) – are important new experimental observations against which to test theoretical models of the large-$S$ kagomé lattice. In the present case, the fact that the angle between ordered spins along the $x$ axis is very close to $120^{\circ}$ (the value at 0.1 K is $129^{\circ}$) is consistent with the magnetic moments being confined along the two-fold axis, which suggests that magnetocrystalline anisotropy may play an important role.

Preliminary Hubbard-corrected density functional theory (DFT+U) calculations 333We used the projector augmented wave method and GGA-PBE implemented in the Vienna ab initio simulations package (VASP), and considered different $U_{\textrm{eff}}=(U-J)$ values to include electron correlation effects stemming from interactions in the Fe[3d] shell Kresse and Furthmüller (1996); Kresse and Joubert (1999). Electronic and magnetic states were stabilized for $U_{\textrm{eff}}>4$ eV, with a self-consistent field energy convergence breakdown condition set to $10^{-5}$ eV. for the non-collinear striped ground state of \ceFe4Si2Sn7O16 reproduced the zero net moment on the Fe-3f(1) sites. However, the $q=0$ state in the same $1\times 2\times 2$ supercell was still found to be energetically lower. Noting that even the definition of a “ground state” in this system is problematic, dedicated detailed theoretical treatments are clearly required. In this context we note that Iqbal et al. Iqbal et al. (2015) and Gong et al. Gong et al. (2016) recently treated the $S=\frac{1}{2}$ kagomé lattice using high-level renormalization group theory. They identified the dominant magnetic interaction as AFM exchange through the long diagonals of the hexagons, labeled $J_{d}$ in Fig. 3; the ground state then depends on the balance between nearest-neighbor ($J_{1}$) and second-nearest-neighbor ($J_{2}$) exchange. For $|J_{1}|<|J_{2}|$, they obtain the “cuboc1” phase, which is consistent with the vertical components of the spin vectors as shown in Fig. 3; while for $|J_{1}|>|J_{2}|$, they obtain the “cuboc2” phase, which is consistent with the horizontal components. The experimental magnetic structure of \ceFe4Si2Sn7O16 can thus be described as a linear combination of cuboc1 and cuboc2. Although this solution was not explicitly predicted for $|J_{1}|\approx|J_{2}|$, our experimental case has additional features, notably much bigger spins which make a VBC state highly unlikely, and the presence of $\sim 90^{\circ}$ Fe–O–Fe superexchange pathways in addition to Fe–Fe direct exchange. A comparably detailed theoretical study is beyond the scope of the present work, but may represent a productive way forward.

Finally, we note an intriguing experimental comparison to \ceGd2Ti2O7. The topology of the magnetic Gd3+ lattice in this pyrochlore-type compound can be described as four sets of inter-penetrating kagomé planes, the triangles of which meet to form tetrahedra. Below $T_{N}=1.1$ K, it adopts the partially ordered “1-$k$” structure, in which one of those four sets of kagomé planes (involving $\frac{3}{4}$ of the spins) is $q=0$ long-range AFM ordered, while the remaining $\frac{1}{4}$ of the spins between those planes remain frustrated. Champion et al. (2001) If the spins in Fig. 3 are collectively rotated about the $x$-axis, the magnetic structure of \ceFe4Si2Sn7O16 becomes equivalent to one of the three other kagomé planes in \ceGd2Ti2O7, which cut through the frustrated spin. Below $T^{\prime}=0.7$ K, the frustrated spin in \ceGd2Ti2O7 may order weakly into the “4-$k$” structure, Stewart et al. (2004) although this has been disputed Paddison et al. and muon-spin relaxation ($\mu$SR) data show that fluctuations continue down to at least 20 mK. Yaouanc et al. (2005) Future neutron diffuse scattering and/or $\mu$SR experiments at dilution temperatures might therefore provide similar insights into \ceFe4Si2Sn7O16, and the reasons for which it adopts the partially ordered striped state in preference to the fully ordered $q=0$ one.

The authors received financial support from the Australian Research Council (DP150102863), the School of Chemical Sciences, University of Auckland (FRDF Project 3704173), the Natural Sciences and Engineering Research Council of Canada, and the Fonds Québécois de la Recherche sur la Nature et les Technologies.