Coupled frustrated quantum spin-1/2 chains with orbital order in volborthite CuVO(Oh)2hO
We present a microscopic magnetic model for the spin-liquid candidate volborthite CuVO(OH)2HO. The essentials of this DFT-based model are (i) the orbital ordering of Cu(1) and Cu(2) (ii) three relevant couplings , and (iii) the ferromagnetic nature of and (iv) frustration governed by the next-nearest-neighbor exchange interaction . Our model implies magnetism of frustrated coupled chains in contrast to the previously proposed anisotropic kagome model. Exact diagonalization studies reveal agreement with experiments.
pacs:71.20.Ps, 75.10.Jm, 75.25.Dk, 91.60.Pn
The search for new magnetic ground states (GS) is a major subject in solid state physics. Magnetic monopoles in the spin ice system DyTiO (Ref. Castelnovo et al., 2008; Jaubert and Holdsworth, 2009; Morris et al., 2009), the metal-insulator transition in the spin-Peierls compound TiOCl Zhang et al., 2008 and the quantum critical behavior in LiZrCuO Drechsler et al., 2007; Schmitt et al., 2009 are among recent discoveries that demonstrate the power of combining precise experimental techniques with modern theory. However, for a rather large number of problems experiment and theory don’t keep abreast, since it is often tricky to find a real material realization for a well studied theoretical model. The most remarkable example is the concept of a “resonating valence bond”Anderson (1973) — a magnetic GS formed by pairs of coupled spin-singlets lacking the long range magnetic order (LRO). Subsequent studies revealed a fascinating variety of disordered GS,Greedan (2001); Moessner and Ramirez (2006) commonly called “spin liquids” in order to emphasize their dynamic nature, and even raised the discussion of their possible applications.Vedral (2008)
Following the common belief that the spin liquid GS may emerge from the interplay of low-dimensionality, quantum fluctuations and magnetic frustration, considerable effort has been spent on the search for spin-1/2 Heisenberg magnets with kagome geometry. The synthesis of herbertsmithite CuZn(OH)Cl Shores et al., 2005, the first inorganic spin-1/2 system with ideal kagome geometry, and subsequent studies revealed besides the desired absence of magnetic LROHelton et al. (2007) (i) intrinsic Cu/Zn structural disorder and (ii) the presence of anisotropic interactions complicating the spin physics.Zorko et al. (2008) The recently synthesized kapellasiteColman et al. (2008) was predicted to imply modified kagome physics due to an additional relevant coupling.Janson et al. (2008)
Since the search for a system representing the pure kagome model is far from being completed, it is natural to consider systems with lower symmetry where the distortion is small enough to keep the essential physics.Tsirlin and Rosner (2009b) This way, the attention has been drawn to the mineral volborthite CuVO(OH)2HO, where the Cu sites form a slightly distorted kagome network.Hiroi et al. (2001) However, the local environment of two independent Cu sites is essentially different: Cu(1) forms dumbbells of two short Cu–O bonds (and four long Cu–O bonds; “2+4”), while Cu(2) resides in a plaquette formed by four short bonds (Fig. 1, top). Recently, DFT studies of CuSbO, implying the “2+4” local environment of Cu atoms, revealed that orbital ordering (OO) drastically changes the nature of the magnetic coupling from three-dimensional (3D) to one-dimensional(1D).Kasinathan et al. (2008) The crucial impact of OO onto magnetism of volborthite will be in the focus of this paper.
The availability of a pure powder and negligible structural disorder in volborthite inspired thorough experimental studies of its magnetic properties. The magnetic GS was recently investigated by V nuclear magnetic resonance (NMR),Yoshida et al. (2009) following earlier NMR,Bert et al. (2005) high-field electron spin resonanceOkubo et al. (2001) and muon spin relaxationBert et al. (2004) studies. The GS is characterized by the absence of magnetic LRO, high density of low energy excitations and two distinct scales of spin fluctuations. At 4.5 Tesla, volborthite undergoes a transition to another magnetic phase. The fingerprint of this transition is a step-like feature in the magnetization curve. Similar features are observed at 25 and 46 Tesla, hinting at a series of successive transitions.Yoshida et al. (2009) Between 60 and 70 Tesla, the slope of magnetization diminishes indicating a possible onset of a magnetization plateau.footnote 1 () Magnetic susceptibility measurements yield a broad maximum at temperatures much smaller than the Curie-Weiss temperature, indicating strong frustration.
Extensive experimental information stimulated theoretical studies aiming to find a consistent description for magnetism of volborthite. Since the pure kagome model doesn’t account for the experimental data, the studies were focused on the GS and thermodynamical properties of the anisotropic kagome model (AKM). However, attempts to reach consistency by varying the degree of anisotropy were not satisfying so far. The most striking disagreement is the deviation of the theoretical magnetic susceptibility even at rather high temperatures ().Sindzingre (2007b)
This disagreement originates from the choice of the AKM which was based on geometry only, while the structural peculiarities of volborthite were not considered. A standard tool to treat such peculiarities properly is density functional theory (DFT) calculations that can provide a reliable microscopically based model.Zhang et al. (2008); Drechsler et al. (2007); Janson et al. (2008); Tsirlin and Rosner (2009b); Janson et al. (2010) Here, we show that DFT calculations yield an unexpected microscopic magnetic model for volborthite, moving away from the kagome model. Moreover, we reveal strong similarities to the physics of frustrated coupled chains due to OO. Our subsequent simulations of the microscopic model evidence an improved agreement with the experimental data.
Ii DFT calculations
The DFT calculations have been performed in the local density approximation (LDA) using the full potentialfootnote 8 () code fplo8.65-32.Koepernik and Eschrig (1999) For the scalar relativistic calculations, the Perdew and Wang parametrizationPerdew and Wang (1992) of the exchange-correlation potential has been used. All calculations have been performed on well-converged -meshes.footnote 9 ()
The reliability of DFT calculations depends crucially on the accuracy of the experimental structural data used as input. The chemical composition of volborthite hampers structural studies due to the considerable content of V and H atoms, which are poor scatterers of neutrons and x-rays, respectively. Therefore, prior to investigations of subtle electronic effects, the structural data should be addressed. Among several structural data sets available we have chosen a structural model (exp) based on joint x-ray and neutron diffraction studies. Although such combination improves the reliability of the resulting data, the statistics (number of reflections) is not sufficient for fully conclusive results. Therefore, we have carried out a structural optimization relaxing the atomic coordinates and minimizing the forces, since LDA calculations for cuprates usually yield accurate and consistent results.Janson et al. (2008); Tsirlin and Rosner (2009b); Janson et al. (2009) Moreover, to evaluate the influence of the different structural models, we perform calculations for the experimental as well as for the optimized structure.
The LDA-optimized crystal structure (opt) yields considerably lower total energy and atomic forces. Although individual bond lengths and angles change up to several percents (O–H distance increased by 10%), the overall structural motive and the different local environment of Cu(1) and Cu(2) are inherited from the original model.
LDA yields a valence band width of 7 eV (Fig. 2) typical for cuprates, and a metallic GS in contrast to the green, transparent samples. This well-known problem of the LDA originates from the underestimation of strong on-site correlations for a Cu configuration. Nevertheless, LDA is a reliable tool to evaluate the relevant orbitals and couplings.footnote 9 () For most cuprates, an effective one-band model is well justified by a band complex at Fermi level () formed by antibonding bands, where is the number of Cu atoms per cell. In contrast, in volborthite ( = 3), six bands in vicinity of (Fig. 2) evidence a sizable hybridization of two different orbitals at each Cu site that need to be included into the modeling.
The relevant Cu orbitals are revealed by projecting the density of states (DOS) onto local orbitals. The resulting orbital-resolved DOS is shown in Fig. 2. For both Cu(1) and Cu(2), the and states are relevant and hybridized with each other. To evaluate the relevant couplings, we consider two orbitals ( and ) per Cu atom and fit the six bands using the Wannier functions (WFs) technique.Eschrig and Koepernik (2009); footnote 9 ()
Prior to evaluation of the relevant couplings, the correct orbital GS should be found. LDA yields an essentially different filling of the orbitals: for Cu(1) the is close to half-filling and the is almost filled (Fig. 2, left), while for Cu(2) it is the other way round (Fig. 2, right). However, the closer proximity to half-filling in the LDA picture does not necessarily provide the correct answer, as revealed for the related system CuSbO.Kasinathan et al. (2008) Thus, we cross-check the LDA result by LSDA+ calculations. In agreement with the LDA, the latter yield the magnetically active Cu(1) and Cu(2) (see details below).
The relevant transfer integrals can be extracted from the WFs considering the hoppings between the GS orbitals (Cu(1) and Cu(2) ). The leading terms are and , which coincide with two NN couplings in the AKM (Fig. 1, bottom). Surprisingly, the next-nearest-neighbor (NNN) coupling is also sizable, while other couplings are considerably smaller. A small value of the inter-layer coupling supports the 2D nature of magnetism.
The correct description of the orbital GS requires an appropriate description of correlations in the Cu shell, which can be treated in a mean-field way using the LSDA+ scheme. By stabilizing solutions comprising different orbital occupations and a subsequent comparison of their total energies, we evaluate the orbital GS. The separation between the orbital GS and the lowest lying excited orbital state ( for Cu(1) and Cu(2)) exceeds 500 meV (6000 K), almost two orders of magnitude larger than the magnetic exchange (100 K). Therefore, orbital and spin degrees of freedom are mostly decoupled and can be analyzed separately.
The leading exchange integrals , and are obtained mapping the results of LSDA+ total energy calculations onto a Heisenberg model.footnote 2 () A careful analysis of the results shows that the individual values of exchange integrals are sensitive to (i) the structural model, (ii) the Coulomb repulsion and (iii) the double-counting correction (DCC) scheme.footnote 3 () The crucial influence of these parameters is visualized in Fig. 3, where the results for the experimental and optimized structures are shown. For each structural model, we use the limiting cases for the DCC — around-the-mean-field (AMF) and the fully localized limit (FLL)Czyżyk and Sawatzky (1994) and vary within a reasonable range.footnote 4 () Depending on these parameters, we obtain = 80 K, = 35 K and = 100 K for the experimental and = 65 K, = 45 K and = 100 K for the optimized lattice.
The substantially ferromagnetic nature of , in accord with Goodenough-Kanamori-Anderson rules, and the relatively small uncertainties of its strength disregarding the parameters used give strong evidence that the pure kagome model is inappropriate for volborthite. The AKM can be ruled out since and support each other and do not give rise to frustration. Our microscopic insight evidences that an essentially different model with frustration governed by NNN exchange (competing with both and ) should be used for volborthite. Despite sizable scattering of the values, important general trends can be established. First, the optimized structure has an enhanced / ratio compared to the experimental structure. Second, FLL yields considerably smaller and somewhat larger values for than AMF.
Based on DFT calculations, we obtain a microscopic magnetic model and determine the parameters , and . Although we find the relevant region in the phase space, the complexity of volborthite impedes a more accurate estimate of individual exchange integrals, especially . In this case, refining the parameters by numerical simulations of measured physical properties and subsequent comparison to experimental data is an appropriate way towards a deeper understanding.
Iii Exact diagonalization
To realize a guided search for a consistent set of exchange integrals, the GS of the -- model should be investigated. We explore the phase space by considering a Heisenberg model with , , . On a classical level, we find two GS: a ferrimagnetic (fM) phase with magnetization and an incommensurate helical (H) phase with spiral correlations along - frustrated chains, similar to those of edge shared quasi 1D cuprates (see e.g. Ref. Drechsler et al., 2007). The transition from the fM-phase to the H-phase is driven by the frustrating NNN in-chain coupling and occurs at . To discuss the GS of the quantum model we use Lanczos exact diagonalization of finite lattices up to sites.Honecker et al. (2004); Richter and Schulenburg (2010) For the quantum model, the fM state competes with a singlet GS with , and the transition is given by .footnote 5 () The transition line together with the DFT-derived exchange integrals are plotted in Fig. 3 (left). Since the experiments evidence zero magnetization of the GS, the fM solutions can be ruled out and the analysis can be restricted to the singlet GS.
To understand the nature of the GS, we consider spin correlations as a sensitive probe for magnetic ordering. While the correlations between the chains are similar to the standard kagome Heisenberg antiferromagnet, they are completely different along the - frustrated chains (Fig. 3, right). These in-chain correlations fit to a spiral state with a pitch angle very close to the classical model. Hence, our data suggest well pronounced in-chain spiral correlations together with weaker inter-chain correlations. We mention, however, that these statements are restricted to short-range correlations.
Since the magnetic correlations along the chains are strongest, one could argue that the model exhibits an effectively 1D low-temperature physics as has been discussed previously for other 2D models such as the crossed-chain model,Starykh et al. (2002) anisotropic triangular lattice,Coldea et al. (2001) as well as for modified kagome compounds.Janson et al. (2008) However, this issue as well as a conclusive answer to the question of helical LRO need further investigation.
To comprise the experimental magnetization curve, we add the magnetic field term to the Heisenberg Hamiltonian and simulate the dependence. For the boundary of the fM and singlet GS, we find a wide 1/3 magnetization plateau starting at . However, the modification of the exchange parameters, in particular increasing of and decreasing of and , according to the limits set by DFT calculations leads to a significant increase of and to a drastic diminishing of the plateau width. Close to the DFT-boundary (/ = 2, / = 1.1, = 100 K), we obtain = 22 T, which is still smaller than the experimentally observed value. We should note that this deviation originates from the minimalistic character of the model and considerable finite size effects. Nevertheless, a slightly modified ratio / = 1.6 yields = 55 T (Fig. 4, bottom) in excellent agreement with the experiment. It should be mentioned that the nature of spin correlations in the 1/3-plateau phase is substantially different compared to the kagome model.Cabra et al. (2005) Small magnetization jumps seen experimentallyYoshida et al. (2009) can not be resolved with present lattice sizes and might be related to anisotropic exchange.
We also calculate the temperature dependence of magnetic susceptibility using two different latticesfootnote 9 () up to sites. We obtain a good fit down to 50 K (Fig. 4, top),footnote 7 () the resulting = 2.16 and = 100.5 K are in excellent agreement with experimentsOkubo et al. (2001) and our estimates from DFT.
To summarize, we suggest a new magnetic model for volborthite: assuming that DFT calculations are applicable to volborthite as they are for a plethora of compounds, the kagome model can be safely ruled out. Instead, the magnetism of volborthite is accounted for by a -- model reminiscent of coupled frustrated chains. For the proposed model, the orbital order of Cu(1) and Cu(2) orbitals is crucial.
We suggest new experiments to challenge our model: resonant x-ray scattering measurements measurements to study orbital effects and measurements in high magnetic fields (70 Tesla) to get an access to the magnetization plateau. Additional investigations of the -- model itself by alternative simulation methods are desirable to clarify the influence of finite size effects, intrinsic for exact diagonalization.
We thank Z. Hiroi for providing us with data. We acknowledge fruitful discussions and valuable comments from Z. Hiroi, F. Mila, G. Nilsen and A. Tsirlin.
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|cell||Sp. gr.||functional||Cu atoms||basis vectors||-mesh|