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# Magnetic nanopantograph in the SrCu_{2}(BO_{3})_{2} Shastry–Sutherland lattice

Edited by Zachary Fisk, University of California, Irvine, CA, and approved January 1, 2015 (received for review November 7, 2014)

## Significance

The spins of the unpaired electrons in a solid tend to align along an applied magnetic field. In the case of antiferromagnetic materials having competing interactions (frustration) it is common to observe that the magnetization increases, exhibiting complicated structures with discrete jumps and plateaus. SrCu_{2}(BO_{3})_{2} is one of these materials, for which we find experimentally that its macroscopic physical dimensions also change with the magnetic field, mimicking the behavior in the magnetization. Using quantum mechanics, we show quantitatively that due to the orthogonal arrangement of the magnetic Cu^{2+} dimers acting as pantographs, minute deformations allow significant reduction in the effective interactions responsible for the antiferromagnetism. This drop is sufficient to compensate the elastic energy loss in the lattice deformation.

## Abstract

Magnetic materials having competing, i.e., frustrated, interactions can display magnetism prolific in intricate structures, discrete jumps, plateaus, and exotic spin states with increasing applied magnetic fields. When the associated elastic energy cost is not too expensive, this high potential can be enhanced by the existence of an omnipresent magnetoelastic coupling. Here we report experimental and theoretical evidence of a nonnegligible magnetoelastic coupling in one of these fascinating materials, SrCu_{2}(BO_{3})_{2} (SCBO). First, using pulsed-field transversal and longitudinal magnetostriction measurements we show that its physical dimensions, indeed, mimic closely its unusually rich field-induced magnetism. Second, using density functional-based calculations we find that the driving force behind the magnetoelastic coupling is the ^{2+} dimers acting as pantographs, can shrink significantly (0.44%) with minute (0.01%) variations in the lattice parameters. With this original approach we also find a reduction of ∼10% in the intradimer exchange integral *J*, enough to make predictions for the highly magnetized states and the effects of applied pressure on SCBO.

- magnetostriction
- high magnetic fields
- spin-lattice coupling
- density functional theory
- Shastry–Sutherland

It has long been understood that magnetoelastic coupling can move magnetic materials phase boundaries in temperature and field and even change the order and/or universality class of magnetic transitions (1). Model Hamiltonians with effective exchange interactions are the common theoretical tool to tackle the complex behaviors of quantum magnet systems (2) and, indeed, these effective exchange interactions depend on subtleties of the electronic structure that, in turn, are naturally linked to the structural degrees of freedom of the system. When magnetoelastic effects are present, structural changes can also modify the macroscopic magnetic state via changes in the effective parameters of the model Hamiltonian. Simply posed, it is challenging to interpret experimental results at a point of high interest in a predicted (H,T) phase diagram of a magnetic material under consideration for fundamental studies or applications without knowing the effects that unavoidable lattice changes have in the exchange interactions as we drive our system toward such a point. Hence, it is highly desirable to be able to quantify such lattice effects.

SrCu_{2}(BO_{3})_{2} (SCBO) is an especially fascinating example of a low-dimension, frustrated quantum antiferromagnetic system. It crystallizes in a tetragonal structure (3) in which layers of [CuBO_{3}]^{−} (Fig. 1*A*) are stacked along the *c* axis and separated by planes of Sr^{2+} ions (Fig. 1*B*). The magnetically active Cu^{2+} ions form a 2D arrangement of mutually orthogonal dimers (Fig. 1*A*). The magnetic properties of this compound can be closely described through a 2D Heisenberg Hamiltonian (4):*J* and _{3}]^{−} planes. Although *J* and **1**) admits the direct product of dimer singlet states as an exact eigenstate. It has been first shown that this

The first indications of a significant lattice involvement in this system were evidenced in the sound velocity measurements by Wolf et al. (13), subsequently studied using X-ray diffraction at low temperature and under high magnetic field by Narumi et al. (14). These were followed by neutron diffraction experiments carried out by Vecchini et al. (15) and Haravifard et al. (16). In the former, structural changes due to the thermal excitation of the magnetic degrees of freedom were directly measured. In the latter the increased lifetime (inelastic linewidths) in a subset of the acoustic phonons has been shown to correlate with the formation of the singlet ground state near 10 K in zero field. In a recent study, Jaime et al. (17) performed magnetostriction experiments to 100 T, showing that the crystallographic *c* axis is an extraordinarily sensitive witness to the magnetic structure and superstructure. However sensitive, the *c*-axis results do not allow for a direct examination of the spin-lattice correlations within the Cu-dimer planes (Fig. 1*A*).

## Experimental Results

Using an experimental setup modified from one previously discussed (17), where an optical fiber Bragg grating (FBG) sensor is placed perpendicular to the applied magnetic field, we obtained the first high-resolution (better than 10^{−6}) in-plane magnetostriction at *T* = 1.36 K (Fig. 2). In our experiment we observe a contraction of the *a* axis that is sensitive to, and shows features at, the magnetic fields identified as onset of the 1/8, 1/4, and 1/3 magnetization plateaus and a concomitant expansion of the *c* axis with an FBG sensor in the conventional configuration parallel to the applied field. Our new data are consistent with *a*-axis vs. field data from Narumi et al. (14), although the X-ray experiment did not resolve individual plateaus. The observed

## Model

From a magnetic point of view, one can expect that the system will respond to the external magnetic field by modifying the internal coordinates of ions playing a role in the magnetic interactions. The modification of any other internal coordinates would have an elastic energy cost without magnetic energy gain. As already pointed out (14, 15), the main internal parameter related with the magnetic properties in this compound is the *α*) mediating the intradimer superexchange interaction. This angle is, indeed, expected to decrease toward 90° with the applied magnetic field as this distortion would weaken the AFM superexchange interaction (18⇓⇓–21) through a reduction of the Cu-3*d*/O-2*p* hopping.

The orthogonal arrangement of the dimers in SCBO allows for a variation of the angle with a minimum lattice deformation. This double-pantograph (22) effect can be most easily visualized in Fig. 3 by considering the atoms moving as sketched. Each pantograph magnifies the decrease of the internal superexchange angles, but the two do not substantially change the overall length of the array. This argument can be demonstrated in a more rigorous way. Indeed, neglecting the buckling, the tetragonal unit cell parameter and the angle can be written as follows in terms of interatomic distances in the [CuBO_{3}]^{−} layer (Fig. 3):*x* the O–O and *y* the Cu–Cu distances, their variations can be written as

Under an increasing magnetic field, the system can reduce the magnetic contribution to the total energy by decreasing the superexchange angle

## Theoretical Results

### Structure Relaxation—Angle.

To gain insight into the modification of the internal structural parameters and their consequences on the electronic structure and magnetic interactions with a change of the magnetic order, total energy calculations have been performed with the Quantum Espresso (23) code.

As, in our problem, magnetostriction involves only very weak structural variations, it would be of interest to compare the structural relaxation of two extreme magnetic states, namely the ground state configuration (a product of dimer singlets) and the ferromagnetic (FM) state expected at saturation. Unfortunately, the ground state solution cannot be written as a single Kohn–Sham determinant and is therefore not accessible with state-of-the-art density functional theory (DFT) codes for periodic systems. For this reason, we have optimized the lattice parameters and internal atomic coordinates of a Néel-like AFM ordered structure and of the FM state (Fig. 1 *C*–*F*). Using the Quantum Espresso code (23), we obtained lattice parameters (^{−4}) indeed falls within the limit of accuracy of DFT when trying to calculate the structural changes between two magnetic orders with different space symmetries.

The difference in *α* and the O–O and Cu–Cu distances (respectively *x* and *y*) between the two magnetic structures are more easy to assess with DFT. A variation of the angle of

### Effective Exchange Interactions from Energy Differences.

The computation of magnetic couplings was first carried out within the broken symmetry formalism, i.e., by mapping total energies corresponding to various collinear spin arrangements within a supercell onto a Heisenberg Hamiltonian,^{2+} ions located at sites *i* and *j*, respectively. It is straightforward to show that the expectation value of the Hamiltonian (6) on a DFT state *d* transition metal oxides, Eq. **7** can be used to model a large set of spin configurations. A numerical evaluation of the couplings is thus obtained through a least-squares minimization of the difference between DFT and Ising relative energies of the set of spin configurations (25⇓–27).

In the case of SCBO the determination of the magnetic couplings up to the fifth nearest neighbor (in the 3D lattice) can be performed using a 44-atom tetragonal cell. Taking crystal and spin reversal symmetries into account, this leads to a total of 22 distinct spin configurations. The calculations using the internal parameters of the relaxed AFM and FM ordered structures give

### Effective Exchange Interactions from Band Structure–Wannier Functions.

An alternative evaluation of the AFM contribution to the magnetic couplings can be achieved through a mapping of the paramagnetic band structure onto a single-band Hubbard model at half filling, which eventually reduces to an Heisenberg model in the strongly correlated limit. In this framework, the effective antiferromagnetic interaction is given by *Inset* shows the maximally localized Wannier functions (MLWF) calculated for the FM internal coordinates. Besides the central Cu-*p* character mediating the dominant in-plane superexchange antiferromagnetic interactions are also clearly visible. Note that a MLWF calculated for the AFM internal coordinates would be indistinguishable at this scale. The hopping integrals *t* and

There is a small but measurable decrease of the nearest-neighbors hopping interaction *t* but no difference in the next-nearest-neighbors interaction _{3} group. The variation of the nearest-neighbors effective exchange interaction estimated through this approach is

## Discussion

It is possible to establish a relation between the intradimer **4** and **5** to obtain^{−1} ^{−1} (Fig. S1) can be deduced from temperature-dependent neutron diffraction experiments (15) to obtain*x*, *y*, and *k* can be obtained for *SI Text*). Finally, replacing *x*, *y*, and *a* by their experimental values (15), we obtain*c* and *a* axes, respectively. These changes imply a drop in *α* of 0.28° and relative variations *c* axis of *a* axis and a change in *α* of *α* is the driving mechanism behind the magnetoelastic coupling in SCBO.

Coming back to DFT calculations, we can now confirm that the energetic arguments used to justify the pantograph effect are correct. Indeed, from the variation of *J* for a tetragonal unit cell, we can estimate the magnetic energy gain due to the magnetoelastic coupling to be about −10 K or *Materials and Methods*).

Moreover, our calculations predict a strong effect (∼10% reduction) in the intradimer exchange integral *J* as *α* decreases, with no detectable variation in the interdimer counterpart

Our results could also shed light on 2D models used so far to predict magnetization plateaus, as the expected changes in *J* can have different effects on different models and help to sort them out according to prediction power. As we show, the reduction in *J* comes alongside a drop in the volume of the crystallographic unit cell of SCBO that is usually the effect of applied pressures. In simpler words, our results also suggest that pressures of the order of a few gigapascals might be used to control the _{2}(BO_{3})_{2} (32). Among these, the

In summary, our original approach and results shed light on the issue of tuning magnetic ground states of matter with the use of external magnetic fields and pressure, by means of a clever combination of state-of-the-art experimental and computational tools.

## Materials and Methods

### Experimental.

The high-quality single-crystal samples used in this work were prepared by the optical floating zone image furnace technique using self-flux as described elsewhere (34), oriented, and cut into ∼2 × 2 × 3-mm^{3} pieces for the magnetostriction experiment. Magnetostriction measurements were carried out using an optical fiber, furnished with a FBG, attached to the sample to detect length variations as a rapidly varying external magnetic field is applied (17). The reflection of light by the FBG at the Bragg wavelength shifts when the grating spacing changes. In our experimental setup, two different configurations were used to measure in-plane and out-of-plane magnetostriction. In the latter case, *c* axis whereas in the former case, the fiber is bent around a circle of 0.6-inch diameter and kept in place with a stainless steel fine diameter tube to reach the sample at 90° from the *c* axis (and magnetic field H) and thus obtain

### Computational.

Total energy calculations were performed with the Quantum Espresso (23) code. This code is based on DFT and uses the pseudopotential plane-wave method. The calculations were performed using ultrasoft pseudopotentials (35). For exchange and correlation, we used the GGA-PBE (36) augmented by a Hubbard *U* term to improve the treatment of strongly correlated Cu-3*d* electrons. A value of the effective Hubbard *U* = 10.3 eV was determined in a 44-atom tetragonal cell, using the experimental structure determined at 300 K (3) with a ferromagnetic order and following the approach proposed in ref. 37. For structural relaxations, a plane-wave cutoff of 60 Ry and a 6 × 6 × 6 Monkhorst–Pack (38) grid for the first Brillouin zone sampling were used. At the end of the relaxation procedure the total pressures were lower than 0.35 kbar and the forces were less than 10^{−4} Ry/bohr.

The mapping of the paramagnetic band structure onto a single-band Hubbard model at half filling was performed first by computing a set of four maximally localized Wannier functions following the method of Marzari and Vanderbilt (39) and spanning the four bands of dominant Cu-

To check the hypothesis that the lattice distortion is fairly inexpensive in SCBO, due to the opposing pantographs in the crystallographic unit cell, we compute the energy involved in the magnetostriction at saturation (ferromagnetic order). If we consider that the total energy of the system can be decomposed in Zeeman, Heisenberg, and elastic contributions, the energy difference between the deformed and undeformed systems can be compared. First, one can suppose that the Zeeman term does not depend on the deformation. Second, for a tetragonal unit cell the energy gain due to the magnetostriction is

where ^{3} are the elastic constants of SCBO calculated with Quantum Espresso (23) in GGA + *U* for the FM unit cell. The elastic energy loss is then

## Acknowledgments

M.J. acknowledges useful discussions with Prof. B. D. Gaulin, McMaster University and Cristian D. Batista, Los Alamos National Laboratory (LANL). The National High Magnetic Field Laboratory Pulsed-Field Facility is supported by the National Science Foundation (NSF), the US Department of Energy (DOE), and the State of Florida through NSF Cooperative Grant DMR-1157490. Work at LANL was supported by the US DOE Basic Energy Science project “Science at 100 Tesla.” This work was granted access to the High Performance Computing resources of Institut du Développement et des Ressources en Informatique Scientique under the allocations 2014-100384 made by Grand Equipement National de Calcul Intensif.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: saul{at}cinam.univ-mrs.fr.

Author contributions: G.R., A.S., and M.J. designed research; G.R., A.S., and M.J. performed research; G.R., A.S., H.A.D., M.B.S., and M.J. analyzed data; G.R., A.S., M.B.S., and M.J. wrote the paper; and H.A.D. provided the high-quality single crystals.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1421414112/-/DCSupplemental.

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