Far-infrared spectroscopy of spin excitations and Dzyaloshinskii-Moriya interactions in the Shastry-Sutherland compound $\mathrm{Sr}{\mathrm{Cu}}_2{\left(\mathrm{B}{\mathrm{O}}_3\right)}_2$

We have studied spin excitation spectra in the Shastry-Sutherland model compound $\mathrm{Sr}{\mathrm{Cu}}_2{\left(\mathrm{B}{\mathrm{O}}_3\right)}_2$ in magnetic fields using far-infrared Fourier spectroscopy. The transitions from the ground singlet state to the triplet state at $24{\mathrm{cm}}^{-1}$ and to several bound triplet states are induced by the electric field component of the far-infrared light. To explain the light absorption in the spin system we invoke a dynamic Dzyaloshinskii-Moriya (DM) mechanism where light couples to a phonon mode, allowing the DM interaction. Two optical phonons couple light to the singlet to triplet transition in $\mathrm{Sr}{\mathrm{Cu}}_2{\left(\mathrm{B}{\mathrm{O}}_3\right)}_2$. One is $a$-polarized and creates an intradimer dynamic DM along the $c$ axis. The other is $c$-polarized and creates an intradimer dynamic DM interaction, it is in the $\left(ab\right)$ plane and perpendicular to the dimer axis. Singlet levels at 21.5 and $28.6{\mathrm{cm}}^{-1}$ anticross with the first triplet as is seen in far-infrared spectra. We used a cluster of two dimers with a periodic boundary condition to perform a model calculation with scaled intra- and interdimer exchange interactions. Two static DM interactions are sufficient to describe the observed triplet state spectra. The static interdimer DM in the $c$-direction $d_1=0.7{\mathrm{cm}}^{-1}$ splits the triplet state sublevels in zero field [Cépas et al., Phys. Rev. Lett. 87, 167205 (2001)]. The static intradimer DM in the $\left(ab\right)$ plane (perpendicular to the dimer axis) $d_2=1.8{\mathrm{cm}}^{-1}$, allowed by the buckling of $\mathrm{Cu}\mathrm{B}{\mathrm{O}}_3$ planes, couples the triplet state to the $28.6{\mathrm{cm}}^{-1}$ singlet as is seen from the avoided crossing.

Figure 1

Cluster with two dimers (1,2) and (3,4). (a) Dimer (3,4) and four nearest-neighbor dimers. The thin dashed line shows the two dimer cluster boundary. Thin solid lines show the distortion of Cu-Cu superexchange bonds due to the buckling of Cu-O-B planes. Thick solid and dashed lines are the inter- and intradimer superexchange constants $j_1$ and $j_2$; interdimer DM vectors (${\mathbf{d}}_1$, solid arrow) are in the $c$ direction and intradimer DM vectors (${\mathbf{d}}_2$, empty arrow) in the $\left(ab\right)$ plane along $a$ and $b$ axis. (b) The two dimer model after the periodic boundary condition has been applied; interdimer interactions have doubled.

Figure 2

Differential absorption in ${\mathbf{E}}_1\perp \mathbf{c}$ (two upper curves) and ${\mathbf{E}}_1\Vert \mathbf{c}$ (lower curve) polarization. Spectra have been offset in the vertical direction.

Figure 3

Differential absorption spectra in magnetic field ${\mathbf{B}}_0\Vert \mathbf{a}$ at $4.4\mathrm{K}$. Vertical offset equals the magnetic field value in Tesla.

Figure 4

(Color online) Magnetic field dependence of line positions and line areas in ${\mathbf{E}}_1\Vert \mathbf{c}$ polarization at $4.4\mathrm{K}$; (a), (b) ${\mathbf{B}}_0\Vert \mathbf{c}$; (c), (d) ${\mathbf{B}}_0\Vert \mathbf{a}$. Solid lines are the results of the calculation based on the two dimer model: $j_1=24{\mathrm{cm}}^{-1}$, $2j_2=9.8{\mathrm{cm}}^{-1}$, $2d_1=1.4{\mathrm{cm}}^{-1}$, and $d_2=1.8{\mathrm{cm}}^{-1}$. Dashed lines in panels (a) and (c) are fits with parameters given in Table .

Figure 5

(Color online) Magnetic field dependence of line positions and line areas in ${\mathbf{E}}_1\Vert \mathbf{a}$ polarization at $4.4\mathrm{K}$; (a), (b) ${\mathbf{B}}_0\Vert \mathbf{c}$; (c), (d) ${\mathbf{B}}_0\Vert \mathbf{a}$. Solid lines are the results of the calculation based on the two dimer model: $j_1=24{\mathrm{cm}}^{-1}$, $2j_2=9.8{\mathrm{cm}}^{-1}$, $2d_1=1.4{\mathrm{cm}}^{-1}$, and $d_2=1.8{\mathrm{cm}}^{-1}$. Dashed lines in panels (a) and (c) are fits with parameters given in Table . The solid line in panel (b) is the sum of two theoretical line areas of $S_0$ to $T_{0m}\left(0\right)$ and to $T_{0p}\left(0\right)$ transitions shown by dashed lines. Dashed lines in (d) are eye guides (see text). The $18\mathrm{T}$ point a panels (a) and (b) was measured at $1.8\mathrm{K}$.

Figure 6

(Color online) Line positions (a) and line areas (b) in ${\mathbf{E}}_1\Vert \mathbf{a}$ and ${\mathbf{B}}_0\Vert \mathbf{b}$ configuration at $4.4\mathrm{K}$. The lines are results of the calculation based on the two dimer model and dynamic DM interaction. The additional splitting of triplet components (triangles) is caused by the magnetic field ${\mathbf{B}}_0$ being misaligned by 9° out of the $\left(ab\right)$ plane (Ref. 26). In panel (b) the line area (triangles up or triangles down) is a sum of line areas of split components.

Figure 7

Intradimer dynamic DM interactions. A lattice distortion with the normal coordinate $\mathbf{Q}$ (solid arrow) creates an intradimer DM interaction ${\mathbf{d}}_3$ (empty arrow). The $c$-axis phonon creates a dynamic DM interaction on both dimers while the $a$-axis phonon affects the dimer (1,2) only.