Infrared spectra of the low-dimensional quantum magnet ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$: Measurements and ab initio calculations

The reflectance of the insulating quasi-two-dimensional quantum magnet ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$ has been examined over a wide temperature and frequency range for light polarized parallel ($a$ axis) and perpendicular ($c$ axis) to the copper- and boron-oxygen sheets. The spectra have been measured for temperatures below the structural phase transition $T_s=395\text{K}$ for both polarizations; above $T_s$ a limited study of the in-plane properties was undertaken in the far-infrared region only. Several new modes appear in the reflectance just below $T_s$ along the $a$ and $c$ axes, while others are visible only for $T⪡T_s$. Below $T_s$, the intensity of some of the new modes displays little or no temperature dependence, while the intensity of some vibrations increases dramatically with decreasing temperature. Ab initio calculations have been performed for the room-temperature phase using density-functional theory, and the frequencies and atomic characters of the infrared-active phonons at the zone center were obtained using the direct method. The agreement between the calculated and experimentally observed frequencies is quite good, and assignments of the modes are discussed. The vibrational features that are observed only at low temperature appear to be magnetic in origin.

Figure 1

(Color online) The unit cell of ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$ in the (a) high-temperature $I4/mcm$ phase with coplanar copper and boron sheets, and (b) the low-temperature body-centered-tetragonal $I\overline{4}2m$ setting with $a=b=8.982\text{Å}$ and $c=6.643\text{Å}$, showing the buckling of the copper- and boron-oxygen planes and loss of inversion symmetry (detailed structural parameters for the low-temperature phase may be found in Table ). The sheets describe the $a\text{−}b$ planes, while the orthogonal direction is the $c$ axis. The Sr atoms (yellow) lie between the Cu (blue) and B (green) sheets; the four-fold coordinated Cu atoms form a distorted square with the O(1) atoms (red), while the three-fold coordinated B also lies on the corner of a distorted square formed by the Cu and O(2) atoms (magenta).

Figure 2

(Color online) The temperature dependence of the reflectance of ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$ versus the log of the frequency for (a) light polarized in the $a\text{−}b$ planes and (b) light polarized along the $c$ axis in the region of the infrared-active vibrations.

Figure 3

(Color online) The detailed temperature dependence of (a) the reflectance of ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$ above and below the structural phase transition at $T_s=395\text{K}$ for light polarized along the $a$ axis in the region of the three low-frequency $E$ lattice vibrations and (b) the imaginary part of the dielectric function. The asterisk in the upper panel denotes the appearance of a shoulder in the reflectance at low temperature. The experimental resolution is $2{\text{cm}}^{-1}$, except below 100 K, where it is $1{\text{cm}}^{-1}$. Note: the frequency intervals in the upper and lower panels are different.

Figure 4

(Color online) The temperature dependence of the fitted vibrational parameters of ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$ for light polarized along the $a$ axis for the (a) fundamental frequencies ${\omega }_j$, (b) widths ${\gamma }_j$, and (c) effective plasma frequencies ${\omega }_{p,j}$ for the in-plane modes in the $120–170{\text{cm}}^{-1}$ region. The solid circles track the mode visible in the high-temperature phase designated as the $141{\text{cm}}^{-1}$, while the open squares and circles represent vibrational modes at 125 and $157{\text{cm}}^{-1}$, respectively, that are observed below $T_s$ (dashed line). The dotted line represents the sum of the oscillator strengths obtained through addition in quadrature of the effective plasma frequencies for the two modes at 141 and $157{\text{cm}}^{-1}$. The errors associated with the fitted results are comparable to the size of the symbols.

Figure 5

(Color online) The detailed temperature dependence of (a) the imaginary part of the dielectric function of ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$ for light polarized in the $a\text{−}b$ planes in the region of the mode at $443{\text{cm}}^{-1}$ below 50 K and (b) the relative intensity of the mode at $443{\text{cm}}^{-1}$ below 35 K.

Figure 6

Calculated total energy for the formula unit ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$ in the low-temperature $I\overline{4}2m$ space group as a function of volume with constrained $c/a$ ratio. The calculated equilibrium volume of $260{\text{Å}}^3/\text{f}\text{.u}\text{.}$ is in good agreement with experimental value of $263{\text{Å}}^3$ (Ref. 36). The dotted line is a fit to the Murnaghan equation of state (Ref. 47).

Figure 7

(Color online) The calculated relative atomic displacements in the $a\text{−}b$ planes for the 16 infrared-active $E$ modes in ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$; only the upper copper-boron sheet in the unit cell is shown. The observed frequency is shown below the displacements, with the calculated frequency indicated in parenthesis. The color scheme used to depict the atoms is taken from Fig. 1.