Magnetic excitations in the spin-$\frac{1}{2}$ tetramer substance ${\mathrm{Cu}}_2{}^{114}\mathrm{Cd}{}^{11}{\text{B}}_2{\mathrm{O}}_6$ obtained by inelastic neutron scattering experiments

We performed inelastic neutron scattering experiments on ${\mathrm{Cu}}_2{}^{114}\mathrm{Cd}{}^{11}{\text{B}}_2{\mathrm{O}}_6$ powder. The magnetic excitations at low temperatures are similar to those of the interacting spin-$\frac{1}{2}$ tetramers in the ordered state. The weak excitations existing above 3 meV suggest that the Higgs mode appears in ${\mathrm{Cu}}_2{\mathrm{CdB}}_2{\mathrm{O}}_6$ at ambient pressure and zero magnetic field. We evaluated $J_1=27.3\pm 1.0$ and $J_2=-14.0\pm 1.4$ meV for the intratetramer interactions and $J_3=-0.4\pm 0.2$ and $J_4=1.4\pm 0.2$ meV for the intertetramer interactions. The spin gap in the isolated spin tetramer was calculated to be 1.6 meV, which is less than the effective intertetramer interaction value $(3.6\pm 0.8$ meV). Therefore, antiferromagnetic long-range order is possible, although the ground state of the isolated spin tetramer is the spin-singlet state. We discuss the temperature dependence of the magnetic excitations.

Figure 1

(a) Schematic drawing of ${\mathrm{Cu}}^{2+}$ ion positions having spin-$\frac{1}{2}$ in ${\mathrm{Cu}}_2{\mathrm{CdB}}_2{\mathrm{O}}_6$. Two crystallographic Cu sites (Cu1 and Cu2) exist. Red and blue circles represent Cu1 and Cu2 sites, respectively. The $J_1$ and $J_2$ exchange interactions form spin tetramers. Two kinds of tetramers (I and II) exist, although the two are equivalent to each other as a spin system. We take the $J_3$ and $J_4$ intertetramer interactions into account. Arrows indicate ordered magnetic moments below $T_{\mathrm{N}}=9.8$ K. The space group is monoclinic $P{2}_1/c$ (No. 14) [26]. The lattice constants at 15 K are $a=3.4047\left(5\right)\AA ,b=15.140\left(2\right)\AA ,c=9.298\left(1\right)\AA$, and $\beta =92.80\left(1\right){}^{\circ }$ [27]. (b) Energies of excited states from the GS in the isolated spin tetramer formed by the $J_1$ and $J_2$ interactions $(+$ symbol) and those in the tetramer in the ordered state (other symbols). We evaluated the exchange interaction parameters that reproduce the magnetic excitations observed by inelastic neutron scattering measurements. Their values are listed in Table 1. $S^{\mathrm{T}}$ is the value of the sum of the spin operators in the tetramer. The eigenstates $|ij\rangle$ of the isolated tetramer are explicitly given in Ref. [30]. In the isolated tetramer, the GS is the spin-singlet $|02\rangle$ state. Circles, squares, and triangles indicate states with $S_z^{\mathrm{T}}=0,1$, and 2, respectively, in the ordered state. (c) An illustration of the isolated tetramer to explain the $|02\rangle$ and $|13\rangle$ states. Details are described in the Appendix.

Figure 2

INS intensity $I(Q,\omega )$ maps in the $Q-\omega$ plane for the ${\mathrm{Cu}}_2{}^{114}\mathrm{Cd}{}^{11}{\text{B}}_2{\mathrm{O}}_6$ powder at several temperatures. The energy of incident neutrons $E_{\mathrm{i}}$ is 7.74 meV. The right vertical key shows the INS intensity in arbitrary units.

Figure 3

$\omega$ dependence of $\mathrm{Im}\chi (Q,\omega )\equiv I(Q,\omega )*(1-e^{-\omega /k_{\mathrm{B}}T})$ for the ${\mathrm{Cu}}_2{}^{114}\mathrm{Cd}{}^{11}{\text{B}}_2{\mathrm{O}}_6$ powder. The energy of incident neutrons $E_{\mathrm{i}}$ is 7.74 meV. The horizontal bar indicates the energy resolution of 0.6 meV at $\omega =0$ meV. (a) $\mathrm{Im}\chi (Q,\omega )$ summed in the $Q$ range of $0.40–0.60{\AA }^{-1}$ at $T\le 9.0$ K and $0.45–0.65{\AA }^{-1}$ at 9.9 K. (b) $\mathrm{Im}\chi (Q,\omega )$ summed in the $Q$ range of $0.45–0.65{\AA }^{-1}$ at $T\ge 10.8$ K. The inset shows the $T$ dependence of the peak energy of $\mathrm{Im}\chi (Q,\omega )$. (c) $\mathrm{Im}\chi (Q,\omega )$ summed in the $Q$ range of $1.05–1.15{\AA }^{-1}$ at 5.3 K.

Figure 4

$Q$ dependence of the INS intensity for the ${\mathrm{Cu}}_2{}^{114}\mathrm{Cd}{}^{11}{\text{B}}_2{\mathrm{O}}_6$ powder. The energy of incident neutrons $E_{\mathrm{i}}$ is 7.74 meV. The data show the INS intensity summed in the $\omega$ range of $2.20–2.40$ meV $(1.25–1.65$ meV) at 5.3 K (12.5 K). The two lines indicate the calculated results described in the text.

Figure 5

$\omega$ dependence of the INS intensity for the ${\mathrm{Cu}}_2{}^{114}\mathrm{Cd}{}^{11}{\text{B}}_2{\mathrm{O}}_6$ powder. The energy of incident neutrons $E_{\mathrm{i}}$ is 42.1 meV. The data show the intensity summed in the $Q$ range of $1.8–2.2{\AA }^{-1}$. Red circles, green squares, and black diamonds show the data at 5.3, 9.9, and 50.3 K, respectively.

Figure 6

Dispersion relations of magnetic excitations (lines) along several symmetric axes calculated for the model of ${\mathrm{Cu}}_2{\mathrm{CdB}}_2{\mathrm{O}}_6$ shown in Fig. 1 using extended Holstein-Primakoff theory. The neutron scattering intensities are also depicted. The values of the exchange interactions are $J_1=27.3,J_2=-14.0,J_3=-0.4p$, and $J_4=1.4p$ meV with $p=1,0.7,0.47$, and 0.3 in (a), (b), (c), and (d), respectively. We used a broadening factor of $\mathrm{\Gamma }=0.5$ meV for the Gaussian function $exp[-\frac{{\{\omega -{\omega }_d\left(\mathbf{Q}\right)\}}^2}{{\mathrm{\Gamma }}^2}]$, where ${\omega }_d\left(\mathbf{Q}\right)$ is the excitation energy at $\mathbf{Q}$. In the ordered states (a) and (b), there are two transverse branches depicted as ${\mathrm{T}}_0$ and ${\mathrm{T}}_Q$, reflecting the two (I and II) tetramers in a unit cell, both of which are doubly degenerate. There are two L modes, ${\mathrm{L}}_0$ and ${\mathrm{L}}_Q$. In the critical state (c), the L and T modes become degenerate. The two lines reflect the two tetramers. In the paramagnetic state (d), the spin gap exists.

Figure 7

(a) INS intensity $I(Q,\omega )$ map in the $Q-\omega$ plane for the ${\mathrm{Cu}}_2{}^{114}\mathrm{Cd}{}^{11}{\text{B}}_2{\mathrm{O}}_6$ powder at 5.3 K. The energy of incident neutrons $E_{\mathrm{i}}$ is 7.74 meV. The right vertical key shows the intensity in arbitrary units. (b) INS intensity map in the $Q-\omega$ plane calculated for the model of ${\mathrm{Cu}}_2{\mathrm{CdB}}_2{\mathrm{O}}_6$ shown in Fig. 1 using extended Holstein-Primakoff theory. The values of the exchange interactions are listed in Table 1. We used a broadening factor of $\mathrm{\Gamma }=0.5$ meV.