Static and dynamic magnetic properties of the spin-$\frac{1}{2}$ inequilateral diamond-chain compounds ${\mathit{A}}_3{\mathbf{Cu}}_3{\mathbf{AlO}}_2{\left({\mathbf{SO}}_4\right)}_4(\mathit{A}=\mathbf{K},\mathbf{Rb},\mathbf{Cs})$

Spin-$\frac{1}{2}$ compounds $A_3{\mathrm{Cu}}_3{\mathrm{AlO}}_2{\left({\mathrm{SO}}_4\right)}_4(A=\mathrm{K},\mathrm{Rb},\mathrm{and}\mathrm{Cs})$ have one-dimensional (1D) inequilateral diamond chains. We analyze the temperature dependence of the magnetic susceptibility and determine the magnetic exchange interactions. In contrast to azurite, a dimer is formed on one of the sides of the diamond. From numerical analyses of the proposed model, we find that the dimer together with a nearly isolated 1D Heisenberg chain characterize magnetic properties including magnetization curve and magnetic excitations. This implies that a dimer-monomer composite chain without frustration is a good starting point for describing these compounds.

Figure 1

(a) Crystal structure of $A_3{\mathrm{Cu}}_3{\mathrm{AlO}}_2{\left({\mathrm{SO}}_4\right)}_4(A=\mathrm{K},\mathrm{Rb},\mathrm{and}\mathrm{Cs})$. The gray, purple, light blue, and red circles denote Cu, $A$, Al, and O atoms, respectively. The tetrahedrons with the red dots at the corners denote (${\mathrm{SO}}_4$). The inequilateral diamond chains run along the $a$ axis. (b) Effective spin model of $A_3{\mathrm{Cu}}_3{\mathrm{AlO}}_2{\left({\mathrm{SO}}_4\right)}_4$. The circles represent ${\mathrm{Cu}}^{2+}$ ions with spin $\frac{1}{2}$. The blue broken, dark blue solid, black broken, black dashed-dotted, red thick solid, red thin solid, red dashed, and red dotted lines denote the exchange interactions $J_1,J_2,J_3,J_4,J_5,J_{\mathrm{m}},J_{\mathrm{d}}$, and $J_{\mathrm{d}}^{\prime }$, respectively.

Figure 2

Temperature dependence of spin susceptibility in $A_3{\mathrm{Cu}}_3{\mathrm{AlO}}_2{\left({\mathrm{SO}}_4\right)}_4$ for (a) $A=\mathrm{K}$ and (b) $A=\mathrm{Rb}$ and Cs. The red solid lines show the experimental data. The black dashed lines represent the fitted data obtained by the FTL method for a 24-spin inequilateral diamond chain. Note that the error of the FTL is within the width of the line. In (a), an ED result for an 18-spin chain is denoted by the brown dot-dashed line, and the Bethe-ansatz solution for the infinite Heisenberg chain is plotted by the green dotted line. The parameters are listed in Table 1.

Figure 3

Magnetization curve for ${\mathrm{K}}_3{\mathrm{Cu}}_3{\mathrm{AlO}}_2{\left({\mathrm{SO}}_4\right)}_4$. The red jagged solid line is a calculated curve by DMRG at zero temperature for a 120-site periodic chain with the exchange interactions estimated from $\chi$. The blue solid line represents the experimental result under the magnetic field up to 72 T at 4.2 K [11]. The green dashed line is the exact magnetization curve for the 1D Heisenberg model. The inset is a schematic of the spin configuration at the 1/3 plateau with the dimers formed by $J_5$ and the 1D chain with $J_{\mathrm{d}}$ whose spins are ferromagnetically aligned with the direction of the applied magnetic-field $H$.

Figure 4

Dynamical spin structure factor $S(q,\omega )$ obtained by dynamical DMRG for a 240-site periodic chain with the exchange interactions for ${\mathrm{K}}_3{\mathrm{Cu}}_3{\mathrm{AlO}}_2{\left({\mathrm{SO}}_4\right)}_4$. The red dashed line represents $(\pi /2)J|sinq|$ with $J=J_{\mathrm{d}}$, whereas the green dashed line represents ${\omega }_q=J_5+J_{\mathrm{m}}cosq+\frac{1}{4}{J}_{\mathrm{m}}^2/J_5(3-cos2q)$.