Magnetization-step spectra of ${\left(\mathrm{C}{\mathrm{H}}_3\text{−}\mathrm{N}{\mathrm{H}}_3\right)}_2{\mathrm{Mn}}_x{\mathrm{Cd}}_{1-x}{\mathrm{Cl}}_4$ at $20\mathrm{mK}$: Fine structure and the second-largest exchange constant

The compounds ${\left(\mathrm{C}{\mathrm{H}}_3\text{−}\mathrm{N}{\mathrm{H}}_3\right)}_2{\mathrm{Mn}}_x{\mathrm{Cd}}_{1-x}{\mathrm{Cl}}_4$ are among the better physical realizations of a diluted Heisenberg antiferromagnet on the square lattice. The magnetization of three powder samples, with $x=0.063$, 0.067, and 0.157, was measured at temperatures $T\cong 20\mathrm{mK}$ in magnetic fields $B$ up to $17\mathrm{T}$. Magnetization-step (MST) spectra were obtained with a much higher resolution than in the earlier MST study at $T=0.6\mathrm{K}$. The earlier study uncovered only two spectral lines, near 6.6 and $13\mathrm{T}$. These lines were attributed to nearest-neighbor (NN) pairs. The higher-resolution richer spectrum observed at $T\cong 20\mathrm{mK}$ is interpreted using a theory which includes two exchange constants: the largest exchange constant, $J^{\left(1\right)}$, and the second-largest, $J^{\left(2\right)}$. A summary of the relevant results of this theory is given. The main result of the present work is the determination of $J^{\left(2\right)}$ by two independent methods. The first method used a series of MSTs that was observed at $T\cong 20\mathrm{mK}$ in magnetic fields below $2\mathrm{T}$. These MSTs, interpreted as the MSTs from $J^{\left(2\right)}$ pairs, gave $J^{\left(2\right)}∕k_{\mathrm{B}}=-0.227\pm 0.010\mathrm{K}$. The second method used the fine-structure (FS) splitting of the spectral line near $6.6\mathrm{T}$. It gave $J^{\left(2\right)}∕k_{\mathrm{B}}=-0.208\pm 0.006\mathrm{K}$. The FS splitting of the spectral line near $13\mathrm{T}$ is smaller, as expected, and is consistent with the same value of $J^{\left(2\right)}$. Attempts to identify the neighbor (in the cation lattice) that is associated with $J^{\left(2\right)}$ were unsuccessful. This failure is attributed to previously observed deviations from a random Mn distribution. The field separation between the strongest spectral line in the FS near $13\mathrm{T}$ and the strongest line near $6.6\mathrm{T}$ gave $J^{\left(1\right)}∕k_{\mathrm{B}}=-4.39\pm 0.05\mathrm{K}$, in excellent agreement with the earlier MST study. Much earlier measurements of the spin-wave dispersion curves in ${\left(\mathrm{C}{\mathrm{H}}_3\text{−}\mathrm{N}{\mathrm{H}}_3\right)}_2\mathrm{Mn}{\mathrm{Cl}}_4$ identified $J^{\left(1\right)}$ as the NN exchange constant $J_1$. Comments about the theoretical interpretation of the present data are relegated to two appendixes.

Figure 1

Skeletons and decorations in 18 of the cluster types of the $J_1\text{−}{J}_2$ cluster model. Solid and empty circles correspond to skeleton spins and decoration spins, respectively. Solid and dashed lines correspond to $J_1$ bonds and $J_2$ bonds, respectively. The upper part of the figure is for 11 of these cluster types in which the skeleton is simple. The lower part is for the seven cluster types with a compound skeleton.

Figure 2

(Color online) The spectra of all cluster types of the $J_1\text{−}{J}_2$ model that contribute to the ${\left(2\right)}_{\mathrm{S}}$ monoskeleton FS when $n_{\max }=5$. Spectral lines in the ${\left(2\right)}_{\mathrm{S}}$ monoskeleton FS are darker (dark blue). The lower part of the figure shows the spectra from cluster types with a simple skeleton of type ${\left(2\right)}_{\mathrm{S}}$. The upper part of the figure shows the spectra from cluster types with a compound skeleton containing a fragment of type ${\left(2\right)}_{\mathrm{S}}$. The labels for the cluster types follow Fig. 1. The abscissa is the primary reduced magnetic field $b_1=g{\mu }_{\mathrm{B}}B∕\mid J_1\mid$. All line intensities are chosen to be equal. These results are for $S=5∕2$, $J_2∕J_1=0.028$, and $T=0$.

Figure 3

(Color online) Magnetization curves of the three samples over the full range of the magnetic field $B$. The ordinate $m_{17}$ is the magnetization $M$ normalized to its value at $17\mathrm{T}$. The very-low-field region, the low-field region, and the field regions $n=1$ and $n=2$, are indicated.

Figure 4

(Color online) Experimental derivative $d{m}_{17}∕dB$ of the normalized magnetization $m_{17}=M∕M$$\left(17\mathrm{T}\right)$ in the low-field region.

Figure 5

(Color online) Experimental derivative (EXP) $d{m}_{17}∕dB$ for $x=0.067$ and simulations based on the $J_1\text{−}{J}_3$ model (see Ref. 20). One simulation is for the actual experimental temperature, $T=20\mathrm{mK}$. The other is for an effective temperature $T=100\mathrm{mK}$. The simulations use the exchange constants $J_1∕k_{\mathrm{B}}=J^{\left(1\right)}∕k_{\mathrm{B}}=-4.39\mathrm{K}$, and $J_3∕k_{\mathrm{B}}=J^{\left(2\right)}∕k_{\mathrm{B}}=-0.227\mathrm{K}$. These results are for the LF region.

Figure 6

(Color online) Magnetization curves in the field region $n=1$. The vertical gains for the three curves were adjusted so that between 6.0 and $8.0\mathrm{T}$ the change of the ordinate is the same for all samples.

Figure 7

(Color online) Derivatives of the magnetization curves in Fig. 6, for the field region $n=1$.

Figure 8

(Color online) Magnetization curves between 10 and $15\mathrm{T}$. This field interval includes the field region $n=2$. The vertical gains were adjusted so that the change in the ordinate between 10 and $15\mathrm{T}$ is the same for all three curves.

Figure 9

(Color online) Derivatives of the magnetization curves in Fig. 8, which includes the field region $n=2$.

Figure 10

Calculated difference $Y=g{\mu }_{\mathrm{B}}(B_n^{\prime }-B_n)∕\mid J^{\left(1\right)}\mid$ between the primary reduced fields at the mixed-triplet line and at the pure-pair line, plotted as a function of $X=J^{\left(2\right)}∕J^{\left(1\right)}$. The two curves are for the field regions $n=1$ and $n=2$.

Figure 11

(Color online) Comparison of the full magnetization curve for $x=0.067$ and a simulation based on the $J_1\text{−}{J}_3$ model. The simulation uses the exchange constants $J_1∕k_B=J^{\left(1\right)}∕k_B=-4.39\mathrm{K}$, $J_3∕k_B=J^{\left(2\right)}∕k_B=-0.208\mathrm{K}$, and an effective temperature $T=100\mathrm{mK}$.

Figure 12

(Color online) (a) Two simulations of magnetization curve in the field region $n=1$, assuming that $x=0.067$ and $T=20\mathrm{mK}$. One simulation uses the $J_1-J_2$ model, and the other the $J_1\text{−}{J}_3$ model. The exchange constants are $J_1∕k_B=-4.39\mathrm{K}$ and $J^{\left(2\right)}∕k_B=-0.208\mathrm{K}$, where $J^{\left(2\right)}$ is either $J_2$ or $J_3$. (b) Comparison of the experimental magnetization curve (EXP) for $x=0.067$ at $T=20\mathrm{mK}$ with simulations based on the $J_1\text{−}{J}_2$ and $J_1\text{−}{J}_3$ models. An effective temperature of $100\mathrm{mK}$ is used in these simulations.