Magnetic properties of a metal-organic antiferromagnet on a distorted honeycomb lattice

For temperatures $T$ well above the ordering temperature $T_{*}=3.0\pm 0.2\mathrm{K}$ the magnetic properties of the metal-organic material $\mathrm{Mn}{\left[{\mathrm{C}}_{10}{\mathrm{H}}_6\left(\mathrm{O}\mathrm{H}\right)\left(\mathrm{C}\mathrm{O}\mathrm{O}\right)\right]}_2\times 2{\mathrm{H}}_2\mathrm{O}$ built from ${\mathrm{Mn}}^{2+}$ ions and 3-hydroxy-2-naphthoic anions can be described by an $S=5∕2$ quantum antiferromagnet on a distorted honeycomb lattice with two different nearest-neighbor exchange couplings $J_2\approx 2J_1\approx 1.8\mathrm{K}$. Measurements of the magnetization $M(H,T)$ as a function of a uniform external field $H$ and of the uniform zero-field susceptibility $\chi \left(T\right)$ are explained within the framework of a modified spin-wave approach which takes into account the absence of a spontaneous staggered magnetization at finite temperatures.

Figure 1

(Color online) View along the $b$ axis of the metal-organic quantum magnet $\mathrm{Mn}{\left[{\mathrm{C}}_{10}{\mathrm{H}}_6\left(\mathrm{O}\mathrm{H}\right)\left(\mathrm{C}\mathrm{O}\mathrm{O}\right)\right]}_2\times 2{\mathrm{H}}_2\mathrm{O}$. Bold lines show exchange paths $\mathrm{Mn}\text{−}\mathrm{O}\text{−}\mathrm{C}\text{−}\mathrm{O}\text{−}\mathrm{Mn}$. The unit cell, denoted by the parallelogram, contains four crystallographically equivalent ${\mathrm{Mn}}^{2+}$ ions.

Figure 2

Chemical formula of $\mathrm{Mn}{\left[{\mathrm{C}}_{10}{\mathrm{H}}_6\left(\mathrm{O}\mathrm{H}\right)\left(\mathrm{C}\mathrm{O}\mathrm{O}\right)\right]}_2\times 2{\mathrm{H}}_2\mathrm{O}$.

Figure 3

(Color) View on the $\left(bc\right)$ plane of the metal-organic quantum magnet $\mathrm{Mn}{\left[{\mathrm{C}}_{10}{\mathrm{H}}_6\left(\mathrm{O}\mathrm{H}\right)\left(\mathrm{C}\mathrm{O}\mathrm{O}\right)\right]}_2\times 2{\mathrm{H}}_2\mathrm{O}$.

Figure 4

Distorted honeycomb lattice. The interactions between spins are displayed as solid lines. The underlying magnetic sublattice is a Bravais lattice, and its primitive cell can be chosen to be the dashed parallelogram. The corresponding primitive vectors are ${\mathit{a}}_1=a_1{\widehat{\mathit{e}}}_x$ and ${\mathit{a}}_2=a_2\cos \phi {\widehat{\mathit{e}}}_x+a_2\sin \phi {\widehat{\mathit{e}}}_y$.

Figure 5

Spin configuration $⟨{\mathit{S}}_i⟩$ in the classical ground state of a two-sublattice antiferromagnet. The dashed arrows represent a uniform magnetic field $B{\widehat{\mathit{e}}}_x$ in the $x$ direction and a staggered magnetic field ${\zeta }_i{B}_s{\widehat{\mathit{e}}}_z$ in the $z$ direction. The small solid arrows represent the vectors of a “comoving” basis that matches the direction defined by the local magnetization $⟨{\mathit{S}}_i⟩$. Not shown are the basis vectors ${\widehat{\mathit{e}}}_{\mathrm{A}}^1={\widehat{\mathit{e}}}_{\mathrm{B}}^1={\widehat{\mathit{e}}}_y$ which point into the plane of the paper.

Figure 6

Reduced Brillouin zone of the distorted honeycomb lattice. The primitive vectors are ${\mathit{b}}_1=(2\pi ∕a_1\sin \phi )(\sin \phi {\widehat{\mathit{e}}}_x-\cos \phi {\widehat{\mathit{e}}}_y)$ and ${\mathit{b}}_2=(2\pi ∕a_2\sin \phi ){\widehat{\mathit{e}}}_y$, where $a_1$, $a_2$, and the angle $\phi$ are defined in Fig. 4.

Figure 7

Normalized uniform magnetization $m\left(h\right)$ for $T=0$ and $h_s=0$. The solid line is the zero-temperature magnetization curve for the honeycomb lattice with $S=1∕2$ and $J_1=J_2$. For comparison we also show the corresponding curve for a square lattice and exact results for a linear antiferromagnetic chain (Ref. 19). However, the $S=1∕2$ chain is critical, so it is not surprising that it is poorly described by means of the spin-wave theory. Note that for $h_s=0$ the classical equation (3.5) is simply $m_0=h$.

Figure 8

Normalized staggered magnetization $n\left(h\right)$ at $T=0$ for honeycomb (solid line) and square lattices with $S=1∕2$ (dotted line). The classical equation $n_0=\sqrt{1-h^2}$ is plotted for comparison. We also show the curves for the anisotropic cases $J_1∕J_2=10,100$.

Figure 9

Uniform magnetization $m\left(h\right)$ for the honeycomb lattice with $S=5∕2$ and $J_1=J_2$ for two values of $T$.

Figure 10

Magnetization $m\left(H\right)$ of $\mathrm{Mn}{\left[{\mathrm{C}}_{10}{\mathrm{H}}_6\left(\mathrm{O}\mathrm{H}\right)\left(\mathrm{C}\mathrm{O}\mathrm{O}\right)\right]}_2\times 2{\mathrm{H}}_2\mathrm{O}$ up to field $H=5\mathrm{T}$. Experimental data are indicated by squares $\left(2\mathrm{K}\right)$, circles $\left(8\mathrm{K}\right)$, and triangles $\left(20\mathrm{K}\right)$. Theoretical magnetization curves for honeycomb lattice with $S=5∕2$ and $J_2=2J_1=1.95\mathrm{K}$ are denoted by lines.

Figure 11

Susceptibility $\chi \left(T\right)$ of $\mathrm{Mn}{\left[{\mathrm{C}}_{10}{\mathrm{H}}_6\left(\mathrm{O}\mathrm{H}\right)\left(\mathrm{C}\mathrm{O}\mathrm{O}\right)\right]}_2\times 2{\mathrm{H}}_2\mathrm{O}$. Circles are experimental data in a field of $2\mathrm{T}$. A theoretical fit for honeycomb lattice with $J_2=2J_1$ (solid line) gives $J_2=1.66\mathrm{K}$.