Magnetic order in quasi-two-dimensional molecular magnets investigated with muon-spin relaxation

We present the results of a muon-spin relaxation (${\mu }^{\text{+}}$SR) investigation into magnetic ordering in several families of layered quasi-two-dimensional molecular antiferromagnets based on transition-metal ions such as $S=\frac{1}{2}$ Cu${}^{\text{2+}}$ bridged with organic ligands such as pyrazine. In many of these materials magnetic ordering is difficult to detect with conventional magnetic probes. In contrast, ${\mu }^{\text{+}}$SR allows us to identify ordering temperatures and study the critical behavior close to $T_{\mathrm{N}}$. Combining this with measurements of in-plane magnetic exchange $J$ and predictions from quantum Monte Carlo simulations we may assess the degree of isolation of the 2D layers through estimates of the effective inter-layer exchange coupling and in-layer correlation lengths at $T_{\mathrm{N}}$. We also identify the likely metal-ion moment sizes and muon stopping sites in these materials, based on probabilistic analysis of the magnetic structures and of muon-fluorine dipole-dipole coupling in fluorinated materials.

Figure 1

Bridging ligands used in the compounds described in this paper: (a) pyrazine (N${}_{\text{2}}$C${}_{\text{4}}$H${}_{\text{4}}$, abbreviated pyz); and (b) pyridine-$N$-oxide (C${}_{\text{5}}$H${}_{\text{5}}$NO, abbreviated pyo). Dashed lines indicate where the ligands bond to other parts of molecular structures.

Figure 2

Example $A\left(t\right)$ spectra with fits above ($T=25$ K) and below ($T=0.35$ K) the magnetic transition temperature $T_{\mathrm{N}}=1.4$ K for [Cu(HF${}_{\text{2}}$)(pyz)${}_{\text{2}}$]BF${}_{\text{4}}$. Note the approximately equal initial asymmetry, and the oscillations and “$\frac{1}{3}$ tail” observed in the ordered phase. The slow oscillation observed for $T>T_{\mathrm{N}}$ is due to muon-fluorine dipole-dipole oscillations.

Figure 3

The structure of [Cu(HF${}_{\text{2}}$)(pyz)${}_{\text{2}}$]PF${}_{\text{6}}$, as an example of the [$M$(HF${}_{\text{2}}$)(pyz)${}_{\text{2}}$]$X$ series. Copper ions are joined in a 2D square lattice by pyrazine ligands to form Cu(pyz)${}_{\text{2}}^{\text{2+}}$ sheets; the 2D layers are joined in the third dimension by HF${}_{\text{2}}^{-}$ groups, making a pseudocubic 3D structure; and this structure is stabilized by a PF${}_{\text{6}}^{-}$ anion at the center of each cubic pore. For clarity, hydrogen atoms attached to pyrazine rings have been omitted, and only one PF${}_{\text{6}}^{-}$ anion is shown.

Figure 4

Data and the results of fits to Eq. (3) for [Cu(HF${}_{\text{2}}$)(pyz)${}_{\text{2}}$]$X$ magnets with tetrahedral anions $X$${}^{-}$. From left to right: (a) and (d) show sample asymmetry spectra $A\left(t\right)$ for $T<T_{\mathrm{N}}$ along with a fit to Eq. (3); (b) and (e) show frequencies as a function of temperature [no data points are shown for the second line because this frequency ${\nu }_2$ was held in fixed proportion to the first, ${\nu }_1$ (see text)]; and (c) and (f) show relaxation rates ${\lambda }_i$ as a function of temperature. In the $\nu \left(T\right)$ plot, error bars are included on the points but in most cases they are smaller than the marker being used. The solid line representing ${\nu }_1$ in (b) and (c) corresponds to the filled circles in the third column of graphs [(c) and (f)] for that component’s relaxation, ${\lambda }_1$, while the dashed line and unfilled circles correspond to ${\nu }_2$ and ${\lambda }_2$, respectively. The filled triangles correspond to the fast relaxation ${\lambda }_3$.

Figure 5

Example data and fits for [Cu(HF${}_2$)(pyz)${}_{\text{2}}$]$X$ magnets with octahedral anions $X$${}^{-}$. From left to right: (a), (d), (g), (j), and (m) show sample asymmetry spectra $A\left(t\right)$ for $T<T_{\mathrm{N}}$ along with a fit to Eq. (3); (b), (e), (h), (k), and (n) show frequencies as a function of temperature [no data points are shown for the second line because this frequency ${\nu }_2$ was held in fixed proportion to the first, ${\nu }_1$ (see text)]; and (c), (f), (i), (l), and (o) show relaxation rates ${\lambda }_i$ as a function of temperature. In the $\nu \left(T\right)$ plot, error bars are included on the points but in most cases they are smaller than the marker being used. The solid line representing ${\nu }_1$ in (b), (e), (h), (k), and (n) corresponds to the filled circles in the third column of graphs [(c), (f), (i), (l), and (o)] for that component’s relaxation, ${\lambda }_1$, while the dashed line and unfilled circles correspond to ${\nu }_2$ and ${\lambda }_2$, respectively.

Figure 6

Sample TF ${\mu }^{\text{+}}$SR data measured for [Cu(HF${}_{\text{2}}$)(pyz)${}_{\text{2}}$]BF${}_{\text{4}}$ in an applied field of 2 T are shown in (a) and (b). Data are shown in the “rotating reference frame,” rotating at ${\gamma }_{\mu }\times 1.9\mathrm{T}=257\mathrm{MHz}$, nearly canceling out spin precession induced by the $2\mathrm{T}$ applied transverse field. (c) The evolution of the magnetic broadening $\sigma$ with $T$, showing a magnetic transition at 1.98 K in 2 T. (d) The $B$-$T$ phase diagram from Ref. 12 showing the nonmonotonic behavior at low applied magnetic field. In the key, HC is heat capacity, theory represents the results of computational modeling, and ${\mu }^{\text{+}}$SR shows our results from TF measurements (see main text).

Figure 7

Data taken at $T=15\mathrm{K}\gg T_{\mathrm{N}}=1.4\mathrm{K}$ for [Cu(HF${}_{\text{2}}$)(pyz)${}_{\text{2}}$]BF${}_{\text{4}}$, showing F$\mu$ oscillations along with a fit to Eq. (11). The inset shows the energy levels present in a simple system of two $S=\frac{1}{2}$ spins, along with the allowed transitions.

Figure 8

(a) Muon-fluorine dipole-dipole oscillations in [Cu(HF${}_2$)(pyz)${}_2$]BF${}_4$ over a temperature range $2\le T\le 300\mathrm{K}$. Asymmetry spectra are displaced vertically so as to approximately align with the temperature scale on plots (b) and (c). Ticks on the $y$ axis of (a) denote 1% asymmetry. Plot (b) shows the fitted value for $r_{\mathrm{F}-\mu }$ as a function of temperature. The line shown is a fit to the low-$T$ points with a $T^2$ scaling law, Eq. (14). The upper $x$ axis shows values of the dipole frequency, ${\nu }_{\mathrm{d}}$, which correspond to the lower $x$-axis values of $r_{F-\mu }$. Plot (c) shows fitted relaxation rates ${\lambda }_{\mathrm{F}-\mu }$ and ${\lambda }_0$, referring to the relaxation of the F$\mu$ function $D_z\left(t\right)$ in Eq. (11), and the pure relaxation in Eq. (13). Shaded regions indicate temperatures where $Q=2\pi {\lambda }_{\mathrm{F}-\mu }/{\omega }_{\mathrm{d}}>2$, roughly parametrizing the disappearance of the oscillations. (d), (e), and (f) follow (a), (b), and (c), but show data for [Cu(HF${}_2$)(pyz)${}_2$]ClO${}_4$. (g), (h), and (i) similarly, but for [Cu(HF${}_2$)(pyz)${}_2$]SbF${}_6$ over the range $5\le T\le 250\mathrm{K}$. The $5\mathrm{K}$ $A\left(t\right)$ plot is omitted because the background is raised substantially by approach to the transition to LRO.

Figure 9

Probability density functions of muon precession frequencies at positions close to likely muon stopping sites in [Cu(HF${}_{\text{2}}$)(pyz)${}_{\text{2}}$]$X$ with Cu${}^{\text{2+}}$ moments $\mu ={\mu }_{\mathrm{B}}$. The graphs show dipole fields near the fluorine or oxygen atoms in the negative anions; near the fluorine atoms in the bifluoride ligands; and near the carbon and nitrogen atoms, as a proxy for proximity to the pyrazine ring. The type of line indicates the compound for which the calculation was performed. The muon site is constrained to be close to particular atoms, indicated by line color. The shaded areas indicate ranges of fitted frequencies as $T\to 0$; two frequencies ${\nu }_{1,2}^{{\mathrm{BF}}_4}$ represent those observed where $X$${}^{-}$ $=$ BF${}_{\text{4}}^{-}$, and ${\nu }_{1,2}^{{\mathrm{PF}}_6}$ those observed in the $X$${}^{-}$ $=$ PF${}_{\text{6}}^{-}$ analog.

Figure 10

Probability density functions for muons in the putative oscillating sites near the pyrazine rings in [Cu(HF${}_{\text{2}}$)(pyz)${}_{\text{2}}$]BF${}_{\text{4}}$ (a) for muon precession frequency $\nu$ assuming that the moment on the copper site $\mu ={\mu }_{\mathrm{B}}$, created from a histogram of dipole fields evaluated using Eq. (16) at points satisfying the constraints detailed in the text; and (b) for moment on the copper sites given the frequencies actually observed, evaluated using the pdfs in (a) and Eq. (18). Lines represent trial magnetic structures. All exhibit antiferromagnetic coupling through both the bifluoride and pyrazine exchange paths, while (i) has copper moments pointing at ${45}^{\circ }$ to the pyrazine grid, (ii) has moments along one of the pyrazine grid directions ($\mathit{a}$ or $\mathit{b}$), and (iii) has copper moments pointing along the bifluoride axis ($\mathit{c}$).

Figure 11

Structure of [Cu(pyz)${}_{\text{2}}$(pyo)${}_{\text{2}}$](BF${}_{\text{4}}$)${}_{\text{2}}$. Copper ions lie in 2D square layers, bound by pyrazine rings. Pyridine-$N$-oxide ligands protrude from the coppers in a direction approximately perpendicular to these layers. Tetrafluoroboride ions fill the pores remaining in the structure. Ion sizes are schematic; copper ions are shown twice as large for emphasis, and hydrogens have been omitted for clarity.

Figure 12

Example data and fits for [Cu(pyz)${}_{\text{2}}$(pyo)${}_{\text{2}}$]$Y$${}_{\text{2}}$. From left to right: (a) and (d) show sample asymmetry spectra $A\left(t\right)$ for $T<T_{\mathrm{N}}$ along with a fit to Eq. (19); (b) and (e) show the frequency $\nu$ as a function of temperature; and (c) and (f) show relaxation rates ${\lambda }_i$ as a function of temperature. The relaxation rates ${\lambda }_1$, associated with the oscillation, and ${\lambda }_3$, the fast-relaxing initial component, do not vary significantly with $T$ and are not shown. Only a slight trend in ${\lambda }_2$ in the $Y$${}^{-}$ $=$ PF${}_{\text{6}}^{-}$ material [graph (f)] is observed.

Figure 13

Structure of [Cu(pyo)${}_{\text{6}}$](BF${}_{\text{4}}$)${}_{\text{2}}$, viewed along the threefold ($\mathit{c}$) axis, after Ref. 45. Copper ions are surrounded by octahedra of six oxygens, each part of a pyridine-$N$-oxide ligand; [Cu(pyo)${}_{\text{6}}$]${}^{\text{2+}}$ complexes space-pack with BF${}_{\text{4}}^{-}$ stabilizing the structure. Ion sizes are schematic; copper ions are shown twice as large for emphasis, and hydrogens have been omitted for clarity. As indicated in the top left, the $\mathit{a}$ and $\mathit{b}$ directions lie in the plane of the paper, separated by $\gamma ={120}^{\circ }$, while the $\mathit{c}$ direction is out of the page.

Figure 14

Example data and fits for [Cu(pyo)${}_{\text{6}}$](BF${}_{\text{4}}$)${}_{\text{2}}$. From left to right: (a) shows sample asymmetry spectra $A\left(t\right)$ for $T<T_{\mathrm{N}}$ and $T>T_{\mathrm{N}}$, along with fits to Eqs. (19) and (20), respectively; (b) shows frequency as a function of temperature; and (c) shows relaxation rate ${\lambda }_2$ as a function of temperature; ${\lambda }_1$ was held fixed during the fitting procedure, and ${\lambda }_3$ persists for $T>T_{\mathrm{N}}$. In the $\nu \left(T\right)$ plot, error bars are included on the points but in most cases they are smaller than the marker being used.

Figure 15

Example data and fits for [Cu(pyo)${}_{\text{6}}$](ClO${}_{\text{3}}$)${}_{\text{2}}$. Representative $A\left(t\right)$ spectra for $T<T_{\mathrm{N}}$ and $T>T_{\mathrm{N}}$, along with fits to Eq. (21), are shown in (a), while the value of the relaxation rate, $\lambda$, as a function of temperature is shown in (b).

Figure 16

Sample asymmetry spectrum for [Cu(pyo)${}_{\text{6}}$](PF${}_{\text{6}}$)${}_{\text{2}}$ measured at $T=0.02\mathrm{K}$. The spectra remain indistinguishable from this across the range of temperatures examined, $0.02\mathrm{K}\le T\le 1\mathrm{K}$.

Figure 17

Example data and fits for Ag(pyz)${}_{\text{2}}$(S${}_{\text{2}}$O${}_{\text{8}}$). From left to right: (a) shows sample asymmetry spectra $A\left(t\right)$ for $T<T_{\mathrm{N}}$ along with a fit to Eq. (22); (b) shows the frequency as a function of temperature; and (c) shows relaxation rates ${\lambda }_i$ as a function of temperature. Filled circles show the relaxation rate ${\lambda }_1$, which relaxes the oscillation. The filled triangles correspond to the relaxation ${\lambda }_2$.

Figure 18

Example data and fits for $M$ $=$ Ni magnets. From left to right: (a) and (d) show sample asymmetry spectra $A\left(t\right)$ for $T<T_{\mathrm{N}}$ along with a fit to Eq. (3); (b) and (e) show frequencies as a function of temperature [no data points are shown for the second line because this frequency ${\nu }_2$ was held in fixed proportion to the first, ${\nu }_1$ (see text)]; and (c) and (f) show relaxation rate ${\lambda }_3$ as a function of temperature. In the $\nu \left(T\right)$ plot, error bars are included on the points but in most cases they are smaller than the marker being used. The inset to (f) shows ${\nu }_2\equiv \nu$ plotted against ${\lambda }_3\equiv \lambda$ on a log-log scale. The black line shows $\lambda ={\nu }^2$.

Figure 19

The asymmetry at late times $A(t>5\mu$s) measured in Ni(HF${}_{\text{2}}$)(pyz)${}_{\text{2}}$PF${}_{\text{6}}$ along with a fit to Eq. (24). Overlaid are regions corresponding to the range of values found by fitting the frequency of oscillations as a function of temperature with Eq. (4) ($5.5\mathrm{K}\le T_{\mathrm{N}}\le 6.2\mathrm{K}$), and the value of the $\lambda$-like anomaly in specific-heat capacity (SHC) measurements with a representative error of $0.1\mathrm{K}$ ($6.1\mathrm{K}\le T_{\mathrm{N}}\le 6.3\mathrm{K}$). The darker area represents the overlap between these regions.

Figure 20

Quantification of the two-dimensionality of the materials in this paper and comparison with other notable 2DSLQHA systems. Filled circles show materials investigated in this paper; open circles show other molecular materials; and filled triangles show inorganic systems. The low dimensionality is parametrized first with the directly experimental ratio $T_{\mathrm{N}}/J$, and then with predictions from quantum Monte Carlo simulations for both the exchange anisotropy, $J_{\perp }/J$ [Eq. (6)], and the correlation length of an ideal 2D Heisenberg antiferromagnet with the measured $J$ at a temperature $T_{\mathrm{N}}$ [Eq. (7)]. The graying-out of the axis for $J_{\perp }/J$ indicates where Eq. (6) is extrapolated beyond the range for which it was originally derived (Ref. 27). Values of $T_{\mathrm{N}}/J$ for the other materials were evaluated from Refs. 50, 51, 52.

Figure 21

Critical exponent $\beta$, as commonly extracted from Eq. (4), plotted against the experimental ratio $k_{\mathrm{B}}{T}_{\mathrm{N}}/J$, indicative of exchange anisotropy. Bright filled circles indicate [Cu(HF${}_{\text{2}}$)(pyz)${}_{\text{2}}$]$X$ materials examined in Sec. 3, while darker circles indicate other materials studied in this work: [Cu(pyz)${}_{\text{2}}$(pyo)${}_{\text{2}}$]BF${}_{\text{4}}$ (Sec. 4); [Cu(pyo)${}_{\text{6}}$](BF${}_{\text{4}}$)${}_{\text{2}}$ (Sec. 5); Ag(pyz)${}_{\text{2}}$(S${}_{\text{2}}$O${}_{\text{8}}$) (Sec. 6). Open circles indicate other molecular materials: CuF${}_{\text{2}}$(H${}_{\text{2}}$O)${}_{\text{2}}$(pyz) (Ref. 54); [Cu(pyz)${}_{\text{2}}$](ClO${}_{\text{4}}$)${}_{\text{2}}$ (Ref. 8).