Spin diffusion in the low-dimensional molecular quantum Heisenberg antiferromagnet $\mathrm{Cu}\left(\mathrm{pyz}\right){\left({\mathrm{NO}}_3\right)}_2$ detected with implanted muons

We present the results of muon-spin relaxation measurements of spin excitations in the one-dimensional quantum Heisenberg antiferromagnet $\mathrm{Cu}\left(\mathrm{pyz}\right){\left({\mathrm{NO}}_3\right)}_2$. Using density-functional theory we propose muon sites and assess the degree of perturbation the muon probe causes on the system. We identify a site involving the muon forming a hydroxyl-type bond with an oxygen on the nitrate group that is sensitive to the characteristic spin dynamics of the system. Our measurements of the spin dynamics show that in the temperature range $T_{\mathrm{N}}<T<J$ (between the ordering temperature $T_{\mathrm{N}}$ and the exchange energy scale $J$) the field-dependent muon spin relaxation is characteristic of diffusive transport of spin excitations over a wide range of applied fields. We also identify a possible crossover at higher applied fields in the muon probe's response to the fluctuation spectrum, to a regime where the muon detects early-time transport with a ballistic character. This behavior is contrasted with that found for $T>J$ and that in the related two-dimensional system $\mathrm{Cu}{\left(\mathrm{pyz}\right)}_2{\left({\mathrm{ClO}}_4\right)}_2$.

Figure 1

Low-energy muon sites in $\mathrm{Cu}\left(\mathrm{pyz}\right){\left({\mathrm{NO}}_3\right)}_2$. Translucent spheres represent the ionic positions in the unit cell without the muon. (a) The nitrate site, first configuration (bonding oxygen moved by 0.4 Å; other ions moved by $\le 0.2$ Å). (b) The nitrate site, second configuration, with the nearly ${90}^{\circ }$ rotation of the nitrate group around the $b$ axis (oxygen ions moved by 1.5, 1.2, and 0.3 Å). (c) The N(pyz) site (Cu ion moved by 1.4 Å).

Figure 2

Example $A\left(t\right)$ spectra for $\mathrm{Cu}\left(\mathrm{pyz}\right){\left({\mathrm{NO}}_3\right)}_2$ at (a) $T=0.33$ K and $B=0$ mT, (b) $T=0.33$ K and $B=0.5$ mT, (c) $T=0.33$ K and $B=11$ mT, and (d) $T=1.40$ K and $B=11$ mT. Solid lines represent the fits described in the text.

Figure 3

(a)–(c) Relaxation rates $\lambda$ for $\mathrm{Cu}\left(\mathrm{pyz}\right){\left({\mathrm{NO}}_3\right)}_2$. Solid lines: power law fit; dotted lines: global fit to ballistic behavior; dashed lines: fit to ballistic behavior between 50 mT and 450 mT with $J$ fixed at 7.2 T. (c) Dot dash line: an example power-law function with fixed $n=0.5$. (d) Relaxation rates $\lambda$ for $\mathrm{Cu}{\left(\mathrm{pyz}\right)}_2{\left({\mathrm{ClO}}_4\right)}_2$; dashed line: constant fit function.

Figure 4

Calculated spin density isosurfaces drawn at $0.004/a_0^3$ ($a_0$ is the Bohr radius). Yellow shading is positive; blue is negative. (a) Unperturbed $\mathrm{Cu}\left(\mathrm{pyz}\right){\left({\mathrm{NO}}_3\right)}_2$ without a muon showing the quasi-one-dimensional exchange via the pyz rings along the $a$ axis. (b) The nitrate site with a rotated nitrate group, performed for a neutral supercell. The muon's electron has been donated to the nearest-neighbor Cu ions, turning it into diamagnetic ${\mathrm{Cu}}^{+}$. (c) The N(pyz) site for a neutral supercell. The nearest-neighbor Cu ion is displaced considerably and, additionally, its moment is destroyed through the donation of the muon's electron. (d) The same N(pyz) site but for a charged supercell. The Cu ion is displaced by a similar amount but it retains a magnetic moment, leading to some magnetic overlap between neighboring Cu–pyz–Cu chains via a nitrate group.