Study of one-dimensional nature of $S=1∕2$ ${(\mathrm{Sr},\mathrm{Ba})}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ and $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$ via ${}^{31}\mathrm{P}$ NMR

The magnetic behavior of the low-dimensional phosphates ${(\mathrm{Sr},\mathrm{Ba})}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ and $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$ was investigated by means of magnetic susceptibility and ${}^{31}\mathrm{P}$ nuclear magnetic resonance (NMR) measurements. We present here the NMR shift, the spin-lattice $(1∕T_1)$, and spin-spin $(1∕T_2)$ relaxation-rate data over a wide temperature range $0.02\mathrm{K}⩽T⩽300\mathrm{K}$. The temperature dependence of the NMR shift $K\left(T\right)$ is well described by the $S=1∕2$ Heisenberg antiferromagnetic chain model [D. C. Johnston, R. K. Kremer, M. Troyer, X. Wang, A. Klümper, S. L. Bud’ko, A. F. Panchula, and P. C. Canfield, Phys. Rev. B 61, 9558 (2000)] with an intrachain exchange of $J∕k_B\simeq 165$, 151, and $108\mathrm{K}$ in ${\mathrm{Sr}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, ${\mathrm{Ba}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, and $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$, respectively. Deviations from Johnston’s expression are seen for all these compounds in the $T$ dependence of $K\left(T\right)$ at low temperatures. ${}^{31}\mathrm{P}$ is located symmetrically between the Cu ions and fluctuations of the staggered susceptibility at $q=\pi ∕a$ should be filtered out due to vanishing of the geometrical form factor. However, the qualitative temperature dependence of our ${}^{31}\mathrm{P}$ NMR $1∕T_1$ indicates that relaxation due to fluctuations around $q=\pi ∕a$ (but $\ne \pi ∕a$) have the same $T$ dependence as those at $q=\pi ∕a$ and apparently dominate. Our measurements suggest the presence of magnetic ordering at $0.85\mathrm{K}$ in $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$ $(J∕k_B\simeq 108\mathrm{K})$ and a clear indication of a phase transition (divergence) in $1∕T_1\left(T\right)$, $1∕T_2\left(T\right)$, and a change of the line shape is observed. This enables us to investigate the one-dimensional (1D) behavior over a wide temperature range. We find that $1∕T_1$ is nearly $T$ independent at low temperatures $(1\mathrm{K}⩽T⩽10\mathrm{K})$, which is theoretically expected for 1D chains when relaxation is dominated by fluctuations of the staggered susceptibility. At high temperatures, $1∕T_1$ varies nearly linearly with temperature, which accounts for contribution of the uniform susceptibility.

Figure 1

Schematic diagram of ${\left[\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2\right]}_{\infty }$ linear chains (a) propagating along $b$ direction for ${(\mathrm{Ba}∕\mathrm{Sr})}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ and (b) along the $c$ direction for $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$. The possible interaction paths are shown by solid lines between atoms.

Figure 2

Magnetic susceptibility $(M∕H)$ vs temperature $T$ for ${\mathrm{Sr}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, ${\mathrm{Ba}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, and $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$ in an applied field of $5\mathrm{kG}$. The solid lines are best fits of the data to Eq. (1).

Figure 3

${}^{31}\mathrm{P}$ shift $K$ vs temperature $T$ for (a) ${\mathrm{Sr}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, (b) ${\mathrm{Ba}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, and (c) $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$. The solid lines are fits of Eq. (2) in the temperature range, $10\mathrm{K}⩽T⩽300\mathrm{K}$ and then extrapolated down to $0.01\mathrm{K}$. Inset shows $K$ vs $T$ on a logarithmic temperature scale for improved visualisation of the low-$T$ data.

Figure 4

${}^{31}\mathrm{P}$ shift $K$ vs spin susceptibility ${\chi }_{\text{spin}}$ for $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$. The solid line shows the linear fit.

Figure 5

Low-field $(H\simeq 4\mathrm{kG})$ ${}^{31}\mathrm{P}$ NMR spectra at different temperatures $T$ for (a) ${\mathrm{Sr}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ and (b) ${\mathrm{Ba}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$. The insets of (a) and (b) contain the high-field and low-field spectra at $1\mathrm{K}$. (c) and (d) show high-field and low-field spectra, respectively, for $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$ at various temperatures below $1\mathrm{K}$, showing the sudden change in line width. Also shown in (d) is the spectrum in case of IC order using Eq. (6) and parameters given in the text.

Figure 6

Spin-lattice relaxation rate $1∕T_1$ (both high and low fields) vs temperature $T$ for (a) ${\mathrm{Sr}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, (b) ${\mathrm{Ba}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, and (c) $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$. In the inset of (a), the normalized nuclear magnetization at high field is plotted as a function of pulse separation $t$ (at 6 and $2\mathrm{K}$) and the solid line is a single-exponential fit for ${\mathrm{Sr}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$. The inset of (b) displays the relaxation rate data for ${\mathrm{Ba}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ on a linear temperature scale. In the inset of (c), the temperature dependence of $1∕\left(K{T}_{1}T\right)$ is presented for the three compounds.

Figure 7

Spin-echo decays are plotted as a function of $t^2$ at two different temperatures for (a) ${\mathrm{Sr}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, (b) ${\mathrm{Ba}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, and (c) $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$. The solid lines show the fitting to a Gaussian function [Eq. (4)]. In the insets, $1∕T_{2G}$ is plotted as a function of temperature $T$.

Figure 8

Distribution function $g\left(x\right)$ which represents the NMR spectrum is plotted for (a) Cu site and (b) P site. In the insets of (a) and (b), ${(-1)}^l{\chi }_{\mathrm{alt}}\left(l\right)$ and ${(-1)}^l{\chi }_{\mathrm{alt}}\left(l\right)+{(-1)}^{l+1}{\chi }_{\mathrm{alt}}(l+1)$, respectively, are plotted as a function of the site index $l$ from the chain end at two different temperatures.

Figure 9

Normalized experimental data (open symbols) as well as theoretical curves (solid lines) of ${(1∕T_1)}_{\mathrm{norm}}$ are plotted vs $k_{B}T∕J$ for ${\mathrm{Sr}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, ${\mathrm{Ba}}_2\mathrm{Cu}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$, and $\mathrm{Ba}\mathrm{Cu}{\mathrm{P}}_2{\mathrm{O}}_7$ at low temperatures, $0.1\mathrm{K}⩽T⩽10\mathrm{K}$. The data and the theoretical curves have been scaled to 1 at $10\mathrm{K}$.