Spin dilution in frustrated two-dimensional $S=\frac{1}{2}$ antiferromagnets on a square lattice

${}^7\mathrm{Li}$ and ${}^{29}\mathrm{Si}$ nuclear magnetic resonance, muon spin relaxation ($\mu$SR), and magnetization measurements in ${\mathrm{Li}}_2{\mathrm{V}}_{1-x}{\mathrm{OTi}}_x{\mathrm{SiO}}_4$, for $0⩽x⩽0.2$, are presented. The $x=0$ compound is a prototype of frustrated two-dimensional Heisenberg antiferromagnet on a square-lattice with competing nearest- $\left(J_1\right)$ and next-nearest- $\left(J_2\right)$ neighbor exchange interactions. ${\mathrm{Ti}}^{4+}(S=0)$ for ${\mathrm{V}}^{4+}(S=1∕2)$ substitution yields the spin dilution of the antiferromagnetic layers. The analysis of the magnetization and of the nuclear spin-lattice relaxation rate shows that spin dilution not only reduces the spin stiffness by a factor $\simeq {(1-x)}^2$, but also causes the decrease of the effective ratio $J_2\left(x\right)∕J_1\left(x\right)$. Moreover, the sublattice magnetization curves derived from zero-field $\mu$SR measurements in the collinear phase point out that, at variance with nonfrustrated two-dimensional Heisenberg antiferromagnets, spin dilution affects the low-temperature staggered magnetization only to a minor extent. This observation is supported also by the $x$ dependence of the collinear ordering temperature. The results obtained for the Ti doped samples are discussed in the light of the results previously obtained in the pure $x=0$ compound and in nonfrustrated two-dimensional Heisenberg antiferromagnets with spin dilution.

Figure 1

Temperature dependence of the susceptibility in ${\mathrm{Li}}_2{\mathrm{V}}_{1-x}{\mathrm{OTi}}_x{\mathrm{SiO}}_4$ for different Ti contents, in a magnetic field of 1 kG.

Figure 2

(Top) Doping dependence of $T_{\chi }^m$, $\Theta$, and $T_c$ as derived from susceptibility and NMR measurements. Lines track the initial suppression relation $-d{T}_c\left(x\right)∕dx=C{T}_c\left(0\right)$ described in the text, with $C=3.2$ (solid line) and $C=2$ (dotted line). (Bottom) The spin stiffness ${\rho }_s\left(x\right)$, derived from Eq. (5) (see text), and the ratio $T_{\chi }^m\left(x\right)∕\Theta \left(x\right)$ are reported for different Ti contents. The solid line passing through ${\rho }_s\left(x\right)∕{\rho }_s\left(0\right)$ data is the function ${(1-x)}^2$. The one through $T_{\chi }^m\left(x\right)∕\Theta \left(x\right)$ points is a guide to the eye.

Figure 3

Temperature dependence of ${}^7\mathrm{Li}$ nuclear spin-lattice relaxation rate in ${\mathrm{Li}}_2{\mathrm{V}}_{1-x}{\mathrm{OTi}}_x{\mathrm{SiO}}_4$ powders for different doping amounts $x$. The $x=0$ data (Refs. 6, 7) were measured on a single crystal with $\vec{H}\Vert c$. In order to compare them with the powder sample data they were divided by a factor 1.18 owing to the different hyperfine coupling involved. The lines are guide to the eye.

Figure 4

Temperature dependence of ${}^{29}\mathrm{Si}$ nuclear spin-lattice relaxation rate in ${\mathrm{Li}}_2{\mathrm{V}}_{1-x}{\mathrm{OTi}}_x{\mathrm{SiO}}_4$ for different doping amounts $x$. The lines are guide to the eye.

Figure 5

$\mu$SR asymmetry in ${\mathrm{Li}}_2{\mathrm{V}}_{1-x}{\mathrm{OTi}}_x{\mathrm{SiO}}_4$ for $x=0.05$ at 2.8 K (top) and 0.5 K (bottom). The solid line in the lower plot is the fit according to Eq. (2) in the text, while the one in the upper plot is the fit according to Eq. (3).

Figure 6

Temperature dependence of the local field at the muon site $H_{\mathrm{int}}$ in ${\mathrm{Li}}_2{\mathrm{V}}_{1-x}{\mathrm{OTi}}_x{\mathrm{SiO}}_4$ for $x=0.05$, obtained from Eq. (2) in the text. Data for ${\mathrm{Li}}_2{\mathrm{VOSiO}}_4$ (from Ref. 20) are reported for comparison. Fields and temperatures are normalized by $H_{\mathrm{int}}\left(0\right)=309.2\mathrm{G}$ and $T_c=2.6\mathrm{K}$ for $x=0.05$ and $H_{\mathrm{int}}\left(0\right)=313\mathrm{G}$ and $T_c=2.86\mathrm{K}$ for $x=0$. For the $x=0.11$ a value for $H_{\mathrm{int}}\left(0\right)=297\pm 5\mathrm{G}$ was derived.

Figure 7

Temperature dependence of the correlation time for the low frequency fluctuations in ${\mathrm{Li}}_2{\mathrm{VOSiO}}_4$ (top) and ${\mathrm{Li}}_2{\mathrm{V}}_{1-x}{\mathrm{OTi}}_x{\mathrm{SiO}}_4$ for $x=0.05$ (bottom).

Figure 8

Correlation length $\xi$ extracted from ${}^7\mathrm{Li}$ NMR $1∕T_1$ in ${\mathrm{Li}}_2{\mathrm{VOSiO}}_4$ by inverting Eq. (7) in the text, in the assumptions that $z=1$ or that $z=2$. The data are compared with theoretical calculations, for different values of $J_2∕J_1$, made in the framework of pure-quantum self-consistent harmonic approximation (Ref. 26). The vertical dotted line indicates the transition to the collinear phase.

Figure 9

Comparison of ${}^{63}\mathrm{Cu}$ $1∕T_1$ in ${\mathrm{La}}_2{\mathrm{Cu}}_{1-x}{\mathrm{Zn}}_x{0}_4$ and ${}^7\mathrm{Li}$ $1∕T_1$ in ${\mathrm{Li}}_2{\mathrm{V}}_{1-x}{\mathrm{OTi}}_x{\mathrm{SiO}}_4$ as a function of the scaled temperature $T∕{(1-x)}^2$.