Phase diagram of diluted Ising ferromagnet ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$

We present a systematic study of the phase diagram of ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$ ($0.25\le x\le 1$) Ising ferromagnets obtained from neutron scattering measurements and mean-field calculations. We show that while the thermal phase transition decreases linearly with dilution, as predicted by mean-field theory, the critical transverse field at the quantum critical point is suppressed much faster. This behavior is related to competition between off-diagonal dipolar coupling and quantum fluctuations that are tuned by doping and applied field, respectively. In this paper, we quantify the deviation of the experimental results from mean-field predictions, with the aim that this analysis can be used in future theoretical efforts towards a quantitative description.

Figure 1

(a) Crystal rotation scans through $\mathbf{Q}=\left(200\right)$ for the $x=0.46$ at three different temperatures. The red lines are fits described in the text. The inset demonstrates the enhancement of critical scattering around $T_{\mathrm{c}}(x=0.46)\approx 0.68$ K. (b) Scattering from long-range magnetic order (circles) and the critical scattering (squares) as a function of temperature. The lines are the power-law fits to the data.

Figure 2

Temperature dependence of the magnetic intensity of ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$ for $0.25\le x\le 1$. To ease comparison, for each compound we normalize the ratio of magnetic to nuclear intensity by the ratio of magnetic to nuclear intensity of ${\mathrm{LiHoF}}_4$ at base temperature. Solid lines show power-law fits from which the critical temperature was extracted as described in the text.

Figure 3

Intensity of the magnetic Bragg peak $\left(200\right)$ close to the phase transition in ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$ for $x=0.46$ and 1. The critical exponent $\beta$ is shown for each scan and composition.

Figure 4

(a) The magnetic phase diagram of ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$ as a function of ${\mathrm{Ho}}^{3+}$ content $x$. The Curie temperature $T_{\mathrm{c}}$ from neutron scattering (ns) is compared with magnetic susceptibility data ($\chi$) from Ref. [15]. The blue line denotes the mean-field prediction. At low ${\mathrm{Ho}}^{3+}$ concentrations the emergence of spin-glass (SG) and spin-liquid (SL) phases was reported. (b) Extracted $T_{\mathrm{c}}$ for each compound normalized to its corresponding mean-field value. The dashed line is a guide to the eye.

Figure 5

(a) Crystal rotation scans through $\mathbf{Q}=\left(200\right)$ for the $x=0.46$ sample at three different fields. The solid lines are fits as described in the text. The critical scattering is strongest around $H_{\mathrm{c}}$ (inset). (b) Field dependence of the magnetic Bragg peak intensity (circles) and critical scattering (squares). The plotted lines are power-law fits to the data.

Figure 6

Bragg peak intensity as a function of $H$ in ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$ at 100 mK. To ease the comparison, the curves are normalized to the maximum and minimum counts measured. The solid lines are power-law fits to the Bragg peak. The data has been displaced vertically.

Figure 7

Phase diagram of ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$ obtained from neutron diffraction (symbols).

Data for $x=1$ are from ac-susceptibility, thermal expansion, and magnetostriction measurements in Refs. [33, 37]; the values of $T_{\mathrm{c}}$ and $H_{\mathrm{c}}$ obtained from neutron scattering measurements on $x=1$ are plotted by solid black triangles. The solid (dashed) lines are mean-field calculations with (without) the hyperfine interaction. The mean-field curves are scaled (see Fig. 8) to match the low-temperature range of the phase diagram.

Figure 8

Critical field behavior as a function of ${\mathrm{Ho}}^{3+}$ content $x$ in ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$ obtained from neutron scattering (blue circles). The fields at $T_{\mathrm{c}}/2$ represent hyperfine-free values of $H_{\mathrm{c}}$ (green squares). The difference, $\mathrm{\Delta }$, between these two values demonstrates the hyperfine effect (red diamonds). The solid lines are mean-field calculations in the presence (absence) [blue (green) solid line] of hyperfine interactions, and the solid red line is the subtraction of the two curves. The dashed lines are $H_{\mathrm{c}}$ values which are obtained by rescaling the critical field by $x$.

Figure 9

The ratio of the magnetic to nuclear intensities as a function of ${\mathrm{Ho}}^{3+}$ content $x$ in ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$ at base temperature and $H=0$ (red circles), normalized to the value for $x=1$. The red solid line is the calculated ratio of the neutron scattering cross sections. The effective magnetic moment per ${\mathrm{Ho}}^{3+}$ ion is plotted as blue squares with a dashed line serving as a guide to the eye.