Spin dynamics of the pyrochlore magnets ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Gd}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ in the paramagnetic state

The strongly correlated disordered phase of two highly frustrated pyrochlore magnets ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Gd}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ is probed using electron-spin resonance in the temperature range $1.3–30\mathrm{K}$. The deviation of the absorption line from the paramagnetic position $\nu =\gamma H$ observed in both compounds below the Curie-Weiss temperature ${\Theta }_{\mathrm{CW}}\simeq 10\mathrm{K}$ suggests an opening up of a gap in the excitation spectra. On cooling to $1.3\mathrm{K}$ (which is above the ordering transition $T_N\simeq 1.0\mathrm{K}$) the resonance spectrum is transformed into a wideband of excitations with the gap amounting to $\Delta \simeq 26\mathrm{GHz}$ $\left(1.2\mathrm{K}\right)$ in ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and $18\mathrm{GHz}$ $\left(0.8\mathrm{K}\right)$ in ${\mathrm{Gd}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$. The gaps increase linearly with the external magnetic field. For ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ this branch coexists with an additional nearly paramagnetic line absent in ${\mathrm{Gd}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$. These low-lying excitations with gaps, which are preformed above the ordering transition, may be interpreted as collective spin modes split by the single-ion anisotropy.

Figure 1

Magnetization curves at $T=1.75\mathrm{K}$ scaled to the saturation magnetization $M_{\mathrm{sat}}=g{\mu }_{B}S$ in ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ (◻) and ${\mathrm{Gd}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ (○) powder samples; solid lines are linear fits of the low-field parts of the curves.

Figure 2

(Color online) The field dependence of the absorption in ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ single crystals for $H\parallel \left[111\right]$ at the frequencies $\nu =9.6\mathrm{GHz}$ (sample No. 1, left panel), $\nu =36.1\mathrm{GHz}$ (sample No. 1, middle panel), and $\nu =69.6\mathrm{GHz}$ (sample No. 2, right panel). Solid lines are two-component Lorentzian fits as described in the text; the dashed and dash-dotted lines show line I and line II of the spectrum separately; out-of-curve points are diphenylpicrylhydrazyl (DPPH) labels used as a reference.

Figure 3

(Color online) The temperature dependence of the integrated intensity $I=W\Delta H$ of two resonance lines determined at various frequencies from fitting the transmitted signal by a single or double-Lorentzian form (as described in the text) and scaled to the value obtained for $20\mathrm{K}$: ˟, $\nu =9.6\mathrm{GHz}$ (line I); ◻ and ◼, $36.1\mathrm{GHz}$; ○ and ●, $69.6\mathrm{GHz}$ (respectively, lines I and II). The inset represents the inverse values of the integrated intensity in the whole temperature range, the solid line is a paramagnetic fit with the Curie-Weiss temperature $\Theta =10\mathrm{K}$.

Figure 4

(Color online) The field dependence of the resonance absorption in ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ powder sample for various temperatures at $\nu =96.7\mathrm{GHz}$ (upper panel) and at different frequencies at $T=1.3\mathrm{K}$ (lower panel); solid, dashed, and dash-dotted lines are the same as in Fig. 2; narrow signal and out-of-curve points are DPPH labels.

Figure 5

(Color online) The field dependence of the resonance absorption in ${\mathrm{Gd}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ powder sample for various frequencies at $T=4.2\mathrm{K}$ (left panel) and $1.3\mathrm{K}$ (right panel); solid lines are single-Lorentzian fits to the experimental points.

Figure 6

(Color online) Field dependence of the resonance absorption and its derivative (upper and lower left panels, respectively) at the frequency $\nu =27.1\mathrm{GHz}$ for the microwave field polarized along the direction of the external magnetic field and the corresponding data for the perpendicular microwave field polarization (upper right panel). Lower right corner inset shows the temperature evolution of the crossover into the ground state with destroyed “soft modes"; the solid line is a guide to the eye.

Figure 7

The field dependences of the absorption at $\nu =69.4\mathrm{GHz}$ in sample No. 1 recorded at different temperatures for $H\parallel \left[111\right]$ (left panel) and at $T=4.2\mathrm{K}$ for different sample orientations (right panel); narrow peaks at all curves have a paramagnetic DPPH label.

Figure 8

The comparison of absorption spectra of sample No. 1 (lower panel) and sample No. 2 (upper panel) taken at $T=4.2\mathrm{K}$ for several frequencies; the upper curve for the large sample and the lower curve for the small sample are obtained at the same frequency of the cylindrical cavity.

Figure 9

(Color online) The frequency-field diagram of the obtained ESR spectra. Upper panel: ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ powder sample (◻ and ○) and single crystal with $H\parallel \left[111\right]$ (◼, ●, and ˟) at different temperatures. Lines are $\nu =\Delta +\gamma H$ ($\gamma =g{\mu }_B∕h$ with $g=2.0$) fits to the data points: dashed, dash-dotted, and solid lines correspond to $T=20\mathrm{K}$, $4.2\mathrm{K}$, and $1.3\mathrm{K}$, respectively, with the gaps $\Delta =0$ (a typical paramagnet), $3.5\mathrm{GHz}$, $5\mathrm{GHz}$, and $26\mathrm{GHz}$. The inset represents the temperature dependence of the value $\nu -\gamma H_{\mathrm{res}}$ for the two resonance modes at various frequencies. Lower panel: analogous results for the powder sample of ${\mathrm{Gd}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ at $T=4.2\mathrm{K}$ (○) and $1.3\mathrm{K}$ (◻) described by gapped linear dependences with $\Delta =2.5\mathrm{GHz}$ (dash-dotted line) and $18\mathrm{GHz}$ (solid line); the inset is the same as for the ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ panel, but with the single-resonance line (error bars in both panels correspond to $\Delta H∕5$).