Geometrically frustrated ${\mathrm{GdInO}}_3$: An exotic system to study negative thermal expansion and spin-lattice coupling

In this article, we report negative thermal expansion and spin frustration in hexagonal ${\mathrm{GdInO}}_3$. Rietveld refinements of the x-ray diffraction patterns reveal that the negative thermal expansion in the temperature range of 50–100 K stems from the triangular lattice of ${\mathrm{Gd}}^{3+}$ ions. The downward deviation of the low-temperature inverse susceptibility $\left({\chi }^{-1}\right)$ versus $T$ plot from the Curie-Weiss law and the large value of the ratio, $|{\theta }_{\text{CW}}|/T_N>28$, where ${\theta }_{\text{CW}}$ and $T_N$ are respectively Curie-Weiss and Neel temperature, indicate a strong spin frustration, which inhibits long-range magnetic ordering down to 1.8 K. Magnetostriction measurements clearly demonstrate a spin-lattice coupling in the system. Low-temperature anomalous phonon softening, as obtained from temperature-dependent Raman measurements, also reveals the same. Our experimental observations are supported by first-principles density functional theory calculations of the electronic and phonon dispersion in ${\mathrm{GdInO}}_3$. The calculations suggest that the ${\mathrm{GdInO}}_3$ lattice is highly frustrated at low temperature. Further, the calculated normal mode frequencies of the Gd-related $\mathrm{\Gamma }$ point phonon modes reveal significant magnetoelastic coupling in this system. The competitive role of magnetic interaction energy and thermal stabilization energy in determining the change in interatomic distances is the possible origin for the negative thermal expansion in ${\mathrm{GdInO}}_3$ over a limited range of temperature.

Figure 1

Temperature dependence of the relative length change $\mathrm{\Delta }L/L\left(0\right)$. Blue shaded zone marks the temperature range for negative thermal expansion.

Figure 2

Rietveld refined pattern of ${\mathrm{GdInO}}_3$ at 50, 100, and 300 K. The red lines are the fitted patterns. The green bars show the positions of Bragg reflection peaks and the blue lines are the difference between the experimental and calculated patterns in each panel.

Figure 3

(a) Variation in $a(\circ ),c(*)$, and cell volume $\left(V,•\right)$ with temperature. The standard deviation of the refined parameters, as obtained from Rietveld analysis, are shown as error bars. The error bars are within the size of the symbols. (b) Variation of $\left(\mathrm{\Delta }\mathrm{Gd}=d_1-d_2\right)$ with temperature. The added standard deviations of $d_1$ and $d_2$ are shown as the error bars for each data point.

Figure 4

(a) Crystal structure of hexagonal ${\mathrm{GdInO}}_3$ and (b) triangular arrangement of Gd ions viewed along c-axis.

Figure 5

(a) $M-H$ curve at 2 K. Inset of the figure shows the first quadrant $M-H$ plots at 20 K and 40 K, (b) $M-T$ and ${\chi }^{-1}-T$ plots at $H$=50 Oe(green and black solid lines, respectively). The dashed line corresponds to the Curie-Weiss law. Inset of the figure shows $C_p/T$ vs $T$ plot.

Figure 6

Magnetostriction plotted as $\mathrm{\Delta }L\left(H\right)/L\left(0\right)$ vs. $H$ at 1.8, 20, 75, and 150 K.

Figure 7

Temperature-dependent Raman spectra of ${\mathrm{GdInO}}_3$ in the spectral range of 200 to $700{\mathrm{cm}}^{-1}$ after linear baseline correction.

Figure 8

Calculated atomic displacement patterns for experimentally observed Raman modes involving Gd1 (magenta spheres), Gd2 (blue spheres), In (grey spheres), and O (red spheres). The experimental values of the corresponding phonon wave numbers at room temperature are available in each panel. The green arrows mark the directions of the vibration of the atoms involved.

Figure 9

Evolution of Raman modes (mentioned in the text) with temperature for ${\mathrm{GdInO}}_3$.

Figure 10

Schematic diagram of all AFM configurations in Gd triangular lattice. The relative stabilization energy of different AFM configurations with respect to the FM configuration has been indicated at the bottom of each AFM configuration.

Figure 11

Calculated phonon spectra of ${\mathrm{GdInO}}_3$ for FM (shown by the filled area) and all three AFM (shown by the solid line) configurations at 0 K. Dashed lines mark the shift in phonon wave numbers between two magnetic states.