EXAFS investigation of the local structure in ${\mathrm{URu}}_{2-x}{\mathrm{Fe}}_x{\mathrm{Si}}_2$: Evidence for distortions below 100 K

X-ray absorption measurements at the U $L_{\mathrm{III}}$, Ru $K$, and Fe $K$ edges are reported for the hidden order (HO) material ${\mathrm{URu}}_{2-x}{\mathrm{Fe}}_x{\mathrm{Si}}_2$ ($x=0$, 0.05, 0.08, 0.10, 0.12, 0.15, and 0.20) as a function of $x$ and temperature $T$. When Fe is substituted for Ru, the local structure about Fe shrinks slightly and the first neighbor Fe-Si bond length decreases by $\approx 0.05\AA$. More importantly excess disorder is observed below 80–100 K (the coherence temperature $T^{*}$) in plots of the Debye-Waller factor ${\sigma }^2$ ($\sigma$ is the width of the pair distribution function); at low $T$ the data deviate from the usual Einstein or correlated-Debye model plots. This excess disorder is most prominent for the Ru-Si bond, and ${\sigma }^2$ actually increases below 80 K. These results suggest a local orthorhombic distortion with $B_{1g}$-like symmetry that develops below 80–100 K. A model that describes these local distortions is presented, and discussed in terms of other measurements that indicate a breaking of fourfold symmetry at low $T$. In addition, the square root of the difference between ${\sigma }^2\left(T\right)$ for the Ru-Si pair and a Debye fit to these data serves as an order parameter for this orthorhombic distortion, in the temperature range below 100 K. This quantity is a length related to $a\text{−}b$, the difference between the $a$ and $b$ lattice constants in the orthorhombic phase, and provides a connection between this distortion and $T^{*}$. X-ray absorption near edge structure (XANES) measurements also show that there are no changes in the edge positions down to 0.1 eV for any edge as a function of $x$, for $T$ in the HO regime.

Figure 1

The unit cell for ${\mathrm{URu}}_2{\mathrm{Si}}_2$; the U atoms (largest spheres, teal) occupy a simple body-centered tetragonal lattice. Around each U atom are eight Ru atoms (intermediate-size spheres, white) that form two square planar arrays and eight Si atoms (smallest spheres, blue) that form a cagelike structure. From Ref. [6]. Space group: $I4/mmm$; atom sites: U (0, 0, 0), Ru (0, 0.5, 0.25), Si (0, 0, 0.371); $a=4.13076\AA$, $c=9.5785\AA$.

Figure 2

The $k$-space data $k\chi \left(k\right)$ for the U $L_{\mathrm{III}}$ edge for each sample at 4 K. Traces are offset vertically by 0.3 for clarity, with $x=0$ at the bottom and 0.2 at the top. Note plots are very similar but the amplitude is slightly higher at high $k$ for the low Fe concentration samples.

Figure 3

(a) Plots of the U $L_{\mathrm{III}}$ edge XANES as a function of Fe concentration at 4 K. The energy for each sample has been corrected slightly using the U $L_{\mathrm{III}}$ edge data for the ${\mathrm{UO}}_2$ reference sample (see text). The inset provides an expanded view near the half-height point, showing that the spread of positions is $\le 0.1$ eV. (b) Similar plots for the Ru $K$ edge XANES for each sample, at 7 K. Likewise, the energy has been corrected using a Ru foil reference sample. Any edge shifts at the half-height point are $\le 0.1$ eV. The inset for these data magnifies the small structure near 22 132 eV, and shows that the tiny amplitude changes are less than 0.3%. (c) Plots of the Fe $K$ edge XANES (fluorescence), energy calibrated with an Fe foil, for the Fe doped samples; noise level is higher because of the low Fe concentrations. Within the noise all traces overlap well except for $x=0.05$. For that scan (red line), the peak near 7121 eV is slightly higher, and the structure shown in the inset for the tiny feature at 7106.5 eV has a lower amplitude. Again, shifts are $\le 0.1$ eV. The possible shifts for all edges are at the limit of our ability to correct energy shifts using reference data.

Figure 4

$r$-space data for the Ru $K$ and U $L_{\mathrm{III}}$ edges ($k$ weighting) as a function of $T$ for the $x=0.10$ sample. For the Ru $K$ edge, the FT range is 3.6–14.5 ${\AA }^{-1}$ while that for the U $L_{\mathrm{III}}$ edge is 3.5–14.8 ${\AA }^{-1}$. The first three peaks for the Ru $K$ edge data correspond roughly to Ru-Si, Ru-Ru, and Ru-U. For U the large peak near 2.85 Å arises from the overlap of the peaks for the U-Si and U-Ru pairs, and includes a tiny contribution from a weak U-Fe pair.

Figure 5

The Debye-Waller factor for the first Ru-Ru pair as a function of $T$ for each concentration. The solid line is a fit to the correlated-Debye model. The solid data points represent the first set of data, while the open data points represent the second set of data, for $x=0.0$, 0.1, and 0.2. The correlated-Debye curve was fit across all data points from 4–300 K, and fit both the low and high temperature data within the error of the data. Error bars for the points are comparable to the symbol size; errors in the Debye temperature are $\pm 6$ K, and comparable to the variation between samples. The data for each value of $x$ are successively offset vertically by 0.0003 ${\AA }^2$.

Figure 6

Left: The Debye-Waller factor ${\sigma }^2$ as a function of temperature for the first Ru-Si peak, at each concentration $x$. The solid line shows a fit to the correlated-Debye model. The final fit to this Debye model was fit across all data points from 100 to 300 K as discussed in the text. The solid data points represent the first set of data, while the open data points represent the second set of data, for $x=0.0$, 0.1, and 0.2, in both graphs. The Debye curves for Ru-Si deviate significantly from the low temperature data (by 0.0004 ${\AA }^2$), below $\approx 70$–80 K, providing evidence of excess static disorder. The ${\sigma }^2\left(T\right)$ data above 100 K increase slowly with $T$ leading to a very high Debye temperature of order 750 K. Three examples of an Einstein fit are shown as black dashes for $x=0.0$, 0.10, and 0.20; they deviate slightly from the Debye fit, but only at low $T$. Right: The Debye-Waller factor for the Ru-U pair as a function of $T$ for each concentration; the solid line is a fit to the Debye model, from 100 to 320 K. The Debye curve, for samples with many data points at higher $T$, deviates from the low temperature data (see insets), providing evidence of excess static disorder below 80 K. Note the much larger $y$ scale for Ru-U; although the deviation is comparable in magnitude to that for Ru-Si, it appears small on this scale. The thermal disorder for Ru-U grows much faster with $T$, consistent with a third neighbor pair and a low Debye temperature. The error bars for individual points are comparable to the symbol size. The errors for the Debye temperatures are $\pm 30$ K for Ru-Si and $\pm 5$ K for Ru-U.

Figure 7

${\sigma }^2$ as a function of $T$ for the U-U atom pair, at each value of $x$. For each data set the solid line is a fit to a correlated-Debye model over the restricted temperature range 100–300 K. For fits over the entire $T$ range from 4–300 K, the data near 100 K were always below the Debye curve while near 300 K the data were always above the curve. Below $\approx 80$ K the data are above the Debye curve (as observed for Ru-U) indicating excess disorder at low $T$. Error bars on individual points are comparable to the symbol size; the errors for the Debye temperatures are $\pm 5$ K. For each sample the plots are successively displaced vertically by 0.0015 Å.

Figure 8

Changes in the (a) Ru-Ru pairs and (b) the U-U and U-Si pairs under an orthorhombic distortion, as viewed along the $c$ axis. The Ru atoms are yellow (open symbols with crosshatch after the distortion). In (b) the Si atom is blue and the U atoms are cyan; the displaced U atoms are light cyan and crosshatched. The pair distances for the nearest Ru-Ru and U-Si pairs remain constant, while the pair distribution function for U-U splits slightly, with the longer pair distance along the horizontal and shorter distance along the vertical.

Figure 9

(a) and (b) Orthorhombic distortion of the Ru-Si pairs. The Si atoms (blue) will either move together or move apart, dictating the displacement of the Ru atom (yellow) along the $c$ axis (vertical). (a) A contraction of the distance between two Si atoms along the $a$ axis ($a\text{−}b$ plane), pushing the Ru atom upward; similarly, the separation of Si atoms along the $b$ axis also shifts the Ru atom up. (b) A compression along $b$ and expansion along $a$ which displaces the Ru atom downward along the $c$ axis. (c) Top half of the unit cell for ${\mathrm{URu}}_2{\mathrm{Si}}_2$ under the orthorhombic distortion, as discussed in the text. Again, the cyan spheres are U, the yellow spheres are Ru, and the blue spheres are Si. The arrows show the direction of compression and elongation when the orthorhombic distortion is present, in this case a compression of the $a$ axis. The square cross section of the tetragonal unit cell is transformed into a rectangle as shown previously in Fig. 8. Irrespective of which axis is compressed, locally there will always be two Ru atoms displaced upwards and two downwards.

Figure 10

Possible nanoscaled domains along the 1, 1, 0 direction when small orthorhombic distortions develop along the $a$ or $b$ axes, as show in Fig. 9. Here small, $1\times 4$ unit cell regions, are stacked to make larger domains. The open regions are elongated along the $a$ axis while the cyan colored regions are elongated along the $b$ axis, as shown by the horizontal and vertical arrows. This arrangement minimizes strain along both $a$ and $b$. Along either axis, there are alternating regions with long and short lattice constants; the average lattice constant will be nearly unchanged from the tetragonal value. In addition, because of the small orthorhombic distortions of these small domains, the unshaded regions will have a shear strain along the $a$ axis, while the cyan shaded regions will have a shear strain along the $b$ axis. The 110 planes become rotated slightly under the orthorhombic distortion, and the diffraction patterns for a scan along 1, 1, 0, will then be slightly different for the unshaded and shaded regions.

Figure 11

The square root of the average of the differences between the ${\sigma }^2$ data and a Debye fit for the Ru-Si pair. The blue squares are an average over all seven samples in the first data set, while the red solid circles are an average over samples with three Fe concentrations $x=0.0$, 0.1, and 0.2, in the second data set. This averaged quantity provides an order parameter for the small orthorhombic distortion observed, which is proportional to that used by Choi et al. [15]. The black dotted line is a power-law fit as a guide to the eye from 0–110 K.