Tuneable correlated disorder in alloys

Understanding the role of disorder, and the correlations that exist within it, is one of the defining challenges in contemporary materials science. However, there are few material systems, devoid of other complex interactions, that can be used to systematically study the effects of crystallographic conflict on correlated disorder. Here, we report extensive diffuse x-ray scattering studies on the epitaxially stabilized alloy ${\text{U}}_{1-x}{\text{Mo}}_x$, showing that a new form of intrinsically tuneable correlated disorder arises from a mismatch between the preferred symmetry of a crystallographic basis and the lattice upon which it is arranged. Furthermore, combining grazing incidence inelastic x-ray scattering and state-of-the-art ab initio molecular dynamics simulations, we discover strong disorder-phonon coupling. This breaks global symmetry and dramatically suppresses phonon lifetimes compared to alloying alone, providing an additional design strategy for phonon engineering. These findings have implications wherever crystallographic conflict can be accommodated, and they may be exploited in the development of future functional materials.

Figure 1

“As-quenched” metastable phase diagram for the U-Mo system with the samples discussed in this paper included. Molybdenum content ($\text{at.}\%$ Mo) was determined from the relaxed lattice parameter ($a_0$) using the empirical fit (red line) to the data from Dwight [26] (open circles). “As-quenched” $\alpha$ and $\gamma$ metastable fields are marked by red and white regions, respectively, separated by a thick dashed red line [27]. A thin dashed red line separates the ${\alpha }^{\prime }$ and ${\alpha }^{\prime \prime }$ metastable structures [27]. The light gray region corresponds to the off-stoichiometric stability limits of ${\gamma }^{\prime }$, the tetragonal compound ${\text{U}}_2\text{Mo}$ [28, 29]. Note that this region is superimposed onto the $\gamma \text{-field}$ and does not represent a phase boundary. A dashed blue line marks the solubility limit of molybdenum in uranium [26], after which a complex mixture of phases exists, including the metastable ${\gamma }^{\mathrm{s}}$ phase as well as the stable ${\gamma }^{\prime }$ and $\text{bcc}\left(\varepsilon \right)\text{Mo}$ phases. As metastable phase boundaries are hard to define, all boundaries are indicative and based on reported experimental values. This is particularly relevant in the $\gamma \text{-field}$, and as such no boundary between the ${\gamma }^{\mathrm{o}}$ and ${\gamma }^{\mathrm{s}}$ phase is reported. In general, the ${\gamma }^{\mathrm{o}}$ phase is formed at lower Mo concentrations, whereas the ${\gamma }^{\mathrm{s}}$ phase dominates at higher concentrations. All samples in this paper are ${\gamma }^{\text{s}}$ and shown as blue solid points with error bars smaller than point size in all cases. Inset: A schematic of the film structure.

Figure 2

Reciprocal space reconstructions of the ${\left(hk2\right)}_{\mathrm{s}}$ (left) and ${\left(h1l\right)}_{\mathrm{s}}$ (right) planes. All three samples investigated are shown with relevant $\text{at.}\%$ Mo indicated for each quadrant. A superstructure unit cell with ${\mathit{b}}_{\mathrm{s}}\equiv {\mathit{b}}_{\mathrm{p}}$ and a clockwise rotation was used for all reconstructions. Reflections are categorized and indexed in the bottom right quadrant, with indices in superstructure notation. Reflections are indicated in black (parent Bragg peaks), blue (N, domain 1), green (N, domain 2), and orange (H). Nb buffer reflections appear as doublets outside of the parent bcc reflections. Powder rings arise from the polycrystalline Nb cap, and narrow intense peaks correspond to substrate Bragg reflections, neither of which are included schematically. The ${\left\{132\right\}}_{\mathrm{s}}$ reflection, marked with an asterisk, is highlighted in all data sets to show mirror symmetry relations. (Right) Reconstructions for the ${\left(h1l\right)}_{\mathrm{s}}$ plane highlighting the evolution of the H-type signal. The H point is marked with a circle. $31\text{at.}\%$ peaks are indexed for ${\text{U}}_2\text{Mo}$ with $\mathit{c}$ parallel to ${\mathit{a}}_{\mathrm{p}}$ (red) and ${\mathit{c}}_{\mathrm{p}}$ (blue).

Figure 3

Relationship between parent and superstructures. (a) Orientational relationship (see the supplemental material VI [37] for transformation matrices) between parent (blue) and superstructure (red) unit cells for one of six possible domains, with $\left|\delta \right|=0.1$ for clarity; see the main text for details. All atoms in the “shear plane” (highlighted red) move collinearly with the direction of motion indicated by arrows on the plane edge. Alternate planes, demarcated by I, I, III, $\cdots$, move in antiphase. (b) Top-down view showing the ${45}^{\circ }$ relationship between the parent and superstructure. The directional relationship of ${\text{TA}}_1{\left[110\right]}_{\mathrm{p}}$ and ${\text{TA}}_2{\left[110\right]}_{\mathrm{p}}$ mode included in the bottom right. (c) Schematic of the atomic motions in a “phonon plane.” Blue dashed and red dotted lines refer to interatomic bonding in the parent and superstructure unit cells, respectively.

Figure 4

$\text{TA}{\left[001\right]}_{\mathrm{p}}$ intensity as a function of $\mathit{q}$ along $\mathrm{\Gamma }\to \mathbf{H}$, for $23\text{at.}\%$ Mo. The strong elastic response at $\mathit{q}=0.15$ has been omitted, and the lower-resolution (see methods) data, indicated by asterisks, are multiplied by 4 for clarity. Data points are shown as open squares, and total fit is shown as solid curves with the phonon contribution highlighted by the shaded regions. Fitted phonon energies are projected on the plane as gray points, with a gray dashed spline as a visual guide of the dispersion.

Figure 5

Phonon energy and linewidth dispersions. Top panel: Experimental ($23\text{at.}\%$ Mo) and theoretical ($25\text{at.}\%$ Mo) phonon dispersion curves. Transverse (longitudinal) acoustic modes are shown as blue squares (red circles), and theoretical results from a virtual crystal approximation are shown as dashed white lines. The full spectral function is plotted as a ${log}_{0.6}$ color map to rescale the intensity divergence at gamma. All directions are within the parent BZ. Bottom panel: Raw linewidths, ${\mathrm{\Gamma }}_{\mathrm{o}}$, are shown as gray squares (TA) and circles (LA) with deconvoluted linewidths ${\mathrm{\Gamma }}_{\mathrm{d}}$ shown by dashed blue (TA) and red (LA) trend lines. The smoothing methodology is described in the main body of the text. Errors were determined by smoothing raw errors through the same algorithm and are shown as confidence bands.

Figure 6

Comparison of theoretical and experimental linewidth dispersions. Smoothed experimental LA and TA linewidths, ${\mathrm{\Gamma }}_{\mathrm{o}}$, shown as open red circles and blue squares, respectively. Error bars are taken directly from the unsmoothed data and were determined from an in-house fitting algorithm. Theoretical LA and TA linewidths, ${\mathrm{\Gamma }}_{\mathrm{m}}$, shown as open dark gray circles and light gray squares, respectively.

Figure 7

Relationship between the activation enthalpy for self-diffusion, $Q$, in bcc metals and the square of the $\text{LA-}2/3{\langle 111\rangle }_{\mathrm{p}}$ frequency, ${\left({\nu }_{2/3}\right)}^2$, adapted from Köhler and Herzig [48]. Universal scaling is achieved by normalizing $Q$ to the melting temperature ${\mathit{\text{T}}}_{\mathrm{m}}$ and ${\nu }_{2/3}$ to ${\nu }_{0.1}$, the latter adjusting for lattice stiffness. Squared frequency ratios for both our experimental results and different theoretical approaches are shown as colored spheres. These data sit on the universal curve by construction, and errors are propagated from uncertainty in the linear fit. Inset: Schematic bcc unit cell with a central monovacancy; a plane containing a hopping saddle point is highlighted in blue.