Understanding the Complex Phase Diagram of Uranium: The Role of Electron-Phonon Coupling

We report an experimental determination of the dispersion of the soft phonon mode along [100] in uranium as a function of pressure. The energies of these phonons increase rapidly, with conventional behavior found by 20 GPa, as predicted by recent theory. New calculations demonstrate the strong pressure (and momentum) dependence of the electron-phonon coupling, whereas the Fermi-surface nesting is surprisingly independent of pressure. This allows a full understanding of the complex phase diagram of uranium and the interplay between the charge-density wave and superconductivity.

Figure 1

IXS data at ambient temperature taken for $\mathbf{q}\approx (0.5,0,0)$. Data show both the energy-loss (Stokes) and energy-gain (anti-Stokes) response, as well as the central elastic line at the lowest (a) and highest (b) pressure measured. (c) Zoom on the Stokes peak for intermediate pressures.

Figure 2

Phonon energy: (a) Experimental (data points with error bars) and theoretical (solid lines) dispersion of the ${\Sigma }_4$ optic component along [100]. (b) Soft-mode energy as a function of pressure (lines are guides for the eyes).

Figure 3

Results of calculations at $T=0\mathrm{K}$ using the experimental volume (see the Supplemental Materials, Part II [16]): (a) Energy of the ${\Sigma }_4$ phonon mode along [100]. (b) The $e$-ph coupling, ${\lambda }_{\mathbf{q},\nu }$. (c) The phonon linewidth due to $e$-ph interaction, ${\gamma }_{\mathbf{q},\nu }$. The different pressures are 0, 0.8, 2.5, 5.1, 10.3, 16, and 25 GPa from bottom to top in (a) and from top to bottom in (b),(c).

Figure 4

The Fermi-surface topology for the $\alpha$-U structure calculated at (a) ambient pressure and (b) at 20 GPa. In each case, the diagram shows the [100] and [010] axes at a fixed position $z=1/2$ along [001]. The different colors correspond to the different sheets of the Fermi surface. The arrow indicating the nesting vector {$\approx 1/2(2\pi /a)$ along [100]} has been drawn the same length in both (a) and (b).

Figure 5

$T\mathrm{\text{−}}P$ phase diagram of uranium in the superconducting region. The vertical dashed line (at 1.2 GPa) indicates the calculated transition between the CDW phase and the normal $\alpha$-U structure in good agreement with experiment. The experimental $T_c$ values, taken from Fig. 3.3 of Ref. 4, are indicated by open triangles and squares, and the solid line is a guide for the eyes. The calculated values are shown as solid squares (${\alpha }_1$) and solid circles ($\alpha$-U) and were scaled at 3 GPa (see text). The inset shows the phonon dispersion curves in the CDW state (${\alpha }_1$ structure) for zero pressure (dashed lines) and 5 GPa (solid lines). The thicker lines are the branches corresponding to the ${\Sigma }_4$ mode in the $\alpha$-U phase.