Lattice dynamics of the heavy-fermion compound ${\mathrm{URu}}_2{\mathrm{Si}}_2$

We report a comprehensive investigation of the lattice dynamics of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ as a function of temperature using Raman scattering, optical conductivity, and inelastic neutron scattering measurements as well as theoretical ab initio calculations. The main effects on the optical phonon modes are related to Kondo physics. The ${\mathrm{B}}_{1g}$ (${\Gamma }_3$ symmetry) phonon mode slightly softens below $\sim 100\mathrm{K}$, in connection with the previously reported softening of the elastic constant, $C_{11}-C_{12}$, of the same symmetry, both observations suggesting a ${\mathrm{B}}_{1g}$ symmetry-breaking instability in the Kondo regime. Through optical conductivity, we detect clear signatures of strong electron-phonon coupling, with temperature-dependent spectral weight and Fano line shape of some phonon modes. Surprisingly, the line shapes of two phonon modes, ${\mathrm{E}}_u$(1) and ${\mathrm{A}}_{2u}$(2), show opposite temperature dependencies. The ${\mathrm{A}}_{2u}$(2) mode loses its Fano shape below 150 K, whereas the ${\mathrm{E}}_u$(1) mode acquires it below 100 K, in the Kondo crossover regime. This may point to momentum-dependent Kondo physics. By inelastic neutron-scattering measurements we have drawn the full dispersion of the phonon modes between 300 and 2 K. No remarkable temperature dependence has been obtained, including through the hidden order transition. Ab initio calculations with the spin-orbit coupling are in good agreement with the data except for a few low-energy branches with propagation in the ($a,b$) plane.

Figure 1

Atomic displacements sketches of the IR-active optical modes (up) and of the Raman-active optical modes (bottom) of ${\mathrm{URu}}_2{\mathrm{Si}}_2$. From left to right, the modes have increasing energy. The red arrows indicate the displacements with very low atomic intensities as calculated from ab initio studies (see Sec. 6).

Figure 2

Raman spectra of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ for sample 1 after polishing along the ($a,c$) plane and annealing. Inset: Raman spectra for sample 2 after polishing along the ($a,a$) plane, annealing, and for another sample cleaved along the ($a,a$) plane.

Figure 3

Typical Raman spectra measured at 4 K in cross- and parallel-polarization of the light. A scheme of sample 1 with the orientation of the polarizations of the incident ${\vec{e}}_i$ and scattering ${\vec{e}}_s$ light is shown.

Figure 4

Temperature dependence of the position of the ${\mathrm{A}}_{1g},{\mathrm{E}}_g$(1), ${\mathrm{E}}_g$(2), and ${\mathrm{B}}_{1g}$ phonon modes normalized to their value at 300 K. ${\mathrm{B}}_{1g}$ phonon exhibits an unusual softening when cooling down below 100 K. Error bars have been extracted from the Lorentzian fits. Lines are fits with a single anharmonic model [37, 38]. The green dot line is a guide to the eyes. Inset: zoom of the data at low temperature (hidden order transition is marked by the vertical black dashed line). No particular anomaly is measured at $T_0$.

Figure 5

Temperature dependence of the FWHM of ${\mathrm{A}}_{1g},{\mathrm{E}}_g$(1), ${\mathrm{E}}_g$(2), and ${\mathrm{B}}_{1g}$ phonon modes. Lines are fits with a single anharmonic model [37, 38]. Dot lines are guides to the eyes.

Figure 6

Optical conductivity centered around each phonon of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ at selected temperatures. Panels (a) and (b) are for light polarized on the (a,a) plane, and panels (c) and (d) are for measurements along the $c$ direction. Symbols are the data and the lines are Drude-Lorentz fits as described in the text. Inset (e) shows the highest frequency ${\mathrm{E}}_u$(2) mode at 20 K described by either a Lorentz oscillator (dashed line) or a Fano mode (solid line). Panel (f) is the equivalent to panel (e) for the highest frequency ${\mathrm{A}}_{2u}$(2) mode at 300 K.

Figure 7

Spectral weight of each phonon normalized by the spectral weight up to $2000{\mathrm{cm}}^{-1}$ calculated at room temperature.

Figure 8

Temperature dependence of the frequency of ${\mathrm{A}}_{2u}$(2), ${\mathrm{E}}_u$(1), and ${\mathrm{E}}_u$(2) phonon modes normalized to their value at 300 K. Lines are fits with a single anharmonic model (see Sec. 3c). The vertical black dashed line marks the transition into the hidden order.

Figure 9

Temperature dependence of Fano-Wigner-Breit $q^{-2}$ parameter for both ${\mathrm{E}}_u$ and the highest frequency ${\mathrm{A}}_{2u}$ phonons. When an electron-phonon coupling occurs, the Fano-Wigner-Breit $q^{-2}$ parameter becomes different from zero.

Figure 10

Inelastic neutron scattering phonon spectra measured for (a) $\mathbf{Q}$ = (0, 0, 5.25) at $T=2$, 60, and 300 K performed on IN8, and (b) $\mathbf{Q}=(2.5,1.5,0)$ at $T=2$ and 30 K performed on IN22. L and T, A and O stand for longitudinal and transverse, acoustic and optic, respectively. The TO mode belongs to the branch that corresponds to the ${\mathrm{B}}_{1g}$ mode at $\mathbf{Q}=0$. Lines are fits of the data with an harmonic oscillator line shape and sloppy background. The neutron intensity is given for a normalized incident flux (M), which corresponds to a counting time of approximately 80 s.

Figure 11

Inelastic neutron scattering phonon spectra at $\Gamma$ point measured for $\mathbf{Q}$ = (2, 0, 2) and $\mathbf{Q}$ = (0, 0, 6) at $T$ = 300 K. The ${\mathrm{E}}_u$ and ${\mathrm{E}}_g$ phonon modes have been observed for $\mathbf{Q}$ = (2, 0, 2) and ${\mathrm{A}}_{2u}$ and ${\mathrm{A}}_{1g}$ phonon modes for $\mathbf{Q}$ = (0, 0, 6). The neutron intensity is given for a normalized incident flux (M), which corresponds to a counting time of approximately 80 s for ${\mathrm{E}}_u,{\mathrm{E}}_g,{\mathrm{A}}_{2u}$ and 110 s for ${\mathrm{A}}_{1g}$. Lines are fits of the data with an harmonic oscillator line shape.

Figure 12

Polarized inelastic neutron scattering spectra measured for $\mathbf{Q}$ = (1.5, 1.5, 0) in the non-spin-flip (NSF) and spin-flip (SF) channels for $T$ = 2 K. Full lines are fits of the data with functions described in the text. The dashed line indicates the background. The neutron intensity is given for a normalized incident flux, which corresponds to a counting time (M) of approximately 25 min.

Figure 13

(a) Brillouin zone of the body-centered tetragonal crystal structure of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ in the paramagnetic state. The arrows indicate the direction of the dispersion for the inelastic neutron scattering (INS) measurements, as reported below. (b) Phonons and magnetic excitations (full red diamonds) dispersion along the $\Gamma \mathrm{Z}$ ([0,0,1]), $\mathrm{Z}\Sigma \Gamma$ ([1,0,0]), and $\Gamma \mathrm{X}$ ([1,1,0]) directions. Full circles and full upward triangles are INS data for the optical and acoustic branches, respectively. Each reported energy is the energy average of all measured temperature for each phonon mode (see text). Empty symbols are data obtained in the zone center at $\Gamma$, with cross, open triangles, and squares for INS, IR, and Raman measurements, respectively. Dashed lines are guides to the eyes for the magnetic excitations dispersion. Solid lines corresponds to the $\mathit{ab}\mathit{initio}$ GGA calculation with spin-orbit coupling and $u=0.06\AA$ (see Sec. 6). When the dispersion curve has a mixed character (for instance from LO (in $\Gamma$) to TO-z (in $Z$) along [1,0,0]), the solid lines are bicolored. The black arrow points to the ${\mathrm{E}}_u\left(1\right)$ phonon mode measured by IR at $\Gamma$ point. The green arrows indicate the calculated branch, which corresponds to the measured points (green full circles). ${\mathrm{Q}}_0$ and ${\mathrm{Q}}_1$ correspond to the minima in magnetic excitation dispersion, known as commensurate and incommensurate excitations, respectively.

Figure 14

Ab initio calculation of the phonon dispersion curves along $\Gamma Z,\Gamma \Sigma Z$, and $\Gamma X$ at 0 K. The parameter $u$ sets for the displacements of the atoms. Increasing $u$ from 0.03 Å (red line) to 0.06 Å (green line) is used to probe the effects of anharmonicity. Black line shows the calculation with spin-orbit coupling (SOC). The results of this last calculation ($u=0.06\AA$ and SOC) are the closest to the experimental results and are the ones reported in Fig. 13.