Detailed optical spectroscopy of hybridization gap and hidden-order transition in high-quality ${\mathrm{URu}}_2{\mathrm{Si}}_2$ single crystals

We present a detailed temperature and frequency dependence of the optical conductivity measured on clean high-quality single crystals of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ of $ac\text{−}$ and $ab\text{-plane}$ surfaces. Our data demonstrate the itinerant character of the narrow $5f$ bands, becoming progressively coherent as the temperature is lowered below a crossover temperature $T^{*}\sim 75$ K. $T^{*}$ is higher than in previous reports as a result of a different sample preparation, which minimizes residual strain. We furthermore present the density-response (energy-loss) function of this compound, and determine the energies of the heavy-fermion plasmons with $a\text{−}$ and $c\text{-axis}$ polarization. Our observation of a suppression of optical conductivity below 50 meV along both the $a$ and $c$ axes, along with a heavy-fermion plasmon at 18 meV, points toward the emergence of a band of coherent charge carriers crossing the Fermi energy and the emergence of a hybridization gap on part of the Fermi surface. The evolution towards coherent itinerant states is accelerated below the hidden order temperature $T_{\mathrm{HO}}=17.5$ K. In the hidden order phase the low-frequency optical conductivity shows a single gap at $\sim 6.5$ meV, which closes at $T_{\mathrm{HO}}$.

Figure 1

Room temperature (290 K) optical conductivity (top) and dielectric function (bottom) of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ with electric field polarized along the $a$ axis (black) and $c$ axis (red). These data were obtained on the $ac\text{-plane}$ surface of a ${\mathrm{URu}}_2{\mathrm{Si}}_2$ single crystal combining spectroscopic ellipsometry along two orthogonal planes of reflection.

Figure 2

Reflectivity of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ single crystal in $a\text{-axis}$ (upper panel) and $c\text{-axis}$ (lower panel) polarizations for selected temperatures.

Figure 3

Optical conductivity (top) and energy loss function (bottom) of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ single crystal in $a\text{-axis}$ and $c\text{-axis}$ polarizations for selected temperatures.

Figure 4

Detailed color map of the optical conductivity as a function of temperature and frequency for $a\text{-axis}$ (upper panel) and $c\text{-axis}$ (lower panel) polarizations. In the $a$ axis, the optical conductivity becomes suppressed below $T^{*}$ in the form of a domelike shape while in the $c\text{-axis}$ direction, it is observed as a narrowing Drude peak as temperature is lowered below $T^{*}$. The HO gap appears as a domelike shape suppression of the conductivity in both axes.

Figure 5

The spectral weight ${\omega }_p^2$ (upper panel) and the scattering rate $\gamma$ (lower panel) of the Drude contribution to the optical conductivity obtained from fitting our reflectivity data to the Drude-Lorentz model (full symbols). ${\omega }_p^2$ decreases below $T^{*}$ with almost the same temperature dependence for both $a$ axis and $c$ axis, however with higher values for the $c$ axis. $\gamma$ is drastically reduced at low temperatures and shows a clear change of slope as a function of temperature in the $a\text{-axis}$ data compared to the $c\text{-axis}$ data. Below $T_{\mathrm{HO}}$ the Drude peak becomes too narrow for the reflectivity data, that begin at 2.5 meV. The parameters in this temperature range (open symbols) were obtained by fitting the low-frequency part (below 4 meV) of ${\sigma }_1\left(\omega \right)$ (Fig. 3) instead of fitting the reflectivity data.

Figure 6

Normalized spectral weight obtained by integrating the conductivity from zero to the cutoff frequency $\omega$ indicated in the figure legend for the $a\text{-axis}$ (upper panel) and $c\text{-axis}$ (lower panel) data. The spectral weights are normalized to $W$(200 K).

Figure 7

Summary of the three fit parameters of the high energy phonon modes of the $a$ axis and $c$ axis to a Fano-shape resonance where $\mathrm{\Delta }\omega =\omega \left(T\right)-\omega \left(0\right)$ and ${\omega }_0$ is the oscillator's central frequency at room temperature.

Figure 8

Low-energy optical conductivity (top panels) and loss function (bottom panels) at selected temperatures close to and below $T_{\mathrm{HO}}$ for $a\text{-axis}$ and $c\text{-axis}$ polarizations.

Figure 9

Detailed temperature and frequency dependence of the optical conductivity at the HO phase transition. The gap closes and follows a BCS gaplike temperature dependence with $T_{\mathrm{HO}}$ of 17.5 K and $2{\mathrm{\Delta }}_{\mathrm{HO}}=6.5$ meV (dashed white line) for both axes (upper panel, $a$ axis; lower panel, $c$ axis).

Figure 10

Reflectivity (upper panel) and optical conductivity (lower panel) simulations of an $a\text{-axis}$ and $c\text{-axis}$ admixture. The 20% $a\text{-axis}$ inclusions depicted in our simulations by a polarization leakage is showing a spurious multiple-gap structure which results in reduced conductivity and an additional low-frequency slope change compared to that of the clean $c\text{-axis}$ data.

Figure 11

$\mathrm{\Psi }\left(\omega \right)$ and $\mathrm{\Delta }\left(\omega \right)$ spectra at 300 K for two planes of reflection of the $ac$ plane of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ measured at two angles of incidence. Experimental data are shown together with a simultaneous fit of all eight curves to the Drude-Lorentz parametrization of the $a\text{-axis}$ and $c\text{-axis}$ dielectric functions. Experimental data and fitted curves cannot be distinguished because they fully overlap.

Figure 12

The real (upper panel) and imaginary (lower panel) parts of the dielectric function $\epsilon \left(\omega \right)a\text{-axis}$ (black) and $c\text{-axis}$ (red) polarizations obtained from a fit to the Drude-Lorentz model and based on the ellipsometry measurements shown in Fig. 11.