Optical conductivity of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ in the Kondo liquid and hidden-order phases

We measured the polarized optical conductivity of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ from room temperature down to 5 K, covering the Kondo state, the coherent Kondo liquid regime, and the hidden-order phase. The normal state is characterized by an anisotropic behavior between the $ab$ plane and $c$-axis responses. The $ab$-plane optical conductivity is strongly influenced by the formation of the coherent Kondo liquid: a sharp Drude peak develops and a hybridization gap at 12 meV leads to a spectral weight transfer to mid-infrared energies. The $c$-axis conductivity has a different behavior: the Drude peak already exists at 300 K and no particular anomaly or gap signature appears in the coherent Kondo liquid regime. When entering the hidden-order state, both polarizations see a dramatic decrease in the Drude spectral weight and scattering rate, compatible with a loss of about 50% of the carriers at the Fermi level. At the same time a density-wave-like gap appears along both polarizations at about 6.5 meV at 5 K. This gap closes respecting a mean-field thermal evolution in the $ab$ plane. Along the $c$ axis it remains roughly constant and it “fills up” rather than closing.

Figure 1

Reflectivity of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ above and below the hidden-order transition for (a) $E\parallel a$ and (b) $E\parallel c$ polarizations. Panels (c) and (d) detail the very far-infrared reflectivity in the hidden-order phase for both polarizations.

Figure 2

Optical conductivity of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ for (a) $E\parallel a$ and (b) $E\parallel c$. The $E\parallel c$ dc limit of ${\sigma }_1$ is roughly three times higher than its value for $E\parallel a$ (note the difference in the optical conductivity scale), in accordance with the strongly anisotropic transport behavior of ${\mathrm{URu}}_2{\mathrm{Si}}_2$. In the 20–40-meV range we observe a decrease of spectral weight related to a narrowing of the Drude peak. For $E\parallel a$ this effect happens mostly below the Kondo temperature. For $E\parallel c$ the spectral weight decreases continuously from room temperature. Below the hidden order a strong peak appears at roughly 8 eV with a strong decrease of ${\sigma }_1$ at lower energies, characteristic of a partial gap at the Fermi level. The inset shows the resistivity from Refs. [4, 47, 54] (see text). The vertical dashed lines indicate the hidden-order transition and the boundary of the coherent Kondo liquid regime.

Figure 3

Real part of the dielectric function for (a) $E\parallel a$ and (b) $E\parallel c$. At very low energies a negative divergence, due to the Drude contribution to Eq. (1), dominates the spectral response and allows us to determine the plasma frequency and scattering rate. The peaks around 15 meV correspond to polar phonons. In the hidden-order state another large peak appears around 5 meV. Note that a well defined negative divergence is present at all temperatures for $E\parallel c$ but only below 75 K for $E\parallel a$.

Figure 4

Drude parameters extracted from the low-energy real part of the dielectric function (Fig. 3). Panel (a) shows the plasma frequency and panel (b) the scattering rate. There is no well defined Drude peak above $\sim 100$ K for data with $E\parallel a$. The vertical dotted lines indicate the position of $T_{\mathit{HO}}$ and $T_{\mathit{KL}}$. The dashed lines are guides to the eye. For clarity the data are only shown up to 150 K along $E\parallel c$. Above that temperature ${\mathrm{\Omega }}_p$ is roughly constant and $1/\tau$ increases slightly. Error bars are estimated by utilizing different models for the hidden-order and phonon excitations as well as by varying the spectral range allowed for a least-squares fitting of the Drude term.

Figure 5

(a) Restricted spectral weight, normalized to its value at 300 K, calculated with ${\omega }_0=20$ meV ($160 {\text{cm}}^{-1}$) and ${\omega }_1=40$ meV ($320 {\text{cm}}^{-1}$). The solid lines are guides to the eye. The vertical dashed lines indicate the hidden-order transition and the boundary of the coherent Kondo liquid regime. Error bars are obtained by varying the cutoff frequencies by $\pm 10$%. Optical conductivity for (b) $E\parallel a$ and (c) $E\parallel c$ in the midinfrared range. Phonons were subtracted for clarity. For $E\parallel a$, part of the spectral weigh lost in the 20–40-meV range (see Fig. 2) is transferred to the 0.5-eV region, indicating the opening of a gap above $T_{\mathit{HO}}$. Along $E\parallel c$ no such transfer is observed, within the accuracy of our data.

Figure 6

Real part of the optical conductivity in the hidden-order phase for (a) $E\parallel a$ and (b) $E\parallel c$. The large peak around 8 meV indicates the opening of a gap. In both cases, we can estimate (from the inflexion point) a gap $2{\mathrm{\Delta }}_{HO}\approx 6.5$ meV at 5 K. The opening of the gap transfers spectral weight from low (just below $2{\mathrm{\Delta }}_{HO}$) to high (just above $2{\mathrm{\Delta }}_{HO}$) energies. Note that for $E\parallel a$ the gap closes, i.e., its energy continuously decreases upon increasing temperature. For $E\parallel c$ the gap fills up rather than closes. Also note a small shoulder (indicated by the arrows) at the leading edge of the hidden-order peak along $c$, showing the presence of a second gap in this orientation. In both panels, the sharp peaks around 14 meV are polar phonons.

Figure 7

Temperature dependence of the hidden-order gap energy, taken as the inflexion point of the leading edge of the peak in Fig. 6. Blue symbols correspond to the $E\parallel a$ optical conductivity and red symbols to the $c$ axis. The dashed line is the result of a mean-field BCS calculation and the solid line is a guide to the eye.