Evidence for strongly coupled charge-density-wave ordering in three-dimensional $R_5{\text{Ir}}_4{\text{Si}}_{10}$ compounds from optical measurements

We report optical spectra of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ and ${\text{Er}}_5{\text{Ir}}_4{\text{Si}}_{10}$, exhibiting the phenomenon of coexisting superconductivity or antiferromagnetism and charge-density wave (CDW) order. We measure the maximum value of the charge-density wave gap present on part of the Fermi surface of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$, corresponding to a ratio $2\Delta /k_B{T}_{\text{CDW}}\simeq 10$, well above the value in the limit of weak electron-phonon coupling. Strong electron-phonon coupling was confirmed by analyzing the optical conductivity with the Holstein model describing the electron-phonon interactions, indicating the coupling to phonons centered at 30 meV, with a coupling constant $\lambda \approx 2.6$. Finally we provide evidence that approximately 16% of the Fermi surface of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ becomes gapped in the CDW state.

Figure 1

(Color online) The crystal structure of ${\text{RE}}_5{\text{Ir}}_4{\text{Si}}_{10}$. The rare-earth atoms are represented with large (red) spheres, Ir atoms with medium (blue) spheres and Si atoms with small (light-gray) spheres. With thick (blue) lines are represented schematically the five chains extending along the $c$ axis which seems to be responsible for the quasimonodimensional behavior of the system and with thick (red) lines the RE-RE network separating the five chains.

Figure 2

(Color online) Optical reflectivity with $a$- and $c$-axis polarized light of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ (dashed curves: green for $a$ axis and magenta for $c$ axis) and of ${\text{Er}}_5{\text{Ir}}_4{\text{Si}}_{10}$ (solid curves: blue for $a$ axis and black for $c$ axis). The spectra are quite similar for both materials. The reflectivity in the infrared range is much higher along the $c$ axis than along the $a$ axis for both materials, confirming the more metallic characteristics along the $c$ axis in this class of materials.

Figure 3

(Color online) The real (bottom) and imaginary (top) parts of the dielectric function of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$. The strong anisotropy in the screened plasma frequency ${\epsilon }_1\left({\omega }_p^{\ast }\right)=0$, observed for $E\parallel a$ (red circles) and $E\parallel c$, reveals a larger screening effect along $a$-axis due to interband contributions.

Figure 4

(Color online) (a) dc resistivity extrapolated from optical data as a function of temperature for $a$- (red circles) and $c$ axis (black circles). The error bars are calculated considering an error in the absolute value of reflectivity of $\pm 0.5\%$. (b) Four terminal resistivity from Ref. 15.

Figure 5

Reflectivity of ${\text{Er}}_5{\text{Ir}}_4{\text{Si}}_{10}$ at 186 meV measured along the $c$ axis during cooling (closed symbols) and heating (open symbols). The measurement shows clearly the two CDW transitions at 148 and 53 K. The low-temperature CDW transition at 53 K is characterized by a reflectivity change of only 0.05%. The hysteresis for this transition while cooling and heating the sample testifies to the first-order nature of this phase transition.

Figure 6

Temperature dependence of the reflection coefficient of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ for $E\parallel a$ and $E\parallel c$ at selected frequencies. Both axes show a softened effect for $\hslash \omega <12.4\text{meV}\approx 100{\text{cm}}^{-1}$.

Figure 7

Temperature dependent dielectric constants of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$. Left hand panels: Real (full circles) and imaginary (open circles) part of the dielectric constant for $E\parallel a$. Right hand panels: The same quantities for $E\parallel c$.

Figure 8

(Color online) Optical conductivity with $a$- and $c$-axis polarized light at 290 K of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ (green for $a$ axis and magenta for $c$ axis) and of ${\text{Er}}_5{\text{Ir}}_4{\text{Si}}_{10}$ (blue for $a$ axis and black for $c$ axis).

Figure 9

The frequency-dependent effective mass $1+\lambda \left(\omega \right)$ of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ for $E\parallel c$ (left panel) and $E\parallel ab$ (right panel). Data are shown for the case of 290 K (full line), 110 K (dashed line), 65 K (dotted line), and 30 K (dash-dot line). Shaded areas are the energy region where the value of $\lambda \left(\omega \right)$ may be affected by unaccounted for interband transitions.

Figure 10

The frequency-dependent scattering rate of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ $1/\tau \left(\omega \right)$ for $E\parallel c$ (left panel) and $E\parallel ab$ (right panel). Data are shown for 290 K (full line), 110 K (dashed line), 65 K (dotted line), and 30 K (dash-dot line). The data shows for both crystal orientations a rigid increase in scattering rate when temperature is lowered below $T_{\text{CDW}}$. Shaded areas are the energy region where the value of $1/\tau \left(\omega \right)$ may be affected by unaccounted for interband transitions.

Figure 11

(Color online) The top left and right panels present a comparison between the experimental scattering rate of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ measured at 110 and 65 K and the results of the fitting routine implemented starting from Eq. (5) for $c$ and $a$ axes, respectively. In the lower panels the resulting ${\alpha }^{2}F\left(\omega \right)$ are displayed. The results in the shaded areas should be interpreted with caution: the features in these regions may be artifacts resulting from unaccounted for interband transitions.

Figure 12

(Color online) Top and middle panels: Optical conductivity of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ along the $a$-axis (red curves) and the $c$-axis (black curves) for $T=60\text{K}$ (dashed lines) and $T=110\text{K}$ (full line). In the lower panel the ratio ${\sigma }_1\left(T=60\text{K}\right)/{\sigma }_1\left(T=110\text{K}\right)$ reveals a clear reduction of the conductivity below 80 meV for both crystal orientations indicating a partial gapping of the Fermi surface.

Figure 13

(Color online) The effective number of carriers of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ as a function of the cutoff frequency ${\omega }_c$ for both crystal orientations for $T=290\text{K}$, $T=110\text{K}$ and $T=65\text{K}$. In the inset $N_{\text{eff}}$ is presented over the full measurement range at room temperature and for both crystal orientations.

Figure 14

The spectral weight $W\left(T\right)$ of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ along the $a$ (lower panel) and the $c$ axis (upper panel) calculated using a cutoff frequency ${\omega }_c=0.31\text{eV}$. The spectral weight behavior for $\omega >{\omega }_c$ $\left(\omega <{\omega }_c\right)$ is shown with full (empty) markers.

Figure 15

The percentage increase of SW of ${\text{Lu}}_5{\text{Ir}}_4{\text{Si}}_{10}$ as defined in Eq. (8) plotted versus cutoff frequency. The comparison between $c$ axis (black-dotted line) and $a$ axis (red-dotted line) reveals a saturation region below 45 meV. The recovery of SW extends up to 3 eV for both optical axis. Inset: spectral weight integral between 0 and $1000{\text{cm}}^{-1}$ as a function of temperature.