Impact of disorder in the charge density wave state of Pd-intercalated ${\mathrm{ErTe}}_3$ revealed by the electrodynamic response

It is a general notion that disorder, introduced by either chemical substitution or intercalation as well as by electron irradiation, is detrimental to the realization of long-range charge density wave (CDW) order. We study the disorder-induced suppression of in-plane CDW orders in two-dimensional Pd-intercalated ${\mathrm{ErTe}}_3$ compositions by exploring the real part of the optical conductivity with light polarized along the in-plane $a$ and $c$ axes. Our findings reveal an anisotropic charge dynamics with respect to both incommensurate unidirectional CDW phases of ${\mathrm{ErTe}}_3$, occurring within the $ac$ plane. The anisotropic optical response gets substantially washed out with Pd intercalation, hand in hand with the suppression of both CDW orders. The spectral weight analysis, though, advances the scenario, for which the CDW phases evolve from a (partially) depleted Fermi surface already above their critical onset temperatures. We therefore argue that the long-range CDW orders of ${\mathrm{ErTe}}_3$ tend to be progressively dwarfed by Pd intercalation, which favors the presence of short-range CDW segments for both crystallographic directions persisting in a broad temperature $\left(T\right)$ interval up to the normal state, and being suggestive of precursor effects of the CDW orders as well as possibly coexisting with superconductivity at low $T$.

Figure 1

$T$ dependence of the optical anisotropy, defined by the ratio ${\sigma }_1^c(\omega ,T)/{\sigma }_1^a(\omega ,T)$, for (a) $x=0$, (b) $x=0.004$, and (c) $x=0.012{\mathrm{Pd}}_x{\mathrm{ErTe}}_3$ from the FIR up to the MIR energy scales $(1\mathrm{eV}=8.06548\times {10}^3{\mathrm{cm}}^{-1})$. The dotted lines refer to $T>T_{\text{CDW}1}$, steady lines to $T_{\text{CDW}2}<T<T_{\text{CDW}1}$, and dashed lines to $T<T_{\text{CDW}2}$. The insets show ${\sigma }_1\left(\omega \right)$ for both the $c$ and $a$ axes at 300 K and 5 or 10 K, as the highest and lowest measured $T$, respectively, covering the entire spectral range up to the visible frequencies (see the Appendix for further data and details).

Figure 2

$T$ dependence of the dc transport anisotropy, defined by ${\sigma }_{\text{dc}}^c\left(T\right)/{\sigma }_{\text{dc}}^a\left(T\right)$ [with ${\sigma }_{\text{dc}}\left(T\right)=\frac{1}{{\rho }_{\text{dc}}\left(T\right)}]$ [25], compared to the anisotropy of the optical SW given by ${\text{SW}}^c({\omega }_c,T)/{\text{SW}}^a({\omega }_c,T)$ (from Fig. 12 in the Appendix) at selected cutoff frequencies ${\omega }_c$ [26], representing the FIR (i.e., ${\omega }_c\sim 10–50{\mathrm{cm}}^{-1}$, therefore significant for the dc limit of the optical response) and MIR (i.e., ${\omega }_c\sim 600–2400{\mathrm{cm}}^{-1}$) spectral range for (a) $x=0$, (b) $x=0.004$, and (c) $x=0.012{\mathrm{Pd}}_x{\mathrm{ErTe}}_3$. The vertical dotted lines mark $T_{\text{CDW}1}$ and $T_{\text{CDW}2}$ [8]. See text about the origin of the error bars in ${\text{SW}}^c({\omega }_c,T)/{\text{SW}}^a({\omega }_c,T)$.

Figure 3

$T$ dependence of the integrated spectral weight ratio ${\text{SW}}^{c,a}\left(T\right)/{\text{SW}}^{c,a}(300\mathrm{K}$) along the $a$ (lower panels) and $c$ (upper panels) axes for (a), (b) $x=0$; (c), (d) $x=0.004$; and (e), (f) $x=0.012{\mathrm{Pd}}_x{\mathrm{ErTe}}_3$. The dotted lines refer to $T>T_{\text{CDW}1}$, steady lines to $T_{\text{CDW}2}<T<T_{\text{CDW}1}$, and dashed lines to $T<T_{\text{CDW}2}$. The range, within which the depletion of the SW ratio occurs at the lowest $T$ across the FIR-MIR energy interval, is noted by the grey areas. It roughly indicates the amount of the FS gapping. The horizontal red-dashed lines represent an estimation [27, 28] of the amount of the FS gapping from ${\rho }_{\text{dc}}\left(T\right)$ [8, 16, 25]. The insets show the integrated ${\text{SW}}^{c,a}\left(T\right)$ at 5 or 10 and 300 K (in units of ${\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-2}$ [26]) for each crystallographic axis. Main panels and insets cover the energy spectral range from zero up to ${\omega }_c\sim {10}^4{\mathrm{cm}}^{-1}(1\mathrm{eV}=8.06548\times {10}^3{\mathrm{cm}}^{-1}$), since above this energy scale ${\text{SW}}^{c,a}\left(T\right)$ is fully recovered and conserved at any $T$ [17].

Figure 4

(a) The original crystal structure of ${\mathrm{ErTe}}_3$ with the relevant axes is reproduced from Ref. [8]. (b)–(d) $T$ dependence of the dc resistivity $\left[{\rho }_{\text{dc}}\left(T\right)\right]$ normalized at 300 K along both the $a$ and $c$ axes for (b) $x=0$, (c) $x=0.005$, and (d) $x=0.019{\mathrm{Pd}}_x{\mathrm{ErTe}}_3$, reproduced from Refs. [8, 16]. The vertical dashed lines mark $T_{\text{CDW}1}$ and $T_{\text{CDW}2}$.

Figure 5

$R\left(\omega \right)$ along the $c$ (a) and $a$ (b) axes [Fig. 4] of ${\mathrm{ErTe}}_3$ at 300 and $10\mathrm{K}(1\mathrm{eV}=8.06548\times {10}^3{\mathrm{cm}}^{-1}$), emphasizing the plasma edge feature and the high-frequency spectrum. The insets show the corresponding $T$ dependence at MIR energy scales.

Figure 6

$R\left(\omega \right)$ along the $c$ (a) and $a$ (b) axes [Fig. 4] of $x=0.004{\mathrm{Pd}}_x{\mathrm{ErTe}}_3$ at 300 and $10\mathrm{K}(1\mathrm{eV}=8.06548\times {10}^3{\mathrm{cm}}^{-1})$, emphasizing the plasma edge feature and the high-frequency spectrum. The insets show the corresponding $T$ dependence at MIR energy scales.

Figure 7

$R\left(\omega \right)$ along the $c$ (a) and $a$ (b) axes [Fig. 4] of $x=0.012{\mathrm{Pd}}_x{\mathrm{ErTe}}_3$ at 300 and $5\mathrm{K}(1\mathrm{eV}=8.06548\times {10}^3{\mathrm{cm}}^{-1})$, emphasizing the plasma edge feature and the high-frequency spectrum. The insets show the corresponding $T$ dependence at MIR energy scales.

Figure 8

Low frequency $R\left(\omega \right)$ at 300 and 5 or 10 K for both $a$ and $c$ axes [Fig. 4] at the smooth crossover between the measured data in FIR and the HR extrapolation: (a), (b) $x=0$; (c), (d) $x=0.004$; and (e), (f) $x=0.012{\mathrm{Pd}}_x{\mathrm{ErTe}}_3$. The vertical dashed lines indicate the energy interval between 50 and 100 ${\mathrm{cm}}^{-1}$, where the numerical interpolation of data and HR extrapolation takes place.

Figure 9

$T$ dependence of ${\sigma }_1\left(\omega \right)$ along the $c$ (a) and $a$ (b) axes [Fig. 4] of ${\mathrm{ErTe}}_3(1\mathrm{eV}=8.06548\times {10}^3{\mathrm{cm}}^{-1})$, emphasizing the MIR-NIR spectral range. The insets show the corresponding spectra at 300 and 10 K over the whole investigated spectral range (please note the logarithmic frequency (energy) scale).

Figure 10

$T$ dependence of ${\sigma }_1\left(\omega \right)$ along the $c$ (a) and $a$ (b) axes [Fig. 4] of $x=0.004{\mathrm{Pd}}_x{\mathrm{ErTe}}_3(1\mathrm{eV}=8.06548\times {10}^3{\mathrm{cm}}^{-1})$, emphasizing the MIR-NIR spectral range. The insets show the corresponding spectra at 300 and 10 K over the whole investigated spectral range (please note the logarithmic frequency (energy) scale).

Figure 11

$T$ dependence of ${\sigma }_1\left(\omega \right)$ along the $c$ (a) and $a$ (b) axes [Fig. 4] of $x=0.012{\mathrm{Pd}}_x{\mathrm{ErTe}}_3(1\mathrm{eV}=8.06548\times {10}^3{\mathrm{cm}}^{-1})$, emphasizing the MIR-NIR spectral range. The insets show the corresponding spectra at 300 and 5 K over the whole investigated spectral range (please note the logarithmic frequency (energy) scale).

Figure 12

$T$ dependence of the ratio ${\text{SW}}^c({\omega }_c,T)/{\text{SW}}^a({\omega }_c,T)$ for the integrated SW (insets Fig. 3) as a function of the cutoff frequency ${\omega }_c$ (see text) at selected $T$ for $x=0$ (a), $x=0.004$ (b), and $x=0.012$ (c) ${\mathrm{Pd}}_x{\mathrm{ErTe}}_3(1\mathrm{eV}=8.06548\times {10}^3{\mathrm{cm}}^{-1})$.

Figure 13

$T$ dependence of the Drude scattering rate ${\mathrm{\Gamma }}_D$ (upper panels) and plasma frequency ${\omega }_{p,D}$ (lower panels) fit parameters [Eq. (A1)] for all ${\mathrm{Pd}}_x{\mathrm{ErTe}}_3$ along the $c$ and $a$ axes: (a), (b) $x=0$; (c), (d) $x=0.004$; and (e), (f) $x=0.012$. The vertical dotted lines mark $T_{\text{CDW}1}$ and $T_{\text{CDW}2}$ [8, 16]. The error bars are estimated numerically within the nonlinear least-squares fit technique.

Figure 14

$T$ dependence of the Lj-HOs resonance frequency ${\omega }_j$ (upper panels), damping ${\gamma }_j$ (middle panels), and strength ${\mathrm{\Omega }}_j$ (lower panels) fit parameters [Eq. (A1)] for all ${\mathrm{Pd}}_x{\mathrm{ErTe}}_3$ along the $a$ axis: (a)–(c) $x=0$, (d)–(f) $x=0.004$; and (g)–(i) $x=0.012$. The vertical dotted lines mark $T_{\text{CDW}1}$ and $T_{\text{CDW}2}$ [8, 16]. The error bars are estimated numerically within the nonlinear least-squares fit technique.

Figure 15

$T$ dependence of the Lj-HOs resonance frequency ${\omega }_j$ (upper panels), damping ${\gamma }_j$ (middle panels), and strength ${\mathrm{\Omega }}_j$ (lower panels) fit parameters [Eq. (A1)] for all ${\mathrm{Pd}}_x{\mathrm{ErTe}}_3$ along the $c$ axis: (a)–(c) $x=0$; (d)–(f) $x=0.004$; and (g)–(i) $x=0.012$. The vertical dotted lines mark $T_{\text{CDW}1}$ and $T_{\text{CDW}2}$ [8, 16]. The error bars are estimated numerically within the nonlinear least-squares fit technique.

Figure 16

Left column: $T$ dependence of the spectral weight relative variation $\mathrm{\Delta }\text{SW}\left(T\right)=\text{SW}\left(T\right)-\text{SW}$(300 K), calculated for the grouped contributions given by the Drude term and HO L1 (i.e., $\text{SW}\sim {\omega }_{p,D}^2+{\mathrm{\Omega }}_1^2$), by HOs L2 and L3 (i.e., $\text{SW}\sim {\mathrm{\Omega }}_2^2+{\mathrm{\Omega }}_3^2$), as well as by HO L4 (i.e., $\text{SW}\sim {\mathrm{\Omega }}_4^2$) with respect to 300 K. The error bars in $\mathrm{\Delta }\text{SW}\left(T\right)$ correspond to the direct propagation of the error in the HOs strength, estimated numerically within the nonlinear least-squares fit technique. Right column: ${\sigma }_1\left(\omega \right)$ at 300 K compared to its Drude-Lorentz fit [Eq. (A1)], showing its single constituent components (please note the logarithmic frequency (energy) scale). The rounded arrows mimic the direction of the SW reshuffling upon decreasing $T$, which is stronger with thicker arrows. All quantities are shown along the $a$ axis [Fig. 4] for all ${\mathrm{Pd}}_x{\mathrm{ErTe}}_3$: (a), (b) $x=0$; (c), (d) $x=0.004$; and (e), (f) $x=0.012$. The vertical dashed lines in (a), (c), and (e) mark $T_{\text{CDW}1}$ and $T_{\text{CDW}2}$ [8, 16].

Figure 17

Left column: $T$ dependence of the spectral weight relative variation $\mathrm{\Delta }\text{SW}\left(T\right)=\text{SW}\left(T\right)-\text{SW}$(300 K), calculated for the grouped contributions given by the Drude term and HO L1 (i.e., $\text{SW}\sim {\omega }_{p,D}^2+{\mathrm{\Omega }}_1^2$), by HOs L2 and L3 (i.e., $\text{SW}\sim {\mathrm{\Omega }}_2^2+{\mathrm{\Omega }}_3^2$), as well as by HO L4 (i.e., $\text{SW}\sim {\mathrm{\Omega }}_4^2$) with respect to 300 K. The error bars in $\mathrm{\Delta }\text{SW}\left(T\right)$ correspond to the direct propagation of the error in the HOs strength, estimated numerically within the nonlinear least-squares fit technique. Right column: ${\sigma }_1\left(\omega \right)$ at 300 K compared to its Drude-Lorentz fit [Eq. (A1)], showing its single constituent components [please note the logarithmic frequency (energy) scale]. The rounded arrows mimic the direction of the SW reshuffling upon decreasing $T$, which is stronger with thicker arrows. All quantities are shown along the $c$ axis [Fig. 4] for all ${\mathrm{Pd}}_x{\mathrm{ErTe}}_3$: (a), (b) $x=0$; (c), (d) $x=0.004$; and (e), (f) $x=0.012$. The vertical dashed lines in (a), (c), and (e) mark $T_{\text{CDW}1}$ and $T_{\text{CDW}2}$ [8, 16].