Optical absorption in amorphous tungsten oxide ($\textit{a}\mathrm{WO}_{3}$), for photon energies below that of the band gap, can be rationalized in terms of electronic transitions between localized states. For the study of this phenomenon, we employed the differential coloration efficiency concept, defined as the derivative of the optical density with respect to the inserted charge. We also made use of its extension to a complex quantity in the context of frequency-resolved studies. Combined $\textit{in situ}$ electrochemical and optical experiments were performed on electrochromic $\textit{a}\mathrm{WO}_{3}$ thin films for a wide lithium intercalation range using an optical wavelength of $810~\mathrm{nm}$ ($1.53~\mathrm{eV}$). Quasi-equilibrium measurements were made by chronopotentiometry (CP). Dynamic frequency-dependent measurements were carried out by simultaneous electrochemical and color impedance spectroscopy (SECIS). The differential coloration efficiency obtained from CP changes sign at a critical intercalation level. Its response exhibits an excellent agreement with a theoretical model that considers electronic transitions between $\mathrm{W}^{4+}$, $\mathrm{W}^{5+}$, and $\mathrm{W}^{6+}$ sites. For the SECIS experiment, the low-frequency limit of the differential coloration efficiency shows a general trend similar to that from CP. However, it does not change sign at a critical ion insertion level. This discrepancy could be due to degradation effects occurring in the films at high $\mathrm{Li}^+$ insertion levels. The methodology and results presented in this work can be of great interest both for the study of optical absorption in disordered materials and for applications in electrochromism.

中文翻译：

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作为嵌入水平函数的电致变色非晶氧化钨的差异着色效率：理论与实验的比较

对于低于带隙的光子能量，无定形氧化钨 ($\textit{a}\mathrm{WO}_{3}$) 中的光吸收可以根据局域态之间的电子跃迁合理化。为了研究这种现象，我们采用了微分着色效率概念，定义为光密度相对于插入电荷的导数。我们还在频率分辨研究的背景下利用了它对复数的扩展。结合 $\textit{in situ}$ 电化学和光学实验在电致变色 $\textit{a}\mathrm{WO}_{3}$ 薄膜上进行，使用 $810~\mathrm 的光学波长，以获得宽的锂嵌入范围{nm}$ ($1.53~\mathrm{eV}$)。通过计时电位法 (CP) 进行准平衡测量。通过同步电化学和颜色阻抗光谱 (SECIS) 进行动态频率相关测量。从 CP 获得的差异着色效率在临界嵌入水平改变符号。它的响应与考虑 $\mathrm{W}^{4+}$、$\mathrm{W}^{5+}$ 和 $\mathrm{W}^{ 之间的电子跃迁的理论模型非常吻合6+}$ 个站点。对于 SECIS 实验，差异着色效率的低频极限显示出与 CP 相似的总体趋势。然而，它不会在临界离子插入水平上改变符号。这种差异可能是由于在高 $\mathrm{Li}^+$ 插入水平下薄膜中发生的降解效应。