Mathematical Modeling of Urbach Tail States in Semiconductors With Iso-valent Impurity Concentration

Abstract

The electronic disorder in a semiconductor, commonly quantified in terms of Urbach energy ((Formula presented.)), is known to be influenced by both temperature and impurity concentration ((Formula presented.)). The temperature dependence of (Formula presented.) has been extensively investigated in various research articles, whereas a quantitative understanding of (Formula presented.) as a function of (Formula presented.) remains limited. In this work, we present an analytical theory of (Formula presented.) in iso-valent-doped semiconductors within the Edwards and Halperin–Lax theoretical framework, explicitly incorporating the screening Length ((Formula presented.)) to derive a scaling relation (Formula presented.). This theoretical formalism is experimentally verified through Hf-doped BaTiO3 and Cr-doped PrFeO3 samples. Additionally, the developed theoretical model enables estimation of the carrier effective mass (m∗) as a function of (Formula presented.), yielding good agreement with experimental results. The obtained results are consistent with prior research, supporting the validity of this approach. Overall, this study provides a quantitative framework for understanding electronic disorders and their implications for semiconductor performance. © 2026 Wiley-VCH GmbH.