Crystals 2023, 13, 83. https://doi.org/10.3390/cryst13010083 www.mdpi.com/journal/crystals Article Impurity Properties of Inversion Layers with Electronic and Substrate Quantum Screening Kamo Aharonyan 1 , Ninel Kokanyan 2,3, * and Edvard Kokanyan 4,5 1 Department of Physics, National Polytechnic University of Armenia, Yerevan 0009, Armenia 2 Chaire Photonique, Laboratoire Matériaux Optiques Photonique et Systèmes (LMOPS), CentraleSupélec, 57070 Metz, France 3 Laboratoire Matériaux Optiques Photonique et Systèmes (LMOPS), Université de Lorraine, 57070 Metz, France 4 Department of Physics, Armenian State Pedagogical University after Kh. Abovyan, Yerevan 0010, Armenia 5 Institute for Physical Researches, National Academy of Sciences of Armenia, Ashtarak-2 0204, Armenia * Correspondence: ninel.kokanyan@centralesupelec.fr Abstract: In this paper, the combined effect of electronic and substrate screening on impurity states in inversion layers is investigated theoretically. An explicit expression of the screened impurity in- teraction potential with an effective screening parameter, depending on the material and structural parameters, is established analytically for the first time. The main physical results are (a) an en- hancement of the carrier saturation effect and (b) the dependence of the nature of the screening mechanism on the dielectric type (low-κ and high-κ) of the oxide layer. An experimentally measur- able impurity binding energy is studied and numerically presented for realistic InSb/SiO2/SiO2/metal (ll-) and InSb/S(sulfur)/HfO2/metal (lh-κ type) multi-layer structures. A sub- stantial enhancement of the binding energy is obtained with the non-degenerate Q2D EG for the ll- κ-type structure, reaching an almost fourfold value of the InSb bulk sample (~0.66 meV). Keywords: impurity; electronic screening; substrate screening; inversion layer 1. Introduction Silicon-based nanoelectronics have now reached their physical limits. The ongoing trend towards the further optimization of related devices currently emphasizes the im- portance of both semiconductor active channels with mobile two-dimensional electron gas (2D EG) and dielectric gate layers with a nano-scale-equivalent oxide thickness [1,2]. In this regard, semiconductors of the III–V group [3], metal dichalcogenides and few-layer graphenes [4] are currently the most promising candidate materials due to their high switching speed and low power consumption, which are important requirements for nanodevices. In turn, high-κ dielectric gate oxides, such as Al2O3 [5] and HfO2 [6], as well as HfO-AlO [7,8], ZrO-AlO [8], HfO-LaO [9] and HfO-chalcogenide [10] nanolaminate alloy forms, are currently of key interest. The aforementioned active channel materials possess a narrow energy band gap, resulting in the easy generation of defect states, for example, in the multi-layer quantum structure [7]. Certainly, such states can directly affect transport and optical properties, and clarifying the specific roles of defect states, impurity states in particular, in the noted system is fundamental [11]. In the present work, the task is to investigate the effect of screening mechanisms on impurity states in a multi-layer quantum structure by studying the impurity optical char- acteristics, namely, both the calculable and measurable quantities of the impurity binding energy. As for the study of transport characteristics (the scattering rate and mobility) un- der these conditions, we strictly intend to address this in our forthcoming article. Citation: Aharonyan, K.; Kokanyan, N.; Kokanyan, E. Impurity Properties of Inversion Layers with Electronic and Substrate Quantum Screening. Crystals 2023, 13, 83. https://doi.org/10.3390/ cryst13010083 Academic Editor: Dmitri Donetski Received: 23 November 2022 Revised: 22 December 2022 Accepted: 27 December 2022 Published: 2 January 2023 Copyright: © 2023 by the authors. Li- censee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and con- ditions of the Creative Commons At- tribution (CC BY) license (https://cre- ativecommons.org/licenses/by/4.0/).