Citation: Yadav, P.; Chakraborty, S.; Moraru, D.; Samanta, A. Variable-Barrier Quantum Coulomb Blockade Effect in Nanoscale Transistors. Nanomaterials 2022, 12, 4437. https://doi.org/10.3390/ nano12244437 Academic Editor: Antonio Di Bartolomeo Received: 18 November 2022 Accepted: 9 December 2022 Published: 13 December 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). nanomaterials Article Variable-Barrier Quantum Coulomb Blockade Effect in Nanoscale Transistors Pooja Yadav 1,† , Soumya Chakraborty 1,† , Daniel Moraru 2 and Arup Samanta 1,3, * 1 Quantum/Nano-Science and Technology Lab, Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, India 2 Research Institute of Electronics, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu 432-8011, Japan 3 Centre of Nanotechnology, Indian Institute of Technology Roorkee, Roorkee 247667, India * Correspondence: arup.samanta@ph.iitr.ac.in † These authors contributed equally to this work. Abstract: Current–voltage characteristics of a quantum dot in double-barrier configuration, as formed in the nanoscale channel of silicon transistors, were analyzed both experimentally and theoretically. Single electron transistors (SET) made in a SOI-FET configuration using silicon quantum dot as well as phosphorus donor quantum dots were experimentally investigated. These devices exhibited a quantum Coulomb blockade phenomenon along with a detectable effect of variable tunnel barriers. To replicate the experimental results, we developed a generalized formalism for the tunnel-barrier dependent quantum Coulomb blockade by modifying the rate-equation approach. We qualitatively replicate the experimental results with numerical calculation using this formalism for two and three energy levels participated in the tunneling transport. The new formalism supports the features of most of the small-scaled SET devices. Keywords: quantum dot; donor atom transistor; single electron transistor; Coulomb blockade; variable tunnel barrier 1. Introduction Advancement in nano-fabrication techniques for the development of silicon (Si) nanoscale devices has provided a valuable platform for the realization and investigation of sophisticated devices that can transfer electrons with higher efficiency and accuracy than the typical metal-oxide-semiconductor field-effect transistors (MOSFETs) [1–6]. Some of these devices, namely the single-electron transistors (SETs), exploit the physics of Coulomb blockade (CB) as the basic operational principle. These exotic devices can have target functionalities towards logic circuits [7], single-electron memories [8], single-charge sens- ing [9], charge- and spin-based quantum computing [10,11], single electron pump [12], single photon detector [13], highly sensitive biosensors [14], etc. A double-barrier quantum dot (QD) geometry formed within such SETs can periodically suppress single electron transfer due to the subsequent charging energy. This phenomenon is generally known as the Coulomb blockade [15–18]. Initially, SETs were studied in metallic QDs, where the discreteness of the energy levels within the QD can be ignored [19–23]. However, in nanoscale semiconductor-based SETs, where the energy separation between successive discrete energy levels within the QD is comparable to or higher than the thermal energy, the scenario is different from the classical Coulomb blockade and is known as quantum Coulomb blockade (QCB). In this QCB regime, single electron passes through discrete energy levels of the QD. Such phenomenon is generally observed for nano-scaled SET devices fabricated in two-dimensional electron gas (2DEG) systems, semiconductor QDs and dopants as QDs [24–29]. The initial theoretical framework for QCB had been put forward by C.W.J. Beenakkar [30], which is valid mainly for the linear-response regime. The general procedure to analyze the Nanomaterials 2022, 12, 4437. https://doi.org/10.3390/nano12244437 https://www.mdpi.com/journal/nanomaterials