Single particle entropy stability and the temperature-entropy diagram in quantum dot transistors

Single and double quantum dot (QD) transistors have been used to investigate entropy transitions in the single particle limit. Precisely controlled QD electron states allow a few-particle thermodynamic system to be defined. Charge stability diagrams are calculated to find the Gibbs entropy S vs bias voltage, providing a framework to define single particle entropy diagrams. The calculation method is applied to experimental dopant atom QD transistor characteristics. As multiple states become occupied, S increases in a stepwise manner towards S=klnω, where ω is the total number of microstates, retaining the Boltzmann interpretation of entropy. The T-S diagram vs gate voltage, where T is temperature, reflects underlying single particle state transitions and enables the definition of heat cycles. These diagrams approximate the behavior of macroscopic phase changes in magnetic, liquid-vapor, and superconducting systems.