Reduced Master Equation as a Novel and Fast Method for Modeling of Spin Transport in Ferromagnetic Single Electron Transistors

Abstract

I. INTRODUCTIONA ferromagnetic single electron transistor (FMSET) is a novel device that offers further miniaturization of electrical device components which is a general need owing to the fact that high density memories and small and accurate sensors are essential to achieve more voluminous and faster chips [1]. A F/F/F FMSET includes a small central ferromagnetic electrode (island) coupled by tunnel barriers, to ferromagnetic source and drain leads as well as external nonmagnetic gate. The structure of such transistors is F/F/F as source/gate/drain [2]. When magnetic configuration of leads and island are parallel or anti parallel alignment, the junction resistance is low and high respectively as shown in Fig. 1. Fig. 2 is equivalent circuit of F/F/F SETs. There are three significant capacitors, first and second junctions (c 1 and c 2 ) and the gate capacitor (c g ), and also There are three different transport regimes: First sequential tunneling, which electrons tunnel consecutively one by one through barriers, second Co-tunneling that electrons flow following higher-order tunneling processes consisting tunneling of one, two or more electrons [1].The third tunneling regime is strong coupling that electrons propagate from the source to the drain and cannot localize on the island. These electrons will coherently move through the system, so perturbation theory fails [1]. If the resistances of tunnel junctions are much larger than the quantum resistance [14]. In this condition, the current in F/F/F SETs can be controlled under the influence of gate charge by the tuning gate voltage. When we keep the bias voltage for adding a charge in or removing a charge from the island, the charging energy c E is required. When the gate voltage is tuned, two adjacent charge states become degenerate, so the electrons can be added to or removed from the island without extra energy cost and sequential tunneling current is produced [20].Modeling of Ferromagnetic single-electron transistors is based on (I) Master Equation method, (II) the Monte Carlo method, and (III) Deterministic method. They are based high numerical intensive and need several hours to get result. However, the reduced master equation method as a novel and fast simulation method of spin dependent transport in ferromagnetic single electron transistors (F/F/F SETs) has not been mentioned in no paper, this method for modeling F/F/F SETs is more simplified and improves the speed of numerical simulation.The essential formulations for modeling following the full Generally, statistical mechanics and stochastic processes are required to specify the mechanism of many electrons tunneling at a junction. When an electron is added to the island, the system energy will be increased by c E . So, the system state is defined according to the number of electrons in the island. The steady state master equation tells us about current situation of the system by calculating occupation probability of each one of discrete set of states. In the Sequential tunneling regime, the electrons tu...