INTRODUCTION The current-voltage characteristics is of prime concern in the study of semiconductor devices with light entering as a third variable in optoelectronics devices.The external characteristics of the device is determined by the interplay of the following internal variables: 1. 2. 3. 4. 5. Electron and hole currents Potential Electron and hole density Doping Temperature Semiconductor equations The semiconductor equations relating these variables are given below: Carrier density: where is the electron quasi Fermi level and equations lead to In equilibrium = is the hole quasi Fermi level. These two = Constant Current: There are two components of current; electron current density There are several mechanisms of current flow: (i) Drift (ii) Diffusion (iii) thermionic emission (iv) tunneling and hole current density . The last two mechanisms are important often only at the interface of two different materials such as a metal-semiconductor junction or a semiconductor-semiconductor junction where the two semiconductors are of different materials. Tunneling is also important in the case of PN junctions where both sides are heavily doped.
In the bulk of semiconductor , the dominant conduction mechanisms involve drift and diffusion. The current densities due to these two mechanisms can be written as where constants. are electron and hole mobilities respectively and are their diffusion Potential: The potential and electric field within a semiconductor can be defined in the following ways: All these definitions are equivalent and one or the other may be chosen on the basis of convenience.The potential is related to the carrier densities by the Poisson equation: - where the last two terms represent the ionized donor and acceptor density. Continuity equations These equations are basically particle conservation equations: Where G and R represent carrier generation and recombination rates.Equations (1-8) will form the basis of most of the device analysis that shall be discussed later on. These equations require models for mobility and recombination along with models of contacts and boundaries. Analysis Flow Like most subjects, the analysis of semiconductor devices is also carried out by starting from
simpler problems and gradually progressing to more complex ones as described below: (i) Analysis under zero excitation i.e. equilibrium. (ii) Analysis under constant excitation: in other words dc or static characteristics. (iii) Analysis under time varying excitation but with quasi-static approximation dynamic characteristics. (iv) Analysis under time varying excitation: non quasi-static dynamic characteristics. Even though there is zero external current and voltage in equilibrium, the situation inside the device is not so trivial. In general, voltages, charges and drift-diffusion current components at any given point within the semiconductor may not be zero. Equilibrium in semiconductors implies the following: (i) steady state: Where Z is any physical quantity such as charge, voltage electric field etc (ii) no net electrical current and thermal currents: Since current can be carried by both electrons and holes, equilibrium implies zero values for both net electron current and net hole current. The drift and diffusion components of electron and hole currents need not be zero. (iii) Constant Fermi energy: The only equations that are relevant (others being zero!) for analysis in equilibrium are:
Poisson Eq: In equilibrium, there is only one independent variable out of the three variables : If one of them is known, all the rest can be computed from the equations listed above. We shall take this independent variable to be potential. The analysis problem in equilibrium is therefore determination of potential or equivalently, energy band diagram of the semiconductor device. This is the reason why we begin discussions of all semiconductor devices with a sketch of its energy band diagram in equilibrium. Energy Band Diagram This diagram in qualitative form is sketched by following the following procedure: 1. The semiconductor device is imagined to be formed by bringing together the various distinct semiconductor layers, metals or insulators of which it is composed. The starting point is therefore the energy band diagram of all the constituent layers. 2. The band diagram of the composite device is sketched using the fact that after equilibrium, the Fermi energy is the same everywhere in the system. The equalization of the Fermi energy is accompanied with transfer of electrons from regions of higher Fermi energy to region of lower Fermi energy and viceversa for holes. 3. The redistribution of charges results in electric field and creation of potential barriers in the system. These effects however are confined only close to the interface between the layers. The regions which are far from the interface remain as they were before the equilibrium Analysis in equilibrium: Solution of Poisson’s Equation with appropriate boundary conditions Non-equilibrium analysis: • The electron and hole densities are no longer related together by the inverse relationship of Eq. (5) but through complex relationships involving all three variables Y , , p