Basic Semiconductor Equations
Poisson’s equation is a very general differential equation governing the operation of IC devices and is based on Maxwell’s field equation that relates the charge density to the electric field potential. We know that the electric field £ in a semiconductor is equal to the negative gradient of the electrostatic potential ф such that
Mathematically, Poisson’s equation (for silicon) is stated as or, using Equation 2.70,
p(.v) is the net charge density at any point x
£0 (= 8.854 x 10"14 F cm') is the permittivity of free space
Ksj(= 11.8) is the relative permittivity of silicon
Now, if n and p are the free electron and hole concentrations, respectively, corresponding to Nj and N,~ ionized donor and acceptor concentrations, respectively, in silicon, then we can express Equation 2.72 as
Assuming complete ionization of dopants, Nd = Nd and N~ = Na, we can write Poisson’s equation as
Equation 2.74 is a one-dimensional (ID) equation and can easily be extended to three-dimensional (3D) space. The ID Poisson equation is adequate for describing most of the basic device operations. However, for small geometry advanced devices like FinFETs, the 2D (two-dimensional) or 3D Poisson’s equation must be used.
Another form of Poisson’s equation is Gauss’ law, which is obtained by integrating Equation 2.71 and is given by
It is to be noted that the semiconductor as a whole is charged neutral, that is, p must be zero. However, when the space charge neutrality does not apply, Poisson’s equation must be used to describe the distribution of charge and electrostatic potentials in the semiconductor.
In Section 18.104.22.168, we have shown that the electron current density J,hdrifl due to drift of electrons by an applied electric field is given by Equation 2.45. On the other hand, the electron diffusion current density Jndiff due to the concentration gradient in a semiconductor as described in Section 22.214.171.124 is given by Equation 2.55. Thus, when an electric field is present in addition to a concentration gradient, both the drift and diffusion currents will flow through the semiconductor. The total electron current density Jn at any point x is then simply equal to the sum of the drift and diffusion currents, that is, J„ (= J„dri(, + Jn>d,sf). Therefore, the total electron current in a semiconductor is given by
Similarly, the total hole current density Jp (= Jpdrif, + dp(tifS) 's given by
so that the total current density J = Jn + Jp. The current Equations 2.76 and 2.77 are often referred to as the transport or drift-diffusion equations of current carriers.
Under thermal equilibrium, no current flows inside the semiconductor and therefore, J„ = Jp = 0. However, under the non-equilibrium conditions, J„ and Jp can be written in terms of quasi-Fermi potentials ф„ and ф;, for electric field, E in Equations 2.76 and 2.77, respectively, to get
All parameters have their usual meanings as defined earlier.
When carriers diffuse through a certain volume of a semiconductor, the current density leaving the volume may be smaller or larger depending upon the recombination or generation taking place inside the volume. Let us consider a small length Ax of a semiconductor shown in Figure 2.13 with cross-sectional area A in the xy plane.
From Figure 2.13, the hole current density entering the volume A.Ax is Jp(x), whereas that leaving is a Jp(x + Да). From the conservation of charge, the rate of change of hole concentration in the volume is the sum of: (a) net holes flowing out of the volume; and (b) net recombination rate. That is,
The negative sign is due to the decrease of holes due to recombination; and Gp and Rp are the generation and recombination rates of holes in the volume, respectively. Then from Equation 2.79 we can show
Similarly, for electrons we can show where:
G„ and Rn are the generation and recombination rates of electrons, respectively
Equations 2.80 and 2.81 are called the continuity equations for holes and electrons, respectively, and describe the time-dependent relationship between current density, recombination and generation rates, and space. They are used for solving transient phenomena and diffusion witli recombination-generation of carriers.
Equations 2.74, 2.78, 2.80, and 2.81 constitute a complete set of ID equations to describe carrier, current, and field distributions in a semiconductor; however, they can easily be extended to 3D space. Given appropriate boundary conditions, we can solve them for any arbitrary device structure. Generally, we will be able to simplify them based on physical approximations.
FIGURE 2.13 Current continuity in a semiconductor: Jp(x) is the hole currents flowing into an elemental length Д.г of the semiconductor and Jp(x + A.v) is the net current flowing out after carrier generation-recombination processes inside the element; U is the net recombination rate.