8.1 Drift diffusion modelBasic theory1 Basic Semiconductor Equations
The device modeling represents the problem of charge transport in macroscopic scale in comparison with the semiconductor lattice dimensions. In this approach we assume that the movement of electron
between two collisions is described by Newton law and only the interactions with lattice are described by quantum mechanics. So, the carriers can be described statistically thought the distribution function that is given by
solving of Boltzman transport equation. However, this is a difficult task to accomplish, thus through several idealistic simplification of Boltzman equation we obtain the practical system of equations
(8.1A.1) called the driftdiffusion model. These equations are also called the basic semiconductor equations. They can be easily deduced from Maxwells equations.

$\nabla \cdot \left(\nabla \Psi \right)=\frac{q}{\epsilon}(npC)$,

( 8.1A.1a )


$\nabla \cdot ({\mathbf{J}}_{n})q\frac{\partial n}{\partial t}=qR$,

( 8.1A.1b )


$\nabla \cdot ({\mathbf{J}}_{p})+q\frac{\partial p}{\partial t}=qR$,

( 8.1A.1c )


${\mathbf{J}}_{n}=qn{\mu}_{n}\mathbf{E}+q{D}_{n}\nabla n$,

( 8.1A.1d )


${\mathbf{J}}_{p}=qp{\mu}_{p}\mathbf{E}q{D}_{p}\nabla p$.

( 8.1A.1e )

Equation (8.1A.1a) is a Poisson equation for electric potential Ψ, where the charge density is
defined by elementary charge q and the concentration of negatively
charged electrons n, the concentration of positively charged holes p
and fixed charge concentration C. ε in this equation
denotes permittivity of used semiconductor. Values of relative permittivity for
the most common materials are listed in tab. 8.1A.1.
Tab. 8.1A.1  Typical values of relative permittivity [44]. 

Materiál  ε_{r} [] 
Si  11,7 
SiO_{2}  3,9 
Si_{3}N_{4}  7,2 
GaAs  12,5 
Ge  16,1 

Continuity equation (8.1A.1b) and (8.1A.1c) express that the sources and sinks
of current density are fully compensated by the time change of free charge and joint by function R, which expresses the rate of generation and
recombination of electrons and holes. J_{n} denotes current density caused by electrons, J_{p} current density
caused holes and t is the time. Expression of the rate R requires a good knowledge of the physical mechanisms of generation and recombination in
semiconductor materials and is one of the key parameters for obtaining the relevant results of simulations of semiconductor structures.
In both current densities equations (8.1A.1d) and (8.1A.1e) the
first term on the right side is a component of current density caused by the Lorenz force, taking into account only the influence
of the electric field E (the effect of magnetic induction is neglected), and establishes the effective mobility of
electrons μ_{n} and holes μ_{p} Their value is generally defined by empirical
relationships. Mobility of charge carriers is a quantity which has a large influence on the properties of semiconductor components and therefore its correct modeling
is very important. The second term on the right side of equations (8.1A.1d) and (8.1A.1e) represents
the effect of carrier diffusion in the direction of the gradient of concentration and introduces Einstein diffusion constants

${D}_{n}={\mu}_{n}\frac{kT}{q}$,

( 8.1A.2a )


${D}_{p}={\mu}_{p}\frac{kT}{q}$,

( 8.1A.2b )

where k is the Boltzmann constant and T is the temperature.
2 Physical Parameters
The equations (8.1A.1) constitute the basic equation system for analyzing and simulation of most semiconductor devices.
However, there may be cases when become important physical phenomena that are not sufficiently covered by these equations. That is the case of tunneling
currents through very thin layers, which have a significant effect on the properties of the devices. Usually are used the empirical corrections than more
complex system of equations to describe such phenomena. Physical parameters of these equations define the geometry of the device, kind of semiconductor material and used manufacturing technology.
Their modeling will be discussed in the following paragraphs.
Doping Profile
Distribution of impurities in the semiconductor volume, resulting in N type or P type semiconductor, determines the geometry and function of a semiconductor device. It is therefore essential
input information to achieve a accurate results of simulations, so the manufacturing processes such as ion implantation, diffusion, thermal oxidation, epitaxial accretion and others should be properly
modeled. The scope of following paragraphs will be restricted only to processes of ion implantation and diffusion of impurities in volume of semiconductor
substrate. The resulting profiles of these processes are often modeled by Gaussian function. An example of doping profile of PN diode is shown in fig. 8.1A.1.

Fig. 8.1A.1  Example of PN diode doping profile 

Carriers Mobility
The movement of carriers in semiconductors is disturbed by collisions with the crystal lattice, impurities
and defects, therefore, track of the carriers movement between two points is not a straight line. In order of easily imaginable description this phenomenon was introduced the mobility, which is modeled with respect to the mechanisms
that affect the carriers collision. These mechanisms may be caused by
 atomic lattice material
 ionized impurities
 carriercarrier collisions
 neutral impurities
 intensity of the electric field
Often can be used the value ${\mu}_{n,p}^{300}$
specified for the given material at a temperature of 300K presented in tab. 8.1A.2.
Tab. 8.1A.2  Typical values of carriers mobility at temperature 300K [44]. 

materiál  ${\mu}_{n}^{300}$ [m^{2}V^{1}s^{1}]  ${\mu}_{p}^{300}$ [m^{2}V^{1}s^{1}] 
Si  0,14  0,04 
GaAs  0,8  0,04 
Ge  0,38  0,18 

Generation and Recombination of Carriers
In the previous paragraphs was mentioned function R, which describes the rate of generation and recombination of electrons and holes. The dominant mechanism is ShockleyReadHall recombination and
generation, which is modeled as a trap between valence and conductive band of semiconductor. The resulting function has the following form

${R}^{SRH}=\frac{np{n}_{i}{}^{2}}{{\tau}_{p}(n+{n}_{i})+{\tau}_{n}(p+{p}_{i})}$,

( 8.1A.3 )

where, τ_{n} and τ_{p} are carriers lifetime and n_{i} is the intrinsic concentration.
3 Analysis of Basic Equations and Boundary Conditions
In the first section was given a set of basic equations (8.1A.1). It is important to note that in the current densities in the relations
(8.1A.1d) and (8.1A.1e) have been omitted the currents caused by a narrowing of the forbidden zone and temperature gradient, because their effect is considered to be negligible.
However, the (8.1A.13) may become incorrect if any of these phenomenon become significant. Using (8.1A.1a) and by substituting the current density
equations (8.1A.1d) and (8.1A.1e) into the continuity equation (8.1A.1b) and (8.1A.1c) we obtain a system of three partial differential equations (8.1A.4) with
variables Ψ, n a p.

$\nabla \cdot \left(\nabla \psi \right)\frac{q}{\epsilon}(npC)=0$,

( 8.1A.4a )


$\nabla \cdot ({D}_{n}\nabla nn{\mu}_{n}\nabla \psi )R=\frac{\partial n}{\partial t}$,

( 8.1A.4b )


$\nabla \cdot ({D}_{p}\nabla p+p{\mu}_{p}\nabla \psi )R=\frac{\partial p}{\partial t}$.

( 8.1A.4c )

For mathematical analysis we need to know the initial estimation of Ψ, n and p and boundary conditions in analyzed domain. The bounded
domain D is generally threedimensional, like practically all semiconductor structures. However, in many cases, this domain can be considered as twodimensional, or even onedimensional, which make the analyzed the
problem significantly easier.
Lets ∂D denotes partial boundary of domain D. It can be divided into two parts

$\partial D=\partial {D}_{p}\cup \partial {D}_{a}$

( 8.1A.5 )

where ∂D_{p} denotes those parts of the boundaries that are real, physical boundaries, such as contacts and interfaces. ∂D_{a} indicates artificial
interfaces that are introducing, for example, to exclusion of substructure on the large substrate or to the separation of neighboring devices on a common substrate.

Fig. 8.1A.2  2D geometry of planar MOSFET transistor [44] 

The illustrative description of idealized 2D geometry of MOS transistor is depicted in fig. 8.1A.2. The entire domain is represented by polygon ABCDEFGHA.
Equation (8.1A.4) is valid only in the subdomain ABEFGHA. Insulator BCDEB can be characterized either by the Laplace equation for
electrostatic potential (8.1A.6) or by assumption of absence of charge carriers (8.1A.7).

$\nabla \cdot \nabla \left(\Psi \right)=0$

( 8.1A.6 )

However, the use of these equations makes it impossible to determine the current passing through the base and the influence of the charge on insulating oxide layer.
The boundaries AB, EF, CD and BE can be considered as physical boundaries representing three contacts and interface between the semiconductor and insulator.
Boundaries AH, BC, DE, FG and GH are considered as artificial boundaries.
Physical boundaries can be divided into three categories.

$\partial {D}_{P}=\partial {D}_{O}\cup \partial {D}_{S}\cup \partial {D}_{I}$,

( 8.1A.8 )

where ∂D_{O} indicate the ohmic contact, ∂D_{S} denotes Schottky contact and ∂D_{I} is the insulator interface.
Summary of boundary conditions for all three dependent variables of the system (8.1A.4) is given in the following tab. 8.1A.3, where
n denotes normal vector perpendicular to ∂D and u_{n,p} denotes the speed of thermal recombination.
Tab. 8.1A.3  Boundary conditions typical semiconductor interface structures. 

 ohmic contact  Shottky contact  semiconductor  insulator  artifical interface 
electric potential  ${\psi}_{b}=\frac{kT}{q}ar\mathrm{sinh}\left(\frac{C}{2{n}_{i}}\right)$ 
${\psi}_{b}=\frac{kT}{q}ar\mathrm{sinh}\left(\frac{C}{2{n}_{i}}\right)$ 
$\frac{\partial \psi}{\partial \mathbf{n}}=0$ 
$\frac{\partial \psi}{\partial \mathbf{n}}=0$ 
electron concentration  $n=\frac{\sqrt[]{{C}^{2}+4{n}_{i}{}^{2}}+C}{2}$ 
${\mathbf{J}}_{n}\mathbf{n}=q{u}_{n}\left(n\frac{\sqrt{{C}^{2}+4{n}_{i}{}^{2}}+C}{2}\right)$ 
${\mathbf{J}}_{n}\mathbf{n}=0$ 
$\frac{\partial n}{\partial \mathbf{n}}=0$ 
hole concentration  $p=\frac{\sqrt[]{{C}^{2}+4{n}_{i}{}^{2}}C}{2}$ 
${\mathbf{J}}_{p}\mathbf{n}=q{u}_{p}\left(p\frac{\sqrt{{C}^{2}+4{n}_{i}{}^{2}}C}{2}\right)$ 
${\mathbf{J}}_{p}\mathbf{n}=0$ 
$\frac{\partial p}{\partial \mathbf{n}}=0$ 

