From Equation 3-49 the
np product at equilibrium is given by:
For the density-gradient formulation, instead of Equation 3-49, one similar to
Equation 3-53 is used, and the following development is similar but does not result in exactly the same formulas (
Ref. 46).
Here Eg=Eg0-ΔEg=Ec-Ev is the effective band gap that includes the material band gap
Eg0 and changes in the band gap (
ΔEg), which result from effects such as
band gap narrowing.
Equation 3-91 is known as the
mass action law. Note that in an undoped semiconductor, where charge conservation at equilibrium implies
n = p, the carrier concentrations are equal to the intrinsic concentration,
ni, since band gap narrowing only occurs in heavily doped semiconductors.
Equation 3-91 is frequently used in recombination models to define the equilibrium carrier concentration.
In considering recombination processes, it is useful to note that photon momentum is usually negligible in comparison to the carrier momentum, and that the energy of phonons (lattice vibrations) is typically small compared to the band gap. Thus photon mediated transitions are typically vertical in an E-k diagram and phonon mediated transitions are frequently horizontal. Direct transitions involving both a photon and a phonon are usually so unlikely that they do not contribute significantly to recombination — in indirect semiconductors recombination is usually mediated by traps (impurities with energies close to the midgap). The following mechanisms are common in practical materials:
The Semiconductor interface has features to add Auger Recombination,
Direct Recombination, and
Trap-Assisted Recombination to a semiconducting domain.
Impact Ionization Generation is also available.
User-Defined Recombination or
User-Defined Generation (use a negative recombination rate for generation) can also be added. Note that the recombination and generation features are additive, so it is possible to model several processes simultaneously.
The recombination process is easiest to treat phenomenologically. Consider electrons in the conduction with energy E. A certain fraction of these electrons decay to states in the valence band with energy
E’. This process contributes an amount
drc to the total recombination process:
where f(E) is the Fermi-Dirac function,
gc(E) is the density of states in the valence band,
gv(E) is the density of states in the conduction band and
cc(
E,E’) is the rate constant for decay between states
E and
E’. If it is assumed that the rate constant does not vary significantly in the vicinity of the band edges then
cc(
E,E’)~
Cc and the expression can be directly integrated to yield:
Where Ce is a rate constant (dependent on the wavelength and the intensity of the incident light). These approximations do not always apply, and in circumstances where a more detailed model is appropriate, the Semiconductor interface includes the
Optical Transitions feature.
where C=Cc is a material constant (SI unit: m
3/s).
For indirect band gap semiconductors such as silicon and germanium, C is effectively zero. In GaAs (a widely used direct band gap semiconductor)
C is approximately 1·10
-10 cm
3/s.
In an indirect band gap semiconductor at low fields, trap-assisted recombination is usually the dominant contributor to Un and
Up. This recombination mechanism involves the trapping of an electron or hole followed by re-emission into the valence or conduction band (see
Ref. 21 and
Ref. 22). The details of this process are described in the
Traps section, and COMSOL Multiphysics provides features to model the traps explicitly, solving for the occupancy of the traps. For less detailed modeling it is common to use the original model of Shockley, Read, and Hall, in which steady-state conditions are assumed for traps located at a single energy level.
Equation 3-87 determines the occupancy factor for the state,
ft. In the steady state the time derivative is zero and the following occupancy factor is obtained by solving the equation:
Finally note that Cn and
Cp can be written in terms of the thermal velocity of the electrons and holes, respectively (
vn,th/
vp,th), as well as their average capture cross sections (
<σn>/
<σp>):
where Cn and
Cp are material constants (SI unit: m
6/s).
For silicon Cp≈2.8·10
-31 cm
6/s and
Cp≈9.9·10
-32 cm
6/s. In practice these coefficients are weakly dependent on temperature and doping level.
For the values of αn and
αp, the Semiconductor interface allows user-defined expressions or using the model of Okuto and Crowell (
Ref. 23):
Where E||,n and
E||,p are the components of the electric field parallel to the electron and hole currents, respectively, and
Tref,
an,
ap,
bn,
bp,
dn, and
dp are material properties (see
Ref. 23 for values of these properties for silicon, germanium, gallium arsenide, and gallium phosphate).
The User-Defined Generation feature allows the carrier generation rate to be defined for both the electrons and the holes.