COMSOL 6.2 - Recombination and Generation

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. 48).

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 3-93 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 3-93 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:

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Direct recombination: Important for direct band gap semiconductors such as gallium arsenide. An electron–hole pair recombines with the emission of a photon (of energy close to the band gap).

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Trap-assisted, or Shockley–Read–Hall recombination: Important in indirect band gap semiconductors, such as silicon and germanium. A defect (usually with an energy close to the midgap) is involved in the recombination process. An electron or hole is first trapped by the defect and then emitted into the valence/conduction band, resulting in a reduction in the number of available carriers. The carrier energy is typically converted to heat.

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Auger recombination: In Auger recombination three carriers are involved. A collision between two like carriers (for example, electrons) results in the recombination of one of the electrons with a hole. The energy released by the transition is transferred to the surviving electron. The resulting highly energetic electron subsequently loses energy as it undergoes collisions.

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Impact ionization: Impact ionization occurs when, for example, an energetic electron undergoes a collision in which it loses sufficient energy to promote an electron in the valence band to the conduction band, resulting in an additional electron–hole pair being produced. It is the mechanism responsible for avalanche breakdown.

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.

Direct recombination is usually the dominant recombination mechanism in direct band gap semiconductors. The recombination rate can be derived phenomenologically from the two process that contribute to the net recombination rate: recombination of a conduction band electron with a hole in the valence band (caused by the emission of a photon by the electron) and generation of an electron hole pair by a valence band electron adsorbing a photon and moving into the conduction band. The corresponding capture (c) and emission (e) processes, are shown in Figure 3-10.

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:

The generation process is more difficult to treat phenomenologically since it involves photon mediated transitions that must occur vertically in the band structure with a fixed energy difference (corresponding to the wavelength of the photons). However for a given band structure and frequency, provided that the semiconductor is nondegenerate, the photon transitions occur at a rate approximately independent of the carrier concentrations for a given illumination intensity. Thus a constant generation rate is a reasonable approximation under these circumstances:

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.

In thermal equilibrium these rates must be equal so:

The net rate of direct recombination is therefore given by:

where C=Cc is a material constant (SI unit: m3/s).

For the density-gradient formulation, the second term on the right hand side is replaced with the product of equilibrium concentrations given by one similar to Equation 3-53.

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 cm3/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 reemission into the valence or conduction band (see Ref. 23 and Ref. 24). 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:

For the density-gradient formulation, the second term in the numerator on the right hand side is replaced with the product of equilibrium concentrations given by one similar to Equation 3-53.

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>):

Auger recombination becomes important at high nonequilibrium carrier densities, since the process requires three carriers. For example, when two electrons collide the collision can result in the recombination of one of the electrons with a hole. The energy released by the transition is transferred to the surviving electron, which subsequently returns to equilibrium as it undergoes collisions with the lattice. The recombination rate is given by:

where Cn and Cp are material constants (SI unit: m6/s).

For the density-gradient formulation, the second term in the second factor on the right hand side is replaced with the product of equilibrium concentrations given by one similar to Equation 3-53.

For silicon, Cn≈2.8·10−31 cm6/s and Cp≈9.9·10−32 cm6/s. In practice, these coefficients are weakly dependent on temperature and doping level.

Impact ionization becomes important at high electric fields. When the carriers are accelerated by the electric field between collisions to velocities where their energies are greater than the gap energy, they can dissipate enough energy during collisions that additional electron hole pairs can be generated. Impact ionization is responsible for the phenomenon of avalanche breakdown. The carrier generation rate due to impact ionization is given by:

For the values of αn and αp, the Semiconductor interface allows user-defined expressions or using the (Ref. 25):

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, cn, cp, dn, and dp are material properties (see Ref. 25 for values of these properties for silicon, germanium, gallium arsenide, and gallium phosphate).