Recombination Channels

High Injection Approximations for Recombination Rates

Since optoelectronic devices usually are made to produce plenty of light, the deviation of the carrier concentrations from equilibrium in the recombination zonemust be large to obtain large recombination rates and thus light

If we write the concentrations as n_e,h = n_e,h(equ) + Dn_e,h, we now may use the simplest possible approximation called high injection approximation:

Dn_e,h	>>	n_min(equ)

i.e. the minority carrier concentration is far above equilibrium.

The surplus carriers contained in Dn_e,h are always injected into the volume under consideration (called recombination zone or recombination volume), usually by forward currents across a junction. They always must come in equal numbers, i.e. in pairs to maintain charge neutrality; otherwise large electrical fields would be generated that would restore neutrality. We thus have

Dn_e	=	Dn_h

The recombination volume usually is the space charge region of a junction or an other volume designed to have low carrier concentrations in equilibrium, cf. the picture in the backbone. Since the equilibrium concentration of both carrier types in the SCR is automatically very low, we may easily reach the high injection case.

The surplus concentration of carriers decays with a characteristic lifetime t_hi. For the recombination rate R we have in analogy to "normal" recombination more close to equilibrium:

R	=	Dn t_hi

The only difference is that the high-injection life time t_hi can be quite different from the equilibrium minority carrier life time. If t_i is the (high-injection) life time of recombination channel No. i, t_hi is given by

1 t_hi	=	S _i		1 t_i

The important thing for optimizing LED's and Laser is that the t_i are not constants but depend on the degree of injection as we will see.

Some Specifics of Recombination Channels

Let's repeat the picture from the backbone to have a listing of the more important recombination channels

Now let's look at the more mportant recombination channels and their dependence on the injection ratio, i.e. the carrier concentration

The band-band recombination channel is easy to understand:

A large number of electrons and holes finds themselves in some volume of a semiconductor at concentrations far above equilibrium. They are running around in a random matter and every now and then a hole and an electron get real close on their perambulations and recombine. The probability for this to happen is clearly proportional to the concentration n_e,h electrons and holes which as we have postulated above is Dn for both carrier types.

the recombination rate R_b-b have for the band-band recombination channel is thus given by

R_b-b = B_b-b · n²

The proportionality constant B is occasionally also called a recombination coefficient.

If we now look at the recombination channel via defects (also called "deep level" recombination because the defect must have a energy level deep in the bandgap).

The story was that an electron on its random migration might encounter a defect, e.g. an impurity atom with an energy level somewhere in the bandgap which it occupies and now is trapped and mellow (low in energy, at least for some time). A hole, somewhat later, also finds the impurity atom plus the electron unable to run away, and happily recombines with the electron. In other words: a girl, wandering around at random finds an irresistible café and sits down for a while. A boy, coming accidentally by the café, seeing the girl trapped there and in a mellow mood, knows what to do... This also means that no light is produced

We obtain a rather simple relation for the recombination rate R_defect

R_defect	=	B_dl · n_dl · n

With B_dl = recombination coefficient for this case. We have a proportionality to the density of defects and the density of carriers, giving the rate that an electron (or hole) is trapped by a defect. The rest, recombination with the opposite carrier, happens "automatically"

If we omit the recombination from donor and acceptor energy levels (which is quite similar to band-band recombination anyway) and the "exotic" recombination via excitons, we only have to consider "Auger recombination".

In this case the energy of the recombination event is transferred to another electron in the conduction band, which then looses its surplus energy by "thermalization", i.e. by transferring it to the phonons of the lattice. This means that no light is produced.

It also means that we have two have two electrons and one hole at the same place at the same time or that the Auger recombination rate R_A is given by

R_A = B_A · n³

Taken everything together we see that we have recombination rates for the major recombination channels that depend on the first, second and third power of the carrier concentration n

If we plot the total recombination rate as a function of carrier density (in a double log plot) with the proper proportionality constants (wherever they come from) and some assumption for defect densities for GaAs and Si, we obtain the following highly informative picture.

Recombination rates as function of the carrier density

The recombination rate in Si is generally far lower than in GaAs and for carrier concentrations <» 10¹⁸ cm^–3 dominated by defect recombination.

In both cases we can increase the relative strength of radiative band-band recombination by decreasing defect recombination, i.e. by making the semiconductor more pure and perfect and by not allowing too large carrier concentrations.

Here we have the explanation for the fact that very good solar cells, having by definition very low defect recombination rates, will show measurable luminescence if a large carrier concentration is introduced via an intensive flash of light. This effect is presently (2008) exploited for the characterization of solar cells.

With frame

9.1.1 Semiconductors and Light

9.2.1 Light Emitting Diodes