So far we have seen that several mechanisms for non-radiative recombination exist, including Shockley-Read, Auger, and surface recombination. Even though non-radiative recombination can be reduced, it can never be totally eliminated. For example, surface recombination can be drastically reduced by device designs that spatially separate the active region from any surfaces. However, even if the separation is large, a few carriers will still diffuse to the surface and recombine there.

Just as for surface recombination, non-radiative bulk recombination and Auger recombination can never be totally avoided. Any semiconductor crystal will have some native defects. Even though the concentration of these native defects can be low, it is never zero. Thermodynamic considerations predict that if an energy Ea is needed to create a specific point defect in a crystal lattice, the probability that such a defect does indeed form at a specific lattice

site, is given by the Boltzmann factor, i. e. exp (- Ea / kT ). The product of the concentration of lattice sites and the Boltzmann factor gives the concentration of defects. A native point defect or extended defect may form a deep state in the gap and thus be a non-radiative recombination center.

Exercise: Concentration of point defects. Assume that the energy required to move a substitutional lattice atom into an interstitial position is Ea = 1.1 eV. What is the equilibrium concentration of interstitial defects of a simple cubic lattice with lattice constant a{) = 2.5 A?

Solution: The concentration of lattice atoms of a simple cubic lattice is given by N = a0-3 =

6.4 x 1022 cm-3. The concentration of interstitial defects under equilibrium conditions at room temperature is then given by

Ndefect = N exp (-Ea /kT) = 2.7 x 104 cm-3 .

Note that the calculated concentration of defects is small when compared to the typical concentrations of electrons and holes. If the defect discussed here forms a level in the gap, non-radiative recombination through the defect level can occur.

Another issue is the chemical purity of semiconductors. It is difficult to fabricate materials with impurity levels lower than the parts per billion (ppb) range. Thus, even the purest semiconductors contain impurities in the 1012 cm-3 range. Some elements may form deep levels and thus reduce the luminescence efficiency.

In the 1960s, when the first III-V semiconductors had been demonstrated, the internal luminescence efficiencies at room temperature were very low, typically a fraction of 1%. At the present time, high-quality bulk semiconductors and quantum well structures can have internal efficiencies exceeding 90%, and in some cases even 99%. This remarkable progress is due to improved crystal quality, and reduced defect and impurity concentrations.

Next, we calculate the internal quantum efficiency in a semiconductor with non-radiative recombination centers. If the radiative lifetime is denoted as Tr and the non-radiative lifetime is denoted as Tnr, then the total probability of recombination is given by the sum of the radiative and non-radiative probabilities:

-1 -1 , -1

т — т r + т nr

The relative probability of radiative recombination is given by the radiative probability over the total probability of recombination. Thus the probability of radiative recombination or internal quantum efficiency is given by

т

nint — — =r. (2 41)

тг + тпг

The internal quantum efficiency gives the ratio of the number of light quanta emitted inside the semiconductor to the number of charge quanta undergoing recombination. Note that not all photons emitted internally may escape from the semiconductor due to the light-escape problem, reabsorption in the substrate, or other reabsorption mechanisms.