Absorption of belowbandgap light in semiconductors
To obtain high lightextraction efficiency and avoid absorption of light, all semiconductor layers
other than the active region should have a bandgap energy larger than the photon energy. This
can be done in different ways, for example by using double heterostructures, window layers, and other structures that will be discussed below. In this section, we discuss the absorption of light if the energy of the light is below the energy gap of the semiconductor.
Naively, one would assume that a semiconductor can absorb light only if the photon energy is higher than the bandgap energy and that the semiconductor is transparent for photon energies
below the bandgap. However, semiconductors do absorb belowbandgap light, although with a much lower absorption coefficient.
Fig. 9.1. Absorption coefficient of a semiconductor with bandgap Eg versus energy. The “Urbach tail” dominates absorption near but below the bandgap. Absorption further below the bandgap is dominated by freecarrier absorption.
The absorption coefficient versus energy of an idealized semiconductor and a real semiconductor is shown schematically in Fig. 9.1. In the idealized semiconductor at low temperatures, the absorption coefficient versus energy is given by (Pankove, 1971)
а к (E  Eg)1/2 (direct gap) (9.2a)
а к (E  Eg)2 (indirect gap) . (9.2b)
The idealized semiconductor has a zero bandtoband absorption coefficient at the bandgap energy (E = Eg). The absorption strength in a real semiconductor, for belowbandgap light, can be expressed in terms of an exponentially decaying absorption strength. In this absorption tail, called the Urbach tail, the absorption coefficient versus energy is given by
а = agexp[ (E  Eg)/EUrbach] (93)
where аg is the experimentally determined absorption coefficient at the bandgap energy and EUrbach is the characteristic energy (here called the Urbach energy), which determines how rapidly the absorption coefficient decreases for belowbandgap energies.
Urbach (1953) measured the absorption tail for different temperatures and showed that the Urbach energy is approximately kT, the thermal energy. The temperature dependence of the Urbach tail led Knox (1963) to the conclusion that the belowbandgap transitions are phonon  assisted transitions. Thus, the Urbach energy is given by
EUrbach = kT.
(9.4) 
The Urbach tail can be caused by mechanisms other than phononassisted absorption. Any
mechanism introducing a potential fluctuation will lead to local variations of the semiconductor band edges. As a result of these fluctuations, the bandgap energy fluctuates as well, and below  bandgap transitions can occur.
The most common potential fluctuations are fluctuations caused by random dopant distribution and local variations of the chemical composition of a ternary or quaternary alloy semiconductor.
Potential fluctuations caused by random dopant distribution can be calculated using Poisson statistics. The magnitude of these fluctuations is given by (see, for example, Schubert et al.,
rL e3/4 
1997)
(9.5)
where rs is the screening radius.
ґ 1/2 dEg f x (1 ~ x) 1 
(9.6) 
^alloy 
where x is the alloy composition of the ternary semiconductor alloy, a0 is the semiconductor lattice constant, and Vexc is the excitonic volume of the electronhole pair. It depends on the specific case at hand, which of the different physical effects dominates in the formation of the Urbach tail. Generally, binary semiconductors such as GaP or GaAs have a smaller tail than alloys such as AlGaAs or GaAsP. Furthermore, lightly doped semiconductors have a smaller Urbach tail than heavily doped semiconductors. For energies sufficiently lower than the bandgap energy, absorption due to the Urbach tail is negligibly small and freecarrier absorption becomes the dominant absorption mechanism. As the name suggests, a free carrier is excited to a higher energy by absorption of a photon. The absorption transition must conserve momentum. Whereas photons have very small momenta, 
Potential fluctuations caused by random compositional fluctuations can be calculated using binomial statistics. The magnitude of these fluctuations is given by (see, for example, Schubert et al, 1984)
electrons must undergo a momentum change when excited higher within a parabolic band. This momentum change is provided by acoustic phonons, optical phonons, or by impurity scattering.
Freecarrier absorption is proportional to the freecarrier concentration, since a free carrier is required for an absorption event. Theoretical considerations in terms of the classical Drude free  electron model further show that freecarrier absorption increases as the square of the incident wavelength (Pankove, 1971). Thus the proportionalities
22 afc к n X and afc к p X (9.7)
are valid for ntype and ptype semiconductors, respectively. Theoretical considerations in terms of a quantum mechanical treatment show that the absorption coefficient is proportional to X32, X52, and X72 depending upon whether acoustic phonon scattering, optical phonon scattering, or ionized impurity scattering is involved in the momentum conservation process, respectively (Swaminathan and Macrander, 1991).
In ntype and ptype GaAs, the roomtemperature freecarrier absorption coefficient near the bandgap energy (X « 950 nm) can be expressed as (Casey and Panish, 1978)
afc = 3 cm_1 —78^ r + 7 cm_1 —— . (9 8)
1018 cm3 1018 cm3
Inspection of the equation indicates that the freecarrier absorption coefficient can be of the order of 10 cm1 at high carrier concentrations. Approximate values for the freecarrier absorption coefficient in several compound semiconductors are given in Table 9.1.
Table 9.1. Freecarrier absorption coefficient (afc) of ntype semiconductors. (a) After Ioffe (2002). (b) After Wiley and DiDomenico (1970). (c) After Casey and Panish (1978). (d) After Kim and Bonner (1983) and Walukiewicz et al. (1980). (e) Data are extrapolated using the proportionality afc к n X2.

In LEDs, freecarrier absorption can affect the intensity of waveguided modes radiating out of the side of the chip. Freecarrier absorption also plays a role in LEDs with transparent
semiconductor substrates. Such transparent substrates have a typical thickness greater than 100 ^m. If the doping concentration of a transparent substrate is high, freecarrier absorption will reduce the lightoutput power. If it is low, the substrate becomes resistive. Thus a compromise needs to be made between the different doping requirements of a transparent substrate. For thin layers, such as confinement layers, freecarrier absorption effects are negligibly small if the optical path length within the layer is short.