Absorption coefficient

The one-electron Hamiltonian in the presence of electromagnetic radiation discussed in Section 3.1.1 (Eqn. 3.4) is the relevant Hamiltonian again. Using Fermi’s golden rule as above, the number of transitions from an initial to a final state per unit volume and time due to photon absorption

N(hν) = 2π

~ X

c,v

|Mf i|2δ(Ef(kf)−Ei(ki)−hν) (3.18)

where the delta-function ensures initial and final states are separated by an energy equal to the photon energy. The transition matrix element

Mf i =hc|E·r|vi (3.19)

where the initial and final states, |vi and |ci, are Bloch states of the valence and conduction bands, respectively, and the electric dipole approximation is valid as the photon wavevector is small [55]. In practice, the magnitude of the matrix element can often be treated as independent ofkover the range of the measurement, yielding

|Mf i|2 ∝ |Pcv|2δ(q−kc+kv) (3.20)

where Pcv is a constant representing the coupling strength of the initial and final

states, and the delta-function ensures that these states have the same wavevector, neglecting the small photon wavevector,q, yielding so-calleddirecttransitions shown schematically in Fig. 3.6(a). The absorption of light is defined as the energy removed from the beam, as a fraction of the incident flux

α(hν) = hνN(hν) εω2_A2 0/2 ∝ 1 hν X c,v |Pcv|2δ(Ef(kf)−Ei(ki)−hν)δ(q−kc+kv). (3.21)

3.2. Optical absorption 45 EF (a) (b) (c) |c |v hν _{= Eg} hν _{= Eg±ħ}Ω ph hν = Eg+EF

Figure 3.6: Schematic representation of (a) direct and (b) indirect absorption, and (c) the Moss- Burstein shift.

TreatingE as a continuous function, changing the sum to an integral [89], Eqn. 3.21 can be written α(hν)∝ 1 hν |Pcv| 2_g j(Ec−Ev) (3.22) where gj(Ec−Ev) = k2 π2 · dEc(k) dk − dEv(k) dk ¸₋₁ (3.23) is the joint density of states over the conduction and valence bands. Assuming parabolic dispersion relations for the conduction and valence bands, the coefficient of direct interband absorption

α(hν)∝(hν−Eg)1/2. (3.24)

The band gap of the semiconductor can, in this limit, be determined by a linear extrapolation ofα2 _{to the background level. This expression is modified, and in fact}

an analytical solution can often not be given, for non-parabolic bands, where the non-parabolic dispersions and potentially k-dependent matrix elements should be considered [90]. However, a linear extrapolation ofα2_{can often still yield sufficiently}

accurate results.

In an indirect semiconductor, the direct transition model discussed above suggests that there will be no onset of absorption until the photon energy is equal to the direct band gap energy, rather than the fundamental band gap energy. However, indirect transitions can occur, where a change of momentum is provided by either emission or absorption of a phonon, as shown schematically in Fig. 3.6(b). In this case, the energy separation of initial and final states is given by the photon energy, hν, plus (absorption) or minus (emission) the phonon energy, ~Ωph, causing the

3.2. Optical absorption 46

delta-function in Eqn. 3.18 to becomeδ(Ef(kf)−Ei(ki)±~Ωph−hν), and momentum

conservation is ensured by modifying the delta function in Eqn. 3.20 toδ(q−kc+ kv+kph). Considering the density of phonons within Bose-Einstein statistics, it can

be shown [91] that the absorption coefficient for indirect transitions

αi(hν)∝(hν−Eg±~Ωph)2. (3.25)

However, this process requires both the absorption of a photon and absorption or emission of a phonon, and so the strength of absorption due to indirect transitions is rather low. Consequently, a pronounced increase in absorption coefficient is still observed when the photon energy becomes large enough to allow direct interband transitions.

In the above, it has been assumed that all of the valence band states are filled, and all of the conduction band states empty. However, for heavily doped semicon- ductors, the Fermi level can move into one of the bands. Consider a degenerately doped n-type semiconductor. Now, the lowest energy corresponding to direct inter- band transitions from initially occupied valence band states to initially unoccupied conduction band states is no longer given by the band gap energy, but rather by the energy separating the valence band at the Fermi wavevector and the Fermi level, as shown schematically in Fig. 3.6(c). Consequently, the absorption edge is shifted to higher energies compared to non-degenerate samples. Such an effect was first observed independently by Moss [92] and Burstein [93] in InSb, and so the shift of the absorption edge due to band-filling is commonly termed the Moss-Burstein shift. Formally, it can be accounted for by including conduction and valence band Fermi functions in the matrix element discussed above.

Another effect can become important for heavily doped materials, where charged impurities can induce a local change of the band edge potential due to Coulombic interactions. Additionally, a change in local strain state due to the presence of an impurity atom or vacancy in the host lattice can induce a similar perturbation of the band edge energies. As these impurities/vacancies are randomly distributed in space, the spatial variations in band edge potential can be averaged into an exponential-like tailing of the density of states of the conduction and valence

3.2. Optical absorption 47

bands into the band gap. Consequently, optical transitions can occur from or into band-tail states, leading to an exponential increase in absorption coefficient below the fundamental onset of absorption [89], known as an Urbach tail [94]. Also, in heavily doped materials, phonon-assisted transitions can occur from an electron in a partially occupied band to an empty state higher in that band. This can lead to a peak in the absorption coefficient at photon energies significantly below the band gap. This ‘free-carrier absorption’ will be considered in more detail in Section 3.3.