Fundamental optical transitions

Since the PL emission requires that the system be in a nonequilibrium condition, and some means of excitation is needed to act on the semiconductor to produce hole-electron pairs. We consider the fundamental transitions, those occurring at or near the band edges.

The ground state of the electronic system of a perfect semiconductor is a completely filled valance band and a completely empty conduction band. We can define this state as the “zero” energy or “vacuum” state. If we start from the above-defined ground state and excite one electron to the conduction band, we simultaneously create a hole in the valance band. In this sense an optical excitation is a two-particle transition. The same is true for the recombination process. An electron in the conduction band can return radiatively or nonradiatively into the valance band only if there is a free space, i.e., a hole. Two quasi-particles are annihilated in the recombination process. What we need for the understanding of the optical properties of the electronic system of a semiconductor is therefore a

description of the excited states of N-particle problem. The quanta of these excitations are called “excitons”. Here we will consider the so-called Wannier excitons more specifically. In Wannier excitons, the Bohr radius (i.e. the mean

distance between electron and hole) is larger in comparison to the length of the lattice unit cell. This condition is met in most II-VI, III-V, and column IV semiconductors.

2.3.1.1 Wannier excitons

Using the effective mass approximation, Fig. 2-5 suggests that the Coulomb interaction between electron and hole leads to a hydrogen-like problem with a

Coulomb potential term ²

Indeed excitons in semiconductors form, to a good approximation, a hydrogen or positronium like series of states below the gap. For a simple parabolic band in a direct-gap semiconductor one can separate the relative motion of electron and hole and the motion of the center of mass. This leads to the dispersion relation of exciton as shown in Fig. 2-6 exciton binding energy, M=me+mh,and K=ke+kh are translational mass and wave vector of the exciton, respectively.

Fig. 2-5 The exciton dispersion in a two-particle (electron-hole) excitation diagram of the entire crystal. The crystal ground state (zero energy and zero momentum) is the point at the origin. Different parabolas represent the kinetic energy bands associated with different terms of the excitonic series.[14]

2.3.1.2 Bound excitons

A real crystal is never perfect. Imperfections such as ion vacancies, interstitials, or substitutional atoms (either native or intentionally introduced) exist in densities ranging from ni < 10¹² cm^-3 in ultrapure crystals. The imperfections can attract excitons that become localized at the defect sites to form bound excitons. The binding energy of the exciton to the defect is often quite small, typically a few meV.

Therefore, the bound excitons are best observed at very low temperatures.

An exciton may be bound to a donor, which is a substitutional atom with a higher number of valance electrons compared with the host atom, or to an acceptor, a substitutional atom with a lower number of valance electrons. Donors contribute excess electrons to the crystal, while acceptors tend to capture electrons or

K

Energy

n= 1 n= 2 n= 3

n= ∞ Conduction band

Valance band

equivalently donate holes. Donor or acceptor atoms may be electrically charged or neutrals. When the donor atom has given away its initial extra valence electrons, it becomes positively charged and it referred to as an ionized donor. Similarly, when an acceptor atom has captured an electron (or equivalently released a hole), it has a negative charge and is called an ionized acceptor. In contrast, a neutral donor or acceptor has no charge, since it has kept its original number of valance electrons.

Excitons may get bound to either an ionized donor or acceptor, or a neutral donor or acceptor by forming complexes represented schematically in Fig. 2-6. In many crystals, the binding energy of the exciton to a neutral donor or acceptor is close to tenth of the donor or acceptor ionization energy, which is the energy required to free the extra valence electron of a neutral donor, or the energy to free a hole (to accept an electron) in a neutral acceptor. Bound excitons are characterized by more sharply peaked emission which occurs at a lower energy than the corresponding free exctions, which is due to reduced kinetic broadening, since the bound exciton is spatially localized at an impurity. The emitted energy of bound energy E_BE is

F B

BE g E E

E =E −E −E , (2-10) where E is energy necessary to bind the exciton to the defect center and _E^B E is the _E^F binding energy of the free exciton. Therefore, luminescence of bound exciton typically dominates the near band edge emission and occurs on the low energy side of the free exciton emission.

Fig. 2-6 Visualization of (a) an exciton bound to an ionized donor, (b) a neutral donor, and (c) a neutral acceptor. [14]

2.3.1.3 Donor-Acceptor Pairs (DAP)

Donors and acceptors can form pairs and act as stationary molecules imbedded in the host crystal. The coulomb interaction between a donor and an acceptor results in a lowering of their binding energies. In the donor-acceptor pair case it is convenient to consider only the separation between the donor and the acceptor level:

Epair =Eg – (ED +EA) + q2

εr, (2-11) where r is the donor-acceptor pair separation, ED and EA are the respective ionization energies of the donor and the acceptor as the isolated impurities.

2.3.1.4 Deep transitions

By deep transition we shall mean either the transition of an electron from the conduction band to an acceptor state or a transition from a donor to the valence band in Fig. 2-7. Such transition emits a photon hν=Eg – Ei for direct transition and hν=Eg – Ei - Ep, if the transition is indirect and involves a phonon of energy Ep. Hence the deep transitions can be distinguished as (1) conduction-band-to-acceptor transition which produces an emission peak at hν = Eg – EA, and (2) donor-to-valence-band transition which produces emission peak at the higher photon energy hν=Eg – ED.

C

V D

A

Fig. 2-7 Radiative transition between a band and an impurity state.