Optical properties of semiconductors cannot be fully understood without discussing excitonic states. For a discussion of bulk excitons reference the classic work of Ref. [41]. For nice reviews of the properties of excitons in NSs, see Refs. [35] and [42].
Excitons are typically formed in semiconductors through optical excitations, leading to a bound state of an electron and hole. They can also be formed by carrier injection. The exciton bound state is understood with the inclusion of the Coulomb interaction between the electron- hole pair. It is valid to treat excitons in a hydrogenic type model for the correlation energy and thus the binding energy of the exciton. The exciton is truly hydrogenic in the case where the hole mass is much larger than the electron, whereas, when they are approximately the same, the exciton is more positronium like. In the case of organics, one observes tightly bound excitons, or Frenkel excitons. In the case of semiconductors, one observes loosely bound excitons, or Wannier-Mott excitons, which have been shown to by hydrogen-like [42]. In this framework, the exciton energy, EX, is written as:
EX(n,k) = EG−R′y 1 n2 + ~2k2 2µ , R′y = 1 ε2 µ mo 13.6 eV; (1.11)
where n is the principal quantum number, k is the electron plus hole wavevector (k=ke+kh),
µis the reduced mass in terms of the electron and hole effective mass 1 µ = 1 m∗ e + 1 m∗ h , R′y is the exciton Rydberg energy, EG is the gap energy, and ε is the dielectric constant. The exciton Rydberg energy in Eq. (1.11) is simply corrected by the reduced mass of the exciton and the dielectric constant. The exciton Bohr radius is given by:
aXB = aHBεmo
µ , for n=1; (1.12)
where aH
B is the Bohr radius in the hydrogen atom. The Coulomb energy of the exciton state lowers the energy of the exciton ground state with respect to the free electron-hole pair. There- fore, one can observe exciton absorption by the fact that absorption will happen below the band gap edge. Fano resonances are also an important consideration for exciton absorption [35]. In
addition, the Coulomb energy tends to increase the probability of finding the electron and hole in the same place [43].
The magnitude of the Coulomb energy is typically on the order of 10→40 meV, which is much less than the gap energy [13]. In the case of strong confinement, the confinement potential is much stronger than the Coulomb interaction. Therefore, one can treat the Coulomb energy as a perturbation [43]. Typically, excitons do not play a significant role in Si NSs because the excitation energy is much larger than the Coulomb energy for very small sizes of the NSs. Ge has a larger Bohr radius, meaning these structures encompass a larger regime of QC where the Coulomb effects can be more important. In addition, the thermal energy must also be considered when thinking of excitonic effects [44]. Coulomb energy is significant around a few Kelvin, where exciton effects have been observed [45, 46].
In the above discussion, the effect of system dimension is handled through the evaluation of the exciton energy as a function of size. The system dimension can have another important effect on the energy. In an infinitely thin QW, the principal quantum number becomes n→n−12, the general form of this expression is:
n→ n+ de f f−3 2 , de f f =3−exp −L 2aXB ; (1.13)
where de f f is the effective dimension ranging between 3 and 2, and L is the QW thickness [44]. The Bohr radius is thus reduced by a half, the exciton energy increases by four, and the oscillator strength increases by a factor of eight for the infinitely thin well. The renormalization of the principle number is numerically treated in Ref. [7] Figs. 6.4 and 6.5. The effect is usually ignored because the system is not truly a 2D system.
Of the features discussed so far, by far the most important feature is the correct treatment of the dielectric constant. This point does not have a clear solution and is also critical in correctly determining the exchange interaction (Sec. 1.2.4). Typically, the Coulomb energy is screened by the bulk dielectric constant. However, there is some variation in the correct value that should be used. In the work of Ref. [47], the Penn model was used to calculate the variation of the dielectric constant with dimension starting from several different values
1.2. Overview of nanostructure properties 15
for the matrix dielectric constant, which all yield essentially the same result. A comparison of the Penn model and a pseudopotential calculation yield very different results [48]. In the work of Ref. [49], the bulk dielectric constant is used. Dielectric constant corrections lead to corrections in the Sommerfeld factor, which leads to an increase in the absorption [8].
Two limiting cases for the dielectric function can be easily understood. If the exciton bind- ing energy is less than the optical phonon energy (i.e. the exciton Bohr radius is greater than the polaron radius), then the static dielectric constant can be used below the phonon energy. In the case that the exciton energy is comparable with the optical phonon energy, then a dielec- tric constant between the bulk value (high frequency) and the static dielectric constant can be used [44]. In this second case, one can use the Haken potential [50]. Neither of these cases can be rigorously justified.
A more accurate treatment of the screening of the electron-hole interaction should contain a contribution from the induced surface polarization charge. To this end, the nature of the interface plays a large role on the details of the dielectric function, whereby, an inner dielectric function can be defined by separating the surface contribution. Defining an image charge in this way, allows one to consider surface self energy corrections based on the difference between the inner and outer dielectric functions [6]. In the case where the inner and outer dielectric functions are of the same order, image charge corrections are not as significant. Generally, one finds that the dielectric function is replaced by the dielectric function in the Thomson- Fermi approximation, which ignores surface polarization effects. This approximation is valid for energies lower than the plasmon energy.
Finally, there are a few fine points to consider in the study of the exciton. Mass re- normalization will also effect the binding energy of an exciton. In a non-rigid lattice, polaron effects couple to the effective mass through Fr¨ohlich couplings. The binding energy of the exciton partly renormalizes this effect. Generally, these couplings are not as important in the case of lower gap materials where they tend to lower the gap energy [44, 51, 52].
Biexciton complexes are also a fascinating subject in the study of NSs. A biexciton is simply the bound state between two excitons. This condition is usually observed in a highly excited NS system, i.e. when the exciton density is high enough where bound states can form (see Refs. [11, 53] for a general discussion). Biexcitons create a situation of interesting decay
dynamics due to the fact that they change the decay scheme to a biexcition decaying to an exciton and then to the ground state [54]. This situation creates an opportunity to observe exciton condensation experimentally [55]. As exciton observation is generally complicated in the case of Si and Ge structures it is even more difficult to observe biexcitons and very little research has been conducted in this area. The problem in Si and Ge is that biexciton lines can be very broad and hard to distinguish [56].