4. EXCITATION OF PHOTOLUMINESCENCE

Many materials are capable of emitting visible luminescence when subjected to some form of excitation such as UV light (photoluminescence), nuclear radiation such as rays and and particles (scintillation), mechanical shock (triboluminescence), heat (thermoluminescence), chemical reactions (chemiluminescence), and electric fields (electroluminescence). In this section we will restrict our attention to photoluminescence, since the interstellar dust luminescence, as manifested by the Extended Red Emission ubiquitously seen in interstellar environments, is believed to be a photon-driven process (Witt & Schild 1985; Duley 1985; Smith & Witt 2002; Witt & Vijh 2004).

There are two pre-requisites for luminescence: (1) the luminescent material must have a semiconductor structure with a nonzero band gap E_g (e.g. metals do not luminesce since they have no band gap); (2) energy must be imparted to this material before luminescence can take place. The mechanism of photoluminescence in semiconductors is schematically illustrated in Figure 8, which plots the E-k diagrams for a direct band gap material (left) and an indirect gap material (right), where E and k are respectively the kinetic energy and wave vector (or "momentum vector") of the electron or hole (E = k^{2 2} / 2m_*, where h / 2 is the Planck constant h divided by 2, and m_* is the electron or hole effective mass). The direct and indirect gap materials are distinguished by their relative positions of the conduction band minimum and the valence band maximum in the Brillouin zone (the volume of k space containing all the values of k up to / a where a is the unit lattice cell dimension). In a direct gap material, both the conduction band minimum and the valence band maximum occur at the zone center where k = 0. In an indirect gap material, however, the conduction band minimum does not occur at k = 0, but rather at some other values of k which is usually at the zone edge or close to it (see Fox 2001).

Figure 8. Schematic band diagrams for the photoluminescence processes in a direct gap material (left) and an indirect gap material (right). The shaded states at the bottom of the conduction band and the empty states at the top of the valence band respectively represent the electrons and holes created by the absorption of photons with energy exc > Eg. The cascade of transitions within the conduction and valence bands represents the rapid thermalization of the excited electrons and holes through phonon emission. In a direct gap material (left), the conduction band minimum and the valence band maximum occur at the same k values. Both the photon absorption and emission (i.e. the electron-hole recombination) processes can conserve momentum without the assistance of phonons, since the momentum of the absorbed or emitted photon is negligible compared to the momentum of the electron. We therefore represent photon absorption and emission processes by vertical arrows on E-k diagrams. In an indirect gap material (right), the conduction band minimum and the valence band maximum occur at different k values. As a result, to conserve momentum, the photon absorption process must involve either absorption (indicated by a "+" sign) or emission (indicated by a "-" sign) of a phonon, while the PL process requires the emission of a phonon. Since the energy of a phonon (~ 0.01 eV) is much smaller than the energy of the PL photon, for an indirect gap material, the peak energy of the PL also roughly reflects its band gap.

Upon absorption of an UV or visible photon with an energy _exc exceeding the band gap E_g (the gap in energy between the valence band and the conduction band) of the material, an electron-hole pair is created and the electron (hole) is excited to states high up in the conduction (valence) band (see Fig. 8).

During a photon absorption process in semiconductors, we must conserve both energy and momentum. In a direct band gap material, the conduction band minimum and the valence band maximum have the same k values (i.e., _i = _f, where _i and _f are respectively the wave vectors of the initial and final electron states; this implies that the electron wave vector should not change significantly during a photon absorption process), conservation of momentum is guaranteed for the photoexcitation of the electron which only involves a UV or visible photon: _i + _{phot i} = _f, since _phot, the wave vector of the absorbed photon (which is in the order of 2 / ~ 10⁵ cm^-1), is negligible compared to the electron wave vector (which is related to the size of the Brillouin zone / a ~ 10⁸ cm^-1, where the unit cell dimension a is in the order of a few angstroms). This implies that in a direct band gap material, the electron wave vector does not change significantly during a photon absorption process. We therefore represent photon absorption processes by vertical arrows in the E-k diagrams (see Fig. 8).

In contrast, for an indirect band gap material of which the conduction band minimum and the valence band maximum have different k values (see Fig. 8), conservation of momentum implies that the photon absorption process must be assisted by either absorbing or emitting a phonon (a quantum of lattice vibration), because the electron wave vector must change significantly in jumping from the valence band in state (E_i, _i) to a state (E_f, _f) in the conduction band, and the absorption of a photon alone can not provide the required momentum change since |_phot| << |_i - _f|.

The excited electron and hole will not remain in their initial excited states for very long; instead, they will relax very rapidly (~ 10^-13 s) to the lowest energy states within their respective bands by emitting phonons. When the electron (hole) finally arrives at the bottom (top) of the conduction (valence) band, the electron-hole pair can recombine radiatively with the emission of a photon (luminescence), or nonradiatively by transferring the electron's energy to impurities or defects in the material or dangling bonds at the surface.

Just like the photon absorption process discussed above, the electron-hole recombination in a direct band gap material does not involve any phonons since there is no need for momentum change for the electron. In contrast, in an indirect gap material, the excited electron located in the conduction band needs to undergo a change in momentum state before it can recombine with a hole in the valence band; conservation of momentum demands that the electron-hole recombination must be accompanied by the emission of a phonon, since it is not possible to make this recombination by the emission of a photon alone. Compared to the photon absorption process in an indirect gap material for which conservation of momentum can be fulfilled by either absorption or emission of a phonon, in the electron-hole recombination process phonon absorption becomes negligible, whereas phonon emission becomes the dominant momentum conservation mediator because (1) the number of phonons available for absorption is small and is rapidly decreasing at lower temperatures, whereas the emission of phonons by electrons which are already at a high-energy state is very probable; and (2) an optical transition assisted by phonon emission occurs at a lower photon energy E_g - h _phon than the gap energy, whereas phonon absorption results in a higher photon energy of at least E_g + h _phon, which can be more readily re-absorbed by the semiconductor nanoparticle. But we note that the energy of a phonon (h _phon) is just in the order of ~ 0.01 eV, much smaller than the energy of the electron-hole recombination luminescence photon. Also because prior to the recombination, the electrons and holes respectively accumulate at the bottom of the conduction band and the top of the valence band, the energy separation between the electrons and the holes approximately equals to the energy of the band gap. Therefore, the luminescence emitted by both types of semiconductors occurs at an energy close to the band gap E_g.

The PL efficiency is determined by the competition between radiative and nonradiative recombination. For an indirect gap material, the PL process, which requires a change in both energy and momentum for the excited electron and hence involves both a photon and a phonon, is a second-order process with a long radiative lifetime (~ 10^-5 -10^-3 s), and therefore a relatively small efficiency because of the competition with nonradiative combination. In contrast, in a direct gap material, the emission of a PL photon does not need the assistance of a phonon to conserve momentum. Therefore, the PL process in a direct gap material is a first-order process with a much shorter radiative lifetime (~ 10^-9 -10^-8 s) and a much higher PL efficiency in comparison with an indirect gap material.

However, for particles in the nanometer size domain, we would expect substantial changes in both the efficiency and the peak energy of the photoluminescence due to the quantum confinement effect. This can be understood in terms of the Heisenberg uncertainty principle. Unlike in bulk materials the electrons and holes are free to move within their respective bands in all three directions, in nanoparticles the electrons and holes are spatially confined and hence their motion is quantized in all three dimensions. The spatial confinement of a particle of mass m to a region in a given direction (say, along the x axis) of length x would introduce an uncertainty in its momentum / x and increase its kinetic energy by an amount E_QC ~ ( p_x)² / 2m ~ ² / 2m( x)². A simple particle-in-a-box analysis, using the Schrödinger's equation and the effective mass approximation, shows that the ground state quantum confinement energy would be E_QC ~ (3 ² / 8 m^*) ( / a)², where m^* m_e^* m_h^* / (m_e^* + m_h^*) is the reduced effective mass of the electron-hole pair (m_e^* and m_h^* are respectively the effective mass of the electron and hole) (Fox 2001). For nano-sized particles, the quantum confinement effect becomes significant since the confinement energy E_QC would be comparable to or greater than their thermal energy E_th ~ 3/2 k_BT at the temperature range expected for nanoparticles in the diffuse ISM (see Figs.7, 8 of Draine & Li 2001 and Fig. 3 of Li & Draine 2002a). For example, a silicon grain with m_e^* 0.98m_e and m_h^* 0.52 m_e (m_e is the free electron mass) smaller than ~ 8 nm would exhibit quantum effects at a temperature T ~ 100 K (which is expected for nano-sized silicon dust in the diffuse ISM; see Fig.3 of Li & Draine 2002a) with E_QC 0.83(a / nm)^-2 eV > E_th 0.013(T / 100 K) eV. More detailed studies show that silicon nanocrystals exhibit an a^-1.39 gap-size dependence: E_g E₀ + 1.42(a / nm)^-1.39 eV, where E₀ 1.17 eV is the bulk silicon band gap (Delerue, Allan, & Lannoo 1993). For nanodiamonds, an a^-2 gap-size dependence was derived from the X-ray absorption spectrum measurements (Chang et al. 1999; also see Raty et al. 2003): E_g E₀ + 0.38(a / nm)^-2 eV, where E₀ 5.47 eV is the bulk diamond band gap.

It is apparent, therefore, the quantum confinement effect would lead to a progressive widening of the band gap of a nano-sized semiconductor as its size is reduced, along with a broadening of the electron-hole pair state in momentum space (i.e. an increased overlap between the electron and hole wavefunctions), and a decreasing probability for the pair to find a nonradiative recombination center, provided that the surface dangling bonds are passivated which would otherwise act as traps for the carriers and quench the PL. While the former would shift the PL peak to higher energies, the latter two effects would greatly enhance the electron-hole radiative recombination probability and result in a higher PL efficiency. This is one of the main reasons why silicon nanocrystals are proposed recently by Witt and his coworkers (Witt et al. 1998, Smith & Witt 2002) and by Ledoux et al. (1998, 2001) as the ERE carrier since they are capable of luminescing very efficiently (with essentially 100% quantum efficiency) in the energy range (~ 1.4-2.4 eV) over which the ERE has been observed in astronomical sources, while bulk silicon, an indirect gap semiconductor with a band gap of ~ 1.17 eV at T = 0 K, does not luminesce.