Electrons and holes possessing large kinetic energies as a result of generation by high-energy photons can dissipate their kinetic energy in two ways. One is by elastic collisions with the lattice atoms, in which energy is transferred in small portions to the lattice atoms until thermal equilibrium with the lattice is established. The other is by inelastic collisions with the lattice atoms in which, by impact ionization, another electron is knocked off its chemical bond or, in other words, in which a free electron and a free hole are produced, as shown in Figure 8.8. Both processes take place in parallel and compete with each other. With elastic collisions the excitation of lattice vibrations is at the expense of the energy of the electron-hole system, while the number of the electrons and holes remains constant. With impact ionization the absorbed energy remains in the electron-hole system, but is more uniformly distributed over a larger number of electrons and holes than were originally generated by the absorption of the photons. Impact ionization, therefore, looks very promising for solar energy conversion because some of the energy removed from the electrons and holes during thermalization is used to generate additional electron-hole pairs.
In order to examine the efficiency of the impact ionization process, we will exclude the competing process, namely the interaction of electrons and holes with the lattice vibrations, which leads to thermalization at constant concentrations. The electrons and holes are then isolated from the lattice vibrations. They do not “know” about the temperature of the lattice and cannot come into thermal equilibrium with the lattice. Collisions between electrons and holes are, however, permitted. This ensures that electrons and holes have a uniform temperature 7a, although this is not the same as the lattice temperature. Finally, according to the Principle of Detailed Balance, we must expressly consider Auger recombination as the inverse process of impact ionization.
We will now examine the temperature and electrochemical potentials of the electrons and holes under these conditions. The simplest answer, unfortunately, may be the most difficult to understand. It is based on the difference between thermalization and impact ionization: while during thermalization, by scattering with phonons, no electrons (or holes) are annihilated or created, so that their number remains constant, during impact ionization and its inverse process, Auger recombination, the number of electrons and holes changes. This has significant consequences for the values of the electrochemical potentials of the electrons and holes. The change of the particle numbers by impact ionization and Auger recombination is unrestricted, except for the condition to establish a minimum of the Free Energy of the electrons and the holes. Thus
dF = … + T|e dTVe + Tjh dA/h +… = 0 .
With impact ionization and Auger recombination, electrons and holes are always created or annihilated in pairs, that is dNc = tWh = dN and
dF = … + (r|e+r|h)cW+… = 0 .
Because the number of particles does not remain constant and is not tied to other particle numbers as in a chemical reaction, 6N Ф 0, so that
Г|е + тЪ = 0.
We will attempt to make this result more plausible. Let us assume that the Free Energy of the electrons and holes describes a state with T|e +T|h > 0. With a reduction in the number of particles due to Auger recombination, i. e., with dN < 0 and therefore dF <0, the Free Energy can be further reduced. With the reduction in the number of particles, Tle +r)h also decreases, until for Г(е +T|h = 0 a further reduction of the particle number no longer reduces the Free Energy and equilibrium is established between impact ionization and Auger recombination. Since the total energy of the electron-hole system is preserved, by reducing the particle number, Auger recombination leads to an increase in the energy per particle, i. e., in the mean kinetic energy of the electrons and holes. This in turn leads to an increase in their temperature, and in the number of electrons and holes capable of participating in impact ionization, until the rate of impact ionization is exactly the same as the rate of Auger recombination and Tie + T|h = 0.
If, however, scattering with the lattice atoms and Auger recombination and impact ionization, all occur at high rates, the temperature of the electrons and holes will be the same as the lattice temperature, and in equilibrium with impact ionization and Auger recombination, the only possible state is one with T& = Tq and T|e +T|h = 0, which does not permit the conversion of energy. It is therefore very important that thermalization and impact ionization, together with its inverse process, do not occur simultaneously and with similar probabilities. It would not improve the efficiency of solar cells, but in fact reduce it, if the probability for impact ionization (and Auger recombination, inevitably) were slightly increased while the interaction with phonons predominates.
We thus establish the fact that the interaction with the lattice vibrations maintains the temperature of the electron-hole system constant at the lattice temperature Tq and, on exposure to light, produces a state with rje + Ль > 0. Impact ionization and Auger recombination, on the other hand, maintain a state with no separation of the Fermi energies, T|e + "Hh = 0, but in the absence of interaction with the lattice vibrations and on exposure to light, produce a state with T > To.
The problem of how to obtain electrical energy from hot electrons and holes then remains to be solved. Energy conversion by means of impact ionization first produces hot electrons and holes with no chemical energy. Chemical energy, and finally electrical energy, must be obtained in subsequent steps.