Energy conversion with hot electrons and holes

For conventional solar cells based on thermalization of electrons and holes in the absorber, complete conversion of chemical energy into electrical energy was achieved by membranes, n-type for the transport of electrons to one contact and p-type for the transport of holes to the other contact. This type of membrane is not sufficient for hot carriers. In addition to the selec­tive transport of electrons and holes, the membranes must now also serve the thermodynamic function of producing chemical energy from the heat of the electrons and holes by cooling them down to the temperature of the environment.

We will discuss this problem for the electrons, the solution can then easily be applied to the holes as well. If we would allow an exchange of electrons between absorber and membrane for all electron energies in the absorber, the thermalization of the electrons in the membrane would lead to a large energy loss, from kT in the absorber to кТц in the membrane (less, however, than in a conventional solar cell). Secondly, with the unimpeded exchange of elec­trons between the absorber and the membrane, the electrons in the absorber would be cooled as well and would no longer be capable of impact ionization. This would result in a state with rje + T]h = 0 and TA = 7b. However, as we have seen earlier, the entire energy loss caused by thermalization can be prevented, if the electrons in the membrane can only occupy states over a narrow range Дєе at the energy єе, as shown in Figure 8.9.

For Дєе < kTo, the occupation of the electron states in the membrane cannot significantly change by interaction with the phonons. As a result, the entropy of the electrons also remains unchanged and thermalization takes place isentropically. Since the number of particles re – mains constant during thermalization in the membrane, the electrochemical potential of the electrons increases. Figure 8.9 demonstrates that the same process takes place with the holes at an energy £h in the hole membrane. The isentropic cooling therefore produces the chemical and electrochemical energy per electron-hole pair,

Me + Mh = Ле + Ль = (єе + eh)0 — To/Ta) ■ (8.8)

The arrangement of Figure 8.9 is a working solar cell. The voltage is

V = (Ле+Ль)/е, (8.9)

and the current is

JQ — &{jE, absorbed jE. emitted)/(^e + ^h) – (8.10)

We can visualize its operation by increasing the current from zero, the open-circuit situation, where the emitted energy current equals the absorbed energy current. With increasing current the energy in the electron-hole system decreases and with it the electron-hole temperature TA in the absorber. Due to the lower temperature, the emitted energy current is reduced and also the voltage. The current rises until, at zero voltage, the short-circuit situation is obtained. A still larger current may be withdrawn by applying a negative voltage, spending energy from a battery, which would cool the remaining electrons and holes down to 7 < Го-

It is interesting that the open-circuit voltage is determined by the energies at which the electrons and holes are removed and not by the absorber material. If the rates of removal of electrons and holes are small compared with the rate of impact ionization/Auger recombina­tion and of carrier-carrier scattering, which we assume to be the case, their equilibrium will hardly be affected by the removal process. Electrons and holes removed at the energies £e and £h will quickly be replenished by impact ionization/Auger recombination and carrier-carrier scattering. Although the charge current and the voltage depend on the energies with which electrons and holes are removed, it is very surprising that the energy current delivered by the cell, obtained by multiplying Eqs. (8.9) and (8.10), is independent of the removal energies. Large removal energies give a large voltage and a small current and small removal energies give a large current and a small voltage, both resulting in the same energy current.

The efficiency is maximum at maximum concentration of the solar radiation, when the temperature 7д of the electrons and holes is equal to the temperature 7s of the sun at open – circuit. Since in the absence of interactions with the lattice vibrations the absorbed energy remains in the electron-hole system, it is advantageous to absorb as much as possible by reducing the band gap £g of the absorber material to zero. The electron-hole system is then a black body and, according to Eq. (2.24), absorbs the energy current oT$ and emits the energy

current оГд, at the temperature Гд. With Eqs. (8.9) and (8.10), the efficiency with which

electrical energy is delivered is

image453Подпись: 1image455Подпись: (8.11)jQV _a(Ts4-T*)

Подпись: <5T;

Подпись: eG/eV image459

"S VJ’S к rT^y vs

and is therefore identical with the efficiency of the ideal solar heat engine in Eq. (2.52) in Section 2.1.1 and of the thermo-photovoltaic conversion process discussed in the previous section. The efficiency has its maximum value of qmax — 0.85 at a temperature of the electron – hole system of Гд = 2478 К, if we assume a temperature of Ts — 5800K for the sun.

Figure 8.10 shows that the efficiency at full concentration falls off with increasing energy gap, because of the decreasing absorption. Without concentration, for Q. = Q.$, that is for the ЛМ0 spectrum, however, a non-zero energy gap is preferable, otherwise more photons would be emitted than absorbed at small photon energies. For Єс > 0 the balance becomes more favourable.

An earlier proposal for a hot-carrier solar cell by Ross and Nozik9 which did not account for impact ionization and Auger recombination, finds even higher efficiencies for narrow – gap semiconductors under less then full concentration of the solar radiation. Similar to a conventional solar cell, the process of carrier-carrier scattering, while leading to a uniform temperature, is assumed to leave the number of electron-hole pairs unchanged, increasing by one for each absorbed photon and decreasing by one for each emitted photon. This assumption leads to high temperatures and negative chemical potentials of the electron-hole pairs for less than full concentration. Since electrons and holes are withdrawn through mono-energy contacts, scattering of the carriers with each other is necessary to replenish the carriers in the energy range from where they are withdrawn. A problem is that a distinction between scattering events which keep the carrier concentrations constant and impact ionization and Auger recombination which don’t, is physically impossible in narrow gap semiconductors. The problem becomes obvious for the case where electron-hole pairs are withdrawn with an

^R. T. Ross, A. J. Nozik. J. Appl. Phys. 53 (1982) 3813.

energy that is smaller than the average energy of the absorbed photons. The more electron – hole pairs are withdrawn, the more energy per pair is piling up for the remaining electron – hole pairs. As a result, their temperature increases beyond any reasonable limit, far beyond the sun’s temperature. No such problems are encountered when impact ionization and Auger recombination are taken into account.

We thus see that impact ionization and Auger recombination allow ideal energy conver­sion, provided that interaction with the lattice vibrations is excluded. However, no material in bulk form is known in which these conditions are even approximately fulfilled. In a (very) thin film, however, one can imagine that electrons and holes can be removed in much less than 10~12s, long before they are thermalized.

Updated: July 1, 2015 — 7:58 am