Photon current density

Even more important for quantum energy converters, e.g., solar cells, than the energy current density is the photon current density, because it determines the rate at which electrons are excited. Neglecting impact ionisation effects, the excitation of one electron requires at least one absorbed photon. 

The photon current density is derived from (4.1) by dividing by the photon energy huj. Hence,... 

Fig. 4.2. Photon current densities per photon energy of AMO (dotted line) and AM 1.5 (solid, line) solar radiation, calculated from data in Fig. 4.1...

Absorption of Radiation. The probability for an incident photon of energy huj to be absorbed per unit length in matter is given by the absorption coefficient a(hu). The rate drabs at which photons with energy between tko and huj + d(tko) are absorbed (number of photons per unit volume and unit time) at a given location x is proportional to the photon current density djj(hu, x) at this location ... 

The number of integrated carriers, iV, is QA-Iwhere q is the electron charge. Because dark current, is a combination of thermal excitation processes, neglecting avalanche and tunneling, ideal performance occurs when the photon-induced current density Jp is greater than Fluctuations of N are the... 

Fig. 4.13 Number of observed charge carriers per absorbed photon as a function of the current density. The photoinduced current at n-type electrodes in HF (squares) is increased compared to a photodiode or p-type electrode...

The photocurrent doubling discussed above can be understood as a consequence of the divalent dissolution reaction as shown in Fig. 4.3. Dissolution for current densities below JPS is initiated by a hole in step 1 and proceeds under injection of an electron in step 2. For the case of an n-type electrode, one photon is required to generate one hole, but the electron injected in the dissolution process doubles the current without consumption of another photon. Hence the resulting current density is twice as large as observed at a reference photodiode. Because step 2 of the reaction depicted in Fig. 4.3 is independent of type of doping it can be concluded that electron injection also takes place at p-type electrodes. There is, however, no simple way to detect these injected electrons because the electrode is under depletion in this regime, as discussed in Section 3.2. 

Photon flux density and the average power of the incident beam were varied in a wide range that exceeded the current possibihties. Highest estimates of the photon-pair yields from SPDC sources, given in the hterature, are in the megahertz range [74]. Therefore we assiune an overestimated pair rate of 10 s which corresponds to the photon flux density of 4 x 10 cm s for a circiflar area with diameter of 0.8 pm. 

In the previous section we presented the semi-classical electron-electron interaction we treated the electrons quantum mechanically but assumed that they interact via classical electromagnetic fields. The Breit retardation is only an approximate treatment of retardation and we shall now consider a more consistent treatment of the electron-electron interaction operator that also provides a bridge to relativistic DFT, which is current-density functional theory. For the correct description we have to take the quantization of electromagnetic fields into account (however, we will discuss only old, i.e., pre-1940 quantum electrodynamics). This means the two moving electrons interact via exchanged virtual photons with a specific angular frequency u>... 

The present chapter is devoted mainly to one of these new theories, in particular to its possible applications to photon physics and optics. This theory is based on the hypothesis of a nonzero divergence of the electric field in vacuo, in combination with the condition of Lorentz invariance. The nonzero electric field divergence, with an associated space-charge current density, introduces an extra degree of freedom that leads to new possible states of the electromagnetic field. This concept originated from some ideas by the author in the late 1960s, the first of which was published in a series of separate papers [10,12], and later in more complete forms and in reviews [13-20]. 

The Lagrangian (824), which is the same as the Lagrangian (839), gives the inhomogeneous equation (826) using the same Euler-Lagrange equation (843). Therefore the photon mass can be identified with the vacuum charge-current density as follows (in SI units) ... 

The Lagrangian (850) shows that 0(3) electrodynamics is consistent with the Proca equation. The inhomogeneous field equation (32) of 0(3) electrodynamics is a form of the Proca equation where the photon mass is identified with a vacuum charge-current density. To see this, rewrite the Lagrangian (850) in vector form as follows ... 

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