## THE MATHEMATICAL SYSTEM

THE MATHEMATICAL SYSTEM
Collecting all our previous work, we can now write down the full mathematical system that describes all the physics taking place in a solar cell in steady state. This mathematical system is made up of, unfortunately, a large set of coupled, non-linear equations. This set includes Eqs. 2.19 and 2.28, the equations of Section 2.2.5, the continuity equations for electrons and holes, and Poisson's equation. The solution to this set for n, p, Jn, Jp, etc., must be consistent with the boundary conditions imposed by the interface transport discussed in Section 2.3.2. We have to obtain this solution in order to establish the cell's current density-voltage （J-V） characteristic （see Fig. 1.3）， which is required for cell evaluation, design, and optimization. Since we have been staying with one dimension, this total conventional current density J of the J-V characteristic is a constant and is obtained from
where the components Jn and Jp are evaluated at the same plane （any plane） in the cell. The terminal voltage V produced by the cell at some operating point while delivering the current density J is given by the integral over the structure of the difference between the electric field distribution present at the selected operating point ￡，（x） and the electric field distribution present in TE which we term ￡，0（x）； i.e.,
It must be stressed that this expression is valid regardless of whether the cell has an exciton-producing absorber or free electron-hole-producing absorber or is any cell structure from a dye sensitized solar cell to a conventional p-n junction device. Put succinctly, it is valid for all types of cells. Equation 2.47 is of such general validity because it calculates the relative shift of the contact Fermi levels present at the selected operating point. This shift is the voltage V at this operating point. The sign convention assumed in writing Eq. 2.47 is that the cathode is the left-hand contact.
As we will see, establishing the current densities Jn and Jp—and therefore the J-V characteristic—for a solar cell can be done analytically in some situations, with the help of assumptions. We will explore this approach in Chapters 4-7. However, turning a computer loose to tackle the full mathematical system, and not making any of those assumptions, is very powerful and insightful and we will do that too in Chapters 4-7. In the computer analysis, we will make extensive use of numerical solutions to the full mathematical system. In this numerical modeling, the specific versions of the equations of the mathematical system that will be utilized are the following:
In these equations, R can be any of the unimolecular or bimolecular recombination mechanisms of Section 2.2.5. In our numerical modeling we employ the full non-linearized formulations discussed in Section 2.2.5 for whichever R is assumed. For example, we will often assume R is controlled by S-R-H recombination and use
summed over some gap state distribution. For Eq. 2.48e the trapped charges nTandpTat energy Easwellasthe ionizeddopantconcen-
trations Na and ND are computed using Fermi-Dirac statistics, as discussed in Appendix C. As seen in that Appendix, these quantities become functions of n and p in steady state. Keeping that in mind, we see that set 2.48 has 5 equations and 5 unknowns.
The quantity Gph（\, x） in this set is the photogeneration function of Section 2.2.6. In the case of free electron-hole pair-caused absorption, it is modeled in the numerical simulations by using a（\）I0（X）e——“（X）x （see Eq. 2.2）； i.e., the absorption mechanism is taken to follow the Beer-Lambert law. In the case of exciton-caused absorption, Gph（\, x） is modeled by a delta-function-like distributions at the region of exciton dissociation. This point is discussed further in Chapter 5.
In the numerical results presented in this text, the computer solves the equations subject to the boundary conditions
for the right （R） contact. These are taken from Eq. 2.40 and assume a coordinate system with x increasing from left to right. As we establish in Chapter 4, these boundary conditions can be made inconsequential by various layers we can place adjacent to contacts. By simultaneously solving this whole set of equations numerically, the computer then assembles the J-V characteristics for no illumination （dark J-V） and illumination （light J-V） situations in our discussions of Chapter 4-7. When the results are tied to the device coordinate system, the conventional current density is taken as positive when it flows in the positive x-direction of the device coordinates. Except where noted, voltage is taken as positive if the Fermi level of the right contact is above that of the left contact. In discussions tied to plots of J versus V, J i s taken as negative in the power quadrant. The resulting numerical solutions give us the J-V behavior but also allow us to peer inside a cell as it is operating and thereby allow us to explore the roles of diffusion, electrostatic fields, effective fields, drift, recombination, trapping, interfaces, etc.