1. INTRODUCTION
One way to increase the efficiency of the dyesensitized solar cells is to add metal nanoparticles to the titanium dioxide porous layer.
At present, a number of experimental studies have been published [14] in which the effect of metal nanoparticles on the work of cells, in particular on the efficiency, is shown. A unique feature of metallic nanoparticles is the generation of very strong local electric fields (the socalled near field or field in the near zone) when light quanta of a certain frequency are absorbed (plasmon resonance).
In the Graetzel cells, the key element of the construction is the dye molecules. They absorb photons and inject photoelectrons into the conduction band of titanium dioxide. The various optical characteristics [58], in particular the absorption cross section, vary to a great extent in the dye molecule entering the zone of the amplified electric field.
In our opinion, a change in the absorption cross section of dye molecules by metal nanoparticles is a key effect in the modeling of plasmon solar cells of Graetzel. The basic idea of increasing the dyesensitized solar cells efficiency is shown in Figure 1.
2. MATHEMATICAL MODEL OF A DYESENSITIZED SOLAR CELL BASED ON THE DIFFUSION EQUATION
The work of Graetzel cells in stationary and nonstationary regimes can be described using the diffusion equation with the generation and recombination terms:
with initial and boundary conditions:
open circuit mode  short circuit mode  work mode 



where,
The speed of photoelectrons generation in equation (1) is determined by the BouguerLambertBeer law:
(2)
where,
The rate of recombination of photoelectrons in a Graetzel solar cell can be described by expression (3):
(3)
where
The values of the parameters for modeling are taken from the works [912].
3. EFFECT OF SILVER NANOPARTICLES ON THE ABSORPTION CAPACITY OF ORGANIC DYE MOLECULES
For a mathematical description of the metal nanoparticles effect on the organic dye molecules absorbing capacity, a model developed by Professor M.G. Kucherenko was used [13].
The change in the rate of photons absorption by the dye molecule is associated with an additional dipole moment, which arises on the dye molecule due to the electromagnetic field reflected by the nanoparticle.
The expression for the probability of an electronic transition in a dye molecule with a dipole moment p in the presence of a metal nanoparticle is written as follows [14]:
(4)
where
The transition probability depends on the coordinate r, the angles
It should be noted that the dependence (4) is characteristic for one pair of "nanoparticledye". For a macroscopic system, for example, for a dyed porous layer of titanium dioxide with metallic nanoparticles, it is necessary to average over the angles and the radial coordinate.
However, a complete account of all the microparameters of the solar cell model with metal nanoparticles is extremely complicated in view of the very complex geometry of the system (Fig. 2). To simplify further calculations, we shall perform an angular and spatial averaging of the transition probability in the approximation of the homogeneous and isotropic distribution of dye molecules with respect to silver nanoparticles.
The concentration of metal nanoparticles in a porous layer of titanium dioxide can be related to the average distance between nanoparticles
(5)
where,
Further calculations were used three concentrations of metal nanoparticles:
4. RESULTS OF CALCULATIONS
Increase in the absorptivity of dye molecules using metal nanoparticles
In the calculations, two types of nanoparticles were used: gold and silver. Experimentally measured complex refractive indices were used to simulate their plasmon properties [15].
Figure 3 shows the normalized absorption spectra of anthocyanindyed titanium dioxide porous electrodes with the addition of metal nanoparticles. On the graphs there are two expressed maxima: the first arises from the absorption of photons by dyed titanium dioxide, and the second is due to absorption by the metal nanoparticles (for the samples with gold nanoparticles the maxima practically coincide).
Note that the absorption capacity at the maximum of the anthocyanin absorption band (509 nm) is increases more than 2 times when added to the porous structure of metal particles with concentration
Effect of metal nanoparticles on the photoelectrons generation
Plasmon amplification of the dye molecules absorption results in the injection of more photoelectrons into the conduction band of TiO
Due to the fact that the absorbing capacity of dye molecules increases with the addition of metal to the structure of nanoparticles, more photoelectrons are generated in the nearsurface regions of the anode layer. This leads to the fact that at high concentrations of metallic nanoparticles, light penetrates less into the depth of the sample. In addition, it is worth noting that, due to a better combination of the spectral properties of anthocyanin and silver nanoparticles, the latter have a stronger effect on the generation of photoelectrons.
Effect of plasmon nanoparticles on the stationary mode of operation of Graetzel cells
Figures 5 and 6 shows the calculated data on the spatial distribution of photoelectrons in the anode without and with the addition of noble metal nanoparticles in the stationary mode of operation of the solar cell. Calculations have been made for the cases when the cells operate in the shortcircuit mode (Figure 5) and in the idle mode (Fig. 6). It should be noted that, as in the previous calculations, silver nanoparticles exert a greater influence on the work of solar cells of Graetzel than gold ones.
On the graphs of the spatial distribution of photoelectrons in the anode layer in the shortcircuit mode (Fig. 5), despite the fact that the amplitude of the curve decreases with increasing concentration of metal nanoparticles, the gradient of the photoelectron concentration in the nearsurface region increases. This results in the cell generating a larger shortcircuit current.
As noted above, this effect is associated with an increase in the absorption of photons in the nearsurface regions of the anode layer due to plasmon nanoparticles.
When considering the spatial distribution of photoelectrons in the anode layer in open circuit mode (Fig. 6), one should note a monotonous increase in the concentration of injected photoelectrons with an increase in the number of plasmon nanoparticles in the sample under study. Calculations show that for the used concentrations of silver nanoparticles, an increase of 2 times the concentration of photoelectrons in the surface region of the anode porous layer of titanium dioxide (Figure 6a). For samples with gold nanoparticles, the concentration of photoelectrons increases to 1.3 times (Fig. 6b). The increase in the concentration of photoelectrons in the anode layer obtained during calculations as a result of interaction with plasmon nanoparticles should be reflected in an increase in the value of the open circuit voltage of the cell samples under study.
Using the obtained data on the effect of plasmon nanoparticles on the absorptivity of dye molecules, the currentvoltage characteristics of the Graetzel cells shown in Fig. 7 were calculated. It can be seen that as the concentration of silver nanoparticles increases in the structure of the solar cell, the shortcircuit current density increases from 1.33 mA/cm
For a more detailed analysis of the effect of plasmon nanoparticles on the stationary mode of operation of Gratzel cells, the dependences of the relative efficiency and filling factor on the concentration of metal nanoparticles in the anode layer of solar cells were calculated. The graphs are shown in Figures 8 and 9 respectively. Calculations give an increase in efficiency to 20% in the case of using gold nanoparticles and an increase of 2 times for silver nanoparticles.
5. COMPARISON OF SIMULATION RESULTS WITH EXPERIMENT
Figure 10 shows the experimentally measured currentvoltage characteristics of the Graetzel cells and the relative efficiency with the addition of different amounts of silver nanoparticles to the design. The graphs are taken from [1].
Comparison of the obtained results with the results presented in Fig. 10 and in [24] gives a good qualitative and quantitative agreement.
As a result of the research it was shown that one of the mechanisms of the plasmon nanoparticles effect on the photocells operation parameters is the plasmon amplification of the dye molecules absorptive capacity.