**5.3. Finding the photoelectric characteristics of solar cells**

integrand *f*(*x*,*V*) and its half-width Δ*x*1/2. By calculating Δ*x*1/2 as the difference of the values for which *f*(*x*,*V*) is equal to ½, one can obtain under conditions that *qV* approaches not very close

j

Eq. (56)

*qV E qV qV E qV* (55)

*qV E qV kT* (56)

Eq. (51)

 m

 m

bi g bi g

So Eq. (53) in the integral form yields the exponential dependence of the recombination current

m

Fig. 21 shows a comparison of the voltage dependence of the recombination currents in CIGS

exp 1 . ( )( 2 ) <sup>2</sup>

0 0.2 0.4 0.6 0.8

*V* (V)

**Figure 21.** Comparison of the recombination currents in CIGS solar cell with 1.14 eV bandgap calculated from Eqs. (51)

As seen in Fig. 21, the *J–V* curves practically coincide. Currents calculated from the exact Eq. (51) exceeds those calculated from Eq. (56) by no more than 5–6% at low voltages and no more than 8–10% at higher voltages. When *V* > 0.7 V the curves are separated from one another

é ù æ ö » - ê ú ç ÷ - -D- ë û è ø

*qn W kT qV <sup>J</sup>*

solar cell, for example, with *E*g=1.14 eV calculated using Eq. (51) and (56).

<sup>2</sup> 2arccosh2 <sup>2</sup> . 2( ) <sup>2</sup> ( )( 2 )

D » ç ÷ = - -D- - -D- è ø

*E qV kT x W <sup>W</sup>*

to *E*g – 2Δμ:

and (56).

1/2

m

1/2

on the applied voltage (sinh(*qV*/2*kT*)=exp(*qV*/2*kT*)/2) [37, 38]:

 j

10<sup>4</sup>

10<sup>2</sup>

10–2

*J*

(mA/cm

)

2

10<sup>0</sup>

10–4

no po bi g

æ ö -D-

tt

i

j

g

38 Solar Cells - New Approaches and Reviews

First consider an applicability of the above theory of generation–recombination in the SCR to the experimental data discussed in Section 5.2.

The dark *J–V* curves (circles) along with the calculated results (solid lines) using Eq. (51) are shown in Fig. 22. In the calculations, we used the exact Eq. (51) in order to reflect the deviation of the forward current from the expression *J* ∝ exp(*qV*/2*kT*), when *qV* approaches *φ*bi. For CIS solar cell (*E*g=1.04 eV) the experimentally obtained data is shown without modification (shunting is virtually absent), whereas for CIGS with the absorber bandgaps 1.14 and 1.36 eV the presented experimental data is obtained by subtracting the current through the shunts *R*sh and taking into account the voltage drop across the series resistance *R*s. As mentioned previously, at *V* < 0.1–0.2 V, the values of current under illumination *J*IL and the short-circuit current *J*sc are very close to each other, therefore the dark current cannot be determined with a proper accuracy and the experimental points are not shown for low voltages for the three solar cells. For solar cell with bandgap of the absorber *E*g=1.5 eV fabricated at elevated temperature the measured data of the dark current is shown for low voltages as well.

**Figure 22.** Comparison of the dark forward characteristics of CIGS solar cells extracted from the *J–V* curves under illu‐ mination [8] and the generation current (solid lines) calculated from Eq. (51).

As seen in Fig. 22, the calculated results agree well with the experimental data for all solar cells. One point to be mentioned is that in order to obtain a fit with the experimental data of solar cell with *E*g=1.5 eV fabricated at elevated temperature, the effective carrier lifetime in the SCR was taken about an order of magnitude less than that for other solar cells. This explains why the curves with so much different bandgaps, *E*g=1.36 and 1.5 eV, are located close to each other. It should also be emphasized that reducing the carrier lifetime results in the lowering of open-circuit voltage by ∼ 0.06 V, which apparently is one of the negative aspects of the growth and post-growth processing at elevated temperature [36].

Knowing the dark *J–V* characteristics and the short-circuit current densities, it is not difficult to calculate the *J–V* curves under illumination as the dependences of *J*IL=*J – J*sc vs. *V*, which are shown in Fig. 23a by bashed lines. For the short-circuit current densities we use the data given in Fig. 18a, although they can be obtained using the spectral distribution of the quantum efficiency and solar radiation (for the studied samples the discrepancy between the *J*sc values calculated by these two methods does not exceed 3–4%). Also shown in Fig. 23a by circles are the measured results of the *J–V* curves for the samples with the absorber bandgaps 1.04, 1.14 and 1.36 eV.

Fig. 23b shows the dependences of the electrical power in the solar cell circuit *P*=(*J – J*sc)*V* as a function of voltage *V*. The solid lines in both figures are the calculated results corresponding to the presence of shunts in cells, whereas dashed lines are the results obtained by subtracting the current through the shunt from the measured current.

35

much different bandgaps, *E*g = 1.36 and 1.5 eV, are located close to each other. It should also be emphasized that reducing the carrier lifetime results in the lowering of open-circuit voltage by 0.06 V, which apparently is one of the negative aspects of the growth and post-growth processing at elevated

Knowing the dark *J–V* characteristics and the short-circuit current densities, it is not difficult to calculate the *J–V* curves under illumination as the dependences of *J*IL = *J – J*sc vs. *V*, which are shown in Fig. 23a by bashed lines. For the short-circuit current densities we use the data given in Fig. 18a, although they can be obtained using the spectral distribution of the quantum efficiency and solar radiation (for the studied samples the discrepancy between the *J*sc values calculated by these two method does not exceed 3–4%). Also shown in Fig. 23a by circles are the measured results of the *J–V*

Fig. 23b shows the dependences of the electrical power in the solar cell circuit *P* = (*J – J*sc)*V* as a

curves for the samples with the absorber bandgaps 1.04, 1.14 and 1.36 eV.

temperature [36].

*E*g *=* 1.36 eV

*E*g *=* 1.5 eV

*E*g *=* 1.14 eV

0 0.2 0.4 0.6 0.8

*V* (V)

**Figure 22.** Comparison of the dark forward characteristics of CIGS solar cells extracted from the *J–V* curves under illu‐

As seen in Fig. 22, the calculated results agree well with the experimental data for all solar cells. One point to be mentioned is that in order to obtain a fit with the experimental data of solar cell with *E*g=1.5 eV fabricated at elevated temperature, the effective carrier lifetime in the SCR was taken about an order of magnitude less than that for other solar cells. This explains why the curves with so much different bandgaps, *E*g=1.36 and 1.5 eV, are located close to each other. It should also be emphasized that reducing the carrier lifetime results in the lowering of open-circuit voltage by ∼ 0.06 V, which apparently is one of the negative aspects of the

Knowing the dark *J–V* characteristics and the short-circuit current densities, it is not difficult to calculate the *J–V* curves under illumination as the dependences of *J*IL=*J – J*sc vs. *V*, which are shown in Fig. 23a by bashed lines. For the short-circuit current densities we use the data given in Fig. 18a, although they can be obtained using the spectral distribution of the quantum efficiency and solar radiation (for the studied samples the discrepancy between the *J*sc values calculated by these two methods does not exceed 3–4%). Also shown in Fig. 23a by circles are the measured results of the *J–V* curves for the samples with the absorber bandgaps 1.04, 1.14

Fig. 23b shows the dependences of the electrical power in the solar cell circuit *P*=(*J – J*sc)*V* as a function of voltage *V*. The solid lines in both figures are the calculated results corresponding to the presence of shunts in cells, whereas dashed lines are the results obtained by subtracting

*E*g *=* 1.04 eV

10–2

10<sup>0</sup>

10–4

10–6

mination [8] and the generation current (solid lines) calculated from Eq. (51).

growth and post-growth processing at elevated temperature [36].

the current through the shunt from the measured current.

*J*

and 1.36 eV.

(mA/cm

)

2

40 Solar Cells - New Approaches and Reviews

10<sup>2</sup>

illumination *I*IL (a) [8], and the electrical power *P* in the circuit of solar cell (b). The measured results are shown by circles, the results of calculations considering the presence of shunts and without them are shown by solid and dashed lines, respectively. **Figure 23.** Comparison of voltage dependences of the measured current in CIGS solar cells under illumination *I*IL (a) [8], and the electrical power *P* in the circuit of solar cell (b). The measured results are shown by circles, the results of calculations considering the presence of shunts and without them are shown by solid and dashed lines, respectively.

Figure 23. Comparison of voltage dependences of the measured current in CIGS solar cells under

As seen in Fig. 23a, the calculated curve for sample with the bandgap 1.04 eV practically coincides with experimental curve, but in the cases of the absorbers with bandgaps 1.14 and 1.36 eV the calculations give overestimated values of the current. However, if the shunts are taken into account, the calculated and experimental curves for all samples practically coincide. It follows that a comparison of the theory and experiment gives the *quantitative* information on the electrical losses in solar cells. These losses manifest themselves clearly on the voltage dependences of the electrical power *P* produced by solar cell, *P* = (*J – J*sc)*V* shown in Fig. 23b (as in Fig. 23a, the solid lines are the calculated results with the presence of a shunt in cell, whereas dashed line are the results obtained by subtracting the current through the shunts from the measured currents.). As seen in Fig. 23a, the calculated curve for sample with the bandgap 1.04 eV practically coincides with experimental curve, but in the cases of the absorbers with bandgaps 1.14 and 1.36 eV the calculations give overestimated values of the current. However, if the shunts are taken into account, the calculated and experimental curves for all samples practically coincide. It follows that a comparison of the theory and experiment gives the *quantitative* information on the electrical losses in solar cells. These losses manifest themselves clearly on the voltage dependences of the electrical power *P* produced by solar cell, *P*=(*J – J*sc)*V* shown in Fig. 23b (as in Fig. 23a, the solid lines are the calculated results with the presence of a shunt in cell, whereas dashed line are the results obtained by subtracting the current through the shunts from the measured currents.).

As seen in Fig. 23, the calculated results agree well with the experimental data and indicate a noticeable negative effect of shunting in the studied cells. The effect of shunting is higher when the bandgap of the CIGS absorber is large (shunting does not reveal itself in CIS solar cell). As expected, the shunting does not practically vary the open circuit voltage but reduces the fill factor and the energy conversion efficiency. For the cell with absorber bandgap 1.14 eV, shunting leads to decreasing the fill factor from 0.73 to 0.70 and the efficiency from 13.3 to 12.7%. For cell with the absorber bandgap 1.36 eV, these values are 0.77 to 0.70 and 14.3 to 13.0%, respectively. The fill factor and efficiency of CuInSe2 solar cell are equal to 0.67 and 11.3%, respectively.

Note that the efficiency of studied CIGS solar cells is in the range of 11-14% which is comparable with the efficiency of the modules produced in large volume, but much inferior to record efficiency of small area CIGS solar cells achieved so far.
