**3.2. Effect of multiple reflection in ZnO and CdS layers**

In the calculations outlined in Section 3.1, we omitted the multiple reflections and interference effects in the ZnO and CdS layers, although they occur like those in antireflection coating (oscillations in CIGS are suppressed by strong absorption of the material). It should be noted that the periodic oscillations in the quantum efficiency spectra of the CIGS solar cells, origi‐ nating from the interference effects, in many cases are not observed. To explain this fact, we calculated the optical transmission of the ZnO/CdS layered structure taking into consideration multiple reflections [9] and using the formula for the double-layer antireflection coatings [16]:

$$R\_{\rm dl} = \frac{r\_1^2 + r\_2^2 + r\_3^2 + r\_1^2 r\_2^2 r\_3^2 + 2r\_1 r\_2 (1 + r\_3^2) \cos(2\beta\_1 \beta\_1) + 2r\_2 r\_3 (1 + r\_1^2) \cos(2\beta\_2) + 2r\_1 r\_3 \cos(2\beta\_1 + 2\beta\_2)}{1 + r\_1^2 r\_2^2 + r\_2^2 r\_3^2 + r\_1^2 r\_3^2 + r\_1^2 r\_3^2 + 2r\_1 r\_2 (1 + r\_3^2) \cos(2\beta\_1) + 2r\_2 r\_3 (1 + r\_1^2) \cos(2\beta\_2) + 2r\_1 r\_3 \cos(2\beta\_1 + 2\beta\_2)},\tag{10}$$

where *r*1, *r*2, and *r*3 are the amplitude reflection coefficients for the air/ZnO, ZnO/CdS and CdS/ CIGS interfaces:

$$r\_1^2 = \frac{\left(1 - \nu\_2\right)^2 + \left(\kappa\_2\right)^2}{\left(1 + \nu\_2\right)^2 + \left(\kappa\_2\right)^2},\tag{11}$$

$$r\_2^2 = \frac{\left(\mu\_2 - \mu\_3\right)^2 + \left(\kappa\_2 - \kappa\_3\right)^2}{\left(\mu\_2 + \mu\_3\right)^2 + \left(\kappa\_2 + \kappa\_3\right)^2} \tag{12}$$

$$r\_3^2 = \frac{\left(n\_3 - n\_4\right)^2 + \left(\kappa\_3 - \kappa\_4\right)^2}{\left(n\_3 + n\_4\right)^2 + \left(\kappa\_3 + \kappa\_4\right)^2},\tag{13}$$

*n*2, *n*3, *n*4 and *n*2, *n*3, *n*4 are the refractive indices and extinction coefficients of ZnO, CdS and CIGS, respectively,

**Origin of optical losses**

12 Solar Cells - New Approaches and Reviews

**Total optical losses with grid**

CIGS interfaces:

**Absorption losses** Absorption in ZnO Absorption in CdS Insufficient absorptivity Total absorption losses

Total reflection losses 4.7% (1.9 mA/cm2

**shading** 17.4% (9.4 mA/cm2

2.9% (1.5 mA/cm2

5.2% (2.7 mA/cm2

0.6% (0.1 mA/cm2

8.1% (4.3 mA/cm2

**Table 1.** Optical and the corresponding photocurrent losses in CIGS solar cells

2 2 2 222 2 2

*r*

*r*

*r*

dl 22 22 22 2 2

**3.2. Effect of multiple reflection in ZnO and CdS layers**

)

)

)

)

**Losses in solar cell**

) 4.0% (1.6 mA/cm2

2.2% (1.1 mA/cm2

5.6% (2.9 mA/cm2

0.2% (0.1 mA/cm2

7.8% (4.1 mA/cm2

) 16.0% (9.0 mA/cm2

In the calculations outlined in Section 3.1, we omitted the multiple reflections and interference effects in the ZnO and CdS layers, although they occur like those in antireflection coating (oscillations in CIGS are suppressed by strong absorption of the material). It should be noted that the periodic oscillations in the quantum efficiency spectra of the CIGS solar cells, origi‐ nating from the interference effects, in many cases are not observed. To explain this fact, we calculated the optical transmission of the ZnO/CdS layered structure taking into consideration multiple reflections [9] and using the formula for the double-layer antireflection coatings [16]:

> b

1 2 3 1 2 3 12 3 1 23 1 2 13 1 2

1 2 2 3 1 3 12 3 1 23 1 2 13 1 2 2 (1 )cos(2 ) 2 (1 )cos(2 ) 2 cos(2 2 ) , <sup>1</sup> 2 (1 )cos(2 ) 2 (1 )cos(2 ) 2 cos(2 2 ) *r r r r r r rr r rr r rr <sup>R</sup> r r r r r r rr r rr r r r* (10)

where *r*1, *r*2, and *r*3 are the amplitude reflection coefficients for the air/ZnO, ZnO/CdS and CdS/

k

2 2

k

k k

2 2

k k

k k

2 2

k k

+++ + + + + + + <sup>=</sup> ++++ + + + + +

b


2 2 2 1 2 2 2 2 (1 ) ( ) , (1 ) ( ) *n*

*n*



2 34 34 3 2 2 34 34 ( )( ) , ( )( )

*n n*

*n n*

*n n*

*n n*

2 23 23 2 2 2 23 23 ( )( ) , ( )( )

**CuInSe2 CuIn0.69Ga0.31Se2 CuIn0.34Ga0.66Se2**

)

)

)

)

b

b

) 3.5% (1.3 mA/cm2

1.9% (0.7 mA/cm2

7.7% (2.6 mA/cm2

0.4% (0.1 mA/cm2

9.6% (3.4 mA/cm2

) 17.5% (7.3 mA/cm2

 bb

 bb )

)

)

)

)

)

(11)

(12)

(13)

$$
\beta\_1 = \frac{2\pi}{\lambda} n\_2 d\_{\rm ZrO'} \tag{14}
$$

$$
\beta\_2 = \frac{2\pi}{\lambda} n\_3 d\_{\rm CdS}.\tag{15}
$$

Considering a grid contact at the front surface of the ZnO layer and using Eq. (10), one can write the expression for the transmission of the ZnO and CdS layers:

$$T(\mathcal{X}) = T\_{\text{gr}}(1 - R\_{\text{dl}})\_{\text{\textquotedblleft}} \tag{16}$$

where *T*gr takes into account the shadowing by grid on the front surface of ZnO.

The curve labeled *1* in Fig. 9 shows the calculated transmittance using Eq. (10) and (16) for CuIn0.7Ga0.3Se2 solar cell, and the used parameters are *T*gr=0.95, *d*ZnO=300 nm, *d*CdS=40 nm. The oscillations in the transmission spectrum are clear though the amplitude is small. The small amplitude may be due to the low reflectance from the interfaces since the refractive indices of the contacting layers are close to each other. This is confirmed by the fact that the amplitude of oscillations increases almost twice when the transmission is calculated without antireflective layer on the ZnO front surface.

Eq. (10) describes the oscillations in the reflection spectra due to interference effects in the ZnO and CdS layers, however it does not take into account absorption in these layers. When extinction coefficient *κ* is small the oscillations due absorption in the layers are visible, but for higher *κ* the oscillations cannot be observed at all. We can take into account the absorption of solar radiation for one passage in the ZnO and CdS layers by entering in the right side of Eq. (16) the exponential factors appearing in Eqs. (3) and (8):

$$T(\lambda) = T\_{\rm gr}(1 - R\_{\rm dl}) \exp(-a\_2 d\_2) \exp(-a\_3 d\_3). \tag{17}$$

The results of calculations using Eq. (17) are shown in Fig. 9 as curve *2*. As seen, the oscillation amplitude in the spectral range *λ* > 500 nm, where the absorption in ZnO and CdS is weak has remained almost unchanged, but in the range of the interband transitions in CdS at *λ* < 500 nm, the amplitude decreased significantly.

The spectrum *3* in Fig. 9 was obtained by using Eq. (3) with *R*arc=*R*12, i.e., without considering the multiple reflections in the ZnO and CdS layers. As it is clear from figure, when multiple reflections are considered (spectrum 2) only minor periodic deviations occur from spectrum *3*. When calculating the photocurrent density of a solar cell, which is the sum of the current over the entire spectral range, ignoring multiple reflections should not cause significant errors. In fact, photocurrent density calculated using Eqs. (8) and (17) for *T*(*λ*) shows only a difference less than 0.5% for the studied solar cells.

**Figure 9.** Transmission spectra of ZnO/CdS layers calculated by taking into account the multi-reflections but without considering absorption (line *1*), line 2 is the spectrum when the absorption in the ZnO and CdS layers is taken into account, and line 3 corresponds to case when multi-reflections are neglected [9].

It should also be borne in mind that the Eq. (17) has been deduced for flat, perfect and parallel interfaces air/ZnO, ZnO/CdS and CdS/CIGS. But real interfaces will be far away from the ideal conditions; hence the oscillations in the transmission spectra can be less than 0.5% or even not visible. In contrast, if the interfaces are perfect as mentioned above, the periodic variations in the transmission and photoresponse spectra of the devices will be clear [17].
