**4. Recombination losses**

*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

*2 3* 

400 600 800

l(nm)

**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

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

In the previous sections while discussing the photocurrent, we assumed 100% light-to-electric conversion in the CIGS layer. However, there are losses related to the incomplete absorption of light in this layer. Considering this fact, while estimating the exact values of *integrated* absorptive capacity of the CIGS layer the spectral distribution of solar radiation and absorption coefficient of the material must be taken into account rather than determining the absorptive

It should be noted that in the photon energy range from *E*<sup>g</sup> to 4.1 eV electron-hole pairs arise independently of the energy absorbed. In other words, the energy needed for the band to band transition is less than the absorbed energy. So while considering the dependence of the absorptivity of solar radiation power *A*<sup>Φ</sup> (rather than photon flux *A*hν) on the thickness of the semiconductor this fact must be considered. For this reason, the number of electron-hole pairs,

1000 1200

0

account, and line 3 corresponds to case when multi-reflections are neglected [9].

**3.3. Absorptive capacity of the CIGS absorber layer**

capacity for one wavelength in the range *hν* > *E*g.

the transmission and photoresponse spectra of the devices will be clear [17].

0.2

0.4

*T*(l) 0.6

0.8

*1* 

1.0

less than 0.5% for the studied solar cells.

14 Solar Cells - New Approaches and Reviews

Electrons and holes (electron-hole pairs) created as a result of photon absorption in the space charge region (SCR) of CdS/CIGS heterostructure move apart under the influence of the junction built-in voltage. The electrons move towards to CdS side and reach the front metal grid contact after passing through the ZnO layer, while holes move to the neutral part of the CIGS layer and reach the Mo back contact. Not all the photogenerated carriers contribute to the current; some of the carriers recombine in the SCR before reaching the contacts. Recombi‐ nation can also occur at the front surface of the absorber contacting with the CdS film. Minority carriers (electrons), which are generated outside the SCR, also take part in the formation of the photocurrent, when electrons reach the SCR as a result of diffusion. This process competes with the recombination of electrons with the majority carriers (holes) in the neutral part of the absorber. Finally, recombination of electrons can occur on the back surface of the absorber, i.e., at the CIGS/metal interface.

For a given thickness of the absorber, the recombination losses depend mostly on the carrier lifetime and the width of the SCR, which in the case of a semiconductor containing both acceptor and donor type impurities are determined by the concentration of uncompensated acceptors *N*a – *N*d. Therefore, investigating the influence of the material parameters on the recombination losses, we will calculate the dependence of photocurrent *J* on *N*a – *N*d for several values of carrier lifetimes.

Consideration of the statistics in a nonequilibrium state leads to the conclusion that the lifetimes of electrons *τ*no and holes *τ*po in the SCR are approximately equal to the lifetime of minority carriers in heavily doped respectively p-and n-type materials [19]. We assume that in the CIGS solar cell, the absorber is a heavily doped semiconductor (hole concentration is about 1015–1016 up to 1017 cm–3 [8, 21].

The recombination losses can be judged by the value of photocurrent density *J* using the formula:

$$J = q \sum\_{i} \frac{\Phi\_i(\mathcal{Z})}{h\nu\_i} \ T(\mathcal{Z}) \eta\_{\text{int}}(\mathcal{Z}) \Delta \mathcal{Z}\_{\text{i}}.\tag{19}$$

where *T*(*λ*) is the optical transmission of the ZnO and CdS layers, and *η*int(*λ*) is the internal quantum efficiency of photoelectric conversion.

The CdS/CIGS solar cell is generally treated as an abrupt asymmetric p-n heterostructure, in which the SCR (depletion layer) is practically located in the p-CIGS and the photoelectric conversion takes place almost in this layer (see [8] and references therein). The potential and field distributions in abrupt asymmetric p-n junction are practically the same as in a Schottky diode, therefore, further consideration of the processes in CdS/CIGS solar cells can be studied on the basis of the concepts developed for Schottky diodes.

The exact expression for the photovoltaic quantum efficiency of a p-type semiconductor Schottky photodiode obtained from the continuity equation taking into account the recombi‐ nation at the front surface has the form [21]:

$$\eta(\alpha) = \frac{1 + \mathcal{S}\_{\text{f}} / D\_{\text{p}} \exp(-\mathcal{W}^{2} / \mathcal{W}\_{\text{o}}^{2}) \ \ [A(\alpha) - D\_{\text{p}} \Delta n / \Phi] }{1 + (\mathcal{S}\_{\text{f}} / D\_{\text{p}}) \ \exp(-\mathcal{W}^{2} / \mathcal{W}\_{\text{o}}^{2}) \ B} - \frac{\exp(-\alpha \mathcal{W})}{1 + \alpha L\_{\text{n}}} - \frac{D\_{\text{n}}}{L\_{\text{n}}} \frac{\Delta n}{\Phi} \,\prime \,\tag{20}$$

where *α* is the absorption coefficient, *S*<sup>f</sup> is the recombination velocity at the front surface, *D*<sup>n</sup> and *D*p are the diffusion coefficients of electrons and holes, respectively, *L*n is the diffusion length of electrons, Δ*n* is the excess concentration of electrons at *x*=*W*, which is equal to

$$
\Delta \mathfrak{m} = \Phi \frac{F\_5}{D\_\mathbb{P}}.\tag{21}
$$

In Eqs. (20) and (21)

the current; some of the carriers recombine in the SCR before reaching the contacts. Recombi‐ nation can also occur at the front surface of the absorber contacting with the CdS film. Minority carriers (electrons), which are generated outside the SCR, also take part in the formation of the photocurrent, when electrons reach the SCR as a result of diffusion. This process competes with the recombination of electrons with the majority carriers (holes) in the neutral part of the absorber. Finally, recombination of electrons can occur on the back surface of the absorber, i.e.,

For a given thickness of the absorber, the recombination losses depend mostly on the carrier lifetime and the width of the SCR, which in the case of a semiconductor containing both acceptor and donor type impurities are determined by the concentration of uncompensated acceptors *N*a – *N*d. Therefore, investigating the influence of the material parameters on the recombination losses, we will calculate the dependence of photocurrent *J* on *N*a – *N*d for several

Consideration of the statistics in a nonequilibrium state leads to the conclusion that the lifetimes of electrons *τ*no and holes *τ*po in the SCR are approximately equal to the lifetime of minority carriers in heavily doped respectively p-and n-type materials [19]. We assume that in the CIGS solar cell, the absorber is a heavily doped semiconductor (hole concentration is

The recombination losses can be judged by the value of photocurrent density *J* using the

lh l l

where *T*(*λ*) is the optical transmission of the ZnO and CdS layers, and *η*int(*λ*) is the internal

The CdS/CIGS solar cell is generally treated as an abrupt asymmetric p-n heterostructure, in which the SCR (depletion layer) is practically located in the p-CIGS and the photoelectric conversion takes place almost in this layer (see [8] and references therein). The potential and field distributions in abrupt asymmetric p-n junction are practically the same as in a Schottky diode, therefore, further consideration of the processes in CdS/CIGS solar cells can be studied

The exact expression for the photovoltaic quantum efficiency of a p-type semiconductor Schottky photodiode obtained from the continuity equation taking into account the recombi‐

> a

1 / exp( / ) [ ( ) / ] exp( ) ( ) , 1 ( / ) exp( / ) 1 *SD W W A Dn W D n*

+ - - DF - <sup>D</sup> <sup>=</sup> - - + - + F

f p o p n 2 2

f p o n n

( ) () () , *<sup>i</sup>*

*Jq T <sup>h</sup>* (19)

a

a

*SD W WB L L* (20)

l

<sup>F</sup> <sup>=</sup> å <sup>D</sup> int i

n

*i*

i

at the CIGS/metal interface.

16 Solar Cells - New Approaches and Reviews

values of carrier lifetimes.

formula:

h a

about 1015–1016 up to 1017 cm–3 [8, 21].

quantum efficiency of photoelectric conversion.

nation at the front surface has the form [21]:

on the basis of the concepts developed for Schottky diodes.

2 2

$$F\_5 = \frac{F\_4 - F\_3 \exp(-aW) / (1 + aL\_n)}{1 + F\_3 / L\_n} \,\tag{22}$$

$$F\_3 = \bigcap\_{o}^{W} \exp\{-\left[\left(\mathbf{x} - \mathbf{W}\right) / W\_o\right]^2\} \text{d}\mathbf{x}\_{\prime} \tag{23}$$

$$F\_4 = \bigcup\_{o}^{W} \exp\{-ax - \left[\left(x - W\right) / W\_o\right]^2\} dx,\tag{24}$$

and the following notations are used:

$$\mathcal{W}\_{\rm o} = \sqrt{\frac{2\varepsilon\varepsilon\_{\rm o}kT}{q^2(N\_{\rm a} - N\_{\rm d})}},\tag{25}$$

$$F\_4 = \bigcap\_{o}^{W} \exp\{-\alpha x + [\{x - W\} / W\_o]^2\} \text{dx},\tag{26}$$

$$B = \int\_0^W \exp[\left(\mathbf{x} - \mathbf{W}\right) / W\_o]^2 d\mathbf{x},\tag{27}$$

where the *W*o / 21/2 value is the Debye length.

In a CIS or CIGS with both acceptor and donor impurities, the SCR width *W* given in the above equations is determined by the expression [22]:

$$W = \sqrt{\frac{2\varepsilon x\_o(q\_{\rm bi} - qV)}{q^2(N\_\text{a} - N\_\text{d})}} \,\text{}\,\text{}\,\tag{28}$$

where *ε* is the relative permittivity of CIGS (*ε*=13.6), *ε*<sup>o</sup> is the permittivity of vacuum, and *N*<sup>a</sup> – *N*d is the concentration of uncompensated acceptors (not the free hole density).

For the convenience in analyzing the dependence of *η* on the parameters of the diode structure, the expression (20) can be simplified.

In a solar cell, CIS or CIGS, with the barrier height from the semiconductor side φbi is of the order of 1 eV and the width of the SCR is about 1 μm, i.e., the electric field is close to 104 V/cm. At the boundary between the depletion layer and the neutral region (*x=W*), the photogenerated electrons are pulled into the SCR by strong electric field, and without causing error, one can assume that Δ*n*=0 and disregard the terms containing Δ*n* in Eq. (20) [21].

The integrand *f*(*x*) in Eq. (24) decreases exponentially as *x* increases. Therefore, we can replace the integration by the multiplication of the maximum value of the function exp(*W*/*W*o) 2 by its "half-width", which is determined by the value of *x*2 at the point where the value of *f*(*x*) is smaller than the peak value by a factor *e=*2.71.

Thus, we can find the value of *x* that satisfies this condition from the equality:

$$\exp\{-ax\_2 + \left[\left(x\_2 - W\right)/W\_o\right]^2\} = \exp\{W/W\_o\}^2 e^{-1} = \exp\left[\left(W/W\_o\right)^2 - 1\right],\tag{29}$$

which is reduced to the quadratic equation

$$\mathbf{x}\_2^{-2} - (\alpha W\_o^2 + 2W)\mathbf{x}\_2 + W\_o^2 = \mathbf{0} \tag{30}$$

with the solution

$$\propto\_2 = \frac{a\mathcal{W}\_\text{o}^2 + 2\mathcal{W}}{2} [1 - \sqrt{1 - 4\mathcal{W}\_\text{o}^2 / \left(a\mathcal{W}\_\text{o}^2 + 2\mathcal{W}\right)^2}].\tag{31}$$

Since the second term under the square root is much less than unity, 1− *x* ≈1− *x* / 2,

$$\mathbf{x}\_2 \approx \left[\boldsymbol{\alpha} + 2\boldsymbol{\mathcal{W}} / \boldsymbol{\mathcal{W}}\_{\boldsymbol{\alpha}}\right]^{-1} = \left[\boldsymbol{\alpha} + 2(\boldsymbol{\varphi}\_{\text{bi}} - qV) / 2kT\boldsymbol{\mathcal{W}}\right]^{-1}.\tag{32}$$

Similarly, by replacing the integration in Eq. (23), we can obtain the expression for *x*2 in the form

$$\text{tr}\_2 \approx \left[ \Im(\phi\_{\text{bi}} - qV) / \Im kT \mathcal{W} \right]^{-1}. \tag{33}$$

So, considering all the simplifications made for the photoelectric quantum yield it is possible to substitute Eq. (20) with the following expression:

where *ε* is the relative permittivity of CIGS (*ε*=13.6), *ε*<sup>o</sup> is the permittivity of vacuum, and *N*<sup>a</sup>

For the convenience in analyzing the dependence of *η* on the parameters of the diode structure,

In a solar cell, CIS or CIGS, with the barrier height from the semiconductor side φbi is of the order of 1 eV and the width of the SCR is about 1 μm, i.e., the electric field is close to 104

At the boundary between the depletion layer and the neutral region (*x=W*), the photogenerated electrons are pulled into the SCR by strong electric field, and without causing error, one can

The integrand *f*(*x*) in Eq. (24) decreases exponentially as *x* increases. Therefore, we can replace

"half-width", which is determined by the value of *x*2 at the point where the value of *f*(*x*) is


22 2 2 o 2o *x W Wx W* - + += ( 2) 0

= -- +

Since the second term under the square root is much less than unity, 1− *x* ≈1− *x* / 2,

a

2 o o

j

o 222

<sup>2</sup> [1 1 4 / ( 2 ) ]. <sup>2</sup>

 j- - »+ =+ - 2 1 <sup>1</sup>

Similarly, by replacing the integration in Eq. (23), we can obtain the expression for *x*2 in the


a

2 2 1 2 2 2 <sup>o</sup> <sup>o</sup> exp{ [( ) / ] } exp( / ) exp ( / ) 1 , *<sup>o</sup> x x W W WW e WW* (29)

a

*x W WW* (31)

<sup>2</sup> <sup>o</sup> bi *x WW* [ 2 / ] [ 2( ) / 2 ] . *qV kTW* (32)

2 bi *x qV kTW* [2( ) / 2 ] . (33)

(30)

the integration by the multiplication of the maximum value of the function exp(*W*/*W*o)

Thus, we can find the value of *x* that satisfies this condition from the equality:

V/cm.

2 by its

– *N*d is the concentration of uncompensated acceptors (not the free hole density).

assume that Δ*n*=0 and disregard the terms containing Δ*n* in Eq. (20) [21].

the expression (20) can be simplified.

18 Solar Cells - New Approaches and Reviews

smaller than the peak value by a factor *e=*2.71.

which is reduced to the quadratic equation

a

a +

*W W*

2

a

with the solution

form

$$\eta\_{\rm int} = \frac{1 + S\_{\rm f} \,/\, D\_{\rm p} \,\, \left[ a + (2/\text{W}) (\phi\_{\rm bi} - qV) / kT \right]^{-1}}{1 + S\_{\rm f} \, /\, D\_{\rm p} \,\, \left[ (2/\text{W}) (\phi\_{\rm bi} - qV) / kT \right]^{-1}} - \frac{\exp(-aW)}{1 + aL\_{\rm n}}.\tag{34}$$

As an example comparison of the curves *η*int(*λ*), calculated using the exact Eq. (20) and Eq. (34) is shown in Fig. 10 in the case of the spectrum of CuInSe2 solar cell.

**Figure 10.** The quantum efficiency spectra of CuInSe2 solar cell calculated by the exact formula (20) (solid line) and the simplified formula (34) (dotted line).

In the calculations we used the absorption curve *α*(*λ*) and typical parameters for this cell (see Table 2): barrier height φbi, the diffusion coefficients of electrons *D*<sup>n</sup> and holes *D*p, their mobility μn and μp, the electron lifetime *τ*n (*L*n=(*τ*n*D*n) 1/2), the surface recombination velocity *S*<sup>f</sup> , and the concentration of uncompensated acceptors *N*a –*N*<sup>d</sup> which varied in the range 1014-1016 cm–3. As seen, variation of *N*a – *N*<sup>d</sup> modifies the shape of the *η*int(*λ*) spectra, but in all cases the curves calculated by Eqs. (20) and (34) are very close, i.e., Eq. (34) represents well the results of the exact calculation.

It should be bear in mind that Eq. (34) takes into consideration both the drift and diffusion components of the quantum efficiency but does not take into account recombination at the back surface of the absorber layer [21] which can result in significant losses in the case of a thin-film solar cell. In the case of CdS/CdTe cell, for example, it is possible to neglect the recombination at the back surface if the thickness of the absorber exceeds 4–5 μm. However, solar cells with thinner layers are also of considerable interest. In this case, Eq. (34) is not valid, and the drift and diffusion components of the quantum efficiency should be separated.

To find an expression for the drift component of the photoelectric quantum yield one can use Eq. (34). Indeed, the absence of recombination at the front surface (*S*<sup>f</sup> =0) transforms this equation into the known Gartner formula:

$$\eta = 1 - \frac{\exp(-aW)}{1 + aL\_n}.\tag{35}$$

The above equation also ignores recombination at the back surface of the absorbed layer. Eq. (34) does not take into account recombination inside the SCR, therefore, subtracting the absorptive capacity of the SCR layer 1 – exp(–*αW*) from the right side of Eq. (34), we obtain the expression for the diffusion component of the quantum yield for a thick solar cell exp(–*αW*) *αL*n/(1+*αL*n) which ignores recombination at the back surface. Upon eliminating the diffusion component from the right side of Eg. (34) we come to the expression for the drift component of the quantum efficiency which take into account the surface recombination at the front surface of the absorber:

$$\eta\_{\rm drift} = \frac{1 + \mathcal{S}\_{\rm f} / D\_{\rm p} \left( a + 2(\rho\_{\rm bi} - qV) / \mathcal{W}kT \right)^{-1}}{1 + \mathcal{S}\_{\rm f} / D\_{\rm p} \left( 2(\eta\_{\rm ext} \rho\_{\rm bi} - qV) / \mathcal{W}kT \right)^{-1}} - \exp(-aW). \tag{36}$$

For the diffusion component of the photoelectric quantum efficiency taking into account surface recombination at the back surface of the absorber layer, one can use the exact expression obtained for the p-layer in a solar cell with p-n junction [22]:

$$\begin{split} \eta\_{\text{dfl}} &= \frac{a L\_{\text{n}}}{a^{2} L\_{\text{n}}^{2} - 1} \exp(-a \mathcal{W}) \times \\ &\times \left\{ a L\_{\text{n}} - \frac{\mathcal{S}\_{\text{b}} L\_{\text{n}} / D\_{\text{n}} \left[ \cosh[\left( d - \mathcal{W} \right) / L\_{\text{n}}] - \exp\left( -a \left( d - \mathcal{W} \right) \right) \right] + \sinh[\left( d - \mathcal{W} \right) / L\_{\text{n}}] + a L\_{\text{n}} \exp(-a \left( d - \mathcal{W} \right))}{\left( \mathcal{S}\_{\text{b}} L\_{\text{n}} / D\_{\text{n}} \right) \sinh[\left( d - \mathcal{W} \right) / L\_{\text{n}}] + \cosh[\left( d - \mathcal{W} \right) / L\_{\text{n}}]} \right\}, \end{split} \tag{37}$$

where *d* is the thickness of the absorber layer and *S*b is the recombination velocity at the back surface.

The internal quantum efficiency of photoelectric conversion in the absorber layer is the sum of the two components:

$$
\eta\_{\rm int} = \eta\_{\rm drift} + \eta\_{\rm dif} \tag{38}
$$

whereas the external quantum efficiency can be written in the form

$$
\eta\_{\rm ext} = T(\mathcal{Z})(\eta\_{\rm drift} + \eta\_{\rm diff}) \,\prime \tag{39}
$$

where *T*(*λ*), as mentioned previously, is the optical transmission of the ZnO and CdS layers determined by Eq. (8).

solar cells with thinner layers are also of considerable interest. In this case, Eq. (34) is not valid, and the drift and diffusion components of the quantum efficiency should be separated.

To find an expression for the drift component of the photoelectric quantum yield one can use

a

*W*

a

The above equation also ignores recombination at the back surface of the absorbed layer. Eq. (34) does not take into account recombination inside the SCR, therefore, subtracting the absorptive capacity of the SCR layer 1 – exp(–*αW*) from the right side of Eq. (34), we obtain the expression for the diffusion component of the quantum yield for a thick solar cell exp(–*αW*) *αL*n/(1+*αL*n) which ignores recombination at the back surface. Upon eliminating the diffusion component from the right side of Eg. (34) we come to the expression for the drift component of the quantum efficiency which take into account the surface recombination at the front

> ( ) ( )

1 / 2( ) / exp( ).

For the diffusion component of the photoelectric quantum efficiency taking into account surface recombination at the back surface of the absorber layer, one can use the exact expression

( )

<sup>ì</sup> é ù - - -- + - + -- <sup>ü</sup> <sup>ï</sup> ë û <sup>ï</sup> ´ - <sup>í</sup> <sup>ý</sup> - +- <sup>ï</sup> <sup>ï</sup> <sup>î</sup> <sup>þ</sup>

/ cosh[( ) / ] exp ( ) sinh[( ) / ] exp( ( )) , ( / )sinh[( ) / ] cosh[( ) / ]

bn n n n

*SL D d W L d W d W L L d W*

*SL D d W L d W L*

where *d* is the thickness of the absorber layer and *S*b is the recombination velocity at the back

The internal quantum efficiency of photoelectric conversion in the absorber layer is the sum

 h

a

bn n n n n

hh

whereas the external quantum efficiency can be written in the form

*S D qV WkT <sup>W</sup>*

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

 j

*S D qV WkT*

h j

1 / 2( ) /


=0) transforms this

(36)

(37)

*<sup>L</sup>* (35)

 a

a

= + int drift dif , (38)

 a


1

Eq. (34). Indeed, the absence of recombination at the front surface (*S*<sup>f</sup>

h

a

f p bi drift 1 f p ext bi

+ -

obtained for the p-layer in a solar cell with p-n junction [22]:

equation into the known Gartner formula:

20 Solar Cells - New Approaches and Reviews

surface of the absorber:

a

*L*

n dif 2 2 n

a

a

n

of the two components:

*L*

h

surface.

h

 a

exp( )

= -´ -

*<sup>L</sup> <sup>W</sup>*

1

Fig. 11 shows a comparison of the measured quantum efficiency spectra of CuInSe2, CuIn0.69Ga0.31Se2 and CuIn0.34Ga0.66Se2 solar cells taken from [8] with the results of calculations using Eq. (39). Note that the data on the extinction coefficients (and hence the absorption coefficient *α*) used in this and following calculations were measured for *E*g=1.02, 1.16 eV and *E*g=1.38 eV [12] rather than 1.04, 1.14 and 1.36 eV, respectively [8] that explains the slight blue shift (10-20 nm) in the calculated long-wavelength edge of the spectra in Fig. 11.

**Figure 11.** Comparison of the measured (circles) [8] and calculated (solid lines) quantum efficiency spectra of CuInSe2 (*E*g=1.04 eV), CuIn0.76Ga0.24Se2 (*E*g=1.14 eV) and CuIn0.39Ga0.61Se2 (*E*g=1.36 eV) solar cells.

According to the data in the above reference [8], in the calculations, the thicknesses of the CIGS, ZnO, CdS and MgF2 layers were assumed to be 2000, 300, 20–50 and 100 nm, respectively. The parameters of the absorber layer were varied within the limits reported in the literature. It should be noted that for polycrystalline CIGS, there is a large spread in mobility values of electrons and holes. At room temperature, the values of the hole mobility are most often indicated in the range from 1–5 to 30–50 cm2 /(Vs) and from 1 to 100 cm2 /(Vs) for the electron mobility [23–25]. Unlike this, it was found in [26] that the electron and hole mobilities in Cu(In,Ga)Se2 are much lower than 1 cm2 /(Vs). As mentioned in Section 2, we believe that such low mobilities refer to the charge transport in the sub-band joined with the conduction band (valence band) due to high doping or/and disorder in the crystal lattice. In contrast, electrons and holes arising as a result of absorption of photons with the energy *hν* ≥ *E*<sup>g</sup> and involving in the photocurrent formation are moving in the conduction and valence band, where their mobilities are much higher.

The lifetime of minority carriers (electrons), determining their diffusion length *L*n=(*τ*n*D*n) 1/2 has also a significant impact on the efficiency of solar cells. As reported back in 1996 [27], the lifetimes of electrons in CuInSe2 are in the range of tens of picoseconds to a few nanoseconds, which was subsequently confirmed in [23, 28]. Lifetime of minority carriers affects the diffusion component of photocurrent and therefore reveals itself in the long-wavelength range (*λ* > 600-700 nm) of the spectrum.

Yet another important parameter determining the quantum efficiency spectra of solar cells is the concentration of uncompensated acceptors *N*a – *N*d in the absorber, which according to Eq. (28) determines the width of the SCR. At high *N*a – *N*d, the width of the SCR amounts to a small portion of the absorber thickness. With decreasing *N*a – *N*d, the efficiency of cell increases since more and more part of the radiation is absorbed in the SCR, where the photogenerated electrons and holes move apart in the opposite directions by the electric field and reach the contacts without recombination. However, an increase of the quantum efficiency due to expansion of the SCR occurs only to a certain extent, because with increasing *W*, the electric field in the SCR is weakened and, therefore, recombination at the front surface of the absorber layer is intensified. As a result, the quantum efficiency in the low wavelength region reduces.

The velocity of recombination at the front surface *S*<sup>f</sup> affects the efficiency in a wide range of wavelengths (excluding the long wavelength part) and more stronger in a wider SCR.

Our calculations in section 4.3 show that recombination at the rear surface of 2 μm thick absorber layers of CuInSe2, CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se2 manifests itself very weakly and only for long lifetimes of electrons.

It follows that the main parameters of the absorber layer affect the QE spectrum of the solar cell *differently* [29]. This facilitates the choice of the parameter values for better agreement between the calculated and experimental data and virtually eliminates getting the same spectral curve for different combinations of parameters. The data of calculations presented in Fig. 12 confirm the above statement showing how the spectrum of CdS/CuInSe2 solar cell is modified when one of the parameters is varied.

As can be seen, a deviation upward or downward from the optimum value of the minority carrier lifetime (*τ*n=2 ns) substantially affect the spectral distribution of the quantum efficiency only at wavelengths longer than ∼ 700 nm. In contrast, the deviation of the recombination velocity at the front surface of the Cu(In,Ga)Se2 layer from the value *S*<sup>f</sup> =2×105 cm/s results in significant changes in the short wavelength region (shorter than ∼ 800 nm). Judging the curves in Fig. 12b, it might give the impression that the recombination at the front surface has more or less the same effect on the quantum efficiency over the entire spectrum, and it can induce doubt in the correctness of the physical model used. However, this is not so, a simple calcu‐ lation shows that for *S*<sup>f</sup> =107 cm/s in the spectral range 600-800 nm, surface recombination reduces the quantum efficiency only by 2-4% but as high as ∼ 22% in the range 350-400 nm.

The concentration of uncompensated acceptors *N*a – *N*d impacts the entire spectrum, and is more appreciable. Moreover, when concentration of *N*a – *N*d is deviated by an order from the combinations of parameters. The data of calculations presented in Fig. 12 confirm the above statement showing how the spectrum of CdS/CuInSe2 solar cell is modified when one of the A Theoretical Description of Thin-Film Cu(In,Ga)Se2 Solar Cell Performance http://dx.doi.org/10.5772/59363 23

19

weakened and, therefore, recombination at the front surface of the absorber layer is intensified. As

The velocity of recombination at the front surface *S*f affects the efficiency in a wide range of

Our calculations in section 4.3 show that recombination at the rear surface of 2 m thick absorber layers of CuInSe2, CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se2 manifests itself very weakly and only for

It follows that the main parameters of the absorber layer affect the QE spectrum of the solar cell *differently* [29]. This facilitates the choice of the parameter values for better agreement between the calculated and experimental data and virtually eliminates getting the same spectral curve for different

wavelengths (excluding the long wavelength part) and more stronger in a wider SCR.

a result, the quantum efficiency in the low wavelength region reduces.

long lifetimes of electrons.

parameters is varied.

the photocurrent formation are moving in the conduction and valence band, where their

also a significant impact on the efficiency of solar cells. As reported back in 1996 [27], the lifetimes of electrons in CuInSe2 are in the range of tens of picoseconds to a few nanoseconds, which was subsequently confirmed in [23, 28]. Lifetime of minority carriers affects the diffusion component of photocurrent and therefore reveals itself in the long-wavelength range (*λ* >

Yet another important parameter determining the quantum efficiency spectra of solar cells is the concentration of uncompensated acceptors *N*a – *N*d in the absorber, which according to Eq. (28) determines the width of the SCR. At high *N*a – *N*d, the width of the SCR amounts to a small portion of the absorber thickness. With decreasing *N*a – *N*d, the efficiency of cell increases since more and more part of the radiation is absorbed in the SCR, where the photogenerated electrons and holes move apart in the opposite directions by the electric field and reach the contacts without recombination. However, an increase of the quantum efficiency due to expansion of the SCR occurs only to a certain extent, because with increasing *W*, the electric field in the SCR is weakened and, therefore, recombination at the front surface of the absorber layer is intensified. As a result, the quantum efficiency in the low wavelength region reduces.

wavelengths (excluding the long wavelength part) and more stronger in a wider SCR.

Our calculations in section 4.3 show that recombination at the rear surface of 2 μm thick absorber layers of CuInSe2, CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se2 manifests itself very weakly

It follows that the main parameters of the absorber layer affect the QE spectrum of the solar cell *differently* [29]. This facilitates the choice of the parameter values for better agreement between the calculated and experimental data and virtually eliminates getting the same spectral curve for different combinations of parameters. The data of calculations presented in Fig. 12 confirm the above statement showing how the spectrum of CdS/CuInSe2 solar cell is

As can be seen, a deviation upward or downward from the optimum value of the minority carrier lifetime (*τ*n=2 ns) substantially affect the spectral distribution of the quantum efficiency only at wavelengths longer than ∼ 700 nm. In contrast, the deviation of the recombination

significant changes in the short wavelength region (shorter than ∼ 800 nm). Judging the curves in Fig. 12b, it might give the impression that the recombination at the front surface has more or less the same effect on the quantum efficiency over the entire spectrum, and it can induce doubt in the correctness of the physical model used. However, this is not so, a simple calcu‐

reduces the quantum efficiency only by 2-4% but as high as ∼ 22% in the range 350-400 nm. The concentration of uncompensated acceptors *N*a – *N*d impacts the entire spectrum, and is more appreciable. Moreover, when concentration of *N*a – *N*d is deviated by an order from the

cm/s in the spectral range 600-800 nm, surface recombination

velocity at the front surface of the Cu(In,Ga)Se2 layer from the value *S*<sup>f</sup>

1/2 has

affects the efficiency in a wide range of

=2×105 cm/s results in

The lifetime of minority carriers (electrons), determining their diffusion length *L*n=(*τ*n*D*n)

mobilities are much higher.

22 Solar Cells - New Approaches and Reviews

600-700 nm) of the spectrum.

The velocity of recombination at the front surface *S*<sup>f</sup>

and only for long lifetimes of electrons.

lation shows that for *S*<sup>f</sup>

modified when one of the parameters is varied.

=107

Figure 12. Effect of the electron lifetime n (a), the surface recombination velocity *S*f (b), the concentration of uncompensated acceptors *N*a – *N*d (c) and the thickness of the CdS film (d) on the spectral distribution of the quantum efficiency of CuInSe2 solar cell. **Figure 12.** Effect of the electron lifetime *τ*n (a), the surface recombination velocity *S*<sup>f</sup> (b), the concentration of uncom‐ pensated acceptors *N*a – *N*d (c) and the thickness of the CdS film (d) on the spectral distribution of the quantum effi‐ ciency of CuInSe2 solar cell.

optimum value 5×1015 cm–3, qualitative changes are observed in the spectrum, notably in the range *λ*=550-850 nm, increasing the efficiency with wavelength is replaced by its decay when *N*a – *N*d is increased.

Finally, the variation of the CdS layer thickness manifests itself only in the range of the fundamental absorption of this semiconductor, i.e., when *λ* < 500 nm.

Parameters giving best match for the spectral distribution of calculated and experimental QE of the studied cells are summarized in Table 2. Notice that in the investigated solar cells, the SCR width (0.4–0.6 μm) amounts to a small part of the thickness of the absorber layer and the recombination velocity *S*<sup>f</sup> is equal to ∼ 105 cm/s, i.e., *S*<sup>f</sup> is low as compared to thin-film CdS/ CdTe heterostructure. This is explained by the known peculiarity of CdS/CIGS heterojunction.

Back in the late 1980s, it was shown that the CIGS solar cells are insensitive to defects caused by a lattice mismatch or impurities at the CdS/CIGS interface. In fact, the lattice mismatch is rather small for CdS and CuInSe2 (∼ 1%) and weakly increases with the Ga content. In addition, the deposition of CdS on the treated and cleaned surface of the CIGS layer is characterized by pseudo-epitaxial growth, and the intermixing of the heterojunction constituents is observed even at relatively low-temperature processes [30, 31].


**Table 2.** Parameters of CuInSe2, CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se2 solar cells giving better match between measured and calculated data

A comparison of measured and calculated results presented in Fig. 11 shows that the theoret‐ ical model describes in detail the spectral distribution of the quantum efficiency of CIGS solar cells, which is important for further analysis of recombination losses. But the question arises, how this model can be applicable in polycrystalline material, with its inhomogeneity, recom‐ bination at the grain boundaries, etc. A possible explanation for the applicability of the model in question to efficient solar cells based on polycrystalline CIGS is that during the growth of the absorber layer and post-growth processing, recrystallization leading to grain growth and their coalescence occur. Also no less important is the fact that a structure in the form of ordered columns oriented perpendicular to the electrodes is created in the CIGS layer (see reviews [32] and references therein). One can assume that in a layer of columnar structure, collection of photogenerated charge occurs without crossing the grain boundaries. In addition, the scatter‐ ing and recombination on the lateral surfaces of the columns also have no significant effect due to the strong electric field in the barrier region.

Indeed, the width of the SCR in the studied solar cell is about 0.5 μm, and the electric field at a barrier height φbi of about 1 eV is higher than 104 V/cm. Under such conditions, the drift length of charge carriers with the mobility of 20-30 cm2 /(Vs), and lifetimes 10–9 s is several microns which is significantly greater than the width of SCR and makes recombination improbable. Outside the SCR, where the diffusion component of photocurrent is formed, the electric field does not exist, but due to the high absorption capacity of CIGS the vast majority of solar radiation is absorbed in the SCR. This is illustrated in Fig. 13 with the example of CuIn0.39Ga0.61Se2 (*E*g=1.36 eV) solar cell, where the quantum efficiency spectra along with the drift and diffusion components are shown. As expected, the diffusion component falls mostly on the long-wavelength part of spectra (*λ* longer than ∼ 600 nm) and its contribution to the quantum efficiency is quite insignificant.

Calculation given by Eq. (39) shows that for the parameters listed in Table 2, the contributions of the diffusion component in the photocurrent of CuInSe2, CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se solar cells is about 2, 4 and 8%, which are far inferior to the drift component. This significantly weakens the role of recombination at the grain boundaries.

**Figure 13.** Drift and diffusion components of the quantum efficiency of CuIn0.39Ga0.61Se2 solar cell and their sum.
