**4.1. Recombination losses at the absorber front surface**

pseudo-epitaxial growth, and the intermixing of the heterojunction constituents is observed

*τ***n (ns)** *N***a –***N***<sup>d</sup> (cm–3)**

*S***f (cm/s)**

V/cm. Under such conditions, the drift

/(Vs), and lifetimes 10–9 s is several

*d***CdS (nm)**

**φbi (eV)**

CuInSe2 1.04 0.60 5 2×1015 1×105 45 CuIn0.3Ga0.7Se 1.14 0.75 20 9×1015 5×105 50 CuIn0.67Ga0.3Se 1.36 0.95 2 5×1015 2×105 40

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

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

Indeed, the width of the SCR in the studied solar cell is about 0.5 μm, and the electric field at

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

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

even at relatively low-temperature processes [30, 31].

due to the strong electric field in the barrier region.

a barrier height φbi of about 1 eV is higher than 104

quantum efficiency is quite insignificant.

length of charge carriers with the mobility of 20-30 cm2

weakens the role of recombination at the grain boundaries.

*E***g (eV)**

**Solar cell**

24 Solar Cells - New Approaches and Reviews

and calculated data

To determine the recombination losses, we will calculate the photocurrent density by varying the parameters of solar cells such as the recombination velocity at the absorber surface, concentration of uncompensated acceptors and carrier lifetimes in the material, as well as the thickness of the absorber. Photocurrent density will be calculated by Eq. (19) using for *T*(*λ*) Eq. (39).

Until now, the internal quantum efficiency was calculated under zero bias, since it was necessary for comparison with the spectra measured at *V*=0. However, recombination losses should be calculated in the operating mode of the solar cell, i.e., at the voltage *V*m, which maximizes the generated electrical power. As will be shown in Section 5.3, these voltages *V*<sup>m</sup> are 0.36, 0.46 and 0.63 V for CuInSe2, CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se2 solar cells, respec‐ tively.

Consider how the photocurrent density *J* varies depending on the concentration of uncom‐ pensated acceptors at different recombination velocities at the front surface of the CIGS absorber. Note that according to Eq. (36), the recombination does not depend on the carrier lifetime in this case. Fig. 14 shows for CuIn0.39Ga0.61Se2 solar cell the dependences of *J* on *N*a – *N*d in the range 5×1014–1018 cm–3 and *S*<sup>f</sup> in the range 104 –107 cm/s and also at *S*<sup>f</sup> =0. The lower range of *N*a – *N*<sup>d</sup> is limited to 5×1014 cm–3, because at this values the width of the SCR becomes greater than the thickness of the absorber layer *d*=2 μm.

As seen in Fig. 14a, if the concentration of uncompensated acceptors exceeds 1017 cm–3, surface recombination does not reveals itself, but at lowering the *N*a – *N*d, the decrease in *J* becomes appreciable. At *S*<sup>f</sup> =104 cm/s, recombination practically does not reduce the current and slightly lowers it at the real values of *N*a – *N*d=5×1015 cm–3 and *S*<sup>f</sup> =2×105 cm/s for this solar cell. As our calculations show in Fig. 14b, for CuIn0.39Ga0.61Se solar cell, surface recombination reduces *J* by 0

1

2

Δ

*J*<sup>o</sup> / *J* (%) 3

4

60 80 100 120 140 *d*arc (nm)

Figure 6.

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

CuInSe2

160

30 (a) <sup>n</sup> = 1000 ns (b) 60 <sup>n</sup> = 0.5 ns **Figure 14.** (a) Photocurrent density *J* as a function of the concentration of uncompensated acceptors *N*a – *N*d at different recombination velocities *S*<sup>f</sup> at the front surface of the CIGS absorber in CuIn0.39Ga0.61Se solar cell. (b) Reduction in the photocurrent density, expressed as in percentage.

Figure 14.

1.8% for *N*a – *N*d=5×1015 cm–3. Similar calculations carried out for CuInSe2 and CuIn0.76Ga0.24Se2 solar cells show that recombination losses for them are equal to 2.1% and 1.9%, respectively. Relatively small recombination losses are explained by the above mentioned fact that the CIGS solar cells are insensitive to defects at the CdS/CIGS interface due to pseudo-epitaxial growth of the CIGS and the intermixing of the heterojunction constituents. 25 *J* (mA/cm2 ) 2 ns 10 ns 5 ns *J* /*J*(%) 20 40 2 ns

#### **4.2. Recombination losses in the SCR** <sup>n</sup> = 0.5 ns

20 ns

20

27

*J*

(mA/cm2

)

29

(a)

Recombination of photogenerated charge carriers in the SCR can be taken into account, using the well-known Hecht equation [33]: 15 10<sup>14</sup> 10<sup>15</sup> 10<sup>17</sup> 10<sup>16</sup> *N*<sup>a</sup> – *N*<sup>d</sup> (cm–3 10<sup>14</sup> 10<sup>15</sup> 10<sup>17</sup> 10<sup>16</sup> *N*<sup>a</sup> – *N*<sup>d</sup> (cm–3 ) 0 20 ns

)

*d* (m)

<sup>n</sup> = 1 ns

$$\eta\_{\rm H}(\mathbf{x}) = \frac{\lambda\_{\rm n}}{W} \left[ 1 - \exp\left( -\frac{\mathbf{x}}{\lambda\_{\rm n}} \right) \right] + \frac{\lambda\_{\rm p}}{W} \left[ 1 - \exp\left( -\frac{W - \mathbf{x}}{\lambda\_{\rm p}} \right) \right] \tag{40}$$

10 ns 5 ns

where *x* is the coordinate (*x* is measured from the CdS/CIGS interface), *λ*n and *λ*p are the drift lengths of electrons and holes in the SCR: 28 <sup>n</sup> = 20 ns 1.0 <sup>n</sup> = 20 ns <sup>n</sup> = 10 ns

$$
\mathcal{A}\_{\rm n} = \mu\_{\rm n} F \tau\_{\rm no} \,\prime \tag{41}
$$

$$
\mathcal{A}\_{\mathsf{p}} = \mu\_{\mathsf{p}} \mathsf{F} \tau\_{\mathsf{p}\circ\prime} \tag{42}
$$

*d* (m)

<sup>n</sup> = 2 ns

*F* is the electric field strength; *τ*no and *τ*po are the lifetimes of electrons and holes in the SCR, respectively, which we will accept to be equal to *τ*n assuming that the CIGS absorber is a highly doped semiconductor. 0 1 2 3 4 5 26 0 1 2 3 4 5 0

Figure 16.

In CdS/CIGS heterostructure, the electric field is not uniform, but the problem of the nonuni‐ formity is simplified, since the field strength *F* decreases in a Schottky diode linearly with the *x* coordinate. In this case, *F* in the expressions (41) and (42) can be replaced by the average values of *F* in the sections (0,*x*) and (*x*,*W*) for electrons and holes, respectively, i.e.,

$$F(0, \chi) = \frac{q\_{\rm bi} - qV}{qW} \left( 2 - \frac{\chi}{W} \right) \tag{43}$$

$$F(\mathbf{x}, \mathcal{W}) = \frac{q\_{\rm bi} - qV}{q\mathcal{W}} \left(1 - \frac{\mathbf{x}}{\mathcal{W}}\right). \tag{44}$$

Evidently, charge collection efficiency in the *dx* interval of the SCR is *η*H*α*exp(–*αx*)*dx* and then the charge-collection efficiency for the wavelength *λ*<sup>i</sup> is determined by expression:

$$\mathfrak{h}\_{\rm H}(\mathscr{A}\_{\rm i}) = \int\_{0}^{W} \eta\_{\rm H}(\mathbf{x}) \alpha\_{\rm i} \exp(-\alpha\_{\rm i} \cdot \mathbf{x}) d\mathbf{x}.\tag{45}$$

Fig. 15 shows the photocurrent density *J* in CuIn0.39Ga0.61Se2 as a function of the concentration of uncompensated acceptors *N*a – *N*d at different carrier lifetimes given by the equation:

1.8% for *N*a – *N*d=5×1015 cm–3. Similar calculations carried out for CuInSe2 and CuIn0.76Ga0.24Se2 solar cells show that recombination losses for them are equal to 2.1% and 1.9%, respectively. Relatively small recombination losses are explained by the above mentioned fact that the CIGS solar cells are insensitive to defects at the CdS/CIGS interface due to pseudo-epitaxial growth

*J* /*J*(%)

(a)

Figure 14.

**Figure 14.** (a) Photocurrent density *J* as a function of the concentration of uncompensated acceptors *N*a – *N*d at different

0

5

*J*/*J* (%) 10

15

*S*<sup>f</sup> = 107 cm/s

106 cm/s

105 cm/s

104 cm/s

at the front surface of the CIGS absorber in CuIn0.39Ga0.61Se solar cell. (b) Reduction in the

60 80 100 120 140 *d*arc (nm)

Figure 6.

(a)

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

CuInSe2

160

10<sup>14</sup> 10<sup>15</sup> 10<sup>16</sup> 10<sup>17</sup> 10<sup>18</sup>

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

<sup>n</sup> = 0.5 ns

10<sup>14</sup> 10<sup>15</sup> 10<sup>17</sup> 10<sup>16</sup>

<sup>n</sup> = 20 ns

> <sup>n</sup> = 10 ns

> > <sup>n</sup> = 5 ns

> > > <sup>n</sup> = 2 ns <sup>n</sup> = 1 ns

)

(b)

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

)

(b)

(b)

Recombination of photogenerated charge carriers in the SCR can be taken into account, using

l

Figure 15.

where *x* is the coordinate (*x* is measured from the CdS/CIGS interface), *λ*n and *λ*p are the drift

*F* is the electric field strength; *τ*no and *τ*po are the lifetimes of electrons and holes in the SCR, respectively, which we will accept to be equal to *τ*n assuming that the CIGS absorber is a highly

Figure 16.

*J/J* (%)

0

0.5

1.0

n p

1.5

0

20

40

60

é ù é ù æ ö æ ö - = --+ -- ê ú ê ú ç ÷ ç ÷ ç ÷ ê ú ç ÷ ê ú ë û è ø ë û è ø

( ) 1 exp 1 exp , *<sup>x</sup> W x*

 l

2 ns

= n n no *F* , (41)

= p p po *F* , (42)

0 1 2 3 4 5

*d* (m)

*W W* (40)

10 ns 5 ns

20 ns

l

n p

l mt

<sup>n</sup> = 20 ns

> l mt

of the CIGS and the intermixing of the heterojunction constituents.

)

<sup>n</sup> = 5 ns

0

10<sup>14</sup> 10<sup>15</sup> 10<sup>16</sup> 10<sup>17</sup> 10<sup>18</sup>

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

)

*S*<sup>f</sup> = 107 cm/s

106 cm/s

105 cm/s 104 cm/s

26 Solar Cells - New Approaches and Reviews

*S*<sup>f</sup> = 0

1

2

Δ

*J*<sup>o</sup> / *J* (%) 3

4

**4.2. Recombination losses in the SCR**

photocurrent density, expressed as in percentage.

<sup>n</sup> = 0.5 ns

2 ns

10 ns 5 ns

20 ns <sup>n</sup> = 1000 ns

20

15

20

*J*

(mA/cm2

)

25

30

recombination velocities *S*<sup>f</sup>

22

24

*J*

(mA/cm2

)

26

28

30

the well-known Hecht equation [33]:

h

(a)

doped semiconductor.

26

27

*J*

(mA/cm2

)

28

29

H

*x*

lengths of electrons and holes in the SCR:

l

0 1 2 3 4 5

<sup>n</sup> = 1 ns

*d* (m)

10<sup>14</sup> 10<sup>15</sup> 10<sup>17</sup> 10<sup>16</sup>

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

$$J = q \sum\_{\mathbf{i}} \frac{\Phi\_{\mathbf{i}}(\boldsymbol{\lambda}\_{\mathbf{i}})}{hv\_{\mathbf{i}}} T(\boldsymbol{\lambda}\_{\mathbf{i}}) \eta\_{\mathbf{H}}(\boldsymbol{\lambda}\_{\mathbf{i}}) \Delta \boldsymbol{\lambda}\_{\mathbf{i}}.\tag{46}$$

As can be seen in Fig. 15a, when *N*a – *N*d > 3×1016 cm–3 (*W* < 0.2-0.3 μm), the curves calculated for different lifetimes of charge carriers *τ*n=*τ*no=*τ*po coincide, but as the SCR width increases, the curves diverge and become more pronounced when *τ*n is less. The reduction of *J* due to recombination can be found by subtracting the current at a given *τ*<sup>n</sup> from the value of current *j* <sup>o</sup> when recombination does not occur. The latter is possible for a sufficiently large lifetime of charge carriers. A simple calculation shows that when *τ*n > 500-600 ns, the current becomes independent of *τ*n, therefore the values of *j* o at different *N*a – *N*d were found for *τ*n=1000 ns (the curve *j* o vs. *N*a – *N*d is also shown in Fig. 15a).

Fig. 15b shows the reduction in *J* due to recombination in the SCR and its dependence on the *N*a – *N*d and *τ*n. As seen, when *N*a – *N*d < 1016 cm–3 (*W* > 1 μm), the recombination losses increase rapidly up to 60% with decreasing *N*a – *N*d and *τ*n. The data presented in this figure allow determining the recombination losses for any values of *N*a – *N*d and *τ*n. For the real carrier lifetime of 2 ns and *N*<sup>a</sup> *– N*d=5×1015 cm–3, which are characteristic of CuIn0.39Ga0.61Se solar cell, the recombination losses amount to 1.0%, whereas for CuIn0.76Ga0.24Se2 solar cell the losses is *S*<sup>f</sup> = 107 cm/s

106 cm/s

105 cm/s 104 cm/s

*S*<sup>f</sup> = 0

20

22

24

*J*

)

(a)

(mA/cm2

)

26

28

30

0

)

1

2

Δ

*J*<sup>o</sup> / *J* (%) 3

4

Figure 14.

0

5

*J*/*J* (%) 10

15

*S*<sup>f</sup> = 107 cm/s

106 cm/s

105 cm/s

104 cm/s

60 80 100 120 140 *d*arc (nm)

Figure 6.

(a)

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

CuInSe2

160

10<sup>14</sup> 10<sup>15</sup> 10<sup>16</sup> 10<sup>17</sup> 10<sup>18</sup>

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

)

<sup>n</sup> = 5 ns (b)

29 1.5 (b) **Figure 15.** (a) Photocurrent density *J* as a function of the concentration of uncompensated acceptors *N*a – *N*d for differ‐ ent carrier lifetimes *τ*<sup>n</sup> in the CuIn0.39Ga0.61Se2 absorber. (b) Reduction of the photocurrent density expressed in percent‐ age.

Figure 15.

only 0.1% because of higher values *τ*n=20 ns and *N*<sup>a</sup> *– N*d*=*9×1015 cm–3. For CuInSe2 solar cell, the recombination losses due to recombination in the SCR amount to 0.7%. 28 <sup>n</sup> = 20 ns 1.0 <sup>n</sup> = 20 ns <sup>n</sup> = 10 ns

#### **4.3. Recombination losses at back surface and neutral part of absorber** (mA/cm2<sup>n</sup> = 5 ns (%)

Useful information about the effect of recombination at the back surface of the solar cell and the neutral part of the absorber on the photocurrent *J* can be obtained by studying the dependence of *J* on the absorber thickness *d*. If *d* is large, the effect of recombination at the back surface is imperceptible, but when the rear surface is close to the SCR by a distance comparable to the diffusion length of electrons, the role of recombination increases. This leads also to a decrease in the diffusion component of the photocurrent. 0 1 2 3 4 5 26 27 *J* <sup>n</sup> = 1 ns 0 1 2 3 4 5 0 0.5 *J/J*<sup>n</sup> = 2 ns <sup>n</sup> = 1 ns

Subtracting the currents calculated for the recombination velocity *S*b=107 cm/s and *S*b=0, one can obtain the change of *J* due to recombination at the back surface. Obviously, for longer minority-carrier lifetime, recombination at the back surface of the absorber is intensified and manifests itself also at larger absorber thickness. *d* (m) *d* (m) Figure 16.

Fig. 16a shows the dependences of photocurrent *J* on the thickness *d* of the CuIn0.39Ga0.61Se2 absorber calculated for *S*b=107 cm/s and *S*b=0 and at different lifetimes of electrons. As it is shown by dashed lines, when the CuIn0.39Ga0.61Se2 layer becomes thinner, the photocurrent for *S*b=0 first slightly grows and then rapidly decreases. The observed current growth can be explained by the fact that the diffusion component of the photocurrent is determined by the derivative of the excess concentration of photogenerated electrons Δ*n/dx* [22]. The derivative may be larger than that in the absence of recombination at the surface, but when *d* approaches the cross section *x=W*, the diffusion component eventually decreases due to reduced the Δ*n* absolute value and next Δ*n*=0 when *x=W*. This also explains why the dependence of Δ*J / J* on *d* shown in Fig. 16b is described by a curve with a maximum.

10<sup>14</sup> 10<sup>15</sup> 10<sup>16</sup> 10<sup>17</sup> 10<sup>18</sup>

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

<sup>n</sup> = 0.5 ns

2 ns

10 ns 5 ns

20 ns

)

(b)

(b)

Figure 15.

0

10<sup>14</sup> 10<sup>15</sup> 10<sup>16</sup> 10<sup>17</sup> 10<sup>18</sup>

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

)

)

*S*<sup>f</sup> = 107 cm/s

106 cm/s

105 cm/s 104 cm/s

*S*<sup>f</sup> = 0

60 80 100 120 140 *d*arc (nm)

Figure 6.

0

5

*J*/*J* (%)

Figure 14.

*J* /*J*(%)

20

40

60

(a)

10

15

*S*<sup>f</sup> = 107 cm/s

106 cm/s

105 cm/s

104 cm/s

(a)

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

CuInSe2

160

1

2

Δ

20

15

20

<sup>n</sup> = 0.5 ns

2 ns

10 ns 5 ns

20 ns <sup>n</sup> = 1000 ns

*J*

(mA/cm2

)

25

30

22

24

*J*

(mA/cm2

)

26

28

30

*J*<sup>o</sup> / *J* (%) 3

4

**Figure 16.** (a) Dependences of the photocurrent density on thickness of the CuIn0.39Ga0.61Se2 layer calculated for differ‐ ent electron lifetimes with and without taking into account recombination at the back surface of the absorber (solid and dashed lines, respectively), (b) Decrease in photocurrent density due to recombination at the back surface calculated for different electron lifetimes.

Figure 16.

only 0.1% because of higher values *τ*n=20 ns and *N*<sup>a</sup> *– N*d*=*9×1015 cm–3. For CuInSe2 solar cell,

Figure 15.

**Figure 15.** (a) Photocurrent density *J* as a function of the concentration of uncompensated acceptors *N*a – *N*d for differ‐ ent carrier lifetimes *τ*<sup>n</sup> in the CuIn0.39Ga0.61Se2 absorber. (b) Reduction of the photocurrent density expressed in percent‐

0

20

40

60

*J* /*J*(%)

Useful information about the effect of recombination at the back surface of the solar cell and the neutral part of the absorber on the photocurrent *J* can be obtained by studying the dependence of *J* on the absorber thickness *d*. If *d* is large, the effect of recombination at the back surface is imperceptible, but when the rear surface is close to the SCR by a distance comparable to the diffusion length of electrons, the role of recombination increases. This leads also to a

*J/J* (%)

Subtracting the currents calculated for the recombination velocity *S*b=107 cm/s and *S*b=0, one can obtain the change of *J* due to recombination at the back surface. Obviously, for longer minority-carrier lifetime, recombination at the back surface of the absorber is intensified and

Figure 16.

0

0.5

1.0

1.5

Fig. 16a shows the dependences of photocurrent *J* on the thickness *d* of the CuIn0.39Ga0.61Se2

shown by dashed lines, when the CuIn0.39Ga0.61Se2 layer becomes thinner, the photocurrent for *S*b=0 first slightly grows and then rapidly decreases. The observed current growth can be explained by the fact that the diffusion component of the photocurrent is determined by the derivative of the excess concentration of photogenerated electrons Δ*n/dx* [22]. The derivative may be larger than that in the absence of recombination at the surface, but when *d* approaches the cross section *x=W*, the diffusion component eventually decreases due to reduced the Δ*n* absolute value and next Δ*n*=0 when *x=W*. This also explains why the dependence of Δ*J / J* on

cm/s and *S*b=0 and at different lifetimes of electrons. As it is

the recombination losses due to recombination in the SCR amount to 0.7%.

**4.3. Recombination losses at back surface and neutral part of absorber**

<sup>n</sup> = 5 ns

<sup>n</sup> = 20 ns

decrease in the diffusion component of the photocurrent.

0 1 2 3 4 5

<sup>n</sup> = 1 ns

*d* (m)

0

10<sup>14</sup> 10<sup>15</sup> 10<sup>16</sup> 10<sup>17</sup> 10<sup>18</sup>

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

10<sup>14</sup> 10<sup>15</sup> 10<sup>17</sup> 10<sup>16</sup>

)

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

)

*S*<sup>f</sup> = 107 cm/s

106 cm/s

105 cm/s 104 cm/s

*S*<sup>f</sup> = 0

60 80 100 120 140 *d*arc (nm)

Figure 6.

0

5

*J*/*J* (%)

Figure 14.

(a)

10

15

*S*<sup>f</sup> = 107 cm/s

106 cm/s

105 cm/s

104 cm/s

(a)

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

CuInSe2

160

10<sup>14</sup> 10<sup>15</sup> 10<sup>16</sup> 10<sup>17</sup> 10<sup>18</sup>

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

<sup>n</sup> = 0.5 ns

2 ns

10 ns 5 ns

20 ns

10<sup>14</sup> 10<sup>15</sup> 10<sup>17</sup> 10<sup>16</sup>

<sup>n</sup> = 20 ns

> <sup>n</sup> = 10 ns

> > <sup>n</sup> = 5 ns

> > > <sup>n</sup> = 2 ns <sup>n</sup> = 1 ns

0 1 2 3 4 5

*d* (m)

)

(b)

*N*<sup>a</sup> – *N*<sup>d</sup> (cm–3

)

(b)

(b)

1

2

Δ

20

15

26

27

*J*

(mA/cm2

)

age.

28

29

20

<sup>n</sup> = 0.5 ns

(a)

2 ns

10 ns 5 ns

20 ns <sup>n</sup> = 1000 ns

28 Solar Cells - New Approaches and Reviews

*J*

(mA/cm2

)

25

30

22

24

*J*

(mA/cm2

)

26

28

30

*J*<sup>o</sup> / *J* (%) 3

4

*d* shown in Fig. 16b is described by a curve with a maximum.

manifests itself also at larger absorber thickness.

absorber calculated for *S*b=107

As can be seen from Fig. 16b, the decrease in photocurrent does not exceed 1.5% even with lifetimes of electrons 20 ns. With such low recombination losses, the creation of a heavily doped layer adjacent to the back contact as it is proposed in CdS/CdTe solar cells [7] or to form a bandgap gradient outside the SCR in the CIGS absorber [8] to reduce the negative impact of recombination at the rear surface of the absorber seems to be unreasonable. As previously mentioned, a very small fraction of carriers taking part in the photocurrent formation (2–8%) falls on the neutral part of the studied CIGS solar cells that also should be borne in mind.

Of course, recombination of photogenerated carriers happens not only at the back surface of the absorber, but also in whole neutral part (outside the SCR). The losses due to such recom‐ bination can be found as the difference between photocurrent at: (i) real electron lifetime and recombination velocity at the rear surface and (ii) large value of the electron lifetime when recombination can be ignored (*τ*n=1000 ns) and *S*b=0. As the calculations show, such losses are 2.5, 5.0 and 2.9% for CuInSe2, CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se2 solar cells, respectively. It is necessary to emphasize that recombination in the neutral part of the absorber and at the rear surface are not additive. But the calculation results for the neutral part, for example, with and without recombination at the rear surface of the absorber differ slightly. Nevertheless the determination of the combined influence of recombination in the neutral part of the absorber and at its rear surface is more correct.

The calculated values of *all types of recombination losses* along with the corresponding decrease in the photocurrent *J* (given in brackets) are summarized in Table 3. Analysis of obtained results allows us to formulate some recommendations to improve the photoelectric conversion efficiency in the solar cells studied.

The total recombination losses in CuInSe2, CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se2 solar cells amount to 5.4, 7.0 and 4.1%, respectively. It can be assumed that the charge collection efficiency of the photogenerated charge in the absorber is 94.6, 93.0 and 95.9%, respectively. Having these data, it is worth to analyze the possibility of reducing the recombination losses and improving the charge collection efficiency that we will make with an example of CuIn0.76Ga0.24Se2 solar cell.


**Table 3.** Recombination and the corresponding *J* losses in CIGS solar cells

According to Eq. (36), the recombination losses at the front surface can be lowered by increasing the hole diffusion coefficient *D*p=*kT*μp/*q*, i.e., increasing the mobility of holes μp. The results of calculation from Eq. (19) show that increasing the hole mobility from 30 to 300 cm2 /(Vs) (this is real according to the literature data) reduces such losses from 1.9 to 0.3%.

Significant improvement of the charge collection efficiency can be achieved by increasing the lifetime of electrons, which is equivalent to an increase of electron mobility since the diffusion length *L*n is equal to (*τ*n*D*n) 1/2. Increase in electron mobility also by an order of magnitude from 20 to 200 cm2 /(V s) leads to a reduction of the recombination losses in the neutral part of the absorber and at its rear surface from 5.0 to 1.1%.

The recombination losses in the SCR 0.1% at large electron and holes mobilities approach to zero since the drift lengths of carries are proportional to the their mobilities. Thus, due to a real increase of the mobility of electrons and holes by one order of magnitude the charge collection efficiency improves from 93.0 to 98.6%.

An even greater improvement of the charge collection efficiency can be achieved by extending the SCR, which leads to absorption of a greater part of the radiation in the SCR and hence the better collection of the photogenerated charge. However, one should keep in mind that when the electron lifetime and mobility increase, the diffusion length becomes longer than the absorber thickness. This weakens the desired effect and in fact the charge collection efficiency for *d*=2 μm is 96.2%.

There is also a positive impact for a higher carrier lifetime and expanded SCR since it leads to a decrease in the forward recombination current. Analyzing the electrical characteristics of the solar cell, it is not difficult to show that this causes enhancing the open circuit voltage (see the next sections).

The total recombination losses in CuInSe2, CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se2 solar cells amount to 5.4, 7.0 and 4.1%, respectively. It can be assumed that the charge collection efficiency of the photogenerated charge in the absorber is 94.6, 93.0 and 95.9%, respectively. Having these data, it is worth to analyze the possibility of reducing the recombination losses and improving the charge collection efficiency that we will make with an example of CuIn0.76Ga0.24Se2 solar

2.2% (0.9 mA/cm2

0.7% (0.4 mA/cm2

2.5% (1.0 mA/cm2

0.2% (< 0.1 mA/cm2

)

)

)

)

According to Eq. (36), the recombination losses at the front surface can be lowered by increasing the hole diffusion coefficient *D*p=*kT*μp/*q*, i.e., increasing the mobility of holes μp. The results of

Significant improvement of the charge collection efficiency can be achieved by increasing the lifetime of electrons, which is equivalent to an increase of electron mobility since the diffusion

The recombination losses in the SCR 0.1% at large electron and holes mobilities approach to zero since the drift lengths of carries are proportional to the their mobilities. Thus, due to a real increase of the mobility of electrons and holes by one order of magnitude the charge

An even greater improvement of the charge collection efficiency can be achieved by extending the SCR, which leads to absorption of a greater part of the radiation in the SCR and hence the better collection of the photogenerated charge. However, one should keep in mind that when the electron lifetime and mobility increase, the diffusion length becomes longer than the absorber thickness. This weakens the desired effect and in fact the charge collection efficiency

There is also a positive impact for a higher carrier lifetime and expanded SCR since it leads to a decrease in the forward recombination current. Analyzing the electrical characteristics of the

calculation from Eq. (19) show that increasing the hole mobility from 30 to 300 cm2

is real according to the literature data) reduces such losses from 1.9 to 0.3%.

**Losses in solar cell CuInSe2 CuIn0.76Ga0.24Se2 CuIn0.39Ga0.61Se2**

)

)

)

)

0.2% (< 0.1 mA/cm2

1.0% (0.3 mA/cm2

2.9% (1.0 mA/cm2

) 4.1% (1.4 mA/cm2

0.1% (< 0.1 mA/cm2

)

)

/(Vs) (this

)

)

)

1.9% (0.7 mA/cm2

5.0% (1.8 mA/cm2

1.0% (0.4 mA/cm2

1/2. Increase in electron mobility also by an order of magnitude from

/(V s) leads to a reduction of the recombination losses in the neutral part of the

) 7.0% (2.5 mA/cm2

0.1% (< 0.1 mA/cm2

cell.

Front surface Space-charge region Neutral part of absorber and back surface Only back surface

**Origin of recombination losses**

30 Solar Cells - New Approaches and Reviews

length *L*n is equal to (*τ*n*D*n)

20 to 200 cm2

for *d*=2 μm is 96.2%.

Total recombination losses 5.4% (2.3 mA/cm2

absorber and at its rear surface from 5.0 to 1.1%.

collection efficiency improves from 93.0 to 98.6%.

**Table 3.** Recombination and the corresponding *J* losses in CIGS solar cells

Some useful information can be obtained from the spectral distribution of the reflection, absorption and recombination losses. Fig. 17 shows the distribution of these losses over the whole spectrum for CuIn0.76Ga0.24Se2 solar cell obtained from the results given above.

**Figure 17.** Illustration of spectral distribution of the reflection, absorption and recombination losses for CuIn0.76Ga0.24Se2 solar cell (*E*g=1.14 eV).

As clearly seen, the recombination losses are considerably less than those caused by reflection and absorption in the ZnO layer and especially in the CdS layer. The radiation in the range *hν* < *E*g gives a small contribution to the solar cell efficiency due to insufficient absorptivity of CIGS in this range, whereas at *hν* > *E*g slight decrease of the absorptivity compared to 100% takes place only when the photon energy is very close to *E*g.
