**5.2. Generation–recombination in space–charge region theory**

As mentioned previously, the theory of generation–recombination of charge carriers in the space-charge region (SCR) was developed for the linearly graded silicon p-n junction by Sah et al. [19] and modified and adapted to a Schottky diode taking into account the distribution of the potential and free carrier concentrations in the SCR [37,38].

The generation–recombination rate through a single level trap for nonequilibrium but steadystate conditions in the cross section *x* of the SCR at voltage *V* is determined by the Shockley-Read-Hall statistics:

$$\text{LI}(\mathbf{x}, V) = \frac{\mathbf{n}(\mathbf{x}, V)p(\mathbf{x}, V) - n\_{\mathrm{i}}^2}{\tau\_{\mathrm{po}}[n(\mathbf{x}, V) + n\_{\mathrm{I}}] + \tau\_{\mathrm{no}}[p(\mathbf{x}, V) + p\_{\mathrm{I}}]},\tag{47}$$

where *n*(*x,V*) and *p*(*x,V*) are the carrier concentrations in the conduction and valence bands within the SCR, respectively; *n*<sup>i</sup> is the intrinsic carrier concentration in the semiconductor; *τ*no=1/*σ*n*υ*th*N*<sup>t</sup> and *τ*po*=1/σ*p*υ*th*N*<sup>t</sup> are respectively the effective lifetimes of electrons and holes in the SCR (*τ*no and *τ*po are the *shortest* lifetimes for given *N*<sup>t</sup> and carrier capture cross sections of electrons and holes *σ*n and *σ*p, respectively [19]); *υ*th is the charge-carrier thermal velocity; *N*<sup>t</sup> is the concentration of generation–recombination centers. The quantities *n*1 and *p*1 in Eq. (47) are determined by the ionization energy of the generation–recombination center *E*<sup>t</sup> : *n*1=*N*cexp[−(*E*<sup>g</sup> − *E*<sup>t</sup> )/*kT*] and *p*1=*N*vexp(−*E*<sup>t</sup> /*kT*), where *N*c*=*2(*т*п*кТ/*2*πћ* <sup>2</sup> ) 3/*2* and *N*v*=*2(*m*р*кТ/*2*πћ* <sup>2</sup> ) 3/2 are the effective densities of state in the conduction and valence band, *m*n and *m*<sup>p</sup> are the effective electron and hole masses, respectively.

The expressions for the electron and hole concentrations in the SCR of Schottky diode involved in Eq. (47) in the chosen reference system take the form [39]

$$m(\mathbf{x}, V) = \mathbf{N}\_{\mathbf{v}} \exp\left[ -\frac{E\_{\mathcal{S}} - \Delta\mu - \varrho\left(\mathbf{x}, V\right) - qV}{kT} \right],\tag{48}$$

$$p(\mathbf{x}, V) = N\_{\mathbf{v}} \exp\left[-\frac{\Delta\_{\mathbf{v}} + \varrho(\mathbf{x}, V)}{kT}\right],\tag{49}$$

where ∆*µ* denotes the energy difference between the Fermi level and the top of the valence band in the bulk part of the CIGS layer, φ(*x*,*V*) is the potential distribution in the SCR at voltage *V*, given by the expression:

$$
\langle \boldsymbol{\varrho}(\mathbf{x}, \boldsymbol{V}) \rangle = (\boldsymbol{\varrho}\_{\text{bi}} - \boldsymbol{q} \boldsymbol{V}) \left( \mathbf{1} - \frac{\boldsymbol{x}}{\boldsymbol{W}} \right)^{2}, \tag{50}
$$

φbi is the height of the potential barrier in equilibrium for holes from the CIGS side related to the built-in (diffusion) voltage *V*bi by the equality *φ*o=*qV*bi, *W* is the width of the SCR given by Eq. (28).

The recombination current density under forward bias and the generation current under reverse bias are found by the integration of *U*(*x,V*) throughout the entire depletion layer

$$J = q \int\_0^W U(\mathbf{x}, V)d\mathbf{x}.\tag{51}$$

From the above equations one can obtain the exponential voltage dependence of the recom‐ bination current under forward bias.

Substitution of Eqs. (48)–(50) into Eq. (47) and simple manipulation yield the following expression for the generation–recombination rate:

$$U = \frac{n\_i (\tau\_{\rm no} \tau\_{\rm po})^{-1/2} \sinh(qV/2kT)}{\exp(-qV/2kT)\cosh[(E\_\rm g^\*-2E\_t)/2kT] + \cosh[(E\_\rm g^\*-2\Delta\mu - qV - 2q\text{(x,V)})/2kT]},\tag{52}$$

where *E*<sup>g</sup> \* =*E*<sup>g</sup> + *kT* ln (*τ*po*N*c)/ (*τ*no*N*v) .

We assume that CIGS contains a lot of shallow and deep impurities (defects) of donor and acceptor types. According to the Shockley-Read-Hall statistics the most effective generation– recombination centers are those whose levels are located near the middle of bandgap. Taking into account this, one can neglect the first term in the denominator of Eq. (52) and obtain the following expression for the current density using Eq. (51):

$$J = qn\_i \sinh(qV / 2kT) \Big| \int\_0^W f(\mathbf{x}, V)d\mathbf{x}.\tag{53}$$

where

cells with wide bandgap for the absorber (up to 1.5 eV) can be fabricated without shunting at

As mentioned previously, the theory of generation–recombination of charge carriers in the space-charge region (SCR) was developed for the linearly graded silicon p-n junction by Sah et al. [19] and modified and adapted to a Schottky diode taking into account the distribution

The generation–recombination rate through a single level trap for nonequilibrium but steadystate conditions in the cross section *x* of the SCR at voltage *V* is determined by the Shockley-

> t

po 1 no 1 (, )(, ) (, ) , [(, ) ] [(, ) ]

where *n*(*x,V*) and *p*(*x,V*) are the carrier concentrations in the conduction and valence bands

electrons and holes *σ*n and *σ*p, respectively [19]); *υ*th is the charge-carrier thermal velocity; *N*<sup>t</sup>

the concentration of generation–recombination centers. The quantities *n*1 and *p*1 in Eq. (47) are

effective densities of state in the conduction and valence band, *m*n and *m*<sup>p</sup> are the effective

The expressions for the electron and hole concentrations in the SCR of Schottky diode involved

é ù -D - m j = -ê ú ë û <sup>v</sup>

> é ù +j

2 i

and *τ*po*=1/σ*p*υ*th*N*<sup>t</sup> are respectively the effective lifetimes of electrons and holes in

) 3/*2*

Δ () ( ) exp , (49)

*nxV n pxV p* (47)

is the intrinsic carrier concentration in the semiconductor;

and carrier capture cross sections of

and *N*v*=*2(*m*р*кТ/*2*πћ* <sup>2</sup>

*kT* (48)

is

: *n*1=*N*cexp[−(*E*<sup>g</sup>

3/2 are the

)


*nxV pxV n UxV*

determined by the ionization energy of the generation–recombination center *E*<sup>t</sup>

( ) *n x,V*

, ( ) N exp , *<sup>g</sup> E x V qV*

*μ x,V p x,V N kT*

 = -ê ú ë û <sup>v</sup>

where ∆*µ* denotes the energy difference between the Fermi level and the top of the valence band in the bulk part of the CIGS layer, φ(*x*,*V*) is the potential distribution in the SCR at voltage

low voltages and the voltage drop across the series resistance at high currents.

**5.2. Generation–recombination in space–charge region theory**

of the potential and free carrier concentrations in the SCR [37,38].

t

the SCR (*τ*no and *τ*po are the *shortest* lifetimes for given *N*<sup>t</sup>

)/*kT*] and *p*1=*N*vexp(−*E*<sup>t</sup> /*kT*), where *N*c*=*2(*т*п*кТ/*2*πћ* <sup>2</sup>

in Eq. (47) in the chosen reference system take the form [39]

Read-Hall statistics:

36 Solar Cells - New Approaches and Reviews

*τ*no=1/*σ*n*υ*th*N*<sup>t</sup>

− *E*<sup>t</sup>

within the SCR, respectively; *n*<sup>i</sup>

electron and hole masses, respectively.

*V*, given by the expression:

$$f(\mathbf{x}, V) = \left\{ \cosh[(E\_g^\* - 2\Delta\mu - qV - 2\rho(\mathbf{x}, V))/2kT] \right\}^{-1}.\tag{54}$$

The function *f*(*x*,*V*) has a symmetric bell-shaped form, therefore without incurring noticeable errors, the integration in Eq. (53) may be replaced by the product of the maximum value of the 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 to *E*g – 2Δμ:

$$
\Delta \mathbf{x}\_{1/2} \approx W \left( \frac{\mathbf{E\_g} - 2\Delta \mu - qV}{2(\varrho\_{\rm bi} - qV)} \right)^{1/2} \cdot \frac{2 \arccos \mathbf{h} \mathbf{2}}{\mathbf{E\_g} - 2\Delta \mu - qV} = W \frac{2kT}{\sqrt{(\varrho\_{\rm bi} - qV)(\mathbf{E\_g} - 2\Delta \mu - qV)}}.\tag{55}
$$

So Eq. (53) in the integral form yields the exponential dependence of the recombination current on the applied voltage (sinh(*qV*/2*kT*)=exp(*qV*/2*kT*)/2) [37, 38]:

$$J \approx \frac{qn\_iW}{\sqrt{\tau\_{\rm no}\tau\_{\rm po}}} \frac{kT}{\sqrt{(\rho\_{\rm bi} - qV)(E\_{\rm g} - 2\Delta\mu - qV)}} \bigg[ \exp\left(\frac{qV}{2kT}\right) - 1\right].\tag{56}$$

Fig. 21 shows a comparison of the voltage dependence of the recombination currents in CIGS solar cell, for example, with *E*g=1.14 eV calculated using Eq. (51) and (56).

**Figure 21.** Comparison of the recombination currents in CIGS solar cell with 1.14 eV bandgap calculated from Eqs. (51) and (56).

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 because in deriving Eq. (56) the effect associated with *V* approaching to φbi/*q* did not take into account. Thus, when analyzing the *J–V* characteristic at higher voltages, only exact Eq. (51) should be applied.

In addition to the results shown in Fig. 21, it is appropriate to draw attention to the rather important fact. If we express the voltage dependence of the recombination current as *J*=*J*oexp(*qV*/*AkT*), the ideality factor *A* turns out to be slightly different from 2, close to 1.9. It is also appropriate to note that as early as the mid-1990s, the well-known scientists and experts came to the conclusion that the forward current in efficient thin-film CdS/CdTe solar cells is caused by recombination in the SCR of the absorber layer and it was shown that the ideality factor *A* in these devices coincides with the value 1.9 [40]. Comprehensive analysis and generalization of the experimental results obtained over many years confirm these results [41].

The difference of *A* from 2 is explained by the fact that according to Eq. (56), the dependence of the recombination current on the voltage is determined not only by the exponent exp(*qV*/ 2*kT*) but also by the pre-exponential factor (*E*g – 2Δμ – *qV*) –1/2, which somewhat accelerates an increase in current with *V*, and thus lowers the ideality factor.
