*3.4.2. Open-circuit voltage (Voc) calculation*

2 2

*s a*

 n

*<sup>U</sup> <sup>f</sup> U U* <sup>=</sup> <sup>+</sup> (44)

(45)

(47)

(48)

n n

 n

(43)

(46)

2

*g*

¥

n

ò

1

With:

p

66 Solar Cells - New Approaches and Reviews

2

n n

exp(( ) / ) 1

radiative and radiative recombination rates respectively, *fRR* is defined by:

*RR*

f

2

*c*

 p

 p

Therefore the current density expression becomes:

*s*

j

*a*

j

*c*

j

2

*c*

 p 2

*g*

Under dark condition and zero bias the current density must be null, then:

*<sup>J</sup> <sup>f</sup>* =Þ = j

*<sup>q</sup> J V qCf <sup>V</sup> <sup>f</sup>* = -- ff

*<sup>d</sup> V c*

ò

¥

n

*g*

n

ò

*g*

<sup>=</sup> -

<sup>=</sup> -

¥

¥

n

ò

*d*

*RR c*

n

<sup>ù</sup> - ´ <sup>ú</sup> - - úû

n

*f h qV kT*

( ) (2 / ) (1 ) exp( / ) 1 exp( / ) 1

<sup>=</sup> <sup>é</sup> ´ +- ´ <sup>ë</sup> - -

ò ò

*d d JV q c Cf C f h kT h kT*

n n

n

*g g*

with reference to the solar cell configuration shown in figure 3, *Ts*, *Ta* and *Tc* are the respective temperatures of sun, ambient background and solar cell. As defined previously, *C* and *f* are the concentration factor and the sun geometrical factor, while *fRR* represents the fraction of the radiative recombination rate or radiative recombination efficiency. If *UNR* and *URR* are the non-

*RR*

*RR NR*

The current density formula (43) can be rewritten in a more compact form as follows:

( ) (1 ) ( ) *s ac*

 ff

2

n n

exp(( ) / ) 1

*RR*

 f

 j

*h qV kT*

*c*

 f

*<sup>q</sup> J V qCf q C f <sup>V</sup>*

2

n n

n

n n

n

*d*

(2 / ) a exp( / ) 1

*s*

(2 / ) b exp( / ) 1

*a*

( ) (2 / ) c

<sup>1</sup> (0) 0 (0) *a c*

( ) ( ) ( ) (0) ( ) *sa c c RR*

n

2

*h kT*

<sup>=</sup> - -

*h kT d*

= +- -

*RR*

*f*

¥ ¥

At open circuit condition electron-hole pairs are continually created as a result of the photon flux absorption, the only mechanism to counter balance this non-equilibrium condition is

**Figure 11.** The maximum short-circuit current density against the energy band-gap of the solar cell, using the AM1.5G spectrum with the blackbody spectrum at *Ts*=6000°*K*.

recombination. Non-radiative recombination could be eliminated, whereas radiative recom‐ bination has a direct impact on the cell efficiency and particularly on the open circuit voltage. The radiative current as the rate of radiative emission increases exponentially with the bias subtracts from the current delivered to the load by the cell. At open circuit condition, external photon emission is part of a necessary and unavoidable equilibration process [15]. The maximum attainable *Voc* corresponds to the condition where the cell emits as many photons as it absorbs. The open circuit voltage of a solar cell can be found by taking the band gap energy and accounting for the losses associated with various sources of entropy increase. Often, the largest of these energy losses is due to the entropy associated with radiative recombination.

In the case where *qV* is several *kT* smaller than *h νg*, *φc(V)* could be approximated by:

$$\phi\_c(V) \equiv (2\pi/c^2) \left( \int\_{\nu\_c}^{\nu} \frac{\nu^2 d\nu}{\exp(\langle h\nu \rangle / kT\_c) - 1} \right) \exp\left(\frac{qV}{kT\_c}\right) = \phi\_c(0) \exp\left(\frac{qV}{kT\_c}\right) \tag{50}$$

The current density expression becomes then:

$$f(V) = q \mathbb{C} f(\phi\_s - \phi\_a) - \frac{q}{f\_{RR}} \phi\_c(0) \left( \exp\left(\frac{qV}{kT\_c}\right) - 1 \right) \tag{51}$$

The open circuit voltage can be deduced directly from this expression as:

Theoretical Calculation of the Efficiency Limit for Solar Cells http://dx.doi.org/10.5772/58914 69

$$0 = C f(\phi\_s - \phi\_a) - \frac{1}{f\_{RR}} \phi\_c(0) \left( \exp\left(\frac{qV\_{\alpha c}}{kT\_c}\right) - 1 \right) \tag{52}$$

$$V\_{oc} = \frac{kT\_c}{q} \ln\left(\frac{\mathbb{C}f(\phi\_s - \phi\_a)}{\phi\_c(0)} f\_{RR} + 1\right) \tag{53}$$

The open circuit voltage is determined entirely by two factors; the concentration rate of solar radiation *C* (*C* ≥1) and radiative recombination rate *fRR* (*fRR* ≤1).

$$\begin{split} V\_{oc} & \approx \frac{kT\_c}{q} \ln \left( \frac{\mathbb{C}f(\phi\_s - \phi\_a)}{\phi\_c(0)} f\_{RR} \right) \\ &= \frac{kT\_c}{q} \ln \left( \frac{f(\phi\_s - \phi\_a)}{\phi\_c(0)} \right) - \frac{kT\_c}{q} \ln \left( \frac{1}{C} \right) - \frac{kT\_c}{q} \ln \left( \frac{1}{f\_{RR}} \right) \end{split} \tag{54}$$

Radiative recombination has a critical role to play, if the created photons are re-emitted out of the cell, which corresponds to low optical losses, the open circuit voltage and consequently the cell efficiency approach the SQ limit. Therefore the limiting factor for high *Voc* (efficiency) is the external fluorescence efficiency of the cell as far as radiative recombination is concerned. Since the escape cone is in general low, efficient external emission involves repeated escape attempts and this is ensured by perfect light trapping techniques. In this case the created photons are allowed to be reabsorbed and reemitted again until they coincide with the escape cone, reaching high external fluorescence efficiency [15, 18-19].

recombination. Non-radiative recombination could be eliminated, whereas radiative recom‐ bination has a direct impact on the cell efficiency and particularly on the open circuit voltage. The radiative current as the rate of radiative emission increases exponentially with the bias subtracts from the current delivered to the load by the cell. At open circuit condition, external photon emission is part of a necessary and unavoidable equilibration process [15]. The maximum attainable *Voc* corresponds to the condition where the cell emits as many photons as it absorbs. The open circuit voltage of a solar cell can be found by taking the band gap energy and accounting for the losses associated with various sources of entropy increase. Often, the largest of these energy losses is due to the entropy associated with radiative recombination.

**Figure 11.** The maximum short-circuit current density against the energy band-gap of the solar cell, using the AM1.5G

In the case where *qV* is several *kT* smaller than *h νg*, *φc(V)* could be approximated by:

*<sup>d</sup> qV qV V c*

( ) ( ) (0) exp 1 *sa c*

 f

æ ö æ ö = -- ç ÷ ç ÷ - ç ÷ è ø è ø

*<sup>q</sup> qV J V qCf <sup>f</sup> kT*

ff

The open circuit voltage can be deduced directly from this expression as:

*cc c*

@ = ç ÷ ç÷ ç÷ - è ø èø èø <sup>ò</sup> (50)

f

(51)

*h kT kT kT*

*RR c*

æ ö ¥ æö æö

2 <sup>2</sup> ( ) (2 / ) exp (0)exp *<sup>g</sup>* exp(( ) / ) 1 *c c*

n n

n

n

 p

spectrum with the blackbody spectrum at *Ts*=6000°*K*.

68 Solar Cells - New Approaches and Reviews

The current density expression becomes then:

f

So dominant radiative recombination is required to reach high *Voc* and this is not sufficient to reach the SQ limit, the other barrier is to get the generated photons out of the cell and this is limited by the external fluorescence efficiency *ηfex*. Hence, the external fluorescence efficien‐ cy*ηfex*, is introduced in the expression of *Voc* and *fRR* is multiplied by *ηfex*, (*ηfex* ≤ 1) then:

$$V\_{oc} = \frac{kT\_c}{q} \ln\left(\frac{f(\phi\_s - \phi\_a)}{\phi\_c(0)}\right) - \frac{kT\_c}{q} \ln\left(\frac{1}{C}\right) - \frac{kT\_c}{q} \ln\left(\frac{1}{f\_{RR}\eta\_{fix}}\right) \tag{55}$$

If we define a maximum ideal open circuit voltage value *Voc,max* for fully concentrated solar radiation (*C*=*Cmax*=1/*f*) and when the radiative recombination is the only loss mechanism with maximum external fluorescence efficiency (i.e, *fRRηfex*=1), then:

$$V\_{oc} = V\_{oc,\text{max}} - \frac{kT\_c}{q} \ln\left(\frac{\mathcal{C}\_{\text{max}}}{\mathcal{C}}\right) - \frac{kT\_c}{q} \ln\left(\frac{1}{f\_{\text{RR}} \eta\_{\text{fix}}}\right) \tag{56}$$

It is worth mentioning that (56) is not an exact evaluation of *Voc*. As shown in figure 12.b equation (56) for narrow band-gap semiconductors yields wrong values of *Voc,max* (above *Eg/q* line), acceptable values are obtained only for *Eg* greater than 2 eV, where it coincides with the result obtained when solving numerically (45) for *J*(*V*)=0.

From this figure one can say that taking *Voc,max*=*Eg/q* is a much better approximation; thus, instead of (56) we can use the following approximation:

$$V\_{oc} = \frac{E\_g}{q} - \frac{kT\_c}{q} \ln\left(\frac{C\_{\text{max}}}{C}\right) - \frac{kT\_c}{q} \ln\left(\frac{1}{f\_{R\text{R}}\eta\_{f\text{rx}}}\right) \tag{57}$$

A more accurate value of *Voc* is obtained after numerical resolution of equation (45) for *J*(*V*)=0, the results are plotted in figure12a and 12.b.

The other type of entropy loss degrading the open-circuit voltage is the photon entropy increase due to isotropic emission under direct sunlight. This entropy increase occurs because solar cells generally emit into 2π steradian, while the solid angle subtended by the sun is only 6.85×10−5 steradian.

The most common approach to addressing photon entropy is a concentrator system. If the concentration factor *C* of sun radiation is increased, this is generally achieved by optical means, a significant increase of *Voc* is obtained (as shown in figure 12.b). The calculation is carried out assuming a dominant radiative recombination and with maximum external fluorescence efficiency (*fRRηfex*=1). In this case we can notice that as *C* is increased *Voc* approaches its ultimate value *Eg*/*q*, for example GaAs (*Eg*=1.43 eV) for *C=CMax* we get *Voc*=0.99×*Eg*/*q*=1.41 V, which corresponds to an efficiency limit of approximately 38.5% (blackbody spectrum at 6000*K*). This theoretical limit shows the importance of dealing with entropy losses associated with angle of acceptance of photons from the sun and emission of photons from the cell efficiently. This value is well above the predicted SQ limit, where the concentration factor was considered.

With reference to table 1 we can clearly see that the record open circuit voltage under one-sun condition (*C*=1) of gallium arsenide solar cell (1.12 *V*) is already close to the SQ limit (1.17 *V*) while silicon solar cell is still behind with a record *Voc* of 0.706 *V* compared to a limit of 0.893 *V*, this difference is due to the fact that GaAs has a direct band gap, which means that it can be used to absorb and emit light efficiently, whereas Silicon has an indirect band gap and so is relatively poor at emitting light. Although Silicon makes an excellent solar cell, its internal fluorescence yield is less than 20%, which prevents Silicon from approaching the SQ limit [20]. On the other hand It has been demonstrated that efficiency in Si solar cells is limited by Auger recombination, rather than by radiative recombination [20-22].
