*3.4.1. Short-circuit current density (Jsc) calculation*

The solar cell is assumed entirely surrounded by the sun and maintained at *Tc*=0*K* as a first approximation and to get the maximum energy transfer from the sun. The total incident energy

*h kT x*

= = - - ò ò (40)

(41)

= (42)

 p

In accordance with the definition of the ultimate efficiency [6, 17], as the rate of the generated photon energy to the input energy density, its expression can be evaluated as a function of *Eg*

density coming from the sun at *Ts* and falling upon the solar cell, *Pin* is given by

n n

n

p

64 Solar Cells - New Approaches and Reviews

as follows:

( ) 3 3 2 4 32 0 0 2 / 2( )/ exp( / ) 1 exp( ) 1 *in <sup>s</sup> s <sup>d</sup> x dx P c kT h c*

> 5 4 32 2 ( ) / 15 *in <sup>s</sup> P kT h c* <sup>=</sup> p

> > ( ) *g s*

*in*

This expression is plotted in figure 10, so the maximum efficiency is approximately 43.87% corresponding to *Eg*=1.12 eV, this energy band-gap is approximately that of crystalline silicon. Similar calculation of the ultimate efficiency taking the solar spectrum AM1.5 G (The standard global spectral irradiance, ASTM G173-03, is used [18]) is shown in figure 10 gives a slightly higher value of 49%. If one compares this efficiency to the aforementioned thermodynamic efficiency limits, most of them approach the Carnot limit for the special case where the converter's temperature is absolute zero, this ultimate efficiency limit is substantially lower (44% or 49%) than the Carnot limit (95%). In quantum converters it is obvious that more than 50% of the solar radiation is lost because of the spectral mismatch. Therefore, non-absorption of photons with less energy than the semiconductor band-gap and the excess energy of

The detailed balance limit efficiency for an ideal solar cell, consisting of single semiconducting absorber with energy band-gap *Eg*, has been first calculated by Shockley and Queisser (SQ) [6]. The illumination of a *pn* junction solar cell creates electron-hole pairs by electronic transition due to the fundamental absorption of photons with *hν* > *Eg*, which is basically a quantum process. The photogenerated pairs either recombine locally or circulate in an external circuit and can transfer their energy. Their approach reposes on the following main assumptions; the probability that a photon with energy *hν* > *Eg* incident on the surface of the cell will produce a hole-electron pair is equal to unity, while photons of lower energy will produce no effect, all photogenerated electrons and holes thermalize to the band edges (photons with energy greater than *Eg* produce the same effect), all the photogenerated charge carriers are collected at short-

*h N P* n

*u g*

h n

photons, larger than the band-gap, are the two main losses.

**3.4. Detailed balance efficiency limit**

¥ ¥

Now we consider a more realistic situation of a solar cell, depicted in figure 3. Three factors will be taken into account, namely; the view factor of the sun seen from the solar cell, the background radiation is represented as a blackbody at ambient temperature *Ta*, and losses due to recombination (radiative and non-radiative).

In steady state condition the current density *J*(*V*) flowing through an external circuit is the algebraic sum of the rates of increase of electron-hole pairs corresponding to the absorption of incoming photons from the sun and the surrounding background, in addition to recombination (radiative and non-radiative). This leads to a general current voltage characteristic formula:

$$\begin{split} f(V) &= q(2\pi \, / \, c^2) \Big[ \big- \mathbb{C} \, f \times \Big|\_{\nu\_s} \frac{\nu^2 d\nu}{\exp(\hbar \nu \, / kT\_s) - 1} + (1 - \mathbb{C} \, f) \times \Big|\_{\nu\_s}^{\nu} \frac{\nu^2 d\nu}{\exp(\hbar \nu \, / kT\_a) - 1} \\ &- \frac{1}{f\_{RR}} \times \int\_{\nu\_s}^{\nu} \frac{\nu^2 d\nu}{\exp(\hbar \nu \, -qV) / kT\_c - 1} \Bigg] \end{split} \tag{43}$$

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

$$f\_{RR} = \frac{\mathcal{U}\_{RR}}{\mathcal{U}\_{RR} + \mathcal{U}\_{NR}}\tag{44}$$

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

$$f(V) = q \mathbf{C} f \phi\_s + q(1 - \mathbf{C} \, f) \phi\_a - \frac{q}{f\_{RR}} \phi\_c(V) \tag{45}$$

With:

2 2 (2 / ) a exp( / ) 1 *g s s d c h kT* n n n j p n ¥ <sup>=</sup> ò

$$\rho\_a = (2\pi/c^2) \int\_{\nu\_\odot}^{\nu\_\odot} \frac{\nu^2 d\nu}{\exp(h\nu/kT\_a) - 1} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\tag{46}$$

$$\left\|\rho\_c(V) = (2\pi/c^2)\right\|\_{\nu\_{\vec{x}}}^{\infty} \frac{\nu^2 d\nu}{\exp(\left(h\nu - qV\right)/kT\_c) - 1} \qquad \text{or}$$

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

$$J(0) = 0 \Longrightarrow \varphi\_a = \frac{1}{f\_{RR}} \varphi\_c(0) \tag{47}$$

Therefore the current density expression becomes:

$$f(V) = q \mathbb{C}f(\phi\_s - \phi\_a) - \frac{q}{f\_{RR}} \left(\phi\_c(V) - \phi\_c(0)\right) \tag{48}$$

From the above *J (V)* expression we can obtain the short-circuit current density (*Jsc*=*J*(*<sup>V</sup>*=0)) as follows:

$$J\_{sc} = q \mathbb{C} f(\phi\_s - \phi\_a) \tag{49}$$

In the ideal case the short-circuit current density depends only on the flux of impinging photons from the sun and the product *Cf*, recombination has no effect. The term *φa* in *Jsc*, representing the radiation from the surrounding ambient is negligible. The total photogener‐ ated carriers are swept away and do not recombine before reaching the external circuit where they give away their electrochemical energy. Figure 11 illustrates the maximum short-circuit current density to be harvested against band-gap energy according to (49) for a blackbody spectrum at *Ts*=6000°*K* normalised to a power density of 1000W/m2 and a spectral photon flux corresponding to the terrestrial AM1.5G spectrum. Narrow band-gap semiconductors exhibit higher photocurrents because the threshold of absorption is very low, therefore most of the solar spectrum can be absorbed. For power extraction this is not enough, the voltage is equally important and more precisely, the open circuit voltage.

The currently achieved short-circuit current densities for some solar cells are very close to predicted limits. Nevertheless, further gain in short-circuit current can therefore still be obtained, mainly by minimising the cell surface reflectivity, while increasing its thickness, so as to maximize the photon absorption. For thin film solar cells the gain in *Jsc* can be obtained by improving light trapping techniques to enhance the cell absorption.

For instance crystalline silicon solar cells with an energy band-gap of 1.12 eV at 300K has already achieved a *Jsc* of 42.7 mA/cm2 compared to a predicted maximum value of 43.85 mA/cm2 for an AM1.5 global spectrum (only 39.52 mA/cm2 for a normalised blackbody spectrum at 6000°*K*), while for GaAs with *Eg=*1.43eV a reported maximum *Jsc* of 29.68 mA/cm2 compared to 31.76 mA/cm2 (only 29.52 mA/cm2 for a normalised blackbody spectrum at 6000°*K*) [14].


**Table 1.** Summary of the reported records [14] and the calculated limits of Si and GaAs solar cells performances under the global AM1.5 spectrum.
