**3.2. Monochromatic solar cell**

It is interesting to examine first an ideal monochromatic converter illuminated by photons within a narrow interval of energy around the bandgap *h ν<sup>g</sup>* = *Eg*. In the ideal case each absorbed photon yields an electron-hole pair. This cell also prevents the luminescent radiation of energy outside this range from escaping. According to the blackbody formula of the Plank distribu‐ tion, the number of photons incident from the sun within an interval of frequency *dν* per unit area per second is:

$$dN\_S = \frac{2\pi}{c^2} \frac{\nu\_{\mathcal{S}}^2}{\exp\left(\frac{h\nu\_{\mathcal{S}}}{kT\_s}\right) - 1} d\nu$$

The number of created electron-hole pairs, in the assumption that each absorbed photon yields an electron-hole pair, could be simply represented by: *Cf Acd NS* , where *C* is the solar radiation concentration factor, *f* is a geometrical factor, taking into account the limited angle from which the solar energy falls upon the solar converter and *Ac* is the converter projected area. In monochromatic cell only photons with proper energy are allowed to escape from the cell (*h ν<sup>g</sup>* ≈ *Eg*) as a result of recombination. To obtain the efficiency of monochromatic ideal quantum converter we assume that only radiative recombination is allowed. Using the generalised Planck radiation law introduced by Würfel [7], the number of photons emitted by the solar cell per unit area per second within an interval of frequency *dν* is:

$$dN\_c = \frac{2\pi}{c^2} \frac{\nu\_{\rm g}^2}{\exp\left(\frac{h\nu\_{\rm g} - \mu\_{\rm eh}}{kT\_c}\right) - 1} d\nu$$

Where *µeh* is the emitted photon chemical potential, with *µeh*=*Fn*-*Fp*, (*Fn*, *Fp* are the quasi Fermi levels of electrons and holes respectively). The cell is assumed in thermal equilibrium with its surrounding ambient, then *Tc*=*Ta*.

This expression describes both the thermal radiation (for *µeh*=0) and emission of luminescence radiation (for *µeh* ≠ 0). The number of emitted photons from the cell is then: *f <sup>c</sup> Acd Nc*, *fc* is a geometrical factor depending on the external area from which the photons are leaving the solar converter and is equal to 1 if only one side of the cell is allowed to radiate and 2 if both sides of the cell are luminescent (*fc* is chosen equal to one in the following calculation). For a spatially constant chemical potential of the electron-hole pairs *µeh*=*Const*., the net current density of extracted electron-hole pairs, in the monochromatic cell, *dJ*, (the area of the cell is taken equal to unity, *Ac*=1) is:

$$\mathcal{L}\,d\mathcal{J} = q \frac{2\pi}{c^2} \left| \frac{\mathcal{C}\,f}{\exp\left(\frac{\hbar\nu\_g}{kT\_s}\right) - 1} - \frac{1}{\exp\left(\frac{\hbar\nu\_g - \mu\_{ch}}{kT\_c}\right) - 1} \right| \nu\_g^2 d\nu \tag{30}$$

This equation allows the definition of an equivalent cell temperature *Teq* from:

tion, the number of photons incident from the sun within an interval of frequency *dν* per unit

2

*g*

n

exp 1

n<sup>=</sup> æ ö

*kT*

The number of created electron-hole pairs, in the assumption that each absorbed photon yields an electron-hole pair, could be simply represented by: *Cf Acd NS* , where *C* is the solar radiation concentration factor, *f* is a geometrical factor, taking into account the limited angle from which the solar energy falls upon the solar converter and *Ac* is the converter projected area. In monochromatic cell only photons with proper energy are allowed to escape from the cell (*h ν<sup>g</sup>* ≈ *Eg*) as a result of recombination. To obtain the efficiency of monochromatic ideal quantum converter we assume that only radiative recombination is allowed. Using the generalised Planck radiation law introduced by Würfel [7], the number of photons emitted by

2

*g*

n

*dN d c h*

n m

<sup>=</sup> æ ö - ç ÷ -

exp 1

*kT*

è ø

Where *µeh* is the emitted photon chemical potential, with *µeh*=*Fn*-*Fp*, (*Fn*, *Fp* are the quasi Fermi levels of electrons and holes respectively). The cell is assumed in thermal equilibrium with its

This expression describes both the thermal radiation (for *µeh*=0) and emission of luminescence radiation (for *µeh* ≠ 0). The number of emitted photons from the cell is then: *f <sup>c</sup> Acd Nc*, *fc* is a geometrical factor depending on the external area from which the photons are leaving the solar converter and is equal to 1 if only one side of the cell is allowed to radiate and 2 if both sides of the cell are luminescent (*fc* is chosen equal to one in the following calculation). For a spatially constant chemical potential of the electron-hole pairs *µeh*=*Const*., the net current density of extracted electron-hole pairs, in the monochromatic cell, *dJ*, (the area of the cell is taken equal

*g eh c*

n

2

*g*

n n

*g s*

ç ÷ - ç ÷ è ø

n

(28)

(29)

(30)

2 2

the solar cell per unit area per second within an interval of frequency *dν* is:

2 2

p

*c*

2

p

2 1

*c h h*

n

exp 1 exp 1

è ø èø è ø

æ ö ç ÷

*kT kT*

*C f dJ q <sup>d</sup>*

<sup>=</sup> - æö æ ö - ç÷ ç ÷ - -

*g g eh s c*

 nm

surrounding ambient, then *Tc*=*Ta*.

to unity, *Ac*=1) is:

p

*dN d c h*

*S*

area per second is:

60 Solar Cells - New Approaches and Reviews

$$\frac{h\nu\_{\mathcal{g}}-\mu\_{eh}}{kT\_{\mathcal{c}}}=\frac{h\nu\_{\mathcal{g}}}{kT\_{eq}}\Longrightarrow\mu\_{eh}=h\nu\_{\mathcal{g}}\left(1-\frac{T\_{\mathcal{c}}}{T\_{eq}}\right)\tag{31}$$

When a solar cell formed by a juxtaposition of two semiconductors *p*- and *n*-types is illuminated, an electrical voltage *V* between its terminals is created. This voltage is simply the difference in the quasi Fermi levels of majority carriers at Ohmic contacts for constant quasi Fermi levels and ideal Ohmic contacts, this voltage is equal to the chemical poten‐ tial: *qV*=*µeh*=*Fn*-*Fp*.

At open circuit condition, the voltage *Voc* is given by a simple expression (deduced from (31)):

$$V\_{oc} = \frac{E\_{\mathcal{g}}}{q} \left( 1 - \frac{T\_c}{T\_{eq,oc}} \right) \tag{32}$$

The density of work delivered to an external circuit (density of extracted electrical power) *dW* is:

$$d\mathcal{W} = d\mathcal{I}.\\V = \frac{2\pi}{c^2} \left( \frac{\mathcal{C}\,f}{\exp\left(\frac{h\nu\_{\mathcal{S}}}{kT\_{\mathcal{S}}}\right) - 1} - \frac{1}{\exp\left(\frac{h\nu\_{\mathcal{S}}}{kT\_{eq}}\right) - 1} \right) \nu\_{\mathcal{S}}^2 d\nu \,h\nu\_{\mathcal{S}} \left(1 - \frac{T\_c}{T\_{eq}}\right) \tag{33}$$

The incoming energy flow from the sun can be written as:

$$dQ\_1 = \frac{2\pi}{c^2} \left(\frac{\mathcal{C}f}{\exp\left(\frac{h\nu\_{\mathcal{S}}}{kT\_s}\right) - 1}\right) \nu\_{\mathcal{S}}^2 d\nu\_{\mathcal{S}} \tag{34}$$

The emitted energy density from the solar cell in a radiation form (radiative recombination) is:

$$dQ\_2 = \frac{2\pi}{c^2} \left( \frac{1}{\exp\left(\frac{h\nu\_{\mathcal{S}}}{kT\_{eq}}\right) - 1} \right) \nu\_{\mathcal{S}}^2 d\nu\_{\mathcal{S}} \tag{35}$$

The efficiency of this system is:

$$d\eta = \frac{d\mathcal{W}}{dQ\_1} = \frac{dQ\_1 - dQ\_2}{dQ\_1} = \left(1 - \frac{T\_c}{T\_{eq}}\right) \tag{36}$$

The work extracted from a monochromatic cell is similar to that extracted from a Carnot engine. The equivalent temperature of this converter is directly related to the operating voltage. At short-circuit condition it corresponds to the ambient temperature (*µeh*=0 then *Teq,sc=Tc*=*Ta*) whereas at open circuit condition and for fully concentrated solar radiation, the equivalent temperature is that of the sun, (*dJ*=0 then *Teq,oc=Ts*). For non-concentrated radiation (*C* ≠ *CMax*) at open circuit voltage *Teq,oc* is obtained by solving the equation *dJ* =0, neglecting the 1 compared to the exponential term, we find then:

$$\frac{1}{T\_{eq,\alpha c}} = \frac{1}{T\_s} - \frac{k}{h\nu\_{\mathcal{g}}} \ln(C.f) \tag{37}$$

It is therefore possible to consider a monochromatic solar cell as reversible thermal engine (Carnot engine) operating between *Teq,oc* and *Ta*.

We can see that an ideal monochromatic cell, which only allows radiative recombination, represents an ideal converter of heat into electrical energy.

In order to find the maximum efficiency of such a cell as a function of the monochromatic photon energy (*hνg)*, the *dW*=*dJ×V* versus *V* characteristic is used. We search for the point (*dJmp*, *Vmp*) corresponding to the maximum extracted power. The maximum chemical energy density (*dWmp=dJmp*× *Vmp*) is divided by the absorbed monochromatic energy gives the efficiency *ηmono*(*Eg*) as a function of the bandgap.

The monochromatic efficiency is considerable, particularly in the case of fully concentrated radiation (*C*=*CMax*), and rises with the energy bandgap, as shown in figure 9. In theory the connection of a large number of ideal monochromatic absorbers will produce the best solar cell for the total solar spectrum. To calculate the overall efficiency numerically, a fine discre‐ tization of the frequency domain is made; the sum of the maximum power density over the solar spectrum divided by the total absorbed energy density. The efficiencies resulting from this calculation are respectively 67.45% and 86.81% for non-concentrated and fully concen‐ trated radiation.

To cover the whole solar energy spectrum an infinite number of monochromatic absorbers, each for a different photon energy interval, are needed. Each absorber would have its own Carnot engine and operate at its own optimal temperature, since for a given voltage the cell equivalent temperature depends on the photon energy (*hνg)*. Finally this model can not be directly used to describe semiconductor solar cells where the electron-hole pairs are generated in bands and not discrete levels, besides if the cell is considered as a cascade of tow-level converters; the notion of effective or equivalent temperature is no longer valid, since for each set of levels a different equivalent temperature is defined.

**Figure 9.** The monochromatic efficiency against the photon energy corresponding to the energy band-gap of the cell for non-concentrated (*C*=1) and fully concentrated (*C*=*Cmax*) solar spectrum.

### **3.3. Ultimate efficiency**

The efficiency of this system is:

62 Solar Cells - New Approaches and Reviews

to the exponential term, we find then:

*ηmono*(*Eg*) as a function of the bandgap.

trated radiation.

(Carnot engine) operating between *Teq,oc* and *Ta*.

represents an ideal converter of heat into electrical energy.

1 2

1 1 ln( . )

It is therefore possible to consider a monochromatic solar cell as reversible thermal engine

We can see that an ideal monochromatic cell, which only allows radiative recombination,

In order to find the maximum efficiency of such a cell as a function of the monochromatic photon energy (*hνg)*, the *dW*=*dJ×V* versus *V* characteristic is used. We search for the point (*dJmp*, *Vmp*) corresponding to the maximum extracted power. The maximum chemical energy density (*dWmp=dJmp*× *Vmp*) is divided by the absorbed monochromatic energy gives the efficiency

The monochromatic efficiency is considerable, particularly in the case of fully concentrated radiation (*C*=*CMax*), and rises with the energy bandgap, as shown in figure 9. In theory the connection of a large number of ideal monochromatic absorbers will produce the best solar cell for the total solar spectrum. To calculate the overall efficiency numerically, a fine discre‐ tization of the frequency domain is made; the sum of the maximum power density over the solar spectrum divided by the total absorbed energy density. The efficiencies resulting from this calculation are respectively 67.45% and 86.81% for non-concentrated and fully concen‐

To cover the whole solar energy spectrum an infinite number of monochromatic absorbers, each for a different photon energy interval, are needed. Each absorber would have its own Carnot engine and operate at its own optimal temperature, since for a given voltage the cell equivalent temperature depends on the photon energy (*hνg)*. Finally this model can not be directly used to describe semiconductor solar cells where the electron-hole pairs are generated in bands and not discrete levels, besides if the cell is considered as a cascade of tow-level

*eq oc s g <sup>k</sup> C f T Th*n

The work extracted from a monochromatic cell is similar to that extracted from a Carnot engine. The equivalent temperature of this converter is directly related to the operating voltage. At short-circuit condition it corresponds to the ambient temperature (*µeh*=0 then *Teq,sc=Tc*=*Ta*) whereas at open circuit condition and for fully concentrated solar radiation, the equivalent temperature is that of the sun, (*dJ*=0 then *Teq,oc=Ts*). For non-concentrated radiation (*C* ≠ *CMax*) at open circuit voltage *Teq,oc* is obtained by solving the equation *dJ* =0, neglecting the 1 compared

*dW dQ dQ T dQ dQ T*


1 *<sup>c</sup> eq*

ç ÷ è ø

= - (37)

(36)

1 1

,

h

The total number of photons of frequency greater than *ν<sup>g</sup>* (*Eg*=*hνg*) impinging from the sun, assumed as a blackbody at temperature *Ts*, in unit time and falling upon the solar cell per unit area *Ns* is given by:

$$N\_s = \left(2\pi \left/ c^2\right)\right)\_{\nu\_s}^{\nu} \frac{\nu^2 d\nu}{\exp(h\nu \,/\, kT\_s) - 1} \tag{38}$$

This integral could be evaluated numerically.

In the assumption that each absorbed photon will produce a pair of electron-hole, the maxi‐ mum output power density that could be delivered by a solar converter will be:

$$P\_{out} = h\nu\_{\chi} \times \mathcal{N}\_s \tag{39}$$

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

$$P\_{in} = \left(2\pi \,/\,c^2\right) \int\_0^\infty \frac{\nu^3 d\nu}{\exp(h\nu \,/\, kT\_s) - 1} = 2\pi (kT\_s)^4 \,/\, h^3 c^2 \int\_0^\infty \frac{\nu^3 d\mathbf{x}}{\exp(\mathbf{x}) - 1} \tag{40}$$

$$P\_{\rm in} = 2\pi^5 (kT\_s)^4 \,/\, 15h^3c^2 \tag{41}$$

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* as follows:

$$\eta\_{\mu}(\nu\_{\ g}) = \frac{h\nu\_{g}N\_{s}}{P\_{in}} \tag{42}$$

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