**2. Solar cell as a heat engine**

When the solar cell is supposed a blackbody converter absorbing radiation from the sun itself a blackbody, without creating entropy, we obtain an efficiency of about 93 % known as the

Whilst a solar cell is assumed as an endoreversible system [4], the energy conversion efficiency is limited to 85.7% this figure is obtained where the sun is assumed fully surrounding the cell (maximum concentration). If we bear in mind that in a real situation the solar cell does not operate always in maximum concentration and the solid angle under which the cell sees the sun is in fact only a minute fraction of a hemisphere, the maximum efficiency is not larger than 12.79%, which is actually lower than most recently fabricated solar cells. However, we can conclude that solar cell is a quantum converter and cannot be treated as a simple solar radiation

Semiconductor *pn* junction solar cell is a quantum converter where the energy band-gap of semiconductor material is the most important and critical factor controlling efficiency. Incident photons with energy higher than the energy gap can be absorbed, creating electron-hole pairs,

In the ideal model of a monochromatic cell incident photons are within a narrow interval of energy, while the cell luminescence outside this range is prevented. The overall resulting efficiency upper limit for an infinite number of monochromatic cells is 86.81% for fully

The ultimate efficiency of a single band gap *pn* junction for an AM1.5 G solar spectrum gives a value of 49%, this maximum efficiency, if compared to Carnot efficiency limit, is substantially lower. Therefore in quantum converters it is obvious that more than 50% of the solar radiation

To represent a more realistic picture of a solar cell, three other fundamental factors should be taken into account, namely; the view factor of the sun seen from the solar cell position, the background radiation which could be represented as a blackbody at ambient temperature, and

In the detailed balance efficiency limit calculation first suggested in 1961 by Shockley and Queisser (SQ) in their seminal paper [6]. It is assumed that illumination of semiconductor *pn* junction by a blackbody source creates electron-hole pairs due to the fundamental absorption of photons with energies greater than the band-gap. These pairs either recombine locally if they are not separated and extracted along different paths to perform work in an external circuit. They assumed that recombination in the semiconductor is partly radiative and the maximum efficiency is attained when radiative recombination in dominant. The Shockley-Queisser model has been extended and completed to account for more physical phenomena [7-13]. For instance, the generalised Planck radiation law introduced by Würfel [7], the effect of background radiation has been included and elaborate numerical techniques has been used

in order avoid mathematical approximations which would yield erroneous results.

The currently achieved short-circuit current densities for some solar cells are very close to predicted limits [14]. Nevertheless, further gain in short-circuit current can therefore still be

while those with lower energy are not absorbed, either reflected or transmitted.

Landsberg efficiency limit, which is slightly lower than Carnot efficiency.

converter [5].

concentrated sun radiation.

48 Solar Cells - New Approaches and Reviews

is lost because of spectral mismatch.

losses due to recombination, radiative and non-radiative.

### **2.1. Solar cell as a reversible heat engine**

Thermodynamics has widely been used to estimate the efficiency limit of energy conversion process. The performance limit of solar cell is calculated either by thermodynamics or by detailed balance approaches. Regardless of the conversion mechanism in solar cells, an upper efficiency limit has been evaluated by considering only the balances for energy and entropy flux rates. As a first step the solar cell was represented by an ideally reversible Carnot heat engine in perfect contact with high temperature *Ts* reservoir (the sun) and low temperature *Ta* reservoir representing the ambient atmosphere. In accordance with the first law of thermo‐ dynamics the extracted work, the cell electrical power output, is represented as the difference between the net energy input from the sun and the energy dissipated to the surrounding environment. The model is illustrated in figure 1. Where *Q1* is the incident solar energy impinging on the cell, *Q2* is the amount of energy flowing from the converter to the heat sink and *W* is the work delivered to a load in the form of electrical energy (*W=Q1 – Q2*). The efficiency of this system is defined using the first thermodynamic law as:

$$
\eta\_c = \frac{W}{Q\_1} = \frac{Q\_1 - Q\_2}{Q\_1} = 1 - \frac{Q\_2}{Q\_1} \tag{1}
$$

for a reversible engine the total entropy is conserved, *S* =*S*<sup>1</sup> −*S*<sup>2</sup> =0, then;

$$\frac{Q\_1}{T\_1} - \frac{Q\_2}{T\_2} = 0\tag{2}$$

Hence the Carnot efficiency could be represented by:

$$\eta\_c = 1 - \frac{T\_a}{T\_s} \tag{3}$$

This efficiency depends simply on the ratio of the converter temperature, which is equal to that of the surrounding heat sink, to the sun temperature. This efficiency is maximum (*ηc*=*1*) if the converter temperature is *0 K* and the solar energy is totally converted into electrical work, while *ηc*=*0* if the converter temperature is identical to that of the sun *Ts* the system is in thermal equilibrium so there is no energy exchange. If the sun is at a temperature of *6000K* and the ambient temperature is *300K*, the Carnot efficiency is *95*% this value constitutes an upper limit for all solar converters.

**Figure 1.** A schematic diagram of a solar converter represented as ideal Carnot engine.

Another way of calculating the efficiency of a reversible heat engine where the solar cell is assumed as a blackbody converter at a temperature *Tc*, absorbing radiation from the sun itself a blackbody at temperature *Ts*, without creating entropy, this efficiency is called the Landsberg efficiency[16].

Under the reversibility condition the absorbed entropy from the sun *Sabs* is given off in two ways one is emitted back to the sun *Semit* and the second part goes to the ambient thermal sink *Sa*. Under this condition, the solar cell is called reversible if:

$$\mathbf{S}\_{abs} - \mathbf{S}\_{emit} - \mathbf{S}\_a = \mathbf{0} \tag{4}$$

In accordance with the Stefan–Boltzmann law of black body, the absorbed heat flow from the sun is:

Theoretical Calculation of the Efficiency Limit for Solar Cells http://dx.doi.org/10.5772/58914 51

$$Q\_{\rm abs} = \sigma \, T\_s^4 \tag{5}$$

For a blackbody radiation, the absorbed density of entropy flow is:

$$S\_{abs} = \frac{4}{3}\sigma \, T\_s^3 \tag{6}$$

The energy flow emitted by the converter at a temperature *Tc* is:

$$Q\_{emit} = \sigma \, T\_c^4 \tag{7}$$

And the emitted entropy flow is:

This efficiency depends simply on the ratio of the converter temperature, which is equal to that of the surrounding heat sink, to the sun temperature. This efficiency is maximum (*ηc*=*1*) if the converter temperature is *0 K* and the solar energy is totally converted into electrical work, while *ηc*=*0* if the converter temperature is identical to that of the sun *Ts* the system is in thermal equilibrium so there is no energy exchange. If the sun is at a temperature of *6000K* and the ambient temperature is *300K*, the Carnot efficiency is *95*% this value constitutes an upper limit

**Figure 1.** A schematic diagram of a solar converter represented as ideal Carnot engine.

*Sa*. Under this condition, the solar cell is called reversible if:

Another way of calculating the efficiency of a reversible heat engine where the solar cell is assumed as a blackbody converter at a temperature *Tc*, absorbing radiation from the sun itself a blackbody at temperature *Ts*, without creating entropy, this efficiency is called the Landsberg

Under the reversibility condition the absorbed entropy from the sun *Sabs* is given off in two ways one is emitted back to the sun *Semit* and the second part goes to the ambient thermal sink

In accordance with the Stefan–Boltzmann law of black body, the absorbed heat flow from the

0 *abs emit a SS S* - -= (4)

for all solar converters.

50 Solar Cells - New Approaches and Reviews

efficiency[16].

sun is:

$$S\_{emit} = \frac{4}{3}\sigma \ T\_c^3 \tag{8}$$

In this model the blackbody source (sun) surrounds entirely the converter at *Tc* which is assumed in a contact with a thermal sink at *Ta* then *Tc*=*Ta*. Therefore the entropy transferred to the thermal sink is:

$$\mathcal{S}\_a = \mathcal{S}\_{abs} - \mathcal{S}\_{emit} = \frac{4}{3}\sigma (T\_s^3 - T\_c^3) \tag{9}$$

And the transferred heat flow is:

$$\mathcal{Q}\_a = T\_c \mathcal{S}\_a = \frac{4}{3} \sigma T\_c (T\_s^3 - T\_c^3) \tag{10}$$

The entropy-free, utilizable work flow is then:

$$\mathcal{W} = \mathbb{Q}\_{abs} - \mathbb{Q}\_{emit} - \mathbb{Q}\_a \tag{11}$$

Therefore the Landsberg efficiency can be deduced as:

$$\eta\_L = \frac{W}{Q\_{abs}} = 1 - \frac{4}{3}\frac{T\_c}{T\_s} + \frac{1}{3} \left(\frac{T\_c}{T\_s}\right)^4 \tag{12}$$

The actual temperature of the converter *Tc* depends on the operating point of the converter and is different from the ambient temperature *Ta*, (*Tc* ≠ *Ta*). To maintain the same assumption as the Landsberg efficiency calculation, the entropy transferred to the ambient thermal sink is rewritten as:

$$\mathcal{Q}\_a = T\_a \mathcal{S}\_a = \frac{4}{3} \sigma \, T\_a (T\_s^3 - T\_c^3) \tag{13}$$

We arrive to a more general form of the Landsberg efficiency *η'*:

$$\boldsymbol{\eta}^{\prime}\_{\perp} = \mathbf{1} - \left(\frac{T\_c}{T\_s}\right)^4 - \frac{4}{3}\frac{T\_a}{T\_s}\left(\mathbf{1} - \frac{T\_c^3}{T\_s^3}\right) \tag{14}$$

Both forms of Landsberg efficiency (*ηL* and *η'L*) are plotted in Figure 2, Carnot efficiency curve is added for comparison. At 300*K ηL* and *η'<sup>L</sup>* coincide at 93.33 % which is slightly lower than Carnot efficiency. When the temperature of the converter is greater than the ambient temper‐ ature (*Tc* > *Ta*) there is less heat flow from the converter to the sun (in accordance with Landsberg model). This means that much work could be extracted from the converter leading to a higher efficiency. As *Tc* approaches the sun temperature, the net energy exchange between the sun and the converter drops, therefore the efficiency is reduced and finally goes to zero for *Tc*=*Ts*.

**Figure 2.** Landsberg and Carnot Efficiency limits of a solar converter versus ambient temperature.

In the Landsberg model the blackbody radiation law for the sun and the solar cell has been included, unlike the previous Carnot engine.

The actual temperature of the converter *Tc* depends on the operating point of the converter and is different from the ambient temperature *Ta*, (*Tc* ≠ *Ta*). To maintain the same assumption as the Landsberg efficiency calculation, the entropy transferred to the ambient thermal sink is

> <sup>4</sup> 3 3 ( ) <sup>3</sup> *Q TS T T T a aa a s c* = = s

<sup>4</sup> ' 1 <sup>1</sup>

**Figure 2.** Landsberg and Carnot Efficiency limits of a solar converter versus ambient temperature.

4 3

3 *c ac*

æ ö æ ö =- - - ç ÷ ç ÷ ç ÷ ç ÷ è ø è ø

*s s s T TT T T T*

Both forms of Landsberg efficiency (*ηL* and *η'L*) are plotted in Figure 2, Carnot efficiency curve is added for comparison. At 300*K ηL* and *η'<sup>L</sup>* coincide at 93.33 % which is slightly lower than Carnot efficiency. When the temperature of the converter is greater than the ambient temper‐ ature (*Tc* > *Ta*) there is less heat flow from the converter to the sun (in accordance with Landsberg model). This means that much work could be extracted from the converter leading to a higher efficiency. As *Tc* approaches the sun temperature, the net energy exchange between the sun and the converter drops, therefore the efficiency is reduced and finally goes to zero for *Tc*=*Ts*.

3

We arrive to a more general form of the Landsberg efficiency *η'*:

*L*

h

(13)

(14)

rewritten as:

52 Solar Cells - New Approaches and Reviews

This figure represents an upper bound on solar energy conversion efficiency, particularly for solar cells which are primarily quantum converters absorbing only photons with energies higher or equal to their energy bandgap. On the other hand in the calculation of the absorbed solar radiation the converter was assumed fully surrounded by the source, corresponding to a solid angle of 4π.

Using the same approach it is possible to split the system into two subsystems each with its own efficiency; the Carnot engine that include the heat pump of the converter at *Tc* and the ambient heat sink at *Ta*, with an efficiency *ηc* (ideal Carnot engine).

$$\eta\_c = 1 - \frac{T\_a}{T\_c} \tag{15}$$

The ambient temperature is assumed equal to 300 *K*, therefore high efficiency is obtained when the converter temperature is higher then the ambient temperature.

The second part is composed of the sun as an isotropic blackbody at *Ts* and the converter reservoir assumed as a blackbody at a temperature *Tc*. The energy flow falling upon *Qabs* and emitted by the solar converter *Qemit* are given by:

$$Q\_{\rm abs} = \mathbb{C} \int \sigma \, T\_s^4 \text{ and } Q\_{\rm emit} = \sigma \, T\_c^4 \tag{16}$$

In which *f* is a geometrical factor taking into account the limited solid angle from which the solar energy falls upon the converter. In accordance with the schematic representation of figure 3, where the solar cell is represented as a planar device irradiated by hemisphere surrounding area and the sun subtending a solid angle *ω<sup>s</sup>* at angle of incidence *θ, f* is defined as the ratio of the area subtended by sun to the apparent area of the hemisphere:

$$\int \cos \theta \, d\phi$$

$$f = \frac{a\_s}{\int \cos \theta \, d\phi} = \frac{a\_s}{\pi} \tag{17}$$

*ω<sup>s</sup>* being the solid angle subtended by the sun, where *ωs*=6.85×10-5 sr. The concentration factor *C* (*C* > 1) is a measure of the enhancement of the energy current density by optical means (lens, mirrors…).The maximum concentration factor is obtained if we take *Ts*=6000°*K*:

$$
\sigma \, T\_s^4 = \mathbb{C}\_{\text{max}} \int \sigma \, T\_s^4 \tag{18}
$$

Then

$$C\_{max} = 1 / f \approx 46200$$

The case of maximum concentration also corresponds to the schematic case where the sun is assumed surrounding entirely the converter as assumed in previous calculations.

Similar situation can be obtained if the solid angle through which the photons are escaping from the cell (emission angle) is limited to a narrow range around the sun. This can be achieved by hosing the cell in a cavity that limits the angle of the escaping photons.

**Figure 3.** A schematic representation of a solar converter as a planar cell irradiated by the sun subtending a solid angle *ωs* at angle of incidence *θ*.

The efficiency of this part of the system (isolated) is given by:

$$\eta\_{abs} = 1 - \frac{Q\_{emit}}{Q\_{abs}} \tag{19}$$

the resulting efficiency is simply the product:

$$
\eta\_{ac} = \eta\_c \cdot \eta\_{abs} = \left(1 - \frac{T\_c^4}{C \int T\_s^4}\right) \left(1 - \frac{T\_a}{T\_c}\right) \tag{20}
$$

This figure represents an overall efficiency of the entropy-free energy conversion by blackbody emitter-absorber combined with a Carnot engine. The temperature of the surrounding ambient *Ta* is assumed equal to 300*K*. When *f* is taken equal to *ωs/π*=2.18×10-5 and without concentration (*C*=1) we obtain a very low efficiency value (about 6.78%), as shown in figure 4. The efficiency for concentrations of 10, 100 and full concentration (46200) is found respectively 31.36, 53.48 and 85.38%. In this model the solar cell is not in thermal equilibrium with its surrounding (*Tc* ≠ *Ta*), then it exchanges radiation not only with the sun but also with the ambient heat sink. Therefore, energy can be produced or absorbed from the surrounding acting as a secondary source. Neglecting this contribution naturally decreases the efficiency of the converter, particularly at *C*=1. The second explanation of the low efficiency value is the under estimation of the re-emitted radiation from the cell, at operating conditions the solar cell re-emits radiation efficiently especially at open circuit point.

**Figure 4.** Efficiency *ηac* for different concentration rates (*C*=1, 10, 100 and *Cmax*) with Landsberg and Carnot Efficiency limits of a solar converter as a function of ambient temperature.

### **2.2. Solar cell as an endoreversible heat engine**

Then

*Cmax* =1 / *f* ≈ 46200

54 Solar Cells - New Approaches and Reviews

*ωs* at angle of incidence *θ*.

The case of maximum concentration also corresponds to the schematic case where the sun is

Similar situation can be obtained if the solid angle through which the photons are escaping from the cell (emission angle) is limited to a narrow range around the sun. This can be achieved

**Figure 3.** A schematic representation of a solar converter as a planar cell irradiated by the sun subtending a solid angle

1 *emit*

*Q Q*

*abs*

4 <sup>4</sup> .1 1 *c a*

This figure represents an overall efficiency of the entropy-free energy conversion by blackbody emitter-absorber combined with a Carnot engine. The temperature of the surrounding ambient

è øè ø

æ öæ ö = =- - ç ÷ç ÷

*s c*

*T T CfT T*

= - (19)

(20)

*abs*

h

The efficiency of this part of the system (isolated) is given by:

*ac c abs*

h hh

the resulting efficiency is simply the product:

assumed surrounding entirely the converter as assumed in previous calculations.

by hosing the cell in a cavity that limits the angle of the escaping photons.

A more realistic model has been introduced by De Vos et al. [8] in which only a part of the converter system is reversible, endoreversible system. An intermediate heat reservoir is inserted at the temperature of the converter *Tc*, this source is heated by the sun at *Ts* (blackbody radiation) and acting as a new high temperature pump in a reversible Carnot engine. In this system the entropy is generated between the *Ts* reservoir and the converter, the temperature *Tc* is fictitious and is different from the ambient temperature *Ta*. The effective temperature *Tc* depends on the rate of work production. In this model the solar converter is assumed to behave like the Müser engine, itself a particular case of the Curzon-Ahlborn engine, as shown in Fig. 5, the sun is represented by blackbody source at temperature *T1*=*Ts* the solar cell includes a heat reservoir assumed as blackbody at *T3*=*Tc* (the converter temperature) and an ideal Carnot engine capable of producing utilizable work (electrical power), however *Tc* is related to the converter working condition. This engine is in contact with an ambient heat sink at *T2* representing the ambient temperature *T2*=*Ta*. In addition to the absorbed energy from the sun, the converter absorbs radiation from ambient reservoir assumed as a blackbody at *Ta*.

**Figure 5.** A schematic diagram of a solar converter represented as an endoreversible system.

The net energy flow input to the converter, including the incident solar energy flow *f σ Ts* 4 , the energy flow (1− *f* ) *σ Ta* 4 from the surrounding and the energy flow emitted by the converter is then:

$$Q\_1 = f \,\sigma \, T\_s^4 + (1 - f) \,\sigma \, T\_a^4 - \sigma \, T\_c^4 \tag{21}$$

The Müser engine efficiency (Carnot engine):

$$\eta\_M = \frac{W}{Q\_1} = 1 - \frac{T\_a^4}{T\_c^4} \tag{22}$$

The converter temperature can be extracted from *ηM*:

$$T\_c = \frac{T\_a}{1 - \eta\_M} \tag{23}$$

And the solar efficiency is defined as ratio of the delivered work to the incident solar energy flux:

$$\eta\_S = \frac{W}{f\sigma \, T\_s^4} \tag{24}$$

hence

4 ,

engine capable of producing utilizable work (electrical power), however *Tc* is related to the converter working condition. This engine is in contact with an ambient heat sink at *T2* representing the ambient temperature *T2*=*Ta*. In addition to the absorbed energy from the sun,

the converter absorbs radiation from ambient reservoir assumed as a blackbody at *Ta*.

**Figure 5.** A schematic diagram of a solar converter represented as an endoreversible system.

s

*M*

h

4

The Müser engine efficiency (Carnot engine):

The converter temperature can be extracted from *ηM*:

the energy flow (1− *f* ) *σ Ta*

56 Solar Cells - New Approaches and Reviews

is then:

The net energy flow input to the converter, including the incident solar energy flow *f σ Ts*

4 44

4 4

*c*

1 *<sup>a</sup>*

*M*

h

*W T Q T*

 ss

<sup>1</sup> (1 ) *Q fT f T T s ac* = +- -

1

1 *a*

*T*

*c*

*T*

from the surrounding and the energy flow emitted by the converter

= = - (22)

<sup>=</sup> - (23)

(21)

$$\eta\_S = \eta\_M \left[ 1 + \frac{\left(1 - f\right)\left(1 - \eta\_M\right)^4 - 1}{\left(1 - \eta\_M\right)^4} \frac{T\_a^4}{fT\_s^4} \right] \tag{25}$$

The maximum solar efficiency is then a function of two parameters; the Müser efficiency and the surrounding ambient temperature. From the 3d representation at figure 6 of the solar efficiency (*ηs*) against Müser efficiency *ηM* and the surrounding temperature *Ta*, the efficiency is high as the temperature is very low and vanishes for very high temperature (as *Ta* approaches the sun temperature). For *Ta*=289.23K the efficiency is 12.79% if the sun's temperature is 6000°*K*.

**Figure 6.** The solar efficiency surface *ηs* (*ηM*, *Ta*), the sun as a blackbody at *Ts=*6000°*K*.

A general expression of solar efficiency of the Müser engine is obtained when the solar radiation concentration factor *C* is introduced:

$$\eta\_S = \eta\_M \left[ 1 + \frac{(1 - \mathbb{C} \, f)(1 - \eta\_M)^4 - 1}{\left(1 - \eta\_M\right)^4} \frac{T\_a^4}{\mathbb{C} \, f \, T\_s^4} \right] \tag{26}$$

Compared to Carnot efficiency engine the Müser engine efficiency, even when *C* is maximal, remains low.

If the ratio *Ta* <sup>4</sup> / *f Ts* 4 is fixed to 1/4, as in [5], which gives a good approximation for ambient temperature, *Ta*=289.23*K*, with *Ts*=6000*K*.

Hence, the corresponding *ηS* becomes:

$$\eta\_S = \eta\_M \left[ 1 + \frac{(1 - C\,f)(1 - \eta\_M)^4 - 1}{4(1 - \eta\_M)^4} \right] \tag{27}$$

In the assumption of *Ta*=289.23*K* the maximum efficiency without concentration, i.e. the solar cell sees the sun through a solid angle *ω<sup>s</sup>* is 12.79% which is better than the predicted value of Würfel [7] but still very low, as shown in figure 8. For concentration equal to 10, 100 and *CMAX*, the efficiency reaches 33.9, 54.71 and 85.7% respectively.

**Figure 7.** The maximum solar efficiency using Müser engine for different concentration rates (*C*=1, 10, 100 and *Cmax*) with Carnot Efficiency limit as a function of ambient temperature.

**Figure 8.** The solar efficiency using Müser engine for different concentration rates (*C*=1, 10, 100 and *Cmax*) as a function of Müser efficiency.
