# Effect of external applied electric field on the silicon solar cell’s thermodynamic efficiency

### Abstract

This paper presents a possible solution to improve the efficiency of photovoltaic solar cells. An external electric field is applied on a silicon photovoltaic solar cell, inducing band-trap ionization of charge carriers. Output current is then monitored and the thermodynamic efficiency is calculated. Results show on the one hand a significant increase in efficiency for a certain margin of applied electric field, and on the another hand the instabilities of efficiency. A simple approach is then suggested for the implementation of these results. An efficiency of 67% has been reached for an applied electric of 1586 V/Cm.

## Keywords

Improve efficiency External applied electric field Band-trap ionization of charge carriers Silicon solar cell## Abbreviation

*h*Planck’s constant (6.626 × 10

^{−34}Js)*c*Photons propagation speed (3 × 10

^{8}m/s)*q*Charge of electron (1.6 × 10

^{−19}C)*V*Voltage (V)

*k*Boltzmann’s constant: (1.38 × 10

^{−23}J/K)*T*_{c}Cell’s temperature (K)

*T*_{s}Sun’s temperature (6000 °K)

*μ*_{p}Hole mobility

*E*_{G}Gap energy (for silicon

*E*_{G}= 1.12 eV)*τ*_{n},*τ*_{p}Recombination life time of electrons and holes, respectively (s)

*μ*_{n}Electron mobility

*L*_{n}*, L*_{p}Electron diffusion length and hole diffusion length, respectively

*S*Area of surface subject to the radiation (Cm

^{2})*n*_{i}Intrinsic concentration of electrons and holes (

*n*_{i}= 1.45 × 10^{10}Cm^{−3}for silicon)*f*_{ω}Geometrical factor

*Q*_{s}Number of quanta

*t*_{s}Probability that incident photon will produce a hole-electron pair

*E*_{o}External applied electric field (V/Cm)

- C.B
Conduction band

- V.B
Valence band

## Introduction

Developing new concepts to improve the efficiency of photovoltaic solar cells is a well-known challenge for the scientific community. In 1961 Shockley and Queisser [1] brought out the theoretical limit of a photovoltaic solar cell. The results of their study are worldwide recognized as theoretical limit of efficiency for a single *pn*-junction solar cell. After them, many studies have been carried out to explore the possibilities of exceeding this limit. Different technologies and methods were used for that purpose [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Among these approaches, we could cite tandem cells, concentrator cell, carrier multiplication, down conversion, hot carriers, etc.

In 1997, by considering the impact ionization phenomenon to generate hot electrons, Würfel [11] found a maximum efficiency of 85% for a vanishing band gap of the solar cell. In 1993, Landsberg et al. [12] reported an efficiency of 60.3% at *E* _{G} = 0.8 eV for a solar cell submitted to band–band impact ionization effect. By considering the impact ionization effects on the efficiency of intermediate band solar cells, Gorji [16] has obtained a thermodynamic efficiency of 81.2%, which was higher than the maximum efficiency of 63.2% for an intermediate band without impact ionization mechanism. All these results show that the impact ionization phenomenon is very promising for the improvement of solar cell efficiency.

A single *pn*-junction of the solar cell is considered in this work. In Ref. [17] it has been shown that outside an electron diffusion length *L* _{ n } to the right or a hole diffusion length *L* _{ p } to the left of the *pn*-junction, the charge current through a *pn*-junction is a pure electron current in the *n*-region and a pure hole current in the *p*-region. This charge current is then given by integrating over the contributions to the electron current (alternatively, the contributions to the hole current). Knowing the number of free electrons in *n*-region (or free holes in *p*-region) could be sufficient to evaluate the charge current through a *pn*-junction and thus the open circuit voltage. In this paper, the number of free electrons in *n*-region of the *pn*-junction is evaluated by solving the reaction diffusion equation. This equation is solved with the factorization method. The total hole-electron generation rate due to the solar radiation is evaluated by following the Shockley–Queisser approach [1].

## Studied model

*n*and

*p*are the number of free electrons and free holes, respectively. The variables

*X*

_{1}and

*X*

_{2}are band-trap impact ionization coefficients (to generate additional electrons and holes, respectively) which depend on the applied electric field.

*Y*and

*B*are the band–band generation coefficient and the band–band recombination coefficient, respectively.

*Y*is a photo-generation parameter due to the illumination of solar cell. This variable is evaluated through Eq. (43). The constants

*N*

_{ D }

^{*}and

*N*

_{t}are the effective donor density and the trap density, respectively.

*P*

_{ D }=

*N*

_{t}−

*N*

_{ D }

^{*}.

*pn*-junction is a pure electron current in the

*n*-region, the knowledge of the free electrons number in the

*n*-region is requested to determine the total charge current through the

*pn*-junction. Considering only the transverse direction and neglecting the transverse electric field, the number of free electrons can be determined using the reaction–diffusion equation presented in Eq. (2).

*D*

_{ n }is the electron diffusion coefficient defined by:

## Thermodynamic efficiency calculation

### Resolution of the reaction–diffusion equation

The factorization method [24, 25] is used in this work to solve the reaction–diffusion equation. The equation to solve is Eq. (3).

*K*

_{1}is an arbitrary constant to be determined.To determine the constants

*K*

_{1}let us consider Eq. (9). This equation can be developed and leads to,

#### First case

*α*and

*β*by their expressions in Eq. (25) one obtains:

#### Second case

### Charge current density through the *pn*-junction

*pn*-junction could be expressed by:

*θ*

_{1}

*, θ*

_{2}

*, θ*

_{3}and

*θ*

_{4}leads to:

*j*

_{sc}(when

*V*= 0) is defined as:

*j*

_{sc}= 0 and for large negative voltages (where \({ \exp }\left( {\frac{qV}{{kT_{\text{c}} }}} \right) \ll 1\)), one gets the reverse saturation current

*j*

_{s}as:

*j*

_{Q}=

*0*) is deducted as:

*Y*is defined by:

## Discussion

Typical materials parameters corresponding to α-si near room temperature for the g–r process of band-trap impact ionization [18]

Parameters | Value |
---|---|

| 3 × 10 |

| 10 |

| 2 × 10 |

| 3 × 10 |

| 1 |

| |

*E*

_{o}= 1586 V/Cm. From

*E*

_{o}> 1586 V/Cm an efficiency fluctuation (increasing and decreasing) is noted. This fluctuation could be due to current instabilities which emerge from solar cells (which are semiconductors) when the applied electric field is increasing [26]. Figure 3 has been plotted for 0 <

*E*

_{o}< 10

^{6}V/Cm and shows that the solar cell could reach a high efficiency for strong electric field as it is the case for

*E*

_{o}= 8 × 10

^{5}V/Cm where an efficiency of 72.7% has been reached. In this case, the problem is that, for the same value of electric field, there are different values of efficiency because of fluctuations. Therefore, it could be difficult to determinate exactly the high efficiency of the solar cell subjected to impact ionization (induced by an external applied electric field) in the margin of which efficiency fluctuates and is unstable. The results obtained shows that the band-trap impact ionization of charge carriers induced by an applied electric field could be an interesting solution to reach a high efficiency of the photovoltaic solar cells. However, it is very important to know and avoid applying electric field belonging to the range which induces efficiency instabilities. Theoretically a high efficiency of solar cell could be reached even for applied electric fields of average intensity (67% has been reached at

*E*

_{o}= 1586 V/Cm).

## Conclusion

In this paper, the effect of an external applied electric field on the thermodynamic efficiency of a silicon photovoltaic solar cell has been studied. Theoretically, it has been shown that an auxiliary applied electric field could be a very promising solution to reach a high efficiency of the solar cells. However, it is not always the stronger electric field which is necessary to induce the higher efficiency. There are efficiency instabilities for strong applied electric field to solar cells.

## Notes

### Acknowledgements

The authors thank Taku Agbor Junior for his useful comments.

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