# Theoretical Study of Proton Radiation Influence on the Performance of a Polycrystalline Silicon Solar Cell

Authors: Tchouadep Guy Serge; Zouma Bernard; Korgo Bruno; Soro Boubacar; Savadogo Mahamadi; Zoungrana Martial; Zerbo Issa
Journal: International Journal of Photoenergy
 (2019)

---

## Abstract

The aim of this work is to study the behaviour of a silicon solar cell under the irradiation of different fluences of high-energy proton radiation (10 MeV) and under constant multispectral illumination. Many theoretical et experimental studies of the effect of irradiation (proton, gamma, electron, etc.) on solar cells have been carried out. These studies point out the effect of irradiation on the behaviour of the solar cell electrical parameters but do not explain the causes of these effects. In our study, we explain fundamentally the causes of the effects of the irradiation on the solar cells. Taking into account the empirical formula of diffusion length under the effect of high-energy particle irradiation, we established new expressions of continuity equation, photocurrent density, photovoltage, and dynamic junction velocity. Based on these equations, we studied the behaviour of some electronic and electrical parameters under proton radiation. Theoretical results showed that the defects created by the irradiation change the carrier distribution and the carrier dynamic in the bulk of the base and then influence the solar cell electrical parameters (short-circuit current, open-circuit voltage, conversion efficiency). It appears also in this study that, at low fluence, junction dynamic velocity decreases due to the presence of tunnel defects. Obtained results could lead to improve the quality of the junction of a silicon solar cell.


---

## Body

## 1. Introduction

In order to improve the efficiency of solar cells embedded in space, mainly by satellites, deep studies about silicon solar cells in radiation environment have been carried with the main focus to build theoretical models for silicon space solar cell damage [1], leading experimental study to understand damage effect induced by gamma rays [2]. The study of semiconductor device behaviour under high-energy particles is extremely important. Several work articles have investigated the theoretical external factors dependence of solar cell output parameters and the analysis provided important information. The study of the influence of 1 MeV electron irradiation on electrical parameters of solar cell showed that radiation-induced defects are considered as recombination centers and lead to degradation of the open-circuit photovoltage Voc, the short-circuit photocurrent density Jsc, and the maximum power Pm [3–5]. The study of the impact of irradiation on shunt resistance showed that it is reduced with increasing of particle fluences [6]. It has been shown that radiation induced damage which manifests itself in important ways [7]: increase in leakage current, caused by the formation of mid-gap generation and recombination centers; generation of recombination centers proportionally to the increase of particle fluence; change of effective doping concentration of the cell; and reduction of charge collection efficiency due to charge carrier trapping in defect states within the bandgap. Some studies [8, 9] also show that at the first order, the degradation rate of the short-circuit current (Isc) of a solar cell is the function of the absorption coefficient of its composing material, while the degradation rate of the open-circuit voltage (Voc) depends on the regime of study, carrier’s diffusion or recombination. These two phenomena lead to the degradation of the solar cell maximum power.In these studies, the authors point out the effect irradiation on the behaviour (increase, decrease) of the solar cell electrical parameters but do not explain the causes of these effects. In this paper, we present a fundamental study that explains the causes of the effects of the irradiation on the solar cells. Firstly, we explain fundamentally the causes of the effects of the defects created by the irradiation on the carrier distribution in the bulk of the base. Secondly, we make the connection between the carrier distribution and the defects created by the irradiation of the solar cell to explain fundamentally the behaviour of the dynamic of the charge carriers at the junction of the solar cell, and then, on its electrical parameters.


## 2. Materials and Methods

### 2.1. Analytical Formulation

This study considers a solar cell irradiated with a low fluence (less than 1013 cm-2) of 10 MeV protons under multispectral illumination. In this range of energy, distribution of defects is uniform [10], so we can consider a simplified model in which diffusion length is sufficient to characterize the degradation of the solar cells under high-energy particles [11, 12]. The study has been done in the theory of quasineutral basis (QNB) hypothesis [13].Under these conditions, the equation of charge carrier distribution in the bulk of the base is given as follows:(1)∂2δx∂x2−δxLn∗2=−GxDn.In this expression,Gx=∑i=13ai⋅exp−bi⋅x is the carrier generation rate at x position [14, 15], Ln∗is the difusion length of the minority charge carriers, Dn is the minority carrier diffusion coefficient before irradiation, δx is the photogenerated minority carrier density at the depth x in the base, ai and bi are the coefficients deduced from modelling of the generation rate considered for overall the solar radiation spectrum under Air Mass 1.5 standard conditions [14].Ln∗ is given as follows:
(2)1Ln∗2=1L02+Kl⋅Φ,where L0 is the diffusion length before irradiation, Ф is the fluence, and Kl is the damage rate related to the material for a given energy of radiation (Kl=5.6.10−7) [16].By injecting Equation (2) into Equation (1), we obtained a general continuity equation of minority charge carriers which is a second-order differential equation with constant coefficients.The general solution of Equation (1) is given by [17–19]:
(3)δx,Φ=A∗coshα∗⋅x+B∗sinhα∗⋅x+∑i=13aiDnα∗2−bi2⋅exp−bi⋅x,with α∗=1/Ln∗ and α∗2−bi2≠0.The following boundary conditions were used [17, 19]:
(i)
At the junction (x=0)
(4)Dn⋅∂δx,Φ∂xx=0=Sf⋅δ0,Φ
(ii)
At the back surface (x=H)
(5)Dn⋅∂δx,Φ∂xx=H=−Sb⋅δH,ΦThe coefficients A∗ and B∗ can be determined by using the previous boundary Conditions (4) and (5). In Equations (4) and (5), the expressions Sf and Sb are, respectively, the junction dynamic velocity and the back surface recombination velocity.The junction dynamic velocitySf characterizes carrier behaviour at the solar cell junction trough two phenomena [20]:
(i)
The carrier intrinsic losses at the solar cell junction interface, characterised by the carrier intrinsic junction recombination velocitySf0
(ii)
The carrier diffusion through the solar cell junction imposed by the external load and characterized by the carrier diffusion velocitySfjThe junction dynamic velocitySf, which is a carrier collection rate at the junction, is the sum of these two components: Sf=Sf0+Sfj.


### 2.2. Expression of Electronic and Electrical Parameters

#### 2.2.1. Photocurrent Density

Photocurrent density expression is given as follows [17–19]:
(6)JphSf,Φ=qDn∂δx,Sf,Φ∂xx=0.The short-circuit current density is given by the relationship:(7)Jsc=limSf→∞Jph(8)Jsc=qDn⋅∑i=13aiDn⋅α∗2−bi2⋅α∗⋅Sb−Dn⋅bi⋅coshα∗⋅H−e−bi⋅H+Dn⋅α∗2−Sb⋅bi⋅sinhα∗⋅HSb⋅sinhα∗⋅H+Dn⋅α∗⋅coshα∗⋅H.


#### 2.2.2. Intrinsic Junction Recombination Velocity

The intrinsic junction recombination velocity is obtained by solving the equation as follows [17, 19]:
(9)∂Jph∂Sb=0forSb⟶∞.The solution of Equation (9) gives:
(10)Sf0=Dn⋅α∗⋅∑i=13ai/Dn⋅α∗2−bi2⋅α∗⋅sinhα∗⋅H+bi⋅coshα∗⋅H⋅e−bi⋅H−bi∑i=13ai/Dn⋅α∗2−bi2⋅α∗⋅coshα∗⋅H+bi⋅sinhα∗⋅H⋅e−bi⋅H−α∗.


#### 2.2.3. Photovoltage

The Boltzmann law leads to an expression of the photovoltage of the solar cell which depends on the charge carrier concentration at the junction. Its expression is as follows [17, 19]:
(11)VphSf=VT⋅lnδ0,Sfn0+1,with n0=ni2/NBand VT=kB.T/q.n0 is the density of the electrons at thermodynamic equilibrium; ni is the intrinsic concentration of the electrons (ni=1010cm−3); q is the elementary charge; NB is the doping density of the base (NB=1016cm−3); VT is the thermal voltage (VT=26mV) at T=300K; kB is the Boltzmann constant.
(12)Vph=VT⋅ln1n0⋅∑i=13aiDn⋅α∗2−bi2⋅Dn⋅α∗⋅Dn⋅bi−Sb⋅e−bi⋅H−coshα∗⋅HDn⋅α∗2+Sf⋅Sb⋅sinhα∗⋅H+Dn⋅α∗⋅Sf+Sb⋅coshα∗⋅H+Dn⋅Dn⋅α∗2−Sb⋅bi⋅sinhα∗⋅HDn⋅α∗2+Sf⋅Sb⋅sinhα∗⋅H+Dn⋅α∗⋅Sf+Sb⋅coshα∗⋅H+1.The open-circuit voltage is obtained for the dynamic junction velocity equal to zero:(13)Voc=VphSf=0.


#### 2.2.4. Conversion Efficiency

The solar cell conversion efficiency which is the relationship between the output power and the incident light power is given by the following equation:(14)η=PelSfmaxPinc,where Pincis the incident light power 1000W.m−2.


## 2.1. Analytical Formulation

This study considers a solar cell irradiated with a low fluence (less than 1013 cm-2) of 10 MeV protons under multispectral illumination. In this range of energy, distribution of defects is uniform [10], so we can consider a simplified model in which diffusion length is sufficient to characterize the degradation of the solar cells under high-energy particles [11, 12]. The study has been done in the theory of quasineutral basis (QNB) hypothesis [13].Under these conditions, the equation of charge carrier distribution in the bulk of the base is given as follows:(1)∂2δx∂x2−δxLn∗2=−GxDn.In this expression,Gx=∑i=13ai⋅exp−bi⋅x is the carrier generation rate at x position [14, 15], Ln∗is the difusion length of the minority charge carriers, Dn is the minority carrier diffusion coefficient before irradiation, δx is the photogenerated minority carrier density at the depth x in the base, ai and bi are the coefficients deduced from modelling of the generation rate considered for overall the solar radiation spectrum under Air Mass 1.5 standard conditions [14].Ln∗ is given as follows:
(2)1Ln∗2=1L02+Kl⋅Φ,where L0 is the diffusion length before irradiation, Ф is the fluence, and Kl is the damage rate related to the material for a given energy of radiation (Kl=5.6.10−7) [16].By injecting Equation (2) into Equation (1), we obtained a general continuity equation of minority charge carriers which is a second-order differential equation with constant coefficients.The general solution of Equation (1) is given by [17–19]:
(3)δx,Φ=A∗coshα∗⋅x+B∗sinhα∗⋅x+∑i=13aiDnα∗2−bi2⋅exp−bi⋅x,with α∗=1/Ln∗ and α∗2−bi2≠0.The following boundary conditions were used [17, 19]:
(i)
At the junction (x=0)
(4)Dn⋅∂δx,Φ∂xx=0=Sf⋅δ0,Φ
(ii)
At the back surface (x=H)
(5)Dn⋅∂δx,Φ∂xx=H=−Sb⋅δH,ΦThe coefficients A∗ and B∗ can be determined by using the previous boundary Conditions (4) and (5). In Equations (4) and (5), the expressions Sf and Sb are, respectively, the junction dynamic velocity and the back surface recombination velocity.The junction dynamic velocitySf characterizes carrier behaviour at the solar cell junction trough two phenomena [20]:
(i)
The carrier intrinsic losses at the solar cell junction interface, characterised by the carrier intrinsic junction recombination velocitySf0
(ii)
The carrier diffusion through the solar cell junction imposed by the external load and characterized by the carrier diffusion velocitySfjThe junction dynamic velocitySf, which is a carrier collection rate at the junction, is the sum of these two components: Sf=Sf0+Sfj.


## 2.2. Expression of Electronic and Electrical Parameters

### 2.2.1. Photocurrent Density

Photocurrent density expression is given as follows [17–19]:
(6)JphSf,Φ=qDn∂δx,Sf,Φ∂xx=0.The short-circuit current density is given by the relationship:(7)Jsc=limSf→∞Jph(8)Jsc=qDn⋅∑i=13aiDn⋅α∗2−bi2⋅α∗⋅Sb−Dn⋅bi⋅coshα∗⋅H−e−bi⋅H+Dn⋅α∗2−Sb⋅bi⋅sinhα∗⋅HSb⋅sinhα∗⋅H+Dn⋅α∗⋅coshα∗⋅H.


### 2.2.2. Intrinsic Junction Recombination Velocity

The intrinsic junction recombination velocity is obtained by solving the equation as follows [17, 19]:
(9)∂Jph∂Sb=0forSb⟶∞.The solution of Equation (9) gives:
(10)Sf0=Dn⋅α∗⋅∑i=13ai/Dn⋅α∗2−bi2⋅α∗⋅sinhα∗⋅H+bi⋅coshα∗⋅H⋅e−bi⋅H−bi∑i=13ai/Dn⋅α∗2−bi2⋅α∗⋅coshα∗⋅H+bi⋅sinhα∗⋅H⋅e−bi⋅H−α∗.


### 2.2.3. Photovoltage

The Boltzmann law leads to an expression of the photovoltage of the solar cell which depends on the charge carrier concentration at the junction. Its expression is as follows [17, 19]:
(11)VphSf=VT⋅lnδ0,Sfn0+1,with n0=ni2/NBand VT=kB.T/q.n0 is the density of the electrons at thermodynamic equilibrium; ni is the intrinsic concentration of the electrons (ni=1010cm−3); q is the elementary charge; NB is the doping density of the base (NB=1016cm−3); VT is the thermal voltage (VT=26mV) at T=300K; kB is the Boltzmann constant.
(12)Vph=VT⋅ln1n0⋅∑i=13aiDn⋅α∗2−bi2⋅Dn⋅α∗⋅Dn⋅bi−Sb⋅e−bi⋅H−coshα∗⋅HDn⋅α∗2+Sf⋅Sb⋅sinhα∗⋅H+Dn⋅α∗⋅Sf+Sb⋅coshα∗⋅H+Dn⋅Dn⋅α∗2−Sb⋅bi⋅sinhα∗⋅HDn⋅α∗2+Sf⋅Sb⋅sinhα∗⋅H+Dn⋅α∗⋅Sf+Sb⋅coshα∗⋅H+1.The open-circuit voltage is obtained for the dynamic junction velocity equal to zero:(13)Voc=VphSf=0.


### 2.2.4. Conversion Efficiency

The solar cell conversion efficiency which is the relationship between the output power and the incident light power is given by the following equation:(14)η=PelSfmaxPinc,where Pincis the incident light power 1000W.m−2.


## 2.2.1. Photocurrent Density

Photocurrent density expression is given as follows [17–19]:
(6)JphSf,Φ=qDn∂δx,Sf,Φ∂xx=0.The short-circuit current density is given by the relationship:(7)Jsc=limSf→∞Jph(8)Jsc=qDn⋅∑i=13aiDn⋅α∗2−bi2⋅α∗⋅Sb−Dn⋅bi⋅coshα∗⋅H−e−bi⋅H+Dn⋅α∗2−Sb⋅bi⋅sinhα∗⋅HSb⋅sinhα∗⋅H+Dn⋅α∗⋅coshα∗⋅H.


## 2.2.2. Intrinsic Junction Recombination Velocity

The intrinsic junction recombination velocity is obtained by solving the equation as follows [17, 19]:
(9)∂Jph∂Sb=0forSb⟶∞.The solution of Equation (9) gives:
(10)Sf0=Dn⋅α∗⋅∑i=13ai/Dn⋅α∗2−bi2⋅α∗⋅sinhα∗⋅H+bi⋅coshα∗⋅H⋅e−bi⋅H−bi∑i=13ai/Dn⋅α∗2−bi2⋅α∗⋅coshα∗⋅H+bi⋅sinhα∗⋅H⋅e−bi⋅H−α∗.


## 2.2.3. Photovoltage

The Boltzmann law leads to an expression of the photovoltage of the solar cell which depends on the charge carrier concentration at the junction. Its expression is as follows [17, 19]:
(11)VphSf=VT⋅lnδ0,Sfn0+1,with n0=ni2/NBand VT=kB.T/q.n0 is the density of the electrons at thermodynamic equilibrium; ni is the intrinsic concentration of the electrons (ni=1010cm−3); q is the elementary charge; NB is the doping density of the base (NB=1016cm−3); VT is the thermal voltage (VT=26mV) at T=300K; kB is the Boltzmann constant.
(12)Vph=VT⋅ln1n0⋅∑i=13aiDn⋅α∗2−bi2⋅Dn⋅α∗⋅Dn⋅bi−Sb⋅e−bi⋅H−coshα∗⋅HDn⋅α∗2+Sf⋅Sb⋅sinhα∗⋅H+Dn⋅α∗⋅Sf+Sb⋅coshα∗⋅H+Dn⋅Dn⋅α∗2−Sb⋅bi⋅sinhα∗⋅HDn⋅α∗2+Sf⋅Sb⋅sinhα∗⋅H+Dn⋅α∗⋅Sf+Sb⋅coshα∗⋅H+1.The open-circuit voltage is obtained for the dynamic junction velocity equal to zero:(13)Voc=VphSf=0.


## 2.2.4. Conversion Efficiency

The solar cell conversion efficiency which is the relationship between the output power and the incident light power is given by the following equation:(14)η=PelSfmaxPinc,where Pincis the incident light power 1000W.m−2.


## 3. Results and Discussion

### 3.1. Influence of Irradiation on Electronic Parameters

#### 3.1.1. Influence of Irradiation on the Junction Dynamic Velocity

The influence of the irradiation on the solar cell manifests itself through its influence on the quality of the solar cell material, and then, on the quantity of carriers that escape from the recombination to produce a current. The junction dynamic velocitySfSf=Sf0+Sfj characterizes both carrier intrinsic recombination at the junction (Sf0) and the carrier diffusion through the junction (Sfj) [16, 21]. For a given illumination, and a given external load (Sfj), the quantity of carriers that can be collected at the junction of the solar cell (related to Sf) will vary with the quality of the material of the solar cell, and then, with the carrier intrinsic recombination at the junction (related to Sf0). We can then deduce that the influence of the particle fluence on the junction dynamic velocity results in its influence on the intrinsic recombination velocity Sf0 of the solar cell.The curve in Figure1 shows the effect of the increase of particle’s fluence on the solar cell junction dynamic velocity Sf=Sf0+Sfj. In the expression of the junction dynamic velocity (Sf), Sf0 is the intrinsic junction recombination velocity given in Equation (10), and we consider an intermediate operating value of carrier diffusion velocity Sfj: Sfj=104cm.s−1.Figure 1
Junction dynamic velocity versus fluence.It appears on this curve that with the increase of the fluence, the junction dynamic velocity has two behaviours.For a low value of the fluence (Φ<1010P.cm‐2), we observe a decrease of the junction recombination velocity. When we increase the particle fluence, we increase also the defects in the materials of the solar cell and at its junction.With the increase of the defects at the junction of the solar cell, the quantity of carriers that must be collected at the junction will be reduced, and then will lead to a reduction of the carrier diffusion velocity (Sfj) and beyond a reduction of the junction dynamic velocity (Sf). For a given operating point, the increase of the fluence leads to a situation where the quantity of defects created at the junction of the solar cell recombines all the carriers that must cross this junction. So, the carrier diffusion velocity is null (Sfj=0). This situation corresponds to the minimum of the curve and for this particular value of the fluence of particles (Φ=1010P.cm‐2), the junction dynamic velocity (Sf) equals to intrinsic recombination velocity at the junction (Sf0). Beyond this value of the fluence (Φ=1010P.cm‐2), the quantity of defects created at the junction is more than the quantity of carriers that must cross the junction, and then, the excess of the defects leads to an increase of the intrinsic recombination velocity (Sf0). For these values of the fluence, the increase of the junction dynamic velocity is only the consequence of the increase of the intrinsic recombination velocity due to the material ionisation by the fluence of particles.


#### 3.1.2. Influence of Irradiation on Carrier Density Profile in the Base

The curves in Figure2 show the effect of the particle’s fluence on the variation of carrier density along the base depth of the solar cell in short-circuit situation.Figure 2
Carrier density versus base depth for different values of the fluence (L0=0.02cm; Sb=4.104cm.s−1; Sf=8.108cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).We observe on these curves that the maxima of the carrier density decreases with the increase of the fluence of particles. The decrease of the carrier density maxima characterizes a reduction of photogenerated carriers in each positionx in the bulk of the base. This situation can be explained by the fact that the increase of the fluence leads to a generation of more defects which acts mainly as recombination centers [22].Indeed, the increase of the fluence means an increase of the quantity of particles that reaches on the solar cell. These particles create displacements and ionisation defects in the lattice of the material [23] and then lead to the increase of carrier recombination in the base. The decrease of carrier density with the increase of the fluence should result in the decrease of the solar cell photocurrent and therefore to an increase of its series resistance [24].We also note that with the increase of the fluence, the peaks of the curves move toward the junction of the solar cell. This phenomenon is interpreted as the consequence of the increase of the defects with the increase of the fluence and that leads to a reduction of the quantity of carriers which cross the junction to participate to the photocurrent. This phenomenon is also the consequence of the fact that radiation increases the depth of the space charge region by producing diffusion of phosphorus atoms in the emitter [25].The curves in Figure3 show the effect of the increase of particle’s fluence on the charge carrier density for various positions in the base of the solar cell.Figure 3
Carrier density versus particle’s flux at various positionx in the base (L0=0.02cm; Sb=4.104cm.s−1; Sf=8.108cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).We observe on these curves that at various position in the bulk of the base, carrier density decreases with the increase of the particle’s fluence reaching on the solar cell. This phenomenon is due to the increase of the defects which acts as recombination centers.We also note that for the values of the particle’s fluence less than 7.107 P.cm-2 (Φ<7.107P.cm‐2) the carrier densities at the base depth near the junction (x<0.004) are smaller than those of the farthest depths of the base (x>0.004). This situation can be explained by the fact that the effect of the small values of the fluence (Φ<7.107P.cm‐2) on the carrier density is not noticeable. So, all the carriers photogenerated near the junction cross this region to participate to the photocurrent, and this situation explains the small values of the carrier densities in the region near the junction. For more remote positions of the junction (x>0.004), the carriers have the time to accumulate, and this situation explains the high values of the carrier density at this position of the junction.We observe also that for the values of fluence higher than 7.107 P.cm-2 (Φ>7.107P.cm‐2), the carrier density decreases as we go far in the depth of the base.These results are also in agreement with the inversion of the curve of junction dynamic velocity observed in Figure1 while the fluence is equal to Φ=7.107P.cm‐2, but also with the decrease of carrier density in the base depth with the increase of particle’s fluence observed on the curve in Figure 3.Indeed, the high values of the carrier density observed for the low fluence of the particles (Φ<7.107P.cm‐2) at the region near the junction (x<0.004) are the consequence of the reduction of the carrier diffusion velocity (Sfj) with the increase of the fluence observed in Figure 1. The decrease of the carrier density in the deeper regions of the base is the consequence of the increase of carrier recombination in the base, due to the increase of defects in the lattice of the material with the increase of the fluence of the particles observed in Figure 3.


### 3.2. Influence of Irradiation on Electrical Parameters

#### 3.2.1. Influence of Irradiation on Short-Circuit Current Density

The curve in Figure4 shows the effect of the increase of particle’s fluence on the short-circuit current density of the solar cell.Figure 4
Short-circuit current density versus fluence (L0=0.02cm; Sb=4.104cm.s−1; Sf=8.108cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).We observe on this curve that with the increase of the fluence of the particles on the solar cell, the short-circuit current density decreases. The decrease of the short-circuit current density results from the reduction of the quantity of carriers which cross the junction of the solar cell to participate to the photocurrent, and then, is and accordance none the less with the decrease of junction diffusion velocity [24] but also with the decrease of carrier density when the fluence of the particles on the solar cell increases.


#### 3.2.2. Influence of Irradiation on Open-Circuit Voltage

The curve in Figure5 shows the effect of the increase of particle’s fluence on the open-circuit voltage of the solar cell.Figure 5
Open-circuit voltage versus fluence (L0=0.02cm; Sb=4.104cm.s−1; Sf=0cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).The photovoltage is directly linked to the carrier concentration in the bulk of the base. As the increase of the fluence of the particles leads to an increase of the material defects by ionisation, then to an increase of carrier recombination in the base, it will result a reduction of carrier concentration in the base. That explains the decrease of the open-circuit voltage with the increase of the fluence of the particles on the solar cell. This is also in accordance with the decrease of carrier’s density and the increase of diffusion velocity with the increase of the fluence of the particles on the solar cell. Indeed, these two situations lead to a reduction of carrier concentration in the base.


#### 3.2.3. Influence of Irradiation on the Efficiency

The curve in Figure6 shows the effect of the increase of the fluence of the particles on the conversion efficiency of the solar cell.Figure 6
Efficiency versus fluences (L0=0.02cm; Sb=4.104cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7; Sf=4.104cm.s−1).It appears on this figure that the conversion efficiency of the solar cell decreases when the fluence of the particles on the solar cell increases. This result is in accordance with the behavior of the short-circuit current density and the open-circuit voltage under the effect of the intensification of the irradiation. Indeed, as the conversion efficiency is function of the electric power, and then of the photocurrent and the photovoltage, their decrease will lead to a decrease of the conversion efficiency.We present in Table1 the values of the short-circuit current density, the open-circuit voltage, the maximum electric power, and the conversion efficiency of the solar cell for five different values of fluence.Table 1
Effect of the irradiation on the solar cell electrical parameters.
Ф (P.cm-2)
Jsc(A.cm-2)
Voc (V)
Pm(W.cm-2)
η%
0
0.031176
0.60206
0.015561
15.561
109
0.030976
0.5996
0.015399
15.399
1010
0.029629
0.58653
0.014385
14.385
1011
0.025494
0.55851
0.011668
11.668
1012
0.020029
0.52351
0.00844926
6.589We observe on the results of this table that all the electrical parameters (short-circuit current density, open-circuit voltage, maximum electric power, and conversion efficiency) decrease with the increase of the intensity of the fluence of the particles on the solar cell. The irradiation affects then strongly the electrical parameters of the solar cell. For an increase of the proton radiation fluence from 0 to 1012 P.cm-2, the efficiency drops from 15.561% to 6.589%.


## 3.1. Influence of Irradiation on Electronic Parameters

### 3.1.1. Influence of Irradiation on the Junction Dynamic Velocity

The influence of the irradiation on the solar cell manifests itself through its influence on the quality of the solar cell material, and then, on the quantity of carriers that escape from the recombination to produce a current. The junction dynamic velocitySfSf=Sf0+Sfj characterizes both carrier intrinsic recombination at the junction (Sf0) and the carrier diffusion through the junction (Sfj) [16, 21]. For a given illumination, and a given external load (Sfj), the quantity of carriers that can be collected at the junction of the solar cell (related to Sf) will vary with the quality of the material of the solar cell, and then, with the carrier intrinsic recombination at the junction (related to Sf0). We can then deduce that the influence of the particle fluence on the junction dynamic velocity results in its influence on the intrinsic recombination velocity Sf0 of the solar cell.The curve in Figure1 shows the effect of the increase of particle’s fluence on the solar cell junction dynamic velocity Sf=Sf0+Sfj. In the expression of the junction dynamic velocity (Sf), Sf0 is the intrinsic junction recombination velocity given in Equation (10), and we consider an intermediate operating value of carrier diffusion velocity Sfj: Sfj=104cm.s−1.Figure 1
Junction dynamic velocity versus fluence.It appears on this curve that with the increase of the fluence, the junction dynamic velocity has two behaviours.For a low value of the fluence (Φ<1010P.cm‐2), we observe a decrease of the junction recombination velocity. When we increase the particle fluence, we increase also the defects in the materials of the solar cell and at its junction.With the increase of the defects at the junction of the solar cell, the quantity of carriers that must be collected at the junction will be reduced, and then will lead to a reduction of the carrier diffusion velocity (Sfj) and beyond a reduction of the junction dynamic velocity (Sf). For a given operating point, the increase of the fluence leads to a situation where the quantity of defects created at the junction of the solar cell recombines all the carriers that must cross this junction. So, the carrier diffusion velocity is null (Sfj=0). This situation corresponds to the minimum of the curve and for this particular value of the fluence of particles (Φ=1010P.cm‐2), the junction dynamic velocity (Sf) equals to intrinsic recombination velocity at the junction (Sf0). Beyond this value of the fluence (Φ=1010P.cm‐2), the quantity of defects created at the junction is more than the quantity of carriers that must cross the junction, and then, the excess of the defects leads to an increase of the intrinsic recombination velocity (Sf0). For these values of the fluence, the increase of the junction dynamic velocity is only the consequence of the increase of the intrinsic recombination velocity due to the material ionisation by the fluence of particles.


### 3.1.2. Influence of Irradiation on Carrier Density Profile in the Base

The curves in Figure2 show the effect of the particle’s fluence on the variation of carrier density along the base depth of the solar cell in short-circuit situation.Figure 2
Carrier density versus base depth for different values of the fluence (L0=0.02cm; Sb=4.104cm.s−1; Sf=8.108cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).We observe on these curves that the maxima of the carrier density decreases with the increase of the fluence of particles. The decrease of the carrier density maxima characterizes a reduction of photogenerated carriers in each positionx in the bulk of the base. This situation can be explained by the fact that the increase of the fluence leads to a generation of more defects which acts mainly as recombination centers [22].Indeed, the increase of the fluence means an increase of the quantity of particles that reaches on the solar cell. These particles create displacements and ionisation defects in the lattice of the material [23] and then lead to the increase of carrier recombination in the base. The decrease of carrier density with the increase of the fluence should result in the decrease of the solar cell photocurrent and therefore to an increase of its series resistance [24].We also note that with the increase of the fluence, the peaks of the curves move toward the junction of the solar cell. This phenomenon is interpreted as the consequence of the increase of the defects with the increase of the fluence and that leads to a reduction of the quantity of carriers which cross the junction to participate to the photocurrent. This phenomenon is also the consequence of the fact that radiation increases the depth of the space charge region by producing diffusion of phosphorus atoms in the emitter [25].The curves in Figure3 show the effect of the increase of particle’s fluence on the charge carrier density for various positions in the base of the solar cell.Figure 3
Carrier density versus particle’s flux at various positionx in the base (L0=0.02cm; Sb=4.104cm.s−1; Sf=8.108cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).We observe on these curves that at various position in the bulk of the base, carrier density decreases with the increase of the particle’s fluence reaching on the solar cell. This phenomenon is due to the increase of the defects which acts as recombination centers.We also note that for the values of the particle’s fluence less than 7.107 P.cm-2 (Φ<7.107P.cm‐2) the carrier densities at the base depth near the junction (x<0.004) are smaller than those of the farthest depths of the base (x>0.004). This situation can be explained by the fact that the effect of the small values of the fluence (Φ<7.107P.cm‐2) on the carrier density is not noticeable. So, all the carriers photogenerated near the junction cross this region to participate to the photocurrent, and this situation explains the small values of the carrier densities in the region near the junction. For more remote positions of the junction (x>0.004), the carriers have the time to accumulate, and this situation explains the high values of the carrier density at this position of the junction.We observe also that for the values of fluence higher than 7.107 P.cm-2 (Φ>7.107P.cm‐2), the carrier density decreases as we go far in the depth of the base.These results are also in agreement with the inversion of the curve of junction dynamic velocity observed in Figure1 while the fluence is equal to Φ=7.107P.cm‐2, but also with the decrease of carrier density in the base depth with the increase of particle’s fluence observed on the curve in Figure 3.Indeed, the high values of the carrier density observed for the low fluence of the particles (Φ<7.107P.cm‐2) at the region near the junction (x<0.004) are the consequence of the reduction of the carrier diffusion velocity (Sfj) with the increase of the fluence observed in Figure 1. The decrease of the carrier density in the deeper regions of the base is the consequence of the increase of carrier recombination in the base, due to the increase of defects in the lattice of the material with the increase of the fluence of the particles observed in Figure 3.


## 3.1.1. Influence of Irradiation on the Junction Dynamic Velocity

The influence of the irradiation on the solar cell manifests itself through its influence on the quality of the solar cell material, and then, on the quantity of carriers that escape from the recombination to produce a current. The junction dynamic velocitySfSf=Sf0+Sfj characterizes both carrier intrinsic recombination at the junction (Sf0) and the carrier diffusion through the junction (Sfj) [16, 21]. For a given illumination, and a given external load (Sfj), the quantity of carriers that can be collected at the junction of the solar cell (related to Sf) will vary with the quality of the material of the solar cell, and then, with the carrier intrinsic recombination at the junction (related to Sf0). We can then deduce that the influence of the particle fluence on the junction dynamic velocity results in its influence on the intrinsic recombination velocity Sf0 of the solar cell.The curve in Figure1 shows the effect of the increase of particle’s fluence on the solar cell junction dynamic velocity Sf=Sf0+Sfj. In the expression of the junction dynamic velocity (Sf), Sf0 is the intrinsic junction recombination velocity given in Equation (10), and we consider an intermediate operating value of carrier diffusion velocity Sfj: Sfj=104cm.s−1.Figure 1
Junction dynamic velocity versus fluence.It appears on this curve that with the increase of the fluence, the junction dynamic velocity has two behaviours.For a low value of the fluence (Φ<1010P.cm‐2), we observe a decrease of the junction recombination velocity. When we increase the particle fluence, we increase also the defects in the materials of the solar cell and at its junction.With the increase of the defects at the junction of the solar cell, the quantity of carriers that must be collected at the junction will be reduced, and then will lead to a reduction of the carrier diffusion velocity (Sfj) and beyond a reduction of the junction dynamic velocity (Sf). For a given operating point, the increase of the fluence leads to a situation where the quantity of defects created at the junction of the solar cell recombines all the carriers that must cross this junction. So, the carrier diffusion velocity is null (Sfj=0). This situation corresponds to the minimum of the curve and for this particular value of the fluence of particles (Φ=1010P.cm‐2), the junction dynamic velocity (Sf) equals to intrinsic recombination velocity at the junction (Sf0). Beyond this value of the fluence (Φ=1010P.cm‐2), the quantity of defects created at the junction is more than the quantity of carriers that must cross the junction, and then, the excess of the defects leads to an increase of the intrinsic recombination velocity (Sf0). For these values of the fluence, the increase of the junction dynamic velocity is only the consequence of the increase of the intrinsic recombination velocity due to the material ionisation by the fluence of particles.


## 3.1.2. Influence of Irradiation on Carrier Density Profile in the Base

The curves in Figure2 show the effect of the particle’s fluence on the variation of carrier density along the base depth of the solar cell in short-circuit situation.Figure 2
Carrier density versus base depth for different values of the fluence (L0=0.02cm; Sb=4.104cm.s−1; Sf=8.108cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).We observe on these curves that the maxima of the carrier density decreases with the increase of the fluence of particles. The decrease of the carrier density maxima characterizes a reduction of photogenerated carriers in each positionx in the bulk of the base. This situation can be explained by the fact that the increase of the fluence leads to a generation of more defects which acts mainly as recombination centers [22].Indeed, the increase of the fluence means an increase of the quantity of particles that reaches on the solar cell. These particles create displacements and ionisation defects in the lattice of the material [23] and then lead to the increase of carrier recombination in the base. The decrease of carrier density with the increase of the fluence should result in the decrease of the solar cell photocurrent and therefore to an increase of its series resistance [24].We also note that with the increase of the fluence, the peaks of the curves move toward the junction of the solar cell. This phenomenon is interpreted as the consequence of the increase of the defects with the increase of the fluence and that leads to a reduction of the quantity of carriers which cross the junction to participate to the photocurrent. This phenomenon is also the consequence of the fact that radiation increases the depth of the space charge region by producing diffusion of phosphorus atoms in the emitter [25].The curves in Figure3 show the effect of the increase of particle’s fluence on the charge carrier density for various positions in the base of the solar cell.Figure 3
Carrier density versus particle’s flux at various positionx in the base (L0=0.02cm; Sb=4.104cm.s−1; Sf=8.108cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).We observe on these curves that at various position in the bulk of the base, carrier density decreases with the increase of the particle’s fluence reaching on the solar cell. This phenomenon is due to the increase of the defects which acts as recombination centers.We also note that for the values of the particle’s fluence less than 7.107 P.cm-2 (Φ<7.107P.cm‐2) the carrier densities at the base depth near the junction (x<0.004) are smaller than those of the farthest depths of the base (x>0.004). This situation can be explained by the fact that the effect of the small values of the fluence (Φ<7.107P.cm‐2) on the carrier density is not noticeable. So, all the carriers photogenerated near the junction cross this region to participate to the photocurrent, and this situation explains the small values of the carrier densities in the region near the junction. For more remote positions of the junction (x>0.004), the carriers have the time to accumulate, and this situation explains the high values of the carrier density at this position of the junction.We observe also that for the values of fluence higher than 7.107 P.cm-2 (Φ>7.107P.cm‐2), the carrier density decreases as we go far in the depth of the base.These results are also in agreement with the inversion of the curve of junction dynamic velocity observed in Figure1 while the fluence is equal to Φ=7.107P.cm‐2, but also with the decrease of carrier density in the base depth with the increase of particle’s fluence observed on the curve in Figure 3.Indeed, the high values of the carrier density observed for the low fluence of the particles (Φ<7.107P.cm‐2) at the region near the junction (x<0.004) are the consequence of the reduction of the carrier diffusion velocity (Sfj) with the increase of the fluence observed in Figure 1. The decrease of the carrier density in the deeper regions of the base is the consequence of the increase of carrier recombination in the base, due to the increase of defects in the lattice of the material with the increase of the fluence of the particles observed in Figure 3.


## 3.2. Influence of Irradiation on Electrical Parameters

### 3.2.1. Influence of Irradiation on Short-Circuit Current Density

The curve in Figure4 shows the effect of the increase of particle’s fluence on the short-circuit current density of the solar cell.Figure 4
Short-circuit current density versus fluence (L0=0.02cm; Sb=4.104cm.s−1; Sf=8.108cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).We observe on this curve that with the increase of the fluence of the particles on the solar cell, the short-circuit current density decreases. The decrease of the short-circuit current density results from the reduction of the quantity of carriers which cross the junction of the solar cell to participate to the photocurrent, and then, is and accordance none the less with the decrease of junction diffusion velocity [24] but also with the decrease of carrier density when the fluence of the particles on the solar cell increases.


### 3.2.2. Influence of Irradiation on Open-Circuit Voltage

The curve in Figure5 shows the effect of the increase of particle’s fluence on the open-circuit voltage of the solar cell.Figure 5
Open-circuit voltage versus fluence (L0=0.02cm; Sb=4.104cm.s−1; Sf=0cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).The photovoltage is directly linked to the carrier concentration in the bulk of the base. As the increase of the fluence of the particles leads to an increase of the material defects by ionisation, then to an increase of carrier recombination in the base, it will result a reduction of carrier concentration in the base. That explains the decrease of the open-circuit voltage with the increase of the fluence of the particles on the solar cell. This is also in accordance with the decrease of carrier’s density and the increase of diffusion velocity with the increase of the fluence of the particles on the solar cell. Indeed, these two situations lead to a reduction of carrier concentration in the base.


### 3.2.3. Influence of Irradiation on the Efficiency

The curve in Figure6 shows the effect of the increase of the fluence of the particles on the conversion efficiency of the solar cell.Figure 6
Efficiency versus fluences (L0=0.02cm; Sb=4.104cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7; Sf=4.104cm.s−1).It appears on this figure that the conversion efficiency of the solar cell decreases when the fluence of the particles on the solar cell increases. This result is in accordance with the behavior of the short-circuit current density and the open-circuit voltage under the effect of the intensification of the irradiation. Indeed, as the conversion efficiency is function of the electric power, and then of the photocurrent and the photovoltage, their decrease will lead to a decrease of the conversion efficiency.We present in Table1 the values of the short-circuit current density, the open-circuit voltage, the maximum electric power, and the conversion efficiency of the solar cell for five different values of fluence.Table 1
Effect of the irradiation on the solar cell electrical parameters.
Ф (P.cm-2)
Jsc(A.cm-2)
Voc (V)
Pm(W.cm-2)
η%
0
0.031176
0.60206
0.015561
15.561
109
0.030976
0.5996
0.015399
15.399
1010
0.029629
0.58653
0.014385
14.385
1011
0.025494
0.55851
0.011668
11.668
1012
0.020029
0.52351
0.00844926
6.589We observe on the results of this table that all the electrical parameters (short-circuit current density, open-circuit voltage, maximum electric power, and conversion efficiency) decrease with the increase of the intensity of the fluence of the particles on the solar cell. The irradiation affects then strongly the electrical parameters of the solar cell. For an increase of the proton radiation fluence from 0 to 1012 P.cm-2, the efficiency drops from 15.561% to 6.589%.


## 3.2.1. Influence of Irradiation on Short-Circuit Current Density

The curve in Figure4 shows the effect of the increase of particle’s fluence on the short-circuit current density of the solar cell.Figure 4
Short-circuit current density versus fluence (L0=0.02cm; Sb=4.104cm.s−1; Sf=8.108cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).We observe on this curve that with the increase of the fluence of the particles on the solar cell, the short-circuit current density decreases. The decrease of the short-circuit current density results from the reduction of the quantity of carriers which cross the junction of the solar cell to participate to the photocurrent, and then, is and accordance none the less with the decrease of junction diffusion velocity [24] but also with the decrease of carrier density when the fluence of the particles on the solar cell increases.


## 3.2.2. Influence of Irradiation on Open-Circuit Voltage

The curve in Figure5 shows the effect of the increase of particle’s fluence on the open-circuit voltage of the solar cell.Figure 5
Open-circuit voltage versus fluence (L0=0.02cm; Sb=4.104cm.s−1; Sf=0cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7).The photovoltage is directly linked to the carrier concentration in the bulk of the base. As the increase of the fluence of the particles leads to an increase of the material defects by ionisation, then to an increase of carrier recombination in the base, it will result a reduction of carrier concentration in the base. That explains the decrease of the open-circuit voltage with the increase of the fluence of the particles on the solar cell. This is also in accordance with the decrease of carrier’s density and the increase of diffusion velocity with the increase of the fluence of the particles on the solar cell. Indeed, these two situations lead to a reduction of carrier concentration in the base.


## 3.2.3. Influence of Irradiation on the Efficiency

The curve in Figure6 shows the effect of the increase of the fluence of the particles on the conversion efficiency of the solar cell.Figure 6
Efficiency versus fluences (L0=0.02cm; Sb=4.104cm.s−1; Dn=27cm2.s−1; H=0.03cm; Kl=5.6.10−7; Sf=4.104cm.s−1).It appears on this figure that the conversion efficiency of the solar cell decreases when the fluence of the particles on the solar cell increases. This result is in accordance with the behavior of the short-circuit current density and the open-circuit voltage under the effect of the intensification of the irradiation. Indeed, as the conversion efficiency is function of the electric power, and then of the photocurrent and the photovoltage, their decrease will lead to a decrease of the conversion efficiency.We present in Table1 the values of the short-circuit current density, the open-circuit voltage, the maximum electric power, and the conversion efficiency of the solar cell for five different values of fluence.Table 1
Effect of the irradiation on the solar cell electrical parameters.
Ф (P.cm-2)
Jsc(A.cm-2)
Voc (V)
Pm(W.cm-2)
η%
0
0.031176
0.60206
0.015561
15.561
109
0.030976
0.5996
0.015399
15.399
1010
0.029629
0.58653
0.014385
14.385
1011
0.025494
0.55851
0.011668
11.668
1012
0.020029
0.52351
0.00844926
6.589We observe on the results of this table that all the electrical parameters (short-circuit current density, open-circuit voltage, maximum electric power, and conversion efficiency) decrease with the increase of the intensity of the fluence of the particles on the solar cell. The irradiation affects then strongly the electrical parameters of the solar cell. For an increase of the proton radiation fluence from 0 to 1012 P.cm-2, the efficiency drops from 15.561% to 6.589%.


## 4. Conclusions

This work put in evidence the effect of the particle (protons) irradiation on the performance of a silicon solar cell. By solving the continuity equation with the assumption that proton damage results in a modification of the diffusion length, we obtained new expressions of the excess minority carrier density, junction dynamic velocity, photocurrent density, and photovoltage. It appears through the study of the radiation influence on these parameters that the carrier density and the junction dynamic velocity decrease with the increase of the fluence of the particles in all levels of the base depth. Fundamentally, this situation can be explained by the fact that the increase of the irradiation increases the ionisation of the solar cell materials by the incident particles and then leads to the increase of the defects in the bulk of the base and at the solar cell junction. The increase of the defects increases the recombination of the carriers photogenerated in the base and then leads none the less to the reduction of the carrier density in the base but also to the reduction of carrier diffusion through the junction. This study put in evidence also a critical value of the irradiation (Φ=1010P.cm‐2) for which the junction dynamic velocity (Sf) equals to intrinsic recombination velocity at the junction (Sf0). For this value of the fluence, all the carriers that must cross the junction are recombined intrinsically at the junction (Sj=0). We observe also in this study that in accordance with the reduction of carrier density in the base of the solar cell with the increase of the fluence, the short-circuit current density, the open-circuit voltage, the maximum electric power, and the conversion efficiency decrease also. These results show the strong influence of the irradiation on the performance of the silicon solar cell.



---

*Source: 8306492-2019-12-29.xml*