1. Introduction
To improve (or control) the quality (performance) of solar cells [1] [2], especially silicon, the recombination parameters of minority carriers, in the bulk (volume) and on interfaces, are the subject of theoretical and experimental investigations [3] [4] [5].
The determination of the recombination in the bulk (lifetime) of minority carriers in the solar cell base [6] [7] is influenced by:
1) The theoretical 1D or 3D study model (crystallography, grain size and thickness of different regions) [8] [9] [10].
2) Recombination at the interfaces, i.e., at the front of the n^{+} emitter (Se), at junction n^{+}/p or SCR (Sf), on the rear side p/p^{+} (Sb) of the base [11] [12] [13] [14] [15].
3) The solar cell’s operating regime under dark or illumination, can be: steady state [16] [17], transient [18] [19] or frequency dynamics [20] [21].
4) The equivalent electric model associated with the solar cell, according to the operating regime [22] [23].
5) External conditions applied to solar cell i.e.: mono or polychromatic illumination [24], temperature (T) [25], electromagnetic field (E, B) [26], irradiation flow (ϕp) by charged particles [27].
It is therefore clear that it is important to carry out the investigations, highlighting, the physical mechanisms of recombination (volume or surface) in each case, and in each region of the solar cell, taking into account the geometric parameters (thickness), in order to dissociate their contribution [28] [29] [30] [31].
Some studies have focused on both the lifetime and the AC back surface recombination velocity of excess minority carriers in the base of the silicon solar cell, in order to dissociate their effects under different external conditions [32] [33].
Our study brings, an exploration by the diagrams of Bode and Nyquist, the AC back surface recombination velocity of minority carriers’ expression [34] [35], deduced on a silicon solar cell maintained at temperature (T), illuminated by the front (n^{+}/p) of the base of thickness (H), by a modulated monochromatic light of short wavelength (α(λ)).
2. Theoretical Modele
The structure of the n^{+}pp^{+} silicon solar cell under front monochromatic illumination [11] [36] in frequency modulation, is given by Figure 1.
The excess minority carriers’ density
$\delta \left(x,t\right)$ generated in the base of the solar cell at T temperature and under modulated monochromatic illumination, obeys to the continuity equation [37] [38] [39] given as:
$D\left(\omega ,T\right)\times \frac{{\partial}^{2}\delta \left(x,t\right)}{\partial {x}^{2}}\frac{\delta \left(x,t\right)}{\tau}=G\left(x,\omega ,t\right)+\frac{\partial \delta \left(x,t\right)}{\partial t}$ (1)
The excess minority carriers’ density expression in the (p) base, can be written, according to the space coordinates (x) and the time t, as:
$\delta \left(x,t\right)=\delta \left(x\right)\cdot {\text{e}}^{j\omega t}$ (2)
 AC carrier generation rate
$G\left(x,t\right)$ is given by the relationship:
$G\left(x,t\right)=g\left(x\right)\cdot {\text{e}}^{j\omega t}$ (3)
With the space component [40] written as:
Figure 1. Structure of front illuminated silicon solar cell with monochromatic light.
$g\left(x\right)=\alpha \left(\lambda \right)\cdot {I}_{0}\left(\lambda \right)\cdot \left(1R\left(\lambda \right)\right)\cdot {\text{e}}^{\alpha \left(\lambda \right)\cdot x}$ (4)
I_{0}, is the incident monochromatic flux,
$\alpha \left(\lambda \right)$ and
$R\left(\lambda \right)$ are both the absoption and reflection coefficients of the Si material.

$D\left(\omega ,T\right)$ is the complex diffusion coefficient of excess minority carrier in the base at T temperature. Its expression is given by the relationship [41]:
$D\left(\omega ,T\right)=D\left(T\right)\times \left(\frac{1j\cdot {\omega}^{2}\cdot {\tau}^{2}}{1+{\left(\omega \tau \right)}^{2}}\right)$ (5)
$D\left(T\right)$ is the temperaturedependent diffusion coefficient given by Einstein’s relationship:
$D\left(T\right)=\frac{\mu \left(T\right)\cdot {K}_{b}\cdot T}{q}$ (6)
T is the temperature in Kelvin, K_{b} is the Boltzmann constant:
${K}_{b}=1.38\times {10}^{23}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\text{m}}^{2}\cdot \text{kg}\cdot {\text{s}}^{1}\cdot {\text{K}}^{1}$
The mobility coefficient is an important electronic parameter, determinated under many external conditions i.e., temperature [42], magnetic field [43] [44], radiation damage by charged particules [45] [46], doping rate [47]. Thus for electrons, mobility is temperature dependent and expressed by [48] [49]:
$\mu \left(T\right)=1.43\times {10}^{19}{T}^{2.42}$ (7)
By replacing Equations (2) and (3) in Equation (1), the continuity equation for the excess minority carriers’ density in the base is reduced to the following relationship:
$\frac{{\partial}^{2}\delta \left(x,\omega \right)}{\partial {x}^{2}}\frac{\delta \left(x,\omega \right)}{{L}^{2}\left(\omega ,T\right)}=\frac{g\left(x\right)}{D\left(\omega ,T\right)}$ (8)
$L\left(\omega ,T\right)$ is the complex diffusion length of excess minority carriers in the base [41] given by:
$L\left(\omega ,T\right)=\sqrt{\frac{D\left(\omega ,T\right)\tau}{1+j\omega \tau}}$ (9)
$\tau $ is the excess minority carriers lifetime in the base.
The solution of Equation (8) is:
$\delta \left(x,\omega ,T\right)=A\cdot \mathrm{cosh}\left[\frac{x}{L\left(\omega ,T\right)}\right]+B\cdot \mathrm{sinh}\left[\frac{x}{L\left(\omega ,T\right)}\right]+K\cdot {\text{e}}^{\alpha \cdot x}$ (10)
With
$K=\frac{\alpha \cdot {I}_{0}\cdot \left(1R\right)\cdot {\left[L\left(\omega ,T\right)\right]}^{2}}{D\left(\omega ,T\right)\left[L{\left(\omega ,T\right)}^{2}\cdot {\alpha}^{2}1\right]}$ and
$L{\left(\omega ,T\right)}^{2}\cdot {\alpha}^{2}\ne 1$ (11)
Coefficients A and B are determined through the boundary conditions:
• At the junction (n^{+}/p) (x = 0)
${D\left(\omega ,T\right)\frac{\partial \delta \left(x,T\right)}{\partial x}}_{x=0}=Sf\cdot {\frac{\delta \left(x,T\right)}{D\left(\omega ,T\right)}}_{x=0}$ (12)
• On the back side (p/p+) in the base (x = H)
${D\left(\omega ,T\right)\frac{\partial \delta \left(x,T\right)}{\partial x}}_{x=H}=Sb\cdot {\frac{\delta \left(x,T\right)}{D\left(\omega ,T\right)}}_{x=H}$ (13)
Sf and Sb are excess minority carrier recombination velocity respectively at the junction and at the back surface.
The variation of recombination velocity Sf, through Equation (12) describes the solar cell operating point that is imposed by the external load [12] [14]. Intrinsic Sf component describing the carrier losses, is then associated with the shunt resistor though the solar cell electrical equivalent model [50] [51] [52].
The excess minority carrier recombination velocity Sb on the back surface is associated with the p/p^{+} junction which generates an electric field, for throwing back the charge carrier toward the junction [14] [15] [36] and then increases their collection.
3. Results and Discussions
3.1. Photocurrent
The density of photocurrent at the junction is obtained from the density of minority carriers in the base and is given by the following expression:
${J}_{ph}\left(Sf,Sb,\omega ,T\right)=qD\left(\omega ,T\right){\frac{\partial \delta \left(x,Sf,Sb,\omega ,T\right)}{\partial x}}_{x=0}$ (14)
where q is the elementary electron charge.
Figure 2 shows AC photocurrent versus the junction surface recombination velocity for different temperature.
3.2. AC Back Surface Recombination Velocity Sb
For a given frequency, the representation of AC photocurrent density versus junction minority carrier’s recombination velocity shows the shortcircuit current density (J_{phsc}) for very large Sf values, where obviously we can write [13] [14] [35]:
${\frac{\partial {J}_{ph}\left(Sf,Sb,\omega ,T,\alpha \left(\lambda \right)\right)}{\partial Sf}}_{Sf\ge {10}^{5}\text{cm}\cdot {\text{s}}^{1}}=0$ (15)
Figure 2. Photocurrent density versus junction surface recombination velocity under temperature influence. (ω = 10^{5} rad/s; H = 0.025 cm; α = 6.2 cm^{−1}).
The solution of this Equation (15) leads to expressions of the AC recombination velocity in the back surface, given by [53]:
$Sb1\left(\omega ,T,\alpha \left(\lambda \right)\right)=\frac{D\left(\omega ,T\right)}{L\left(\omega ,T\right)}\cdot \left[\frac{\alpha \left(\lambda \right)\cdot L\left(\omega ,T\right)\cdot \left(\mathrm{exp}\left(\alpha \left(\lambda \right)\cdot H\right)\mathrm{cosh}\left(\frac{H}{L\left(\omega ,T\right)}\right)+\mathrm{sinh}\left(\frac{H}{L\left(\omega ,T\right)}\right)\right)}{\mathrm{exp}\left(\alpha \left(\lambda \right)\cdot H\right)\mathrm{cosh}\left(\frac{H}{L\left(\omega ,T\right)}\right)+\alpha \left(\lambda \right)\cdot L\left(\omega ,T\right)\cdot \mathrm{sinh}\left(\frac{H}{L\left(\omega ,T\right)}\right)}\right]$ (16)
$Sb2\left(\omega ,T\right)=\frac{D\left(\omega ,T\right)}{L\left(\omega ,T\right)}\cdot \mathrm{tanh}\left(\frac{H}{L\left(\omega ,T\right)}\right)$ (17)
3.3. Amplitude and Phase (Bode Diagrams)
Previous studies have focused on the second solution given to the Equation (17) [32] [33] [34]. Our study will consider the second solution (Equation (16) [54] whose module and phase are represented versus logarithm of the modulation frequency by Figure 3 and Figure 4 for different temperature and long wavelength (λ) corresponding to low absorption coefficient value (α = 6.02 cm^{−1}), characterized by deep penetration in the base (
$\alpha L\left(\omega \right)\ll 1$ ) [16] [24] [53] [55].
Sbampl (ω, T) and ϕ(ω, T) correspond, for a given temperature T, to the amplitude and phase component of Sb. At low frequencies (≤10^{4} rad/s), the stationary regime is observed and gives constant amplitudes that decrease with temperature T (Figure 3).
The (Ac) Sb recombination velocity at the rear face in complex form (real and imaginary components, with a complex number (J)) is presented by analogy with the MaxwellWagnerSillars model (MWS) [56] and can be written as:
$Sb\left(\omega ,T\right)=S{b}^{\prime}\left(\omega ,T\right)+J\cdot S{b}^{\u2033}\left(\omega ,T\right)$ (18)
The alternative phase (Figure 4) for a given temperature, is written:
Figure 3. Module of Sb versus frequency for different temperature (H = 0.025 cm; α = 6.2 cm^{−1}).
Figure 4. Phase of Sb versus frequency for different temperature (H = 0.025cm; α = 6.2 cm^{−1}).
$\mathrm{tan}\left(\varphi \left(\omega ,T\right)\right)=\frac{S{b}^{\u2033}\left(\omega ,T\right)}{S{b}^{\prime}\left(\omega ,T\right)}$ (19)
The phase of the recombination velocity is negative at low values of the pulse. At large frequencies (ω less than 10^{5} rad/s), it is presented as a damped sine wave, with amplitude and resonant frequency decreasing with temperature.
The positive and negative semicircles correspond respectively to small and large diameters of the Nyquist diagram and allow to conclude on the equivalent electrical model characterizing the AC Sb recombination velocity [33] [34] [35] [57].
3.4. Niquyst Diagram of the Recombination Velocity
The Nyquist diagram which is the representation of the imaginary part of Sb as a function of the real part, for different temperatures.
Figure 5 and with a zoom represented by Figure 6 show semicircles, of different diameters, which decrease with temperature. The semicircles corresponding to Sb’’ positive imaginary (ReSb(ind)) are of smaller diameters than those corresponding to Sb’’ negative imaginary (ImSb(cap)).
The quantities (ReSb(cap)) and (ImSb(ind)) represent the inductive and capacitive effects (dominant effect) of the recombination velocity of the minority charge carriers, for each temperature.
An intersection point (Sb’) of each semicircle with the real (horizontal) axis of Sb is observed. This offset (shift) from the origin of the axes narrows with temperature. This difference is the real part of the recombination velocity of the minority charge carriers, for each temperature represents the resistive part [33] [34] [35] [51] [57] [58].
The quantities (ImSb(cap)), (ImSb(ind)) and (Sb’) are extracted, for each temperature and presented in the Table 1.
Figure 5. Imaginary component versus real component of Sb for different temperature (H = 0.025 cm; α = 6.2 cm^{−1}).
Figure 6. Imaginary component versus real component of Sb for different temperature (H = 0.025 cm; α = 6.2 cm^{−1}).
Table 1. Shif part, maximum amplitude of both the imaginary and real parts of Sb for different temperature values.
Figures 711, are drawn from the Table 1. Figure 7 shows the representation of (Re(Sb)), the real part (Sb’) as a function of temperature, which reflects the resistive (ohmic) effect associated with the recombination velocity Sb of minority carriers.
$Re\left(Sb\right)=0.69\times T\left(\text{K}\right)+3.4\times {10}^{2}$ (20)
The modeling expression in Figure 7 shows the decreasing line of Re(Sb) with temperature. This quantity is associated with the resistive behavior of the recombination velocity on the rear face [50] [51] [59]. The increase in temperature reduces the loss of minority carriers and reinforces the BSF character of the junction (p/p^{+}) on the rear face.
Figure 8 and Figure 10 give the reciprocal of both, the real parts Sb(cap) and Sb(ind), respectively of Sb’’, as a function of temperature. While Figure 9 and Figure 11, produce the reciprocal representations of the imaginary parts of Sb(cap) and Sb(ind), as a function of temperature.
$1/Re\left(Sb\right)cap=2.3\times {10}^{6}\times T\left(\text{K}\right)0.00025$ (21)
$1/Im\left(Sb\right)cap=4.7\times {10}^{6}\times T\left(\text{K}\right)0.00049$ (22)
$1/Re\left(Sb\right)ind=2.5\times {10}^{6}\times T\left(\text{K}\right)0.0002$ (23)
$1/Im\left(Sb\right)cap=5.1\times {10}^{6}\times T\left(\text{K}\right)0.0005$ (24)
Figures 811 show increasing lines with the rise in temperature associated with the Umklapp process which acts on the diffusion coefficient of minority carriers [33] [60] [61] [62]. Modeling expressions are given through Equations (21)(24).
Figure 8 and Figure 10 show that the capacitive and inductive effects resulting from the imaginary part of Sb are not perfect and therefore reflect the ohmic losses (or leaks).
On the other hand, Figure 9 and Figure 11 are associated respectively with a purely capacitive and inductive behavior of the minority carrier recombination velocity, by storage or discharge towards the junction (n^{+}/p).
The AC recombination velocity (Sb), can be presented, through its equivalent electric model like a pure resistance (associated with Re(Sb)), in series with both imperfect capacitor (capacitor in parrallel with resistor) and inductance (inductance in parallel with a resistor) undergoing the effects of the temperature [25] [34] [63].
Figure 7. Real of Sb versus temperature.
Figure 8. Reciprocal of Sb real (capacitance) versus temperature.
Figure 9. Reciprocal of Sb imaginary (capacitance) versus temperature.
Figure 10. Reciprocal of Sb real (inductance) versus temperature.
Figure 11. Reciprocal of Sb imaginary (inductance) versus temperature.
4. Conclusions
This study of the monofacial silicon solar cell (n^{+}/p/p^{+}) under temperature and under monochromatic illumination in frequency modulation, made it possible to extract the theoretical expression AC of the recombination velocity of minority carriers on the rear face (p/p^{+}), at long wavelengths giving deep penetration (low absorption coefficient) of the wave.
The analysis of this AC recombination velocity, at different temperatures, through the diagrams of Boode (amplitude and phase) and Nyquist, led to an equivalent electrical model, suggesting, a series resistance associated with both imperfect capacitor and an inductive winding in series. At low frequencies (static regime), whatever the temperature, the resistive effect of the AC Sb is preponderant.