The techniques of characterization of solar cell for quality control    improve the fabrication processes, go through experimental  and theoretical  studies in static  or dynamic transient    and frequency    .
The phenomenological parameters  , which allow this quality control, are the carrier recombination velocity:
1) in the bulk  , defined by the diffusion length (L) and coefficient (D), lifetime (τ) of excess minority carrier.
2) on the surfaces   , i.e., recombination velocity (Sf) at the junction (n-p)   , (Sb) at the junction (p-p+)   , and (Sg) at the grain boundaries in the 3-dimensional   model.
The solar cell is placed under different operating modes  and for different illuminations: monochromatic  , polychromatic  , constant multspectral  .
The solar cell can be maintained under different experimental conditions while varying: temperature   , electric field   , magnetic field   , or irradiation energy of particles    .
To achieve low cost solar concentrator cell, vertical multi-junction (VMJ) cells have been manufactured  . There are two types of VMJ    , according to the connection between cell units, in view to improve, either charge carrier current collection or tension. Thus, series-connected VMJ  and parallel-connected VMJ  have been process, allowing poor minority carrier diffusion length to be collected, by use of silicon material regardless of crystal orientation (multi-crystalline or ribbon).
In our study, the structure of the series-connected vertical junction solar cell  with different lamella widths (H), is investigated in order to determine the recombination velocity of the excess minority carrier at the back surface. This new expression of the ac recombination velocity is analyzed through Bode and Nyquist diagrams, and is shown to depend strongly upon the lamella widths of the base (H).
Figure 1 shows series-connected vertical multi-junction solar cells where each base (lamella) is framed by two emitters. Between emitter (n+) and base (p) we have the space charge region (SCR), called the junction. And at the back side of each base region, there is a high doping layer (p+) giving rise to a back surface field (BSF), which induced the back surface recombination velocity (Sb) (Figure 2).
In this series-connected architecture, each solar cell unitis separated on both sides by metal contacts    .
The continuity equation at which the density of minority charge carriers in excess obeyed at the position x in the base, in an instant t, is given by  :
The ac component of the excess minority density is in the following form:
j is the complex notation.
With is the steady state minority carrier density position dependent.
The expression of the ac generation rate of the minority carrier at depth z, is given by :
Figure 1. Schematic of a series-connected vertical multi-junction solar cell.
Figure 2. Across section of the vertical junction.
I (λ) is the intensity of the monochromatic illumination of wavelength λ. α is the absorption coefficient of the monochromatic light incident on the cell and R (λ) its reflectance coefficient.
D (ω) and L (ω) are respectively, the excess ac minority carrier diffusion coefficient and diffusion length in the base subjected to illumination in frequency modulation (ω).
L (ω) and D (ω) ac expressions are giving by  :
where D denotes the diffusion constant and τ the bulk lifetime in steady state.
By substituting Equation (1) together with Equation (2) and Equation (3), leads to:
Thus the resolution of Equation (6) gives the excess minority carrier density in the base through the following expression:
with coefficients A et B are deduced from the boundary conditions:
1) At the junction (x = 0), the expression of the photocurrent    is given by:
2) On the back side in the base at x = H.
Sf and Sb are respectively the recombination velocities of the excess minority carrier at the junction and at the back surface. The recombination velocity Sf reflects the charge carrier velocity of passage at the junction, in order to participate in the photocurrent. It is then imposed, by the external load which fixes the solar cell operating point    . It has an intrinsic component, which represents the carrier losses associated with the shunt resistor in the solar cell electrical equivalent model    . The excess minority carrier recombination velocity Sb on the back surface is associated with the presence of the p+ layer, which generates an electric field for throwing back the charge carrier toward the junction  .
After calculation, the following expression of the ac excess minority carrier density is obtained by:
3. Results and Discussions
The excess minority carrier recombination velocity at the back surface is deduced from the resolution of the following equation  :
Solving Equation (13) leads to two solutions     . The intrinsic solution, i.e., that which is not a function of the absorption coefficient, is retained and its expression is given by:
Sb in complex form (real and imaginary components) is presented by analogy of the effect of Maxwell-Wagner-Sillars (MWS) model    and can be written as:
We define the ac phase as following equation:
and correspond to the amplitude and phase component of Sb.
We represent in Figure 3 the spectra of the excess minority carrier recombination velocity at the back surface for different lamella H thickness values.
Figure 3 shows a decrease in the amplitude of the excess minority carrier recombination at the back surface when (H) the thickness of the lamella increases.
The large (H) thicknesses give weak oscillation periods. Thus whatever the lamella width, the oscillation is around a fixed (Sb0) value of the excess minority carrier recombination velocity.
Figure 4 gives the profile of Sb the amplitude of the recombination velocity as a function of H the base thickness, for given values of the modulation frequency of the illumination.
Figure 3. Recombination velocity of the minority charge carriers at the back surface Sb versus frequency for different lamella width values.
Figure 4. Back surface Recombination velocity as function of lamella width for different frequencies. D = 35cm2/s; τ = 2 × 10−6 s.
For high range frequencyvalues (ωτ > 1) we observe a damped sinusoid (aperiodic response), to then give the appearance of an exponential growth. For large lamella width, these curves tend towards an asymptote (Sb0 = 4184 cm/s), whatever the frequency.
On the frequency axis, the region corresponding to the frequencies below 104 rd/s (i.e. ), constitutes the steady state.
In this zone the amplitude of SbAmpl believes with the thickness H. The phase remains constant and obviously equal to zero.
In high frequency region 104 rd/s < ω, tarts the dynamic regime i.e. ), showing a sinusoid of amplitude (SbAmpl), oscillating around Sb0, the asymptotical recombination velocity, with periods TSb decreasing with H (see Table 1).
The phase spectrum shows regular sinusoids with constant amplitudes fampl, for each given H lamella widths, but decreases with H. The period Tfampl of these oscillations decreases with the lamella thickness H (see Table 2).
Figure 6 produces the representation of the imaginary part as a function of the real part of the recombination velocity Sb.
The circles obtained have for their center Sb0, on the axis of the reals. The radius of the circles increases when the lamella thickness H decreases. According to the spectroscopy techniques   , the intersections of the circles with the real axis, at high frequency range, indicate the presence of a series resistance which decreases with H the lamella thickness. The negative part of circles indicates a capacitive phenomenon (C), while the positive part, an inductive
Figure 5. Bode diagram of Sb (a) and its phase (b) for different lamella widths.
Table 1. Sb periods for different lamella widths.
Table 2. Phase period for different lamella widths.
Figure 6. Imaginary component versus real component of Sb for different H lamella widths.
phenomenon (Lh), both decrease with the lamella thickness H. The ac equivalent circuit of Sb, suggests that, the capacitor and the inductor are associated in parallel and connected in series with a resistance (obtained for large frequency)    .
The series-connected vertical multi-junction silicon solar cell was studied under frequency modulated illumination and yielded the determination of the ac back surface recombination velocity of the excess minority carrier. It is expressed as dependent of both lamella thickness and illumination frequency. The excess minority carrier recombination is investigated through the Bode diagrams of its amplitude and phase. The study also showed through the Nyquist diagram, the capacitive, inductive and resistive responses of the ac recombination velocity Sb, as well as the effect of the thickness of the lamella based solar cell.
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