New Simplified Model of Back Surface Field Polycrystalline Silicon Solar Cells

The development of solar cells is essentially restricted by the cost and efficiency [1, 2].

For photovoltaic applications, crystalline silicon based solar cells are generally employed, but although their high conversion efficiency, these classical components are considered as expensive. A cost – efficient alternative to Si wafers is Polycrystalline silicon thin films, due to their easier, lower-cost fabrication methods and the ability to deposit them on inexpensive substrates like glass or plastic.

However, Poly-Si based Solar cells have shown very low efficiency as a result of the existence of grain boundaries (GBs), which serve as sites of recombination for minority excess carriers [3, 4] affecting negatively the open-circuit voltage of the cell.

In practice, the preparation method can minimize the impact of grain boundaries by reducing their number as pointed of by Taretto [5].

A passivation or a preferential doping (PD) can also be used to lower the electrical activity of GBs [6, 7].

Low device efficiency can be caused also by defective regions across the cell's back surface.

In order to avoid voltage losses, one must be able to manage recombination at the cell surfaces. The use of a heavily doped back surface field (BSF) layer is one of the most prevalent way thus resulting in a low- high (LH) junction. Its inclusion enables the minority carriers to be repelled towards the junction, hence reducing recombination and improving the cell's conversion efficiency.

The increase in the open circuit voltage is found to be the source of that improvement [8].

Many authors replicated the BSF effect in solar cells. According to their findings, these devices display much greater open circuit voltages and increased short circuit currents. But up until 1978, all of these investigations made the assumption of abrupt low-high junctions and a uniform doping of the p+ region.

 Building on this, the presence of a BSF was modeled in terms of an effective recombination velocity Seff at the edge of a back pp+ junction as reported by Godlewski et al. [9] and Fossum [10], Del Alamo et al. [11] established a new model for Seff which considers an arbitrary profile for p+ region and includes mobility and lifetime that are position dependent. Del Alamo et al. [11] obtained a new general non-linear first order differential equation for the recombination velocity and solved it using numerical techniques.

Thus, most prior analytical models of back surface field (BSF) solar cells rely on simplified representations. This limits the ability to fully characterize BSF effects and cannot represent the case of polycrystalline silicon solar cells. So, there is a need for a new method that can account for non-uniform, doping-dependent electric fields within the BSF region and their impact on carrier transport.

Basis on this, the present work is focused to the 2D-modelling of back junction thin cells, assuming a gaussian doping profile in the high region.

More precisely, the model considers a p type base which is free field region, while a gaussian doping profile was considered for p+ region, thus inducing a position dependent electric field.

Here, we consider the field's average value which is supposed to be effective across some distance as mentioned by Blouke et al. [12]. This assumption makes easy the solution of continuity equations that determines the distribution of excess carriers in the device.

The proposal model permits us to investigate the influence of parameters such as internal electric field, grain size, back and grain boundary recombination velocities on the solar’s cell output parameters in order to provide insights into optimizing the back surface field design.

Finally, to validate the model, our results are compared with those obtained by Diallo et al. [13-15].

The new model supposes that the solar cell has both a field free region and a thin region near to the back surface. This last region is characterized by an average constant field and a distance Wbsf. Depending on its direction, this field accelerates the photocarriers away or towards the back surface.

To explore the back surface field effect on electron distribution in high-low junction, we need to solve the continuity equations, which are as follows:

$\frac{{{\partial }^{2}}\Delta n(x,y)}{{{\partial }^{2}}x}+\frac{{{\partial }^{2}}\Delta n(x,y)}{{{\partial }^{2}}y}-\frac{1}{{{L}^{2}}}\Delta n\left( x,y \right)=-\frac{G(y)}{{{D}_{n}}}$               (1)

$\begin{align}  & \frac{{{\partial }^{2}}\Delta {{n}_{bsf}}(x,y)}{{{\partial }^{2}}x}+\frac{{{\partial }^{2}}\Delta {{n}_{bsf}}(x,y)}{{{\partial }^{2}}y}+\frac{{{\mu }_{nbsf}}}{{{D}_{nbsf}}}E\frac{\partial \Delta {{n}_{bsf}}(x,y)}{\partial y} \\ & -\frac{1}{L_{bsf}^{2}}\Delta {{n}_{bsf}}(x,y)=-\frac{G(y)}{{{D}_{nbsf}}} \\ \end{align}$(2)

where, Dn(x, y), Dn_bsf(x, y) are the excess minority carrier density, D_n, D_nbsf the diffusion constant, and L, L_bsf the diffusion length in p and p+ region respectively; µ_nbsf is the carrier mobility in BSF region.

E is the average constant field due to impurity gradient given by:

$E=\frac{1}{{{w}_{bsf}}}\int\limits_{{{w}_{e}}+{{w}_{sc}}+{{w}_{b}}}^{H}{\frac{kt}{q}}\frac{1}{N(y)}\frac{dN(y)}{dy}dy$                (3)

where, N(y) is the doping density in p⁺ region expressed as:

$N\left( y \right)={{N}_{a}}+{{N}_{bsf}}\exp \left( -{{\left( \frac{y-H}{\sigma } \right)}^{2}} \right)$                    (4)

where, N_a, N_bsf are respectively the base and BSF doping levels, whereas σ denotes the standard deviation [17].

G is the generation rate written as [13]:

$G\left( y \right)=\sum\limits_{i=1}^{3}{{{g}_{i}}{{e}^{-{{\alpha }_{i}}y}}}$                 (5)

where, parameters g_i and α_i are generation rate and absorption coefficient in AM1.5 G illumination condition [18].

The solutions of (1) and (2) are expressed as [13]:

$\Delta n(x,y)=\sum\limits_{k}{{{X}_{k}}(y)}.\cos \left( {{c}_{k}}x \right)$                         (6)

$\Delta {{n}_{bsf}}(x,y)=\sum\limits_{k}{{{Y}_{k}}(y).\cos ({{s}_{k}}x)}$                        (7)

The carrier density is computed for k range from 1 to 10, which has been found to be sufficient to get a good convergence whatever the grain size and the recombination velocity as reported by Kolsi et al. [19], while the coefficients c_k, s_k are determined from the following boundary conditions. These boundary conditions are expressed in terms of the convenient balance of currents, by equating the minority carrier diffusion current and the grain boundary recombination current:

${{\left[ \frac{\partial \Delta n(x,y)}{\partial x} \right]}_{x=\pm \frac{\lg }{2}}}=\frac{\mp {{S}_{gb}}}{{{D}_{n}}}\Delta n(\pm \frac{\lg }{2},y)$                           (8)

${{\left[ \frac{\partial \Delta {{n}_{nbsf}}(x,y)}{\partial x} \right]}_{x=\pm \frac{\lg }{2}}}=\frac{\mp {{S}_{GB}}}{{{D}_{nbsf}}}\Delta {{n}_{nbsf}}(\pm \frac{\lg }{2},y)$                        (9)

here, S_gb and S_GB represent the grain boundary recombination velocity in p and p+ regions respectively and lg the grain size.

Replacing Dn(x, y) and Dn_bsf(x, y) by their expressions (6) and (7) respectively into the above two boundary conditions leads to the following transcendental equations (see Appendix):

${{c}_{k}}\tan \left( {{c}_{k}}\frac{\lg }{2} \right)=\frac{{{S}_{gb}}}{{{D}_{n}}}$                         (10)

${{s}_{k}}\tan \left( {{s}_{k}}\frac{\lg }{2} \right)=\frac{{{S}_{GB}}}{{{D}_{nbsf}}}$                         (11)

where, c_k and s_k are Eigen values of these transcendental equations are solved graphically.

Replacing the expressions of Dn(x, y) and Dn_bsf(x, y) in the continuity Eqs. (1) and (2) respectively, we get after some simplifications the following differential equations:

$\frac{{{\partial }^{2}}{{X}_{k}}(y)}{\partial {{y}^{2}}}-\frac{1}{L_{k}^{2}}{{X}_{k}}(y)=-\frac{1}{{{D}_{k}}}G(y)$                        (12)

$\begin{align} & \frac{{{\partial }^{2}}{{Y}_{k}}(y)}{\partial {{y}^{2}}}+\frac{{{\mu }_{nbsf}}}{{{D}_{nbsf}}}E\frac{\partial {{Y}_{k}}(y)}{\partial y}-\frac{1}{Lb_{k}^{2}}{{Y}_{k}}(y) \\ & =-\frac{1}{D{{b}_{k}}}G(y) \\ \end{align}$                          (13)

where,

${{L}_{k}}=\frac{1}{\sqrt{c_{k}^{2}+\frac{1}{{{L}^{2}}}}}$                        (14)

${{D}_{k}}=\frac{{{D}_{n}}(\sin ({{c}_{k}}\lg )+{{c}_{k}}\lg )}{4\sin ({{c}_{k}}\frac{\lg }{2})}$                       (15)

$L{{b}_{k}}=\frac{1}{\sqrt{s_{k}^{2}+\frac{1}{L{{b}^{2}}}}}$                               (16)

$D{{b}_{k}}=\frac{{{D}_{nbsf}}(\sin ({{s}_{k}}\lg )+{{s}_{k}}\lg )}{4\sin ({{s}_{k}}\frac{\lg }{2})}$                                   (17)                         

where, L and L_bsf are respectively the diffusion length in the base and BSF regions given by:

$L=\sqrt{{{D}_{n}}.{{\tau }_{n}}}$                        (18)

${{L}_{bsf}}=\sqrt{{{D}_{nbsf}}.{{\tau }_{nbsf}}}$                                

The minority carrier mobility and lifetime as a function of doping N in the base area, can be represented by the following relations [19]:

${{\mu }_{n}}=232+\frac{1180}{1+{{\left( \frac{N}{{{8.10}^{16}}} \right)}^{0.9}}}$                 (20)

${{\tau }_{n}}=\frac{1}{3.45.*{{0}^{-12}}N+0.95*{{10}^{-31}}{{N}^{2}}}$                (21)

In the BSF region, the same above expressions are used. Nevertheless, it is important to consider here the impact of the large values of the average longitudinal electric field on the electron mobility.

In this regard, we utilize the following relation [17]:

${{\mu }_{nbsf}}={{\mu }_{n}}\sqrt{\frac{1}{1+{{\left( \frac{{{\mu }_{n}}.E}{Vsat} \right)}^{2}}}}$                         (22)

In this expression, μ_n is the carrier mobility at low electric field given by Eq. (20), and V_sat the saturation velocity equal to 10⁷ cm.s^-1.

It is important to mention that the carrier mobility is not altered for a given electric field of moderate strength. Nevertheless, it is decreased as a result of scattering caused by an increase in the longitudinal electric field.

The solutions X_k(y) and Y_k(y) of Eqs. (12) and (13) respectively can be written in the form [13]:

${{X}_{k}}(y)={{A}_{k}}{{e}^{\frac{y}{{{L}_{k}}}}}+{{B}_{k}}{{e}^{-\frac{y}{{{L}_{k}}}}}-\sum\limits_{i=1}^{3}{\left( {{G}_{i}}{{e}^{-{{\alpha }_{i}}y}} \right)}$                            (23)

${{Y}_{k}}(y)={{C}_{k}}{{e}^{\frac{{{M}_{k}}y}{L{{b}_{k}}}}}+{{F}_{k}}{{e}^{-\frac{{{N}_{k}}y}{L{{b}_{k}}}}}-\sum\limits_{i=1}^{3}{\left( {{R}_{i}}{{e}^{-{{\alpha }_{i}}y}} \right)}$                       (24)

with:

${{G}_{i}}=\frac{L_{k}^{2}{{g}_{i}}}{(L_{k}^{2}{{\alpha }_{i}}^{2}-1){{D}_{k}}}$                                (25)

${{R}_{i}}=\frac{Lb_{k}^{2}{{g}_{i}}}{D{{b}_{k}}(1+{{\alpha }_{i}}(\frac{{{\mu }_{nbsf}}}{{{D}_{nbsf}}}E-{{\alpha }_{i}})Lb_{k}^{2})}$                           (26)

${{M}_{k}}=\frac{\frac{{{\mu }_{nbsf}}}{{{D}_{nbsf}}}E*L{{b}_{k}}+\sqrt{{{\left( \frac{{{\mu }_{nbsf}}}{{{D}_{nbsf}}}E \right)}^{2}}Lb_{k}^{2}+4}}{2L{{b}_{k}}}$                        (27)

${{N}_{k}}=\frac{\frac{{{\mu }_{nbsf}}}{{{D}_{nbsf}}}E*L{{b}_{k}}-\sqrt{{{\left( \frac{{{\mu }_{nbsf}}}{{{D}_{nbsf}}}E \right)}^{2}}Lb_{k}^{2}+4}}{2L{{b}_{k}}}$                             (28)

Constants A_k, B_k, C_k and F_k are determined using Maple software and by mean of the following boundary conditions:

${{X}_{k}}(y={{w}_{e}}+{{w}_{sc}})=0$                              (29)

At pp+ interface, it can be stated that the current and the electron concentration are continuous [9].

So, we can write:

${{X}_{k}}(y={{w}_{e}}+{{w}_{sc}}+{{w}_{b}})={{Y}_{k}}(y={{w}_{e}}+{{w}_{sc}}+{{w}_{b}})$                                (30)

$\begin{align}  & {{D}_{n}}{{\left[ \frac{\partial {{X}_{k}}}{\partial y} \right]}_{y={{w}_{e}}+{{w}_{sc}}+{{w}_{b}}}}= \\ & {{\mu }_{nbsf}}E.{{Y}_{k}}({{W}_{e}}+{{W}_{sc}}+{{W}_{b}})+ \\ & {{D}_{nbsf}}{{\left[ \frac{\partial {{Y}_{k}}}{\partial y} \right]}_{y={{w}_{e}}+{{w}_{sc}}+{{w}_{b}}}} \\ \end{align}$                    (31)

At y=H

${{D}_{nbsf}}{{\left[ \frac{\partial {{Y}_{k}}}{\partial y} \right]}_{y=H}}=-{{Y}_{k}}(Sb+{{\mu }_{nbsf}}E)$                            (32)

The calculated photogenerated carrier density permits us to evaluate the solar cell’s output parameters.

3.1 Photocurrent density

In the base region of the solar cell, the photocurrent density is computed as follows [13]:

$Jp{{h}_{b}}=\frac{q{{D}_{n}}}{\lg }\int\limits_{-\frac{\lg }{2}}^{\frac{\lg }{2}}{{{\left[ \frac{\partial \Delta n(x,y)}{\partial y} \right]}_{y={{w}_{e}}+{{w}_{sc}}}}}dy$                     (33)

where, q the electron charge.

In the BSF region, the generated photocurrent density can be written as:

$\begin{align}  & Jp{{h}_{bsf}}=\frac{q{{D}_{nbsf}}}{\lg }\int\limits_{-\frac{\lg }{2}}^{\frac{\lg }{2}}{{{\left[ \frac{\partial \Delta {{n}_{bsf}}(x,y)}{\partial y} \right]}_{y={{w}_{e}}+{{w}_{sc}}+{{w}_{b}}}}}dx \\ & +\frac{q{{\mu }_{nbsf}}E}{\lg }\int\limits_{-\frac{\lg }{2}}^{\frac{\lg }{2}}{\Delta {{n}_{bsf}}{{(x,y)}_{y={{w}_{e}}+{{w}_{sc}}+{{w}_{b}}}}}dx \\\end{align}$                  (34)

The Eq. (35) is used to calculate the photocurrent in the space charge layer [19]:

${{J}_{sc}}=\sum\limits_{i=1}^{3}{\frac{q.{{g}_{i}}}{{{\alpha }_{i}}}}.{{e}^{-{{\alpha }_{i}}{{w}_{e}}}}(1-{{e}^{-{{\alpha }_{i}}.{{w}_{sc}}}})$                            (35)

W_sc is the space region thickness given by:

${{w}_{sc}}=\sqrt{\frac{2.\varepsilon }{q}\left( \frac{1}{{{N}_{a}}}+\frac{1}{{{N}_{d}}} \right)0.026\ln \left( \frac{{{N}_{a}}Nd}{{{n}_{i}}^{2}} \right)}$                           (36)

Taking into account the homogenous doping level in the emitter region, the current density Jph_e generated in this region at w_e can be expressed as follows using the same method as in the base region.

$Jp{{h}_{e}}=\frac{q{{D}_{p}}}{\lg }\int\limits_{-\frac{\lg }{2}}^{\frac{\lg }{2}}{\mathop{\left[ \frac{\partial \Delta p(x,y)}{\partial y} \right]}_{y={{w}_{e}}}}$                              (37)

where, Dp(x, y) and D_p are respectively the excess minority carrier density and diffusion constant in n⁺ region.

The cell's total photocurrent density (Jph) is given by:

${{J}_{ph}}=Jp{{h}_{e}}+{{J}_{sc}}+Jp{{h}_{b}}+Jp{{h}_{bsf}}$                            (38)

3.2 Open-circuit voltage

The open-circuit voltage can be linked to the total photocurrent density Jph and to the diode saturation current density Js by the following formula [9]:

${{V}_{oc}}=\frac{KT}{q}\ln (\frac{Jph}{Js}+1)$                               (39)

To compute the saturation current, the calculation of the dark minority carrier concentration is performed under the assumption that the generation rate G is equal to zero [20].

3.3 Efficiency

Solar-cell conversion efficiency refers to the portion of energy in the form of sunlight that can be converted into electricity. It is given by Eq. (40) [21]:

$\eta =\frac{P\max }{Pinc}$                                 (40)

where, Pinc=0.1 W.cm^-2 represents standard AM1.5 solar illumination conditions which models the spectral irradiance of sunlight on the earth's surface. This power density of 0.1 W/cm² corresponds to 1 sun intensity, and Pmax is the maximum power point.

In this paper, a new model of back surface field poly-Si cells is proposed. It considers that a solar cell consists of three regions, a field free emitter and base, while the back region has a gaussian doping profile inducing consequently, an electric field there. 

In this model, the built-in electric field is represented by an average value. 

This assumption makes easy the solution of the 2D continuity equations to evaluate the distribution of excess minority carrier and the photovoltaic parameters of the device.

The results show a behavior that would be expected. For positive fields, the excess carriers show a depletion near the rear surface, whereas negative fields drive the photogenerated carriers towards the rear surface displaying an enhancement there. 

Calculations of performance parameters reveal that positive and modest field E prevents any impact of surface recombination for values less or equal to 5.10³ cm.s^-1 while higher back surface recombination velocities indicate Sb limited performance.

For E higher than 10⁴V.cm^-1, the photovoltaic parameters become substantially less sensitive to back surface recombination velocities, indicating bulk-lifetime-limited performance. 

Furthermore, the performance of the cell with the BSF structure is superior to that of the classical n+p structure. Depending on lg, the improvement in conversion efficiency can be as much as 7.2%. Our findings and those from studies published before [13-15] are fairly compatible.

Finally, we consider that the present model can be applied to any thin film silicon cell. The only assumption made is that of an average internal electric field to simplify the solution of the 2D continuity equations. We believe that it nevertheless doesn't alter the results.

However, additional factors could be considered in future studies such as expanding the model to 3D to better account for non-uniformities and edges effects in real solar cell geometries, exploring different non-uniform doping profiles beyond gaussian for the back surface field region, and validating the model predictions against experimental results across a wider range of fabrication processes and materials.