A Theoretical Model for Voltage-Dependent Photocurrent Collection in CdTe Solar Cells

: The classic solar cell model assumes that the photo-generated current is a constant, independent of the cell’s output voltage. Experimental data of CdTe solar cells, however, show that the photocurrent collection efﬁciency decreases with the increase of the cell’s output voltage. In this work, we proposed a theoretical model for the CdTe thin-ﬁlm cell, which assumes that the loss of photocurrent in the CdTe absorber is primarily due to the minority carrier recombination in the neutral region and at the back contact. By solving the neutral region’s diffusion equation, with proper boundary conditions, we have obtained the analytical expressions of the photocurrent collection efﬁciency and the cell’s J-V performance. Our theoretical results agree well with the experimental data. According to our theoretical model, the CdTe thin-ﬁlm solar cell has an optimized p-doping level. A higher doping density may not be always good for a CdTe solar cell due to the reduced depletion width and decreased photocurrent at normal operation voltage, although the higher doping density can improve the open-circuit voltage by increasing built-in voltage.


Introduction
In thin-film solar cells, photocurrent collection losses can be extremely significant due to their high absorption coefficient, short absorption lengths, small depletion widths, and short diffusion lengths [1]. Voltage-dependent photocurrent collection losses have been observed in all thin-film solar cells, including those based on Cu 2 S, CuInSe 2, CdTe, organic P3HT/PCBM, and inorganic III-V solar cells [2][3][4][5][6]. The physical model of classic silicon solar cells assumes that the photo-generated current is a constant, independent of the cell's output voltage. Unlike their crystalline silicon counterpart, the principle of superposition and the shift of dark current downwards by a constant photocurrent do not work for these thin-film solar cells [7][8][9]. Therefore, the classical five-parameter model fails to describe the J-V characteristics of CdTe solar cells, especially in the range of forward bias close to V oc [10]. The actual J-V curve of a CdTe thin-film solar cell shows: (1) in the low forward bias region, the current changes with photon flux, (2) the photo-carrier collection depending on the voltage applied, and (3) in some of the devices with large ohmic series resistance and poor performance, the S-kink behavior [11,12]. To unveil the reason, there is much literature that reports the bulk and interfacial recombination from both experiments and calculations [13][14][15], involving complicated and interdependent physical elements.
To obtain a more direct view and find a better theoretical model, in this work, we introduced a very simple analytical expression for photocurrent collection whereby recombination in the neutral region is taken into account. We assumed that the loss of photocurrent in the CdTe absorber is primarily due to the minority carrier recombination in the neutral region and at the back contact. The most important parameters in our model are the absorption length, depletion region width, and diffusion length. We solved the diffusion equation with boundary conditions in CdTe solar cells and obtained the analytical expressions of the bias-dependent photocurrent collection and the cell's J-V performance.
ical expressions of the bias-dependent photocurrent collection and the cell's J-V performance. The diffusion length/doping density dependence of the photocurrent was validated against both simulation and experimental data in forward bias.

CdTe Solar Cell Structure
In this work, the structure of a typical CdTe solar cell is based on the superstrate concept [16]. The incident sunlight hits the glass substrate, followed by the transparent front contact (i.e., TCO), n-type CdS (window layer), p-type CdTe (absorber), and metal back contact. Accordingly, we define and summarize parameters used in our model, as shown in Figure 1. CdTe solar cell structure with a defined thickness for each layer, that is, tgl, tTCO, twindow, and H represent the thickness of glass, TCO layer, CdS window layer, and CdTe absorber layer, respectively. x is defined as any position in the neutral region, thus x = 0 is set to be at the depletion region edge, and x = H − W at the metal back contact.

Optical Condition in CdTe Solar Cells
The spectrum of the sunlight incident on the terrestrial solar cell or module is expressed by standard θ = , θ being the incident angle of the sunlight with respect to the normal of the Earth's surface. Therefore, the photon flux density of the incident light with photon energy greater than the absorber's bandgap is expressed as The photon flux where the superstrate glass reflectance R and optical absorption length of the glass, TCO, and window layers Lgl, LTCO, and Lwindow are averaged through the photon energy spectrum. CdTe solar cell structure with a defined thickness for each layer, that is, t gl , t TCO , t window , and H represent the thickness of glass, TCO layer, CdS window layer, and CdTe absorber layer, respectively. x is defined as any position in the neutral region, thus x = 0 is set to be at the depletion region edge, and x = H − W at the metal back contact.

Optical Condition in CdTe Solar Cells
The spectrum of the sunlight incident on the terrestrial solar cell or module is expressed by standard I AM1.5 (E), which has the dimension of W/m 2 eV. Air Mass 1.5 (AM 1.5) indicates cos θ = 1 1.5 , θ being the incident angle of the sunlight with respect to the normal of the Earth's surface. Therefore, the photon flux density of the incident light with photon energy greater than the absorber's bandgap is expressed as The photon flux φ AM1.5 is partially reflected, and absorbed by the glass, TCO, and window layers, and the photon flux density, with E ≥ E G , incident on the CdTe absorber surface is where the superstrate glass reflectance R and optical absorption length of the glass, TCO, and window layers L gl , L TCO , and L window are averaged through the photon energy spectrum.

Electrical Conditions in CdTe Solar Cells
Our model, developed for CdTe solar cells, represents a solution to the continuity equation assuming complete photocurrent collection in the depletion region. Thus, the Energies 2021, 14, 1615 3 of 9 main recombination is considered to occur in the neutral region of the absorber and possibly at interfaces, which will be further discussed in Section 3.
Based on the low injection assumption, the photo-generated electron density n in the neutral region of CdTe obeys the diffusion equation: where the optical absorption length L a is the weighted average of the photon energy dependent absorption length, D n the diffusivity, and τ n the lifetime of charge carrier. The dependence of depletion width W on the output voltage V is described as With ε and ε r being the dielectric constant, V bi the built-in voltage of the CdS-CdTe heterojunction, and the number of ionized dopants N A − [17]. Considering the fact that the activation energies of the dopant levels are non-shallow in CdTe [18], the dopants are partially ionized and the number of ionized dopants N A − can be expressed by [19]: where E A is the transition energy level.
where E g is the bandgap of CdTe with a value of 1.5 eV [17,20] Table 1 for CdTe solar cells with different CdTe doping densities N A ranged from 10 14 to 10 17 cm −3 . Due to the non-shallow activation energy levels of the Cu dopants in CdTe, the hole density is a few orders lower than the acceptor concentration. The finding is further supported by the calculation of p-doping limit and donor compensation in CdTe polycrystalline thin-film solar cells [19], which showed in a p-type semiconductor with non-shallow acceptor levels, such as p-CdTe with CuCd impurities, the hole density was 5.6 × 10 14 cm −3 under the acceptor concentration of 1.0 × 10 17 cm −3 [19]. The dependence of W on V at various doping densities is also shown in Figure 2.  It can be found that at the solar cell operation voltage of approximately 0.7 V, the depletion width is about 1.8 um in the low doping CdTe film, which can be lowered to 0.1 um in the high doping CdTe film. As the absorption coefficient for CdTe is higher than 10 6 m −1 above the absorption edge, and most of the incident light will be absorbed within 1 um depth or less.

General Solution and Boundary Conditions
The general solution of Equation (3) in the p-type absorber's neutral region is where A and B are constants, dependent on the two boundary conditions of the neutral region. We plotted the individual numerical item and the relationship between ∆n and x of Equation (7) for a better understanding, as shown in Figures 3 and 4. It can be found that at x = 0 (depletion region edge), the dominant parts are cosh and exp n a It can be found that at the solar cell operation voltage of approximately 0.7 V, the depletion width is about 1.8 um in the low doping CdTe film, which can be lowered to 0.1 um in the high doping CdTe film. As the absorption coefficient for CdTe is higher than 10 6 m −1 above the absorption edge, and most of the incident light will be absorbed within 1 um depth or less.

General Solution and Boundary Conditions
The general solution of Equation (3) in the p-type absorber's neutral region is where A and B are constants, dependent on the two boundary conditions of the neutral region. We plotted the individual numerical item and the relationship between ∆n and x of Equation (7) Here we replace B with 1 + b and use the expression L n = √ D n τ n . Assuming that L n and H are of the same order, and H >> L a , we can neglect the third term of higher order in the expression of ∆n and d∆n dx , and have Therefore, from ∆n = 0, or equivalently S n = ∞ at the back x = H − W, we have The photocurrent density contributed from the neutral p-region J NR is the value of −qD n d∆n dx at x= 0. Thus, Assuming there is no photo-electron loss in the depletion region, the photo-generated carrier collection efficiency in the neutral region is The dependence of neutral region photo-carrier collection coefficient on CdTe's diffusion length L n is shown in Figure 5, assuming H, W, and L a are 2.5 µm, 0.1 µm, and 0.2 µm, respectively.  ( ) ( ) ( ) ( ) 1 exp exp coth n n a total a a n a n a n It is assumed that the collection coefficient of photo-generated current in the depletion region W J is 100% since there is a strong electric field in the depletion region, which drives the photo-carriers swiftly without any recombination. The light generated photocurrent density is not constant L J , or SC J , but bias voltage dependent as described in Equation (16). Thus, the ideal polycrystalline thin-film CdTe solar cell's photo-carrier collection efficiency η total at various doping densities is as shown in Figure 6   ( ) ( ) ( ) ( ) 1 exp exp coth n n a total a a n a n a n  It is assumed that the collection coefficient of photo-generated current in the depletion region J W is 100% since there is a strong electric field in the depletion region, which drives the photo-carriers swiftly without any recombination. The light generated photocurrent density is not constant J L , or J SC , but bias voltage dependent as described in Equation (16).
We also plot the photocurrent at different doping densities of p-type CdTe in Figure 7, and it shows the trend of a much heavier dependence of photocurrent on bias voltage at higher doping density while the photocurrent seems to keep constant at a low doping density of 10 14 cm −3 .
Energies 2021, 14, x FOR PEER REVIEW 8 of 10 We also plot the photocurrent at different doping densities of p-type CdTe in Figure  7, and it shows the trend of a much heavier dependence of photocurrent on bias voltage at higher doping density while the photocurrent seems to keep constant at a low doping density of 10 14 cm −3 .

Comparison of Theory and Experiment
Following Equation (16), we obtain the ideal CdTe solar cell's J-V curve without consideration of any series or shut resistance, which is depicted by The model is further validated by using experimental J-V curves from both our lab and James R. Sites [21], as shown in Figures 8 and 9, respectively. It can be found that our model fits very well on James R. Sites' experimental data, but not so good on our data, which may be attributed to the relatively poor device performance compared with James R. Sites (efficiency 13% of our lab vs 18% of James R. Sites [21]).

Comparison of Theory and Experiment
Following Equation (16), we obtain the ideal CdTe solar cell's J-V curve without consideration of any series or shut resistance, which is depicted by The model is further validated by using experimental J-V curves from both our lab and James R. Sites [21], as shown in Figures 8 and 9, respectively. It can be found that our model fits very well on James R. Sites' experimental data, but not so good on our data, which may be attributed to the relatively poor device performance compared with James R. Sites (efficiency 13% of our lab vs. 18% of James R. Sites [21]). We also plot the photocurrent at different doping densities of p-type CdTe in Figure  7, and it shows the trend of a much heavier dependence of photocurrent on bias voltage at higher doping density while the photocurrent seems to keep constant at a low doping density of 10 14 cm −3 .

Comparison of Theory and Experiment
Following Equation (16), we obtain the ideal CdTe solar cell's J-V curve without consideration of any series or shut resistance, which is depicted by The model is further validated by using experimental J-V curves from both our lab and James R. Sites [21], as shown in Figures 8 and 9, respectively. It can be found that our model fits very well on James R. Sites' experimental data, but not so good on our data, which may be attributed to the relatively poor device performance compared with James R. Sites (efficiency 13% of our lab vs 18% of James R. Sites [21]).   The model simulations reveal that the photocarrier collection can be improved by increasing the charge carrier diffusion length Ln. For long τn and big Dn, a higher total η is obtained, and the collection becomes more effective. Surprisingly, a higher doping density may not be always good for a CdTe solar cell due to the reduced depletion width and decreased photocurrent JL at normal operation voltage, although the higher doping density can improve the Voc by increasing Vbi as indicated in Table 1. When the doping density is extremely big, the depletion width becomes very narrow, leading to a more severe photo-carrier collection loss, since the photo-carriers should be generated primarily within, or very near to the depletion region when the solar cell is biased at high bias voltage. The more recent experiments from First Solar involved various combinations of ptype back contact materials, buffer layers, and front window layers, in order to create an energy barrier or other obstacle to the movement of electrons, and therefore reduce the electron diffusion to the back contact. Our model is one of the most classic device configurations of CdTe thin-film solar cells, and in the future, it can be extended by adding energy barriers in the front and back sides. Furthermore, the surface recombination velocity Sn at the metal back contact is assumed to be infinite in our model, which can be changed to a non-infinite value in the existence of a back surface field, that is, by the deposition of ZnTeCu.

Conclusions
A theoretical model for CdTe solar cells is developed to describe the voltage-dependent charge carrier collection phenomenon. We solved the diffusion equations with boundary conditions in the neutral region and obtained the analytical expression of the biasdependent photocurrent collection in terms of depletion width, diffusion length, and absorption length. Moreover, we validated our model by employing both simulation and experimental data in forward bias. It was found that a longer charge carrier diffusion length can improve the carrier collection efficiency, and a higher doping density may not be good for CdTe solar cells due to the reduced depletion width and decreased photocurrent. This model can be used further to understand the device physics, predict the device performance and calculate the device parameters including the surface recombination speed in the back contact. The model simulations reveal that the photocarrier collection can be improved by increasing the charge carrier diffusion length L n . For long τ n and big D n , a higher η total is obtained, and the collection becomes more effective. Surprisingly, a higher doping density may not be always good for a CdTe solar cell due to the reduced depletion width and decreased photocurrent J L at normal operation voltage, although the higher doping density can improve the V oc by increasing V bi as indicated in Table 1. When the doping density is extremely big, the depletion width becomes very narrow, leading to a more severe photo-carrier collection loss, since the photo-carriers should be generated primarily within, or very near to the depletion region when the solar cell is biased at high bias voltage.
The more recent experiments from First Solar involved various combinations of p-type back contact materials, buffer layers, and front window layers, in order to create an energy barrier or other obstacle to the movement of electrons, and therefore reduce the electron diffusion to the back contact. Our model is one of the most classic device configurations of CdTe thin-film solar cells, and in the future, it can be extended by adding energy barriers in the front and back sides. Furthermore, the surface recombination velocity S n at the metal back contact is assumed to be infinite in our model, which can be changed to a non-infinite value in the existence of a back surface field, that is, by the deposition of ZnTeCu.

Conclusions
A theoretical model for CdTe solar cells is developed to describe the voltage-dependent charge carrier collection phenomenon. We solved the diffusion equations with boundary conditions in the neutral region and obtained the analytical expression of the biasdependent photocurrent collection in terms of depletion width, diffusion length, and absorption length. Moreover, we validated our model by employing both simulation and experimental data in forward bias. It was found that a longer charge carrier diffusion length can improve the carrier collection efficiency, and a higher doping density may not be good for CdTe solar cells due to the reduced depletion width and decreased photocurrent. This model can be used further to understand the device physics, predict the device performance and calculate the device parameters including the surface recombination speed in the back contact.