𝐶-𝑉 Calculations in CdS/CdTe Thin Films Solar Cells with a CdS_𝑥Te_1−𝑥 Interlayer

Abstract

In CdS/CdTe solar cells, chemical interdiffusion at the interface gives rise to the formation of an interlayer of the ternary compound CdS_𝑥CdTe_1−𝑥. In this work, we evaluate the effects of this interlayer in CdS/CdTe photovoltaic cells in order to improve theoretical results describing experimental 𝐶-𝑉 (capacitance versus voltage) characteristics. We extended our previous theoretical methodology developed on the basis of three cardinal equations (Castillo-Alvarado et al., 2010). The present results provide a better fit to experimental data obtained from CdS/CdTe solar cells grown in our laboratory by the chemical bath deposition (for CdS film) and the close-spaced vapor transport (for CdTe film) techniques.

1. Introduction

Polycrystalline solar cells based on the thin films technology are outstanding candidates for an aggressive expansion of the photovoltaic industry. During the past few years, despite the world economic recession, the photovoltaic market has continued to grow. In Europe and Asia, the market has expanded from the Mega Watt (MW) to the Giga Watt (GW) scale. PV installations grew to 7.3 GW in 2009, which represents 20% from the previous year. The various forecast scenarios predict an increase in the demand from 15.4 to 37 GW in 2014, more than five times the size of the 2009 market [1, 2]. Today, there is much interest and research activity concerning second generation solar cells based on polycrystalline cadmium telluride CdTe thin films and cadmium sulfide CdS as “window” layer. Capacitance versus voltage is one of the most important techniques for the device characterization. In this work, we extend the theoretical model used in a previous paper [3] in order to take into account the formation of a layer of the ternary compound CdS_𝑥Te_1−𝑥 (𝑥, the concentration) at the CdS/CdTe interface generated by interdiffusion processes [4]. During manufacturing of these solar cells, the CdTe/CdS interface is subjected to relatively high temperatures giving rise to atomic interdiffusion and the formation of an interfacial region of composition Cd_𝑥Te_1−𝑥. The understanding of this ternary CdS_𝑥Te_1−𝑥 interlayer will help to conceive improvements in the efficiency of the photovoltaic cell. In this work, we use a 𝐶-𝑉 theoretical method for the determination of the interface charge density 𝜎 in the CdS/ternary/CdTe heterojunction and the band discontinuity Δ𝐸_𝑣 simultaneously. The methodology is based on three cardinal equations as discussed in the theory below. For comparison with our calculations, we used solar cells of maximum efficiency (12.4%) fabricated in our group [5].

2. Theoretical Method

We present the methodology for the simultaneous calculations of the valence band offset Δ𝐸_𝑣 and the interface charge density𝜎. It is assumed that these two quantities are independent of the bias voltage. This assumption is justified because we are interested in heterojunctions with fixed defect densities.

The clue for obtaining the energy bands at both sides of the interfaces are (a) the two interfacial potentials (𝜑_𝑠1, 𝜑_𝑠2which give the band bending at each side of the interface), (b) the energy increment of the valence band, and (c) the interface charge density. In addition to these, three cardinal equations originating from the displacement of the band line-up equation, charge neutrality, and the total capacitance of the interface are necessary. This method makes no use of any approximations and provides simultaneous determination of the discontinuities of the valence band and the interfacial charge density.

Assuming that the heterojunction is like a parallel plate capacitor, the capacitance-voltage characteristics for low bias voltages𝑉_𝑎are given by 𝐶=𝑞𝜀_𝑆1𝜀_𝑆2𝑁_𝑑𝑁_𝑎2(𝜀_𝑆1𝑁_𝑎+𝜀_𝑆2𝑁_𝑑)^1/2∗𝑉_𝑑−𝑉_𝑎^−1/2, (1) where𝑉_𝑑is the potential well in the heterojunction,𝑉_𝑎is the bias voltage, 𝑞 is the electron charge,𝑁_𝑎is the acceptor concentration, 𝑁_𝑑is the donor concentration, and𝜀_𝑆1and𝜀_𝑆2are the dielectric constants of n-type and p-type semiconductor [6]. Therefore, in our case, we have a linear dependence of 1/𝐶² versus 𝑉_𝑎close to𝑉_𝑎=0. We solve numerically the band energy in any site of the heterojunction using the interface according to the experimental data [7]; the spectral response of champion CdS/CdTe solar cells has an increase in a photon energy close to the band gap CdS. Thus, we have assumed in our equations that the physical properties of the ternary compound are closer to those of CdS for each part of the ternary compound, which are the total bending on each side of the two heterojunction (CdTe/CdS_𝑥Te_1−𝑥).

We note that the total band curvature at each side of the interface depends on the bias voltage 𝑉_𝑎, that is, 𝜑_𝑆1(𝑉_𝑎) and 𝜑_𝑆2(𝑉_𝑎). We assume that the discontinuities of the valence band and the charge density at the interface are independent of the applied voltage, which is a usual assumption for low voltages. Validity of this assumption is validated by a good agreement with experiment.

Because there are a fixed equal number of separated positive and negative charges at the interface between different materials, we consider that the region of spatial charge behaves like a parallel plate capacitor.

2.1. Cardinal Equations for the CdTe/CdS_𝑥Te_1−𝑥 Interface (𝐶-𝑉 Matching Method)

The band line-up equation in this case is (Figure 1) 𝐸_𝐹𝑉1−𝐸_𝐹𝑉2−Δ𝐸_{𝑉(ternary)}−𝑞𝑉_𝑎=𝑞𝜑_𝑆2𝑉_𝑎−𝜑_𝑆1𝑉_𝑎, (2) where “𝑥” is the ternary concentration and 𝐸_𝐹𝑉1 and 𝐸_𝐹𝑉2 are the differences between the quasi-Fermi energy levels with subscripts 1 and 2 corresponding to CdTe and CdS_𝑥Te_1−𝑥, respectively. The valence band offsetΔ𝐸_{𝑉(ternary)}and the respective bulk valence band levels are given by 𝐸_𝐹𝑉1=𝐸_𝐹𝑛1−𝐸_𝑉1=𝐸_𝑔12+34𝑚𝑘𝑇ln^∗ℎ1𝑚^∗_𝑒1−𝑞𝜑_𝐹1,𝐸_𝐹𝑉2=𝐸_𝐹𝑝2−𝐸_𝑉2=𝐸_𝑔22+34𝑚𝑘𝑇ln^∗ℎ2𝑚^∗_𝑒2−𝑞𝜑_𝐹2. (3) The separation between the quasi-Fermi energy levels is determined for the bias voltage: 𝐸_𝐹𝑛1=𝐸_𝐹𝑝2+𝑞𝑉_𝑎, (4) where 𝑉_𝑎 is the bias voltage in the p-n junction.

Figure 1 

Energy band (band lineup) diagram of two n-p heterojunctions under reversible bias. The reference level for the potential of each semiconductor is at the intrinsic energy level. We assume that the ternary behaves like an n-type semiconductor for union with CdTe.

The charge neutrality equation under nonequilibrium is given by 𝑄₁𝜑_𝑆1𝑉_𝑎+𝑄₂𝜑_𝑆2𝑉_𝑎+𝑞𝜎=0, (5) where the expressions for the semiconductor charge under nonequilibrium, 𝑄₁ and 𝑄₂, are the half semiconductor charges (per unit area) which are given by [8] 𝑄₁=sign𝜑_𝑆1𝑉_𝑎⋅√2𝜀₀𝜀_𝑆1𝛽𝐿_𝐷1⋅𝑒^Δ𝑢1/2×𝜑_𝑆1𝑉_𝑎⋅sinh𝑢^∗₁+cosh𝑢^∗₁−𝑢_𝑆1𝑉_𝑎−cosh𝑢^∗₁^1/2,𝑄₂=sign𝜑_𝑆2𝑉_𝑎⋅√2𝜀₀𝜀_𝑆2𝛽𝐿_𝐷2⋅𝑒^Δ𝑢2/2×𝜑_𝑆2𝑉_𝑎⋅sinh𝑢^∗₂+cosh𝑢^∗₂−𝑢_𝑆2𝑉_𝑎−cosh𝑢^∗₂^1/2, (6) where sign=+1,𝑢<0−1,𝑢>0. (7) Here, 𝑢^∗=𝛽((𝜑_𝐹1+𝜑_𝐹2)/2) and𝛽=𝑞/𝑘𝑇.

The third cardinal equation is obtained from the expression of the half capacitance per unit area of the device: 1𝐶_CdTe−ter=1𝐶₁𝜑_𝑆1𝑉_𝑎+1𝐶₂𝜑_𝑆2𝑉_𝑎, (8) where 𝐶₁=𝜀₀𝜀_𝑆1²𝛽𝐿²_𝐷1⋅𝑄₁⋅𝑒^▵𝑢1/2⋅sinh𝑢^∗₁−sinh𝑢^∗₁−𝑢_𝑆1𝑉_𝑎=𝑞𝜀₀𝜀_𝑆1𝑄₁𝑁₁+2𝑛_𝑖1⋅𝑒^▵𝑢1/2⋅sinh𝑢^∗₁−𝑢_𝑆1𝑉_𝑎,𝐶₂=𝜀₀𝜀_𝑆2²𝛽𝐿²_𝐷2⋅𝑄₂⋅𝑒^▵𝑢2/2⋅sinh𝑢^∗₂−sinh𝑢^∗₂−𝑢_𝑆2𝑉_𝑎=𝑞𝜀₀𝜀_𝑆2𝑄₂𝑁₂+2𝑛_𝑖2⋅𝑒^▵𝑢2/2⋅sinh𝑢^∗₂−𝑢_𝑆2𝑉_𝑎1𝐶_Total≈1𝐶_CdTe−ter. (9)(10) It is important to note that the energy gap of the ternary is not a simple linear combination but a more complex function of 𝑥 [9]: 𝐸_𝑔(𝑥)=(1−𝑥)𝐸_𝑔(CdS)+𝑥𝐸_𝑔(CdTe)−𝑏𝑥(1−𝑥), (11) where 𝑏 is the “optical bowing coefficient” given by 1.69 eV and the valence band offset Δ𝐸_𝑉.

3. Results

We find the ternary concentration value of 0.75 with the best fitting of the curve, which is in good agreement with the results described by Cediel et al. [10]. The various quantities or constants in the cardinal equations were taken from the previous paper [3]. We show the result of the 𝐶-𝑉 fitting method in Figure 2 for a voltage range from 0 to 0.5 V. A much better fit is obtained by assuming the formation of a ternary compound at the CdTe/CdS interface.

Figure 2 

Calculated 𝐶-𝑉 characteristics of a CdTe/CdS PV cell assuming the formation of a ternary compound at the CdTe/CdS interface. Comparison is made with 𝐶-𝑉 measurements of a 12.4% efficient PV cell [5] and with calculated results within a model without the assumption of the ternary compound at the interface [3].

In addition, we have also obtained the values of the valence band offset and the interface charge density simultaneously, namely, Δ𝐸_𝑉=0.99eV,𝜎=1×10¹³cm⁻², (12) which are in good agreement with the reported values [9, 11, 12].

4. Conclusions

In summary, we have calculated the capacitance versus applied voltage of CdS/CdTe thin film solar cells considering the formation of a CdS_𝑥Te_1−𝑥 interlayer using the 𝐶-𝑉 matching method. Our results are in better agreement with experimental data than our previous theoretical results that did not assume the presence of the CdS_𝑥Te_1−𝑥 interlayer. We may also say that the ternary layer acquires the properties of CdS and behaves like the type n semiconductor instead of the CdS itself. This could be taken as evidence of the existence of a ternary compound that plays an important role in solar cell devices. This can be taken as evidence of the existence of the ternary interlayer which plays an important role in the solar cell.

Acknowledgments

F. L. Castillo-Alvarado, J. Ortiz-Lopez, and G. Contreras-Puente gratefully acknowledge fellowships granted by COFAA-IPN, EDI-IPN, and EDD-IPN. This work was partially supported by CONACyT (Mexico).