# C-V Calculations in CdS/CdTe Thin Films Solar Cells with a CdSxTe1-x Interlayer

Authors: A. Gonzalez-Cisneros; F. L. Castillo-Alvarado; J. Ortiz-Lopez; G. Contreras-Puente
Journal: International Journal of Photoenergy
 (2013)

---

## Abstract

In CdS/CdTe solar cells, chemical interdiffusion at the interface gives rise to the formation of an interlayer of the ternary compoundCdSxCdTe1-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.


---

## Body

## 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 CdSxTe1-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 CdxTe1-x. The understanding of this ternary CdSxTe1-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 ΔEv 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 offsetΔEv 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 (φs1, φs2which 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 voltagesVaare given by
(1)C=[qεS1εS2NdNa2(εS1Na+εS2Nd)]1/2*(Vd-Va)-1/2,
whereVdis the potential well in the heterojunction,Vais the bias voltage, q is the electron charge,Nais the acceptor concentration, Ndis the donor concentration, andεS1andεS2are the dielectric constants of n-type and p-type semiconductor [6]. Therefore, in our case, we have a linear dependence of 1/C2 versus Vaclose toVa=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/CdSxTe1-x).We note that the total band curvature at each side of the interface depends on the bias voltageVa, that is, φS1(Va) and φS2(Va). 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 theCdTe/CdSxTe1-x Interface (C-V Matching Method)

The band line-up equation in this case is (Figure1)
(2)EFV1-EFV2-ΔEV(ternary)-qVa=q[φS2(Va)-φS1(Va)],
where “x” is the ternary concentration and EFV1 and EFV2 are the differences between the quasi-Fermi energy levels with subscripts 1 and 2 corresponding to CdTe and CdSxTe1-x, respectively. The valence band offsetΔEV(ternary)and the respective bulk valence band levels are given by
(3)EFV1=EFn1-EV1=Eg12+34kTln(mh1*me1*)-qφF1,EFV2=EFp2-EV2=Eg22+34kTln(mh2*me2*)-qφF2.
The separation between the quasi-Fermi energy levels is determined for the bias voltage:
(4)EFn1=EFp2+qVa,
where Va 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 withCdTe.The charge neutrality equation under nonequilibrium is given by(5)Q1[φS1(Va)]+Q2[φS2(Va)]+qσ=0,
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]
(6)Q1=sign[φS1(Va)]·2ε0εS1βLD1·eΔu1/2×{φS1(Va)·sinh(u1*)+cosh[u1*-uS1(Va)]-cosh(u1*)}1/2,Q2=sign[φS2(Va)]·2ε0εS2βLD2·eΔu2/2×{φS2(Va)·sinh(u2*)+cosh[u2*-uS2(Va)]-cosh(u2*)}1/2,
where
(7)sign={+1,u<0-1,u>0.
Here, u*=β((φF1+φF2)/2) andβ=q/kT.The third cardinal equation is obtained from the expression of the half capacitance per unit area of the device:(8)1CCdTe-ter=1C1[φS1(Va)]+1C2[φS2(Va)],
where
(9)C1=(ε0εS1)2βLD12·Q1·e∆u1/2·{sinh(u1*)-sinh[u1*-uS1(Va)]}=qε0εS1Q1{N1+2ni1·e∆u1/2·sinh[u1*-uS1(Va)]},C2=(ε0εS2)2βLD22·Q2·e∆u2/2·{sinh(u2*)-sinh[u2*-uS2(Va)]}=qε0εS2Q2{N2+2ni2·e∆u2/2·sinh[u2*-uS2(Va)]}(10)1CTotal≈1CCdTe-ter.
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]:
(11)Eg(x)=(1-x)Eg(CdS)+xEg(CdTe)-bx(1-x),
where b is the “optical bowing coefficient” given by 1.69 eV and the valence band offset ΔEV.


## 2.1. Cardinal Equations for theCdTe/CdSxTe1-x Interface (C-V Matching Method)

The band line-up equation in this case is (Figure1)
(2)EFV1-EFV2-ΔEV(ternary)-qVa=q[φS2(Va)-φS1(Va)],
where “x” is the ternary concentration and EFV1 and EFV2 are the differences between the quasi-Fermi energy levels with subscripts 1 and 2 corresponding to CdTe and CdSxTe1-x, respectively. The valence band offsetΔEV(ternary)and the respective bulk valence band levels are given by
(3)EFV1=EFn1-EV1=Eg12+34kTln(mh1*me1*)-qφF1,EFV2=EFp2-EV2=Eg22+34kTln(mh2*me2*)-qφF2.
The separation between the quasi-Fermi energy levels is determined for the bias voltage:
(4)EFn1=EFp2+qVa,
where Va 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 withCdTe.The charge neutrality equation under nonequilibrium is given by(5)Q1[φS1(Va)]+Q2[φS2(Va)]+qσ=0,
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]
(6)Q1=sign[φS1(Va)]·2ε0εS1βLD1·eΔu1/2×{φS1(Va)·sinh(u1*)+cosh[u1*-uS1(Va)]-cosh(u1*)}1/2,Q2=sign[φS2(Va)]·2ε0εS2βLD2·eΔu2/2×{φS2(Va)·sinh(u2*)+cosh[u2*-uS2(Va)]-cosh(u2*)}1/2,
where
(7)sign={+1,u<0-1,u>0.
Here, u*=β((φF1+φF2)/2) andβ=q/kT.The third cardinal equation is obtained from the expression of the half capacitance per unit area of the device:(8)1CCdTe-ter=1C1[φS1(Va)]+1C2[φS2(Va)],
where
(9)C1=(ε0εS1)2βLD12·Q1·e∆u1/2·{sinh(u1*)-sinh[u1*-uS1(Va)]}=qε0εS1Q1{N1+2ni1·e∆u1/2·sinh[u1*-uS1(Va)]},C2=(ε0εS2)2βLD22·Q2·e∆u2/2·{sinh(u2*)-sinh[u2*-uS2(Va)]}=qε0εS2Q2{N2+2ni2·e∆u2/2·sinh[u2*-uS2(Va)]}(10)1CTotal≈1CCdTe-ter.
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]:
(11)Eg(x)=(1-x)Eg(CdS)+xEg(CdTe)-bx(1-x),
where b is the “optical bowing coefficient” given by 1.69 eV and the valence band offset ΔEV.


## 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
CalculatedC-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,(12)ΔEV=0.99eV,σ=1×1013cm-2,
which are in good agreement with the reported values [9, 11, 12].


## 4. Conclusions

In summary, we have calculated the capacitance versus applied voltage ofCdS/CdTe thin film solar cells considering the formation of a CdSxTe1-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 CdSxTe1-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.



---

*Source: 513217-2013-09-11.xml*