A First-Principles Study of Copper-Deficient Layer and Its Effect in Chalcopyrite-Based Solar Cells: Carrier Dynamics Characteristics

Abstract

CuIn₅Se₈ is reported as a remarkable copper-deficient layer that contains ordered vacancy compounds (OVCs) for high-efficiency chalcopyrite-based solar cells; however, the understanding of its carrier characteristics has remained limited. OVCs could naturally form on the surface of chalcopyrite absorber. In this study, the carrier dynamics characteristics of OVCs were investigated by constructing a junction consisting of chalcopyrite absorber and CdS buffer layer. At first, the band structure of CuIn₅Se₈ was studied to determine the bandgap properties. Then, thermodynamic stability, defect formation energy, defects and carrier concentration, defect transition energy level of CuIn₅Se₈ and its Cd doping state (caused by CdS) were comparatively studied. The results suggest that Cd doping has different effects on the defect and carrier characteristics of OVCs with various chemical potentials. However, the OVC always remains n-type under the whole thermodynamically stable region, with contribution from the hallow-level In_Cu donor defect. Finally, the OVC’s carrier dynamics characteristics were assessed using the collected defect and carrier data. It is indicated that the OVC layer may contribute to the formation of a p-n homojunction in solar cells. Under selenium-rich conditions, the OVC layer increases the carrier density on the n-type side of p-n junction nearly 30-fold, which helps reduce the difference in carrier density and minority current density between two sides of the p-n junction. The conversion efficiency of the solar cell with OVC shows a 7.25% improvement when compared to the control. The distinct behavior of OVCs may serve as a valuable reference for the creation or improvement of a related functional film layer or device.

1. Introduction

Chalcopyrite-based CuIn_xGa_1−xSe₂ (known as Cu(In,Ga)Se₂ or CIGS) solar cells are excellent compound thin film solar cells [1,2,3,4,5,6,7]. They are stable and have a high light absorption coefficient of up to 10⁵ cm⁻¹. The greater part of incident sunlight can be therefore absorbed by a covering of only a few hundred nanometers in thickness. Its bandgap can be continuously adjusted between 1.04 and 1.70 eV by controlling the Ga/In ratio (according to a recent study of chalcopyrite solid solutions, by using the many body perturbation theory, the band adjustment range could be wider, with an upper limit of 3 eV [8]), which makes it possible to generate a greater current. Its module efficiency ranges from 13% to 18.6%, but its solar cell conversion efficiency in the laboratory is as high as 23.6%, showing excellent performance. Additionally, soda-lime glass can be used as the substrate for CIGS solar cells, which reduces production costs compared to traditional crystalline silicon solar cells. To enhance carrier separation and transport efficiency, CIGS is typically formed into a p-n junction with an n-type compound such as CdS, which has been proven to be effective for high-efficiency CIGS solar cells [1]. A space charge region at the interface of the p-n junction can be created with an internal electric field that points from the n region to the p region. When sunlight shines, the photon energy excites the CIGS absorption layer, generating electron hole pairs. Under the built-in electric field, electrons move towards the n-type CdS side, and holes move towards the p-type CIGS side. When linked to an external circuit load, the electrodes gather the separated charge carriers to create a photocurrent, which produces electrical energy. During this process, CdS serves as a crucial so-called buffer layer in the CIGS solar cells. Typically, the CdS buffer layer is deposited using the chemical bath approach right after the CIGS layer was fabricated but before the window layer was prepared. During the solar cell manufacturing process, the CdS buffer layer not only completely encapsulates the rough CIGS surface, passivates the surface and reduces interfacial carrier recombination, but also protects the CIGS absorption layer from damage during the subsequent sputtering process of the ZnO window layer, thereby eliminating the possibility of cell shorting caused by sputtering. Furthermore, CdS has a bandgap of approximately 2.4 eV, which can reduce the conduction band mismatch between the CIGS and ZnO window layers, thereby improving carrier transport in solar cells.

On the surface of CIGS, the formation of ordered vacancy compounds (OVCs) such as Cu(In, Ga)₃Se₅ or Cu(In, Ga)₅Se₈ has been widely accepted. As mentioned above, n-type CdS is an ideal buffer layer for CIGS solar cells. Numerous studies have shown that, during the construction of a solar cell, Cd atoms in CdS can migrate into the surface of CIGS, slightly doping it and therefore enhancing the heterojunction performance in the solar cell. By using the evaporation technology, Ishizuka S. and colleagues [9] constructed the solar cell structure of alkali-containing soda-lime glass or alkali-free zirconia substrate/Mo/CIGS/CuInSe₂-based OVC/CdS/intrinsic and Al-doped ZnO, and investigated the effect of glass substrate and alkali-halide RbF postdeposition treatment (PDT) on the properties of the fabricated solar cell with various OVC thicknesses. It is indicated that both the alkali-containing soda-lime glass and RbF PDT can be benefit to the properties of the CIGS solar cell with thick OVC. And the use of a thick OVC can lead to an enhanced effect of RbF PDT by decreasing the defect recombination, which indicates the OVC is of great significance in suppressing the carrier recombination in the solar cell. By using the evaporation process, Schmid D. and colleagues [10] constructed the Mo/CuInSe₂/OVC structure and conducted a thorough study of the CuInSe₂ and CdS interfaces. It has been suggested that the OVC layer could enhance the CIGS’s optical absorption in visible light. Additionally, the valence band offsets (VBOs) between CuInSe₂ and OVC and conduction band offsets (CBOs) between CuInSe₂ and OVC are −0.28 eV and −0.02 eV, respectively. Using epitaxial samples and in situ synchrotron-based photoelectron spectroscopy, Hofmann A. and colleagues [11] examined the electronic structure and band alignment of the interface between the CuInSe₂ and CuIn₃Se₅ OVC. The top valence band of 0.29 eV for CuIn₃Se₅ is found to be lower than that of CuInSe₂. And the CBO is similarly small, as reported by Schmid D. and colleagues, at −0.04 eV. A theoretical investigation into the structure and energetics of Cd insertion and its connection to Cu depletion in CuIn₅Se₈ OVC was conducted by Janos K. and et al. [12]. It has been discovered that Cd atoms prefer predominantly to sit on Cu sites or Cu vacancy sites. Furthermore, they found that the addition of Cd to the CuInSe₂ materials may enhance the formation of Cu-poor phases. Even at very low temperatures, the CuIn₅Se₈ OVC would spontaneously incorporate Cd up to concentrations of the order of 1% if the Cd chemical potential was fixed by a metallic Cd reservoir. In addition, the characteristic of Cd doping at PVD-CdS/CuInGaSe₂ heterojunctions has also been investigated by Rockett A. and et al. [13]. It is believed that Cd from CdS was driven onto the CIGS surface by an exchange with Cu from CIGS. It is reported that, at the interface of PVD-CdS/CIGS, the concentration of Cd may reach up to 10%. It is worth noting that the reference also suggested that a p-n homojunction could be formed in the PVD-CdS/CIGS solar cells with the help of an OVC layer. The roles of Cu- and Ag-deficient layers in ABX₂ chalcopyrite solar cells were calculated from first principles by Varley J. B. and et al. [14]. It is suggested that a decreased p-d interaction between the X and A atoms in OVC results in lower energy valence and conduction bands compared to the parent ABX₂ chalcopyrite. They discovered that the valence band offsets of OVCs in comparison to parent ABX₂ chalcopyrite are favorable, whereas the conduction band offsets of chalcopyrites beyond CuInSe₂-based absorbers may be harmful in terms of band arrangement. These published studies unequivocally show that the OVC layer does indeed exist on the CIGS surface. It is also usually doped with a certain amount of Cd from the CdS buffer layer and offers good band alignment for CIGS solar cells. However, the understanding of its carrier dynamics characteristics has remained limited. There is little known about the OVC layer’s carrier characteristics and how they relate to experimental synthesis conditions. Additionally, there are still unanswered questions regarding whether Cd doping causes severe carrier recombination centers and finally how the OVC layer affects the carrier dynamics of the fabricated solar cell.

In this study, the carrier and Cd doping characteristics of the CuIn₅Se₈ OVC, which have drawn interest as the key components for high-efficiency CIGS solar cells, were examined by using a first-principles calculation [15,16,17,18]. First, the OVC’s thermodynamic stability was investigated. When determining the chemical potential of the stable OVC, competitive secondary phases were fully considered. Then, the defect formation energy, ionization energy, carrier concentration, Fermi energy level and deep-level defects were systematically investigated within the thermodynamically stable region. Additionally, the non-radiative recombination characteristics of the Cd doped OVC was also examined. Finally, the quantitative defect and carrier characteristics of the OVC were collected, and the effects of the OVC layer together with the p-n junction on the carrier dynamic characteristics of the constructed device were comparatively investigated.

2. Results and Discussion

2.1. Band Structure of the OVC

The band structure of the OVC CuIn₅Se₈ was calculated and the result is shown in Figure 1. The Figure provides valuable insights into the band structure of the OVC and is also related to the charge-transport characteristics. As can be seen from the Figure, the horizontal dashed line stands for the Fermi level. The vertical black thick dashed lines represent high-symmetry Brillouin Zone points in momentum space (K-space). The gap between the conduction band minimum (CB) and valence band maximum (VB) is called bandgap. The bandgap can be either direct or indirect, and it can define the classification of materials, such as insulators, conductors, or semiconductors. The calculation result indicates that the conduction and valence bands separates clearly, which demonstrates the OVC CuIn₅Se₈ is a semiconductor. The bands of CBM are composed of Se sp–In sp states, and the bands of VBM are composed of hybridized Cu d–Se p states, which is similar to its parent CuInSe₂ [14]. CBM possess a dispersive band, indicating that the effective mass of electrons should be small. VBM is relatively flat, indicating that the effective mass of holes would be relatively larger than that of electrons. With the Hartree–Fock exchange value α (AEXX) of about 0.3371 that is mentioned in the Section 3, the OVC turns out a direct bandgap at г high-symmetry points with a bandgap value of 1.33 eV, as indicated in the Figure with a blue arrow. The obtained bandgap value was employed for the subsequent study.

2.2. Thermodynamically Stability of the OVC

The stability of CuIn₅Se₈ compound is a prerequisite for studying its properties. During the crystal growth process of the CuIn₅Se₈, the change in the elemental chemical potential should satisfy several thermodynamic conditions to make the synthesized crystal stable. As indicated in the following Equation (1), under equilibrium growth conditions, the formation energy of the CuIn₅Se₈ should be equal to the chemical potential weighted sum of its component elements, where ${\mu }_i$ stands for the chemical potential of the element, and i represents the most stable phases of the involved elements. ${\mu }_{{{CuIn}_{5}Se}_8}$ represents the formation energy of CuIn₅Se₈. The formation energy of compounds can be calculated from its total energy and the elemental standard states of its component elements, with an AEXX of 0.3371. Additionally, the CuIn₅Se₈ has competing secondary phases such as InSe, In₂Se₃, CuSe₂, CuSe and CuIn₃Se₅. The formation or coexistence of these related binary and ternary compounds and the elemental phases Cu, In and Se should all be avoided to synthesize pure-phase CuIn₅Se₈. Therefore, their corresponding formation energies (the formation energies of elemental phases are 0) should be greater than the chemical potential weighted sum of their component elements. As a result, the relationships in Equation (2) should also be satisfied as the following: where ${\mu }_{InSe}$, ${\mu }_{{{In}_{2}Se}_3}$, ${\mu }_{{CuSe}_2}$, ${\mu }_{CuSe}$, and ${\mu }_{{{CuIn}_{3}Se}_5}$ represent the formation energies of InSe (R3m, NO. 160), In₂Se₃ (P2_1, NO. 4), CuSe₂ (Pnnm, NO. 58), CuSe (Cmcm, NO. 63) and CuIn₃Se₅ (P1, NO. 1):

$${\mu }_{Cu}+{5\mu }_{In}+{8\mu }_{Se}={\mu }_{{{CuIn}_{5}Se}_8}=-11.1874 \mathrm{e}\mathrm{V}$$

(1)

$${\mu }_{In}+{\mu }_{Se}<{\mu }_{InSe}=-1.6995 \mathrm{e}\mathrm{V},$$

$${2\mu }_{In}+{3\mu }_{Se}<{\mu }_{{{In}_{2}Se}_3}=-4.1372 \mathrm{e}\mathrm{V},$$

$${\mu }_{Cu}+{2\mu }_{Se}<{\mu }_{{CuSe}_2}=-0.2390 \mathrm{e}\mathrm{V},$$

$${\mu }_{Cu}+{\mu }_{Se}<{\mu }_{CuSe}=-0.3734 \mathrm{e}\mathrm{V},$$

$${\mu }_{Cu}+3{\mu }_{In}+{5\mu }_{Se}<{\mu }_{{{CuIn}_{3}Se}_5}=-7.0208 \mathrm{e}\mathrm{V}.$$

(2)

where ${\mu }_i$ stands for the chemical potential of an element or compound, and i represents the most stable phases of the involved elements. Here, ${\mu }_{Cu}$ = 0, ${\mu }_{In}$ = 0 and ${\mu }_{Se}$ = 0 indicates that these elements are so rich that they would exist as a pure solid phase. As listed above, it can be concluded that the considered competitive secondary phases of CuIn₅Se₈ are three elements and five additional compounds. Under these constraints, according to the calculation result, the CuIn₅Se₈ that formed on the surface of the chalcopyrite absorber is stable. The chemical potential range of Cu, In and Se that stabilizes the CuIn₅Se₈ compound is bound to a polyhedron in the ${\mu }_{Cu}$, ${\mu }_{In}$ and ${\mu }_{Se}$ space. In order to clearly show the chemical potential boundary points, the top view of the polyhedron is depicted in Figure 2.

Table 1. The chemical potential boundary points p1–p4 that correspond to Figure 2.

Position	Cu/eV	In/eV	Se/eV
p1	−0.3873	−0.932	−0.7675
p4	−0.4752	−0.9613	−0.7382
p2	−0.7711	−2.0832	0.0
p3	−0.8443	−2.0686	0.0

lists the chemical potential boundary points p1–p4 that correspond to Figure 2. In the following sections, all discussion is focused on the stable region of CuIn₅Se₈ with boundary points p1–p4, as indicated in Table 1. Using the same methodology as before, the stability of the Cd doped CuIn₅Se₈ was also computed. It is found that the chemical potential region where Cd doped CuIn₅Se₈ is stable against the formation of four constituent elements (Cu, In, Se and Cd) and six competitive compounds (InSe, In₂Se₃, CuSe₂, CuSe, CuIn₃Se₅ and CdIn₂Se₄) is similar to that of the CuIn₅Se₈. Except for the chemical potential of Cd at p1 (−1.0162 eV), p4 (−1.0748 eV), p2 (−1.7838 eV) and p3 (−1.813 eV), its boundary points are the same as those of CuIn₅Se₈.

2.3. Defect and Carrier Characteristics in OVC

In ternary CuIn₅Se₈ compound, the selenium chemical potential (or experimental conditions) could have a significant impact on the properties (especially defect characteristics) of the compounds [19]. In the study, the calculation was conducted for the entire stable chemical potential region (boundary points p1–p4). And the calculation results were analyzed in terms of the chemical potentials in the sequence of p1-p4-p2-p3, with the chemical potential transition of Se from poor to rich. Meanwhile, as can be seen from the Figure, the chemical potential of Cu and In would generally decreases during this transition. The differential change trend between the chemical potential of the constituent elements Cu, In and Se could have a significant impact on the defect characteristics of CuIn₅Se₈, as can be seen in the following study. It is also noted that the chemical potential of Cd gradually decreases from p1 (−1.0162 eV) to p4 (−1.0748 eV), p2 (−1.7838 eV) and p3 (−1.813 eV).

Figure S1 in the Supplementary Information shows the formation energy of the intrinsic defects in CuIn₅Se₈ as a function of the Fermi energy (E_F, also Fermi level) in the p1-p4-p2-p3 sequence, with Se chemical potential transition from poor to rich, as indicated in Table 1. At the beginning point p1, Se is poor. The In_Cu is the most likely defect to form in this situation in the copper-deficient OVC CuIn₅Se₈ if it is just assessed in terms of chemical trends, no matter whether the E_F is on the VBM side or CBM side. According to Table 1, while moving to p4, Se becomes comparatively richer, but Cu and In’s chemical potentials also become relatively low. In this instance, the In_Cu is still the most possible defect when the E_F is on the VBM side. However, the defect Cu_In with Cu occupying another nonequivalent In in the CuIn₅Se₈ will become dominant when the E_F is on the CBM side. The result shows that, when the Se is still poor and E_F is on the CBM side, the dominant defect in the CuIn₅Se₈ compound could be vulnerable to the change in chemical potentials. This suggests that the synthesis conditions for CuIn₅Se₈ growth should be closely monitored in the experiment. With regard to p2, although Se has become so rich approaching its solid-state precipitation, together with In, it has become much poorer at this point; as shown in Table 1, the defect characteristics of CuIn₅Se₈ compound do not show significant change in comparison to those in p4. Finally, when it is at p3, the Se is still approaching its solid-state precipitation but Cu becomes poorer (with In almost remaining the same as that in p2), the Cu_In defect disappeared, and the defect characteristics of CuIn₅Se₈ compound are similar to that in p1. The formation energy of doping defects in Cd doped CuIn₅Se₈ as a function of the Fermi energy was also studied in the same sequence of p1-p4-p2-p3, and the result is shown in Figure S2 (displayed in corresponding order in Figure S2a–d). It is noted that, in this sequence, the chemical potential of doping element Cd switches from rich to poor, as indicated above. The Figure shows that the doping defects Cd_Cu or Cd_Se would be the most likely defects when the E_F is at the VBM side. The doping defects Cd_Cu or Cd_In would take center stage when the E_F was at the CBM side. The above discussion shows that the defect features of the OVC are strongly influenced by the chemical potentials of the constituent elements, which has important reference value for the controllable growth of OVC in experiments.

Additionally, as illustrated in Figure 3, the transition levels of the intrinsic and Cd doping defects in CuIn₅Se₈ were also extracted. As shown, the decisive defects in CuIn₅Se₈ and Cd doped CuIn₅Se₈ are anticipated to be In_Cu, Cu_In, V_In, Se_In, Se_Cu, In_Se, Cu_Se, In_i, Cu_i, Se_i, Cd_Cu, Cd_Se, Cd_In and Cd_i from the perspective of carrier characteristics and the impact degree of the defect. For CuIn₅Se₈, as displayed in Figure 3a, both V_Cu and V_In are acceptor defects, and they are all found on the VBM side, with corresponding placement positions of 0.045 eV and 0.027 eV above. In_Cu (1+/0, 2+/1+), Se_Cu (1+/0), In_i (1+/0) and Cu_i (1+/0) locate on the CBM side, with location positions of (0.127 eV, 0.137 eV), 0.006 eV, 0.011 eV, and 0.053 eV below, respectively, and they are all donor defects. Among all these shallow-level defects, In_Cu (1+/0, 2+/1+) and Cu_i (1+/0) have comparatively low formation energies and they might have significant impacts on the carrier concentration of OVC CuIn₅Se₈. The OVC could be n-type, since these important shallow-level defects are donor. In addition, a number of deep-level defects, as indicated by the red arrow in Figure 3 (upward denoting acceptor and downward denoting donor), may serve as carrier recombination centers (RCs) in the OVC. For Cd doped CuIn₅Se₈, the important Cd_Cu, Cd_Se, Cd_In and Cd_i defects with low defect formation energy, as indicated in Figure S2, are discussed here. As can be seen in Figure 3b, Cd_Cu (2+/1+) is a deep-level donor, whereas Cd_Cu (1+/0) is a shallow-level donor with a location position of 0.079 eV below the CBM. Cd_Se (2+/0) is a deep-level donor. Cd_In (2+/0) and Cd_In (0/1−) are a deep-level donor and acceptor, respectively, while Cd_In (0/1−) in another nonequivalent position is a shallow-level acceptor with a location position of 0.035 eV above VBM. Cd_i (4+/2+) is deep-level donor, whereas Cd_i (1+/0, 2+/1+) are shallow-level donors with location positions of (0.13 eV, 0.145 eV) below CBM.

Then, the carrier concentration was calculated, and Figure 4a shows the findings. When the chemical potential transforms in the sequence of p1-p4-p2-p3 with Se progressively transition from poor to rich, the carrier properties also undergo a corresponding modification. Owing to the donor defect In_Cu, as shown in Figure S1 (also discussed above), the CuIn₅Se₈ always remains n-type in the whole stable chemical potential region. Although CuIn₅Se₈’s n-type feature has been documented in numerous published works, its carrier source has hardly ever been studied [12,13,14,20,21]. However, understanding the relationship between defects and charge carriers in OVCs with the chemical potentials is of great significance for its performance regulation and future application. As displayed in the Figure, at p1, the n₀ and p₀ of CuIn₅Se₈ are at 4.28 × 10¹⁶ and 0.39 cm⁻³, respectively. The n₀ is much higher than p₀, indicating the n-type semiconductor of the OVC. The n-type OVC CuIn₅Se₈ with high intrinsic carrier concentration may be very beneficial for use in semiconductors and electronics. When at p4, the n₀ of the CuIn₅Se₈ decreases to 3.28 × 10¹⁶ cm⁻³. And the n₀ further decreases to 1.11 × 10¹⁴ cm⁻³ and 1.54 × 10¹⁴ cm⁻³ when coming to p2 and p3, respectively. This indicates the carrier concentration in OVC is decreased when the Se becomes richer. The concentration characteristics of deep-level defects inside the bandgap of CuIn₅Se₈ were also investigated and the result is shown in Figure 5. The decreased n₀ could be due to the deep-level defects with positively charged In_Cu²⁺ increasing and negatively charged Cu_i¹⁻ decreasing, as can be seen in Figure 5a, with Se changing from poor to rich. Meanwhile, during this whole change process, the p₀ maintains the same comparatively low level, indicating the donor always maintains an absolute advantage in the OVC. A change in chemical potential or experimental synthesis circumstances leads to a change in the concentration of charged defects, which in turn induces a change in the concentration of carriers. As shown in Figure 4a, the doping of Cd does not significantly alter the concentration of charge carriers in the OVC. Nevertheless, as compared to undoped OVC, it doubles the n₀ to 3.18 × 10¹⁴ cm⁻³ at p3 and decreases it by half to 1.99 × 10¹⁶ cm⁻³ at p1. The results suggest that Cd doping has different effects on the defect and carrier characteristics of the OVC under different chemical potentials. Furthermore, the E_F of CuIn₅Se₈ was calculated and is pictured in Figure 4b, which provides a clearer illustration of the semiconductor type alterations. And it is evident that the CuIn₅Se₈ together with its Cd doping state are an n-type semiconductor in the entire stable chemical potential region. At p1 and p4, the E_F locates beneath the CBM. When at p2 and p3, the E_F still sits below the CBM but slightly away from it when compared to that of p1 and p4, which may be due to the decrease in the electron carrier concentration, as analyzed above.

In chalcopyrite solar cells with a CdS buffer layer, Cd doping is an important aspect for the OVC. As indicated in Figure 5b, Cd enters into CuIn₅Se₈, resulting in some deep-level defects with high concentrations (highest one Cd_Cu¹⁺ can reach 10²¹ cm⁻³). This is consistent with the published reference that the defect Cd_Cu in Cd doped CuIn₅Se₈ would have a very high concentration [12]. In this study, the non-radiative recombination characteristic of the deep-level defect Cd_Cu¹⁺ located at 1.227 eV below the CBM (also 0.06959 eV above the VBM) with high concentration was investigated. Based the recombination cycle, the recombination process would start from Cd_Cu¹⁺, first capturing one hole from the VBM, becoming 2+ charged. In the second step, the Cd_Cu²⁺ then returns to Cd_Cu¹⁺ with absorbing one electron from the CBM. One step that limits the pace is the minority hole carrier capture. Thus, the total recombination rate will be determined by the hole capture between the 1+ charge (q1) and 2+ charge (q2) states. Therefore, for the Cd doped CuIn₅Se₈ system, the donor defect Cd_Cu (2+/1+) with the capture of a minority carrier hole from a 1+ charge (q1) to 2+ charge (q2) state was extensively studied. And Figure S3 shows the result. Figure S3a shows the configuration coordinate (CC) diagram for 1+/2+ transition of Cd_Cu¹⁺, where the red curve is the ground state potential energy surface (PES) of Cd_Cu¹⁺, and the blue curve is the excited state PES of it. According to the calculation, the non-radiative process barrier for the 1+ charge (q1) and 2+ charge (q2) state transition level is 0.02683 eV, which is reasonable for the transition level value of 0.06959 eV for Cd_Cu (2+/1+), as indicated in Figure 3b. Afterwards, the $W_{ \mathrm{if} }$ and other parameters, as indicated in the Supplementary Information, were also calculated. Finally, the temperature-dependent capture coefficient for the minority hole carrier was obtained, as shown in Figure S3b. It is shown that the non-radiative capture coefficient for the minority hole carrier at 300.0 K is relatively small at 1.16487 × 10⁻¹⁵ cm³/s. The majority (electron) carrier capture coefficient was also studied and it is extremely modest, at 8.8966 × 10⁻²⁵ cm³/s under 300.0 K. According to the analysis above, even though the capture coefficient for the majority (electron) carriers is lower than that of the hole, the concentration of electron carriers is significantly higher than that of the holes. Consequently, the minority carrier hole actually determines the carrier recombination properties.

2.4. Carrier Dynamics Characteristics of the OVC in Chalcopyrite-Based Solar Cells

With the above collected defect and carrier data, the carrier dynamics characteristics of the OVC in the chalcopyrite CuInSe₂ solar cell were studied by employing a solar cell capacitor simulator from the University of Gent in Belgium. The schematic diagram of the OVC-related junction between the CuInSe₂ absorber layer and the CdS buffer layer is shown in Figure 6. The main functionality of the simulator is to solve the one-dimensional equations including Poisson’s equation and continuity equations for the electron and hole carriers [22,23,24], as indicated in the Section 3. For comparative study, the carrier dynamics properties of the junction with or without the OVC layer (CuIn₅Se₈ and Cd doped CuIn₅Se₈ layer) were studied. And the carrier and defect features of the OVC at p3, common conditions for Se abundance in the experiment, were chosen for the comparative simulation analysis. The essential input parameters were from our calculations and published references [14,19,20]. Table S1 provides a summary of the parameters used for the modeling of the solar cell device. The current–voltage characteristic of the solar cell with and without the OVC layer was thus performed, and the result is shown in Figure 7 (the “+” symbol on the upper right side of the marks stands for the solar cell with the OVC layer). The results indicate that the solar cell without the OVC shows a V_oc of 0.5212 V, J_sc of 43.1743 mA cm⁻², FF of 77.89% and conversion efficiency of 17.53%. In contrast, the solar cell with the OVC has a V_oc of 0.6332 V, J_sc of 37.0438 mA cm⁻², FF of 79.29% and conversion efficiency of 18.60%. The enhanced V_oc and FF could be the primary factors contributing to the absolute gain of 1.27 percentage points in conversion efficiency (also a 7.25% relative improvement when compared to the 17.53% conversion efficiency of the solar cell without the OVC) [13,20]. It is noted that the absolute value of 18.60% is modest for the CIGS context (state-of-the-art >23%, as mentioned in the Section 1), which may be due to the fact that this is the modeling result for idealized structures, without consideration of Ga grading, alkali effects, etc.

For further exploration of the improved V_oc in the solar cell with the OVC layer, the Mott–Schottky plot is implemented to compare the variation in the built-in potential (V_bi) for the solar cell with and without the OVC layer. As shown in Figure 8a, the solar cell with the OVC layer possesses a higher V_bi of 0.86 V than that of the solar cell without the OVC layer (0.81 V), indicating a more efficient separation and transport of photogenerated charger carriers for the solar cell with the OVC layer. A higher V_bi would result in a higher V_oc for the solar cell with the OVC layer, as shown above. And the higher V_oc in the solar cell with the OVC layer could be attributed to the suppressed defect recombination. In Figure 8b, the comparative trend of V_oc dependence on illumination intensity is shown for the solar cell with and without the OVC layer. The V_oc-illumination study reveals that the V_oc has a good linear relationship with the illumination intensity, regardless of the presence or absence of the OVC layer. However, at all illumination intensities, the V_oc of solar cells with the OVC layer is higher than that of solar cells without the OVC layer. Figure 9 shows the nyquist impedance characteristic of the solar cell with and without the OVC layer. The nyquist impedance plot displays decreased impedance as the OVC layer appears in the solar cell. And a single semicircle with greatly decreased radius is observed when the OVC appears in the solar cell, suggesting that the low real part of impedance for the solar cell with the OVC layer could be obtained when compared to that without the OVC layer. This indicates that the carrier recombination in the solar cells with the OVC layer may be suppressed, consistent with the result of the improved V_oc and FF, as demonstrated above.

To gain a more comprehensive understanding of OVC functionality, the carrier density of the solar cell without and with the OVC layer, as well as the associated band diagram, were also studied, as displayed in Figure 10. It can be seen from Figure 10a,b that, in the device with the OVC, the carrier density on the p-type side of the solar cell p-n junction is 1.34 × 10²⁰ cm⁻³, which is higher than that in the device without the OVC at 1.22 × 10¹⁹ cm⁻³. In addition, in the device with the OVC, the carrier density on the n-type side of the p-n junction is 1.10 × 10¹⁶ cm⁻³, which is much higher than that in the device without the OVC at 3.55 × 10¹⁴ cm⁻³, indicating the OVC can increase carrier density on the n-type side by nearly 30-fold. Therefore, the difference in carrier density between two sides of the p-n junction with the OVC layer decreased from 3.43 × 10⁴ times (1.22 × 10¹⁹ cm⁻³ to 3.55 × 10¹⁴ cm⁻³) to 1.22 × 10⁴ times (1.34 × 10²⁰ cm⁻³ to 1.10 × 10¹⁶ cm⁻³) when compared to that without the OVC layer. The improved carrier behavior could be contributed to the OVC layer and the possible formation of the p-n homojunction between the CuInSe₂ and the CuIn₅Se₈ [13]. Figure 10c is the corresponding band diagram of the device under working. The solar cell with the OVC has a smoother E_c band at the right side of the junction, which improves the E_c band continuity between the CuInSe₂ and the CdS and thus enhances the carrier transport. Figure 11 shows the current density of the minority hole in the solar cell without and with the OVC. It is evident that the solar cell’s p-n junction with the OVC likewise has a well-balanced minority hole, with the difference in minority carriers between two sides of the p-n junction decreased (from 6000 mA cm⁻² to 30 mA cm⁻²) when compared to that of the p-n junction without the OVC (from 3.97 × 10⁶ mA cm⁻² to near zero). The well-balanced minority hole between two sides of the p-n junction could benefit from the speculative homojunction that may form between the p-type CuInSe₂ absorber layer and n-type OVC layer. Figure 12 shows the occupation probability of deep-level defects for electrons in the solar cell with and without the OVC layer. From Figure 12a, the electrons that may be present at the n side of the p-n heterogeneous junction could be from CdS with neutral. However, when the OVC layer was added into the solar cell, the electrons that may be present at the n side of the speculative p-n homogeneous junction could be from CuIn₅Se₈ (3+/2+) and Cd doped CuIn₅Se₈ (0/−1, 2+/1+), with higher occupation probability, as shown in Figure 12b. This could be responsible for the improved balanced minority hole between the two sides of the p-n junction. The reduced difference in carrier concentration between the two sides of the p-n junction may help form a more balanced carrier concentration between the two sides of p-n junction, which may contribute to the reduced carrier recombination at the p-n interface. As illustrated in Figure 7, the reduced carrier recombination at the p-n junction interface of the solar cell with the OVC layer (which may contribute to the enhanced V_oc and FF, as discussed above) could be the main reason for the increased conversion efficiency in the solar cell [13]. Therefore, further improvement in defect passivation and carrier balance (also with Ga grading, alkali treatment and more complex interface chemistry regulation) is expected to achieve higher conversion efficiency in solar cells.

3. Method of Calculation

Using VASP (version 6.4.2), first-principles calculations were performed to study the electronic structure of the tetragonal OVC CuIn₅Se₈ (P-42m, NO. 111) [25,26,27] and its Cd doped state (the concentration of Cd is around 0.5%, with one Cd atom added into the 224-atom supercell with a substitution or interstitial site, soon to be mentioned) [12]. The total energy was calculated in VASP using density functional theory with a plane wave energy cutoff of 479 eV (approximately 1.3-fold the maximum ENMAXs in the pseudopotential files, proved sufficiently converged) [28]. Based on the data from the Materials Project database (the structures along with the reciprocal space meshes and PAW potentials of the elemental components and secondary phases), the stable chemical potentials of the compound were studied. For calculation, all structures were first relaxed at the PBE level, and then were calculated with the Heyd–Scuseria–Ernzerhof hybrid functional (HSE06). The defect characteristics of the OVC were investigated by employing a supercell of 224 atoms with an optimized lattice constant a of 17.666 Å, b of 21.257 Å and c of 17.666 Å. Due to the large size of the supercell, the size of the reciprocal space is significantly reduced, resulting in highly localized defect states. And considering the high computational power required for HSE hybrid functional calculations, this study set the reciprocal space meshes for the 224-atom supercell calculation at 1 × 1 × 1 Г-centered k-point grid sampling, which proved sufficiently converged after comparison with results at 2 × 2 × 2 Г-centered k-point grid sampling. The calculation was stopped with forces below 0.01 eV/Å. The defect corrections (including potential alignment for charged defects) were also performed [29]. To determine the formation energy and transition energy levels of the defects, HSE06 and finite-size electrostatic correction schemes were employed [30,31,32]. The bandgap value of OVC CuIn₅Se₈ varies between approximately 1.2–1.3 eV. And CuIn₅Se₈ is considered as a copper-deficient state of CuInSe₂. Based on the influence of Cu deficiency on the bandgap value of CuInSe₂ (also reported in reference NO. 14), and considering that Cd doping may also exacerbate the degree of Cu deficiency (the concentration for Cd doping could reach as high as 10%, as reported in reference NO. 13), the bandgap of the OVC in this study was set at 1.33 eV with a Hartree–Fock exchange value α (AEXX) of about 0.3371.

In order to better study the carrier dynamics characteristic of the OVC layer, at first, some essential and important parameters, such as quantitative data in defect formation energy, defect charge transition level, deep-level defects, defect and carrier concentration, and carrier capture coefficient, should be determined. The calculation processes for these parameters can be found in the references [32,33,34,35,36,37,38,39] (also outlined in Supplementary Information). And the calculated results for the CuIn₅Se₈ and its Cd doping state were also comparatively analyzed in the Section 2. Then, the carrier dynamics characteristics of the OVC layer in the chalcopyrite CuInSe₂ solar cell were simulated by using a solar cell capacitor simulator from the University of Gent in Belgium. The main functionality of the simulator is to solve the one-dimensional equations including Poisson’s Equation (3) and continuity equations for electron (4) and hole (5) carriers with the following approximate boundary conditions [23]:

$$\nabla \cdot \mathsf{ϵ}\nabla \phi =-\mathrm{q}\left(\mathrm{p}-\mathrm{n}+{\mathrm{N}}_{\mathrm{D}}^{+}-{\mathrm{N}}_{\mathrm{A}}^{-}\right)$$

(3)

$$\nabla \cdot {\mathrm{J}}_{\mathrm{n}}=\mathrm{q}\left(\mathrm{R}-\mathrm{G}\right)+\mathrm{q}{\displaystyle \frac{\partial \mathrm{n}}{\partial \mathrm{t}}}$$

(4)

$$\nabla \cdot {\mathrm{J}}_{\mathrm{p}}=-\mathrm{q}\left(\mathrm{R}-\mathrm{G}\right)+\mathrm{q}{\displaystyle \frac{\partial \mathrm{p}}{\partial \mathrm{t}}}$$

(5)

where ε and φ are the dielectric constant and electrostatic potential, respectively. N_D⁺ and N_A⁻ are the density of ionized donor and acceptor, respectively. n and p are the free carrier concentrations for the electron and hole, respectively. J_n and J_p are the electron and hole current density, respectively. R and G are the recombination and generation rate, respectively.

4. Conclusions

The defect and carrier characteristics of the OVC CuIn₅Se₈ have been systematically studied via first-principles calculations by employing a 224-atom supercell with HSE06 hybrid functional and finite-size effect correction. The stable chemical potential range of CuIn₅Se₈ was determined. The stable characteristics of the Cd doped OVC was also studied and it is similar to that of OVC CuIn₅Se₈. The chemical potential boundary points were then determined. The formation energy of intrinsic defects in CuIn₅Se₈ and Cd doped CuIn₅Se₈ was studied in the sequence of Se chemical potential transition from poor to rich. It is indicated that the CuIn₅Se₈ always remains n-type in the entire stable chemical potential region, which could be due to the shallow-level donor defect of In_Cu. A deep-level defect Cd_Cu (2+/1+) with extremely high concentrations up to 10²¹ cm⁻³ is present in the Cd doped CuIn₅Se₈. Fortunately, the non-radiative capture coefficient for the minority hole carrier of the deep-level defect Cd_Cu (2+/1+) is relatively small at 1.16487 × 10⁻¹⁵ cm³/s, indicating its defect tolerance characteristics for the Cd doping to the OVC CuIn₅Se₈.

Based on the carrier and defect data, the carrier dynamics characteristics of the OVC layer were further examined. According to research, the OVC may help solar cells generate p-n homojunctions, which greatly reduces the difference in carrier density (from 3.43 × 10⁴ times to 1.22 × 10⁴ times) and minority current density (from 10⁶ mA cm⁻² to 6000 mA cm⁻²) between the two sides of the p-n junction. With decreased carrier recombination in the junction interface, this contributes to a 7.25% improvement in conversion efficiency when compared to the control. Additionally, the investigation of the OVC’s defect and carrier properties helps to clarify how it functions in chalcopyrite-based solar cells, which is beneficial for the further optimization and development of these solar cells. The study is also of crucial importance for the further development and application of a relevant photoelectric device with such a unique functional thin film.