A hybrid functional study of native point defects in Cu₂SnS₃: implications for reducing carrier recombination

Abstract

The native point defects in the earth-abundant solar material Cu₂SnS₃ are studied using the hybrid functional. To generate more accurate formation energies of defects, the extended Freysoldt, Neugebauer, and Van de Walle (FNV) method is used for finite-size corrections in the charged supercell calculations. According to the calculated defect energetics, it is found that the usual experimental conditions can lead to abundant deep centers that deteriorate solar cell performance. To reduce the carrier recombination caused by the deep centers, Sn-rich and S-poor conditions should be attempted. The present calculations also give satisfactory explanations for a recent experimental work on the defect levels in Cu₂SnS₃.

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1. Introduction

Recently, there has been increasing interest in searching for earth-abundant solar materials such as Cu₂ZnSn(S,Se)₄,¹ ZnSnP₂,² and Cu₂SnS₃.³ Photovoltaic device efficiencies greater than 4% and 6% have been achieved based on Cu₂SnS₃ and its alloy, respectively.^4–6 The bandgap of Cu₂SnS₃ is usually reported at approximately 0.9–1 eV,^6–10 and can be tuned by alloying with Si or Ge on the Sn site.^5,11 The conversion efficiency of Cu₂SnS₃ solar cells depends on many factors such as the p–n junction interface and properties of the Cu₂SnS₃ absorber including morphology, secondary phases, and point defects.¹² While morphology and secondary phases have been relatively well controlled, the understanding and control of point defects are still insufficient for Cu₂SnS₃.¹² Point defects are difficult to experimentally measure, and thus, theoretical calculations are important and valuable.¹³ There are some theoretical works regarding the defect chemistry of Cu₂SnS₃ in the literature,^3,14 but only the functionals at the local density approximation (LDA)/generalized gradient approximation (GGA) level are used, and thus, their reliability is questionable. In addition, finite-size corrections are needed for charged defects, but the proper correction method has long been debated.

Komsa et al.¹⁵ compared several major methods and found that the scheme proposed by Freysoldt, Neugebauer, and van de Walle (FNV)¹⁶ provides overall superior performance. Later, Kumagai et al. extended the FNV method to anisotropic systems by using the atomic site potential.¹⁷ It is noted that both the original FNV method and the extended method have been implemented in high-throughput defect calculations.¹⁸ The aim of the present work is to determine more accurate defect energetics of Cu₂SnS₃ by using hybrid functionals and the extended FNV method, which is important for controlling the point defects and the conversion efficiency of Cu₂SnS₃.

2. Computational details

All first-principles calculations in the present work are based on density functional theory (DFT),^19,20 using the projector augmented wave (PAW)²¹ method implemented in the Vienna Ab initio Simulation Package (VASP).^22,23 The (semi)local functionals such as LDA and GGA²⁴ usually underestimate the bandgap and cannot correctly describe point defects; thus the Heyd–Scuseria–Ernzerhof (HSE) hybrid functional²⁵ is used due to its improved performance with regard to these aspects. The HSE functional mixes 25% of screened Hartree–Fock exchange with the screening parameter being 0.2 Å⁻¹, and is used in conjunction with PBEsol²⁶ for a more accurate description of solids, as well as the D3 correction for the van der Waals force, which is important for correctly predicting thermochemical quantities in sulfur-bearing systems.²⁷ This functional combination can be termed as HSEsol + D3.²⁸

For the defect calculations, a 2 × 1 × 2 supercell (constructed from the conventional unit cell) is used that consists of 96 atoms when the supercell is defect-free and a 2 × 2 × 2 mesh of Monkhorst–Pack k-points.²⁹ An energy cutoff of 280 eV is used for plane-wave basis expansion in all the calculations, which is similar to the cutoff used in a hybrid functional study of point defects in Cu₂ZnSnS₄.³⁰ A larger cutoff of 370 eV leads to changes below 5 meV per atom in the formation energies of compounds in the Cu–Sn–S system according to the test calculations, which are sufficiently small for the purposes of the present work. The residual forces after geometry optimization are less than 0.02 eV Å⁻¹. For the finite-size corrections for charged-defect supercell calculations, the extended FNV method proposed by Freysoldt et al.¹⁶ and modified by Kumagai¹⁷ is used.

The formation energy of a point defect is calculated as²

where E[D^q] and E_P are the total energies of the supercell with a defect D in the charge state q and the perfect supercell, respectively. E_corr[D^q] is the finite-size correction for the charged-defect supercell. n_i and μ_i are the number of removed atoms i due to the defect formation and its chemical potential, respectively. ε_F is the Fermi level.

The thermodynamic transition level between the charge states q and q′ of the defect D referenced to the valence band maximum (VBM) is calculated as²

where ε_VBM is the energy of VBM.

Finite-size corrections are also needed for the defect single-particle levels. The following corrections^2,15,31 are used:

where E^PC_corr is the point charge correction energy, i.e., the Madelung energy for a periodic array of point charges immersed in a neutralizing background charge, which can be estimated by an Ewald summation.^2,17,31

3. Results and discussion

3.1. Properties of the perfect crystal

Several structures of Cu₂SnS₃ have been previously reported, including cubic, tetragonal, monoclinic, and triclinic.^5,32,33 All these structures are varieties of the zincblende structure with different degrees of Cu/Sn disorder. In the present work, only the monoclinic structure illustrated in Fig. 1 was studied for simplicity; it has 24 atoms in the conventional unit cell with the space group Cc. The calculated lattice parameters listed in Table 1 are in good agreement with the experimental values. The slightly smaller experimental lattice parameters may be partly caused by abundant thermal vacancies due to their low formation energies, which will be shown later. Because the dielectric constants are important in the finite-size corrections for charged defects, they were also calculated and are listed in Table 1. Due to the Cc symmetry, the dielectric tensor has four independent components, showing weak anisotropy. The total dielectric tensor ε^tot is the sum of the ion-clamped dielectric tensor ε^ele and the ionic contribution ε^ion. It can be seen that ε^ion much less contributed to ε^tot than ε^ele from Table 1, because the ionicity of Cu₂SnS₃ is not very strong. The band structure is plotted in Fig. 2, clearly showing that Cu₂SnS₃ is a direct bandgap semiconductor, with both VBM and conduction band minimum (CBM) located at the Γ point. The calculated bandgap is 0.80 eV, which is the same as the value obtained from another HSE study,³⁴ and is also in reasonable agreement with the experimental values at approximately 0.9–1 eV.^6–10

Fig. 1 Crystal structure of the monoclinic Cu₂SnS₃. The Cu, Sn, and S atoms are represented by the red, grey, and yellow spheres, respectively.

Table 1 Calculated lattice parameters and dielectric constants of the monoclinic Cu₂SnS₃

	Lattice parameter	ε ^ele	ε ^ion	ε ^tot
Cu2SnS3 Cc (#9) 24 atoms	a = 6.601 Å	ε 11 = 10.16	ε 11 = 3.71	ε 11 = 13.87
	b = 11.425 Å	ε 22 = 10.86	ε 22 = 3.69	ε 22 = 14.55
	c = 6.607 Å	ε 33 = 11.48	ε 33 = 4.18	ε 33 = 15.66
	β = 109.44°	ε 13 = −1.61	ε 13 = 0.38	ε 13 = −1.23
Expt.	a = 6.449 Å	NA	NA	NA
	b = 11.319 Å
	c = 6.428 Å
	β = 108.37°

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Fig. 2 Band structure of the monoclinic Cu₂SnS₃.

3.2. Chemical potential diagram

The stability of Cu₂SnS₃ is limited by the other phases in the Cu–Sn–S system. The formation energies of the compounds in the Cu–Sn–S system were calculated using HSEsol + D3 and are listed in Table 2. An overall agreement between the calculations and the experiments^35–38 was found, although it is still difficult to achieve chemical accuracy (approximately 0.04 eV per atom) by DFT at the current stage even when using chemically similar references.³⁹ For comparison, the results calculated by PBEsol + D3 are also listed in Table 2, showing poorer agreement with the experimental values than those calculated by HSEsol + D3. Using the formation energies calculated by HSEsol + D3, the chemical potential diagram of the Cu–Sn–S system is generated and visualized in Fig. 3. Cu₂SnS₃ is surrounded by Cu, Cu₂S, Cu₅Sn₂S₇, Cu₄Sn₇S₁₆ and SnS, which are therefore the possible secondary phases in Cu₂SnS₃. Within the stable region of Cu₂SnS₃, the variation of Cu chemical potential is small, while both Sn and S have large variation in their chemical potential, which has an important consequence in the point defects, as will be discussed later. It should be noted that the phase equilibria is dependent on the temperature, which has been revealed by our previous work,⁴⁰ although we focus on the 0 K first-principles calculations in the present work. It is also noted that Cu₄SnS₄ and Cu₃SnS₄ do not appear in the chemical potential diagram because they are not thermodynamically stable against other phases, according to the present calculations.

Table 2 Formation energies of compounds in the Cu–Sn–S system calculated by PBEsol + D3 and HSEsol + D3, compared with experimental values. The energies are in eV per formula. The reference states are fcc Cu (Fm3̄m), tetragonal Sn (I4₁/amd), and orthorhombic S (Fddd)

Compound	Space group	E f
		PBEsol + D3	HSEsol + D3	Expt.
Cu2S	P21/c	−0.45	−0.76	−0.83^35
CuS	P63/mmc	−0.45	−0.54	−0.55^36
SnS	Pnma	−1.00	−0.99	−1.13^37
SnS2	P3̄ml	−1.36	−1.47	−1.57^37
Cu2SnS3	Cc	−2.21	−2.80	−2.52^38
Cu4SnS4	Pnma	−2.46	−3.51	−3.40^38
Cu4Sn7S16	R3̄m	−11.17	−13.03	NA
Cu3SnS4	I4̄2m	−3.04	−3.39	NA
Cu5Sn2S7	C2	−5.34	−6.29	NA

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Fig. 3 Chemical potential diagram of the Cu–Sn–S system based on 0 K HSEsol + D3 calculations.

3.3. Native point defects

The native point defects of Cu₂SnS₃ studied in the present work include V_Cu, V_Sn, V_S, Cu_Sn, and Sn_Cu. The sulfur atoms have two different coordinations, consisting of “three Cu plus one Sn” and “two Cu plus two Sn,” respectively, therefore leading to two types of S vacancies, V_S1 and V_S2. The thermodynamic transition levels between different charge states of defects are listed in Table 3, including both the uncorrected values and the corrected ones. The extended FNV method employed here is not applicable to the defects with the perturbed host state (PHS), which is very delocalized and spills out from the supercell, making the correction method invalid.^2,17 The defect with PHS has small thermal ionization energy, typically less than 0.1 eV.² In Table 3, three uncorrected levels are close to the band edge, including V_Cu (0/−1), V_Sn (0/−1) and Cu_Sn (0/−1), and are thus related to small thermal ionization energy and possibly PHS. The holes in V⁰_Cu, V⁰_Sn, and Cu⁰_Sn are possible PHS. One method for determining if a defect level is PHS is based on its scaling behavior with the size of the supercell,¹⁵ which may be not efficient. Here, the squared wavefunction of the studied defect state⁴¹ is used for this purpose. Fig. 4 shows the squared wavefunction of the lowest unoccupied states of V⁰_Cu, V⁰_Sn, and Cu⁰_Sn. It can be clearly seen that the hole in V⁰_Cu is very delocalized and spills out from the supercell, while for V⁰_Sn and Cu⁰_Sn, the holes are relatively localized and contained in the supercell, despite a certain extent of delocalization. Therefore, it is justified to treat V⁰_Sn and Cu⁰_Sn in the framework of the extended FNV method, i.e., the correction is zero due to the zero charge, but this is not true for V⁰_Cu due to its acceptor-like PHS. To address this issue, it is simply approximated that ε^th(V_Cu,0/−1) ≈ 0, as the treatment in the literature.² After the corrections, the thermodynamic transition levels are deeper, and some of them are removed from the bandgap, with the remaining deep levels being V_Sn (0/−1), V_Sn (−1/−2), and Cu_Sn (0/−1). Both V_S1 and V_S2 have no thermodynamic transition levels within the bandgap, regardless of whether the corrections are applied or not. A comparative example is the sulfur vacancy in Cu₂ZnSnS₄ (CZTS), which also has no transition levels in the bandgap.³⁰ It is noted that a previous GGA level study reports the charge states within the bandgap ranging from [−1,0], [−4,0], [−3,0], and [0,3] for V_Cu, V_Sn, Cu_Sn, and Sn_Cu, respectively,³ which is similar to the present uncorrected results.

Table 3 Thermodynamic transition levels of point defects in the monoclinic Cu₂SnS₃ with and without the extended FNV corrections¹⁷

Defect	q/q′	ε ^th (uncorrected)	ε ^th (corrected)
V Cu	0/−1	−0.067	0
V Sn	0/−1	0.093	0.470
	−1/−2	0.120	0.689
	−2/−3	0.146	>CBM
	−3/−4	0.382	>CBM
CuSn	0/−1	−0.049	0.241
	−1/−2	0.314	>CBM
	−2/−3	0.384	>CBM
SnCu	+1/+2	0.121	<VBM
	+2/+3	0.541	<VBM

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Fig. 4 Squared wavefunction of the lowest unoccupied state of (a) neutral Cu vacancy V⁰_Cu, (b) neutral Sn vacancy V⁰_Sn, and (c) neutral antisite Cu⁰_Sn within the (001) plane through the position of the defect in the monoclinic Cu₂SnS₃. The unit of the squared wavefunction is a₀⁻³, where the Bohr radius a₀ = 0.529 Å. The contour lines are spaced by 0.005 a₀⁻³. Each figure contains four supercells. The position of the defect is indicated by the red dashed circle.

The local structures and defect single-particle levels of selected defects are illustrated in Fig. 5. For V_Cu⁻¹, the distances between the vacancy and the neighboring S atoms are 2.19–2.27 Å [Fig. 5(a)], while in the perfect crystal, the Cu atom forms three bonds of 2.28 Å and one bond of 2.33 Å with the neighboring S atoms. V_Cu⁻¹ has no in-gap states [Fig. 5(a)]. As shown in Fig. 5(b) and (c), V⁰_Sn and V_Sn⁻² have rather different geometric and electronic structures. In the perfect crystal, the Sn atom forms two bonds of 2.36 Å and two bonds of 2.47 Å with the neighboring S atoms. For V⁰_Sn, the distances between the vacancy and the neighboring S atoms are 2.21–2.31 Å, and there are two unoccupied defect single-particle levels slightly above VBM. For V_Sn⁻², one S atom has a significantly larger distance of 2.35 Å with the vacancy, compared to the corresponding value of 2.25 Å for V⁰_Sn. The two defect levels of V_Sn⁻² are within the upper part of the bandgap, with the lower one fully occupied. From Fig. 5(d) and (e), it can be seen that Cu_Sn⁻¹ has two Cu–S bonds of 2.36 and 2.40 Å, larger than those of 2.33 and 2.36 Å in Cu⁰_Sn, respectively. The two defect levels of Cu⁰_Sn are within the lower part of the bandgap, with the lower one at VBM half occupied. Cu_Sn⁻¹ has only one in-gap defect level, which is in the upper part of the bandgap and unoccupied. Sn_Cu⁺¹ has Sn–S bonds of 2.61–2.73 Å, and has no in-gap states but has an occupied defect level 0.11 eV below VBM [Fig. 5(f)].

Fig. 5 Local structures and defect single-particle levels of selected point defects in the monoclinic Cu₂SnS₃. The position of a vacancy is defined as the geometric center of the neighboring atoms. The distances between the defect and its neighboring atoms are in units of Å. The corrected defect single-particle levels are labeled with their positions relative to VBM in eV. The defect single-particle levels located in the valence or conduction band before the correction are schematically represented by dashed lines, because they are not clearly defined.

The formation energies of point defects in Cu₂SnS₃ are calculated under two different conditions of (μ(Cu), μ(Sn), μ(S)) represented by A = (−0.606, −1.303, −0.092) eV and B = (0, −0.081, −0.905) eV in Fig. 3, respectively. The secondary phases at A are Cu₅Sn₂S₇ and Cu₄Sn₇S₁₆, while those at B are Cu and SnS. The results are shown in Fig. 6. No matter under which condition, the most favorable charged defect is V_Cu⁻¹, making Cu₂SnS₃ p-type the intrinsic type with the minority carriers being electrons. Under condition A, the pinning level of V_Cu⁻¹, where its formation energy becomes zero, is far away from CBM, making n-type doping very difficult under this condition. Cu_Sn and V_Sn are deep centers, which have defect levels deep in the bandgap and are detrimental for solar cell performance because they can accelerate electron–hole recombination. Cu_Sn should be more abundant than V_Sn due to its lower formation energy. In the p-type conditions, i.e., when the Fermi level is near VBM, the formation energy of Cu⁰_Sn is only approximately 0.94 eV, leading to a concentration of approximately 10¹⁶ cm⁻³ at a typical growth temperature of 550 °C. Such abundant Cu⁰_Sn can effectively capture electrons, severely limiting the minority carrier life and reducing the conversion efficiency. Similarly, Cu_Sn is also found in a deep center in Cu₂ZnSnS₄, but the formation energy is higher.³⁰ Under condition B, V_Cu is less favorable compared with the situation under A, and no pinning level occurs within the bandgap, making n-type doping possible. The deep centers Cu_Sn and V_Sn are also less favorable. At first glance, this seems slightly counterintuitive for Cu_Sn, because B has such a high Cu chemical potential that Cu metals form. However, B also has the highest Sn chemical potential within the stable region of Cu₂SnS₃. Because Sn has much larger variation in its chemical potential than Cu, it plays a dominant role in the formation of Cu_Sn. Therefore, it is seen that the phase stability has a profound influence on the defect chemistry.

Fig. 6 Formation energies of native point defects in the monoclinic Cu₂SnS₃ under condition (a) A and condition (b) B. The Fermi level is referenced to the VBM.

In the experiments, usually S-rich conditions are employed, which are closer to condition A, resulting in abundant Cu_Sn deep centers according to the present calculations. This may be one of the reasons why the conversion efficiency of pure Cu₂SnS₃ solar cells has not exceeded 5%⁴ thus far. Condition B can reduce Cu_Sn and V_Sn, but the drawback is the existence of Cu, which may reduce the shunt resistivity and be detrimental for solar cell performance. It may be more optimal to slightly decrease the Cu chemical potential from B and enter into the single phase Cu₂SnS₃ region or the Cu₂SnS₃/SnS mixture region to avoid the formation of Cu while maintaining low concentrations of the Cu_Sn deep center. This condition is relatively Sn-rich and S-poor. Admittedly, the solar cell performance is also determined by many factors in addition to the concentration of deep centers, although we are focusing on point defects here.

According to the present calculations, the two most abundant transition levels are ε^th(V_Cu,0/−1) ≈ 0 and ε^th(Cu_Sn,0/−1) = 241 meV, while the other transition levels are associated with much higher formation energies and with more difficult detection. A recent experimental study found two trap levels with activation energies of 41 ± 0.4 meV and 206 ± 7 meV based on impedance spectroscopy,⁴² which are in excellent agreement with our results. The satisfactory explanation of experimental facts demonstrates the soundness of the methods used in the present study. It is worth noting that there are large discrepancies in previous theoretical works employing GGA level calculations.^3,14 For instance, in ref. 3, the formation energy of Cu⁰_Sn is above 2.5 eV, but in ref. 14, it is below 1 eV and can be even less than −0.5 eV under some conditions. Such discrepancies indicate that the GGA level calculations may not be sufficiently reliable, and additional sophisticated tools such as the hybrid functionals are necessary.

4. Conclusion

In the present work, the native point defects in the earth-abundant solar material Cu₂SnS₃ are studied using the hybrid functional and the extended FNV method for finite-size corrections in the supercell calculations. First, the properties of the perfect crystal are calculated and the stable region of Cu₂SnS₃ is determined based on the chemical potential diagram of the Cu–Sn–S system. Then, the thermodynamic transition levels of the studied defects are calculated and corrected based on the extended FNV method. Prior to performing the corrections, the neutral Cu vacancy is identified as the defect with the perturbed host state based on the squared wavefunction analysis, which cannot be addressed with the extended FNV method. The number of charge states within the bandgap is much less than that obtained in the previous study. Third, the local structures and defect single-particle levels are analyzed for selected defects. Fourth, the defect formation energies are calculated under different conditions, indicating that the typical experimental conditions may lead to a high concentration of deep centers that are detrimental for solar cell performance. The Sn-rich and S-poor condition is proposed to reduce deep centers. Last, it is found that the present calculations provide satisfactory explanations for a recent experimental work on defect levels. The present work provides useful insights into the defect chemistry of Cu₂SnS₃, which are indispensable for understanding and improving its properties.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

This work was financially supported by the NSF with Grant No. CHE-1230924 and CHE-1230929. First-principles calculations were carried out on the LION clusters at Pennsylvania State University. Calculations were also carried out on the CyberStar cluster funded by the NSF through Grant No. OCI-0821527. The crystal structure and the squared wavefunction were plotted using VESTA.⁴³ Visualization of the chemical potential diagram was performed with CHESTA.⁴⁴

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