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Elucidating Structural Disorder and the Effects of Cu Vacancies on the Electronic Properties of Cu2ZnSnS4 Kuang Yu, and Emily A. Carter Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.5b04351 • Publication Date (Web): 21 Jan 2016 Downloaded from http://pubs.acs.org on January 27, 2016

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Chemistry of Materials

Elucidating Structural Disorder and the Effects of Cu Vacancies on the Electronic Properties of Cu2ZnSnS4 Kuang Yu1 and Emily A. Carter1,2,* 1

Department of Mechanical and Aerospace Engineering and 2Program in Applied and Computational

Mathematics and Andlinger Center for Energy and the Environment, Princeton University, Princeton, New Jersey 08544-5263 *

E-mail: [email protected]

ABSTRACT Although a promising photovoltaic material that is inexpensive and easy to manufacture, Cu2ZnSnS4 (CZTS) suffers from a low open circuit voltage thought to be due to local potential fluctuations caused by a disordered Cu/Zn sublattice. The disordered character of CZTS is difficult to study experimentally and has been universally neglected in computational studies. Here, we develop a cluster expansion model that enables simulation of the order-disorder phase transition in CZTS for the first time. With a proper atomic structure of the disordered phase in hand, we investigate the temperature-dependent voltage loss in CZTS, illustrating intrinsic limitations of existing synthesis methods and suggesting an optimal annealing temperature. We offer one explanation why Cu-poor CZTS is optimally efficient, as Cu vacancies increase the band gap via interactions between free carriers and the disordered nature of as-grown CZTS. Accordingly, increasing carrier concentrations may be an effective strategy to flatten the fluctuating potentials. 1

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Thin film photovoltaics (PVs) are promising alternatives to the widely used silicon-based solar cells. Among the myriad choices of photo-sensitizing materials, the chalcogenide family (e.g., CdTe and CuInxGa1-xSe4 (CIGS)) features great efficiencies (> 20%),1 thus attracting broad attention from both industrial and scientific communities. However, the currently commercialized chalcogenides contain either expensive or toxic heavy metal elements (e.g., Cd, In or Ga) that present economic and/or environmental challenges. A possible solution is to replace Ga and In in CIGS with cheaper and nontoxic Zn and Sn, leading to Cu2ZnSnSxSe4-x (CZTSSe, or CZTS for pure sulfide). Although low production costs appear to be achievable, the current efficiency record for CZTSSe is only 12.6 %,2 much lower than the state-of-the-art crystalline silicon solar cell (25.6 %).1 The prevailing explanation for this low efficiency is associated with Cu/Zn sublattice disorder that produces open circuit voltages lower than expected (vide infra).

Figure 1. The crystal structure of the kesterite phase of CZTS, where the Cu/Zn and the Cu/Sn (001) planes are labeled by blue and red colors, respectively. In Figure 1, we show the most stable (kesterite) phase of CZTS,3 a derivative of the zinc blende phase of ZnS, that has alternating Cu/Zn and Cu/Sn planes stacked along the [001] direction. According to previous density functional theory (DFT) studies,4-9 formation energies of CuZn and ZnCu antisites are 2


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fairly low, suggestive of instability in the ordered Cu/Zn sublattice. Temperature-dependent Raman spectra show abrupt changes around 533 K, indicating an order-disorder phase transition around this temperature.10 Partial Cu/Zn disorder is indeed observed in CZTS samples by neutron/X-ray diffraction11 and electron energy loss spectroscopy measurements.12 Previous photoluminescence experiments13 measured local potential fluctuations within the CZTS sample that may lower the band gap and the overall performance of the device. This local potential fluctuation phenomenon is thought to be a direct consequence of the Cu/Zn inhomogeneity in disordered CZTS. One recent experiment14 demonstrated that the band gap decreases with elevating temperatures, highlighting the effects of atomic disorder on electronic structure. Despite the clear influence of structural and compositional disorder on the electronic properties of this compound in real devices, most previous calculations focused on ordered CZTS or isolated point defects in the ordered phase. Existing computational reports on disordered phases are very preliminary: either the construction of the disordered structures was arbitrary,15 or the electronic structure of the disorder phase was not thoroughly investigated.16 Important questions regarding the disordered phase remain unanswered: for example, is it a random-kesterite phase with disorder that only appears in the Cu/Zn planes, or is it a completely random phase with disorder in both Cu/Zn and Cu/Sn planes? Unfortunately, conflicting experiments have been unable to offer conclusive answers.11, 12 Difficulties in identifying reasonable atomic structures for disordered CZTS further prevented theorists from studying the character of its electronic structure at the microscopic level. Consequently, the details of the local potential fluctuations (i.e., the fluctuation magnitude and its spatial distribution) and how they affect the open circuit voltage are unclear, and the temperature dependence of the electronic structure also becomes difficult to simulate. Experiments cannot elucidate the physical reason for the observed compositional inhomogeneity: is it due to the intrinsic limitations of thermodynamics or merely a consequence of slow kinetics due to imperfections in the fabrication procedure? Finding answers to 3



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these questions is not only important to help design new manufacturing methods for better CZTS-based solar cells, but also brings about a new paradigm on how we should think about modeling disorder in materials from first principles. In this work, we address the aforementioned problems by combining DFT and Monte Carlo (MC) simulation techniques. By developing a cluster expansion model based on first-principles data, we are able to perform the first direct simulations of the order-disorder phase transition process. Using this model, we are able to determine the temperature dependence of both the atomic and electronic structures of CZTS, which is critical to design better annealing procedures. We also analyze the interplay between disordered structures and stoichiometry of the system; this then reveals unique behaviors that only exist in the disordered phase. Most importantly, we provide a novel physical insight into the role of shallow point defects (e.g., VCu) in CZTS, which has not been identified previously.

RESULTS AND DISCUSSION Cluster Expansion Model We start our work by developing an energy model for the disordered CZTS system, using a cluster expansion formula fitted to DFT results. In this model, the total energy of a particular supercell is expanded to include up to trimeric neighbor interactions, expressed as:

E − Ekesterite = ∑ nijε ij + i> j

∑n

ε

ijk ijk

(1)

i > j >k

Here, E and Ekesterite are the energies of a stoichiometric disordered and ordered supercell, respectively. nij ( k ) represents the number of interacting dimers and trimers within the supercell and ε ij ( k ) are the corresponding energies fitted to DFT results. We consider four different interacting dimers here, including the nearest Cu-Cu pairs within the same (001) plane or residing in two adjacent planes as well as the nearest Cu-Zn pairs within the same planes 4


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or two adjacent planes. For trimers, we only consider four different Zn-Zn-Zn configurations, including: perpendicular Zn trimers, linear Zn trimers, symmetric Zn trimers sharing one S atom, and ring-like symmetric Zn trimers (see Figure S1 for geometries). It can be proved that this is a complete model at the trimer level, considering the constraints enforced by stoichiometry (e.g., the number of Cu trimers can be determined by the number of Zn trimers, Cu-Zn pairs, and the stoichiometry, and thus is not included in the model as an independent degree of freedom). This model is then fitted to PBE+U energies of one hundred random supercells, and then tested against another two hundred either completely random or random-kesterite supercells with the same size (see Figure S2 for fitting results). All the training and testing supercells have the correct stoichiometry but random Cu/Zn cation distributions in either the entire Cu/Zn sublattice or just the Cu/Zn planes. The fitted model features positive energies for Cu-Cu pairs and negative energies for Cu-Zn pairs, in agreement with the preferred distribution of Cu and Zn ions in the most stable kesterite phase in which they are spatially alternating. Both the model and DFT results predict lower energies on average for the random-kesterite phase compared to the completely random phase (Figure S2). However, there is a large overlap between the energy distributions of both phases; thus, using only the internal energies as the criterion for stability is inadequate. The effects of configurational entropy must therefore be explicitly included via MC simulations. Order-Disorder Phase Transition Utilizing the cluster expansion model, we perform MC simulations on a 4×4×2 512-atom supercell at different temperatures. The temperature-dependent energies are then fit to a 10th order polynomial and the heat capacities ( CV (T) ), i.e., the first derivative of the fit, are derived accordingly (Figure 2a).

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Figure 2. a) Temperature-dependent energies and heat capacities (inset); b) temperature-dependent order parameters, where the black curve represents O1 and red curve represents O2. The heat capacity features a maximum located at 530 K, indicating a phase transition in this temperature region. This prediction is in excellent agreement with the Raman spectroscopy experiments (533±10 K),10 validating our methodology. We also examined the size convergence of the simulation box using three different supercells (2×2×1, 4×4×2, and 6×6×3, see Figure S3 for results). The 2×2×1 supercell used in previous studies16 overestimates the transition temperature by over 100 K and is thus unreliable. The 4×4×2 supercell used in our simulation presented above generates results in reasonable agreement with a 6×6×3 supercell (within 10-20 K for the phase transition temperature). We therefore conclude that at least a 4×4×2 512-atom supercell is needed to fully account for the configurational entropy in disordered CZTS, the size of which is adopted here for all subsequent calculations. To examine the atomic structure of disordered CZTS in further detail, we can define an order parameter φl for each (001) plane l . Considering the site occupation correlation between a disordered structure with the ordered kesterite phase, we have:

φl = 4 ( fil − 0.5) ( fil0 − 0.5)

i

6


(2)

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Here, the discrete occupation function fil equals 1 if the i'th site within plane l is occupied by Cu and equals 0 when occupied by Zn. fil0 is the same occupation function defined for ordered kesterite and

... i means average over all sites i within plane l. Global order parameters can be defined by taking averages of φl for different subsets of planes ( {l} ): O = φl

{l}

(3)

Here, {l} includes all the Cu/Zn planes for the first order parameter O1 , while {l} includes all the Cu/Sn planes for O2 . Therefore, O1 and O2 represent the order within the Cu/Zn and Cu/Sn planes, respectively. According to this definition, in the ideal random-kesterite limit, we have O1 ~ 0.0, O2 ~ 1.0 . In the completely random limit, we have O1 ~ 0.0, O2 ~ 0.11 (see supporting information for more explanation). In the typical annealing temperature range (from 500 to 800 K), the values of the two order parameters approach neither limit, indicating neither the random-kesterite phase nor the completely random phase accurately represents the equilibrium structures (Figure 2b). The order-disorder phase transition is rather smooth, and the disorder appears in both types of planes. Although demonstrating significant disorder, the distribution of Cu and Zn are still statistically correlated with lattice site positions, and capturing this correlation is critical in describing the temperature-dependent electronic structure of CZTS (vide infra). This analysis highlights the importance of conducting proper thermodynamic sampling to obtain correct atomic structures for disordered solid solutions. From this analysis, we select the structures that best represent the phase of CZTS as a function of temperature. From these structures we determine the local and phase averaged electronic structures which we will describe in the following sections. Electronic Structure of Disordered CZTS

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With the atomic structures from the MC simulations in hand, we computed the electronic structure of disordered CZTS using PBE+U theory. Note that DFT and DFT+U systematically underestimate the band gap, and therefore none of the band gap values presented here should be directly compared to experiments. Instead, we focus on the qualitative trend of the band gaps with respect to different structures but do not aim to generate quantitative predictions of the open circuit voltage. We found that a typical disordered CZTS structure features a band gap much lower than the ordered CZTS (Figure S4). However, it still exhibits a finite band gap at the PBE+U level of theory, in comparison to the metallic behavior of a completely random structure. It is clear that both the disorder character and residual correlations of the atom distributions within the supercell affects the electronic structure of CZTS. The interplay between these two factors is responsible for the temperature dependence of the band gap, especially around the phase transition temperature region. To quantify this effect, we randomly pick structures (see method section for details) from each MC ensemble at a given temperature and then compute the corresponding averaged band gaps (Figure 3).

Figure 3. Averaged band gaps of disordered CZTS as a function of temperature. Standard deviations (σ) are labeled in the figure using vertical error bars. Below the phase transition temperature, the band gap measured from the density of states (DOS) plot is about 0.2 eV lower than the perfectly ordered phase (~1.0 eV), due to the band tails at the valence 8


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band edge associated with the shallow Cu/Zn antisite states. Within the order-disorder phase transition region, the band gap undergoes a steep decrease (~0.3 eV) between 500 and 700 K, indicating a significant voltage loss. This prediction is qualitatively consistent with experimental measurements of CZTSe,14 which in turn is very similar to the CZTS simulated here. CZTS is typically annealed during synthesis at relatively arbitrary temperatures though generally above 300 °C (573 K), based on various considerations.17, 18 From a thermodynamic perspective, our results demonstrate that at least 0.2 eV of voltage loss is inevitable in this temperature range due to the Cu/Zn sublattice disorder. Therefore, using annealing temperatures below 500 K is a promising direction to consider when designing new fabrication techniques for CZTS to alleviate voltage losses. In addition to the band gaps, DFT(+U) also allows us to investigate the detailed character of electronic bands that is otherwise challenging to measure. The spatial distributions of the valence and conduction band edges (VBE and CBE, including, on average, six bands per k-point within the first Brillouin zone) of a representative disordered structure are shown in Figure 4a.

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Figure 4. The partial density profiles of the VBE (cyan) and CBE (orange) for: a) a stoichiometric disordered structure; and b) a disordered structure with 5% copper vacancies. The disordered structure plotted in this figure was selected from the 750 K ensemble. c) schematically depicts the effect of spatial potential fluctuations, demonstrated in a and b, and the band opening effects due to the local distribution of the free carriers (electrons, −, and holes, +). In ordered structures, both the VBE and CBE should be distributed uniformly in all unit cells because of translational symmetry (see Figure S5). In contrast, the VBE and CBE of disordered structures are localized in nanometer sized domains, indicating potential fluctuations on the same length scale, which are introduced by the ion distribution fluctuations. This finding is consistent with previous observations using aberration-corrected scanning transmission electron microscopy,12 which also indicates a compositional inhomogeneity on the 1.5 - 5 nm length scale. We emphasize that these compositional and potential inhomogeneities are not related to the structures arising from slow formation kinetics or 10


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imperfections in the annealing processes (which are not simulated here) but are simply due to thermodynamically favored configurational entropy. Therefore, newly designed fabrication procedures that avoid high temperature annealing will be necessary to avoid formation of these local carrier traps. C. Cu Vacancy Effects

All of the supercells considered so far have been perfectly stoichiometric, although it is well-known that optimal performance of the CZTS PV cell is achieved under Cu-poor conditions.19 Therefore, we randomly remove five or ten percent of Cu atoms from the stoichiometric structures found above, and then investigate the effects of Cu vacancies (VCu) on the electronic structure.

Figure 5. Density of states (DOS) of the stoichiometric supercell (black lines) in comparison with the Cu-poor supercells (colored lines): a) results for ordered kesterite; and b) results for one typical disordered structure taken from the 750 K ensemble. In this figure, all Cu-poor supercells contain 5% of VCu. For comparison purposes, the DOS of stoichiometric supercells are shifted such that its VBEs are aligned with nonstoichiometric supercells. For the ordered phase, no significant changes in the band gap are observed (Figure 5a). This is expected as the VCu is a very shallow defect that introduces free holes but no extra defect states in the band tail. Although the Cu states comprise the valence band edge of CZTS, the low concentration of VCu only perturbs the band gap by ~0.03 eV. Similarly, comparing Figure 4b with Figure 4a, spatial distributions of the VBE and CBE are also not significantly perturbed by VCu in the disordered phase. However, the band gap of the Cu-poor disordered structure significantly increases when compared to the 11



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corresponding stoichiometric case (Figure 5b). Without changing the chemical character of the frontier states or introducing new states, the VCu opens the band gap by 0.2~0.3 eV, which is an order of magnitude larger than the vacancy defect-state effects found in the ordered phase and much more visible in Figure 5. In practice, this nonstoichiometric effect alleviates the band decrease effects caused by local potential fluctuations. Note that in this single case, the positions of VCu were assigned randomly and arbitrarily; a rigorous sampling of the vacancy configurations requires a more complicated MC model, which is the subject of future research. Here, we verify this finding is not associated with a specific supercell and vacancy structure by performing similar calculations on many more disordered structures; their calculated band gaps are shown in Figure 6.

Figure 6. Band gaps of the Cu-poor structures versus the band gaps of the corresponding stoichiometric structures. The red crosses are structures with 5% VCu while the blue squares are structures with 10% VCu. The diagonal line represents the (unrealized) case in which band gaps are unaffected by copper vacancies. The band gap opening effect of VCu is quite a general phenomenon for disordered structures, though it is much smaller in the ordered phase. The physical reason for this effect is closely related to the band fluctuations displayed in the previous section. Because of the trapping effects of the spatially localized band edges, the free holes introduced by VCu tend to aggregate in the spatial regions where the VBE 12


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localizes. This creates a local positive electrostatic potential that lowers the valence band energy and opens the band gap (Figure 4c). The mutual repulsive interaction between the carriers essentially fills the caves in the band structure caused by compositional inhomogeneity. Thus the presence of free carriers, rather than the Cu vacancy defect states, is the crucial factor to improve performance. In our simulations, we can also introduce free charges without point defects to test this hypothesis further, via removing or adding a corresponding number of electrons. Very similar band opening effects are observed in these charged supercells (Figure S6), verifying our interpretation. This finding thus explains the critical role of VCu in the CZTS system, and also suggests new strategies to improve the efficiency of CZTS. Any method that introduces more free carriers (e.g., gating voltages, doping in other shallow defects, or more intense radiation sources) would help flatten the fluctuating bands and therefore achieve better performance. CONCLUSIONS In this work, we used a first-principles based cluster expansion model to simulate the order-disorder phase transition in CZTS and accurately reproduced the experimental phase transition temperature. With rigorous thermodynamic sampling, we also obtained reasonable atomic structures for disordered CZTS. Further investigation of the electronic structure demonstrated a sharp band gap decrease beyond 500 K, indicating significant voltage loss. We therefore suggest that experimentalists consider developing new synthesis methodologies with lower annealing temperatures. Our calculations confirmed the postulated local potential fluctuation phenomenon that creates nanometer sized traps for free carriers. Based on our calculations, we also found a new explanation of the band gap opening effects of VCu, which is correlated with localized hole-hole interactions in disordered CZTS. We therefore also suggest increasing free carrier concentrations in the material to flatten the fluctuating bands and improve the overall efficiency of CZTS solar cells.

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METHODS All first-principles quantum mechanics calculations were performed with the VASP 5.3 program20-23 using the projector-augmented-wave method.24,

25

We utilized the Perdew-Burke-Ernzerhof (PBE)

exchange-correlation functional26 in conjunction with ab initio-derived U corrections identical to those used in our previous work.27 For all calculations, the Gaussian smearing method was employed with a smearing width of 0.05 eV. The first principles data utilized to fit the cluster expansion energy model was obtained using 2×2×1 64-atom supercells. For these calculations, we used a kinetic energy cutoff of 520 eV, and a Γ-point-centered 3×3×3 k-point mesh (both settings converge the energy within 1 meV/atom). The fitting and testing structure sets were generated completely randomly and geometry optimizations were performed using a force threshold of 0.01 eV/Å. Once we obtained the cluster expansion model, the MC simulations and the electronic structure calculations were carried out using 4×4×2 512-atom supercells. For the first-principles calculations on these supercells, we used the same settings as the 64-atom supercell calculations, except the kinetic energy cutoff was decreased to 400 eV to make the calculations more affordable. All the 512-atom supercells were constructed using the occupation configurations from the MC simulation and the atom site coordinates from the optimized unit cell of the ordered kesterite. Since the Cu(I) and Zn(II) ions have similar sizes and the Cu/Zn antisites are not likely to induce severe lattice distortions, no further geometry optimizations were performed for the 512-atom supercells. All MC simulations were conducted using a discretized lattice model, in which each lattice site represents a Cu/Zn site in the crystal and can be occupied by a Cu or a Zn ion. In each MC move, a random Cu site is swapped with a random Zn site, and each sampling consists of 100 million moves. The MC samplings were conducted in NVT ensemble using standard Metropolis algorithm,28 with temperatures ranging from 350 K to 1500 K. For each temperature, the averaged energy was computed

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first and ten structures with the averaged energy were randomly selected for further electronic structure calculations.

Supporting Information Available Supporting Information contains the detailed explanations to the definition of the order parameters and Figure S1-S6 contains extended results.

ACKNOWLEDGMENTS E.A.C. thanks the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DE-SC0002120 for funding this project. We also thank Dr. John Mark Martirez for helping to revise the manuscript.

References (1) Green, M. A.; Emery, K.; Hishikawa, Y.; Warta, W.; Dunlop, E. D. Solar Cell Efficiency Tables (version 46). Prog Photovoltaics 2015, 23, 805-812. (2) Wang, W.; Winkler, M. T.; Gunawan, O.; Gokmen, T.; Todorov, T. K.; Zhu, Y.; Mitzi, D. B. Device Characteristics of CZTSSe Thin-Film Solar Cells with 12.6% Efficiency. Adv Energy Mater 2014, 4, 1301465. (3) Schorr, S.; Hoebler, H. J.; Tovar, M. A Neutron Diffraction Study of the Stannite-Kesterite Solid Solution Series. Eur J Mineral 2007, 19, 65-73. (4) Chen, S. Y.; Gong, X. G.; Walsh, A.; Wei, S. H. Defect Physics of the Kesterite Thin-Film Solar Cell Absorber Cu2ZnSnS4. Appl Phys Lett 2010, 96, 021902. (5) Chen, S. Y.; Yang, J. H.; Gong, X. G.; Walsh, A.; Wei, S. H. Intrinsic Point Defects and Complexes in the Quaternary Kesterite Semiconductor Cu2ZnSnS4. Phys Rev B 2010, 81, 245204. (6)

Maeda, T.; Nakamura, S.; Wada, T. First Principles Calculations of Defect Formation in In-Free Photovoltaic

Semiconductors Cu2ZnSnS4 and Cu2ZnSnSe4. Jpn J Appl Phys 2011, 50, 04DP07.

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