Defects properties and vacancy diffusion in Cu2MgSnS4

Kin Fai Tse; Shengyuan Wang; Man Hoi Wong; Junyi Zhu

doi:10.1088/1674-4926/43/2/022101

Abstract

Cu₂ZnSnS₄ (CZTS) is a promising photovoltaic absorber material, however, efficiency is largely hindered by potential fluctuation and a band tailing problem due to the abundance of defect complexes and low formation energy of an intrinsic Cu_Zn defect. Alternatives to CZTS by group I, II, or IV element replacement to circumvent this challenge has grown research interest. In this work, using a hybrid (HSE06) functional, we demonstrated the qualitative similarity of defect thermodynamics and electronic properties in Cu₂MgSnS₄ (CMTS) to CZTS. We show Sn_Mg to be abundant when in Sn- and Cu-rich condition, which can be detrimental, while defect properties are largely similar to CZTS in Sn- and Cu-poor. Under Sn- and Cu-poor chemical potential, there is a general increase in formation energy in most defects except Sn_Mg, Cu_Mg remains as the main contribution to p-type carriers, and Sn_Mg may be detrimental because of a deep defect level in the mid gap and the possibility of forming defect complex Sn_Mg+Mg_Sn. Vacancy diffusion is studied using generalized gradient approximation, and we find similar vacancy diffusion properties for Cu vacancy and lower diffusion barrier for Mg vacancy, which may reduce possible Cu-Mg disorder in CMTS. These findings further confirm the feasibility of CMTS as an alternative absorber material to CZTS and suggest the possibility for tuning defect properties of CZTS, which is crucial for high photovoltaic performance.

1. Introduction

Kesterite Cu₂ZnSnS₄ (CZTS) is an earth-abundant, non-toxic absorber material for photovoltaics with a direct band gap tunable from 1.5 eV in pure CZTS^[1-3] to 1.0 eV in pure Cu₂ZnSnSe₄^{[1, 4]} by anion mixing of S and Se and a high band edge absorption coefficient larger than 10⁴ cm^–1^[4]. The record efficiency of alloyed CZTS_xSe_1–x (CZTSSe) is at 12.6%^[5] using solution process, and a 11% efficient pure CZTS absorber device is recently achieved^[6].

Comparing CZTS to the Cu(In/Ga)S₂ (CIGS) absorber material, CZTS often suffers from a high open-circuit voltage (V_OC) deficit^[7-9] larger than 0.6 eV, low external quantum efficiency (EQE) in the long wavelength region, low carrier lifetime, and thus fill factor^[10]. These observations are commonly attributed to the abundance of point defects and defect complexes^{[11, 12]}, band edge or electrostatic potential fluctuation induced by defect compensation^[13], interface defects^{[14, 15]}, unfavorable band alignment^{[16, 17]} and secondary phases due to composition inhomogeneity, and back contact decomposition^{[15, 18]}. Defect abundance in CZTS originates from the intrinsically low formation energy of point defects in thermodynamic limit^{[8, 19]}. Moreover, the formation of defect complexes and clusters, such as Cu_Zn+Zn_Cu, is favored energetically and entropically^{[20, 21]}, resulting in a transition to Cu–Zn disordered kesterite at a low temperature of 200–260 °C^[22]. The high density of defects in disordered kesterite may introduce a defect band within the band gap^[8] that contributes to band tailing and electrostatic potential fluctuation^[15]. Additionally, CZTS shows strong Cu/Zn compositional fluctuation due to low lattice mismatch to secondary phases^{[23, 24]}. Also a back-contact decomposition reaction is suggested to be responsible for secondary phases segregation near the back contact^{[25, 26]}. These challenges of CZTS were extensively studied in literature and various processing strategies were proposed to mitigate the problems, including annealing at low-temperature^[8] and high sulphurization pressure^{[25, 27]}, surface etching^{[25, 28]}, interface passivations^[29-31], alternative buffer layers^[32-35] and back-contact passivations^{[26, 36-40]}.

Despite the theoretical and experimental efforts in the development of processing techniques, an empirically lower EQE and V_OC still persists in CZTSSe, contributing to the efficiency gap compared to record efficiency of higher than 20% in other thin film materials such as CIGS. It is suggested that intrinsic phase stability and electronic properties play an important role in the construction of high efficiency kesterite solar cells^{[15, 16]}. Element substitution or alloying starting from CZTS can provide a large number of candidate material alternatives with similar but modified structural and electronic properties^{[41, 42]}. It is thus a promising strategy for tuning defect properties in CZTS. Since the band edge positions of CZTS are determined by Cu-d/S-p and Sn-s/S-p hybridization at the valence band (VB) and conduction band (CB) respectively^[1], band alignment and defect properties are mainly determined by the choice of group I, IV, VI elements. A large portion of the phase diagram in CZTS is governed by the phase boundary due to decomposition reaction into ZnS and Cu₂SnS₃^[41]. As a result, the phase stability of CZTS may be tuned independently by the substitution of group II elements. A number of element substitutions were investigated in detail from theoretical or experimental approaches. Anion S/Se mixing is used to adjust the band edge position and band gap^[16]. Ag₂ZnSnS₄ suppresses p-type defect and widen the band gap due to low VB and become weakly n-type^[43], and Ag composition grading is proposed to reduce Fermi level pinning^{[43, 44]}. Up to 40% Cd substitution to Zn is found to be beneficial to device performance attributed to the reduction of ZnS phase^[45], and 40%–50% of Ge substitution can reduce V_OC deficit and increase the bandgap by up to 0.3 eV^{[46, 47]}.

Cu₂MgSnS₄ (CMTS) is also a viable alternative that is less investigated in literature. Early work based on semi-local exchange-correlational functional predicts CMTS to have a small instability with respect to phase segregation into Cu₂SnS₄ and MgS^[41]. However, later experiments had reported successful synthesis of CMTS^[48-50], which contradict early conclusions. Agawane et al. synthesized Cu₂Zn_xMg_1–xSnS₄ by pulsed laser deposition covering x = 0 to x = 1, showing an increasing band gap with a fraction of Mg substitution and improved grain morphology at 50% Zn replaced with Mg^[49]. Existence of CMTS phase is confirmed by x-ray diffraction and Raman spectroscopy for pure Mg sample without addition of Zn^[49]. Improved carrier mobility and reduced secondary phases is reported by an Mg/Zn alloying strategy^{[51, 52]}. A device using a pure CMTS absorber layer with 0.78% efficiency was fabricated only recently^[50].

CMTS is different from CZTS in that 1) a more ionic and larger covalent radius Mg replaces the role of Zn in CZTS; 2) the competing secondary phase MgS is of halite structure instead of the zinc-blende structure; and 3) the ground-state structure of CMTS is stannite, where Cu and Sn/Zn ions are organized into layers^[42]. These differences may lead to major consequences in the phase diagram, type and concentration of ionizable defects and deep traps, and diffusion kinetics of defects. Understanding CMTS differences and similarities compared to CZTS can guide important design choices in the fabrication process. Although a large body of literature is available from first-principles calculations and experiments for CZTS, to our knowledge, only a handful of experiment studies are available^[48-52], and theoretical studies are largely absent for CMTS. We investigate the defect thermodynamics, electronic properties and defect diffusion kinetics of CMTS, and provide a direct comparison with CZTS^{[11, 12]}. Our result shows that CMTS shares a significant similarity in major carrier providers (Cu_Mg), a lower diffusion barrier for Mg, and a reduction in defect density and detrimental defects, which can be beneficial to device performance. We identified a change in dominant detrimental defect in CMTS to Sn_Mg related defects. We also examined favorable thermodynamic conditions and strategies to control this Sn-related defect based on our first-principles results.

2. Electronic and thermodynamic properties

The phase diagram is calculated following the methodology of our earlier study for ease of discussion. The atomic chemical potential corresponding to the growth window of CMTS is calculated by requiring that, under the chemical potential window specified, CMTS is energetically stable while other secondary phases are unstable. Detailed thermodynamic discussions of CMTS are referred to in Ref. [42]. Formation energies for binary secondary phases including halite and zinc-blende MgS, and ternary secondary phases in CMTS as listed in other related works^{[11, 53]} are calculated with reference to element fcc-Cu, hcp-Mg, grey Sn and S₈ phases. In the CMTS phase, a sizable growth window exists when calculated under a hybrid functional, indicated by the colored region of Fig. 1. The range of zero-temperature chemical potentials, at which CMTS is energetically more stable than secondary phases, are used as an estimation to the range of free energy at which CMTS can be formed spontaneously. The shape of the CMTS phase diagram resembles that of CZTS, and the growth window is largely limited by the relative stability between CMTS and MgS and Cu₂SnS₃ phases. Cu poor chemical potential is in general S richer within the growth window, indicating that a higher sulphur pressure is thus preferred for CMTS growth under Cu deficit.

Figure 1.(Color online) Phase diagram of CMTS calculated from hybrid functional (HSE06).

We calculated a direct band gap of 1.61 eV for CMTS, showing a hybridization of Cu-3d (Sn-5s) and S-3p orbitals that are composed of the VB (CB). Meanwhile, Mg is not involved in the composition of VB or CB. Band alignment of CMTS with reference to CZTS is calculated through a CMTS(010)/CZTS(010) hetero-junction following the methodology of Himuma et al.^[54]. The increase in band gap is almost entirely due to a higher CB by +198 meV.

3. Intrinsic defect formation

The formation of defect is governed by both the growth condition (atomic chemical potential) and the Fermi energy level, at equilibrium, the formation energy of a defect D with charge q can be expressed as^[55-57]

$ \Delta H\left(D,q\right)=\left[E\left(D,q\right)-{E}_{0}\right]+{E}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}\left(D,q\right)+\sum _{i}\left[{n}_{i}\right({E}_{i}+{\mu }_{i}\left)\right]+q({\varepsilon}_{\mathrm{V}\mathrm{B}\mathrm{M}}+{E}_{\mathrm{F}}) , $  (1)

where the third term sums over the energy required to remove (negative for add) atom of species i to corresponding element reservoir, and the last term accounts for the energy required to remove q electrons to an electron reservoir equivalent to the host cell. represents the Freysoldt−Neugebauer−Van de Walle (FNV) finite-size correction^[56]. First-principles simulation using Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional is performed by a periodic supercell containing 64 atoms per defective atom, for 64 atom supercell, a Г-centered 2 × 2 × 2 is chosen and a 600 eV plane-wave cut-off is used. The accuracy in defect calculation is set to 10^–4 eV/atom, while ionic relaxation will stop when the forces reach 10^–2 eV/Å. During relaxation with a hybrid functional, Fock exchange potential is evaluated on a reduced k-grid by a factor of 2, such reduction is tested to recover accurate defect structure while mitigating the stiff computational requirement of the hybrid functional. The detailed convergence tests are available in the supplementary information. We first discuss the effect of growth condition on defect populations in CMTS. The neutral defect formation energy of intrinsic defects in stannite CMTS are shown in Fig. 2, and only the lowest energy of symmetrically distinct configurations is reported.

Figure 2.(Color online) Neutral defect formation energy for (a) intrinsic defects and (b) defect complexes in CMTS at chemical potentials ranging from Cu poorest (A) to Cu richest (G).

The local bonding environment of CMTS largely follows that of CZTS, in which one S atom is bonded directly to 2 Cu, 1 group II and 1 Sn atom, and each cation is bonded to 4 S atoms. We therefore expect that defect properties will be somewhat similar to CZTS. On the other hand, the difference in electronic structure and local stress can also enhance or suppress defect formation selectively where a larger size difference leads to an increase in defect formation energy^{[58, 59]}. In the literature of CZTS, defects with formation energy below 1.5–2.0 eV^[11], which will be highlighted below, are considered to have a concentration high enough to modify device performance. The formation energy of Cu_Mg is lower than 1.0 eV at all chemical potential similar to CZTS, indicating a relatively high concentration of Cu_Mg that can be formed under most growth conditions. V_Cu has a formation energy of 1.10 eV in the Cu poorest limit, and the formation energy increases as increases. Sn_Mg has a lower neutral defect formation energy compared to Sn_Zn in CZTS, which can be explained by the lower donor level that is fully occupied. The neutral defect formation energies from point A' to G' are qualitatively similar to point A to G, with only minor shift in formation energy due to slight difference in the choice of chemical potential.

Several defects including Cu_i, Mg_Cu, Sn_Mg and V_S have lower formation energy in Cu rich limit (points E, F and G), while the formation energy of acceptor defects V_Cu and Cu_Mg becomes higher. As a result, the Cu rich condition may lead to stronger charge compensation and higher concentration of donor defects that may deteriorate device performance. At chemical potential point B and D, neutral formation energy of all point defects except Cu_Mg and V_Cu is above 1.6 eV. These chemical potentials correspond to a mildly Cu poor and Sn poor growth condition. As a comparison, the formation energy of a number of donor defects in CZTS are always between 1–1.5 eV at any chemical potential^{[11, 12]}. There is also a general increase in defect formation energy for all point defects calculated except Sn_Mg. From a quasi-chemical model, the concentration of defect D can be related to its free energy by

$ \left[D\right]={N}_{i}g{\rm e}^{-\Delta {H}_{\rm f}/{k}_{\rm b}T} , $  (2)

where is the density of site (Cu = 1.2 × 10²², Mg/Sn = 6.0 × 10²¹, S = 2.4 × 10²² cm^–3 in CMTS) and account for structural and electronic degeneracy. A higher defect formation energy for deep defect can help reduce detrimental defect density. Thus by fine tuning the metallic precursor ratio^{[34, 60]} towards Cu and Sn poor, the density of most defects can be reduced. Moreover, if defects in CMTS equilibrate at a lower temperature of 400 °C instead of 550 °C, the Boltzmann factor in Eq. (2) with = 1.6 eV can be further reduced by more than two orders of magnitude, suggesting that careful the control of temperature profile or the annealing of CMTS film at a lower temperature may help reduce bulk defect density. These growth strategies can be beneficial to carrier lifetime.

4. Defect complexes

Defect complexes can have significant influence on band edge positions^[12], so that device performance can be affected if some defect complex can potentially exist in abundance. The binding energy ( ) of a defect complex is defined as the energy gained by forming defect complex from separated point defects^[53], i.e., . The binding energy is independent of chemical potential as it reflects only the energy difference between two geometrical arrangements of defects. We calculated the binding energy of defect complex involving various lower energy point defects, and the result is summarized in Table 1, its chemical potential dependence is shown in Fig. 2.

Table Infomation Is Not Enable

The binding energy of defect complex can be negative because of the charge or stress compensation, and also Coulombic attraction between defects^[55], thus it is expected that partial or fully compensated defect complexes may have a highly negative binding energy hence enhancing total defect population.

Literature in CZTS showed that Cu–Zn disorder is partly responsible to V_oc loss, which may be explained by the low formation energy of Cu–Zn related defect complexes. In CMTS, we also show that fully compensated defect complexes Cu_Mg+Mg_Cu, (2Cu)_Mg and Mg_(2Cu) have a low formation energy less than 1 eV. The formation energy for Cu_Mg+Mg_Cu is 0.24 eV, similar to Cu_Zn+Zn_Cu in CZTS. Because of the low formation energy, the concentration of Cu_Mg+Mg_Cu at equilibrium can be as high as >10 ²⁰ cm^–3, which may not be desirable. Similarly to CZTS, these low energy defect complexes can possibly induce a Cu–Mg disorder phase in CMTS, and thus a careful control of growth temperature may also be required for the growth of CMTS.

Defect complexes involving Sn atom has a large binding energy due to stronger charge compensation between p- and n-type defects. In CZTS, Zn_Sn+Sn_Zn can induce strong band gap shrinking, which can be a significant contributor to band gap fluctuation because of its low formation energy of 0.86 eV^[12].

The formation energy of Mg_Sn+Sn_Mg in CMTS is higher by +0.49 eV, possibly lowering its concentration by several orders of magnitude. For other defect complexes involving Sn, the absolute formation energy of Sn related defect complexes is not lowered significantly past that of an isolated point defect, together with the large binding energy. Such a mechanism may also help convert deep intrinsic defects into defect complexes that are less electrically active.

5. Electronic properties of defects

In this part, we examine the dependence of charge transition level on the Fermi energy level. The charged defect formation energy at chemical potential point B is shown in Fig. 3. In CMTS, Cu_Mg shows properties analogous to Cu_Zn in CZTS. Cu_Mg can be ionized to –1 charge state, the ionized state is energetically more favorable when Fermi energy level is higher than VBM + 0.21 eV, slightly deeper than Cu_Zn in CZTS^[61]. Regardless of charge state, Cu_Mg is always the most favorable acceptor defect in CMTS, while V_Cu^– is at least 0.5 eV higher. We expect a slightly lower but comparable carrier concentration in CMTS, which is sufficient as a photovoltaic absorber. Mg_Cu and Cu_i are the dominant donor type defects in the CMTS system. However, both of them are ionized at all Fermi energy level within the band gap, and the formation energy of these ionized single donor defects are further lowered at VBM, at 0.10 and 0.65 eV, respectively, similar to CZTS. In general, the Fermi energy level pinning can be approximated by the Fermi energy level at which the formation energies of the lowest energy donor and acceptor are equal. We find this value in CMTS to be 0.27 eV by Cu_Mg and Mg_Cu, indicating that CMTS grown under Sn poor and Cu mildly poor condition is intrinsically p-type. However, at chemical potential point C, formation energy for donor defects are generally lower while higher for acceptor defects. At this growth condition, although Cu_Mg and Mg_Cu remain as the dominant p-type and n-type defects in the system, the Fermi energy pinning will shift to 0.6 eV, indicating possible lowering of p-type carrier concentration due to higher charge compensation, which may not be desirable.

Figure 3.(Color online) Charged defect formation energy of point defects at chemical potential B, Fermi energy is varied from VBM to CBM.

Our calculations show that anion defects in CMTS do not show charge transition within the band gap. Acceptor defects Mg_Sn and Cu_Sn can have multiple transitions into lower energy negatively charged states as the Fermi energy level moves towards CBM. Around the pinned Fermi energy level, these defects remain charge neutral and has a high formation energy above 2 eV. However, we find that the Sn_Mg defect has a deep (0/2+) charge transition at VBM + 0.55 eV near the mid gap with relatively low formation energy of 1.77 eV in a neutral state at chemical potential point B, which can serve as carrier traps. Such a negative-U center is more detrimental than Sn_Zn in CZTS because of relatively low formation energy^[61]. In turn, the control of the Sn_Mg defect is crucial to fabricate a CMTS photovoltaic device with high efficiency.

We then explore the scenario where defect concentration is high. At high defect concentration, defect bands may be formed leading to band tailing, which is commonly regarded as a major contribution to loss of open circuit voltage in CZTS^[15]. We calculated the band edge shift for compensated defect complexes in CMTS and aligned with the host cell using core energy levels further away from the defect (Table 2). We find that antisite pairs, both Cu_Mg+Mg_Cu and Mg_Sn+Sn_Mg, may induce a sizable band tailing. For Cu_Mg+Mg_Cu, an upward shifting of VBM by 0.1 eV is found, while for Mg_Sn+Sn_Mg, a significant shrink of band gap by 0.45 eV is found, mainly attributed to the lowering of CBM. Controlling defect density of Mg_Sn+Sn_Mg is thus crucial to reduce possible band tailing in CMTS.

Table Infomation Is Not Enable

6. Vacancy-mediated diffusion

Lastly, we show our results regarding the diffusion kinetics of a few point defects in CMTS. Defect thermodynamics provide a good estimation of bulk defect density when spatial variation of atom stoichiometry can be minimized, that is, when the diffusion is relatively quick, it is thus important also to understand the diffusion kinetics in CMTS. As we have already showed that the formation energy of Cu and S vacancies are relatively low in CMTS, the density of these defects should be relatively high, thus our current result focuses on the diffusion of atoms in the presence of Cu or S vacancy. Moreover, the activation energy of diffusions through V_Cu is found to be relatively low in both CZTS and CIGS. Therefore, our result on CMTS can provide qualitative insight into the diffusion properties of CMTS. The chosen diffusion paths are shown in Fig. 4. Diffusion simulation is conducted via the nudged elastic band method with a climbing image using seven images between optimized initial and final defect configurations to provide smooth interpolation to the reaction energetics and good estimation of activation energy^[62]. Due to much higher computational complexity of diffusion calculation, diffusion calculations are conducted using a Perdew−Burke−Ernzerhof (PBE) functional^[63]. We checked our calculated activation energy to agree within 0.1 eV in CZTS and CIGS compared to previous literature values.

Figure 4.(Color online) Illustration of vacancy diffusion paths investigated in the current study.

Relative energy along minimum energy path (MEP) is shown in Fig. 5. The activation energy for the diffusion of V_Cu is 1.23 eV in CMTS, which is similar to Cu diffusion via vacancies in CZTS and CIGS system. For Mg diffusion into V_Cu, the formation energy is 0.18 eV lower (1.05 eV). In comparison, Zn diffusion into V_Cu in CZTS is 0.13 eV higher than diffusion of copper vacancy, indicating that Mg is also a relatively mobile atom species in CMTS. On the other hand, Sn diffusion into V_Cu has a much higher activation energy. This is partly due to the higher formation energy of Sn defect and relatively large atomic size of Sn. This high diffusion barrier may also help to prevent the formation of Sn antisite defect, which we considered detrimental in CMTS. V_S represents the vacancy-mediated diffusion of S atom in CMTS with all initial and final configuration of V_S being equivalent, thus the energy profile along the MEP should be symmetric. Despite this, we have not enforced such symmetry during the calculation, and our result shows an almost symmetric energy profile, indicating that the images are well-converged to minimal energy. For all S vacancy diffusion in CMTS, the activation energy is at the minimal 1.86 eV with a strong asymmetry. When S atoms diffuse across Sn atoms, additional energy is required to lose the stronger S–Sn bonding, which results in the diffusion barrier of S across the Cu layer to be significantly lower. Our result shows calculated activation energy for Sn and S diffusion has an activation energy comparable to In diffusion in CISe^[64], suggesting diffusion of Sn and S through our calculated MEP is difficult. It should be noted that S or Sn diffusion can be facilitated by the presence of grain boundaries or off-stoichiometry conditions, which are not modelled by the current calculation.

Figure 5.(Color online) Energy relative to V_Cu for cation and V_S for anion diffusion along the MEP, energies of calculated images are represented by markers and solid lines are interpolation.

7. Conclusion

We presented a systematic study of point defects and a few defect complexes in CMTS using the HSE06 functional within the density functional theory framework. Our calculation shows that the growth window of CMTS is similar to that of CZTS, and there is an increase in formation energy for most defects compared to CZTS hence a lower defect density. Such an elevation reduces the concentration of Mg_Sn and Cu_Sn defects with multiple charge states. Then we predict a p-type behavior due to ionization of Cu_Mg to Cu_Mg^–, showing the Fermi energy located around 0.2 eV above the valence band due to compensation with a Mg_Cu defect. However, we still unearth a strong Cu/Mg compositional fluctuation due to low formation energy for a Cu_Mg+Mg_Cu defect complex, analogous to CZTS results. And we identify Sn_Mg double donor defect as the likely detrimental defect in CMTS system because of high density at Sn-rich or Cu-rich chemical potential, and a deep donor level in the mid gap. The profile of the Sn_Mg defect is inferior than Sn_Zn in CZTS. To avoid a high concentration of the Sn_Mg defect, we propose tuning the chemical potential to Sn-poor and Cu mildly poor condition or anneal at lower temperature. Finally, we justify our defect finding with a comparison of activation energy of vacancy diffusion in CMTS and CZTS. We find the Cu and Mg diffusion through V_Cu to be relatively quick, and the activation energy of these diffusion mechanisms are similar to previous reported values in CZTS and CIGS.

Acknowledgements

The authors would like to acknowledge support from Research Grants Council of Hong Kong (under GRF/14319416, GRF/14301318, and Direct Grant).

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