The kesterite Cu₂ZnSn(S,Se)₄ (CZTSSe) based compound solar cells has attracted a great number of attentions because of the non-toxicity and abundance of the constituent elements^[1]. To date, solar cells with efficiency of 12.6% was demonstrated via a hydrazine-based solution approach^[2]. Despite this progress, performance of CZTSSe solar cells remains far lower than the Shockely- Quiser theoretical limit^[3]. The existence of abundant antisite defects (Cu_Zn or Zn_Cu) in the CZTSSe absorber are considered to be an important factor that deteriorates the solar cell performances^[4]. Therefore, a wide variety of research has been done to find the alternatives for kesterite CZTSSe and an approach to substitute other elements for Cu or Zn to suppress the formation of antisite defects and thus improve the device performance. Substitution of Cd, Fe and Mn for Zn to form Cu₂CdSnS₄^[5], Cu₂FeSnS₄^[6] and Cu₂MnSnS₄^[7] thin film and Cr doping^[8] had been experimentally achieved. Whereas, these elements are toxic and may bring undesirable magnetism into device. By contrast, Mg element is environmentally friendly, and Mg-doped ZnO window layer was used in CIGS thin film solar cell to improve short circuit current^[9].

Recently, Mg-doped CZTS(Se) materials have been synthesized by many techniques such as ultrasonic co-spray method^[10], pulsed laser deposition method (PLD)^[11], and a liquid-phase reactive sintering technique^[12]. In these works, structural, electronic and optical properties of Mg-doped CZTS(Se) were studied. Nevertheless, Mg-doped CZTS deposited by PLD showed p-type conductivity behavior while Mg-doped CZTSe prepared by reactive sintering technique exhibited n-type electrical conductivity. To the best of our knowledge, no experimental or theoretical work has been done to clarify which element in CZTS(Se) is more likely substituted by Mg and what kind of defects might be introduced into CZTS(Se) by Mg doping. Rapidly-developing first-principles calculation methods make it possible to study substitution of Zn in CZTS(Se) theoretically^[13,14,15]. Therefore, in this work we studied the formation energies of Mg impurities in kesterite CZTS via hybrid functional calculations. On the basis of our results, we identified the energetics of the most likely positions in CZTS for the incorporation of Mg extrinsic defects, which may help improving the further application of CZTS(Se) solar cells.

In the present work, all calculations have been performed based on density functional theory (DFT) as implanted in the VASP code^[16]. The interaction between ions and electrons is described by the projector augmented wave (PAW) method^[17] with a plane-wave cutoff energy of 400 eV. For defect calculations, a 2×2×1 supercell containing 64 atoms is adopted, as shown in Fig. 1. A 2×2×2 Monkhorst-Pack k-point mesh is used for the Brillouin-zone integration of the 64-atom cell.

Screened Coulomb hybrid functional Heyd-Scuseria- Ernzerhof (HSE06)^[18] is used to calculate the electronic structures and defect properties. The range-separation screening parameter μ is set to 0.02 nm^-1 and the amount of Hartree-Fock exchange is set to 0.25 in the HSE06 functional. Our calculations are done in two steps. In the first step we optimize the atomic structure by using GGA in the Perdew-Burke-Ernzerhof (PBE) form^[19]. In this step, the lattice parameters were optimized through the minimization of total energy. In the second step we do the static electronic calculation by the HSE06 hybrid functional. All lattice vectors and atomic positions were fully relaxed by minimizing the quantum mechanical stresses and forces. It shows that the band gap obtained by this two-step procedure is in excellent agreement with the experimental value in previous work^[20]. These parameters and methods ensure the calculated band gap for CZTS close to the experimental values.

Mg-related point defects including three antisites Mg_Cu, Mg_Zn and Mg_Sn are considered (one Mg atom substitutes Cu, Zn and Sn site, respectively). The defect formation energies are calculated as^[21]

(1) $$$\begin{array}{rl}& \mathrm{\Delta }{H}_{\alpha ,\text{q}}(E_{\text{F}},{\mu }_i)=(E_{\alpha ,\text{q}}-E_{\text{h}})+\\ & \sum _i{n}_i(E_i+{\mu }_i)+q({\epsilon }_{\text{VBM}}+E_{\text{F}})\end{array}$$$

where E_α,_q and E_h is the total energy of the supercell with and without a defect α, respectively. E_i is the total energy of the component element i in its pure phase; n_i is the number of atoms i removed from the supercell in forming the defect with the chemical potential μ_i in forming the defect α. The μ_i is referenced to the total energy E_i of the elemental solid and μ_i=0 means the element is so rich that the pure element phase can form. E_F is the Fermi energy, which varies from valence band maximum (VBM) of the host, denoted by ε_VBM, to the conduction band minimum (CBM) for non-degenerate semiconductors. Because of the finite size of supercells, the Lany and Zunger correction method^[22] is used to correct the image-charge interaction with a relative dielectric constant (ε_r) of 8.1^[23] of the monopole correction. And potential alignment correction is applied by aligning the core-averaged electrostatic potentials far from the defect^[24].

The predicted lattice parameters and bandgap results by using two types of exchange-correlation functionals are presented in Table 1. HSE06 functional predicts a bandgap of 1.45 eV for CZTS, in consistent with the experimental values. By contrast, the PBE functional largely underestimate the bandgap. Therefore, HSE06 is used to calculate the electronic structures and defect properties of Mg-doped CZTS.

A defect often produces states within the band gap, with the stronger the gap states, the more localization of the defect charge distribution. The charge localization feature can be seen from the band structures, as shown in Fig. 2. The charge states of Mg_Cu and Mg_Sn are similar to Zn_Cu and Zn_Sn which are both charge delocalized defects^[26]. For these defects, defect-induced states hybridize with the conduction band or the valence band and are difficult to be separated, thus there is invisible defect- induced levels within the band gap^[27]. The electrons on these charge delocalized defects are loosely bounded and easy to be ionized and will not produce deep defect levels within the band gap.

Whether a defect level could significantly affect the solar cell performance also depends on its concentration which is related to the formation energy, a function of the elemental chemical potentials and Fermi level according to eq. (1). And the stable region of chemical potential is needed to be determined before performing the calculation of formation energies of the Mg-related defects. A series of thermodynamic conditions must be satisfied by the chemical potentials:

(I) The sum of the chemical potentials of the component elements should maintain a stable host compound, which means:

where ΔH_f (Cu₂ZnSnS₄) relates to the formation energy of Cu₂ZnSnS₄.

(II) The formation of pure elemental phase should be avoided. Thus, the atomic chemical potentials in CZTS should be smaller than that of the corresponding elemental solid. That is:

(III) The formation of all other secondary compounds including CuS, Cu₂S, ZnS, SnS, SnS₂, Cu₂SnSe₃ and MgS should be avoided, as described by the following relations:

${{\mu }_{\text{Cu}}}\text{+}{{\mu }_{\text{S}}}\text{}\Delta {{H}_{\text{f}}}(\text{CuS})=-0.\text{53 eV}$

$2{{\mu }_{\text{Cu}}}\text{+}{{\mu }_{\text{S}}}\text{}\Delta {{H}_{\text{f}}}(\text{C}{{\text{u}}_{\text{2}}}\text{S})=-0.94\text{ eV}$

${{\mu }_{\text{Zn}}}\text{+}{{\mu }_{\text{S}}}\text{}\Delta {{H}_{\text{f}}}(\text{ZnS})=-1.96\text{eV}$

${{\mu }_{\text{Sn}}}\text{+}{{\mu }_{\text{S}}}\text{}\Delta {{H}_{\text{f}}}(\text{SnS})=-0.84\text{ eV}$

$2{{\mu }_{\text{Cu}}}\text{+}{{\mu }_{\text{Sn}}}\text{+}3{{\mu }_{\text{S}}}\text{}\Delta {{H}_{\text{f}}}(\text{C}{{\text{u}}_{\text{2}}}\text{Sn}{{\text{S}}_{3}})=-2.64\text{ eV}$

${{\mu }_{\text{Mg}}}\text{+}{{\mu }_{\text{S}}}\text{}\Delta {{H}_{\text{f}}}(\text{MgS})=-3.14\text{ eV}$

With the thermodynamic conditions established, the stable chemical potential region of CZTS is confined in a three-dimensional space. As a result of Equation 2, there can be only three independent variables for CZTS, such as μ_Cu, μ_Zn and μ_Sn. Here we take the chemical potential range determined by Zhang et al.^[28] for CZTS. When Mg induced secondary phase MgS is considered, the stable chemical potential region is surrounded by ABCD, as shown in Fig. 3. The chemical potential values for each element at points A-D are listed in Table 2. Point D is more Zn-rich in growth condition compared to that of points A-C.

To figure out the microscopic mechanism of Mg-doping, the formation energies as a function of E_F for Mg-related defects at points A-D are calculated. Whereas we only show the results at point D since Zn-rich growth condition is preferred by CZTS solar cells with high efficiency. As shown in Fig. 4, the formation energies of Mg_Cu⁺ and Mg_Zn⁰ are lower than 2 eV in the whole E_F range, which means these defects can exist in large amount in CZTS. By contrast, the formation energy of Mg_Sn^2- is higher than 2 eV when E_F is closed to the VBM.

It is also found that the formation energy of Mg_Zn⁰ is even smaller than that of Mg_Cu⁺ in large E_F range, which indicates that the Mg atom prefers to substitute the Zn atomic site in CZTS. Since Mg and Zn are isovalent, the Mg_Zn will not affect the conduction type. Moreover, the existence of Mg_Zn makes it easy for other Cu atoms to be substituted by Mg atom. The comparison of the formation energy of Mg_Cu in the supercell with and without a Mg dopant on the Zn site is shown in Fig. 5. With Mg dopant, the formation energy of Mg_Cu⁺ decreases by 52 meV than that without Mg atom. The Bader charge shows that the amount of transferred electron from Mg to S is larger than that of Zn in CZTS, as shown in Fig. 6. Thus, it compensates electrons transferred from other Cu atoms to obey the Octet rule which makes the Coulomb attraction between Cu-S weaker than that in pure CZTS. As a result, it is easier for Cu atom to be substituted by Mg atom, and Mg doping promotes the population of Mg_Cu donor.

A buried p-n junction in Cu(InGa)Se₂ (CIGS) film facilitates electron-hole separation of photogenerated carriers. Some groups attempted doping of Cd into CIGS films to convert the near-interface region from p-type CIGS to n-type to improve the performances^[29]. However, intentional doping of CZTS to n-type through extrinsic elements is challenging, and may be one reason CZTS suffers from poor efficiencies^[30]. In addition to nontoxicity of Mg element, Mg_Cu donor has a relatively low formation energy which can be further lowered with the existence of Mg_Zn. Therefore, we propose the doping of Mg at the CZTS surface need to be further studied by a theoretical and experimental approach. Besides the Cu-poor and Zn-rich condition which are preferred in the high efficiency solar cells, previous work^[26] has discussed that Sn-rich growth condition is important in the CZTS synthesis procedure to suppress Cu_Sn. We propose the Sn-rich growth condition is also important to suppress Mg_Sn when E_F is close to the CBM. As a result, Mg prefers to substitute the Cu atomic site and acts as shallow donor in CZTS. Hence, Mg is a promising dopant to further improve the performance of Cu₂ZnSnS₄-based solar cell.

In this work, we have carried out hybrid functional computations on the Mg-related defect properties in CZTS. According to our results, all Mg-related defects are not charge localized defects and will not produce deep defect levels within the bandgap. Furthermore, Mg_Cu and Mg_Zn have very low formation energies in large E_F range. Besides the well-established Cu-poor and Zn-rich growth conditions, Sn-rich growth condition is proposed to suppress Mg_Sn. Under these conditions, Mg prefers to occupy the Cu atomic site and acts as shallow donor. Finally, we suggest that Mg doping on the surface might be an effective method to further improve the efficiency of Cu₂ZnSnS₄-based solar cells.