Lone-pair effect on carrier capture in Cu$_2$ZnSnS$_4$ solar cells

The performance of kesterite thin-film solar cells is limited by a low open-circuit voltage due to defect-mediated electron-hole recombination. We calculate the non-radiative carrier-capture cross sections and Shockley-Read-Hall recombination coefficients of deep-level point defects in Cu$_2$ZnSnS$_4$ (CZTS) from first-principles. While the oxidation state of Sn is +4 in stoichiometric CZTS, inert lone pair (5$s^2$) formation lowers the oxidation state to +2. The stability of the lone pair suppresses the ionization of certain point defects, inducing charge transition levels deep in the band gap. We find large lattice distortions associated with the lone-pair defect centers due to the difference in ionic radii between Sn(II) and Sn(IV). The combination of a deep trap level and large lattice distortion facilitates efficient non-radiative carrier capture, with capture cross-sections exceeding $10^{-12}$ cm$^2$. The results highlight a connection between redox active cations and `killer' defect centres that form giant carrier traps. This lone pair effect will be relevant to other emerging photovoltaic materials containing n$s^2$ cations.


Introduction
In a semiconductor subject to above-band-gap illumination, the lifetime of charge carriers is determined by the kinetics of electron-hole recombination: radiative, Auger, and trapassisted processes. 1,2 Radiative and Auger recombination usually only become signicant at high carrier concentrations such as in light-emitting diodes or solar cells using concentrated sunlight. In most photovoltaic technologies, defects limit carrier lifetimes and device efficiencies by acting as nonradiative electron-hole recombination centers. 3,4 Thin-lm solar cells offer advantages over traditional siliconbased solar cells as they require less raw materials and energy to produce, and open up new application areas such as buildingintegrated photovoltaics. As an alternative to the current thin-lm light absorbers such as CdTe and Cu(In,Ga)Se 2 (CIGS) whose constituting elements are vulnerable to decreases in supply, kesterite minerals such as Cu 2 ZnSnS 4 (CZTS) and Cu 2 -ZnSnS 4 (CZTSe) (see Fig. 1(a)), have attracted much attention due to the earth-abundance of Cu, Zn, and Sn. [5][6][7] In 2014, an alloyed CZTSSe kesterite solar cell reached a record light-toelectricity conversion efficiency of 12.6%. 8 Recently, 11% efficiency is achieved in a pure sulde CZTS solar cell. 9 However, this technology suffers from a large open-circuit voltage (V OC ) decit. [10][11][12] The performance of current kesterite-based solar cells falls far below the Shockley-Queisser limit of $30%. 13,14 One likely origin of the V OC decit is a short minority carrier (electron) lifetime of below few ns due to fast non-radiative recombination pathways. 12,15 Thus, it is important to identify dominant recombination centers and to control their concentrations. According to Shockley-Read-Hall (SRH) statistics, 16,17 a deep level in the band gap of a semiconductor acts as an efficient recombination channel that facilitates the sequential capture of minority and majority carriers. In addition to deep thermodynamic charge transition levels, large lattice distortions are required to achieve fast recombination rates. 3,18 However, due to the strong interactions between an impurity and a host material, it is hard to predict the properties (charge transition level and lattice distortion) of impurities a priori or to nd general trends in various host materials. Identication of recombination centers has relied on individual experimental or theoretical studies. If a simple criterion existed, then we could identify detrimental defects limiting the efficiency more easily and screen candidate optoelectronic materials more efficiently.
In this work, we argue that defects in semiconductors involving heavy post-transition metals are likely to act as fast non-radiative recombination centers not only due to their deeplevel nature but also the large lattice distortions that accompany a change in oxidation state. As a representative case, we study the native point defects in CZTS containing multivalent ions of Sn and Cu. Through analysis of carrier capture rates from rstprinciples, we nd that the dominant non-radiative recombination centers (V S , V S -Cu Zn and Sn Zn ) are associated with Sn 5s 2 lone-pair congurations. These defects produce deep donor levels due to the Sn double reduction, and the recombination processes involve large structure distortions because of the change in the ionic radius of Sn during the carrier capture. We expect to nd similar behaviour for other lone pair cations including Bi and Sb.
Here, we use the notion X Y q for a defect where X is the species (e.g. Cu, Zn, Sn, S or V for a vacancy), Y is the lattice site, and q is the charge state if specied. For example, V S represents the sulfur vacancy and Sn Zn is Sn-on-Zn antisite. Defect complexes are represented by symbols connected by a dash as in V S -V Cu .

Non-radiative carrier capture
The phenomenon of carrier capture in semiconductors via multiphonon emission has been extensively studied following pioneering work by Huang and Rhys, 19 and Henry and Lang. 20 The initial excited state of system, for example, a positively charged donor (D + ) with an electron in the conduction band (e À ), vibrates around the equilibrium geometry. Owing to the electron-phonon coupling, the deformation of the structure causes the electronic energy level of a state localized around the defect to oscillate. As the defect level approaches the conduction band, the probability for the defect to capture an electron increases signicantly. When an electron is captured, the donor becomes neutral (D 0 ) and relaxes to a new equilibrium geometry by emitting multiple phonons as shown in Fig. 2. To describe and predict this process, quantitative accounts of the electronic and atomic structures as well as the vibrational properties of the defect are essential. Recently, approaches have been developed for rstprinciples calculations of capture rates within a certain set of approximations. 21,22 We have adopted a one-dimensional conguration coordinate for the effective vibrational wave function and the static coupling theory for electron-phonon coupling matrix elements as proposed by Alkauskas et al. 22,23 We described the degree of deformation using a onedimensional conguration coordinate Q dened by where m a and DR a,i are the atomic mass and the displacement along the direction i from the equilibrium position of atom a, respectively. The vibrational wave function of excited (x im ) and ground (x fn ) states, and associated frequencies u i and u f were obtained by solving the one-dimensional Schrödinger equation for potential energy surfaces around the equilibrium geometries. The capture coefficient is given by where V, g and W if 2 are the volume of supercell, the degeneracy factor and the electron-phonon coupling matrix element of initial and nal states, respectively. w m is the occupation number of the excited vibrational state x im , and DE corresponds to the difference in energy of excited and ground states.
The Coulomb interaction at temperature T between a carrier with charge q and a defect in a charge state Q is accounted by the Sommerfeld factor hsi; 24,25  where k B is the Boltzmann constant. E R ¼ m*q 4 /(2ħ 2 3 2 ) is an effective Rydberg energy where m* and 3 are an effective mass of the carrier and a low-frequency dielectric constant, respectively. For an attractive center, Z ¼ Q/q is negative, while Z is positive for a repulsive center.
Based on the principle of detailed balance, the steady-state recombination rate R via a defect with electron-capture cross section s n and hole-capture cross section s p is given by 16,17 Here, n, p and N T denote concentrations of electrons, holes and defects, respectively. n 1 and p 1 represent the densities of electrons and holes, respectively, when the Fermi level is located at the trap level. The thermal velocities of electrons from the effective masses of electron ðm * e Þ and hole ðm * h Þ in the electronic band structure. C n and C p are the capture coefficients for electron and hole, respectively. Finally, the Shockley-Read-Hall recombination coefficient A is given by We assume an excess carrier concentration Dn ¼ 1 Â 10 14 cm À3 , which results in an open-circuit voltage V OC of 0.9 V. The Shockley-Read-Hall coefficient is not sensitive to the excess carrier concentration as long as Dn is much lower than the majority carrier concentration.

Electronic structure theory
The atomic and electronic structure of defects were calculated from rst-principles within the framework of density functional theory (DFT). 26,27 We employed the projector-augmented wave (PAW) method 28 and the hybrid exchange-correlation functional of Heyd-Scuseria-Ernzerhof (HSE06), 29 as implemented in VASP. 30 The wave functions were expanded in plane waves up to an energy cutoff of 380 eV. A Monkhorst-Pack k-mesh 31 with a grid spacing less than 2p Â 0.03Å À1 was used for Brillouin zone integration. The atomic coordinates were optimized until the residual forces were less than 0.02 eVÅ À1 . The lattice vectors were relaxed until stress was below 0.5 kbar. For defect formation, a 2 Â 2 Â 1 supercell expansion (64 atoms) of the conventional cell was employed.
We calculated the formation energy DE form (D q ) of a defect D in the charge state q which is given by 32 where E tot (bulk) and E tot (D q ) are the total energies of a bulk supercell and a supercell containing the defect D q , respectively. In the third term on the right-hand side, m i and n i are the chemical potential and number of atoms i added to the supercell, respectively. E F is the Fermi level, and E corr is a correction term to account for the articial electrostatic interaction due to periodic boundary conditions. 33,34 The formation energy is a function of the Fermi level, while the Fermi level is determined by the concentration of charged defects. Thus, we calculated the equilibrium concentration of defects and the Fermi level self-consistently, under the constraint of charge neutrality condition for overall system of defects and charge carriers, using the SC-FERMI code. 35

Results and discussion
Equilibrium phase diagram A challenge to achieving high efficiency from kesterite thin-lm solar cell is to synthesize homogeneous CZTS without unintentional formation of secondary phases. 15,[36][37][38][39] The thermodynamic chemical potential m of each element depends on the growth environment including partial pressures and temperature. We compare the DFT total energies of CZTS and its competing phases in the chemical potential space ( Fig. 1(b)), showing the range of chemical potentials that favors the formation of CZTS, using CPLAP. 40 The narrow range and complex shape of the phase diagram implies that it is hard to get a single-phase and homogeneous CZTS sample without the secondary phases. Even 'pure' CZTS is expected to contain an equilibrium population of point defects whose concentrations are controlled by the chemical potentials. We calculate the formation energies of the native defects under S-poor and S-rich conditions depicted in the phase diagram ( Fig. 1(b)).

S-poor growth environment
Under S-poor conditions, which could be realized by annealing in a low sulfur partial pressure, the most dominant native defects are Cu Zn and Zn Cu antisites which are shallow and responsible for the p-type behaviour with a Fermi level close to the valence band (see Fig. 3(a)). At the Fermi level of 0.22 eV determined self-consistently, we predict high concentrations of V S (1.3 Â 10 16 cm À3 ), V S -Cu Zn (3.0 Â 10 17 cm À3 ) and Sn Zn (1.1 Â 10 18 cm À3 ). Here, we assume the growth and annealing temperature of 853 K resulting in defect populations. The operating temperature of a solar cell is a complex function of a level of irradiation, wind speed, humidity, and mounting type. 41 We assumed a typical operating temperature of 330 K to equilibrate the Fermi level. Previously, we have shown that V S can act as an efficient nonradiative recombination center in CZTS. 42 However, for electron capture, V S needed to be activated. Firstly, as the ground state of V 0 S involving Sn(II) is neutral and produces a state resonant within the valence band, thermal excitation is required to access V S + . As shown in Fig. 4a, the hole capture barrier for V S + is so high that the thermal motion can not overcome it. Instead, the optical absorption can trigger the vertical transition from V S + to V S 2+ , which corresponds to Sn(III) to Sn(IV) oxidation.
Here, we nd that V S can also be activated by forming a defect complex with Cu Zn . In (V S -Cu Zn ) 0 , the electronic wave function is localized around the Sn 5s lone-pair orbital similar to that of V S 1+ (Fig. 5(a) and (b)), suggesting that the ionized acceptor Cu Zn 1À ionizes the neutral donor V 0 S . Thus, Sn(III)  becomes the ground-state electronic conguration in the neutral V S -Cu Zn complex, indicating, unlike the isolated V S , thermal excitation is not necessary. We further nd that optical excitation is not required for hole capture by the V S -Cu Zn complex. As a stronger Coulomb force binds the negatively charged acceptor Cu Zn 1À , the formation energy difference between Sn(III) and Sn(IV) is reduced in the V S -V Cu complex ( Fig. 4(b)). Accordingly, the reduced barrier for hole capture facilitates carrier recombination without optical excitation. The subsequent electron capture process will be fast due to the negligible energy barrier (see Fig. 4(b)). The V S -V Cu complex shows similar behavior, but its concentration is low under standard growth conditions. Activation by passivation. It has been suggested that donoracceptor complexes passivate deep donors in kesterite CZTS 43 and chalcopyrite CIGS, 44 which make them more tolerant to defects. However, we show that the neutral donor V 0 S , which is deactivated by double Sn reduction, can be reactivated by forming complexes with the ionized acceptor Cu Zn 1À and thus become an efficient recombination center. This is partially because the dominant defect-defect interaction is the classical Coulomb attraction instead of quantum mechanical level repulsion as is oen considered. 45 We also examine recombination pathways via the donor levels of Sn Zn . Fig. 5(c) shows the defect charge density of Sn Zn is well localized around the Sn lone pair, suggesting the transitions involving Sn reduction and oxidation could trigger the carrier recombination similar to those in V S and V S -Cu Zn . The recombination path involving the double donor level Sn Zn (2+/ 1+) has a relatively high electron capture barrier of 0.23 eV (Fig. 4(c) and Table 1). On the other hand, Sn Zn (1+/0)corresponding to the transition between Sn(III) and Sn(II)has a smaller energy barrier of 0.05 eV, implying a faster recombination process.
In Fig. 6(a), we present the capture cross section calculated within the static coupling approximation. 22 V S -V Cu and Sn Zn can be classied as a giant electron trap whose electron capture cross section ($10 4Å2 ) far exceeds the size of its atomic structure. 18 The calculated capture cross sections of the native defects in CZTS are orders of magnitudes larger than extrinsic transition metal impurities in silicon solar cells including Ti, V, Cr, Mo, Fe, Au and Zn whose cross sections range from 10 À1Å2 to 10 3 A 2 . 46,47 This analysis suggests that V S -Cu Zn and Sn Zn are the main sources of non-radiative recombination that limit the efficiency of CZTS solar cells (see Table 1). Note that due to the small energy barrier of V S -Cu Zn , the recombination is expected to be fast even at low temperature. At high temperature, the slight decrease in the capture cross section is attributed to the high Landau-Zener velocity; 48 the faster the defect level crosses the conduction bands, the less likely the defect captures electrons. The calculated capture cross section of Sn Zn 1+ is an order of magnitude higher than that of Sn Zn 2+ ( Fig. 6 and Table 1).
In an operating solar cell, the recombination rate due to Sn Zn may depend on the spatial position because the distance from the interface between CZTS and CdS determines the Fermi level (electronic band bending) and, hence, the charge state of Sn Zn .   In the undepleted region (d > d(2 +/1+)) in Fig. 6(b)), most of Sn Zn is in the form of +2 charge state which is a slower recombination channel. However, in the depletion region ((d < d(2 +/1+)), Sn Zn favors a +1 charge state which has much larger capture cross section. In this case, a recombination pathway is activated by band bending in a photovoltaic device.

S-rich growth environment
Under S-rich conditions, the formation of V S and V S -Cu Zn is strongly suppressed (see Fig. 3(b)). We associate this with the experimentally observed increase in V OC under a high S partial pressure during the annealing of a photovoltaic device. 49 However, even under S-rich conditions, a considerable concentration of Sn Zn is still expected, which can limit the lifetime of carriers to below 7.1 ns (see Table 1). This shows good agreement with the reported photoluminescence (PL) decay times of kesterite materials which range from 1 ns to 10 ns. 9,50 Moreover, the electron paramagnetic resonance (EPR) signal in CZTS 51 supports the existence partially oxidized Sn(III) with an unpaired electron (5s 1 ), which is the active state in the proposed recombination pathways.

Inert-pair effect
The heavy post-transition metals (elements in groups 13, 14, 15 and 16) oen exhibit oxidation states two less than the group valency, referred to as the inert-pair effect. The inert-pair effect is explained by the insufficient screening by d 10 electrons resulting in the s 2 lone-pair electrons tightly bound to the ion. [52][53][54][55] However, the role of the inert-pair effect on the properties of defects in semiconductors has not been fully explored.
Deep defect nature. We nd that the inert-pair effect of Sn produces deep defects, consistent with the previous theoretical studies. 56,57 The ability of Sn to accommodate excess charges stabilizes the neutral state over the ionized state. In a multivalent compound, such as CZTS, the variation of Madelung potential between cation sites with formal +1, +2 and +4 oxidation states favours particular defect charge states, so defect ionization is suppressed relative to elemental or binary semiconductors.
Large lattice distortion. Sn also produces defects with large distortions during carrier capture. Electron addition or removal from V S , V S -Cu Zn and Sn Zn are followed by the oxidation or reduction of Sn and are therefore accompanied by large structure change. We nd, in the Sn-related defects in CZTS, large lattice distortion quantied by a Huang-Rhys factor S [ 1. 19 Especially, in V S and V S -Cu Zn , a radiative transition pathway is impossible due to the very large lattice distortion where the minimum of the excited state (Sn(IV)) is located outside of the potential energy surface of the ground-state (Sn(III)). 58 Thus, the inert-pair effect in Sn is responsible for both the deep charge transition levels and the large lattice distortion of V S , V S -Cu Zn and Sn Zn , which make them efficient non-radiative recombination centers. Similarly, we nd the deep acceptor level of Cu Sn owing to the oxidation of Cu. However, the electron capture rates by Cu Sn (1À/2À) are low ( Table 1). The multivalency of Cu is due to the change in the occupation of 3d orbital: from 3d 10 in Cu(I) to 3d 9 in Cu(II). Thus, the local relaxation aer the oxidation is not signicant. The small lattice distortion in Cu Sn produces an electron capture barrier above 1.6 eV, making capture unlikely ( Fig. 6(a)).
Emergence of deep defects induced by the formation of lone pairs has also been reported in CIGS. Extrinsic dopants of Bi and Sb are deep in CuInSe 2 due to the lone-pair s 2 states. 59 Han et al. 60 have also found that the formation of lone pairs in amorphous oxide semiconductors, such as InGaZnO 4 and ZnSnO 3 , is responsible for electron trapping.
While Sn reduction captures electrons, in lone-pair compounds whose cations, such as Tl(I), Pb(II) and Bi(III), have occupied s 2 orbitals in the stoichiometric compounds, the oxidation of cations could capture holes. Several EPR measurements show that Pb(III) is responsible for the hole traps in lead halides. [61][62][63][64] The efficiency and lifetime of optoelectronic devices can be severely damaged by a defect with fast non-radiative recombination. Stoneham 18 suggests several characteristics of such killer centers, including: (1) defects producing many and closely spaced electronic levels; (2) defects with large lattice distortions. The rst type can be directly related to the transition metal impurities with partially lled d orbitals (e.g. Ni in GaP). While a simple vacancy center was suggested as a candidate for the second-type, 18 a wide variety of vacancies are not recombination centers in photovoltaic materials (e.g. V Cu is a shallow acceptor in CZTS, and V I is a shallow donor in CH 3 NH 3 PbI 3 .) Because of the strong interaction between impurities and host materials, it is difficult to nd a general trend of the properties of defects in the absence of detailed calculations. On the other hand, we nd that the inert lone-pair of Sn is the origin of the large crosssection of an wide range of defects and not signicantly altered by a specic conguration or an electronic state of the defect. Thus, we speculate that the inert-pair effect could likely cause killer centers with the ability to act as giant carrier traps in a broad range of semiconductors.
Many photovoltaic materials with band gaps close to the theoretical optimum of 1.3-1.5 eV (ref. 13) show poor performances; in particular a low V OC . The record efficiency of Cu 2 SnS 3 , whose band gap is around 1 eV, is still low (4.63%) even with the high current density J SC of 37.3 mA cm À2 mainly because of low V OC of 283 mV. 65 The carrier lifetime of Cu 2 SnS 3 was reported to be very short (0.1-10 ps). 66 Sb 2 Se 3 solar cells also exhibit low V OC with short carrier lifetime of 1.3 ns. 67 The rst light-to-electricity conversion efficiency of Cu 3 BiS 3 solar cell (0.17%) has been achieved only recently with V OC of 190 meV, 68 and, to our knowledge, a successful operation of CuBiS 2 solar cell has not been reported. The presence of lone-pair cations is a common feature in these technologies.

Conclusion
The lone-pair effect associated with Sn is responsible for both the deep defect levels and large lattice distortions that facilitate rapid electron-hole recombination in the semiconductor Cu 2 -ZnSnS 4 . By employing a rst-principles approach to predict the defect levels, concentrations, and capture rates, we can distinguish between active and inactive defect centres. For a material grown under S-poor conditions, V S , V S -Cu Zn and Sn Zn act as dominant recombination centers, while Sn Zn limits the minority carrier lifetime under S-rich conditions. We propose that a similar mechanism could underpin the low performance of other emerging photovoltaic compounds. We emphasise the need for further experimental and theoretical investigation of defects in semiconductors composed of heavy post-transition metals to further evaluate the general role of the inert-pair effect on the non-radiative electron-hole recombination process.

Conflicts of interest
There are no conicts to declare.