Upper limit to the photovoltaic efficiency of imperfect crystals

The Shockley-Queisser (SQ) limit provides a convenient metric for predicting light-to-electricity conversion efficiency of a solar cell based on the band gap of the light-absorbing layer. In reality, few materials approach this radiative limit. We develop a formalism and a computational method to predict the maximum photovoltaic efficiency of imperfect crystals from first principles. Our scheme includes equilibrium populations of native defects, their carrier-capture coefficients, and the associated recombination rates. When applied to kesterite solar cells, we reveal an intrinsic limit of 20% for Cu2ZnSnSe4, which falls far below the SQ limit of 32%. The effects of atomic substitution and extrinsic doping are studied, leading to pathways for an enhanced efficiency of 31%. This approach can be applied to support targeted-materials selection for future solar-energy technologies.

Sunlight is the most abundant source of sustainable energy. Similar to Carnot efficiency of heat engines, the maximum efficiency for photovoltaic energy conversion is determined by thermodynamics and can be as high as 86% owing to the high temperature of the sun. 1,2 However, in practical solar cells with single p-n semiconductor junctions, large irreversible energy loss occurs mainly through hot-carrier cooling and low light absorption below the band gap. 3 The Shockley-Queisser (SQ) limit describes the theoretical sunlight-to-electricity conversion efficiency of a single-junction solar cell. 3 The SQ limit (33.7% under AM 1.5 illumination) and its variations, including spectroscopic limited maximum efficiency (SLME), 4 determine the maximum efficiency of a solar cell based on the principle of detailed balance between the absorption and emission of light. The amount of photon absorbed determines the short-circuit current density J SC , and, hot-carrier cooling and radiative recombination limit the maximum carrier concentration and hence the opencircuit voltage V OC .
In the SQ limit, the predicted efficiency is a function of the semiconductor band gap, which is a trade-off between light absorption (current generation) and energy loss due to hot-carrier cooling. This analysis secured the band gap as a primary descriptor when searching for new photovoltaic compounds, often within a 1-1.5 eV target window. Unfortunately, few materials approach the SQ limit. Less than 10 classes of materials have achieved conversion efficiency greater than 20%. 5 Many emerging technologies struggle to break the 10% efficiency threshold.
Kesterites are a class of emerging materials for thinfilm photovoltaic applications. Although a lot of progress has been made during the past few decades, the certified champion efficiency of 12.62% 6 has been increased by less than 0.1% since 2013. 7 The main bottleneck is the low open-circuit voltage, which is far below the SQ limit. 8 Many routes to engineer compositions and architectures have been considered, but it is not clear which process dominates. 9 One of the biggest questions in the field is if there is an intrinsic problem with kesterite semiconductors that prevent them achieving the radiative limit. [10][11][12] The discrepancy between the SQ limit and efficiencies of real solar cells results from the extra irreversible processes such as electron-hole nonradiative recombination. While Shockley and Queisser studied the effect of the nonradiative recombination, it has been treated as a parameter of radiative efficiency and often a radiative efficiency of 100% is assumed, which is unrealistic for real materials.
The rate of nonradiative recombination mediated by traps is described by Shockley-Read-Hall statistics. 13,14 The steady-state recombination rate is determined by the detailed balance where the net electron-capture rate is equal to the net hole capture rate. A microscopic theory of carrier capture was proposed by Henry and Lang in 1977. 15 The thermal vibration of the defect together with the electron-phonon coupling causes charge transfer from a delocalized free carrier to a localized defect state. Thus the carrier capture coefficient heavily depends on the electron and phonon wave functions associated with a defect, which are difficult to probe experimentally. Instead, the microscopic processes in materials, including nonradiative carrier capture, have been inferred from macroscopic responses such as a capacitance transient. 15 Macroscopic properties of solar cells (e.g. open-circuit voltage and device efficiency) and microscopic processes in the material (e.g. carrier capture coefficient) have not been directly connected yet. Therefore, although theories of solar cells are well known, the theoretical approaches have failed to provide a priori predictions of photovoltaic efficiencies of real materials. Each material has a fundamental limit of its radiative efficiency because the material contains a certain amount of native defects. Their concentrations in ther-mal equilibrium are intrinsic properties of the materials, and the 'soup' of defects determines the maximum radiative efficiency. Recently, a first-principles method based on density functional theory (DFT) has been developed to calculate the nonradiative carrier capture, [16][17][18] which opens up the possibility for studying the theoretical upper-bound of photovolataic efficiency of a real material limited by both the radiative and the nonradiative recombination.
In this work, we propose a first-principles method of the trap-limited conversion efficiency (TLC) to calculate the upper-limit of photovolatic efficiency of a material containing the number of native defects in thermal equilibrium. To take into account both radiative and nonradiative processes, we perform a series of calculations for kesterites. The absorption and the emission of light are calculated in the framework of Shockley and Queisser. To obtain the nonradiative recombination rate, we calculate the carrier capture coefficients and equilibrium concentrations of native defects. The workflow for our method is shown in Fig. 1. We conclude that kesterite solar cells suffer from significant nonradiative recombination and are unable to reach the SQ limit even under optimal growth conditions. Strategies to overcome such rapid recombination rates are suggested.

A. Radiative Recombination
The short-circuit current J SC of a solar cell whose absorber thickness is W is given by the absorbed photon flux multiplied by an elementary charge q: where Φ sun (E) and a(E; W ) are the solar spectrum and the absorptivity at a photon energy E, respectively. Following the SQ limit, we assume that an absorbed photon generates one electron-hole pair. The radiative recombination rate for the solar cell at temperature T is given by where V is a bias voltage serving a chemical potential of the electron-hole pair. At the short-circuit condition, the solar cell and ambient are in equilibrium: the radiative recombination rate R rad (0) is equal to the absorption rate from the ambient irradiation. The net current density   J rad limited by the radiative recombination is given by where the saturation current J rad 0 = qR rad (0). In the SQ limit, an absorptivity is assumed to be a step function being 1 above the band gap E g and 0 otherwise, while a real material has a finite absorptivity with a tail near the band gap E g which depends on the sample thickness. Rau et al. 19 defined a photovoltaic band gap using the absorption edge spectrum and found that, in inorganic solar cells, the effect of the finite absorption tail on the open-circuit voltage loss is small. 19 However, the band tail due to the disorder can cause serious reduction in V OC .

B. Nonradiative Recombination
A material in thermal equilibrium will contain a population of native defects. Defect processes are unavoidable and define the upper limit of performance of optoelectronic devices. The nonradiative recombination at charge carriers via defects is often a dominant source of degradation of solar cells and should be carefully controlled. 20 Based on the principle of detailed balance 13,14 , the steady-state recombination rate R SRH via a defect with electron-capture coefficient C n and hole-capture cross coefficient C p is given by where Here, n, p, and N T denote concentrations of electrons, holes, and defects, respectively. n i is an intrinsic carrier concentration (n 2 i = n 0 p 0 where n 0 and p 0 are intrinsic electron and hole concentrations). n t and p t represent the densities of electrons and holes, respectively, when the Fermi level is located at the trap level E T . The capture cross section (σ n for electron and σ p for hole) is commonly used in experimental studies, and can be calculated taking the thermal velocities of electron v th,n and hole v th,p to be 10 7 cm s −1 .
For doped semiconductors, minority carrier lifetime often determines the rate of the total recombination process. For example, in a p-type semiconductor where the acceptor concentration, p 0 , is much higher than the photoexcited carrier density, the R SRH due to a deep defect is proportional to the (photoexcited) excess carrier density ∆n: 21 In case of a material containing many types of recombination centers, the total recombination rate R SRH is the sum over all independent centers. The calculation of R SRH requires three properties of a defect (concentration N T , defect level E T , and capture coefficient C n/p ) and the carrier concentrations n and p, as well as the intrinsic doping density n 0 or p 0 in the bulk host, as explained in the following subsections.

Equilibrium defect concentrations
Phase diagram: The growth environment of a crystal including elemental ratio, partial pressures, and temperature determines the properties of the material including concentrations of the native defects. In a theoretical framework, the growth conditions can be expressed using the thermodynamic chemical potential µ of each element. We compare the energies of kesterites and their competing secondary phases, showing a range of chemical potentials that favors the formation of kesterites, using CPLAP. 27 We can avoid the formation of the secondary phases by a careful choice of synthesis conditions. However even 'pure' kesterites without secondary phases will contain native defects whose concentrations are controlled by this choice of chemical potentials.
Formation energy of a defect: We calculated the formation energy ∆E f (D q ) of a defect D with the charge state q as given by 28 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, N i is the number of atoms i added to the supercell, and µ i is its chemical potential which is limited by the aforementioned phase diagram. E F is the Fermi level, and E corr is a correction term to account for the spurious electrostatic interaction due to periodic boundary conditions. 29,30 Self-consistent Fermi level: For a given synthesis condition (set of atomic chemical potentials), the formation energy is a function of the Fermi level as shown in Eq. 7, while the Fermi level is determined by the concentrations of charged defects and carriers. Thus we calculate the equilibrium concentrations of defects and carriers, and the Fermi level self-consistently under the constraint of charge neutrality condition for overall system of defects and charge carriers using SC-FERMI 31 .
For a given Fermi level, the equilibrium concentration of a defect N (D q ) is given by where N site and g are the number of available sites per unit volume and the degeneracy of the defect, respectively. In the dilute limit, the competition between defects is negligible. The partition function is approximated as 1 (i.e. the majority of lattice sites are regular). Note that we use the internal energy of formation to calculate the defect density, neglecting the vibrational entropy change. Thus the estimated defect densities are lower bounds. 32 The concentrations of holes p 0 and electrons n 0 are determined by the effective density of states of valence band N V and conduction band N C : Here, E VBM and E CBM are the reference energies of the valence band maximum and conduction band minimum, respectively.
The net charge of defects should be compensated by the net charge of electrons and holes: Thus, we iteratively update the Fermi level until the charge neutrality condition (Eq. 10) is satisfied. First, we determined the equilibrium concentration of defects at high temperature (T an = 800 K) and equilibrate their charge states at room temperature (T op = 300 K) with a fixed concentrations of defects.

Defect levels
A defect can change its charge state by capturing or emitting carriers. The recombination process requires that defects are electrically active with more than one charge state. The energy required to change the charge state of the defect level is often referred to as a thermal activation energy or a charge-transition-level. In modern defect theory, the defect level D is calculated as the position of Fermi level where the formation energies with two charge states of q 1 and q 2 are equal: FIG. 3. Growth condition. Calculated phase diagrams of Cu 2 ZnSnSe 4 (a) and Ag 2 ZnSnSe 4 (b) where µi = 0 represents the chemical potential of element i in its elemental state. Each plane represents a phase boundary with the secondary phase. Blue and red circles indicate Se-poor and Se-rich conditions, respectively.

Carrier capture coefficient
The nonradiative carrier capture via a defect is triggered by the lattice vibration accompanying the electronphonon coupling between the localized trap state and the delocalized free carriers in the conduction and valence bands. The initial excited state, for example, a positively charged donor (D + ) with an electron in the conduction band (e − ), vibrates around the equilibrium geometry. Owing to the electron-phonon coupling, the deformation of the structure causes the electronic energy level of the trap state to oscillate. As the energy level approaches the conduction band, the probability for the defect to capture the electron increases significantly. When the electron is captured, the donor becomes neutral D 0 and relaxes to a new equilibrium geometry by emitting multiple phonons. To describe and predict such a process, quantitative accounts of the electronic and atomic structures, as well as vibrational properties of the defect are essential.
The carrier capture coefficient C can be expressed using the electron-phonon coupling W ct and the overlap of phonon wave functions ξ im |∆Q|ξ f n , 17,18 which is given by where Ω and g denote the volume of supercell and the degeneracy of the defect, respectively. ψ and ξ are electron and phonon wave functions, respectively, and the subscripts c and t specify the free carrier and trap states. In this formalism, the temperature-dependence is determined by the thermal occupation number w m of the initial vibrational state. In the following discussions, we calculate the capture coefficients at room temperature (T op = 300 K). We use an effective configuration coordinate ∆Q for the phonon wave functions and adopt the static coupling theory for W ct . The Coulomb attraction and repulsion between charged defects and carriers are accounted by using Sommerfeld factor. 33,34 See Supplementary information for details.

Steady-state illumination
Under illumination or bias voltage, the steady-state electron and hole concentrations deviate from those determined by the equilibrium Fermi level. The amount of applied voltage V is the difference between the electron and hole quasi -Fermi levels (E F,n for electron and E F,p for hole) which are functions of an additional carrier concentration ∆n: where we ignore the voltage drop due to a series resistance and a shunt across the device. One can rewrite Eq. 13 for ∆n as a function of V : where Accordingly, the steady-state concentrations of electron n and hole p under applied voltage V are given by

C. Trap limited conversion efficiency
By taking into account the carrier annihilation due to both radiative recombination (Eq. 3) and nonradiative recombination (Eq. 4), the trap-limited current density J under a bias voltage V is given by The voltage-dependent nonradiative recombination rate R SRH is obtained by combining Eq. 4, 8, 11, 12, and 15. Finally, we evaluate the photovoltaic maximum efficiency: Shockley-Queisser limit: In the SQ limit under 1sun (AM 1.5G) illumination, the maximum efficiency of CZTSe with a band gap of 1 eV is 31.6% (see Fig. 2) with a V OC of 0.77 V. Next, we calculate the nonradiative recombination rate due to the native defects.
Growth conditions: Stoichiometric CZTSe is formed when the chemical potential of the elements are in the phase field of CZTSe as shown in Fig. 3a. The phase diagram of CZTSe has a small volume with a narrow window of available chemical potentials, which the stability of ZnSe is largely responsible for. At high Zn-ratio, Zn atoms tend to form ZnSe rather than to incorporate at their lattice sites in CZTSe. Later, we will show that this poor incorporation of Zn results in high concentrations of antisite defects: Cu Zn and Sn Zn , which are responsible to the p-type Fermi level and the low carrier lifetime, respectively.
Defect levels: Point defects introducing defect levels close to the band edge are categorized as shallow and generate free carriers. 20 On the other hand, deep defects are often responsible for carrier trapping and nonradiative recombination, limiting the efficiency of solar cells. 20 The band structure of CZTSe is composed of the antibonding Sn 5s-Se 4p * state at the CBM and the antibonding Cu 3d -Se 4p * state at the VBM. According to recent models for defect tolerance, 35,36 the Cu dangling bond would produce a shallow level while a deep level can be introduced by the Sn dangling bond. Moreover, the cation antisites, especially (Cu,Zn) Sn and Sn (Cu,Zn) are expected to be deep due to the large difference in the site Madelung potentials. 37 Admittance spectroscopy (AS) measurements identified several shallow acceptors in Cu 2 ZnSn(S,Se) 4 , CZTSSe, CZTSe and CZTS at an energy range between 0.05-0.17 eV 38-43 which were attributed to V Cu and Cu Zn . They also found a deep level close to the midgap (E T = 0.5 eV). A series of deep-level transient spectroscopy (DLTS) experiments also revealed the presence of the shallow levels as well as a broad spectrum of deep levels around the mid gap. [44][45][46] Transient photocapacitance (TPC) spectra showed sub-band-gap absorption via deep defects near 0.8 eV with broad bandwidth. 47,48 Theoretical calculations 37,49-51 revealed the atomic origins of shallow defects: acceptors V Cu and Cu Zn and a donor Zn Cu . Several atomic models for the deep defects have been proposed such as (Cu 3 ) Sn , Sn Zn ,V S , V S -Cu Zn , and Sn Zn -Cu Zn . 37,[49][50][51] First, we find shallow acceptors (V Cu and Cu Zn ) and a shallow donor (Zn Cu ) (see Fig. 4a and Supplementary Table 2). Due to the similar ionic radii of Cu and Zn, the energy cost for the formation of Cu Zn and Zn Cu is very low. The very low formation energy of Cu Zn for every set of chemical potentials is largely responsible for the p-type Fermi level around 0.2 eV. We find that the decrease in oxidation state of Sn found in V Se , Sn Zn and V Se -Cu Zn produces deep levels, similar to those found in CZTS. 37,[49][50][51] The deep donor Sn Zn becomes shallow when it combines with Cu Zn because of the Coulomb attraction between the ionized donor and acceptor. 50 Capture coefficients: As Cu-based kesterites are intrinsic p-type semiconductors, the carrier lifetime is deter- mined by the electron-capture processes via deep defects. We calculate electron-capture coefficients of the selected deep defects: V Se -Cu Zn and Sn Zn , satisfying the criterion Due to the Sn reduction associated with these defects, they exhibit not only a deep level, but also a large lattice relaxation which leads to large electron-capture coefficients. 37,50 Fig. 5a shows the configuration coordinate for Sn Zn (2+/1+), illustrating that the carriercapture barrier is small due to the large lattice relaxation, the horizontal shift of the potential energy surface of Sn 1+ Zn with respect to that of Sn 2+ Zn . Thus, we find that Sn Zn (2+/1+) has a large electron-capture coefficient of 9 × 10 −7 cm 3 s −1 (corresponding to the capture cross section of 9.29 × 10 −14 cm 3 s −1 ), which classify them as killer centers. 52 Note that the minority-carrier capture coefficient of these native defects in CZTSe are of a similar order of magnitude of the most detrimental extrinsic impurities in Si solar cells. 53,54 We also find a large electron-capture coefficient of V Se -Cu Zn which is listed in Supplementary Table 2.
Equilibrium concentration: The concentration of native point defects can be tuned using different chemical environments. However, we find that it is difficult to reduce the concentration of the killer centers in CZTSe. For example, to reduce the concentration of Sn Zn , we need: i) to increase Zn incorporation, ii) to decrease Sn incorporation, or iii) to decrease hole concentration. These are difficult to achieve due to the narrow thermal equilibrium phase diagram. First, the high-Zn incorporation is difficult to achieve because of the aforementioned high stability of ZnSe. On the other hand, the incorporation can be tuned to decrease the concentration of Sn Zn . The low Sn incorporation, together with the low Zn incorporation, will, however, result in the formation of the highly conductive secondary phases of CuSe and Cu 2 Se (see Fig. 3a), which can electrically short the device. 55 Thus, the low Sn incorporation should actually be avoided. We also find the hole concentrations are high under all conditions due to the high concentrations of Cu Zn , which is also the consequence of the poor Zn incorporation. Therefore, it is difficult to decrease the concentrations of Sn Zn in thermal equilibrium. Fig. 6a shows the equilibrium concentrations of native defects under Se-poor and Se-rich conditions (see Fig. 3a). Under Se-poor conditions, we find high concentration of V Se -Cu Zn , which is an efficient recombination center. While their concentrations can be significantly decreased through Se incorporation, the concentration of Sn Zn can not be decreased below 10 14 cm −3 , which limits the maximum performance of CZTSe solar cells.
Finally, we stress that the capture cross section and defect concentrations of the dominant recombination center in CZTSe (Sn Zn ) are in good agreement with experiments. 40,56 Our previous admittance spectroscopy 40 revealed a deep defect level located at 0.5 eV. Based on the thermal emission prefactors of up to 5 × 10 12 cm s −1 at room temperature, we estimate the capture cross section as 1 × 10 −13 cm 2 which agrees well with our calculation of 9 × 10 −14 cm 2 (see Supplementary Table 2). We also find the longest minority-carrier lifetime achievable is less than 5.5 ns in CZTSe which closely agrees with the previous assessment of the real minority-carrier lifetime of below 1 ns based on timeresolved photoluminescence. 56,57 Trap limited conversion efficiency: We calculate the current-voltage characteristic (Eq. 16) of a CZTSe solar cell containing the equilibrium concentrations of native point defects under the Se-rich condition (See Fig. 7a). We used the a film thickness of 2 µm. The overall power-conversion efficiency is 20.3%, which is below two thirds of the SQ limit of 31.6% (see Fig. 2 and Table I).
Sulfide kesterite: Cu 2 ZnSnS 4 (CZTS) also suffers from nonradiative recombination due to the redox activity of Sn and the narrow phase space limited by the high stability of ZnS. Similar to Sn Zn in CZTSe, we find the large lattice relaxation in Sn Zn in CZTS which causes fast carrier capture. Moreover, although the defect complex Sn Zn -Cu Zn is a shallow donor in CZTSe, in CZTS having the larger band gap of 1.5 eV, Sn Zn -Cu Zn produces the deep donor level at E T = 0.90 eV as shown in Fig. 4a and b. Thus, the recombination pathways in CZTS are not only through the isolated Sn Zn but also the Sn Zn bound to the acceptor Cu Zn , which agrees well with a previous theoretical study 51 . We find that the similar behavior for Ge Zn in Cu 2 ZnGeSe 4 which will be discussed in detail in the following subsection. We calculate a nonradiative V OC loss of 0.39 V, corresponding to an achievable V OC of 0.84 V and a maximum TLC of 20.9% for CZTS, which is similar to that of CZTSe.

B. Cu 2 ZnGeSe 4
As the redox activity of Sn is the culprit that reduces the open-circuit voltage and the efficiency of CZTSe and CZTS, we can suppress the nonradiative recombination by substituting Sn with other cations such as Si with a more stable 4+ oxidation state. However, the SQ limit of Cu 2 ZnSiSe 4 is below 16% because of its large band gap of 2.33 eV. 58 On the other hand, Cu 2 ZnGeSe 4 (CZGSe) has the optimal band gap of 1.36 eV whose SQ limit is 33.6%. However, we find that the similar redox activity of Ge in CZGSe causes significant nonradiative recombination and limits the V OC .
Ge also exhibit inert-pair effect with large ionization energy for the 4s orbital. Thus, Ge-related defects (Ge Zn , Ge Zn -Cu Zn , V Se and V Se -Cu Zn ) introduce deep donor levels in the band gap. Ge Zn exhibits the similar potential energy surfaces to those of Sn Zn in CZTSe (Fig. 5b). However, Ge Zn has a deeper donor level than that of Sn Zn due to the larger band gap of CZGSe (see supplementary  Table 1). As shown in Fig. 5, because the electroncapture processes due to Sn Zn and Ge Zn are in the socalled "Marcus inverted region", 59 the deeper donor level of Ge Zn results in a higher energy barrier for electroncapture (0.62 eV). We find a several orders of magnitude smaller electron-capture coefficient for Ge Zn (2+/1+) as compared to that of Sn Zn (2+/1+), implying that the recombination due to the isolated Ge Zn is unlikely to happen (see Supplementary Table 2).
However, the nonradiative recombination rate in CZGSe is still high due to a defect complex. The abundant acceptor Cu Zn tends to form a defect complex with donors such as Ge Zn . The Coulomb attraction between the ionized donor and acceptor further promote the formation of the complex. Moreover, the donor-acceptor complex makes the defect level shallower (E T = 0.87 eV). 50 We find that the electron-capture barrier is 71 meV for Ge Zn -Cu Zn (1+/0), which is the dominant recombination pathway in CZGSe. Although, we considered only the Ge Zn and Cu Zn pair bound at the closest site, in reality, there are a variety of complexes with a wide range of distances between Sn Zn and Cu Zn . Such a spectrum of complexes are partially responsible for the broad defect levels in kesterites measured in photocapacitance spectroscopies. 47,48 Owing to the formation of defect complexes, we find significant nonradiative loss in CZGSe. The maximum efficiency is predicted to be 21.9% with large non-radiative open-circuit voltage loss of 0.29 V (see Fig. 2 and Table I). Therefore, we conclude that an attempt to turn harmful defects into benign defects by isovalent cation substitution alone, i.e. replacement of Zn by Ge, is not feasible.
C. Hydrogen and alkali-metal doping, and Ag 2 ZnSnSe 4 Next, as an additional lever to tune the defect profiles, we consider extrinsic doping. As shown in Eq. 7, the formation energy, and hence the concentration, of a defect depends on the chemical potential of an electron (Fermi level). In CZTSe, CZTS, and CZGSe, the intrinsic Fermi levels are pinned c.a. 0.2 eV above the valence band (Fig.  4a, b, and c), promoting the formation of deep donors. As illustrated in Fig. 8a, such high concentrations of donors arise at high (growth) temperature and remain after cooling because they are mostly immobile vacancies and antisites. While n-type doping can increase the Fermi level, this conventional doping will not increase the V OC (efficiency) for a material with limited minority carrier lifetime, because the n-type doping will decrease the p-type conductivity. Instead, we predict that hydrogen and alkali-metal doping is helpful to increase the efficiency. At high temperature during the thin-film growth or thermal annealing, the incorporation of the hydrogen or alkali metal at the interstitial sites increases the Fermi level as they act as donors in p-type semiconductors. 60 The high Fermi level decreases the hole concentration and the formation of donor type defects as well (see Fig. 8c). Since hydrogen and alkali-metals are mobile, they tend to diffuse easily and segregate to the grain boundary or outgas, when the thin-film cools down to the room temperature (see Fig. 8d). The final thin-film will exhibit an increased hole concentration and longer carrier lifetime, consistent with the experiments. 61 This is indeed the mechanism behind the success of hydrogen-codoping in nitride semiconductors. 62,63 We calculate the concentrations of defects in CZTSe with a n-type doping concentration of 10 20 cm −3 at T = T an . Once the dopants are removed, the hole concentration increases by an order of magnitude at T = T op , and the concentration of Sn Zn is significantly lowered (see Fig. 6a). Thus, the maximum efficiency increases up to 23.7 % (Fig. 2 and Fig. 7a). This requires a high level of doping to gain a noticeable improvement due to the high concentration of native donors and acceptors, and the self-compensation mechanism via them. Alkali-metal elements may be less effective dopants due to their low solubility. 61 The low formation energies and the high concentra-tions of Cu Zn and Zn Cu originate from the similar ionic radii of Cu 1+ and Zn 2+ . We may decrease their concentrations by exploiting Ag substituting Cu or Cd substituting Zn. 64 Ag substitution for Cu gives Ag 2 ZnSnSe 4 (AZTSe) which also has a narrow phase diagram as shown in Fig. 3b. However, we find several orders of magnitude lower concentrations of the dominant acceptor and donor, Ag Zn and Zn Ag (see Fig. 6b). AZTSe is an intrinsic semiconductor under Se-rich conditions while n-type Fermi level was found under Se-poor conditions. Under the Se-rich condition, the calculated selfconsistent Fermi-level is located at 0.55 eV above the valence band. Due to the low hole concentration in AZTSe, Eq. 6 is not valid, and the hole-capture process is the bottleneck in the recombination process owing to the high hole-capture barrier of 0.20 eV as compared to the electron-capture barrier of 0.11 eV. However, due to the high Fermi level in AZTSe or even ntype conductivity, Ag-based solar cells based on the commonly used thin-film architecture for Cu-based kesterites (Mo/kesterite/CdS/ZnO/ITO), are expected and have been found to exhibit very limited device performance so far. 23,65,66 Notwithstanding these practical challenges, we predict that Ag-based kesterites should show much lower non-radiative recombination and thus possess a significantly larger efficiency potential than the previously discussed Cu-or Ge-based kesterites. Indeed, increased photoluminescence quantum yields (PLQY) have been recently observed for Ag-substituted kesterites. 67 A n-type doping at the level of 10 20 cm −3 during growth can lead to to a lower (p-type) Fermi-level of 0.18 eV at room temperature. As shown in Fig. 6b, this causes the concentration of Sn Zn to decrease below 10 14 cm −3 , enhancing the maximum efficiency up to 30.8% (see Fig. 2 and Fig. 7), implying that co-doped AZTSe is a promising material as a p-type absorber if the synthesis and processing be appropriately controlled.

D. Calculation of optoelectronic parameters
The achievable solar cell parameters estimated for four types of kesterite materials using our first-principle approach are summarized in Table I, and compared with the (defect-free) Shockley-Queisser limit, as well as current champion devices. It can be seen that the Ge-and Agbased materials so far significantly underperform, and that big leaps in efficiency appear possible by the proposed codoping of these materials as discussed above. Since device performance can be limited by a number of non-idealities such as non-optimized functional layers, wrong band line-ups, as well as interface recombination, it is helpful to consider also the main (absorber layer) optoelectronic parameters, which are experimentally accessible even without building devices. Among the most relevant ones to judge potential device performance are the carrier lifetime, the net doping density, and the external PLQY, which indicates the ratio of ra-diative recombination over the total recombination, typically dominated by non-radiative processes. The PLQY can be estimated from non-radiative voltage loss using ∆V nonrad OC = k B T ln(PLQY). 68 A summary of these parameters, calculated from first-principles, are listed in Table II, indicating small PLQY and lifetimes for CZTS and large PLQY and long lifetimes for co-doped AZTSe. The small predicted PLQY for CZTS is in agreement with observations that the luminescence yield of this material is consistently below the detection limit (ca. 1 × 10 −4 %). Also, the PLQY value of 1 × 10 −2 % is consistent with recent reports of 1.5 × 10 −3 % measured on a CZTSe single crystal 57 and of 3 × 10 −3 % on 11.6% efficient Li-doped CZTSSe solar cells. 69 In these solar cells the lifetime did not change significantly with Li-doping, while the PLQY and net doping density increased, again inline with our predictions. With regard to the calculated minority carrier lifetimes, we point out that the small estimated lifetimes for CZTS and CZTSe are in good agreement with recent findings indicating that reported carrier lifetimes for kesterites are often overestimated and that (typical) real lifetimes are in fact below 1 ns. 56 Although, our approach significantly advances firstprinciples theories for solar cells over previous models, some limitations should be noted. Our method inherits some of the limitations of the SQ approach. 70 TLC does not take into account parasitic absorption effects in the buffer layer, window layer, or additional recombination at the interfaces. Also, high series resistance can further reduce the efficiency of kesterite solar cells. 9 Although we assume the formation of defects in thermal equilibrium, in reality, their concentrations may not reach equilibrium especially when processed rapidly at low temperature. Kesterite thin-films are not homogeneous, and often lateral variations of the stoichiometry are found in the samples. Therefore it has been pointed out that the fluctuations of the band gap and the electrostatic potential can further reduce the open-circuit voltage beyond our predictions. 71 Secondary phases may decrease shunt resistance or induce parasitic absorption. 9 However, it is widely accepted that the high open-circuit voltage deficit owing to a short carrier life is the main culprit for the low efficiency in kesterite solar cells. 9,56 In commercial photovolatic solar cells, J SC and FF approach the SQ limit, while the main efficiency-limiting factor is V OC 70,72 which we tackle. Therefore, our method can provide a new direction for searching for promising photovoltaic materials by providing a realistic upper limit on expected performance.

III. CONCLUSIONS
In summary, we have compiled the physics of solar cells and the modern defect theory from first-principles to assess the efficiency limit of solar cells. We have included the thermal equilibrium concentrations of native defects of the absorber material which reduces the carrier life- is the VOC loss due to the nonradiative recombination, p0 is the intrinsic hole concentration, τSRH is the ShockleyReadHall lifetime and PLQY is the external photoluminescence quantum yield at 1-sun equivalent conditions. time and have proposed the first-principles method to calculate the maximum efficiency limited by recombination centers. Sn-based kesterites suffer from severe nonradiative recombination due to native defects. The fast nonradiative recombination can be mitigated by extrin-sic doping and Ag-alloying, reducing the concentration of recombination centres, thereby increasing their performance up to 29%. The TLC approach can be used as part of screening procedures to select viable candidates from the pool of emerging materials for photovoltaics. Finally, we emphasise that, to assess the genuine potential of real materials as photovoltaics, one should consider not only the thermodynamics of light and electrons, but also the thermodynamics of crystals.

IV. DATA AVAILABILITY
The data that support the findings of this study are available in Zenodo repository with the identifier doi:10.5281/zenodo.XXXXXXX.