Upper limit to the photovoltaic efficiency of imperfect crystals
UUpper limit to the photovoltaic efficiency of imperfect crystals: the case of kesteritesolar cells
Sunghyun Kim, Jos´e A. M´arquez, Thomas Unold, and Aron Walsh
1, 3 Department of Materials, Imperial College London, Exhibition Road, London SW7 2AZ, UK Helmholtz-Zentrum Berlin f¨ur Materialien und Energie GmbH,Department Structure and Dynamics of Energy Materials,Hahn-Meitner-Platz 1, D-14109 Berlin, Germany Department of Materials Science and Engineering, Yonsei University, Seoul 03722, Korea ∗ (Dated: January 30, 2020)The Shockley-Queisser (SQ) limit provides a convenient metric for predicting light-to-electricityconversion 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 computational methodto predict the maximum photovoltaic efficiency of imperfect crystals from first principles. Ourscheme includes equilibrium populations of native defects, their carrier-capture coefficients, andthe associated recombination rates. When applied to kesterite solar cells, we reveal an intrinsiclimit of 20% for Cu ZnSnSe , which falls far below the SQ limit of 32%. The effects of atomicsubstitution and extrinsic doping are studied, leading to pathways for an enhanced efficiency of31%. This approach can be applied to support targeted-materials selection for future solar-energytechnologies. Sunlight is the most abundant source of sustainableenergy. Similar to the Carnot efficiency of heat engines,the maximum efficiency for photovoltaic energy conver-sion is determined by thermodynamics and can be ashigh as 86% owing to the high temperature of the sun.
However, in practical solar cells with single p-n semi-conductor junctions, large irreversible energy loss occursmainly through hot-carrier cooling and low light absorp-tion below the band gap. The Shockley-Queisser (SQ) limit describes the the-oretical sunlight-to-electricity conversion efficiency of asingle-junction solar cell. The SQ limit (33.7% underAM1.5g illumination) and its variations, including spec-troscopic limited maximum efficiency (SLME), deter-mine the maximum efficiency of a solar cell based onthe principle of detailed balance between the absorptionand emission of light. The amount of photons absorbeddetermines the short-circuit current density J SC , and,hot-carrier cooling and radiative recombination limit themaximum carrier concentration and hence the open-circuit voltage V OC .In the SQ limit, the predicted efficiency is a functionof the semiconductor band gap, which is a trade-off be-tween light absorption (current generation) and energyloss due to hot-carrier cooling. This analysis secured theband gap as a primary descriptor when searching for newphotovoltaic compounds, often within a 1–1.5 eV targetwindow. Unfortunately, few materials approach the SQlimit. Less than 10 classes of materials have achievedconversion efficiency greater than 20%. Most emergingtechnologies struggle to break the 10% efficiency thresh-old.Kesterites are a class of quaternary materials studiedfor thin-film photovoltaic applications. Although a lot ofprogress has been made during the past few decades, thecertified champion efficiency of 12.6% has been increasedby less than 0.1% since 2013. The main bottleneck is the low open-circuit voltage, which is far below the SQlimit. Many routes to engineer compositions and archi-tectures have been considered, but it is not clear whichprocess dominates. One of the biggest questions in thefield is if there is an intrinsic problem with kesterite semi-conductors that prevent them approaching the radiativelimit.
The discrepancy between the SQ limit and efficienciesof real solar cells results from the extra irreversible pro-cesses such as electron-hole nonradiative recombination.While Shockley and Queisser studied the effect of thenonradiative recombination, it has been treated as a pa-rameter of radiative efficiency and often a radiative effi-ciency of 100% is assumed, which is unrealistic for realmaterials.The rate of nonradiative recombination mediatedby traps can be described by Shockley-Read-Hallstatistics.
The steady-state recombination rate is de-termined by the detailed balance where the net electron-capture rate is equal to the net hole capture rate. Amicroscopic theory of carrier capture was proposed byHenry and Lang in 1977. The thermal vibration ofthe defect, together with the electron-phonon coupling,causes charge transfer from a delocalised free carrier toa localised defect state. Thus the carrier capture coeffi-cient heavily depends on the electron and phonon wavefunctions associated with a defect, which are difficultto probe experimentally. Instead, the microscopic pro-cesses in materials, including nonradiative carrier cap-ture, have been inferred from macroscopic responses suchas a capacitance transient. Macroscopic properties ofsolar cells (e.g. open-circuit voltage and device efficiency)and microscopic processes in the material (e.g. carriercapture coefficient) are rarely connected. Therefore, al-though theories of solar cells are well known, the theoreti-cal approaches have failed to provide a priori predictionsof photovoltaic efficiencies of real materials. a r X i v : . [ phy s i c s . c o m p - ph ] J a n Each material has a fundamental limit of radiative effi-ciency because the material contains a certain amount ofnative defects. Their concentrations in thermal equilib-rium are intrinsic properties of the materials, and the re-sulting ‘soup’ of defects determines the maximum radia-tive efficiency. Recently, first-principles methods basedon density functional theory (DFT) have been devel-oped to calculate the nonradiative carrier capture, which opens up the possibility for studying the theoreti-cal upper-bound of photovolataic efficiency of a real ma-terial limited by both the radiative and the nonradiativerecombination.In this work, we propose a first-principles method ofthe trap-limited conversion efficiency (TLC) to calculatethe upper-limit of photovolatic efficiency of a materialcontaining the number of native defects in thermal equi-librium. To take into account both radiative and nonra-diative processes, we perform a series of calculations forkesterites. The absorption and the emission of light arecalculated in the framework of Shockley and Queisser. Toobtain the nonradiative recombination rate, we calculatethe carrier capture coefficients and equilibrium concen-trations of native defects. The workflow for our methodis shown in Fig. 1. We conclude that kesterite solarcells suffer from significant nonradiative recombinationand are unable to reach the SQ limit even under optimalgrowth conditions. Strategies to overcome such rapid re-combination rates are suggested.
I. THEORYA. Radiative Recombination
The short-circuit current J SC of a solar cell whose ab-sorber thickness is W is given by the absorbed photonflux multiplied by an elementary charge q: J SC ( W ) = q (cid:90) ∞ a ( E ; W ) Φ sun ( E ) d E, (1)where Φ sun ( E ) and a ( E ; W ) are the solar spectrum andthe absorptivity at a photon energy E , respectively. Fol-lowing the SQ limit, we assume that an absorbed photongenerates one electron-hole pair.The radiative recombination rate for the solar cell attemperature T is given by R rad ( V ) = 2 πc h (cid:90) ∞ a ( E ; W ) (cid:104) e E − q V / k B T − (cid:105) − E d E ≈ πc h e q VkBT (cid:90) ∞ a ( E ; W ) (cid:104) e E / k B T − (cid:105) − E d E = R rad (0)e q VkBT , (2)where V is a bias voltage serving a chemical potential ofthe electron-hole pair. At the short-circuit condition, thesolar cell and ambient are in equilibrium: the radiative C p C n E g a p V B C B E g a p E n e r g y Configuration coordinate p n ΔE f E F E F E T N V p +Δnn +Δn DOS N C μ A μ B B ABA E F,p qVE
F,n
J VC n : ElectroncaptureC p : Holecapture V B + V A − V B V A E F h + e − J SC J (e qV/ k B T − qR SRH
W η max
Bandstructure E gap , N C , N V Growthcondition μ i Formationenergy E f , E T ConfigurationcoordinateCapturecoefficient C n/p Self-consistentFermi level E F , N T , n , p SRHrecombination rateDevice simulationTrap-limitedconversion efficiency R SRH η Radiative limit J SC , J J=J SC +J r ad ( − e eV/k B T )−qR SRH W
11 22 3 345 6 6 77 88 85 54
FIG. 1.
Diagram for the calculation of trap-limitedconversion efficiency.
The dependent calculations are con-nected by lines (upper panel). For each numbered step, thecalculated quantities are appended. The red and blue boxesrepresent calculations for radiative and nonradiative electron-hole recombination, respectively. The combined device simu-lations are marked in green. The corresponding physical pro-cesses are drawn in the electronic and the atomic structures(lower panel). recombination rate R rad (0) is equal to the absorption ratefrom the ambient irradiation. The net current density J rad limited by the radiative recombination is given by J ( V ; W ) = J SC ( W ) + J rad0 ( W )(1 − e q VkBT ) , (3)where the saturation current J rad0 = q R rad (0).In the SQ limit, an absorptivity is assumed to be astep function being 1 above the band gap E g and 0 oth-erwise, while a real material has a finite absorptivity witha tail near the band gap E g , which depends on the samplethickness. Rau et al. defined a photovoltaic band gapusing the absorption edge spectrum and found that, ininorganic solar cells, the effect of the finite absorption tailon the open-circuit voltage loss is small. However, theband tail due to the disorder can cause serious reductionin V OC . B. Nonradiative Recombination
A material in thermal equilibrium will contain a popu-lation of native defects. Defect processes are unavoidableand define the upper limit of performance of optoelec-tronic devices. The nonradiative recombination at chargecarriers via defects is often a dominant source of degra-dation of solar cells and should be carefully controlled. Based on the principle of detailed balance , thesteady-state recombination rate R SRH via a defect withelectron-capture coefficient C n and hole-capture cross co-efficient C p is given by R SRH = np − n i τ p ( n + n t ) + τ n ( p + p t ) , (4)where τ − n = N T C n = N T σ n v th ,n ,τ − p = N T C p = N T σ p v th ,p . (5)Here, n , p , and N T denote concentrations of electrons,holes, and defects, respectively. n i is an intrinsic carrierconcentration ( n i = n p , where n and p are intrinsicelectron and hole concentrations). n t and p t representthe densities of electrons and holes, respectively, whenthe Fermi level is located at the trap level E T . The cap-ture cross section ( σ n for electron and σ p for hole) iscommonly used in experimental studies, and can be cal-culated taking the thermal velocities of electron v th ,n andhole v th ,p to be 10 cm s − .For doped semiconductors, minority carrier lifetime of-ten determines the rate of the total recombination pro-cess. For example, in a p -type semiconductor where theacceptor concentration, p , is much higher than the pho-toexcited carrier density, the R SRH due to a deep defect isproportional to the (photoexcited) excess carrier density∆ n : R SRH ≈ ∆ nτ n = ∆ nN T C n . (6)In case of a material containing many types of recombi-nation centers, the total recombination rate R SRH is thesum over all independent centers.The calculation of R SRH requires three properties of adefect (concentration N T , defect level E T , and capturecoefficient C n/p ) in addition to the carrier concentrations n and p , as well as the intrinsic doping density n or p inthe bulk host, as explained in the following subsections. Equilibrium defect concentrations
Phase diagram:
The growth environment of a crystalincluding elemental ratio, partial pressures, and temper-ature determines the properties of the material includ-ing concentrations of the native defects. In a theoreti-cal framework, the growth conditions can be expressed E g (eV) E g (eV)0204060 J S C ( m A / c m ) a bc d V O C ( V ) FF ( % ) P C E ( % ) C Z G S e A Z T S e C Z T SS e S Q li m i t S Q li m i t S Q l i m i t S Q li m i t E g FIG. 2.
Shockley-Queisser limit and trap-limited-conversion efficiency. a , Short-circuit current density J SC , b , open-circuit voltage V OC , c , fill factor F F , and d , effi-ciency η . Filled symbols represent the trap-limited conversion(TLC), while a black line is the SQ limit. TLCs with doping(triangles) show better performances as compared to TLCswithout doping (circles). Plus signs indicate experimentaldata for kesterite solar cells taken from Ref. using the thermodynamic chemical potential µ of eachelement. We compare the energies of kesterites and theircompeting secondary phases, showing a range of chem-ical potentials that favors the formation of kesterites,using CPLAP . We can avoid the formation of the sec-ondary phases by a careful choice of synthesis conditions.However even ‘pure’ kesterites without secondary phaseswill contain native defects whose concentrations are con-trolled by this choice of chemical potentials.
Formation energy of a defect:
We calculated the for-mation energy ∆ E f (D q ) of a defect D with the chargestate q as given by ∆ E f (D q ) = E tot (D q ) − E tot (bulk) − (cid:88) i N i µ i + qE F + E corr , (7)where E tot (bulk) and E tot (D q ) are the total energies of abulk 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 isits chemical potential which is limited by the aforemen-tioned phase diagram. E F is the Fermi level, and E corr isa correction term to account for the spurious electrostaticinteraction due to periodic boundary conditions. Self-consistent Fermi level:
For a given synthesis con-dition (set of atomic chemical potentials), the formationenergy is a function of the Fermi level as shown in Eq. 7,while the Fermi level is determined by the concentrationsof charged defects and carriers. Thus we calculate theequilibrium concentrations of defects and carriers, andthe Fermi level self-consistently under the constraint ofcharge neutrality condition for overall system of defectsand charge carriers using
SC-FERMI .For a given Fermi level, the equilibrium concentrationof a defect N (D q ) is given by N (D q ) = N site g e − ∆ E f (D q )/ k B T , (8)where N site and g are the number of available sites perunit volume and the degeneracy of the defect, respec-tively. In the dilute limit, the competition between de-fects is negligible. The partition function is approximatedas 1 (i.e. the majority of lattice sites are regular). Notethat we use the internal energy of formation to calcu-late the defect density, neglecting the vibrational entropychange. Thus the estimated defect densities are lowerbounds. The concentrations of holes p and electrons n aredetermined by the effective density of states of valenceband N V and conduction band N C : p = N V e − E F − E VBM / k B T ,n = N C e − E CBM − E F / k B T . (9)Here, E VBM and E CBM are the reference energies of thevalence band maximum and conduction band minimum,respectively.The net charge of defects should be compensated bythe net charge of electrons and holes: (cid:88) i,j q j N (D q j i ) = p − n . (10)Thus, we iteratively update the Fermi level until thecharge neutrality condition (Eq. 10) is satisfied. First,we determined the equilibrium concentration of defects athigh temperature ( T an = 800 K) and equilibrated theircharge states at room temperature ( T op = 300 K) with afixed concentration of defects. Defect levels
A defect can change its charge state by capturing oremitting carriers. The recombination process requiresthat defects are electrically active with more than onecharge state. The energy required to change the chargestate of the defect level is often referred to as a thermal activation energy or a charge-transition-level. In moderndefect theory, the defect level D is calculated as the po-sition of Fermi level where the formation energies withtwo charge states of q and q are equal: E T ( q /q ; D) = ∆ E f ( E F = 0; D q ) − ∆ E f ( E F = 0; D q ) q − q . (11) Ag SnSe Ag Ag SnSe ZnSeSnSeSnSe Se CuSeCu SeCu Cu SnSe SeZnSeSnSe SnSe ₂ μ S n μ Z n μ C u μ S n μ Z n μ A g a b FIG. 3.
Growth condition.
Calculated phase diagrams ofCu ZnSnSe ( a ) and Ag ZnSnSe ( b ) where µ i = 0 representsthe chemical potential of element i in its elemental state. Eachplane represents a phase boundary with the secondary phase.Blue and red circles indicate Se-poor and Se-rich conditions,respectively. Carrier capture coefficient
Nonradiative carrier capture via a defect is triggeredby a vibration and the associated electron-phonon cou-pling between the localised trap state and the delocalisedfree carriers. The initial excited state, for example, a pos-itively charged donor (D + ) with an electron in the con-duction band (e − ), vibrates around the equilibrium ge-ometry. The deformation of the structure causes the elec-tronic energy level of the trap state to oscillate. As theenergy level approaches the conduction band, the prob-ability for the defect to capture the electron increasessignificantly. When the electron is captured, the donorbecomes neutral D and relaxes to a new equilibrium ge-ometry by emitting multiple phonons. To describe andpredict such a process, quantitative accounts of the elec-tronic and atomic structures, as well as vibrational prop-erties of the defect are essential.The carrier capture coefficient C can be expressed us-ing the electron-phonon coupling W ct and the overlap ofphonon wave functions (cid:104) ξ cm | ∆ Q | ξ tn (cid:105) , which is givenby C =Ω g π (cid:126) | W ct | (cid:88) m,n w m | (cid:104) ξ tn | ∆ Q | ξ cm (cid:105)| × δ (∆ E + (cid:15) cm − (cid:15) tn ) (12)where Ω and g denote the volume of supercell and thedegeneracy of the defect, respectively. ψ and ξ are elec-tron and phonon wave functions, respectively, and thesubscripts c and t specify the free carrier and trap states.In this formalism, the temperature-dependence is deter-mined by the thermal occupation number w m of the ini-tial vibrational state. In the following discussion, we cal-culate the capture coefficients at room temperature. Weemploy an effective configuration coordinate ∆ Q for thephonon wave functions and adopt static coupling theoryfor W ct . The Coulomb attraction and repulsion betweencharged defects and carriers are accounted for by theSommerfeld factor. See Supplementary informationfor details.
Steady-state illumination
Under illumination or bias voltage, the steady-stateelectron and hole concentrations deviate from those de-termined by the equilibrium Fermi level. The amountof applied voltage V is the difference between the elec-tron and hole quasi -Fermi levels ( E F,n for electron and E F,p for hole) which are functions of an additional carrierconcentration ∆ n :q V (∆ n ) = E F,n (∆ n ) − E F,h (∆ n )= E CBM + k B T ln (cid:18) n + ∆ nN C (cid:19) − E VBM + k B T ln (cid:18) p + ∆ nN V (cid:19) = E g + k B T ln (cid:18) ( n + ∆ n )( p + ∆ n ) N C N V (cid:19) , (13)where we ignore the voltage drop due to a series resis-tance and a shunt across the device. One can rewrite Eq.13 for ∆ n as a function of V :∆ n ( V ) = 12 (cid:34) − n − p + (cid:114) ( n + p ) − n i (cid:16) − e q VkBT (cid:17)(cid:35) , (14)where n i = n p = N C N V e − EgkBT . Accordingly, thesteady-state concentrations of electron n and hole p un-der applied voltage V are given by n ( V ) = n + ∆ n ( V ) ,p ( V ) = p + ∆ n ( V ) . (15) C. Trap limited conversion efficiency
By taking into account the carrier annihilation due toboth radiative recombination (Eq. 3) and nonradiativerecombination (Eq. 4), the trap-limited current density J under a bias voltage V is given by J ( V ; W ) = J SC ( W ) + J rad0 ( W )(1 − e q VkBT ) − qR SRH ( V ) W. (16)The voltage-dependent nonradiative recombination rate R SRH is obtained by combining Eq. 4, 8, 11, 12, and15. Finally, we evaluate the photovoltaic maximum effi-ciency: η = max V (cid:32) JVq (cid:82) ∞ E Φ sun ( E ) d E (cid:33) . (17) II. RESULTS
We apply our scheme to kesterite solar cells(Cu ZnSnSe , Cu ZnSnS , Cu ZnGeSe , andAg ZnSnSe ), with details presented in the Meth-ods section and Supplementary Table 1. A. Cu ZnSnSe and Cu ZnSnS Shockley-Queisser limit:
In the SQ limit under 1-sun (AM1.5g) illumination, the maximum efficiency ofCZTSe 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 nonradia-tive recombination rate due to native defects. Growth conditions:
Single-phase CZTSe is formedwhen the chemical potential of the elements are in thephase field of CZTSe as shown in Fig. 3a. The phase di-agram of CZTSe has a small volume with a narrow win-dow of available chemical potentials, which the stabilityof ZnSe is largely responsible for. At high Zn-ratio, Znatoms tend to form ZnSe rather than to incorporate attheir lattice sites in CZTSe. Later, we will show that thispoor incorporation of Zn results in high concentrationsof antisite defects: Cu Zn and Sn Zn , which are responsi-ble to the p -type Fermi level and the low carrier lifetime,respectively. Defect levels:
Point defects introducing defect levelsclose to the band edge are categorized as shallow andgenerate free carriers. On the other hand, deep defectsare often responsible for carrier trapping and nonradia-tive recombination, limiting the efficiency of solar cells. The band structure of CZTSe is composed of antibond-ing Sn 5 s -Se 4 p ∗ state at the lower conduction band andantibonding Cu 3 d -Se 4 p ∗ state at the upper valenceband. According to models for defect tolerance, theCu dangling bond would produce a shallow level, whilea deep level can be introduced by the Sn dangling bond.Moreover, the cation antisites, especially (Cu , Zn) Sn andSn (Cu,Zn) are expected to be deep due to the large differ-ence in the site electrostatic (Madelung) potentials. Admittance spectroscopy (AS) measurements iden-tified several shallow acceptors in Cu ZnSn(S , Se) ,CZTSSe, CZTSe and CZTS at an energy range between0.05-0.17 eV , which were attributed to V Cu andCu Zn . They also found a deep level close to the midgap( E T = 0 . Transient photocapac-itance (TPC) spectra showed sub-band-gap absorption via deep defects near 0 . Theoretical calculations revealed the atomic ori-gins of shallow defects: acceptors V Cu and Cu Zn and adonor Zn Cu . Several atomic models for the deep defectshave been proposed such as (Cu ) Sn , Sn Zn ,V S , V S -Cu Zn ,and Sn Zn -Cu Zn . E n e r g y ( e V ) a b c d (+/0)(+/0)(2+/+)(+/0)(0/−)(0/−) (0/+) (0/−) V Se Sn Zn V Se -Cu Zn Sn Zn -Cu Zn Zn Cu Cu Zn V Cu E F (0/−)(+/0)(+/0)(2+/+)(+/0)(0/−)(0/−) (0/+) (+/−) V S Sn Zn V S -Cu Zn Sn Zn -Cu Zn Zn Cu Cu Zn V Cu E F (0/−)(+/0)(+/0) (+/0)(2+/+)(+/0)(0/−)(0/−) (0/+) (0/−) V Se Ge Zn V Se -Cu Zn Ge Zn -Cu Zn Zn Cu Cu Zn V Cu E F (0/−)(+/0)(2+/+) (+/0)(+/0)(+/0)(2+/0)(0/−)(0/−) (0/+) (0/−) V Se Sn Zn V Se -Ag Zn Sn Zn -Ag Zn Zn Ag Ag Zn V Ag E F w/ doping E F CZTSe CZTS CZGSe AZTSe
FIG. 4.
Defect levels of native defects.
Donor (red) and acceptor (blue) levels of native point defects of Cu ZnSnSe ( a ), Cu ZnSnS ( b ), Cu ZnGeSe ( c ), and Ag ZnSnSe ( d ). Blue and red bands represents valence and conduction bands,respectively. Fermi levels are shown in gray lines. The black line in d represents the Fermi level of Ag ZnSnSe with a dopingdensity of 10 cm − . First, we find shallow acceptors (V Cu and Cu Zn ) anda shallow donor (Zn Cu ) (see Fig. 4a and SupplementaryTable 2). Due to the similar ionic radii of Cu and Zn,the energy cost for the formation of Cu Zn and Zn Cu isvery low. The very low formation energy of Cu Zn forevery set of chemical potentials is largely responsible forthe p -type Fermi level around 0.2 eV. We find that thedecrease in oxidation state of Sn found in V Se , Sn Zn andV Se -Cu Zn produces deep levels, similar to those foundin CZTS. The deep donor Sn Zn becomes shallowwhen it combines with Cu Zn because of the Coulombattraction between the ionized donor and acceptor. Capture coefficients:
As Cu-based kesterites are intrin-
Zn2+ + h + + e − Sn Zn1+ + h + Sn Zn2+ Ge Zn2+ + h + + e − Ge Zn1+ + h + Ge Zn2+
021 Q (amu
Å) Q (amu Å) E n e r g y ( e V ) a b E b E b FIG. 5.
Configuration coordinate diagram for carriercapture.
Potential energy surfaces for the vibrations of Sn Zn (2+/+) in Cu ZnSnSe ( a ) and Ge Zn (2+/+) in Cu ZnGeSe ( b ). The solid circle represents the relative formation energycalculated using DFT, and the line is a spline fit. E b repre-sents the electron-capture barrier. sic p -type semiconductors, the carrier lifetime is deter-mined by the electron-capture processes via deep defects.We calculate electron-capture coefficients of the selecteddeep defects: V Se -Cu Zn and Sn Zn , satisfying the criterion E CBM − E T > E VBM − E F + 0 . n t (cid:28) p at T =300 K, and N T > cm − .Due to the Sn reduction associated with these de-fects, they exhibit not only a deep level, but also alarge structural relaxation that leads to large electron-capture coefficients. Fig. 5a shows the configura-tion coordinate for Sn Zn (2+/1+), illustrating that thecarrier-capture barrier is small due to the large latticerelaxation, the horizontal shift of the potential energysurface of Sn with respect to that of Sn . Thus, wefind that Sn Zn (2+/1+) has a large electron-capture co-efficient of 9 × − cm s − (corresponding to the cap-ture cross section of 9 . × − cm s − ), which classifythem as killer centers. Note that the minority-carriercapture coefficient of these native defects in CZTSe areof a similar order of magnitude of the most detrimental extrinsic impurities in Si solar cells.
We also find alarge electron-capture coefficient of V Se -Cu Zn , which islisted in Supplementary Table 2. Equilibrium concentration:
The concentration of na-tive point defects can be tuned through the chemical en-vironment. However, we find that it is difficult to reducethe concentration of the killer centers in CZTSe. Forexample, to reduce the concentration of Sn Zn , we need:i) to increase Zn incorporation, ii) to decrease Sn incor-poration, or iii) to decrease hole concentration. Theseare difficult to achieve due to the narrow thermal equi-librium phase diagram. First, the high-Zn incorporationis difficult to achieve because of the aforementioned highstability of ZnSe. On the other hand, the incorporationcan be tuned to decrease the concentration of Sn Zn . Thelow Sn incorporation, together with the low Zn incorpo- C o n ce n t r a t i o n ( c m - ) Se-poor Se-rich p S n Z n V S e - C u Z n V S e Z n S n V Z n Sn Cu C u Z n Z n C u S n Z n - C u Z n V C u Doping concentration (cm −3 ) p D o p i n g Sn Zn V Se -Ag Zn V Se V Zn S n A g Cu Zn Zn Ag Sn Zn -Ag Zn V Cu a b FIG. 6.
Concentrations of native defects. a , The con-centrations of native defects in CZTSe. The dashed lines rep-resent the concentration with the doping during the growth(see text for details). b , The concentrations of native defectsin AZTSe with the doping during the growth. The dasheddiagonal line represents the doping concentration. ration, will, however, result in the formation of the highlyconductive secondary phases of CuSe and Cu Se (see Fig.3a), which can electrically short the device. Thus, thelow Sn incorporation should actually be avoided. We alsofind the hole concentrations are high under all conditionsdue to the high concentrations of Cu Zn , which is also theconsequence of the poor Zn incorporation. Therefore, itis difficult to decrease the concentrations of Sn Zn in ther-mal equilibrium.Fig. 6a shows the equilibrium concentrations of na-tive defects under Se-poor and Se-rich conditions (seeFig. 3a). Under Se-poor conditions, we find high concen-tration of V Se -Cu Zn , which is an efficient recombinationcenter. While their concentrations can be significantlydecreased through Se incorporation, the concentration ofSn Zn can not be decreased below 10 cm − , which limitsthe maximum performance of CZTSe solar cells.Finally, we stress that the capture cross sectionand defect concentrations of the dominant recombi-nation center in CZTSe (Sn Zn ) are in good agree-ment with experiments. Our previous admittancespectroscopy revealed a deep defect level located at0 . × cm s − at room temperature, we estimate thecapture cross section as 1 × − cm which agrees wellwith our calculation of 9 × − cm (see Supplemen-tary Table 2). We also find the longest minority-carrierlifetime achievable is less than 5 . real minority-carrier lifetime of below 1 ns based on time-resolved photoluminescence. Trap limited conversion efficiency:
We calculate thecurrent-voltage characteristic (Eq. 16) of a CZTSe solarcell containing the equilibrium concentrations of nativepoint defects under the Se-rich condition (See Fig. 7a). We used the a film thickness of 2 µ m. The overall power-conversion efficiency is 20.3%, which is below two thirdsof the SQ limit of 31.6% (see Fig. 2 and Table I). Sulfide kesterite: Cu ZnSnS (CZTS) also suffers fromnonradiative recombination due to the redox activity ofSn and the narrow phase space limited by the high stabil-ity of ZnS. Similar to Sn Zn in CZTSe, we find the largestructural relaxation for Sn Zn that causes fast carriercapture. Moreover, although the defect complex Sn Zn -Cu Zn is a shallow donor in CZTSe, in CZTS having thelarger band gap of 1 . Zn -Cu Zn produces the deepdonor level at E T = 0 .
90 eV as shown in Fig. 4a andb. Thus, the recombination pathways in CZTS are notonly through the isolated Sn Zn but also the Sn Zn boundto the acceptor Cu Zn , which agrees well with a previoustheoretical study . We find that the similar behaviorfor Ge Zn in Cu ZnGeSe which will be discussed in de-tail in the following subsection. We calculate a nonradia-tive 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 ZnGeSe As the redox activity of Sn is one culprit that reducesthe voltage and efficiency of CZTSe and CZTS devices,we can suppress the nonradiative recombination by sub-stituting Sn with other cations such as Si with a morestable 4+ oxidation state. However, the SQ limit ofCu ZnSiSe is below 16% because of its large band gapof 2.33 eV. On the other hand, Cu ZnGeSe (CZGSe)has an optimal band gap of 1.36 eV with an SQ limit of33.6%. However, we find that the similar redox activityof Ge in CZGSe causes significant nonradiative recombi-nation and limits the V OC .Ge also exhibits an inert-pair effect with large ionisa- Voltage (V) S Q li m i t N d = c m – S Q li m i t N d = c m – w / o d o p i n g a C u rr e n t d e n s i t y ( m A / c m ) Voltage (V) b FIG. 7.
Current-voltage simulation. J - V curves forCZTSe ( a ) and AZTSe ( b ) solar cells based on the propertiesof the bulk absorber materials and not including inferfacialprocesses. Green lines represent the TLCs with various dop-ing concentrations up to 10 cm − . The SQ limit is shownin the blue curve. tion energy for the 4 s orbital. Thus, Ge-related defects(Ge Zn , Ge Zn -Cu Zn , V Se and V Se -Cu Zn ) introduce deepdonor levels in the band gap. Ge Zn exhibits the simi-lar potential energy surfaces to those of Sn Zn in CZTSe(Fig. 5b). However, Ge Zn has a deeper donor level thanthat of Sn Zn due to the larger band gap of CZGSe (seesupplementary Table 1). As shown in Fig. 5, becausethe electron-capture processes due to Sn Zn and Ge Zn arein the so-called “Marcus inverted region”, the deeperdonor level of Ge Zn results in a higher energy barrier forelectron-capture (0.62 eV). We find a several orders ofmagnitude smaller electron-capture coefficient for Ge Zn (2+/1+) as compared to that of Sn Zn (2+/1+), imply-ing that the recombination due to the isolated Ge Zn isunlikely to happen (see Supplementary Table 2).However, the nonradiative recombination rate inCZGSe is still high due to defect complexation. Theabundant acceptor Cu Zn tends to form a defect com-plex with donors such as Ge Zn . The Coulomb attrac-tion between the ionized donor and acceptor furtherpromote the formation of the complex. Moreover, thedonor-acceptor complex makes the defect level shallower( E T = 0 .
87 eV). We find that the electron-capture bar-rier is 71 meV for Ge Zn -Cu Zn (1+/0), which is the dom-inant recombination pathway in CZGSe. Although, weconsidered only the Ge Zn and Cu Zn pair bound at theclosest site, in reality, there are a variety of complexeswith a wide range of distances between Sn Zn and Cu Zn .Such a spectrum of complexes are partially responsiblefor the broad defect levels in kesterites measured in pho-tocapacitance spectroscopies. By taking into account the formation of defect com-plexes, we find significant nonradiative loss in CZGSe.The maximum efficiency is predicted to be 21 .
9% withlarge non-radiative open-circuit voltage loss of 0.29 V(see Fig. 2 and Table I).
C. Hydrogen and alkali-metal doping, andAg ZnSnSe As an additional lever to tune the defect profiles, weconsider extrinsic doping. The formation energy, andhence concentration, of a defect depends on the chemicalpotential of an electron (Fermi level). In CZTSe, CZTS,and CZGSe, the intrinsic Fermi levels are pinned ∼ n -typedoping can increase the Fermi level, this type of dopingwill not increase the V OC (efficiency) for a material withlimited minority carrier lifetime, because n -type dopingwill decrease the p -type conductivity.Instead, we predict that hydrogen and alkali-metaldoping is helpful to increase the efficiency. At high tem-perature during the thin-film growth or thermal anneal- D + e − h + H i + High temperature Low temperature a Without doping b With doping
FIG. 8.
The effect of hydrogen/alkali-metal doping onkesterites.
Schematics for the formation of defects withoutdoping ( a ) and with doping ( b ). During thermal annealing,the native defects are formed at high temperature (left panel),whose populations remain the same when the sample is cooleddown to low temperature (right panel). A high concentrationof hole (white circle) promote the formation of donors (bluecircle). Dopants are marked as yellow circles. For the clarity,the acceptors are not drawn. ing, the incorporation of the hydrogen or alkali metalsat the interstitial sites will increase the Fermi level asthey act as donors in p -type semiconductors. The highFermi level decreases the hole concentration and the for-mation of donor type defects as well (see Fig. 8c). Sincehydrogen and alkali-metals are mobile, they tend to dif-fuse easily and segregate to the grain boundary or out-gas, when the thin-film cools down to the room tem-perature (see Fig. 8d). The final thin-film will exhibitan increased hole concentration and longer carrier life-time, consistent with the experiments. This is indeedthe mechanism behind the success of hydrogen-codopingin nitride semiconductors.
We calculate the concentrations of defects in CZTSewith a n -type doping concentration of 10 cm − at T = T an . Once the dopants are removed, the hole concentra-tion 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 increasesup to 23 . On the other hand, the previouscalculations have shown that the formation energies ofH i in kesterites are low at p -type Fermi-level, suggest-ing high solubility of H in kesterites. We also noted thatSon et al. formed a S-Se grading in the current championdevice using H S gas, which may introduce the H-dopingunintentionally and be responsible for the high efficiency.The low formation energies and the high concentra-tions of Cu Zn and Zn Cu originate from the similar ionicradii of Cu and Zn . We may decrease their concen-trations by exploiting Ag substituting Cu or Cd substi-tuting Zn. Ag substitution for Cu gives Ag ZnSnSe (AZTSe), which also has a narrow phase diagram asshown in Fig. 3b. However, we find several orders ofmagnitude lower concentrations of the dominant accep-tor and donor, Ag Zn and Zn Ag (see Fig. 6b). AZTSeis an intrinsic semiconductor under Se-rich conditions,while n -type Fermi level was found under Se-poor condi-tions.For a set of atomic chemical potentials determinedunder Se-rich conditions, the calculated self-consistentFermi-level is 0 .
55 eV above the valence band. Dueto the low hole concentration in AZTSe, Eq. 6 isnot valid, and the hole-capture process becomes thebottleneck in the recombination process owing to thehigh hole-capture barrier of 0 .
20 eV as compared tothe electron-capture barrier of 0 .
11 eV. However, dueto the high Fermi level in AZTSe or even n-typeconductivity, Ag-based solar cells based on the com-monly used thin-film architecture for Cu-based kesterites(Mo/kesterite/CdS/ZnO/ITO), have been found to ex-hibit limited device performance.
Notwithstand-ing these practical challenges, we predict that Ag-basedkesterites should show much lower non-radiative recom-bination and thus possess a significantly larger efficiencypotential than the previously discussed Cu- or Ge-basedkesterites. Indeed, increased photoluminescence quan-tum yields (PLQY) have been recently observed for Ag-substituted kesterites. An extrinsic n -type doping level of 10 cm − duringgrowth can lower the room temperature Fermi-level to0 .
18 eV. As shown in Fig. 6b, this causes the concentra-tion of Sn Zn to decrease below 10 cm − , enhancing themaximum efficiency up to 30 .
8% (see Fig. 2 and Fig. 7),implying that co-doped AZTSe is a promising materialas a p -type absorber if the synthesis and processing beappropriately controlled. D. Calculation of optoelectronic parameters
The achievable solar cell parameters estimated for fourtypes of kesterite materials using our first-principles ap-proach are summarized in Table I, and compared withthe (defect-free) Shockley-Queisser limit, as well as cur-rent champion devices.The Ge- and Ag-based materials so far significantlyunderperform, and that big leaps in efficiency appearpossible by the proposed co-doping strategy. Device per-formance can be limited by a number of non-idealitiessuch as non-optimised functional layers, wrong band line-ups, as well as interface recombination. It is thereforehelpful to consider the main (absorber layer) optoelec- tronic parameters that are experimentally accessible evenwithout building devices. Among the most relevant tojudge potential device performance are carrier lifetime,net doping density, and external PLQY, which indicatesthe ratio of radiative recombination over the total recom-bination, typically dominated by non-radiative processes.The PLQY can be estimated from non-radiative voltageloss using ∆ V nonradOC = k B T ln(PLQY). A summary of these parameters, calculated from first-principles, are listed in Table II, indicating small PLQYand lifetimes for CZTS and large PLQY and long life-times for co-doped AZTSe. The small predicted PLQYfor CZTS is in agreement with observations that the lu-minescence yield of this material is consistently belowthe detection limit (ca. 1 × − %). Also, the PLQYvalue of 1 × − % is consistent with recent reports of1 . × − % measured on a CZTSe single crystal andof 3 × − % on 11.6% efficient Li-doped CZTSSe solarcells. In these solar cells the lifetime did not change signif-icantly with Li-doping, while the PLQY and net dop-ing density increased, again inline with our predictions.With regards to the calculated minority carrier lifetimes,we point out that the small estimated lifetimes for CZTSand CZTSe are in good agreement with recent findingsindicating that reported carrier lifetimes for kesterites areoften overestimated and that (typical) real lifetimes arein fact below 1 ns. E gap η J SC V OC FF Reference(eV) (%) (mA / cm ) (V) (%)CZTS 1.50 32.1 28.9 1.23 90.0 SQ limitCZTSe 1.00 31.6 47.7 0.77 85.7 SQ limitCZGSe 1.36 33.3 34.3 1.10 89.1 SQ limitAZTSe 1.35 33.7 34.7 1.09 89.0 SQ limitCZTS 1.50 20.9 28.9 0.84 86.4 TLCCZTSe 1.00 20.3 47.7 0.53 81.0 TLCCZGSe 1.36 24.1 34.3 0.81 86.2 TLCCZTS:H 1.50 23.1 28.9 0.91 87.4 TLCCZTSe:H 1.00 23.7 47.7 0.60 82.7 TLCCZGSe:H 1.36 27.9 34.3 0.93 87.5 TLCAZTSe:H 1.35 30.8 34.7 1.01 88.1 TLCCZTS 1.50 11.0 21.7 0.73 69.27 Exp. CZTSe 1.00 11.6 40.6 0.42 67.3 Exp. CZTSSe 1.13 12.6 35.4 0.54 65.9 Exp. CZTGSe 1.11 12.3 32.3 0.53 72.7 Exp. CZGSe 1.36 7.6 22.8 0.56 60 Exp. AZTSe 1.35 5.2 21.0 0.50 48.7 Exp. ACZCTS 1.40 10.1 23.4 0.65 66.2 Exp. TABLE I. Device performance parameters of selected Cu andAg kesterite solar cells and predicted by Shockley-Queisserlimit and trap-limited conversion efficiency and found exper-imentally (Exp). E gap ∆ V nonradOC p τ SRH
PLQY(eV) (V) (cm − ) (ns) (%)CZTS 1.50 0.39 3 . × . × − CZTSe 1.00 0.24 1 . × . × − CZGSe 1.36 0.29 9 . × . × − AZTSe 1.35 1 . × CZTS:H 1.50 0.32 3 . × . × − CZTSe:H 1.00 0.17 1 . × . × − CZGSe:H 1.36 0.17 4 . × . × − AZTSe:H 1.35 0.08 1 . × . V nonradOC is the V OC loss due to the nonradiative recombination, p is the in-trinsic hole concentration, τ SRH is the ShockleyReadHall life-time and PLQY is the external photoluminescence quantumyield at 1-sun equivalent conditions.
III. CONCLUSIONS
We have combined the physics of solar cells with mod-ern first-principles defect theory to assess the efficiencylimit of solar cells. We have included the thermal equi-librium concentrations of native defects of the absorbermaterial, which reduces carrier lifetime, and have pro-posed a first-principles method to calculate the maxi-mum efficiency limited by recombination centers. Sn-based kesterites suffer from severe nonradiative recombi-nation due to native point defects. The fast nonradiativerecombination can be mitigated by extrinsic doping andAg-alloying, reducing the concentration of recombinationcentres, thereby increasing the performance threshold to29%.Although, our approach advances first-principles ap-proaches for solar cells, its limitations should be noted.We are pushing defect theory to its limits of applicabil-ity and note that inaccuracies, e.g. through finite-sizedcorrections or choice of exchange-correlation functional,will become magnified in the predictions of defect con-centrations and capture cross-sections. The method in-herits some of the limitations of the SQ approach. It isbased on bulk properties are therefore does not take intoaccount surface or interface recombination. Parasitic ab-sorption effects in the buffer or window layers are alsoignored.In the case of kesterite solar cells, although it is widelyaccepted that a short carrier life is the main performancebottleneck, high series resistance can further reduceefficiency. Thin-films are often inhomogeneous with lat- eral variations in stoichiometry. Therefore, fluctuationsof the band gap and the electrostatic potential can reducethe open-circuit voltage beyond our predictions. The TLC metric should be considered as an upperbound, based on the bulk properties of the absorber, thatcan be achieved when losses through other degradationpathways are minimal. In commercial photovolatic solarcells, J SC and FF approach the SQ limit. The mainefficiency-limiting factor is V OC 71,73 , which we tackle.Therefore, our method can provide a new direction forsearching for promising photovoltaic materials by pro-viding a realistic upper limit on expected performance.It can be used as part of screening procedures to selectviable candidates. Finally, we emphasise that to assessthe genuine potential of real materials for photovoltaics,one should consider not only the thermodynamics of lightand electrons, but also the thermodynamics of crystals.
IV. DATA AVAILABILITY
The data that support the findings of this studyare available in Zenodo repository with the iden-tifier doi:10.5281/zenodo.XXXXXXX. The codeused in this study is available in Zenodo repositorywith the identifier doi:10.5281/zenodo.YYYYYYY.We make use of the open-source packages https://github.com/WMD-group/CarrierCapture.jl (capture cross-sections), https://github.com/jbuckeridge/sc-fermi (equilibrium concentrations)and https://github.com/jbuckeridge/cplap (chemi-cal fields).
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