Issue 
EPJ Photovolt.
Volume 8, 2017
Topical Issue: Theory and modelling



Article Number  85504  
Number of page(s)  8  
Section  Theory and modelling  
DOI  https://doi.org/10.1051/epjpv/2017005  
Published online  16 June 2017 
https://doi.org/10.1051/epjpv/2017005
High absorption coefficients of the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys enable highefficient 100 nm thinfilm photovoltaics
^{1} Department of Materials Science and Engineering, KTH Royal Institute of Technology, SE100 44 Stockholm, Sweden
^{2} Centre for Materials Science and Nanotechnology, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway
^{3} Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway
^{a}
email: Rongzhen.Chen@gmail.com
Received: 29 January 2017
Accepted: 4 May 2017
Published online: 16 June 2017
We demonstrate that the bandgap energies E_{g} of CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} can be optimized for high energy conversion in very thin photovoltaic devices, and that the alloys then exhibit excellent optical properties, especially for tellurium rich CuSb(Se_{1−x}Te_{x})_{2}. This is explained by multivalley band structure with flat energy dispersions, mainly due to the localized character of the Sb/Bi plike conduction band states. Still the effective electron mass is reasonable small: m_{c} ≈ 0.25m_{0} for CuSbTe_{2}. The absorption coefficient α(ω) for CuSb(Se_{1−x}Te_{x})_{2} is at ħω = E_{g} + 1 eV as much as 5–7 times larger than α(ω) for traditional thinfilm absorber materials. Auger recombination does limit the efficiency if the carrier concentration becomes too high, and this effect needs to be suppressed. However with high absorptivity, the alloys can be utilized for extremely thin inorganic solar cells with the maximum efficiency η_{max} ≈ 25% even for film thicknesses d ≈ 50 − 150 nm, and the efficiency increases to ∼30% if the Auger effect is diminished.
R.Z. Chen and C. Persson, published by EDP Sciences, 2017
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Chalcopyrite Cu(In,Ga)Se_{2} (CIGS) has become a commercialized absorber for thinfilm photovoltaics (PV), and laboratory devices has reached the solar conversion efficiency of η = 22.6% by the ZSW in BadenWürttemberg [1]. Cu_{2}ZnSn(S,Se)_{4} (CZTSSe) has the last decade attracted much attention, and the material is considered as a promising emerging inorganic PV material. CZTSSe has not only earthabundant and lowcost elements but, similar to CIGS, also tunable bandgap energy and large absorption coefficient. The maximum efficiency for CZTSSe based solar cell is today 12.6% by the IBM in New York [2]. However, the device efficiency together with the material usage and manufacturing costs of CIGS and CZTSSe technologies must be considered. The total world energy consumption is estimated to be double by the year of 2050 referring to the current level [3]. In order to meet these future needs, it is important to explore novel materials as well as to develop different types of solar energy technologies, especially for large scale PV production. Finding alternative absorber materials for lowcost PV technologies is an ongoing research. CIGS and CZTSSe devices are produced with the absorber thickness of typically d ≈ 1000nm. Devices with thinner films reduce the material usage as well as decrease the path for carrier collection, but thinner films imply less absorption of the sunlight and that effect reduces the conversion efficiency considerably as the absorptivity depends exponentially on the thickness. Therefore, developing “ultrathin” (d< 100nm) inorganic PV devices requires also tailormaking new materials with higher absorption coefficients. One way to enable ultrathin devices is to utilize nanostructures, like lightscattering nanoparticles or quantum dots, but that involves additional growth processes. Instead, using traditional pn junction technologies, a higher capacity to absorb light is required. Cu(Sb/Bi)(S/Se)_{2} compounds (i.e., CuSbS_{2}, CuSbSe_{2}, CuBiS_{2}, and CuBiSe_{2}) have been demonstrated to have much higher absorption coefficients than CIGS and CZTSSe alloys [4, 5, 6, 7, 8, 9, 10]. These Sb/Bi containing chalcostibites have fairly inexpensive and earthabundant elements, and the materials are thus interesting for thin PV technologies. The compounds have indirectgap energies, however the difference between indirect and direct bandgap energies is only 0.1–0.3 eV due to the rather flat lowest conduction band (CB) [4, 5, 11]. The experimental values of the bandgaps energies are ~1.5 eV [12], ~1.65 eV [13], and ~1.09 eV [14] for CuSbS_{2}, CuBiS_{2}, and CuSbSe_{2}, respectively. No experimental gap is available for CuBiSe_{2}, but the calculated gap energy is ~1.13 eV [4, 11]. The conversion efficiency is around 3% for devices based on CuSbS_{2} or CuSbSe_{2}[15, 16], so further investigations of the materials are required as well as development of the devices.
In this work, the electronic and optical properties of CuSb(Se_{1−x}Te and CuBi(S_{1−x}Se for the alloy compositions x = 0, 0.25, 0.50, 0.75 and 1 are explored theoretically, using hybrid functional approach based on the density functional theory (DFT). The maximum conversion efficiency η_{max} is calculated with the ShockleyQueisser (SQ) model, using the DFT absorption coefficient α(ω) and considering different thicknesses d of the absorber films. The calculations reveal that all considered compounds have indirect gaps, and the fundamental gap energies are between 0.9 and 1.3 eV. By alloying the gap energies can be tuned in for optimized device efficiency. The lowest CBs are rather flat with a large DOS at the CB edge, especially for the CuSb(Se,Te)_{2} alloys, and the DOS is large also near the valence band (VB) edge. Still, however, the effective electron masses are relatively small for both CuSbTe_{2} and CuBiSe_{2}. Due to the flat energy dispersion, the absorption efficiencies are much higher in these alloys compared with those of CIGS and GaAs, especially for photon energies in the energy region E_{g} to E_{g} + 1eV; this region is of special importance for the efficiency in thin film PV technologies. Moreover, we find that the absorption for CuSb(Se_{1−x}Te with x = 0.50 − 0.75 is slightly higher than for the other alloys. These tellurium containing alloys have also suitable relation between the direct and the fundamental indirect E_{g} band gaps, implying that these alloy compositions will have highest maximum conversion efficiency of the considered alloys. For these indirectgap materials we find that the Auger effect plays an important role, where small variations in the intrinsic carrier concentration easily can change the efficiency by (2–4)%. Due to the Auger effect the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys have some 5% smaller efficiency compared to directgap materials for thick films (d> 1000nm). However, for very thin films (d< 400nm) the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys have significantly higher efficiency than the traditional absorbers. The CuSb(Se,Te)_{2} alloys remain a maximum efficiency of η_{max} ≈ (23 − 28)% for d = 50 − 200nm. Thus, CuSb(Se_{1−x}Te and CuBi(S_{1−x}Se can be alternative compounds for ultrathin inorganic PV devices.
2 Computational methods
The calculations of the electronic structure and the optical properties were based on a groundstate DFT, employing the projector augmented wave basis set of the VASP package [17, 18, 19] and a hybrid functional (HSE06) [20]. The alloys (x = 0, 0.25, 0.50, 0.75, and 1) were modelled by small 16atom unit cells, constructed from the ternary compounds with the space group Pnma (no. 62). We justify the small cells of the alloys by the more accurate HSE06 calculation of the optical properties regarding band dispersion and the kspace integration. The atomic positions and the lattice constants were relaxed with the quasiNewton algorithm until a maximum force of 10 meV/Å of each atom was reached. The local, atomic and angularresolved DOS (LDOS) was obtained from the modified tetrahedron integration method with a Γcentered 5 × 8 × 2kmesh in the irreducible Brillouin zone. We excluded the spin orbit coupling (SOC) because we have found that there no degeneracies at the band edges that can be split by the SOC. In addition, SOC narrows the bandgap energy by only 0.1–0.2 eV, it does not change the materials from being indirect to direct, and it has only a moderate effect on the DOS.
The optical property of the alloys was analyzed by the means of the absorption coefficient α(ω) = ωc^{1}(2  ε(ω)  − 2ε_{1}(ω))^{1 / 2}, determined directly from the complex dielectric function ε(ω) = ε_{1}(ω) + iε_{2}(ω), where c is the speed of light. Here, the tensor ε_{2}(ω) with components α and β (i.e., )) was calculated within the linear response of optical transition probability between occupied and unoccupied states (see Refs. [17, 18, 19] for details) and performing similar kspace integration as for the LDOS. Subsequently, ) was obtained via the KramersKronig transformation with a broadening of 3 meV for the equidistant energy grid with step size of 2 meV up to the maximum energy of 65 eV. All DFTbased calculations were performed with HSE06 and with a cutoff energy of 420 eV. A large kmesh can be important to describe details in the absorption coefficient [21, 22]. However, from test calculations we have found that the effect is not crucial for the considered materials, and we use HSE06 (which avoids incorrect resonance across the underestimated gaps for local potentials [23, 24]) and with the same kmesh as for the calculations of the LDOS.
The maximum energy conversion efficiency η_{max} = P_{max}/P_{in} of a singlejunction cell was modelled within the SQ theory [25] assuming air mass 1.5 solar spectral irradiance I_{sun}(ω) and an absorptivity A(ω) = 1 − exp( − 2·α(ω)·d) in the absorber film with thickness d. P_{in} is the Sun’s total irradiance which is ~1000 W/m^{2}. The maximum efficiency is reached for P_{max} = J_{opt}·V_{opt} where J_{opt} and V_{opt} are the current (per unit area) and the device voltage optimized for maximum power. Here, the current J and the device voltage V are balanced within the Shockley ideal diode equation [25, 26, 27]: ) where q is the elementary charge, k_{B} is the Boltzmann constant, and T_{c} is the device temperature set to 300 K. The first term on the right hand side is the maximum short circuit current J_{sc} = ℏ·q·_{∫}A(ω)·I_{sun}(ω)∂ω. The second term is the thermal recombination current which can involve both radiative and nonradiative recombinations. Since we analyze the maximum efficiency of the materials, recombinations via defects are not considered in this study.
Fig. 1 Electronic band structure and LDOS of (a) CuSbSe_{2}, (b) CuSbSeTe, i.e., 50% alloying, (c) CuSbTe_{2}, (d) CuBiS_{2}, (e) CuBiSSe, 50% alloying, and (f) CuBiSe_{2}. The energy refers to the VBM (dashed lines). The atom and angular momentum resolved densityofstates has been scaled with 1/(2l + 1) for better visualization, and it is presented with a 0.20 eV Lorentzian broadening. One can notice the strong Sb/Bi plike states in the lower energy region of the conduction bands. 
3 Results
The crystal structure of the ternary compounds CuSb(Se/Te)_{2} and CuBi(S/Se)_{2} was fully relaxed with the hybrid functional. The lattice constants are, as normally for HSE06, slightly overestimated but in agreement within 4% with available experimental results: a = 6.31Å, b = 3.97 Å, and c = 14.65 Å for CuSbSe_{2}, a = 6.76 Å, b = 4.26 Å, and c = 15.45 Å for CuSbTe_{2}, a = 6.16 Å, b = 3.88 Å, and c = 14.20 Å for CuBiS_{2}, and a = 6.47 Å, b = 4.08 Å, and c = 14.64 Å for CuBiSe_{2}. Relaxation of the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} quaternary alloys results in lattice parameters that follow, despite rather small unit cells, linearly within 0.5% with respect to the alloy composition x.
The electronic band structures of the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys (Fig. 1) show strikingly flat energy dispersions for both the CBs and VBs. Overall, all compounds have similar band structure with indirect fundamental band gap E_{g}, where CB minima (CBM) tend to be along (100) direction while VB maxima (VBM) is along the (001) direction. The main difference between the compounds is the energy of this fundamental gap. CuSbSe_{2} with the lighter groupV cation atom has gap energy E_{g} = 1.21 eV, which is 0.14 eV larger than for CuBiSe_{2} with E_{g} = 1.07eV. Also, the lighter groupVI anion Se atom in CuSbSe_{2} yields a 0.34 eV larger gap compared with CuSbTe_{2} with E_{g} = 0.87eV. Similarly, the lighter S atom in CuBiS_{2}(E_{g} = 1.30eV) yields a 0.23 eV larger gap compared with CuBiSe_{2}; this energy difference is somewhat smaller than corresponding value of ~0.5 eV for both Cu_{2}ZnSn(S/Se)_{4} and CuIn(S/Se)_{2} when comparing their groupVI anion atoms [28, 29]; heavier groupV atoms yields smaller energy difference. Thus, the heavier Bi atom yields the smallest variation of E_{g} with respect to anion alloying. This is true also when comparing CuSbSe_{2} with CuSbS_{2} (not presented in this work). The bandgap energies for the alloys vary fairly linearly with alloy composition x (Fig. 2); the small deviation from this is mainly due to cell sizes. In CuSb(Se,Te)_{2} the direct gap at the Γpoint is about 0.4 eV larger than the indirect gap E_{g}, while the corresponding energy difference is about 0.7 eV in CuBi(S,Se)_{2}. In both the CuSb(Se,Te)_{2} alloys and the CuBi(S,Se)_{2} alloys, the direct gap energy is only ~0.2 eV larger than the fundamental indirect E_{g}. This is a rather small energy difference, and that is important for enabling efficient photovoltaics. The direct gap governs the sunlight absorption and mainly also the radiative recombination, while the indirect gap governs the Auger nonradiative recombination. Therefore, optimizing the directgap energy by forming bands with small energy difference is the route for tailormaking highefficient PV materials with large joint DOS for energies just above . The values of the bandgap energies are summarized in Table 1.
Fig. 2 Energies of the fundamental gap (circles), the direct gap (rhomboids), and the direct Γpoint gap (squares) for the CuSb(Se,Te)_{2} alloy (red marks) and the CuBi(S,Se)_{2} alloy (blue marks). 
The fundamental (E_{g}), direct (, and Γpoint ( gap energies, as well as the highfrequency dielectric constant (ε_{∞}); we present the average dielectric constant since the anisotropy is rather small. The maximum solar conversion efficiency (η_{max}) is presented for two film thicknesses (d), and it includes both radiative and Auger recombinations (values in parenthesis are results when the Auger effect is neglected).
The error bar of the HSE06 bandgap energies is estimated to be typically about 0.1–0.2 eV. Our values agree well with earlier published calculated values for the Cu(Sb/Bi)(S/Se)_{2} ternaries [4, 5, 11]. The experimental gap energies are not fully established, but the available roomtemperature data [12, 13, 14] suggest that our theoretical values might be off by about 0.1 to 0.3 eV. However, since the directgap energies of these alloys are between about 1.0 and 1.6 eV and the optimized gap within the SQ theory is 1.2–1.4 eV, it is possible to find an alloy composition with the proper direct bandgap energy.
The LDOS in Figure 1 also demonstrates that all the compounds have comparable electronic structures (apart from the values of the gap energies). The CBs are dominated by relatively localized Sb/Bi plike states and that generates a strong DOS close to the CB edges. This effect yields rather flat lowest CBs. Flat energy dispersion can imply large effective masses which are disadvantageous for the lifetime of the carries. However, the effective electron masses at the CBM are relatively small for these compounds, for instance: m_{x} ≈ 0.23m_{0}, m_{y} ≈ 0.16m_{0}, and m_{z} ≈ 0.40m_{0} for CuSbTe_{2}, and m_{x} ≈ 0.19m_{0}, m_{y} ≈ 0.83m_{0}, and m_{z} ≈ 0.50m_{0} for CuBiSe_{2}. These values are comparable to the corresponding calculated effective masses for CIGS ((0.08–0.13)m_{0}[30]) and Si (m_{x} = m_{y} ≈ (0.19 − 0.22)m_{0}; m_{z} ≈ (0.96 − 1.02)m_{0}[23, 24]), while considerably larger than that of GaAs ((0.06−0.09)m_{0}[23, 24]). Interestingly, two of the mass components (m_{x} and m_{z}) are similar for the two compounds, while the third compound (i.e., m_{y}) is significantly different. This may indicate that small variation in the crystal structure (e.g., strain, defects in resonance with the CBs) can easily change the band dispersion and thereby change the values of the effective masses, and this can thus be a way to optimize the material properties with respect to the effective masses. It is worth mentioning that the mass values are valued only in the vicinity of the CBM where the band is parabolic (at most 50 meV above CBM in these materials). However, even with small effective electron mass of ~0.1m_{0} the quasiFermi level is only ~20 meV above the CBM even for a high free electron concentration of 10^{17}cm^{3}[31]. Furthermore, the VBs in these Cubased chalcostibites have the characteristic Cud–anionp hybridization that helps forming localized band structures near the valence band edges. Since the topmost VBs are flat, also the DOS near the valenceband edges are strong if one compare with the DOS of CIGS and CZTSSe. Thus, the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} compounds have flat CBs and VBs, and that is an advantage for a strong onset of the optical absorption.
Fig. 3 (a) The absorption coefficients α(ω) of CuSbSe_{2}, CuSbSeTe, CuSbTe_{2}, CuBiS_{2}, CuBiSSe, and CuBiSe_{2} on the energy scale ( and with a 0.30 eV Lorentzian broadening. These compounds exhibit much stronger neargapenergy absorption compared to the traditional absorbers GaAs (E_{g}~1.41 eV with HSE06) and CIGSe (50% alloying: E_{g}~1.30 eV). (b) The corresponding maximum solar energy efficiency η_{max} demonstrates high efficiency of CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys for very thin films of d< 400 nm (see also Tab. 1). 
Indeed, the absorption coefficients for these compounds (Fig. 3a) are significantly larger than corresponding calculated results for the traditional PV absorbers CIGS and GaAs; the figure shows α(ω) on the energy scale in order to easier compare materials with different gap energies. This larger α(ω) is especially notable in the energy region to eV. For instance, at the photon energy eV, the CuSb(Se,Te)_{2} alloys have ~7 times larger absorption coefficients than those of CIGS and GaAs. One notices also that the CuSb(Se,Te)_{2} alloys have slightly better absorption than the CuBi(S,Se)_{2} alloys, however considering also the proper bandgap energies (Fig. 2) both the CuSb and CuBibased alloys have sufficiently good optical properties for thin film applications. The highfrequency dielectric constant (ε_{∞}; see Tab. 1) is rather high, above 10 for all compounds; this can be compared with our calculated value 10.0 for GaAs and 7.0 for CIGS. The CuSb(Se,Te)_{2} alloys have higher values (13.0 ≤ ε_{∞} ≤ 17.2) compared with the CuBi(S,Se)_{2} alloys (10.5 ≤ ε_{∞} ≤ 12.7), and as expected the dielectric constant increases with larger alloy composition x, as the bandgap energy decreases. The related refractive index is a material property that needs to be considered when designing device with these compounds.
From the absorption coefficients and the gap energies, the device efficiency for the absorber material with thickness d is determined within the SQ theory; see previous section. The maximum energy conversion efficiency η_{max} is modelled by the Shockley ideal diode equation, considering both radiative recombination and nonradiative Auger recombination: . The radiative recombinations depend on the blackbody spectrum of the device: . The Auger recombinations in a ptype absorber depends mainly on the intrinsic free carrier concentrations n_{i} and the acceptor concentration N_{A}: [26] where C_{a} is the total ambipolar Auger coefficient. This coefficient is rather independent of material (typically 10^{32}–10^{30}cm^{6}/s and it is in the order of 10^{31} for indirect transition in Si and Ge and GaAs [32, 33, 34, 35]. We want to estimate the Auger effect using an approximate model that is material independent, and we therefore set C_{a} = 10^{31}cm^{6}/s for all considered materials. The acceptor concentration is set to N_{A} = 10^{16}cm^{3} which is a normal doping concentration. The square of the intrinsic carrier concentration can be expressed as where n_{a} = 2·(m_{a}k_{B}T_{c}/ (2πℏ^{2}))^{3 / 2} with the effective DOS mass where γ_{e} is the number of CBM and is the effective DOS mass of a single CB minimum (and corresponding for the holes in the VB maximum). Again, to construct a material independent expression, we set m_{a} = 1m_{0}, which is a good value for indirect material like for instance Si (with six equivalent CBM) and somewhat overestimated for Ge (with three equivalent CBM). The value is however much larger than for direct materials, like GaAs and CIGS, but since the Auger effect is much weaker for these materials, using m_{a} = 1m_{0} also for these directgap materials does not affect the results in this work.
The maximum efficiency η_{max} for different film thicknesses d< 1000 nm is displayed in Figure 3b. One observes the typical drop in the efficiency for d< 500 nm for the traditional, directgap materials CIGS and GaAs. For d larger than ~500 nm CIGS and GaAs has higher efficiency than the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys. If the thickness increases to even further (i.e., d → ∞) then η_{max} will become independent of the absorption coefficient, and it will only depend on the gap energy with η_{max} = 33% for both CIGS and GaAs. Also the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys would have had similar high efficiency if they were directgap materials (e.g., then η_{max} would have been 34% for CuSbTe_{2} and 33% for CuBiSe_{2}). However, as being indirectgap materials the Auger effect decreases their efficiencies by about 3–5%. This decrease due to the nonradiative recombination effect is fairly constant with respect to thickness d, and therefore the Auger effect is less dominant for smaller thicknesses where instead the efficiency drop is due to the lower absorptivity. That is, for thin films with d< 500 nm, the absorption coefficient in the low photonenergy region (E_{g} to about E_{g} + 1 eV) is the important material property for retaining a high efficiency. One notices clearly that the CuSb(Se,Te_{2}) and CuBi(S,Se_{2}) alloys keep their efficiencies when the film thickness narrows to some 10–200 nm. Here, CuBiS_{2} has a too large gap energy (eV), and its efficiency is significantly lower than the other alloy compounds, but the other compounds have efficiencies close to 25% down to 100 nm. Further, the excellent optical properties of the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys yield remarkable high efficiency even for d = 50 − 100 nm, especially for the CuSbbased compounds. The maximum efficiency is presented in Table 1 for two specific film thicknesses. The CuSb(Se_{1−x}Te with x ≈ 0.5 − 0.75 has the highest efficiency. For instance η_{max} = 28% for d = 200nm and η_{max} = 24% for d = 50nm for x = 0.75. The reason for high efficiency of this alloy (i.e., CuSb(Se_{1−x}Te with x = 0.75) is that the energy difference is only 0.11 eV. Here, the smaller Δ_{g} for this alloy might be due to using too small unit cell, and further investigation is required before establishing a result. Nonetheless, it demonstrates the sensitivity of η_{max} with respect to small variation in the band structure. Moreover, the overall results demonstrate that CuSb(Se_{1−x}Te and CuBi(S_{1−x}Se can be alternative compounds for “ultrathin” (d< 100nm) inorganic photovoltaics based on traditional single p  n junction technique. The advantage with such thin film absorber layer is the shorter path for the minority carriers. However many other material and device properties related to defects/doping and interface physics have to be controlled and optimized.
The values within brackets for η_{max} in Table 1 show the efficiency if the Auger effect is neglected, demonstrating that this effect lowers the efficiency by ~4%. One can significantly reduce the Auger recombination by decreasing the energy difference Δ_{g}. If Δ_{g} goes from ~0.2 eV to 0 eV, then the Auger effect is suppressed by the radiative recombination that determine the voltage drop. Then the maximum efficiency increases by 3–5% and reaching the efficiency very close to having no Auger effect. Alternatively, the Auger effect can be reduced by minimizing the carrier concentration since . That can be achieved by using materials with large fundamental gap, but too large gap will be a disadvantage for the optical absorption. For ptype material, the efficiency is increased by ~(1–2)% if the doping concentration decreases to N_{A} = 10^{15}cm^{3}. However, for intrinsic materials, the Auger model becomes[26] that depends even stronger on Δ_{g} and on the voltage drop, and this can result in even smaller efficiency. Alternatively, the carrier concentration can be decreased by smaller effective DOS masses. The employed effective DOS mass in this work is m_{a} = 1m_{0} which is estimated to be somewhat too large, especially for CuSbTe_{2}. Instead, if using a smaller mass of m_{a} = 0.4m_{0} (and with Δ_{g} ≈ 0.2eV) the maximum efficiency of CuSbTe_{2} increases by ~1.5%. However, trying to optimize the electronic band structure to obtain even smaller DOS masses also means more dispersion of the band structure and thereby smaller absorption coefficient. Therefore, multivalley engineering (with large joint DOS and large α(ω)) balanced with bandedge engineering for small effective masses (with low carrier concentration and long lifetime) is the concept to tailormake materials for even thinner PV devices. Here, better models that include nonparabolic energy dispersion together with better description for the absorption/recombination are required for more accurate describing the material behavior.
4 Conclusions
We have analyzed the electronic and optical properties of CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys for highefficient thinfilm PV technologies. The directgap energy can be tuned from ~1.0 to 1.6 eV, and the fundamental indirectgap energy is typically 0.2 eV smaller. The materials exhibit excellent optical properties, with absorption coefficients ~6 times larger than for the traditional solar cell materials CIGS and GaAs in the low photon energy region. The main reason for the high absorption coefficients of these alloys is the strong Sb/Bi plike character in the lower energy region of the CBs, forming flat multivalley bands, and thereby yielding extraordinary strong absorption for photon with energies to eV. Interestingly, even though the lowest CBs are flat, the corresponding effective electron masses of the CBM are reasonable small for both CuSbTe_{2} (DOS mass is for a single CB minimum) and CuBiSe_{2} (≈0.43m_{0}). These DOS masses are heavier than for GaAs (~0.07m_{0}), but using thinner absorber layers can compensate the larger masses.
For these indirectgap materials we find that the Auger effect plays an important role. Small variations in the intrinsic carrier concentration can easily change the efficiency by ~3%. Due to the Auger effect the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys have some 4% smaller efficiency compared to directgap materials for relatively thick film (d> 1000nm). The affect due to the Auger effect depends on the balance between the diode’s voltage drop and the free carrier concentration. If the electron DOS mass is decreased from 1.0m_{0} to 0.4m_{0}, then the intrinsic carrier concentration decreases by a factor of 2 and the efficiency increases by about 1.5%. Moreover, if the energy difference between the indirect and directgap energies goes to zero, then the Auger effect is negligible and the efficiency is increased by (3–5)%. Also Yu et al. [36] discuss this relation for indirect versus directgap, using the SQ model, though with a slightly different model for the Auger effect. Thus, the concept to optimize the device efficiency relies on creating materials with flat multivalley band edges with direct energy gaps, where the carrier channels have high energy dispersions (i.e., small effective masses). Nonetheless, since the Auger effect is less sensitive to the film thickness, while the absorptivity depends strongly on the absorption coefficient for d< 500nm, the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys have significantly higher efficiency than the traditional absorbers for thinner film thicknesses. The CuSb(Se,Te)_{2} with composition x ≈ 0.5 − 0.75 has the largest absorption coefficient (α(ω) ≈ 1 × 10^{5}/cm already for the photon energy ħω = E_{g} + 0.4eV) and proper gap energies eV and E_{g} = 0.91 − 1.03eV, and this compound exhibits highest efficiency for thin film thicknesses of d< 400nm. The tellurium rich alloys remain an efficiency of η_{max} ≈ (25 − 27)% when film thickness decreases to d = 100 nm. Including the exciton effect in the calculation can improve the efficiency further by a stronger absorption coefficient, though in bulk materials the effect is expected to be weaker at operating temperature.
Altogether, due to the outstanding optical properties we believe that CuSb(Se_{1−x}Te and CuBi(S_{1−x}Se can be alternative compounds for “ultrathin” (d< 100nm) inorganic photovoltaics. However, concern regarding material quality and modifications in the band dispersion is needed in order to minimize the nonradiative recombination. That is, to tailormake materials for very thin, inorganic PV technologies one shall consider (i) an optimized bandgap energy for the specific film thickness; (ii) multivalley band edges; and (iii) rather flat energy dispersions to achieve high absorption coefficient for the low photon energies ħω = E_{g} to E_{g} + 0.5 eV. However, (iv) the band structure shall have direct fundamental gap to avoid Auger recombinations; thus both CBM and VBM shall then be located at the same kpoint but away from the Γpoint. (v) Small DOS electron and hole masses reduce the intrinsic carrier concentration, which in turn reduces the Auger effect. Preferable, the material (vi) shall have small masses for achieving high minority carrier mobility. In addition to this, the material shall of course, as for regular thinfilm PV, (vii) involve only earth abundant, nontoxic, and inexpensive elements, and the devices shall be inexpensive regarding fabrication and handling. The material must also be (viii) thermodynamic stable with no devastating ingap defect states, be able to be doped, and to have suitable band alignment at the interfaces.
Acknowledgments
We thank the Research Council of Norway (project 243642) as well as the Swedish Foundation for Strategic Research for financial support. We acknowledge access to HPC resources Abel at UiO in Norway operated by USIT, and allocation provided through NOTUR, Beskow at KTH operated by PDC, Triolith at LiU operated by NSC, both in Sweden and the allocation was provided through SNIC. We also acknowledge PRACE for awarding access to resource MareNostrum based in Spain at BSCCNS, access to the DECI resource Archer based in the UK Edinburgh with support from the PRACE aisbl.
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Cite this article as: Rongzhen Chen, Clas Persson, High absorption coefficients of the CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys enable highefficient 100 nm thinfilm photovoltaics, EPJ Photovoltaics 8, 85504 (2017).
All Tables
The fundamental (E_{g}), direct (, and Γpoint ( gap energies, as well as the highfrequency dielectric constant (ε_{∞}); we present the average dielectric constant since the anisotropy is rather small. The maximum solar conversion efficiency (η_{max}) is presented for two film thicknesses (d), and it includes both radiative and Auger recombinations (values in parenthesis are results when the Auger effect is neglected).
All Figures
Fig. 1 Electronic band structure and LDOS of (a) CuSbSe_{2}, (b) CuSbSeTe, i.e., 50% alloying, (c) CuSbTe_{2}, (d) CuBiS_{2}, (e) CuBiSSe, 50% alloying, and (f) CuBiSe_{2}. The energy refers to the VBM (dashed lines). The atom and angular momentum resolved densityofstates has been scaled with 1/(2l + 1) for better visualization, and it is presented with a 0.20 eV Lorentzian broadening. One can notice the strong Sb/Bi plike states in the lower energy region of the conduction bands. 

In the text 
Fig. 2 Energies of the fundamental gap (circles), the direct gap (rhomboids), and the direct Γpoint gap (squares) for the CuSb(Se,Te)_{2} alloy (red marks) and the CuBi(S,Se)_{2} alloy (blue marks). 

In the text 
Fig. 3 (a) The absorption coefficients α(ω) of CuSbSe_{2}, CuSbSeTe, CuSbTe_{2}, CuBiS_{2}, CuBiSSe, and CuBiSe_{2} on the energy scale ( and with a 0.30 eV Lorentzian broadening. These compounds exhibit much stronger neargapenergy absorption compared to the traditional absorbers GaAs (E_{g}~1.41 eV with HSE06) and CIGSe (50% alloying: E_{g}~1.30 eV). (b) The corresponding maximum solar energy efficiency η_{max} demonstrates high efficiency of CuSb(Se,Te)_{2} and CuBi(S,Se)_{2} alloys for very thin films of d< 400 nm (see also Tab. 1). 

In the text 
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