Open Access Article

This Open Access Article is licensed under a

Creative Commons Attribution 3.0 Unported Licence

Kenneth P.
Marshall
^{a},
Shuxia
Tao
^{b},
Marc
Walker
^{c},
Daniel S.
Cook
^{a},
James
Lloyd-Hughes
^{c},
Silvia
Varagnolo
^{a},
Anjana
Wijesekara
^{a},
David
Walker
^{c},
Richard I.
Walton
^{a} and
Ross A.
Hatton
*^{a}
^{a}Department of Chemistry, University of Warwick, CV4 7AL, Coventry, UK. E-mail: Ross.Hatton@warwick.ac.uk
^{b}Center for Computational Energy Research, Department of Applied Physics, Technische Universiteit Eindhoven, P.O. Box 513 5600 MB, Eindhoven, The Netherlands
^{c}Department of Physics, University of Warwick, CV4 7AL, Coventry, UK

Received
9th April 2018
, Accepted 4th May 2018

First published on 13th June 2018

We show that films of the 3-dimensional perovskite Cs_{1−x}Rb_{x}SnI_{3} can be prepared from room temperature N,N-dimethylformamide solutions of RbI, CsI and SnCl_{2} for x ≤ 0.5, and that for x ≤ 0.2 film stability is sufficient for utility as the light harvesting layer in inverted photovoltaic (PV) devices. Electronic absorption and photoluminescence spectroscopy measurements supported by computational simulation, show that increasing x increases the band gap, due to distortion of the lattice of SnI_{6} octahedra that occurs when Cs is substituted with Rb, although it also reduces the stability towards decomposition. When Cs_{0.8}Rb_{0.2}SnI_{3} perovskite is incorporated into the model inverted PV device structure; ITO|perovskite|C_{60}|bathocuproine|Al, an ∼120 mV increase in open-circuit is achieved which is shown to correlate with an increase in perovskite ionisation potential. However, for this low Rb loading the increase in band gap is very small (∼30 meV) and so a significant increase in open circuit-voltage is achieved without reducing the range of wavelengths over which the perovskite can harvest light. The experimental findings presented are shown to agree well with the predictions of density functional theory (DFT) simulations of the stability and electronic structure, also performed as part of this study.

The V_{oc} in inverted perovskite PV devices depends strongly on a number of factors including the energetics at the perovskite-ETL interface^{7} and the degree of crystallinity in the ETL.^{11} Additionally, for PV devices using B-γ CsSnI_{3} as the light harvesting layer, we have shown that SnCl_{2} is an effective additive for reducing the density of tin vacancy defects in the band gap and reducing the reverse saturation current,^{8} both of which help to improve V_{oc}. However, V_{oc} is ultimately limited by the relatively small ionisation potential (I_{p}) of B-γ CsSnI_{3} of ∼4.9 eV,^{9,12} which is approximately half an electron volt smaller than that of lead halide perovskites.^{13} Fortunately, similar to lead halide perovskites, B-γ CsSnI_{3} is amenable to substitution of iodide ions with bromide ions, which results in an increase in I_{p} and band gap which translates to an increase in V_{oc} in PV devices.^{14} However, this benefit must be balanced against the inevitable reduction in short circuit current density (J_{sc}) resulting from fewer photons having sufficient energy to excite electrons across the larger band gap. An alternative strategy for tuning the electronic structure of B-γ CsSnI_{3} is substitution of A-site Cs cation with the smaller Rb cation to form Cs_{1−x}Rb_{x}SnI_{3}. Unlike the case of halide ion substitution, the orbitals of the A-site metal cation do not contribute directly to the conduction and valance band edges,^{4,15} but still indirectly affect the band gap as a result of the distortion of the lattice of SnI_{6} octahedra via tilting that occurs when reducing the size of the A-site ion.^{16,17} RbSnI_{3} has a tolerance factor of 0.840 which is very close to the value that allows for the likely formation of a 3D structure; 0.85,^{18} which is consistent with the fact that it has only been reported in the one-dimensional yellow phase.^{19}

Herein we report the facile room temperature preparation of thin films of 3-dimensional (3D) Cs_{1−x}Rb_{x}SnI_{3} and show how partial exchange of Cs with Rb can be used to increase I_{p} and band gap. Notably, for low levels of substitution it is possible to achieve a significant increase in I_{p} with only a very small increase in band gap which, in model inverted PV devices, results in a sizable increase in V_{oc} with no significant adverse impact on the light harvesting capability. The experimental findings presented are shown to agree well with the predications of density functional theory (DFT) simulations of the stability and electronic structure, also performed as part of this study.

Fig. 1 shows the X-ray diffraction (XRD) patterns for ∼80 nm thick Cs_{1−x}Rb_{x}SnI_{3} films supported on glass at room temperature, along with simulated patterns for B-γ CsSnI_{3} and RbSnI_{3}. It is evident from these patterns that for x = 0 and 0.2 the crystal structure is similar to that of B-γ CsSnI_{3}. For x = 0.5 the films were very unstable, with the colour evolving from a deep red to yellow in the few minutes taken to load the sample into the X-ray diffractometer.

Lattice parameter fitting; ESI† (Fig. S2), shows there is no significant or systematic change in lattice parameters between B-γ CsSnI_{3}, Cs_{0.9}Rb_{0.1}SnI_{3} and Cs_{0.8}Rb_{0.2}SnI_{3}, entirely consistent with the computational simulations of Jung et al.^{16} and DFT simulations performed as part of this study (ESI,† Fig. S3 and S4), both of which predict only a small change in lattice parameters between the 3D perovskite B-γ CsSnI_{3} and the hypothetical 3D RbSnI_{3}, due to a much higher degree of tilting between the corner sharing SnI_{6} octahedra in the latter. Simulation of the X-ray diffraction pattern based on a crystallographic information file of CsSnI_{3} in which 20% of Cs atoms have been replaced with Rb atoms shows that there should be some difference in the peak intensities (Fig. 2): the simulation predicts that the (101) and (020) reflections should have increased intensity relative to the (202) and (040) reflections when Cs is partially substituted with Rb. Comparing the measured patterns of Cs_{1−x}Rb_{x}SnI_{3} there is no significant trend in the relative ratios of the (101) + (020):(202) + (040) intensities, although both samples incorporating Rb have higher relative intensities of (101) and (020) compared with (202). Additionally, before the sample degraded the Cs_{0.5}Rb_{0.5}SnI_{3} sample has a very intense peak at 14.5° which is assigned to (101) and (020) Miller planes for perovskite material. By comparison, the fully degraded Cs_{0.5}Rb_{0.5}SnI_{3} has peaks at 10.0° and 13.3°, characteristic of the yellow phase.^{2,19} It should be noted that preferred orientation effects may also modify the relative intensities of reflections, indeed we have previously shown that B-γ CsSnI_{3} films deposited in the same way as used in the current study can exhibit substrate specific preferred crystallite orientation.^{9} Since the degree of orientation may also conceivably be affected by the compositions of the samples this analysis cannot be quantified.

Fig. 2 XRD patterns of Cs_{1−x}Rb_{x}SnI_{3} where x = 0, 0.1, or 0.2, with simulated patterns of B-γ CsSnI_{3}^{2} and of CsSnI_{3} in which 20% of Cs atoms are replaced by Rb. |

To determine how the stability of the 3D perovskite films in ambient air depends on Rb content, the evolution of the absorption spectrum of Cs_{1−x}Rb_{x}SnI_{3} films with a thickness of ∼50 nm on glass substrates was monitored as a function of time in ambient air. It is evident from Fig. 3 that incorporation of Rb into the lattice destabilises the film towards oxidation in air since there is a faster degradation of the absorption spectra with time with increasing Rb content, although for low Rb content (10%) this effect is relatively small. The observed reduction in stability with increasing substitution of Cs with Rb is consistent with the results of DFT calculation (Fig. 4), which show that the formation energy of the yellow phase of Cs_{1−x}Rb_{x}SnI_{3} from the 3D perovskite is slightly exothermic and becomes increasingly exothermic with increasing Rb content, facilitating the formation of the yellow phase. DFT simulations also show that the yellow phase of Cs_{1−x}Rb_{x}SnI_{3} will spontaneously convert to (Cs_{1−x}Rb_{x})_{2}SnI_{6} upon exposure to O_{2} in air due to the large negative formation energies of the oxidised products. The reduction in stability with increasing Rb content is attributed to the increased octahedral tilting that results from substitution of Cs by Rb, leading to increased strain. Direct evidence for increased octahedral tilting of the lattice upon incorporation of Rb, based on measurement of the optical band gap and DFT simulation, is discussed below. Corroborating evidence for a distortion of the Sn–I–Sn bonds with increasing Rb content is also provided by core level photoelectron spectroscopy; ESI,† Fig. S5(a), which shows a continuous increase in the binding energy of the Sn3d peaks with increasing Rb substitution. Notably, scanning electron microscopy images of films with x = 0 and x = 0.2 and x = 0.5 (ESI,† Fig. S7) show that the film porosity significantly increases with increasing Rb content, which may also partially account for the differences in film stability, since more porous films have an increased surface area to volume ratio.

Fig. 3 (a–e) UV/vis/NIR spectra as a function of time in air (measurements made every 5 minutes) for Cs_{1−x}Rb_{x}SnI_{3}:SnI_{2} with x = 0, 0.1, 0.2, 0.3, and 0.5. (f) Evolution of normalised absorbance at a wavelength of 500 nm for data shown in (a–e) as a function of time in ambient air. Data for x = 40 are given in ESI,† Fig. S6. |

Fig. 5 shows the photoluminescence spectra for encapsulated Cs_{1−x}Rb_{x}SnI_{3} films with increasing Rb content, from which it is evident that the band gap increases with increasing x from ∼1.34 eV to ∼1.50 eV. Films with x = 0.5 were found to be unstable even with encapsulation under nitrogen and so the spectrum shown in Fig. 5 is a sample before complete degradation. The increase in band gap with increasing Rb substitution is consistent with the electronic structure calculations performed as part of this study using the DFT-1/2 method: Fig. 6. The DFT-1/2 method has the advantage of improved accuracy in band gap calculation by introducing a half-electron/half-hole occupation, and has been successfully applied for the accurate prediction of band gaps of several metal halide perovskites including CsSnI_{3}.^{20} The smaller size of Rb compared with Cs increases tilting of the SnI_{6} octahedra, which reduces Sn–I orbital overlap leading to a larger band gap.^{16,17} The increase in band gap is also compelling evidence that phase separation between domains of RbSnI_{3} and B-γ CsSnI_{3} does not occur, since RbSnI_{3} is predicted to have a much larger band gap than B-γ CsSnI_{3} (Fig. 6 and ref. 16) and so the photoluminescence spectrum of a film with phase separated domains of RbSnI_{3} and B-γ CsSnI_{3} would still have a significant peak at 1.34 eV. It is evident from Fig. 6 that for x = 0.2 the measured band gap is significantly smaller than predicted. This disparity is attributed to the relatively small structural model used in the DFT calculations which will overestimate the distortion of the lattice and the octahedral tilting for small x; ESI,† Fig. S3 and S4, giving rise to an overestimate of the band gap. However, the simulation correctly predicts the trend of increasing band gap with increasing Rb content, and for larger x the simulation and experiment are quantitatively in close agreement.

Further evidence of the uniform inclusion of Rb into the perovskite lattice is provided by the Cs:Rb elemental ratio estimated from the XPS peak intensities: ESI,† Table S1. The Cs:Rb ratios for those compositions that result in a stable perovskite structure (i.e. Rb ≤ 50% substitution) are in close agreement with the ratio used in the preparative solution. Since 95% of the XPS signal originates from the top ∼8 nm of the perovskite film,^{8} this can only be the case if the Rb is uniformly incorporated into the perovskite lattice. Taken together with the photoluminescence spectroscopy, valence band photoelectron spectroscopy and prediction of simulation, this provides compelling evidence for the inclusion of Rb in to the perovskite lattice.

The potential of these materials as the light harvester in PV devices was tested in the model inverted device architecture: ITO|Cs_{1−x}Rb_{x}SnI_{3}:10 mol% SnI_{2}|C_{60}|BCP|Al, for x = 0, 0.2, or 0.5. Whilst using C_{60} as the ETL is known to give a lower V_{oc} than can be achieved using PCBM, due to its lower lying lowest unoccupied molecular orbital (LUMO),^{21} it was used in the first instance because C_{60} can be deposited in a very controlled and highly reproducible way by vacuum deposition, rendering it well suited to this fundamental study. We have recently shown that B-γ CsSnI_{3} PV devices with an inverted planar device architecture exhibit the best efficiently and stability when not using a hole-transport layer.^{8,9} For this reason we have used this simplified device architecture as a test bed for these new perovskite materials that are closely related to B-γ CsSnI_{3}.

Fig. 7 shows representative current–voltage (J–V) characteristics in the dark and under 1 sun simulated illumination. The full data set is given in ESI,† Table S2. Most striking is the large increase in open-circuit voltage (V_{oc}) with increasing Rb content, which increases by ∼50% from 0.31 V to 0.48 V when the Rb content is increased from x = 0 to x = 0.5. Given that Rb inclusion into the B-γ CsSnI_{3} lattice increases the band gap, the simplest explanation for the increase in V_{oc} is a commensurate increase in the perovskite I_{p}, since for this device architecture the maximum V_{oc} is expected to scale with the energy difference between the valence band edge in the perovskite and the LUMO level of the fullerene ETL. To verify this hypothesis the change in I_{p} of Cs_{1−x}Rb_{x}SnI_{3} samples with increasing x was measured using ultra-violet photoelectron spectroscopy: Fig. 6 and ESI,† Fig. S8. From these measurements it is not possible to determine the absolute I_{p} in each case, because the excess SnI_{2} used during the preparation of the perovskite films is accumulated at the film surface^{8} where it inevitably modifies the surface potential contribution to the measured I_{p}.

However, the direction and magnitude of the change in I_{p} can be deduced on the assumption that the excess SnI_{2} is distributed in a similar way in all of the perovskite films, giving rise to a comparable perturbation of the surface potential contribution to the I_{p} measurement for all of the samples. Fig. 6 shows that substitution of Cs with Rb lowers the energy of the valence band edge with respect to the vacuum level (i.e. increases I_{p}) and the magnitude scales with increasing Rb content. Indeed, the magnitude of the change in I_{p} with increasing Rb content correlates closely with the increase in device V_{oc}.

The change in band gap with Rb inclusion into the perovskite lattice is also evident from the change in device external quantum efficiency (EQE) spectra: Fig. 7(lower). It is estimated from the EQE spectrum for x = 0 and 0.2 (ESI,† Fig. S9(a)) that the band gap is increased by ∼ 20 meV for x = 0.2 which is in close agreement with the photoluminescence measurements (Fig. 5). Given that the magnitude of this increase is comparable to the thermal energy of an electron at room temperature it is barely significant, and so for x = 0.2 the energy of both the valence and conduction band edges must be decreased by approximately the same amount, enabling a significant increase in V_{oc} of ∼120 meV with only a very small increase in band gap. For x = 0.5 the increase in band gap compared with x = 0 becomes significant, increasing by ∼160 meV from ∼1.34 to ∼1.5 eV. For x = 0.5 the change in energy of the low energy edge in the EQE spectrum is, again, in close agreement with the photoluminescence measurements. The reduction in device J_{sc} with increasing x can be partially explained by the differences in perovskite film coverage, which reduces from ∼98% for x = 0, to ∼94% for x = 0.2 and ∼84% for x = 0.5 (Fig. S7, ESI†). For x = 0.5 the significant reduction in band gap is also a plausible reason for the reduction in J_{sc}, since fewer long wavelength photons can be harvested.

To test the generality of this result, PV devices were also fabricated using perovskite films prepared using SnCl_{2} as the source of excess Sn and a PCBM ETL in place of C_{60} (Fig. 8), since we have previously shown that B-γ CsSnI_{3} devices using SnCl_{2} in conjunction with PC_{61}BM achieves substantially higher fill factor and V_{oc}, which is most pronounced after a few days storage in an inert atmosphere.^{8}

Fig. 8 (upper) J–V characteristics of PV devices with the structure ITO|Cs_{1−x}Rb_{x}SnI_{3} + 10 mol% SnCl_{2}|PC_{61}BM|BCP|Al (x = 0, 0.2 or 0.5) tested in the dark and under 1 sun simulated solar illumination immediately after fabrication (solid lines) and after 12 days storage under nitrogen (<1 ppm O_{2} and H_{2}O) (dashed lines). Full data set given in ESI,† Table S3. (lower) Corresponding EQE spectra. |

It is evident from the data in Fig. 8 that the correlation between V_{oc} and the Rb content for devices using SnCl_{2} as the source of excess Sn is consistent with that observed for devices using SnI_{2}. For freshly made devices, the magnitude of the increase is in very close agreement (Table S3, ESI†): using SnI_{2} with x = 0.2 and 0.5 the increase in V_{oc} is ∼120 mV and ∼170 mV respectively. Using SnCl_{2} with x = 0.2 and 0.5 the increase in V_{oc} is ∼130 mV and ∼160 mV respectively. Interestingly, Fig. 8 and Table S3 (ESI†) show that the difference in V_{oc} for devices with and without Rb incorporated into the perovskite lattice reduces from ∼130 mV to ∼90 mV after 12 days storage in an inert atmosphere due to an increase in the V_{oc} of devices using CsSnI_{3}, which does not occur in devices incorporating Rb. The reason why the V_{oc} in devices with Rb incorporated into the perovskite lattice does not also exhibit an increase in V_{oc} with storage time is not yet understood. However, it is important to note that for the case of Cs_{1−x}Rb_{x}SnI_{3} prepared using SnCl_{2} as the source of excess Sn, the material system is complicated by the possible presence of Cs_{1−x}Rb_{x}SnI_{3−y}Cl_{y} and/or phase separated RbSnCl_{3}. Whilst the difference in J_{sc} for devices with 0% and 20% Rb substitution is small, consistent with the very small difference in band gap, the J_{sc} (Table S3, ESI†) for x = 0.5 Rb substitution is greatly reduced. The latter is attributed to partial decomposition of the perovskite film in situ in the device, since even with encapsulation under nitrogen films with x = 0.5 are unstable.

Immediately after UV/O_{3} treatment the slides were transferred into a dry nitrogen filled glovebox for CsSnI_{3} film deposition, followed by deposition of a PC_{61}BM film from 15 mg ml^{−1} chlorobenzene solution using a spin speed of 1500 rpm, or C_{60} which was deposited by thermal evaporation at a rate of 0.4 Å s^{−1} to a thickness of 40 nm. This was followed by thermal evaporation of 6 nm at 0.5 Å s^{−1} of bathocuproine (BCP) and then 50 nm of Al at 1 Å s^{−1}. Thermal evaporation was performed at a pressure of ≤1 × 10^{−5} mbar (with substrate rotation). The Al electrode was deposited through a shadow mask to make six devices per slide, each with an area of 6 mm^{2}.

J–V and EQE measurements were made using custom LabVIEW programs.

Powder X-ray diffraction patterns were analysed using the TOPAS software^{23} with the Pawley method used to fit the measured profile and refine lattice parameters.

UPS was performed in the same vacuum system as for XPS using a He 1α source at 21.22 eV.

DFT–LDA underestimates the lattice parameters of the orthorhombic CsSnI_{3} by about 3%. The deviations of the band gaps (compared to those with experimental lattice constants) are in the range of Cs_{1−x}Rb_{x}SnI_{3} about 250 meV to 350 meV when using LDA-optimized lattice constants. To be consistent, all the electronic structure calculations were performed with corrected lattice parameters by expanding the lattice parameters proportionally (to match experimental volume of the cells) while keeping the LDA-optimized shape of the cells. This procedure has shown to keep the ab initio aspects of the approach without compromising accuracy. The subsequent electronic structure calculations were performed using the DFT-1/2 method. The DFT-1/2 method stems from Slater's proposal of an approximation for the excitation energy, a transition state method,^{28,29} to reduce the band gap inaccuracy by introducing a half-electron/half-hole occupation. Ferreira et al.^{30} extended the method to modern DFT and particularly to solid-state systems. Recently, Tao et al.^{20} has reported the successful application of the DFT-1/2 method in predicting accurate band gaps of several metal halide perovskites including CsSnI_{3} with a calculated band gap of 1.33 eV. Fortunately, the computational effort is the same as for standard DFT, with a straightforward inclusion of spin–orbit coupling when coupled with VASP. In this work, the DFT-1/2 method with the same setting is used to looking at band gap evolution of Cs_{1−x}Rb_{x}SnI_{3} by substituting Cs by Rb. The DFT simulations in Fig. 4 assume the following reaction pathway: 2Cs_{1−x}Rb_{x}SnI_{3} + O_{2} → SnO_{2} + (Cs_{1−x}Rb_{x})_{2}SnI_{6}.

- B. Wu, Y. Zhou, G. Xing, Q. Xu, H. F. Garces, A. Solanki, T. W. Goh, N. P. Padture and T. C. Sum, Adv. Funct. Mater., 2017, 27, 1604818 CrossRef.
- I. Chung, J.-H. Song, J. Im, J. Androulakis, C. D. Malliakas, H. Li, A. J. Freeman, J. T. Kenney and M. G. Kanatzidis, J. Am. Chem. Soc., 2012, 134, 8579–8587 CrossRef PubMed.
- Z. Chen, C. Yu, K. Shum, J. J. Wang, W. Pfenninger, N. Vockic, J. Midgley and J. T. Kenney, J. Lumin., 2012, 132, 345–349 CrossRef.
- L.-Y. Huang and W. R. L. Lambrecht, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 88, 165203 CrossRef.
- M. H. Kumar, S. Dharani, W. L. Leong, P. P. Boix, R. R. Prabhakar, T. Baikie, C. Shi, H. Ding, R. Ramesh, M. Asta, M. Graetzel, S. G. Mhaisalkar and N. Mathews, Adv. Mater., 2014, 26, 7122–7127 CrossRef PubMed.
- T.-B. Song, T. Yokoyama, S. Aramaki and M. G. Kanatzidis, ACS Energy Lett., 2017, 2, 897–903 CrossRef.
- K. P. Marshall, R. I. Walton and R. A. Hatton, J. Mater. Chem. A, 2015, 3, 11631–11640 Search PubMed.
- K. P. Marshall, M. Walker, R. I. Walton and R. A. Hatton, Nat. Energy, 2016, 1, 16178 CrossRef.
- K. P. Marshall, M. Walker, R. I. Walton and R. A. Hatton, J. Mater. Chem. A, 2017, 5, 21836–21845 Search PubMed.
- Q. Wang, X. Zheng, Y. Deng, J. Zhao, Z. Chen and J. Huang, Joule, 2017, 1, 371–382 CrossRef.
- Y. Yan, Nat. Energy, 2016, 1, 15007 CrossRef.
- I. Chung, B. Lee, J. He, R. P. H. Chang and M. G. Kanatzidis, Nature, 2012, 485, 486–489 CrossRef PubMed.
- J. Endres, D. A. Egger, M. Kulbak, R. A. Kerner, L. Zhao, S. H. Silver, G. Hodes, B. P. Rand, D. Cahen, L. Kronik and A. Kahn, J. Phys. Chem. Lett., 2016, 7, 2722–2729 CrossRef PubMed.
- D. Sabba, H. K. Mulmudi, R. R. Prabhakar, T. Krishnamoorthy, T. Baikie, P. P. Boix, S. Mhainsalkar and N. Mathews, J. Phys. Chem. C, 2015, 119, 1763–1767 Search PubMed.
- L. Lang, J.-H. Yang, H.-R. Liu, H. J. Xiang and X. G. Gong, Phys. Lett. A, 2014, 378, 290–293 CrossRef.
- Y. K. Jung, J. H. Lee, A. Walsh and A. Soon, Chem. Mater., 2017, 29, 3181–3188 CrossRef PubMed.
- R. Prasanna, A. Gold-Parker, T. Leijtens, B. Conings, A. Babayigit, H. G. Boyen, M. F. Toney and M. D. McGehee, J. Am. Chem. Soc., 2017, 139, 11117–11124 CrossRef PubMed.
- C. Li, X. Lu, W. Ding, L. Feng, Y. Gao and Z. Guo, Acta Crystallogr., Sect. B: Struct. Sci., 2008, 64, 702–707 CrossRef PubMed.
- G. Thiele and B. R. Serr, Z. Kristallog., 1995, 210, 64 Search PubMed.
- S. X. Tao, X. Cao and P. A. Bobbert, Sci. Rep., 2017, 7, 14386 CrossRef PubMed.
- B. W. Larson, J. B. Whitaker, X.-B. Wang, A. A. Popov, G. Rumbles, N. Kopidakis, S. H. Strauss and O. V. Boltalina, J. Phys. Chem. C, 2013, 117, 14958–14964 Search PubMed.
- C. F. Macrae, I. J. Bruno, J. A. Chisholm, P. R. Edgington, P. McCabe, E. Pidcock, L. Rodriguez-Monge, R. Taylor, J. Van De Streek and P. A. Wood, J. Appl. Crystallogr., 2008, 41, 466–470 CrossRef.
- A. A. Coelho, J. Appl. Crystallogr., 2000, 33, 899–908 CrossRef.
- W. Kohn and L. J. Sham, Phys. Rev., 1965, 140, A1133 CrossRef.
- G. Kresse and J. Furthmüller, Comput. Mater. Sci., 1996, 6, 15–50 CrossRef.
- J. P. Perdew and A. Zunger, Phys. Rev. B: Condens. Matter Mater. Phys., 1981, 23, 5048–5079 CrossRef.
- G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758–1775 CrossRef.
- J. C. Slater, Advances in Quantum Chemistry, Academic Press, 1972, vol. 6, pp. 1–92 Search PubMed.
- J. C. Slater and K. H. Johnson, Phys. Rev. B: Condens. Matter Mater. Phys., 1972, 5, 844–853 CrossRef.
- L. G. Ferreira, M. Marques and L. K. Teles, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 125116 CrossRef.

## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8qm00159f |

This journal is © the Partner Organisations 2018 |