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Spin–orbit coupling effect on energy level splitting and band structure inversion in CsPbBr3

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Abstract

The band structures and density of states (DOS) of all the three structural configurations of CsPbBr3 without spin–orbit coupling (SOC = 0) and with the addition of spin–orbit coupling (SOC ≠ 0) effects were calculated, using density functional theory. Upon the inclusion of the spin–orbit coupling, the bandgaps exhibit reductions of 1.27 eV, 1.16 eV and 1.08 eV for the cubic, tetragonal and orthorhombic phases, respectively. These calculations provide a positive split-off energy value of Δso = 1.69 eV for the simple cubic phase. For the lower symmetry phases, the p-like fourfold degenerate \(\varGamma_{8v}^{(4)}\) band has been observed to split to form two bands, in addition to the \(\varGamma_{6v}^{(2)}\) split-off band. The calculated splitting energies between these bands are found to be in close agreement with previous experimentally measured values. The calculated electronic band structures show that CsPbBr3 has a negative ‘inversion energy’ (Δi < 0). The magnitude of the inversion energy for the cubic phase is 2.36 eV for SOC = 0, which increased by 0.4–2.76 eV with the addition of the spin–orbit coupling. The arrangement of Bloch levels in the band structure of CsPbBr3 has been found to resemble that of a typical topological semimetal, but with a nonzero bandgap opening, due to the presence of the inversion asymmetry within its molecular structure.

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References

  1. Kang J, Wang L (2017) High defect tolerance in lead halide perovskite CsPbBr3. J Phys Chem 8(2):489–493

    CAS  Google Scholar 

  2. Dong Q, Fang Y, Shao Y, Mulligan P, Qiu J, Cao L, Huang J (2015) Electron-hole diffusion lengths > 175 μm in solution-grown CH3NH3PbI3 single crystals. Science 347:967–969

    CAS  Google Scholar 

  3. Shi D, Adinolfi V, Comin R, Yuan M, Alarousu E, Buin A, Chen Y, Hoogland S, Rothenberger A, Katsiev K, Losovyj Y, Zhang X, Dowben PA, Mohammed OF, Sargent EH, Bakr OM (2015) Low trap-state density and long carrier diffusion in organolead trihalide perovskite single crystals. Science 347:519–522

    CAS  Google Scholar 

  4. Hu M, Bi Cheng, Yuan Y, Bai Y, Huang J (2015) Stabilized wide bandgap MAPbBrxI3-x perovskite by enhanced grain size and improved crystallinity. Adv Sci 3:1500301–1500306

    Google Scholar 

  5. Zhao B, Jalebi MA, Tabachnyk M, Glass H, Kamboj VS, Nie W, Pearson AJ, Puttisong Y, Gödel KC, Beere HE, Ritchie DA, Mohite AD, Dutton SE, Friend RH, Sadhanala A (2017) High open-circuit voltages in tin-rich low band gap perovskite-based planar heterojunction photovoltaics. Adv Mater 29:1604744–1604751

    Google Scholar 

  6. Walsh A (2015) Principles of chemical bonding and band gap engineering in hybrid organic–inorganic halide perovskites. J Phys Chem C 119:5755–5760

    CAS  Google Scholar 

  7. Richter JM, Jalebi MA, Sadhanala A, Tabachnyk M, Rivett JPH, Outón LMP, Gödel KC, Price M, Deschler F, Friend RH (2016) Enhancing photoluminescence yields in lead halide perovskites by photon recycling and light out-coupling. Nat Commun 7:13941–13948

    CAS  Google Scholar 

  8. Shin SS, Yang WS, Noh JH, Suk JH, Jeon NJ, Park JH, Kim JS, Seong WM, Seok S (2015) High-performance flexible perovskite solar cells exploiting Zn2SnO4 prepared in solution below 100°C. Nat Commun 6:7410–7417

    CAS  Google Scholar 

  9. Yong ZJ, Guo SQ, Ma JP, Zhang JY, Li ZY, Chen YM, Zhang BB, Zhou Y, Shu J, Gu JL, Zheng LR, Bakr OM, Sun HT (2018) Doping-enhanced short-range order of perovskite nanocrystals for near-unity violet luminescence quantum yield. J Am Chem Soc 140:9942–9951

    CAS  Google Scholar 

  10. Nedelcu G, Protesescu L, Yakunin S, Bodnarchuk MI, Grotevent MJ, Kovalenko MV (2015) Fast anion-exchange in highly luminescent nanocrystals of cesium lead halide perovskites (CsPbX3, X = Cl, Br, I). Nano Lett 15:5635–5640

    CAS  Google Scholar 

  11. Akkerman QA, D’Innocenzo V, Accornero S, Scarpellini A, Petrozza A, Prato M, Manna L (2015) Tuning the optical properties of cesium lead halide perovskite nanocrystals by anion exchange reactions. J Am Chem Soc 137:10276–10281

    CAS  Google Scholar 

  12. Ahmed GH, Yin J, Bose R, Sinatra L, Alarousu E, Yengel E, AlYami NM, Saidaminov MI, Zhang Y, Hedhili MN (2017) Pyridine-induced dimensionality change in hybrid perovskite nanocrystals. Chem Mater 29:4393–4400

    CAS  Google Scholar 

  13. Yang HZ, Zhang YH, Pan J, Yin J, Bakr OM (2017) Room-temperature engineering of all-inorganic perovskite nanocrsytals with different dimensionalities. Chem Mater 29:8978–8982

    CAS  Google Scholar 

  14. Peng LC, Dutta A, Xie RG, Yang WS, Pradhan N (2018) Dot-wire-platelet-cube: step growth and structural transformations in CsPbBr3 perovskite nanocrystals. ACS Energy Lett 3:2014–2020

    CAS  Google Scholar 

  15. Deng Y, Zheng X, Bai Y, Wang Q, Zhao J, Huang J (2018) Surfactant-controlled ink drying enables high-speed deposition of perovskite films for efficient photovoltaic modules. Nat Energy 3:560–566

    CAS  Google Scholar 

  16. NREL (2019) Best research-cell efficiencies. National Renewable Energy Laboratory

  17. Akkerman QA, Raino G, Kovalenko MV, Manna LG (2018) Challenges and opportunities for colloidal lead halide perovskite nanocrystals. Nat Mater 17:394–405

    CAS  Google Scholar 

  18. Pan J, Shang YQ, Yin J, Bastiani MD, Peng W, Dursun I, Sinatra L, El-Zohry AM, Hedhili MN, Emwas AH, Mohammed OF, Ning Z, Bakr OM (2018) Bidentate ligand-passivated CsPbI3 perovskite nanocrystals for stable near-unity photoluminescence quantum yield and efficient red light-emitting diodes. J Am Chem Soc 140:562–565

    CAS  Google Scholar 

  19. Chen QS, Wu J, Ou XY, Huang BL, Almutlaq J, Zhumekenov AA, Guan XW, Han SY, Liang LL, Yi ZG (2018) All-inorganic perovskite nanocrystal scintillators. Nature 561:88–93

    CAS  Google Scholar 

  20. Isarov M, Tan LZ, Bodnarchuk MI, Kovalenko MV, Rappe AM, Lifshitz E (2017) Rashba effect in a single colloidal CsPbBr3 perovskite nanocrystal detected by magneto-optical measurements. Nano Lett 17:5020–5026

    CAS  Google Scholar 

  21. Zhang XY, Lin H, Huang H, Reckmeier C, Zhang Y, Choy WCH, Rogach AL (2016) Enhancing the brightness of cesium lead halide perovskite nanocrystal based green light-emitting devices through the interface engineering with perfluorinated lonomer. Nano Lett 16:1415–1420

    CAS  Google Scholar 

  22. Pan J, Sarmah SP, Murali B, Dursun I, Peng W, Parida MR, Liu J, Sinatra L, Alyami N, Zhao C (2015) Air-stable surface-passivated perovskite quantum dots for ultra-robust, single- and two-photon-induced amplified spontaneous emission. J Phys Chem Lett 6:5027–5033

    CAS  Google Scholar 

  23. Zhang Y, Sun R, Ou X, Fu K, Chen Q, Ding Y, Xu LJ, Liu L, Han Y, Malko AV (2019) Metal halide perovskite nanosheet for X-ray high-resolution scintillation imaging screens. ACS Nano 13:2520–2525

    CAS  Google Scholar 

  24. Li Z, Yang M, Park JS, Wei SH, Berry J, Zhu K (2016) Stabilizing perovskite structures by tuning tolerance factor: formation of formamidinium and cesium lead iodide solid-state alloys. Chem Mater 28:284–292

    Google Scholar 

  25. Li X, Cao F, Yu D, Chen J, Sun Z, Shen Y, Zhu Y, Wang L, Wei Y, Wu Y, Zeng H (2017) All inorganic halide perovskites nanosystem: synthesis, structural features, optical properties and optoelectronic applications. Small 13:1603996–1604019

    Google Scholar 

  26. Liang J, Wang C, Wang Y, Xu Z, Lu Z, Ma Y, Zhu H, Hu Y, Xiao C, Yi X, Zhu G, Lv H, Ma L, Chen T, Tie Z, Jin Z, Liu J (2016) All-inorganic perovskite solar cells. J Am Chem Soc 138:15829–15839

    CAS  Google Scholar 

  27. Stasio DF, Christodoulou S, Huo NJ, Konstantatos G (2017) Near-unity photoluminescence quantum yield in CsPbBr3 nanocrystal solid-state films via postsynthesis treatment with lead bromide. Chem Mater 29:7663–7667

    CAS  Google Scholar 

  28. Stoumpos CC, Malliakas CD, Peters JA, Liu Z, Sebastian M, Im J, Chasapis TC, Wibowo AC, Chung DY, Freeman AJ, Wessels BW, Kanatzidis MG (2013) Crystal growth of the perovskite semiconductor CsPbBr3: a new material for high energy radiation detection. Cryst Growth Des 13:2722–2727

    CAS  Google Scholar 

  29. Manser JS, Christians JA, Kamat PV (2016) Intriguing optoelectronic properties of metal halide perovskites. Chem Rev 116:12956–13008

    CAS  Google Scholar 

  30. Li G, Rivarola FWR, Davis NJLK, Bai S, Jellicoe TC, Peña FDL, Hou S, Ducati C, Gao F, Friend RH, Greenham NC, Tan ZK (2016) Highly efficient perovskite nanocrystal light-emitting diodes enabled by a universal crosslinking method. Adv Mater 28:3528–3534

    CAS  Google Scholar 

  31. Dursun I, Shen C, Parida MR, Pan J, Sarmah SP, Priante D, Alyami N, Liu J, Saidaminov MI, Alias MS, Abdelhady AL, Ng TK, Mohammed OF, Ooi BS, Bakr OM (2016) Perovskite nanocrystals as a color converter for visible light communication. ACS Photonics 3:1150–1156

    CAS  Google Scholar 

  32. Gmitra M, Fabian J (2016) First-principles studies of orbital and spin-orbit properties of GaAs, GaSb, InAs, and InSb zinc-blende and wurtzite semiconductors. Phys Rev B 94:165202–165211

    Google Scholar 

  33. Kim M, Im J, Freeman AJ, Ihm J, Jin H (2014) Switchable S = 1/2 and J = 1/2 Rashba bands in ferroelectric halide perovskites. PNAS 111:6900–6904

    CAS  Google Scholar 

  34. Kepenekian M, Robles R, Katan C, Sapori D, Pedesseau L, Even J (2015) Rashba and Dresselhaus effects in hybrid organic-inorganic perovskites: from basics to devices. ACS Nano 9:11557–11567

    CAS  Google Scholar 

  35. Niesner D, Wilhelm M, Levchuk I, Osvet A, Shrestha S, Batentschuk M, Brabec C, Fauster T (2016) Giant Rashba splitting in CH3NH3PbBr3 organic–inorganic perovskite. Phys Rev Lett 117:126401–126406

    Google Scholar 

  36. Belykh VV, Yakovlev DR, Glazov MM, Grigoryev PS, Hussain M, Rautert J, Dirin DN, Kovalenko MV, Bayer M (2019) Coherent spin dynamics of electrons and holes in CsPbBr3 perovskite crystals. Nat Commun 10:673–678

    CAS  Google Scholar 

  37. Brivio F, Butler KT, Walsh A (2014) Relativistic quasiparticle self-consistent electronic structure of hybrid halide perovskite photovoltaic absorbers. Phys Rev B 89:155204–155209

    Google Scholar 

  38. Becker MA, Vaxenburg R, Nedelcu G, Sercel PC, Shabaev A, Mehl MJ, Michopoulos JG, Lambrakos SG, Bernstein N, Lyons JL, Stöferle T, Mahrt RF, Kovalenko MV, Norris DJ, Rainò G, Efros AL (2018) Bright triplet excitons in caesium lead halide perovskites. Nature 553:189–193

    CAS  Google Scholar 

  39. Richard SB, Katan C, Traoré B, Scholz R, Jancu JM, Even J (2016) Symmetry-based tight binding modeling of halide perovskite, semiconductors. J Phys Chem Lett 7:3833–3840

    Google Scholar 

  40. Canneson D, Shornikova EV, Yakovlev DR, Rogge T, Mitioglu AA, Ballottin MV, Christianen PCM, Lhuillier E, Bayer M, Biadala L (2017) Negatively charged and dark excitons in CsPbBr3 perovskite nanocrystals revealed by high magnetic fields. Nano Lett 17:6177–6183

    CAS  Google Scholar 

  41. Ahmad M, Rehman G, Ali L, Shafiq M, Iqbal R, Ahmad R, Khan T, Asadabadi SJ, Maqbool M, Ahmad I (2017) Structural, electronic and optical properties of CsPbX3 (X = Cl, Br, I) for energy storage and hybrid solar cell applications. J Alloys Compd 705:828–839

    CAS  Google Scholar 

  42. Birch F (1947) Finite elastic strain of cubic crystals. Phys Rev 71:809–824

    CAS  Google Scholar 

  43. Hedin L (1965) New method for calculating 1-particle greens function with application to electron-gas problem. Phys Rev 139(3A):A796–A823

    Google Scholar 

  44. Salpeter EE, Bethe HA (1951) A relativistic equation for bound-state problems. Phys Rev 84(6):1232–1242

    Google Scholar 

  45. Even J, Pedesseau L, Jancu JM, Katan C (2013) Importance of spin-orbit coupling in hybrid organic/inorganic perovskites for photovoltaic applications. J Phys Chem Lett 4:2999–3005

    CAS  Google Scholar 

  46. Becke AD, Johnson ER (2006) A simple effective potential for exchange. J Chem Phys 124:221101–221105

    Google Scholar 

  47. Tran F, Blaha P (2009) Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential. Phys Rev Lett 102:226401–226404

    Google Scholar 

  48. Perdew JP, Yue W (1986) Accurate and simple density functional for the electronic exchange energy: generalized gradient approximation. Phys Rev B 33:8800–8802

    CAS  Google Scholar 

  49. Even J, Pedesseau L, Dupertuis MA, Jancu JM, Katan C (2012) Electronic model for self-assembled hybrid organic/perovskite semiconductors: reverse band edge electronic states ordering and spin-orbit coupling. Phys Rev B 86:205301–205304

    Google Scholar 

  50. Whitcher TJ, Gomes LC, Zhao D, Bosman M, Chi X, Wang Y, Carvalho A, Hui HK, Chang Q, Breese MBH, Neto AHC, Wee ATS, Sun HD, Chia EEM, Rusydi A (2019) Dual phases of crystalline and electronic structures in the nanocrystalline perovskite CsPbBr3. NPG Asia Mater 11:70–81

    Google Scholar 

  51. Tamarat PFM, Huang H, Even J, Rogach AL, Lounis B (2017) Neutral and charged exciton fine structure in single lead halide perovskite nanocrystals revealed by magneto-optical spectroscopy. Nano Lett 17(5):2895–2901

    Google Scholar 

  52. Yang F, Wang C, Pan Y, Zhou X, Kong X, Ji W (2019) Surface stabilized cubic phase of CsPbI3 and CsPbBr3 at room temperature. Chin Phys B 28:056402–056409

    CAS  Google Scholar 

  53. Ghaithan HM, Alahmed ZA, Qaid SMH, Hezam M, Aldwayyan AS (2020) Density functional study of cubic, tetragonal, and orthorhombic CsPbBr3 perovskite. ACS Omega. https://doi.org/10.1021/acsomega.0c00197

    Article  Google Scholar 

  54. Jiang LQ, Guo JK, Liu HB, Zhu M, Zhou X, Wu P, Li CH (2006) Prediction of lattice constant in cubic perovskites. J Phys Chem Solids 67:1531–1536

    CAS  Google Scholar 

  55. Rodova M, Brozek J, Knızek K, Nitsch K (2003) Phase transitions in ternary cesium lead bromide. J Therm Anal Calorim 71:667–673

    CAS  Google Scholar 

  56. Jaroenjittichai AP, Laosiritaworn Y (2018) Band alignment of cesium-based halide perovskites. Ceram Int 44:S161–S163

    CAS  Google Scholar 

  57. Cottingham P, Brutchey RL (2016) On the crystal structure of colloidally prepared CsPbBr3 quantum dots. Chem Commun 52:5246–5249

    CAS  Google Scholar 

  58. Moller CK (1959) The structure of perovskite-like caesium plumbo trihalides. Mater Fys Medd Dan Vid Selsk 32:1–27

    CAS  Google Scholar 

  59. Zhao YQ, Ma QR, Liu B, Yu ZL, Cai MQ (2018) Pressure-induced strong ferroelectric polarization in tetra-phase perovskite CsPbBr3. Phys Chem 20:14718–14724

    CAS  Google Scholar 

  60. Tomanová K, Čuba V, Brik MG, Mihóková E, Turtos M, Lecoq R, Auffray P, Nikl E (2019) On the structure, synthesis, and characterization of ultrafast blue-emitting CsPbBr3 nanoplatelets. APL Mater 7:011104–011116

    Google Scholar 

  61. Linaburg MR, McClure ET, Majher JD, Woodward PM (2017) Cs1−XRbxPbCl3 and Cs1−XRbxPbBr3 solid solutions: understanding octahedral tilting in lead halide perovskites. Chem Mater 29:3507–3514

    CAS  Google Scholar 

  62. Atourki L, Vega E, Mollar E, Marí M, Kirou B, Bouabid H, Ihlal K (2017) Impact of iodide substitution on the physical properties and stability of cesium lead halide perovskite thin films CsPbBr3-xIx(0 ≤ x ≤ 1). J. Alloys Compd. 702:404–409

    CAS  Google Scholar 

  63. Tanaka K, Takahashi T, Ban T, Kondo T, Uchida K, Miura N (2003) Comparative study on the excitons in lead-halide-based perovskite-type crystals CH3NH3PbBr3 CH3NH3PbI3. Solid State Commun 127(9–10):619–622

    CAS  Google Scholar 

  64. Vidal J, Zhang X, Stevanović V, Luo J-W, Zunger A (2012) Large insulating gap in topological insulators induced by negative spin-orbit splitting. Phys Rev B 86(7):075316–075321

    Google Scholar 

  65. Ramade J, Andriambariarijaona LM, Steinmetz V, Goubet N, Legrand L, Barisien T, Bernardot F, Testelin C, Lhuillier E, Bramati A, Chamarro M (2018) Fine structure of excitons and electron-hole exchange energy in polymorphic CsPbBr3 single nanocrystals. Nanoscale 10(14):6393–6401

    CAS  Google Scholar 

  66. Chantis AN, van Schilfgaarde M, Kotani T (2006) Ab initio prediction of conduction band spin splitting in zinc blende semiconductors. Phys Rev Lett 96:086405–086408

    Google Scholar 

  67. Luo JW, Bester G, Zunger A (2009) Full-zone spin splitting for electrons and holes in bulk GaAs and GaSb. Phys Rev Lett 102:056405–056408

    Google Scholar 

  68. Heidrich K, Schäfer W, Schreiber M, Söchtig J, Trendel G, Treusch J, Grandke T, Stolz HJ (1981) Photoemission spectra, and vacuum-ultraviolet optical spectra of CsPbCl3 and CsPbBr3. Phys Rev B 24:5642–5649

    CAS  Google Scholar 

  69. Li Y, Duan J, Yuan H, Zhao Y, He B, Tang Q (2018) Lattice modulation of alkali metal cations doped Cs1−XRXPbBr3 halides for inorganic perovskite solar cells. Sol RRL 2:1800164–1800171

    Google Scholar 

  70. Akkerman Q, Motti A, Kandada SGS, Mosconi AR, D’Innocenzo E, Bertoni V, Marras G, Kamino S, Miranda BA, De Angelis L (2016) Solution synthesis approach to colloidal cesium lead halide perovskite nanoplatelets with monolayer-level thickness controls. J Am Chem Soc 138:1010–1016

    CAS  Google Scholar 

  71. Yaffe O, Guo Y, Tan LZ, Egger DA, Hull T, Stoumpos CC, Zheng F, Heinz TF, Kronik L, Kanatzidis MG (2016) The nature of dynamic disorder in lead halide perovskite. Crystals arXiv:1604.08107

  72. Zhang L, Zeng Q, Wang K (2017) Pressure-induced structural and optical properties of inorganic halide perovskite CsPbBr3. J Phys Chem Lett 8:3752–3758

    CAS  Google Scholar 

  73. Paul T, Chatterjee BK, Maiti S, Sarkar S, Besra N, Das BK, Panigrahi KJ, Thakur S, Ghorai UK (2018) Tunable cathodoluminescence over the entire visible window from all-inorganic perovskite CsPbX3 1D architecture. J Mater Chem C 6:3322–3333

    CAS  Google Scholar 

  74. Bir GL, Pikus GE (1974) Symmetry and strain-induced effects in semiconductors: ch. 23. Wiley, Hoboken

    Google Scholar 

  75. Borriello I, Cantele G, Ninno D (2008) Ab initio investigation of hybrid organic-inorganic perovskites based on tin halides. Phys Rev B Condens Mater Phys 77:235214–235222

    Google Scholar 

  76. Chabot J, Cote M, Briere J, (2004) American Physical Society, March Meeting

  77. Qi XL, Zhang SC (2011) Topological insulators and superconductors. Rev Mod Phys 83:1057–1110

    CAS  Google Scholar 

  78. Thaller B (1992) The Dirac equation. Springer, New York

    Google Scholar 

  79. Zhang H, Liu CX, Qi XL, Dai X, Fang Z, Zhang SC (2009) Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat Phys 5:438–442

    CAS  Google Scholar 

  80. Qi XL, Zhang SC (2010) The quantum spin Hall effect and topological insulators. Phys Today 63(1):33–38

    CAS  Google Scholar 

  81. Zhu Z, Cheng Y, Schwingenschlog U (2012) Band inversion mechanism in topological insulators: a guideline for materials design. Phys Rev B 85:235401–235405

    Google Scholar 

  82. Cardona M (2010) A fine point on topological insulators. Phys Today 63(8):10–12

    Google Scholar 

  83. Feng W, Xiao D, Ding J, Yao Y (2011) Three-dimensional topological insulators in I–III–VI2 and II–IV–V2 chalcopyrite semiconductors. Phys Rev Lett 106:016402–016405

    Google Scholar 

  84. Yaffe O, Guo Y, Tan LZ, Egger DA, Hull T, Stoumpos CC, Zheng F, Heinz TF, Kronik L, Kanatzidis MG, Owen JS, Rappe AM, Pimenta MA, Brus LE (2017) Local polar fluctuations in lead halide perovskite crystals. Phys Rev Lett 118:136001(1)–136001(6)

    Google Scholar 

  85. Mosconi E, Etienne T, De Angelis F (2017) Rashba band splitting in organohalide lead perovskites: bulk and surface effects. J Phys Chem Lett 8:2247–2252

    CAS  Google Scholar 

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Acknowledgements

We would like to thank the Higher Education Commission, Pakistan, for supporting this work through the fund No: 7294/Balochistan/NRPU/R&D/HEC/2017. Also, we would like to acknowledge DAAD for their funding under the project titled ‘EXCIPLAS: Time-resolved studies of bound exciton–plasmon coupling in wide-bandgap semiconductor nanostructure–metallic nanoparticle composites’.

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  1. Center for Micro and Nano Devices, Department of Physics, COMSATS University Islamabad, Islamabad, 44000, Pakistan

    Mujtaba Hussain, Faisal Saeed & A. S. Bhatti

  2. Department of Physics, COMSATS University Islamabad, Islamabad, 44000, Pakistan

    Muhammad Rashid

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Hussain, M., Rashid, M., Saeed, F. et al. Spin–orbit coupling effect on energy level splitting and band structure inversion in CsPbBr3. J Mater Sci 56, 528–542 (2021). https://doi.org/10.1007/s10853-020-05298-8

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