A Generic Slater-Koster Description of the Electronic Structure of Centrosymmetric Halide Perovskites
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J a n A Generic Slater-Koster Description of the Electronic Structure ofCentrosymmetric Halide Perovskites
Ravi Kashikar, Mayank Gupta, and B. R. K. Nanda a) Condensed Matter Theory and Computational Lab, Department of Physics, Indian Institute of Technology Madras,Chennai - 36, India (Dated: 22 January 2021)
The halide perovskites have truly emerged as efficient optoelectronic materials and show the promise of ex-hibiting nontrivial topological phases. Since the bandgap is the deterministic factor for these quantum phases,here we present a comprehensive electronic structure study using first-principle methods by considering nineinorganic halide perovskites CsBX (B = Ge, Sn, Pb; X = Cl, Br, I) in their three structural polymorphs(cubic, tetragonal and orthorhombic). A series of exchange-correlations (XC) functionals are examined to-wards accurate estimation of the bandgap. Furthermore, while thirteen orbitals are active in constructingthe valence and conduction band spectrum, here we establish that a four orbital based minimal basis setis sufficient to build the Slater-Koster tight-binding model (SK-TB), which is capable of reproducing thebulk and surface electronic structure in the vicinity of the Fermi level. Therefore, like the Wannier basedTB model, the presented SK-TB model can also be considered as an efficient tool to examine the bulk andsurface electronic structure of halide family of compounds. As estimated by comparing the model study andDFT band structure, the dominant electron coupling strengths are found to be nearly independent of XCfunctionals, which further establishes the utility of the SK-TB model. I. INTRODUCTION
Halide perovskites of the form ABX (where A is anorganic or inorganic monovalent entity, B is a divalentcation such as Pb, Sn, Ge and X is a halogen (Cl,Br, I and F)), have brought a paradigm shift in thephotovoltaic applications because of parity allowed di-rect transitions between the band extrema . In re-cent times along with the optical properties, these per-ovskites have shown the ferroelectrically driven spin-texture, and topological quantum phase transition un-der external forces in both centrosymmetric and noncen-trosymmetric phases . While the entity A predom-inantly decides the structural stability and centrosym-metricity, B and X governs the electronic properties ofthese compounds . The cubic phase of halide per-ovskites exhibit structural phase transition with tem-perature and pressure, and the resulted lower symmetrycrystal structures are characterized by in-plane as wellas out-of-plane octahedral rotations . The schematicrepresentation of the orbital overlap in high symmetricand lower symmetric phases is illustrated in the Fig. 1.Over the last decade, both experimental and densityfunctional theory (DFT) methods have been employedto unravel the orbital and crystal interplay to study theoptoelectronic and other intriguing properties of theseperovskite materials. The previous electronic structurestudies on these compounds suggest that the band spec-trum consists of anti-bonding and bonding states, alongwith the X-p dominated non-bonding states arise out ofstrong covalent hybridization between B- { s, p } and X-pstates . Nature of the band structure remains sim- a) Electronic mail: [email protected] ilar for a particular phase with varying bandwidth for afamily of halide perovskites. Beyond this basic observa-tions, there are several issues which need to be addressedto construct a comprehensive picture of the electronicstructure of this important class of compounds. For ex-ample choice of exchange correlation functional is one ofthe debatable issue .As the bandgap varies widely in this class of com-pounds and the materials applicability in optoelectronicdevices as well as in inducing non-trivial topologicalphases primarily depend on the bandgap, and we knowthat the bandgap highly sensitive to type of exchange-correlation functionals employed within the DFT formal-ism. Therefore, it is pertinent that a relation betweenthe bandgap and the type of exchange correlation func-tional be established, which could capture the experimen-tal observations. Furthermore, this class of compoundsexhibit three temperature dependent structural phasesgoverned by anionic displacements . The bandgap ofthese phases differ significantly from each other. There-fore, it is crucial to identify the chemical interactions thatgovern the bandgap in this family. Also, it has been ob-served that electronic structure is sensitive to both B andX. For example if B is Sn for any X, we observe a lowerbandgap as compared Pb and Ge and the reason hasnot been able to explain through parameter-free densityfunctional calculations. As a whole comprehensive firstprinciples calculations and formulation of model Hamil-tonians are required to provide a generic description ofthe electronic structure of the halide perovskites.In the literature, from the model Hamiltonian per-spective, only a handful of studies have examined theband structure of halide perovskites. Boyer-Richard etal., envisaged the fourteen orbital basis based tight-binding (TB) Hamiltonian without incorporating the sec-ond neighbour interaction and Jin et al. have studied
FIG. 1. Schematic representation of chemical bonding of B- { s,p } -X-p orbitals in various polymorphs of halide perovskites. the continuum model based on the four orbital basis TBHamiltonian . In our recent work, the four orbital ba-sis model is employed to lower symmetry polymorphs toexamine the band topology of these perovskites systems .However, all these studies were carried out on individualmembers, and so far, there has been no systematic studyof TB Hamiltonian for all the members of the halide per-ovskites family to understand sheer variety of propertieswith varying B and X elements. This is the first attemptto bring the generic picture of halide perovskites, andfor that, we have considered the nine members and threestructural phases to analyze the role of each entity ofinorganic ABX compounds. II. COMPUTATIONAL DETAILS
In the present work, we have employed both pseu-dopotential method with the plane-wave basis set as im-plemented Vienna Abinitio Simulation Package (VASP),and full-potential linearized augmented plane wave (FP-LAPW) method as implemented in WIEN2k simulationtool for DFT calculations . In both the methods,the Perdew-Burke-Ernzerhof (PBE) exchange-correlationfunctional within the generalized gradient approxima-tion (GGA) is considered. However, the PBE underesti-mates the bandgap severely as compared to the experi-mental bandgap. Therefore, the PBE-GGA approxima-tion along with modified Becke-Johnson (mBJ) poten-tial as well as hybrid functionals are used to take intoaccount the exchange-correlation effect and for compari-son purpose . All the band structures in the presentwork are presented from PBE+Tran-Blaha mBJ poten-tial. The self-consistent field (SCF) calculations compriseof augmented plane waves of the interstitial region andlocalized orbitals: B- { ns, np } (B = Ge, Sn, Pb) and X-p(X = Cl, Br, I). Here, n and m vary with B and X. TheR XMT set to 7.0 for all the compounds. The Brillouinzone integration is carried out with a Monkhorst-Packgrid. We used a k -mesh of 10 × ×
10 (yielding 35 irre- σ ∗ s−p σ s−p E F E F E F α −Phase β −Phase −Phase γ M M M R R R X X Γ Z Z Γ Γ σ∗ p−p π∗ p−p π p−p σ p−p Γ E ne r g y ( e V ) M X M R X Γ B−s X−pB−p (a) (b)(c) (d)3 Pb−pPb−sI−p4−1−24210−1−2 E ne r g y ( e V ) Γ X Γ Z A Z Z RXY Γ FIG. 2. (a) Molecular orbital picture of halide perovskitesenvisaged from the B- { s, p } -X-p atomic orbitals, producesthe bonding and antibonding orbitals along with the non-bonding orbitals. (b-d) Band structure of cubic, tetragonaland orthorhombic band structure of representative compoundCsPbI . ducible points), 6 × × × × α , β and γ -phases respectively. To build a Slater-Koster tightbinding (SK-TB) Hamiltonian, we consider the appropri-ate basis set out of the orbitals dominating the bands atthe Fermi level and this information is obtained from thefirst-principle based density functional calculations. Thedetails of the Hamiltonian and orbital basis is discussedin details in section IIIB. III. RESULTS AND DISCUSSIONA. DFT study
To start with, we will first discuss the electronic struc-ture of CsPbI and explain it through a generic molecularorbital picture as shown in Fig. 2. Here, we outline someof the standard features of band spectrum of halide per-ovskites: (I) The band spectrum in general consist of fourantibonding and bonding bands along with the five non-bonding bands arising due to the B- { s, p } -X-p covalenthybridization in BX octahedron. Thus, the eigenfunc-tions of antibonding and bonding states are the linearcombination of B-s, B-p and X-p orbitals. Their strength TABLE I. Structural parameters of various CsBX perovskites as obtained from DFT calculations and in comparison with theexperimental results. The compounds exist in different crystal polymorphs at different temperature ranges. The θ ab and θ c is180 ◦ for α phase. Phase B X Lattice Parameter (˚A) Octahedral angle Temperature (K) Ref.DFT Expt. θ ab , θ c Cl 5.34 5.47 443,449
Ge Br 5.6 5.69 543 I 6.0 6.05 573 Cl 5.62 5.55 293 α Sn Br 5.88 5.8 292,300 I 6.27 6.21 426 Cl 5.71 5.6 320 Pb Br 5.98 5.87 403 I 6.38 6.29 554 Sn Br 8.27 8.27 5.92 8.18, 8.18, 5.82 162.8 ◦ , 180 ◦ β I 8.81 8.81 6.31 8.77, 8.77, 6.26 162.6 ◦ , 180 351-426 Pb Br 8.35, 8.35, 6.04 156.3 ◦ , 180 ◦ I 8.89 8.89 6.44 8.82, 8.82, 6.3 155.9 ◦ , 180 ◦ Sn Br 8.15 8.37 11.77 8.19, 11.58, 8.02 157.4 ◦ , 163.5 ◦ γ I 8.66 8.92 12.53 8.68, 8.64, 12.37 154.6 ◦ , 161.4 ◦ Pb Br 8.24 8.47 11.93 8.21, 8.25, 11.76 153.4 ◦ , 160.6 ◦ I 8.76 9.04 12.7 8.62, 8.85, 12.5 152.3 ◦ , 159.2 ◦ varies with the type of atom at B, halogens as well ascrystal symmetry. The nonbonding states are from X-porbitals. The molecular orbital picture presented in Fig.2a is for O h point group symmetry which exists at R andΓ points of cubic Brillouin zones. The X-p orbitals un-dergo symmetry adapted linear combinations and formthe molecular orbitals with B- { s, p } orbitals. (II) TheFermi level in the electronic spectrum is solely deter-mined by the valence electron count (VEC) which is thetotal number of valence electrons available per formulaunit. In single halide perovskites, the atom at A con-tributes one electron, while atoms B and X contributesfour and five electrons respectively. Thus, the VEC forhalide perovskites turns out to be 20. This makes thestates up to σ ∗ s − p occupied and hence the Fermi level liesin the gapped region. (III) All of the inorganic halideperovskites have direct bandgap in nature. The bandgapvalue varies with the crystal symmetry and chemical com-position of ABX and spin-orbit coupling (SOC) strengthof B site atom.(IV) The parities of the valence band andconduction band edges for three polymorphs in terms ofKoster notations is denoted in Fig. 2. The analysis indi-cates that all the halide perovskites exhibit parity allowedtransitions due to the opposite of parity of band edges.The instability of the high-temperature cubic phaseintroduces the octahedral rotations in the unitcell, andthere occurs the crystal phase transition from the cubicto tetragonal to orthorhombic phases with temperatureas listed in Table I. The tetragonal phase is character-ized by only inplane octahedral rotation, whereas the or-thorhombic phase is characterized both inplane and out of plane octahedral rotations. These rotations increasesthe unitcell volume as well as the number of atoms. TheGe based halide perovskites exhibit single phase tran-sition and crystallize in rhombohedral unitcell in roomtemperature phase.In Fig. 3 and Fig. 4 we have compared the bandgap ofcubic and lower symmetry halide perovskites for variousexchange-correlation functionals. It is known that GGA-PBE underestimates the bandgap significantly, the cor-rection is added through several other approximations.The modified Becke-Johnson (mBJ) proposed by Tran-Blaha improves the bandgap significantly in most of thehalide perovskites. The hybrid functionals with mixingparameter α = 0 .
25 (HSE06) and α = 0 . . In few cases;CsGeI and CsSnI hybrid functional minimizes the er-ror as compared to Jishi-mBJ. The bandgap for lowersymmetry polymorphs for PBE. mBJ and Jishi-mBJ areshown in Fig. 4, in comparison with available experimen-tal values. It is observed that the gap decreases from Clto Br to I, irrespective of atoms at A and B sites, andSn-based halide perovskites have lower bandgap values ascompared to the Pb and Ge based perovskites. A broadexplanation to this end can be given by comparing thefree atomic energy eigenvalues. The difference betweenthe onsite energies of Sn valence orbitals ( E p / − E s / = 6.35 eV) is lowest as compared to that of Pb (7.04 eV) FIG. 3. Bar chart representation of bandgap of cubic halideperovskites under various exchange correlation functionals.Here, HSE06 and HSE are calculated with pseudopotentialmethods with mixing parameter 0.25 and 0.3, respectively.FIG. 4. Bar chart representation of bandgap of tetragonal(a) and orthorhombic (b) halide perovskites under variousexchange correlation functionals. The experimental bandgapvalues of tetragonal and orthorhombic systems are obtainedfrom . and Ge (7.53 eV). Whereas the onsite energy halogen va-lence p orbital decreases from I (-7.84 eV) to Br (-8.96eV) to Cl (-9.96 eV) The bandgap also increases as wemove from cubic to tetragonal to orthorhombic phasesdue to octahedral rotations which decrease the orbitalsoverlap as shown in Fig. 1. The valence band maxi-mum (VBM) and conduction band minimum (CBM) arepredominantly of B-s and B-p character. The VBM andCBM occurs at R (0.5, 0.5, 0.5), Z (0, 0, 0.5) and Γ (0, 0,0) k -points of cubic, tetragonal and orthorhombic Bril-louin zone respectively. The DFT calculations offers alimited quantitative understanding to establish a genericdescription of the electronic structure. Specifically, thetype and strength of the interactions that govern the va-lence and conduction bands in the vicinity of the Fermilevel needs to be determined. Therefore, in the nextsection, we build the Slater-Koster based tight-bindingHamiltonian using a thirteen orbitals (one B-s, three B-pand nine X-p) basis set. Subsequently, the basis set will be reduced to four. B. Model Hamiltonian for Halide Perovskites
The appropriate SK-TB Hamiltonian for the family ofCsBX , in the second quantization notation is expressedas H = X i,m ǫ im c † im c im + X hh ij ii ; m,n t imjn ( c † im c jn + h.c )+ λ L · S . (1)Here, i ( j ) and α ( β ) are site and the orbitals indicesrespectively. The parameters ǫ iα and t iαjβ respectively,represent the on-site energy and hopping integrals. Thespin-orbit coupling (SOC) is included in the third term ofthe Hamiltonian with λ denoting the SOC strength. Theinclusion of SOC doubles the Hilbert space. Adoptinga two centre integral approach J. C. Slater and G. F.Koster, expressed the t iαjβ with direction cosines (DCS)( l , m , n ) of the vector joining site ( i ) to other site ( j ) .For s and p orbitals, which forms the basis for halideperovskites, the generic expressions are provided below E s,s = t ssσ (2) E s,p x = lt spσ E s,p y = mt spσ E s,p z = nt spσ E p x ,p x = l t ppσ + (1 − l ) t ppπ E p x ,p m = lmt ppσ − lmt ppπ Using Eq. 1 and 2, we now develop TB Hamiltonian forthe halide perovskites. Thus, the spin independent TBHamiltonian matrix, with the basis set in the order {| s B i , | p Bx i , | p By i , | p Bz i , | p X x i , | p X y i , | p X z i , | p X x i , | p X y i , | p X z i , | p X x i , | p X y i , | p X z i} , can be written as H F BT B = M B − B × M B − X × ( M B − X × ) † M X − X × . (3)Here, FB indicates the full basis. The individual blocksof this matrix are as follows, The M B − B × is given by ǫ s + h ( ~k ) 2 it B − Bspσ S x it B − Bspσ S y it B − Bspσ S z − it B − Bspσ S x ǫ p + h ( ~k ) 0 0 − it B − Bspσ S y ǫ p + h ( ~k ) 0 − it B − Bspσ S z ǫ p + h ( ~k ) (4) M B − X × = (cid:16) M × M × M × (cid:17) (5) M B − X × = t B − Xspσ S x t B − Xppσ C x t B − Xppπ C x
00 0 t B − Xppπ C x (6)From the Fig. 2 it is observed that X-p dominated bandsare very narrow ( < . { p } -X- { p } second interactions. Hence the block M X − X × canbe approximated as M X − X × = ǫ p I × (7)Here, ǫ s , ǫ p and ǫ p are on-site energies of B-s, B-p andX-p orbitals respectively. The terms C x and S x are shortnotations for 2 cos( k x a/
2) and 2 i sin( k x a/
2) respectively.The dispersion term g i ( i = 1 , , , h ( ~k ) = 2 t B − Bss ( cos ( k x a ) + cos ( k y a ) + cos ( k z a )) h ( ~k ) = 2 t B − Bppσ cos ( k x a ) + 2 t B − Bppπ [ cos ( k y a ) + cos ( k z a )] h ( ~k ) = 2 t B − Bppσ cos ( k y a ) + 2 t B − Bppπ [ cos ( k x a ) + cos ( k z a )](8) h ( ~k ) = 2 t B − Bppσ cos ( k z a ) + 2 t B − Bppπ [ cos ( k x a ) + cos ( k y a )]The analytical expression for eigenvalues at time rever-sal invariant momentum (TRIM) R ( πa , πa , πa ), which is ofparticular interest as the valence band maximum (VBM)and conduction band minimum (CBM) are observed atthis point, and are given as E [1] = ( E Xp + E Bs )2 − t B − Bss − ηE [8] = E Xp E [3] = E Bp − t B − Bppσ − t B − Bppπ E [1] = ( E Xp + E Bs )2 − t B − Bss + η. (9)Where η = q ( E Xp − E Bs + 6 t B − Bss ) + 48( t B − Xsp ) H soc = λ L.S on B- p orbital basis with the order ( p Bx ↑ , p By ↑ , p Bz ↓ , p Bx ↓ , p By ↓ , p Bz ↑ ), we get the following matrix H SOC = λ − i i − i − i i − i i i . (10)As a case study, the thirteen band model is applied tonine cubic CsBX (B = Ge, Sn, Pb; X = Cl, Br, I) CsGeI CsGeCl CsGeBr CsSnBr CsSnCl CsSnI CsPbBr CsPbI CsPbCl Γ M XM Γ X M R X M Γ X M R X62−2−6−1062−2−6−106−2−6−10 E ne r g y ( e V ) TBDFT
FIG. 5. Thirteen orbital basis based TB band structureof CsBX in comparison with DFT band structure obtainedfrom GGA+mBJ exchange-correlation functional. perovskites and corresponding bands are fitted with DFTbands to obtain onsite and hopping interactions, whichare listed in Table II. The DFT and TB bands are shownin Fig. 5, suggesting the excellent agreement betweeneach other. The table infers that (I) tin-based halideperovskites exhibit higher onsite energies (lowest E s -E p energy), agrees with the atomic energies mentioned insection-II. A similar trend is observed for halogen orbitalenergies. The hopping parameter t σsp varies from Cl to Brto I indicating a decrease in the bandwidth of uppermostvalence band for all the compounds as we move from Cl toBr to I. The onsite energies provided in Table II agreesqualitatively well with the recent work of Hoffmann etal. . The other hopping parameters t B − Xsp and t B − Bppσ governs the bandwidth of uppermost valence band andconduction band, and these parameters increase with Cl-Br-I. The other parameters remain more or less constantfor all the perovskites.Having understood about the full band spectrum ofcubic halide perovskites, now we aim to apply for lowersymmetry polymorphs. However, looking at the size andnumber atoms in the lower symmetry crystal structures,it is difficult to trace and analyze a large number of in-teractions in these systems. Therefore, it is necessary todevelop minimal basis Hamiltonian having less number ofinteraction without losing essential physics. The bandsforming the VBM and CBM at high symmetry points areprimarily created by the four B- { s, p } orbitals. From ourcalculations, we found that the contribution B-p orbital TABLE II. On-site energy, hopping parameters and GGA-mBJ bandgap of CsBX perovskites in units of eVB X E B − s E B − p E X − p t B − Xsp t B − Xppσ t B − Xppπ t B − Bss t B − Bspσ t B − Bppσ t B − Bppπ λ Cl -3.17 5.35 -0.18 -1.20 1.94 -0.58 0.02 -0.16 0.31 0.03 0.07Ge Br -3.55 4.76 0.40 -1.16 1.94 -0.56 0.02 -0.15 0.32 0.02 0.07I -3.97 4.29 0.92 1.00 1.92 -0.52 0.02 -0.15 0.37 0.02 0.07Cl -0.71 6.42 -0.06 -1.29 1.94 -0.52 -0.08 -0.2 0.24 0.06 0.16Sn Br -1.43 5.71 0.36 -1.27 1.96 -0.53 -0.07 -0.19 0.34 0.05 0.16I -2.34 4.79 0.92 -1.12 1.9 -0.53 -0.02 -0.17 0.38 0.01 0.14Cl -1.61 7.63 0.25 -1.18 1.88 -0.57 -0.03 -0.13 0.31 0.06 0.53Pb Br -3.15 5.84 0.43 -1.13 1.86 -0.53 -0.02 -0.12 0.27 0.04 0.53I -4.11 4.75 0.96 -0.94 1.82 -0.45 0.01 0.12 0.25 0.02 0.5FIG. 6. Schematic representation hopping interaction in thir-teen orbital basis based TB model and four orbital basis basedTB model. Here, Fermi level is set to zero.TABLE III. Orbital weight in-terms of % at conduction bandminimum and valence band maximum of different halide per-ovskites. CBM VBMB X B-p B-s X-pCl 100 65 35Ge Br 100 63 37I 100 65 35Cl 100 69 31Sn Br 100 67 33I 100 67 33Cl 100 66 34Pb Br 100 61 39I 100 60 40 at conduction band minimum is 100%. Whereas the va-lence band maximum is made up of a linear combinationof X-p and B-s characters as listed in Table III. As wecan see, the contribution of X-p orbitals is approximatelyhalf of B-s. Thus, there is room to minimize the number of interactions. Therefore, we choose the basis set fromB- { s, p } , where the interactions between B-X-B will beincluded in B-B interactions as shown in Fig. 6. Thus,the SOC incorporated four-band TB Hamiltonian in thematrix form can be written as H MBT B = H ↑↑ H ↑↓ H †↓↑ H ↓↓ ! , (11) H ↑↑ = ǫ s + h ( ~k ) 2 i ( t xsp ) AA S x i ( t xsp ) AA S y i ( t zsp ) AA S z − i ( t xsp ) AA S x ǫ xp + h ( ~k ) − iλ − i ( t xsp ) AA S y iλ ǫ xp + h ( ~k ) 0 − i ( t zsp ) AA S z ǫ zp + h ( ~k ) (12) Here, MB refers to minimal basis. The ǫ A s are bandcentres of the anti-bonding bands, and t A s are the secondneighbour hopping integrals. The dispersion functions f i ( i = 1 , , ,
4) are expressed in Eq. 8. The TB bands ob-tained from this four-band model are shown in the Fig. 7for nine cubic halide perovskites and fitting parametersare listed in Table IV, and they completely agree withDFT bands along the broad M-R-X k -path of the cu-bic Brillouin zone. Thus, it validates the minimal basisset based model which captures the essential features ofhalide perovskites around the Fermi level with the lessparametric quantities. In the next section, we furthervalidate this minimal basis set based TB model to lowersymmetry polymorphs.In the present section, we extend our minimal basisset TB model to tetragonal and orthorhombic systems.In the tetragonal phase, the in-plane rotation of the oc-tahedra produces two in-equivalent B atoms. We de-note them as B A and B B . Now the basis set includeeight eigenstates, viz , | s B A i , | p B A x i , | p B A y i , | p B A z i , | s B B i , | p B B x i , | p B B y i , | p B B z i . The corresponding SOC included16 ×
16. Thus, the spin-orbit coupled TB Hamiltonianis envisaged on 16 orbital basis set. For the completeHamiltonian ref, . As the tetragonal phase having apseudo-cubic structure and creates anisotropic interac-tions in in-plane and out of plane directions. Thus, thereare two sets of TB parameters exist corresponding to in-plane and out of plane directions. The DFT fitted TB CsGeCl CsGeBr CsGeI CsSnCl CsSnBr CsSnI CsPbCl CsPbBr CsPbI XM X M X E ne r g y ( e V ) M R R RTBDFT240−2420−2420−2
FIG. 7. Band structure of various halide perovskites ob-tained from four orbital based TB Hamiltonian and comparedwith the DFT band structure. Here, Fermi level is set to zero.TABLE IV. Interaction parameters ( ǫ A s and t A s) and SOCstrength λ in units of eV.B X ǫ s ǫ p t ss t sp t ppσ t ppπ λ Cl 1.17 6.47 -0.26 0.47 0.75 0.09 0.07Ge Br 1.46 6.10 -0.23 0.48 0.84 0.09 0.06I 1.70 5.55 -0.16 0.48 0.86 0.09 0.06Cl 2.08 8.68 -0.25 0.45 0.74 0.10 0.52Sn Br 1.73 7.11 -0.21 0.50 0.77 0.11 0.52I 1.68 6.23 -0.15 0.48 0.83 0.10 0.49Cl 2.46 7.57 -0.31 0.49 0.72 0.10 0.16Pb Br 2.36 6.88 -0.30 0.52 0.79 0.11 0.16I 2.18 6.05 -0.22 0.48 0.85 0.10 0.14 band structures of tetragonal phase for various halideperovskites are listed in Table V. As can be seen, thein-plane interactions’ strength is weak compared to outof plane interactions. Similarly, in the case of the or-thorhombic system, the rotation along c direction createsinequivalence of B atoms, and thus the Hamiltonian ma-trix is designed for 32 orbitals basis set. In Fig. 8, wehave presented the DFT fitted TB band structure andfitting parameters are listed in Table V.The table infersthat the interaction strength has decreased from that ofthe cubic phase, due to octahedral rotations. TABLE V. Interaction parameters ( ǫ and t ) for unstainedequilibrium configurations in the units of eV. The SOCstrength ( λ ) is estimated to be 0.14 eV.Phase Interaction ǫ s ǫ p t ss t sp t ppσ t ppπ Path β − CsSnBr ˆ x ,ˆ y -1.52 2.99 -0.27 0.47 0.8 0.09ˆ z β − CsSnI ˆ x ,ˆ y -1.24 2.59 -0.21 0.45 0.8 0.085ˆ z β − CsPbBr ˆ x ,ˆ y -1 4.2 -0.18 0.38 0.76 0.08ˆ z β − CsPbI ˆ x ,ˆ y z γ − CsSnBr ˆ x ,ˆ y -1.63 3.07 -0.26 0.4 0.68 0.07ˆ z -0.24 0 0.665 0.06 γ − CsSnI ˆ x ,ˆ y -1.23 2.66 -0.18 0.38 0.64 0.08ˆ z -0.24 0 0.665 0.06 γ − CsPbBr ˆ x ,ˆ y -0.9 4.23 -0.14 0.4 0.68 0.05ˆ z -0.16 0.04 0.7 0.07 γ − CsPbI ˆ x ,ˆ y -0.75 3.59 -0.12 -0.32 0.64 0.04ˆ z -0.14 0.02 0.66 0.07 C. Effect of Exchange-Correlation functionals on TBparameters.
Having validated about the minimal basis set TBmodel, now we examine how the TB parameters are sen-sitive towards the XC functionals employed for DFT cal-culations. From the Eq. 9, the bandgap can be definedas E -E , which is thus the function of TB parameters.In the minimal basis set TB model, it turns out to be aE g = E Bp − E Bs − t Bppσ − t Bppπ + 6 t Bss . It can be seen fromthe Fig. 3 and 4 that the bandgap of halide perovskites ishighly sensitive to the type of exchange-correlation func-tionals used in performing DFT calculations. Thus, wewould like to examine the the effect of XC functionalson these effective parameters. In Fig. 9, we have shownthe effective TB parameters for three different XC viz.,
GGA, mBJ and Jishi-mBJ, functional for cubic halideperovskites to know their dependency on XC functionals.The figure infers that difference between the onsite en-ergies increases with the bandgap and exhibit maximumvalues for Jishi-mBJ XC functional. Among all the hop-ping interactions, t ss influence is profound on the bandspectrum and small change in the values affect the bandtopology significantly. M Γ Γ
Z X Z Γ XZ Y R
Γ Γ
XZ Y R
Γ Γ
XZ Y R
Γ Γ
XZ Y R Γ CsSnI CsPbI CsSnBr CsPbBr Γ Γ
Z X Z M
Γ Γ
Z X Z M
Γ Γ
Z X Z(a) (b) (c) (d)(e) (f) (g) (h) E ne r g y ( e V ) −1−2−2−1 DFT TB FIG. 8. DFT fitted TB band structure of tetragonal and orthorhombic crystal phases of various halide perovskites. Here,Fermi level is set to zero. − ε s ε ppp t σ ss t0246800.20.400.40.81.2 Cl Br I Cl Br I Cl Br ISn PbGe PBE PBE+mBJ PBE+J−mBJ E ne r g y ( e V ) FIG. 9. Comparison of various TB parameters of cubichalide perovskites obtained from fitting with the DFT bandstructure under various exchange-correlation functionals.
D. Role of A Site Atom on the Electronic Structure.
To know the role of A site atoms, we have investigatedthe band structure of cubic RbPbX by placing the Rb M R X E CsPbI RbPbI E F FIG. 10. Band structure of CsPbI and RbPbI perovskitesfor the similar lattice parameter. atom at Cs site without relaxing the structure and sameis shown in the Fig. 10. Our results show a tiny increasein the bandgap value of 0.02 eV for RbPbI as comparedwith the CsPbI and no significant changes have beenobserved in band spaghetti. Structural relaxation car-ried out on the RbPbX shows the reduced lattice pa-rameters of the cubic lattice as compared to CsPbX . Incontrast, the structural relaxation study with the organicmolecule at A site shows the larger volume as comparedto the CsPbX . However, the organic molecule intro-duces structural distortion in the octahedral cage, andcompounds show noncentrosymmetric nature and this,in turn, affects the band structure profoundly. E. Surface Band Structure
The bulk electronic structure study has enabled usto analyze the bulk properties of the halide perovskites.However, in many cases, the study requires the surfaceelectronic structure to understand the surface and in-terface phenomena of these systems. The DFT basedcalculations on slab structure require a huge amount ofcomputational time and memory to analyze these prop-erties. Thus to overcome such difficulty, we build SlabTB model and obtain the surface electronic structure.The slab TB model is envisaged by taking four orbitalbasis of the proposed bulk TB model and according thesurface bands are estimated from the bulk Hamiltonianby using the bulk TB parameters.Here, the slab consists of alternative stacking of CsIand BI layers. However, as discussed in section-II, thereare no contribution X-p orbitals at the Fermi level, themodel is restricted to B- { s, p } orbital basis. The appro-priate Hamiltonian for the slab consisting of n unit cellsalong (001) direction. We employ bulk TB parametersand construct the Hamiltonian in the desired direction.The matrix form the TB Hamiltonian is given by H SlabT B = H ↑↑ H ↑↓ H ↓↑ H ↓↓ ! . (13)Where, H ↑↑ = H H . . .H H H . . . H H H . . . ... . . . . . . . . . . . . H n − n − H n − n − H n − n . . . H nn − H nn (14)Here, H jj is the Hamiltonian for j th layer of the slab,and it is given by, H jj = ǫ s + f it xsp S x it xsp S y − it xsp S x ǫ xp + f − iλ − it xsp S y iλ ǫ xp + f
00 0 0 ǫ zp + f (15)Here, f = 2 t xss cos ( k x a ) + 2 t xss cos ( k y a ) ,f = 2 t xppσ cos ( k x a ) + 2 t xppπ cos ( k y a ) ,f = 2 t xppσ cos ( k y a ) + 2 t xppπ cos ( k x a ) ,f = 2 t xppπ cos ( k x a ) + 2 t xppπ cos ( k y a ) . (16)The block H jj − describe the interaction between layer j and j-1. H j − j = ( H jj − ) T = t zss t zsp t zppπ t zppπ − t zsp t zppσ (17)The off diagonal block emerging from the SOC is of theform H ↑↓ = G . . . G . . . G . . . ... . . . . . . . . . . . . G n − n − . . . G nn (18) G jj = λ − iλ λ − iλ . (19)The band spectrum obtained from the diagonalizationsurface TB matrix for cubic, tetragonal and orthorhom-bic phases are shown in Fig. 11 and is compared withWannier functions based TB band structure. IV. CONCLUSIONS
In conclusion, the present work, we have analyzedthe family of halide perovskites using both parameterfree Density functional calculations and parametric tight-binding (TB) model through the Slater-Koster descrip-tion. The analysis has unravelled the dominant orbitaloverlapping interactions that govern the bandgap andplay an important role in exploring the optoelectronicapplications and topological phases. Various exchange-correlation (XC) functionals (PBE, PBE+mBJ, HSE06,HSE and PBE+J-mBJ) were employed on nine halideperovskites in three structural polymorphs. While HSEand PBE+mBJ underestimate the bandgap by a similarmagnitude, PBE+J-mBJ either gives bandgap close tothe experimental value or overestimate it . The cor-rections mBJ and J-mBJ adopts the same formalism, butwith varying parameters. Furthermore, the study revealsthat, though thirteen orbitals are involved in the chemi-cal bonding, a four orbital based tight-binding model isgood enough to capture energy dispersion in the momen-tum space in the vicinity of the Fermi level. The success-ful extension of the present minimal basis TB model toother lower symmetry crystal polymorphs and slab struc-ture for various experimentally synthesized compoundsprovides the effectiveness of this model. Interestingly,the strength of the dominant electron hopping integrals0 Γ M X X Γ X X Γ X E ne r g y ( e V ) Cubic Tetragonal Orthorhombic −110
FIG. 11. Surface band structure of CsPbI polymorphs obtained from the SK-TB formalism (upper panel) and comparedwith band structure of Wannier function based TB model (lower panel). are found to be nearly independent of the XC functionaladopted and therefore, the proposed TB model has be-come more universal. Also, since it excellently repro-duces the surface band structures, it can be further im-proved to study the transport phenomenon as the Wan-nier formalism does. Acknowledgement:
The work is funded by the De-partment of Science and Technology, India, throughGrant No. CRG/2020/004330. We acknowledge the useof the computing resources at HPCE, IIT Madras.
Data availability:
The data that support the find-ings of this study are available from the correspondingauthor upon reasonable request. D. Trots and S. Myagkota, “High-temperature structuralevolution of caesium and rubidium triiodoplumbates,”Journal of Physics and Chemistry of Solids , 2520 – 2526 (2008). Y. Fujii, S. Hoshino, Y. Yamada, and G. Shirane,“Neutron-scattering study on phase transitions of cspbcl ,”Phys. Rev. B , 4549–4559 (1974). S. Hirotsu, J. Harada, M. Iizumi, andK. Gesi, “Structural phase transitions in cspbbr ,”Journal of the Physical Society of Japan , 1393–1398 (1974). R. X. Yang, J. M. Skelton, E. L. da Silva, J. M. Frost, andA. Walsh, “Spontaneous octahedral tilting in the cubic inorganiccesium halide perovskites cssnx and cspbx (x = f, cl, br, i),”The Journal of Physical Chemistry Letters , 4720–4726 (2017). K. Yang, W. Setyawan, S. Wang, M. Buongiorno Nardelli,and S. Curtarolo, “A search model for topological in-sulators with high-throughput robustness descriptors,”Nature Materials , 614–619 (2012). S. Liu, Y. Kim, L. Z. Tan, and A. M. Rappe, “Strain-inducedferroelectric topological insulator,” Nano Letters , 1663–1668(2016). R. Kashikar, B. Khamari, and B. R. K. Nanda, “Second-neighbor electron hopping and pressure induced topologicalquantum phase transition in insulating cubic perovskites,”Phys. Rev. Materials , 124204 (2018). H. Jin, J. Im, and A. J. Freeman, “Topological insulator phase inhalide perovskite structures,” Phys. Rev. B , 121102 (2012). W.-J. Shi, J. Liu, Y. Xu, S.-J. Xiong, J. Wu, and W. Duan, “Con-verting normal insulators into topological insulators via tuningorbital levels,” Phys. Rev. B , 205118 (2015). G. Song, B. Gao, G. Li, and J. Zhang, “First-principles studyon the electric structure and ferroelectricity in epitaxial cssni films,” RSC Adv. , 41077–41083 (2017). M. Kepenekian, R. Robles, C. Katan, D. Sapori, L. Pedesseau,and J. Even, “Rashba and dresselhaus effects in hybrid or-ganic–inorganic perovskites: From basics to devices,” ACS Nano , 11557–11567 (2015). W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronicsfrom inversion symmetry breaking,” Phys. Rev. B , 235406(2008). I. Borriello, G. Cantele, and D. Ninno, “Ab initio investiga-tion of hybrid organic-inorganic perovskites based on tin halides,”Phys. Rev. B , 235214 (2008). R. X. Yang, J. M. Skelton, E. L. da Silva, J. M. Frost, andA. Walsh, “Spontaneous octahedral tilting in the cubic inorganiccesium halide perovskites cssnx and cspbx (x = f, cl, br, i),”The Journal of Physical Chemistry Letters , 4720–4726 (2017). C. Yu, Y. Ren, Z. Chen, and K. Shum, “First-principles study of structural phase transitions in cssni ,”Journal of Applied Physics , 163505 (2013). R. X. Yang, J. M. Skelton, E. L. da Silva, J. M. Frost, andA. Walsh, “Spontaneous octahedral tilting in the cubic inorganic cesium halide perovskites cssnx and cspbx (x = f, cl, br, i),”The Journal of Physical Chemistry Letters , 4720–4726 (2017). L.-y. Huang and W. R. L. Lambrecht, “Electronic band structuretrends of perovskite halides: Beyond pb and sn to ge and si,”Phys. Rev. B , 195211 (2016). R. A. Jishi, O. B. Ta, and A. A. Sharif, “Modelingof lead halide perovskites for photovoltaic applications,”The Journal of Physical Chemistry C , 28344–28349 (2014),https://doi.org/10.1021/jp5050145. B. Traor´e, G. Bouder, W. Lafargue-Dit-Hauret, X. Roc-quefelte, C. Katan, F. Tran, and M. Kepenekian, “Effi-cient and accurate calculation of band gaps of halide per-ovskites with the tran-blaha modified becke-johnson potential,”Phys. Rev. B , 035139 (2019). S. Boyer-Richard, C. Katan, B. Traor´e, R. Scholz, J.-M. Jancu,and J. Even, “Symmetry-based tight binding modeling of halideperovskite semiconductors,” The Journal of Physical ChemistryLetters , 3833–3840 (2016). G. Kresse and J. Furthm¨uller, “Efficient iterative schemes forab initio total-energy calculations using a plane-wave basis set,”Phys. Rev. B , 11169–11186 (1996). D. R. Hamann, “Semiconductor charge densi-ties with hard-core and soft-core pseudopotentials,”Phys. Rev. Lett. , 662–665 (1979). P. Blaha, K. Schwartz, G. Madsen, D. Kvasnicka, and J. Luitz,
WIEN2k An Augmanted Plane Wave+Local Orbitals Programfor Calculating Crystal Properties (Karlheinz Schwartz, Tech.Universitt Wien, Austria, 2001). J. P. Perdew, K. Burke, and M. Ernzerhof,“Generalized gradient approximation made simple,”Phys. Rev. Lett. , 3865–3868 (1996). F. Tran and P. Blaha, “Accurate band gaps of semiconductorsand insulators with a semilocal exchange-correlation potential,”Phys. Rev. Lett. , 226401 (2009). G. Thiele, H. W. Rotter, and K. D. Schmidt, “Kristall-strukturen und Phasentransformationen von Caesiumtri-halogenogermanaten(II) CsGeX3 (X = Cl, Br, I),”ZAAC - Journal of Inorganic and General Chemistry , 148–156 (1987). K. Yamada, K. Isobe, T. Okuda, and Y. Furukawa,“Successive Phase Transitions and High Ionic Con-ductivity of Trichlorogermanate(II) Salts as Stud-ied by 35Cl NQR and Powder X-Ray Diffraction,”Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences , 258–266 (1994). R. X. Yang, J. M. Skelton, E. L. Da Silva, J. M. Frost, andA. Walsh, “Spontaneous octahedral tilting in the cubic inorganiccesium halide perovskites CsSnX3 and CsPbX3 (X = F, Cl, Br, I),” Journal of Physical Chemistry Letters , 4720–4726 (2017). D. H. Fabini, G. Laurita, J. S. Bechtel, C. C. Stoumpos,H. A. Evans, A. G. Kontos, Y. S. Raptis, P. Falaras, A. derVen, M. G. Kanatzidis, and R. Seshadri, “Dynamic Stereo-chemical Activity of the Sn2+ Lone Pair in Perovskite CsSnBr3,”Journal of the American Chemical Society , 11820–11832 (2016). K. Yamada, S. Funabiki, H. Horimoto, T. Matsui, T. Okuda,and S. Ichiba, “Structural phase transitions of the poly-morphs of cssni3 by means of rietveld analysis of thex-ray diffraction,” Chemistry Letters , 801–804 (1991),https://doi.org/10.1246/cl.1991.801. Y. Fujii, S. Hoshino, Y. Yamada, and G. Shirane,“Neutron-scattering study on phase transitions of CsPbCl3,”Physical Review B , 4549–4559 (1974). S. Hirotsu, J. Harada, M. Iizumi, andK. Gesi, “Structural phase transitions in cspbbr3,”Journal of the Physical Society of Japan , 1393–1398 (1974),https://doi.org/10.1143/JPSJ.37.1393. A. Marronnier, G. Roma, S. Boyer-Richard, L. Pedesseau, J. M.Jancu, Y. Bonnassieux, C. Katan, C. C. Stoumpos, M. G.Kanatzidis, and J. Even, “Anharmonicity and Disorder in theBlack Phases of Cesium Lead Iodide Used for Stable InorganicPerovskite Solar Cells,” ACS Nano , 3477–3486 (2018). http://savrasov.physics.ucdavis.edu/mindlab/MaterialResearch/Databases/indexatoms.htm. L. Y. Huang and W. R. Lambrecht, “Electronic bandstructure, phonons, and exciton binding energies ofhalide perovskites CsSnCl3, CsSnBr3, and CsSnI3,”Physical Review B - Condensed Matter and Materials Physics , 1–12 (2013),arXiv:1510.01834. S. Tao, I. Schmidt, G. Brocks, J. Jiang, I. Tranca,K. Meerholz, and S. Olthof, “Absolute energy levelpositions in tin- and lead-based halide perovskites,”Nature Communications , 1–10 (2019), arXiv:1902.06646. D. B. Straus, S. Guo, and R. J. Cava, “KineticallyStable Single Crystals of Perovskite-Phase CsPbI3,”Journal of the American Chemical Society , 11435–11439 (2019). J. C. Slater and G. F. Koster, “Simplified lcao method for theperiodic potential problem,” Phys. Rev. , 1498–1524 (1954). M. G. Goesten and R. Hoffmann, “Mir-rors of bonding in metal halide perovskites,”Journal of the American Chemical Society , 12996–13010 (2018),pMID: 30207152, https://doi.org/10.1021/jacs.8b08038. R. Kashikar, B. Khamari, and B. R. K. Nanda, “Second-neighbor electron hopping and pressure induced topologicalquantum phase transition in insulating cubic perovskites,”Phys. Rev. Materials2