Tight-binding description of inorganic lead halide perovskites in cubic phase
TTight-binding description of inorganic lead halide perovskites in cubic phase
M. O. Nestoklon
Ioffe Institute, St. Petersburg 194021, Russia
Band structure of inorganic lead halide perovskites is substantially different from the band struc-ture of group IV, III-V and II-VI semiconductors. However, the standard empirical tight-bindingmodel with sp d s ∗ basis gives nearly perfect fit of band structure calculated in the density func-tional theory. The tight-binding calculations of ultrathin CsPbI layer show good agreement withthe corresponding DFT calculations. The parameters corrected for the experimental data allow forthe numerically cheap atomistic calculations of inorganic perovskite nanostructures. I. INTRODUCTION
Lead halide perovskites attracted a lot of attention lastdecade due to excellent optical properties and simple pro-cessing [1–4]. Clearly, design of new perovskite-basedstructures and devices calls for theoretical models whichwould adequately describe their properties. While mod-ern methods of electron structure calculations allow oneto compute the band structure of bulk semiconductorswith reasonable precision [5], the use of density functionaltheory (DFT) for nanostructure calculations is challeng-ing, and more computationally accessible approaches arerequired. For large structures and qualitative analysis,effective mass approximation [6] is the method of choice.For intermediate size nanostructures, the attraction istoward atomistic empirical methods [7, 8]. Empiricaltight-binding method (ETB) is one of the simplest ap-proximations suitable for an accurate description of theband structure of conventional semiconductors with low-est possible computational cost [8].The band structure of inorganic perovskites and hy-brid organic-inorganic perovskites is substantially differ-ent from the band structure of widely used group IV,III–V, and II–VI semiconductors. First of all, bulk per-ovskites have a complex sequence of structural phasetransitions. To make the analysis simpler, it is natural tofirst consider the high-symmetry cubic structure and thendescribe energy bands of lower symmetry crystal phasesas originating from a folded and distorted band structureof the most symmetric phase [9]. The band structure ofthe cubuc phase is also important by itself: recently ithas been demonstrated that, in small nanocrystals, thecubic phase is stabilized by the surface up to room tem-perature [10]. In cubic perovskites, the direct band gap islocated at the R point of the Brillouin zone which has O h point symmetry. The top of the valence band is formedby the states which transform under Γ − representation(in Koster notation [11]), while the bottom of the con-duction band is formed by Γ +6 states [12]. The differencewith III–V and II–VI direct band gap semiconductors isin the formation of these bands: in III–V and II-VI com-pounds, the spin-orbit interaction forms the valence bandstates and is negligible in the conduction band while, inperovskites, the conduction band is formed by the largespin-orbit interaction. In perovskites, both the valenceband top and conduction band bottom are two-fold de- generate while the top of the valence band in III-V andII-VI compounds is four-fold degenerate which results inmuch simpler selection rules for the optical transitions.The empirical tight-binding method with sp d s ∗ ba-sis in the nearest neighbor approximation gives a pre-cise description of the band structure of bulk III–V [13]and group IV [14] semiconductors. It has a long historyof application to nanostructures [15–20]. Parameters ofthe method may be in principle extracted from the DFTcalculations [21–23] which makes this method an atom-istic interpolation of the underlying DFT model. It maybe extended to describe other materials as well but, forperovskites, ETB has been used mostly for a qualitativeanalysis of the band structure [12, 24]. Below we showthat it can be used to describe the band structure of in-organic perovskites with a meV-range precision.The empirical methods of band structure calculationsheavily rely on the band structure they are assumed tofit. For this purpose we use the band structure resultingfrom DFT calculations with the modified Becke-Johnsonexchange-correlation potential [25] in Jishi parametriza-tion [26]. While the band-gap energy is known to be un-derestimated in most DFT calculations, this exchange-correlation potential is almost free from this deficiency[27] while it may give a noticeable error in masses andpositions of secondary maxima in the conduction band[28]. However, the standard DFT calculations givethe “bare” electron structure without account on theelectron-phonon interaction. In many semiconductors,the renormalization of the band structure by the electron-phonon interaction is within few tens meV [29], whichis beyond standard precision of DFT calculations. How-ever, for lead halide perovskites this effect is estimated tobe of the order of hundreds meV [30]. This value is alsolarge in other perovskites [31, 32]. To account for thissystematic error, we use the approach which used to bethe “golden standard” when high-quality electron struc-ture calculations were unavailable. First, we obtain thebest possible fit of the band structure obtained in DFTcalculations. Next, we consider a thin perovskite slab andcompare the in-plane dispersion of the electron states inthis structure calculated in DFT and ETB. Nearly per-fect agreement between the two approaches shows thatETB may be used for the calculations of the perovskitenanostructures. Finally, we revise this parametrizationin order to introduce corrections aimed to describe the a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n experimental data on the material band structure. Thisprocedure gives the two sets of parameters: one is suit-able for a direct comparison with the DFT calculations,the other one may be used in nanostructure calculationsto be compared with the experimental data. II. DENSITY FUNCTIONAL CALCULATIONSOF CUBIC CsPbI For the band structure calculations we use theWIEN2k package [33]. For better comparison with ex-perimental data, we use the modified Becke-Johnsonexchange-correlation potential [25] in Jishi parametriza-tion [26]. The lattice constant of cubic CsPbI is set to a = 6 .
289 ˚A. All parameters of the convergence are de-fault except R MT · K max = 9, l vns = 6, and the FFTmesh enhancement factor 4. In addition, for spin-orbitinteraction RLO [34] is added on Pb atoms and emax =10is used. We use k -mesh 14 × ×
14. The energy andcharge convergence were set to 10 − and 10 − , respec-tively. With these convergence parameters, for the bandgap we obtain the value E g = 1 .
017 eV. Note that theexperimental value of the band gap of CsPbI in the cu-bic phase is E g = 1 .
65 eV [35]. The large differencebetween DFT results and experimental data for this ma-terial is usually attributed to the renormalization of theband structure by the electron-phonon interaction [29].For lead halide perovskites this effect is estimated to beof the order of hundreds meV [30].In WIEN2k, Kohn-Sham equations are solved in thebasis which is obtained as a numerical combination ofSlater-like numerical orbitals near atoms and plane wavesbetween the spheres around atoms. For the valence bandstates, most of electron density is located near the atomswhich allows for an unambiguous definition of the angu-lar momentum-resolved partial density of states. This isin line with the general idea of the tight-binding methodwhich assumes that the band structure may be obtainedin the basis of (in principle, unknown) functions local-ized near atoms and corresponding to a certain value ofthe orbital angular momentum [36]. From the analysisof DFT calculations, one may conclude that the valenceband is mostly constructed from functions with orbitalangular momentum 1 (“ p -orbitals” in ETB) on halogen(I in case of CsPbI ) atoms with significant admixtureof functions with zero momentum (“ s -orbitals”) on Pb.Conduction band is constructed almost exclusively fromfunctions with orbital angular momentum 1 on Pb. Csatoms give few flat bands deep inside valence band andwe drop them from the ETB parametrization, followingRef. [12]. This analysis gives a good starting point forthe ETB parametrization without explicit interpolationbetween DFT and ETB wave functions [22]. M R Γ X M Γ − E ne r g y ( e V ) FIG. 1. Comparison of CsPbI band structure calculated inDFT and in empirical tight-binding method using sp modelwith parameters from Table I. Green dashed lines show theDFT results, black solid lines ETB results. III. INTERPOLATION OF DFT BANDSTRUCTURE OF CUBIC CsPbI IN EMPIRICALTIGHT-BINDINGA. ETB parametrization of DFT results in sp basis The use of complete sp d s ∗ model is critically impor-tant to obtain the correct energy positions of the bandsin all high-symmetry points of the Brillouin zone, see be-low. However, it is instructive to start from the minimalmodel and check if the band structure may be fit to be inagreement with the DFT results. To fit the ETB param-eters, we begin with a straightforward minimization ofthe difference between DFT and ETB energies in specialpoints of the Brillouin zone starting from the parame-ters of Ref. [12]. The starting parameters are importantto have the correct order of the bands. In this case theprocedure works even without special attention to localminima. After this general fit, which indeed gives bandstructure similar to the DFT results, we refine the ETBparameters to exactly reproduce few selected energies inthe special points calculated in the DFT. First we fit thetop of the valence band in R point, then the bottom ofthe conduction band in R point, then position of Pb s -band in R point (band near − p -band in Γ point (band near − M point.The result of the fit is presented in Fig. 1. The spin-orbitinteraction is taken from the sp d s ∗ fit, see below.Within the minimal sp model, it is impossible to fur-ther increase the number of points to be fitted exactly.In particular, note the incorrect position of the bottom ofthe conduction band in Γ point which arises from the ab-sence of higher bands in the minimal ETB model. How-ever, the overall description of the band structure is sur-prisingly good. In particular, this model “automatically”reproduces the position of the valence band maximum in M point and the splitting of bands in the region from TABLE I. Tight-binding parameters used in calculations.In addition to parameters presented in the table, s ∗ s ∗ σ = s c s ∗ a σ = s ∗ a d c σ = p c d a σ = p a d c π = p c d a π = 0. sp sp d s ∗ expt. corrected E sa − . − . − . E sc − . − . − . E s ∗ a . . E s ∗ c . . E pa − . − . − . E pc .
774 3 . . E da . . E dc . . ssσ . − . − . s a s ∗ c σ . . s a p c σ .
627 0 . . s c p a σ .
048 1 . . s ∗ a p c σ . . s ∗ c p a σ − . − . s a d c σ . . s c d a σ . . s ∗ c d a σ . . ppσ − . − . − . ppπ .
635 0 . − . p a d c σ . . ddσ − . − . ddπ . . ddδ − . − . a / .
300 0 . . c / .
433 0 . . − − − − sp d s ∗ model, but abrief comment should be added concerning the use of anintermediate sp s ∗ model. Similar to simple direct bandgap semiconductors [37], the sp s ∗ model allows one tofit the position of the conduction band bottom in Γ point.However, the behavior of the second conduction band isnot reproduced even qualitatively and we skip a detaileddiscussion of this model. B. ETB parametrization of DFT results in sp d s ∗ basis The minimal sp model does not reproduce the bandstructure of the bands near the band gap far from the R point. This is of critical importance for an accuratedescription of the nanostructures as the details of theband structure in the whole Brillouin zone are impor-tant [38]. We follow the strategy proved to be efficientfor III-V semiconductors [13]: we start from the sp pa- M R Γ X M Γ − E ne r g y ( e V ) FIG. 2. Comparison of CsPbI band structure calculatedin DFT and in empirical tight-binding method. The DFTcalculations are shown in green dashed lines, tight-bindingresults are in thin black lines. rameters fitted to the DFT band structure and add s ∗ and d orbitals in the model with energies close to the en-ergies expected from the free-electron dispersion. Then,we reoprimize the parameters allowing for the interactionbetween the sp and d s ∗ orbitals. In this case, generaloptimization of the parameters is not necessary and wemay start directly from the exact fit of few selected spe-cial points. The result of this improved parametrizationis shown in Fig. 2. The resulting fit is nearly perfect,with the exception of the bands in the region from − − sp s d s ∗ model. To show this, onemay numerically evaluate the Jacobian matrices ∂E i ( k k ) ∂p j of the derivatives of energies in the special points of theBrillouin zone with respect to the ETB parameters andsee that there are no parameters which would allow oneto split these bands in the correct order. It may be shownthat this is not the local optimum by considering small,but not negligible change of parameters. For the exactETB description of these bands the use of second neigh-bor interaction or a generalization of the Slater-Kosterscheme [36] is necessary. IV. TIGHT-BINDING DESCRIPTION OFSURFACE OF CUBIC CsPbI The main aim of the current study is the ETB param-eters which are suitable for the description of nanostruc-tures from inorganic halide perovskites. In the previoussection it has been demonstrated that the description ofthe band structure of bulk CsPbI with nearest neighbor sp d s ∗ ETB is possible. For the nanostructures, the de-scription of the surface is also important. To check thatthe description of the surface is reasonable, one shouldcompare the tight-binding results with DFT calculations
R X M Γ Z − E ne r g y ( e V ) FIG. 3. Band dispersion in CsPbI nanoplatelet calculatedin DFT following the procedure outlined in Sec. II is shown ingreen dashed line and the dispersion calculated in ETB withthe parameters obtained in Sec. III B is shown in black solidlines. To the right, 3D view [41] of arrangement of atoms inelementary cell of CsPbI platelet used for the comparison ofDFT and ETB. for a structure with a surface.For this purpose, we calculate the dispersion in aCsPbI nanoplatelet in DFT using the same approachand convergence parameters we used for the band struc-ture calculations of bulk CsPbI in Section II. An elemen-tary cell of the calculated structure is shown in Fig. 3.To model the surface we add 14 ˚A of vacuum in the “ z ”direction. We set constant parameters c [25] taken fromthe converged bulk band structure calculation.Properties of the halide perovskite surface are sensi-tive to details of the surface passivation [40]. Withoutpassivation, a structure with an iodine surface in DFT ismetallic. However, a structure with a surface abruptlycut at the PbI plane demonstrates the band structure onewould expect from a quantum-well like structure withquantum confined states localized in the middle of theplatelet. The band structure of such a system calcu-lated in DFT is shown in Fig. 3. To simplify comparisonwith the ETB results we neglect the structure relaxation.In DFT calculations with structure relaxation (not pre-sented here), the surface atoms are noticeably shifted,but the band structure remains qualitatively the same,with some band gap change. To compare the DFT re-sults for the relaxed structure with ETB calculations weneed to incorporate the deformation in TB model, whichis beyond the scope of the current paper.The band structure of this slab calculated using ETBwith the parameters obtained in the previous Section arealso presented in Fig. 3. As seen from the comparisonof the two calculations, ETB qualitatively correctly de-scribes the surface. For a detailed model of a realisticsystem with a particular passivation the procedure pre-sented here should be significantly extended allowing forthe passivation within the ETB model. For any particu-lar passivation one might need a separate set of the sur- face ETB parameters. However, this would heavily relyon the DFT calculations of the passivated surface whichis complicated by the fact that most of the standard pas-sivation agents used in experiments (oleic acid, etc. [40])are relatively large organic molecules. We also neglectthe effect of dielectric confinement [42]. Note, however,that the effect of dielectric confinement is usually can-celed out by the excitonic effects [43, 44]. V. ETB PARAMETERS CORRECTED FORTHE EXPERIMENTAL DATA
In previous sections, it has been demonstrated thatETB gives nearly perfect fit of cubic CsPbI band struc-ture calculated in DFT. In addition, even without specialparametrization, it gives a reasonable description of theelectronic structure of the surface of this material. How-ever, the agreement between resulting ETB and experi-mental data is not satisfactory, due to the large differencebetween the underlying DFT results and experiment. Tobe able to compare ETB results with experimental data,we alter the ETB parameters obtained from the fit of theDFT band structure to reproduce experimental data onthe energy bands of cubic CsPbI .For the experimental data we refer to Ref. [35], in thiswork authors associate the three peaks in optical absorp-tion of α -phase (cubic) CsPbI at 1.65 eV, 2.75 eV and3.40 eV with the direct transitions between the valenceand conduction band in R point, in M point and in Xpoint of the Brillouin zone, respectively. In our DFT cal-culations, the spin-orbit splitting of the conduction bandin R point is 1.48 eV. The value of spin-orbit splitting israther stable against any perturbations including a bandgap change, which means that the direct transition inR point should be accompanied by a transition which is1.48 eV above. It allows us to assume that the thirdpeak observed in the experiment [35] corresponds to thedirect transition in R point from the top of the valenceband to the second conduction band. Then, the energiesof all the direct transitions in our DFT calculations areunderestimated for 400 ÷
600 meV as compared with theexperimental data which we associate with neglect of theelectron-phonon interaction [29]. Based on this consid-eration, we change the target position of the conductionband bottom in R and M to have the direct band gaps inthese points equal to E Rg = 1 .
65 eV and E Mg = 2 .
75 eV.We set the new values of energies in special points of theBrillouin zone and change the parameters accordingly us-ing the procedure outlined above. The results are pre-sented in Fig. 4. For comparison, we included also ETBcalculations using parameters from Ref. [12].It should be noted, that the experimental data inRef. [35] were obtained at relatively high temperature:bulk CsPbI undergoes phase transitions to lower sym-metry phases at 260 ◦ C and 160 ◦ C. Our main goal is tofind the parameters suitable for the ETB calculations ofperovskite quantum dots at room temperature. It has
M R Γ X M Γ − E ne r g y ( e V ) FIG. 4. The band structure calculated using the ETB pa-rameters fit to DFT results and corrected for the experimentaldata. Grey dotted line shows the band structure calculatedusing parameters from Ref. [12]. been demonstrated experimentally that, in small QDs, a(meta)stable cubic phase of CsPbI may exist down toroom temperature and below [10]. This stabilization isassociated with the balance between the bulk and sur-face energies of different phases [10, 45]. We hypothe-size that the mechanism of the cubic phase stabilizationshould have an effect on CsPbI band gap similar to thatof the elevated temperature. Under this assumption, weuse the experimental band gap of the cubic perovskite athigh ( ∼ ◦ C) temperature to fit the ETB band struc- ture corrected for the experiment. Also note that in thesematerials the change of the band gap with the temper-ature is surprisingly small as compared with the zero-temperature correction.
VI. CONCLUSIONS
In conclusion, the band structure of the inorganic cu-bic perovskite CsPbI calculated using the density func-tional approach has been fitted near the band gap with ameV precision by the standard nearest-neighbor sp d s ∗ empirical tight-binding model. Description of complexcrystal-field splitting of few bands formed from the p -orbitals of halogen atoms is problematic for the nearest-neighbor model and an extension of the model is neces-sary for a precise description of these flat valence bands.It has been demonstrated that the empirical tight-binding gives a qualitatively correct description of thePbI [001] surface of CsPbI without passivation, whichopens the way to numerically cheap simulations of inor-ganic perovskite based nanostructures. A set of empiricaltight-binding parameters carefully fit to the available ex-perimental data has been obtained. ACKNOWLEDGMENTS
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