1

Analysis of multi-valley and multi-bandgap

absorption and enhancement of free carriers related

to exciton screening in hybrid perovskites

Jacky Even,*† Laurent Pedesseau,†and Claudine Katan*‡

†Université Européenne de Bretagne, INSA, FOTON, UMR 6082, 35708 Rennes, France

‡CNRS, Institut des Sciences Chimiques de Rennes, UMR 6226, 35042 Rennes, France

CONTENTS:

Computational details and optical matrix elements...…………… P2

Symmetry analysis………………………………………..………... P4

Band structures of CH3NH3PbI3 and CH3NH3PbCl3…................... P5

Absorption spectra..……………...........................……………...….. P6

Gap (exciton) switching..….…………………………...……………. P8

Dielectric properties.…………………………………...…………… P8

References…………..……………...…………...……………….…... P10

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Computational details and optical matrix elements

The first principles calculations were carried out using the DFT (ref. S1) implementation available in the ABINIT pseudo-potential packageS2 with LDA or GGA (ref. S3) for exchange-correlation. Relativistic, norm-conserving, separable, dual-space Gaussian-type pseudopotentials of Goedecker, Teter and Hutter (HGH) have been used for all atoms in order to include the spin-orbit coupling (SOC)S4. The electronic wave-functions are expanded onto a plane-wave basis set with an energy cut-off of 680 eV. This level of theory has already been checked, especially against all-electron computations and for the respective number of core and valence electrons in our previous worksS5-S8. Optical absorption spectra were computed from the imaginary part

(ε2(ω)) of the macroscopic dielectric functionS9 with a 16x16x16 grid except for the cubic phase for which a 24x24x24 grid was chosen. The Bethe-Salpeter equation (BSE) was solved for the cubic phase with a 12x12x12 gridS6,S9. All calculations have been performed with experimental crystal structures: CH3NH3PbI3

(refs S10,S11), CH3NH3PbCl3 (refs S12,S13).

Moreover, it is well known that the DFT (LDA or GGA) bandgaps are underestimatedS9. We have shown in earlier worksS5,S6 that the nice agreement between DFT bandgaps and values deduced from experiments is fortuitous and stems from error cancellations: both SOC and GW (ref. S9) corrections are large and act in opposite directions. Unfortunately, taking both these effects into account is beyond available computational resources for large systems. Nevertheless, the systematic bandgap increase observed upon halogen substitution (I, Br, Cl) is well accounted for within DFT both with and without SOC (Table S1). Thus, while we believe that the qualitative shape of the reported band structures calculated with SOC should be correct, the reader should keep in mind that GW corrections will not necessarily induce simple rigid shifts over the whole Brillouin Zone (BZ). Indeed renormalization is expected to be larger at critical points (e. g. R in the cubic phase)S9.

Table S1. Comparison between calculated electronic bandgaps for CH3NH3PbX3 (X=I, Br, Cl), and optical bandgaps measured at room temperature (RT). Calculated values are reported for the low temperature (LT) orthorhombic Pnma phase and room temperature (RT) cubic Pm3m phase. Values in brackets have been obtained without SOC.

Eg (eV) X=I X=Br X=Cl

LT (GGA) 0.5 (1.5) 0.8 (1.9) -

LT (LDA) 0.4 (1.4) - (1.9) 1.2 (2.3)

RT (LDA) 0.2 (1.3) 0.5 (1.5) 0.6 (1.8)

RT (exp.) 1.5 (ref. S14) 2.3 (ref. S14) 3.1 (ref. S15)

Considering two-particle wave function and effective mass equations for electron and hole, optical absorption spectra (Figures 2 and S5) have been computed for bound and continuum statesS16 (see expressions (1)-(3) of the manuscript). The energy and absorption of the excitonic bound states are related to the Rydberg energy of the transition. The absorption of the continuum states is enhanced by the so-called Sommerfeld factor. The whole absorption spectrum was computed for different dielectric constants in order to highlight screening effects.

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The strength of the optical transitions are evaluated from energy parameters for the optical matrix

elements, obtained for each pair of [CB,VB] states from the expressionS7,S16: 22VBCB

ei

mψψψψψψψψ ∇− h .

Values given Tables S2-S5 take into account the level degeneracies (depicted Figure 1c), spin degeneracies and the threefold contribution stemming from the fact that the star of kM contains three arms (see symmetry analysis). In the case of the fundamental optical transition these values can be related to the Kane energyS7,S16.

Table S2. Energy parameters quantifying the optical matrix elements and transition energies (eV) with respect to the bandgap energy (Eg) for the cubic phase of CH3NH3PbI3 calculated at the LDA level of theory without SOC. Energy parameters account for level degeneracies (depicted Figure 1c), spin degeneracies and the threefold contribution at M.

Eg (eV) RuT1 M

uE

RgA1 40 (Eg) 0

MgA1 0 120 (Eg+1.0)

RgT1 16 (Eg+0.8) 0

Table S3. Energy parameters quantifying the optical matrix elements and transition energies (eV) with respect to the bandgap energy (Eg) for the cubic phase of CH3NH3PbI3 calculated at the LDA level of theory with SOC. Energy parameters account for level degeneracies (depicted Figure 1c), spin degeneracies and the threefold contribution at M.

Eg (eV) RuE 2/1 M

uE 2/1 RuF 2/3

RgE 2/1 17 (Eg) 0 18 (Eg+1.6)

MgE 2/1 0 60 (Eg+1.4) 0

RgF 2/3 5 (Eg+0.8) 0 10 (Eg+2.4)

Table S4. Energy parameters quantifying the optical matrix elements and transition energies (eV) with respect to the bandgap energy (Eg) for the cubic phase of CH3NH3PbCl3 calculated at the LDA level of theory without SOC. Energy parameters account for level degeneracies (depicted Figure 1c), spin degeneracies and the threefold contribution at M.

Eg (eV) RuT1 M

uE

RgA1 38 (Eg) 0

MgA1 0 132 (Eg+1.3)

RgT1 26 (Eg+1.9) 0

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Table S5. Energy parameters quantifying the optical matrix elements and transition energies (eV) with respect to the bandgap energy (Eg) for the cubic phase of CH3NH3PbCl3 calculated at the LDA level of theory with SOC. Energy parameters account for level degeneracies (depicted Figure 1c), spin degeneracies and the threefold contribution at M.

Eg(eV) RuE 2/1 M

uE 2/1 RuF 2/3

RgE 2/1 14 (Eg) 0 23 (Eg+1.7)

MgE 2/1 0 63 (Eg+1.7) 0

RgF 2/3 6 (Eg+1.9) 0 20 (Eg+4.2)

Symmetry analysis The symmetry analysisS17 of the electronic states at M and R of the BZ leads to the interpretation of the SOC induced splitting and optically allowed transitions. Point groups of vectors kR (1/2,1/2,1/2) and kM (1/2,1/2,0) are Oh and D4h, respectively. Alternative spin-orbitals notations for the E1/2g, F3/2g, E1/2u, F3/2u irreducible representations (IR) at R may be usedS5,S17. E1/2g, F3/2g IR are respectively associated to combinations of

g2/1,2/1 ± and

g2/1,2/3 ±

g2/3,2/3 ± ,

whereas E1/2u, F3/2u IR are respectively associated to combinations of u

2/1,2/1 ± and

u2/1,2/3 ±

u2/3,2/3 ± . All the cross products between IR of the CB and VB at R, given in

Figure 1c, contain the T1u vectorial representation, indicating allowed and isotropic transitions. The cross products between IR at M (Figure 1c) contain the Eu vectorial representation, indicating allowed and transverse isotropic transitions (in the (x,y) plane for kM). Isotropic optical transitions are predicted because the star of kM contains three arms: (1/2,1/2,0), (0,1/2,1/2) and (1/2,0,1/2) which all contribute to the observed absorption.

5

Band structures of CH3NH3PbI3 and CH3NH3PbCl3

Figure S1. Electronic band structure for the high temperature cubic Pm3m phase of CH3NH3PbI3 with SOC at the LDA level of theory. An upward energy shift of 1.4eV has been applied to match the experimental bandgap at R.

Figure S2. Electronic band structure for the high temperature cubic Pm3m phase of CH3NH3PbI3 without SOC at the LDA level of theory. An upward energy shift of 0.3eV has been applied to match the experimental bandgap at R.

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Figure S3. Electronic band diagrams for the high temperature cubic Pm3m phase of CH3NH3PbCl3 at the LDA level of theory. a, without spin orbit coupling (SOC). b, with SOC.

Absorption spectra

Figure S4. Optical absorption spectra for the high temperature cubic Pm3m phase of CH3NH3PbCl3 with (BSE, red) and without (RPA, black) the excitonic interaction. Onsets of optical transitions at the R (RgE ) and M ( M

gE ) and exciton binding energy (bxE ) are indicated.

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Figure S5. Computed optical absorption spectra of CH3NH3PbI3 for bound and continuum pair states, considering two-particle wave function and effective mass equations for electron and hole for various effective dielectric constants. It shows the influence of exciton screening, both on the bound pair states below the bandgap, and on the continuum of pair states above the bandgapS16. Expression (3) is used with eV03.0=γγγγ and em16.0=µµµµ . The same energy gap (1.685 eV) is considered for all spectra (no gap switching). Monoelectronic states dispersions are described in the parabolic (effective mass) approximation.

Regarding the comment on exciton binding energies deduced from magnetoabsorption spectroscopy measurements at liquid helium temperatureS18-S20, we further detail some technical points. In these experiments, the exciton binding energy Ry , the reduced massµµµµ and the Bohr

radius Ba for the 1S bound state have been deduced for CH3NH3PbX3 based on a 3D Wannier exciton model equivalent to the one used in this work (expressions (1)-(3)), with an additional contribution related to magnetic coupling. The relations used by the authors to determineRy , µµµµ

and Ba are:

3/1

4

=

c

c

RyRy

eff

H

H εεεε,

3/1

,

=

H

eff

HB

B

c

c

a

a εεεεand

3/12

=

c

c

mHeff

e

εεεεµµµµ, where c is the

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diamagnetic shift of the CH3NH3PbI3 or CH3NH3PbBr3 crystals and, HRy , HBa , , Hc are relevant

parameters for the hydrogen atom. Exciton binding energies deduced for CH3NH3PbI3 amount to 37 (ref. S18) 45 (ref. S19) and 50 (ref. S20) meV. All these values have been obtained using

∞= εεεεεεεε eff and differ due to different experimental determination of the diamagnetic shift c .

Here, we underline that effεεεε is an additional parameter that also strongly affects evaluation of

exciton binding energies.

Gap (exciton) switching

We briefly comment on the so-called “exciton-switching” phenomenon reported for 2D hybrid perovskites at the monoclinic-orthorhombic structural phase transitionS19,S21. In 2D hybrids, exciton-switching has been discussed in relation with the shift of the emission spectrum. A similar shift has been observed experimentally at Tc=162K for CH3NH3PbI3 (ref. S21), but the exciton peak is clearly screened (Figure 2 and S5) and becomes washed out as compared to the narrow exciton peak of 2D hybrids. Our DFT studiesS6-S8 have shown, for both 2D and 3D hybrids, that this shift is primarily related to a shift in energy of the monoelectronic states close to the gap. This energy shift results from a coupling to lattice strain and in-plane and out-of-plane tilts of the inorganic octahedraS5-S8,S22.

Dielectric properties Available experimental data on dielectric properties of CH3NH3PbI3, CH3NH3PbCl3 and CsPbCl3

crystals are summarized in Table S6. These values have been used to draw schematically the variation of the dielectric constant as a function of frequency (Figure 4).

Table S6. Experimental data of dielectric constants reported for various frequency ranges for CH3NH3PbI3, CH3NH3PbCl3 and CsPbCl3. Temperature is given in parentheses and RT denotes room temperature.

Eg (eV) CH3NH3PbI3 CH3NH3PbCl3 CsPbCl3

RT Phase

I4cmS11 Pm3mS12 Pm3mS24

εs (1KHz) 62(RT)/36(20K)S23 45(RT)/17(20K)S23 -

ε90GHz (90GHz) 32(RT)/30(100K)S12 26(RT)/19(100K)S12 -

ε600GHz (<600GHz) - - 25(358K)S24

ε∞ 6.5S18 - (>10THz) 2(358K)S24

First, the high frequency limit optical dielectric constant of CH3NH3PbI3 reported in Ref. S18, ε∞ ≈6.5, is consistent with present DFT calculations, from which we deduce an average value of 5.5 for the I4cm phase, and earlier DFT results by Brivio et al.S25

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Next, in the medium frequency range (90GHz), the dielectric constant increases to about 30 both at room and low temperature. In a recent Raman scattering study of CH3NH3PbI3, low frequency optical phonons have been observed and attributed to vibrational modes of the inorganic latticeS26. Similar results have also been evidenced in CH3NH3PbCl3 (ref. S27). A detailed analysis of such modes has been performed for the high temperature Pm3m cubic phase of CsPbCl3

S24,S28,S29, an all-inorganic analogue for which a similar increase of the dielectric constant has been reportedS28 (Table S6). For CsPbCl3, the change of the dielectric constant in the far infrared frequency range has been attributed to odd-type phononsS24. A straightforward symmetry analysis of the Brillouin zone center phonons yields 4T1u+T2u decomposition into 5 three-fold degenerate and non-Raman active modes. One of the T1u modes is related to acoustic phonons and the three remaining are infrared active. The T2u optical mode is inactive, both in Raman and infrared spectroscopiesS28,S29. Kramers-Kroenig analysis of reflectivity measurements indicates that the increase of the dielectric constant is due to the three T1u infrared active modesS24. These modes are mainly associated to distortions of the CsPbCl3 inorganic latticeS29. From this, we can conclude that the dielectric constants increase in the medium to high frequency range of CH3NH3MX 3 stems from polar modes of the inorganic lattice.

In the static limit, a further increase of the dielectric constants shows up at room temperature (Table S6, Figure 4). Below the tetragonal to orthorhombic phase transition, this increase is suppressed and dielectric constants are practically temperature independentS23 (Table 6, Figure 4). For Poglitsch and Weber, “the dielectric measurements reveal a picosecond relaxation process which corresponds to a dynamic disorder of the methylammonium group in the high-temperature phases of the trihalogenoplumbates” S12. Onoda et al. have extended the complex permittivity measurements towards low frequencies and lower temperatures.S26 They conclude : “methylammonium ions are fully disordered” at room temperature and “undergo successive ordering at the respective transitions on cooling”. Furthermore, from 2H and 14N NMR spectra, Wasylishen et al. have shown that rotations of the C-N axis are restricted in the low temperature phasesS30. For all these reasons, we attribute the dielectric increment at low frequency and room temperature to the correlated tumbling of the C-N axis of the methylammonium cations. A proper description of the phase transition could be performed using the model developed for “ translation-rotation coupling, phase transitions, and elastic phenomena in orientationally disordered crystals” by Lynden-Bell and MichelS31. In this work, the authors state : “if the orientational order parameter corresponding to the ordered phase couples with a phonon, the result is a mixed order-disorder/displacive phase transition”. In order to describe properly “ lattice vibrations or elastic constants, it is necessary to define collective orientational functions that contain the translational symmetry of the crystal as well as the site molecule symmetry”S31,

rather then a local pseudo-spin model implemented by Poglitsch and WeberS12.

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