Relativistic GW Calculations on CH3NH3PbI3 and CH3NH3SnI3 Perovskites for Solar Cell ApplicationsPaolo Umari,a,b Edoardo Mosconi,c Filippo De Angelis c,*

a Dipartimento di Fisica e Astronomia, Università di Padova, via Marzolo 8, I-35131 Padova, Italy.

b CNR-IOM DEMOCRITOS, Theory@Elettra Group, c/o Sincrotrone Trieste, Area Science Park, Basovizza, I-34012 Trieste, Italy.

c Computational Laboratory for Hybrid/Organic Photovoltaics (CLHYO), CNR-ISTM, Via Elce di Sotto 8, I-06123, Perugia, Italy.

SUPPLEMENTARY INFORMATION

Theoretical and Computational Details

Many body-perturbation theory, within the GW approximation, permits an accurate calculation of quasi-particle energy levels.1 The method takes its name from approximating the electronic self-energy operator as a product of the one body Green’s function G with the screened interaction W. GW calculations are usually performed in a perturbative way from a starting DFT calculation. At the first perturbative level, usually referred to as the one-shot G0W0 scheme, the quasi-particle amplitudes are approximated with the Kohn-Sham (KS) DFT orbitals φi, while the corresponding quasi-particle energies Ei are obtained through the solution of the following self-consistent one-variable equation:

(1)

where is the DFT energy level of the i-th state, is exchange part of the self-energy and is its correlation part. corresponds to the Fock exchange operator which reads:

(2)

where the index v runs over the occupied valence states. is expressed as the convolution of the DFT one-body Green’s function G0 with the screened Coulomb interaction W0, which is found from the random-phase approximation:

(3)

where η is a positive infinitesimal.

With respect to DFT calculations even the simple G0W0 approach becomes much more computationally demanding, due to the appearance of large matrices for representing the operators and to the appearance of sums over a large (in principle infinite) number of unoccupied KS orbitals.2 To cope with this issues, we have recently introduced a new approach which permits on one side to reduce the basis sets used for representing operators, such as the polarizability operators,3 and on the other side it completely eliminates sums over unoccupied KS orbitals.4 Our GW code is implemented in the Quantum-Espresso5 suite of codes. Recently, such method has been successfully applied to investigating the alignment of energy levels in dye-sensitized TiO2, proving to be very accurate against experimentally available quantities.6

The study of systems incorporating heavy atoms requires, already at the DFT level, the explicit treatment of spin-orbit coupling through the introduction of 2 dimensional spinors for describing the electronic wave functions.7, 8 Here, we have extended the relativistic DFT scheme of Ref. 9, in which the spin-orbit coupling is modeled by pseudopotentials and wave-functions and charge densities are developed on plane-waves basis sets, to our GW scheme. The two dimensional spinor exchange operator is expressed as:

(4)

where the index αand α’ run over the two spinor components of the occupied relativistic KS states . For evaluating the self-energy we have considered the suggestion given in Ref. 10 of approximating the screened relativistic coulomb interaction with that obtained from a scalar relativistic calculation :

(5)

As we have observed at the DFT level that for the systems we want to address the full relativistic treatment does not change significantly the conduction electronic levels but those close to the conduction band minimum, for calculating the relativistic correlation part of the self-energy we can calculate the DFT relativistic Green’s function considering explicitly only the lowest relativistic states:

(6)

where for simplicity in the scalar relativistic calculation we have considered doubly occupied states. In this way we still avoid sums over unoccupied KS states which would be particularly cumbersome when dealing with large model structures.

Scalar relativistic GW calculations were performed using norm-conserving PBE pseudopotentials with an energy cutoff of 70 Ryd defining the plane-waves used for representing the wave-functions. For the full-relativistic GW calculations we used PBE ultrasoft 11 pseudopotentials and energy cut-offs of 45 and 280 for the wave-functions and charge densities, respectively.[1] The scalar relativistic GW calculations were performed developing polarizability operators on a basis sets obtained as explained in Ref. 4 using an energy cutoff 3 Ry and selecting the 2000 most important basis vectors. The self-energy expectation values are first obtained on imaginary frequency and then analytically continued on the real frequency axis fitting with a two poles expansion.12 For the GW scalar relativistic case, we have checked that a basis for the polarizability consisting of 1000 vectors would lead to energies of frontier orbitals within 0.05 eV. The full relativistic GW calculations were performed feeding eq. 6 with the first 400 KS states, the first 200 of them are fully occupied. We have checked that if only 240 states are considered (200 valence, 40 conduction), the energy levels of frontier orbitals change of less than 0.06 eV while the energy gap changes of less than 0.03 eV.

Geometry optimizations and electronic structure analysis have been carried out using the PWSCF code as implemented in the Quantum-Espresso program package.5 Electron-ion interactions were described by ultrasoft pseudopotentials with electrons from Pb 5d, 6s, 6p; N and C 2s, 2p; H 1s; I 5s, 5p; Br 4s, 4p; Cl 3s, 3s, shells explicitly included in the calculations. A 4x4x4 Monkhorst–Pack grid13 was chosen for sampling the Brillouin zone. Plane-wave basis set cutoffs for the smooth part of the wave functions and the augmented density of 25 and 200 Ry, respectively, were used. Geometry optimizations were performed for all structures employing available lattice parameters, and checking their adequacy by performing atomic and lattice parameters optimizations.

Calculation of the GW DOS and optical properties

Our GW scheme has been implemented considering the sampling of the Brillouin’s zone (BZ) only at the Γ-point, although denser k-points sampling is used for the starting DFT calculation and for evaluating the long range parts of the dielectric matrix. It should be noted that our simulation cells are quite large comprising 48 atoms. However, we observed that the evaluation of the electronic DOS at the DFT level required at least a 4x4x4 k-points mesh. As in weakly correlated materials, the GW method almost completely retains the band ordering at different k-points, we have envisaged a scheme for introducing GW corrections to DFT energy levels calculated at an arbitrary k-point considering only the GW levels calculated at theΓ-point. Let be the KS energy of the i-th KS state at the k point in the BZ. The corresponding approximated GW energy is given by:

(7)

where we indicate with the GW quasi-particle energy level calculated at the Γ-point for the j-th state, and we have chosen that precise j so that:

It is always possible to find such aj as both the VBM and the CBM are at the Γ-point. It should be note that we have , for every i. For evaluating the optical properties we have first evaluated the frequency dependent complex dielectric functions:

(8)

where is the volume of the simulation cell, is the total number of k-points in the BZ, is the velocity operator, η is an opportune broadening factor, and the indices v and c run over the occupied and unoccupied states, respectively.

We tested this scheme in bulk Si, considering a simulation cell of only 8 atoms and, only at the DFT level, a mesh of 4x4x4 k-points in the BZ. We observed that the calculated is very close to the corresponding GW-RPA line calculated in Ref.19.

The frequency dependent absorption coefficient is then given by:

(9)

It should be noted that in our treatment we omitted any evaluation of the electron-hole coupling. Such effects can be calculated in principle through a scheme based on the Bethe-Salpeter equation for the two-body Green’s function20 although such a scheme has not yet been developed for full-relativistic calculations. However, in the case of small band gap semiconductor neglecting electron-hole coupling still leads to spectra in quite a reasonable agreement with experiment.19 In our calculation of the DOSs and absorption spectra we considered a mesh of 4x4x4 k-points and 95 unoccupied states.

Alignment of the GW-calculated DOS

Figure S1. Calculated SOC-GW DOS for MAPbI3 (blue) and MASnI3 (red) aligned at the carbon 2p peak, found at ca. -8 eV below the valence band maximum in MASnI3.

Figure S2. SOC-DFT calculated electronic band structure for MAPbI3 (blue, panel A) and MASnI3 (red, panelB).

References:

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S1

[1]Note that we have implemented our method for the general case of ultrasoft pseudopotentials. Details will be given elsewhere.