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Arthur Marronnier LPICM, CNRS, Ecole Polytechnique, Université Paris-Saclay, 91128 Palaiseau, France arthur.marronnier@polytechnique.edu    Guido Roma DEN - Service de Recherches de Métallurgie Physique, CEA, Université Paris-Saclay, 91191 Gif sur Yvette, France    Marcelo Carignano Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, P.O. Box 5825, Doha, Qatar    Yvan Bonnassieux LPICM, CNRS, Ecole Polytechnique, Université Paris-Saclay, 91128 Palaiseau, France    Claudine Katan Univ Rennes, ENSCR, INSA Rennes, CNRS, ISCR (Institut des Sciences Chimiques de Rennes) – UMR 6226, F-35000 Rennes, France    Jacky Even Univ Rennes, INSA Rennes, CNRS, Institut FOTON — UMR 6082, F-35000 Rennes, France    Edoardo Mosconi Computational Laboratory for Hybrid/Organic Photovoltaics (CLHYO), CNR-ISTM, Via Elce di Sotto 8, I-06123 Perugia, Italy    Filippo De Angelis Computational Laboratory for Hybrid/Organic Photovoltaics (CLHYO), CNR-ISTM, Via Elce di Sotto 8, I-06123 Perugia, Italy
Abstract

Doping organic metal-halide perovskites with cesium could be the best solution to stabilize highly-efficient perovskite solar cells. The understanding of the respective roles of the organic molecule, on one hand, and the inorganic lattice, on the other, is thus crucial in order to be able to optimize the physical properties of the mixed-cation structures. In particular, the study of the recombination mechanisms is thought to be one of the key challenges towards full comprehension of their working principles. Using molecular dynamics and frozen phonons, we evidence sub-picosecond anharmonic fluctuations in the fully inorganic perovskite. We reveal the effect of these fluctuations, combined with spin-orbit coupling, on the electronic band structure, evidencing a dynamical Rashba effect which could reduce recombination rates in these materials. Our study show that under certain conditions space disorder can quench the Rashba effect. As for time disorder, we evidence a dynamical Rashba effect which is similar to what was found for and which is still sizable despite temperature disorder, the large investigated supercell, and the absence of the organic cations’ motion.

\altaffiliation

D3-CompuNet, Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy achemso demonstration] Influence of Disorder and Anharmonic Fluctuations on the Dynamical Rashba Effect in Purely Inorganic Lead-Halide Perovskites

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Keywords

inorganic perovskite solar cells, anharmonicity, cesium, phonons, DFT, molecular dynamics, Rashba

1 Introduction

Fully inorganic metal-halide perovskites have attracted more and more attention in the past two years as they have showed promising efficiencies (record efficiency of 13.4% for quantum dot devicesSanehiraeaao4204) and as cesium doping has proven to be a good way to improve the environmental stability of hybrid metal-halide perovskites saliba2016cesium. Moreover, a better understanding of the physical properties of fully inorganic halide perovskites is needed in order to further understand, by contrast, the role of the organic cation in their hybrid cousins.

In general, the enthusiasm for metal-halide perovskites can be explained by their exceptional optoelectronic properties, whether it be their optical propertieschiarella2008combined, ogomi2014ch3nh3sn, Eperon2014, Stoumpos, the long lifetimes both electrons and holeswehrenfennig2014charge, zheng2015rashba, ponseca2014organometal and the high mobility in these materialscahen2014elucidating, ponseca2014organometal. But what is probably the most remarkable feature of these materials is the fact that they present on the one hand good absorption and charge generation propertiesxing2013long, stranks2013electron, and very low recombination rates on the other hand. If the former could be explained in particular by the materials’ direct band gap, the latter is more surprising as one should expect high values for both the radiative (direct band gap) and non-radiative recombination (high density of defects). As for defects, one should note that charge separation could be actually eased in these materials through halide ionic migrationlee2017direct, yang2015significance which could either favour exciton screeningeven2014analysis or give birth to local screening domainsma2014nanoscale, Quarti2016struct.

This apparent paradox could potentially be explained by the consequences of the interplay of spin and orbital degrees of freedom, which are of important magnitude in these materials because of the presence of the heavy lead atoms. In particular, the giant spin-orbit coupling (SOC) that was reported in these materials even2013importance is expected to be at the origin of Rashba-like splittingsKim6900, Even_Rashba, Amat, Brivio_PRB14. Such splittings correspond to the lift of the electronic bands’ spin degeneracy in the presence of SOC and time reversal symmetry when the inversion symmetry is broken in the crystalzhang2014hidden, Kepenekian. Assuming long-range polar distortions of the perovskite lattice, it can be shown that these band splittings can drastically impact the recombination rates by limiting direct transitions between the valence and conduction bands. This impact was theoretically estimated by Zheng et al. to contribute to a reduction of the recombination rates reaching up to two orders of magnitude zheng2015rashba. However, the existence of long-range polar distortions of iodide-based perovskite lattices is still debated, the influence of Rashba-like spinor band splittings could be more subtle and rather related to local lattice distortions.

Etienne et al. Etienne investigated by DFT-based molecular dynamics the interplay of electronic and nuclear degrees of freedom in the prototype perovskite and revealed a dynamical Rashba effect. They reported the influence of temperature and found a spatially local Rashba effect with fluctuations at the subpicosecond time scale, that is to say on the scale of the MA cation motion. It is worth pointing out that this numerical demonstration was based on MAPI structures preserving centrosymmetry on the average. They noticed that the Rashba splitting can be quenched when reaching room temperature but also when using larger supercells (up to 32 MAPI units, i.e., 3 nanometers cells) representing a higher and more realistic spatial disorder.

The local and dynamical nature of polar distortions may weaken the influence of Rashba-like spinor splittings by comparison to long range and static polar distortions. However the lack of long range correlations between local polar distortions could be compensated by the unusually strong amplitudes of the atomic motions. The strong anharmonicity of the perovskite lattice is a general feature of this new class of semiconductorsKatan_riddles, that was pointed out experimentally very early by inelastic neutron scattering in the context of inorganic halide perovskitesNeutron_Cl and that shall give rise to at least two characteristic experimental signatures: large and anisotropic Debye-Waller factors in diffraction studiesTrots, hutton1979high and a so-called quasielastic central peak observed either in inelastic neutron or Raman scattering studies simultaneously with highly damped phononsEven_disorder, yaffe2017local. However, the strong perovskite lattice anharmonicity is not expected to affect only zone center polar optical modes, but also acoustic modes or optical modes located at the edges or the Brillouin zone and related to non-polar antiferrodistortionsKatan_riddles. The previously mentionned experimental signatures (Debye-waller factors, phonon damping and central peaks by inelastic neutron or Raman scattering studies) can hardly be considered as unambiguous experimental proofs of the existence of strongly anharmonic polar fluctuations. Nowadays direct experimental investigations of the dielectric response give useful indications about the influence of lattice polar distortionsMarronnier2018179, anusca2017dielectric and the importance of the Fröhlich interactionSender_Mater for electron-phonon coupling processesfu2018unraveling, but do not directly probe the anaharmonicity of polar distortions. Numerical simulations are therefore still useful tools that already allowed showing the presence of anharmonicity features in Marronnier, Marronnier_ACSNano, leading to symmetry breaking minimum structures in the high temperature phases both at the edges and at the center of the Brillouin zone. MD simulations for also suggested that the fluctuations in this material are mostly due to head-to head Cs motion and Br face expansion happening on a few hundred femtosecond time scaleyaffe2017local.

In that sense, large polar fluctuations of the perovskite lattice at the local scale may lead to two main effects: Rashba-like spinor splittings and strongly anharmonic polarons related to the Fröhlich interaction. We focus in the present contribution on the former aspect.

2 Results and Discussion

In this article, we aim to further analyze the Rashba effect induced by the anharmonic double well and the symmetry breaking for the cubic phase of . We have shown in our previous worksMarronnier, Marronnier_ACSNano using the frozen phonon method that the highly symmetric cubic phase can be distorted to form two lower-symmetry structures with a slightly lower total energy (by a few meV). These two distorted structures, that we will call in the rest of the article and , have no inversion symmetry and correspond to the two minimum structures of the double well-instability recalled on Figure 1a.

Figure 1: a. Potential energy surface from frozen phonon calculations of cubic along the eigenvector of the unstable optical phonon at (b) as a function of displacement parameter . The 3N dimensional displacement needed to reach the new minimum corresponds to around , including a displacement for the cesium atom. The blue, purple and green atoms respectively denote Cs, Pb and I. We chose to label as "x" the axis parallel to which this chosen distortion mostly occurs. and represent the direct (DFPT) and frozen phonons estimations of the second derivative at (yellow and blue-dotted curve). For further details see Ref. Marronnier.

They correspond to opposite distortions ( and ) along the eigenmode represented in Figure 1b. Note that with rotational symmetry similar studies could be done on the two other eigenmodes corresponding to distortions along the two remaining Cartesian axes (y and z given our labeling).

The first aim of the study here is to look at the possible formation of " domains" and " domains", both in space (supercells) and in time (Car-Parrinello molecular dynamics "CPMD").

Then, we analyze in detail the dynamical Rashba effect induced by the time dynamics of the oscillations between structure and structure through the highly symmetric structure S (). This study is done on CPMD trajectories obtained from Ref. carignano2017critical

2.1 Spatial disorder

First, the aim is to investigate the influence of spatial / domains on the electronic band structure, in particular in terms of Rashba effect. Given that the eigenvector under study mostly corresponds to a distortion along one axis (x as labeled in Figure 1b), we built supercells by doubling the single cell both in the x direction, and in the z direction. These 211 and 112 supercells are built putting together 2 single cells: 1 single cell in configuration (or x up), and one configuration in configuration (or x down). These supercells, representing modulated structures with the smallest possible period, are schematically shown in Figure 2a.

Figure 2: a. Simplified representations of the supercells used to study the influence of spatial domains / or "x up"/"x down" on the electronic band structure. Electronic band structure (including SOC) of the (b) 112 supercell and (c) 211 supercell. As the cell is slightly orthorhombic (see the Methods section), we use here the orthorhombic q-point convention.c.

The Rashba splitting obtained at the R point for a single cubic cell, in the minimum, symmetry-breaking structure is shown in Figure 3. When doubling the cell along z (resp. x), the R point folds onto the S point (resp. T point), using the orthorhombic convention. For this ordered, static reference we find energy splittings of 57 meV and 40 meV between the point and respectively the conduction band minimum and valence band maximum. In order to give an estimate of the Rashba splitting taking into account the effect both in energy and in the k-space, we calculated the commonly used parameter as defined in Ref. Etienne:

(1)

where is the energy difference between the first (resp. last) two bands of the conduction (resp. valence) bands and the splitting of the minimum (resp. maximum) in the k-space. For the reference static, highly ordered structure we found values of 4.09 eV. and 2.01 eV. for respectively the conduction and the valence bands. These values are comparable to those in Ref. Etienne in the case of polar : 3.17 eV. and 1.17 eV. respectively. Note that so far the highest values found in ferroelectric materials for the Rashba parameter are 4.2-4.8 eV. for GeTe alpha_max1, alpha_max2.

Figure 3: Electronic band structure (including SOC) of the unit cell of cubic .

The results for the two modulated structures are shown in Figures 2b and 2c. Whereas no Rashba effect is found in the case of a modulation orthogonal to the direction of symmetry breaking, a band splitting around the valence band maximum and the conduction band minimum is found for a modulation parallel to the direction of symmetry breaking. We found values of 0.58 eV. and 2.18 eV. for the conduction and valence bands respectively.

In general, the Rashba splitting in the band structure of a two-dimensional system results from the combined effect of atomic spin-orbit coupling and asymmetry of the potential in the direction (here x) perpendicular to the two-dimensional plane, causing a loss of inversion symmetry. In the case of a modulation orthogonal to the direction of symmetry breaking (112 supercell), the symmetry along x is respected on average: the inversion symmetry is kept and the Rashba splitting is quenched. We expect then that the quenching of the Rashba effect results from a competition between parallel and orthogonal modulations: the former keeps the Rashba effect, while the latter tends to cancels it.

2.2 Dynamical structural fluctuations

Next, we analyze in detail CPMD trajectories of cubic in the light of our findings on the double well potential energy surface. The trajectories were computed by Carignano et al. in the framework of a study carignano2017critical of the anharmonic motion of the iodine atoms in and , where they showed that, at variance with , these two perovskite structures are expected to have a deviation from the perfect cubic unit cell at any time of the MD, with a probability very close to 1. This hints towards the interpretation that the symmetry can be seen as a time average, including for . This phenomenon had already been reported for in earlier studies Quarti2016struct, where it was evidenced that the system strongly deviates from the perfectly cubic structure in the sub-picosecond time scale.

The molecular dynamics simulation were performed at 370 K under NPT-F conditions, which allow volume fluctuations by changing the supercell edges and angles. The temperature was controlled by a Nose-Hoover thermostat with three chains, and the pressure was controlled by the Martyna’s barostat martyna1996explicit. The time constant for both, the thermostat and barostat, was set at 50 fs. The system they used for has 320 atoms (444 supercells).

Figure 4: a. Lattice parameters fluctuations along the CPMD trajectory at 370 K. b. Fluctuations of the distance to the average pseudocubic structure, as defined in Eq. 2.

In Figures 4a we show the lattice parameters fluctuations versus time. In particular, from this first simple analysis we can infer that the structure fluctuates around a cubic structure: the difference between the lattice parameters stays below 3%. Even though the structure is not perfectly cubic on average (see Table 1), the distance to the average pseudocubic lattice structure (Figure 4b) is smaller than 1%. This distance d is obtained as:

(2)

where are the 3 lattice parameters and their time average over the whole trajectory.


In Angstroms
370 K 450 K
a 6.358 6.372
b 6.338 6.391
c 6.358 6.361
Table 1: Average lattice parameters (in Angstroms) along the CPMD trajectories at 370 K and 450 K.

In order to analyze the MD trajectories in the light of the aforementioned double well instability, we project these trajectories onto two kind of structures: the perfectly cubic symmetric structure ("S") and the symmetry breaking structures and . The chosen approach is to study the radial distribution function of the cesium-lead pairs during the MD simulation and to compare it to our two reference structures.

Figure 5 focuses on averages over 0.5 ps intervals. At this time scale, our double well references seem to explain very well how the system explores the energy landscape. Whereas some intervals show a distance peak corresponding to the distance in the average pseudocubic structure S, for instance the [11-11.5 ps] interval show two peaks centered on both minimum structures and . This means that within 0.5 ps the structure has enough time to explore the whole double well. We think that this is the most appropriate time-scale to evidence the double well instabilities.

Figure 5: Distribution function of the cesium-lead pairs’ distances along the MD trajectory. Here the references (vertical lines) correspond to the distances for structures S, and , weighted by the ratio between the lattice parameters.

2.3 Dynamical Rashba effect

We now focus on the dynamical Rashba effect possibly ensuing from the nuclear dynamics exposed above. We expect to find in an effect similar to what was evidenced for for which the spatially local Rashba splitting was found to fluctuate on the subpicosecond time scale typical of the methylammonium cation dynamics Etienne.

To investigate this effect, we calculate the electronic band structure, including spin-orbit coupling, at different snapshots along the trajectory. Given the results of the Pb-Cs distance analysis, we chose to focus these calculations on the [10-15 ps] interval in which we chose 50 regularly distributed snapshots (hence separated by 100 fs from each other) in order to better capture the sub pico-second dynamics. For each snapshot, we used the MD structure of the 444 supercells (we remind the reader that the cell’s atomic positions, lattice parameters and angles vary) and derived its electronic band structure (see the Methods section). These calculations for 444 supercells follow the guidelines of those previously done for Quarti2016struct.

The electronic band structure calculations are done at 7 q points of the Brillouin zone (Table 2).


Point number
x y z
1 0.1 0 0
2 0.05 0 0
3 () 0 0 0
4 0 0.05 0
5 0 0.1 0
6 0 0 0.05
7 0 0 0.1
Table 2: q points of the Brillouin zone used in the 50 band structure calculations in units of where are the lattice parameters for snapshot i in one of the three carthesian direction, x, y or z.

In Figure 6 we plot for each snapshot i of the 50 structures chosen in the MD trajectory and for each q point the normalized energy difference :

(3)

where CBM is the conduction band minimum and VBT the valence band top. This is necessary as the cell is variable along the trajectory: the fluctuations on the gap value, which are large with respect to the Rashba splitting, would mask it otherwise.

These results show that 100 fs is a good estimate of the timescale of the Rashba effect dynamics. Moreover, on average we see a band gap shift to the Y direction, the band gap being reduced by 1.3 meV on average. Taking the extreme case, we can infer that the amplitude of the oscillations in the 5 ps timescale is around 10 meV.

Figures S1 and S2 show that this is mostly due to a Rashba splitting happening at the CBM rather than at the VBT. This is coherent with the fact that the most relativistic atom, Pb, is mostly contributing to the conduction band. This is coherent with what was previously reported for MAPI Quarti2016struct.

Figure 6: Differences between the gap at finite q and at for the 50 snapshots chosen along the MD trajectory. This difference is 0 at by construction (see Eq. 3). The Figures in blue represent the average values over the 50 points of the trajectory.

Figure 7a represent the Rashba effect oscillations in this interval through the previously defined parameter. This result confirms that the Rashba effect is much more substantial for the conduction band than for the valence bands, and oscillates with values close to 1 eV.. Even though this is smaller than in the static case (values around 4 eV.), this means that the effect is still sizable despite the disorder induced by temperature and the large investigated supercell.

Figure 7: a. parameter for the conduction and valence bands versus time. b. Fourier analysis of ’s oscillations.

It is interesting to further look into these oscillations through a Fourier analysis (Figure 7b) which revealed the existence of two main frequency components :

  • A 2.5 ps component which could correspond to the jump through double well and thus to the Cs slow motion

  • A 0.8 ps component corrresponding to the phonon modes usually associated to the Pb-I stretching (around 20-40 )

We think it is of interest to compare the order of magnitudes of these oscillations to those obtained in a similar study lead on hybrid perovskite by Etienne et al. Etienne. We report in Table 3 the corresponding values for the apolar structure of Ref. Etienne, because Cs atom has no permanent dipole moment. Note that nevertheless, for the polar structure, the largest value reported in Ref. Etienne is even smaller (10.36). One needs to keep in mind that we have here very large supercells compared to what was used in that study. The conclusion we can draw from this comparison is that we observe a sizable dynamical Rashba effect even with large supercells and the absence of the organic molecule, which in general is a possible source of symmetry breaking in these halide perovskite structures.


Number of formula units
for hybrid MAPI for inorganic
from Ref. Etienne (eV.) from our results (eV.)
1 12.48
4 3.86
32 2.19
64 0.96
Table 3: Maximum value of the oscillations.

3 Methods

3.1 Density functional theory for band structure calculations

In this study we used for the minimum reference structures the ones obtained in RefMarronnier after relaxation performed with LDA (cutoff of 70 Ry) letting both the lattice parameters and the atomic positions relax, keeping the cell’s angles fixed. These structures are thus slightly orthorhombic (0.1 and 0.3 % distortion). The total energy of these relaxed structures is 11.3 meV under the maximum, high-symmetry cubic structure. (If the angles are not constrained, the energy difference reads 12.1 meV).

For these reference structures, geometry optimizations and force calculations were performed with the Local Density Approximation (LDA). Non-relativistic (scalar-relativistic for Pb) and norm-conserving pseudopotentials were used, with the Cs , I and Pb electrons treated as valence states. The choice of 14 electrons for Pb and 9 for Cs was made after testing the influence of semi-core electrons on the potential energy surface (see an example in Figure 5 of our previous article Marronnier).

In a second step:

  • as for the study of the spatial domains, the band structure calculations of the constructed 211 and 112 supercells were performed with fully relativistic pseudopotentials (for Pb and I) with LDA.

  • in order to study the dynamical Rashba effect from CPMD, the band structure calculations were performed with fully relativistic US pseudopotentials (for Pb, I and Cs) datasets using the PBE xc functional (with the same number of valence electrons as in the scalar or non-relativistic case). This was done in order to be coherent with the CPMD calculations from which MD trajectories were taken carignano2017critical, which were done using PBE as well (with the CP2K code). We carefully checked that symmetry breaking is present even when using PBE, see Ref Marronnier_ACSNano.

For all the calculations, the Brillouin zone was sampled with -centered Monkhorst-Pack meshes monkhorst1976special with subdivisions of 888 k-points.

4 Concluding remarks

In summary, we investigated the effect of spatial and temporal disorder on the Rashba splitting in the cubic phase of inorganic halide perovskite . The analysis focused on the fluctuations of the Rashba -parameter —a measure of the Rashba band splitting— along a molecular dynamics trajectory for a large supercellcarignano2017critical.

Our results highlight a dynamical Rashba effect similar to the one previously observed for hybrid organic-inorganic halide perovskitesEtienne, which persists in spite of the quenching effect coming from spatial disorder in this relatively large simulation cell (320 atoms).

Some low-frequency vibrational modes of the system, and in particular the anharmonic behavior, which has been shown to originate from the double well potential energy landscape of a polar optical phononMarronnier, contribute to the spatial extension of the Rashba effect. This is confirmed by the Fourier analysis of the Rashba -parameter fluctuations.

An expected consequence of the Rashba effect is the reduction of the carriers recombination ratezheng2015rashba and consequent enhancement of their lifetime ; therefore, our results are a promising piece of information for the applications of fully inorganic halide perovskite as photovoltaic devices.

5 Associated Content

The authors declare no competing financial interest.

\acknowledgement

Dr. Arthur Marronnier’s PhD project was funded by the Graduate School of École des Ponts ParisTech and the French Department of Energy (MTES). HPC resources of TGCC and CINES were used through allocation 2017090642 and x20170906724 GENCI projects. The work at FOTON and ISCR was funded by the European Union’s Horizon 2020 program, through a FET Open research and innovation action under the grant agreement No 687008.

{suppinfo}

We provide in the supporting information additional results on the effect of dynamical disorder on the conduction band minimum and valence band top throughout the chosen 5 ps interval of study for our band structure calculations.

References

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