Dynamical Origin of the Rashba Effect in Organohalide Lead Perovskites

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Dynamical Origin of the Rashba Effect in Organohalide Lead Perovskites: A Key to Suppressed Carrier Recombination in Perovskite Solar Cells? Thibaud Etienne,*,† Edoardo Mosconi,†,‡ and Filippo De Angelis*,†,‡ †

Computational Laboratory for Hybrid/Organic Photovoltaics (CLHYO), CNR-ISTM, via Elce di Sotto, I-06123 Perugia, Italy CompuNet, Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy

‡

S Supporting Information *

ABSTRACT: The presence of a Rashba band-splitting mechanism mediated by spin−orbit coupling and breaking of inversion symmetry has been suggested as a possible cause for the reduced recombination rates observed in organohalide perovskites. Here, we investigate the interplay of electronic and nuclear degrees of freedom in defining the Rashba splitting in realistic MAPbI3 models. Our simulations disclose a “dynamical Rashba ef fect”, allowing for a quantification of its magnitude under thermal conditions. We find that even in globally centrosymmetric structures the dynamics of the coupled inorganic−organic degrees of freedom give rise to a spatially local Rashba effect which fluctuates on the subpicosecond time scale typical of the methylammonium cation dynamics. This effect is progressively quenched in globally centrosymmetric structures, likely representing the MAPbI3 perovskite at room temperature, on increasing the probed spatial scale up to 32 MAPbI3 units (∼3 nm size) because of the incoherent nuclear thermal motion mediated by the disorder of the organic cations.

O

photophysics of MAPbI3, due to the paucity of deep levels introduced into the MAPbI3 band gap by typical ion vacancies and interstitials.20,27,28 Also, the presence of mobile (dipolar) organic cations has been related to the charge separation efficiency, by virtue of an exciton screening mechanism29 or by locally creating screening localization domains.23,30 The presence of lead, in addition, imparts the material a giant spin−orbit coupling (SOC),31 which was found to be responsible for the band gap variation compared to the analogous tin perovskites,32 band gap modulation with distortions from the cubic symmetry,33 and reduction of the carrier effective masses.34 Notably, the peculiar interplay of spin and orbital degrees of freedom could underlie the observed spin-dependent charge recombination and spin-polarized carrier dynamics reported in MAPbI3,35,36 opening the way to spintronic applications of organohalide perovskites. A property intimately connected to SOC is the so-called Rashba band splitting, which has received considerable attention from a theoretical point of view.32−37 The Rashba effect16,38−40 is the consequence of the breaking of inversion symmetry in the crystal in a direction orthogonal to a k-point

rganohalide perovskites hold the potential of revolutionizing the landscape of photovoltaic and optoelectronic applications.1−10 These materials exhibit a unique combination of desired properties which are hard to find even in the best single-crystalline semiconductors, despite being made by solution processing techniques. Among them, we mention their optical11−14 and charge generation and transfer properties,9,10 the long carrier lifetimes,15−17 and their mobility.17,18 Perhaps the most surprising property of organohalide lead perovskites is the repeatedly reported low recombination rate of photogenerated carriers found in methylammonium lead iodide (MAPbI3) and related compounds, despite the unavoidable presence of defects in such solution-processed polycrystalline films. This clearly contrasts with the picture of a typical semiconductor in which scattering at defects or impurities severely limits the carrier lifetime and diffusion length.19−25 Understanding the reasons behind the reduced carrier recombination rate in MAPbI3, beating the predicted Langevin limit by 5 orders of magnitude,26 is one of the hottest topics in current perovskite research, and it is probably the main reason behind the impressive rise of photovoltaic efficiency of perovskite solar cells, now exceeding 22%. It is obvious to speculate that such a property is inherent to the combination of the constituent materials, i.e. lead, iodine (or different halide combinations), and the organic cations. Various reports have disclosed the nondetrimental impact of native defects into the © 2016 American Chemical Society

Received: March 11, 2016 Accepted: April 8, 2016 Published: April 9, 2016 1638

DOI: 10.1021/acs.jpclett.6b00564 J. Phys. Chem. Lett. 2016, 7, 1638−1645

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The Journal of Physical Chemistry Letters sampling plane, and it is described by the so-called Bychkov− Rashba Hamiltonian: /̂ =

p2 2m

− α′

Scheme 1. Schematic Representation of Rashba Splitting for Parabolic Bands, along with Relevant Interaction Parametersa

ℏ (σ ·p ) × ∇⊥V 4m02c 2

where we find the Rashba primary correlation factor, α′, the electrostatic potential, V, and p = x̂px + ŷpy ;

∑

σ=

(σξ̂ )ξ ;̂

ξ = x, y, z

∇⊥ = z ̂

∂ × ∂z

that is, the linear momentum (defining the momentum space sampling, which is orthogonal to the gradient vector), the Pauli spin matrices, and the gradient operator, respectively. According to the scalar and cross products construction rules and the electric field projection (Ez) expression a · ( b × c ) = (a × b ) · c ;

∇⊥V = z ̂

∂V = −(Ez)z ̂ ∂z

a

See text for definitions. The valence and conduction bands (VB and CB) in the presence (absence) of Rashba splitting are the solid red and blue (dashed) lines, respectively. The Δk-shifted absorption and radiative recombination processes are also illustrated.

we can rewrite the Hamiltonian /̂ =

p2 2m

+

α′ℏEz 4m02c 2  

(σ × p ) · z ̂

While this Rashba-mediated mechanism is surely an elegant framework to explain, at least in part, the reduced carrier recombination in organohalide lead perovskites, it requires in principle a noncentrosymmetric crystal symmetry. However, the assignment of the space group symmetry of MAPbI3 is quite debated. Poglitsch and Weber proposed the I4/mcm space group,41 while Stoumpos et al. proposed the noncentrosymmetric I4cm space group,14 which may allow for polar (ferroelectric) distortions. It is worth mentioning, however, that the C3v symmetry of the methylammonium cation prevents the formation of genuinely symmetric structures belonging to either space group. Despite many attempts, the crystal structure of MAPbI3 in the tetragonal phase stable at room temperature has not been fully solved, including the position of the organic cations, although it is likely that the material does not exhibit a macroscopic ferroelectric behavior at room temperature.42 This would in principle prevent the Rashba band-splitting mechanism from being operative. However, Rappe and coworkers have put forward the hypothesis that structural fluctuations, possibly accessible at room temperature, may give rise to local electric fields extending beyond the characteristic Rashba length scale (1−2 nm),16 thus not strictly requiring the presence of macroscopic lack of inversion symmetry for this effect to be at work. Motivated by the huge interest in understanding carrier recombination in organohalide perovskites, here we investigate the interplay of electronic (via first-principles band-structure calculations) and nuclear (via ab initio molecular dynamics simulations) degrees of freedom in defining the Rashba splitting in realistic MAPbI3 models under thermal conditions. Our simulations disclose the temporal and spatial scale of the “dynamical Rashba ef fect” in MAPbI 3 , allowing for a quantification of the magnitude of this effect under realistic conditions. We find that even in a globally centrosymmetric structure, likely representing the average MAPbI3 crystal at room temperature, the fluctuations of the coupled inorganic− organic degrees of freedom give rise to a spatially local Rashba effect which fluctuates on the subpicosecond time scale typical of the methylammonium cation dynamics. This is in turn

α /ℏ

where the electric field projection is now included into the general Rashba interaction coefficient, hereafter α. ε±(k) =

ℏ2 2 k ± αk 2m

The eigenvalues difference at a particular k-point sets the α value: ε≠ = 2αΔk ⇒ α =

ε≠ 2Δk

±Δk indicates the respective position of the two ε± wells vertices for energy curves crossing at the k origin, and ε≠ defines the difference between the two energy curves at the vertex positions, see Scheme 1 for a definition of relevant parameters. It is important to notice here that a single (parabolic) band will split into two bands separated in k-space by Δk and in energy by ε≠ in the presence of SOC (α′) and of an internal electric field (Ez), i.e., in absence of inversion symmetry in the crystal. As such, the Rashba effect is intimately connected to the presence of heavy nuclei, e.g., lead and possibly iodine, which have a high SOC constant, and to the presence of internal electric fields such as those due to the presence of polar (ferroelectric) structures. In MAPbI3, the valence (conduction) bands (VB and CB in Scheme 1) are mainly contributed by iodine (lead); thus, they undergo a different Rashba splitting, by virtue of the different SOC characterizing lead and iodine. This introduces a different Rashba splitting for the VB and CB, Δ(Δk) in Scheme 1. The connection to recombination in a perovskite solar cell is then clear: as the photogenerated carriers thermalize at the band extrema which are located at a different point in k-space, radiative recombination, which preserves k, will be limited as in an indirect band gap semiconductor. Furthermore, the MAPbI3 VB and CB may have different curvature; thus, a different density of states, and spin helicity, would further slow recombination.16 1639

DOI: 10.1021/acs.jpclett.6b00564 J. Phys. Chem. Lett. 2016, 7, 1638−1645

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The Journal of Physical Chemistry Letters consistent with the organic cation motion representing the driving symmetry breaking element in the MAPbI3 crystal. Notably, the time average of this local Rashba effect is nonzero on the investigated ∼2 ps time scale, longer than the expected time scale typical of electronic transitions. We also illustrate how this effect is progressively quenched in globally centrosymmetric structures on increasing the probed spatial scale to 32 units (∼3 nm size) due to the incoherent nuclear thermal motion mediated by the disorder of the organic cations. Our results shed light on the impact of the observed dynamical Rashba effect in contributing to reduced carrier recombination in MAPbI3, a property likely lying at the heart of perovskite solar cell efficiency. To simulate the MAPbI 3 perovskite under thermal conditions, we performed Car−Parrinello molecular dynamics simulations on a 32 MAPbI3 supercell containing 2 × 2 × 2 replica of the original tetragonal unit cell (Figure 1), setting the temperature at 330 ± 20 K. Here, we consider two starting arrangements of the organic cations, corresponding to globally polar and apolar structures, hereafter 1 and 2, respectively.43

Structure 1 is our most stable calculated structure for tetragonal MAPbI3, showing the MA cations aligned along the c axis forming an average angle with the ab plane (hereafter ϕ) of ca. −27° (Figure 1). This structure belongs to the I4cm space group,14 with a calculated polarization of 4.4 μC/cm2,43,44 and it likely corresponds to the low-temperature tetragonal structure reported by Weller and co-workers.45 Structure 2, only 0.07 eV per unit cell (4 MAPbI3 units) less stable than 1, formally belongs to the I4/mcm space group and is employed here to mimic the disordered room-temperature structure. As previously reported,43 the polar structure 1 shows a clear Rashba band splitting along the Γ → M direction of the Brillouin zone, i.e., the direction perpendicular to the polar c axis, while in the apolar structure 2 no such splitting is observed, see the Supporting Information. The two starting structures containing 32 MAPbI3 units, hereafter 1-32 and 2-32, were subjected to CPMD simulations for a total of ca. 12 ps after a few picoseconds of thermalization. To investigate the effect of local asymmetry in smaller material portions, we selected from each supercell a four-MAPbI3 cell, 14 and 2-4, respectively, and again from each of these structures we individually investigated the four MAPbI3 units composing the unit cell. Hereafter, we focus on one of such units extracted from 1-4 and 2-4, labeled 1-1 and 2-1 in Figure 1; see the Supporting Information for the full data set. We then selected a time frame of ca. 2.5 ps from the central part of the CPMD simulation, after thermalization, and for structures 1-4, 2-4, 1-1, and 1-2 we performed 800 band structure calculations including SOC, sampling ca. 1 structure/0.003 ps. We stress here that the smaller subsystems 1-4, 2-4, 1-1, and 1-2 are used only to represent what the Rashba effect would be if the entire material behaved as the smaller subsystems in each time f rame, periodically repeated. Because the subsystems are extracted from the bigger cells 1-32 and 2-32 and periodicity is enforced a posteriori, the values extracted for such systems are purely indicative. For the larger 1-32 and 2-32 systems, only a few structures from the same dynamics data set were subjected to band structure calculations; these results are reliable because the band structure is calculated in this case on exactly the same system on which periodic ab initio molecular dynamics simulations are performed. The time evolution of the ϕ dihedral angle, characterizing the average MA cation orientation during the dynamics of systems 1 and 2, is reported in Figure 2. As can be seen, both structures fluctuate around their initial ϕ value with a time average close to −27 and 0° for 1-32 and 232, respectively; thus, they on average remain polar and apolar, respectively, during the investigated time frame. Notably, higher fluctuations are observed within the subunits 1-4 and 2-4 extracted from the bigger systems, corresponding to the size of the tetragonal unit cell. This behavior is related to the disordered and possibly uncoupled nuclear motion expected at room temperature when sampling increasingly larger space regions. We also notice that when considering a single MAPbI3 unit, no compensation from neighboring MA cations is included; thus, the effect of structural asymmetry is expectedly maximized. We now examine the impact of local asymmetry on the Rashba effect by investigating the time evolution of the Rashba interaction coefficient, α, calculated for structures 1-4, 1-1, 2-4, and 2-1 in Figure 3; see the Supporting Information for the full data set. Here, we report data calculated along the R → M and Γ → M directions for 1-1/2-1 and 1-4/2-4, respectively. While

Figure 1. Structural models of MAPbI3 employed to sample different space scales. The systems containing 1, 4, and 32 MAPbI3 units are shown. The notation 1-n and 2-n (n = 1, 4, and 32) refers to globally polar and apolar structures, respectively, in relation to the starting orientation of the methylammonium cations, whose dipoles are schematically represented by the arrows on the right side of second panel. The ϕ dihedral angle is illustrated schematically. The Brillouin zones for the tetragonal and cubic lattices corresponding to 1-4/2-4 and 1-1/2-1 are also shown. 1640

DOI: 10.1021/acs.jpclett.6b00564 J. Phys. Chem. Lett. 2016, 7, 1638−1645

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Figure 2. Time evolution of the ϕ dihedral angle, characterizing the average MA cation orientation during the dynamics of systems 1-32 and 2-32, and on the smaller scale represented by systems 1-4 and 2-4, right side, along with the ϕ average values. The shaded area on the left panels correspond to the time scale of the right panels. The vertical arrows indicate the structure analyzed on larger spatial scales in Table 1 and Figure 4.

Figure 3. Time evolution of the Rashba interaction coefficient, α, calculated for the conduction (upper panel) and valence (bottom panel) bands for systems 1-1 and 2-1 (left panels) and 1-4 and 2-4 (right panels). Notice the different scale used for systems 1-1/2-1 and 1-4/2-4. Average α values calculated during the reported time span are also reported and depicted (dashed lines).

Γ → M is the only relevant direction to be examined for the tetragonal unit cells, with the direct band gap at Γ, for the pseudocubic single MAPbI3 cells 1-1 and 1-2, with the band gap at R, a Rashba effect is observed also along the R → Γ direction (Figure 1). Once again, we stress that the data for the smaller subsystems are purely indicative and are reasonably exaggerated because periodicity has been enforced here a posteriori. This data is however useful to catch the spatial scale of the investigated effect and to discuss the relative importance

of the effect on the VB and CB, which is expectedly captured by the model. As anticipated, the VB Rashba coefficient is smaller than the corresponding CB value, Figure 3, in relation to the different lead/iodine CB/VB composition and thus to the different SOC due to the different nuclei. Also, the time evolution of α for the VB and CB are not strictly correlated, see for example the maximum at ∼5.2 ps for the CB of 1-1 which corresponds to a minimum in the VB (Figure 3), so that on average the CB 1641

DOI: 10.1021/acs.jpclett.6b00564 J. Phys. Chem. Lett. 2016, 7, 1638−1645

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The Journal of Physical Chemistry Letters

Figure 4. Calculated SOC-DFT parabolic band structure of the globally polar and apolar structures 1 and 2, probed on different spatial scales, up to the largest 32 MAPbI3 units supercells (1-32 and 2-32). The dashed arrows represent the evolution of ε≠ and Δk when increasing the probed size of the supercell. Notice the different scale of the two panels.

Table 1. Rashba Interaction Coefficient, α, Calculated for a Selected Structural Frame along the Γ → M Direction for the Tetragonal Systems (1-32, 1-4, 2-32, and 2-4) and along the R → M Direction for Structures 1-1 and 2-1a

always features a stronger splitting than the VB. The same picture is maintained when probing the four different MAPbI3 units constituting the tetragonal unit cell and when looking at the R → Γ direction; see the Supporting Information. As it can be noticed, structures 1-4 and 1-1, which originated from the polar structure 1-32, show a significant average value of α, which increases when looking at the evolution of a single MAPbI3 cell (compare 1-1 and 1-4 in Figure 3). The reduced α value found for 1-4 compared to 1-1 is due to the increased disorder in the former, leading to partial quenching of the internal electric field. Notably, also for the globally nonpolar structure 2-4, characterized by a close-to-zero average of the ϕ dihedral angle in the investigated time span (Figure 2), we find a significant Rashba effect. This suggests that thermal fluctuations may induce local electric fields which provide a nonzero instantaneous and time average Rashba effect on the MAPbI3 tetragonal unit cell. However, the question is what happens on larger scales, e.g., when moving to the entire simulation supercell. We thus looked at a single structural frame within the investigated data set and calculated the band structure of the 2 × 2 × 2 supercell systems 1-32 and 2-32. The results obtained at different scales on the same structure are summarized in Figure 4 and Table 1, where the Rashba parabolic bands (obtained from the SOC density functional theory (DFT) calculated electronic band structure on the considered systems) are reported. While no significant α variation was found for the polar 1-4 and 1-32 structures, in line with the long-range order characterizing these structures on the investigated time scale, for the globally apolar structure 2, the α value drops by ca. a factor 2 when probing the larger 2-32 system (Table 1). This situation clearly corresponds to the increased disorder sampled on larger spatial scales for the apolar structures, whereby local fluctuations tend to cancel out when increasing the probed space region. Our simulations thus predict the Rashba effect to persist up to 8 tetragonal unit cells,

Δk (×10−3 Å−1) 1-32 1-4 1-1

2-32 2-4 2-1

ε≠ (meV)

α (eV Å)

VB

CB

VB

CB

VB

CB

2.19 2.03 2.34 (5.03) VB 1.09 2.07 12.20 (6.18)

3.08 3.76 6.44 (4.58) CB 1.59 3.16 10.25 (7.40)

12 8 12 (31) VB 2 7 118 (42)

31 36 133 (85) CB 7 24 256 (157)

2.82 2.06 2.66 (3.11) VB 1.12 1.67 4.82 (2.79)

5.02 4.75 10.36 (8.78) CB 2.19 3.86 12.48 (10.34)

a

For the latter, the maximum and average (in parentheses) values calculated over for four single MAPbI3 cells are reported.

i.e., on volumes exceeding ∼8 nm3. We notice, however, that our 2 × 2 × 2 ab initio molecular dynamics results could lead to slightly overcorrelated methylammonium dynamics,46 possibly impacting the calculated Rashba values for 1-32 and 2-32, which should be taken as upper bounds. While so far we have intuitively correlated the presence of internal electric fields required to observe the Rashba effect to the orientation of the MA cations, we now wish to probe whether this is the sole origin of the observed effect. We have thus resorted to a model 2 × 2 × 2 cubic MAPbI3 system in which the Pb and I atoms were kept fixed at their cubic sites and the MA cations were allowed to fluctuate in time during a CPMD simulation.47 The calculated α values for this rigid system are about 1 order of magnitude smaller than what was calculated for the fully dynamical system (Supporting Information), suggesting that while the motion of the MA 1642

DOI: 10.1021/acs.jpclett.6b00564 J. Phys. Chem. Lett. 2016, 7, 1638−1645

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simulations have been performed with an integration time step of 5 au, for a total simulation time of ca. 12 ps. The fictitious masses used for the electronic degrees of freedom is 500 au, and we set the atomic masses to their real values except for hydrogen atoms, for which a 2.00 a.m.u. is used. Initial ion position randomization has been used to reach a temperature of ∼350 K, without further applying any thermostat. The electronic structure analysis has been carried out using the PWscf code with Plane-wave basis set cutoffs for the smooth part of the wave functions, and the augmented density were 25 and 200 Ry. Spin−orbit coupling interaction has been included using ultrasoft PBE-GGA pseudo potentials. For systems 1-1 and 1-2, an 8 × 8 × 8 k-point grid has been used; for 1-4 and 24, a 4 × 4 × 4 k-point grid has been used; while for the 1-32 and 2-32, systems a Γ point calculation has been performed. PWscf calculation was followed by a band structure calculation sampling the Brillouin’s zone in the direction Γ → M and Γ → Z for the 1-4, 2-4, 1-32, and 2-32 models, and along the R → M and R → Γ directions for the 1-1 and 2-1 models.

cations likely drives the structural distortions which originate the effect, it is the interplay of the inorganic and organic moieties that provides the actual value of the Rashba interaction coefficient. This conclusion is in agreement with Zheng et al., who identified significant contributions from lead and iodine displacement to the observed effect;16 with the work by Stroppa et al., who found that the possible low-temperature ferroelectric behavior of MAPbI3 is the result of both the organic and inorganic moieties; 44 and with the symmetry analysis performed on MAPbI3 by Even et al.48 Finally, we wish to comment on the magnitude of the observed effect. We calculate significant ε≠ values for the smallest selected investigated structures of Table 1 and Figure 4, with CB splitting values for the tetragonal unit cells 1-4 and 2-4 of ∼20−40 meV. While similar values are calculated for the polar 1-32 system, for the realistic 2-32 system, likely representing the MAPbI3 perovskite at room temperature, ε≠ values of 7 and 2 meV are calculated for the CB and VB, respectively. On the basis of the calculated values and adopting the model developed in ref 16, one predicts up to 2 orders of magnitude reduction of the carriers recombination rate at the level of a single MAPbI3 unit, dropping to 1 order of magnitude at the level of the tetragonal unit cells and further decreasing for the largest considered supercells. In summary, we have shown that a dynamical Rashba effect exists in MAPbI3 regardless of whether the crystal is globally centrosymmetric or not. While in globally polar systems, possibly corresponding to the tetragonal MAPbI3 phase at low temperature, i.e., just above the transition to the orthrombic phase, the effect extends throughout the entire crystal; in the disordered structure expected at room temperature, a dynamical Rashba effect is found to locally operate in crystal portions extending for a few nanometers. This local Rashba effect fluctuates with a subpicosecond time scale, in relation to fluctuations of the methylammonium cations, providing a locally nonzero effect in space and time. Notably, the effect is not solely due to the organic cations, because the inorganic lattice provides significant contributions to the local distortions originating the effect. This is likely due to the soft nature of the lead−iodine bond, which can easily be deformed under the symmetry-breaking driving force due to fluctuation of the organic cations. The band splitting associated with the observed dynamical Rashba effect, shifting the valence and conduction band populations at different positions in k space, may thus lead to reduced carrier recombination in perovskite solar cells, especially if a local mechanism can be envisioned to protect the charge carriers. Based on current estimates on the largest spatial scales amenable to our simulations, however, additional effects should be sought to further account for the surprisingly small recombination rates found in organohalide perovskites.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.6b00564. Additional figures and data (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: fi[email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research leading to these results has received funds from the European Union Seventh Framework Programme [FP7/ 2007%2013] under Grant Agreement No. 604032 of the MESO project.

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REFERENCES

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COMPUTATIONAL METHODS For all the calculations, the experimental cell parameters reported by Poglitsch et al.41 have been used. CPMD simulations have been carried out with the Quantum Espresso package49 along with the GGA-PBE50 functional. For all calculations, electron−ion interactions were described by scalar relativistic ultrasoft pseudopotentials with electrons from O, N, and C 2s, 2p; H 1s; Ti 3s, 3p, 3d, 4s; I 5s, 5p; and Pb 6s, 6p, 5d shells explicitly included in the calculations. Plane-wave basis set cutoffs for the smooth part of the wave functions and the augmented density were 25 and 200 Ry, respectively. For the tetragonal MAPbI3 2 × 2 × 2 models 1 and 2, CPMD 1643

DOI: 10.1021/acs.jpclett.6b00564 J. Phys. Chem. Lett. 2016, 7, 1638−1645

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DOI: 10.1021/acs.jpclett.6b00564 J. Phys. Chem. Lett. 2016, 7, 1638−1645