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The Role of Ferroelectric Nanodomains in the Transport Properties of Perovskite Solar Cells Alessandro Pecchia, Desirée Gentilini, Daniele Rossi, Matthias Auf der Maur, and Aldo Di Carlo Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b03957 • Publication Date (Web): 22 Dec 2015 Downloaded from http://pubs.acs.org on December 25, 2015
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The Role of Ferroelectric Nanodomains in the Transport Properties of Perovskite Solar Cells Alessandro Pecchia†*, Desirée Gentilini§, Daniele Rossi§, Matthias Auf der Maur§, Aldo Di Carlo§ † Consiglio Nazionale delle Ricerche, ISMN, Via Salaria km 29.300, 00017 Monterotondo, Italy § Dipartimento Ing. Elettronica, Università di Roma ‘Tor Vergata’, Via del Politecnico 1, 00133 Roma, Italy ABSTRACT: Metropolis Monte Carlo simulations are used to construct local dipoles current pathways minimal energy configurations by electrostatic coupling of rotating dipoles associated to each unit cell of a perovskite CH3NH3PbI3 crystal. Short-range anti-ferroelectric order is found, whereas at scales of 8-10 nm we observe the formation of nanodomains, strongly influencing the electrostatics of the device. The models are coupled to drift-diffusion simulations in order to study the actual role of nanodomains in the I-V characteristics, especially focusing on charge separation and recombination losses. We demonstrate that holes and electrons separate into different nanodomains following different current pathways. From our analysis we can conclude that even antiferroelectric ordering can ultimately lead to an increase of photoconversion efficiencies thanks to a decrease of trap-assisted recombination losses and the formation of good current percolation patterns along domain edges. KEYWORDS: solar cells, perovskite, halides, ferroelectric domains Lead halide perovskites have been introduced by Kojima and coworkers [1] in 2009, as a new class of dye-sensitized solar cells (DSSC). Initially, methylammonium (MA) PbI3 compounds (or similar variants using Cl, I, Br ions) where employed as a new sensitizer of a nanoporous TiO2 layer, substituting organic dyes. Few years later Im et al. demonstrated 6.5% conversion efficiency [2], but the material was quickly degraded by iodine corrosion from the I3-/I- electrolyte. After these first studies, the field has literally exploded and a fast escalation of record efficiencies were demonstrated [3]-[7]. The original DSSC architecture has been abandoned in favor of a more standard p-in junction solar cell, where typically the n-side is occupied by a conducting oxide, and the p-side by an organic hole conductor. Solar cells completely based on perovskite were also demonstrated. Currently the certified record of conversion efficiency of 20,1% is held at KRICT (Korea) [8]. Among the many exciting properties of this new material, certainly the most intriguing is that even cheap solution processing, starting from CH3NH3+-I- and PbI2 precursors [9], produces good quality films, leading to the highest efficiencies. A typical perovskite structure is obtained, with the organic molecule playing the role of a coordinating cation of the cubic cage made of PbI3. At room temperature the stable crystalline structure is tetragonal, obtained from a √2×√2×2 supercell of the original cubic cell, shown in Figure 1, with a transition to cubic symmetry at
327.4 K. Theoretical calculations give small formation energies of MA-I and MA-PbI3 vacancies, supporting the idea that the material has a quite high defect concentration (1017-1020 cm-3) [10]. These values are larger than traditional inorganic solar cells, hence the reason for the high efficiencies is still debated, since recombination losses should dominate in these materials. Among the different explanations, the absence of deep level traps associated to the vacancy defects might be satisfactory [10]. Another intriguing argument for the high efficiency points to the ferroelectric properties of the perovskite crystals [11], that may favor charge separation and reduce recombination.
Figure. 1. CH3NH3PbI3 perovskite tetragonal crystal structure (right) and the slightly deformed original cubic cell (left).
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Indeed, similarly to metal-oxide perovskites like BiFeO3, also MA-PbI3 are expected to display ferroelectric properties. In ABO3 crystals the spontaneous polarization originates from a cubic to orthorhombic deformation of the crystalline structure. In MA-PbI3, instead, at the origin of the dipole moment is the interplay between the charge transfer of 1e from the N+ to the PbI3- cage, and the non centro-symmetric position of the charge centers. Additionally, the isolated molecule possess a dipole moment of 2.2 Debye. A heated debate has broken out in the field about the possible formation and role of ferroelectric domains in these materials. Strong hysteresis and a large dielectric permittivity were initially ascribed to ferroelectricity [12]. However recent studies have been unable to detect any macroscopic ferroelectric domain, showing instead that the hysteretic behavior is rather due to mobile ions and charge traps [13]-[14]. Nevertheless, local dipoles can play a relevant role in the photovoltaic behavior of these materials. It has been suggested that ferroelectric domains may enhance electronhole separation and help achieve above-band-gap open circuit voltages [11],[15]. Spectroscopic measurements indeed indicate the presence of nanoscale structural domains [12]. Berry-Phase calculations give a quantitative estimate of the spontaneous polarization of MAPbI3. Depending on the authors, the calculated value ranges from a maximum of P=0.13 C/m2 [12], to a minimum of P=0.03 C/m2, obtained accounting just for the molecular dipole moment [11]. The correct value is most likely in between, at around P=0.08 C/m2, as obtained in ref [16]. Clearly, this would translate to a macroscopic polarization, only for a perfectly ordered crystal. In practice at room temperature the molecules are relatively free to rotate or tumble between discrete orientations, with their axes mainly oriented along the faces of the cubic cell [12],[17] with a rotation time-scale of 3 ps [18]. A consequence is that molecules may prefer to reduce electrostatic energy assuming opposite (anti-ferroelectric) orientations. In order to take into account for this electrostatic energy we built a model of the crystal in which every cubic cell (evidenced in Fig 1) is associated a dipole moment. In doing so we approximate the orthorhombic cells in which a=0.624 nm and c=0.632 nm, to be cubic (a≈c). This approximation has no effect on our analysis. The magnitude of the associated dipole moment is therefore p=Pa3. When a molecule rotates changing orientation of the polar head (NH3), we assume it induces a negative charge on the iodine ions, such as to rotate the whole cell dipole. In practice every cell possess a dipole of constant magnitude p that can randomly orient itself. In order to find stable configurations controlled by the dipole-dipole interaction energy,
r r U = ∑ pi ⋅ Eext +
1 4πε 0ε r
∑
r r pi ⋅ p j r3
−
r r 3( pi ⋅ rˆ)( p j ⋅ rˆ) r3
, (1) we performed metropolis Monte Carlo (MMC) runs, similarly to those introduced in ref. [12]. In this analysis elastic i
i, j
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energies originating from lattice deformations associated to molecular rotations are neglected. This may be justified on the basis that the molecule can easily change orientation at room temperature, hence elastic energy must be small. An important parameter in (1) is the dielectric screening constant of the material. The static permittivity, however, is not well known, since it is experimentally hindered by mobile charges, measuring values as high as 3 εr =10 [19]. Neighboring dipoles are rather screened by the dielectric response of the crystal, for which we expect εr≈10, a typical value for semiconductors of similar gaps. Values found in the literature indeed ranges between 5.6 and 8.2 [20]-[21], and recently GW calculations obtained εr =7.1 [22]. In this work we have analyzed cell behavior for values εr =6.1 and εr =10.0, as an upper bound.
Figure. 2. Diagram representing a solar cell with selective contacts. Polarization map in the perovskite (xcomponent) obtained using P=0.13 C/m2, εr=6.1 for T=0. In red are dipoles pointing to +x and in blue are pointing to -x. White areas corresponds to dipoles pointing to ±y.
a)
b) Figure. 3.Polarization maps (x-component) at T=300 K, ob2 2 tained using a) P=0.13 C/m , εr=6.1 and b) P=0.03 C/m , εr=10. Color codes as in Figure 2.
An example of the MMC simulations is shown in Figure 2. In this case a two dimensional (2D) model of the device was considered, with a lateral size of 130 cells (84.5 nm) and 384 cells in length (249.6 nm). The energy converged configuration was obtained after 108 MMC steps. From Figure 2 it is possible to appreciate the tendency of local anti-ferroelectric order, with alternate rows of dipoles oriented along +x (red) and -x (blue) direction. The dipole arrangements at T=0 do not depend on the dipole magnitude, p (hence on P), or the value of εr. In Figure 2 we have reported an example for P=0.13 C/m2 and εr =6.1.
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At T=300 K, instead, disorder and dipole-dipole interactions compete, leading to very different maps, depending on the parameters. As shown in Fig. 3a, for P=0.13 C/m2 and εr=6.1, anti-ferroelectric nanodomains are formed, with a typical size of 8-10 nm. In the opposite case of low dipoles (P=0.03 C/m2) and εr=10, the orientations tend to be rather random, as shown in Fig. 3b. MMC simulations on 3D models are easily performed, leading to domains of similar size. Results are shown in the supporting information (SI). 2D simulations where chosen essentially as a trade-off for computational complexity of the coupled transport calculations (see below). In order to compute solar cell characteristics, we consider intrinsic perovskite with band edges, EV=-5.4 eV, EC=-3.85 eV (Eg=1.55 eV) and selective Schottky contacts with Fermi levels, Ef= -5.35 eV at the cathode and Ef =-3.9 eV at the anode. This configuration mimics the presence of electron and hole transport layers. Transport is considered by solving a drift-diffusion (DD) model on top of the frozen 2D polarization maps. The coupling is performed on a non self-consistent manner. This is justified since free carrier densities leads to a correction of the electrostatic field of the order of 5% (SI). The DD equations are discretized with finite elements (FEM) using TiberCAD simulation tool [23]. The complete system of equations couple the Poisson and continuity equations:
∇ ⋅ jn = ∇ ⋅ ( µn n∇φn ) = − R + G ∇ ⋅ j p = ∇ ⋅ ( µ p p∇ φ p ) = R − G ∇ ⋅ (ε∇ϕ − P ) = − ρ
taken care consistently in the term ∇⋅P and ε is the dielectric constant of the material, that we choose consistent to equation (1). An important point to remark here is that polarization charges arise when ∇⋅P≠0, a condition occurring only across domain boundaries where the normal component of p changes. This produces potential variations at nanodomain interfaces as high as 0.5 eV, that superimpose to the band edges and confine electron and holes on different regions. To the contrary, antiferroelectric ordering does not give rise to potential variations, since locally ∇⋅P=0. An example of the charge density map can be seen in Figure 4, obtained for the dipole map of Figure 3a, clearly showing a separation between electrons and hole densities in regions having the typical size of the nanodomains. The charge separation of Figure 4, corresponds to preferential patterns for electron and hole currents, shown in Figure 5.
(2)
The first two equations represents continuity condition for the electron and hole currents, n and p are the electron and hole densities, µn=µp=10 cm2V-1s-1 are the carrier mobility, R and G are the recombination and generation rates, respectively. The generation is computed from the Lambert-Beer model and R accounts for recombination losses, either defect-mediated (Shockley-Read-Hall) or direct.
Figure. 4. Electron (blue) and hole (red) densities. Colorbar scales are in cm-3. The last equation in (2) is the Poisson equation, solving for the potential, ϕ, according to the total density, ρ, including free carrier densities as well as trap distributions, ionized donors and acceptors. The polarization field is
Figure. 5. Electron (top) and hole (bottom) current paths at short-circuit. Colorbar scales normalized to maximum current density.
A comment about the use of classical DD down to the scale of the crystal lattice is also mandatory. Clearly, this is an approximation, giving though the correct qualitative picture. Quantum delocalization and quantization is smearing out the classical charge. On the other hand quantum confinement enhances relative charge density inside the nano-regions, reducing density at domain boundaries. Quantum-corrected DD, on the other hand, is extensively used in device modeling giving even quantitative current densities. Another important issue concerns the competition between the dipole rotation timescale (3 ps) [18] and the average time the carrier takes to migrate to the electrodes. Assuming a purely diffusive motion we obtain for our device of 249.6 nm a maximal transverse time of 4.8 ns. This is much larger than the switching time and it is tempting to conclude that the polarization domains have no effects on average. However, for a mobility of 10 cm2/Vs, during
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the typical rotation timescale of 3 ps, the carriers diffuse a distance of about 9 nm which is of the same size as the nanodomains. Therefore, the electrostatic landscape remains frozen for a sufficiently long time for the electron and hole paths to separate on average. Additionally, in the SI we show that the domain boundaries move slowly giving time to the carrier to adapt their trajectories as they move to the electrodes. We conclude that domain orientation could guide potential variations and carrier separation over much larger distances than the crystal unit cell. In the following we analyze the impact on solar cell performance due to this carrier separation.
Figure. 6. SRH (top) and Direct (bottom) recombi-3 -1 nation patterns. Colorbar scales in cm s .
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constant throughout the cell, being proportional to np=ni2exp[β(Fn-Fp)] (Fn and Fp are the quasi Fermi levels). Deviations are possible only inside some domain, where the charge accumulation of one carrier become degenerate, exceeding the simple mass-action law. Hence, as seen in figure 6, radiative recombination displays hot spots whereas SRH a more fine-grained texture. The color scales of Figure 6 show that with this parameter choice, recombination at Voc is still dominated by defectmediated SRH processes. Since this is still much debated, as for instance a Langevin type recombination, often used in these materials, gives k2=qµ/ε=2.8⋅10-6 cm3s-1 (using µ=8 cm2V-1s-1), we have considered an alternative scenario in which recombination is dominated by direct processes. This is obtained by increasing the quadratic recombination rate to k2=10-3 cm3s-1, keeping τn and τp unchanged. A comparison between the SRH (k2=10-9 cm3s-1) vs Directdominated scenario (k2=10-3 cm3s-1) is displayed in Figures 7 and 8. Figure 7 compares the I-V characteristics in the two cases (left and right panels, respectively) for increasing magnitude of P at εr=6.1. Figure 7 is obtained by averaging over 10 different MMC realizations. The homogeneous case is plotted as a term of comparison. We notice that the (anti)ferroelectric domains typically induce a decrease of Isc and an increase of Voc. The latter is much larger in the case of SRH-dominated cells. The decrease of current is due to the increased disorder induced by the potential variations, and is higher for smaller values of the cell dipoles. In practice small dipoles generate a random noisy potential, detrimental for current paths. When P increases, domains are formed and good current percolation patterns can form at domain boundaries, as also seen in Fig 5.
The most important point we want to emphasize is how recombination losses are modified by the presence of nanodomains inducing charge separations. We consider SRH for trap levels close to midgap and direct recombination, given respectively by,
RSRH =
np − ni2 τ n ( p + pi ) + τ p (n + ni )
(3)
RDIR = k2 ( np − ni2 ) On the other hand we neglect Auger recombination (cubic in the carrier density) as this is known to be unimportant in perovskite cells [24]. We set τn =τp = 10-10 s and k2=10-9 cm3s-1 for the recombination parameters. These values are compatible with carrier lifetimes and two photon spectroscopy measurements [24]. At low carrier densities, hence close to Isc, linear SRH recombination dominates, whereas close to Voc carrier density increase and the quadratic recombination becomes important. The mathematical form of RSRH and RDIR in (3) allows to understand the different recombination patterns at Voc, shown in Figure 6. The SRH recombination reaches the largest values at domain boundaries, whenever n≈p>>ni. This is seen in (3) since when n≈p then RSRH ~ n/2τn, when n>>p, RSRH ~p/τn or vice versa when p>>n, RSRH ~n/τn. On the other hand the direct recombination should be rather
Figure. 7. I-V characteristics of the perovkite solar cell for increasing dipole magnitude at T=300K, εr =6.1, τn =τp -10 -9 -3 -1 -3 -3 -1 = 10 s and (left) k2=10 cm s ; (right) k2=10 cm s .
Figure. 8. (left) Open-circuit voltage vs dipole magnitude when SRH or Direct recombination dominate (see text). (right) Solar cell efficiencies vs dipole magnitude for the same cases.
The increase of Voc with respect to the homogeneous case is related to the overall decrease of recombination losses.
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In our model we assume local vertical absorption, hence we assume nanodomains have no effect on generation. Since no current flows at Voc, from equation (2) it must be R≡G. The only way to fulfill this condition is by an increase of carrier densities, or, equivalently, by a further splitting of the quasi Fermi levels (Voc = Fn - Fp). The increase of Voc is up to 40 mV (See Figure 8), a value certainly not large but significant in understanding and optimizing solar cells. The right panel of Figure 7 shows the same I-V characteristics for the direct-dominated cell. In this case the increase of Voc is negligible. This is consistent with the fact that direct recombination is essentially constant in the cell, except at few hot spots. Therefore, the condition R=G is reached for values of the quasi Fermi levels similar to the homogeneous case. Finally, in Figure 8 (right) we have reported the computed conversion efficiency, η. The presence of dipoles can increase η when P>0.08 C m-2. The values of η we obtain are lower than reported records because of the reduced thickness. Indeed, optimal device lengths are larger (~300 nm) including a metal reflector [25]. Scaling proportionally, we expect Isc ≈20 mA/cm2, from which we would get η=15%. A further improvement can be obtained by tuning Voc~1.1 V, however in this work we did not focus on the recombination details at the Schottky contacts affecting Voc. Nevertheless, our calculations point out a general behavior, relevant in understanding this type of hybrid solar cells. A final issue concerns the 2D modelling presented here, compared to realistic 3D simulations. This is especially important in connection to percolation pathways that are notoriously sensitive to dimensionality. We emphasize that most of our discussion is focused on Voc that is found insensitive to MMC samples and percolations pathways (see SI). Further, the shift of Voc found in the 2D models is supported by 3D calculations. On the other hand, in well performing solar cells (almost) all carriers are collected at the electrodes and Isc is fixed by the generation rate. For this reason, Isc should slightly increase when recombination decrease, as indeed shown by the 3D simulations. In the reduced 2D model, Isc exhibits sample to sample variations, depending on the formation of efficient percolation paths. The problem is reduced when increasing lateral thickness, that was set to 84.5 nm as a trade-off with computational complexity. In this respect, the efficiencies shown in Figure 8 are slightly underestimated. In conclusion, we have presented a Metropolis MC and drift diffusion calculation of the steady state properties of a perovskite solar cell in the presence of a pattern of orientated dipoles. The pattern was obtained from MC runs by energy minimization of the dipole-dipole interaction. Short-range anti-ferroelectric order was found, whereas at scales of 8-10 nm we observed the formation of nanodomains, strongly influencing the electrostatic of the device. We have analyzed the parameter space in terms of dipole magnitude and permittivity that leads to the formation of nanodomains. Then we have coupled the obtained models to drift-diffusion simulations in order to
study the actual role of nanodomains in the I-V characteristics, especially focusing on charge separation and recombination losses. We demonstrate that holes and electrons accumulates within the nanodomains and can follow very different current pathways at opposite boundaries. From our analysis we can conclude that dipoles can ultimately lead to an increase of photoconversion efficiencies thanks to a decrease of SRH recombination losses and the formation of good current percolation patterns along nanodomain edges.
ASSOCIATED CONTENT Supporting Information contain validation of the presented work with three dimensional simulations “This material is available free of charge via the Internet at http://pubs.acs.org.”
AUTHOR INFORMATION Corresponding Author CNR- ISMN, Via Salaria km 29.300, 00017 Monterotondo, Italy. e-mail:
[email protected] ACKNOWLEDGMENT We acknowledge CHOSE organization at University of ‘Tor Vergata’ for financial support. The authors declare no competing financial interest.
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