Erratum: Photovoltage from ferroelectric domain walls in BiFeO$_3$

TL;DR

This paper corrects previous simulations showing that ferroelectric domain walls in BiFeO3 produce negligible photovoltage, contradicting earlier claims that they could explain the observed photovoltage due to delocalized excitons.

Contribution

It provides a corrected methodology demonstrating that excitons in BiFeO3 are self-trapped, leading to minimal domain-wall photovoltage, thus revising prior interpretations of experimental results.

Findings

Self-trapped excitons dominate in BiFeO3

Domain-wall photovoltage is too small to explain measurements

Previous delocalization predictions were due to methodological errors

Abstract

In the original article a mistake in the methodology lead to an incorrect prediction of exciton delocalization below a critical exciton density. This unrealistic delocalized exciton state yielded a sizable domain-wall photovoltage. When done correctly, the simulations yield self-trapped (localized) excitons at all considered exciton densities, without any evidence for a transition to a delocalized exciton. This realistic self-trapped exciton state yields only a negligible domain-wall photovoltage. The original conclusion that ferroelectric domain walls could be responsible for the measured photovoltage if carrier lifetime and diffusion length are higher than expected is incorrect. Correct is: The domain-wall photovoltage in BiFeO3 is much too small to explain the measured photovoltage. The original analysis is meaningful for ferroelectrics without carrier self-trapping, not for BiFeO3.

Equations

$$V_{\mathrm{photo}}(s)=\left[V_e^X(s)-V_e^{\mathrm{GS}}(s)\right]/(-|e|),$$

$$V_{\mathrm{photo}}^{\mathrm{DW}}=V_{\mathrm{photo}}\left(s_{\mathrm{DW}}^{-}\right)-V_{\mathrm{photo}}\left(s_{\mathrm{DW}}^{+}\right),$$

$$n_X=g\tau =\frac{I_{\mathrm{light}}(1-R)\tau }{E_{\mathrm{photon}}{d}_{\mathrm{film}}},$$

Full text

Domain walls, Excitons, Ferroelectric domains, Photovoltaic effect, Ferroelectrics, Oxides, Perovskite, DFT+U, Density functional calculations, First-principles calculations

Introduction

It has long been debated if and to what extent ferroelectric domain walls contribute to the photovoltaic effect (PVE) in ferroelectric oxides like BaTiO3 and BiFeO3. One possible origin of the PVE in BiFeO3 is the so-called bulk photovoltaic effect (BPVE) Fridkin and Popov (1978); Kraut and von Baltz (1979); von Baltz and Kraut (1981); Fridkin (2001); Young and Rappe (2012); Young et al. (2012). The BPVE is a phenomenological term describing any photovoltaic effect that takes place in the homogeneous interior of the material, as opposed to interface effects. The BPVE in ferroelectrics such as BiFeO3 and BaTiO3 has been ascribed to noncentrosymmetric scattering or relaxation of electrons and holes after photoexcitation in noncentrosymmetric crystals, resulting in a net shift of charge carriers (“shift current”) Kraut and von Baltz (1979); von Baltz and Kraut (1981); Young and Rappe (2012); Young et al. (2012). The BPVE depends on the polarization direction of the incoming light Fridkin (2001). This dependence was indeed observed in the case of BiFeO3, and it was hence concluded that the BPVE is at the origin of the photovoltaic effect in BiFeO3 Bhatnagar et al. (2013); Yang et al. (2017).

Alternatively a domain-wall driven PVE (DW-PVE) has been proposed Seidel et al. (2009); Yang et al. (2010); Seidel et al. (2011). The argument in favor of the DW-PVE is a variation in the ferroelectric polarization of the atomic lattice at the domain walls. It has been suggested that this polarization variation gives rise to internal electrostatic fields at the domain walls. If true, this would mean that ferroelectric domain walls could separate photogenerated charge carriers in the same way as $p$-$n$ junctions, without the need for $p$- and $n$-type doping, and it would be possible to align arbitrarily many such junctions in series and add up the individual voltages created by each single junction. However, the DW-PVE theory needs to postulate a local ferroelectric polarization not only of the spatially discrete atomic lattice, but also for the continuum of the valence electrons. But is it possible to define and determine the electric polarization of an arbitrary section of a crystal? Only then can this local polarization induce an electric field. Such approach was adopted in previous first-principles studies of voltage steps/drops at ferroelectric domain walls in PbTiO3 Meyer and Vanderbilt (2002) and BiFeO3 Seidel et al. (2009) and yielded electrostatic potential drops (electronic potential steps) ranging from 0.02 to 0.2 V per domain wall for the dark state (without illumination). We will demonstrate below that this polarization-based approach does not yield the correct sign of the potential step/drop and the photoinduced charge density at domain walls in BiFeO3. In contrast to the BPVE, the DW-PVE is independent of the polarization direction of the incoming light.

Some studies Young et al. (2012); Inoue et al. (2015) take the middle ground by assuming that the BPVE and the DW-PVE may exist simultaneously and can be cooperative or antagonistic depending on the system geometry. The argumentation here is that, besides the characteristic angular dependence of the BPVE with respect to the polarization of the incoming light, there is a polarization-independent offset in the photocurrent, which might be ascribed to a DW-PVE originating in an electrostatic field at the ferroelectric domain walls Young et al. (2012), and/or in a locally modified BPVE, caused by the local crystal-structure modifications at the domain walls Inoue et al. (2015).

Whereas the atomistic structure of ferroelectric domain walls in BiFeO3 is accurately known thanks to electron microscopy Wang et al. (2013), experimental spectroscopy of photoelectrons at ferroelectric domain walls with atomic resolution is to our knowledge not possible today, but is perfectly within reach of first-principles calculations based on density-functional theory (DFT). In fact, optical excitations and the spatial distributions of the photoexcited charge carriers in molecules and solids are nowadays routinely investigated by means of many-body perturbation theory Blase et al. (2018); Spataru et al. (2004); Cudazzo et al. (2016); Rödl et al. (2008), such as the Bethe-Salpeter equation, or time-dependent density-functional theory. In the case of ferroelectric domain walls such studies would be hard to undertake because of the large system size needed to model a domain wall. However, there are approximate methods with a favorable balance between accuracy and computational cost, such as the excitonic $\Delta$-self-consistent-field ($\Delta$SCF) method, which yields more than qualitative agreement with highly accurate many-body methods Artacho et al. (2004), but can be performed at the same computational cost as a DFT ground-state calculation. The excitonic $\Delta$SCF approach has been applied to study excitons in various systems, including organic dyes Kowalczyk et al. (2011), polymers Artacho et al. (2004); Ceresoli et al. (2004), and surfaces Pankratov and Scheffler (1995). Here we use it to directly determine the magnitude of the domain-wall driven photovoltaic effect in BiFeO3, as given by the electronic potential induced by excitons localized at the domain walls.

Methods

We focus on the 71°  and the 109° domain wall, for which the direction of the ferroelectric polarization in adjacent domains differs by about 71° and by 109°, respectively. In rhombohedral perovskites like BiFeO3 there exists also a 180° domain wall; however, due to its symmetry it should be photovoltaically inactive, and hence we do not consider it here. The DFT calculations were performed with the vasp code Kresse and Furthmüller (1996), using the projector-augmented wave (PAW) method and pseudopotentials with 5 (Bi), 16 (Fe), and 6 (O) valence electrons, respectively. We employed the local-spin-density approximation, and corrected the $3d$ states of Fe with a Hubbard-$U$ of 5.3 eV following Dudarev’s scheme Dudarev et al. (1998). The $U$ value for Fe was taken from the “materials project” database Jain et al. (2013) and it is optimized for oxide formation energies, but it also yields an optical band gap of 2.54 eV close to the experimental one of $\approx$2.7–2.8 eV Basu et al. (2008); Hauser et al. (2008); Ihlefeld et al. (2008); Kumar et al. (2008); Železný et al. (2010); Sando et al. (2018). This computational setup yields structural properties of BiFeO3 in reasonable agreement with experiment Moreau et al. (1971); Palewicz et al. (2007, 2010); Fischer et al. (1980); Lebeugle et al. (2007); Note (1). The reciprocal space was sampled with $2\times 5\times 3$ $k$-points for the 71° wall and with $2\times 5\times 5$ $k$-points for the 109° wall. Plane-wave basis functions with energies up to 520 eV were used. We employed a supercell approach with periodic boundary conditions, such that each supercell contained 120–280 atoms and two domain walls. Both the atomic positions and cell parameters were allowed to relax until the energy difference between subsequent ionic relaxation steps fell below 0.1 meV. Our calculated structural properties of the domain walls are similar to those in previous theoretical Diéguez et al. (2013); Chen et al. (2017); Ren et al. (2013); Wang et al. (2013) as well as experimental Wang et al. (2013) works Note (1). Excitons were modeled with the excitonic $\Delta$SCF method Kowalczyk et al. (2011), namely by occupying the valence states with altogether $(N-N_{X})$ electrons and the conduction states with $N_{X}$ electrons, where $N$ and $N_{X}$ are the total number of electrons and excitons in the supercell, respectively. This was done in every iteration step of the electronic self-consistency cycle, using the same density functional for minimizing the energy as in a ground-state calculation. The geometries of the systems in the presence of an exciton were optimized when considering excitonic polaron states. Supercells containing 120 atoms were used in this case. The excitonic $\Delta$SCF method is suitable for exploring low-lying excited states only, whereas access to higher-lying excited states such as Rydberg states can be obtained using constrained DFT Ramos and Pavanello (2018); roychoudhury:2020:neutral or by maximizing the similarity of the excited-state orbitals to reference orbitals Cheng et al. (2008); Gilbert et al. (2008). In order to obtain the photovoltage per domain wall, $V_{\mathrm{photo}}^{\mathrm{DW}}$, we first determine the photovoltage profile, $V_{\mathrm{photo}}$, which is given by the difference in electronic potential between the excited state, $V_{e}^{X}$, and the ground state, $V_{e}^{\mathrm{GS}}$:

[TABLE]

where $s$ is the coordinate perpendicular to the domain wall. The domain-wall contribution $V_{\mathrm{photo}}^{\mathrm{DW}}$ is then equal to the amplitude of the spatial variation of the photovoltage profile at the domain wall (compare Fig. 6),

[TABLE]

where $s_{\mathrm{DW}}^{+/-}$ are positions to the right and to the left of the domain wall. We calculate both $V_{e}^{X}$ and $V_{e}^{\mathrm{GS}}$ in the ground-state structure. By doing so, the electronic screening is already included. We still need to consider the screening by the lattice, which we calculate ab initio Note (1). Note that, since we are working with DFT, $V_{e}$ is the Kohn-Sham potential. Furthermore, the use of periodic boundary conditions implies that we are calculating the short-circuit potential. The open-circuit potential is then obtained from the short-circuit potential by adding a constant gradient that compensates the potential slope in the domain interior Meyer and Vanderbilt (2002). The photovoltages are extrapolated Note (1); Wolfgang P. Schleich (2001) to avoid finite-size effects.

Results and discussion

We begin the discussion with the electronic potential at the ferroelectric domain walls in the ground state (without excitons). Figure 1 depicts the ferroelectric polarization, the electronic potential energy per electron, $V_{e}$, and the polarization-based potential energy per electron, $V_{e}^{\mathrm{pb}}$, as calculated from the polarization variation at the wall including screening Note (1). If the polarization-based approach was valid, $V_{e}^{\mathrm{pb}}$ should yield the same potential step/drop at the wall as $V_{e}$. $V_{e}$ has a potential minimum on the right-hand side of the wall. Accordingly, we expect excess electrons to accumulate on the right-hand side of the wall. The magnitude of the electronic potential drop (the electrostatic potential step) at the wall is 0.13 eV (extrapolated Note (1)). $V_{e}^{\mathrm{pb}}$ exhibits an electronic potential step at the wall instead of a potential drop, and the magnitude of the potential variation, 45 meV, is too small. For the 109° domain wall (Fig. 2) the potential has a pronounced minimum inside the domain wall, and a very small slope in the domain interior corresponding to an electronic potential drop of about 17 meV Note (1). $V_{e}^{\mathrm{pb}}$ lacks the minimum inside the wall, and exhibits an electronic potential step instead of a potential drop, and the magnitude of the potential variation, 56 meV, is too large. We conclude that the polarization-based approach fails to provide the correct sign and magnitude of potential steps/drops at domain walls in BiFeO3.

Next, we will add an electron-hole pair (an exciton) to our supercell. Depending on the exciton density, two different types of exciton polarons form, as depicted in Fig. 3. For low exciton densities, a large exciton forms with photoelectrons and -holes localized on opposite sides of the domain wall resulting in an excitonic dipole moment. Note that the sign of the photoinduced charge density (electron on the right-hand side of the wall, hole on the left) is the same as one would expect based on the electronic potential drop (the electrostatic potential step) in the ground state, but it is opposite to what one would expect based on the polarization-based potential (electron on the left-hand side of the wall, hole on the right). The same is true for individual electrons and holes Körbel et al. (2018). At high exciton densities a small, almost concentric exciton polaron forms (a self-trapped exciton) with a negligible dipole moment. Figure 4 shows the densities of excess electrons and holes for different densities of exciton polarons in the 120-atom supercell, and the excitation energy per exciton of the large and the small exciton polaron. There is a critical exciton density above which the large exciton polaron transforms into a small exciton polaron (above which the formation energy of the small exciton polaron becomes lower than that of the large exciton polaron), and a coexistence region in which either the large or the small exciton polaron is stable and the other one is metastable. Since only the large exciton configuration will lead to a sizable photovoltage, we determine next under which experimental conditions the large exciton forms.

In Ref. Alexe, 2012 it was found that the photocurrent in BiFeO3 follows a rate equation, $\dot{n}=g-n/\tau,$ where $n$ is the photocarrier density, $g$ is the photocarrier generation rate, and $\tau\approx 75~{}\mu$s is the photocarrier lifetime. Here we adopt the same rate equation for the exciton density $n_{X}$, which in the steady state is given by

[TABLE]

where $I_{\mathrm{light}}$ is the intensity of the light in W/cm2, $d_{\mathrm{film}}$=100 nm is the thickness of the experimentally studied films in Refs. Bhatnagar et al., 2013 and Seidel et al., 2011, $E_{\mathrm{photon}}$=3.06 eV is the photon energy of the laser used in the experiment in Ref. Bhatnagar et al., 2013, and $R\approx 0.27$ is the reflectivity calculated at this photon energy from first principles. In the following we assume that 100% of the penetrating light is absorbed 111See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.102.081304 for figures and tables with calculation results for crystal structure, domain-wall properties, and photovoltage, compared with the literature where available, and technical details. , in line with experiment Alexe and Hesse (2011). We assume further that all photocarriers within a distance $l_{\mathrm{diff}}$ (the diffusion length) from the domain wall reach the domain wall. Figure 4(b) shows that the critical exciton density from the supercell calculation is $n_{X}^{\mathrm{crit}}\approx 0.22$ $X$ per supercell, corresponding to an planar exciton density of $\approx 5.2\times 10^{13}/$cm2. This is equal to a photoelectron density of 3.7$\times 10^{18}$/cm3 if one assumes a diffusion length of 140 nm. From Eq. (3) we obtain the critical light intensity for the transition between large and small exciton polarons Note (1), which is depicted in Fig. 5.

In the case of natural sunlight and a diffusion length near typical domain-wall spacings of a few hundred nanometers, the large exciton forms, which should give rise to a photovoltage, whereas in the case of a thousandfold concentrated sunlight, such as in a concentrator solar-cell setup, the small exciton polaron forms, which should contribute less to the photovoltage, if at all. The small exciton polaron might be detectable with photoluminescence spectroscopy as a state inside the band gap, or as a drop in the photovoltage at illumination intensities of the order of 0.1–1 W/cm2 or higher, depending on the carrier diffusion length.

In the following we consider moderate light intensities, at which the large exciton forms. The domain-wall photovoltage is the spatial potential variation induced by the large exciton at the domain wall, depicted in Fig. 6. The potential generated by the excitons (partially) compensates the electronic potential drop (the electrostatic potential step) at the domain wall. The relation between exciton density and light intensity is the same as before, now we use the parameters of the experimental photovoltage measurement from Ref. Seidel et al., 2011 in Eq. (3) ($E_{\mathrm{photon}}=3.31$ eV, $R=0.25$, domain-wall spacing $d_{\mathrm{DW}}$=140 nm). These parameters are similar to those used above to determine the critical exciton density of the phase transition. As an upper limit we consider a diffusion length equal to the domain-wall spacing, $l_{\mathrm{diff}}=d_{\mathrm{DW}}=140$ nm Note (1). We also consider an estimated lower limit of $l_{\mathrm{diff}}$=10 nm similar to that assumed in Ref. Seidel et al., 2011, and a shorter carrier lifetime of 1 ns, similar to that reported in Ref. Sheu et al., 2012. The resulting open-circuit photovoltage contribution per domain wall, together with the experimental results from Ref. Seidel et al., 2011, are depicted in Fig. 7 and extrapolated to lower light intensities using a power law Note (1). In the case of the 109° domain wall we can only give a possible range (shaded area) Note (1). The experimental photovoltage is the total photovoltage of a film with 71° domain walls, consisting of all photovoltaic effects (bulk and domain-wall effects), divided by the number of domains. The experimental conditions are well inside the range in which the large exciton forms (marked by the vertical solid line), for which our approach should be valid. The calculated domain-wall photovoltage matches the experimentally measured one only in the most optimistic scenario [Fig. 7(a)], in which we assume a photocarrier density of $10^{16}$/cm3 to $10^{18}$/cm3, a carrier lifetime of 75 $\mu$s, and a carrier diffusion length of 140 nm. For comparison, in Ref. Seidel et al., 2011 a carrier density of about $10^{12}$/cm3 to $10^{13}$/cm3, a carrier diffusion length of 8 nm, and a lifetime of 35 ps were assumed, and in Ref. Sheu et al., 2012 a lifetime of 1 ns. If we assume such conditions [Figs. 7(b) and 7(c)], the DW-PVE is orders of magnitude too small to account for the major part of the measured photovoltage in BiFeO3. The domain-wall photovoltage may be further reduced through screening by free charge carriers and/or point defects that accumulate at the domain walls.

Conclusion

We have analyzed the contribution of ferroelectric domain walls to the photovoltage in BiFeO3 using first principles methods. In general we find that the ferroelectric polarization profile does not allow one to determine the correct sign and magnitude of the electrostatic potential at the domain walls. Instead the electronic potential should be directly determined from ab initio calculations. The domain-wall driven photovoltages can be as large as the experimentally measured total photovoltages (up to $\approx$10 mV per domain wall), and may therefore be responsible for a large portion of the photovoltaic effect in BiFeO3. This, however, is true only if the carrier lifetimes and carrier diffusion lengths are of the order of $\approx 100~{}\mu$s and $\approx 100$ nm, respectively, which is orders of magnitude larger than previously assumed. Otherwise, the domain-wall photovoltage is orders of magnitude smaller, and then the major fraction of the photovoltage should originate from other effects, for example bulk effects. Furthermore, there is a transition from a large to a small exciton polaron at high illumination intensities of the order of 0.1–1 W/cm2 or higher, which might be experimentally detected as a drop in the photovoltage at high light intensities, or as a state inside the band gap in photoluminescence spectroscopy.

Acknowledgement

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 746964. Computation time and support provided by the Trinity Centre for High Performance Computing funded by Science Foundation Ireland is gratefully acknowledged. We also thank the Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support. We are grateful to Subhayan Roychoudhury (Trinity College Dublin) for sharing his knowledge of excited-state density-functional theory. Graphics were made using gnuplot and vesta Momma and Izumi (2011).