# Photogalvanic Shift Currents in BiFeO3 –LaFeO3 Superlattices

Francesco Delodovici, Charles Paillard

► To cite this version:

Francesco Delodovici, Charles Paillard. Photogalvanic Shift Currents in BiFeO3 –LaFeO3 Superlattices. ACS Applied Energy Materials, 2025, 8 (3), pp.1716-1721. 10.1021/acsaem.4c02857. hal-04952725

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Distributed under a Creative Commons Attribution 4.0 International License
Photogalvanic shift currents in BiFeO₃-LaFeO₃
superlattices

Francesco Delodovici*,† and Charles Paillard*,‡

†Université Paris-Saclay, CentraleSupélec, CNRS, Laboratoire SPMS, 91190,
Gif-sur-Yvette, France

‡Smart Ferroic Materials center, Institute of Nanosciences & Engineering and Department
of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA

E-mail: francesco.delodovici@centralesupelec.fr; paillard@uark.edu

Abstract

Designing materials with controlled photovoltaic response may lead to improved
solar cells or photosensors. In this regard, ferroelectric superlattices have emerged as a
rich platform to engineer functional properties. In addition, ferroelectrics are naturally
endowed with a bulk photovoltaic response stemming from non-thermalized photo-
excited carriers, which can overcome the fundamental limits of current solar cells. Yet,
their photovoltaic output has been limited by poor optical absorption and poor charge
collection or photo-excited carrier mean free path. We use Density Functional Theory
and Wannierization to compute the so-called Bulk Photovoltaic shift current and the
optical properties of BiFeO3/LaFeO3 superlattices. We show that, by stacking these
two materials, not only the optical absorption is improved at larger wavelengths (due
to LaFeO3 smaller bandgap), but the photovolgavanic shift current is also enhanced
compared to that of pure BiFeO3, by suppressing the destructive interferences occurring
between different wavelengths.
## Keywords

Photogalvanic effect, shift-current, materials design, superlattices, $BiFeO_3$.

# 1 Introduction

The bulk photovoltaic (or more accurately, photogalvanic) effect (BPVE)¹ has harbored the promise of breaking the fundamental thermodynamic limit which impedes the yield of current junction-based solar cells, known as the Shockley-Queisser limit.² One of the first requirement for the BPVE is the absence of inversion symmetry. Therefore, ferroelectrics, which show a spontaneous electrical polarization that can be switched by electric fields, have been deemed particularly promising photovoltaic materials as they naturally break inversion symmetry. In fact, Spanier *et al*. showed that the Shockley-Queisser barrier could be overcome in a thin film of barium titanate,³ a prototypical ferroelectric materials. However, it has proven difficult to reach power conversion efficiencies beyond a few percents with ferroelectric-only based solar cells. This is due, in part, to the large bandgap of most ferroelectrics which start absorbing in the ultraviolet part of the solar spectrum, as well as the usually low carrier mobility exhibited by conventional ferroelectrics.⁴,⁵

To circumvent these problems, many have attempted to partially modify the chemical structure of the active absorbing element. For instance, by making a solid solution involving ferroelectric $KNbO_3$ and oxygen deficient $BaNi_{1/2}Nb_{1/2}O_{3-\delta}$, the bandgap of $KNbO_3$ is strongly reduced down to 1.4
eV.⁶ Alternatively, co-doping methods, used to improve optical absorption while limiting leakage and maintaining ferroelectric properties, have been explored but often result in localized intragap states with very limited dispersion.⁷ Perhaps most promisingly, Nechache *et al.* reached a 8.1% power conversion efficiency by stacking 3 layers of Bi(Fe,Cr)O₃ thin films with varying Cr content to enhance light absorption.⁸ This latter study, combined with the improved photovoltaic efficiency of barium titanate thin films⁹ and demonstrated enhanced collection of photo-induced charges by Atomic Force Microscopy tips,³,¹⁰ strongly indicate that superlattice architectures, which are periodical repeated stackings of nanolayers, may help (1) enhancing optical absorption by combining materials with different bandgaps and (2) increase the collection of photocurrents created by the bulk photovoltaic effect.

In this work, we investigate the BPVE in superlattices (SLs) of BiFeO₃ (BFO) and LaFeO₃ (LFO) by means of Density Functional Theory (DFT) calculations. The choice of these two perovskites is dictated by several natural reasons. Firstly, BFO is one of the few ferroelectrics which absorb visible light with its typical bandgap of 2.7 eV, showing for instance good piezophotocatalytic¹¹ or photovoltaic responses,¹²,¹³ as well as a predicted high shift current response compared to other classical ferroelectrics.¹⁴ Secondly, LFO has a smaller bandgap of about 2.34 eV,¹⁵ has been grown in superlattices with BFO¹⁶⁻¹⁸ exhibiting ferroelectric features, in particular at short periods.¹⁹ We thus expect that BFO/LFO SLs should exhibit some degree
of improved visible absorption and BPVE thanks to the complementarity of the BFO and LFO nanoscale layers.

## 2 Systems and methods

The linear BPVE contains two main contributions:¹ a ballistic current, caused by asymmetry in the distribution of photo-induced carriers in momentum space, and a shift current²⁰ caused by asymmetry in the spatial location of photo-induced carriers after photo-excitation. Young et al.¹⁴,²¹ have developed methods to calculate the latter from DFT. A third order tensor, $\sigma_{ijk}$, describes the relationship between the shift photogalvanic current density in direction $i$, $j_i$, the intensity $I$ and polarization vector $e$ of a linearly polarized electromagnetic wave of pulsation $\omega$,

$$j_i = \sigma_{ijk} I e_j e_k \quad (1)$$

To calculate this tensor in BFO/LFO SLs, we performed DFT spin-polarized calculations as implemented in Quantum Espresso²² in the projected augmented wave (PAW) scheme²³ at the PBEsol²⁴ level of approximation. We included the rotationally invariant U correction, in the simplified Dudarev approach, with $U = 4.5$ eV¹⁶ to represent the on-site Coulomb interaction on Fe-ions belonging to the BFO layer, and $U = 3.65$ eV on the Fe-ions belonging to the LFO layer. This choice allows to maintain a similar bandgap
difference between bulk BFO (which we calculate at 2.26 eV here) and bulk
LFO (1.83 eV calculated bandgap). We relaxed pure BFO and LFO until
the maximum force was smaller than $10^{-3}$ eV/Å, sampling the Brillouin zone
with a Monkhorst-Pack mesh with a density of at least 110 kpoints×Å³. With
this choice of U, the relaxed ground state for BFO and LFO are described
respectively by a pseudo-cubic lattice parameter $a = 3.977$ Å and an angle
of $\alpha = 89.65$ degree and by an orthorhombic Pbnm phase whose primitive
vectors are $a = 5.531$ Å, $b = 5.591$ Å, $c = 7.838$ Å, in good agreement with
the literature.¹⁶

We then considered the superlattices as if they were epitaxially grown
on (001)-SrTiO₃, which is a common substrate for growing BFO films and
BFO/REFO superstructures.²⁵⁻²⁷ We create the SL by alternating layers of
BFO and LFO along the c direction of LFO. In this orientation, the in-plane
pseudo-cubic vectors of BFO form a 45 degree angle with the in-plane prim-
itive vectors of the superlattice, and the polarization is now directed along
the [101] direction, as reported in Figure 1. The superlattice primitive vec-
tors in Figure 1 are chosen as the lab reference frame, and all subsequent
tensor components (optical dielectric permittivity, shift current tensor) refer
to these axes. The mechanical effect of the substrate was simulated by fixing
the in-plane lattice constant of the multi-layer to the one of cubic SrTiO₃
(previously relaxed with DFT, lattice parameter of 3.891 Å in its cubic unit
cell), allowing all the remaining degrees of freedom to relax. In this con-
figuration the BFO in-plane pseudo-cubic lattice parameters are strained by
-2% when comparing to the BFO relaxed bulk structure, while the **a** and **b**
lattice parameters of LFO are strained by -0.6% and -1.6% respectively. It is
known that STO substrates inhibit the cycloidal order in BFO films²⁸ leading
rather to G-type antiferromagnetic (AFM) order. We thus performed spin-
polarized simulations without spin-orbit coupling (SOC) to correctly describe
the antiferromagnetic G-type magnetic ordering of the two pure perovskites.

Past theoretical studies have shown that good agreement of the shift cur-
rent with experiment could be obtained without SOC.²⁹ After relaxation,
the self-consistent charge densities were employed as input to obtain Kohn-
Sham wave functions from non-self consistent calculation on denser k-point
meshes (of the order of 10³ kpoints×Å³). The resulting Kohn-Sham wave
functions were then processed through Wannier90³⁰ to obtained the corre-
sponding set of maximally-localized Wannier orbitals. A mixture of chem-
ically sound atomic orbitals (O *p*, Fe *p* and *d*, Bi *p*, La *p*) were employed
as an initial guess to derive the Wannier orbitals. An example of the band-
structure reconstructed from Wannier orbitals is shown in Figure 2. We then
obtained the electrons transport properties by post-processing the Wannier
orbitals using the method described in Ref.³¹ and extracted the shift-current
contribution to the BPVE in these SLs, as well as the Wannier-orbital based
Kubo-formulation to obtain optical conductivity and dielectric response. We
focus, in the following, on BFO₁/LFO₁ SLs structures.
# 3 Results

## 3.1 Superlattice structure

Let us first describe the relaxed structure of the different components of the BFO₁/LFO₁ SLs, compared to pure BiFeO₃ and LaFeO₃. As can be observed in Figure 1, BFO bi-axially strained at the STO lattice constant exhibits its well-known R-phase structure,²⁸ characterized by a total polarization lying along the diagonal of the perovskite cube (which corresponds, in our chosen reference frame, to the [101/2] direction) estimated to be 81.9 µC/cm² and an oxygen octahedra tilt pattern $a^-a^-a^-$ in Glazer notation.³² On the other hand, strained LFO (see right panel in Figure 1) retains its *Pnma* structure, characterized by antipolar displacements along [100] direction and a tilt pattern of the form $a^-a^-c^+$. When relaxing the BFO₁/LFO₁ superlattice, the most stable phase shows $a^-a^-c^+$ tilt patterns, in agreement with previous reports,¹⁹,³³ with no out-of-plane polarization but in-plane polarization along the [100] axis due to the difference in the amplitude of cationic motion of La and Bi ions (see Figure 1). The space group of the relaxed superlattice is *Pc*.

## 3.2 Electronic bandstructure and optical properties

Next, we focus on the electronic bandstructure and ensuing optical properties, calculated in the Independent Particle Approximation (see Methods and Systems section). In Figure 2a, we plot the bandstructure of BFO₁/LFO₁.
<figure><img src="image_2.png" /><figcaption>Figure 1: Calculated cells for strained BFO (left), BFO₁/LFO₁ (middle) and LFO (right); oxygen atoms are depicted in red, iron atoms in gold, bismuth in purple, and lanthanum atoms are depicted in green.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: (a) Band structure of BFO₁/LFO₁ SL calculated from DFT (black solid lines) and from Wannier interpolation (red dashed lines). (b) Layer-by-layer decomposed Density of States in BFO₁/LFO₁ SL: dashed lines corresponds to orbitals in the LFO layer, continuous lines to those in BFO layer. (c) imaginary part of the longitudinal dielectric relative permittivity shows that the onset of BFO/LFO (yellow plain line) absorption occurs, as in LFO (black dashed-dotted lines), at lower photon energy compared to bulk BFO (purple dashed lines).</figcaption></figure>
It shows that the electronic bandgap of 1.88 eV is indirect between the *T* and Γ point, although an almost equally large direct bandgap is present at the *T* point. Inspection of the atomic-decomposed Density of States (DOS) indicates that the bottom of the conduction band at Γ mainly consists of Fe *d* states belonging to the FeO₂ plane closer to the LaO plane (i.e. to the LFO layer). In contrast, the top of the valence band, located at the *T* point in the Brillouin zone, is primarily made of O 2*p* states belonging to the FeO₂ planes. Pure compressed BFO and LFO have indirect band gap too, as reported in Supplementary Information and consistent with the literature.<sup>15,34</sup> The conduction (valence) bands minimum (maximum) sit at Γ (along the S-X line) in BFO, and at Γ (X) in LFO. The orbital character is preserved when the SL is formed.

Figure 2c compares the optical dielectric permittivity of BFO₁/LFO₁ SL, strained LFO and BFO bulk. Strained LFO, as expected, starts significantly absorbing at lower photon energies (near about 1.9 eV) compared to bulk BFO (which starts absorbing near 2.25 eV). Interestingly, the optical absorption of BFO₁/LFO₁ SL is dominated by the LFO layer at low photon energies (from 1.9 eV to 2.25 eV). Therefore, the SL architecture successfully manages to improve optical absorption at lower photon energies compared to pure BFO. In contrast, at larger photon energies (> 2.3 eV), the imaginary part of the relative dielectric permittivity is significantly degraded compared to pure BFO, indicating that optical absorption will be poorer than BFO at
large photon energies. Yet, the presence of BFO improves optical absorption over LFO in BFO/LFO SLs.

### 3.3 Shift current photogalvanic tensor

We now focus on the BPVE response of our BFO/LFO SLs. In Figure 3, we plot the shift current tensor $\sigma_{ijk}$ (which relates the photogalvanic current density to the incident light intensity and polarization, see Eq. 1) for $BFO_1/LFO_1$ SL and bulk BFO (LFO is centrosymmetric, and thus does not exhibit a BPVE response). Interestingly, as already observed in the dielectric permittivity in Figure 2c, the shift current photo-response in $BFO_1/LFO_1$ SL starts at lower photon energies. For instance, in Figure 3a, the $\sigma_{yyy}$ component of the shift current tensor has a finite value (near 2-6 $\mu$A/V$^2$), which is 20 times as large as the response of BFO for photon energies in the range 2.1-2.3 eV, i.e. right before the absorption edge of pure BFO.

In contrast, for photon energy larger than the calculated bandgap of BFO ($\approx$ 2.26 eV), the magnitude of the BPVE shift current tensor in $BFO_1/LFO_1$ is on par or slightly smaller than BFO. To further probe the photovoltaic response of $BFO_1/LFO_1$, we depict polar plots in Figure 3b representing the absolute value of the photogalvanic shift current density response at selected photon energies. We assume that the visible light is polarized in the (xy) plane, i.e. that the polarization vector of the light is of the form $\mathbf{e} = (\cos \phi, \sin \phi, 0)$, which essentially corresponds to the case of a through-
<figure><img src="image_4.png" /><figcaption>Figure 3: (a) Shift current tensor components (in $\mu$A·V$^{-2}$) of BFO$_{1}$/LFO$_{1}$ (yellow) and bulk BFO (purple) corresponding to light linearly polarized along the lab reference frame axes. (b) polar plot showing the magnitude of the current density response $j_i$ to light polarized in the (xy) lab reference frame for BFO$_{1}$/LFO$_{1}$ (yellow) and bulk BFO (purple).</figcaption></figure>
thickness device. The results in Figure 3b clearly show that up to 2.3 eV, the shift current response of the SL is superior to that of pure BFO. This is no longer the case for photon energies of 2.5 or 2.7 eV. At further photon energies, such as 2.9 eV, a strong peak in the shift current components $\sigma_{xxx}$, $\sigma_{xyy}$ and $\sigma_{zzz}$ of the SL yields a large photocurrent response compared to BFO, where destructive interferences strongly reduce the photovoltaic tensor component magnitude. Note that the shift current response is shifted by about 0.7 eV when realizing calculations with the HSE hybrid functional$^{35}$ (see Supplementary Information).

Interestingly, various components of the BPVE shift current tensor no longer alternate sign for photon energies in the range 2-3 eV (see for instance $\sigma_{xxx}$ or $\sigma_{zzz}$ in Figure 3a). This is particularly interesting since, in the case of BFO, several components such as $\sigma_{xxx}$ change sign with different photon energies, resulting in destructive interferences which limit the BPVE output. To further quantify the potential advantage of BFO/LFO SLs, we compute the weighted average of each components of the BPVE, with the Planck distribution as a weighted average. Specifically, we compute

$$ \bar{\sigma}_{ijk}(\hbar\omega) = \frac{2}{c\varepsilon_0} \int_0^{\hbar\omega} dEB(E, T)\sigma_{ijk}(E), \quad (2) $$

where $\varepsilon_0$ is the vacuum permittivity, c the speed of light, $B(E, T) = \frac{2}{\hbar^3 c^2} \frac{E^3}{e^{k_B T} - 1}$ is the black body spectral radiance in W.m$^{-2}$.sr$^{-1}$.J$^{-1}$, h the Planck constant, $k_B$ the Boltzmann constant and $T \approx 5,777$ K the temperature of
the sun, approximated as a black body. Essentially, $\bar{\sigma}_{ijk}$ acts as a sort of Fig-
ure of Merit which integrates the shift current tensor to appropriately sum
positive or destructive interferences in the photogalvanic response at differ-
ent photon energies. Note that technically, we apply a small scissor shift
correction to $\sigma_{ijk}(E)$ when computing $\bar{\sigma}_{ijk}(\hbar\omega)$, as we know that the cal-
culated bandgaps of BFO and LFO are underestimated by approximately
0.5 eV compared to experimental values. The quantity $\bar{\sigma}_{ijk}(\hbar\omega)$, in the
limit $\hbar\omega \to \infty$, approximates the integrated output photogalvanic current
density (see Supplementary Information), and thus integrates the destruc-
tive/constructive interferences occurring due to the summation over multiple
wavelengths. To further simplify the matter, we depict, in Figure 4, the quan-
tities $\bar{j}_i(\hbar\omega) = \frac{1}{2}(\bar{\sigma}_{ixx}(\hbar\omega) + \bar{\sigma}_{iyy}(\hbar\omega))$, which approximate the photogalvanic
current density under an unpolarized light traveling in the (001) direction
of the SL. Figure 4 shows an important feature: the photogalvanic outputs
$\bar{j}_x$ and $\bar{j}_y$ (i.e. for in-plane current densities generated by unpolarized light)
are almost twice as large as those in pure BFO. This is due, in part, to the
fact that the shift current tensor components $\sigma_{xxx}$ and $\sigma_{yyy}$ BFO$_1$/LFO$_1$ do
not change sign over a significant range of photon energy where the Planck
distribution. One may wonder whether this is related to the smaller in-plane
polarization in BFO$_1$/LFO$_1$ (about 10 $\mu$C/cm$^2$, calculated by the Berry phase
method$^{36}$) compared to bulk BFO (whose polarization has been measured to
exceed 100 $\mu$C/cm$^{237}$). Although there is no straightforward relationship
between the electrical polarization and the shift current tensor,³⁸,³⁹ previous
theoretical works have shown how some superlattice-like structures reduce the
polar ionic cations and may lead to a strong enhancement of the integrated
BPV shift current output.⁴⁰ Thus, the smaller polarization and anisotropic
distortion of our BFO₁/LFO₁ SL likely favor constructive interferences com-
pared to BFO. In contrast, the current generated in the out-of-plane direction
of the SL is much smaller than in pure BFO. As our SL features no out-of-
plane polarization component in the *z* direction, we essentially recover a mir-
ror symmetry along *z* which would prevent the emergence of a photogalvanic
current when light polarized along an in-plane direction is absorbed, as pre-
viously noted in a different nanolayered ferroelectric oxide in Ref.⁴⁰ Note also
that the giant bulk photovoltaic effect found in the prototypical ferroelectric
BaTiO₃ reported photocurrent density smaller than 10 µA.cm⁻².⁹ In contrast,
BFO and LFO are both known to have smaller band gaps, and our calculated
shift current tensor components such as σ<sub>xxx</sub> in Figure 3 reach larger magni-
tude at lower photon energy compared to previous theoretical works³⁹ if one
accounts for the difference in bandgaps. We thus anticipate that, given the
shape of the black-body weighting function in Equation 2, BFO<sub>1</sub>/LFO<sub>1</sub> would
yield superior photovoltaic current density output compared to BaTiO<sub>3</sub> thin
films.

Finally, one may wonder whether other BFO<sub>n</sub>/LFO<sub>m</sub> may yield larger
photocurrent density than BFO<sub>1</sub>/LFO<sub>1</sub>. We argue that this is unlikely. In-
<figure><img src="image_5.png" /><figcaption>Figure 4: Integrated shift photogalvanic current using the Planck distribution as a weighting function in BFO₁/LFO₁ (yellow) and BFO (dashed purple), assuming unpolarized light.</figcaption></figure>

deed, previous works indicate that for $m \ge 2$, computationally tractable BFOₙ/LFOₘ superlattices are centrosymmetric,¹⁶ thus preventing the shift current effect due to the presence of inversion symmetry. On the other hand, one may want to explore BFOₙ/LFO₁ superlattices. Yet, it can be expected that, for large n, the photocurrent density will not be significantly improved due to the lower proportion of LFO, and thus poorer absorption at large wavelengths.

# 4 Conclusion

Using first-principle calculations, we show how stacking nanolayers of a ferroelectric, BiFeO₃, with a dielectric absorbing at lower photon energy (LaFeO₃)
not only increases optical absorption but may also improve the Bulk Photo-
voltaic shift current output. The latter improvement results from the subtle
reduction in cationic displacements which, as in Ref.,³⁸ leads to the emer-
gence of a larger range of photon energies where the shift current maintains
the same direction. Our work further highlights how ferroelectric nanos-
tructures, such as superlattices, may be used to engineer and improve the
photovoltaic response of ferroelectrics. We hope that the current theoretical
work will stimulate experimentalists to explore the shift current photovoltaic
response of ferroelectric superlattices. The material-based, nanostructuring
strategy adopted in this work is a first step toward evaluating the potential
of ferroelectric superlattices for photogalvanic conversion. Future works, in-
cluding the effect of bi-axial strain, temperature, as well as a system approach
involving the nature of electrical contacts are needed to further optimize the
photovoltaic output of multiferroic-based superlattices.

**5 Supporting information**

The supporting information contains: the absorption coefficient and the Glass
coefficient;⁴¹ the electrons bands and the atomic-orbital projected DOS of
pure perovskites; the study of the electronic band gap computed with Hy-
brid functional; the comparison of the full shift-conductivity tensors of pure
BiFeO₃ and BFO₁/LFO₁ for different percentages of exact exchange and Hub-
bard U corrections applied.
Acknowledgement

This work was supported by Agence Nationale de la Recherche under grant agreement no. ANR-21-CE24-0032 (SUPERSPIN). This work was performed using computational resources from GENCI- TGCC (grant 2024-A0150912877) as well as from the “Mésocentre” computing center of Université Paris-Saclay, CentraleSupélec and École Normale Supérieure Paris-Saclay supported by CNRS and Région Île-de-France (https://mesocentre.universite-paris-saclay.fr/).

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