Abstract
A bulk material without inversion symmetry can generate a direct current under illumination. This interfacefree current generation mechanism, referred to as the bulk photovoltaic effect (BPVE), does not rely on pn junctions. Here, we explore the shift current generation, a major mechanism responsible for the BPVE, in singleelement twodimensional (2D) ferroelectrics represented by phosphorenelike monolayers of As, Sb, and Bi. The strong covalency, small band gap, and large joint density of states afforded by these elemental 2D materials give rise to large shift currents, outperforming many stateoftheart materials. We find that the shift current, due to its topological nature, depends sensitively on the details of the Bloch wave functions. It is crucial to consider the electronic exchangecorrelation potential beyond the generalized gradient approximation as well as the spinorbit interaction in density functional theory calculations to obtain reliable frequencydependent shift current responses.
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Introduction
Highperforming photoelectric conversion is essential to the solar cell technology. Traditional photovoltaic cells based on pn junctions need the builtin electric field at the interface to separate electronhole pairs and the efficiency is constrained by the Shockley–Queisser limit^{1}. The bulk photovoltaic effect (BPVE) is the direct conversion of solar energy into direct current (DC), which has been considered as a promising alternative source of photocurrent^{2,3,4,5,6,7}. As the name suggests, the presence of the BPVE does not require a complicated interface that often demands precisely controlled heterostructure fabrication process to minimize unwanted impurities and electric resistance. Singlephase bulk materials with broken inversion symmetry can generate steady photocurrent and abovebandgap photovoltage under uniform illumination in the absence of external bias^{8,9,10,11,12}, potentially enabling the implementation of the whole bulk material for photoelectric conversion^{13,14,15,16}.
In the clean limit, the BPVE response can be obtained from the quadratic response theory using the density matrix method^{17,18} or from the perspective of divergent susceptibilities^{19,20,21} where the light is treated as a classical electromagnetic field interacting with Bloch electrons. Ferroelectrics intrinsically exhibit the BPVE because of the fundamental requirement of spontaneous inversion symmetry breaking. Shift current is one of the most important mechanisms responsible for the BPVE^{22,23,24}, and was first observed in ferroelectrics experimentally^{25}. As a zerobias topological photocurrent^{26,27}, shift current is intimately related to the change in the phases of Bloch wave functions during the photoexcitation of an electron from the valence to the conduction band^{28}. A large shift current is desirable for photoelectric conversion. Previous experimental and theoretical investigations of shift current in a wide range of noncentrosymmetric materials systems such as bulk perovskite ferroelectrics^{29,30,31,32}, twodimensional (2D) materials^{33,34,35,36,37,38}, nanotubes^{39}, conjugated polymers^{40}, and topological insulators^{41,42,43,44,45} have led to two general design principles for enhancing the current density. First, lowdimensional materials with large joint density of states (JDOS) tend to present large shift current responses upon photoexcitation^{46}. The delocalization of electronic states is another important parameter that affects the shift current magnitude, and covalently bonded materials characterized by large longrange electron hopping amplitudes could give rise to large shift currents^{6,29,40}.
Recently, a family of phosphorenelike 2D elemental groupV (As, Sb, and Bi) monolayers was predicted to possess spontaneous switchable inplane polarization^{47} arising from the outofplane atomiclayer buckling. We propose that singleelement 2D ferroelectrics are promising candidates to support large shift currents because of the strong covalency intrinsic to the homoelement bonding and pronounced singularities in the density of states common to 2D materials. In this work, we explore the shift current in these newly predicted elemental ferroelectrics with density functional theory (DFT) calculations and find that they can generate large shift currents over a wide range of wavelengths including the visible light spectrum. In addition, our work highlights the importance of spinorbit coupling (SOC) and electronic correlation effect on the shift current spectrum even in 2D systems containing light elements.
Results and discussion
Structure
The structure of singleelement groupV monolayer is displayed in Fig. 1. The puckered lattice structure without centrosymmetry (space group Pmn2_{1}) can be viewed as a distorted phosphorenelike structure. The outofplane atomic buckling causes charge accumulations at the outmost groupV atoms, leading to spontaneous inplane polarization along the y axis^{47}. We compute the Born effective charges (see inset table in Fig. 1) and confirm the bucklinginduced charge accumulation/depletion mechanism. The dynamic stability of ferroelectric groupV monolayer has been confirmed by the computed phonon spectrum that has no imaginary phonon modes over the whole Brillouin zone^{47}. Xu et al. recently simulated the polarizationelectric field hysteresis loop for monolayer As^{48}, and the predicted inplane spontaneous polarization is 0.42 × 10^{−10} C/m, comparable with ferroelectric monolayer SnTe^{49}. We compute the energy evolution as a function of buckling parameter h (Fig. 1a) in monolayer As and Sb, respectively, each revealing a double well potential landscape, typical for a ferroelectric phase (Fig. 1c). It is worth noting that groupV monolayers, similar to phosphorene, have already been experimentally synthesized for Sb and Bi^{50,51,52}. We only consider inplane shift currents (σ^{xbc} and σ^{ybc}) in this work. Since the mirror symmetry \({{{{\mathcal{M}}}}}_{x}:x\to x\) leaves the monolayer invariant, only three components of the response tensor, σ^{yxx}, σ^{yyy}, and σ^{yzz}, can be nonzero due to the symmetry constraint that holds at any photon frequency. Here, we focus on the responses under incident light perpendicular to the 2D sheet, namely, σ^{yxx} and σ^{yyy}.
Monolayer arsenic
The DFT band structures and the corresponding shift current spectra for monolayer As computed with PBE, PBE+SOC, HSE, and HSE+SOC are presented in Fig. 2, revealing several intriguing features. The band gap predicted by PBE is 0.15 eV, and the peak in the σ^{yyy} spectrum exceeds 1000 μA/V^{2} for photon energies near the band edge. In comparison, HSE predicts a larger band gap of 0.47 eV, whereas the peak value of σ^{yyy} drops considerably to only 150 μA/V^{2}. Similarly, the bandedge value of σ^{yxx} estimated by PBE is >3000 μA/V^{2} but the HSE predicts a much smaller value of 250 μA/V^{2}. Such pronounced reduction in the peak response is unexpected as the HSE band structure looks like a rigid shift of the PBE band structure without substantial changes in the band dispersions. This highlights the importance of treating the exchangecorrelation potential beyond the semilocal approximation in DFT calculations.
Moreover, the inclusion of the SOC effect, though inducing little impact on the band gap, causes drastic changes in the shift current spectra for both PBE and HSE. At the PBE level, the band gap reduction due to SOC is merely 0.01 eV, but the bandedge values of σ^{yyy} and σ^{yxx} computed with SOC reduce to 450 and 2200 μA/V^{2} from their PBE values of 1000 and 3000 μA/V^{2}, respectively; the PBE and PBE + SOC spectra are almost identical for higher photon frequencies. Interestingly, compared to the HSE values of 150 and 250 μA/V^{2}, the peak values of σ^{yyy} and σ^{yxx} estimated with HSE + SOC increase to 300 and 400 μA/V^{2}, respectively. In addition, the HSE and HSE + SOC spectrum profiles are notably different for both σ^{yyy} and σ^{yxx} in the frequency range between 1.5 and 3.0 eV.
To get a better understanding of the subtle changes in the shift current responses due to SOC, we analyze the spectrum of σ^{yyy} by plotting two BZintegrated quantities: the energy averaged shift vector \({\bar{R}}^{a,b}\) and the transition intensity \({\varepsilon }_{2}^{bb}\) defined as
and
respectively. For a given photon frequency ω, \({\bar{R}}^{a,b}(\omega )\) is a measure of aggregate contributions of shift vectors \({R}_{nm}^{a,b}\) at all k points in the BZ; \({\varepsilon }_{2}^{bb}(\omega )\) reflects the photon absorption strength. By comparing the spectra of \({\bar{R}}^{y,y}\) and \({\varepsilon }_{2}^{yy}\) computed with PBE, PBE + SOC, HSE, and HSE + SOC (Fig. 3), we find that the integrated shift vectors are comparable at the lowfrequency region but these four methods predict rather different transition intensities near the band edge. Specifically, PBE predicts a larger magnitude of \({\varepsilon }_{2}^{yy}\) than PBE + SOC, whereas HSE yields a lower transition intensity than HSE+SOC. Therefore, the SOC effect suppresses the transition at the PBE level but promotes the transition in HSE. According to eq. (2), the magnitude of \({\varepsilon }_{2}^{yy}\) depends on the interband Berry connections between every conduction and valence band pair that has the energy difference matching the energy of the incident light. The presence of SOC, regardless the strength, will lift the spin degeneracy and lead to level anticrossing at some k points. These changes in electronic bands albeit localized in the momentum space could strongly affect the interband Berry connections. Therefore, the pronounced SOC effect on the spectrum of ε_{2} and then σ is a manifestation of the topological nature of shift current that exhibits highly nontrivial dependence on the Berry connections of a bundle of Bloch bands (valence and conduction bands relevant to photon excitation).
Previous studies have demonstrated that the shift current response of 2D materials can exist a significant dependence on the layer number^{15,53,54}. We examine the layer stacking impact on the response functions. Our HSE+SOC calculations indicate that bilayer As remains polar and possesses a small band gap of 0.36 eV (Fig. 4a), whereas trilayer As becomes centrosymmetric and metallic. The shift current spectrum is presented in Fig. 4b, revealing a peak value of 125 μA/V^{2} for σ^{yyy} and −135 μA/V^{2} for σ^{yxx}; these magnitudes are smaller than those in monolayer, consistent with those observed experimentally in CuInP_{2}S_{6} where the photocurrent density decreases drastically when the thickness exceeds ≈40 nm^{15}.
We believe HSE+SOC is the most reliable method among the ones employed in the current work and most previous works, and the predicted peak shift current response in monolayer As is 400 μA/V^{2} for σ^{yxx}, much higher than previous reports for 2D materials, i.e., 100 μA/V^{2} in GeS. This highlights the potential of elemental ferroelectric 2D materials for photoelectric conversion.
Monolayer antimony
The band structures and shift current spectra of monolayer Sb computed with four different methods are displayed in Fig. 5. The direct band gap values based on the band structures plotted along highsymmetry lines are 0.31, 0.32, 0.41, and 0.29 eV for PBE, PBE+SOC, HSE, and HSE + SOC, respectively. Despite yielding comparable band structures, these four methods predict distinct shift current spectra. Similar to the case of monolayer As, we observe a reduction of bandedge response magnitude of σ^{yyy} (σ^{yxx}) from −1000 (4000) μA/V^{2} in PBE (Fig. 5c, e) to −300 (1700) μA/V^{2} in HSE (Fig. 5d, f), reaffirming the importance of including exact exchange in DFT calculations.
The inclusion of the SOC effect completely changes the spectrum profiles of σ^{yyy} and σ^{yxx} at the PBE level (Fig. 5c, e). The most striking result is the reversal of the current direction for lowfrequency excitations. For example, PBE predicts a current running against the polarization (σ^{yyy} < 0) but PBE+SOC predicts a current flowing along the polarization (σ^{yyy} > 0). The sign of the shift current is determined by the integrated shift vector as defined in eq. (1). Indeed, as shown in Fig. 6a, the sign of \({\bar{R}}^{y,y}\) is the same as the sign of σ^{yyy}, and SOC causes a sign change. For a given photon energy ω, the value of \({\bar{R}}^{a,b}\) depends on the topological quantity \({R}_{nm}^{a,b}\) at every k point in the BZ that supports resonant excitation. Following our previous argument regarding the effect of SOC on the transition intensity, the intraband Berry connection (\({{{\mathcal{A}}}}\)) of some bands may be altered drastically by SOC, particularly around the k point where SOC induces a hybridization gap. Therefore, it is physically plausible to have a signchanging situation due to SOC, as demonstrated in the case of monolayer Sb when treated with PBE. We further compute kresolved \({\bar{R}}^{y,y}\) at a photon energy of 0.2 eV with PBE and PBE+SOC, respectively, with results ploted in Fig. 6e and f. It is found that all kresolved \({\bar{R}}^{y,y}\) computed with PBE have negative values whereas those computed with PBE+SOC become mostly positive.
At the HSE level, SOC causes a redshift of the bandedge peak to ≈0.1 eV for both σ^{yyy} and σ^{yxx} (Fig. 5d, f), though the band gap value taken from the HSE+SOC band structures is 0.29 eV, higher than the onset photon energy. We further decompose σ^{yyy} into \({\bar{R}}^{y,y}\) and \({\varepsilon }_{2}^{yy}\) and find that the magnitudes of \({\bar{R}}^{y,y}\) computed with HSE and HSE+SOC are comparable at the lowfrequency region (Fig. 6c). However, the \({\varepsilon }_{2}^{yy}\) spectrum of HSE+SOC has a sharp peak at a much lower frequency of 0.1 eV (Fig. 6d), consistent with the HSE+SOC spectrum of σ^{yyy} (Fig. 5d). This seems to suggest forbidden optical excitations with ingap photon frequencies. To resolve this puzzle, we perform a diagnostic analysis on the electronic structures of monolayer Sb obtained with HSE and HSE+SOC. The zoomedin band structures along ΓYS presented in Fig. 7a show that the inclusion of SOC gives a smaller band gap of 0.29 eV than the HSE band gap of 0.4 eV, and breaks the spin degeneracy. We compute the JDOS defined as
with results plotted in Fig. 7b. The onset frequency of the HSE JDOS is 0.4 eV, as expected from the HSE band gap. However, the JDOS computed with HSE+SOC acquires nonzero values staring at a lower frequency of 0.1 eV. As the JDOS is obtained by integrating over the whole BZ while the band structure only shows the band energies along the highsymmetry BZ boundary paths, the JDOS of HSE + SOC hints at a smaller gap at a generic kpoint. We thus map out the energy difference between the highest valence band and the lowest conduction band over the whole BZ (Fig. 7c, d) with HSE and HSE+SOC, respectively. Indeed, HSE gives a gap of 0.38 eV at a k point very close to the highsymmetry line ΓY (Fig. 7c); HSE+SOC yields a band gap of 0.1 eV at a generic kpoint ([0.04, 0.35] in reduced coordinates) away from the zone boundary (Fig. 7d), consistent with the onset photon frequency for σ^{yyy} and σ^{yxx} predicted by HSE+SOC (Fig. 5d, f). It is noted that the peak value of σ^{yxx} in monolayer Sb estimated with HSE+SOC reaches 2000 μA/V^{2}, even higher than monolayer As. Moreover, the value of σ^{yyy} reaches 600 μA/V^{2} at ω = 1.8 eV, suitable for visible light absorption.
It is noted that we have compared the shift current spectrum and JDOS of monolayer GeS, a model material for investigating BPVE in 2D, with monolayer Sb. The peak shift current in GeS is ≈100 μA/V^{2}, one order of magnitude smaller than that in monolayer Sb. We find that monolayer Sb exhibits much larger JDOS than GeS. Nevertheless, because GeS has a much larger band gap than Sb and the shift current scales inversely with the band gap according to velocity gauge formalism^{17}, we suggest that the giant shift current in monolayer Sb could be attributed to multiple factors including small band gap and large JDOS.
Monolayer bismuth
Since Bi has a larger atomic number, the SOC effect is more pronounced in monolayer Bi as demonstrated from the notably different band structures between PBE (HSE) and PBE + SOC (HSE + SOC), as shown in Fig. 8a, b. Without SOC, PBE and HSE predict similar spectrum profiles of σ^{yyy} and σ^{yxx} and large bandedge responses. The magnitude of the peak response of σ^{yxx} computed with HSE is close to 6500 μA/V^{2}. In comparison, the spectra obtained with SOC are qualitatively different. In general, the spectra of PBE+SOC and HSE+SOC share similar peak structures with the latter predicting lower peak values (Fig. 8c–f). For example, PBE+SOC predicts a main peak of −2000 μA/V^{2} at 1.0 eV for σ^{yyy} while the HSE + SOC spectrum of σ^{yyy} has the highest peak of −500 μA/V^{2} located at 1.4 eV. We note that the bandedge response is no longer the strongest. The spectrum of \({\varepsilon }_{2}^{yy}\) (Fig. 9b) confirms that the SOC interaction reduces the bandedge photon absorption than that in HSE. Given the substantially different band structures predicted by HSE and HSE+SOC, it is not surprising to obtain drastically different spectra of \({\bar{R}}^{y,y}\) (Fig. 9a). For monolayer Bi, the inclusion of SOC is crucial to acquire correct shift current spectrum.
Strain effect
Because reduceddimensional structures can sustain much larger strains than their bulk counterparts, it has become common to use strain to modulate the structural, electronic, and optical properties of 2D materials^{55,56,57,58}. Here, we explore the effect of uniaxial strain (η_{x}) on the shift current response. Taking monolayer As for example, we find that a tensile strain along the x direction can effectively reduce the band gap and promote the current density. As shown in Fig. 10, a 3% tensile strain enhances both σ^{yyy} and σ^{yxx} by nearly fourfold. In contrast, a compressive strain (η_{x} = − 3 %) increases the band gap hence reduces the bandedge response. Interestingly, the peak of σ^{yyy} at a higher photon energy of 1.6 eV gets enhanced by the compressive strain of −3%. Similar tensile strainpromoted response is also found in monolayer Sb. In particular, the magnitude of σ^{yxx} gets enhanced to 2800 μA/V^{2} when stretching the monolayer by 2% along x (Fig. 10d).
Finally, we compare the shift current responses of different materials including bulk polar materials (e.g., PbTiO_{3}, BaTiO_{3}, and GaAs) and 2D materials (e.g., GeSe and CrI_{3}). As summarized in Fig. 11a, the magnitude of the response tensor roughly scales inversely with the band gap, and the stretched monolayer Sb has the highest response of 2800 μA/V^{2}. Note that most previous results were based on PBE calculations, which generally overestimate the response. To compare with experimental data, we further estimate the shift photocurrent density based on the computed response tensor and a light intensity of 0.1 W/cm^{2} following the method in ref. ^{33}. It is evident from Fig. 11b that elemental 2D ferroelectrics represented by Sb, As, and Bi outperform conventional bulk ferroelectrics and 2D layered materials such as CuInP_{2}S_{6} over a wide range of wavelengths including the visible light spectrum. It is worth noting that TaAs exhibits a strong response at a wavelength of ≈10,000 nm, which is much larger than the wavelengths of visible light. Given TaAs is a Weyl semimetal, it is not surprising that its shift current response is large because of the gapless band dispersion and topological nature. The midinfrared response is likely more relevant to applications such as optical detectors and sensors. In comparison, monolayers As and Sb exhibit large shift current response across a wide range of wavelengths including the visible light spectrum, which would be advantageous for utilizing most of the solar spectrum. Additionally, since the peak responses of groupV elemental ferroelectrics are consistent with lightinduced terahertz emission, the large current magnitude suggests their potential applications in terahertz source platforms.
In this work, we investigate the shift current responses in singleelement twodimensional ferroelectrics represented by monolayer As, Sb, and Bi in the space group of Pmn2_{1} using firstprinciples density functional theory calculations. We find that PBE and HSE yield qualitatively different shift current spectra, demonstrating the importance of exact exchange potential for reliable predictions of optical responses. Moreover, the spinorbit coupling, largely overlooked in previous studies, can substantially affect the magnitude, sign, and spectral profile of shift current even for light elements such as arsenic. This highlights the topological nature of shift current that has nontrivial dependence on both intraband and interband Berry connections of a bundle of valence and conduction bands. Regarding computational materials by design for new solar materials that can generate large shift currents, we suggest that it is essential to treat the electronic exchange–correlation interaction beyond the generalized gradient approximation and to include the spinorbit interaction in density functional theory calculations. Based on the results predicted by HSE+SOC, we propose that elemental 2D ferroelectrics can support large shift currents, outperforming many stateoftheart materials. This work unravels the potential of elemental 2D ferroelectrics for photovolatic and optoelectronic applications.
Method
We follow the formalism adopted in Refs. ^{59,60} to compute the photon frequencydependent shift current spectrum. The shift current density (J_{2}) is regarded as a secondorder optical response to the electromagnetic field E of frequency ω,
where the thirdrank tensor σ^{abc}(0; ω, − ω) is the shift current response tensor; a, b, and c are Cartesian indexes, and the index a specifies the direction of the generated DC current, while b and c are polarization directions of an incident light. Within the length gauge, the shift current tensor is derived as
where n and m are band indexes, and k is the wavevector of the Bloch wave function. f_{nm} = f_{n} − f_{m} is the difference in the FermiDirac occupation number between bands n and m; ω_{nm} = ω_{n} − ω_{m} represents the band energy difference. \({r}_{nm}^{b}=i\langle n \partial {k}^{b} m\rangle\) is the dipole matrix element (interband Berry connection). \({r}_{nm;a}^{b}\) is the generalized derivative expressed as \({r}_{nm;a}^{b}=\frac{\partial {r}_{nm}^{b}}{\partial {k}^{a}}i{r}_{nm}^{b}({{{{\mathcal{A}}}}}_{n}^{a}{{{{\mathcal{A}}}}}_{m}^{a})\) with \({{{{\mathcal{A}}}}}_{n}^{a}=i\langle n \partial {k}^{a} n\rangle\) the intraband Berry connection.
Under a linearly polarized light (b = c), the shift current response tensor can be reformulated into a more compact expression,
where \({R}_{nm}^{a,b}=\frac{\partial {\phi }_{nm}^{b}}{\partial {k}^{a}}+{{{{\mathcal{A}}}}}_{n}^{a}{{{{\mathcal{A}}}}}_{m}^{a}\) is the shift vector with ϕ_{nm} being the phase factor of the dipole matrix element \({r}_{nm}^{b}= {r}_{nm}^{b} {e}^{i{\phi }_{nm}^{b}}\). The shift vector has a unit of length and represents the average displacement of the coherent photoexcited carriers during their lifetimes. The \({r}_{nm}^{b}{r}_{mn}^{b}\) term measures the transition intensity which describes the optical absorption strength for the transition from band m to band n. Therefore, the response tensor can be viewed as the product of shiftvector and optical transition intensity.
The structural parameters of monolayer As, Sb, and Bi are optimized using Quantum Espresso^{61,62} with Garrity–Bennett–Rabe–Vanderbilt (GBRV) ultrasoft pseudopotentials^{63}. The exchangecorrelation functional is treated within the generalized gradient approximation of Perdew–Burke–Ernzerhof (PBE) type^{64}. We use a planewave kinetic energy cutoff of 50 Ry, a charge density cutoff of 250 Ry, a 12 × 12 × 1 MonkhorstPack kpoint mesh for Brillouin zone (BZ) integration, an ionic energy convergence threshold of 10^{−5} Ry, and a force convergence threshold of 10^{−4} Ry in structural optimizations. Maximally localized Wannier functions to fit DFT electronic structures are obtained using the Wannier90 code^{65}, and then the shift current response tensor is calculated in the Wannier basis as described in ref. ^{59}. We use a numerical smearing parameter of 20 meV and a dense kpoint grid of 1000 × 1000 × 1 to ensure the spectrum convergence. In addition, the 3Dlike response is estimated by rescaling the calculated 2D response with the effective thickness of monolayer^{33}. The spinorbit coupling (SOC) is taken into account at the fully relativistic level with normconserving pseudopotentials provided by the PseudoDoJo project^{66}. We also compute the shift current using the HeydScuseriaErnzerhof (HSE) hybrid functional^{67} with a 8 × 8 × 1qpoint grid during the selfconsistentfield cycles. The HSE band structure is obtained via Wannier interpolation^{68} using Wannier90 interfaced with Quantum Espresso. We employ the Wannier Berri code^{69,70} to compute the shift vector and transition intensity (see discussions below) to analyze the shift current spectrum. It is noted that all shift current calculations are based on the independentparticle approximation and do not take the exciton effect into account.
Data availability
The data that support the findings of this study are included in this article and are available from the corresponding author upon reasonable request.
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Acknowledgements
Z.Q. and S.L. acknowledge the supports from Westlake Education Foundation. We acknowledge Dr. JaeMo Lihm for useful suggestions regarding the usage of Wannier90 and Wannier Berri. Z.Q. acknowledges the help from Yudi Yang during the preparation of the manuscript. The computational resource is provided by Westlake HPC Center. J.Z. acknowledges National Natural Science Foundation of China under Grant No. 11974270.
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S.L. conceived and led the project. Z.Q. performed calculations and data analysis. All authors contributed to the discussion and the manuscript preparation.
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Qian, Z., Zhou, J., Wang, H. et al. Shift current response in elemental twodimensional ferroelectrics. npj Comput Mater 9, 67 (2023). https://doi.org/10.1038/s41524023010263
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DOI: https://doi.org/10.1038/s41524023010263
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