Ferroelectric polarization switching with a remarkably high activation-energy in orthorhombic GaFeO3 thin films

Orthorhombic GaFeO3 (o-GFO) with the polar Pna21 space group is a prominent ferrite by virtue of its piezoelectricity and ferrimagnetism, coupled with magneto-electric effects. Herein, we unequivocally demonstrate a large ferroelectric remanent polarization in undoped o-GFO thin films by adopting either a hexagonal strontium titanate (STO) or a cubic yttrium-stabilized zirconia (YSZ) substrate. The polarization-electric-field hysteresis curves of the polar c-axis-grown o-GFO film on a SrTiO3/STO substrate show the net switching polarization of ~35 {\mu}C/cm2 with an unusually high coercive field of +-1400 kV/cm at room temperature. The PUND measurement also demonstrates the switching polarization of ~26 {\mu}C/cm2. The activation energy for the polarization switching, as obtained by density-functional theory calculations, is remarkably high, 1.05 eV per formula unit. This high value accounts for the observed stability of the polar Pna21 phase over a wide range of temperature up to 1368 K.


INTRODUCTION
Multiferroics are an interesting group of materials that simultaneously exhibit ferroelectricity and magnetic ordering with coupled electric, magnetic and structural orders. Multiferroic materials with a pronounced degree of magnetoelectric (ME) coupling at room temperature are of great scientific and technological importance for use in various types of electronic devices that include sensors, actuators and electric-field-controllable magnetic memories. 1-3 Among all known multiferroics, BiFeO 3 is most extensively studied because of its large room-temperature-spontaneous polarization with improved magnetic properties in epitaxially strained thin-film forms. [4][5][6][7] Polar orthorhombic GaFeO 3 (o-GFO) is another prominent multiferroic oxide by virtue of its room-temperature piezoelectricity (possibly ferroelectricity as well), near-room-temperature ferrimagnetism and pronounced low-temperature ME effects. Since a linear ME effect was first reported in the 1960s by Rado,8 magnetization-induced second-harmonic generation, 9 the optical ME effect 10 and other interesting phenomena [11][12][13][14][15][16][17][18][19][20][21][22] continue to renew our research attention to this system. GFO crystallizes into the polar orthorhombic Pna2 1 (equivalently, Pc2 1 n) space group with a ferrimagnetically ordered spin structure. The four Fe 3+ ions in a unit cell are antiferromagnetically coupled along the a axis in the Pna2 1 setting. However, the intermixed Fe 3+ ions occupying the Ga sites (in other words, different Fe occupations at the Fe1 and Fe2 sites) can lead to a ferromagnetic order with Fe 3+ ions at the Fe sites, which gives rise to ferrimagnetic ordering at a temperature of~230 K. 11,12 In contrast to ferrimagnetic ordering and low-temperature ME effects, 11 much less is known on the ferroelectricity of o-GFO. In principle, o-GFO should exhibit ferroelectricity up to high temperatures as (or because) the polar Pna2 1 phase (with the corresponding point group of 2 mm or C2v in Schönflies notation) remains stable up to at least 1368 K. 22 According to Stoeffler,20,23 the computed ab initio polarization value of the undoped o-GFO is as high as 25 μC cm − 2 . In spite of these predictions, however, there has been no experimental demonstration of room-temperature ferroelectric polarization switching with a polarization value compatible with the ab initio predictions. Several groups have studied the P-E (polarization-electric field) response of GFO ferroelectrics, mostly using thin-film forms. [13][14][15][16] However, all of these studies revealed that the remanent polarization (P r ) of GFO is o0.5 μC cm − 2 with an unrealistically small value of the coercive field (E c ⩽ 5 kV cm − 1 ) as well. More recently, Oh et al. 17 reported the epitaxial growth of o-GFO film on an SrRuO 3 (111)/SrTiO 3 (111) substrate with a remarkably enhanced E c of~150 kV cm − 1 . However, their P r value still remains at~0.05 μC cm − 2 . Moreover, Mukherjee et al. 18 recently reported room-temperature ferroelectricity of an o-GFO thin film grown on a indium tin oxide (ITO) (001)/yttrium-stabilized zirconia (YSZ) (001) substrate based on their observation of a 180°phase shift in the piezoresponse. However, the 180°phase shift (or switching of the piezoelectric phase) does not necessarily indicate ferroelectric polarization switching across the barrier of a double-well potential.
To date, the only unequivocal experimental demonstration of reversible polarization switching in o-GFO thin films was made by Thomasson et al. 19 According to their study, the 2% Mg-doped GFO film exhibits a well-saturated P-E switching curve with a negligible tendency of the non-switching polarization. 19 However, the measured P r value is as small as 0.2 μC cm − 2 , 19 which is o1% of the predicted P r value (~25 μC cm − 2 ) of the undoped GFO. Moreover, this system is not a stoichiometric o-GFO (i.e., not an orthoferrite with Ga: Fe = 0.6:1.4) in addition to 2% Mg doping. 19 Thus, all reported P r values of o-GFO are in the range of 0.05-0.5 μC cm − 2 , which is unacceptably small to be compatible with the ab initio polarization of 25 μC cm − 2 . 20,23 In this article, we have clarified a puzzling discrepancy between the observed P r value (⩽0.5 μC cm − 2 ) and the ab initio prediction (25 μC cm − 2 ) and unequivocally demonstrated ferroelectric polarization switching with a net switching polarization value of~30 μC cm − 2 by using o-GFO thin films preferentially grown along the polar c axis in Pna2 1 setting (b axis in Pc2 1 n setting). In the present study, we adopt either a hexagonal or a cubic substrate to demonstrate the room-temperature polarization switching: (i) a SrRuO 3 (SRO) (111)-buffered hexagonal strontium titanate (STO) (111) substrate and (ii) an ITO (001)-buffered cubic YSZ (001) substrate.

Experimental methods
The o-GFO film and the SRO bottom-electrode layer were grown on a Ti 4+ -single-terminated STO (111) substrate by pulsed laser deposition with a KrF excimer laser (λ = 248 nm) operated at 3 and 10 Hz, respectively. GFO films were deposited at 800°C in an oxygen ambient atmosphere (200 mTorr) with a fluence of 1 J cm − 2 focusing on a stoichiometrically sintered o-GFO target while 30-nm-thin SRO bottom-electrode layers were grown at 680°C in a 100 mTorr oxygen atmosphere with a fluence of 2 J cm − 2 . After the deposition, the SRO layer was cooled to room temperature under the same oxygen pressure used in the o-GFO film deposition. We observed the thickness fringes of SRO (222) around the two-theta (2θ) value of STO (222), which strongly indicates an epitaxial growth of the SRO bottom-electrode layer. The thickness of this SRO layer (30 nm) was also determined from the positions of the interference fringes (see Supplementary Information for details). The o-GFO film and the ITO bottom-electrode layer were grown on a cubic YSZ (001) substrate by pulsed laser deposition with operating ac frequencies of  Analysis of the domain orientation and phase formation in the films was performed by using a high-resolution x-ray diffractometer (D8 discover; Bruker, Billerica, MA, USA) under Cu Kα radiation. The domain structures were investigated in detail by using high-resolution transmission electron microscopy (JEM-2100F with a probe CS corrector; JEOL, Tokyo, Japan). Z-contrast high-angle annular dark-field scanning transmission electron microscopy (STEM) image and selected-area electron diffraction experiments were carried out under 200-kV acceleration voltage.
For dielectric-ferroelectric measurements of the c-axis-grown o-GFO film, a Pt top electrode with a diameter of 100 μm was deposited using a dc sputtering system. Current and voltage (I-V) curves were recorded using a Keithley 2400 source meter (Keithley, Cleveland, OH, USA). Polarization-electric field hysteresis loops (P-E curves) and positive-up and negative-down (PUND) pulse sequences were measured using a precision LC ferroelectric tester (Radiant Technologies, Albuquerque, NM, USA). A commercial atomic force microscope (DC-EFM in XE-100; Park Systems, Suwon, Korea) was used for vertical piezoelectric force microscopy (vPFM) study to map the piezoelectric properties of thin films. A Pt/Ir-coated tip was used for probing the signals. The input modulation voltage V ac (with an amplitude in the range of 0.1-2 V and an ac frequency of 15 kHz) was applied between the conductive tip and the bottom electrode using a function generator. The resulting oscillations of the cantilever were read out with a lock-in amplifier. The magnetic properties were measured by superconducting quantum interference device magnetometry (MPMS; Quantum Design, San Diego, CA, USA) from 10 to 300 K under various applied magnetic fields.

Computational methods
We have performed DFT calculations of o-GFO based on the generalized gradient approximation 24 and generalized gradient approximation+U methods 25 implemented with projector augmented-wave pseudopotential 26,27 using the Vienna ab initio Simulation Package. 28 All DFT calculations were performed by adopting (i) an 11 × 6 × 6 Monkhorst-Pack k-point mesh 29 centered at the Γ-point, (ii) 650-eV plane-wave cutoff energy and (iii) the tetrahedron method with Blöchl corrections for the Brillouin-zone integrations. 30 The structural optimizations were performed for the 40-atom cell that corresponds to an orthorhombic unit cell consisting of eight formula units. The ions were relaxed until the Hellmann-Feynmann forces on them were o0.01 eV Å − 1 . A Hubbard U eff of 4 eV and an intra-atomic exchange parameter (J) of 0.89 eV for the Fe 3d orbital were chosen based on the previous work. 20

In-plane domain orientation of the polar c-axis-grown film
To fabricate the polar c-axis-grown GFO film with a simpler domain configuration, we have carefully chosen a hexagonal STO (111) substrate as an alternative to a cubic YSZ (001) substrate. To implement this, we adopt an SRO (111) bottom electrode by considering its compatibility with the STO substrate and fatigue and imprinting resistance. 31,32 This scheme of the o-GFO film growth on a hexagonal substrate enables us to achieve a substantial simplification in the domain configuration, from 12 orientations (on a cubic YSZ (001) substrate 18 ) to 6 in-plane orientations. Figure 1a shows that the [001]-oriented GFO thin film is preferentially grown on an SRO (111)-buffered STO(111) substrate. The calculated c-axis parameter using this θ − 2θ XRD pattern is 9.3996 Å, which essentially coincides with the bulk c-axis parameter. 13 This suggests that the present GFO film is fully relaxed along the growth direction. For the [001]oriented film grown on an STO(111), the most probable domain orientation that minimizes the in-plane lattice mismatch is represented by three in-plane domain configurations (D 1 , D 2 and D 3 ), as schematically depicted in Figure 1b.
To examine the validity of the proposed domain configuration, we have measured the in-plane XRD phi(φ)-scan, and the result is presented in Figure 1c. These φ-scan spectra were obtained by maintaining the Bragg angle at (013) for the o-GFO film (orange line) and at (110) and (100) for the STO (blue and green lines, respectively). In the case of the o-GFO (013), six peaks appear (orange color), each of which is separated from the two neighboring GFO (013) peaks by 60°. However, as shown in Figure 1c, these six (013) peaks are separated from the STO (110) and (100) peaks by 30°. These two observations clearly indicate that the GFO (013) has six in-plane orientations with a successive tilting angle of 60°when projected onto an STO (111) surface, and each of these six in-plane orientations is rotated by 30°from the STO (111) projection of the STO (110) (Figure 1b). The result of our structural analysis is very similar to the previous result on ε-Fe 2 O 3 , which is isostructural with the o-GFO grown on an STO (111). 33 In contrast, the o-GFO film grown on an ITO/YSZ (001) substrate is characterized by a total of 12 in-plane orientations with a successive in-plane tilting angle of 30°(see Supplementary Information for details).
However, there exists a considerable degree of lattice mismatch (Δ), although the (001)-oriented GFO film possesses six in-plane orientations to reduce the lattice mismatch between the film and substrate. According to our estimate, Δa = 5.8654% ða GFO ffiffi ffi 2 p a STO cos 60 o ð Þ). 13,30 This indicates that the o-GFO film is unable to show cube-on-cube type growth on the unit-cell basis. However, if the o-GFO film growth is preceded by the formation and deposition of a supercell with the dimensions of 16a GFO × 18b GFO , the lattice mismatch between the GFO layer and the substrate surface can be effectively removed (with Δa = 0.01%, Δb = 0.08%). We will examine this proposition by looking into scanning transmission electron microscopy (STEM) images.
A bright-field STEM image is shown in Figure 2a for the crosssection of the [001]-oriented GFO film grown on an SRO(111)buffered STO(111) substrate. According to this cross-sectional image, the thicknesses of the GFO layer and the SRO electrode are 200 and 30 nm, respectively. The selected-area electron diffraction pattern shown in Figure 2b is indexed by considering the superposition of the diffracted peaks along the zone axes [010] and 310 Â Ã of the GFO and the peaks along the STO 110 Â Ã zone axis (red rectangle) used as the standard. The diffracted peaks along the GFO [010] zone axis (yellow rectangle) correspond to the D 1 domain configuration (Figure 1). In contrast, the peaks along the GFO 310 Â Ã zone axis (white rhombus) correspond to the D 2 domain configuration. The selected-area electron diffraction patterns also indicate epitaxial growth of the GFO film with the polar axis along the growth direction. One can notice two separated peaks by carefully examining the area encircled by green dots. This demonstrates that the GFO film is fully relaxed rather than epitaxially strained.
High-resolution high-angle annular dark-field STEM images clearly show structural domain boundaries that are formed by distinct in-plane orientations. A high-angle annular dark-field STEM image of the area surrounded by a yellow rectangle is magnified and presented on the right-hand side of Figure 2c to visualize clearly the structural domain boundary. The three fast Fourier-transformed images (upper regions of the magnified STEM image) indicate that the in-plane orientation of the central domain is different from those of the two neighboring domains. Figure 2c indicates that the width of the central domain is~8 nm, which nearly coincides with the a-axis dimension of the supercell proposed previously (16a GFO × 18b GFO ). Thus, the central domain in Figure 2c represents the D 1 domain configuration, as depicted in Figure 1.
PFM images of the polar c-axis-grown film vPFM is a suitable method for measuring ferroelectricity or piezoelectricity for a small local region. In Figures 3a and b, we present vPFM amplitude and phase images, respectively, acquired over an area of 1 μm × 1 μm (or simply 1 μm 2 ). The phase-contrast image shown in Figure 3b can be interpreted as the existence of ferroelectric domains with two antiparallel polarizations. However, this interpretation would be false if these domains were not maintained at a particular polarization state (i.e., remain in either up or down state) after turning off the bias E-field, which had been used for the polarization switching. To resolve clearly this critical issue, we have chosen a particular region of the o-GFO film and applied an alternative dc voltage from +10 V to − 10 V. Thus, the corresponding E-field is ± 500 kV cm − 1 . The two applied voltages and the corresponding regions are marked in Figures 3c and dfor the amplitude and phasecontrast images, respectively. The two lower-resolution vPFM images (10 μm × 10 μm) shown in Figures 3c and d were taken immediately after turning off these dc voltages. Figure 3d clearly indicates that the 180°phase-shifted domains by the bias E-field return to the initial unshifted state as soon as the bias E-field is turned off. Thus, the present vPFM results do not show any evidence of ferroelectricity with a nonzero remanent polarization up to a bias E-field of ± 500 kV cm − 1 .
Ferroelectric polarization switching of the polar c-axis-grown films Having failed to obtain clear evidence of the ferroelectricity up to ± 500 kV cm − 1 , we have carried out P-E hysteresis measurement by applying much stronger E-fields. This has been performed because our optimally processed [001]-oriented GFO/SRO/STO (111) film capacitors are characterized by I-V current densities as low as~10 − 6 A cm − 2 even at ± 500 kV cm − 1 (i.e., at ± 10 V; see   Supplementary Information for details). This low leakage current is a remarkable improvement over those of other reported GFO films. For instance, the current density of the previously reported GFO/SRO/ STO (111) film capacitor (with 0% MgO doping) is in the order of a few amps per square centimeter for the same thickness of 200 nm (i.e., at ± 500 kV cm − 1 ). 34 Thus, our film capacitor processed by the optimized layer-by-layer growth of SRO shows a remarkable 10 6 -fold improvement in the leakage current density (10 − 6 vs 10 0 ). Similarly, the reported current density of the GFO/Pt/YSZ (111) film capacitor at 300 K is as high as 1.4 × 10 − 2 A cm − 2 (as estimated using 300 μm as the diameter of the top electrode) for the same thickness of 200 nm (i.e., at ± 500 kV cm − 1 ). 35 Again, our film capacitor shows a remarkable~10 4 -fold improvement in the leakage current density.
The Pt/o-GFO (001)/SRO (111) film capacitor grown on an STO (111) shows ferroelectric switching behavior with a remarkably high coercive field (E c ) of ± 1400 kV cm − 1 at room temperature ( Figure 4a). Thus, the E max value used in the PFM measurements is only 1/3 of the minimum electric field needed for polarization switching (E c ). Thus, ferroelectric polarization switching cannot be attained by PFM techniques (with E max of ± 10 V), although a 180°p hase shift of the piezoresponse is readily observed. As shown in the inset of Figure 4a, P r tends to reach its saturated value with increasing E max . This type of saturation tendency in P r is also observed with increasing E c (Supplementary Figure S8). Thus, the net switching polarization (2P r ) estimated from Figure 4a is~35 μC cm − 2 , which is much larger than the previously reported values of 2P r (between 0.1 and 1.0 μC cm − 2 ). [14][15][16] A more reliable value of the switching polarization can be obtained by using PUND (positive-up and negative-down) pulse test. Figure 4b presents the PUND result of the 200-nm-thin GFO film capacitor obtained using the pulse delay time of 100 ms. We have found that the switching polarization decreases with increasing delay time and reaches a saturated value for pulse delay times longer than~80 ms. The net switching polarization (2P r ) is evaluated using the following relation: 2P r = (± P*) − ( ± P Λ ) The 2P r value obtained from Figure 4b is~26 μC cm − 2 . Comparing this value with the 2P r value obtained from Figure 4a, one can conclude that the P-E curves overestimate the 2P r value substantially. This suggests that the ac frequency of 1 kHz used in the P-E measurement is not sufficiently high to eliminate completely the responses of mobile space charges, which tend to be significant at lower ac frequencies.
To examine the universality of the present finding of a large 2P r , we have also examined the polarization switching characteristics of the polar c-axis-oriented GFO film grown on a conducting ITO-buffered cubic YSZ (001) substrate. As shown in Figure 5a Compared with the 2P r of the Pt/GFO/SRO/STO (111) capacitor, the 2P r of the Pt/GFO/ITO/YSZ (001) capacitor is thus substantially reduced:~35 vs~30 μC cm − 2 according to their P-E results. As stated previously, the c-axis-grown o-GFO film on an ITO/YSZ (001) substrate is characterized by the six crystallographic variants with a total of 12 in-plane orientations with a successive in-plane tilting angle of 30°, in spite of slight lattice mismatch between the GFO film and the YSZ substrate (see Supplementary Information for details). The reduced 2P r value can possibly be correlated with this increase in the crystallographic variants (i.e., the degree of complexity in the domain configuration) upon replacing the SRO/STO (111) substrate with the ITO/YSZ (001) substrate. However, further in-depth studies should be made before clearly resolving the origin of the substrate-dependent 2P r .

DISCUSSION
For an in-depth understanding of the atomic-scale origin of the observed polarization switching, we have calculated the DFT polarization of the polar Pna2 1 phase of o-GFO using the Berry-phase method. 36,37 To obtain a correct evaluation of the ferroelectric polarization, a centrosymmetric prototypic phase of GFO should be identified first. To accomplish this, we have used the pseudocode of the Bilbao crystallographic server, 38 which allows one to determine the nearest supergroup structure for an input arbitrary structure based on lattice dynamics calculations. In the present case, four centrosymmetric supergroup structures are examined: Pnna, Pccn, Pbcn and Pnma. Among these four, the nearest reference structure, in terms of the total displacement of all atoms, is the Pnna phase, which is consistent with the previously reported result. 20 The optimized lattice parameters of the ferroelectric Pna2 1 and prototypic Pnna phases were subsequently obtained by calculating the Kohn-Sham (K-S) energy as a function of the unit-cell volume and finding its minimum that corresponds to the ground state K-S energy in the absence of any external pressure. The optimized lattice parameters are (i) a = 5.1647 Å, b = 8.8197 Å and c = 9.5079 Å for the Pna2 1 structure, and (ii) a = 5.2672 Å, b = 8.9193 Å and c = 9.5684 Å for the Pnna structure. The optimized structures of the Pnna and Pna2 1 phases of o-GFO are depicted in Figure 6a.
According to group theoretical analysis, 39,40 there exists only one conceivable transition path that connects the prototypic Pnna phase to the ferroelectric Pna2 1 phase. We have decomposed the atomic displacements that relate the nonpolar Pnna phase to the polar Pna2 1 phase into the symmetry-adapted mode of the prototypic phase. The resulting symmetry-adapted mode is exclusively given by G À 4 Thus, the Pnna-to-Pna2 1 phase transition should be mediated by freezing in the zone-center G À 4 polar phonon. Let us define the displacement amplitude of the polar G À 4 phonon as Q G À 4 . In Figure 6b, the K-S energy and Berry-phase polarization are plotted as functions of the mode amplitude Q G À 4 . 41 Here, the polarization (lower panel of Figure 6b) is given by the product of Q G À 4 and the Born effective charge tensor. As shown in Figure 6b, the computed K-S energy exhibits a double-well-type potential, which demonstrates the relative stability of the ferroelectric Pna2 1 phase over the prototypic nonpolar Pnna phase with the energy difference of 1.05 eV per formula unit (f.u.). Herein, the equilibrium ferroelectric polarization of the Pna2 1 phase is given by the computed polarization values (lower panel) at the two K-S energy minima-namely, ± 25.67 μC cm − 2 -which correspond to Q G À 4 of ± 1, respectively. Because polarization switching is expected to occur along the Pna2 1 -to-Pnna phase-transition path-that is, G À 4 -the activation free energy of the polarization switching between the double wells can be obtained from Figure 6b, which is~1.05 eV per f.u. This value lies within the two extreme ab initio values (0.52 and 1.30 eV per f.u.) previously obtained by Stoeffler 20 using ABINIT and FLAPW/FLEUR codes. The activation barrier of 1.05 eV per f.u. is~2.5 times larger than that of BiFeO 3 , the most extensively studied multiferroic, and 20 times larger than that of Pb(Zr,Ti)O 3 , the most widely used displacive ferroelectric. 42 This unusually high activation barrier indicates that the polar Pna2 1 phase in o-GFO is very stable against thermally activated random dipole switching across the centrosymmetric Pnna barrier.
We will then theoretically correlate the reported high ferroelectric transition temperature (41368 K; Mishra et al. 22 ) with this remarkably high activation free energy. According to the transition-state theory of rate processes, 43 the frequency (v) of the polarization switching across the Pnna potential barrier can be written as where Φ 0 is the barrier height of the polarization switching, which can be treated as the difference in the K-S energy between the Pnna and Pna2 1 phases (1.05 eV per f.u.). Strictly speaking, Equation (1) is valid for describing the dipole switching rate of order-disorder ferroelectrics. For sufficiently high temperatures near the phase transition point (T c ), however, Equation (1) is also applicable to the dipole switching rate of displacive ferroelectrics, 44 which include the present Pna2 1 -to-Pnna transition. As temperature increases and approaches T c , the dipole switching rate becomes so rapid that the mean residence time (τ 0 ) of the bound polarization in one of the two ferroelectric double wells becomes shorter than a certain critical time for the experimental observation. Under this condition, the net ferroelectric polarization effectively disappears because of the switching average of two opposite polarizations across the centrosymmetric barrier (Pnna state in the present case; Figure 6b). Let v 0 be the frequency at which the net bound polarization first begins to disappear. In the vicinity of T c , Φ 0 ok B T for a fixed value of Φ 0 . Under this condition, v 0 can be approximated by the following expression: where T 0 denotes the temperature that corresponds to the critical switching frequency v 0 . Because T c is expressed by E is a small positive number (including zero), T c can be correlated with Φ 0 by the following relation for a fixed value of Φ 0 : Equation (3) clearly indicates that T c is linearly proportional to Φ 0 . This explains the observed high T c of o-GFO in terms of the unusually high activation energy barrier (Φ 0 ) as predicted by ab initio calculations. Finally, we will correlate both the observed large polarization and the high coercive field (E c ) with the high activation energy barrier, Φ 0 by exploiting the phenomenological Landau theory. To accomplish this, let us write the elastic Gibbs free energy density (at a constant pressure or stress) of the polar Pna2 1 phase of o-GFO in terms of the polarization as a relevant order parameter (up to P 4 -term): where ξ40 for the P 4 expansion and χ = (T − T c )/Co0 for temperatures below T c , where C and T c denote the Curie-Weiss constant and Curie-Weiss temperature, respectively. Starting from Equation (4), one can derive the following expression of the equilibrium polarization (P eq ) in terms of the barrier height of the ferroelectric double-well potential (Φ 0 ; 1.05 eV as shown in Figure 6b): Similarly, the following relation between the coercive field and the barrier height can be derived (detailed derivations are given in the Supplementary Information): According to Equation (6), the coercive field is proportional to the square root of the barrier height (Φ 0 ). It is now clear that this high value of Φ 0 is responsible for both the observed large equilibrium polarization (Equation (5)) and the high coercive field (Equation (6)). Although the phenomenological Landau theory is strictly valid for a single-domain ferroelectric, the conclusions drawn above can be applied to the present polar c-axis grown o-GFO thin film, at least qualitatively.

CONCLUSIONS
We have clarified a puzzling discrepancy between the observed P r values (⩽0.5 μC cm − 2 ) and the ab initio prediction and have unequivocally demonstrated ferroelectric polarization switching with net switching polarization (2P r ) of~30 μC cm − 2 by using the two distinct types of o-GFO thin films preferentially grown along the polar c-axis in Pna2 1 setting (b-axis in Pc2 1 n setting). The estimated activation energy for the polarization switching is~1.05 eV per f.u. This high value accounts for the extraordinarily high coercive field (41100 kV cm − 1 ) and the stability of the polar Pna2 1 phase over the centrosymmetric Pnna phase for a wide range of temperatures up to 1368 K.