Controlling ferroelectric hysteresis offsets in PbTiO$_{3}$ based superlattices
Simon Divilov, Hsiang-Chun Hsing, Mohammed Humed Yusuf, Anna Gura, Joseph A. Garlow, Myung-Geun Han, Massimiliano Stengel, John Bonini, Premala Chandra, Karin M… 展开作者
Simon Divilov, Hsiang-Chun Hsing, Mohammed Humed Yusuf, Anna Gura, Joseph A. Garlow, Myung-Geun Han, Massimiliano Stengel, John Bonini, Premala Chandra, Karin M. Rabe, Marivi Fernandez Serra, Matthew Dawber
Ferroelectric materials are characterized by degenerate ground states with multiple polarization directions. In a ferroelectric capacitor this should manifest as equally favourable up and down polarization states. However, this ideal behavior is rarely observed in ferroelectric thin films and superlattice devices, which generally exhibit a built-in bias which favors one polarization state over the other. Often this polarization asymmetry can be attributed to the electrodes. In this study we examine bias in PbTiO$_3$-based ferroelectric superlattices that is not due to the electrodes, but rather to the nature of the defects that form at the interfaces during growth. Using a combination of experiments and first-principles simulations, we are able to explain the sign of the observed built-in bias and its evolution with composition. Our insights allow us to design devices with zero built-in bias by controlling the composition and periodicity of the superlattices.
The effect of layer-by-layer heterostructuring and epitaxial strain on lattice instabilities and related ferroelectric properties is investigated from first principles for the [001]-(PbTiO$_3$)$_1$(PbZrO$_3$)$_1$ superlattice and pure PbTiO$_3$ on a cubic substrate. The results for the superlattice show an enhancement of the stability of the monoclinic r-phase with respect to pure PbTiO$_3$. Analysis of the lattice instabilities of the relaxed centrosymmetric reference structure computed within density functional perturbation theory suggests that this results from the presence of two unstable …
Artificial PbTiO$_{3}$/SrTiO$_{3}$ superlattices were constructed using off-axis RF magnetron sputtering. X-ray diffraction and piezoelectric atomic force microscopy were used to study the evolution of the ferroelectric polarization as the ratio of PbTiO$_{3}$ to SrTiO$_{3}$ was changed. For PbTiO$_{3}$ layer thicknesses larger than the 3-unit cells SrTiO$_{3}$ thickness used in the structure, the polarization is found to be reduced as the PbTiO$_{3}$ thickness is decreased. This observation confirms the primary role of the depolarization field in the polarization reduction in thin films. For …
We have fabricated PbTiO$_{3}$/SrRuO$_{3}$ superlattices with ultra-thin SrRuO$_{3}$ layers. Due to the superlattice geometry, the samples show a large anisotropy in their electrical resistivity, which can be controlled by changing the thickness of the PbTiO$_{3}$ layers. Therefore, along the ferroelectric direction, SrRuO$_{3}$ layers can act as dielectric, rather than metallic, elements. We show that, by reducing the concentration of PbTiO$_{3}$, an increasingly important effect of polarization asymmetry due to compositional inversion symmetry breaking occurs. The results are significant as …
Ab initio phonon calculations for the nonpolar reference structures of the (001), (110), and (111) PbZrO_3/PbTiO_3 [1:1] superlattices are presented. The unstable polar modes in the tetragonal (001) and (110) structures are confined in either the Ti- or the Zr-centered layers and display two-mode behavior, while in the cubic (111) case one-mode behavior is observed. Instabilities with pure oxygen character are observed in all three structures. The implications for the ferroelectric behavior and related properties are discussed.
The local structural distortions in polydomain ferroelectric PbTiO3/SrTiO3 superlattices are investigated by means of high spatial and energy resolution electron energy loss spectroscopy combined with high angle annular dark field imaging. Local structural variations across the interfaces have been identified with unit cell resolution through the analysis of the energy loss near edge structure of the Ti-L2,3 and O-K edges. Ab-initio and multiplet calculations of the Ti-L2,3 edges provide unambiguous evidence for an inhomogeneous polarization profile associated with the observed structural dist…
We report first-principles calculations, within the density functional theory, on the structural and energetic properties of 180$^\circ$ stripe domains in (PbTiO$_{3}$)$_{n}$/(SrTiO$_{3}$)$_{n}$ superlattices. For the explored periodicities ($n$=3 and 6) we find that the polydomain structures compete in energy with the monodomain phases. Our results suggest the progressive transition, as a function of $n$, from a strong to a weak electrostatic coupling regime between the SrTiO$_{3}$ and PbTiO$_{3}$ layers. Structurally, they display continuous rotation of polarization connecting 180$^\circ$ do…
In this paper, an efficient computational material design approach (cluster expansion) is employed for the ferroelectric $PbTiO_3$/$SrTiO_3$ system. Via exploring a conguration space including over $3 \times 10^6$ candidates, two special cation ordered configurations: either perfect or mixed 1/1 (101) superlattice, are identified with the mostly enhanced ferroelectric polarization by up to about 100% in comparison with the (001) superlattice. Analyzing these two special configurations reveals the exotic couplings between the antiferrodistortive distortion (AFD) and ferroelectric polarization (…
A set of symmetric and asymmetric superlattices with ferromagnetic La0.6Sr0.4MnO3 and ferroelectric 0.7Pb(Mg0.33Nb0.67)O3 0.3(PbTiO3) as the constituting layers were fabricated on LaNiO3 coated (100) oriented LaAlO3 substrates using pulsed laser ablation. The crystallinity, magnetic and ferroelectric properties were studied for all the superlattices. All the superlattice structures exhibited a ferromagnetic behavior over a wide range of temperatures between 10K and 300K, whereas only the asymmetric superlattices exhibited a reasonably good ferroelectric behaviour. Strong influence of an applie…
We report first-principles calculations on the coupling between epitaxial strain, polarization, and oxygen octahedra rotations in monodomain (PbTiO$_{3}$)$_{n}$/(SrTiO$_{3}$)$_{n}$ superlattices. We show how the interplay between (i) the epitaxial strain and (ii) the electrostatic conditions, can be used to control the orientation of the main axis of the system. The electrostatic constrains at the interface facilitate the rotation of the polarization and, as a consequence, we predict large piezoelectric responses at epitaxial strains smaller than those that would be required considering only s…
We study the emergence of Berezinskii-Kosterlitz-Thouless (BKT) phases in (PbTiO$_3$)$_3$/(SrTiO$_3$)$_3$ superlattices by means of second-principles simulations. Beyond a threshold tensile epitaxial strain of $ε= 0.25 \%$ the local dipole moments within the superlattices are confined to the film-plane, and thus the polarization can be effectively considered as two-dimensional. The analysis of the decay of the dipole-dipole correlation with the distance, together with the study of the density of defects and its distribution as function of temperature, supports the existence of a BKT phase in a…
The defect stability in a prototypical perovskite oxide superlattice consisting of SrTiO$_3$ and PbTiO$_3$ (STO/PTO) is determined using first principles density functional theory calculations. Specifically, the oxygen vacancy formation energies E$_v$ in the paraelectric and ferroelectric phases of a superlattice with four atomic layers of STO and four layers of PTO (4STO/4PTO) are determined and compared. The effects of charge state, octahedral rotation, polarization, and interfaces on the E$_v$ are examined. The formation energies vary layer-by-layer in the superlattices, with E$_v$ being hi…
We present first-principles calculations of the structural phase behavior of the [1:1] \pzta\ superlattice and the \ptoa\ and \pzoa\ parent compounds as a function of in-plane epitaxial strain. A symmetry analysis is used to identify the phases and clarify how they arise from an interplay between different kinds of structural distortions, including out-of-plane and in-plane polar modes, rotation of oxygen octahedra around out-of-plane or in-plane axes, and an anti-polar mode. Symmetry-allowed intermode couplings are identified and used to elucidate the nature of the observed phase transitions.…
First-principles density functional calculations are performed to investigate the interplay between inplane strains and interface effects in 1by1 PbTiO3/SrTiO3 and BaTiO3/SrTiO3 superlattices of tetragonal symmetry. One particular emphasis of this study is to conduct side-by-side comparisons on various ferroelectric properties in short-period superlattices and in constituent bulk materials, which turns out to be rather useful in terms of obtaining valuable insight into the different physics when ferroelectric bulks form superlattices. The various properties that are studied in this work includ…
Charge dipole moment and spin moment rarely coexist in single-phase bulk materials except in some multiferroics. Despite the progress in the past decade, for most multiferroics their magnetoelectric performance remains poor due to the intrinsic exclusion between charge dipole and spin moment. As an alternative approach, the oxide heterostructures may evade the intrinsic limits in bulk materials and provide more attractive potential to realize the magnetoelectric functions. Here we perform a first-principles study on LaAlO$_3$/PbTiO$_3$ superlattices. Although neither of the components is magne…
The spontaneously formed striped polarization nanodomain configuration of a PbTiO${_3}$/SrTiO${_3}$ superlattice transforms to a uniform polarization state under above-bandgap illumination with a time dependence varying with the intensity of optical illumination and a well-defined threshold intensity. Recovery after the end of illumination occurs over a temperature-dependent period of tens of seconds at room temperature and shorter times at elevated temperatures. A model in which the screening of the depolarization field depends on the population of trapped electrons correctly predicts the obs…
We examine the application of short-circuit electrical boundary conditions in density functional theory calculations of ferroelectric thin films. Modeling PbTiO$_3$ films with Pt electrodes, we demonstrate that under periodic boundary conditions of supercells, short-circuit conditions for the electrodes are equivalently satisfied in two repeated slab geometries: a PbTiO$_3$/metal superlattice geometry and a periodic metal/PbTiO$_3$/metal/vacuum geometry, where the metal is Pt or SrRuO$_3$. We discuss the benefits of each geometry in the study of ferroelectricity in thin films.
In the first-principles bulk-layer model the superlattice structure and polarization are determined by first-principles computation of the bulk responses of the constituents to the electrical and mechanical boundary conditions in an insulating superlattice. In this work the model is extended to predict functional properties, specifically dielectric permittivity and piezoelectric response. A detailed comparison between the bulk-layer model and full first-principles calculations for three sets of perovskite oxide superlattices, PbTiO$_3$/BaTiO$_3$, BaTiO$_3$/SrTiO$_3$ and PbTiO$_3$/SrTiO$_3$, is…
Complex polar superstructure controlled thermal conductivity in ferroelectric PbTiO3/SrTiO3 superlattices
Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学）；Universidad Rey Juan Carlos（胡安卡洛斯国王大学）；Universitat de Barcelona（巴塞罗那大学）；Catalan Institute of Nanoscience and Nanotechnology (ICN2)（加泰罗尼亚纳米科学与技术研究所）；ALBA Synchrotron Light Facility（阿尔巴同步辐射光源设施）
The effect of intermixing at the interface of short period PbTiO$_3$/SrTiO$_3$ superlattices is studied using first-principles density functional theory. The results indicate that interfacial intermixing significantly enhances the polarization within the superlattice. This enhancement is directly related to the off-centering of Pb and Sr cations and can be explained through a discussion of interacting dipoles. This picture should be general for a wide range of multicomponent superlattices and may have important consequences for the design of ferroelectric devices.
Short period ferroelectric/ferroelectric BaTiO3 (BTO)/PbTiO3 (PTO) superlattices are studied using density functional theory. Contrary to the trends in paraelectric/ferroelectric superlattices the polarization remains nearly constant for PTO concentrations below 50%. In addition, a significant decrease in the c/a ratio below the PTO values were observed. Using a superlattice effective Hamiltonian we predict an enhancement in the d33 piezoelectric coefficient peaking at ~75% PTO concentration due to the different polarization-strain coupling in PTO and BTO layers. Further analysis reveals that …
Moire materials, created by lattice-mismatch or/and twist-angle, have spurred great interest in excavating novel quantum phases of matter. Latterly, emergent interfacial ferroelectricity has been surprisingly found in spatial inversion symmetry broken systems, such as rhombohedral-stacked bilayer transition metal dichalcogenides (TMDs). However, the evolution of moire superlattices corresponding to polarization switching and hysteresis is still unclear, which is crucial for giving insight into the interplay between lattice symmetry and band topology, as well as developing optoelectronic memory…
Superlattices composed of ferromagnetic La0.6Sr0.4MnO3 and ferroelectric 0.7Pb(Mg0,33Nb0.66)O3 0.3(PbTiO3) layers were fabricated on (100) LaAlO3 substrates by pulsed laser deposition technique. Ferromagnetic and frequency independent ferroelectric hysteresis characteristics established the biferroic nature of the superlattices. Influence of magnetic field was observed in tuning the P-E characteristics of the superlattices. Similar effect was observed on application of a high DC electric field to the samples. The nature of the observed ferroelectric properties and their modulation by applied m…
In the development of first-principles high-throughput searches for materials with desirable functional properties, there is a clear need for an efficient method to determine the ground state and low-energy alternative structures of superlattices. A method based on a simple strategy -- to generate starting structures based on low-energy structures of the constituent compounds, which are then optimized via structural relaxation calculations -- is proposed. This "stacking method" is demonstrated on the 2:2 PbTiO$_3$/SrTiO$_3$ superlattice, which has been the subject of recent experimental and th…
The polarization response of antiferroelectrics to electric fields is such that the materials can store large energy densities, which makes them promising candidates for energy storage applications in pulsed-power technologies. However, relatively few materials of this kind are known. Here we consider ferroelectric/paraelectric superlattices as artificial electrostatically-engineered antiferroelectrics. Specifically, using high-throughput second-principles calculations, we engineer PbTiO$_{3}$/SrTiO$_{3}$ superlattices to optimize their energy-storage performance at room temperature (to maximi…
Ferroelectrics subject to suitable electric boundary conditions present a steady negative capacitance response. When the ferroelectric is in a heterostructure, this behavior yields a voltage amplification in the other elements, which experience a potential difference larger than the one applied, holding promise for low-power electronics. So far research has focused on verifying this effect and little is known about how to optimize it. Here we describe an electrostatic theory of ferroelectric/dielectric superlattices, convenient model systems, and show the relationship between the negative perm…
Depth-Resolved Evolution of Buried Polar Topologies in a PbTiO3/SrTiO3 Superlattice
Catalan Institute of Nanoscience and Nanotechnology - ICN2 (CSIC & BIST)（加泰罗尼亚纳米科学和技术研究所）；Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons, Forschungszentrum Jülich（于利希研究中心恩斯特·鲁卡电子显微与谱学中心）；CiQUS, Centro Singular de Investigacion en Quimica Bioloxica e Materiais Moleculares, Departamento de Quimica-Fisica, Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学生物化学与分子材料 singular 研究中心）；Department of Materials Science and Engineering, University of Wisconsin-Madison（威斯康星大学麦迪逊分校材料科学与工程系）；Chinese Academy of Sciences Fujian Institute of Research on the Structure of Matter（中国科学院福建物质结构研究所）；School of Aerospace Engineering, Beijing Institute of Technology（北京理工大学航天工程学院）；Institució Catalana de Recerca i Estudis Avançats (ICREA)（加泰罗尼亚研究高级学院）
We present a systematic approach to modeling the electrical and structural properties of charge-mismatched superlattices from first principles. Our strategy is based on bulk calculations of the parent compounds, which we perform as a function of in-plane strain and out-of-plane electric displacement field. The resulting two-dimensional phase diagrams allow us to accurately predict, without performing further calculations, the behavior of a layered heterostructure where the aforementioned building blocks are electrostatically and elastically coupled, with an arbitrary choice of the interface ch…
We study the equilibrium arrangements of polarization vortices in (PbTiO$_3$)$_n$/(SrTiO$_3$)$_n$ superlattices by means of second-principles simulations. We find that, at low temperatures, polarization vortices organize in a regular arrangement in which clockwise and counter-clockwise vortices alternate positions, leading to a crystal-like structure with well defined handedness. This chiral crystal melts at a critical temperature $T_\mathrm{M}$ into a chiral liquid, where long-range order is lost but handedness is preserved. At even higher temperatures, $T_\mathrm{C}$, a second phase transiti…
A key property that drives research in ferroelectric perovskite oxides is their strong piezoelectric response in which an electric field is induced by an applied strain, and vice-versa for the converse piezoelectric effect. We have achieved an experimental enhancement of the piezoelectric response and dielectric tunability in artificially layered epitaxial PbTiO$_{3}$/CaTiO$_{3}$ superlattices through an engineered rotation of the polarization direction. As the relative layer thicknesses within the superlattice were changed from sample to sample we found evidence for polarization rotation in m…
While electrical compatibility constraints normally prevent head-to-head (HH) and tail-to-tail (TT) domain walls from forming in ferroelectric materials, we propose that such domain walls could be stabilized by intentional growth of atomic layers in which the cations are substituted from a neighboring column of the periodic table. In particular, we carry out predictive first-principles calculations of superlattices in which Sc, Nb, or other substitutional layers are inserted periodically into PbTiO$_3$. We confirm that this gives rise to a domain structure with the longitudinal component of th…
Thin films of (1-x)BiFeO<sub>3</sub>-xPbTiO<sub>3</sub> (BF-xPT) with x ~ 0.60 were fabricated on Pt/Si substrates by chemical solution deposition of precursor BF and PT layers alternately in three different multilayer configurations. These multilayer deposited precursor films upon annealing at 700°C in nitrogen show pure perovskite phase formation. In contrast to the equilibrium tetragonal structure for the overall molar composition of BF:PT::40:60, we find monoclinic structured BF-xPT phase of M<sub>A</sub> type. Piezo-force microscopy confirmed ferroelectric switching in the films and revea…
Tetragonal ferroelectric/ferroelectric BaTiO3/PbTiO3 superlattice under uniaxial tensile stress along the c axis is investigated from first principles. We show that the calculated ideal tensile strength is 6.85 GPa and that the superlattice under the loading of uniaxial tensile stress becomes soft along the nonpolar axes. We also find that the appropriately applied uniaxial tensile stress can significantly enhance the piezoelectricity for the superlattice, with piezoelectric coefficient d33 increasing from the ground state value by a factor of about 8, reaching 678.42 pC/N. The underlying mech…
A monoclinic phase was recently discovered near the morphotropic phase boundary in several high-performance piezoelectric perovskite solid solutions, but its properties have not been reported. In this paper the dielectric, piezo- and ferroelectric properties of the monoclinic Pm phase in the (1-x)Pb(Mg1/3Nb2/3)O3 -xPbTiO3 perovskite system are studied. In a (001)-oriented crystal of composition x=0.33, ferroelectric hysteresis loops with remanent polarization of 23 uC/cm^2 are displayed. In poled monoclinic crystals, under unipolar drive up to 10 kV/cm, the domain walls remain unchanged, the p…
In contrast to ordinary ferroelectrics where the temperature, Tm, of the permittivity maximum monotonically increases with bias field, E, in (1-x)PbMg1/3Nb2/3O3-(x)PbTiO3 (0<x<0.35) single crystals, Tm was found to remain constant or decrease with E up to a certain threshold field, Et, above which Tm starts increasing. Et decreases with x and almost disappears at about x=0.4. We explain this field dependence by the quenched random fields and coupling between the orientable dipoles with average polarization. For crystals with 0.06<x<0.13, the T-E phase diagrams are constructed, which include th…
Ferroelectric hysteresis in singly aligned graphene-hBN moiré superlattices
CIC nanoGUNE（纳米科技联合研究中心）；University of the Basque Country (UPV/EHU)（巴斯克大学）；National Institute for Materials Science（物质材料研究机构）；Centro de Física de Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU（物质物理中心（CFM-MPC）CSIC-巴斯克大学联合中心）
The exchange coupling at the interfaces of magnetic superlattices consisting of ferromagnetic $SrRuO_3$ and antiferromagnetic $SrMnO_3$ grown on (001) oriented $SrTiO_3$ is studied with in-plane and out-of-plane orientations, with respect to the substrate plane, of the cooling magnetic field. The magnetization of the in-plane, field cooled hysteresis loop is lower than the corresponding in-plane zero-field-cooled hysteresis loop. The out-of-plane field cooled hysteresis loop is shifted, from the origin, along the graphical magnetization axis. We attribute this irreversible rotation of the mome…
We present a theoretical proposal for the design of a thermal switch based on the anisotropy of the thermal conductivity of PbTiO3 and of the possibility to rotate the ferroelectric polarization with an external electric field. Our calculations are based on an iterative solution of the phonon Boltzmann Transport Equation and rely on interatomic force constants computed within an efficient second-principles density functional theory scheme. We also characterize the hysteresis cycle of the thermal conductivity in presence of an applied electric field and show that the response time would be limi…
Weakly coupled ferroelectric/dielectric superlattice thin film heterostructures exhibit complex nanoscale polarization configurations that arise from a balance of competing electrostatic, elastic, and domain-wall contributions to the free energy. A key feature of these configurations is that the polarization can locally have a significant component that is not along the thin-film surface normal direction, while maintaining zero net in-plane polarization. PbTiO3/SrTiO3 thin-film superlattice heterostructures on a conducting SrRuO3 bottom electrode on SrTiO3 have a room-temperature stripe nanodo…
The interface between graphene and the ferroelectric superlattice $\mathrm{PbTiO_3/SrTiO_3}$ (PTO/STO) is studied. Tuning the transition temperature through the PTO/STO volume fraction minimizes the adsorbates at the graphene-ferroelectric interface, allowing robust ferroelectric hysteresis to be demonstrated. Intrinsic charge traps from the ferroelectric surface defects can adversely affect the graphene channel hysteresis, and can be controlled by careful sample processing, enabling systematic study of the charge trapping mechanism.
Despite the technological significance of the tetragonal PbTiO3 for the piezoelectric transducer industry, its high pressure behaviour is quite controversial as two entirely different scenarios, involving pressure induced (1) morphotropic phase boundary (MPB) like structural transition with concomitant rotation of the ferroelectric polarization vector and (2) antiferrodistortive (AFD) phase transition followed by emergence of a reentrant ferroelectric phase, have been proposed in recent theoretical and experimental studies. We have attempted to address these controversies through a high resolu…
Artificial superlattices of (BiFeO3)m(SrTiO3)m (m= 1 to 10 unit cells) consisting of multiferroic BiFeO3 and insulating SrTiO3 layers were fabricated on (100)-oriented SrTiO3 substrates by pulsed laser ablation. The remnant polarization and leakage current behavior were studied varying the periodicity (8-80A) of the superlattice. The leakage current was reduced by few orders of magnitude on increase of periodicity compared to single layer BiFeO3 thin films. Reduced leakage and intrinsic polarization hysteresis was observed and was confirmed by PUND analysis for periodicities in the range 20-60…
Exchange bias (EB) and the training effects (TE) in an antiferromagnetically coupled La0.7Sr0.3MnO3 / SrRuO3 superlattices were studied in the temperature range 1.8 - 150 K. Strong antiferromagnetic (AFM) interlayer coupling is evidenced from AC - susceptibility measurements. Below 100 K, vertical magnetization shifts are present due to the two remanent states corresponding to the two ferromagnetic (FM) layers at FM and AFM coupling condition. After field cooling (FC), significant decrease in the exchange bias field (HEB) is observed when cycling the system through several consecutive hysteres…
The ideal intrinsic barriers to domain switching in c-phase PbTiO_3 (PTO), PbZrO_3 (PZO), and PbZr_{1-x}Ti_xO_3 (PZT) are investigated via first-principles computational methods. The effects of epitaxial strain on the atomic structure, ferroelectric response, barrier to coherent domain reversal, domain-wall energy, and barrier to domain-wall translation are studied. It is found that PTO has a larger polarization, but smaller energy barrier to domain reversal, than PZO. Consequentially the idealized coercive field is over two times smaller in PTO than PZO. The Ti--O bond length is more sensitiv…
Strain engineering of perovskite oxide thin films has proven to be an extremely powerful method for enhancing and inducing ferroelectric behavior. In ferroelectric thin films and superlattices, the polarization is intricately linked to crystal structure, but we show here that it can also play an important role in the growth process, influencing growth rates, relaxation mechanisms, electrical properties and domain structures. We have studied this effect in detail by focusing on the properties of BaTiO$_{3}$ thin films grown on very thin layers of PbTiO$_{3}$ using a combination of x-ray diffrac…
We have developed and implemented a formalism for computing the structural response of a periodic insulating system to a homogeneous static electric field within density-functional perturbation theory (DFPT). We consider the thermodynamic potentials E(R,eta,e) and F(R,eta,e) whose minimization with respect to the internal structural parameters R and unit cell strain eta yields the equilibrium structure at fixed electric field e and polarization P, respectively. First-order expansion of E(R,eta,e) in e leads to a useful approximation in which R(P) and eta(P) can be obtained by simply minimizing…
A GPU-based Solver for Polarization Dynamics in Ferroelectric Materials
University of Messina（墨西拿大学）；Politecnico di Bari（巴里理工大学）；Luxembourg Institute of Science and Technology (LIST)（卢森堡科学与技术研究所）；University of Luxembourg（卢森堡大学）
