Domain Switching on the Pareto Front: Multi-Objective Deep Kernel Learning in Automated Piezoresponse Force Microscopy
University of Tennessee（田纳西大学）；Institute of Science Tokyo（东京科学大学）；University of California, Berkeley（加州大学伯克利分校）；Rice University（莱斯大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
Yu Liu, Utkarsh Pratiush, Kamyar Barakati, Hiroshi Funakubo, Ching-Che Lin, Jaegyu Kim, Lane W. Martin, Sergei V. Kalinin
Ferroelectric polarization switching underpins the functional performance of a wide range of materials and devices, yet its dependence on complex local microstructural features renders systematic exploration by manual or grid-based spectroscopic measurements impractical. Here, we introduce a multi-objective kernel-learning workflow that infers the microstructural rules governing switching behavior directly from high-resolution imaging data. Applied to automated piezoresponse force microscopy (PFM) experiments, our framework efficiently identifies the key relationships between domain-wall configurations and local switching kinetics, revealing how specific wall geometries and defect distributions modulate polarization reversal. Post-experiment analysis projects abstract reward functions, such as switching ease and domain symmetry, onto physically interpretable descriptors including domain configuration and proximity to boundaries. This enables not only high-throughput active learning, but also mechanistic insight into the microstructural control of switching phenomena. While demonstrated for ferroelectric domain switching, our approach provides a powerful, generalizable tool for navigating complex, non-differentiable design spaces, from structure-property correlations in molecular discovery to combinatorial optimization across diverse imaging modalities.
The extrinsic size effect in Piezoresponse Force Microscopy of ferroelectric and piezoelectric thin films on non-polar dielectric substrate is analyzed. Analytical expressions for effective piezoresponse, object transfer function components, and Rayleigh two-point resolution are obtained. These results can be broadly applied for effective piezoelectric response calculations in thin piezoelectric and ferroelectric films as well as surface polar layers e.g. in organic materials and bio-polymers. In particular, the effective piezoresponse strongly decreases with film thickness whereas the sharpne…
The interpretation of ferroelectric domain images obtained with piezoresponse force microscopy (PFM) is discussed. The influences of an inherent experimental background on the domain contrast in PFM images (enhancement, nulling, inversion) as well as on the shape and the location of the domain boundaries are described. We present experimental results to evidence our analysis of the influence of the background on the domain contrast in PFM images.
The contrast mechanism for ferroelectric domain imaging via piezoresponse force microscopy (PFM) is investigated. A novel analysis of PFM measurements is presented which takes into account the background caused by the experimental setup. This allows, for the first time, a quantitative, frequency independent analysis of the domain contrast which is in good agreement with the expected values for the piezoelectric deformation of the sample and satisfies the generally required features of PFM imaging.
As a consequence of elasticity, mechanical deformations of crystals occur on a length scale comparable to their thickness. This is exemplified by applying a homogeneous electric field to a multi-domain ferroelectric crystal: as one domain is expanding the adjacent ones are contracting, leading to clamping at the domain boundaries. The piezomechanically driven surface corrugation of micron-sized domain patterns in thick crystals using large-area top electrodes is thus drastically suppressed, barely accessible by means of piezoresponse force microscopy.
Signal formation mechanism of Piezoresponse Force Microscopy of piezoelectric surface layers and thin films on stiff and elastically matched substrates is analyzed and thickness dependence of effective piezoelectric response, object transfer function components and Rayleigh two-point resolution are derived. Obtained exact series and simple Pade approximations can be applied for the effective piezoresponse analytical calculations in the case of films capped on different substrates. The effective piezoresponse is thickness dependent for piezoelectric films on substrates with low dielectric permi…
Piezoresponse force microscopy (PFM) was used to investigate the ferroelectric properties of sol-gel derived LiNbO$_3$ nanoparticles. To determine the degree of ferroelectricity we took large-area images and performed statistical image-analysis. The ferroelectric behavior of single nanoparticles was verified by poling experiments using the PFM tip. Finally we carried out simultaneous measurements of the in-plane and the out-of-plane piezoresponse of the nanoparticles, followed by measurements of the same area after rotation of the sample by 90$^{\circ}$ and 180$^{\circ}$. Such measurements bas…
In studies using piezoresponse force microscopy, we observe a non-zero lateral piezoresponse at 180$^\circ$ domain walls in out-of-plane polarized, c-axis-oriented tetragonal ferroelectric Pb(Zr$_{0.2}$Ti$_{0.8}$)O$_3$ epitaxial thin films. We attribute these observations to a shear strain effect linked to the sign change of the $d_{33}$ piezoelectric coefficient through the domain wall, in agreement with theoretical predictions. We show that in monoclinically distorted tetragonal BiFeO$_3$ films, this effect is superimposed on the lateral piezoresponse due to actual in-plane polarization, and…
SrTiO3 ceramics are investigated by piezoresponse force microscopy. Piezoelectric contrast is observed on polished surfaces in both vertical and lateral regimes and depends on the grain orientation varying in both sign (polarization direction) and amplitude. The observed contrast is attested to the surface piezoelectricity due to flexoelectric effect (strain gradient-induced polarization) caused by the surface relaxation. The estimated flexoelectric coefficient is approximately one order of magnitude smaller as compared to those recently measured in SrTiO3 single crystals. The observed enhance…
We present a quantitative investigation of the impact of tip radius as well as sample type and thickness on the lateral resolution in piezoresponse force microscopy (PFM) investigating bulk single crystals. The observed linear dependence of the width of the domain wall on the tip radius as well as the independence of the lateral resolution on the specific crystal-type are validated by a simple theoretical model. Using a Ti-Pt-coated tip with a nominal radius of 15 nm the so far highest lateral resolution in bulk crystals of only 17 nm was obtained.
Electromechanical hysteresis loop formation in piezoresponse force microscopy of thin ferroelectric films is studied with special emphasis on the effects of tip size and film thickness, as well as dependence on the tip voltage frequency. Here, we use a combination of Landau-Ginzburg-Devonshire (LGD) theory for the description of the local polarization reversal with decoupling approximation for the calculation of the local piezoresponse loops shape, coercive voltages and amplitude. LGD approach enables addressing both thermodynamics and kinetics of hysteresis loop formation. In contrast to the …
The image formation mechanism in Piezoresponse Force Microscopy (PFM) of capacitor structures is analyzed. We demonstrate that the spatial resolution is a bilinear function of film and top electrode thicknesses, and derive the corresponding analytical expressions. For many perovskites, the opposite contributions of d31 and d33 components can result in anomalous domain wall profiles. This analysis establishes the applicability limits of PFM for polarization dynamics studies in capacitors, and applies to other structural probes, including focused X-ray studies of capacitor structures.
The presence of electrostatic forces and associated artifacts complicates the interpretation of piezoresponse force microscopy (PFM) and electrochemical strain microscopy (ESM). Eliminating these artifacts provides an opportunity for precisely mapping domain wall structures and dynamics, accurately quantifying local piezoelectric coupling coefficients, and reliably investigating hysteretic processes at the single nanometer scale to determine properties and mechanisms which underly important applications including computing, batteries and biology. Here we exploit the existence of an electrostat…
In piezoresponse force microscopy a lateral signal at the domain boundaries is occasionally observed. In recent years, a couple of experiments have been reported and varying explanations for the origin of this lateral signal have been proposed. Additionally, elaborated theoretical modeling for this particular issue has been carried out. Here we present experimental data obtained on different crystallographic cuts of $\rm LiNbO_3$, $\rm BaTiO_3$, and $\rm KTiOPO_4$ single crystals. We could thereby rule out some of the explanations proposed so far, introduce another possible mechanism, and quan…
Given that a ferroelectric domain is generally a three dimensional entity, the determination of its area as well as its depth is mandatory for full characterization. Piezoresponse force microscopy (PFM) is known for its ability to map the lateral dimensions of ferroelectric domains with high accuracy. However, no depth profile information has been readily available so far. Here, we have used ferroelectric domains of known depth profile to determine the dependence of the PFM response on the depth of the domain, and thus effectively the depth resolution of PFM detection.
At first sight piezoresponse force microscopy (PFM) seems an ideal technique for the determination of piezoelectric coefficients (PCs), thus making use of its ultra-high vertical resolution (<0.1 pm/V). Christman et al. \cite{Chr98} first used PFM for this purpose. Their measurements, however, yielded only reasonable results of unsatisfactory accuracy, amongst others caused by an incorrect calibration of the setup. In this contribution a reliable calibration procedure is given followed by a careful analysis of the encounted difficulties determining PCs with PFM. We point out different approach…
Piezoresponse Force Microscopy contrast in transversally isotropic material corresponding to the case of c+ - c- domains in tetragonal ferroelectrics is analyzed using Green's function theory by Felten et al. [J. Appl. Phys. 96, 563 (2004)]. A simplified expression for PFM signal as a linear combination of relevant piezoelectric constant are obtained. This analysis is extended to piezoelectric material of arbitrary symmetry with weak elastic and dielectric anisotropies. This result provides a framework for interpretation of PFM signals for systems with unknown or poorly known local elastic and…
Combination of local heating and biasing at the tip-surface junction in temperature-assisted piezoresponse force microscopy (tPFM) opens the pathway for probing local temperature induced phase transitions in ferroics, exploring the temperature dependence of polarization dynamics in ferroelectrics, and potentially discovering coupled phenomena driven by strong temperature- and electric field gradients. Here, we analyze the signal formation mechanism in tPFM and explore the interplay between thermal- and bias-induced switching in model ferroelectric materials. We further explore the contribution…
We investigated the time-dependent domain wall motion of epitaxial PbZr0.2Ti0.8O3 capacitors 100 nm-thick using modified piezoresponse force microscopy (PFM). We obtained successive domain evolution images reliably by combining the PFM with switching current measurements. We observed that domain wall speed (v) decreases with increases in domain size. We also observed that the average value of v, obtained under applied electric field (Eapp),showed creep behavior: i.e. <v> ~ exp(-E0/Eapp)^$μ$ with an exponent $μ$ of 0.9 $\pm$ 0.1 and an activation field E0 of about 700 kV/cm.
Ferroelectric domain imaging with piezoresponse force microscopy (PFM) relies on the converse piezoelectric effect: a voltage applied to the sample leads to mechanical deformations. In case of PFM one electrode is realized by the tip, therefore generating a strongly inhomogeneous electric field distribution inside the sample which reaches values up to $10^8 $V/m directly underneath the apex of the tip. Although often assumed, this high electric field does not lead to an enhancement of the piezoelectric deformation of the sample. On the contrary, internal clamping of the material reduces the ob…
The frequency-dependent amplitude and phase in piezoresponse force microscopy (PFM) measurements are shown to be a consequence of the Euler-Bernoulli (EB) dynamics of atomic force microscope (AFM) cantilever beams used to make the measurements. Changes in the cantilever mode shape as a function of changes in the boundary conditions determine the sensitivity of cantilevers to forces between the tip and the sample. Conventional PFM and AFM measurements are made with the motion of the cantilever measured at one optical beam detector (OBD) spot location. A single OBD spot location provides a limit…
Piezoresponse force microscopy (PFM) is a powerful tool for probing nanometer-scale ferroelectric and piezoelectric properties. Hysteretic switching of the phase and amplitude of the PFM response are believed to be the hallmark of ferroelectric and piezoelectric behavior respectively. However, the application of PFM is limited by the fact that similar hysteretic effects may also arise from mechanisms not related to ferroelectricity or piezoelectricity. In this paper we report our studies on regular glass slides that show ferroelectric-like signal without being ferroelectric and frequently used…
We report the development and experimental implementation of the automated experiment workflows for the identification of the best predictive channel for a phenomenon of interest in spectroscopic measurements. The approach is based on the combination of ensembled deep kernel learning for probabilistic predictions and a basic reinforcement learning policy for channel selection. It allows the identification of which of the available observational channels, sampled sequentially, are most predictive of selected behaviors, and hence have the strongest correlations. We implement this approach for mu…
The local hysteresis loop obtained by switching spectroscopy piezoresponse force microscopy (SS-PFM) is usually regarded as a typical signature of ferroelectric switching. However, such hysteresis loops were also observed in a broad variety of non-ferroelectric materials in the past several years, which casts doubts on the viewpoint that the local hysteresis loops in SS-PFM originate from ferroelectricity. Therefore, it is crucial to explore the mechanism of local hysteresis loops obtained in SS-PFM testing. Here we proposed that non-ferroelectric materials can also exhibit amplitude butterfly…
Correlated domain and crystallographic orientation mapping in uniaxial ferroelectric polycrystals by interferometric vector piezoresponse force microscopy
Piezoresponse force microscopy (PFM) has been used extensively for exploring nanoscale ferro/piezoelectric phenomena over the past two decades. The imaging mechanism of PFM is based on the detection of the electromechanical (EM) response induced by the inverse piezoelectric effect through the cantilever dynamics of an atomic force microscopy. However, several non-piezoelectric effects can induce additional contributions to the EM response, which often lead to a misinterpretation of the measured PFM response. This review aims to summarize the non-piezoelectric origins of the EM response that im…
Frequency dependent dynamic behavior in Piezoresponse Force Microscopy (PFM) implemented on a beam-deflection atomic force microscope (AFM) is analyzed using a combination of modeling and experimental measurements. The PFM signal comprises contributions from local electrostatic forces acting on the tip, distributed forces acting on the cantilever, and three components of the electromechanical response vector. These interactions result in the bending and torsion of the cantilever, detected as vertical and lateral PFM signals. The relative magnitudes of these contributions depend on geometric pa…
Piezoresponse Force Microscopy (PFM), as a powerful nanoscale characterization technique, has been extensively utilized to elucidate diverse underlying physics of ferroelectricity. However, the intensive study of conventional PFM has revealed a growing number of concerns and limitations which are largely challenging its validity and application. Herein, we developed a new advanced PFM technique, named Heterodyne Megasonic Piezoresponse Force Microscopy (HM-PFM), which uniquely uses 106 to 108 Hz high-frequency excitation and heterodyne method to measure the piezoelectric strain at nanoscale. W…
Coupling between electrical and mechanical phenomena is a near-universal characteristic of inorganic and biological systems alike, with examples ranging from ferroelectric perovskites to electromotor proteins in cellular membranes. Understanding electromechanical functionality in materials such as ferroelectric nanocrystals, thin films, relaxor ferroelectrics, and biosystems requires probing these properties on the nanometer level of individual grain, domain, or protein fibril. In the last decade, Piezoresponse Force Microscopy (PFM) was established a powerful tool for nanoscale imaging, spect…
Domain dynamics in the Piezoresponse Force Spectroscopy (PFS) experiment is studied using the combination of local hysteresis loop acquisition with simultaneous domain imaging. The analytical theory for PFS signal from domain of arbitrary cross-section is developed and used for the analysis of experimental data on Pb(Zr,Ti)O3 polycrystalline films. The results suggest formation of oblate domain at early stage of the domain nucleation and growth, consistent with efficient screening of depolarization field within the material. The fine structure of the hysteresis loop is shown to be related to t…
Piezoresponse force microscopy (PFM) has been extensively utilized as a versatile and an indispensable tool to understand and analyze nanoscale ferro- /piezoelectric properties by detecting the local electromechanical response on a sample surface. However, it has been discovered that the electromechanical response not only originates from piezoelectricity but also from other factors such as the electrostatic effect. In this study, we explore the dependence of off-field PFM hysteresis loops on the surface-potential-induced electrostatic effect in a prototypical ferroelectric thin film by applyi…
To achieve quantitative interpretation of Piezoresponse Force Microscopy (PFM), including resolution limits, tip bias- and strain-induced phenomena and spectroscopy, analytical representations for tip-induced electroelastic fields inside the material are derived for the cases of weak and strong indentation. In the weak indentation case, electrostatic field distribution is calculated using image charge model. In the strong indentation case, the solution of the coupled electroelastic problem for piezoelectric indentation is used to obtain the electric field and strain distribution in the ferroel…
Domains walls and topological defects in ferroelectric materials have emerged as a powerful new paradigm for functional electronic devices including memory and logic. Similarly, wall interactions and dynamics underpin a broad range of mesoscale phenomena ranging from giant electromechanical responses to memory effects. Exploring the functionalities of individual domain walls, their interactions, and controlled modifications of the domain structures is crucial for applications and fundamental physical studies. However, the dynamic nature of these features severely limits studies of their local …
Piezoresponse Force Microscopy (PFM) has emerged as a primary tool for imaging, domain engineering, and switching spectroscopy on ferroelectric materials. Quantitative interpretation of PFM data including measurements of the intrinsic width of the domain walls, geometric parameters of the domain below the tip in local hysteresis loop measurements, as well as interpretation of switching and coercive biases in terms of materials properties and switching mechanisms, requires reliable knowledge on electrostatic field structure produced by the tip. Using linear imaging theory, we develop a theoreti…
Piezoresponse Force Spectroscopy (PFS) has emerged as a powerful tool for probing polarization dynamics on the nanoscale. Application of a dc bias to a nanoscale probe in contact with a ferroelectric surface results in the nucleation and growth of a ferroelectric domain below the probe apex. The latter affects local electromechanical response detected by the probe. Resulting hysteresis loop contains information on local ferroelectric switching. The self-consistent analysis of the PFS data requires (a) deriving the thermodynamic parameters of domain nucleation and (b) establishing the relations…
Nanoscale polarization switching in ferroelectric materials by Piezoresponse Force Microscopy (PFM) in weak and strong indentation limits is analyzed using exact solutions for electrostatic and coupled electroelastic fields below the tip. It is proposed that the tip-induced domain switching can be mapped on the Landau theory of phase transitions with the domain size as an order parameter. For a point charge interacting with a ferroelectric surface, switching of both first and second order is possible depending on the charge-surface separation. For a realistic tip shape, the domain nucleation p…
A nanometric needle sensor mounted in an Atomic Force Microscopy allows systematic picometer-range distance measurements. This force sensing device is used in Piezoresponse Force Microscopy (PFM) as a distance sensor, by employing the cantilever spring constant as the conversion factor opening a pathway to explore the piezoelectric effect at the nanoscale. The force-distance equivalence is achieved if the force sensor does not disturb the system to study, solely. In this manuscript we report a systematic study in which different Lead Zirconate Titanate (PZT) materials, having different d33 val…
Mitigating parasitic contributions in measured piezoresponse for accurate determination of piezoelectric coefficients in Sc-alloyed-AlN thin films using piezo-response force microscopy
Piezoresponse Force Microscopy is a powerful but delicate nanoscale technique that measures the mechanical response resulting from the application of a highly localized electric field. Though mechanical response is normally due to piezoelectricity, other physical phenomena, especially electrostatic interaction, can contribute to the signal read. We address this problematic through the use of longer ultra-stiff probes providing state of the art sensitivity, with the lowest electrostatic interaction and avoiding working in high frequency regime. In order to find this solution we develop a theore…
Hafnium oxide (HfO2)-based ferroelectrics offer remarkable promise for memory and logic devices in view of their compatibility with traditional silicon CMOS technology, high switchable polarization, good endurance and thickness scalability. These factors have led to steep rise in research on this class of materials over the past number of years. At the same time, only a few reports on the direct sensing of nanoscale ferroelectric properties exist, with many questions remaining regarding the emergence of ferroelectricity in these materials. While piezoresponse force microscopy (PFM) is ideally …
Electromechanical hysteresis loop measurements in Piezoresponse Force Microscopy (Piezoresponse Force Spectroscopy) have emerged as a powerful technique for probing ferroelectric switching behavior on the nanoscale. Interpretation of PFS data requires the relationship between the domain parameters and PFM signal to be established. Here, we analyze the switching process using a modified point charge model for the electric field of the tip. The charge value and position are selected so that its electric field isopotential surface reproduces the tip with definite radius of curvature. Using linear…
Piezoresponse force-microscopy (PFM) has become the standard tool to investigate ferroelectrics on the micro- and nanoscale. However, reliability of PFM signals is often problematic and their quantification is challenging and thus not widely applied. Here, we present a study of the reproducibility of PFM signals and of the so-called PFM background signal which has been reported in literature. We find that PFM signals are generally reproducible to certain extents. The PFM signal difference between 180° domains on periodically-poled lithium niobate (PPLN) is taken as the reference signal in a la…
We present a full analysis of the contrast mechanisms for the detection of ferroelectric domains on all faces of bulk single crystals using scanning force microscopy exemplified on hexagonally poled lithium niobate. The domain contrast can be attributed to three different mechanisms: i) the thickness change of the sample due to an out-of-plane piezoelectric response (standard piezoresponse force microscopy), ii) the lateral displacement of the sample surface due to an in-plane piezoresponse, and iii) the electrostatic tip-sample interaction at the domain boundaries caused by surface charges on…
The thermodynamics and kinetics of tip-induced polarization switching in Piezoresponse Force Microscopy in the presence of surface charge defects is studied using the combination of analytical and numerical techniques. The signature of the defects in hysteresis loop fine structure and Switching Spectroscopy PFM images is identified and compared to experimental observations. An approach for the deconvolution of PFM spectroscopy measurements to extract relevant defect parameters is derived. This methodology is universal and can be extended to switching in other ferroics and in reversible electro…
Detection of dynamic surface displacements associated with local changes in material strain provides access to a number of phenomena and material properties. Contact resonance-enhanced methods of Atomic Force Microscopy (AFM) have been shown capable of detecting ~1-3 pm-level surface displacements, an approach used in techniques such as piezoresponse force microscopy, atomic force acoustic microscopy, and ultrasonic force microscopy. Here, based on an analytical model of AFM cantilever vibrations, we demonstrate a guideline to quantify surface displacements with a high accuracy by taking into …
Piezoresponse force microscopy (PFM) is a powerful tool widely used to characterize piezoelectricity and ferroelectricity at the nanoscale. However, it is necessary to distinguish microscopic mechanisms between piezoelectricity and non-piezoelectric contributions measured by PFM. In this work, we systematically investigate the first and second harmonic apparent piezoresponses of silicon wafer in both vertical and lateral modes, and we show that it exhibits apparent electromechanical response that is quadratic to the applied electric field, possibly arising from ionic electrochemical dipoles in…
This paper presents multiple-modes Scanning Probe Microscopy (SPM) studies on characterize resistance switching (RS), polarization rotation (PO) and surface potential changes in copper doped ZnO (ZnO:Cu) thin films. The bipolar RS behavior is confirmed by conductive Atomic Force Microscopy (c-AFM). The PO with almost 180° phase angle is confirmed by using the vertical and lateral Piezoresponse Force Microscopy (PFM). In addition, it elucidates that obvious polarization rotation behavior can be observed in the sample with increasing Cu concentration. Furthermore, correlation of the RS behavior …
Ferroelectrics are technologically important, with wide application in micromechanical systems, nonlinear optics, and information storage. Recent discoveries of exotic polarisation textures in these materials, which can strongly influence their properties, have brought to the forefront questions about the nature of their domain walls -- long believed to be primarily Ising, with locally null polarisation. Here, combining three complementary techniques -- second harmonic generation microscopy, piezoresponse force microscopy, and transmission electron microscopy - to cover all the relevant length…
We report the first investigation of the piezoelectric response on a nanoscale in the poled ferroelectric crystals with engineered configuration of domains. Piezoresponse force microscopy of tetragonal 0.63PMN-0.37PT relaxor-based ferroelectric crystals reviled that the d33 piezoelectric coefficient is significantly reduced within the distance of about 1 um from the uncharged engineered domain wall. This finding is essential for understanding the mechanisms of the giant piezoresponse in relaxor-based crystals and for designing new piezoelectric materials.
Electromechanical response of solids underpins image formation mechanism of several scanning probe microscopy techniques including the piezoresponse force microscopy (PFM) and electrochemical strain microscopy (ESM). While the theory of linear piezoelectric and ionic responses are well developed, the contributions of quadratic effects including electrostriction and capacitive tip-surface forces to measured signal remain poorly understood. Here we analyze the electrostrictive and capacitive contributions to the PFM and ESM signals and discuss the implications of the dielectric tip-surface gap o…
We report on low-energy electron microscopy imaging of ferroelectric domains with submicron resolution. Periodic strips of 'up' and 'down'-polarized ferroelectric domains in bismuth ferrite -a room temperature multiferroic- serve as a model system to compare low-energy electron microscopy with the established piezoresponse force microscopy. The results confirm the possibility of full-field imaging of ferroelectric domains with short acquisition times by exploiting the sensitivity of ultraslow electrons to small variations of the electric potential near surfaces in the "mirror" operation mode.
Scanning Probe Microscopy is used to study and quantify the nanoscale electric phenomena in the two classes of oxide systems, namely transport at electroactive grain boundaries and surface behavior of ferroelectric materials. Scanning Impedance Microscopy is developed to study the capacitance and local C-V characteristic of the interfaces combining the spatial resolution of traditional SPMs with the precision of conventional electrical measurements. SPM of SrTiO3 grain boundaries in conjunction with variable temperature impedance spectroscopy and I-V measurements allowed to find and theoretica…
