# Force-Electric Coupling of Nanoscale Ferroelectric Domains Based on Piezoelectric Force Microscopy (PFM)

Authors: Xusheng Wang; Chenchen Feng
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

Piezoelectric and ferroelectric materials are widely used in various types of microelectronics due to their excellent mechanical and electrical coupling characteristics. In recent years, piezoelectric force microscopy has developed into a powerful tool for analyzing nanoscale ferroelectric materials. However, quantitative analysis based on PFM is difficult to properly study it. This article studies the related issues of PFM quantitative analysis. First, the relationship between the effective piezoelectric coefficient and piezoelectric coefficient of different materials is analyzed from the PFM nanometer scale to analyze the force-electric coupling effect. The study found that the effective piezoelectric coefficient is closely related to the intrinsic electroelastic constant of the material. Secondly, the analysis of the nanoscale piezoelectric deformation of ferroelectric materials shows that under the conductive SPM probe, as the clustering with the probe increases, the in-plane displacement first increases and then decreases, and the out-of-plane displacement gradually decreases. Finally, the half-width region of the nanoscale ferroelectric response domain was analyzed by PFM. Taking the in-plane and out-of-plane domains of 180° and 90° domains as examples, the relationship between the response boundary domain and the tip radius was analyzed, and the results showed that whether the PFM can effectively solve the problem of detection and analysis depends on the half-width of the response boundary domain, and the resolution of the vertical PFM is higher than that of the lateral PFM.


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## Body

## 1. Introduction

Because PFM has the ability to be highly sensitive to the local detection of piezoelectric sensors, PFM has been widely used to study the microstructure and properties of new ferroelectric materials. By using PFM nanoscale detectors to perform image scanning on ferroelectric materials, not only can the domain structure signal be grasped more efficiently, but also special iron structure domains can be established, and the changes related to the performance of the brake can be studied. In some materials containing more iron elements, because ferroelectric domains and ferromagnetic domains are closely connected, it is important to process iron and manipulate the iron domains through PFM. However, due to the unbalanced characteristics of the electric field, the high-load electric field caused by the PFM conductive probe, and the complex and long-term electromechanical interaction between the test materials, the quantitative analysis of the PFM test is still very difficult. Nanoscale electromagnetic coupling hinders the research of ferroelectric materials and delays the development of electronic components produced by nanoscale electromagnetic coupling.Many scholars at home and abroad have studied the analysis of the electromechanical coupling of nanoscale ferroelectric domains based on piezoelectric force microscopy (PFM) and have achieved certain research results. For example, Chen et al. proposed functional materials, which refer to the continuous or gradual change of single or multiple superimposed properties of materials in a certain direction, which are used to adapt to different environments and develop distinctive functional materials. As the material parameters of functional materials gradually change, the thermal and stress mismatch in the material structure is greatly alleviated. The size and spatial distribution of all constituent materials are optimized, so that the advantages of all constituent materials can be effectively used, and the most advanced materials’ needs for multifunctional technology can be met [1]. But the complex structure of ferroelectric materials can affect the PFM technique to analyze the electromechanical coupling effect. After comparing and analyzing several separate models, Kuerban et al. finally decided to adopt a combined forecasting model. Combined forecasting is the weighted average of forecasts obtained by different forecasting methods to reach the final forecast result. The forecast result is generally better than that the prediction results of each single model are an effective way to improve the prediction accuracy of the model, and the results are more reliable [2]. Xing and other scholars combined the comprehensive load statistics of the formatting module and the overall module load of the module to carry out applied research on the power load and carried out the composite model characteristics, influencing factors and evaluation indicators of the power load indicators of the subindustry of my country’s power system. According to the time-sharing principle, a load composition model for different industries is proposed. According to the characteristics of short-term load forecasting and combined with domestic and foreign engineering experience, various solutions to the load forecasting problem are comprehensively analyzed [3]. In the 1940s, a scientific researcher discovered a piezoelectric ceramic material with excellent performance and easy to make and store. This material can change the chemical composition and polarization phenomenon in a targeted manner. According to its advantages in piezoelectric coefficient, electrical coupling coefficient, and stable performance, the application of piezoelectric ceramics has been rapidly expanded [4]. Capeli et al. conducted a tip crack fracture experiment on single crystal niobium. This experiment observes and examines the morphological characteristics of interface cracks and further measures the fracture hardness and atomic separation requirements. This experiment finally showed that although niobium is a flexible material and has many variants, the cracks at the interface of the two materials remain as sharp atomic cracks, and no extreme cracks will appear [5]. Although researchers have carried out a lot of related work in this area, quantitative analysis based on PFM will encounter many difficulties, that is, the complex structure of ferroelectric materials will affect the PFM technology to analyze the effect of force-electric coupling.This article introduces the characteristics of piezoelectric and ferroelectric materials and analyzes the relationship between the effective piezoelectric coefficient and the piezoelectric coefficient of different piezoelectric materials under the decoupling and full coupling methods based on the working principle of PFM; comparing the effective piezoelectric coefficients of different materials under decoupling and full coupling, comparing the time-induced in-plane and out-of-plane displacement responses of single-domain ferroelectrics with PFM detection. Then, take PZT-4 as an example according to the genetic algorithm optimization; the piezoelectric coefficient is inversely optimized to obtain the optimal piezoelectric coefficient; then, the single-domain and complex ferroelectric domain nanoscale piezoelectric deformation are analyzed, and finally, the BaTiO3 ceramic piezoelectric ceramic material is used as an example to analyze the half-width of the domain boundary response of the in-plane and out-of-plane domains of 180 degrees and 90 degrees varies with the radius of the tip. Through these experiments, it can be known that PFM technology is an important nanoscale ferroelectric material analysis tool in the study of ferroelectric domains.


## 2. Introduction of Piezoelectric Force Microscopy and Performance Analysis of Ferroelectric Domain Materials

### 2.1. PFM Working Principle

When a ferroelectric crystal exerts a mechanical force in a certain direction, confinement charges are formed on the corresponding crystal surface. The amount of charge is related to the amount of mechanical stress applied. This phenomenon is called the positive piezoelectric effect. On the other hand, if an external electric field is applied to the ferroelectric crystal, the ferroelectric crystal will also deform, which is called the piezoelectric inverse effect. PFM is a novel scanning probe microscope (SPM) based on the inverse piezoelectric effect of ferroelectric samples. Its working principle diagram is shown in Figure1.Figure 1
Working principle diagram of piezoelectric force microscope.As shown in Figure1, during PFM operation, an alternating voltage was applied to the SPM conductive probe to generate an electric field between the conductive probe and the bottom electrode. According to the inverse piezoelectric effect, the detection sample changes its shape under the action of an electric field, causing the material surface to displace, thereby pushing the cantilever beam to twist. The twist signal is reflected from the laser to the spectral detector and then sent to the amplifier lock. Computer-generated PFM signals include signal amplitude and phase signals. The amplitude depends on the local mechanical and electrical coupling of the sample, and the phase is related to the internal polarization direction. In the multisample power tracking detection experiment, regions with different polarization directions have different degrees of electric field, which triggers the conduction effect of the probe, so that the piezoelectric images show different response resolutions. On the other hand, the feedback signal determines the distance between the probe and the specimen, so the interaction force between the probe and the model remains constant. At this time, special specimens can be developed according to the size of the specimens [6, 7].


### 2.2. Characteristics and Applications of Piezoelectric and Ferroelectric Materials

Piezoelectric material is a special type of intermediate electrical material. Due to its piezoelectric properties, it can identify the conversion between electricity and electricity. This conversion will produce changes under external loads such as mechanical force, electric field, temperature field, and light, thereby having piezoelectric, ferroelectric, pyroelectric, and nonlinear optical characteristics. When the electric domain changes the electric structure under the action of the medium thermal field, the electric heat generated by the ferroelectric material can provide the ferroelectric material for the infrared detector; when the applied stress field changes, the electric domain structure will also change, and the ferroelectric material will produce the piezoelectric effect which makes the application of ferroelectric materials in sensors and energy maps have a variety of research types. The polarization and conversion of the ferroelectric crystal produce a typical hysteresis model with the change of the applied electric field, which has the characteristics of the reverse polarization electric field and can be applied to the ferroelectric memory. The discovery of these properties has led to the production of ferroelectric materials in various microelectronic components, such as thermal elements and electrical transducers [8].The positive piezoelectricity of piezoelectric materials is simply the ability to convert force into electricity. Specifically, it refers to the internal polarization when the external force is increased in a certain direction by the special structure of the dielectric material, which causes the material structure to deform. At this time, positive and negative charges are generated on the two opposite surfaces. The other is the inverted piezoelectric effect, in layman’s terms, it converts electricity into a force that causes deformation. When the electric field is removed, the deformation of the dielectric will return to its original shape [9]. Therefore, the polarization direction of piezoelectric materials is also a very important research part.Since the polarization phenomenon can occur in the ferroelectric material itself, and the ferroelectric material usually exists in multiple regions, each region has the same polarization direction. The regions with uniform polarization directions are called electrical domains, and the boundaries between domains are called wall domains. Generally speaking, each domain has an “end-to-end” polarity, so that there is no constrained charge at the boundary between domains, and the electrostatic energy in the boundary area is not reduced. The introduction of domains reduces the energy and electrostatic energy of the system, but introduces the energy of the wall domains, making the ferroelectric structure correspond to the minimum static energy in the system [10, 11]. Since the center of the positive and negative charges does not enter the ferroelectric material, the relative arrangement of the positive and negative charges changes under the change of the external thermal field, causing spontaneous changes in polarization. The charge filtered by the surface of the ferroelectric material cannot maintain complete free movement on the road surface and the accumulated material, thereby generating a potential difference between the surface of the ferroelectric material and the surface of the material. This phenomenon is called the thermoelectric effect. Since all four sides of the material are heated at the same time, the positive and negative center displacements of the corresponding symmetry are the same, which is different from the piezoelectric effect. It shows that pyroelectric materials must have a greater range and spontaneous polarity, so pyroelectric materials have piezoelectricity. Driven by an electric field with a strong optical frequency or a DC electric field, ferroelectric materials will produce photoelectric effects and nonlinear optical effects. The refractive index of the ferroelectric material changes with the change of the additional charge, and the secondary fiber effect is generated with the regular change of the electric field. The secondary electro-optic effect refers to the birefringence of the isotropic material under the action of an external electric field, resulting in a change in the refractive index, and the change in the refractive index is proportional to the square of the electric field intensity. The material with central symmetry has no primary electrostatic effect, and the secondary fiber effect is very common in any dielectric material. The refractive index change is not only related to the external electric field but also the refractive index of the ferroelectric material can be changed by increasing the mechanical force. High-energy light is the source point light that radiates ferroelectric materials, generating bipolar electric pulses that can be driven by electromagnetic waves of a certain frequency.


### 2.3. Point Charge and Its Electric Field

In the PFM test, the SPM conductive probe is similar to a moving electrode [12]. By adding a certain voltage to the SPM conductive probe, a nonuniform electric field is induced in the virtual electric field space. Compared with the test sample, the SPM probe is very small, and the tip size can reach the nanometer level. By comparing the different shapes and thicknesses of the SPM probe and the test sample, people can regard the electric field caused by the SPM probe as an electric field caused by a point charge or a series of mirror charges, where the equivalent charge can also be used the equivalent capacitance between the SPM probe, and the sample is judged [13, 14]. The SPM conductive probes used in the experiment have different shapes, but mainly include conical, spherical, and disc shapes.During the working process of PFM, a quantitative voltage is applied to the SPM probe, and an electric field probe is introduced. Compared with the SPM probe, the test material can be considered as a semi-infinite body. Therefore, the spatial electric field shaped by the conductive probe can be equal to the electric field induced by a point charge. According to the charge imaging method, it can be assumed that there is a mirror chargeQ above the test sample from the sample surface. It can be seen from the singularity theorem of the mirror image method that the introduced image charge is outside the required field, and its introduction does not change the boundary conditions of the required field. Therefore, we can use the electric field generated by the charge Q to balance the space electric field shaped by the SPM probe, thereby simulating the distribution of the space electric field [15].During PFM operation, a quantitative voltage is applied to the SPM probe, and an electric field probe is introduced. Compared to the SPM probe, the test material can be regarded as a semi-infinite body. Therefore, the space electric field created by the conductive probe can be equal to the electric field induced by the point charge. According to the charge imaging method, the charge imaging technique refers to the method of monitoring the calcium ion concentration in the tissue using a calcium ion indicator. In nervous system research, calcium ion imaging technology is commonly used to monitor the changes of calcium ions in neurons, indicating neuronal activity. With calcium imaging technology, the originally silent electrophysiological activity has become a vivid, colorful, and flickering spectacular image. It can be assumed that, viewed from the sample surface, there is an image chargeQ above the test sample. It can be seen from the singularity theorem of the mirror image method that the introduced image charge is outside the required field, and its introduction does not change the boundary conditions of the required field. Therefore, we can use the electric field generated by the charge Q to balance the space electric field formed by the SPM probe, thereby simulating the distribution of the space electric field [15].


### 2.4. The Electroelastic Field Method between the Conductive Tip and the Sample

(1)
Decoupling methodThe essential idea of the force electrolytic coupling method is to separate the electrostatic field and the elastic electric field and evaluate them separately. The specific ideas are as follows: first, the finite electrostatic field model is used to calculate the internal piezoelectric electric field distribution of piezoelectric materials without considering the influence of the force-electric coupling; second, the internal pressure is considered to be the pressure generated by the electric field, so elasticity and stress can be solved by the rebound relationship; finally, the Green’s function of the decoupling of the ferroelectric surface is used to obtain the piezoelectric displacement on the surface of the ferroelectric material. For the balanced distribution of ferroelectric materials detected by the SPM conductive probe, assuming that its polarization direction is perpendicular to the horizontal direction, the SPM conductive probe can be treated as a loading point for loadingQ, where h is the height from the sample surface [8]. Using the loading image method, the potential φρ,z distribution of the entire space can be obtained.
(1)φρ,x3=Q2πλ0k+1ρ2+x3+h2,x3≥0,Q4πλ01ρ2+x3+h2−k−1k+11ρ2+x3−h2,x3<0.Among them,ρ=x12+x22 and λ0 are the vacuum permittivity, k=λ11λ33 is the effective pressure permittivity, γ=λ33/λ11 is the dielectric anisotropy factor, which reflects the difference between in-plane and out-of-plane dielectric properties, and x1 represents the horizontal direction in the three-dimensional coordinate axis. In the right direction, x3 represents the vertical downward direction, and x2 is perpendicular to x1 and x3, respectively. The electric field is the gradient of the electric potential, namely,
(2)Ei=−∂φ∂xi.Assuming that the voltage applied to the needle tip isφo and the radius of curvature of the needle tip is Ro, the effective point charge model shows:
(3)Q=2πλoRoφok+1k,(4)d=Rok.The charged probe causes an electric field to form inside the piezoelectric material, and the piezoelectric material produces strain under the action of the electric field due to the inverse piezoelectric effect. This piezoelectric accessory can be seen as an internal filter, namely,(5)smnT=dmnkEk.Among them,dmnk is the piezoelectric tensor component on the three-dimensional image, and the repetition index k (k=1, 2, 3) satisfies the weighted summation. Because piezoelectric tension destroys the dynamic equilibrium state of the system, internal stress emn and stress σjl are generated:
(6)emn=smn−smnT,(7)σjl=Cjlmnsmn−smnT.Among them,Cjlmn is the stiffness sensor component of the elastic component. By comparing the displacements of piezoelectric and elastic structures and introducing half-space elastic Green’s function Gijx,x′, the surface displacement of the material can be obtained as:
(8)uix=∫0∞dx3′∫−∞∞dx2′∫−∞∞dx1′CjlmndmnkEkx′∂Gijx,x′∂xl′.For a uniform ferroelectric material and the polarization orientation are perpendicular to the surface of the material, the Fourier transform can be used to solve the surface displacement to obtain the analytical form of the surface in-plane and out-of-plane displacement:(9)u1ρ,0=Q2πλok+1x1d1g1γ+d2g2γ+d3g3γρ2+h2ρ2+h2+h,(10)u2ρ,0=Q2πλok+1x2d1g1γ+d2g2γ+d3g3γρ2+h2ρ2+h2+h,(11)u3ρ,0=Q2πλok+1d1g1γ+d2g2γ+d3g3γρ2+h2.Among them,d1, d2, and d3 are voltage coefficients, g1γ=2+γ+21+γν/1+γ2, g2γ=‐γ/1+γ2, g3γ=γ/1+γ2, and ν is Poisson’s ratio.
(2)
Fully coupled methodSince the force-electric coupling between the needle tip and the sample is axisymmetric, it can be solved by the Hankel integral transformation method [16, 17]. The Hankel integral transformation pair defined for any function Fr,z is:
(12)F~ξ,z=∫0∞Fr,zrJpξrdr,(13)Fr,z=∫0∞F~ξ,zξJpξrdξ.Among them, in the three-dimensional coordinate axis,r represents the horizontal axis to the right, z represents the vertical axis downward, Jpξr is the first-type Bessel function of order p, and ξ is the polar axis in the Fourier space.


## 2.1. PFM Working Principle

When a ferroelectric crystal exerts a mechanical force in a certain direction, confinement charges are formed on the corresponding crystal surface. The amount of charge is related to the amount of mechanical stress applied. This phenomenon is called the positive piezoelectric effect. On the other hand, if an external electric field is applied to the ferroelectric crystal, the ferroelectric crystal will also deform, which is called the piezoelectric inverse effect. PFM is a novel scanning probe microscope (SPM) based on the inverse piezoelectric effect of ferroelectric samples. Its working principle diagram is shown in Figure1.Figure 1
Working principle diagram of piezoelectric force microscope.As shown in Figure1, during PFM operation, an alternating voltage was applied to the SPM conductive probe to generate an electric field between the conductive probe and the bottom electrode. According to the inverse piezoelectric effect, the detection sample changes its shape under the action of an electric field, causing the material surface to displace, thereby pushing the cantilever beam to twist. The twist signal is reflected from the laser to the spectral detector and then sent to the amplifier lock. Computer-generated PFM signals include signal amplitude and phase signals. The amplitude depends on the local mechanical and electrical coupling of the sample, and the phase is related to the internal polarization direction. In the multisample power tracking detection experiment, regions with different polarization directions have different degrees of electric field, which triggers the conduction effect of the probe, so that the piezoelectric images show different response resolutions. On the other hand, the feedback signal determines the distance between the probe and the specimen, so the interaction force between the probe and the model remains constant. At this time, special specimens can be developed according to the size of the specimens [6, 7].


## 2.2. Characteristics and Applications of Piezoelectric and Ferroelectric Materials

Piezoelectric material is a special type of intermediate electrical material. Due to its piezoelectric properties, it can identify the conversion between electricity and electricity. This conversion will produce changes under external loads such as mechanical force, electric field, temperature field, and light, thereby having piezoelectric, ferroelectric, pyroelectric, and nonlinear optical characteristics. When the electric domain changes the electric structure under the action of the medium thermal field, the electric heat generated by the ferroelectric material can provide the ferroelectric material for the infrared detector; when the applied stress field changes, the electric domain structure will also change, and the ferroelectric material will produce the piezoelectric effect which makes the application of ferroelectric materials in sensors and energy maps have a variety of research types. The polarization and conversion of the ferroelectric crystal produce a typical hysteresis model with the change of the applied electric field, which has the characteristics of the reverse polarization electric field and can be applied to the ferroelectric memory. The discovery of these properties has led to the production of ferroelectric materials in various microelectronic components, such as thermal elements and electrical transducers [8].The positive piezoelectricity of piezoelectric materials is simply the ability to convert force into electricity. Specifically, it refers to the internal polarization when the external force is increased in a certain direction by the special structure of the dielectric material, which causes the material structure to deform. At this time, positive and negative charges are generated on the two opposite surfaces. The other is the inverted piezoelectric effect, in layman’s terms, it converts electricity into a force that causes deformation. When the electric field is removed, the deformation of the dielectric will return to its original shape [9]. Therefore, the polarization direction of piezoelectric materials is also a very important research part.Since the polarization phenomenon can occur in the ferroelectric material itself, and the ferroelectric material usually exists in multiple regions, each region has the same polarization direction. The regions with uniform polarization directions are called electrical domains, and the boundaries between domains are called wall domains. Generally speaking, each domain has an “end-to-end” polarity, so that there is no constrained charge at the boundary between domains, and the electrostatic energy in the boundary area is not reduced. The introduction of domains reduces the energy and electrostatic energy of the system, but introduces the energy of the wall domains, making the ferroelectric structure correspond to the minimum static energy in the system [10, 11]. Since the center of the positive and negative charges does not enter the ferroelectric material, the relative arrangement of the positive and negative charges changes under the change of the external thermal field, causing spontaneous changes in polarization. The charge filtered by the surface of the ferroelectric material cannot maintain complete free movement on the road surface and the accumulated material, thereby generating a potential difference between the surface of the ferroelectric material and the surface of the material. This phenomenon is called the thermoelectric effect. Since all four sides of the material are heated at the same time, the positive and negative center displacements of the corresponding symmetry are the same, which is different from the piezoelectric effect. It shows that pyroelectric materials must have a greater range and spontaneous polarity, so pyroelectric materials have piezoelectricity. Driven by an electric field with a strong optical frequency or a DC electric field, ferroelectric materials will produce photoelectric effects and nonlinear optical effects. The refractive index of the ferroelectric material changes with the change of the additional charge, and the secondary fiber effect is generated with the regular change of the electric field. The secondary electro-optic effect refers to the birefringence of the isotropic material under the action of an external electric field, resulting in a change in the refractive index, and the change in the refractive index is proportional to the square of the electric field intensity. The material with central symmetry has no primary electrostatic effect, and the secondary fiber effect is very common in any dielectric material. The refractive index change is not only related to the external electric field but also the refractive index of the ferroelectric material can be changed by increasing the mechanical force. High-energy light is the source point light that radiates ferroelectric materials, generating bipolar electric pulses that can be driven by electromagnetic waves of a certain frequency.


## 2.3. Point Charge and Its Electric Field

In the PFM test, the SPM conductive probe is similar to a moving electrode [12]. By adding a certain voltage to the SPM conductive probe, a nonuniform electric field is induced in the virtual electric field space. Compared with the test sample, the SPM probe is very small, and the tip size can reach the nanometer level. By comparing the different shapes and thicknesses of the SPM probe and the test sample, people can regard the electric field caused by the SPM probe as an electric field caused by a point charge or a series of mirror charges, where the equivalent charge can also be used the equivalent capacitance between the SPM probe, and the sample is judged [13, 14]. The SPM conductive probes used in the experiment have different shapes, but mainly include conical, spherical, and disc shapes.During the working process of PFM, a quantitative voltage is applied to the SPM probe, and an electric field probe is introduced. Compared with the SPM probe, the test material can be considered as a semi-infinite body. Therefore, the spatial electric field shaped by the conductive probe can be equal to the electric field induced by a point charge. According to the charge imaging method, it can be assumed that there is a mirror chargeQ above the test sample from the sample surface. It can be seen from the singularity theorem of the mirror image method that the introduced image charge is outside the required field, and its introduction does not change the boundary conditions of the required field. Therefore, we can use the electric field generated by the charge Q to balance the space electric field shaped by the SPM probe, thereby simulating the distribution of the space electric field [15].During PFM operation, a quantitative voltage is applied to the SPM probe, and an electric field probe is introduced. Compared to the SPM probe, the test material can be regarded as a semi-infinite body. Therefore, the space electric field created by the conductive probe can be equal to the electric field induced by the point charge. According to the charge imaging method, the charge imaging technique refers to the method of monitoring the calcium ion concentration in the tissue using a calcium ion indicator. In nervous system research, calcium ion imaging technology is commonly used to monitor the changes of calcium ions in neurons, indicating neuronal activity. With calcium imaging technology, the originally silent electrophysiological activity has become a vivid, colorful, and flickering spectacular image. It can be assumed that, viewed from the sample surface, there is an image chargeQ above the test sample. It can be seen from the singularity theorem of the mirror image method that the introduced image charge is outside the required field, and its introduction does not change the boundary conditions of the required field. Therefore, we can use the electric field generated by the charge Q to balance the space electric field formed by the SPM probe, thereby simulating the distribution of the space electric field [15].


## 2.4. The Electroelastic Field Method between the Conductive Tip and the Sample

(1)
Decoupling methodThe essential idea of the force electrolytic coupling method is to separate the electrostatic field and the elastic electric field and evaluate them separately. The specific ideas are as follows: first, the finite electrostatic field model is used to calculate the internal piezoelectric electric field distribution of piezoelectric materials without considering the influence of the force-electric coupling; second, the internal pressure is considered to be the pressure generated by the electric field, so elasticity and stress can be solved by the rebound relationship; finally, the Green’s function of the decoupling of the ferroelectric surface is used to obtain the piezoelectric displacement on the surface of the ferroelectric material. For the balanced distribution of ferroelectric materials detected by the SPM conductive probe, assuming that its polarization direction is perpendicular to the horizontal direction, the SPM conductive probe can be treated as a loading point for loadingQ, where h is the height from the sample surface [8]. Using the loading image method, the potential φρ,z distribution of the entire space can be obtained.
(1)φρ,x3=Q2πλ0k+1ρ2+x3+h2,x3≥0,Q4πλ01ρ2+x3+h2−k−1k+11ρ2+x3−h2,x3<0.Among them,ρ=x12+x22 and λ0 are the vacuum permittivity, k=λ11λ33 is the effective pressure permittivity, γ=λ33/λ11 is the dielectric anisotropy factor, which reflects the difference between in-plane and out-of-plane dielectric properties, and x1 represents the horizontal direction in the three-dimensional coordinate axis. In the right direction, x3 represents the vertical downward direction, and x2 is perpendicular to x1 and x3, respectively. The electric field is the gradient of the electric potential, namely,
(2)Ei=−∂φ∂xi.Assuming that the voltage applied to the needle tip isφo and the radius of curvature of the needle tip is Ro, the effective point charge model shows:
(3)Q=2πλoRoφok+1k,(4)d=Rok.The charged probe causes an electric field to form inside the piezoelectric material, and the piezoelectric material produces strain under the action of the electric field due to the inverse piezoelectric effect. This piezoelectric accessory can be seen as an internal filter, namely,(5)smnT=dmnkEk.Among them,dmnk is the piezoelectric tensor component on the three-dimensional image, and the repetition index k (k=1, 2, 3) satisfies the weighted summation. Because piezoelectric tension destroys the dynamic equilibrium state of the system, internal stress emn and stress σjl are generated:
(6)emn=smn−smnT,(7)σjl=Cjlmnsmn−smnT.Among them,Cjlmn is the stiffness sensor component of the elastic component. By comparing the displacements of piezoelectric and elastic structures and introducing half-space elastic Green’s function Gijx,x′, the surface displacement of the material can be obtained as:
(8)uix=∫0∞dx3′∫−∞∞dx2′∫−∞∞dx1′CjlmndmnkEkx′∂Gijx,x′∂xl′.For a uniform ferroelectric material and the polarization orientation are perpendicular to the surface of the material, the Fourier transform can be used to solve the surface displacement to obtain the analytical form of the surface in-plane and out-of-plane displacement:(9)u1ρ,0=Q2πλok+1x1d1g1γ+d2g2γ+d3g3γρ2+h2ρ2+h2+h,(10)u2ρ,0=Q2πλok+1x2d1g1γ+d2g2γ+d3g3γρ2+h2ρ2+h2+h,(11)u3ρ,0=Q2πλok+1d1g1γ+d2g2γ+d3g3γρ2+h2.Among them,d1, d2, and d3 are voltage coefficients, g1γ=2+γ+21+γν/1+γ2, g2γ=‐γ/1+γ2, g3γ=γ/1+γ2, and ν is Poisson’s ratio.
(2)
Fully coupled methodSince the force-electric coupling between the needle tip and the sample is axisymmetric, it can be solved by the Hankel integral transformation method [16, 17]. The Hankel integral transformation pair defined for any function Fr,z is:
(12)F~ξ,z=∫0∞Fr,zrJpξrdr,(13)Fr,z=∫0∞F~ξ,zξJpξrdξ.Among them, in the three-dimensional coordinate axis,r represents the horizontal axis to the right, z represents the vertical axis downward, Jpξr is the first-type Bessel function of order p, and ξ is the polar axis in the Fourier space.


## 3. Analysis and Research on the Electromechanical Coupling of Nanoscale Ferroelectric Domains Based on Piezoelectric Force Microscopy (PFM)

### 3.1. Research Content

There are three main research contents in this article. First, when using the PFM technology to quantitatively analyze the nanoscale piezoelectric coefficients, it is discussed whether the effective voltage coefficient under the decoupling method and the fully coupled method is compared with the actual voltage coefficient, and whether the two methods will overestimate or underestimate the conductive SPM probe [18] surface displacement caused by the needle. The second is to analyze PFM to detect nanoscale piezoelectric deformation of single-domain ferroelectric materials and complex ferroelectric domains. The third is to analyze the half-width of the domain boundary response half-width of the in-plane domain and the out-of-plane domain of the vertical angle and the lateral angle and discuss the effect of the PFM vertical resolution and the lateral resolution [19].


### 3.2. Research Methods

This article mainly uses the literature data method and the comparison method to obtain the relevant properties and research directions of ferroelectric materials by searching literature, compare the effective piezoelectric coefficients of different materials under decoupling and full coupling, and compare PFM to detect single-domain ferroelectric samples the in-plane and out-of-plane displacement response caused by the time, and the half-width of the domain boundary response of the in-plane and out-of-plane domains of the BaTiO3 ceramic piezoelectric ceramic material with 180 degrees and 90 degrees domains change with the radius of the tip [20].


## 3.1. Research Content

There are three main research contents in this article. First, when using the PFM technology to quantitatively analyze the nanoscale piezoelectric coefficients, it is discussed whether the effective voltage coefficient under the decoupling method and the fully coupled method is compared with the actual voltage coefficient, and whether the two methods will overestimate or underestimate the conductive SPM probe [18] surface displacement caused by the needle. The second is to analyze PFM to detect nanoscale piezoelectric deformation of single-domain ferroelectric materials and complex ferroelectric domains. The third is to analyze the half-width of the domain boundary response half-width of the in-plane domain and the out-of-plane domain of the vertical angle and the lateral angle and discuss the effect of the PFM vertical resolution and the lateral resolution [19].


## 3.2. Research Methods

This article mainly uses the literature data method and the comparison method to obtain the relevant properties and research directions of ferroelectric materials by searching literature, compare the effective piezoelectric coefficients of different materials under decoupling and full coupling, and compare PFM to detect single-domain ferroelectric samples the in-plane and out-of-plane displacement response caused by the time, and the half-width of the domain boundary response of the in-plane and out-of-plane domains of the BaTiO3 ceramic piezoelectric ceramic material with 180 degrees and 90 degrees domains change with the radius of the tip [20].


## 4. Based on Piezoelectric Force Microscopy (Pfm) Nanoscale Ferroelectric Domain Force-Electric Coupling Analysis

### 4.1. Quantitative Analysis of Nanoscale Piezoelectric Coefficients Based on PFM

(1)
Effective piezoelectric coefficient based on decoupling methodIn order to quantitatively analyze the relationship between the effective piezoelectric coefficient and the inherent parameters of the material, select BaTiO3, PbTiO3, PZT (Zr/Ti=48/52), PZT-8, and PMN-42%PT as examples, and the corresponding inherent material parameters are shown in Table 1. In order to clearly see the difference between the effective piezoelectric coefficient and the inherent piezoelectric coefficient, the correction factor f of the material is defined. The results show that the correction factor of PZT-8 is the smallest with a value of 0.72, while the correction factor of PMN-42%PT is the largest at 2.65. There is no obvious trend in the correction factor between different materials. In addition, for different materials, the correction factor may be greater than 1 or less than 1, which indicates that the piezoelectric coefficient of the PFM test may be overestimated or underestimated, which makes the quantitative analysis of the piezoelectric coefficient of the PFM technology more difficult.Table 1
Piezoelectric, dielectric, and elastic constants used in the calculation.
Materiald1 (pm/V)d2 (pm/V)d3 (pm/V)νFBaTiO3-772651870.350.78PbTiO3-6.562480.260.81PZT (Zr/Ti=48/52)421581070.311.24PZT-8-1003203040.300.72PMN-42%PT-27.91418.6101.20.252.65Figure2 shows the relationship between the effective piezoelectric coefficient of different materials and the piezoelectric coefficient and dielectric anisotropy factor. Fixing all other material parameters and changing only the piezoelectric coefficient d3, then the linear relationship between the effective piezoelectric coefficient d∗3 and the piezoelectric coefficient d3 can be obtained. Although there is a linear relationship between d∗3 and d3, the slopes of d∗3 and d3 are different for different materials, which means that the piezoelectric coefficients of other materials cannot be accurately obtained by calibrating with reference to the material. The effective piezoelectric coefficient of different materials and the dielectric anisotropy factor r are obviously nonlinear, especially for PMN-42%PT, the corresponding effective piezoelectric coefficient increases sharply as r increases. The obvious nonlinear characteristics of d∗3 and r in PMN-42%PT are mainly due to its large d2. It is worth noting that although the piezoelectric coefficient is fixed, the effective piezoelectric coefficient still changes with the elastic constant and the dielectric constant. This shows that the quantitative analysis of the piezoelectric coefficient for the PFM technology requires all other material parameters to be known.
(2)
Effective piezoelectric coefficient based on the full coupling methodFigure 2
The relationship between effective piezoelectric coefficient and piezoelectric coefficient and dielectric anisotropy factor of different materials.This paper selects five different piezoelectric materials, namely, PZT-4, PZT-5A, BaTiO3, (BaaCab)TiO3, and PZT (Zr/Ti=48/52), and the comparative analysis is based on the full coupling method and the decoupling method. The difference of effective piezoelectric coefficient, further analyze the relationship between effective piezoelectric coefficient and material inherent parameters. The actual material parameters corresponding to the selected materials are shown in Table 2. The elastic constant of PZT-4 material is 138.67 GPa, the piezoelectric coefficient is 12.8 C/m2, and the dielectric constant is 6.521×10−9 F/m; PZT-5A material the elastic constant is 124.00 GPa, the piezoelectric coefficient is 12.6 C/m2, and the dielectric constant is 8.127×10−9 F/m; the elastic constant of BaTiO3 material is 150.14 GPa, the piezoelectric coefficient is 11.5 C/m2, and the dielectric constant is 9.842×10−9 F/m; the elastic constant of (BaaCab)TiO3 material is 158.33 GPa, the piezoelectric coefficient is 10.8 C/m2, and the dielectric constant is 8.786×10−9 F/m; PZT (Zr/Ti=48/52); the elastic constant of the material is 124.74 GPa, the piezoelectric coefficient is 5.92 C/m2, and the dielectric constant is 4.812×10−9 F/m. Analyzing the difference of the effective piezoelectric coefficients of the two theories, the effective piezoelectric coefficients obtained by the two theories for different materials are summarized as shown in Figure 3. The effective piezoelectric coefficient of PZT-4 material in decoupling mode is 267.24 (pm/V), and the effective piezoelectric coefficient in full coupling mode is 252.13 (pm/V); PZT-5A material is effective in decoupling mode the piezoelectric coefficient is 325.46 (pm/V), the effective piezoelectric coefficient in the fully coupled mode is 305.37 (pm/V); the effective piezoelectric coefficient of the BaTiO3 material in the decoupling mode is 167.95 (pm/V), fully coupled the effective piezoelectric coefficient in the mode is 167.81 (pm/V); the effective piezoelectric coefficient of (BaaCab)TiO3 material in the decoupling mode is 133.14 (pm/V), and the effective piezoelectric coefficient in the fully coupled mode is 132.78 (pm/V); the effective piezoelectric coefficient of PZT (Zr/Ti=48/52) material in decoupling mode is 148.23 (pm/V), and the effective piezoelectric coefficient in full coupling mode is 147.36 (pm/V). It can be seen from these data that compared with the actual piezoelectric coefficient, whether it is a decoupling method or a fully coupled method, the corresponding effective piezoelectric coefficient may be overestimated or underestimated. Moreover, compared with the fully coupled method, decoupling analysis tends to underestimate the surface displacement caused by the conductive SPM probe.
(3)
Inversion of piezoelectric coefficientTable 2
Properties of piezoelectric ceramics.
MaterialElastic constant (GPa)Piezoelectric coefficient (C/m2)Dielectric constant (10-9 F/m)PZT-4138.6712.86.521PZT-5A124.0012.68.127BaTiO3150.1411.59.842(BaaCab)TiO3158.3310.88.786PZT (Zr/Ti=48/52)124.745.924.812Figure 3
Effective piezoelectric coefficients of different materials.There is an obvious nonlinear relationship between the effective piezoelectric coefficient and the intrinsic material constant. Therefore, we propose a method to inversely optimize the electroelastic constant of the material under different test conditions. It is based on this idea, focusing on the inversion of piezoelectric coefficients based on genetic algorithm (GA). The idea of genetic algorithm originates from the most suitable survival process in biological evolution, and its basic process is shown in Figure4. They first randomly generate populations and evaluate the suitability of the main population; then, select parent individuals based on suitability, such as roulette selection method, traversal sampling, and competitive bidding; secondly, transfer or reorganize the selected parent individuals; transfer or after reorganization, in order to prevent the optimization result from appearing a local minimum, the population will change with a small probability; finally, if the newly generated population reaches the optimization of the termination condition, the optimization result is output; otherwise, the second step is returned.Figure 4
Basic flow chart of genetic algorithm.The inversion optimization of piezoelectric coefficients uses PZT-4 material as an example, and the three piezoelectric coefficients are optimized through multistep optimization. The optimization process is shown in Figure5. The first step is to consider the three piezoelectric coefficients e1, e2, and e3, and perform simultaneous optimization. The three piezoelectric coefficients obtained by the first optimization are recorded as e11, e12, e13, and e1=e11 in the second optimization. Only optimize the other two piezoelectric coefficients e2,e3, and then, you can get e22, e23. In the third step of optimization, e3=e23, and e1=e11, and only e3 is optimized separately.Figure 5
Piezoelectric coefficient optimization process.The piezoelectric coefficient results obtained by the three-step optimization are shown in Table3. The intrinsic piezoelectric coefficients e01, e02, and e03 of the three steps are 11.5, -5.4, and 14.7. After the first step of optimization, the optimized voltage coefficients e1, e2, and e3 are 11.486, -5.346, and 14.822, respectively; after the second step of optimization, the optimized voltage coefficients e1, e2, and e3 are 11.487, -5.428, and 14.604, respectively; after three-step optimization, the optimized voltage coefficients e1, e2, and e3 are 11.532, -5.398, and 14.676, respectively.Table 3
Optimal piezoelectric coefficient and intrinsic piezoelectric coefficient of PZT-4.
Intrinsic piezoelectric coefficient (C/m2)Optimized piezoelectric coefficient (C/m2)e01e02e03e1e2e3111.5-5.414.711.486-5.34614.822211.5-5.414.711.487-5.42814.604311.5-5.414.711.532-5.39814.676


### 4.2. Analysis of Nanoscale Piezoelectric Deformation of Ferroelectric Materials

(1)
Nanoscale piezoelectric deformation of single-domain ferroelectric materialsThe surface displacement of a single-domain ferroelectric sample is calculated by the nanoscale piezoelectric response displacement equation. Figure6 shows the spatial distribution of the surface displacement caused by PFM detecting a single-domain ferroelectric sample. (a) shows the in-plane displacement response. (b) The figure shows the out-of-plane displacement response. From the results shown in the figure, it can be seen that the deformation displacement is highly concentrated and uneven. The out-of-plane displacement under the conductive SPM probe has a maximum value of 0.165 nm. As the distance from the conductive SPM probe is farther, the out-of-plane displacement under the conductive SPM probe gradually decreases; while the in-plane displacement is farther away from the conductive SPM. In the process of the probe, it increases first and then decreases, with a maximum value of 0.032 nm at 16 nm away from the probe. The nanoscale piezoelectric deformation of single-domain ferroelectric samples will lay the foundation for the analysis of complex ferroelectric domain structures.
(2)
Nanoscale piezoelectric deformation of complex ferroelectric domainsFigure 6
Spatial distribution of surface displacement caused by PFM detecting single-domain ferroelectric samples: (a) and (b) are the in-plane and out-of-plane displacement responses, respectively.
(a)(b)Ferroelectric materials usually contain complex microdomain structures, such as 180-degree electrical domains and 90-degree electrical structures. Due to this unbalanced microdomain structure, it is difficult to quantitatively analyze the piezoelectric response of nonferrous metal materials, which brings great difficulty to the PFM test and analysis work. Compared with a single nonferrous metal material, the polarization vector of a ferroelectric material is not uniformly distributed in the multiple fusion structure, but three nonuniform distribution dimensions are known, which determines the inherent piezoelectric coefficient of its tensor coefficient and also leads to the local intrinsic piezoelectric coefficient tensor which also presents a complex distribution. However, there is a uniformly distributed intrinsic piezoelectric coefficient tensor in a single electric domain region, and independent and uniform piezoelectric coefficient sensors are internally distributed in individual regions. This can be obtained by the spatial coordinates of the piezoelectric coefficient tensor in a single coordinate domain.


### 4.3. Domain Boundary Response Half-Width Analysis

In this experiment, the BaTiO3 ceramic material was used as an example to compare the change of the half-width of the domain boundary response of 90-degree and 180-degree in-plane domains and out-of-plane domains with the tip radius. An angle of 90 degrees represents PFM vertical detection, and an angle of 180 degrees represents PFM lateral detection. It can be seen from Figure 7 that whether it is a 180-degree domain or a 90-degree domain, whether it is an in-plane or out-of-plane domain, the domain boundary broadening increases with the increase of the tip radius. It shows that the linear curves corresponding to the 180-degree domain and the 90-degree domain are very close, and the anisotropy of the in-plane polarization has little effect on the results of the PFM measurement of the domain boundary thickness. Under the same tip radius, the domain boundary response half-width of a 180-degree in-plane domain is larger. For example, when the tip radius is 25 nm, the corresponding domain boundary response half-width of a 180-degree out-of-plane domain is 11.9 nm, but 180 degrees the domain boundary response half-width corresponding to the in-plane domain is 20 nm, which means that the lateral PFM has a worse resolution than the vertical PFM.Figure 7
Comparison of the half-width of the domain boundary response of the 90-degree and 180-degree in-plane and out-of-plane domains of BaTiO3 ceramics as a function of the tip radius.


## 4.1. Quantitative Analysis of Nanoscale Piezoelectric Coefficients Based on PFM

(1)
Effective piezoelectric coefficient based on decoupling methodIn order to quantitatively analyze the relationship between the effective piezoelectric coefficient and the inherent parameters of the material, select BaTiO3, PbTiO3, PZT (Zr/Ti=48/52), PZT-8, and PMN-42%PT as examples, and the corresponding inherent material parameters are shown in Table 1. In order to clearly see the difference between the effective piezoelectric coefficient and the inherent piezoelectric coefficient, the correction factor f of the material is defined. The results show that the correction factor of PZT-8 is the smallest with a value of 0.72, while the correction factor of PMN-42%PT is the largest at 2.65. There is no obvious trend in the correction factor between different materials. In addition, for different materials, the correction factor may be greater than 1 or less than 1, which indicates that the piezoelectric coefficient of the PFM test may be overestimated or underestimated, which makes the quantitative analysis of the piezoelectric coefficient of the PFM technology more difficult.Table 1
Piezoelectric, dielectric, and elastic constants used in the calculation.
Materiald1 (pm/V)d2 (pm/V)d3 (pm/V)νFBaTiO3-772651870.350.78PbTiO3-6.562480.260.81PZT (Zr/Ti=48/52)421581070.311.24PZT-8-1003203040.300.72PMN-42%PT-27.91418.6101.20.252.65Figure2 shows the relationship between the effective piezoelectric coefficient of different materials and the piezoelectric coefficient and dielectric anisotropy factor. Fixing all other material parameters and changing only the piezoelectric coefficient d3, then the linear relationship between the effective piezoelectric coefficient d∗3 and the piezoelectric coefficient d3 can be obtained. Although there is a linear relationship between d∗3 and d3, the slopes of d∗3 and d3 are different for different materials, which means that the piezoelectric coefficients of other materials cannot be accurately obtained by calibrating with reference to the material. The effective piezoelectric coefficient of different materials and the dielectric anisotropy factor r are obviously nonlinear, especially for PMN-42%PT, the corresponding effective piezoelectric coefficient increases sharply as r increases. The obvious nonlinear characteristics of d∗3 and r in PMN-42%PT are mainly due to its large d2. It is worth noting that although the piezoelectric coefficient is fixed, the effective piezoelectric coefficient still changes with the elastic constant and the dielectric constant. This shows that the quantitative analysis of the piezoelectric coefficient for the PFM technology requires all other material parameters to be known.
(2)
Effective piezoelectric coefficient based on the full coupling methodFigure 2
The relationship between effective piezoelectric coefficient and piezoelectric coefficient and dielectric anisotropy factor of different materials.This paper selects five different piezoelectric materials, namely, PZT-4, PZT-5A, BaTiO3, (BaaCab)TiO3, and PZT (Zr/Ti=48/52), and the comparative analysis is based on the full coupling method and the decoupling method. The difference of effective piezoelectric coefficient, further analyze the relationship between effective piezoelectric coefficient and material inherent parameters. The actual material parameters corresponding to the selected materials are shown in Table 2. The elastic constant of PZT-4 material is 138.67 GPa, the piezoelectric coefficient is 12.8 C/m2, and the dielectric constant is 6.521×10−9 F/m; PZT-5A material the elastic constant is 124.00 GPa, the piezoelectric coefficient is 12.6 C/m2, and the dielectric constant is 8.127×10−9 F/m; the elastic constant of BaTiO3 material is 150.14 GPa, the piezoelectric coefficient is 11.5 C/m2, and the dielectric constant is 9.842×10−9 F/m; the elastic constant of (BaaCab)TiO3 material is 158.33 GPa, the piezoelectric coefficient is 10.8 C/m2, and the dielectric constant is 8.786×10−9 F/m; PZT (Zr/Ti=48/52); the elastic constant of the material is 124.74 GPa, the piezoelectric coefficient is 5.92 C/m2, and the dielectric constant is 4.812×10−9 F/m. Analyzing the difference of the effective piezoelectric coefficients of the two theories, the effective piezoelectric coefficients obtained by the two theories for different materials are summarized as shown in Figure 3. The effective piezoelectric coefficient of PZT-4 material in decoupling mode is 267.24 (pm/V), and the effective piezoelectric coefficient in full coupling mode is 252.13 (pm/V); PZT-5A material is effective in decoupling mode the piezoelectric coefficient is 325.46 (pm/V), the effective piezoelectric coefficient in the fully coupled mode is 305.37 (pm/V); the effective piezoelectric coefficient of the BaTiO3 material in the decoupling mode is 167.95 (pm/V), fully coupled the effective piezoelectric coefficient in the mode is 167.81 (pm/V); the effective piezoelectric coefficient of (BaaCab)TiO3 material in the decoupling mode is 133.14 (pm/V), and the effective piezoelectric coefficient in the fully coupled mode is 132.78 (pm/V); the effective piezoelectric coefficient of PZT (Zr/Ti=48/52) material in decoupling mode is 148.23 (pm/V), and the effective piezoelectric coefficient in full coupling mode is 147.36 (pm/V). It can be seen from these data that compared with the actual piezoelectric coefficient, whether it is a decoupling method or a fully coupled method, the corresponding effective piezoelectric coefficient may be overestimated or underestimated. Moreover, compared with the fully coupled method, decoupling analysis tends to underestimate the surface displacement caused by the conductive SPM probe.
(3)
Inversion of piezoelectric coefficientTable 2
Properties of piezoelectric ceramics.
MaterialElastic constant (GPa)Piezoelectric coefficient (C/m2)Dielectric constant (10-9 F/m)PZT-4138.6712.86.521PZT-5A124.0012.68.127BaTiO3150.1411.59.842(BaaCab)TiO3158.3310.88.786PZT (Zr/Ti=48/52)124.745.924.812Figure 3
Effective piezoelectric coefficients of different materials.There is an obvious nonlinear relationship between the effective piezoelectric coefficient and the intrinsic material constant. Therefore, we propose a method to inversely optimize the electroelastic constant of the material under different test conditions. It is based on this idea, focusing on the inversion of piezoelectric coefficients based on genetic algorithm (GA). The idea of genetic algorithm originates from the most suitable survival process in biological evolution, and its basic process is shown in Figure4. They first randomly generate populations and evaluate the suitability of the main population; then, select parent individuals based on suitability, such as roulette selection method, traversal sampling, and competitive bidding; secondly, transfer or reorganize the selected parent individuals; transfer or after reorganization, in order to prevent the optimization result from appearing a local minimum, the population will change with a small probability; finally, if the newly generated population reaches the optimization of the termination condition, the optimization result is output; otherwise, the second step is returned.Figure 4
Basic flow chart of genetic algorithm.The inversion optimization of piezoelectric coefficients uses PZT-4 material as an example, and the three piezoelectric coefficients are optimized through multistep optimization. The optimization process is shown in Figure5. The first step is to consider the three piezoelectric coefficients e1, e2, and e3, and perform simultaneous optimization. The three piezoelectric coefficients obtained by the first optimization are recorded as e11, e12, e13, and e1=e11 in the second optimization. Only optimize the other two piezoelectric coefficients e2,e3, and then, you can get e22, e23. In the third step of optimization, e3=e23, and e1=e11, and only e3 is optimized separately.Figure 5
Piezoelectric coefficient optimization process.The piezoelectric coefficient results obtained by the three-step optimization are shown in Table3. The intrinsic piezoelectric coefficients e01, e02, and e03 of the three steps are 11.5, -5.4, and 14.7. After the first step of optimization, the optimized voltage coefficients e1, e2, and e3 are 11.486, -5.346, and 14.822, respectively; after the second step of optimization, the optimized voltage coefficients e1, e2, and e3 are 11.487, -5.428, and 14.604, respectively; after three-step optimization, the optimized voltage coefficients e1, e2, and e3 are 11.532, -5.398, and 14.676, respectively.Table 3
Optimal piezoelectric coefficient and intrinsic piezoelectric coefficient of PZT-4.
Intrinsic piezoelectric coefficient (C/m2)Optimized piezoelectric coefficient (C/m2)e01e02e03e1e2e3111.5-5.414.711.486-5.34614.822211.5-5.414.711.487-5.42814.604311.5-5.414.711.532-5.39814.676


## 4.2. Analysis of Nanoscale Piezoelectric Deformation of Ferroelectric Materials

(1)
Nanoscale piezoelectric deformation of single-domain ferroelectric materialsThe surface displacement of a single-domain ferroelectric sample is calculated by the nanoscale piezoelectric response displacement equation. Figure6 shows the spatial distribution of the surface displacement caused by PFM detecting a single-domain ferroelectric sample. (a) shows the in-plane displacement response. (b) The figure shows the out-of-plane displacement response. From the results shown in the figure, it can be seen that the deformation displacement is highly concentrated and uneven. The out-of-plane displacement under the conductive SPM probe has a maximum value of 0.165 nm. As the distance from the conductive SPM probe is farther, the out-of-plane displacement under the conductive SPM probe gradually decreases; while the in-plane displacement is farther away from the conductive SPM. In the process of the probe, it increases first and then decreases, with a maximum value of 0.032 nm at 16 nm away from the probe. The nanoscale piezoelectric deformation of single-domain ferroelectric samples will lay the foundation for the analysis of complex ferroelectric domain structures.
(2)
Nanoscale piezoelectric deformation of complex ferroelectric domainsFigure 6
Spatial distribution of surface displacement caused by PFM detecting single-domain ferroelectric samples: (a) and (b) are the in-plane and out-of-plane displacement responses, respectively.
(a)(b)Ferroelectric materials usually contain complex microdomain structures, such as 180-degree electrical domains and 90-degree electrical structures. Due to this unbalanced microdomain structure, it is difficult to quantitatively analyze the piezoelectric response of nonferrous metal materials, which brings great difficulty to the PFM test and analysis work. Compared with a single nonferrous metal material, the polarization vector of a ferroelectric material is not uniformly distributed in the multiple fusion structure, but three nonuniform distribution dimensions are known, which determines the inherent piezoelectric coefficient of its tensor coefficient and also leads to the local intrinsic piezoelectric coefficient tensor which also presents a complex distribution. However, there is a uniformly distributed intrinsic piezoelectric coefficient tensor in a single electric domain region, and independent and uniform piezoelectric coefficient sensors are internally distributed in individual regions. This can be obtained by the spatial coordinates of the piezoelectric coefficient tensor in a single coordinate domain.


## 4.3. Domain Boundary Response Half-Width Analysis

In this experiment, the BaTiO3 ceramic material was used as an example to compare the change of the half-width of the domain boundary response of 90-degree and 180-degree in-plane domains and out-of-plane domains with the tip radius. An angle of 90 degrees represents PFM vertical detection, and an angle of 180 degrees represents PFM lateral detection. It can be seen from Figure 7 that whether it is a 180-degree domain or a 90-degree domain, whether it is an in-plane or out-of-plane domain, the domain boundary broadening increases with the increase of the tip radius. It shows that the linear curves corresponding to the 180-degree domain and the 90-degree domain are very close, and the anisotropy of the in-plane polarization has little effect on the results of the PFM measurement of the domain boundary thickness. Under the same tip radius, the domain boundary response half-width of a 180-degree in-plane domain is larger. For example, when the tip radius is 25 nm, the corresponding domain boundary response half-width of a 180-degree out-of-plane domain is 11.9 nm, but 180 degrees the domain boundary response half-width corresponding to the in-plane domain is 20 nm, which means that the lateral PFM has a worse resolution than the vertical PFM.Figure 7
Comparison of the half-width of the domain boundary response of the 90-degree and 180-degree in-plane and out-of-plane domains of BaTiO3 ceramics as a function of the tip radius.


## 5. Conclusion

The continuous improvement of the requirements of multifunctional materials has promoted the continuous development of functional composite materials and multifield coupling materials, making more and more materials have multifield coupling characteristics. The continuous development of nanotechnology has led to the emergence of various new SPM technologies, which have prompted everyone to study the multifield coupling behavior caused by various SPM probes. The quantitative analysis of experimental data is very important for the study of nanoscale ferroelectric materials. To this end, this paper established a nanoscale piezoelectric deformation method for analyzing complex ferroelectric domain structure, explained a series of PFM test results of nanoscale deformation of complex ferroelectric domain structure, and analyzed the nanoscale pressure of ferroelectric materials for PFM quantitative test. The electrical response provides a good guide plan and provides a reference for the experimental preparation of ferroelectric materials with strong electrical coupling performance, thereby promoting the application of nanoscale ferroelectric materials.



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