Hysteresis in Ferroelectric Material

Ferroelectric materials exhibit nonlinear polarization behavior such as hysteresis and saturation at large applied electric fields. Many piezoelectric materials are ferroelectric. This model analyzes a simple actuator made of a PZT piezoelectric ceramic material, which is subjected to an applied electric field.

Model Definition

In its ferroelectric phase, the material exhibits spontaneous polarization, so that it is constituted of domains with nonzero polarization even at zero applied field. The application of an electric field can rearrange the domains, resulting in a net polarization in the material. At very large electric fields, the polarization saturates, as all ferroelectric domains in the material are aligned along the direction of the applied field. Domain wall interactions can also lead to a significant hysteresis in the polarization.

The Jiles–Atherton hysteresis model for ferroelectric materials is available in COMSOL Multiphysics. It assumes that the total polarization can be represented as a sum of reversible and irreversible parts. The polarization change is computed from the incremental equation

where the reversibility is characterized by the parameter cr, and the anhysteretic polarization is found from the relation

where Ps is the saturation polarization. The effective electric field is given by

(1)

where α is a material parameter called the inter-domain coupling. The polarization shape is characterized by the Langevin function

where a is a material parameter called the domain wall density.

Finally, the change of the irreversible polarization is computed from the incremental relation

where the pinning loss is characterized by the parameter kp.

The ferroelectric actuator in this model is a simple disk with the radius of 0.5 in and thickness of 0.01 in, which is composed of PZT-5A piezoelectric ceramic material. The following parameter values have been estimated in Ref. 1 based on experimental data:

Table 1: Material properties of PZT-5A.
Material property	Value	Description
Ps	0.49 C/m2	Saturation polarization
a	4.4·105 V/m	Domain wall density
α	3.6·106 m/F	Inter-domain coupling
cr	0.18	Polarization reversibility
kp	1.9·106 V/m	Pinning loss

The upper surface of the actuator is grounded, while the lower one is subjected to an electric potential that can cyclically vary in small increments between −Vmax and +Vmax. The actuator is surrounded by air. Because of the symmetry, it is sufficient to model only the upper half of the setup.

Figure 1: Model geometry. The smaller domain represents the upper half of the ferroelectric actuator. The larger domain is used for modeling the surrounding air.

Because of the actuator aspect ratio, a relatively coarse mesh can be used everywhere except in the regions near the disk edge.

Figure 2: Model mesh.

Results and Discussion

The typical distribution of the electric potential is show in Figure 3.

Figure 3: Electric potential field.

Four full cycles have been computed for each value of Vmax. The polarization variation is studied at the point in the middle of the actuator.

The first cycle includes the initial transient; see Figure 4. The hysteresis loops become fully established after three full cycles; see Figure 5. The results agree well with those presented in Ref. 1.

Figure 4: Hysteresis loop including the initial transient for the maximum applied voltage of 800 V.

Figure 5: Hysteresis loops fully established after three cycles for different values of the maximum applied voltage.

Notes About the COMSOL Implementation

The Ferroelectric dielectric model option becomes available in any Charge Conservation feature under Electrostatics interface as soon as the material type for that feature is set to Solid.

In this example, you study the hysteresis with respect to the incremental variation of the applied electric potential using a stationary parametric study. The same hysteresis model can be also used for time-dependent studies.

Reference

1. R.C. Smith and Z. Ounaies, “A Domain Wall Model for Hysteresis in Piezoelectric Materials,” J. Int. Mat. Sys. Struct., vol. 11, no. 1, pp. 62–79, 2000.

ACDC_Module/Capacitive_Devices/ferroelectric_hysteresis

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

Define parameters for the geometry, material properties, and applied voltage.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
t	0[s]	0 s	Time parameter
H0	0.01[in]	2.54E-4 m	Actiator thickness
R0	0.5[in]	0.0127 m	Actuator radius
Vmax	1600[V]	1600 V	Maximum applied voltage
alpha	3.6e6[m/F]	3.6E6 m/F	Inter-domain coupling
a	4.4e5[V/m]	4.4E5 V/m	Domain wall density
c	0.18	0.18	Polarization reversibility
k	1.9e6[V/m]	1.9E6 V/m	Pinning loss
Ps	0.49[C/m^2]	0.49 C/m²	Saturation polarization

Definitions

Variables 1

In the window, right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
V0	Vmaxsin(2pi*t[1/s])	V	Applied voltage

This variation of the potential with respect to the parameter at one of the actuator boundaries will cause the electric field within the material to change at the frequency of 1 Hz.

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type R0.

Because of the symmetry, it is sufficient to model only the upper half of the actuator.

In the text field, type H0/2.

Rectangle 2 (r2)

Right-click and choose .

In the window for , locate the section.

In the text field, type 1.5*R0.

In the text field, type 5*H0.

Click

Electrostatics (es)

The default Charge Conservation feature will be used for modeling the air domain. Add one more such feature to model the ferroelectric material

Charge Conservation: PZT5A

In the window, under right-click and choose .

You need to set the feature material type to solid to be able to select a ferroelectric type of the constitutive model.

In the window for , type Charge Conservation: PZT5A in the text field.

Select Domain 1 only.

Locate the section. From the list, choose .

Locate the section. Select the check box.

Ground 1

In the toolbar, click

and choose .

Select Boundary 4 only.

Electric Potential 1

In the toolbar, click

and choose .

Because of the symmetry, you set the potential at the symmetry boundary to a half of that applied to the whole actuator.

Select Boundary 2 only.

In the window for , locate the section.

In the V0 text field, type V0/2.

Materials

In the toolbar, click

and choose .

Add Material

Go to the window.

In the tree, select .

Click

Materials

Air (mat1)

In the window, under click .

Select Domain 2 only.

PZT5A

In the window, right-click and choose .

In the window for , type PZT5A in the text field.

Select Domain 1 only.

Define the ferroelectric properties for the material using the parameters.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Saturation polarization	Psat	Ps	C/m²	Ferroelectric
Inter-domain coupling	alphaJAe_iso ; alphaJAeii = alphaJAe_iso, alphaJAeij = 0	alpha	m/F	Ferroelectric
Domain wall density	aJAe_iso ; aJAeii = aJAe_iso, aJAeij = 0	a	V/m	Ferroelectric
Pinning loss	kJAe_iso ; kJAeii = kJAe_iso, kJAeij = 0	k	V/m	Ferroelectric
Polarization reversibility	cJAe_iso ; cJAeii = cJAe_iso, cJAeij = 0	c	1	Ferroelectric