Ferroelectricity
Encyclopedia
Ferroelectricity is a property of certain materials which possess a spontaneous electric polarization
that can be reversed by the application of an external electric field. The term is used in analogy to ferromagnetism
, in which a material exhibits a permanent magnetic moment
. Ferromagnetism was already known when ferroelectricity was discovered in 1920 in Rochelle salt by Valasek. Thus, the prefix ferro, meaning iron, was used to describe the property despite the fact that most ferroelectric materials do not contain iron.
Most materials are polarized linearly by an external electric field; nonlinearities are insignificant. This is called dielectric polarization (see figure). Some materials, known as paraelectric
materials, demonstrate a more pronounced nonlinear polarization (see figure). The electric permittivity
, corresponding to the slope of the polarization curve, is thereby a function of the external electric field. In addition to being nonlinear, ferroelectric materials demonstrate a spontaneous (zero field) polarization (see figure). The distinguishing feature of ferroelectrics is that the direction of the spontaneous polarization can be reversed by an applied electric field, yielding a hysteresis
loop.
Typically, materials demonstrate ferroelectricity only below a certain phase transition temperature, called the Curie temperature, Tc, and are paraelectric above this temperature.
simply consists of a pair of electrodes sandwiching a layer of ferroelectric material. The permittivity of ferroelectrics is not only tunable but commonly also very high in absolute value, especially when close to the phase transition temperature. Because of this, ferroelectric capacitors are small in physical size compared to dielectric (non-tunable) capacitors of similar capacitance.
The spontaneous polarization of ferroelectric materials implies a hysteresis
effect which can be used as a memory function, and ferroelectric capacitors are indeed used to make ferroelectric RAM
for computers and RFID cards. In these applications thin films of ferroelectric materials are typically used, as this allows the field required to switch the polarization to be achieved with a moderate voltage. However, when using thin films a great deal of attention needs to be paid to the interfaces, electrodes and sample quality for devices to work reliably.
Ferroelectric materials are required by symmetry considerations to be also piezoelectric and pyroelectric. The combined properties of memory, piezoelectricity
, and pyroelectricity
make ferroelectric capacitors very useful, e.g. for sensor applications. Ferroelectric capacitors are used in medical ultrasound machines (the capacitors generate and then listen for the ultrasound ping used to image the internal organs of a body), high quality infrared cameras (the infrared image is projected onto a two dimensional array of ferroelectric capacitors capable of detecting temperature differences as small as millionths of a degree Celsius), fire sensors, sonar, vibration sensors, and even fuel injectors on diesel engines.
Another idea of recent interest is the ferroelectric tunnel junction (FTJ) in which a contact made up by nanometer-thick ferroelectric film placed between metal electrodes. The thickness of the ferroelectric layer is small enough to allow tunneling of electrons. The piezoelectric and interface effects as well as the depolarization field may lead to a giant electroresistance (GER) switching effect.
Yet another hot topic is multiferroics
, where researchers are looking for ways to couple magnetic and ferroelectric ordering within a material or heterostructure; there are several recent reviews on this topic.
. A change in the spontaneous polarization of a material in response to a change in temperature is called pyroelectricity
.
Ferroelectric phase transitions are often characterized as either displacive (such as BaTiO3) or order-disorder (such as NaNO2), though often phase transitions will demonstrate elements of both behaviors. In barium titanate
, a typical ferroelectric of the displacive type, the transition can be understood in terms of a polarization catastrophe, in which, if an ion is displaced from equilibrium slightly, the force from the local electric field
s due to the ions in the crystal increases faster than the elastic-restoring forces. This leads to an asymmetrical shift in the equilibrium ion positions and hence to a permanent dipole moment. The ionic displacement in barium titanate concerns the relative position of the titanium ion within the oxygen octahedral cage. In lead titanate
, another key ferroelectric material, although the structure is rather similar to barium titanate the driving force for ferroelectricity is more complex with interactions between the lead and oxygen ions also playing an important role. In an order-disorder ferroelectric, there is a dipole moment in each unit cell, but at high temperatures they are pointing in random directions. Upon lowering the temperature and going through the phase transition, the dipoles order, all pointing in the same direction within a domain.
An important ferroelectric material for applications is lead zirconate titanate
(PZT), which is part of the solid solution formed between ferroelectric lead titanate and anti-ferroelectric lead zirconate. Different compositions are used for different applications; for memory applications, PZT closer in composition to lead titanate is preferred, whereas piezoelectric applications make use of the diverging piezoelectric coefficients associated with the morphotropic phase boundary that is found close to the 50/50 composition.
Ferroelectric crystals often show several transition temperature
s and domain structure hysteresis, much as do ferromagnetic
crystals. The nature of the phase transition
in some ferroelectric crystals is still not well understood.
In 1974 R.B. Meyer used symmetry arguments to predict ferroelectric liquid crystals, and the prediction could immediately be verified by several observations of behavior connected to ferroelectricity in smectic liquid-crystal phases that are chiral and tilted. The technology allows the building of flat-screen monitors. Mass production began in 1994 by Canon. However, the costs were too high, and the production was shut down 1999 (or before) after big losses.
In 2010 David Field
found that prosaic films of chemicals such as nitrous oxide or propane exhibited ferroelectric properties. This new class of ferroelectric materials may have wide ranging applications in device and nano-technology and also influence the electrical nature of dust in the interstellar medium.
in terms of the order parameter, P. If a sixth order expansion is used (i.e. 8th order and higher terms truncated), the free energy is given by:
where Px, Py, and Pz are the components of the polarization vector in the x, y, and z directions respectively, and the coefficients,
must be consistent with the crystal symmetry. To investigate domain formation and other phenomena in ferroelectrics, these equations are often used in the context of a phase field model
. Typically, this involves adding a gradient term, an electrostatic term and an elastic term to the free energy. The equations are then discretized onto a grid using the finite difference method
and solved subject to the constraints of Gauss's law
and Linear elasticity
.
In all known ferroelectrics,
and 
. These coefficients may be obtained experimentally or from ab-initio simulations. For ferroelectrics with a first order phase transition, 
and 
for a second order phase transition.
The spontaneous polarization, Ps of a ferroelectric for a cubic to tetragonal phase transition may be obtained by considering the 1D expression of the free energy which is:
This free energy has the shape of a double well potential with two free energy minima at
, where Ps is the spontaneous polarization. At these two minima, the derivative of the free energy is zero, i.e.:
Since Px = 0 corresponds to a free energy maxima in the ferroelectric phase, the spontaneous polarization, Ps, is obtained from the solution of the equation:
which is:
and elimination of solutions yielding a negative square root (for either the first or second order phase transitions) gives:
If
, using the same approach as above, the spontaneous polarization may be obtained as:
The hysteresis loop (Px versus Ex) may be obtained from the free energy expansion by adding an additional electrostatic term, Ex Px, as follows:
Plotting Ex as a function of Px and reflecting the graph about the 45 degree line gives an 'S' shaped curve. The central part of the 'S' corresponds to a free energy local maximum
(since
). Elimination of this region, and connection of the top and bottom portions of the 'S' curve by vertical lines at the discontinuities gives the hysteresis loop.
Physics
Lists
Polarization density
In classical electromagnetism, polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is...
that can be reversed by the application of an external electric field. The term is used in analogy to ferromagnetism
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
, in which a material exhibits a permanent magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
. Ferromagnetism was already known when ferroelectricity was discovered in 1920 in Rochelle salt by Valasek. Thus, the prefix ferro, meaning iron, was used to describe the property despite the fact that most ferroelectric materials do not contain iron.
Polarization
Paraelectricity
Paraelectricity is the ability of many materials to become polarized under an applied electric field. Unlike ferroelectricity, this can happen even if there is no permanent electric dipole that exists in the material, and removal of the fields results in the polarization in the material returning...
materials, demonstrate a more pronounced nonlinear polarization (see figure). The electric permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
, corresponding to the slope of the polarization curve, is thereby a function of the external electric field. In addition to being nonlinear, ferroelectric materials demonstrate a spontaneous (zero field) polarization (see figure). The distinguishing feature of ferroelectrics is that the direction of the spontaneous polarization can be reversed by an applied electric field, yielding a hysteresis
Hysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...
loop.
Typically, materials demonstrate ferroelectricity only below a certain phase transition temperature, called the Curie temperature, Tc, and are paraelectric above this temperature.
Applications
The nonlinear nature of ferroelectric materials can be used to make capacitors with tunable capacitance. Typically, a ferroelectric capacitorFerroelectric capacitor
Ferroelectric capacitor is a capacitor based on a ferroelectric material. In contrast, traditional capacitors are based on dielectric materials. Ferroelectric devices are used in digital electronics as part of ferroelectric RAM, or in analog electronics as tunable capacitors .In memory...
simply consists of a pair of electrodes sandwiching a layer of ferroelectric material. The permittivity of ferroelectrics is not only tunable but commonly also very high in absolute value, especially when close to the phase transition temperature. Because of this, ferroelectric capacitors are small in physical size compared to dielectric (non-tunable) capacitors of similar capacitance.
The spontaneous polarization of ferroelectric materials implies a hysteresis
Hysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...
effect which can be used as a memory function, and ferroelectric capacitors are indeed used to make ferroelectric RAM
Ferroelectric RAM
Ferroelectric RAM is a random-access memory similar in construction to DRAM but uses a ferroelectric layer instead of a dielectric layer to achieve non-volatility. FeRAM is one of a growing number of alternative non-volatile memory technologies that offer the same functionality as Flash memory...
for computers and RFID cards. In these applications thin films of ferroelectric materials are typically used, as this allows the field required to switch the polarization to be achieved with a moderate voltage. However, when using thin films a great deal of attention needs to be paid to the interfaces, electrodes and sample quality for devices to work reliably.
Ferroelectric materials are required by symmetry considerations to be also piezoelectric and pyroelectric. The combined properties of memory, piezoelectricity
Piezoelectricity
Piezoelectricity is the charge which accumulates in certain solid materials in response to applied mechanical stress. The word piezoelectricity means electricity resulting from pressure...
, and pyroelectricity
Pyroelectricity
Pyroelectricity is the ability of certain materials to generate a temporary voltage when they are heated or cooled. The change in temperature modifies the positions of the atoms slightly within the crystal structure, such that the polarization of the material changes. This polarization change...
make ferroelectric capacitors very useful, e.g. for sensor applications. Ferroelectric capacitors are used in medical ultrasound machines (the capacitors generate and then listen for the ultrasound ping used to image the internal organs of a body), high quality infrared cameras (the infrared image is projected onto a two dimensional array of ferroelectric capacitors capable of detecting temperature differences as small as millionths of a degree Celsius), fire sensors, sonar, vibration sensors, and even fuel injectors on diesel engines.
Another idea of recent interest is the ferroelectric tunnel junction (FTJ) in which a contact made up by nanometer-thick ferroelectric film placed between metal electrodes. The thickness of the ferroelectric layer is small enough to allow tunneling of electrons. The piezoelectric and interface effects as well as the depolarization field may lead to a giant electroresistance (GER) switching effect.
Yet another hot topic is multiferroics
Multiferroics
Multiferroics have been formally defined as materials that exhibit more than one primary ferroic order parameter simultaneously . The four basic primary ferroic order parameters are ferromagnetism, ferroelectricity, ferroelasticity and ferrotoroidicity, the latter still being under debate...
, where researchers are looking for ways to couple magnetic and ferroelectric ordering within a material or heterostructure; there are several recent reviews on this topic.
Materials
The internal electric dipoles of a ferroelectric material are coupled to the material lattice so anything that changes the lattice will change the strength of the dipoles (in other words, a change in the spontaneous polarization). The change in the spontaneous polarization results in a change in the surface charge. This can cause current flow in the case of a ferroelectric capacitor even without the presence of an external voltage across the capacitor. Two stimuli that will change the lattice dimensions of a material are force and temperature. The generation of a surface charge in response to the application of an external stress to a material is called piezoelectricityPiezoelectricity
Piezoelectricity is the charge which accumulates in certain solid materials in response to applied mechanical stress. The word piezoelectricity means electricity resulting from pressure...
. A change in the spontaneous polarization of a material in response to a change in temperature is called pyroelectricity
Pyroelectricity
Pyroelectricity is the ability of certain materials to generate a temporary voltage when they are heated or cooled. The change in temperature modifies the positions of the atoms slightly within the crystal structure, such that the polarization of the material changes. This polarization change...
.
Ferroelectric phase transitions are often characterized as either displacive (such as BaTiO3) or order-disorder (such as NaNO2), though often phase transitions will demonstrate elements of both behaviors. In barium titanate
Barium titanate
Barium titanate is the inorganic compound with the chemical formula BaTiO3. Barium titanate is a white powder and transparent as larger crystals...
, a typical ferroelectric of the displacive type, the transition can be understood in terms of a polarization catastrophe, in which, if an ion is displaced from equilibrium slightly, the force from the local electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
s due to the ions in the crystal increases faster than the elastic-restoring forces. This leads to an asymmetrical shift in the equilibrium ion positions and hence to a permanent dipole moment. The ionic displacement in barium titanate concerns the relative position of the titanium ion within the oxygen octahedral cage. In lead titanate
Lead titanate
Lead titanate is an inorganic compound with the chemical formula PbTiO3. It is the lead salt of titanic acid. Lead titanate is a yellow powder that is insoluble in water....
, another key ferroelectric material, although the structure is rather similar to barium titanate the driving force for ferroelectricity is more complex with interactions between the lead and oxygen ions also playing an important role. In an order-disorder ferroelectric, there is a dipole moment in each unit cell, but at high temperatures they are pointing in random directions. Upon lowering the temperature and going through the phase transition, the dipoles order, all pointing in the same direction within a domain.
An important ferroelectric material for applications is lead zirconate titanate
Lead zirconate titanate
Lead zirconate titanate , also called PZT, is a ceramic perovskite material that shows a marked piezoelectric effect. PZT-based compounds are composed of the chemical elements lead and zirconium and the chemical compound titanate which are combined under extremely high temperatures. A filter is...
(PZT), which is part of the solid solution formed between ferroelectric lead titanate and anti-ferroelectric lead zirconate. Different compositions are used for different applications; for memory applications, PZT closer in composition to lead titanate is preferred, whereas piezoelectric applications make use of the diverging piezoelectric coefficients associated with the morphotropic phase boundary that is found close to the 50/50 composition.
Ferroelectric crystals often show several transition temperature
Transition temperature
Transition temperature is the temperature at which a material changes from one crystal state to another. There are seven crystal systems and every crystallized material exists in one of them....
s and domain structure hysteresis, much as do ferromagnetic
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
crystals. The nature of the phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
in some ferroelectric crystals is still not well understood.
In 1974 R.B. Meyer used symmetry arguments to predict ferroelectric liquid crystals, and the prediction could immediately be verified by several observations of behavior connected to ferroelectricity in smectic liquid-crystal phases that are chiral and tilted. The technology allows the building of flat-screen monitors. Mass production began in 1994 by Canon. However, the costs were too high, and the production was shut down 1999 (or before) after big losses.
In 2010 David Field
David Field (astrophysicist)
David Field is an astrophysicist and author, living in Århus, Denmark.- Education :Field studied Chemistry at Newcastle University, UK, followed by a PhD at the University of Cambridge, and was later awarded the degree of Doctor of Science by the University of Cambridge...
found that prosaic films of chemicals such as nitrous oxide or propane exhibited ferroelectric properties. This new class of ferroelectric materials may have wide ranging applications in device and nano-technology and also influence the electrical nature of dust in the interstellar medium.
Theory
Based on Ginzburg–Landau theory, the free energy of a ferroelectric material, in the absence of an electric field and applied stress may be written as a Taylor expansionTaylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
in terms of the order parameter, P. If a sixth order expansion is used (i.e. 8th order and higher terms truncated), the free energy is given by:
where Px, Py, and Pz are the components of the polarization vector in the x, y, and z directions respectively, and the coefficients,
must be consistent with the crystal symmetry. To investigate domain formation and other phenomena in ferroelectrics, these equations are often used in the context of a phase field modelPhase field models
A phase field model is a mathematical model for solving interfacial problems. It has mainly been applied to solidification dynamics, but it has also been applied to other situations such as viscous fingering, fracture dynamics, vesicle dynamics, etc....
. Typically, this involves adding a gradient term, an electrostatic term and an elastic term to the free energy. The equations are then discretized onto a grid using the finite difference method
Finite difference method
In mathematics, finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.- Derivation from Taylor's polynomial :...
and solved subject to the constraints of Gauss's law
Gauss's law
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
and Linear elasticity
Linear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
.
In all known ferroelectrics,
and . These coefficients may be obtained experimentally or from ab-initio simulations. For ferroelectrics with a first order phase transition, and for a second order phase transition.The spontaneous polarization, Ps of a ferroelectric for a cubic to tetragonal phase transition may be obtained by considering the 1D expression of the free energy which is:
This free energy has the shape of a double well potential with two free energy minima at
, where Ps is the spontaneous polarization. At these two minima, the derivative of the free energy is zero, i.e.:Since Px = 0 corresponds to a free energy maxima in the ferroelectric phase, the spontaneous polarization, Ps, is obtained from the solution of the equation:
which is:
and elimination of solutions yielding a negative square root (for either the first or second order phase transitions) gives:
If
, using the same approach as above, the spontaneous polarization may be obtained as:The hysteresis loop (Px versus Ex) may be obtained from the free energy expansion by adding an additional electrostatic term, Ex Px, as follows:
Plotting Ex as a function of Px and reflecting the graph about the 45 degree line gives an 'S' shaped curve. The central part of the 'S' corresponds to a free energy local maximum
Second derivative test
In calculus, the second derivative test is a criterion often useful for determining whether a given stationary point of a function is a local maximum or a local minimum using the value of the second derivative at the point....
(since
). Elimination of this region, and connection of the top and bottom portions of the 'S' curve by vertical lines at the discontinuities gives the hysteresis loop.See also
- :Category:Ferroelectric materials
- Ferroelectric capacitorFerroelectric capacitorFerroelectric capacitor is a capacitor based on a ferroelectric material. In contrast, traditional capacitors are based on dielectric materials. Ferroelectric devices are used in digital electronics as part of ferroelectric RAM, or in analog electronics as tunable capacitors .In memory...
- FeRAM
- Ferroelectric polymersFerroelectric polymersFerroelectric Polymersare a group of crystalline polar polymers that are also ferroelectric, meaning that they maintain a permanent electric polarization that can be reversed, or switched, in an external electric field....
- Piezoresponse Force MicroscopyPiezoresponse Force MicroscopyPiezoresponse force microscopy is a variant of atomic force microscopy that allows imaging and manipulation of ferroelectric domains...
Physics
- ParaelectricityParaelectricityParaelectricity is the ability of many materials to become polarized under an applied electric field. Unlike ferroelectricity, this can happen even if there is no permanent electric dipole that exists in the material, and removal of the fields results in the polarization in the material returning...
- PiezoelectricityPiezoelectricityPiezoelectricity is the charge which accumulates in certain solid materials in response to applied mechanical stress. The word piezoelectricity means electricity resulting from pressure...
- PyroelectricityPyroelectricityPyroelectricity is the ability of certain materials to generate a temporary voltage when they are heated or cooled. The change in temperature modifies the positions of the atoms slightly within the crystal structure, such that the polarization of the material changes. This polarization change...
- AntiferroelectricityAntiferroelectricityAntiferroelectricity is a physical property of certain materials. It is closely related to ferroelectricity; the relation between antiferroelectricity and ferroelectricity is analogous to the relation between antiferromagnetism and ferromagnetism....
- FerroelasticityFerroelasticityFerroelasticity is a phenomenon in which a material may exhibit a spontaneous strain. In ferroics, ferroelasticity is the mechanical equivalent of ferroelectricity and ferromagnetism. When a stress is applied to a ferroelastic material, a phase change will occur in the material from one phase to an...
- FlexoelectricityFlexoelectricityFlexoelectricity is a property of a dielectric material whereby it exhibits a spontaneous electrical polarization induced by a strain gradient. Flexoelectricity is not the same as Ferroelasticity....
- Condensed matter physicsCondensed matter physicsCondensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
- SpintronicsSpintronicsSpintronics , also known as magnetoelectronics, is an emerging technology that exploits both the intrinsic spin of the electron and its associated magnetic moment, in addition to its fundamental electronic charge, in solid-state devices.An additional effect occurs when a spin-polarized current is...
- CeramicCeramicA ceramic is an inorganic, nonmetallic solid prepared by the action of heat and subsequent cooling. Ceramic materials may have a crystalline or partly crystalline structure, or may be amorphous...
s - MultiferroicsMultiferroicsMultiferroics have been formally defined as materials that exhibit more than one primary ferroic order parameter simultaneously . The four basic primary ferroic order parameters are ferromagnetism, ferroelectricity, ferroelasticity and ferrotoroidicity, the latter still being under debate...
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