Ferroelectricity

Ferroelectricity is a characteristic of certain materials that have a

When most materials are

Typically, materials demonstrate ferroelectricity only below a certain phase transition temperature, called the Curie temperature (T_C) and are paraelectric above this temperature: the spontaneous polarization vanishes, and the ferroelectric crystal transforms into the paraelectric state. Many ferroelectrics lose their pyroelectric properties above T_C completely, because their paraelectric phase has a centrosymmetric crystal structure.^[5]

Applications

The nonlinear nature of ferroelectric materials can be used to make capacitors with adjustable capacitance. Typically, a ferroelectric capacitor simply consists of a pair of electrodes sandwiching a layer of ferroelectric material. The permittivity of ferroelectrics is not only adjustable but commonly also very high, 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

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 is made up by nanometer-thick ferroelectric film placed between metal electrodes.^[8] 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 burgeoning application 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.^[9]

Photoferroelectric imaging is a technique to record optical information on pieces of ferroelectric material. The images are nonvolatile and selectively erasable.^[22]

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 piezoelectricity. A change in the spontaneous polarization of a material in response to a change in temperature is called pyroelectricity.

Generally, there are 230 space groups among which 32 crystalline classes can be found in crystals. There are 21 non-centrosymmetric classes, within which 20 are piezoelectric. Among the piezoelectric classes, 10 have a spontaneous electric polarization which varies with temperature; thus they are pyroelectric. Ferroelectricity is a subset of pyroelectricity, which brings spontaneous electronic polarization to the material.^[23]

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

In 2010 David Field found that prosaic films of chemicals such as nitrous oxide or propane exhibited ferroelectric properties.^[26] This new class of ferroelectric materials exhibit "spontelectric" properties, and may have wide-ranging applications in device and nano-technology and also influence the electrical nature of dust in the interstellar medium.

Other ferroelectric materials used include triglycine sulfate, polyvinylidene fluoride (PVDF) and lithium tantalate.^[27] A single atom thick ferroelectric monolayer can be created using pure bismuth. ^[28]

It should be possible to produce materials which combine both ferroelectric and metallic properties simultaneously, at room temperature.^[29] According to research published in 2018 in Nature Communications,^[30] scientists were able to produce a two-dimensional sheet of material which was both ferroelectric (had a polar crystal structure) and which conducted electricity.

Theory

An introduction to Landau theory can be found here.^[31] 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 expansion 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:

${\displaystyle {\begin{array}{ll}\Delta E=&{\frac {1}{2}}\alpha _{0}\left(T-T_{0}\right)\left(P_{x}^{2}+P_{y}^{2}+P_{z}^{2}\right)+{\frac {1}{4}}\alpha _{11}\left(P_{x}^{4}+P_{y}^{4}+P_{z}^{4}\right)\\&+{\frac {1}{2}}\alpha _{12}\left(P_{x}^{2}P_{y}^{2}+P_{y}^{2}P_{z}^{2}+P_{z}^{2}P_{x}^{2}\right)\\&+{\frac {1}{6}}\alpha _{111}\left(P_{x}^{6}+P_{y}^{6}+P_{z}^{6}\right)\\&+{\frac {1}{2}}\alpha _{112}\left[P_{x}^{4}\left(P_{y}^{2}+P_{z}^{2}\right)+P_{y}^{4}\left(P_{x}^{2}+P_{z}^{2}\right)+P_{z}^{4}\left(P_{x}^{2}+P_{y}^{2}\right)\right]\\&+{\frac {1}{2}}\alpha _{123}P_{x}^{2}P_{y}^{2}P_{z}^{2}\end{array}}}$

where P_x, P_y, and P_z are the components of the polarization vector in the x, y, and z directions respectively, and the coefficients, ${\displaystyle \alpha _{i},\alpha _{ij},\alpha _{ijk}}$ 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

.

In all known ferroelectrics, ${\displaystyle \alpha _{0}>0}$ and ${\displaystyle \alpha _{111}>0}$. These coefficients may be obtained experimentally or from ab-initio simulations. For ferroelectrics with a first order phase transition, ${\displaystyle \alpha _{11}<0}$, whereas ${\displaystyle \alpha _{11}>0}$ for a second order phase transition.

The spontaneous polarization, P_s of a ferroelectric for a cubic to tetragonal phase transition may be obtained by considering the 1D expression of the free energy which is:

${\displaystyle \Delta E={\frac {1}{2}}\alpha _{0}\left(T-T_{0}\right)P_{x}^{2}+{\frac {1}{4}}\alpha _{11}P_{x}^{4}+{\frac {1}{6}}\alpha _{111}P_{x}^{6}}$

This free energy has the shape of a double well potential with two free energy minima at ${\displaystyle P_{x}=P_{s}}$, the spontaneous polarization. We find the derivative of the free energy, and set it equal to zero in order to solve for ${\displaystyle P_{s}}$:

${\displaystyle {\frac {\partial \Delta E}{\partial P_{x}}}=\alpha _{0}\left(T-T_{0}\right)P_{x}+\alpha _{11}P_{x}^{3}+\alpha _{111}P_{x}^{5}}$

${\displaystyle 0={\frac {\partial \Delta E}{\partial P_{x}}}=P_{s}\left[\alpha _{0}\left(T-T_{0}\right)+\alpha _{11}P_{s}^{2}+\alpha _{111}P_{s}^{4}\right]}$

Since the P_s = 0 solution of this equation rather corresponds to a free energy maxima in the ferroelectric phase, the desired solutions for P_s correspond to setting the remaining factor to zero:

${\displaystyle \alpha _{0}\left(T-T_{0}\right)+\alpha _{11}P_{s}^{2}+\alpha _{111}P_{s}^{4}=0}$

whose solution is:

${\displaystyle P_{s}^{2}={\frac {1}{2\alpha _{111}}}\left[-\alpha _{11}\pm {\sqrt {\alpha _{11}^{2}+4\alpha _{0}\alpha _{111}\left(T_{0}-T\right)}}\;\right]}$

and eliminating solutions which take the square root of a negative number (for either the first or second order phase transitions) gives:

${\displaystyle P_{s}=\pm {\sqrt {{\frac {1}{2\alpha _{111}}}\left[-\alpha _{11}+{\sqrt {\alpha _{11}^{2}+4\alpha _{0}\alpha _{111}\left(T_{0}-T\right)}}\;\right]}}}$

If ${\displaystyle \alpha _{11}=0}$, the solution for the spontaneous polarization reduces to:

${\displaystyle P_{s}=\pm {\sqrt[{4}]{\frac {\alpha _{0}\left(T_{0}-T\right)}{\alpha _{111}}}}}$

The hysteresis loop (P_x versus E_x) may be obtained from the free energy expansion by including the term -E_x P_x corresponding to the energy due to an external electric field E_x interacting with the polarization P_x, as follows:

${\displaystyle \Delta E={\frac {1}{2}}\alpha _{0}\left(T-T_{0}\right)P_{x}^{2}+{\frac {1}{4}}\alpha _{11}P_{x}^{4}+{\frac {1}{6}}\alpha _{111}P_{x}^{6}-E_{x}P_{x}}$

We find the stable polarization values of P_x under the influence of the external field, now denoted as P_e, again by setting the derivative of the energy with respect to P_x to zero:

${\displaystyle {\frac {\partial \Delta E}{\partial P_{x}}}=\alpha _{0}\left(T-T_{0}\right)P_{x}+\alpha _{11}P_{x}^{3}+\alpha _{111}P_{x}^{5}-E_{x}=0}$

${\displaystyle E_{x}=\alpha _{0}\left(T-T_{0}\right)P_{e}+\alpha _{11}P_{e}^{3}+\alpha _{111}P_{e}^{5}}$

Plotting E_x (on the X axis) as a function of P_e (but on the Y axis) gives an S-shaped curve which is multi-valued in P_e for some values of E_x. The central part of the 'S' corresponds to a free energy

local maximum

(since ${\displaystyle {\frac {\partial ^{2}\Delta E}{\partial P_{x}^{2}}}<0}$ ). 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 of internal polarization due to an external electric field.

Sliding ferroelectricity

Sliding ferroelectricity is widely found but only in two-dimensional (2D) van der Waals stacked layers. The vertical electric polarization is switched by in-plane interlayer sliding.[32]

See also

References