Dielectric

In general, a material cannot polarise instantaneously in response to an applied field. The more general formulation as a function of time is

$${\displaystyle \mathbf {P} (t)=\varepsilon _{0}\int _{-\infty }^{t}\chi _{e}\left(t-t'\right)\mathbf {E} (t')\,dt'.}$$

That is, the polarisation is a convolution of the electric field at previous times with time-dependent susceptibility given by χ_e(Δt). The upper limit of this integral can be extended to infinity as well if one defines χ_e(Δt) = 0 for Δt < 0. An instantaneous response corresponds to Dirac delta function susceptibility χ_e(Δt) = χ_eδ(Δt).

It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Due to the convolution theorem, the integral becomes a simple product,

$${\displaystyle \mathbf {P} (\omega )=\varepsilon _{0}\chi _{e}(\omega )\mathbf {E} (\omega ).}$$

The susceptibility (or equivalently the permittivity) is frequency dependent. The change of susceptibility with respect to frequency characterises the dispersion properties of the material.

Moreover, the fact that the polarisation can only depend on the electric field at previous times (i.e., χ_e(Δt) = 0 for Δt < 0), a consequence of causality, imposes Kramers–Kronig constraints on the real and imaginary parts of the susceptibility χ_e(ω).

Electric field interaction with an atom under the classical dielectric model

In the classical approach to the dielectric, the material is made up of atoms. Each atom consists of a cloud of negative charge (electrons) bound to and surrounding a positive point charge at its center. In the presence of an electric field, the charge cloud is distorted, as shown in the top right of the figure.

This can be reduced to a simple dipole using the superposition principle. A dipole is characterised by its dipole moment, a vector quantity shown in the figure as the blue arrow labeled M. It is the relationship between the electric field and the dipole moment that gives rise to the behaviour of the dielectric. (Note that the dipole moment points in the same direction as the electric field in the figure. This isn't always the case, and is a major simplification, but is true for many materials.)

When the electric field is removed the atom returns to its original state. The time required to do so is called relaxation time; an exponential decay.

This is the essence of the model in physics. The behaviour of the dielectric now depends on the situation. The more complicated the situation, the richer the model must be to accurately describe the behaviour. Important questions are:

• Is the electric field constant or does it vary with time? At what rate?
• Does the response depend on the direction of the applied field (isotropy of the material)?
• Is the response the same everywhere (homogeneity of the material)?
• Do any boundaries or interfaces have to be taken into account?
• Is the response linear with respect to the field, or are there nonlinearities?

The relationship between the electric field E and the dipole moment M gives rise to the behaviour of the dielectric, which, for a given material, can be characterised by the function F defined by the equation:

$${\displaystyle \mathbf {M} =\mathbf {F} (\mathbf {E} ).}$$

When both the type of electric field and the type of material have been defined, one then chooses the simplest function F that correctly predicts the phenomena of interest. Examples of phenomena that can be so modelled include:

• Refractive index
• Group velocity dispersion
• Birefringence
• Self-focusing
• Harmonic generation

Dipolar polarisation is a polarisation that is either inherent to polar molecules (orientation polarisation), or can be induced in any molecule in which the asymmetric distortion of the nuclei is possible (distortion polarisation). Orientation polarisation results from a permanent dipole, e.g., that arising from the 104.45° angle between the asymmetric bonds between oxygen and hydrogen atoms in the water molecule, which retains polarisation in the absence of an external electric field. The assembly of these dipoles forms a macroscopic polarisation.

When an external electric field is applied, the distance between charges within each permanent dipole, which is related to chemical bonding, remains constant in orientation polarisation; however, the direction of polarisation itself rotates. This rotation occurs on a timescale that depends on the torque and surrounding local viscosity of the molecules. Because the rotation is not instantaneous, dipolar polarisations lose the response to electric fields at the highest frequencies. A molecule rotates about 1 radian per picosecond in a fluid, thus this loss occurs at about 10¹¹ Hz (in the microwave region). The delay of the response to the change of the electric field causes friction and heat.

When an external electric field is applied at infrared frequencies or less, the molecules are bent and stretched by the field and the molecular dipole moment changes. The molecular vibration frequency is roughly the inverse of the time it takes for the molecules to bend, and this distortion polarisation disappears above the infrared.

Ionic polarisation is polarisation caused by relative displacements between positive and negative ions in ionic crystals (for example, NaCl).

If a crystal or molecule consists of atoms of more than one kind, the distribution of charges around an atom in the crystal or molecule leans to positive or negative. As a result, when lattice vibrations or molecular vibrations induce relative displacements of the atoms, the centers of positive and negative charges are also displaced. The locations of these centers are affected by the symmetry of the displacements. When the centers don't correspond, polarisation arises in molecules or crystals. This polarisation is called ionic polarisation.

Ionic polarisation causes the ferroelectric effect as well as dipolar polarisation. The ferroelectric transition, which is caused by the lining up of the orientations of permanent dipoles along a particular direction, is called an order-disorder phase transition. The transition caused by ionic polarisations in crystals is called a displacive phase transition.

Ionic polarisation enables the production of energy-rich compounds in cells (the proton pump in mitochondria) and, at the plasma membrane, the establishment of the resting potential, energetically unfavourable transport of ions, and cell-to-cell communication (the Na+/K+-ATPase).

All cells in animal body tissues are electrically polarised – in other words, they maintain a voltage difference across the cell's plasma membrane, known as the membrane potential. This electrical polarisation results from a complex interplay between ion transporters and ion channels.

In neurons, the types of ion channels in the membrane usually vary across different parts of the cell, giving the dendrites, axon, and cell body different electrical properties. As a result, some parts of the membrane of a neuron may be excitable (capable of generating action potentials), whereas others are not.

Debye relaxation is the dielectric relaxation response of an ideal, noninteracting population of dipoles to an alternating external electric field. It is usually expressed in the complex permittivity ε of a medium as a function of the field's angular frequency ω:

$${\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon _{\infty }+{\frac {\Delta \varepsilon }{1+i\omega \tau }},}$$

where ε_∞ is the permittivity at the high frequency limit, Δε = ε_s − ε_∞ where ε_s is the static, low frequency permittivity, and τ is the characteristic relaxation time of the medium. Separating into the real part ${\displaystyle \varepsilon '}$ and the imaginary part ${\displaystyle \varepsilon ''}$ of the complex dielectric permittivity yields:[10]

$${\displaystyle {\begin{aligned}\varepsilon '&=\varepsilon _{\infty }+{\frac {\varepsilon _{s}-\varepsilon _{\infty }}{1+\omega ^{2}\tau ^{2}}}\\[3pt]\varepsilon ''&={\frac {(\varepsilon _{s}-\varepsilon _{\infty })\omega \tau }{1+\omega ^{2}\tau ^{2}}}\end{aligned}}}$$

Note that the above equation for ${\displaystyle {\hat {\varepsilon }}(\omega )}$is sometimes written with ${\displaystyle 1-i\omega \tau }$ in the denominator due to an ongoing sign convention ambiguity whereby many sources represent the time dependence of the complex electric field with ${\displaystyle \exp(-i\omega t)}$ whereas others use ${\displaystyle \exp(+i\omega t)}$. In the former convention, the functions ${\displaystyle \varepsilon '}$ and ${\displaystyle \varepsilon ''}$ representing real and imaginary parts are given by ${\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon '+i\varepsilon ''}$ whereas in the latter convention ${\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon '-i\varepsilon ''}$. The above equation uses the latter convention.[11]

The dielectric loss is also represented by the loss tangent:

$${\displaystyle \tan(\delta )={\frac {\varepsilon ''}{\varepsilon '}}={\frac {\left(\varepsilon _{s}-\varepsilon _{\infty }\right)\omega \tau }{\varepsilon _{s}+\varepsilon _{\infty }\omega ^{2}\tau ^{2}}}}$$

This relaxation model was introduced by and named after the physicist Peter Debye (1913).[12] It is characteristic for dynamic polarisation with only one relaxation time.

Cole–Cole equation

This equation is used when the dielectric loss peak shows symmetric broadening.

Cole–Davidson equation

This equation is used when the dielectric loss peak shows asymmetric broadening.

Havriliak–Negami relaxation

This equation considers both symmetric and asymmetric broadening.

Kohlrausch–Williams–Watts function

Fourier transform of stretched exponential function.

Curie–von Schweidler law

This shows the response of dielectrics to an applied DC field to behave according to a power law, which can be expressed as an integral over weighted exponential functions.

Djordjevic-Sarkar approximation

This is used when the dielectric loss is approximately constant for a wide range of frequencies.

Charge separation in a parallel-plate capacitor causes an internal electric field. A dielectric (orange) reduces the field and increases the capacitance.

Commercially manufactured capacitors typically use a solid dielectric material with high permittivity as the intervening medium between the stored positive and negative charges. This material is often referred to in technical contexts as the capacitor dielectric.[18]

The most obvious advantage to using such a dielectric material is that it prevents the conducting plates, on which the charges are stored, from coming into direct electrical contact. More significantly, however, a high permittivity allows a greater stored charge at a given voltage. This can be seen by treating the case of a linear dielectric with permittivity ε and thickness d between two conducting plates with uniform charge density σ_ε. In this case the charge density is given by

$${\displaystyle \sigma _{\varepsilon }=\varepsilon {\frac {V}{d}}}$$

and the capacitance per unit area by

$${\displaystyle c={\frac {\sigma _{\varepsilon }}{V}}={\frac {\varepsilon }{d}}}$$

From this, it can easily be seen that a larger ε leads to greater charge stored and thus greater capacitance.

Dielectric materials used for capacitors are also chosen such that they are resistant to ionisation. This allows the capacitor to operate at higher voltages before the insulating dielectric ionises and begins to allow undesirable current.

A dielectric resonator oscillator (DRO) is an electronic component that exhibits resonance of the polarisation response for a narrow range of frequencies, generally in the microwave band. It consists of a "puck" of ceramic that has a large dielectric constant and a low dissipation factor. Such resonators are often used to provide a frequency reference in an oscillator circuit. An unshielded dielectric resonator can be used as a dielectric resonator antenna (DRA).

From 2002 to 2004, the United States Army Research Laboratory (ARL) conducted research on thin film technology. Barium strontium titanate (BST), a ferroelectric thin film, was studied for the fabrication of radio frequency and microwave components, such as voltage-controlled oscillators, tunable filters and phase shifters.[19]

The research was part of an effort to provide the Army with highly-tunable, microwave-compatible materials for broadband electric-field tunable devices, which operate consistently in extreme temperatures.[20] This work improved tunability of bulk barium strontium titanate, which is a thin film enabler for electronics components.[21]

In a 2004 research paper, U.S. ARL researchers explored how small concentrations of acceptor dopants can dramatically modify the properties of ferroelectric materials such as BST.[22]

Researchers "doped" BST thin films with magnesium, analyzing the "structure, microstructure, surface morphology and film/substrate compositional quality" of the result. The Mg doped BST films showed "improved dielectric properties, low leakage current, and good tunability", meriting potential for use in microwave tunable devices.[19]