Permittivity


In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ε, is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material. In electrostatics, the permittivity plays an important role in determining the capacitance of a capacitor.
In the simplest case, the electric displacement field resulting from an applied electric field is
More generally, the permittivity is a thermodynamic function of state. It can depend on the frequency, magnitude, and direction of the applied field. The SI unit for permittivity is farad per meter.
The permittivity is often represented by the relative permittivity which is the ratio of the absolute permittivity and the vacuum permittivity
This dimensionless quantity is also often and ambiguously referred to as the permittivity. Another common term encountered for both absolute and relative permittivity is the dielectric constant which has been deprecated in physics and engineering as well as in chemistry.
By definition, a perfect vacuum has a relative permittivity of exactly 1 whereas at STP, air has a relative permittivity of κair = 1.0006.
Relative permittivity is directly related to electric susceptibility by
otherwise written as

Units

The standard SI unit for permittivity is farad per meter.

Explanation

In electromagnetism, the electric displacement field represents how an electric field influences the organization of electric charges in a given medium, including charge migration and electric dipole reorientation. Its relation to permittivity in the very simple case of linear, homogeneous, isotropic materials with "instantaneous" response to changes in electric field is:
where the permittivity is a scalar. If the medium is anisotropic, the permittivity is a second rank tensor.
In general, permittivity is not a constant, as it can vary with the position in the medium, the frequency of the field applied, humidity, temperature, and other parameters. In a nonlinear medium, the permittivity can depend on the strength of the electric field. Permittivity as a function of frequency can take on real or complex values.
In SI units, permittivity is measured in farads per meter. The displacement field is measured in units of coulombs per square meter, while the electric field is measured in volts per meter. and describe the interaction between charged objects. is related to the charge densities associated with this interaction, while is related to the forces and potential differences.

Vacuum permittivity

The vacuum permittivity is the ratio in free space. It also appears in the Coulomb force constant,
Its value is
where
The constants and were defined in SI units to have exact numerical values until redefinition of SI units in 2019.

Relative permittivity

The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivity . In an anisotropic material, the relative permittivity may be a tensor, causing birefringence. The actual permittivity is then calculated by multiplying the relative permittivity by :
where is the electric susceptibility of the material.
The susceptibility is defined as the constant of proportionality relating an electric field to the induced dielectric polarization density such that
where is the electric permittivity of free space.
The susceptibility of a medium is related to its relative permittivity by
So in the case of a vacuum,
The susceptibility is also related to the polarizability of individual particles in the medium by the Clausius-Mossotti relation.
The electric displacement is related to the polarization density by
The permittivity and permeability of a medium together determine the phase velocity of electromagnetic radiation through that medium:

Practical applications

Determining capacitance

The capacitance of a capacitor is based on its design and architecture, meaning it will not change with charging and discharging. The formula for capacitance in a parallel plate capacitor is written as
where is the area of one plate, is the distance between the plates, and is the permittivity of the medium between the two plates. For a capacitor with relative permittivity, it can be said that

Gauss's law

Permittivity is connected to electric flux through Gauss's law. Gauss's law states that for a closed Gaussian surface, s
where is the net electric flux passing through the surface, is the charge enclosed in the Gaussian surface, is the electric field vector at a given point on the surface, and is a differential area vector on the Gaussian surface.
If the Gaussian surface uniformly encloses an insulated, symmetrical charge arrangement, the formula can be simplified to
where represents the angle between the electric field vector and the area vector.
If all of the electric field lines cross the surface at 90°, the formula can be further simplified to
Because the surface area of a sphere is, the electric field a distance away from a uniform, spherical charge arrangement is
where is Coulomb's constant. This formula applies to the electric field due to a point charge, outside of a conducting sphere or shell, outside of a uniformly charged insulating sphere, or between the plates of a spherical capacitor.

Dispersion and causality

In general, a material cannot polarize instantaneously in response to an applied field, and so the more general formulation as a function of time is
That is, the polarization is a convolution of the electric field at previous times with time-dependent susceptibility given by. The upper limit of this integral can be extended to infinity as well if one defines for. An instantaneous response would correspond to a Dirac delta function susceptibility.
It is convenient to take the Fourier transform with respect to time and write this relationship as a function of frequency. Because of the convolution theorem, the integral becomes a simple product,
This frequency dependence of the susceptibility leads to frequency dependence of the permittivity. The shape of the susceptibility with respect to frequency characterizes the dispersion properties of the material.
Moreover, the fact that the polarization can only depend on the electric field at previous times, a consequence of causality, imposes Kramers–Kronig constraints on the susceptibility.

Complex permittivity

As opposed to the response of a vacuum, the response of normal materials to external fields generally depends on the frequency of the field. This frequency dependence reflects the fact that a material's polarization does not change instantaneously when an electric field is applied. The response must always be causal, which can be represented by a phase difference. For this reason, permittivity is often treated as a complex function of the frequency of the applied field:
. The definition of permittivity therefore becomes
where
The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity :
At the high-frequency limit, the complex permittivity is commonly referred to as . At the plasma frequency and below, dielectrics behave as ideal metals, with electron gas behavior. The static permittivity is a good approximation for alternating fields of low frequencies, and as the frequency increases a measurable phase difference emerges between and. The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate field strength, and remain proportional, and
Since the response of materials to alternating fields is characterized by a complex permittivity, it is natural to separate its real and imaginary parts, which is done by convention in the following way:
where
The choice of sign for time-dependence,, dictates the sign convention for the imaginary part of permittivity. The signs used here correspond to those commonly used in physics, whereas for the engineering convention one should reverse all imaginary quantities.
The complex permittivity is usually a complicated function of frequency, since it is a superimposed description of dispersion phenomena occurring at multiple frequencies. The dielectric function must have poles only for frequencies with positive imaginary parts, and therefore satisfies the Kramers–Kronig relations. However, in the narrow frequency ranges that are often studied in practice, the permittivity can be approximated as frequency-independent or by model functions.
At a given frequency, the imaginary part,, leads to absorption loss if it is positive and gain if it is negative. More generally, the imaginary parts of the eigenvalues of the anisotropic dielectric tensor should be considered.
In the case of solids, the complex dielectric function is intimately connected to band structure. The primary quantity that characterizes the electronic structure of any crystalline material is the probability of photon absorption, which is directly related to the imaginary part of the optical dielectric function. The optical dielectric function is given by the fundamental expression:
In this expression, represents the product of the Brillouin zone-averaged transition probability at the energy with the joint density of states, ; is a broadening function, representing the role of scattering in smearing out the energy levels. In general, the broadening is intermediate between Lorentzian and Gaussian; for an alloy it is somewhat closer to Gaussian because of strong scattering from statistical fluctuations in the local composition on a nanometer scale.

Tensorial permittivity

According to the Drude model of magnetized plasma, a more general expression which takes into account the interaction of the carriers with an alternating electric field at millimeter and microwave frequencies in an axially magnetized semiconductor requires the expression of the permittivity as a non-diagonal tensor..
If vanishes, then the tensor is diagonal but not proportional to the identity and the medium is said to be a uniaxial medium, which has similar properties to a uniaxial crystal.

Classification of materials

Materials can be classified according to their complex-valued permittivity, upon comparison of its real and imaginary components. A perfect conductor has infinite conductivity,, while a perfect dielectric is a material that has no conductivity at all, ; this latter case, of real-valued permittivity is also associated with the name lossless media. Generally, when we consider the material to be a low-loss dielectric, whereas is associated with a good conductor; such materials with non-negligible conductivity yield a large amount of loss that inhibit the propagation of electromagnetic waves, thus are also said to be lossy media. Those materials that do not fall under either limit are considered to be general media.

Lossy medium

In the case of a lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:
where
The size of the displacement current is dependent on the frequency ω of the applied field E; there is no displacement current in a constant field.
In this formalism, the complex permittivity is defined as:
In general, the absorption of electromagnetic energy by dielectrics is covered by a few different mechanisms that influence the shape of the permittivity as a function of frequency:
The above effects often combine to cause non-linear effects within capacitors. For example, dielectric absorption refers to the inability of a capacitor that has been charged for a long time to completely discharge when briefly discharged. Although an ideal capacitor would remain at zero volts after being discharged, real capacitors will develop a small voltage, a phenomenon that is also called soakage or battery action. For some dielectrics, such as many polymer films, the resulting voltage may be less than 1–2% of the original voltage. However, it can be as much as 15–25% in the case of electrolytic capacitors or supercapacitors.

Quantum-mechanical interpretation

In terms of quantum mechanics, permittivity is explained by atomic and molecular interactions.
At low frequencies, molecules in polar dielectrics are polarized by an applied electric field, which induces periodic rotations. For example, at the microwave frequency, the microwave field causes the periodic rotation of water molecules, sufficient to break hydrogen bonds. The field does work against the bonds and the energy is absorbed by the material as heat. This is why microwave ovens work very well for materials containing water. There are two maxima of the imaginary component of water, one at the microwave frequency, and the other at far ultraviolet frequency. Both of these resonances are at higher frequencies than the operating frequency of microwave ovens.
At moderate frequencies, the energy is too high to cause rotation, yet too low to affect electrons directly, and is absorbed in the form of resonant molecular vibrations. In water, this is where the absorptive index starts to drop sharply, and the minimum of the imaginary permittivity is at the frequency of blue light.
At high frequencies, molecules cannot relax, and the energy is purely absorbed by atoms, exciting electron energy levels. Thus, these frequencies are classified as ionizing radiation.
While carrying out a complete ab initio modelling is now computationally possible, it has not been widely applied yet. Thus, a phenomenological model is accepted as being an adequate method of capturing experimental behaviors. The Debye model and the Lorentz model use a first-order and second-order lumped system parameter linear representation.

Measurement

The relative permittivity of a material can be found by a variety of static electrical measurements. The complex permittivity is evaluated over a wide range of frequencies by using different variants of dielectric spectroscopy, covering nearly 21 orders of magnitude from 10−6 to 1015 hertz. Also, by using cryostats and ovens, the dielectric properties of a medium can be characterized over an array of temperatures. In order to study systems for such diverse excitation fields, a number of measurement setups are used, each adequate for a special frequency range.
Various microwave measurement techniques are outlined in Chen et al.. Typical errors for the Hakki-Coleman method employing a puck of material between conducting planes are about 0.3%.
At infrared and optical frequencies, a common technique is ellipsometry. Dual polarisation interferometry is also used to measure the complex refractive index for very thin films at optical frequencies.