Editing Permittivity (section)

== 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

<math display="block">\mathbf{P}(t) = \varepsilon_0 \int_{-\infty}^t \chi\left(t - t'\right) \mathbf{E}\left(t'\right) \, \mathrm{d}t' ~.</math>

That is, the polarization is a [[convolution]] of the electric field at previous times with time-dependent susceptibility given by {{math|''χ''(Δ''t'')}}. The upper limit of this integral can be extended to infinity as well if one defines {{math|''χ''(Δ''t'') {{=}} 0}} for {{math|Δ''t'' < 0}}. An instantaneous response would correspond to a [[Dirac delta function]] susceptibility {{math|''χ''(Δ''t'') {{=}} ''χδ''(Δ''t'')}}.

It is convenient to take the [[continuous Fourier transform|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,

<math display="block">\ \mathbf{P}(\omega) = \varepsilon_0\ \chi(\omega)\ \mathbf{E}(\omega) ~.</math>

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 (optics)|dispersion]] properties of the material.

Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. effectively {{math|''χ''(Δ''t'') {{=}} 0}} for {{math|Δ''t'' < 0}}), a consequence of [[causality]], imposes [[Kramers–Kronig relation|Kramers–Kronig constraints]] on the susceptibility {{math|''χ''(0)}}.

=== Complex permittivity ===
[[Image:Dielectric responses.svg|thumb|right|454px|A dielectric permittivity spectrum over a wide range of frequencies. {{mvar|ε′}} and {{mvar|ε″}} denote the real and the imaginary part of the permittivity, respectively. Various processes are labeled on the image: ionic and dipolar relaxation, and atomic and electronic resonances at higher energies.<ref>{{cite web|url=http://www.psrc.usm.edu/mauritz/dilect.html |title=Dielectric Spectroscopy |access-date=2018-11-20|archive-url=https://web.archive.org/web/20060118002845/http://www.psrc.usm.edu/mauritz/dilect.html |archive-date=2006-01-18 }}</ref>]]

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'' (arising after the applied field), which can be represented by a phase difference. For this reason, permittivity is often treated as a complex function of the [[Angular frequency|(angular) frequency]] {{mvar|ω}} of the applied field:

<math display="block">\varepsilon \rightarrow \hat{\varepsilon}(\omega)</math>

(since [[complex number]]s allow specification of magnitude and phase). The definition of permittivity therefore becomes

<math display="block">D_0\ e^{-i \omega t} = \hat{\varepsilon}(\omega)\ E_0\ e^{-i \omega t}\ ,</math>
where

* {{mvar|D}}{{sub|o}} and {{mvar|E}}{{sub|o}} are the amplitudes of the displacement and electric fields, respectively,
* {{mvar|i}} is the [[imaginary unit]], {{math|''i''<sup>2</sup> {{=}} − 1 }}.

The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity {{mvar|ε}}{{sub|s}} (also {{mvar|ε}}{{sub|DC}}):

<math display="block">\varepsilon_\mathrm{s} = \lim_{\omega \rightarrow 0} \hat{\varepsilon}(\omega) ~.</math>

At the high-frequency limit (meaning optical frequencies), the complex permittivity is commonly referred to as {{mvar|ε}}{{sub|∞}} (or sometimes {{mvar|ε}}{{sub|opt}}<ref>{{cite book |last=Hofmann |first=Philip |date=2015-05-26 |title=Solid State Physics |edition=2 |publisher=Wiley-VCH |page=194 |isbn=978-352741282-2 |url=http://philiphofmann.net/solid-state-book/ |access-date=2019-05-28 |url-status=dead |archive-url=https://web.archive.org/web/20200318074603/https://philiphofmann.net/solid-state-book/ |archive-date=2020-03-18 }}</ref>). 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 {{mvar|δ}} emerges between {{math|'''D'''}} and {{math|'''E'''}}. The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate field strength ({{mvar|E}}{{sub|o}}), {{math|'''D'''}} and {{math|'''E'''}} remain proportional, and

<math display="block">\hat{\varepsilon} = \frac{D_0}{E_0} = |\varepsilon|e^{-i\delta} ~.</math>

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:

<math display="block">\hat{\varepsilon}(\omega) = \varepsilon'(\omega) - i\varepsilon''(\omega) = \left| \frac{D_0}{E_0} \right| \left( \cos \delta  - i\sin \delta  \right) ~.</math>

where
* {{mvar|ε′}} is the real part of the permittivity;
* {{mvar|ε″}} is the imaginary part of the permittivity;
* {{mvar|δ}} is the [[loss angle]].

The choice of sign for time-dependence, {{math|e<sup>−''iωt''</sup>}}, 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 {{mvar|ω}}, since it is a superimposed description of [[dispersion (optics)|dispersion]] phenomena occurring at multiple frequencies. The dielectric function {{math|''ε''(''ω'')}} must have poles only for frequencies with positive imaginary parts, and therefore satisfies the [[Kramers–Kronig relation]]s. 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, {{mvar|ε″}}, leads to absorption loss if it is positive (in the above sign convention) and gain if it is negative. More generally, the imaginary parts of the [[Eigenvalues and eigenvectors|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 {{math|''ε''(''ω'')}}. The optical dielectric function is given by the fundamental expression:<ref name=Cardona>
{{cite book
 |first1=Peter Y. |last1=Yu
 |first2=Manuel   |last2=Cardona
 |title=Fundamentals of Semiconductors: Physics and materials properties
 |year= 2001
 |page=261
 |publisher=Springer
 |location=Berlin
 |isbn=978-3-540-25470-6
 |url=https://books.google.com/books?id=W9pdJZoAeyEC&pg=PA261
}}
</ref>

<math display="block">\varepsilon(\omega) = 1 + \frac{8\pi^2 e^2}{m^2}\sum_{c,v}\int W_{c,v}(E) \bigl( \varphi (\hbar \omega - E) - \varphi( \hbar\omega + E) \bigr) \, \mathrm{d}x ~.</math>

In this expression, {{math|''W''<sub>''c'',''v''</sub>(''E'')}} represents the product of the [[Brillouin zone]]-averaged transition probability at the energy {{mvar|E}} with the joint [[density of states]],<ref name=Bausa>
{{cite book
 |first1=José García |last1=Solé
 |first2=Jose |last2=Solé
 |first3=Luisa |last3=Bausa
 |title=An introduction to the optical spectroscopy of inorganic solids
 |year= 2001
 |at=Appendix&nbsp;A1, p&nbsp;263
 |publisher=Wiley
 |isbn=978-0-470-86885-0
 |url=https://books.google.com/books?id=c6pkqC50QMgC&pg=PA263
}}
</ref><ref name=Moore>
{{cite book
 |first1=John H. |last1=Moore
 |first2=Nicholas D. |last2=Spencer
 |title=Encyclopedia of Chemical Physics and Physical Chemistry
 |year= 2001
 |page=105
 |publisher=Taylor and Francis
 |isbn=978-0-7503-0798-7
 |url=https://books.google.com/books?id=Pn2edky6uJ8C&pg=PA108
}}
</ref>
{{math|''J''<sub>''c'',''v''</sub>(''E'')}}; {{mvar|φ}} is a broadening function, representing the role of scattering in smearing out the energy levels.<ref name=Bausa2>
{{cite book
 |last1 = Solé  |first1 = José García
 |last2 = Bausá |first2 = Louisa E.
 |last3 = Jaque |first3 = Daniel
 |date = 2005-03-22
 |title = An Introduction to the Optical Spectroscopy of Inorganic Solids
 |page=10
 |isbn=978-3-540-25470-6
 |publisher=John Wiley and Sons
 |url=https://books.google.com/books?id=c6pkqC50QMgC&pg=PA10
 |access-date=2024-04-28
}}
</ref>
In general, the broadening is intermediate between [[Lorentzian function|Lorentzian]] and [[List of things named after Carl Friedrich Gauss|Gaussian]];<ref name=Haug>
{{cite book
 |first1 = Hartmut    |last1 = Haug
 |first2 = Stephan W. |last2 = Koch
 |year = 1994
 |title = Quantum Theory of the Optical and Electronic Properties of Semiconductors
 |page = 196
 |publisher=World Scientific
 |isbn=978-981-02-1864-5
 |url=https://books.google.com/books?id=Ab2WnFyGwhcC&pg=PA196
}}
</ref><ref name=Razeghi>
{{cite book
 |first = Manijeh |last = Razeghi
 |year = 2006
 |title = Fundamentals of Solid State Engineering
 |page = 383
 |publisher = Birkhauser
 |isbn = 978-0-387-28152-0
 |url = https://books.google.com/books?id=6x07E9PSzr8C&pg=PA383
}}
</ref>
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:<ref>{{cite journal |last1=Prati |first1=E. |year=2003 |title=Propagation in gyroelectromagnetic guiding systems |journal=Journal of Electromagnetic Waves and Applications |volume=17 |issue=8 |pages=1177–1196 |bibcode=2003JEWA...17.1177P |doi=10.1163/156939303322519810 |s2cid=121509049 }}</ref>

<math display="block">\mathbf{D}(\omega) = \begin{vmatrix}
    \varepsilon_1 & -i \varepsilon_2 & 0 \\
  i \varepsilon_2 &    \varepsilon_1 & 0 \\
    0             &    0             & \varepsilon_z \\
\end{vmatrix} \; \operatorname{\mathbf{E}}(\omega)</math>

If {{mvar|ε}}{{sub|2}} 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 ===
{{Classification of materials based on permittivity}}
Materials can be classified according to their complex-valued permittivity {{mvar|ε}}, upon comparison of its real {{math|''ε''′}} and imaginary {{math|''ε''″}} components (or, equivalently, [[Electric conductivity|conductivity]], {{mvar|σ}}, when accounted for in the latter). A ''[[perfect conductor]]'' has infinite conductivity, {{math|''σ'' {{=}} ∞}}, while a ''[[perfect dielectric]]'' is a material that has no conductivity at all, {{math|''σ'' {{=}} 0}}; this latter case, of real-valued permittivity (or complex-valued permittivity with zero imaginary component) is also associated with the name ''lossless media''.<ref>{{cite book|url=https://rutgers.app.box.com/s/rwzifofsu9slf8xy38f6uwhjd5gmn2q7|title=Electromagnetic Waves and Antennas|chapter=1: Maxwell's Equations|publisher=Rutgers University|first=Sophocles J.|last=Orfanidis}}</ref> Generally, when <math>\frac{\sigma}{\omega \epsilon} \ll 1</math> we consider the material to be a ''low-loss dielectric'' (although not exactly lossless), whereas <math>\frac{\sigma}{\omega \epsilon} \gg 1</math> is associated with a ''good conductor''; such materials with non-negligible conductivity yield a large amount of [[dielectric loss|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 media ===
In the case of a lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:

<math display="block">J_\text{tot}\ =\ J_\mathrm{c} + J_\mathrm{d} = \sigma\ E\ +\ i\ \omega\ \varepsilon'\ E = i\ \omega\ \hat{\varepsilon}\ E\ </math>

where
* {{mvar|σ}} is the [[electrical conductivity|conductivity]] of the medium;
* <math>\ \varepsilon'\ =\ \varepsilon_0\ \varepsilon_\mathsf{r}\ </math> is the real part of the permittivity.
* <math>\ \hat{\varepsilon}\ =\ \varepsilon' - i\ \varepsilon''\ </math> is the complex permittivity

Note that this is using the electrical engineering convention of the [[Mathematical descriptions of opacity#Complex conjugate ambiguity|complex conjugate ambiguity]]; the physics/chemistry convention involves the complex conjugate of these equations.

The size of the [[displacement current]] is dependent on the [[frequency]] {{mvar|ω}} of the applied field {{mvar|E}}; there is no displacement current in a constant field.

In this formalism, the complex permittivity is defined as:<ref>
{{cite book
 |first=John S. |last=Seybold
 |year=2005
 |title=Introduction to RF Propagation
 |publisher=John Wiley & Sons
 |page=22, eq.&nbsp;(2.6)
 |isbn=9780471743682
 |url=https://books.google.com/books?id=4LtmjGNwOPIC
}}
</ref><ref>
{{cite book
 |first=Kenneth L. |last=Kaiser
 |title=Electromagnetic Shielding
 |publisher=CRC Press
 |year=2005
 |pages=1–28, eqs.&nbsp;(1.80) and (1.81)
 |isbn=9780849363726
 |url=https://books.google.com/books?id=bDuOAQDk38gC
}}
</ref>

<math display="block">\ \hat{\varepsilon}\ =\ \varepsilon' \left(\ 1\ -\ i\ \frac{\sigma}{\ \omega \varepsilon'\ } \ \right)\ =\ \varepsilon'\ -\ i\ \frac{\ \sigma\ }{\ \omega\ } </math>

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:

* First are the [[Dielectric relaxation|relaxation]] effects associated with permanent and induced [[Dipole|molecular dipoles]]. At low frequencies the field changes slowly enough to allow dipoles to reach [[wikt:equilibrium|equilibrium]] before the field has measurably changed. For frequencies at which dipole orientations cannot follow the applied field because of the [[viscosity]] of the medium, absorption of the field's energy leads to energy dissipation. The mechanism of dipoles relaxing is called [[dielectric relaxation]] and for ideal dipoles is described by classic [[Debye relaxation]].
* Second are the [[resonance|resonance effects]], which arise from the rotations or vibrations of atoms, [[ion]]s, or [[electron]]s. These processes are observed in the neighborhood of their characteristic [[Absorption (electromagnetic radiation)|absorption frequencies]].

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 capacitor]]s or [[supercapacitor]]s.

=== Quantum-mechanical interpretation ===
In terms of [[quantum mechanics]], permittivity is explained by [[atom]]ic and [[molecule|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 bond]]s. 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 (the absorptive index) of water, one at the microwave frequency, and the other at far [[ultraviolet]] (UV) 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 (optical regime).

At high frequencies (such as UV and above), 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]]'' (that is, first-principles) 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 relaxation|Debye model]] and the [[Lorentz model]] use a first-order and second-order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).