Editing Permittivity (section)

=== 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.