In electromagnetism, absolute permittivity is the measure of the resistance that is encountered
when forming an electric field in a medium. In other words, permittivity is a measure
of how an electric field affects, and is affected by, a dielectric medium. The permittivity
of a medium describes how much electric field is 'generated' per unit charge in that medium.
More electric flux exists in a medium with a high permittivity because of polarization
effects. Permittivity is directly related to electric susceptibility, which is a measure
of how easily a dielectric polarizes in response to an electric field. Thus, permittivity relates
to a material's ability to transmit an electric field.
In SI units, permittivity ε is measured in farads per meter; electric susceptibility
χ is dimensionless. They are related to each other through
where εr is the relative permittivity of the material, and ε0 = 8.8541878176.. × 10−12
F/m is the vacuum permittivity.
Explanation In electromagnetism, the electric displacement
field D represents how an electric field E 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 ε 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 D is measured in units of coulombs per square meter, while
the electric field E is measured in volts per meter. D and E describe the interaction
between charged objects. D is related to the charge densities associated with this interaction,
while E is related to the forces and potential differences.
Vacuum permittivity
The vacuum permittivity ε0 is the ratio D/E in free space. It also appears in the Coulomb
force constant, ke = 1/(4πε0). Its value is
where c0 is the speed of light in free space,
µ0 is the vacuum permeability. Constants c0 and μ0 are defined in SI units
to have exact numerical values, shifting responsibility of experiment to the determination of the
meter and the ampere. Relative permittivity
The linear permittivity of a homogeneous material is usually given relative to that of free
space, as a relative permittivity εr. 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 ε0:
where χ is the electric susceptibility of the material.
The susceptibility is defined as the constant of proportionality relating an electric field
E to the induced dielectric polarization density P such that
where ε0 is the electric permittivity of free space.
The susceptibility of a medium is related to its relative permittivity εr 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 D is related to
the polarization density P by
The permittivity ε and permeability µ of a medium together determine the phase velocity
v = c/n of electromagnetic radiation through that medium:
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 χ(Δt). The upper limit of this integral can be extended
to infinity as well if one defines χ(Δt) = 0 for Δt < 0. An instantaneous response would
correspond to a Dirac delta function susceptibility χ(Δt) = χ δ(Δt).
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 = 0 for Δt < 0), a consequence of causality,
imposes Kramers–Kronig constraints on the susceptibility χ(0).
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 respond instantaneously to an applied
field. 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 ω: . The definition of permittivity therefore becomes
where D0 and E0 are the amplitudes of the displacement
and electric fields, respectively, i is the imaginary unit, i2 = −1.
The response of a medium to static electric fields is described by the low-frequency limit
of permittivity, also called the static permittivity εs:
At the high-frequency limit, the complex permittivity is commonly referred to as ε∞. At the plasma
frequency and above, 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 δ emerges between D and
E. The frequency at which the phase shift becomes noticeable depends on temperature
and the details of the medium. For moderate fields strength, D and E 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 ε’ is the real part of the permittivity,
which is related to the stored energy within the medium;
ε” is the imaginary part of the permittivity, which is related to the dissipation of energy
within the medium; δ is the loss angle.
It is important to realize that the choice of sign for time-dependence, exp(-iωt), 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 ω, since it is a superimposed description of dispersion phenomena occurring
at multiple frequencies. The dielectric function ε(ω) 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 of 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 ε(ω). The optical dielectric function is given by
the fundamental expression:
In this expression, Wc,v(E) represents the product of the Brillouin zone-averaged transition
probability at the energy E with the joint density of states, Jc,v(E); φ 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 ε, upon comparison of its real ε’ and imaginary ε" components.
A perfect conductor has infinite conductivity, σ=∞, while a perfect dielectric is a material
that has no conductivity at all, σ=0; this latter case, of real-valued permittivity is
also associated with the name lossless media. Generally, when σ/(ωε’) ≪ 1 we consider
the material to be a low-loss dielectric, whereas σ/(ωε’) ≫ 1 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 lossy medium, i.e. when the
conduction current is not negligible, the total current density flowing is:
where σ is the conductivity of the medium;
ε’ is the real part of the permittivity. is the complex permittivity
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:
First, are the relaxation effects associated with permanent and induced molecular dipoles.
At low frequencies the field changes slowly enough to allow dipoles to reach 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 effects, which arise from the rotations or vibrations of atoms,
ions, or electrons. These processes are observed in the neighborhood of their characteristic
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 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 1st-order
and 2nd-order lumped system parameter linear representation.
Measurement
The dielectric constant 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 Hz. 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%. Low-frequency time domain measurements
Low-frequency frequency domain measurements Reflective coaxial methods
Transmission coaxial method Quasi-optical methods
Terahertz time-domain spectroscopy Fourier-transform methods
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. See also
Acoustic attenuation Density functional theory
Electric-field screening Green-Kubo relations
Green's function Linear response function
Rotational Brownian motion Electromagnetic permeability
References
Further reading Theory of Electric Polarization: Dielectric
Polarization, C.J.F. Böttcher, ISBN 0-444-41579-3 Dielectrics and Waves edited by von Hippel,
Arthur R., ISBN 0-89006-803-8 Dielectric Materials and Applications edited
by Arthur von Hippel, ISBN 0-89006-805-4. External links
Electromagnetism, a chapter from an online textbook
What's all this trapped charge stuff . . ., A different approach to some capacitor problems
Complex Permittivites of Metals, Plots of the complex permittivity and refractive index
for many different metals.