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Modeling of Materials in Wave Electromagnetics Problems

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0楼发表于: 2015-07-28 09:30:13

Modeling of Materials in Wave Electromagnetics Problems

Walter Frei | May 27, 2015

Whenever we are solving a wave electromagnetics problem in COMSOL Multiphysics, we build a model that is composed of domains and boundary conditions. Within the domains, we use various material models to represent a wide range of substances. However, from a mathematical point of view, all of these different materials end up being handled identically within the governing equation. Let’s take a look at these various material models and discuss when to use them.

What Equations Are We Solving?

Here, we will speak about the frequency-domain form of Maxwell’s equations in theElectromagnetic Waves, Frequency Domain interface available in the RF Module and the Wave Optics Module. The information presented here also applies to the Electromagnetic Waves, Beam Envelopes formulation in the Wave Optics Module.Under the assumption that material response is linear with field strength, we formulate Maxwell’s equations in the frequency domain, so the governing equations can be written as:

图片:latex.jpg

This equation solves for the electric field,

, at the operating (angular) frequency

(

is the speed of light in vacuum). The other inputs are the material properties

, the relative permeability;

, the relative permittivity; and

, the electrical conductivity. All of these material inputs can be positive or negative, real or complex valued numbers, and they can be scalar or tensor quantities. These material properties can vary as a function of frequency as well, though it is not always necessary to consider this variation if we are only looking at a relatively narrow frequency range.Let us now explore each of these material properties in detail.

Electrical Conductivity

The electrical conductivity quantifies how well a material conducts current — it is the inverse of the electrical resistivity. The material conductivity is measured under steadystate (DC) conditions, and we can see from the above equation that as the frequency increases, the effective resistivity of the material increases. We typically assume that the conductivity is constant with frequency, and later on we will examine differen models for handling materials with frequencydependent conductivity.Any material with nonzero conductivity will conduct current in an applied electric field and dissipate energy as a resistive loss, also called Joule heating. This will often lead to a measurable rise in temperature, which will alter the conductivity. You can enter any function or tabular data for variation of conductivity with temperature, and there is also a builtin model for linearized resistivity.Linearized Resistivity is a commonly used model for the variation of conductivity with temperature, given by:

where

is the reference resistivity,

is the reference temperature, and

is the resistivity temperature coefficient. The spatially-varyingtemperature field,

, can either be specified or computed.Conductivity is entered as a realvalued number, but it can be anisotropic, meaning that the material’s conductivity varies in different coordinatedirections. This is an appropriate approach if you have, for example, a laminated material in which you do not want to explicitly model the individual layers. You can enter a homogenized conductivity for the composite material, which would be either experimentally determined or computed from a separate analysis.Within the RF Module, there are two other options for computing a homogenized conductivity: Archie’s Law for computing effective conductivity of nonconductive porous media filled with conductive liquid and a Porous Media model for mixtures of materials.Archie’s Law is a model typically used for the modeling of soils saturated with seawater or crude oil, fluids with relatively higher conductivity compared to the soil.Porous Media refers to a model that has three different options for computing an effective conductivity for a mixture of up to five materials. First, the Volume Average, Conductivityformulation is:

where

is the volume fraction of each material. This model isappropriate if the material conductivities are similar. If the conductivities are quite different, the Volume Average, Resistivityformulation is more appropriate:

Lastly, the Power Law formulation will give a conductivity lying between the other two formulations:

These models are all only appropriate to use if the length scale over which the material properties’ change is much smaller than the wavelength.

Relative Permittivity

The relative permittivity quantifies how well a material is polarized in response to an applied electric field. It is typical to call any material with

a dielectric material, though even vacuum (

) can be called a dielectric. It is also common to use the term dielectric constant to refer to a material’s relative permittivity.A material’s relative permittivity is often given as a complex-valued number, where the negative imaginary component represents the loss in the material as the electric field changes direction over time.Any material experiencing a time-varying electric field will dissipate some of the electrical energy as heat. Known as dielectric loss,this results from the change in shape of the electron clouds around the atoms as the electric fields change. Dielectric loss is conceptually distinct from the resistive loss discussed earlier; however, from a mathematical point of view, they are actually handled identically — as a complex-valued term in the governing equation. Keep in mind that COMSOL Multiphysics follows the convention that a negative imaginary component (a positive-valued electrical conductivity) will lead to loss, while a positive complex component (a negative-valued electrical conductivity) will lead togain within the material.There are seven different material models for the relative permittivity. Let’s take a look at each of these models.Relative Permittivity is the default option for the RF Module. A real- or complex-valued scalar or tensor value can be entered. The same Porous Media models described above for the electrical conductivity can be used for the relative permittivity.Refractive Index is the default option for the Wave Optics Module. You separately enter the real and imaginary part of the refractive index, called

and

, and the relative permittivity is

.This material model assumes zero conductivity and unit relative permeability.Loss Tangent involves entering a real-valued relative permittivity,

, and a scalar loss tangent, The relative permittivity is computed via

, and the material conductivity is zero.Dielectric Loss is the option for entering the real and imaginary components of the relative permittivity

Be careful to note the sign: Entering a positive-valued real number for the imaginary component

when using this interface will lead to loss, since the multiplication by is done within the software. For an example of the appropriate usage of this material model, please see the Optical Scattering off of a GoldNanosphere tutorial.The DrudeLorentz Dispersion model is a material model that was developed based upon the Drude free electron model and the Lorentz oscillator model. The Drude model (when

) is used for metals and doped semiconductors, while the Lorentz modeldescribes resonant phenomena such as phonon modes and interband transitions. With the sum term, the combination of these two models canaccurately describe a wide array of solid materials. It predicts the frequency-dependent variation of complex relative permittivity as:

where

is the high-frequency contribution to the relative permittivity,

is the plasma frequency,

is the oscillator strength,

is the resonance frequency, and

is the damping coefficient. Since this model computes a complexvalued permittivity, the conductivity inside of COMSOL Multiphysics is set to zero. This approach is one way of modeling frequency-dependent conductivity.The Debye Dispersion model is a material model that was developed by Peter Debye and is based on polarization relaxation times. The model is primarily used for polar liquids. It predicts the frequency-dependent variation of complex relative permittivity as:

where

is the high-frequency contribution to the relative permittivity,

is the contribution to the relative permittivity, and is the relaxation time. Since this model computes a complex-valued permittivity, the conductivity is assumed to be zero. This is an alternate way to modelfrequency-dependent conductivity.The Sellmeier Dispersion model is available in the Wave Optics Module and is typically used for optical materials. It assumes zero conductivity and unit relative permeability and defines the relative permittivity in terms of the operating wavelength,

, rather than frequency:

where the coefficients

and

determine the relative permittivity.The choice between these seven models will be dictated by the way the material properties are available to you in the technical literature. Keep in mind that, mathematically speaking, they enter the governing equation identically.

Relative Permeability

The relative permeability quantifies how a material responds to a magnetic field. Any material with

is typically referred to as a magneticmaterial. The most common magnetic material on Earth is iron, but pure iron is rarely used for RF or optical applications. It is more typical to work with materials that are ferrimagnetic. Such materials exhibit strong magnetic properties with an anisotropy that can be controlled by an applied DC magnetic field. Opposed to iron, ferrimagnetic materials have a very low conductivity, so that high-frequency electromagnetic fields are able to penetrate into and interact with the bulk material. This tutorial demonstrates how to model ferrimagnetic materials.There are two options available for specifying relative permeability: The Relative Permeabilitymodel, which is the default for the RF Module, and the Magnetic Losses model. The Relative Permeability model allows you to enter a real- or complex-valued scalar or tensor value. The same Porous Media models described above for the electrical conductivity can be used for the relative permeability. The Magnetic Losses model is analogous to the Dielectric Loss model described above in that you enter the real and imaginary components of the relative permeability as real-valued numbers. An imaginary-valued permeability will lead to a magnetic loss in the material.

Modeling and Meshing Notes

In any electromagnetic modeling, one of the most important things to keep in mind is the concept of skin depth, the distance into a material over which the fields fall off to

of their value at the surface. Skin depth is defined as:

where we have seen that relative permittivity and permeability can be complex-valued.You should always check the skin depth and compare it to the characteristic size of the domains in your model. If the skin depth is much smaller than the object, you should instead model the domain as a boundary condition as described here: “Modeling Metallic Objects in Wave Electroma gnetics Problems“. If the skin depth is comparable to or larger than the object size, then the electromagnetic fields will penetrate into the object and interact significantly within the domain.

A plane wave incident upon objects of different conductivities and hence different skin depths. When the skin depth is smaller than the wavelength, a boundary layer mesh is used (right). The electric field is plotted.If the skin depth is smaller than the object, it is advised to use boundary layer meshing to resolve the strong variations in the fields in the direction normal to th ..

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2015-07-28

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