V&V Article 5: Modeling Dispersive Materials Using EM.Tempo
Contents
Introduction
In this verification & validation (V&V) article, we will demonstrate EM.Tempo’s ability to simulate models and structures containing dispersive materials.
An Overview of Dispersive Materials
In a dispersive material, the constitutive parameters εr and μr of a material vary with frequency. The parameters’ dependence on frequency can be due to the natural properties of the material, or dispersion can be used as a type of macromodeling to abstract the behavior of a certain structure. The latter is often the case in metamaterials, where a structured array of unit elements made of well-behaved material collectively appear to exhibit dispersive properties.
The dispersive properties of a material are often discussed in terms of its poles. In EM.Tempo, materials can be modeled using multiple electric Debye poles,
<img alt="debye pole" src="http://www.emagtech.com/images/debye.png" />
multiple electric Drude poles,
<img alt="debye pole" src="http://www.emagtech.com/images/drude.png" />
or multiple electric Lorentz poles:
<img alt="debye pole" src="http://www.emagtech.com/images/lorentz.png" />
EM.Tempo can also model uniaxial materials with both electric and magnetic dispersion based on any of the pole types listed above. This model is required for representing double negative index materials (DNGs) and other metamaterials.
For more technical information about dispersive materials in EM.Tempo, visit the relevant section in <a href="http://wiki.emagware.com/index.php/EM.Tempo#Dispersive_Materials">EM.Cube Wiki</a>.
The complex permittivity of a Debye material with N poles is given by:
- [math] \varepsilon (\omega) = \varepsilon_\infty + \sum_{p=1}^N \dfrac{\Delta \varepsilon_p}{1 + j\omega \tau_p}, \quad \Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_\infty [/math]
where [math]\varepsilon_{\infty}[/math] is the value of the permittivity at infinite frequency, [math]\tau_p[/math] is the relaxation time corresponding to the pth pole having the unit of seconds, and [math]\varepsilon_{sp}[/math] is the value of the static permittivity (at DC) corresponding to the pth pole. [math]\Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty}[/math] represents the change in permittivity due to the pth pole.
The complex permittivity of a Lorentz material with N poles is given by:
- [math] \varepsilon(\omega) = \varepsilon_{\infty} - \sum_{p=1}^N \dfrac{\Delta \varepsilon_p {\omega_p}^2}{\omega^2 - 2j\omega \delta_p - {\omega_p}^2}, \quad \Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty} [/math]
where [math]\omega _p[/math] and [math]\delta_p[/math] are the angular resonant frequency and angular damping frequency corresponding to the pth pole, respectively, and both are expressed in rad/s. Similar to a Debye material, [math]\Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty}[/math] represents the change in permittivity due to the pth pole.
Drude Plasma Slab
For this project, we will use EM.Tempo's periodic boundary condition to simulate a plasma slab with infinite extents in the X and Y directions, but a finite Z-thickness of 1.5 cm. The Drude model often provides a good abstraction for an unmagnetized plasma. The complex permittivity of a Drude material with N poles is given by:
- [math] \varepsilon(\omega) = \varepsilon_{\infty} - \sum_{p=1}^N \dfrac{{\omega_p}^2}{\omega^2 - j\omega \nu_p} [/math]
where [math]\omega_p[/math] and [math]\nu_p[/math] are the angular plasma frequency and angular collision frequency corresponding to the pth pole, respectively, and both are expressed in rad/s. For an unmagnetized plasma, [math]\varepsilon_{\infty} = 1[/math]. A Drude pole with ωp = 1.803×1011 rad/s, and νc = 2×1010rad/s is used as the dispersive model for this project.
The table below summarizes the simulation parameters:
| Center Frequency | Bandwidth | Grid Spacing | Termination Criterion |
|---|---|---|---|
| 40GHz | 60GHz | 0.1 mm | -100dB |
 |
 |
Note that, due to the infinite nature of ε as frequency approaches zero in the Drude model, the excitation bandwidth is set up to avoid the very bottom of the band. Also, since energy in the low end of the band may take a very long time to dissipate (the magnitude of the conductivity is very high here), we set a relatively stringent termination criteria. Figure 1 shows the results for the reflection coefficient of the plasma slab as computed by EM.Tempo and compares them to the data given by Ref. [1], demonstrating a very good agreement.
Jerusalem-Cross Frequency Selective Surface (FSS)
In the second example, we will model a sandwiched frequency selective surface (FSS) structure described in Ref. [2]. A Jerusalem-cross patterned metallic layer is sandwiched between two Debye material layers. The complex permittivity of a Debye material with N poles is given by:
- [math] \varepsilon (\omega) = \varepsilon_\infty + \sum_{p=1}^N \dfrac{\Delta \varepsilon_p}{1 + j\omega \tau_p}, \quad \Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_\infty [/math]
where [math]\varepsilon_{\infty}[/math] is the value of the permittivity at infinite frequency, [math]\tau_p[/math] is the relaxation time corresponding to the pth pole having the unit of seconds, and [math]\varepsilon_{sp}[/math] is the value of the static permittivity (at DC) corresponding to the pth pole. [math]\Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty}[/math] represents the change in permittivity due to the pth pole. In this project, the Debye pole has parameters τ = 5.27×10-10 and Δε = 1.5.
The table below summarizes the simulation parameters:
| Center Frequency | Bandwidth | Grid Spacing | Termination Criterion |
|---|---|---|---|
| 5GHz | 10GHz | 0.1 mm | -100dB |
The transmission coefficient of the Jerusalem-cross FSS computed by EM.Tempo is shown in Figure 3. The computed data matches the result from Ref. [2]. These data are quite different from the transmission coefficient of a structure where the sandwhich layers have a frequency-independent permittivity.
Figure 3: Transmission coefficient of a Jerusalem-cross FSS. Solid line: results computed by EM.Tempo, symbols: data presented by Ref. [2].
Figure 2: EM.Cube environment for simulating the dispersive Jerusalem-cross FSS.
Nano-Particle-Coated Solar Cell
First, we will demonstrate a periodic solar cell structure. A silicon-dioxide substrate is coated with an array of patterned silver-particle cuboids, which can be modeled with a single Lorentz pole having parameters ωp = 2.20254e12 rad/s, δp = 1.4e13 rad/s, and Δε = 4.8e7. The silver-particles increase the absorption of the solar cell [2]. Some notable parameters for this simulation are listed below:
- Project Frequency: 500 THz
- Project Bandwidth: 500 THz
- Grid Spacing: 1.5nm in all directions
- Termination Criterion: -50 dB
For this simulation, it is important to remember that ε(ω) will have a resonant peak near ωp = 2.20254e12 rad/s. The project bandwidth should be chosen to avoid this peak, since obtaining accurate results at these frequencies can be very difficult. The transmission coefficient for this structure is shown in Figure 5, where the results computed by EM.Tempo are compared with the results of Ref. [2], and a reasonable agreement is observed.
Figure 5: Transmission coefficient of nano-particle-coated solar cell. Solid line: results computed by EM.Tempo, symbols: data presented by Ref. [2].
Figure 4: EM.Cube environment for simulating the nano-particle coated solar cell.
References
[1] K.S. Kunz and R.J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics. CRC Press, 1993.
[2] K. ElMahgoub, F. Yang, and A. Elsherbeni, Scattering Analysis of Periodic Structures Using Finite-Difference Time-Domain Method. Morgan & Claypool Publishers, 2012.
