In a plasma model, the electron energy distribution function as well as the transport properties of the electrons (e.g., electron mobility) are needed. For the simplest cases, a Maxwellian EEDF and a constant value for the electron mobility can be used. The other transport properties are then computed in COMSOL Multiphysics using Einstein’s relation. However, in some cases, it might be advantageous to use a more elaborate form like a Boltzmann EEDF and define the electron transport properties as a function of the electron energy. But how do we obtain this data?

The answer: The *Boltzmann Equation, Two-Term Approximation* interface in COMSOL Multiphysics. Examples illustrating the use of this interface are available in the Model Library, one of which is the Argon Boltzmann Analysis model. To compute the Boltzmann EEDF, parameters such as the ionization degrees of the plasma are required. These parameters are not known *a priori*. Thus, the procedure is an iterative process.

The process begins with making an initial guess for the parameters and solving the Boltzmann equation. Then, the EEDF and transport properties are imported, if needed, to your plasma model. Finally, the plasma model is computed and the Boltzmann equation is resolved with the new volume-averaged parameters from the plasma model. You can continue to repeat these steps until convergence is reached.

Let’s walk through the steps of importing the data into the plasma model.

The figure below shows a Boltzmann EEDF in an argon plasma. The plasma features a gas temperature of 400 K, an electron density of 10^{16} 1/m³, an ionization degree of 10^{-10}, and a mole fraction of excited argon of 0.01. It can be observed that the EEDF is a function of the electron energy. Normally, an array of curves for the mean electron energy as a parameter is required.

*Boltzmann EEDF in an argon plasma.*

The next figure depicts the corresponding reduced electron transport properties computed with the *Boltzmann Equation, Two-Term Approximation* interface. The data is a function of the mean electron energy.

*Reduced electron transport properties in an argon plasma.*

The EEDF has to be imported to the plasma model as a spreadsheet consisting of three rows. The first row (*x*-data) has to be the electron energy (eV), while the second row (*y*-data) has to be the mean electron energy (eV). Meanwhile, the third row must contain the value for the distribution function (eV^(-3/2)). Finally, you need to export a 2D plot that looks like the following image.

*A 2D plot of the Boltzmann EEDF. Here, the *x*-axis indicates the electron energy and the *y*-axis represents the mean electron energy. The color illustrates the value of the distribution function.*

To export the EEDF in the desired format, use the Parametric Extrusion data set. Right-click “Data Sets” and choose *Parametric Extrusion 1D*. Set the level scale factor to 1. Then, right-click the Parametric Extrusion 1D data and choose *Add Data to Export*. Select a file name and click “Export”.

The transport properties can be exported in a straightforward manner from the respective 1D plot. Right-click the Global data and choose *Add Plot Data to Export*. Select a file name and click “Export”.

To import the EEDF to the plasma model, create an Interpolation function. Select *File* as the Data source and enter “2″ in the Number of arguments field. Click “Browse” and import the file.

Once you have done this, you can choose the interpolation function as an EEDF in the *Electron Energy Distribution Function* Settings in the main node of the plasma model. This is illustrated in the screenshot below.

The functions for the transport properties, such as electron mobility and diffusivity, can also be imported as an interpolation function. Here, however, the number of arguments is 1. In the plasma model node, you can use this interpolation function by entering “int2(icp.ebar)”. In this case, int2 is the name of the function, icp is the tag of the interface, and ebar is the mean electron energy.

- Blog post: Electron Energy Distribution Function

The Vivaldi antenna was invented by Peter Gibson. Gibson had a passion for music and he supposedly chose to name the antenna after Antonio Vivaldi, a Baroque composer. Although sources vary on this, the choice was probably due to the antenna resembling an instrument from Vivaldi’s era. The instrument in question may be a violin, cello, or a cross section of a trumpet.

The Vivaldi antenna is made to be malleable. Being created on a thin and flexible substrate gives it the ability to mold itself over a variety of surfaces. This means that it can have many different applications in a range of environments.

There’s a patent that utilizes a double Vivaldi element feed section for use on planes. This works because the antenna is able to form a streamlined shape. Another reason the Vivaldi antenna is a good fit for the airplane is that it can handle velocities up to Mach 2. This antenna is going places…

*Aircraft. ( By Björn — Own work. Licensed under Creative Commons Attribution Share Alike 2.0, via Wikimedia Commons)*

You can also find uses for the Vivaldi antenna inside of a hospital. By using this antenna in conjunction with microwave imaging, doctors may be able to better detect breast and brain cancer.

The breast cancer detection function of the Vivaldi antenna makes use of an interferometric called I-MUSIC. (This is a different kind of music than what the composer Vivaldi made…) I-MUSIC stands for interferometric Multiple Signal Classification and when combined with a Vivaldi antenna, it becomes a powerful method for detecting tumors.

The Vivaldi antenna (both as itself and when modified) has an incredibly wide range of functions. Security workers can use it to detect concealed weapons and military personnel can use it as a high-range radar when it’s placed in an array configuration.

This versatile antenna can do a lot, but first we have to analyze the design to see how well it works.

Using the Vivaldi, Tapered Slot Antenna (TSA) model, we can evaluate the antenna based on its far-field pattern and impedance.

Our model simulates a realistic Vivaldi antenna design by using a thin dielectric substrate for the antenna. On top of this substrate, a tapered slot is patterned with a perfect electric conductor (PEC) ground plane. We build the curves of the tapered slot by using the exponential function *e*^{0.044x}. The tapered slot itself does look something like a trumpet with a wide end curving into a narrow line. However, unlike a trumpet, the thin end connects to a circular slot while the wide end opens out to air, as seen below.

*The geometry of a Vivaldi antenna.*

Flipping the substrate over, there is a shorted 50 Ω microstrip feed line, which is modeled as PEC surfaces. It is a crucial element to the design since the lumped port located on the line is used to excite the antenna.

A perfectly matched layer (PML) is surrounding the whole modeling domain. This functions as an anechoic chamber and absorbs all of the radiated energy.

After setting up and solving our antenna model using COMSOL Multiphysics and the RF Module, we get the results back in various forms. The SWR plot reveals that our model has good wide-band impedance matching. It’s even better than 2:1 in the majority of our simulated range.

*The results of our simulation seen in the form of an SWR plot.*

We also create a model to display the far-field pattern. We know that a Vivaldi antenna forms a directive radiation pattern toward the wide end of its tapered slot. Our model shows that the far-field pattern does display a directional radiation pattern.

*A 3D far-field pattern, displayed at 5.5 GHz.*

Imagine a situation in which you are seeking the orientation of an object that is not within your sight, perhaps hidden behind a wall. For rescue workers, this is often a particular area of concern. Upon entering a building, workers may hear calls for help but locating their sources can often be a challenge.

Researchers at MIT responded to this issue by developing a portable device that, when pointed at walls, documented any movements occurring on the other side of the wall, making it particularly useful for such emergency situations. By emitting Wi-Fi signals, this device (known as *Wi-Vi*) tracked movements toward or away from a wall, picking up on shifts as small as a simple step.

In simulation, cases can also arise in which we want to determine a hidden object’s orientation — in a sense, *see through walls*. With a focus on polarization-dependent scattering, the Detecting the Orientation of a Metallic Cylinder Embedded in a Dielectric Shell model illustrates the use of linearly polarized plane waves in determining the orientation of an object.

The model consists of a metallic cylindrical rod that is embedded in a polystyrene foam dielectric shell surrounded by air, with the orientation of the rod unknown. Perfectly matched layers (PMLs) are used to enclose the model domain.

*A metallic cylinder in a dielectric shell.*

The analysis focuses on the detection of the rod for the polarization angle at which the scattered field is greatest. This is accomplished by evaluating the cylindrical object’s polarization-dependent scattered field as well as completing a parametric sweep that is a function of the polarization angle.

A linearly polarized plane wave — a new background wave type in available in COMSOL Multiphysics version 5.0 — is selected for the background field. A linearly polarized plane wave option provides an easy path for defining the background field without having to worry about the validity of a mathematical definition of a plane wave at any arbitrary angle of incidence.

The predefined initial background wave, \mathbf{E}_0=exp(-jk_xx)\mathbf{z}, is transformed by three successive rotations along the *roll*, *pitch*, and *yaw* angles (in that order). If you are interested in flight dynamics, you are likely already familiar with these angular configurations defining a vehicle’s orientation. The image below shows how these parameters are adapted to define the background field.

*A linearly polarized background wave.*

The roll angle is a right-handed rotation with respect to the +*x*-direction. The default is 0 rad, corresponding to polarization along the +*z*-direction. The pitch angle is a right-handed rotation with respect to the +*y*-direction. In this case, the default is also 0 rad, corresponding to the initial direction of propagation pointing in the +*x*-direction. The yaw angle is a right-handed rotation with respect to the +*z*-direction. In the given example model, the direction of a plane wave is defined by the roll angle and the polarization is determined by the yaw angle’s parameterized value.

To model the copper rod, the Impedance boundary condition is applied, with the inner volume taken out of the model domain. A near-field to far-field transformation enables the computation of the rod’s scattered field. Additionally, there is a perfectly matched layer domain outside of the surrounding air that serves as an absorber of the scattered field.

We will skip ahead to the results here, but you can follow the instructions from the Model Gallery to try modeling this concept yourself.

The first plot below depicts the model’s *radar cross section* in the decibel (dB) scale. The radar cross section is a measurement that refers to an object’s ability to scatter or reflect radio frequency (RF) radiation. When the radar cross section is larger, it indicates that the object can be more easily detected.

From the graph below, we can see that the largest radar cross section occurs at 30º and the lowest at 120º. It is known that the maximum scattering takes place when the polarization and the cylinder are parallel. Thus, we can conclude that the cylinder’s orientation is at 30º.

*A graph of the radar cross section. The results show that the maximum radar cross section is at 30º.*

The second plot illustrates the electric field norm in the *xy*-plane at the highest scattering angle. These results show an electric field that is similar to that of a dipole antenna. An oscillating current is induced along the rod by the background field, with the rod then radiating as a dipole.

*A plot showing the electric field norm at the maximum scattering angle. The electric field is similar to a dipole antenna’s electric field.*

- Download the model: Detecting the Orientation of a Metallic Cylinder Embedded in a Dielectric Shell

When modeling electromagnetic structures (e.g., antennas, waveguides, cavities, filters, and transmission lines), we can often limit our analysis to one small part of the entire system. Consider, for example, a coaxial splitter as shown here, which splits the signal from one coaxial cable (coax) equally into two. We know that the electromagnetic fields in the incoming and outgoing cables will have a certain form and that the energy is propagating in the direction normal to the cross section of the coax.

There are many other such cases where we know the form (but not the magnitude or phase) of the electromagnetic fields at some boundaries of our modeling domain. These situations call for the use of the Lumped Port and the Port boundary conditions. Let us look at what these boundary conditions mean and when they should be used.

We can begin our discussion of the Lumped Port boundary condition by looking at the fields in a coaxial cable. A coaxial cable is a waveguide composed of an inner and outer conductor with a dielectric in between. Over its range of operating frequencies, a coax operates in Tranverse Electro-Magnetic (TEM) mode, meaning that the electric and the magnetic field vectors have no component in the direction of wave propagation along the cable. That is, the electric and magnetic fields both lie entirely in the cross-sectional plane. Within COMSOL Multiphysics, we can compute these fields and the impedance of a coax, as illustrated here.

However, there also exists an analytic solution for this problem. This solution shows that the electric field drops off proportional to 1/r between the inner and outer conductor. So, since we know the shape of the electric field at the cross section of a coax, we can apply this as a boundary condition using the *Lumped Port, Coaxial* boundary condition. The excitation options for this condition are that the excitation can be specified in terms of a cable impedance along with an applied voltage and phase, in terms of the applied current, or as a connection to an externally defined circuit. Regardless of these three options, the electric field will always vary as 1/r times a complex-valued number that represents the sum of the (user-specified) incoming and the (unknown) outgoing wave.

*The electric field in a coaxial cable.*

For a coaxial cable, we always need to apply the boundary condition at an annular face, but we can also use the Lumped Port boundary condition in other cases. There are also a Uniform and a User-Defined option for the Lumped Port condition. The Uniform option can be used if you have a geometry as shown below: a surface bridging the gap between two electrically conductive faces. The electric field is assumed to be uniform in magnitude between the bounding faces, and the software automatically computes the height and width of the Lumped Port face, which should always be much smaller than the wavelength in the surrounding material. Uniform Lumped Ports are commonly used to excite striplines and coplanar waveguides, as discussed in detail here.

*A typical Uniform Lumped Port geometry.*

The User-Defined option allows you to manually enter the height and width of the feed, as well as the direction of the electric field vector. This option is appropriate if you need to manually enter these settings, like in the geometry shown below and as demonstrated in this example of a dipole antenna.

*An example of a User-Defined Lumped Port geometry.*

Another use of the Lumped Port condition is to model a small electrical element such as a resistor, capacitor, or inductor bonded onto a microwave circuit. The Lumped Port can be used to specify an effective impedance between two conductive boundaries within the modeling domain. There is an additional Lumped Element boundary condition that is identical in formulation to the Lumped Port, but has a customized user interface and different postprocessing options. The example of a Wilkinson power divider demonstrates this functionality.

Once the solution of a model using Lumped Ports is computed, COMSOL Multiphysics will also automatically postprocess the S-parameters, as well as the impedance at each Lumped Port in the model. The impedance can be computed for TEM mode waveguides only. It is also possible to compute an approximate impedance for a structure that is very nearly TEM, as shown here. But once there is a significant electric or magnetic field in the direction of propagation, then we can no longer use the Lumped Port condition. Instead, we must use the Port boundary condition.

To begin discussing the Port boundary condition, let’s examine the fields within a rectangular waveguide. Again, there are analytic solutions for propagating fields in waveguide. These solutions are classified as either Transverse Electric (TE) or Transverse Magnetic (TM), meaning there is no electric or magnetic field in the direction of propagation, respectively.

Let’s examine a waveguide with TE modes only, which can be modeled in the 2D plane. The geometry we will consider is of two straight sections of different cross-sectional area. At the operating frequency, the wider section supports both TE10 and TE20 modes, while the narrower supports only the TE10 mode. The waveguide is excited with a TE10 mode on the wider section. As the wave propagates down the waveguide and hits the junction, part of the wave will be reflected back towards the source as a TE10 mode, part will continue along into the narrower section again as a TE10 mode, and part will be converted to a TE20 mode, and then propagate back towards the source boundary. We want to properly model this and compute the split into these various modes.

The Port boundary conditions are formulated slightly differently from the Lumped Port boundary conditions in that you can add multiple types of ports to the same boundary. That is, the Port boundary conditions each *contribute to* (as opposed to the Lumped Ports, which *override*) other boundary conditions. The Port boundary conditions also specify the magnitude of the incoming wave in terms of the power in each mode.

*Sketch of the waveguide system being considered.*

The image below shows the solution to the above model with three Port boundary conditions, along with the analytic solution for the TE10 and TE20 modes for the electric field shape. Computing the correct solution to this problem does require adding all three of these ports. After computing the solution, the software also makes the S-parameters available for postprocessing, which indicates the relative split and phase shift of the incoming to outgoing signals.

*Solution showing the different port modes and the computed electric field.*

The Port boundary condition also supports Circular and Coaxial waveguide shapes, since these cases have analytic solutions. However, most waveguide cross sections do not. In such cases, the Numeric Port boundary condition must be used. This condition can be applied to an arbitrary waveguide cross section. When solving a model with a Numeric Port, it is also necessary to first solve for the fields at the boundary. For examples of this modeling technique, please see this example first, which compares against a semi-analytic case, followed by this example, which can only be solved by numerically computing the field shape at the ports.

*Rectangular, Coaxial, and Circular Ports are predefined.*

*Numeric Ports can be used to define arbitrary waveguide cross sections.*

The last case, when using the Port boundary condition, is appropriate for the modeling of plane waves incident upon quasi-infinite periodic structures such as diffraction gratings. In this case, we know that any incoming and outgoing waves must be plane waves. The outgoing plane waves will be going in many different directions (different diffraction orders) and we can determine ahead of time the directions, albeit not the relative magnitudes. In such instances, you can use the Periodic Port boundary condition, which allows you to specify the incoming plane wave polarization and direction. The software will then automatically compute all the directions of the various diffracted orders and how much power goes into each diffracted order.

For an extensive discussion of the Periodic Port boundary condition, please read this previous blog post on periodic structures. For a quick introduction to the use of these boundary conditions, please see this model of plasmonic wire grating.

We have introduced the Lumped Port and the Port boundary conditions for modeling boundaries at which an electromagnetic wave can pass without reflection and where we know something about the shape of the fields. Alternative options for the modeling of boundaries that are non-reflecting in cases where we do not know the shape of the fields can be found here.

The Lumped Port boundary condition is available solely in the RF Module, while the Port boundary condition is available in the *Electromagnetic Waves* interface in the RF Module and the Wave Optics Module as well as the Beam Envelopes formulation in the Wave Optics Module. This previous blog post provides an extensive description of the differences between these modules.

But what about those boundaries that are not transparent, such as the conductive walls of the waveguide we have looked at today? These boundaries will reflect almost all of the wave and require a different set of boundary conditions, which we will look at next.

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When studying the behavior of transmission lines, the *characteristic impedance* is often analyzed. The characteristic impedance refers to the ratio between the voltage and the current of a wave that propagates along the transmission line. By calculating this impedance, designers can modify the transmission line to improve its power transfer and reduce signal reflection.

The Model Gallery features examples of finding impedance in transmission line structures. Two types highlighted are the coaxial cable and the parallel-wire transmission line.

In each of these models, the transmission line operates in TEM mode, with the electric and magnetic fields normal to the propagation’s direction along the cable. Because of this, modeling a 2D cross section is enough to calculate the fields and impedance, and the models are solved with a Mode Analysis study feature. For both cases, the voltage is analyzed as a line integral of the electric field that exists between conductors. Meanwhile, the current is obtained as a line integral of the magnetic field along either conductor’s boundary or any closed contour that cuts the space between the conductors into two parts.

*In these arrow plots, the surface represents the electric field magnitude and the arrows represent the magnetic field magnitude. The image on the left shows the fields inside a coaxial cable and the image on the right shows the fields around two parallel wires.*

The problem-solving technique used to calculate the impedance in these 2D models has now been extended into the 3D realm with the Numeric TEM port feature. With the addition of settings and subfeatures in COMSOL Multiphysics version 5.0, this feature can now be used to find the impedance in 3D models.

The Notch Filter Using a Split Ring Resonator model provides an introduction on how to use the Numeric TEM ports. This model consists of a split ring resonator that is coupled to a microstrip line. The circuit as a whole acts as a notch, or band-stop, filter.

*A schematic of the circuit.*

To realize a band-stop filter response, the resonator’s split part is bordering and coupled to the microstrip line. The printed split ring resonator, which is on a ground plane, features a number of resonant modes. In this case, we are interested in the frequency near 2.4 GHz. The metal parts of the model are treated as perfect electric conductors and scattering boundary conditions are used on the model domain’s exterior boundaries, with the exception of the ground plane. The remainder is treated as a vacuum domain.

A numeric port is added on each surface of the ends of the microstrip line. These ports are designed to calculate the electric mode field on the structure, which is accomplished through a Boundary Mode Analysis. “Analyzed as a TEM field” is selected within the numeric port setting. This setting requires that the electric and magnetic field integration lines are defined in order to calculate the port’s voltage and current. The information from these integration lines is then used to find the port characteristic impedance.

The ratio of the calculated impedance and the reference impedance is used to scale the port mode field. The electric fields are led between two conductors and the field component in the propagation’s direction, with the normal to the port boundary negligible. Thus, the port mode can be analyzed as a transverse electromagnetic, or TEM, field.

In the first figure below, the electric field norm is illustrated on the *xy*-plane. This plot shows the symmetrical confinement of the electric fields along the split ring resonator at the frequency of interest (2.4 GHz).

*A model of the electric field norm highlights the electric fields’ symmetrical confinement along the split ring resonator at 2.4 GHz.*

The next plot illustrates the device’s frequency response. We can observe that the device behaves as a band-stop filter, as around 2.4 GHz, its S_{11} is nearly 0 dB and its S_{21} is less than -10 dB.

*A plot of the device’s frequency response illustrates that the device does act as an efficient band-stop filter.*

MEMS gyroscopes are found in a wide range of consumer products, from phones to game consoles to cars. These devices detect the rotation of an object, often using a coupling between motion in two directions at right angles induced by the Coriolis force.

*The figure above shows two degenerate modes of a section through a wine glass. The equation is a mathematical expression of the fact that the two modes are orthogonal, or independent of each other. We could construct other degenerate pairs by taking combinations of these two modes that are also orthogonal. They would look similar in form, except for a rotation about the center of the wine glass.*

The principle behind degenerate mode sensors was first discovered by the physicist G.H. Bryan when he struck the edge of a wine glass and listened to its vibrations. He found that when the glass was being rotated about its stem, the tone of vibration undulated, which he recognized as *beats* between two sound waves with slightly different frequencies.

What Bryan had discovered was the symmetry breaking of *degenerate modes* — two different modes of vibration that resonate at identical frequencies. In vibrating objects that are symmetric, such as Bryan’s wine glass, two degenerate mode pairs exist for every degree of rotation (natural number). When a symmetric object is rotating, Coriolis forces break the symmetry between sound waves generated by the object, splitting their frequencies in proportion to the rate of rotation. Measuring the difference in frequency of the broken degenerate modes can tell us the rate of rotation.

This concept is the principle behind the design of some MEMS gyroscopes. When the symmetry of the system is broken by an environmental factor of interest (in the case of a MEMS gyroscope — the Coriolis force), this change causes degeneracy breaking in the modes of vibration that can be used as a means to sense the environmental factor. However, instead of using the Coriolis force to cause degeneracy breaking, we could use other factors that change the symmetry of the device. For example, selectively adding a small amount of material to a part of the object can alter its symmetry and cause degeneracy breaking. Thus, sensors other than gyroscopes can utilize this principle.

In the paper “Degeneracy Breaking, Modal Symmetry and MEMS Biosensors” by H.T.D. Grigg, T.H. Hanley, and B.J. Gallacher, researchers at Newcastle University in the UK leveraged simulation to investigate the effects of material and geometric symmetry breaking in a degenerate mode sensor, a device designed using the same principles behind the MEMS gyroscopes that operate through degeneracy breaking.

Using COMSOL Multiphysics software, the team designed a degenerate Rayleigh SAW device with a symmetric geometry — that initially had degenerate modes. The surface of the sensor is treated so that an analyte (a substance or chemical of interest) will bind selectively to parts of the device, leading to a non-symmetric mass loading. This induces a frequency split in the degenerate modes.

The vibrational mode used can be seen in an image from their COMSOL Conference presentation:

*Image credit: H.T.D. Grigg, T.H. Hanley, B.J. Gallacher, Newcastle University, Newcastle upon Tyne, United Kingdom.*

Since the device is symmetric, the researchers used a quarter of the geometry in order to reduce computation time. The images below show how the device was designed using an anisotropic piezoelectric substrate covered by an isotropic layer. This arrangement of materials creates dispersive, anisotropic Rayleigh waves. The device was modeled with the *Piezoelectric Devices* interface using Frequency Domain and Time Dependent studies in COMSOL Multiphysics. The image below also shows the Perfectly Matched Layer (PML), which the researchers used to simulate a boundary absorber.

In their simulations, the researchers broke the degeneracy of the modes by changing the boundary conditions and/or the material properties in a non-symmetric manner. In a laboratory experiment, the degeneracy would be broken by asymmetric mass loading of the analyte. Because the sensor is symmetric with respect to other environmental factors (such as changes in temperature, etc.), it is generally robust to environmental disturbances — just like the MEMS gyroscopes that operate on the same principle.

*Image credit: H.T.D. Grigg, T.H. Hanley, B.J. Gallacher, Newcastle University, Newcastle upon Tyne, United Kingdom.*

To learn more about the MEMS biosensor, download the paper and presentation from the COMSOL Conference 2013.

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If a plasma is located in a closed system, each process is in equilibrium with its reverse process (detailed balance) and the plasma is in thermodynamic equilibrium. Such a plasma can be characterized with a few parameters: The (typically very high) temperature, pressure, and number densities of each single species. The radiation corresponds to black-body radiation and can be described with Planck’s law. The population of the excited states obeys a Boltzmann distribution, while the velocity distribution is Maxwellian. The Saha equation gives the ionization degree.

In a real plasma, deviations from equilibrium occur. For example, radiation escapes out of the plasma, disturbing the detailed balance. If the equilibrium still holds in a local sense, the plasma is in local thermodynamic equilibrium (LTE). The plasma can still be described (locally) by the above mentioned relations, with the exception of Planck’s law. Local thermodynamic equilibrium can only exist if the radiation processes can be neglected and the plasma is collision-determined. Therefore, a sufficiently high electron density is required.

Electrons are more susceptible to the electric field than the heavy particles, due to their significantly lower mass. Electrons therefore gain more energy than the heavy particles per unit time. If the pressure is low or the electron density is small, the heavy particle temperature remains lower than the electron temperature. The plasma can no longer be described by a single temperature.

The energy difference between the ground state and the first excited state is normally large, whereas the energy difference between the excited states is small, especially in noble gases. If the electron density is sufficiently high, the excited states can be in equilibrium among each other (excluding the ground state). The velocity distribution within the different kind of species can still be Maxwellian (in spite of different temperatures). In such a case, the plasma is in partial local thermodynamic equilibrium (PLTE).

In non-equilibrium plasma, the heavy particle temperature is much lower than the electron temperature. The background gas in which the electrons and ions reside is more or less close to room temperature. There are considerable deviations from equilibrium. None of the aforementioned relations are still valid. In this case, a more elaborate approach is necessary.

The Plasma Module brings with it several different interfaces. Here is a screenshot of the expanded Plasma node, to give you an overview:

The Equilibrium Discharges interfaces are suitable for modeling thermal plasmas. These plasmas can be considered as a conductive fluid mixture and may therefore be modeled using the magnetohydrodynamics (MHD) equations. MHD combines the Navier-Stokes, heat, and Maxwell’s equations to describe the motion of the conducting fluid in an electromagnetic field. The chemical composition of the plasma is neglected.

The following figure shows the Model Builder of an inductively coupled plasma torch. You can find this model in the Model Library. The *Magnetic Fields* interface is used to solve Ampère’s law. The *Heat Transfer* interface solves an energy balance equation and the *Laminar Flow* interface the Navier-Stokes equation. In addition, there is a Multiphysics node for the coupling terms.

The Equilibrium Discharges interfaces assume that the plasma is fully ionized and that the plasma is under local thermodynamic equilibrium (LTE) conditions.

The *Equilibrium DC Discharge* interface is used to study equilibrium discharges that are sustained by a static or slow-varying electric field where induction currents and fluid flow effects are negligible.

The *Combined Inductive/DC Discharge* interface is used to study equilibrium discharges that are sustained by induction currents and/or static or slow-varying electric field, such as in arc welding simulations, for instance.

The *Equilibrium Inductively Coupled Plasma* interface is used to study equilibrium discharges that are sustained by induction currents, for example in inductively coupled plasma torches.

The other interfaces are suitable for modeling non-thermal or non-equilibrium plasmas. Here, a fluid model with drift diffusion approximation is used. A mass conservation equation for each species, an energy balance equation for the electrons, and Poisson equation for the electric field are solved. Here, the plasma chemistry is an essential part of the model.

The following screenshot shows the Model Builder of an inductively coupled non-equilibrium argon plasma. You can find this model in the Model Library, too. The plasma chemistry can be seen. The Plasma Model node contains the equations for the fluid model with drift diffusion approximation. Ampère’s law is needed for the electromagnetic field of the coil.

The *Inductively Coupled Plasma* interface is used to study discharges that are sustained by induction currents.

The *Microwave Plasma* interface is suitable for studying discharges that are sustained by electromagnetic waves (wave-heated discharges).

The *Capacitively Coupled Plasma* interface is used to study discharges that are sustained by a time-varying electrostatic field.

The *DC Discharge* interface is used to study discharges that are sustained by a static electric field.

- Get started creating plasma simulations with a tutorial model from the online Model Gallery
- Contact technical support if you have any questions about your particular case

A *biconical antenna* is a broad-bandwidth antenna that is comprised of two conductive objects, which are cone-shaped. These broadband dipole antennas typically feature a bandwidth of three or more octaves. What allows the antenna to achieve such an extremely wide bandwidth? It can be attributed to its structure. In particular, the antenna’s two symmetrical radiating cones.

Biconical antennas are noted for their use in conducting electromagnetic compatibility (EMC) testing. This testing ensures that products adhere to the electromagnetic compatibility compliance guidelines that are issued by governing organizations. It is very important for these antennas to include the broadband characteristics needed to meet such guidelines.

To ensure that the antenna’s design meets these requirements, we can turn our attention to the power of simulation.

The Biconical Antenna model in our Model Gallery features two conical, metallic, radiating elements, with a dielectric-filled coaxial feed structure at its center. This structure houses a small cylindrical domain that contains the antenna’s power source, which is not included within the modeling domain.

As an alternative, this source can be modeled by implementing a coaxial lumped port boundary condition at the boundary that is facing the coaxial cable. This results in a wave that is launched down the coax. A perfectly matched layer is used to truncate the area of free space surrounding the antenna.

*The biconical antenna model.*

Wires, which are modeled as perfect electric conductors, are used to connect the inner and outer conductors of the coax to the conical radiators. Within each cone, there is a small symmetric cutout that allows for ample clearance for mounting and assembly purposes. The distance between the radiators and the surface area of the cone’s end tips are responsible for the reactance of the antenna’s input port. Modifying this distance can thus change the performance of the antenna.

The first graph highlights the frequency response of the biconical antenna. The results show that the bandwidth is significantly wider than that of a traditional dipole antenna, as S_{11} is less than -10 dB from 1.5 GHz to 3.5 GHz. This introductory example model simulates only a bandwidth of one octave.

*Frequency response.*

We can then shift our focus to the antenna’s radiation pattern. The simulation results show the far-field radiation pattern at 1.9 GHz at the E-plane (shown in blue) and the H-plane (shown in green). The resulting pattern resembles the radiation pattern of a dipole antenna.

*Far-field radiation pattern.*

The model discussed here illustrates that biconical antennas are well suited for applications requiring an omnidirectional radiation pattern as well as a wide bandwidth. In the next release of COMSOL Multiphysics — version 5.1 — we will include a new commercial look of this model, similar to that featured in the picture below. Stay tuned!

*A biconical microwave antenna. (“Small Biconical Microwave Antenna (SBA 9113: 0.5 – 3 GHz, SBA 9112: 1 – 18 GHz)” by Schwarzbeck Mess-Elektronik — Own work. Licensed under Creative Commons Attribution Share-Alike 3.0, via Wikimedia Commons).*

We are often interested in modeling a radiating object, such as an antenna, in free space. We may be building this model to simulate an antenna on a satellite in deep space or, more often, an antenna mounted in an anechoic test chamber.

*An antenna in infinite free space. We only want to model a small region around the antenna.*

Such models can be built using the *Electromagnetic Waves, Frequency Domain* formulation in the RF Module or the Wave Optics Module. These modules provide similar interfaces for solving the frequency domain form of Maxwell’s equations via the finite element method. (For a description of the key differences between these modules, please see my previous blog post, titled “Computational Electromagnetics Modeling, Which Module to Use?“)

Let’s limit ourselves in this blog post to considering only 2D problems, where the electromagnetic wave is propagating in the *x-y* plane, with the electric field polarized in the *z*-direction. We will additionally assume that our modeling domain is purely vacuum, so that the frequency domain Maxwell’s equations reduce to:

\nabla \cdot \left( \mu_r^{-1} \nabla E_z \right) -k_0^2 \epsilon_r E_z= 0

where E_z is the electric field, relative permeability and permittivity \mu_r = \epsilon_r = 1 in vacuum, and k_0 is the wavenumber.

Solving the above equation via the finite element method requires that we have a finite-sized modeling domain, as well as a set of boundary conditions. We want to use boundary conditions along the outside that are transparent to any radiation. Doing so will let our truncated domain be a reasonable approximation of free space. We also want this truncated domain to be as small as possible, since keeping our model size down reduces our computational costs.

Let’s now look at two of the options available within the COMSOL Multiphysics simulation environment for truncating your modeling domain: the scattering boundary condition and the perfectly matched layer.

One of the first transparent boundary conditions formulated for wave-type problems was the Sommerfeld radiation condition, which, for 2D fields, can be written as:

\lim_{ r \to \infty} \sqrt r \left( \frac{\partial E_z}{\partial r} + i k_0 E_z \right) = 0

where r is the radial axis.

This condition is exactly non-reflecting when the boundaries of our modeling domain are infinitely far away from our source, but of course an infinitely large modeling domain is impossible. So, although we cannot apply the Sommerfeld condition exactly, we *can* apply a reasonable approximation of it.

Let’s now consider the boundary condition:

\mathbf{n} \cdot (\nabla E_z) + i k_0 E_z = 0

You can clearly see the similarities between this condition and the Sommerfeld condition. This boundary condition is more formally called the *first-order scattering boundary condition (SBC)* and is trivial to implement within COMSOL Multiphysics. In fact, it is nothing other than a Robin boundary condition with a complex-valued coefficient.

If you would like to see an example of a 2D wave equation implemented from scratch along with this boundary condition, please see the example model of diffraction patterns.

Now, there is a significant limitation to this condition. It is only non-reflecting if the incident radiation is exactly normally incident to the boundary. Any wave incident upon the SBC at a non-normal incidence will be partially reflected. The reflection coefficient for a plane wave incident upon a first-order SBC at varying incidence is plotted below.

*Reflection of a plane wave at the first-order SBC with respect to angle of incidence.*

We can observe from the above graph that as the incoming plane wave approaches grazing incidence, the wave is almost completely reflected. At a 60° incident angle, the reflection is around 10%, so we would clearly like to have a better boundary condition.

COMSOL Multiphysics also includes (as of version 4.4) the second-order SBC:

\mathbf{n} \cdot (\nabla E_z) + i k_0 E_z -\frac{i }{2 k_0} \nabla_t^2 E_z= 0

This equation adds a second term, which takes the second tangential derivative of the electric field along the boundary. This is also quite easy to implement within the COMSOL software architecture.

Let’s compare the reflection coefficient of the first- and second-order SBC:

*Reflection of a plane wave at the first- and second-order SBC with respect to angle of incidence.*

We can see that the second-order SBC is uniformly better. We can now get to a ~75° incident angle before the reflection is 10%. This is better, but still not the best we can achieve. Let’s now turn our attention away from boundary conditions and look at perfectly matched layers.

Recall that we are trying to simulate a situation such as an antenna in an anechoic test chamber, a room with pyramidal wedges of radiation absorbing material on the walls that will minimize any reflected signal. This can be our physical analogy for the perfectly matched layer (PML), which is not a boundary condition, but rather a domain that we add along the exterior of the model that should absorb all outgoing waves.

Mathematically speaking, the PML is simply a domain that has an anisotropic and complex-valued permittivity and permeability. For a sample of a complete derivation of these tensors, please see *Theory and Computation of Electromagnetic Fields*. Although PMLs are theoretically non-reflecting, they do exhibit some reflection due to the numerical discretization: the mesh. To minimize this reflection, we want to use a mesh in the PML that aligns with the anisotropy in the material properties. The appropriate PML meshes are shown below, for 2D circular and 3D spherical domains. Cartesian and spherical PMLs and their appropriate usage are also discussed within the product documentation.

*Appropriate meshes for 2D and 3D spherical PMLs.*

In COMSOL Multiphysics 5.0, these meshes can be automatically set up for 3D problems using the Physics-Controlled Meshing, as demonstrated in this video.

Let’s now look at the reflection from a PML with respect to incident angle as compared to the SBCs:

*Reflection of a plane wave at the first- and second-order SBC and the PML with respect to angle of incidence.*

We can see that the PML reflects the least amount across the widest range. There is still reflection as the wave is propagating almost exactly parallel to the boundary, but such cases are luckily rather rare in practice. An additional feature of the PML, which we will not go into detail about for now, is that it absorbs not only the propagating wave, but also any evanescent field. So, from a physical point of view, the PML truly can be thought of as a material with almost perfect absorption.

Clearly, the PML is the best of the approaches described here. However, the PML does use more memory as compared to the SBC.

So, if you are early in the modeling process and want to build a model that is a bit less computationally intensive, the second-order SBC is a good option. You can also use it in situations where you have a strong reason to believe that any reflections at the SBC won’t greatly affect the results you are interested in.

The first-order SBC is currently the default, for reasons of compatibility with previous versions of the software, but with COMSOL Multiphysics version 4.4 or greater, use the second-order SBC. We have only introduced the plane-wave form of the SBC here, but cylindrical-wave and spherical-wave (in 3D) forms of the first- and second-order SBC’s are also available. Although they do use less memory, they all exhibit more reflection as compared to the PML.

The SBC and the PMLs are appropriate conditions for open boundaries where you do not know much about the fields at the boundaries a *priori*. If, on the other hand, you want to model an open boundary where the fields are known to have a certain form, such as a boundary representing a waveguide, the Port and Lumped Port boundary conditions are more appropriate. We will discuss those conditions in an upcoming blog post.

Originally invented in the late 1940s, bipolar transistors were widely used in the first integrated circuits. Although modern field-effect devices have largely replaced bipolar transistors in digital logic circuits, bipolar transistors remain widely used for analog applications. They are particularly widespread in power regulation circuitry, where they are used as switches and current amplifiers.

In order to ensure that a simulation effectively captures all the required physics to give an accurate result, it is important to understand the processes that need to be included in the model. These could vary depending on the configuration of the device and the situation in which it is intended to operate. It is always a good idea to ensure that a model gives satisfactory reliability and accuracy while minimizing the complexity of the problem.

This is *particularly* important for 3D semiconductor simulations, where models created using non-recommended techniques can take many days to solve or possibly never converge.

In this blog post, I will walk you through the process for setting up a 3D semiconductor model of a bipolar transistor device. First, I will introduce and explain the operation of the device and the important physical processes that must be included. I will also discuss the measures needed to effectively include them within the model.

Like many semiconductor devices, doping is critical to the operation of bipolar transistors. There are two kinds of doping: p-type doping, where a region has extra holes, and n-type doping, where a region has extra electrons.

A bipolar transistor consists of three regions of alternating p-type and n-type doping. Although there are two possible doping structures, n-p-n or p-n-p, we will focus on the n-p-n configuration as this is the most common variety. The n-p-n structure is formed by sandwiching a p-type layer between two n-type layers. The device can be considered to have three different regions known as the emitter, base, and collector. Each region can be individually electrically addressed via three separate metal contacts, which are labeled according to the region that they are connected to.

A schematic of the n-p-n doping structure, device regions, and electrical connections is shown here:

*Geometry and structure of the bipolar transistor device. Top: The geometry of a bipolar transistor device represented in the COMSOL Multiphysics simulation software. Bottom: Cross section through the device taken along the *z-x* plane, highlighted by blue edges in the upper image. The n-p-n doping pattern is labeled, along with the electric contacts that connect to the emitter, base, and collector regions.*

Due to the alternating doping of the three regions, the bipolar transistor forms two back-to-back p-n junctions that share the base region. The behavior of carriers as they encounter these p-n junctions is crucial for the operation of the bipolar transistor.

Throughout the rest of this example modeling process, I will present suitable steps for including all the relevant physics needed to describe this behavior in a computationally efficient way.

The first question to ask yourself when designing a 3D semiconductor model is: “Can I use symmetry to reduce the model size?”

Many kinds of devices have planes of symmetry, rotational symmetry, or even axisymmetric geometry. If at all possible, using axisymmetric designs is recommended, as this can reduce a 3D simulation down to 2D. An example of the axisymmetric modeling of a cylindrical field-effect transistor can be found here.

The device we want to model in this example has two planes of symmetry that bisect the device in the *x-z* and *y-z* planes. This means that we only need to model one quadrant. In the figure below, the upper-right quadrant has been selected:

*Geometry required by the model. Due to the planes of symmetry, which are highlighted in blue, only one quadrant of the entire device needs to be included. This reduces the size of the model, allowing it to be solved in a shorter amount of time and with less memory. The metal contacts are applied to surface boundaries as indicated.*

After creating the smallest model geometry that symmetry allows, the next question for consideration is: “What resolution do I need to reliably capture the physics within the model?”

Obviously, the geometry dimensions of the device need to be sufficiently resolved. However, the required physical processes often take place on a length scale that is much smaller than the geometry features. For semiconductor models, ensuring adequate resolution can be a challenge, as the various physical processes that are involved often require very different length scales. To make things more complicated, the spatial resolution required to adequately describe many processes can vary drastically throughout the device and can even vary as a function of other model parameters, such as the applied voltages.

A good place to start is to ensure that the doping profiles are resolved. This is because in regions where the dopant concentration changes rapidly, other quantities often undergo drastic variation as well. Thankfully, ensuring that the doping profiles are resolved is relatively straightforward, as the dopant distribution is user-controlled and does not vary when other model parameters change.

The COMSOL Multiphysics simulation environment provides convenient tools for easily viewing the doping throughout devices. It is a good practice to use the *Get Initial Value* option available from the Study node, since the dopant concentration is an analytic function that can be computed and visualized without solving the semiconductor equations. The initial value of the doping can then be plotted to help set up a suitable mesh for the full computation.

Below, I have added a 3D volume plot of the doping throughout the model, along with a line cut of the dopant concentration taken along a vertical cut through the center of the device.

*Dopant distribution throughout the bipolar transistor device. Left: 3D visualization of the doping throughout the model geometry. The red area in the front-top corner is the n-type emitter region. Due to the order of magnitude variation in the dopant concentration, which is typical of semiconductor devices, it is difficult to see the three different doping regions. Right: Line cut of the dopant concentration taken along the red line in the left-hand image. The logarithmic scale allows the n-p-n doping pattern to be clearly seen and each region is labeled. Note that the dopant concentration varies within each region but the most rapid changes in dopant concentration occur around the emitter-base and collector-base p-n junctions.*

In COMSOL Multiphysics, it is advantageous to use a *structured swept* mesh for 3D semiconductor simulations. This is where a surface mesh on an exterior face of the geometry is swept through the geometry volume, resulting in mesh elements that are prisms aligned with the direction of the sweep.

This can be seen in the mesh used in our example model, shown below, where a free-triangular mesh has been generated on the top surface and then swept down to the bottom surface. The mesh is *structured* so that the height of the prisms vary throughout the device, tightening the mesh in the *z*-direction in regions that require high resolution.

*The mesh used is a structured swept mesh where the resolution in the *z*-direction is largest around the p-n junctions and near to the electrical contacts on the top and bottom surfaces.*

As discussed above, it is important to refine the mesh around gradients in the doping profile. This has been achieved by using an extra geometry cuboid embedded within the main device domain just under the top surface. The extra cuboid is positioned such that the extra internal boundaries are in regions where the dopant concentration gradient is largest. These internal boundaries are used to control the structure of the mesh to tighten it in the required regions. Additionally, for semiconductor models, it is a good idea to tighten the mesh near electrical contacts in order to resolve the high current densities and electric field effects that are often present in their vicinity. For this reason, the mesh has also been refined near the top and bottom surfaces of the device.

Refining the mesh around doping gradients and near electrostatic boundaries is a good starting point. However, as mentioned earlier, care must be taken to consider the physical processes that are present in the model and modify the mesh to accommodate them.

For the bipolar devices modeled here, the active areas are spread over the p-n junction regions and there are no other special length scales to consider besides ensuring that the p-n doping structure is resolved. However, other devices may need additional thought. For example, field-effect devices often require very fine meshing under surface gate contacts, as the current density is localized very drastically in the thin channel regions under these gates (see here).

When in doubt, a good quantity to consider is the *Debye length*, which is smallest in regions that have high charge density. Finally, as with any numerical simulation, it is important to assess COMSOL Multiphysics models to ensure that solutions are mesh independent. You can achieve this conveniently by parameterizing the mesh density and varying the parameter using an auxiliary sweep.

For 3D semiconductor models, it is advisable to perform some preliminary studies to confirm a suitable mesh resolution before moving on to creating the full model. For example, you could simulate a 2D cross section to get an idea of the Debye length and to test for mesh independence prior to progressing to a full 3D version.

Semiconductor devices can often be operated in a variety of different configurations depending on the intended application. As 3D semiconductor simulations are computationally intensive, it is advisable to carefully consider the relevant application and design the model to compute suitable solutions.

When building a semiconductor model in COMSOL Multiphysics, it is useful to ask yourself: “What physics is important for my device?”

For 3D semiconductor models, including additional physical effects rapidly increases the computation time, so try to limit the model to only the important processes. For example, when modeling a device where the vast majority of current is carried by electrons and holes do not significantly participate, it may be suitable to solve only for electrons rather than both electrons and holes. Alternatively, when modeling a device that is connected to a very effective heat sink, it is perhaps not important to simulate the effects of varying temperature.

Below is a screenshot of the Model Builder for our bipolar transistor example model. In addition to the default nodes, the doping structure has been created using three domain features and the contacts have been assigned to the appropriate boundaries using boundary features. A *Trap-Assisted Recombination* feature has also been added, as this is needed to correctly account for the current flow through such a highly doped device. However, no other physical processes are included.

*Model Builder for the example bipolar transistor model.*

A common operation mode for bipolar transistor devices is for the emitter to be grounded and for voltages to be applied individually to the base and collector regions. This configuration, which is shown in the figure below, is suitable for using the bipolar transistor as a current amplifier. We will model this configuration.

*Circuit diagram showing a bipolar transistor in the common emitter configuration.*

This operation mode results in both the base and collector voltages being measured relative to the grounded emitter, so this set-up is known as the *common emitter configuration*.

In the common emitter configuration, the effective resistance between the emitter and collector can be varied by applying a current to the base. This enables the device to function as a current amplifier. This is because the magnitude of the current that flows between the collector and emitter (at a given collector-emitter voltage) is proportional to the current that flows between the base and the emitter.

The ratio between the current that is output from the collector to the current that is applied to the base is known as the *current gain*. Typical bipolar transistors have current gains of over 100, enabling the current output from the collector to be controlled by an input base current that is over 100 times smaller than the required output current. This makes bipolar transistors attractive across a broad range of power management applications.

A popular application is to use a small current from some sensing circuitry to control the current flow to a more energy-intensive component. For example, a small current produced by a temperature-sensing circuit can be used as an input to control a larger output current needed to power a heating element. With a particular chosen application, we can consider the relevant physical processes to ensure that our model includes the correct features and is adequately resolved.

Bipolar transistors get their name from their reliance on both electron and hole currents in order to function. This is in contrast to unipolar transistors, such as common MOSFET devices, in which current is carried only by one carrier species. Because of this, it is important to model both electron and hole currents when simulating bipolar devices.

In order to understand the operation of an n-p-n bipolar transistor, it is useful to consider the two p-n junctions that form the device and the direction of the bias that is applied to each one. When used as a current amplifier, the voltage at both the base and collector contacts is positive, relative to the grounded emitter contact, and the voltage at the collector contact has a larger magnitude than that at the base contact. This is known as the *forward-active* regime.

In the forward-active regime, a forward bias is applied to the emitter-base junction, while a reverse bias is applied to the collector-base junction. The forward bias at the emitter-base junction allows thermally excited carriers to be injected from the emitter into the base region. Electrons that traverse from the n-type emitter region into the p-type base region are referred to as *minority carriers*, because a p-type region has an abundance of holes but few electrons. The minority carrier electrons *diffuse* through the base region from a high concentration adjacent to the emitter-base junction towards the lower concentration deeper into the p-type base region. Minority electrons that travel near the reverse bias collector-base junction are then driven to the collector contact by the collector-base junction electric field. However, the majority holes in the base region cannot penetrate the reverse-bias junction. The overall result is that the electron current can flow between the emitter and collector contacts, traversing all three regions, while the hole current is confined to the base and emitter regions.

Now that we have an idea of how we expect the device to function, it is possible to check if the important physics will be both included in the model and adequately resolved.

We have already refined the mesh around the doping gradient and in the vicinity of the contacts to account for the expected physics common to most semiconductor models. This should ensure that the p-n junctions — and the associated electric fields — are properly resolved. The only additional physical process to consider is the minority carrier diffusion throughout the base region. This is an important physical process, as it significantly affects the performance of the device and could be of interest when designing or optimizing a bipolar transistor.

For instance, the base region must be thin enough compared to the electron diffusion length, so that electrons can make it into the collector region, but not so thin that electrons and holes tunnel directly from the emitter into the collector. As it happens, carrier diffusion is already included in the standard set of semiconductor equations and the mesh is already fairly fine compared to the electron diffusion length in the base region. Therefore, no special amendments are required for this particular model.

The last thing to do is to configure suitable studies to simulate the device under conditions relevant to its normal operation. Now is the time to ask yourself: “What are the typical operating conditions for my application?”.

Often, semiconductor devices can be wired up in different configurations for different situations. However, due to computation time considerations, it may not be practical to simulate the full range of possible operational conditions. For example, instead of running a voltage sweep that covers every possible combination of voltages, a study can be restricted to only solve for the combinations and ranges that will be encountered during typical operation.

Furthermore, consider what information is needed from the model and design the studies so that this will be convenient to extract from the solutions. The example model has two studies: one that sets the collector voltage to 0.5 V and sweeps the base voltage and another that sets the base current to 2 μA and sweeps the collector voltage. The first study allows the current gain to be calculated with ease. The second study allows the emitter-collector current to be calculated as a function of emitter-collector voltage for a fixed input base current.

The first study can be used to generate a graph known as a *Gummel plot*. This is a standard way of evaluating bipolar transistors for use as current amplifiers. It shows the collector and base currents as a function of the base voltage on a logarithmic *y*-axis scale. The ratio of the collector current to base current gives the current gain, which is a crucial performance parameter for an amplifier device.

Below, you can see a Gummel plot. Judging by the plot, it is clear that the collector current is around two orders of magnitude larger than the base current over the whole base voltage range.

*Gummel plot showing the collector and base currents as a function of the base voltage when a voltage of 0.5 V is applied to the collector.*

Next, we have a figure that shows the current gain as a function of the collector current. The current gain is fairly constant, at ~160, over a collector current range of 9 orders of magnitude. However, for collector currents above 1 mA, the current gain falls drastically. This simulation indicates that the device has an operational limit of around 1 mA if used as a current amplifier in an application with a collector-emitter voltage drop of 0.5 V.

*Current gain of the bipolar transistor as a function of collector current with a collector-emitter voltage drop of 0.5 V.*

It is important to note that the same study and analysis *could* be performed at a range of different collector voltages. However, due to the computational expense involved, it is advisable to narrow this range to the expected operational voltages during normal use, as discussed above.

The second study is used to generate a graph of the collector current as a function of collector voltage when an input base current of 2 μA is applied, as shown below. Initially, the collector current increases very quickly with collector voltage, before saturating towards a value of around 300 μA.

*Collector current as a function of collector voltage when an input current of 2 μA is applied to the base.*

From 2D simulations of a similar device, it can be shown that the saturation current level is controlled by the input base current. This is an example of when a 2D model can be used in conjunction with a 3D model to make device modeling more efficient. It is a good practice to use a 2D model wherever possible to observe general behavior and identify parameter combinations or ranges of interest before moving to a full 3D device simulation.

Finally, using the full 3D simulation, we can visualize the 3D current flow within the device. The last figure shows arrow plots of the electron current (black arrows) and hole current (white arrows) for a collector voltage of 1.5 V with an input base current of 2 μA, which is in the saturation regime. The color plot slice shows the voltage and allows us to see the p-n junctions. As expected, the hole current does not traverse into the collector region, while the electron current predominantly flows between the collector and emitter.

*3D visualization of the current flow throughout the bipolar transistor at a collector voltage of 1.5 V with an input current of 2 μA applied to the base. Electron current is shown with black arrows and hole current with white arrows. The color of the slice represents the voltage throughout the device.*

This example model demonstrates best practices for modeling and analyzing 3D semiconductors using the COMSOL Multiphysics software. Although the device studied here is relatively simple, the same thought process and model building steps can be applied to any semiconductor model. While 3D semiconductor modeling remains a computationally challenging field, the advice in this blog post should take the guesswork out of getting started and creating useful device models.

- For more details on the example bipolar transistor models in the COMSOL Multiphysics Model Library, please see:
- For additional information on some of the physics used for this model, please see:
- Informative Wikipedia page on bipolar transistors
- A good general guide to semiconductor junctions

The dielectrophoretic effect will show up in both DC and AC fields. Let’s first look at the DC case.

Consider a dielectric particle immersed in a fluid. Furthermore, assume that there is an external static (DC) electric field applied to the fluid-particle system. The particle will in this case always be pulled from a region of weak electric field to a region of strong electric field, provided the permittivity of the particle is higher than that of the surrounding fluid. If the permittivity of the particle is lower than the surrounding fluid, then the opposite is true; the particle is drawn to a region of weak electric field. These effects are known as *positive dielectrophoresis* (pDEP) and *negative dielectrophoresis* (nDEP), respectively.

The pictures below illustrate these two cases with a few important quantities visualized:

- Electric field
- Maxwell stress tensor (surface force density)
- Surface charge density

*An illustration of positive dielectrophoresis (pDEP), where the particle permittivity is higher than that of the surrounding fluid \epsilon_p > \epsilon_f. At the surface of the particle, the induced surface charge is color-coded with red representing a positive charge and green a negative charge. Yellow represents a neutral charge.*

*An illustration of negative dielectrophoresis (nDEP), where the particle permittivity is lower than that of the surrounding fluid \epsilon_p < \epsilon_f.*

The Maxwell stress tensor represents the local force field on the surface of the particle. For this stress tensor to be representative of what forces are acting on the particle, the fluid needs to be “simple” in that it shouldn’t behave too weirdly either mechanically or electrically. Assuming the fluid is simple, we can see from the above illustrations that the net force on the particle appears to be in opposite directions between the two cases of pDEP and nDEP. Integrating the surface forces will indeed show that this is the case.

It turns out that if we shrink the particle and look at the infinitesimal case of a very small particle acting like a dipole in a fluid, then the net force is a function of the gradient of the square of the electric field.

Why is the net force behaving like this? To understand this, let’s look at what happens at a point on the surface of the particle. At such a point, the magnitude of the electric surface force density, f, is a function of charge times electric field:

(1)

f \propto \rho E

where \rho is the induced polarization charges. (Let’s ignore for the moment that some quantities are vectors and make a purely phenomenological argument by just looking at magnitudes and proportionality.)

The induced polarization charges are proportional to the electric field:

(2)

\rho \propto \epsilon E

Combining these two, we get:

(3)

f \propto \rho E = \epsilon E^2

But this is just the local surface force density at one point at the surface. In order to get a net force from all these surface force contributions at the various points on the surface, there needs to be a difference in force magnitude between one side of the particle and the other. This is why the net force, \bf{F}, is proportional to the gradient of the square of the electric field norm:

(4)

\mathbf{F} \propto \epsilon \nabla |\mathbf{E}|^2

In the above derivation, we have taken some shortcuts. For example, what is the permittivity in this relationship? Is it that of the particle or that of the fluid or maybe the difference of the two? What about the shape of the particle? Is there a shape factor?

Let’s now address some of these questions.

In a more stringent derivation, we instead use the vector-valued relationship for the force on an electric dipole:

(5)

\mathbf{F} = \mathbf{P} \cdot \nabla \mathbf{E}

where \bf{P} is the electric dipole moment of the particle.

To get the force for different particles, we simply insert various expressions for the electric dipole moment. In this expression, we can also see that if the electric field is uniform, we get no force (since the particle is small, its dipole moment is considered a constant). For a spherical dielectric particle with a (small) radius r_p in an electric field, the dipole moment is:

(6)

\mathbf{P} = 4 \pi r_p^3 k \mathbf{E}

where k is a parameter that depends on the the permittivity of the particle and the surrounding fluid. The factor 4 \pi r_p^3 can be seen as a shape factor.

Combining these, we get:

(7)

\mathbf{F} = 4 \pi r_p^3 k \mathbf{E} \cdot \nabla\mathbf{E} = 2 \pi r_p^3 k \nabla |\mathbf{E}|^2

This again shows the dependency on the gradient of the square of the magnitude of the electric field.

If the electric field is time-varying (AC), the situation is a bit more complicated. Let’s also assume that there are losses that are represented by an electric conductivity, \sigma. The dielectrophoretic net force, \bf{F}, on a spherical particle turns out to be:

(8)

\mathbf{F} = 2 \pi r^3_p k \nabla |\mathbf{E}_{\textrm{rms}}|^2

where

(9)

k = \epsilon_0 \Re\{ \epsilon_f \} \Re \left\{ \frac{\epsilon_p -\epsilon_f}{\epsilon_p + 2 \epsilon_f} \right\}

and

(10)

\epsilon = \epsilon_{\textrm{real}} -j \frac{\sigma}{2 \pi \nu}

is the complex-valued permittivity. The subscripts p and f represent the particle and the fluid, respectively. The radius of the particle is r_p and \bf{E}_{\textrm{rms}} is the root-mean-square of the electric field. The frequency of the AC field is \nu.

From this expression, we can get the force for the electrostatic case by setting \sigma = 0. (We cannot take the limit when the frequency goes to zero, since the conductivity has no meaning in electrostatics.)

In the expression for the DEP force, we can see that indeed the difference in permittivity between the fluid and the particle plays an important role. If the sign of this difference switches, then the force direction is flipped. The factor k involving the difference and sum of permittivity values is known as the *complex Clausius-Mossotti function* and you can read more about it here. This function encodes the frequency dependency of the DEP force.

If the particles are not spherical but, say, ellipsoidal, then you use another proportionality factor. There are also well-known DEP force expressions for the case where the particle has one or more thin outer shells with different permittivity values, such as in the case of biological cells. The simulation app presented below includes the permittivity of the cell membrane, which is represented as a shell.

*The settings window for the effective DEP permittivity of a dielectric shell.*

There may be other forces acting on the particles, such as fluid drag force, gravitation, Brownian motion force, and electrostatic force. The simulation app shown below includes force contributions from drag, Brownian motion, and DEP. In the Particle Tracing Module, a range of possible particle forces are available as built-in options and we don’t need to be bothered with typing in lengthy force expressions. The figure below shows the available forces in the *Particle Tracing for Fluid Flow* interface.

*The different particle force options in the *Particle Tracing for Fluid Flow* interface.*

Medical analysis and diagnostics on smartphones is about to undergo rapid growth. We can imagine that, in the future, a smartphone can work in conjunction with a piece of hardware that can sample and analyze blood.

Let’s envision a case where this type of analysis can be divided into three steps:

- Extract blood using the hardware, which attaches directly to your smartphone, and compute mean platelet and red blood cell diameter.
- Compute the efficiency of separation of the red blood cells and platelets. This efficiency needs to be high in order to perform further diagnostics on the isolated red blood cells.
- Use the computed optimum separation conditions to isolate the red blood cells using the hardware attached to your smartphone.

The COMSOL Multiphysics simulation app focuses on Step 2 of the overall analysis process above. By exploiting the fact that blood platelets are the smallest cells in blood and have different permittivity and conductivity than red blood cells, it is possible to use DEP for size-based fractionation of blood; in other words, to separate red blood cells from platelets.

Red blood cells are the most common type of blood cell and the vertebrate organism’s principal means of delivering oxygen (O_{2}) to the body tissues via the blood flow through the circulatory system. Platelets, also called *thrombocytes*, are blood cells whose function is to stop bleeding.

Using the Application Builder, we created an app that demonstrates the continuous separation of platelets from red blood cells (RBCs) using the *Dielectrophoretic Force* feature available in the *Particle Tracing for Fluid Flow* interface. (The app also requires one of the following: the CFD Module, Microfluidics Module, or Subsurface Flow Module and either the MEMS Module or AC/DC Module.)

The app is based on a lab-on-a-chip (LOC) device described in detail in a paper by N. Piacentini et al., “Separation of platelets from other blood cells in continuous-flow by dielectrophoresis field-flow-fractionation”, from *Biomicrofluidics*, vol. 5, 034122, 2011.

The device consists of two inlets, two outlets, and a separation region. In the separation region, there is an arrangement of electrodes of alternating polarity that controls the particle trajectories. The electrodes create the nonuniform electric field needed for utilizing the dielectrophoretic effect. The figure below shows the geometry of the model.

*The geometry used in the particle separation simulation app.*

The inlet velocity for the lower inlet is significantly higher (853 μm/s) than the upper inlet (154 μm/s) in order to focus all the injected particles toward the upper outlet.

The app is built on a model that uses the following physics interfaces:

*Creeping Flow*(Microfluidics Module) to model the fluid flow.*Electric Currents*(AC/DC or MEMS Module) to model the electric field in the microchannel.*Particle Tracing for Fluid Flow*(Particle Tracing Module) to compute the trajectories of RBCs and platelets under the influence of drag and dielectrophoretic forces and subjected to Brownian motion.

Three studies are used in the underlying model:

- Study 1 solves for the steady-state fluid dynamics and frequency domain (AC) electric potential with a frequency of 100 kHz.
- Study 2 uses a Time Dependent study step, which utilizes the solution from Study 1 and estimates the particle trajectories without the dielectrophoretic force. In this study, all particles (platelets and RBCs) are focused to the same outlet.
- Study 3 is a second Time Dependent study that includes the effect of the dielectrophoretic force.

You can download the model that the app was based on here.

To create the simulation app, we used the Application Builder, which is included in COMSOL Multiphysics® version 5.0 for the Windows® operating system.

The figure below shows the app as it looks like when first started. In this case, we have connected to a COMSOL Server™ installation in order to run the COMSOL Multiphysics app in a standard web browser.

*A biomedical simulation app running in a standard web browser.*

The app lets the user enter quantities, such as the frequency of the electric field and the applied voltage. The results include a scalar value for the fraction of red blood cells separated. In addition, three different visualizations are available in a tabbed window: the blood cell and platelet distribution, the electric potential, and the velocity field for the fluid flow.

The figures below show visualizations of the electric potential and the flow field.

*Screenshot showing the instantaneous electric potential in the microfluidic channel.*

*Screenshot displaying the magnitude of the fluid velocity.*

The app has three different solving options for computing just the flow field, computing just the separation using the existing flow field, or combining the two. A warning message is shown if there is not a clean separation.

Increasing the applied voltage will increase the magnitude of the DEP force. If the separation efficiency isn’t high enough, we can increase the voltage and click on the *Compute All* button, since in this case, both the fields and particle trajectories need to be recomputed. We can control the value of the Clausius-Mossotti function of the DEP force expression by changing the frequency. It turns out that at the specified frequency of 100 kHz, only red blood cells will exit the lower outlet.

The fluid permittivity is in this case higher than that of the particles and both the platelets and the red blood cells experience a negative DEP force, but with different magnitude. To get a successful overall design, we need to balance the DEP forces relative to the forces from fluid drag and Brownian motion. The figure below shows a simulation with input parameters that result in a 100% success in separating out the red blood cells through the lower outlet.

*Successful separation of red blood cells.*

To learn more about dielectrophoresis and its applications, click on one of the links listed below. Included in the list is a link to a video on the Application Builder, which also shows you how to deploy applications with COMSOL Server™.

- Model Gallery: Dielectrophoretic Particle Separation
- Video Gallery: How to Build and Run Simulation Apps with COMSOL Server™ (archived webinar)
- Wikipedia: Dielectrophoresis
- Wikipedia: Maxwell-Wagner-Sillars polarization
- Wikipedia: Clausius-Mossotti relation

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