Resistor

1. Overview

In this example project, a simple uniformly doped n-type silicon resistor is modelled and simulated to demonstrate the basic workflow of semiconductor device simulation. An analytical resistance is calculated from fundamental semiconductor relations based on geometry, doping concentration, and mobility. The simulation includes geometry definition, doping profile assignment, physical model selection, contact specification, and bias application. The extracted current is used to compute the simulated resistance, and the result is compared with the analytical expectation. This example serves both as a simple validation case for the Aquarius TCAD environment and as an instructional reference for users implementing basic electrical simulations of passive devices.

2. Parameters

Parameter	Symbol	Value	Unit	Description
Length	$L$	$1 \times 10^{-3}$	cm	Distance between the two contacts.
Depth	$D$	$1 \times 10^{-4}$	cm	Depth of the resistor (Scaling Factor)
Height	$H$	$1 \times 10^{-4}$	cm	Vertical thickness of the resistor.
Doping Concentration	$N_D$	$1 \times 10^{-16}$	cm$^{-3}$	Uniform donor concentration.
Carrier Mobility	$\mu_n$	1000	cm$^{2}$ /V·s	Nominal electron mobility (constant).

3. Analytical Result

3.1. Resistance Formula

The analytical resistance of a uniformly doped semiconductor resistor can be derived from Ohm’s law and the definition of resistivity in terms of semiconductor transport properties. The total resistance $R$ between two terminals of a rectangular resistor is given by:

$$R = \rho \cdot \frac{L}{A}$$

Where:

• $R$ = resistance (Ω)
• $\rho$ = resistivity of doped silicon (Ω·cm)
• $L$ = length of the resistor (cm)
• $A$ = cross-sectional area = Height × Depth (cm²)

In a doped semiconductor, the resistivity $\rho$ is related to the doping concentration and carrier mobility as follows:

$$\rho = \frac{1}{q \cdot \mu \cdot N}$$

Where:

• $q$ = elementary charge ($\approx$ $1.6 \times 10^{-19}$ C)
• $\mu$ = carrier mobility (cm²/V·s)
• $N$ = doping concentration (cm⁻³)

Combining the two equations, the resistance becomes:

$$R = \frac{L}{q \cdot \mu \cdot N \cdot A}$$

3.2 Example Calculation

Using the parameter values defined above:

$$A = H \cdot D = 10^{-4} \cdot 10^{-4} = 10^{-8} cm^{2}$$

Now substitute into the resistance formula:

$$R = \frac{1.0 \times 10^{-3}}{(1.6 \times 10^{-19}) \cdot 1000 \cdot (1.0 \times 10^{16}) \cdot (1.0 \times 10^{-8})} = \underline{\mathbf{62.5\text{k}\Omega}}$$

4. Resistor in Aquarius

4.1 Creating the Device Model

To create a resistor device model in Aquarius, follow the steps below:

4.1.1. Launch the Device Editor

• Open the Aquarius application.
• On the Start Page, three options are available:
  1. New Device Model
  2. New Simulation
  3. Open Results Visualiser
• Click New Device Model (the first option).
• The Device Editor will open with an empty workspace.

4.1.2. Define the Device Geometry

tip

For more detailed instructions on defining regions, click here.

• After drawing the rectangle, the Exact Coordinates dialog will open automatically:
  • Set the First Vertex to (0, 0).
  • Set the Second Vertex to (10, 1).
  • Click OK to confirm.

• The Region Properties dialog will then open:
  • Set the Region Material to Silicon (Si);
    • Note: the default material properties can be used, so the carrier mobility of 1000 does not need to be changed.
  • Set the Donor Doping to 1E+16.
  • Click OK to confirm.

4.1.3. Define the Device Contacts

With the device geometry and material properties defined, the next step is to specify electrical contacts to allow the device to be connected to a circuit for simulation. In this example, the left and right edges of the region will be used to define contacts A and B, respectively, with the default contact parameters applied.

Define 2 contacts, A on the left of the device and B on the right as can been seen in the image above.

• Define first contact:

  1. From the Menu, select Define → Contact.
  2. Using the cursor, hover the cursor over the left geometric edge that will make up the contact. When the edge is highlighted in green and the cursor changes to indicate a selectable element, left-click to select the edge.
  3. Next right-click anywhere to open the properties dialog for the contact. Use this dialog to set the contact's properties.
  4. In the General section of the Contact Properties window, set the Name to A and the Colour to red (note: Leave the colour as red if it is already red)
  5. Leave all of the other contact properties as the default values.
  6. Click OK.

• Define second contact. Follow steps 1-6 above, but instead:

  1. Select the right geometric edge.
  2. In the Contact Properties window, set the Name to B and the Colour to green.
  3. Click OK.

tip

For more detailed instructions on defining contacts, click here.

4.1.4. Define the Finite Element Mesh

For a simple resistor with uniform material properties and no junctions, a coarse mesh is sufficient. It provides a good balance between simulation accuracy and computational efficiency, avoiding unnecessary refinement that would increase computation time without significant benefit.

Define the Mesh Construction Grid:

• In the menu toolbar at the top select: Mesh → Define Mesh Construction Grid.

• Firstly vertical grid lines will be added.
  • In the Mesh Construction Grid window click Add.
  • The Mesh Grid Lines window will open. Set the below properties:
    • Set the Orientation to Vertical.
    • Set the Interval to Fixed.
    • Set the Coordinates of the box that will contain the mesh line to:
      • X1 = 0
      • X2 = 10
      • Y1 = 1
      • Y2 = 0
    • Set the Spacing Between Lines to 1.0 microns.
  • Click OK
• The grid lines will be set as in the image below:

• Next the horizontal grid lines will be added.
  • In the Mesh Construction Grid window click Add again.
  • Set the properties to the same values as above but change the Orientation to Horizontal.
• The Horizontal grid lines will be added.
• click x in the top right corner to close the Mesh Construction Grid window.

tip

For more detailed instructions on defining the mesh construction grid, click here.

Generate the Finite Element Mesh:

• In the Menu Bar, select Mesh → Generate Finite Element Mesh Model. The Mesh Properties window will open.
• No refinement is required for this model, so click OK to generate the mesh with default settings.

tip

For more detailed instructions on generating the finite element mesh, click here.

4.1.5. Save the Device Model

• In the Menu Bar, select File → Save As.... The Save dialog will open.
• Navigate to the folder you wish to save the device in.
• Specify the filename (e.g. Resistor.sol).
• Click Save to store the device model.

4.2 Steady State Simulation

Next the resistor will be used in a simple steady state example.

• On the Start Page Click New Simulation (the second option).
• The Circuit Editor will open with an empty workspace.

4.2.1 Create Circuit

The circuit must be designed before simulation can begin.

A device (the resistor), a DC Voltage Source and a Ground will be added to the circuit editor and connected together.

4.2.1.1 Add Device

• Select the Device from the tool bar and drag and drop it onto the circuit editor.
• Set the device Properties:
  • Click Get .sdm File, select the device file that you created in the previous steps and click OK.
  • Ensure Scaling Factor (cm) is set to 0.0001. Which sets the device depth to 1$\mu$m depth.
  • Click OK to close the Device Properties.
• Note: The device can be rotated by left clicking on it to select it and then using ctrl + r to rotate it by $90^\circ$.

tip

For more detailed instructions on Adding Components, click here.

4.2.1.2 Add Source & Ground

• Select the DC Voltage Source from the tool bar by dragging and droping it onto the circuit editor.
• Select the Ground from the tool bar by dragging and droping it onto the circuit editor.
• Connect the components togeter as they are in the image below.

tip

For more detailed instructions on Wiring Circuits, click here.

4.2.1.3 Set Source Properties

• Double click on the DC Voltage Source to open its properties.
• Click Add Range to open the Add Range Properties and set to the values below:
  • Start Voltage (V) = 0
  • End Voltage (V) = 100
  • Step (V) = 1
  • Click OK to set the range.
• Click OK to set the DC Voltage Source Properties.

tip

For more detailed instructions on Steady State (DC) Sources, click here.

• Save the simulation at this point. The circuit will be saved in a .sol solution file format.

4.2.2 Run Simulation

To run the Steady State Simulation press the blue run button, alternatively in the menu use Simulation → Steady State.

Note: If the file has not been saved yet, there will be a prompt at this point to save before the simulation can run.

tip

For more detailed instructions on Running a Simulations, click here.

The simulation will begin and the Simulation Status will appear on the screen. Wait until the simultion has completed at which point Simulation Stopped will apear in the bottom left of the status window and the text Aquarius simulator completed execution will be show in the screen.

tip

For more detailed instructions on the Simulation Status output, click here.

4.2.2 Visualise outputs

Next results visualiser will be used to understand the output of the simulation.

• On the Start Page Click Open Results Visualiser (the third option).
• The Results Visualiser will ask the user to select a .res results file.
• Then the Results Visualiser will open with the selected results file loaded.

The resistance was earlier calculated and is expected to be: $R = 62.5k\,\Omega$, therfore, a linear relationship between the voltage and current is expected.

• Click External Plot at the top of the results visualiser.
• Set External Plot Settings:
  • The file should match your results file name.
  • X Axis Contact = DEV1.A
  • X Axis Variable = Voltage
  • Y Axis Contact = DEV1.A
  • X Axis Variable = Total Current
  • Click check box to turn on Show Tick Marks
• Click New Plot to generate a graph showing Total Current (A) at terminal A of DEV1 (the resistor's name) versus the Voltage (V) at the same terminal. The plot demonstrates a linear relationship between voltage and current, as expected.

Using simple Ohm's law the expected current $I$ through the device at a voltage $V$ can be calculated

$$I = \frac{V}{R} = \frac{100V}{62.5k\,\Omega} = \frac{100}{62.5 \times 10^{3}} = 1.6mA$$

It can be seen on the plot that the Total Current at 100V is approimatley $1.6mA$ as expected.

This concludes the Resistor example tutorial.