In this example, we simulate ohmic heating arising from current flow in a wire. The simulation results are compared with analytic calculations.

## Simulation setup

The file heated_wire.ldev has an Aluminum wire connected between two copper leads. The entire structure is embedded inside SiO_{2} (glass). The voltage boundary condition in HEAT is used to apply a bias voltage between the two leads. The voltage results in a current flow through the wire that results in an ohmic power loss in the wire which is dissipated as heat. The voltage on the left lead is swept from 0 to 1 V and its effect on the current flow and on the slabs temperature profile can be observed. The top, bottom surfaces and the left and right surfaces are used to provide a fixed temperature boundary condition to the simulation region by setting them to 300 K.

## Results and discussion

Run the simulation and right-click on the HEAT solver region to visualize the 'thermal' dataset. The 'thermal' dataset will provide the heat (Q) generated in the wire as a function of voltage as well as the 3D temperature profile of the system as a function of voltage. To plot the heat generation and temperature profile on the plane where the wire is placed, go to the chart settings and select 'clipped plane' from the Data Visualization options. The following plots of heat generation and temperature distribution can be observed at different bias voltages.

Heat generated from current flow in the wire (W/m3) under different bias voltages.

Temperature profile on the 2D plane containing the wire under different bias voltages.

The resistance of the wire can be calculated from its dimension and conductivity. The conductivity (sigma) of Aluminum here is ~3.4*10^{7} S/m. The wire has a cross-section (A) of 1 um x 1 um and a length (L) of 50 um. The resistance of the wire is given by,

$$ R=L / {\sigma A} $$

This equates to a resistance of ~1.47 ohm. This value is confirmed by the slope of the I-V characteristics which can be seen by visualizing the 'boundaries' dataset from the results in the HEAT solver region and by selecting the 'I_bias_right' attribute. The same dataset also gives the heat flux flowing out of the simulation region through the boundaries which combined gives the total amount of heat flux generated by the wire as a function of voltage. Use the script file heated_wire_current_power_plot.lsf to add the heat flux at all the boundaries and plot it as a function of voltage. The resulting power curve is quadratic in nature since power lost due to ohmic loss is given by P = V^{2} / R. The script command also plots the analytic heat power generated from ohmic loss in a 1.47 ohm wire.