Designing a Silicon Resistor

To design a silicon resistor of resistance 5 kOhms that can handle a current density of 10 A/cm2 at 5 V, the following steps can be taken:

Determine the Cross-Sectional Area of the Resistor

The current density is given by J = I/A, where I is the current and A is the cross-sectional area of the resistor. We know that the current density should not exceed 10 A/cm2. Therefore, we can calculate the minimum cross-sectional area required for the resistor as follows:

J = 10 A/cm2 (given)
I = 5 V / 5 kOhms = 1 mA (Ohm's Law)
A = I/J = 1 mA / 10 A/cm2 = 0.1 cm2

Calculate the Dimensions of the Resistor

Knowing the cross-sectional area, we can now calculate the dimensions of the resistor. Assuming a rectangular shape for the resistor, we can choose a length and width that will give us the required cross-sectional area. For example, if we choose a length of 1 cm, the width would be 0.1 cm:

A = L x W
0.1 cm2 = 1 cm x W
W = 0.1 cm

Check for Thermal Stability

Finally, we need to check if the resistor will be thermally stable at the maximum current density of 10 A/cm2. The power dissipated by the resistor can be calculated using P = I2R, where R is the resistance of the resistor. Therefore:

P = (1 mA)2 x 5 kOhms = 5 mW

This power dissipation is well below the maximum power the resistor can handle, so it will be thermally stable.

Conclusion

To design a silicon resistor of resistance 5 kOhms that can handle a current density of 10 A/cm2 at 5 V, we need a rectangular resistor with dimensions of 1 cm x 0.1 cm. This resistor will have a minimum cross-sectional area of 0.1 cm2 and will be thermally stable at a power dissipation of 5 mW.