Understanding Drift Current Density Calculation in Silicon

This problem asks us to find the drift current density ($J$) in silicon given its electrical conductivity ($\sigma$) and the applied electric field ($E$). We are provided with the following values:

• Conductivity, $\sigma = 10 \text{ } (\Omega \text{ cm})^{-1}$
• Electric Field, $E = 15 \text{ V/cm}$

Key Concept: Conductivity and Current Density

The drift current density in a material is directly related to its conductivity and the electric field applied across it. This relationship is a fundamental concept in semiconductor physics and is derived from Ohm's Law.

Formula Used

The formula that connects drift current density ($J$), conductivity ($\sigma$), and electric field ($E$) is:

$ J = \sigma E $

Where:

• $J$ is the drift current density (measured in Amperes per square centimeter, $A/cm^2$).
• $\sigma$ is the electrical conductivity of the material (measured in $(\Omega \text{ cm})^{-1}$ or Siemens per centimeter, $S/cm$).
• $E$ is the electric field strength (measured in Volts per centimeter, $V/cm$).

Step-by-Step Calculation

Now, let's substitute the given values into the formula:

1. Identify Given Values:
   • $\sigma = 10 \text{ } (\Omega \text{ cm})^{-1}$
   • $E = 15 \text{ V/cm}$
2. Apply the Formula: Substitute these values into the equation $J = \sigma E$.

   $ J = \left( 10 \text{ } (\Omega \text{ cm})^{-1} \right) \times \left( 15 \text{ V cm}^{-1} \right) $

3. Calculate the Result: Multiply the numerical values and determine the units.

   The numerical calculation is straightforward:

   $ 10 \times 15 = 150 $

   Let's check the units. We know that conductivity $\sigma$ has units of Siemens per centimeter ($S/cm$), where $1 \text{ S} = 1/\Omega$. The electric field $E$ has units of Volts per centimeter ($V/cm$). The unit of current density $J$ is Amperes per square centimeter ($A/cm^2$).

   The unit analysis is as follows:

   $ (\Omega \text{ cm})^{-1} \times \text{V cm}^{-1} = \frac{1}{\Omega \cdot \text{cm}} \times \frac{\text{V}}{\text{cm}} $

   Since Ohm's Law relates Voltage, Current, and Resistance ($V = IR$, so $R = V/I$), the unit of resistance $\Omega$ is $V/A$. Thus, $1/\Omega = A/V$. Using this, the unit becomes:

   $ \frac{1}{\Omega} \times \frac{1}{\text{cm}} \times \frac{\text{V}}{\text{cm}} = \frac{A}{V} \times \frac{1}{\text{cm}} \times \frac{\text{V}}{\text{cm}} = \frac{A}{\text{cm}^2} $

   So, the units correctly result in $A/cm^2$. Combining the numerical value and the units:

   $ J = 150 \text{ A cm}^{-2} $

4. Final Answer: The calculated drift current density is $150 \text{ A/cm}^2$.

Conclusion

By applying the formula $J = \sigma E$, we found that the drift current density in the silicon material under the given electric field is $150 \text{ A/cm}^2$. This calculation highlights how conductivity directly influences the current flow in response to an applied electric field.