Question

The diffusion coefficient for aluminum in silicon is Dₐₗ =4×10⁻¹⁴ cm² /s at 800 K. What is a reasonable value for Dₙₗ at 400 K ?

Answer 1

A reasonable value for the diffusion coefficient
`(\(D_{\text{Al}}\))` of aluminum in silicon at 400 K can be estimated using the Arrhenius equation. Assuming an activation energy (Q) of 1.2 eV, the diffusion coefficient at 400 K
`(\(D_{\text{Al,400K}}\))` is approximately
`\(1 * 10^(-17) \, \text{cm}^2/\text{s}\)`.

Step-by-step explanation:

The Arrhenius equation relates the diffusion coefficient (D) to temperature (T) through the formula
`\(D = D_0 * \exp\left(-(Q)/(RT)\right)\)`, where \(D_0\) is a pre-exponential factor, (Q) is the activation energy, (R) is the gas constant, and (T) is the absolute temperature. To estimate
`\(D_{\text{Al,400K}}\)`, we can rearrange the equation as
`\(D_{\text{Al,400K}} = D_{\text{Al}} * \exp\left((Q)/(R)\left(\frac{1}{400 \, \text{K}} - \frac{1}{800 \, \text{K}}\right)\right)\)`.

Given
`\(D_{\text{Al}} = 4 * 10^(-14) \, \text{cm}^2/\text{s}\)` at 800 K, assuming (Q = 1.2) eV, and using
`\(R \approx 8.314 \, \text{J/(mol} \cdot \text{K)}\)`, we find
`\(D_{\text{Al,400K}}\)` to be approximately
`\(1 * 10^(-17) \, \text{cm}^2/\text{s}\)`. This lower value at 400 K is expected due to the exponential temperature dependence of diffusion coefficients.

In materials science, such calculations are crucial for understanding diffusion processes at different temperatures, providing insights into material behavior and aiding in the design and optimization of electronic devices and other applications.