Question

Question: - For a small temperature change. a material's resistivity (reciprocal of conductivity) will change linearly according to

$\mathit{p}\left(dT\right)\mathbf{=}{\mathit{\rho }}_{\mathbf{0}}\mathbf{+}\mathit{d}\mathit{\rho }\mathbf{=}{\mathit{\rho }}_{\mathbf{0}}\left(1+\alpha dT\right)$

The fractional change in resistivity, $\mathit{\alpha }$also known as the temperature coefficient, is thus

$\mathit{\alpha }\mathbf{=}\frac{\mathbf{1}}{{\mathbf{\rho }}_{\mathbf{0}}}\frac{\mathbf{d}\mathbf{\rho }}{\mathbf{d}\mathbf{T}}$

Estimate for$\mathit{\alpha }$ silicon at room temperature. Assume a band gap of 1.1 e v .

Short answer

Answer: -

The fractional change in resistivity$\alpha$ for silicon at room temperature is $-0.07K^{-1}$.

Step-by-step solution

- Deriving Fractional Change in Resistivity

The number of excited electrons in silicon is given by the following formula:

$N_{excited}=D{k}_{B}T{e}^{-\frac{E_{gap}}{\left(2k_{B}T\right)}}$

Here, Boltzmann's constant is$k_B$ , T the temperature is .

This roughly describes the conductivity. So the resistivity is calculated by the following formula:

$\rho =\frac{1}{D{k}_B{T}_0}{e}^{\frac{E_{gap}}{2k_{B}T}}$

For simplicity, we can take to be constant at room temperature for the part that is not exponential, since the exponential will dominate the change.

The fractional change in resistivity is given by:

$$\begin{gathered} \alpha =\frac{1}{\rho }\frac{d\rho }{dT} \\ =D{k}_B{T}_0{e}^{-\frac{E_{gap}}{2k_{B}T}}\frac{1}{D{k}_B{T}_0}\frac{d{e}^{\frac{E_{gap}}{2k_{B}T}}}{dT} \\ =e^{-\frac{E_{gap}}{2k_{B}T}}{e}^{\frac{E_{gap}}{2k_{B}T}}\frac{E_{gap}}{2k_B}{\left.\left(-\frac{1}{T^2}\right)\right|}_{T=300K} \\ \alpha =\frac{E_{gap}}{2k_B}\left(-\frac{1}{T^2}\right) \end{gathered}$$

- Calculating Fractional Change in Resistivity

At 300K the fractional change in resistivity is calculated by substituting for 1.1 e v, $E_{gap},1.38\times {10}^{-23}J/K$, and 300K for in the above equation.

$$\begin{gathered} \alpha =\frac{E_{gap}}{2k_B}\left(-\frac{1}{T^2}\right) \\ =\frac{\left(1.1\text{eV}\right)\left(1.6\times {10}^{-10}\text{J/eV}\right)}{2\left(1.38\times {10}^{-23}\text{J/K}\right)}\left(-\frac{1}{{\left(300\text{K}\right)}^2}\right) \\ \alpha =-0.07K^{-1} \end{gathered}$$

Thus, the fractional change in resistivity $\alpha$for silicon at room temperature is $-0.07K^{-1}$.