To determine the electric field in a semiconductor with a spatially varying donor concentration $N_d(x)$, we use the relationship derived from the drift-diffusion equation in thermal equilibrium. The electric field $E(x)$ can be calculated using the following formula:

$E(x) = \frac{kT}{q} \frac{d(\ln N_d(x))}{dx}$

Where:

• $k$ is the Boltzmann constant ($1.38 \times 10^{-23}$ J/K),
• $T$ is the temperature (given as 300 K),
• $q$ is the charge of an electron ($1.6 \times 10^{-19}$ C),
• $N_d(x)$ is the donor concentration as a function of position.

Given $N_d(x) = N_0 \exp(-10x)$, we can proceed as follows:

1. First, calculate the logarithm of $N_d(x)$:

$\ln(N_d(x)) = \ln(N_0 \exp(-10x)) = \ln(N_0) - 10x$

2. Take the derivative of $\ln(N_d(x))$ with respect to $x$:

$\frac{d(\ln N_d(x))}{dx} = -10$

3. Substitute the values into the electric field equation:

$E(x) = \frac{kT}{q} \times (-10)$

Given $T = 300$ K, the constants are:

$\frac{kT}{q} = \frac{(1.38 \times 10^{-23} \text{ J/K}) \times (300 \text{ K})}{1.6 \times 10^{-19} \text{ C}} \approx 0.0259 \text{ V}$

Thus,

$E(x) = 0.0259 \times (-10) = -0.259 \text{ V/m}$

The magnitude of the electric field is:

$|E(x)| \approx 0.26 \text{ V/m}$

This corresponds to option (c) 0.26 V/m.

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Do you want any more details or have any questions?

Here are some related questions:

1. How does a varying donor concentration affect the electric field in a semiconductor?
2. What is the significance of thermal equilibrium in semiconductors?
3. How does the temperature influence the electric field in a semiconductor?
4. What are the implications of a negative electric field value in this context?
5. Can you explain the drift-diffusion model in more detail?
6. How would the electric field change if the temperature were different?
7. What would happen to the electric field if the donor concentration had a different spatial variation?
8. How do you calculate the potential difference across the semiconductor from the electric field?

Tip: Always consider the temperature dependence when calculating electric fields in semiconductors, as it plays a crucial role in determining the electric field magnitude.