Consider a semiconductor in thermal equilibrium (no current). Assume that the donor concentration varies as 

Nd(x)Nd(x)

=

N0N0

exp(-10x) over the range 0

≤≤

x

≤≤

0.1 m ,where 

N0N0

 is constant . assume T=300K what is the value of electric field?

(a) 2.6 V/m
 
(b) 2.6 N0 V/m
 
(c) 0.26 V/m
 
(d) 0.52 N0 V/m

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.

Learn how to calculate the electric field in a semiconductor in thermal equilibrium with a varying donor concentration. Includes step-by-step derivation using semiconductor physics principles. Suitable for advanced college-level studies.

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