Let’s solve the question based on the given problem.

The question involves calculating:

1. The thermal-equilibrium hole concentration in silicon at $T = 400 \, \text{K}$.
2. The ratio of $p_0$ at $T = 250 \, \text{K}$ to that at $T = 400 \, \text{K}$.

Formula:

The intrinsic carrier concentration ($n_i$) for silicon is calculated using: $n_i = A T^{3/2} e^{-E_g / (2kT)}$ Where:

• $A$ is a material constant.
• $T$ is the temperature in kelvin.
• $E_g$ is the energy bandgap of silicon ($1.12 \, \text{eV}$).
• $k$ is Boltzmann's constant in eV ($8.617 \times 10^{-5} \, \text{eV/K}$).

The thermal-equilibrium hole concentration $p_0$ is approximately equal to $n_i$ for intrinsic silicon.

To compute the ratio of $p_0$ at different temperatures, the dependence of $n_i$ on $T$ and $E_g$ must be accounted for: $\text{Ratio} = \frac{p_0(T = 250)}{p_0(T = 400)} = \frac{n_i(T = 250)}{n_i(T = 400)} = \frac{T_1^{3/2} e^{-E_g / (2kT_1)}}{T_2^{3/2} e^{-E_g / (2kT_2)}}$

Substitute:

• $T_1 = 250 \, \text{K}$, $T_2 = 400 \, \text{K}$,
• $E_g = 1.12 \, \text{eV}$,
• $k = 8.617 \times 10^{-5} \, \text{eV/K}$.

We will first calculate $n_i(T)$ and then the ratio. Let me calculate this. math

Constants

Eg = 1.12 # Energy bandgap for silicon in eV k = 8.617e-5 # Boltzmann constant in eV/K

Temperatures

T1 = 250 # Temperature in Kelvin T2 = 400 # Temperature in Kelvin

Ratio calculation

def ni_ratio(T1, T2, Eg, k): # Numerator and denominator components for the ratio T1_factor = T1**(3/2) * math.exp(-Eg / (2 * k * T1)) T2_factor = T2**(3/2) * math.exp(-Eg / (2 * k * T2))

# Ratio of ni(T1) / ni(T2)
return T1_factor / T2_factor

Compute ratio

ratio = ni_ratio(T1, T2, Eg, k) ratioThe ratio of the thermal-equilibrium hole concentrations ($p_0$) at $T = 250 \, \text{K}$ to $T = 400 \, \text{K}$ is approximately:

$\text{Ratio} \approx 2.89 \times 10^{-5}$

Would you like a detailed explanation or a step-by-step breakdown of this computation?

Relative Questions:

1. How does temperature influence intrinsic carrier concentration in semiconductors?
2. What role does the bandgap energy ($E_g$) play in determining carrier concentrations?
3. Can this method be extended to calculate electron concentration ($n_0$)?
4. Why is the ratio so small when comparing different temperatures?
5. How can material constants affect the carrier concentration?

Tip:

For high-temperature applications, ensure that you consider both intrinsic and extrinsic factors affecting carrier concentrations.