Math Problem Statement

Consider a silicon semiconductor doped with 5 x 10^16 cm βˆ’ 3 phosphorus atoms at 300K. The intrinsic carrier concentration of silicon is 𝑛 𝑖 = 1.5 Γ— 10^10 cm^ βˆ’3 . Calculate: The electron concentration 𝑛 is _____ cm^-3 . and the position of the Fermi level relative to the intrinsic Fermi level E Fi is _____ eV .

4.5 x 10^3 and 0.388    5 x 10^16 and 0.388    5 x 10^16 and 0.126   4.5 x 10^3 and 0.126

Solution

To solve this problem, we need to calculate two quantities:

  1. The electron concentration nn in the silicon semiconductor.
  2. The position of the Fermi level relative to the intrinsic Fermi level EFiE_{Fi}.

Given Data:

  • Doping concentration ND=5Γ—1016 cmβˆ’3N_D = 5 \times 10^{16} \, \text{cm}^{-3} (phosphorus atoms).
  • Intrinsic carrier concentration ni=1.5Γ—1010 cmβˆ’3n_i = 1.5 \times 10^{10} \, \text{cm}^{-3}.
  • Temperature T=300 KT = 300 \, \text{K}.
  • Silicon is an n-type semiconductor (since it is doped with phosphorus, which donates electrons).

Step 1: Calculate the electron concentration nn.

For an n-type semiconductor, the electron concentration nn can be approximated as:

nβ‰ˆNDn \approx N_D

This is because in an n-type semiconductor, the majority carrier concentration (electrons) is approximately equal to the donor concentration when the doping is not too high and the temperature is not too low.

Thus:

n=5Γ—1016 cmβˆ’3n = 5 \times 10^{16} \, \text{cm}^{-3}

Step 2: Calculate the position of the Fermi level relative to the intrinsic Fermi level EFiE_{Fi}.

The shift in the Fermi level relative to the intrinsic Fermi level EFiE_{Fi} can be calculated using the formula:

EFβˆ’EFi=kTln⁑(nni)E_{F} - E_{Fi} = kT \ln\left(\frac{n}{n_i}\right)

Where:

  • kk is Boltzmann's constant, 8.617Γ—10βˆ’5 eV/K8.617 \times 10^{-5} \, \text{eV/K}.
  • T=300 KT = 300 \, \text{K}.
  • n=5Γ—1016 cmβˆ’3n = 5 \times 10^{16} \, \text{cm}^{-3}.
  • ni=1.5Γ—1010 cmβˆ’3n_i = 1.5 \times 10^{10} \, \text{cm}^{-3}.

Substitute the values:

EFβˆ’EFi=8.617Γ—10βˆ’5Γ—300Γ—ln⁑(5Γ—10161.5Γ—1010)E_{F} - E_{Fi} = 8.617 \times 10^{-5} \times 300 \times \ln\left(\frac{5 \times 10^{16}}{1.5 \times 10^{10}}\right)

Calculate the argument of the logarithm:

5Γ—10161.5Γ—1010=3.33Γ—106\frac{5 \times 10^{16}}{1.5 \times 10^{10}} = 3.33 \times 10^{6}

Now, take the natural logarithm:

ln⁑(3.33Γ—106)β‰ˆln⁑(3.33)+ln⁑(106)=1.203+13.816=15.019\ln(3.33 \times 10^{6}) \approx \ln(3.33) + \ln(10^{6}) = 1.203 + 13.816 = 15.019

Now calculate EFβˆ’EFiE_F - E_{Fi}:

EFβˆ’EFi=8.617Γ—10βˆ’5Γ—300Γ—15.019E_{F} - E_{Fi} = 8.617 \times 10^{-5} \times 300 \times 15.019

EFβˆ’EFiβ‰ˆ0.388 eVE_{F} - E_{Fi} \approx 0.388 \, \text{eV}

Final Answer:

  • Electron concentration nn: 5Γ—1016 cmβˆ’35 \times 10^{16} \, \text{cm}^{-3}
  • Position of the Fermi level EFβˆ’EFiE_{F} - E_{Fi}: 0.388 eV0.388 \, \text{eV}

The correct option from the choices given is:

Option: 5Γ—1016 cmβˆ’35 \times 10^{16} \, \text{cm}^{-3} and 0.388 eV0.388 \, \text{eV}

Would you like any further explanations or details?

Relative Questions:

  1. How does doping concentration affect the position of the Fermi level in a semiconductor?
  2. What is the significance of the intrinsic carrier concentration in semiconductors?
  3. How does the Fermi level shift for p-type semiconductors?
  4. Why is the Fermi level important in understanding the electrical properties of semiconductors?
  5. How does temperature affect the Fermi level in semiconductors?
  6. What assumptions are made when calculating the electron concentration in an n-type semiconductor?
  7. How would the calculations change if the semiconductor were heavily doped?
  8. What happens to the Fermi level in intrinsic semiconductors?

Tip:

In semiconductor physics, always check whether the doping concentration significantly exceeds the intrinsic carrier concentration; this will help you determine if the majority carrier concentration is approximately equal to the doping level.

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Math Problem Analysis

Mathematical Concepts

Semiconductor Physics
Carrier Concentration
Fermi Level

Formulas

Electron concentration in n-type semiconductor
Fermi level shift formula

Theorems

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Suitable Grade Level

Undergraduate