Intrinsic carrier concentration where did I go wrong?

In summary, the conversation discusses the process of finding the free electron concentration in silicon and diamond at room temperature, given the intrinsic carrier concentration and bandgap. The speaker considers using the equation ##n_i = \sqrt{N_cN_v} \exp\left(\frac{-E_g}{2kT}\right)## and discusses the limitations of this approach. They also mention the importance of considering the densities of states at different temperatures. Ultimately, the speaker's professor assures them that the electron concentration and bandgap should be sufficient for solving the problem.
  • #1
Kara386
208
2

Homework Statement


I've looked up the intrinsic carrier concentration of silicon, and what I've got isn't close. The question says given there are ##2\times 10^{22}## electrons per cubic cm in silicon, and the bandgap is ##1.1##eV, what is the free electron concentration at room temperature?

Homework Equations

The Attempt at a Solution


My first thought is that since
##n_i = \sqrt{N_cN_v} \exp\left(\frac{-E_g}{2kT}\right)##
Maybe as T tends to infinity all electrons are free electrons so that ##\sqrt{N_cN_v}=n_i##, but then subbing that back in at room temp would mean ##10^{12}## free electrons per cubic cm, so too high, but then that would be assuming the densities of states are constant at different temperatures, and they're not.

For silicon I can get around this by just looking up ##N_c## and ##N_v##, but I have to repeat the process for diamond and the values aren't available. I do get the right order of magnitude estimate for diamond's carrier concentration just by using the same ##N_c## and ##N_v## as for silicon, presumably that's because they're similar in terms of crystal structure.

However, my professor assures me that all I should need is the electron concentration and the bandgap. I'd appreciate any help, been stuck on this for a while! :)
 
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  • #2
Kara386 said:
Maybe as T tends to infinity all electrons are free electrons so that ##\sqrt{N_cN_v}=n_i##, but then subbing that back in at room temp would mean ##10^{12}## free electrons per cubic cm, so too high, but then that would be assuming the densities of states are constant at different temperatures, and they're not.
How is the situation at T=0?
 

Related to Intrinsic carrier concentration where did I go wrong?

What is intrinsic carrier concentration?

Intrinsic carrier concentration is the concentration of charge carriers (either electrons or holes) that are present in a pure, undoped semiconductor material at thermal equilibrium.

What factors affect intrinsic carrier concentration?

Intrinsic carrier concentration is affected by temperature, bandgap energy, and effective mass of the charge carriers in the material. It also varies depending on the type of semiconductor material.

How is intrinsic carrier concentration calculated?

Intrinsic carrier concentration is calculated using the formula ni = (Nc * Nv) * exp(-Eg/kT), where ni is the intrinsic carrier concentration, Nc is the effective density of states in the conduction band, Nv is the effective density of states in the valence band, Eg is the bandgap energy, k is Boltzmann's constant, and T is the temperature in Kelvin.

What is the significance of intrinsic carrier concentration?

Intrinsic carrier concentration plays a crucial role in determining the electrical properties of a semiconductor material. It affects the material's conductivity, resistivity, and other important characteristics.

Where did I go wrong in calculating intrinsic carrier concentration?

There are several possible sources of error in calculating intrinsic carrier concentration, including incorrect values for Nc, Nv, Eg, or temperature, as well as incorrect use of units or incorrect calculation of the exponential term. It is important to double-check all values and calculations when determining intrinsic carrier concentration.

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