Fraction of unoccupied electron states

In summary, to find the fraction of unoccupied electron states at the top of the valence band and the fraction of occupied states at the bottom of the conduction band, you can use the Fermi-Dirac distribution function with the Fermi energy set to the appropriate energy levels.
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Homework Statement



2) For an intrinsic (undoped) semiconductor at room temperature with the Fermi energy in the center of the 1 eV band gap, find the fraction of unoccupied electron states at the top of the valence band and the fraction of occupied states at the bottom of the conduction band.



Homework Equations



I am looking for a push in the right direction for this problem. I have never encountered this before and we do not have a book as this is a lab class and I had to miss the lecture because my son was sick and had to stay home from school.

I found an equation for the Fermi Energy which is: E_F=E_{N/2}-E_0=(hbar^2 pi^2)/(2 m L^2) (N/2)^2


The Attempt at a Solution



I am not sure how to use the Fermi energy to find the occupied and unoccupied fraction of electron states.

Ok found some more information. The Fermi function is described by:

f(E)=1/(e^[(E-E_f)/kT]+1)

where E_f is the Fermi energy. This equation is supposed to give you the probability that electrons will exist above the Fermi level at a given temperature. I just don't know which values to use for E and E_f. Any suggestions?

I think I figured it out using the population of conduction band equation I found:

N_cb = AT^(3/2)e^(-E_gap/(2kT))

going with this to find the fraction. Sound good to anyone? Ended up getting 2.09x10^-5 as the fraction of unoccupied electron states in the valence band and occupied states in the conduction.
 
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Thank you for sharing your thoughts and attempts at solving this problem. It seems like you have made some good progress in finding the relevant equations and understanding the concept of Fermi energy.

To find the fraction of unoccupied electron states at the top of the valence band and the fraction of occupied states at the bottom of the conduction band, you can use the Fermi-Dirac distribution function. This function describes the probability that a state at a given energy level will be occupied by an electron at a certain temperature.

The general form of the Fermi-Dirac distribution function is:

f(E) = 1 / (e^[(E-E_f)/kT] + 1)

Where E_f is the Fermi energy, E is the energy of the state, k is Boltzmann's constant, and T is the temperature in Kelvin.

To find the fraction of unoccupied states at the top of the valence band, you can use the Fermi-Dirac distribution function with E_f set to the bottom of the conduction band (since the Fermi energy is in the center of the band gap). This will give you the probability that a state at the top of the valence band will be occupied. To find the fraction of unoccupied states, you can subtract this probability from 1.

Similarly, to find the fraction of occupied states at the bottom of the conduction band, you can use the Fermi-Dirac distribution function with E_f set to the top of the valence band (since the Fermi energy is in the center of the band gap). This will give you the probability that a state at the bottom of the conduction band will be unoccupied.

I hope this helps guide you in the right direction. Good luck with your calculations!
 

Related to Fraction of unoccupied electron states

What is the concept of "Fraction of unoccupied electron states"?

The fraction of unoccupied electron states refers to the percentage of energy levels in a material that are not filled with electrons. This quantity is important in understanding the electrical and thermal properties of materials.

How is the fraction of unoccupied electron states calculated?

The fraction of unoccupied electron states is calculated by dividing the number of unoccupied energy levels by the total number of energy levels in the material. This can be determined using mathematical models or experimental techniques such as spectroscopy.

What factors affect the fraction of unoccupied electron states in a material?

The fraction of unoccupied electron states in a material is influenced by several factors, including temperature, the number of available energy levels, and the presence of impurities or defects in the material. Additionally, the band structure and electronic properties of the material can also impact this quantity.

Why is the fraction of unoccupied electron states important in materials science?

The fraction of unoccupied electron states is an important parameter in understanding the electrical and thermal conductivity of materials. It also plays a significant role in determining the material's optical and magnetic properties. Additionally, this quantity is crucial in the design and development of electronic devices and materials for energy applications.

How can the fraction of unoccupied electron states be controlled or manipulated?

The fraction of unoccupied electron states can be controlled and manipulated through various techniques such as doping, alloying, and applying external stimuli like pressure or electric fields. These methods alter the energy levels and band structure, thereby changing the fraction of unoccupied electron states and the material's overall properties.

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