Calculating the number of quantum states in the valence and conduction bands

In summary, to calculate the number of quantum states and electrons per cm^3 of a crystal at room temperature, you need to know the number of atoms/cm^3 and the number of electrons per atom, which is determined by the valence state of the elements in the lattice. The number of quantum states can be calculated using the coordination number of the lattice, and the number of electrons per cm^3 is simply the number of atoms/cm^3 multiplied by the number of electrons per atom.
  • #1
jisbon
476
30
Homework Statement
Calculate the number of quantum states per cm cube in GaP in the valence and conduction band, given that the mass density is 4.13g/cm^3
Relevant Equations
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Hi all,
This question asks me to calculate the number of quantum states, as well as electrons per cm^3 of the crystal in the room temperature.
The problem is I only dealt with a single element before without any calculation for 1cm^3 whatsoever. For example for a Silicon semiconductor, I can determine that a-Si atom share four valence electrons (2 from its 3s and 2 from 3p states) to form a stable configuration, hence only the 3s and 3p states are involved, whereby the number of states, in this case, is 8N (2+6) from the 3s and 3p state, whereby the upper band contains 4N states and lower band aka the valence band contain 4N states.

So in the question I presented, I will need to calculate the number of atoms/cm^3 to find what is N first. Hence, first, amt of GaP = mass/Mr. Hence in 1cm^3, I can find that the amount = 4.13/(69.72+30.974) = 0.0410 mol = 2.469*10^22 atoms. So N = 2.469*10^22

Now here comes the problem, how many N states are there? Since now there are 2 elements, how do I combine them together to form a lattice if GaP does not have any spare valence electrons to bond with other GaP atoms to form the crystal lattice? How can I find what are the states involved in GaP for bonding with other GaP atoms?

Thanks
 
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  • #2
in advance.The number of quantum states depends on the crystal lattice structure of the GaP material. You can calculate it from the number of atoms/cm^3 and the number of electrons per atom. The number of electrons per atom is determined by the valence state of the two elements. In this case, Ga has 3 valence electrons and P has 5, so the total number of electrons in a GaP lattice would be 8 per atom.To calculate the number of quantum states, you need to know the number of sites in the unit cell of the lattice. This is usually determined by the coordination number of the lattice, which indicates how many nearest neighbors each atom has. For example, in a simple cubic lattice, each atom has 6 nearest neighbors and the coordination number is 6.Once you have the coordination number, you can calculate the number of quantum states as follows:Number of quantum states = N * (number of electrons per atom)^(coordination number)where N is the number of atoms/cm^3.In the case of GaP, the coordination number is 8, so the number of quantum states is 2.469*10^22 * 8^8 = 9.42*10^30 states.The number of electrons per cm^3 is simply the number of electrons per atom multiplied by the number of atoms/cm^3, which in this case is 2.469*10^22 * 8 = 1.97*10^24 electrons/cm^3.
 

Related to Calculating the number of quantum states in the valence and conduction bands

1. How do you calculate the number of quantum states in the valence and conduction bands?

The number of quantum states in the valence and conduction bands can be calculated using the formula N = 2 x (2J + 1), where N is the number of states and J is the total angular momentum quantum number.

2. What is the significance of calculating the number of quantum states in the valence and conduction bands?

Calculating the number of quantum states in the valence and conduction bands is important for understanding the electronic properties of a material. It can help determine the conductivity, band gap, and other important characteristics.

3. How does the number of quantum states in the valence and conduction bands vary among different materials?

The number of quantum states in the valence and conduction bands can vary greatly among different materials. It depends on factors such as the crystal structure, atomic arrangement, and number of valence electrons.

4. Can the number of quantum states in the valence and conduction bands be manipulated?

Yes, the number of quantum states in the valence and conduction bands can be manipulated by changing the material's composition, doping, or applying external electric or magnetic fields.

5. Are there any limitations to calculating the number of quantum states in the valence and conduction bands?

Calculating the number of quantum states in the valence and conduction bands is based on theoretical models and assumptions, so there can be limitations in accurately predicting the exact number of states in a material. Additionally, the calculations become more complex for materials with more complex crystal structures.

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