Calculating the Density of States Flashcards
What is the formula for Density Of States?
why is it multiplied by 2?
2 * 1/V * dN/dE
multiplied by 2 to account for spin degeneracy
What is the formula for N (number of available states)?
total k volume / volume of 1 k value
What is the formula for E in E vs k?
E = (h^2 * k^2)/2m
top of valence band and bottom of the conduction band both have parabola
What happens when you get a 3D k-space diagram?
the k-values form a sphere representing the available energy states for energy value E
What is the formula for the volume of the sphere in 3D k-space?
how do you derive the radius?
1/8 * 4pi/3 * k^3
radius is k, derived by rearranging the formula for E
check notes
What condition must be met of a cuboid used to examine the movement of an electron through k-space?
the cuboid must perfectly fit the electron’s wavelength so it doesn’t destructively interfere
how do you derive the volume of 1 k value?
- find the length of the cuboid the electron is moving through: L = n * (lambda/2)
- rearrange for lambda: lambda=2L/n
- plug lambda into the formula k=2pi/lambda: k = (n * pi)/L
- assuming n=1, we can cube this to find k in 3D space: k = pi^3/V
What is the formula for the number of states N between two energy levels?
N = total volume / volume occupied by 1 k value
this relates to the volume of the sphere and volume of k we found before
check notes
How is the density of states found?
2 * 1/V * dN/dE
How and why is the energy shifted in the DOS equation?
conduction band: E-Ec, valence band: Ev-E
for the energy difference between the bottom of CB & E or top of VB & E
How does the dependence on energy for DOS change for 3D, 2D, 1D & 0D?
Three-Dimensional (3D): DOS ∝ √E; increases with energy.
Two-Dimensional (2D): DOS is independent of energy, leading to a flat DOS curve.
One-Dimensional (1D): DOS ∝ 1/√E; inversely related to the square root of energy.
Zero-Dimensional (0D): DOS characterised by discrete spikes, represented as delta functions.
