- #1

cepheid

Staff Emeritus

Science Advisor

Gold Member

- 5,192

- 36

## Homework Statement

Find the density of states for a 2D electron gas.

## Homework Equations

See attempted solution below

## The Attempt at a Solution

Assume that in real space the gas is confined to an area

[tex] A = l_xl_y [/tex]

Write the components of the electron wavevector in terms of the respective principle quantum numbers:

[tex] k_x = \frac{2\pi}{l_x}n_x \ \ \ \ k_y = \frac{2\pi}{l_y}n_y [/tex]

Therefore the number of states associated with an element [itex] d^2\mathbf{k} [/itex] is (supposedly)

[tex] 2dn_xdn_y = \frac{A}{4\pi^2}2dk_xdk_y [/tex]

Again, in an analogous way to what was done in the notes in 3D, I switch to polar coordinates, so that I can get the density of states as a function of [itex] k = |\mathbf{k}| [/itex].

# of states between k and k + dk

[tex] Z(k)dk = \frac{A}{4\pi^2}2(2 \pi k dk) [/tex]

Change variables to convert Z(k) to D(E), the density of states as a function of energy.

[tex] E = \frac{\hbar^2k^2}{2m} \Rightarrow dE = \frac{\hbar^2}{2m}2kdk \Rightarrow dk = \frac{m}{\hbar^2 k}dE [/tex]

[tex] D(E)dE = \frac{A}{4\pi^2} 2(2 \pi k) \frac{m}{\hbar^2 k}dE = \frac{Am}{\pi \hbar^2} dE [/tex]

[tex] \Rightarrow D(E) = \textrm{const.} = \frac{Am}{\pi \hbar^2} [/tex]

This result is drastically different from the 3D case, and later on in the same problem, it leads me to the conclusion that the Fermi energy is independent of temperature for the 2D gas. Is this solution correct, or have I made some egregious error somewhere?