Section 8.6 counted the states of a three-dimensional free-electron gas by filling a…

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure

Condensed Matter Physics, Ch The free-electron gas: what Drude gets right and what only Fermi fixes

Problem

Section 8.6 counted the states of a three-dimensional free-electron gas by filling a sphere in

\mathbf{k}

space. Suppose instead the electrons are confined to a thin two-dimensional sheet of side

L

(a metallic film only a few atoms thick), so the allowed

\mathbf{k}

form a square grid of spacing

2\pi/L

in a plane. Count the states inside a circle of radius

k

the way Section 8.6 counted them inside a sphere, convert to a function of energy using

E=\hbar^2k^2/2m

, and show that the two-dimensional density of states per unit area,

g_{\rm 2D}(E)

, does not depend on

E

at all. Give

g_{\rm 2D}(E)

in terms of

m

and

\hbar

Browse the problem library