Question

2. 2D Density of States (~Kasap 4.9)

Kasap Section 4.5 (in 4th Ed.) derives an expression for the density of states for electrons in a 3D infinite potential well (3D Sommerfeld box). This derivation is similar to the one we did in class.

a. Following the approaches used in the 3D derivations, consider an electron confined to a 2D infinite potential well and derive an expression for the 2D density of states, g_2D, which is the number of states per unit area per unit energy with energy E between E and E+dE.

(Hint: you might start by solving the 2D Schrödinger Eqn. to determine the allowed k-space values, which you could then plot and use to guide your derivation.)

Answer 1

For 2D potential well extanding from 0<x<a, 0<y<b, the Schrodinger's equation is given by:

(inside the well as V=0)

and the wavefuction is zero outside.

Let,

Now, as x and y are independent, E can be written as Ex+Ey

and

Let,

Then the equation will have solution of the type

As at x=0

So, B=0

and for x=a

Hence,

Similarly,

Area occupied in k space upto k= (As kx and ky can have only positive values)

Now, area occupied by each allowed value =

So, no. of allowed states,

where 2 is because of spin degeneracy of electrons

Now,

So, no. of states per unit area per unit energy level