# Solution to Exercise 2.1-2: Density of States for Lower Dimensions

| Calculate the density of states for a one-dimensional semiconductor ("quantum wire") and for the two-dimensional case.
|  | Draw some conclusions from the results.

| The number of states Z(k) up to a wave vector k is generally given by
Z(k) | = | Volume of "sphere" in m dimensions Volume of state
--- | --- | ---
|  | The volume Vm of a "sphere" with radius k in m dimension is
|  | æ ç ç ç ç è | Volume of sphere = 4/3 p · k3 | for m = 3
Vm(k) | = | Volume = area of circle = p · k2 | for m = 2
|  | Volume = length = 2k | for m = 1
| The density of states D(E) follows by substituting the variable k by E via the dispersion relation E(k) and by differentiation with respect to E.
|  | One obtains the following relations:
|  | æ ç ç ç ç è | (E)½ | for m = 3
Dm(E) | µ | const. | for m = 2
|  | (E)–½ | for m = 1
| The consequences can be pretty dramatic. Consider, e.g. the concentration of electrons you can get in the three case for E » 0 eV, i.e close to the band edge.
| The question, of course, is: Are there 1-dim. and 2-dim. semiconductors? The answer is: yes – as soon as the other dimensions are small enough, we will encounter these cases. We will run across examples later in the lecture course.

With frame

Exercise 2.1-1 Free Electron Gas with Constant Boundary Conditions

Exercise 2.1-2: Density of States for Lower Dimensions

Exercise 2.3.5-1

© H. Föll (Semiconductors - Script)