The program from the previous page on Bloch waves in 1-D can be put in a loop to calculate wave vectors $k$ for many energies. This can be used to plot the electron dispersion relation $E\text{ vs. }k$. This is called a band structure calculation. From this data of $E$ and $k$ it is possible to calculate the density of states is,

$$D(E) = \frac{2}{\pi}\frac{dk}{dE},$$

and the group velocity is,

$$v_g= \frac{1}{\hbar}\frac{dE}{dk}.$$

The density of states and the group velocity go to zero in the band gaps. There is also a plot of the $\alpha$ parameter, which is used in the calculation. If $|\alpha| < 2$, the state is in a band, otherwise it is in a band gap.

Notice that the size of the band gap is about the same as the amplitude of the potential. For potentials with small amplitudes, the $E\text{ vs. }k$ dispersion relation is nearly that of the free electron model $E=\frac{\hbar^2k^2}{2m}$.

V(x) [eV] x [nm] The 1-D Potential Initial energy E1 =   [eV]  Final energy E2 =   [eV]  Mass m =   [kg]  Lattice constant a  =   [m]  Potential V(x)  =   [eV]  $E$ [eV] $ka$ $D(E)$ [eV-1 nm-1] $E$ $E$ $\alpha$    $\frac{2ma}{\hbar}v_g$  $E$ $E$ Data in 4 columns: $E$ [eV]  $ka$   $D(E)$ [1/(eV nm)]    $\frac{2mav_g}{\hbar}$

Standard mathematical functions abs(x), acos(x), asin(x), atan(x), cos(x), exp(x), log(x), pi = 3.141592653589793, pow(x,y) = x^y, round(x), sin(x), sqrt(x), tan(x) can be used in the form. In addition, the Heaviside step function H(x) can be used. Multiplication must be specified with a '*' symbol, 3*cos(x) not 3cos(x). Powers are specified with the 'pow' function: x² is pow(x,2) not x^2.

Some potentials that can be pasted into the form are given below.