 Update 12_band_structures_in_higher_dimensions.md - typo

parent 514171d1
 ... ... @@ -36,13 +36,13 @@ When the Fermi level lies in the band gap, the material is called a semiconducto In an insulator every single band is either completely filled or completely empty. What determines if an energy band if fully occupied? To answer this we need to know the number of available states within an energy band, and the number of electrons in the system. We can find the number of states in a band by integrating the density of states \$g(E)\$, but this is hard. Fortunately, we can easily see how many states there are in an energy band by counting the number of \$k\$-states in the first Brillouin zone. What determines if an energy band if fully occupied or only partly? To answer this we need to know the number of available states within an energy band, and the number of electrons in the system. We can find the number of states in a band by integrating the density of states \$g(E)\$, but this is hard. Fortunately, we can easily see how many states there are in an energy band by counting the number of \$k\$-states in the first Brillouin zone. For a single band: \$\$ N_{states} = 2 \frac{L^3}{(2\pi)^3} \int_{BZ} dk_x dk_y dk_z = 2 L^3 / a^3 \$\$ Here, \$L^3/a^3\$ is the number of unit cells in the system, so we see that a single band has space for 2 electrons per unit cell (the factor 2 comes from the spin). Here, \$L^3/a^3\$ is the number of unit cells in the system, so we see that a single band has room for 2 electrons per unit cell (the factor 2 comes from the spin). We come to the important rule: ... ...
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