 Update 12_band_structures_in_higher_dimensions.md - polish

parent ff2ebaea
 ... ... @@ -36,17 +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. How many electrons per unit cell can we expect for an insulator? To answer this we need to know the number of states within an energy band. We can calculate this by integrating the density of states \$g(E)\$, but this is hard. However, 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? 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 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 can host 2 electrons per unit cell (because of spin). If there are no overlapping bands, a system with 2 electrons per unit cell will therefore be an insulator/semiconductor. ??? question "Can we now understand why diamond is an insulator?" Hint: how many atoms per unit cell does diamond have (see exercises Lecture 7)? And how many valence electrons does a carbon atom have? 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). We come to the important rule: ... ... @@ -54,6 +50,7 @@ We come to the important rule: If the material has an even number of electrons per unit cell it may be a semiconductor, but only if the bands are not overlapping (see the figure above). For example: Si, Ge, Sn all have 4 valence electrons. Si (silicon, band gap 1.14 eV) and Ge (germanium, band gap 0.67 eV) are semiconductors, Sn (tin) is a metal. **Interesting feature: the heaviest material is a metal, why?** ## Fermi surface using a nearly free electron model Sequence of steps (same procedure as in 1D, but harder because of the need to imagine a 2D dispersion relation): ... ...
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