 ... ... @@ -18,6 +18,10 @@ _(based on chapter 16 of the book)_ - examine 1D and 2D band structures and argue if you expect the corresponding material to be an insulator/semiconductor or a conductor. - describe how the light absorption spectrum of a material relates to its band structure. ??? info "Lecture video" ## Band structure How are material properties related to the band structure? ... ... @@ -42,7 +46,7 @@ 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 room 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: ... ... @@ -225,10 +229,10 @@ _(based on exercise 15.4 of the book)_ Suppose we have a square lattice with lattice constant $a$, with a periodic potential given by $V(x,y)=2V_{10}(\cos(2\pi x/a)+\cos(2\pi y/a))+4V_{11}\cos(2 \pi x/a)\cos(2 \pi y/a)$. 1. Use the Nearly-free electron model to find the energy of state $\mathbf{q}=(\pi/a, 0)$. ??? hint This is analogous to the 1D case: the states that interact have $k$-vectors $(\pi/a,0)$ and $(-\pi/a,0)$; ($\psi_{+}\sim e^{i\pi x /a}$ ; $\psi_{-}\sim e^{-i\pi x /a}$). 2. Let's now study the more complicated case of state $\mathbf{q}=(\pi/a,\pi/a)$. How many $k$-points have the same energy? Which ones? 3. Write down the nearly free electron model Hamiltonian near this point. 4. Find its eigenvalues. \ No newline at end of file 4. Find its eigenvalues.