From cd102caa037bd904ff1b137dfdb8c12913a7db5e Mon Sep 17 00:00:00 2001
From: Anton Akhmerov <anton.akhmerov@gmail.com>
Date: Thu, 26 Mar 2020 16:15:18 +0100
Subject: [PATCH] add one more lecture video
---
src/12_band_structures_in_higher_dimensions.md | 10 +++++++---
1 file changed, 7 insertions(+), 3 deletions(-)
diff --git a/src/12_band_structures_in_higher_dimensions.md b/src/12_band_structures_in_higher_dimensions.md
index 0047362b..6dbf71f0 100644
--- a/src/12_band_structures_in_higher_dimensions.md
+++ b/src/12_band_structures_in_higher_dimensions.md
@@ -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"
+
+ <iframe width="100%" height="315" src="https://www.youtube-nocookie.com/embed/eTx8FnVQ0pw" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+
## 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.
--
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