From 5a3b040c7a0c0b783771dfc87162b31e1f02a169 Mon Sep 17 00:00:00 2001
From: "T. van der Sar" <t.vandersar@tudelft.nl>
Date: Tue, 24 Mar 2020 21:02:42 +0000
Subject: [PATCH] Update 11_nearly_free_electron_model.md - polish
---
src/11_nearly_free_electron_model.md | 23 +++++++++++++++++------
1 file changed, 17 insertions(+), 6 deletions(-)
diff --git a/src/11_nearly_free_electron_model.md b/src/11_nearly_free_electron_model.md
index 4f91e5ff..c9b7db5d 100644
--- a/src/11_nearly_free_electron_model.md
+++ b/src/11_nearly_free_electron_model.md
@@ -129,25 +129,36 @@ We have seen that in the nearly-free electron model, the electrons behave as pla
Bloch theorem:
> All Hamiltonian eigenstates in a crystal have the form
-> $$ \psi_n(\mathbf{r}) = u_n(\mathbf{r})e^{i\mathbf{k}\cdot \mathbf{r}} $$
-> with $u_n(\mathbf{r})$ having the same periodicity as the lattice potential $V(\mathbf{r})$, and index $n$ labeling electron bands with energies $E_n(\mathbf{k})$.
+> $$ \psi^\alpha(\mathbf{r}) = u^\alpha(\mathbf{r})e^{i\mathbf{k}\cdot \mathbf{r}} $$
+> with $u^\alpha(\mathbf{r})$ having the same periodicity as the lattice potential $V(\mathbf{r})$, and index $\alpha$ labeling electron bands with energies $E^\alpha(\mathbf{k})$.
In other words: any electron wave function in a crystal is a product of a periodic part that describes electron motion within a unit cell and a plane wave. In both the tight-binding and the nearly-free electron models, the wave functions we considered are consistent with Bloch's theorem.
-??? question "Does our nearly-free electron wavefunction $|\psi(x)\rangle = \alpha|k\rangle + \beta|k'\rangle$ satisfy the Bloch theorem? What is $u(x)$ in this case?"
+??? question "Does our nearly-free electron wavefunction $|\psi\rangle = \alpha|k\rangle + \beta|k'\rangle$ satisfy the Bloch theorem? What is $u(x)$ in this case?"
The wave function has a form $\psi(x) = \alpha \exp[ikx] + \beta \exp[i(k - 2\pi/a)x]$
(here $k = \pi/a + \delta k$). Choosing $u(x) = \alpha + \beta \exp(2\pi i x/a)$ we see
that $\psi(x) = u(x) \exp(ikx)$.
### Extra remarks
-The wave function $u_n(\mathbf{r})e^{i\mathbf{k} \cdot \mathbf{r}}$ is called a **Bloch wave**.
+The wave function $u^\alpha(\mathbf{r})e^{i\mathbf{k} \cdot \mathbf{r}}$ is called a **Bloch wave**.
-The $u_n(\mathbf{r})$ part is some unknown function. To calculate it we need to solve the Schrödinger equation. It is hard in general, but there are two limits when $U$ is "weak" and $U$ is "large" that provide us with most intuition.
+The $u^\alpha(\mathbf{r})$ part is some unknown function. To calculate it we need to solve the Schrödinger equation. It is hard in general, but there are two limits when $U$ is "weak" and $U$ is "large" that provide us with most intuition.
-If we change $\mathbf{k}$ by a reciprocal lattice vector $\mathbf{k} \rightarrow \mathbf{k} + h\mathbf{b}_1 + k\mathbf{b}_2 + l\mathbf{b}_3$, and we change $u_n(\mathbf{r}) \rightarrow u_n(\mathbf{r})\exp\left[i(-h\mathbf{b}_1 - k\mathbf{b}_2 - l\mathbf{b}_3)\cdot \mathbf{r}\right]$ (also periodic!), we obtain the same wave function. Therefore energies of all bands $E_n(\mathbf{k})$ are periodic in reciprocal space with the periodicity of the reciprocal lattice.
+If we change $\mathbf{k}$ by a reciprocal lattice vector $\mathbf{k} \rightarrow \mathbf{k} + h\mathbf{b}_1 + k\mathbf{b}_2 + l\mathbf{b}_3$, and we change $u^\alpha(\mathbf{r}) \rightarrow u^\alpha(\mathbf{r})\exp\left[i(-h\mathbf{b}_1 - k\mathbf{b}_2 - l\mathbf{b}_3)\cdot \mathbf{r}\right]$ (also periodic!), we obtain the same wave function. Therefore energies of all bands $E^\alpha(\mathbf{k})$ are periodic in reciprocal space with the periodicity of the reciprocal lattice.
+An alternative way to write the Bloch wave is to formulate $u^\alpha(r)$ as a Fourier series:
+$$
+u^\alpha(r) = \Sum_\mathbf{G} u^{\alpha}_\mathbf{G}e^{i\mathbf{G}\cdot\mathbf{r}}
+$$
+where $u^{\alpha}_\mathbf{G}$ are the Fourier coefficients. Substituing into our experssion for the Bloch wave, we get
+$$
+\psi^\alpha(r) = \Sum_\mathbf{G} u^{\alpha}_\mathbf{G} e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}
+$$
+which shows that each eigenstate can be written as a sum over plane waves that differ by a reciprocal lattice vector.
+??? question "Does the tight-binding wavefunction $|\psi\rangle = \Sum_n e^{ikna}(\phi_0|n,1\rangle+\psi_0|n,2\rangle)$ (see exercise 2 in Lecture 8) satisfy the Bloch theorem? What part of $|\psi\rangle$ describes $u(x)$ in this case?"
+
### Repeated vs reduced vs extended Brillouin zone
There are several common ways to **plot** the same dispersion relation (no difference in physical information).
--
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