add lecture 6

README.md

@@ -22,7 +22,7 @@
   (based on chapters 12–14, up to and including 14.2)  
   Exercises 12.3, 12.4, 13.3, 13.4, 14.2
 
-* Week 6: Tight binding and nearly free electrons  
+* Week 6: [Tight binding and nearly free electrons](lecture_6.md)  
   (based on chapters 15–16)  
   Exercises 15.1, 15.3, 15.4, 16.1, 16.2
 

SUMMARY.md

@@ -5,6 +5,6 @@
 * [Atoms and bonds](lecture_3.md)
 * [Electrons and phonons in 1D](lecture_4.md)
 * [Crystal structure and diffraction](lecture_5.md)
-* Tight binding and nearly free electrons
+* [Tight binding and nearly free electrons](lecture_6.md)
 * Semiconductors
 * Magnetism

lecture_6.md

+# Tight binding and nearly free electrons
+
+Let's summarize what we learned about electrons so far:
+* Free electrons form a Fermi sea ([lecture 2](lecture_1.md))
+* Isolated atoms have discrete orbitals ([lecture 3](lecture_3.md))
+* When orbitals hybridize we det *LCAO* or *tight-binding* band structures ([lecture 4](lecture_4.md))
+
+In this lecture we:
+* Formulate a general way of computing the electron band structure, the **Bloch theorem**.
+* Derive the electron band structure when the interaction with the lattice is weak using the **Nearly free electron model**. This model:
+  - Helps to understand the relation between tight-binding and free electron models
+  - Describes the properties of metals.
+
+All the different limits can be put onto a single scale as a function of the strength of the lattice potential $V(x)$:
+
+![](figures/models.svg)
+
+## Bloch theorem
+
+> All Hamiltonian eigenstates in a crystal have the form
+> $$\psi_n(\mathbf{r}) = u_n(\mathbf{r})e^{i\mathbf{kr}}$$
+> with $u_n(\mathbf{r})$ having the same periodicity as the lattice potential $V(\mathbf{r})$, and index $n$ labeling electron bands with energies $E_n(\mathbf{k})$.
+
+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.
+
+### Extra remarks
+
+The wave function $u_n(\mathbf{r})e^{i\mathbf{kr}}$ 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.
+
+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 $u_n(\mathbf{r}) \rightarrow u_n(\mathbf{r})\exp\left[-h\mathbf{b}_1 - k\mathbf{b}_2 - l\mathbf{b}_3\right]$ (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.
+
+Bloch theorem is extremely similar to the ansatz we used in [1D](lecture_4.md), and to the description of the [X-ray scattering](lecture_5.md).
+
+## Nearly free electron model
+
+In free electron model $E = \hbar^2 k^2/2m$.
+
+* There is only one band
+* The band structure is not periodic in $k$-space
+* In other words the Brillouin zone is infinite in momentum space
+
+Within the **nearly free electron model** we want to start from the dispersion relation of the free electrons and consider the effect of introducing a weak lattice potential. The logic is very similar to getting optical and acoustic phonon branches by changing atom masses (and therefore reducing the size of the Brillouin zone).
+
+In our situation:
+
+![](figures/nearly_free_electron_bands.svg)
+
+The band gaps open where two copies of the free electron dispersion cross.
+
+### Avoided level crossing
+
+*Remark: this is an important concept in quantum mechanics, based on the perturbation theory. You will only learn it later in QMIII, so we will need to postulate some important facts.*
+
+Let's focus on the first crossing. The momentum near it is $k = \pi/a + \delta k$ and we have two copies of the original band structure coming together. One with $\psi_+ \propto e^{i\pi x/a}$, another with $\psi_- \propto e^{-i\pi x/a}$. Near the crossing the wave function is the linear superposition of $\psi_+$ and $\psi_-$: $\psi = \alpha \psi_+ + \beta \psi_-$. We actually used almost the same form of the wave function in LCAO, except instead of $\psi_\pm$ we used the orbitals $\phi_1$ and $\phi_2$ there.
+
+Without the lattice potential we can approximate the Hamiltonian of these two states as follows:
+$$H\begin{pmatrix}\alpha \\ \beta \end{pmatrix} = 
+\begin{pmatrix} E_0 + v \hbar \delta k & 0 \\ 0 & E_0 - v \hbar \delta k\end{pmatrix}
+\begin{pmatrix}\alpha \\ \beta \end{pmatrix}.
+$$
+
+Here we used $\delta p = \hbar \delta k$, and we expanded the quadratic function into a linear plus a small correction.  
+**Question: can you calculate $E_0$ and the velocity $v$?**
+
+Without $V(x)$ the two wave functions $\psi_+$ and $\psi_-$ are independent since they have a different momentum. When $V(x)$ is present, it may couple these two states.
+
+So in presence of $V(x)$ the Hamiltonian becomes
+
+$$H\begin{pmatrix}\alpha \\ \beta \end{pmatrix} = 
+\begin{pmatrix} E_0 + v \hbar \delta k & W \\ W^* & E_0 - v \hbar \delta k\end{pmatrix}
+\begin{pmatrix}\alpha \\ \beta \end{pmatrix},
+$$
+
+Here the coupling strength $W = \langle \psi_+ | V(x) | \psi_- \rangle$ is the matrix element of the potential between two states. *(This where we need to apply the perturbation theory, and this is very similar to the LCAO Hamiltonian)*.
+
+**Question: how does our solution satisfy the Bloch theorem? What is $u(x)$ in this case?**
+
+#### Dispersion relation near the avoided level crossing
+
+We need to diagonalize a 2x2 matrix Hamiltonian. The answer is
+$$ E(\delta k) = E_0 \pm \sqrt{v^2\hbar^2\delta k^2 + |W|^2}$$
+
+In one of the exercises you will encounter the details of this calculation.
+
+#### Physical meaning of $W$
+
+Now we expand the definition of $W$:
+
+$$W = \langle \psi_+ | V(x) | \psi_- \rangle = \int_0^{a} dx \left[e^{i\pi x/a}\right]^* V(x) \left[e^{-i\pi x/a}\right] = \int_0^a e^{2\pi i x /a} V(x) = V_1$$
+
+Here $V_1$ is the first Fourier component of $V(x)$ (using a *complex* Fourier transform).
+
+$$ V(x) = \sum_{n=-\infty}^{\infty} V_n e^{2\pi i n x/a}$$
+
+In the book the *real symmetric* Fourier transform is used:
+
+$$ V(x) = \sum_{n=0}^{\infty} \tilde{V}_n \cos(2\pi n x/a),$$
+
+which can be used *only* when $V(x) = V(-x)$. Then the relation between the two is $\tilde{V}_n = 2 V_n$.
+
+#### Crossings between the higher bands
+
+Everything we did applies to the crossings at higher energies, only there we would get higher Fourier components of $V(x)$: $V_2$ for the crossing between the second and third band, $V_3$ for the crossing between third and 4th, etc.
+
+### Repeated vs reduced vs extended Brillouin zone
+
+All different ways to **plot** the same dispersion relation (no difference in physical information).
+
+Repeated BZ (all possible Bloch bands):
+
+![](figures/nearly_free_electron_bands.svg)
+
+* Contains redundant information
+* May be easier to count/follow the bands
+
+Reduced BZ (all bands within 1st BZ):
+
+![](figures/reduced_nearly_free_electron_bands.svg)
+
+* No redundant information
+* Hard to relate to original dispersion
+
+Extended BZ (n-th band within n-th BZ):
+
+![](figures/extended_nearly_free_electron_bands.svg)
+
+* No redundant information
+* Easy to relate to free electron model
+* Contains discontinuities
+
+## 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):
+
+1. Compute $k_f$ using the free electron model (remember this is our starting point).
+2. Plot the free electron model Fermi surface and the Brillouin zones.
+3. Apply the perturbation where the Fermi surface crosses the Brillouin zone (due to avoided level crossings).
+
+![](figures/nearly_free_FS.svg)
+
+If $V$ is sufficiently weak, the material can be conducting even with 2 electrons per unit cell!
+
+A larger $V$ makes the Fermi surface more square-like and eventually makes the material insulating.
+
+## Summary
+
+* In periodic potential all electron states are **Bloch waves**
+* Electron dispersion is organized into **energy bands** that may overlap, or may be separated by **band gaps**
+* If the number of electrons per unit cell is odd, the material must be conducting.
+* If the lattice potential is weak, the dispersion can be obtained by copying $p^2/2m$ into different Brillouin zones, and opening gaps at every level crossing. Each gap is equal to the Fourier component of the lattice potential.