@@ -58,38 +58,39 @@ Bloch theorem is extremely similar to the ansatz we used in [1D](7_tight_binding
## Nearly free electron model
In the free electron model, the dispersion is $E = \hbar^2 |\mathbf{k}|^2/2m$ and the corresponding eigenfunctions $|\mathbf{k}\rangle$ are plane waves with a real-space representation $\psi=\langle\mathbf{r}|\mathbf{k}\rangle= e^{i\mathbf{k}\cdot \mathbf{r}}$. We note that in the free electron model,
In the free electron model, the dispersion is $E = \hbar^2 |\mathbf{k}|^2/2m$. The corresponding eigenfunctions $|\mathbf{k}\rangle$ are plane waves with a real-space representation $\psi(\mathbf{r}=\langle\mathbf{r}|\mathbf{k}\rangle= e^{i\mathbf{k}\cdot \mathbf{r}}/\sqrt{L^3}$. We note that in the free electron model,
* there is only one band
* the band structure is not periodic in $k$-space
*therefore the Brillouin zone is infinite in $k$-space
*i.e., the Brillouin zone is infinite in $k$-space
Within the **nearly free electron model** we want to start from the dispersion relation of 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 thereby reducing the size of the Brillouin zone). The lattice potential results in a band structure that is periodic in $k$-space, with a period given by the period of the reciprocal lattice:
Within the **nearly free electron model** we start from the dispersion relation of free electrons and analyze 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 thereby reducing the size of the Brillouin zone). The lattice potential results in a band structure that is periodic in $k$-space, with a period given by the period of the reciprocal lattice:
The red curves represent the nearly-free electron dispersion, which differs from the free-electron dispersion (black curves) because of the interaction with the lattice potential, as we will derive in this lecture. We see that gaps in the band structure open where two copies of the free electron dispersion cross. There are no states with an energy within these **band gaps**.
In this figure, the red curves represent the nearly-free electron dispersion, which differs from the free-electron dispersion (black curves) because of the interaction with the lattice potential as we will derive in this lecture. We see that **band gaps** (regions in the band structure where there are no allowed energy states) open where two copies of the free electron dispersion cross.
### Avoided level crossing
### Calculating the avoided crossings
*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.*
*Remark: Avoided crossings are an important concept in quantum mechanics, and are analyzed using **perturbation theory**. You will only learn this theory 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, free-electron 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.
We first formulate a matrix equation describing the two energy eigenstates of the free-electron model near the first crossing. We have two copies of the free-electron band structure coming together with eigenfunctions $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$. We express the eigenfunctions as a linear superposition of $|\psi\rangle \alpha |\mathbf{k}\rangle + \beta |\mathbf{k'}\rangle$. Note that we used almost the same form of the wave function in the LCAO model, except there we used the orbitals $|1\rangle$ and $|2\rangle$ instead of the plane waves $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$.
Without the lattice potential, we can write the Hamiltonian near the crossing as a matrix using $\langle \mathbf{k} |H|\mathbf{k}\rangle = \hbar^2k^2/2m$ and $\langle \mathbf{k'} |H|\mathbf{k'}\rangle = \hbar^2(k-2\pi/a)^2/2m$, while $\langle \mathbf{k'} |H|\mathbf{k'}\rangle = 0$. This yields
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 term plus a small correction.
where we approximated the quadratic dispersion near the crossing $k=\pi/a$ by a linear term plus a small correction.
??? question "calculate $E_0$ and the velocity $v$"
The edge of the Brilloin zone has $k = \pi/a$. Substituting this in the free electron dispersion $E = \hbar^2 k^2/2m$ we get $E_0 = \hbar^2 \pi^2/2 m a^2$, and $v=\hbar k/m=\hbar \pi/ma$.
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.
Note that in this Hamiltonian without $V(x)$, the eigenfunctions are simply $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$ and the corresponding eigenenergies are on the diagonal.
So in presence of $V(x)$ the Hamiltonian becomes
The lattice potential $V(x)$ may couple the states $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$. The corresponding matrix element $W=\langle \mathbf{k} |V| |\mathbf{k'}\rangle$ defines the coupling strength. The Hamiltonian thus becomes
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)*.
*(This where we applied 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?"
The wave function has a form $\psi(x) = \alpha \exp[ikx] + \beta \exp[i(k - 2\pi/a)x]$
