@@ -13,18 +13,19 @@ _(based on chapters 15–16 of the book)_
Before the start of this lecture, you should be able to:
- Write down the dispersion and wavefunction of an electron in free space (by solving the Schrödinger equation).
- Describe how the periodicity of a band structure (=dispersion) is related to the reciprocal lattice.
- Write down a Fourier series representation of a periodic function such as the lattice potential.
- Diagonalize a 2x2 matrix
- Write down the dispersion and wavefunction of an electron in free space (solving the Schrödinger equation).
- Describe how the periodicity of a band structure (=dispersion) is related to the reciprocal lattice.
- Write down a Fourier series representation of a periodic function.
- Diagonalize a 2x2 matrix (i.e., find its eigenvalues and eigenfunctions).
!!! summary "Learning goals"
After this lecture you should be able to:
- derive the electron band structure when the interaction with the lattice is weak using the **Nearly free electron model**.
- formulate a general way of computing the electron band structure - the **Bloch theorem**.
- recall that in a periodic potential, all electron states are Bloch waves.
- Derive the electron band structure when the interaction with the lattice is weak using the **Nearly free electron model**.
- Argue at what locations in the band structure the interaction with the lattice opens gaps.
- Formulate a general way of computing the electron band structure - the **Bloch theorem**.
- Recall that in a periodic potential, all electron states are Bloch waves.
Let's summarize what we learned about electrons so far:
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@@ -58,7 +59,7 @@ 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 $E = \hbar^2 |\mathbf{k}|^2/2m$. 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,
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 $\psi(\mathbf{r}) \propto e^{i\mathbf{k}\cdot \mathbf{r}}$. We note that in the free electron model,
* there is only one band
* the band structure is not periodic in $k$-space
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@@ -68,15 +69,15 @@ Within the **nearly free electron model** we start from the dispersion relation
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.
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. 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. A key goal of this lecture is to understand how the weak interaction with the lattice leads to this modified band structure.
### Calculating the avoided crossings
### Analyzing the avoided crossings
*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.*
*Remark: An avoided crossing is an important concept in quantum mechanics that can be analyzed using **perturbation theory**. You will only learn this theory later in QMIII, so we will need to postulate some important facts here.*
To analyze what happens near the crossings, we first neglect the lattice potential and formulate a matrix equation describing the energy dispersion near the first crossing, concentrating on 1D. Near the crossing, we have two copies of the free-electron dispersion coming together. The corresponding eigenfunctions are $|k\rangle$ and $|k'\rangle$. We express the wavefunction as a linear superposition $|\psi\rangle = \alpha |k\rangle + \beta |k'\rangle$. Note that we used almost the same form of wave function in the LCAO model, except there we used linear combinations of the orbitals $|1\rangle$ and $|2\rangle$ instead of the plane waves $|k\rangle$ and $|k'\rangle$.
To analyze what happens near the crossings, we first neglect the lattice potential and consider the free-electron energy dispersion near the crossing at $k=\pi/a$ in 1D. Near this crossing, we see that two copies of the free-electron dispersion come together (one copy centered at $k=0$, the other at $k=2\pi/a$). We call the corresponding plane-wave eigenfunctions $|k\rangle$ and $|k'\rangle =|k+2\pi/a\rangle$ and express the wavefunction near this crossing as a linear superposition $|\psi\rangle = \alpha |k\rangle + \beta |k'\rangle$. Note that wave function is very similar to that used in the LCAO model, except there we used linear combinations of the orbitals $|1\rangle$ and $|2\rangle$ instead of the plane waves $|k\rangle$ and $|k'\rangle$.
We can formulate the Hamiltonian near the crossing as a matrix using $\langle k |H|k\rangle = E_0 + v \hbar \delta k$ and $\langle k' |H|k'\rangle = E_0 - v \hbar \delta k$, where $\delta k = k-\pi/a$ and we approximated the eigenenergy near the crossing by a linear term plus a small correction. This yields
We formulate the Hamiltonian near the crossing as a matrix, using the $| k \rangle$ and $| k' \rangle$ as the basis states. The matrix elements are given by $\langle k |H|k\rangle = E_0 + v \hbar \delta k$ and $\langle k' |H|k'\rangle = E_0 - v \hbar \delta k$, where $\delta k = k-\pi/a$ and we approximated the eigenenergy near the crossing by a linear term plus a small correction. In matrix form, this yields
$$H\begin{pmatrix}\alpha \\\beta \end{pmatrix} =
\begin{pmatrix} E_0 + v \hbar \delta k & 0 \\ 0 & E_0 - v \hbar \delta k\end{pmatrix}
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@@ -87,9 +88,9 @@ $$
??? 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 $v=\hbar k/m=\hbar \pi/ma$.
Note that this Hamiltonian is diagonal so the eigenenergies are on the diagonal and the eigenfunctions are the $|k\rangle$ and $|k'\rangle$ plane waves.
Note that this Hamiltonian is diagonal so the eigenenergies are on the diagonal and the eigenfunctions are simply the $|k\rangle$ and $|k'\rangle$ plane waves.
As we will see below, the lattice potential $V(x)$ may couple the states $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$. The coupling strength is given by the matrix element $W=\langle \mathbf{k} |V| |\mathbf{k'}\rangle$. Including the coupling into our Hamiltonian, we get
As we will see below, the lattice potential $V(x)$ can couple the states $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$. The coupling betweenthese states is given by the matrix element $W=\langle \mathbf{k} | V | \mathbf{k'}\rangle$. We now including this coupling into the Hamiltonian
*(This where we applied the perturbation theory, and this is very similar to the LCAO Hamiltonian where the coupling strength was given by $-t=\langle 1 | H | 2 \rangle$)*.
*(This is where we applied the perturbation theory, and this is very similar to the LCAO Hamiltonian where the coupling was given by $-t=\langle 1 | H | 2 \rangle$)*.
#### Dispersion near the avoided level crossing
To find the dispersion, we need to diagonalize this 2x2 matrix Hamiltonian. The answer is
To find the dispersion near $k=\pi/a$, we need to diagonalize this 2x2 matrix Hamiltonian. The solutions for the eigenvalues are
$$ V(x) =\sum_{n=-\infty}^{\infty} V_n e^{2\pi i n x/a}$$
Here $V_1$ is the first Fourier component of $V(x)$ (using a complex Fourier transform).
and recall that such a series has Fourier components $V_n$ given by
$$
V_n = frac{1}{a}\int_0^a e^{- i n 2\pi x /a} V(x) dx
$$
We now calculate
$$W = \langle k | V(x) | k' \rangle = \frac{1}{a}\int_0^{a} dx \left[e^{i k x}\right]^* V(x) \left[e^{-i k'x}\right] = \frac{1}{a}\int_0^a e^{-2\pi i x /a} V(x) dx = V_1$$
Here, we have used that $k-k'=2\pi/a$ for the first crossing. We observe that the first component of the Fourier-series representation of $V(x)$ determines the strength of the coupling between the two states near the first crossing.
$$ V(x) = \sum_{n=-\infty}^{\infty} V_n e^{2\pi i n x/a}$$
#### 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.
Everything we did can also be applied to the higher-energy crossings observed in the figure above. All crossings occur between parabola's that are shifted by integer multiples of reciprocal lattice vectors $n 2\pi/a$. The first crossing corresponds to $n=1$, and we found that the magnitude of the gap is given by $V_1$. Similarly, $V_2$ determines the gap between the second and third bands, $V_3$ for the crossing between third and 4th, etc.
### Repeated vs reduced vs extended Brillouin zone
