@@ -75,22 +75,21 @@ In this figure, the red curves represent the nearly-free electron dispersion, wh
*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 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 $|k'\rangle$.
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 $|k'\rangle =|k-2\pi/a\rangle$. We now express the wavefunction near this crossing as a linear superposition $|\psi\rangle = \alpha |k\rangle + \beta |k'\rangle$. Note that this 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 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
We express the Hamiltonian near the crossing as a matrix, using $| k \rangle$ 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 $\delta k = k-\pi/a$ is the distance from the center of the crossing and we approximated the dispersion near the crossing by a linear term. In matrix form, this yields
$$H\begin{pmatrix}\alpha \\\beta \end{pmatrix} =
\begin{pmatrix} E_0 + v \hbar \delta k & 0 \\ 0 & E_0 - v \hbar \delta k\end{pmatrix}
\begin{pmatrix}\alpha \\\beta \end{pmatrix}.
$$
Note that this Hamiltonian is diagonal, so the eigenenergies are on the diagonal and the eigenfunctions are simply the $|k\rangle$ and $|k'\rangle$ basis states.
??? 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 simply the $|k\rangle$ and $|k'\rangle$ plane waves.
As we will see below, the lattice potential $V(x)$ can couple the states $|k\rangle$ and $|k'\rangle$. The coupling between these states is given by the matrix element $W=\langle k | V | k'\rangle$. We now including this coupling into the Hamiltonian
As we will see below, the lattice potential $V(x)$ can couple the states $|k\rangle$ and $|k'\rangle$. The coupling between these states is given by the matrix element $W=\langle k | V | k'\rangle$. Including this coupling into the Hamiltonian:
$$
H\begin{pmatrix}\alpha \\\beta \end{pmatrix} =
...
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@@ -105,7 +104,7 @@ $$
To find the dispersion near $k=\pi/a$, we need to diagonalize this 2x2 matrix Hamiltonian. The solutions for the eigenvalues are
(Check out section 15.1.1 of the book for the details of this calculation.) This equation describes the avoided crossings observed in the plot above. We observe that the gap at $\delta k=0$ is equal to $2W$.
(Check out section 15.1.1 of the book for the details of this calculation). This equation describes the avoided crossing. We observe that the gap that has opened at $\delta k=0$ is equal to $2W$.
??? question "Does our solution $\psi(x)$ satisfy the Bloch theorem? What is $u(x)$ in this case?"
The wave function has a form $\psi(x) = \alpha \exp[ikx] + \beta \exp[i(k - 2\pi/a)x]$
...
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@@ -126,20 +125,20 @@ V_n = \frac{1}{a}\int_0^a e^{- i n 2\pi x /a} V(x) dx
$$
Calculating $W$, we find
$$W = \langle k | V | k' \rangle = \frac{1}{a}\int_0^{a} e^{i k x} V(x) e^{-i k'x} dx = \frac{1}{a}\int_0^a e^{-i 2\pi x /a} V(x) dx = V_1$$
$$W = \langle k | V | k' \rangle = \frac{1}{a}\int_0^{a} e^{-i k x} V(x) e^{i k'x} dx = \frac{1}{a}\int_0^a e^{-i 2\pi x /a} V(x) dx = V_1$$
where we have used that $k'-k =2\pi/a$ because we are analyzing the first crossing. We see that the first component of the Fourier-series representation of $V(x)$ determines the gap near the first crossing.
where we have used that $k-k' =2\pi/a$ because we are analyzing the first crossing. We see that the first component of the Fourier-series representation of $V(x)$ determines the gap near the first crossing.
#### Crossings between the higher bands
Everything we did can also be applied to the higher-energy crossings seen in the figure above. We note that 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 fourth, etc.
The key conclusion is that the lattice potential couples plane-wave states that differ by integer multiples of reciprocal lattice vectors. This coupling alters the band structure most strongly where the free-electron eigenenergies cross, opening up gaps determined by the Fourier components of the lattice potential.
The key conclusion is that the lattice potential couples plane-wave states that differ by integer multiples of reciprocal lattice vectors. This coupling alters the band structure most strongly where the free-electron eigenenergies cross, opening up gaps of which the magnitudes are determined by the Fourier components of the lattice potential.
### Repeated vs reduced vs extended Brillouin zone
All different ways to **plot** the same dispersion relation (no difference in physical information).
There are several common ways to **plot** the same dispersion relation (no difference in physical information).
