which shows that each Bloch wave 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? Try to describe in words how this Bloch wave is built up."
!!! 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? Try to describe in words how this Bloch wave is built up."
### Repeated vs reduced vs extended Brillouin zone
...
...
@@ -297,30 +297,29 @@ Let's consider a 1D crystal with a periodicity of $a$.
Let $k_0$ be any wave number of an electron in the first Brillouin zone.
1. Which $k_n$ are equivalent to $k_0$ in this crystal?
2. First, we assume that the electrons with these $k_n$ are free. In that case, what are the wavefunctions $\phi_n(x)$ and energies $E_n$ of these electrons?
3. Make a sketch of the dispersion relation using a repeated Brillouin zone representation. Indicate some $k_n$ and $E_n$ as well as the first Brillouin zone in your sketch.
We will now introduce a weak periodic potential $V(x) = V(x+na)$ to the system, which causes coupling between eigenstates $\left| \phi_n\right>$ of the free electrons.
In order to find the right eigenstates of the system with that potential, we use an 'LCAO-like' trial eigenstate
3. Make a sketch of the dispersion relation using a repeated Brillouin zone representation.
Indicate some $k_n$ and $E_n$ as well as the first Brillouin zone in your sketch. We will now introduce a weak periodic potential $V(x) = V(x+na)$ to the system, which causes coupling between eigenstates $\left| \phi_n\right>$ of the free electrons. In order to find the right eigenstates of the system with that potential, we use an 'LCAO-like' trial eigenstate $$ \left|\psi\right> = \sum_{n=-\infty}^{\infty}C_n \left|\phi_n\right> $$
4. Using this trial eigenstate and the Schrödinger equation, show that
E C_m = \varepsilon_m C_m+\sum_{n=-\infty}^{\infty}C_{n}V_{m-n},
$$
4. Using this trial eigenstate and the Schrödinger equation, show that $$
E C_m = \varepsilon_m C_m+\sum_{n=-\infty}^{\infty} C_{n}V_{m-n}
,$$
where $V_n$ are the Fourier components of the potential defined [above](#physical-meaning-of-w).
Find an expression for $\varepsilon_m$. _**NB:** This equation is also known as the central equation (in 1D)._
where $V_n$ are the Fourier components of the potential defined [above](#physical-meaning-of-w).
Find an expression for $\varepsilon_m$. _**NB:** This equation is also known as the central equation (in 1D)._
??? hint
- Apply $\left<\phi_m\right|$ to the Schrödinger equation.
- To evaluate $\left<\phi_m\right|\hat{H}\left|\phi_n\right>$, it may be helpful to separate the kinetic energy and potential energy of the Hamiltonian.
??? hint
- Apply $\left<\phi_m\right|$ to the Schrödinger equation.
- To evaluate $\left<\phi_m\right| \hat{H} \left| \phi_n\right>$, it may be helpful to separate the kinetic energy and potential energy of the Hamiltonian.
5. Why is the dispersion relation only affected near $k=0$ and at the edge of the Brillouin zone (see also figures [above](#repeated-vs-reduced-vs-extended-brillouin-zone))?
??? hint
To answer this question, only consider consider two free electron wavefunctions in the Hamiltonian and ignore all the others. Between what two of free electron wavefunctions does the coupling give significant contribution to the energy levels of the free electron wavefunctions?
??? hint
To answer this question, only consider consider two free electron wavefunctions in the Hamiltonian and ignore all the others. Between what two of free electron wavefunctions does the coupling give significant contribution to the energy levels of the free electron wavefunctions?
### Exercise 3: the tight binding model vs. the nearly free electron model
Consider a 1D crystal with a periodic potential given by delta peaks:
...
...
@@ -334,37 +333,41 @@ where $\lambda>0$. In this exercise, we will find the band structure of this cry
- By means of the nearly free electron model explained in this lecture.
- By means of the tight binding model explained in [lecture 7](/7_tight_binding).
1.We first find the band structure using the nearly free electron model. To this end, we consider the effect of the potential on the free electron wavefunctions given by $\psi_1(x) \propto e^{ikx}$ and $\psi_2(x) \propto e^{i[k-2\pi/a]x}$ on the interval $k=[0,\pi/a]$. Derive a dispersion relation of the lower band using the Schödinger equation and the trial eigenstate
We first find the band structure using the nearly free electron model.
$$
\Psi(x) = \alpha \psi_1(x) + \beta \psi_2(x).
$$
1. To this end, we consider the effect of the potential on the free electron wavefunctions given by $\psi_1(x) \propto e^{ikx}$ and $\psi_2(x) \propto e^{i[k-2\pi/a]x}$ on the interval $k=[0,\pi/a]$. Derive a dispersion relation of the lower band using the Schödinger equation and the trial eigenstate
??? hint
$$
\Psi(x) = \alpha \psi_1(x) + \beta \psi_2(x).
$$
Using the Schrödinger equation and the trial eigenstate, first derive a 2×2 eigenvalue problem given by
3. We now use a tight binding model approach to derive the dispersion relation.
We know from that the corresponding dispersion is $$
E = \varepsilon_0 - 2 t \cos (ka). $$
Find an expression for $\varepsilon_0=\left<n\right|\hat{H}\left|n\right>$ and $-t=\left<n-1\right|\hat{H}\left|n\right>$, using the bound state wavefunction around a single $\delta$-peak, centered at site $n$: $$
|n\rangle = \kappa e^{- \kappa | x-na | }
, $$
where $\kappa = -\frac{m \lambda}{\hbar^2}$.
??? hint
3. We now use a tight binding model approach to derive the dispersion relation.
We know from that the corresponding dispersion is
$$
E = \varepsilon_0 - 2 t \cos (ka).
$$
Find an expression for $\varepsilon_0=\left<n\right|\hat{H}\left|n\right>$ and $-t=\left<n-1\right|\hat{H}\left|n\right>$, using the bound state wavefunction around a single $\delta$-peak, centered at site $n$: $$
|n\rangle = \kappa e^{- \kappa | x-na | }
, $$
where $\kappa = -\frac{m \lambda}{\hbar^2}$.
To ease the calculating $\epsilon_0$ and $t$, calculate them for $| n = 0 \rangle $ and $ | n = 1 \rangle $.
??? hint
You may also make use of the results obtained in [exercise 2 of lecture 5](/5_atoms_and_lcao/#exercise-2-application-of-the-lcao-model) or found on the [wikipedia](https://en.wikipedia.org/wiki/Delta_potential).
To ease the calculating $\epsilon_0$ and $t$, calculate them for $| n = 0 \rangle $ and $ | n = 1 \rangle $.
You may also make use of the results obtained in [exercise 2 of lecture 5](/5_atoms_and_lcao/#exercise-2-application-of-the-lcao-model) or found on the [wikipedia](https://en.wikipedia.org/wiki/Delta_potential).
4. Compare the bands obtained in exercise 1 and 2: what are the minima and bandwidths (difference between maximum and minimum) of those bands?
5. For what $a$ and $\lambda$ is the nearly free electron model more accurate? And for what $a$ and $\lambda$ is the tight binding model more accurate?
