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docs/11_nearly_free_electron_model.md
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......@@ -12,7 +12,7 @@ pi = np.pi
```
# The nearly free electron model & Bloch theorem
_(based on chapter 15 of the book)_
_(based on chapter 15 of the book)_
!!! success "Expected prior knowledge"
......@@ -20,15 +20,15 @@ _(based on chapter 15 of the book)_
- 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.
- 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**.
- Describe what momentum states of particles in a crystal may couple through the periodic lattice potential.
- Describe what momentum states of particles in a crystal may couple through the periodic lattice potential.
- 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.
......@@ -38,7 +38,7 @@ Let's summarize what we learned about electrons so far:
* Electrons on isolated atoms live in discrete orbitals ([lecture 5](5_atoms_and_lcao.md))
* When orbitals hybridize we get *LCAO* or *tight-binding* band structures ([lecture 7](7_tight_binding.md))
In this lecture, we will analyze how electrons behave in solids using the *nearly-free electron model*. This model considers electrons as plane waves (as in the free electron model) that are weakly perturbed by the periodic potential associated with the atoms in a solid. This approach is opposite to that of the tight-binding model, where our starting point was that the electrons are strongly bound to the individual atoms and we included hopping to other atoms as a small effect. Perhaps surprisingly, we will find that the nearly-free electron model gives very similar results to the tight binding model: it also leads to the formation of energy bands, and these bands are separated by *band gaps* - regions in the band structure where there are no allowed energy states.
In this lecture, we will analyze how electrons behave in solids using the *nearly-free electron model*. This model considers electrons as plane waves (as in the free electron model) that are weakly perturbed by the periodic potential associated with the atoms in a solid. This approach is opposite to that of the tight-binding model, where our starting point was that the electrons are strongly bound to the individual atoms and we included hopping to other atoms as a small effect. Perhaps surprisingly, we will find that the nearly-free electron model gives very similar results to the tight binding model: it also leads to the formation of energy bands, and these bands are separated by *band gaps* - regions in the band structure where there are no allowed energy states.
## Nearly free electron model
......@@ -125,19 +125,19 @@ $$ E(\delta k) = E_0 \pm \sqrt{v^2\hbar^2\delta k^2 + |W|^2}$$
#### Calculating the magnitude of the gaps
We will now show that $W=\langle k | V |k' \rangle$ represents a Fourier component of the lattice potential. To see this, we express the lattice potential (which is periodic with $V(x)=V(x+a)$) as a Fourier series
We will now show that $W=\langle k | V |k' \rangle$ represents a Fourier component of the lattice potential. To see this, we express the lattice potential (which is periodic with $V(x)=V(x+a)$) as a Fourier series
$$
V(x) = \sum_{n=-\infty}^{\infty} V_n e^{2\pi i n x/a}
$$
and recall that such a series has Fourier components $V_n$ given by
and recall that such a series has Fourier components $V_n$ given by
$$
V_n = \frac{1}{a}\int_0^a e^{- i n 2\pi x /a} V(x) dx
$$
Calculating $W$, we find
Calculating $W$, we find
$$
W = \langle k | V | k' \rangle = \frac{1}{a}\int_0^{a} e^{-i k x} V(x) e^{i k'x} dx = \frac{1}{a}\int_0^a e^{-i 2\pi x /a} V(x) dx = V_1
......@@ -150,10 +150,10 @@ where we have used that $k-k' =2\pi/a$ because we are analyzing the first crossi
Everything we did can also be applied to the higher-energy crossings seen in the figure above. We note that all crossings occur between parabola's that are shifted by integer multiples of reciprocal lattice vectors $n 2\pi/a$. The first crossing corresponds to $n=1$, and we found that the magnitude of the gap is given by $V_1$. Similarly, $V_2$ determines the gap between the second and third bands, $V_3$ for the crossing between third and fourth, etc.
The key conclusion is that the lattice potential couples plane-wave states that differ by integer multiples of reciprocal lattice vectors. This coupling alters the band structure most strongly where the free-electron eigenenergies cross, opening up gaps of which the magnitudes are determined by the Fourier components of the lattice potential.
The key conclusion is that the lattice potential couples plane-wave states that differ by integer multiples of reciprocal lattice vectors. This coupling alters the band structure most strongly where the free-electron eigenenergies cross, opening up gaps of which the magnitudes are determined by the Fourier components of the lattice potential.
??? question "Suppose the lattice potential is $V(x)=A\cos(2\pi/ax)$. At what locations in the dispersion does $V(x)$ lead to the formation of gaps?"
Hint: The Fourier series of $V(x)$ is $V(x)=A(e^{i2\pi/ax}+e^{-i2\pi/ax})/2$, so the only non-zero Fourier components are $V_1=V_{-1} = A/2$.
Hint: The Fourier series of $V(x)$ is $V(x)=A(e^{i2\pi/ax}+e^{-i2\pi/ax})/2$, so the only non-zero Fourier components are $V_1=V_{-1} = A/2$.
## General description of a band structure in a crystal - Bloch theorem
......@@ -204,7 +204,7 @@ $$
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."
### Repeated vs reduced vs extended Brillouin zone
There are several common ways to **plot** the same dispersion relation (no difference in physical information).
......@@ -279,62 +279,47 @@ draw_classic_axes(ax, xlabeloffset=4)
### Exercise 1: Bloch's theorem
Suppose we have a crystal with lattice vectors $\mathbf{a}_ 1$, $\mathbf{a}_ 2$, and $\mathbf{a}_ 3$.
#### Question 1.
What can be said about the symmetry of the Hamiltonian $\hat{H}$ of this crystal?
#### Question 2.
Now define the translation operator $\hat{T}_{\alpha,\beta,\gamma}$ so that
1. What can be said about the symmetry of the Hamiltonian $\hat{H}$ of this crystal?
To describe the translation through the crystal in terms of the lattice vectors, we define the translation operator $\hat{T}_{\alpha,\beta,\gamma}$ in such a way that
$$
\hat{T}_{\alpha,\beta,\gamma} \psi(\mathbf{r}) = \psi(\mathbf{r} - \alpha \mathbf{a}_1 - \beta \mathbf{a}_2 - \gamma \mathbf{a}_3),
$$
where $\alpha$, $\beta$, $\gamma$ are integers.
where $\alpha$, $\beta$, $\gamma$ are integers. Show that $\hat{T}_{\alpha,\beta,\gamma}$ and $\hat{H}$ commute.
#### Question 3.
Show that the Bloch wavefunctions defined in the lecture notes are eigenfunctions of $\hat{T}_{\alpha,\beta,\gamma}$. What are the corresponding eigenvalues? What does this say about the eigenfunctions of $\hat{H}$.
#### Question 4.
By applying $\hat{H}$ to the Bloch wavefunction, show that the Schrödinger equation can be rewritten as
$$
\left[ \frac{\mathbf{\hat{p}}^2}{2m} + \frac{\hbar}{m} \mathbf{k} \cdot \mathbf{\hat{p}} + \frac{\hbar^2 \mathbf{k}^2}{2m} + V(\mathbf{r}) \right] u_{n,\mathbf{k}}(\mathbf{r}) = E_{n,\mathbf{k}} u_{n,\mathbf{k}}(\mathbf{r}),
$$
2. Show that $\hat{T}_{\alpha,\beta,\gamma}$ and $\hat{H}$ commute.
3. Show that the Bloch wavefunctions defined in the lecture notes are eigenfunctions of $\hat{T}_{\alpha,\beta,\gamma}$. What are the corresponding eigenvalues? What does this say about the eigenfunctions of $\hat{H}$.
5. By applying $\hat{H}$ to the Bloch wavefunction, show that the Schrödinger equation can be rewritten as $$
\left[ \frac{\mathbf{\hat{p}}^2}{2m} + \frac{\hbar}{m} \mathbf{k} \cdot \mathbf{\hat{p}} + \frac{\hbar^2 \mathbf{k}^2}{2m} + V(\mathbf{r}) \right] u_{n,\mathbf{k}}(\mathbf{r}) = E_{n,\mathbf{k}} u_{n,\mathbf{k}}(\mathbf{r})
, $$
where $\mathbf{\hat{p}} =-i\hbar\nabla$.
#### Question 5.
What is $u_{n,\mathbf{k}}(\mathbf{r})$ in case of free electrons? Is your answer consistent with the equation above?
5. What is $u_{n,\mathbf{k}}(\mathbf{r})$ in case of free electrons? Is your answer consistent with the equation above?
### Exercise 2: the central equation in 1D
Let's consider a 1D crystal with a period $a$. Let $k_0$ be any wave number of an electron in the first Brillouin zone.
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.
#### Question 1.
What $k_n$ are equivalent to $k_0$ in this crystal?
#### Question 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?
#### Question 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 $V(x) = V(x+na)$ in our system. This causes coupling between eigenstates $\left| \phi_n\right>$ in the free electron case. In order to find the right eigenstates of the system with that potential, we need an 'LCAO-like' trial eigenstate given by
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
$$
\left|\psi\right> = \sum_{n=-\infty}^{\infty}C_n \left|\phi_n\right>
$$
#### Question 4.
Using the trial eigenstate above and the Schrödinger equation, show that
$$
E C_m = \varepsilon_m C_m+\sum_{n=-\infty}^{\infty} V_{n}C_{m-n},
$$
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)._
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)._
??? 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.
#### Question 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))?
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?
......@@ -351,14 +336,14 @@ 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).
#### Question 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
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
$$
\Psi(x) = \alpha \psi_1(x) + \beta \psi_2(x).
$$
??? hint
Using the Schrödinger equation and the trial eigenstate, first derive a 2×2 eigenvalue problem given by
$$
......@@ -367,18 +352,21 @@ $$
What are $\varepsilon_0(k)$, $V_0$ and $V_1$?
#### Question 2.
Make a sketch of the lower band.
#### Question 3.
We now use the tight binding model, where we know that the dispersion relation can be described by
2. Make a sketch of the lower band.
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 $\kappa = -\frac{m \lambda}{\hbar^2}$.
$$
E = \varepsilon_0 - 2 t \cos (ka).
$$
??? hint
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).
Find an expression for $\varepsilon_0=\left<n\right| \hat{H} \left|n\right>$ and $-t=\left<n-1\right| \hat{H} \left| n \right>$, where $|n\rangle$ is the wavefunction of a single $\delta$-peak well at site $n$. You may make use of the results obtained in [exercise 2 of lecture 5](/5_atoms_and_lcao/#exercise-2-application-of-the-lcao-model) or [look up the wavefunction](https://en.wikipedia.org/wiki/Delta_potential).
#### Question 4.
Compare the bands obtained in exercise 1 and 2: what are the minima and bandwidths (difference between maximum and minimum) of those bands?
#### Question 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?
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?
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