@@ -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.
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@@ -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.
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
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@@ -148,12 +148,12 @@ where we have used that $k-k' =2\pi/a$ because we are analyzing the first crossi
#### 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 first and third bands, $V_3$ for the crossing between first and fourth, etc.
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
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@@ -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).
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@@ -283,9 +283,9 @@ Suppose we have a crystal with lattice vectors $\mathbf{a}_ 1$, $\mathbf{a}_ 2$,
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
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 $\hat{H}$.
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@@ -296,14 +296,14 @@ where $\mathbf{\hat{p}} =-i\hbar\nabla$.
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 periodicity of $a$.
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
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
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).
,$$
where $V_n$ are the Fourier components of the potential defined [above](#physical-meaning-of-w).
Find an expression for $\varepsilon_m$. _**NB:** This equation is also known as the central equation (in 1D)._
??? hint
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@@ -343,7 +343,7 @@ $$
$$
??? hint
Using the Schrödinger equation and the trial eigenstate, first derive a 2×2 eigenvalue problem given by
$$
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@@ -358,13 +358,13 @@ 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
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).
