Band structure

Exercises

Exercise 1: 3D Fermi surfaces

Using the periodic table of the Fermi surfaces (or the static images at https://www.phys.ufl.edu/fermisurface/ if 3D does not work for you), answer the following questions:

1. Find 4 elements that are well described by the nearly-free electron model and 4 that are poorly described by it.

   Well described: (close to) spherical.

2. Is the Fermi surface of lithium or potassium better described by the free electron model? What about nearly-free electron model? Why?

   K is more spherical, hence 'more' free electron model. Li is less spherical, hence 'more' nearly free electron model. Take a look at Au, and see whether you can link this to what you learned in lecture 11.

3. Do you expect a crystal with a simple cubic lattice and monovalent atoms to be conducting?

   Yes. Cubic -> unit cell contains one atom -> monovalent -> half filled band -> metal.

4. What Fermi surface shape would you expect the NaCl crystal to have? Explain your answer using both the atomic valences and the optical properties of this crystal.

   With Solid State knowledge: Na has 1 valence electron, Cl has 7. Therefore, a unit cell has an even number of electrons -> insulating.

   Empirical: Salt is transparent, Fermi level must be inside a large bandgap -> insulating.

Exercise 2: Tight-binding in 2D

Consider a rectangular lattice with lattice constants $a_x$ and $a_y$. Suppose the hopping parameters in the two corresponding directions to be $-t_1$ and $-t_2$. Consider a single orbital per atom and only nearest-neighbour interactions.

1. Write down a 2D tight-binding Schrödinger equation (expand to 2D the results of 1D).

   $$E \phi_{n,m} = \varepsilon_0\phi_{n,m}-t_1 \left(\phi_{n-1,m}+\phi_{n+1,m}\right) -t_2 \left(\phi_{n,m-1}+\phi_{n,m+1}\right)$$

2. Formulate the Bloch ansatz for the wave function.

   $$\psi_n(\mathbf{r}) = u_n(\mathbf{r})e^{i\mathbf{k}\cdot\mathbf{r}} \quad \leftrightarrow \quad \phi_{n,m} = \phi_0 e^{i(k_x n a_x + k_y m a_y)}$$

3. Calculate the dispersion relation of this model.

   $$E = \varepsilon_0 -2t_1 \cos(k_x a_x) -2t_2 \cos(k_y a_y)$$

4. What Fermi surface shape would this model have if the atoms are monovalent?

   Monovalent -> half filled bands -> rectangle rotated 45 degrees.

5. What Fermi surface shape would it have if the number of electrons per atom is much smaller than 1?

   Much less than 1 electron per unit cell -> almost empty bands -> elliptical.

   

Exercise 3: Nearly-free Electron model in 2D

(based on exercise 15.4 of the book)

Suppose we have a square lattice with lattice constant $a$, with a periodic potential given by $V(x,y)=2V_{10}(\cos(2\pi x/a)+\cos(2\pi y/a))+4V_{11}\cos(2 \pi x/a)\cos(2 \pi y/a)$.

1. Use the Nearly-free electron model to find the energy of state $\mathbf{q}=(\pi/a, 0)$.

   Hint

   This is analogous to the 1D case: the states that interact have $k$-vectors $(\pi/a,0)$ and $(-\pi/a,0)$; ($\psi_{+}\sim e^{i\pi x /a}$ ; $\psi_{-}\sim e^{-i\pi x /a}$).

   Construct the Hamiltonian with basis vectors $(\pi/a,0)$ and $(-\pi/a,0)$, eigenvalues are

   $$E=\frac{\hbar^2}{2m} \left(\frac{\pi}{a}\right)^2 \pm \left|V_{10}\right|.$$

2. Let's now study the more complicated case of state $\mathbf{q}=(\pi/a,\pi/a)$. How many $k$-points have the same energy? Which ones?

   Four in total: $(\pm\pi/a,\pm\pi/a)$.

3. Write down the nearly free electron model Hamiltonian near this point.

   Define a basis, e.g. $$\begin{align} \left|0\right\rangle &= (\pi/a,\pi/a) \\ \left|1\right\rangle &= (\pi/a,-\pi/a) \\ \left|2\right\rangle &= (-\pi/a,-\pi/a) \\ \left|3\right\rangle &= (-\pi/a,\pi/a) \end{align}$$ The Hamiltonian becomes

   $$\hat{H}= \begin{pmatrix} \varepsilon_0 & V_{10} & V_{11} & V_{10} \\ V_{10} & \varepsilon_0 & V_{10} & V_{11} \\ V_{11} & V_{10} & \varepsilon_0 & V_{10} \\ V_{10} & V_{11} & V_{10} & \varepsilon_0 \\ \end{pmatrix}$$

4. Find its eigenvalues.

   Using the symmetry of the matrix, we try a few eigenvectors: $$\mathbf{v_\pm}= \begin{pmatrix} 1 \\ \pm 1 \\ 1 \\ \pm 1 \\ \end{pmatrix}$$ $$\mathbf{v_\alpha}= \begin{pmatrix} \alpha \\ 1 \\ -\alpha \\ -1 \\ \end{pmatrix}$$

   $$E_\pm = \varepsilon_0 + V_{11} \pm 2 V_{10} \quad \text{and}\quad E_\alpha = \varepsilon_0 - V_{11}$$

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Last update: October 30, 2023

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