17.1. Introduction to the QM of Solids#

17.1.1. Energy bands#

First, the energy level splitting is discussed. Then, two models (Kronig-Penny, and nearly free electron models) are introduced to describe the concept of allowed and forbidden energy bands, starting from Schrodinger equation.

17.1.1.1. Energy level splitting#

This section deal with energy level splitting in a two-element system.

Non-interacting systems

Let \(| L \rangle\), and \(| R \rangle\) two localized eigen-states of non-interacting identical subsystems, with the same energy \(E_0\),

\[\begin{split}\begin{aligned} \hat{H}_0 | L \rangle & = E_0 | L \rangle \\ \hat{H}_0 | R \rangle & = E_0 | R \rangle \\ \end{aligned}\end{split}\]

As these states are localized in space, they’re (approximately?) orthogonal, \(\langle L | R \rangle = 0\).

Interacting subsystems - Perturbation potential

When the two systems are brought close together, they start interacting weakly, so that the Hamiltonian becomes

\[\hat{H} = \hat{H}_0 + \hat{V} \ .\]

The matrix elements of the Hamiltonian in the base \(\left\{ | L \rangle, | R \rangle \right\}\) (where are all the other eigen-functions? We’re not interested in them, right now, but only on the pair of eigenfunctions with the same energy when the syb-systems are not interacting) read

\[\begin{split}\begin{aligned} \langle L | \hat{H} | L \rangle & =: E_0' \\ \langle R | \hat{H} | R \rangle & =: E_0' \\ \langle L | \hat{H} | R \rangle & = \langle R | \hat{H} | L \rangle^* =: -t \\ \end{aligned}\end{split}\]
Energy levels - Eigenvalues of the Hamiltonian operator

If the new eigen-functions of the interacting system can be written as a linear combination of the non-interacting functions, \(| \Psi_{1,2} \rangle= \ell_{1,2} | L \rangle + r_{1,2} | R \rangle\), the eigen-problem becomes

\[\hat{H} | \Psi_{1,2} \rangle = E_{1,2} | \Psi_{1,2} \rangle \ .\]

Projecting onto \(\langle L |\), and \(\langle R |\),

\[\begin{split} \begin{cases} \ell E_0' - r t = E \ell \\ - \ell t + r E_0' = E r \\ \end{cases} \quad \rightarrow \quad \begin{cases} \ell (E_0'-E) - r t = 0 \\ - \ell t + r (E_0'-E) = 0 \\ \end{cases} \end{split}\]

This system has non trivial solutions, if the determinant of the linear system is zero,

\[\begin{split}0 = \left| \begin{matrix} E_0' - E & -t \\ -t & E_0' - E \end{matrix}\right| = ( E_0' - E ) - t^2 \ . \end{split}\]

The energy levels of the interacting states are slightly shifted w.r.t. the non-interacting level,

\[E = E_0' \mp t \ .\]
Eigen-states

The eigen-states are

  • the bonding state, for \(E = E_0' - t\), \(\ell = -r\), and with normalization

    \[| \Psi \rangle_{-} = \frac{1}{\sqrt{2}} \left( | L \rangle + | R \rangle \right) \ \]

  • the anti-bonding state, for \(E = E_0' + t\), \(\ell = r\), and with normalization

    \[| \Psi \rangle_{+} = \frac{1}{\sqrt{2}} \left( | L \rangle - | R \rangle \right) \ \]

17.1.1.2. Kronig-Penny model#

Kronig-Penny potential

Kronig-Penny model is a 1-dimensional model with periodic square wave potential,

\[\begin{split}V(x) = \left\{\begin{aligned} V_0 \quad & , \quad x \in \text{regions II} = [ n a + (n-1) b, n ( a + b )] \\ 0 \quad & , \quad x \in \text{regions I } = [ (n-1) (a+b), n a + (n-1) b] \\ \end{aligned}\right.\end{split}\]
General expression of Bloch states

Following Bloch’s theorem, the eigen-states of an electron (in position base) in a infinite periodic lattice can be written as

\[\Psi(\mathbf{r}) = u(\mathbf{r}) e^{i \mathbf{k} \cdot \mathbf{r}} \ ,\]

with \(u(\mathbf{r}) = u \left(\mathbf{r} + \sum_j n_j \mathbf{a}_i \right)\) with the same periodicity as the lattice, \(n_j \in \mathbb{Z}\), or in 1-dimensional problems,

\[\Psi(x) = u(x) e^{i k x} \ .\]
Eigenvalue problem for the Hamiltonian operator

The eigenvalue problem for the Hamiltonian operator reads

\[E \Psi = \hat{H} \Psi = \left[ \frac{\hat{p}^2}{2m} + V(x) \right] \Psi \ ,\]

with \(\hat{p}\) the momentum operator, whose expression in position base is given by (13.1), and in 1-dimensional problems \(\hat{p} \Psi = - i \hbar \partial_x \Psi\). The operator acting twice gives \(\hat{p}^2 \Psi = - \hbar^2 \partial_{xx} \Psi\).

Introducing the expression of the eigenfunctions from Bloch’s theorem1, it follows

\[E u(x) = - \frac{\hbar^2}{2m} \left( u''(x) + 2 ik u'(x) - k^2 u(x) \right) + V(x) u(x) \ ,\]

or

\[\begin{split}\begin{aligned} 0 & = u'' + 2 i k u'(x) - \left( k^2 - \frac{2m E}{\hbar^2} \right) u(x) && \text{regions I} \\ 0 & = u'' + 2 i k u'(x) - \left( k^2 - \frac{2m ( E - V_0 )}{\hbar^2} \right) u(x) && \text{regions II} \\ \end{aligned}\end{split}\]

Solution of the ODEs.

Continuity of \(\Psi(x)\), and \(\partial_x \Psi(x)\). for matching solutions.**

Energy levels.

17.1.1.3. Nearly free electron model#

1

The derivatives read \(\Psi' = \left( u' + i k u \right) e^{i k x}\), and \(\Psi'' = \left( u'' + 2 ik u' - k^2 u \right) e^{ikx}\).