Models of electronic structures in solids

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🗒️ Note: the motion of valence electrons in a solid is quantum mechanical in nature

💼 Case: consider a many electron system is a solid, steps to solving

Find energy spectrum $E(k)$ of a single electron in the crystal
Use Aufbau principle to obtain the total energy and wave functions of the system
Include fermi distribution to find physical properties at finite temperature

🗒️ Notes:

$E(k)$ is the energy levels $E_{n,l,m}$ of the electron in hydrogen atom.
$\vec k=(k_x,k_y,k_z)$ are good quantum numbers, where $\vec k$-space is reciprocal space, as $\vec p=\hbar k$
Density of states $g(E)$ ($\rm Nb$ of states per unit $V$ in energy interval $\text dE$)

Since Schrödinger's equation is too complicated for multiple electrons we have multiple models:

Valence electrons are free to move through the solid
Ignore interactions between them and the atomic cores (setting potential to $0$)

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The free electron model of metals

🗒️ Note: the “🧽 Assume:” marker will be used to describe both assumptions and approximations

Start with the electron-in-a-box model applied to 3D with a potential well

$$ V(x,y,z)=\left \{ \begin{matrix} 0 & \text{if} \; 0\le x,y,z\le L\\ \infin & \text{otherwise} \end{matrix} \right . $$
We solve the TISE, within $V=0$ using conditions $\psi=0$ when $x=\{0,L\}$

$$ \psi_{n_x,n_y,n_z}(x,y,z)=\left ( \frac{2}{L} \right )^\frac 32 \sin \left ( \frac{n_x\pi x}{L}\right )\sin \left ( \frac{n_y\pi y}{L}\right )\sin \left ( \frac{n_z\pi z}{L}\right ) $$

where $n_x,n_y,n_z\in \N^*$ and the wave number is given by

$$ k_x=\frac{n_x\pi}{L}\qquad k_y=\frac{n_y\pi}{L} \qquad k_z=\frac{n_z\pi}{L} $$