8. Weak potential approximation and tight-binding method

· Tight-binding method is an application of the Bloch theorem
· Electron energy has a gap at the Bragg planes
· Weak potential approximation \( \approx \) Nearly free electron approximation
· Metals found in groups I, II, III, and IV → Conduction electrons can be described as moving in what amounts to an almost constant potential → *Nearly free electron*
· Sommerfeld free electron gas, modified by the presence of a weak periodic potential

﹡ Why the conduction bands of these metals should be so free-electron-like
· Why the strong interactions of the conduction electrons with each other and with the positive ions can have the net effect of a very weak potential

1. Electron-ion interaction is strong, but the conduction electrons are forbidden (Pauli principle) because this region is already occupied by the core electrons
2. Screening effect

· The tight-binding method → Very different point of view from the nearly free electron approximation

8.1. General approach to the Schrödinger equation when the potential is weak

· Zero potential → Plane wave solution
· Weak periodic potential → Expansion of the exact solution in plane waves

$$ \psi  _{\vec{k}  }(\vec{r}) = \sum_{\vec{K}} c_{\vec{k} -\vec{K}   } e^{i\vec{k}-\vec{K}} \cdot\vec{r} \tag{8.1}$$

Substitute into the SE

$$ \left( \frac{\hbar^2}{2m} (\vec{k}-\vec{K})^2 - \varepsilon \right)c_{ \vec{k}- \vec{K} }+\sum _{ \vec{K}'} U_{\vec{K}'-\vec{K}}c_{\vec{k}-\vec{K}'}=0 \tag{8.2} $$

 

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