Graduate Course: Solid State Chemistry

Tight Binding Transitions

From Molecular Orbitals to Periodic Band Structure

L1.01

The Power of \( N \)

Consider a diatomic molecule \( H_2 \). Two orbitals, two states: Bonding and Antibonding.

"What happens to the HOMO-LUMO gap when you connect \( 10^{23} \) atoms together?"

N=2

N=6

N \(\rightarrow \infty\)

Energy quantization transitions to a continuum.

Bloch's Theorem

In a periodic potential \( V(r + R) = V(r) \), the eigenfunctions of the Hamiltonian take the form:

\[ \psi_k(r) = e^{ik \cdot r} u_k(r) \]

\( u_k(r) \) has the same periodicity as the crystal lattice.
\( e^{ik \cdot r} \) is a plane-wave envelope factor.
The wavevector \( k \) defines the phase shift between unit cells.

Concept Check

Why is the magnitude of the wavefunction \( |\psi_k|^2 \) the same in every unit cell?

"The electron probability density must reflect the translational symmetry of the lattice."

k-Space (Reciprocal Space)

While real space describes positions \( r \), reciprocal space describes wavevectors \( k \).

Real Lattice

Periodicity \( a \)

Unit: Meters (m)

Reciprocal Lattice

Periodicity \( 2\pi/a \)

Unit: Inverse length (m\(^{-1}\))

The First Brillouin Zone (1st BZ)

The primitive cell in reciprocal space. For 1D: \( -\pi/a \le k \le \pi/a \).

0

+\(\pi/a\)

-\(\pi/a\)

Energy E(k)

The Tight Binding Result

For a 1D chain of s-orbitals with site energy \( \alpha \) and hopping integral \( \beta \), the band energy is:

\[ E(k) = \alpha + 2\beta \cos(ka) \]

\( \alpha \)

Site Energy

Atomic orbital energy

\( \beta \)

Hopping Integral

Overlap / Interaction strength

\( 4\beta \)

Bandwidth

Total energy range of the band

Key Takeaways

01

Orbitals overlap in solids to form continuous bands of allowed energy states.

02

Bloch's Theorem allows us to solve the Schrödinger equation for a periodic system using \( k \).

03

The dispersion relation \( E(k) \) connects electronic energy to the crystal's periodic lattice.

Up Next:

"How do we represent these bands for real materials? We'll dive into Band Structure diagrams and Density of States (DOS) to classify metals and insulators."

Go to Lesson 2

Quantum Crystals and Band Theory

Tight Binding Transitions

Band Diagrams and DOS Analysis

Lattice Instability and Peierls Distortion

Semiconductor Doping Engineering

Crystal Defects and Deviations

Orbital Overlap Slides

Bloch Functions Worksheet

Spaghetti Plot Slides

DOS Analysis Worksheet

DOS Analysis Key

Lattice Instability Slides

Distortion Dynamics Worksheet

Doping Dynamics Slides

Junction Engineering Worksheet

Crystal Imperfection Slides

Defect Detective Worksheet

Solid State Seminar Project Guide

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