# Learner lab record: Band-structure gap and effective-mass study

Course: Condensed matter, superconductivity, and coherent states

Name: ____________________  Date: ____________________  Group: ____________________

## Investigation question

How do lattice coupling and filling reshape free-particle motion into bands, gaps, and effective quasiparticle response?

## Setup

Use the band-structure laboratory. Establish the uncoupled dispersion, increase the periodic coupling, then compare gap size, curvature, filling, and transport interpretation.

## Variables

| Variable | Role | Unit |
| --- | --- | --- |
| Lattice spacing and coupling | model inputs | length and energy |
| Wavevector | state coordinate | 1/length |
| Band energy and gap | dependent spectrum | eV |
| Band curvature/effective mass | dependent response | energy·length² and mass |

## Predict before changing controls

1. Predict where a periodic potential opens a gap.

   ________________________________________________________________

2. Predict how flatter band curvature changes effective mass magnitude.

   ________________________________________________________________

## Observation table

| lattice spacing | coupling | k point | lower band | upper band | gap | curvature sign |
| --- | --- | --- | --- | --- | --- | --- |
|   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |

## Analyze

1. Which symmetry point hosts the gap?

   ________________________________________________________________

2. How does coupling change the avoided crossing?

   ________________________________________________________________

3. Why does band filling matter for conduction?

   ________________________________________________________________

4. Why does an effective metric analogy not imply modified fundamental spacetime?

   ________________________________________________________________

## Evidence-bounded conclusion

Increasing lattice coupling from ___ to ___ changed the zone-boundary gap from ___ to ___ and local curvature ___; the transport implication is ___.

________________________________________________________________