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Vanderbilt Lecture, Surface Science, 3/4/2013
Surface Science Lecture
Band Structure: Bulk and Surface
David Vanderbilt Rutgers University
Vanderbilt Lecture, Surface Science, 3/4/2013
3D Crystals: Bloch’s theorem
Hamiltonian eigenstates:
Crystal Hamiltonian:
Vanderbilt Lecture, Surface Science, 3/4/2013
Bulk bandstructure
Occupation
Vanderbilt Lecture, Surface Science, 3/4/2013
Bloch’s Theorem
Brillouin zone
Vanderbilt Lecture, Surface Science, 3/4/2013
Bulk bandstructure of diamond (C)
fcc Brillouin
zone
Vanderbilt Lecture, Surface Science, 3/4/2013
Real space and reciprocal space
Brillouin zone
Unit cell
Vanderbilt Lecture, Surface Science, 3/4/2013
Beware alternate notations
Previous a1, a2, a3 now a, b, c
Previous A1, A2, A3 now A, B, C
Vanderbilt Lecture, Surface Science, 3/4/2013
Periodicity in reciprocal space
k’
k
BZ is unit cell in reciprocal space
Vanderbilt Lecture, Surface Science, 3/4/2013
Summary: 3D crystals
Vanderbilt Lecture, Surface Science, 3/4/2013
Surface periodicity
Vanderbilt Lecture, Surface Science, 3/4/2013
Surface Brillouin zone
Vanderbilt Lecture, Surface Science, 3/4/2013
Bulk Bloch state of wavevector k
k⊥: 1D wavenumber
k||: 2D wavevector
Vanderbilt Lecture, Surface Science, 3/4/2013
Bulk band-structure from surface viewpoint
Vanderbilt Lecture, Surface Science, 3/4/2013
Projected bulk bands at surface
Example:
This is the conventional representation
Vanderbilt Lecture, Surface Science, 3/4/2013
Cubic crystal: Tight binding, 1 band
A simple tight-binding model with one s state per site gives:
Vanderbilt Lecture, Surface Science, 3/4/2013
Cubic crystal: Tight binding, 1 band
A simple tight-binding model with one s state per site gives:
Look down
Vanderbilt Lecture, Surface Science, 3/4/2013
Cubic crystal: Tight binding, 1 band
Vanderbilt Lecture, Surface Science, 3/4/2013
Cubic crystal: Tight binding, 1 band
Vanderbilt Lecture, Surface Science, 3/4/2013
Cubic crystal: Tight binding, 1 band
Vanderbilt Lecture, Surface Science, 3/4/2013
Cubic crystal: Add second band
Vanderbilt Lecture, Surface Science, 3/4/2013
Cubic crystal: Redraw
Vanderbilt Lecture, Surface Science, 3/4/2013
Sometimes you will see:
What are these solid and
dashed lines?
Vanderbilt Lecture, Surface Science, 3/4/2013
Surface bandstructure of diamond (111)
Vanderbilt Lecture, Surface Science, 3/4/2013
Bloch’s theorem for a surface
Vanderbilt Lecture, Surface Science, 3/4/2013
Surface states and resonances
Vanderbilt Lecture, Surface Science, 3/4/2013
How do various states look at the surface?
Bulk state
Surface state
Resonance
Bulk Vacuum
Bulk Vacuum
Bulk Vacuum
Vanderbilt Lecture, Surface Science, 3/4/2013
Experiments
Angle Resolved Photo-Emission Spectroscopy
Angle Resolved Inverse
Photo-Emission Spectroscopy
Vanderbilt Lecture, Surface Science, 3/4/2013
Experiments
Angle Resolved Photo-Emission Spectroscopy
Angle Resolved Inverse
Photo-Emission Spectroscopy
occ.
empty
EF
occ.
empty
EF
Vanderbilt Lecture, Surface Science, 3/4/2013
Experiments: Photoemission
Vanderbilt Lecture, Surface Science, 3/4/2013
Experiments: Inverse Photoemission
A. Goldman, V. Dose and G. Borstel
Vanderbilt Lecture, Surface Science, 3/4/2013
Recent: Topological insulators
Spin-Orbit Coupling
Vanderbilt Lecture, Surface Science, 3/4/2013
How do various states look at the surface?
Bulk state
Surface state
Resonance
Bulk Vacuum
Bulk Vacuum
Bulk Vacuum
Vanderbilt Lecture, Surface Science, 3/4/2013
Typically, surface states exist if:
1. Character is similar to a bulk state, but shifted in energy by the surface perturbation
Bulk Vacuum
Vanderbilt Lecture, Surface Science, 3/4/2013
2. State of new character, e.g., a “broken bond” or “adsorbate state”, is created at the surface
Typically, surface states exist if:
Typically the state is near the middle of the gap and is very localized (e.g., within one or a few atom distances from the surface in real space)
Example: Si (111) 1x1:
Vanderbilt Lecture, Surface Science, 3/4/2013
Surface bandstructure of diamond (111)
Vanderbilt Lecture, Surface Science, 3/4/2013
Surface bandstructure of diamond (111)
Vanderbilt Lecture, Surface Science, 3/4/2013
Summary
• Bulk bandstructure
– 3D Real space and reciprocal space
– Bloch’s theorem for 3D crystals
• Surface bandstructure
– 2D Real space and reciprocal space
– Projected bulk bandstructure
– Surface bands
– Resonances
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