### First Principles Calculations of Off-Normal LEEM

```FIRST PRINCIPLES CALCULATION OF
OFF-NORMAL LEEM REFLECTIVITY SPECTRA
OF FEW LAYER GRAPHENE
Collaborators:
Jiebing Sun - Physics, MSU
Karsten Pohl - Physics, UNH
Jian-Ming Tang - Physics, UNH
John McClain, Ph.D. Candidate
Integrated Applied Mathematics Program
University of New Hampshire
Acknowledgements:
Jim Hannon - IBM Watson
Research Center
APS March Meeting: March 3, 2014
Outline

Motivation
 LEEM:
very low-energy I-V
curves
 Need for new I-V analysis

Method
 Density
Functional Theory,
wave-matching

Results
 Normal
Incidence Free-standing
FLG
 General Angle of Incidence FLG
Low-energy Electron Microscopy


Illuminate areas down to
8nm x 8nm
Record I-V curve for
specular/diffracted beam


http://en.wikipedia.org/wiki/
LEEM
Down to very low
energies
Hibino, et al. Phys. Rev.
B 77 (2008)
Compare to curves from
model to determine
structural details
Berger, et al. J. Phys. Chem.
108 (2004)
I-V Curve Calculations

Most methods restricted to muffin tin
scattering potentials (Pendry 1974, Van Hove
1986)



We’ve developed a first principles method



Rely on fitting parameters
Are not valid at very low energies
Using self-consistent potentials
More efficient than other first principles
methods
Other first principles approaches

Flege, Meyer, Falta, and Krasovskii PRB 84
(2011), Self-limited oxide formation in Ni(111)
oxidation.

Feenstra, et al. PRB 87 (2013), Low-energy
electron reflectivity from graphene.
Scattering via Wave Matching with DFT


Our method: Find self-consistent potential and scattering states with DFT
packages for solids
 Introduces a supercell
 Match incoming and outgoing plane waves to Bloch solutions at interfaces
Quantum ESPRESSO (plane wave basis)
Scattering via Wave Matching with DFT



Our method: Find self-consistent potential and scattering states with DFT
packages for solids
 Introduce a supercell
 Match incoming and outgoing plane waves to Bloch solutions at interfaces
Quantum ESPRESSO (plane wave basis)
Specular reflection only; lowest energy range
Scattering via Wave Matching with DFT




Our method: Find self-consistent potential and scattering states with DFT
packages for solids
 Introduce a supercell
 Match incoming and outgoing plane waves to Bloch solutions at interfaces
Quantum ESPRESSO (plane wave basis)
Specular reflection only; lowest energy range
Focus on Free-Standing Graphene
Free-standing FLG Reflectivity:
Normal Incidence
Experimental FLG on SiC
Calculated Free-standing FLG
Hibino, et al. Phys. Rev. B 77 (2008)
McClain, et al. arXiv :1311.2917 (2013)

Also, agrees with findings of
Feenstra, et al. PRB 87 (2013)
Free-standing FLG Reflectivity:
Normal Incidence
Hibino, et al. e-J. Surf. Sci. Nanotech. Vol. 6 (2008)



Oscillations at 15-20 eV likely killed by damping/inelastic effects
Quantum Interference oscillations align with dispersive bands
Reflection peaks align with bulk band gaps: ~10 eV, 25 eV, & 35 eV
Off-Normal Incidence


Why?
 New distinguishing features
Continue to consider only specular reflection
‘
In-plane k-vector vs Angle of Incidence
Fixed Angle ≈ 5°
Fixed k//
M
Г
K
M
Г
Bauer, Carl A. et al.
arXiv:1309.0914
K
General Incidence Reflectivity



Similar
oscillations
 With energy
shifts
3-Way Splitting
of Peak
New layerdependent
oscillations
Near K
M
Г
K
M
Г
K
Band Gaps and Spectra Peaks


Just like we did for
normal incidence, we
can match spectra
peaks to band gaps.
But now we have a
band structure for
each k//.
from
dissertation
of Tesfaye
Alayew
General Incidence Reflectivity
M
Г
K
General Incidence Reflectivity
M
Г
K
General Incidence Reflectivity
M
Г
K
General Incidence Reflectivity
M
Г
K
General Incidence Reflectivity
M
Г
K
General Incidence Reflectivity
M
Г
K
General Incidence Reflectivity
M
Г
K
General Incidence Reflectivity
M
Г
K
General Incidence Reflectivity
M
Г
K
General Incidence Reflectivity
M
Г
K
General Incidence Reflectivity
M
Г
K
CONCLUSIONS

Wave matching approach is able to produce reflection coefficients
for specular reflection for general angles of incidence.

Calculated reflectivities match experimental results for normal
incidence


Free standing graphene matches FLG on SiC
Off-normal Scattering
 Similar quantum-interference
oscillations with energy shifts
 Peak splitting; New layer-dependent
oscillations
 Connection between reflectivity and
bulk graphite band gaps persists
John McClain
Overcoming Artificial Energy Gaps

Different energy
ranges accessed
using different
supercell sizes

4 supercells
cover all but
narrow regions

Difficult to
predict which
supercell sizes
cover which
energies
```