FIRST PRINCIPLES CALCULATION OF OFF-NORMAL LEEM REFLECTIVITY SPECTRA OF FEW LAYER GRAPHENE APS March Meeting: March 3, 2014 John McClain, Ph.D. Candidate Integrated Applied Mathematics Program University of New Hampshire Collaborators: Jiebing Sun - Physics, MSU Karsten Pohl - Physics, UNH Jian-Ming Tang - Physics, UNH Acknowledgements: Jim Hannon - IBM Watson Research Center

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  Down to very low energies  Compare to curves from model to determine structural details Hibino, et al. Phys. Rev. B 77 (2008) Berger, et al. J. Phys. Chem. 108 (2004) en.wikipedia.org/wiki/ LEEM

I-V Curve Calculations  Most methods restricted to muffin tin scattering potentials (Pendry 1974, Van Hove 1986)  Rely on fitting parameters  Are not valid at very low energies  We’ve developed a first principles method  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)

 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 Scattering via Wave Matching with DFT

Free-standing FLG Reflectivity: Normal Incidence Experimental FLG on SiCCalculated Free-standing FLG McClain, et al. arXiv : (2013)Hibino, et al. Phys. Rev. B 77 (2008)  Also, agrees with findings of Feenstra, et al. PRB 87 (2013)

Hibino, et al. e-J. Surf. Sci. Nanotech. Vol. 6 (2008)  Oscillations at 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 Free-standing FLG Reflectivity: Normal Incidence

Off-Normal Incidence  Why?  More information for given energy range  New distinguishing features  Continue to consider only specular reflection ‘

In-plane k-vector vs Angle of Incidence Fixed k // Fixed Angle ≈ 5 ° M Г KM Г K Bauer, Carl A. et al. arXiv:

General Incidence Reflectivity  Similar oscillations  With energy shifts  3-Way Splitting of Peak  New layer- dependent oscillations Near K M Г KM 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 //. Adapted from dissertation of Tesfaye Alayew

General Incidence Reflectivity M ГK

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M ГK

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CONCLUSIONS John McClain  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

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