Quantitative LEED Analysis Methods Using Pre-Existing

Mathematical Software

Agenda

  Introduction to graphene   Motivation for studying graphene with LEED   LEED imaging techniques   Analysis of LEED images   Applications in the classroom

Graphene

  2D carbon crystal with ‘honeycomb’ lattice

  All allotropic forms of carbon derive from graphene   Buckyballs   Nanotubes   Graphite

Image courtesy of http://physics.bu.edu

Graphene Properties

  Linear dispersion curve   Electrons in graphene

are ‘massless’   High electron speeds

even at room temperature

  Low resistivity   Many layers of

graphene have the same properties as a single layer

  Integrated circuit design Image courtesy of GA Tech

Mechanical Exfoliation

  IMPRACTICAL!!   Random chance   Sheets are too small to be useful

+ =

Image courtesy of J.Hass

Epitaxial Growth

  SiC wafers (small and expensive) are acid washed and placed in furnaces

  As wafer heats, Si sublimates, C forms honeycomb lattice

  Si-face graphene is different from C-face graphene!

UHV Furnace Grown

  Si-face growth starts at 1250°C

  C-face growth starts at 1100°C

F. Varchon, et al. PRB 77 165415 (2008) 15

0 x

150

nm

height (Å)

300

x 30

0 nm

Image courtesy of J.Hass

RF Furnace Grown

  C-face growth starts at 1420°C   Best candidate for device construction

9 x 9 µm

Imag

e co

urte

sy o

f J.H

ass

height (Å)

400 x 400 nm

Imag

e co

urte

sy o

f J.H

ass

4000Å X 4000Å

26Å X 26Å

Image courtesy of J.Hass

LEED

  Low-Energy Electron Diffraction   Low energy: 50-300 eV

  Same principle as ‘high school’ diffraction   Low energy electrons do not penetrate into

the sample bulk   Surface structure

  LEED images represent the lattice structure in reciprocal space

LEED Apparatus

  Apparatus is contained in UHV

  A camera (video or digital) is used to take a picture of the phosphor screen

  Spots on image correspond with reciprocal lattice vectors

Si-Face UHV grown

C-Face RF furnace grown

Image courtesy of J.Hass

C-face: 71.2 eV

C-face: 118.0 eV

Image courtesy of J.Hass

LEED Analysis

  Visual qualitative analysis   Image Analysis Software

  Problem: Commercial packages are prohibitively expensive

  Solution: IDL

IDL

  Mathematical analysis software   Mathematica   Matlab

  Programming language allows user to define routines

  We already have it!!

Step One: Regions of Interest

  Identify a location on the image to analyze

Step Two: Define an Origin

  Any three points on a plane define a unique circle!

Step Three: ‘Integrate’ Over Region

  Break region into radial slices, sum intensities over each slice

Step Four: Produce Azimuthal Cuts

  Display intensity profile for a fixed radius

Step Six: Produce Radial Cuts

  Display intensity profile for a fixed angle

Step Seven: Produce Surface Plots

Step Eight: I(V) Profiles

  Spots on LEED images are at a distance from the origin defined by:

  where the numerator is a constant related to the lowest energy

  Can ‘shrink’ a region through several images as the electron energy increases

  By calculating the intensity over each region, a plot of intensity versus energy can be generated

In the future…

  Code will continue to be tested and debugged

  Routines can be added into interface as needed

  Interface can be improved from command line to GUI

  Code can be adapted to analyze STM image stacks

Going Into the Classroom

  At HS level, mathematical analysis is not necessarily engaging

  LAB ACTIVITY: Natural Diffraction Gratings   Students use laser pointers and fabric swatches to

explore non-slit diffraction patterns   Qualitative analysis: Draw conclusions about

relationships between patterns   Quantitative analysis: Measure spacing   Extension: Use a prism to generate different

wavelengths and track patterns

Acknowledgements

  My advisors, Ed Conrad and Phil First, for support and assistance

  Joanna Hass, for providing images for this presentation

  Kevin, Britt, Lee, and Nikhil for discussion and support