Diffraction Techniques

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Clinton Davisson obtained the Nobel prize in 1937 for confirming de Broglie's prediction that electrons behave as waves. (Louis De Broglie was awarded a Nobel Prize for this theory in 1929). Davisson and Germer were doing experiments at Bell Labs in 1925, when the vacuum in their apparatus was lost ruining their Ni sample. When heating the sample to use it again they involuntarily made large crystals. Low energy electrons scattered off these crystals produced diffraction patterns demonstrating that electrons behaved like waves.

The de Broglie wavelength of an electron is:

where h is Planck's constant, mv the electron momentum and k its wave vector (i.e., the wave propagates in direction r as Aexp(ikr), where A is its amplitude. In terms of the electron energy:

Electron diffraction can be seen in a simple form in scattering from two atoms separated by a distance d

wpe1.jpg (5147 bytes) 

The wave scatters from the two centers. There will be constructive interference when the path difference, dsinq is a multiple of the wavelength (destructive interference when it is a multiple of l/2).

When the scattering of the electrons is from a crystalline surface plane, a diffraction pattern is produced, different from the diffraction from the bulk of the crystal (as occurs with high energy electrons (several keV or higher) or X-rays).   Low energy electrons cannot penetrate much into a solid without losing some energy.   This is shown in the graph below, which gives the mean free path for inelastic scattering (energy loss) vs. the electron energy.  At energies around 50-100 eV, the mean free path is only about 1-2 layers.

mfp-luth.GIF (44389 bytes)

Therefore if one observes only electrons which are reflected from a crystal surface without any energy loss, one will find diffraction from the two dimensional ordered array of atoms at the surface and not from the three dimensional atomic array in the bulk of the material. 

In the reflection from a surface plane, the momentum of the electron perpendicular to the surface is not conserved.  The momentum perpendicular to the surface is conserved within a reciprocal lattice vector g.


k-conserve.GIF (52859 bytes)

There are two main used of LEED:

1) quick determination of the crystallinity of the surface by observing the spots produced by the diffracted electron beams on a phosphor screen. Analysis of spot pattern serves to determine symmetry of substrate surface and adsorbates, reconstruction, etc.

2) elaborate determination of positions of surface and surface atoms from measurements of spot intensity and additional computer modeling.

The difficulty in these calculations, compared to the case of X-ray diffraction, is that scattering of electrons by atoms is strong. Thus kinematic theory is insufficient, dynamic theory needed.

LEED measurements involve a low energy electron gun (a typical electron energy is 100 eV, a set of concentric spherical-sector grids (the "LEED optics"), and a phosphor screen, which emits light when struck by an energetic electron. The first gird is at the sample potential to have a field-free region in front of the sample thus avoiding changes in the angle of the trajectories of the backscattered electrons.  Grids biased just below the voltage of the electron source is used to reject inelastically scattered electrons. The electrons are post-accelerated into the phosphor screen by a voltage of several kilovolts, to cause efficient luminescence.  A grounded grid is placed between the grid and the energy analysis grids to avoid penetration of the field. 


The next figures shows schematic LEED patterns at 100 eV for Si[111]1x1 (top) and Si[111]3x3-30o (bottom).  The dots represent the diffracted beams for which the indices are noted.  The reciprocal lattice and directions are indicated. These figures and the one above are taken from  http://dol1.eng.sunysb.edu/expcht1.html, where you can get more examples. 1

In this diagram, the spots (beams) labeled from a to f are 1/3 1/3, 2/3 2/3, 1/3 4/3, 4/3 1/3, 2/3 5/3, and 5/3 2/3 respectively. The solid and dashed lines indicate the reciprocal nets for the ideal solid and the surface superstructure. 


This is a method to obtain the spot patterns.  It uses the reciprocal lattice (Fourier transform of the real surface lattice).

1) draw incident ko vector to terminate at origin of reciprocal lattice

2) draw Ewald sphere of radius |k| with center at start of vector ko (|k| is conserved for elastic scattering.

3) Draw rods from reciprocal lattice positions, perpendicular to surface

4) Find the diffracted beams starting at the start of ko and ending at any point where Ewald sphere intersects a rod. Discard beams moving towards surface.

EWALD.GIF (13077 bytes)

The observation of the pattern of the beam spots (intersection of the diffracted beam with the phosphor screen) serves to determineknow the symmetry of the surface structure.  

Surface structure from I-V curves

This method involves measurements of the intensity of the diffracted beams as a function of energy for different spots and fitting them with calculations based on the assumed surface model.  The steps involved in the analysis are:

  1. Assume structure
  2. Choose electron-atom interaction potentials (they can be measured for free atoms in the gas phase from scattering experiments, but one actually needs potentials for atoms in the solid state.)
  3. Choose the electron-surface interaction potential and the depth of the inner potential (typically about 10 eV.)  The inner potential is the gain in kinetic energy of the electron as it enters the solid.
  4. Calculate cross sections / phase shifts
  5. Calculate the intensities of the diffracted beam in a multiple scattering model.
  6. Compare with experiments.  Go back to step 1 if needed.

More information can be found in the LEED I(V) data repository and the surface structure site at SUNY-Stony Brook.


Ewald Construction

 RHEED2.GIF (8998 bytes)

Sampling depth is electron attenuation length x sin a.

Uses typically 5 - 100 keV electron guns
Electron inelastic mean free path L at these energies is about 100 - 1000 but the sampling depth is shallow due to the glancing angle.
Elastic scattering is strongly forward peaked.
The method is simpler than LEED since it does not require grids and the beam does not need to be accelerated into the phosphor.
Only useful for very smooth and flat surfaces.
RHEED is in general not quantitative.
Roughness: electrons go through asperities and produce "bulk-type" electron diffraction (one-half, pattern is bound by a "shadow edge" parallel to the surface.
RHEED requires rotation of the sample. Changes in periodicity in plane of incidence are not seen.
Width of spots, which give streaks, are not well understood.

RHEED Oscillations

RHEED is used typically to monitor growth of atomic layers during MBE (Molecular beam epitaxy), by measuring the oscillations of RHEED specular intensities.  Defects scatter electrons off the specular direction producing a drop in the intensity. As growth continues, the number of steps on the surface increases and the intensity is damped.  A method of MBE is to find growth conditions that will give minimum damping.

More class notes on LEED can be found here.

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Updated September 14, 2000

Copyright 2002, by Ral Baragiola, University of Virginia. All rights reserved.