1. Tunneling An electron of such an energy will never appear here! classically E kin = 1 eV 0 V-2 Vx

2. Potential barriers and tunneling According to Newtonian mechanics, if the total energy is E, a particle that is on the left side of the barrier can go no farther than x=0. If the total energy is greater than U 0, the particle can pass the barrier.

3. Tunneling – quantum approach Schroedinger eq. for region x>L Solution:

4. Potential barriers and tunneling Two solutions: or Normalization condition: Solution: The probability to find a particle in the region II within

5. Potential barriers and tunneling

6. example Let electrons of kinetic energy E=2 eV hit the barrier height of energy U 0 = 5 eV and the width of L=1.0 nm. Find the percent of electrons passing through the barrier? T=7.1·10 -8 insulator semiconductor metal A If L=0.5 nm.then T=5.2 ·10 -4 !

7. Scanning tunneling electron miscroscope

11. Image downloaded from IBM, Almaden, Calif. It shows 48 Fe atoms arranged on a Cu (111) surface Scanning tunneling electron miscroscope

12.  particle decay Approximate potential - energy function for an  particle in a nucleus.

13. Tunneling Nuclear fusion ( synteza ) is another example of tunneling effect E.g. The proton – proton cycle

14. Young’s double slit experiment   a) constructive interference For constructive interference along a chosen direction, the phase difference must be an even multiple of  m = 0,  1,  2, … d b) destructive interference For destructive interference along a chosen direction, the phase difference must be an odd multiple of  m = 0,  1,  2, …

15. a, b, c – computer simulation d - experiment Electron interference

16. Franhofer Diffraction  a dy Re Im  E R R 

17. Electron Waves Electrons with 20eV energy, have a wavelength of about 0.27 nm This is around the same size as the average spacing of atoms in a crystal lattice These atoms will therefore form a diffraction grating for electron “waves”

18. d Ni =0.215nm diffraction de Broglie C.J.Davisson and L.G.Germer

19. Resolution Rayleigh’s criterion: When the location of the central maximum of one image coincides with the the location of the first minimum of the second image, the images are resolved. For a circular aperture: 

20. Electron Microscope