Tunneling An electron of such an energy will never appear here! classically E kin = 1 eV 0 V-2 Vx.
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Tunneling An electron of such an energy will never appear here! classically E kin = 1 eV 0 V -2 V x
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Tunneling – quantum approach Schroedinger eq. for region x>L Solution:
Transcript of Tunneling An electron of such an energy will never appear here! classically E kin = 1 eV 0 V-2 Vx.
Tunneling
An electron of such an energy will never appear here!
classically
Ekin= 1 eV
0 V -2 V x
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 U0, the particle can pass the barrier.
Tunneling – quantum approach
Schroedinger eq. for region x>L
EUdxd
m 02
22
2 )(2
022
2EUm
dxd
Solution: xAex )(
Potential barriers and tunneling
)(2)(202
202
2 EUmAeEUmeA xx
Two solutions: )(2021 EUm
or )(2
022 EUm
Normalization condition: 1)(0
dxx
Solution: xAex 2)(
The probability to find a particle in the region II within
xxEUmAxpr
002
20 )(22exp)(
x
Potential barriers and tunneling
xxEUmAxpr
002
20 )(22exp)(
xAex 2)(
Potential barriers and tunnelingexample
Let electrons of kinetic energy E=2 eV hit the barrier height of energy U0= 5 eV and the width of L=1.0 nm. Find the percent of electrons passing through the barrier?
LEUm
UE
UE
IIT
pad
trans )(22exp116 000 T=7.1·10-8
insulator
semiconductor
metalA
If L=0.5 nm.then T=5.2 ·10-4!
Scanning tunneling electron miscroscope
LeI 2
)(20 EUmgdzie
Scanning tunneling electron miscroscope
Scanning tunneling electron miscroscope
Scanning tunneling electron miscroscope
Image downloaded from IBM, Almaden, Calif.It shows 48 Fe atoms arranged on a Cu (111) surface
Scanning tunneling electron miscroscope
particle decay
Approximate potential - energy function for an particle in a nucleus.
Tunneling
Nuclear fusion ( synteza ) is another example of tunneling effect
E.g. The proton – proton cycle
Young’s double slit experiment
a) constructive interferenceFor constructive interference along a chosen direction, the phase difference must be an even multiple of
msind m = 0, 1, 2, …
d
b) destructive interference For destructive interference along a chosen direction, the phase difference must be an odd multiple of
21msind m = 0, 1, 2,
…
a, b, c – computer simulation
d - experiment
Electron interference
Franhofer Diffraction
a dy
dysin2d
sin2 a
Re
Im
ER
RmaxER
20E
21I )cos1(R2
21 2
0
2/sin2E 22
max0
2
2max0 2/
2/sinE
22max /sin
/sinsinI
a
a
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”
dNi=0.215nmdiffraction
nm165.0sin d
de Broglie
mpeVba 2
2
nm167.02
bameV
hph
C.J.Davisson and L.G.Germer
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:
D22.1min
Electron Microscope
