midterm - stanford universitydionne.stanford.edu/.../matsci152_2011/lecture11_ppt.pdftunnelling the...
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MidtermMidterm
Can we move it to next Friday, April 29?
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Lecture 11Lecture 11
Electron wavefunctions, tunneling, & uncertaintyy
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Electron wavefunctions: STM Image of an atomic corral (Co atoms on a Cu surface)
From htt // l d d / h i / h / h 6/i / t ifFrom http://www.colorado.edu/physics/phys3220/phys3220_sp06/images/stm.gif
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Electron wavefunctions: STM Image of an elliptical atomic corral (Co atoms on a Cu surface)
From Manorahan et al. Nature, Feb. 3, 2000
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The Quantum Mirage
aphy
Scanning tunneling
Topo
grag g
microscope tip
states)
Co
dI/dV
density
of s
Cu (111)
(d
Mirage
Manoharan, H. C.; Lutz, C. P.; Eigler, D. M. Nature 403 (2000)
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The Quantum Mirage
aphy
Scanning tunneling
Topo
grag g
microscope tip
states)
Co
dI/dV
density
of s
Cu (111)
(d
Mirage
Manoharan, H. C.; Lutz, C. P.; Eigler, D. M. Nature 403 (2000)
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Tunnelling
Classical physics: climbing up the hill
Q hQuantum physics: tunnelling
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AFM Image of Pentacene (1.4 nm long), Science 2009, IBM Zurich
Tunnelling and atomic force microscopy
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Tunnelling and VLSI current leakage
1 μm
500 μm
http://spie.org/Images/Graphics/Newsroom/Imported/0888/0888_fig2.jpg
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The Schrodinger equation describes electron waves…much like Maxwell’s equations describe light waves
Traveling wave description for light
)exp()(~)sin(),( tixEtkxtx oy EE
E(x) = wave expression describing just the spatial behavior( ) p g j p
k=wavevector
c=ω/k = λν, energy of a photon=hν
Experimentally we measure and interpret the intensity of a light Experimentally, we measure and interpret the intensity of a light wave:
22 |)(~|1 txEcI E |),(~|2
txEcI ooE
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Electron Wavefunctions
iEtSteady-state total wavefunction:
iEtxtx, )exp()(
E energy of the electron E=energy of the electron
t=time
ψ(x) = electron wavefunction that describes only the spatially ψ(x) electron wavefunction that describes only the spatially behavior
Experimentally, we measure the probability of finding an electron in a given position at time t (like an intensity):
22 |)(||)(| zy,x,tz,y,x,
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S h di ’ i f di i
Time independent Schrodinger equation
022
VEmd
Schrodinger’s equation for one dimension
022 VEdx
Schrondinger’s equation for three dimensions
0)(222
2
2
2
2
2
VEm
zyx zyx
A mathematical “crank”: we input the potential V of the electron (i.e., the ‘force’ it experiences, F=-dV/dx), and can obtain the
electron energies E and their wavefunctions / probability
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw‐Hill, 2005)
electron energies E and their wavefunctions / probability distributions.
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Example 1: electrons in a 1D box
0222
2
VEmdxd
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Example 1: electrons in a 1D box
0222
2
VEmdxd
For 0<x<a, V=0:
22d 0222
2
Emdxd
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Example 1: electrons in a 1D box
0222
2
VEmdxd
For 0<x<a, V=0:
22d 0222
2
Emdxd
)exp()exp()( ikxBikxAx *k is a constant, to be determined
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Example 1: electrons in a 1D box
)exp()exp()( ikxBikxAx *k is a constant, to be determined
Since ψ(0)=0, then B=-A.
)sin(2)]exp()[exp()( kxAiikxikxAx )sin(2)]exp()[exp()( kxAiikxikxAx
Now, plug this solution back into the Schrodinger equation…
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Example 1: electrons in a 1D box
)sin(2)]exp()[exp()( kxAiikxikxAx
Now plug this solution back into the Schrodinger equationNow, plug this solution back into the Schrodinger equation…
0222
2
Emd 022 Edx
0)sin2()2()(sin2 2 kxAiEmkxAik 0)sin2()()(sin2 2 kxAiEkxAik
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Example 1: electrons in a 1D box
2
f d h f h l
0)sin2()2()(sin2 22 kxAiEmkxAik
We can find the energy of the electron!:
kE22
mE
2
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Example 1: electrons in a 1D box
22
mkE
2
22
To find k, use the boundary condition at x=a:
Since ψ(a)=0, we have:
0)sin(2)( kaAia
,...3,2,1 nnka
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Example 1: electrons in a 1D box
axnAixn
sin2)(
2222
Wavefunction:
2
22
2
22
82)(
manh
manEn
Electron energy in an infinite PE well:
2 )12( nh
Energy separation in an infinite PE well:
21 8)12(
manhEEE nn
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Example 1: electrons in a 1D box
Wavefunction: Probability:
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Heisenberg’s Uncertainty Principle
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Heisenberg’s Uncertainty Principle
We cannot exactly and simultaneously know both the position and momentum of a particle:
Heisenberg uncertainty principle for position and momentum
the position and momentum of a particle:
xpx
Similarly for energy and time:
tEy gy
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw‐Hill, 2005)
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Tunnelling
The roller coaster released from A can at most make it to C, but not to E. Its PE at Ais less than the PE at D When the car is at the bottom its energy is totally KE is less than the PE at D. When the car is at the bottom, its energy is totally KE.
CD is the energy barrier that prevents the care from making it to E.
In quantum theory, on the other hand, there is a chance that the car could tunnel (leak) through the potential energy barrier between C and E and emerge on
the other side of hill at E.
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Tunnelling
The wavefunction for the electron incident on a potential energy barrier (V0).
The incident and reflected waves interfere to give 1(x).
There is no reflected wave in region III There is no reflected wave in region III.
In region II, the wavefunction decays with x because E < V0.
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Tunnelling
( ) A ( k ) A ( k ) 2 E1(x):=A1exp(ikx)+A2exp(-ikx)
2(x):=B1exp(αx)+B2exp(- αx)
22 2
mEk
where )(2 EVm2( ) 1 p( ) 2 p( )
3(x):=C1exp(ikx)+C2exp(-ikx)2
2 )(2
EVm o
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Tunneling Phenomenon: Quantum Leak
Probability of tunneling from region I to region III (transmission coefficient T):
)(sinh11
)(
)(22
1
21
2I
2III
aDAC
x
xT
I
)](4/[2 EVEVD oo
Probability of tunneling through a wide or high barrier, αa>>1
)2exp( aTT o 2
)(16 oo V
EVET where
oV
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How do we go from tunnelling to images like these?
From Manorahan et al. Nature, Feb. 3, 2000
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Quantum tunnelling of electrons from a metal
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Quantum tunnelling of electrons from twointeracting metalsg
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Scanning tunnelling microscopy!
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Scanning tunnelling microscopy!
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STM in 1987
Scanning Tunneling Microscopy (STM) image of a graphite surface where contours represent electron concentrations within the surface, and carbon rings p , g
are clearly visible. Two Angstrom scan. |SOURCE: Courtesy of VeecoInstruments, Metrology Division, Santa Barbara, CA.
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The inventors: Gerd Binning and Heinrich Rohrer (1986 Nobel Prize)( )
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STM image of Ni (100) surface STM image of Pt (111) surface
SOURCE: Courtesy of IBM SOURCE: Courtesy of IBM
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw‐Hill, 2005)
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Quantum mechanics in 3D
Electron confined in three dimensions by a three-dimensional infinite PE box. yEverywhere inside the box, V = 0, but outside, V = . The electron cannot escape
from the box.
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Quantum mechanics in 3D: 3 quantum numbers
Electron wavefunction in infinite PE well
znynxn
czn
byn
axnAz y, x,nnn
321 sinsinsin)(321
Electron energy in infinite PE box
2
22
2
23
22
21
2
88321
NhnnnhE nnn
22 88321 mamannn
23
22
21
2 nnnN 321 nnnN
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Coming soon: Understanding electrons in atoms our first device: lasers! d l dBandstructure in solidsLots of other devices!