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Can we move it to next Friday, April 29?

Lecture 11Lecture 11

Electron wavefunctions, tunneling, & uncertaintyy

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

Electron wavefunctions: STM Image of an elliptical atomic corral (Co atoms on a Cu surface)

From Manorahan et al. Nature, Feb. 3, 2000

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)

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)

Tunnelling

Classical physics: climbing up the hill

Q hQuantum physics: tunnelling

AFM Image of Pentacene (1.4 nm long), Science 2009, IBM Zurich

Tunnelling and atomic force microscopy

Tunnelling and VLSI current leakage

1 μm

500 μm

http://spie.org/Images/Graphics/Newsroom/Imported/0888/0888_fig2.jpg

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

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,

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.

Example 1: electrons in a 1D box

0222

2

VEmdxd

Example 1: electrons in a 1D box

0222

2

VEmdxd

For 0<x<a, V=0:

22d 0222

2

Emdxd

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

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…

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

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

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

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

Example 1: electrons in a 1D box

Wavefunction: Probability:

Heisenberg’s Uncertainty Principle

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)

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.

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.

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

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

How do we go from tunnelling to images like these?

From Manorahan et al. Nature, Feb. 3, 2000

Quantum tunnelling of electrons from a metal

Quantum tunnelling of electrons from twointeracting metalsg

Scanning tunnelling microscopy!

Scanning tunnelling microscopy!

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.

The inventors: Gerd Binning and Heinrich Rohrer (1986 Nobel Prize)( )

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)

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.

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

Coming soon: Understanding electrons in atoms our first device: lasers! d l dBandstructure in solidsLots of other devices!