Lecture 8 - The Dionne Group | Stanford...

Lecture 8Lecture 8

Wave/Particle Duality for Matter: the Nature of Confined Electrons

Photoelectric effect: Light is a particle with energy

Red photons – no current; blue photons – measured currentLight = energy packets (photons) with energy E=hνPhotoemission only occurs when E > workfunction Øy Ø

Ø=hc/eλo, where λo is the longest wavelength for photoemission Work function of a metal keeps the electron in the material

Compton effect: Light also has momentum

khp

Momentum:

Blackbody radiation

2 2hcI

1exp5

kThc

I

452 k

Integrating the irradiance gives the total radiated power PSemitted by a blackbody per unit surface area at a temperature T.

4TP SS 42832

45

K m W 10670.5152

hck

S

The electron is also a wave

Y ’ d bl lit i t ith l t i l l t d t lit i Young’s double-slit experiment with electrons involves an electron gun and two slits in a Cathode ray tube (CRT) (hence, in vacuum).

Electrons from the filament are accelerated by a 50 kV anode voltage to produce a beam thatIs made to pass through the slits. The electrons then produce a visible pattern when they strikep g p p y

A fluorescent screen (e.g., a TV screen), and the resulting visual pattern is photographed.SOURCE: Pattern from C. Jonsson, D. Brandt, and S. Hirschi, Am. J. Physics, 42, 1974, p.9,

figure 8. Used with permission.

De Broglie Relationship

Wavelength of the electron depends on its momentum p (just like photons!)

h

p

Wavelengths of particle-like objects

50 gram golf ball travelling at 20 m/s

M th i (60 Marathon winner (60 kg, 26 miles or 16.2 km)

Electron accelerated by 100 V

h = 6.62610-34 J·s e = 1.60210-19 C

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


0222

2

VEmdxd

For 0<x<a, V=0:

22d 0222

2

Emdxd


0222

2

VEmdxd

For 0<x<a, V=0:

22d 0222

2

Emdxd

)exp()exp()( ikxBikxAx *k is a constant, to be determined


)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…


)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


2

f d h f h l

0)sin2()2()(sin2 22 kxAiEmkxAik

We can find the energy of the electron!:

kE22

mE

2


22

mkE

2

22

To find k, use the boundary condition at x=a (wavefunctions are continuous!):

Since ψ(a)=0, we have:

0)sin(2)( kaAia 0)sin(2)( kaAia,...3,2,1 nnka


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


Wavefunction: Probability:

2 nm2 nm

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


Tunnelling

Classical physics: climbing up the hill

Q hQuantum physics: tunnelling

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)

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

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 tunneling 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!

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


Midterm topics: lectures 1-8

i i b di ilib i b d l h• ionic bonding – equilibrium bond lengths• kinetic molecular theory

•Maxwell’s principle of equipartition of energyp p q p gy•Thermal expansion

•Maxwell-Boltzmann Distribution•Thermally activated processes & diffusionThermally activated processes & diffusion

•Crystals•Electrical conductivity (Drude Model)

T t ffi i t f i ti it•Temperature coefficient of resistivity•Hall Effect

•Thermal conduction in metals & non-metals•Wave/particle duality

•Photon and electron diffraction•The photoelectric effect compton scatteringThe photoelectric effect, compton scattering

•Blackbody radiation•Potential wells

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