lecture 8 - the dionne group | stanford...
TRANSCRIPT
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
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
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 (wavefunctions are continuous!):
Since ψ(a)=0, we have:
0)sin(2)( kaAia 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
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)
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)
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
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.
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)
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