Wave particle duality

1. Chapter 1: Wave-particle duality of
electrons

2. Three approaches for the understanding of the
electronic properties of materials
1. “Continuum theory”
Conclusions are drawn from the empirical laws
No assumptions
Examples: Ohm’s law, the Maxwell equations,
Newton’s law, etc.

3. 2. “classical electron theory”
(Accomplished at the turn to 20th century by introducing atomistic principles)
It postulated that free electrons in metals drift as a response to an external
force and interact with certain lattice atoms.
The principal proponent of this approach: Paul Drude
He developed several equations that are widely today– Drude equations
Drude Model :
electrons
constantly bounce
between heavier,
stationary crystal
ions.
DC field:
E=0,Vd=0 E0, Vd0
This is what you
have already know
from the Solid State
Physics course.
By the way, this physical picture is incorrect.
Metals: Free
electrons from the
outer most shell

4. Microscopic view of Ohm’s law
Drift motion of electrons in an electric field:
Newton’s law: a = F/m=eE/m
 The drift speed: vd=a = eE/m ( : mean free time)
Current density: J = nevd
 J = (ne2/m)E
 = ne2/m
= (e/m)(ne)=(ne)
 = e/m: mobility
Ohm’s law: J =  E
ne: charge density (measured by Hall effect).

5. Hall effect (high school solution)
Magnetic force: Fm=evdB
I = cross-section area * J
= wdnevd, vd=I/(wdne)
Fm= IB/(wdn)
Electric force: Fe=eVH/w
In equilibrium,
Fm= Fe
IB/(wdn) = eVH/w
VH =(1/ne)(IB/d)
RH=1/(ne)
Hall voltage has a different polarity for
positive and negative charge carriers.
vd can be measured with the Hall effect,
~10-3 m/s
Important technique for evaluating electronic properties
of materials: Charge carrier density and sign

6. Magnetotransport (college solution)

7. Classical transport:

8. Experimental observations
• Free electron gas:
• Negative RH – due to the negative charge of electrons
• Inversely proportional to electron density (charge carrier density)
• T and B -independent
RH = -1/ne
• Positive RH observed!
• Another type of carrier with positive charge –
failure of free electron gas, can be explained by
band theory.

9. Experimental observations
Barnard & Rahiem, J. Phys. F: Metal Phys. 10, 2739(1980)
RH of pure Al
Magnetic field dependence Temperature dependence
RH is both field and temperature dependent!
 Due to multiband conduction

10. Other problems with Drude model
Based on classic statistic physics, MB
• Overestimate of electronic heat capacity
• Correction: FD statistics, Sommerfeld model
• Incorrect relaxation mechanism
• Temperature dependence of conductivity
• Correction: quantum theory on lattice vibration

11. 3. “Quantum theory”
(accomplished at the beginning of the 20th centry)
It can interpret important experimental observations
which could not be readily interpreted by “classical
electron theory”, e.g. superconducting phenomenon
and magnetism.

12. Chapter 1: Wave-particle duality of
electrons
Light: wave-particle duality
Wavelike-nature of light: confirmed by many
crucial experiments, such as diffraction, interference,
and dispersion

13. Light: electromagnetic wave

15. Interference of wave
Interference pattern of water wave

17. Light: particle-like nature, photons
The Photoelectric Effect
• Albert Einstein
• Theorized Photons
• Won Nobel prize - 1921
• Photons have an energy equal to:
E = h
• h = Plank’s Constant, and is equal to:
6.6260755 x 10- 34 J sec

18. The Photoelectric Effect
An adjustable voltage
is applied. Voltage
can be forward or
reverse biased (which
slows down the
electrons)
- +
eV = K
V
U = qV

19. Some experimental results
• Maximum kinetic energy of the ejected particles, or:
KEmax = eVstopping
Perplexing Observations:
The intensity of light had no effect on energy
There was a threshold frequency for ejection
 Slope is same for all targets
 y intercept is different for
different target materials
Classical physics failed to explain this,
Lenard won the Nobel Prize in Physics in 1905 (cathode rays).

20. Light: electromagnetic wave

21. Interpretation
 Light comes in energy packets equal to h
 Each packet acts more like a particle than a
wave
 These light “particles” are called photons
 The target is being bombarded by photons
like tiny billiard balls!
Einstein figured out the photoelectric effect in
1905 (the same year he developed the theory of
special relativity and explained Brownian Motion).
This is what he got the Nobel prize for.
Conservation of energy
light energy ejecting electron & KE
KE  eV
h = f + KE = f + eV

22. Electrons: Wave-Particle duality
The particle property of electrons
• Electron
Discovered by J.J. Thomson in Cambridge University in
1897
mass: m0 = 9.110-31 kg
charge: e = -1.6 10-19 C

23. Discovery of electron
• Electron, the first fundamental particle
• Birth of Elementary Particle Physics
e
e

24. Deflexion of Cathode rays by
Electrostatic Field and Velocity
Cathode-ray tube
A, B: Anode and Ground
C: Cathode
D, E: Capacitor plates
Velocity
• qvB = qE or v = E/B
24
Thomson J. J , Cathode Rays, Philosophical Magazine, 44 (1897) 293

25. Determination of e/m
Displacement
y =(1/2) a t2
= (1/2)( Ee/m )( l2/v2 )
Charge to mass ratio
e/m = (2s/E)( v2/l2 )
25

26. Spin: Another property of electrons
e-
That is why it is called “spin”
And of cause this picture is just for understanding, not correct.
Current Loop
Generates a Magnetic Moment

27. Electron: wave-like nature
• DeBroglie hypothesis (1924 in his PhD thesis)
Next, how to experimentally demonstrate?

28. Interference of wave
• Wave-particle duality
of light
Particle property:
photo-electric effect
Wave property:
interference

29. Interference of wave
Pathlength difference

30. Interference of wave
Interference pattern of water wave
So how about electrons?

31. Davisson-Germer Experiment
DeBroglie wavelength:
Bragg's law: 2dsinθ = nλ
Electron diffraction by Nickel crystal

32. Davisson-Germer Experiment
Electron diffraction by Nickel crystal