lecture 6 – photons and other quanta

Lecture 6 – Photons, electrons and other quanta

EECS 598-002 Winter 2006Nanophotonics and Nano-scale Fabrication

P.C.Ku

2EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku

From classical to quantum theory

In the beginning of the 20th century, experiments showed:Particle nature of EM radiation Wave nature of electrons

In 1927, Heisenberg proposed the uncertainly principle which later on became the foundation of modern quantum mechanics (QM):

Not all the physical quantities can be measured at the same time with however precision we would like. E.g. the more precisely we measure the position of a particle, the less precisely we will be able to measure its momentum.


Example: Photoelectric effect

In the experiment, the light shines on a metal and knocks out the electrons on the metal surface. When we measured the stopping voltage V0 to counteract the generated current, we found a cutoff frequency f0. This can not be explained by the Maxwell’s equations.


Quantum state and operators

In the language of QM, the physics of a particle (or a system that is comprising of many particles) can be described by a state (a vector in Hilbert space).

The physical quantities (measurables) can be obtained by applying a suitable operator Ω to the state . This will yield one of the eigenvalues with probability P given by . The measurement will also change the system to the new state .

e.g. In coordinate basis: Position operator Momentum operatorTotal energy operator

( )tΨ

( )tΩ Ψ

2( ) ( )P tω ω= Ψ

ω

2 2

( )2

ri

V rm

− ∇

∇− + Hamiltonian


Dynamics of quantum states

The evolution of the quantum state obeys the Schroedinger equation:

where H is the Hamiltonian.For example, an electron in a time-independent potential V(r) is governed by:

( )( )

d ti H t

dtΨ

= Ψ

( )

2 2

/

2 2

( )( ) ( )

2

( )

( ) 02

iEt

d tV r t i

m dt

t E e

V r E Em

−

Ψ⎡ ⎤∇− + Ψ =⎢ ⎥

⎣ ⎦Ψ =

⎡ ⎤∇⇒ − + − =⎢ ⎥

⎣ ⎦


Comparison b/w QM and EM

( )22

2 ( ) 0m E V r E⎡ ⎤∇ + − =⎢ ⎥⎣ ⎦( )2 2

0 0 0( ) 0E

rH

ω µ ε ε ε⎧ ⎫

⎡ ⎤∇ + + − =⎨ ⎬⎣ ⎦⎩ ⎭

QM EM

Equation

Potential

Eigenstate

Eigenvalue

Wavevector

( )V r− 0( )rε ε−

and E HE

E 2ω

Equation governing QM and EM have lots of mathematicalsimilarities.

( )2

2 ( )m E V r− 0 ( )rω µ ε


First analogy: tunnelingV(z)

zQM

z

( )zε

EM

1. Electron experiences reflection and tunneling through the potential barrier. The wavefunction is exponential decayed in the barrier.

2. The tunneling probability increases when the width of the potential barrier decreases.

E


Slab waveguide vs potential well

kz

ω

V(z)

zQM

z

( )zε

EM

At a fixed K.E. in the x-y plane,the kinetic energy of an electron in the z direction is quantized.


Electrons in crystals

When electron travels in a periodic potential (e.g. in the crystal), its eigenstate satisfies the Bloch’s theorem.

The Bloch function has the same periodicity as the crystal lattice.

The spatial symmetry of the crystal lattice with respect to a fixed point determines the eigenfunctions of the electron.

Because of the periodicity, the energy bandgap exists at certain k values.

( ) ( ) ik rr u r eψ ⋅=

( )u r


Effective mass vs constitutive relation

Similar to the “macroscopic version” of the Maxwell’s equations, we can also derive an “effective-mass”equation for electrons traveling in the lattice with the assumption that the electron interacts weakly with the lattice.

( )2 2

2 2

*

( ) ( ) 02

( ) 02

crystal macroscopic

macroscopic

V r V r E Em

V r E Em

⎡ ⎤∇− + + − =⎢ ⎥

⎣ ⎦⎡ ⎤∇

→ − + − =⎢ ⎥⎣ ⎦


Example of a dispersion relation

Electron in periodic V(r) EM wave in periodic ε(r)


Density of states

Density of states g(E) is the total number of allowed-to-occupy states with frequencies between Ε and Ε+δΕ per unit volume.

where is the total number of states from 0 to Ε. In a potential well:

1 ( )( ) dN Eg EV dE

=

( )N E

( ) ( )

( )

''

' '

2

( ) ( ') ( ') ' ( ')

1 12 ( ') ' 2 ( ') ( ') '

1 2 ( ) 1( ) 2 ( )2

3 dim ( ) 42 dim ( ) 2

1 dim ( ) 10 dim ( ) (

kE E

d dk k k k

d

n d

n

N E g E f E dE f k

f k dk A k f k dkk k

A k dk dkg E A kV dE dEk

A k kA k k

dA k

A k k k

π

ππ

δ

<

< <

= =

= × = ×∆ ∆

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∆

==

==

= −

∑∫

∫ ∫

)

⎧⎪⎪⎨⎪⎪⎩


Density of states (cont.)2 2

2

3/ 2

2 2

For electrons, / 2 . For EM waves (photons), .Note can be discrete due to quantization.

For electrons: (d>0) ; ( ) ( )

1 23 dim ( )2

2 dim ( )

nn

E k m E ckk

dk m g E g EdE k

mg E E

g Ed

ω

π

= = =

= =

⎛ ⎞= ⎜ ⎟⎝ ⎠

=

∑

2

1/ 2

2

( )

1 11 dim ( )2

0 dim ( ) 2 ( )

nn

n n

nn

m E E

mg EE

g E E E

θπ

π

δ

⎧⎪⎪⎪

= −⎪⎪⎨⎪ ⎛ ⎞=⎪ ⎜ ⎟

⎝ ⎠⎪⎪ = −⎪⎩

∑

∑

∑


Density of states for electrons

g(E) = Density of states

Eg E

g(E)

Eg E

g(E)

Eg E

g(E)

Eg E

g(E)

bulk sheet wire dot

3D 2D 1D 0D


Occupation probability

In thermal equilibrium, fermions (e.g. electrons) satisfy the Fermi-Dirac statistics (two fermions cannot stay in the same state):

In thermal equilibrium, bosons (e.g. photons) satisfy the Bose-Einstein statistics:

[ ]1( )

exp ( ) / 1F B

f EE E k T

=− +

[ ]1( )

exp ( ) / 1F B

f EE E k T

=− −


From single-particle to many-particle system

So far we have considered only the quantization of a single particle, namely the electron. This procedure (namely the uncertainly principle) applies to all particles that are governed by Newton’s Laws classically.

Similarly, we can also treat a multi-particle system as a whole and quantize the system at once. To treat the system classically, we can imagine the position and momentum of each particle form a field. The moving of a particle w.r.t. its equilibrium position looks like a perturbation of the field, i.e. wave.


Number operator

ˆ ˆ ˆ

ˆ 1 1

ˆ 1ˆ ˆ ˆ( 1/ 2)

N a a

a n n n

a n n n

H a aω

+

+

+

=

= + +

= −

= +

Creation operator

Annihilation operator

Hamiltonian operator


Photons

Similar to the quantization of a many-particle system, EM wave can be thought as a perturbation of the EM field in space-time.

Quantization of the EM field photons

Properties of photonsMass = 0Charge = 0Energy = hν = Momentum =

ˆ ˆ( , ), ( ', ) ( ')i j ijkk

D r t B r t i r rx

ε δ∂⎡ ⎤ = − −⎣ ⎦ ∂

ω

k


When do we need concepts of photons?

When do we need to treat the EM wave as photons?When the momentum of each photon is comparable to that of the material upon which it impinge.When the number of photons involved in the interaction is very small.

Examples:Photoelectric effectSpontaneous emission


Density of states for photons in 3D

( ) ( )

''

23 3

' '

2 2

2 3

1

( ) ( ') ( ') ' ( ')

1 12 ( ') ' 2 4 ' ( ') '

( )

k

k k k k

dkd c

N g f d f k

f k dk k f k dkk k

k dkgd c

ω ω

ωω ω ω ω

π

ωωπ ω π

<

< <

=

= =

= × = ×∆ ∆

⎛ ⎞= =⎜ ⎟⎝ ⎠

∑∫

∫ ∫


Other quanta

PhononPlasmonSurface plasmon


Classification of quanta

MassMass of the electron = 9.1E-31 kgMass of the photon = 0

ChargeElectrons carry one unit of negative chargePhotons don’t carry charges.

SpinSpin = 1/2 (electrons), 3/2, 5/2, … = fermionsSpin = 0, 1 (photons), 2, … = bosons. E.g. Excitons can carry integer spins.

Dispersion relation Particularly useful for quasi particles

lecture 6 – photons and other quanta

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