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3.23 Electrical, Optical, and Magnetic Properties of Materials
Fall 2007
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3.23 Fall 2007 – Lecture 17
FERMAT’s FIRST THEOREM
Image removed due to copyright restrictions.
Please see: http://en.wikipedia.org/wiki/
Image:Ibn_haithem_portrait.jpg
Abū ʿAlī al-Ḥasan ibn
al-Ḥasan ibn al-Haytham
Pierre-Louis
Moreau de
Maupertuis
Hero
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Last time
1.
2.
3.
4.
5.
Electric field, polarization, displacement, susceptibility
M
Maxwell’s
ll’ equations
ti
Potentials and gauges
Electromagnetic waves (no free charges, currents)
Refractive index, phase and group velocity
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
1
Study
• (mostly read) Fox,
Fox Optical Properties of
Solids: 1.1 to 1.4, 2.1 to 2.2.3, 3.1 to 3.3
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Polarization, transversality of EM fields
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
2
Boundary conditions (Gauss theorem)
∫
r r
∇ ⋅ Bdv =
∫
r r
∇ ⋅ Ddv =
volume
∫
r
ˆ =0
B ⋅ ndS
∫
r
ˆ = 4π
D ⋅ ndS
surface
volume
surface
∫
ρ dv
volume
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Boundary conditions
r r
nˆ ⋅ B2
− B1 = 0
(
)
r
r
ˆn ⋅ D2 − D1 = σ (σ = surface charge
g densityy ))
(
)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
3
Boundary conditions (Stokes theorem)
r r
ˆ =
∇ × E ⋅ ndS
∫
surface
∫
r r
E ⋅ dr
line
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Boundary conditions
r
r
n̂ × E2 − E1 = 0
r
r
r
nˆ × H 2
− H1 = K
r
K = surface current density
(
(
(
)
)
)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
4
Snell’s law
r r
r
Ei eωt −iki ⋅r incident wave
r r
r
Er eωt −ikr ⋅r reflected wave
r r
r
Et eωt −ikt ⋅r transmitted wave
Image from Wikimedia Commons, http://commons.wikimedia.org
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Snell’s law
r
r ωn
ki = k r = 1
c
r ωn
kt = 2
c
Image from Wikimedia Commons, http://commons.wikimedia.org
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
5
Snell’s law
r
( k ⋅ rr )
1
(k
1y
x=0
r r
= k1′ ⋅ r
(
)
x=0
r r
= k2 ⋅ r
(
)
x=0
y + k1z z ) = ( k1y′ y + k1z′ z ) = ( k2 y y + k2 z z ) → k1y = k1y′ = k2 y
and k1z = k1′z = k2 z
(
r r
r r
r r
k1t ⋅ rt = k1t′ ⋅ rt = k2t ⋅
rt
) (
) (
)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Snell’s law
r
r
ω
k1 = k1′ = n1
c
r
ω
k2 = n2
c
kiz = ktz → ki sin θ1 = kt sin θ 2
ω n1
c
sin θ1 =
ω n2
c
sin θ 2
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
6
Snell’s law
r
ω
⎫
k1z = k1
sin θ1 = n1 sin θ1 ⎪
⎪
c
⎬ n1 sin θ1 = n2 sin θ 2
r
ω
k2 z = k2
sin θ 2 = n2 sin θ 2 ⎪
⎪⎭
c
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Principle of least action
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
7
Energy law
r
r
r r r r r r 4π r r 1 r ∂D 1 r ∂B
E ⋅∇ × H − H ⋅∇ × E =
J ⋅E + E⋅
+ H⋅
c
c
∂t c
∂t
r r r r r r
r r r
E ⋅∇ × H − H ⋅∇ × E = −∇ ⋅ E × H
r
r
4π r r 1 r ∂D 1 r ∂B r r r
→
J ⋅E + E⋅
+ H⋅
+∇⋅ E× H = 0
c
c
∂t c
∂t
Apply Gauss's theorem
r
r
r r
⎛ 1 r ∂D 1 r ∂B ⎞
4π r r
⋅
+
⋅
+
⋅
+
J
Edv
E
H
dv
E
∫c
∫ ⎜⎝ c ∂t c ∂t ⎟⎠ ∫ × H ⋅ nˆdS = 0
(
)
(
)
(
)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Energy law
1
4π
1
4π
r r
r
r
r2
r ∂D
1 r ∂ε E
1 ∂ε E
1 ∂ E⋅D
E⋅
=
E⋅
=
=
∂t 4π
∂t
∂t
8π ∂t
8π
r r
r
r ∂B 1 ∂ H ⋅ B
=
H⋅
∂t 8π
∂t
(
(
)
)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
8
Energy conservation
r r
r r r r
r r
∂
ˆ =0
E ⋅ D + H ⋅ B dv + ∫ E × H ⋅ ndS
∫ J ⋅ Edv + ∂t ∫ 14
4244
3
1
424
3
(
)
total energy stored in electrical
and magnetic field
per volume
(
)
energy surface
flux per unit area
r c r r
S=
E×H
4π
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Optical processes
•
•
•
•
Reflection and refraction
Absorption
Luminescence
Scattering
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
9
Optical coefficients
T: ratio of transmitted vs incident power
R+T=1 (no absorption, scattering) Absorption:
Transmission:
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Complex refractive index
n% = n + ik
k
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
10
Complex refractive index
Image removed due to copyright restrictions.
Please see any image of the structure of amorphous silica,
such as http://www.research.ibm.com/amorphous/figure1.gif.
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Modeling Optical Constants with a Damped Harmonic Oscillator
d 2
X
dX
+ m0γ
+ m0ω02
X = −eE ( t )
m0
2
1
424
3
123
2
dt
dt
{
1
424
3 harmonic restoring time
dependent
acceleration
dissipation
force
electric field
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
11
Modeling Optical Constants with a Damped Harmonic Oscillator
X0 =
Presonant = Np = −NeX =
−eE0
m0 (ω − ω 2 − iγω )
2
0
Ne 2
E
m0 (ω02 − ω 2 − iγω )
14442444
3
α
Ne 2
E =εE
m0 (ω02 − ω 2 − iγω )
144424443
D = E + 4π P + 4π Presonantt = E + 4πχ E + 4π
α
Atomic polarizability = α
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Modeling Optical Constants with a Damped Harmonic Oscillator
ε = 1 + 4πχ + 4π
(
Ne 2 (ω02 − ω 2 )
)
m0 (ω02 − ω 2 ) + γ 2ω 2
144444424444443
2
ε1
− i 4π
(
Ne 2γω
γ
)
m0 (ω02 − ω 2 ) + γ 2ω 2
14444244443
2
ε2
2
ε = ( n + ik ) = n12
−3
k 2 + i 2{
nk
123
2
ε1
ε2
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
12
Amorphous silica
ε'
ε"
Figure by MIT OpenCourseWare.
Courtesy of Elsevier, Inc., http://www.sciencedirect.com.
Used with permission.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Optical materials
Image removed
. due to copyright restrictions
Please see: Fig. 1.4 in Fox, Mark. Optical Properties of Solids. Oxford, England: Oxford University Press, 2001.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
13
Infrared active modes
Image removed due to copyright restrictions. Please see Fig. 1a and 2a in Giannozzi, Paolo, et al.
"Ab initio Calculation of Phonon Dispersions in Semiconductors." Physical Review B 43 (March 15, 1991): 7231-7242.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Optical materials
Image removed due to copyright restrictions.
Please see: Fig. 1.7 in Fox, Mark. Optical Properties of Solids. Oxford, England: Oxford University Press, 2001.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
14
Optical materials
Image removed due to copyright restrictions.
Please see: Fig. 1.5 in Fox, Mark. Optical Properties of Solids. Oxford, England: Oxford University Press, 2001.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Transition rate for direct absorption
Image removed due to copyright restrictions.
Please see any diagram of GaAs energy bands,
such as http://ecee.colorado.edu/~bart/book/book/chapter2/gif/fig2_3_6.gif.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
15