1.021, 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2012
Part II – Quantum Mechanical Methods : Lecture 8
Advanced Prop. of Materials:
What else can we do? Jeffrey C. Grossman
Department of Materials Science and Engineering
Massachusetts Institute of Technology
1
Part II Topics
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
It’s a Quantum World:The Theory of Quantum Mechanics
Quantum Mechanics: Practice Makes Perfect
From Many-Body to Single-Particle; Quantum Modeling of Molecules
Application of Quantum Modeling of Molecules: Solar Thermal Fuels
Application of Quantum Modeling of Molecules: Hydrogen Storage
From Atoms to Solids
Quantum Modeling of Solids: Basic Properties
Advanced Prop. of Materials:What else can we do?
Application of Quantum Modeling of Solids: Solar Cells Part I
Application of Quantum Modeling of Solids: Solar Cells Part II
Application of Quantum Modeling of Solids: Nanotechnology
2
Lesson outline
• Brief Review
• Optical properties
• Magnetic properties
• Transport properties
• Vibrational properties
5
energy [eV]
0.000 (A)
0.005 (B)
0.010 (C)
4
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
Γ
X
W
L
Γ
K
X’
0
2
4
DOS
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3
Keeping Relevant
© Bulletin of the Atomic Scientists. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
May/June 2001, Bulletin of the Atomic Scientists
4
Review: inverse lattice
Schrödinger
equation
certain
symmetry
quantum
number
hydrogen
atom
spherical
symmetry
ψn,l,m (r)
[H, L2 ] = HL2 − L2 H = 0
[H, Lz ] = 0
periodic
solid
translational
symmetry
[H, T ] = 0 5
ψn,k (r)
Review: inverse lattice
kz
some G
ky
kx
r)
⇤
k (⇤
⇥
r)
⇤
⇤ (⇤
k+G
E⇤k = E⇤k+G
⇤
6
Review:The band structure
6
E (eV)
G15
G'25
0
EC
EV
X1
S1
-10
G1
Figure by MIT OpenCourseWare.
L
k is a continuous
variable
L
G
D
X
U,K
S
G
k
Image by MIT OpenCourseWare.
7
The Fermi energy
6
Fermi energy
unoccupied
G
E (eV)
15
G'25
0
EC
EV
X1
each band can hold:
S1
2N electrons and you have
(electrons per unit cell)*N
occupied
or
-10
G1
Figure by MIT OpenCourseWare.
L
L
G
D
k
X
U,K
S
two electrons and you have
(electrons per unit cell)
G
Image by MIT OpenCourseWare.
8
Electrical properties
silicon
6
E (eV)
G15
G'25
0
EC
EV
X1
Fermi energy
S1
Number of electrons in unit cell:
EVEN: MAYBE INSULATOR
ODD: FOR SURE METAL
-10
G1
Figure by MIT OpenCourseWare.
L
L
G
D
X
U,K
Are any bands crossing
the Fermi energy?
YES: METAL
NO: INSULATOR
S
G
k
Image by MIT OpenCourseWare.
9
Electrical properties
diamond:
insulator
10
Electron Transport
E
E-field
eE
1
dv
=
− v=0
dt
m
τ
eτE
v=
m
At equilibrium
m, e
ne2 τ
E ≡ σE
j = nev =
m
Electric current
Electrical conductivity
ne2 τ
σ=
m
11
Electron Transport Calculating σ from band structure
Fermi function
2
σ=e τ
dk
4π 3
∂f
−
∂E
v(k)v(k)
6
E (eV)
G15
G'25
0
1
v(k) = ∇k E(k)
�
EC
EV
X1
S1
-10
Curvature of band structure
G1
Figure by MIT OpenCourseWare.
L
L
G
D
X
U,K
S
G
k
12
Image by MIT OpenCourseWare.
Simple optical properties
6
unoccupied
G
15
E (eV)
E=hv G'25
0
photon has almost
no momentum:
only vertical transitions
possible
energy conversation and momentum conversation apply
S1
gap
EC
EV
X1
occupied
-10
G1
Figure by MIT OpenCourseWare.
L
L
G
D
k
X
U,K
S
G
Image by MIT OpenCourseWare.
13
Silicon Solar Cells Have
to Be Thick ($$$)
It’s all in
the bandstructure!
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Commons license. For more information, see http://ocw.mit.edu/fairuse.
14
Simple optical properties
Image in the public domain.
15
Magnetism
S
N
Origin of magnetism: electron spin
An electron has a magnetic moment of μB, Bohr magneton.
μ = μB (n↑ − n↓ )
Spin up
n↑
Spin down
n↓
16
Magnetization
spin-polarized calculation:
separate density for electrons with spin
up
down
Integrated difference between up and down
density gives the magnetization.
17
Magnetism
In real systems, the density of states needs to be
considered.
bcc Fe
DOS (states/eV)
3
2
1
0
-1
-2
-3
-6
-4
-2
0
2
E - EF (eV)
μ = μB
EF
dE[g↑ (E) − g↓ (E)]
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Quantum Molecular Dynamics
…and let us, as nature
directs, begin first with
first principles.
Aristotle (Poetics, I)
9DWLFDQIUHVFR7KH6FKRRORI$WKHQVE\5DSKDHO,PDJHLVLQWKHSXEOLFGRPDLQ
F=ma
Use Hellmann-Feynman!
$ULVWRWOHGHSLFWHGE\5DSKDHOKROGLQJKLV
(WKLFV,PDJHLVLQWKHSXEOLFGRPDLQ
19
Carbon Nanotube Growth
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Commons license. For more information, see http://ocw.mit.edu/fairuse.
20
Silicon Nanocluster Growth
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Commons license. For more information, see http://ocw.mit.edu/fairuse.
21
Water
Henry Cavendish was the first to describe correctly the
composition of water (2 H + 1 O), in 1781.
He reported his findings in terms of phlogiston (later the gas he made was proven to be hydrogen) and dephlogisticated air (later this was proven to be oxygen).
Cavendish was a pretty neat guy.
Image is in the public domain.
A University dropout, he also compared the conductivities of electrolytes and expressed a version of Ohm's law.
His last major work was the first measurement of Newton's gravitational constant, with the mass and density of the Earth.The accuracy of this experiment was not improved for a century.
22
Water
Which of the following is the correct picture for H2O?
Courtesy of Martin Chaplin, London South Bank University. Used with permission.
Cool water site: http://www.lsbu.ac.uk/water/
23
Classical or Quantum?
More than 50
classical
potentials in use
today for water.
Which one is
best?
Courtesy of Martin Chaplin, London South Bank University. Used with permission.
24
Mg++ in Water
Courtesy of Martin Chaplin, London South Bank University. Used with permission.
Important Differences!
25
Vibrational properties
lattice vibrations
are called: phonons
force
What is the frequency of this vibration?
26
Vibrational properties
animated phonons on the web
http://dept.kent.edu/projects/ksuviz/
leeviz/phonon/phonon.html
• sound in solids determined by acoustical phonons
(shock waves)
• some optical properties related to optical
phonons
• heat capacity and transport related to phonons
27
Summary of properties structural properties
electrical properties
optical properties
magnetic properties
vibrational properties
Image of Airbus A380 on Wikimedia Commons. License: CC-BY-SA. Images of circuit board,
phone, hard drive © sources unknown. All rights reserved. This content is excluded from our
Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Aerothermodynamic image of shuttle, courtesy NASA .
28
Literature
• Charles Kittel, Introduction to Solid State
Physics
• Ashcroft and Mermin, Solid State Physics
• wikipedia, “phonons”, “lattice vibrations”, ...
• solar PV: tons of web sites, e.g.: http://
pveducation.org/pvcdrom
29
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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modelling and Simulation
Spring 2012
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