1.021, 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2012 Part II – Quantum Mechanical Methods : Lecture 9 Some Review & Introduction to Solar PV Jeffrey C. Grossman Department of Materials Science and Engineering
Massachusetts Institute of Technology
�
Part II Topics
1.
2.
3.
4.
5.
6.
7.
8.
It’s a Quantum World:The Theory of Quantum Mechanics
10.
11.
Application of Quantum Modeling of Solids: Solar Cells Part II
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: Nanotechnology 2
Lesson outline
• Discussion of PSET
• Review for the Quiz
• Introduction to Solar PV
3
Motivation: ab-initio modeling! ?
mechanical
properties
?
electrical
properties
?
optical
properties
4
Vision without Acton is a Dream
Acton without Vision is a Ni htmare
Japanese proverb
5
Why quantum mechanics? Problems in classical physics that
led to quantum mechanics:
• “classical atom”
• quantization of properties
• wave aspect of matter
• (black-body radiation), ...
6
Wave aspect of matter light
matter
wave character
particle character
_
_
_
_
_
_
_
_
_
_
_
_
_
Image by MIT OpenCourseWare.
Image in public domain. See Wikimedia Commons.
7
Wave aspect of matter particle:E and momentum pk
wave
k
h
k
p
k = nk
k=
λ |k
k|
de Broglie: free particle can be described a as
h
planewave
i(�
k·�
r −ωt)
ψ(k
r , t) = Ae
λ=
with
.
mv
8
Interpretation of a wavefunction ψ(k
r , t)
wave function (complex)
|ψ|2 = ψψ ∗
interpretation as probability to find particle!
�
∞
−∞
�(r, t)
�(r, t)
2
Image by MIT OpenCourseWare.
9
ψψ ∗ dV = 1
Schrödinger equation
H time independent:
in
f˙(t)
f (t)
=
ψ(k
r , t) = ψ(k
r ) · f (t)
Hψ(k
r)
ψ(k
r)
Hψ(k
r ) = Eψ(k
r)
= const. = E
ψ(k
r , t) = ψ(k
r) · e
time independent Schrödinger equation stationary Schrödinger equation ��
− �i Et
The hydrogen atom
stationary
Schrödinger equation
Hψ = Eψ
�
�
e
v
l
o
s
just
�
−
�
T + V ψ = Eψ
�
n
r ) = Eψ(k
r)
−
∇2 + V ψ(k
2m
n2
2m
2
2
∇ −
e2
4π�0 r
��
�
ψ(k
r ) = Eψ(k
r)
The hydrogen atom quantum numbers
n
1
2
2
2
l
0
0
1
1
ml
F(φ)
P(θ)
0
1
2π
1
2
a03/2
0
1
2π
1
2

 -r / 2a
1
r
0
2e
2 2a03/2  a0 
0
1
2π
6
cos θ
2
1
r -r / 2a0
e
3/2 a
2 6 a0
0
1 ±iφ
e
2π
3
sin θ
2
1
r -r / 2a0
e
3/2 a
2 6 a0
0
±1
R(r)
2
e -r/a0
Image by MIT OpenCourseWare.
�2
The hydrogen atom Energies: © R. Nave. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
�3
Atomic units 1 eV =
-19
1.6021765
J
1 Rydberg = 13.605692 eV = 2.1798719-18 J 1 Hartree = 2 Rydberg 1 Bohr =5.2917721-11 m Energies in Ry
Distances in Bohr
Atomic units (a.u.):
Also in use:
-10
1Å=10 m, nm=
�4
-9
10 m
Everything is spinning ...
Stern–Gerlach experiment (1922)
k
F
= −∇E
k
= ∇m
k ·B
Image courtesy of Teresa Knott.
�5
Everything is spinning ... new quantum number: spin quantum number for electrons: spin quantum number can ONLY be up
down �6
Pauli’s exclusion principle Two electrons in a system cannot have the same quantum numbers! hydrogen
...
3s
2s
quantum numbers:
main n: 1,2,3 ...
orbital l: 0,1,...,n-1 magnetic m: -l,...,l spin: up, down
1s
�7
... ...
3p 3d
2p
...
Periodic table of elements This image is in the public domain. Source: Wikimedia Commons.
�8
e­
Next? Helium? r12
e­
+
Hψ = Eψ
⇥
H1 + H2 + W ψ(k
r1 , k
r2 ) = Eψ(k
r1 , k
r2 )
r2
⇥
T1 + V1 + T2 + V2 + W ψ(k
r1 , k
r2 ) = Eψ(k
r1 , k
r2 )
−
n2
2m
∇21 −
e2
4π� 0 r1
−
n2
2m
∇22 −
e2
4π� 0 r2
+
e2
4π� 0 r12
⇥
ψ(k
r1 , k
r2 ) = Eψ(k
r1 , k
r2 )
problem! cannot be solved analytically
�9
Solutions density
functional
theory
quantum chemistry
Moller-Plesset
perturbation theory
MP2
coupled cluster
theory CCSD(T)
2�
The Two Paths Ψ is a wave function of all positions & time. ⇥
∂
2
−
r , t) ψ(k
r , t) = in ψ(k
∇ + V (k
r , t)
2m
∂t
n2
Chemists (mostly)
Ψ = something
simpler
- - - -
-
Physicists (mostly)
H = something
simpler
“mean field” methods
2�
Working with the Density E[n] = T [n] + Vii + Vie [n] + Vee [n]
kinetic
ion-ion
Walter Kohn
© 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/.
n=#
1
10
100
1,000
ion-electron
electron-electron
Ψ(N3n)
8
109
1090
10900
ion potential
22
ρ(N3)
8
8
8
8
Hartree potential
exchange-correlation
potential
Why DFT? 100,000
10,000
Number of Atoms
e
n
Li
c
s
ar
1000
g
n
ali
T
F
D
DFT
MP2
100
QMC
CCSD(T)
10
1
Exact treatment
2003
2007
2011
Year 2015 Image by MIT OpenCourseWare.
23
Density functional theory
E[n] = T [n] + Vii + Vie [n] + Vee [n]
kinetic
ion-ion
ion-electron
electron density n(kr) =
electron-electron
|φi (k
r )|
2
i
Eground state = min E[n]
φ
Find the wave functions that minimize the energy using a functional derivative. 24
Self-consistent cycle Kohn-Sham equations
sc
f lo
op
|φi (k
r )|2
n(k
r) =
i
25
Density functional theory
Only one problem: vxc not known!!! approximations necessary
local density
approximation
LDA
general gradient
approximation
GGA
26
• structure
• bulk modulus
• shear modulus
• elastic constants
• vibrational properties
• sound velocity
binding energies
reaction paths
forces
pressure
stress
...
27
Convergence for molecules g
i
b
x
o
b
y
m
s
a
?
h
W
g
u
o
en
?
f
c
s
e
h
t
t
i
x
e
I
t
d
h
i
g
i
D
r
e
h
t
t
a
loop
?
t
n
i
po
28
as my
basis b
ig
enoug
h?
Crystal symmetries S = simple
BC = body centered
FC = face centered
© Sandeep Sangal/IITK. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
29
The inverse lattice The real space lattice is described by three basis vectors: k = n1k
a1 + n2k
a2 + n3k
a3
R
The inverse lattice is described by three basis vectors: k = m1kb1 + m2kb2 + m3k
G
b3
iG·R
e
ψ(k
r) =
=1
iGj ·r
cj e
j
automatically periodic in R! 3�
The Brillouin zone inverse lattice The Brillouin zone is a special unit cell of the inverse lattice. Image by Gang65 on Wikimedia Commons. License: CC-BY-SA. This content is excluded from our
Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
3�
The Brillouin zone Brillouin zone of the FCC lattice 32
Periodic potentials V (k
r)
k
R
k
R
k
R
© Prof. Dr. Helmut Föll. All rights reserved. This content is excluded from our
Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.
llattice vector
k
V (k
r ) = V (k
r + R)
NEW quantum number k that N
lives in the inverse lattice! Bloch’s theorem ψk (kr) = e
ik·r
uk (k
r)
k)
uk(k
r ) = uk (k
r+R
33
Inverse lattice Results of Bloch’s theorem:
ik·R
k
ψk (k
r + R) = ψk (k
r )e
2
2
k
|ψk (k
r + R)| = |ψk (k
r )|
if solution
r)
⇤
k (⇤
with
r)
⇤
⇤ (⇤
k+G
⇥
E⇤k = E⇤k+G
⇤
34
charge density
is lattice periodic
also solution
Structural properties Etot
finding the
stress/pressure
and the bulk
modulus
V
V0
p=−
∂E
∂V
σbulk = −V
35
∂p
∂V
=V
2
∂ E
∂V 2
Calculating the band structure 1. Find the converged ground state
3-step procedure 2.
3.
density and potential.
For the converged potential calculate
the energies at k-points along lines.
Use some software to plot the band
structure.
6
E (eV)
Kohn-Sham equations
G15
G'25
0
EC
EV
X1
S1
n(k
r) =
|φi (k
r )|
-10
2
G1
Figure by MIT OpenCourseWare.
L
i
L
G
D
X
U,K
S
G
k
36
Image by MIT OpenCourseWare.
Metal/insulator silicon
6
E (eV)
G15
G'25
0
EC
EV
X1
Fermi energy
S1
Number of electron 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.
37
Simple optical properties
6
unoccupied
G
15
E (eV)
E=hv
G'25
0
S1
EC
EV
X1
occupied
-10
G1
Figure by MIT OpenCourseWare.
L
L
G
D
k
X
U,K
S
G
Image by MIT OpenCourseWare.
shortest wave length visible 400 nm ;
corresponds to photon with E=3.1 eV
38
Image in the public domain.
Vibrational properties lattice vibrations are called: phonons force
What is the frequency of this vibration? 39
iron
Magnetization
&control
calculation = 'scf',
pseudo_dir = ''
/
&system
ibrav=3,
celldm(1)=5.25,
nat=1,
ntyp=1,
ecutwfc=25.0,
occupations='smearing'
smearing='gauss',
degauss=0.05,
nspin=1
starting_magnetization(1)=0.0
/
&electrons
conv_thr=1.0d-10
/
ATOMIC_SPECIES
Fe 55.847 iron.UPF
ATOMIC_POSITIONS {crystal}
Fe 0.0 0.0 0.0
K_POINTS {automatic}
444111
nspin=1: non spin-polarized
nspin=2: spin-polarized
starting magnetization
for each atom
perform three calculations and find lowest energy: non spin-polarized
spin-polarized
ferromagnetic
4�
anti-ferromagnetic With the band structure and DOS we find: •
•
•
•
•
•
electrical conductivity (insulator/metal/semiconductor) thermal conductivity
optical properties
magnetization/polarization
magnetic/electric properties
...
4�
Convergence for solids h
s
e
m
k
y
m
?
s
h
a
g
u
W
o
n
e
e
fin
?
f
c
s
e
h
t
t
i
x
e
I
t
d
h
i
g
i
D
r
e
h
t
t
a
loop
?
t
n
i
po
42
as my
basis b
ig
enoug
h?
Summary of properties structural properties
electrical properties
optical properties
magnetic properties
vibrational properties
43
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.
It’s Not Only About Warming: Abundance of Affordable Energy Resources Can Uplift the World
x4
x13
HUMAN WELL-BEING INCREASES WITH INCREASED PER-CAPITA ENERGY USE
© AIP Publishing. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
44
Courtesy:Vladimir Bulovic
It’s Not Only About Warming: Abundance of
Affordable Energy Resources Can Uplift the World
North American Electrical Blackout
08/14/2003
… August 15 …
just 24 hours into blackout
Air Pollution was Reduced
SO2 >90%
O3 ~50%
Light Scattering
Particles ~70%
“This clean air benefit was realized
over much of eastern U.S.”
Marufu et al., Geophysical Research
Letters 2004
New York City Skyline
by 4:13 pm 256 power plants were off-line
Courtesy:Vladimir Bulovic
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Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
45
Map of the World Scaled to Energy Consumption by Country Newman, U. Michigan, 2006
Courtesy of M. E. J. Newman.
• In 2002 the world burned energy at a rate of 13.5 TW
• How fast will we burn energy in 2050?
• (assume 9 billion people)
• If we use energy like in U.S. we will need 102 TW
• Conservative estimate: 28~35 TW
Energy use estimates from Nocera, Daedalus 2006 & Lewis and Nocera, PNAS 2006 46
U.S. Energy Consumption Goal: consume half of
our electricity through
renewable sources by
the year 2050.
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license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
47
Energy from the Sun • Energy released by an earthquake of
magnitude 8 (1017 J):
• the sun delivers this in one second
• Energy humans use annually (1020 J):
• sun delivers this in one hour
• Earth’s total resources of oil (3
trillion barrels, 1022 J):
• the sun delivers this in two days
48
Courtesy of SOHO/EIT (ESA & NASA) consortium.
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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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