1.021, 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2012
Part II – Quantum Mechanical Methods : Lecture 7
Quantum Modeling of
Solids: Basic Properties
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.
2
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
“The purpose of computing
is insight, not numbers.”
! Richard Hamming
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“What are the most important problems in your field? Are
you working on one of them? Why not?”
“It is better to solve the right problem the wrong way than to
solve the wrong problem the right way.”
“In research, if you know what you are doing, then you
shouldn't be doing it.”
“Machines should work. People should think.”
3
...and related: “With great power comes great responsibility.” (Spiderman’s Uncle)
Lesson outline
4
energy [eV]
• Review
• structural properties
• Band Structure
• DOS
• Metal/insulator
• Magnetization
5
5
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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Let’s take a walk through memory lane for a moment...
4
In the Beginning....
There were some strange observations by some very
smart people.
e-
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+
_
_
_
_
_
_
_
_
_
_
_
_
_
Image by MIT OpenCourseWare.
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see http://ocw.mit.edu/help/faq-fair-use/.
5
In the Beginning....
The weirdness just kept going.
Courtesy of Dan Lurie on Flickr.
Creative Commons BY-NC-SA.
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from our Creative Commons license. For more information,
see http://ocw.mit.edu/help/faq-fair-use/.
6
© John Richardson for Physics World, March 1998. All rights reserved. This content is excluded from
our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
It Became Clear...
...that matter behaved like waves (and vice versa).
And that we had to lose our “classical” concepts of
absolute position and momentum.
And instead consider a particle as a wave, whose
square is the probability of finding it.
Ψ(r,t) = Aexp[i(k • r − ωt)]
But how would we describe the
behavior of this wave?
7
Then, F=ma for
Quantum Mechanics
M
m
V
v
Image by MIT OpenCourseWare.
2
2m
8
⇥ + V (r, t)
⇥
2
(r, t) = i
⇤
⇤t
(r, t)
It Was Wonderful
It explained many things.
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license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
hydrogen
...
3s
2s
... ...
3p 3d
2p
...
Courtesy of David Manthey. Used with permission.
Source: http://www.orbitals.com/orb/orbtable.htm.
It gave us atomic
orbitals.
9
1s
It predicted the energy
levels in hydrogen.
It Was Wonderful
It gave us the means to understand much of
chemistry.
This image is in the public domain. Source: Wikimedia Commons.
BUT...
10
Nature Does > 1 electron!
It was impossible to solve for more than a single
electron.
Enter computational quantum mechanics!
But...
11
We Don’t Have The
Age of the Universe
Which is how long it would take currently to solve the
Schrodinger equation exactly on a computer.
So...we looked at this guy’s back.
And started making some approximations.
12
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The Two Paths
Ψ is a wave function of all positions & time.
2
2m
⇥ + V (r, t)
2
Chemists (mostly)
Ψ = something
simpler
13
⇥
(r, t) = i
- - - -
-
⇤
⇤t
(r, t)
Physicists (mostly)
H = something
simpler
“mean field” methods
Working with the Density
E[n] = T [n] + Vii + Vie [n] + Vee [n]
kinetic
ion-ion
Walter Kohn
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excluded fromour Creative Commons license.
For more information,see http://ocw.mit.edu/
help/faq-fair-use/.
n=#
1
10
100
1,000
Ψ(N3n)
ion potential
14
ion-electron
electron-electron
8
109
1090
10900
ρ(N3)
8
8
8
8
Hartree potential
exchange-correlation
potential
Review:Why DFT?
100,000
g
n
ali
10,000
T
F
D
c
s
r
a
Number of Atoms
e
in
L
DFT
1000
MP2
100
QMC
CCSD(T)
10
Exact treatment
1
2003
2007
2011
2015 Year 15
Image by MIT OpenCourseWare.
Review: Self-consistent
cycle
Kohn-Sham equations
Iterations to selfconsistency
sc
f lo
n(⌅
r) =
i
|
r )|
i (⌅
2
3.021j Introduction to Modeling and Simulation - N. Marzari (MIT, May 2003)
16
op
Review: Crystal symmetries
A crystal is built up
of a unit cell and
periodic replicas thereof.
lattice
unit cell
Image of Sketch 96 (Swans) by M.C. Escher
removed due to copyright restrictions.
17
Review: Crystal symmetries
Image of August Bravais
is in the public domain.
Bravais
The most common
Bravais lattices are
the cubic ones
(simple, bodycentered, and facecentered) plus the
hexagonal closepacked arrangement.
...why?
S = simple
BC = body centered
FC = face centered
18
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Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use /.
Reciprocal Lattice &
Brillouin Zone
Associated with each real space lattice, there exists
something we call a reciprocal lattice.
The reciprocal lattice is the set of wave-vectors which
are commensurate with the real space lattice.
Sometimes we like to call it “G”.
It is defined by a set of vectors a*, b*, and c* such that
a* is perpendicular to b and c of the Bravais lattice, and
the product a* x a is 1.
19
Review:The inverse
lattice
real space lattice (BCC)
inverse lattice (FCC)
z
a
a1
a2
x
20
a3
y
Image by MIT OpenCourseWare.
The Brillouin zone
inverse lattice
The Brillouin zone is a special
unit cell of the inverse lattice.
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Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
21
The Brillouin zone
Brillouin zone of the FCC lattice
22
Bloch’s Theorem
The periodicity of the lattice in a solid means that the
values of a function (e.g., density) will be identical at
equivalent points on the lattice.
The wavefunction, on the other hand, is periodic but
only when multiplied by a phase factor.
This is known as Bloch’s theorem.
NEW quantum number k that
lives in the inverse lattice!
r)
⌅
k (⇧
=e
i⌅
k·⌅
r
u⌅k (⇧
r)
⌅)
u⇥k (⌅
r ) = u⇥k (⌅
r+R
23
Periodic potentials
Results of the Bloch theorem:
⇧) =
(⇧
r
+
R
⌅
k
r )e
⌅
k (⇧
⌅
i⌅
k·R
Courtesy Stanford News Service.
License CC-BY.
2
⌅
| ⇥k (⌅
r + R)| = |
if solution
l⇤k (⇤
r)
with
24
r )|
⇥
k (⌅
⇥ l⇤k+G
r)
⇤ (⇤
E⇤k = E⇤k+G
⇤
2
charge density
is lattice periodic
also solution
Periodic potentials
Schrödinger
equation
certain
symmetry
hydrogen
atom
spherical
symmetry
[H, L2 ] = HL2
quantum
number
r)
n,l,m (⇤
L2 H = 0
[H, Lz ] = 0
periodic
solid
25
translational
symmetry
[H, T ] = 0
r)
n,⌅
k (⇤
Origin of band structure
Different wave functions can satisfy the Bloch theorem for the same k:
eigenfunctions and eigenvalues labelled with k and the index n
energy bands
26
From atoms to bands
Atom
Molecule
Solid
Energy
Antibonding p
Conduction band
from antibonding
p orbitals
Antibonding s
Conduction band
from antibonding
s orbitals
p
s
Bonding p
Bonding s
Valence band from
p bonding orbitals
Valence band from
s bonding orbitals
k
Image by MIT OpenCourseWare.
27
The inverse lattice
kz
some G
ky
kx
l⇤k (⇤
r)
⇥ l⇤k+G
r)
⇤ (⇤
E⇤k = E⇤k+G
⇤
28
The inverse lattice
⌅ ) = l0 (⌅
l0 (⌅
r+R
r)
periodic over unit cell
R
⌅
(⌅
r)
lG/2
(⌅
r
+
2
R
) = lG/2
⇤
⇤
periodic over larger domain
R
29
R
The inverse lattice
choose certain
k-mesh e.g. 8x8x8
N=512
number of
k-points (N)
30
unit cells in
the periodic
domain (N)
The inverse lattice
Distribute all electrons over the lowest states.
N k-points
You have
(electrons per unit cell)*N
electrons to distribute!
31
The band structure
Silicon
6
E (eV)
G15
G'25
0
EC
EV
X1
S1
-10
• energy levels in the
Brillouin zone
• k is a continuous variable
32
G1
Figure by MIT OpenCourseWare.
L
L
G
D
X
U,K
S
G
k
Image by MIT OpenCourseWare.
The band structure
Silicon
6
unoccupied
G
E (eV)
15
G'25
0
S1
EC
EV
X1
occupied
-10
• energy levels in the
Brillouin zone
• k is a continuous variable
33
G1
Figure by MIT OpenCourseWare.
L
L
G
D
k
X
U,K
S
G
Image by MIT OpenCourseWare.
The Fermi energy
6
gap: also visible
in the DOS
unoccupied
G
E (eV)
15
G'25
0
S1
EC
EV
X1
Fermi energy
one band can hold two
electrons (spin up and
down)
occupied
-10
G1
Figure by MIT OpenCourseWare.
L
L
G
D
k
X
U,K
S
G
Image by MIT OpenCourseWare.
34
The electron density
electron
density
of silicon
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35
Structural properties
Forces on the atoms can be calculated with the
Hellmann–Feynman theorem:
For λ=atomic position, we
get the force on that atom.
36
Forces automatically in
most codes.
Structural properties
Etot
finding the
equilibrium
lattice constant
alat
mass density
mu=1.66054 10-27 Kg
37
a
Structural properties
finding the
stress/pressure
and the bulk
modulus
Etot
V
V0
p=
38
BE
BV
lbulk = -V
⇥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
G15
E (eV)
Kohn-Sham equations
G'25
0
EC
EV
X1
S1
n(⌅
r) =
i
|
r )|2
i (⌅
-10
G1
Figure by MIT OpenCourseWare.
L
L
G
D
X
U,K
S
G
k
39
Image by MIT OpenCourseWare.
Calculating the DOS
1. Find the converged ground state
3-step procedure 2.
3.
density and potential.
For the converged potential calculate
energies at a VERY dense k-mesh.
Use some software to plot the DOS.
Kohn-Sham equations
n(⌅
r) =
i
40
|
2
(⌅
r
)|
i
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license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Metal/insulator
silicon
6
E (eV)
G15
G'25
0
EC
EV
X1
Fermi energy
Number of electrons in unit cell:
EVEN: MAYBE INSULATOR
ODD: FOR SURE METAL
S1
-10
G1
Figure by MIT OpenCourseWare.
L
L
G
D
X
U,K
S
G
k
Image by MIT OpenCourseWare.
41
Are any bands crossing
the Fermi energy?
YES: METAL
NO: INSULATOR
Metal/insulator
diamond:
insulator
42
Metal/insulator
5
energy [eV]
4
5
0.000 (A)
0.005 (B)
0.010 (C)
BaBiO3:
metal
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
Fermi energy
2
4
DOS
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43
Simple optical properties
ns are emitted from
m electromagnetic
1] The emitted
rons in this context.
ct,[2][3]photon
due to its has
ough the term has
6
unoccupied
G
15
E (eV)
a Photoelectric effect
a Compton scattering
a Pair productionE=hv
G'25
0
gap
EC
EV
X1
S
almost
occupied
no momentum: A diagram illustrating the emission of
-10 plate, requiring energy
only vertical transitions
electrons from a metal
mportant steps in
gained from an incoming photonG to be more
possible
Figure by MIT OpenCourseWare.
than
the
work
function
of
the material.
t and electrons and
1
1
energy conversation and
of wave–particle
momentum conversation apply
L
L
G
D
k
X
U,K
S
G
Image by MIT OpenCourseWare.
44
Silicon Solar Cells Have
to Be Thick ($$$)
It’s all in
the bandstructure!
45
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Commons license. For more information, see http://ocw.mit.edu/fairuse.
Literature
• Charles Kittel, Introduction to Solid State
Physics
• Ashcroft and Mermin, Solid State Physics
• wikipedia,“solid state physics”,“condensed
matter physics”, ...
46
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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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