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
Massac husetts Institute of T ec hnology
Par t II T opics
1. It ’ s a Quantum W orld: The Theor y of Quantum Mechanics
2. Quantum Mechanics: Practice Mak es P erf ect
3. Fr om Man y-Body to Single-Par ticle; Quantum Modeling of Molecules
4. Application of Quantum Modeling of Molecules: Solar Thermal Fuels
5. Application of Quantum Modeling of Molecules: Hydr ogen Storage
6. Fr om Atoms to Solids
7. Quantum Modeling of Solids: Basic Pr oper ties
8. Advanced Pr op . of Materials: What else can w e do?
9. Application of Quantum Modeling of Solids: Solar Cells Par t I
10. Application of Quantum Modeling of Solids: Solar Cells Par t II
11. Application of Quantum Modeling of Solids: Nanotechnolog y
“The purpose of computing is insight, not n umbers . ”
□ Richar d Hamming
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“What ar e the most impor tant pr oblems in your field? Ar e
you working on one of them? Wh y not?”
“It is better to solve the r ight pr oblem the wr ong w ay than to solve the wr ong pr oblem the r ight w a y . ”
“In r esearc h, if you kno w what you ar e doing, then you shouldn't be doing it . ”
“Mac hines should work. P eople should think . ”
Lesson outline
0.000 (A)
0. 005 (B)
0. 010 (C)
5
4
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
• Re vie w
• structural p r oper ties
energy [eV]
• Band Structur e
•
• DOS
Metal/insulator
• Magnetization
X W L K X’
0 2 4
DOS
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Let ’ s ta k e a wal k th r oug h memor y lan e f o r a moment.. .
In the Beginning....
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Ther e w er e some strange obser vations b y some v er y smar t people .
_
_
_
_
_
_
_
_
_
_
_
_ _
e -
+
Image by MIT OpenCourseWare.
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see http://ocw.mit.edu/help/faq-fair- use/ .
In the Beginning....
The w eir dness just k ept g oing.
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Courtesy of Dan Lurie on Flickr. Creative Commons BY-NC-SA.
© John Richardson for Physics World, March 1998. All rights reserved. This content is excluded from
It Became Clea r ...
...that matter beh a v ed li k e w a v es (and vice v ersa).
And that w e had to lose our “classical” concepts of absolute position and momentum.
And instead consider a par ticle as a w a v e , whose squar e is the p r obability of finding it.
( r , t ) A exp[ i ( k r t )]
But ho w w ould w e describe the beh a vior of this w a v e?
Then, F=ma f or Quantum Mechanics
M
m
V
v
k 2
— 2 m D
2
+ V ( → r , t ) G ( → r , t ) = i k G ( → r , t )
6 t
⇥
⇤
Image by MIT OpenCourseWare.
It W as W onderful
It explained man y things.
h ydr ogen
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... ... ... ...
3s 3p 3d
2s 2p
Courtesy of David Manthey. Used with permission. Source: http://www.orbitals.com/orb/orbtable.htm .
It g a v e us atomic 1s
orbitals. It pr edicted the energ y le v els in h yd r ogen.
It W as W onderful
It g a v e us the means to understand m uch of chemistr y .
T his image is in the public domain. Source: Wikimedia Commons .
BU T .. .
Natu r e Does > 1 elect r on!
It was impossible to solv e f or mor e than a single elect r on.
Enter computational quantum mechanics!
But...
W e Don ’ t H a v e The
Age of the Uni v erse
Which is ho w long it w ould ta k e cur r entl y to solv e the Sch r odinger equation exactl y on a compute r .
S o ...w e loo k ed at this guy ’ s back.
And star ted making some a pp r o ximations.
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The T w o Paths
Ψ is a w a v e function of all positions & time .
k 2
— 2 m D
2
+ V ( → r , t ) G ( → r , t ) = i k G ( → r , t )
6 t
⇥
6
Chemists (mostl y)
- - -
P h ysicists (mostl y)
- -
-
- - -
Ψ = something simpler
H = something simpler
Walter Kohn
W orking with the Density
E [ n ] = T [ n ] + V ii + V ie [ n ] + V ee [ n
kinetic ion-electr on
|
n=# |
Ψ (N 3n ) |
ρ (N 3 ) |
|
1 |
8 |
8 |
|
10 |
10 9 |
8 |
|
100 |
10 90 |
8 |
|
1,000 |
10 900 |
8 |
ion-ion electr on-electr on
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ion potential Har tr ee potential exchange-cor r elation
potential
Number of Atoms
Linear scaling DFT
DFT
R e vi e w: W h y DFT?
100,000
10,000
1000
100
MP2
QMC
CCSD(T)
10
Exact treatment
1
2003
2007
201 1
2015
Y ear
Image by MIT OpenCourseWare.
-
R e vi e w: Self-consistent cycle
K ohn-Sham equations
scf loop
n ( → r ) = Σ | ф i ( → r ) | 2
i
R e vi e w: Cr ystal symmetries
A cr ystal is built up of a unit cell and
periodic r eplicas ther eof.
lattice unit cell
Image of Sketch 96 (Swans) by M.C. Escher removed due to copyright restrictions.
R e vi e w: Cr ystal symmetries
Image of August Bravais is in the public domain.
Bra vais
The most common Br a vais lattices ar e the cubic ones (simple , body- center ed, and face- center ed) plus the hexag onal close- pac k ed ar rangement.
...w h y?
S = simple
BC = body center ed FC = face center ed
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Recip r ocal Lattice & Brillouin Zone
Associated with each r eal space lattice , ther e exists something w e call a r ecip r ocal lattice .
The r ecip r ocal lattice is the set of w a v e-v ectors which ar e commensurate with the r eal space lattice .
Sometimes w e li k e to call it “G”.
It is defined b y a set of v ectors a*, b*, and c* such that a* is perpendicular to b and c of the Br a vais lattice , and the p r oduct a* x a is 1.
R e vi e w: The i n v erse lattice
r eal space lattice (BCC) in v erse lattice (FCC)
z
a
a 2
a 1
x
a 3
y
Image by MIT OpenCourseWare.
The Brillouin zone
in v erse lattice
The Brillouin zone is a special unit cell of the in v erse lattice .
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The Brillouin zone
Brillouin zone of the FCC lattice
Bloch ’ s Theo r em
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 w a v efunction, on the other hand, is periodic but onl y when m ultiplied b y a phase facto r .
This is kno wn as Bloch ’ s theor em.
NEW quantum n umber k that liv es in the in v erse lattice!
G ⌅ k
( → r ) = e i ⌅ k · ⌅ r u
( → r )
⌅ k
u ⇥ k ( ⌅ r ) = u ⇥ k ( ⌅ r + R ⌅ )
P eriodic potentials
Results of the Bloch theor em:
G ⌅ k
( → r + R → ) = G ⌅ k
( → r ) e i ⌅ k · R ⌅
Courtesy Stanford News Service .
License CC-BY.
| G ⇥ ( → r + R → ) | 2 = | G ⇥ ( → r ) | 2
k k charge density
is lattice periodic
if solution
l ⇤ k ( ⇤ r ) — → l ⇤ k + G ⇤ ( ⇤ r )
also solution
with
E ⇤ k = E ⇤ k + G ⇤
P eriodic potentials
|
Schrödinger |
certain |
quantum |
|
equation |
symmetry |
number |
hydrogen spherical G
atom symmetry
[ H , L 2 ] = H L 2 — L 2 H = 0
[ H , L z ] = 0
n,l,m
( → r )
periodic translational solid symmetry
[ H , T ] = 0
G n , ⌅ k ( → r )
Origin of band structu r e
Diff er ent w a v e functions can satisfy the Bloch theor em f or the same k : eigenfunctions and eigen values labelled with k and the index n
energ y bands
F r om atoms to bands
Atom
Molecule
Solid
Ener gy
Antibonding p
Conduction band from antibonding p orbitals
p
Antibonding s
Conduction band from antibonding s orbitals
s
Bonding p
V alence band from p bonding orbitals
Bonding s
V alence band from s bonding orbitals
k
Image by MIT OpenCourseWare.
The i n v erse lattice
k z
some G
k y
k x
l ⇤ k ( ⇤ r ) → l ⇤ k + G ⇤ ( ⇤ r )
E ⇤ k
= E ⇤ k + G ⇤
The i n v erse lattice
l 0 ( ⌅ r + R ⌅ ) = l 0 ( ⌅ r )
periodic o v er unit cell
R
l G ⇤ / 2 ( ⌅ r + 2 R ⌅ ) = l G ⇤ / 2 ( ⌅ r )
periodic o v er larger domain
R R
The i n v erse lattice
choose cer tain k-mesh e .g. 8x8x8
N=512
unit cells in
n umber of the periodic
k-p oints (N) domain (N)
The i n v erse lattice
Distribute all elect r ons o v er the lo w est states.
N k-points
Y ou h a v e (electr ons per unit cell)*N
electr ons to distribute!
The band structu r e
Silicon
E (eV)
• energ y le v els in the Brillouin zone
• k is a contin uous variable
6
15
0
'
25
X
1
E C
E
V
S 1
-10
L
k
X
U,K
Image by MIT OpenCourseWare.
The band structu r e
Silicon
E (eV)
• energ y le v els in the Brillouin zone
• k is a contin uous variable
6
u noccupied
15
0
'
25
X
1
E C
E
V
S 1 occupied
-10
L
k
X
U,K
Image by MIT OpenCourseWare.
The F ermi energ y
6
E (eV)
u nocc up i e d
ga p: also visible in the DOS
0
-10
15
E V
X 1
' 25
E C
S 1 occu p ie d
F ermi energ y
one band can hold tw o elect r ons (spin up and do wn)
L
k
X U,K
Image by MIT OpenCourseWare.
The elect r on density
elect r on density of silicon
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Structural p r oper ties
F o r ces on the atoms can be calculated with the Hellmann–F e ynman theor em:
F or λ =atomic position, w e F or ces automaticall y in get the f or ce on that atom. most codes.
Structural p r oper ties
finding the equilibrium lattice constant
E tot
mass density
a lat a
m u =1.66054 10 -27 Kg
Structural p r oper ties
finding the str ess/pr essur e and the bulk modulus
E tot
V 0 V
B E ⇥ p ⇥ 2 E
p = — l bulk = - V = V
B V ⇥ V ⇥ V 2
Calculating the band structu r e
1. Find the con v erged gr ound state
density and potential.
3- step p r ocedur e 2. F or the con v erged potential calculate
the energies at k-points along lines .
3. Use some softwar e to plot the band
6
15
0
'
25
X
E C
1
E
V
S 1
-10
L
k
X U,K
structur e .
n ( → r ) = Σ | ф i ( → r ) | 2
i
E (eV)
K ohn-Sham equations
Calculating the DOS
1. Find the con v erged gr ound state density and potential.
3- step p r ocedur e 2. F or the con v erged potential calculate
energies at a VE R Y dense k-mesh .
3. Use some softwar e to plot the DOS.
n ( → r ) = Σ | ф i ( → r ) | 2
i
K ohn-Sham equations
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Metal/insulator
silicon
6
E (eV)
15
E V
E C
' 25 X 1
0
F e rmi energ y
Ar e an y bands cr ossing the F ermi energ y?
YES: ME T AL NO: INSUL A T OR
S 1
-10
Number of electr ons in unit cell: EVEN: M A YBE INSUL A T OR ODD: FOR SURE ME T AL
L
k
X U,K
Image by MIT OpenCourseWare.
Metal/insulator
diamond: insulator
Metal/insulator
0.000 (A)
0.005 (B)
0.010 (C)
3
2
energy [eV]
1
5
4
3
2
1 F ermi energ y
BaBiO 3 :
metal
0 0
-1 -1
-2 -2
-3 -3
-4 -4
-5 -5 0 2 4
X W L K X’
DOS
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Simple optical p r oper ties
E=hv
photon has almost no momentum:
onl y v er tical transitions possible
energ y con v ersation and momentum con v ersation a ppl y
6
E (eV)
0
-10
L
ie d
1
X
25
'
15
u noccup
E C E V
S 1 occu pied
X
U,K
ga p
k
Image by MIT OpenCourseWare.
Silicon Solar Cells H a v e to Be Thick ($$$)
It ’ s all in the band- structur e!
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Literatu r e
• Charles Kittel, Int r oduction to Solid State P h ysics
• Ashc r oft and Mermin, Solid State P h ysics
• wikipedia, “solid state p h ysics”, “condensed matter p h ysics”, ...
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