3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction...
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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Spring 2008
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1.021/3.021/10.333/18.361/22.00 Introduction to Modeling and Simulation
Timo Thonhauser
Department of Materials Science and Engineering Massachusetts Institute of Technology
Quantum Modeling of Solids: Basic Properties
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5. Quantum Modeling of Solids: Basic Properties
6.
Class outline
2
theory practice
1. It’s A Quantum World: The Theory of Quantum Mechanics
2. Quantum Mechanics: Practice Makes Perfect
4. From Atoms to Solids
3. From Many-Body to Single-Particle; Quantum Modeling of Molecules
Advanced Prop. of Materials; What else can we do?
7. Review session
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Motivation!
3
mechanical propertiespr
electrical propertiespr
optical propertiespr
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Lesson outline
Review
structural properties
Calc. the band structure
Calc. the DOS
Metal/insulator
Magnetization
4
Image removed due to copyright restrictions. Please see:Fig. 3 in Thonhauser, T., and K. M. Rabe. "Fcc Breathing Instability inBaBiO3 From First Principles." Physical Review B 73 (2006): 212106.
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Review: Why DFT?
5
100,000
10,000
1000
100
10
2003 2007 2011 20151
Year
Num
ber o
f Ato
ms
Linear
scalin
g DFT
DFT
QMC
CCSD(T)
Exact treatment
MP2
Figure by MIT OpenCourseWare.
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3.021j Introduction to Modeling and
selfconsistency
Review: Self-consistent cycle
6
Image removed due to copyright restrictions. Please see: Fig. 7 in Payne, M. C., et al. "Iterative Minimization Techniques for ab Initio Total-EnergyCalculations: Molecular Dynamics and Conjugate Gradients." Reviews of Modern Physics 64 (October 1992): 1045-1097.
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A crystal is built up of a unit cell and
periodic replicas thereof.
5
unit celllattice
Image of M. C. Escher's "Mobius with Birds" removed due to copyright restrictions.
Review: Crystal symmetries
7
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Review: Crystal symmetries
8
Image removed due to copyright restrictions.Please see: http://commons.wikimedia.org/wiki/Crystal_structure
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Review: The inverse lattice
real space lattice (BCC) inverse lattice (FCC)
9
xy
a
z
a1a2
a3
Figure by MIT OpenCourseWare.
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Review: Crystal binding
crystalline Ar
Ar
Ar Ar
ArAr
Ar Ar sodium chloride
(van der Waals) (ionic)
Cl-
Na+
Na+
Na+
Cl-
Cl-
Cl- Cl-
Na+
C
C
C
C
C
Na+ Na+
Na+
Na+ Na+
sodium diamond(metallic) (covalent)
10
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ψ�k(�r) −→ ψ�k+ �G(�r)
E�k = E�k+ �G
if solution also solution
with
Review: Periodic potentials
11
ψ�k(�r + �R) = ψ�k(�r)e i�k·�r
Results of the Bloch theorem:
|ψ�k(�r + �R)|2 = |ψ�k(�r)|2 charge density is lattice periodic
!!k("r ) !" !!k +!G("r )
E!k =E!k + !G
if solution also solution
with
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Review: The band structure
12
Atom Molecule Solid Energy
p
s
Antibonding p
Antibonding s
Bonding p
Bonding s
Conduction band from antibondingp orbitals
Conduction band from antibondings orbitals
Valence band from p bonding orbitals
Valence band from s bonding orbitals
k
Figure by MIT OpenCourseWare.
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Review: band structure and DOS
13
Image removed due to copyright restrictions. Please see:Fig. 3 in Thonhauser, T., and K. M. Rabe. "Fcc Breathing Instability inBaBiO3 From First Principles." Physical Review B 73 (2006): 212106. L
U
WK
X
Q
kx
kz
kyΓ
Σ
∆ Σ'Λ
Figure by MIT OpenCourseWare.
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Review: The Fermi energy
14
one band can holdtwo electrons (spin
up and down)
gap: also visible in the DOS
Fermi energy
L
ECEV
X1
-10
0
6
E (e
V)
S1
X
k
U,KΓ Γ
Γ1
Γ'25
Γ15
∆ ΣΛ
unoccupied
occupied
Figure by MIT OpenCourseWare.
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The electron density
electrondensity
of silicon
15
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Structural properties
Forces on the atoms can be calculated with the Hellmann–Feynman theorem:
For �=atomic position, we PWscf calculates theget the force on that atom. forces automatically.
16
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Structural properties
Etot
finding the equilibrium
lattice constant
alat a
mass density
17
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V0 V
Structural properties
Etotfinding the
stress/pressure and the bulk
modulus
∂E ∂p ∂2E p = − σbulk = −V = V
∂V ∂V ∂V 2
18
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Calculating the band structure
3-step procedure
1. Find the converged ground state density and potential.
2.For the converged potential calculate the energies at k-points along lines.
3.Use some software to plot the band structure.
Kohn-Sham equations
n(�r) =�
i
|φi(�r)|2
19
L
ECEV
X1
-10
0
6
E (e
V)
S1
X
k
U,KΓ Γ
Γ1
Γ'25
Γ15
∆ ΣΛ
Figure by MIT OpenCourseWare.
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Calculating the band structure
20
&control calculation = 'scf' pseudo_dir = ”
/
&systemibrav = 2
celldm(1) = 10.20 nat = 2 ntyp = 1 ecutwfc = 18.0
/
&electrons conv_thr = 1.0d-8
/
ATOMIC_SPECIES Si 28.086 silicon.UPF
ATOMIC_POSITIONS Si 0.00 0.00 0.00 Si 0.25 0.25 0.25
K_POINTS AUTOMATIC 8 8 8 1 1 1
step 1 &control
calculation = 'bands' pseudo_dir = ”
/
&systemibrav = 2
celldm(1) = 10.20 nat = 2 ntyp = 1 ecutwfc = 18.0 nbnd = 8
/
&electrons conv_thr = 1.0d-8
/
ATOMIC_SPECIES Si 28.086 silicon.UPF
ATOMIC_POSITIONSSi 0.00 0.00 0.00 Si 0.25 0.25 0.25
K_POINTS51 0.50 0.50 0.50 1.0 0.45 0.45 0.45 1.0 0.40 0.40 0.40 1.0 0.30 0.30 0.30 1.0 0.25 0.25 0.25 1.0 …
step 2 &inputpp
lsym = .true./
step 3
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Calculating the band structure
21
Image removed due to copyright restrictions. Please see the Genepattern MultiplePW module.
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Calculating the DOS1. Find the converged ground state
density and potential.
3-step procedure 2.For the converged potential calculate energies at a VERY dense k-mesh.
3.Use some software to plot the DOS.
22
Kohn-Sham equations
n(�r) =�
i
|φi(�r)|2
Image removed due to copyright restrictions. Please see Fig. 3 in Yu, R., and X. F. Zhang."Platinum Nitride with Fluorite Structure." Applied Physics Letters 86 (2005):121913.
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Calculating the DOS
23
&control calculation ='scf' pseudo_dir = ”
/
&systemibrav = 2
celldm(1) = 7.50 nat = 2 ntyp = 1 ecutwfc = 30.0 nbnd = 8
/
&electrons conv_thr = 1.0d-8
/
ATOMIC_SPECIES C 12.0107 carbon.UPF
ATOMIC_POSITIONS C 0.00 0.00 0.00 C 0.25 0.25 0.25
K_POINTS AUTOMATIC 8 8 8 0 0 0
step 1 &control
calculation ='nscf' pseudo_dir = ”
/
&systemibrav = 2
celldm(1) = 7.50 nat = 2 ntyp = 1 ecutwfc = 30.0 nbnd = 8
/
&electrons conv_thr = 1.0d-8
/
ATOMIC_SPECIES C 12.0107 carbon.UPF
ATOMIC_POSITIONS C 0.00 0.00 0.00 C 0.25 0.25 0.25
K_POINTS AUTOMATIC 16 16 16 0 0 0
step 2 &inputpp
Emin = -10.0 Emax = 20.0 DeltaE = 0.1
/
step 3
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Calculating the DOS
24
Image removed due to copyright restrictions. Please see the Genepattern MultiplePW module.
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Metal/insulator
25
Number of electron in unit cell: EVEN: MAYBE INSULATOR
ODD: FOR SURE METAL
silicon Are any bands crossing
the Fermi energy? YES: METAL
NO: INSULATOR
L
ECEV
X1
-10
0
6
S1
X
k
U,KΓ Γ
Γ1
Γ'25
Γ15
∆ ΣΛ
unoccupied
occupied
Fermi energy
Figure by MIT OpenCourseWare.
E (e
V)
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Metal/insulator
26
diamond: insulator
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Metal/insulator
27
Image removed due to copyright restrictions. Please see:Fig. 3 in Thonhauser, T., and K. M. Rabe. "Fcc Breathing Instability inBaBiO3 From First Principles." Physical Review B 73 (2006): 212106.
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Simple optical properties
28
photon has almost no momentum:
only vertical transitions possible
energy conversation and momentum conversation apply
E=hv gap
L
ECEV
X1
0
0
6
S1
X
k
U,KΓ Γ
Γ1
Γ'25
Γ15
∆ ΣΛ
unoccupied
occupied
Figure by MIT OpenCourseWare.
-1
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Simple optical properties
29
Image removed due to copyright restrictions.Please see: http://commons.wikimedia.org/wiki/Image:Srgbspectrum.png
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Magnetizationspin-polarized calculation:
separate density for electrons with spin
up downIntegrated difference between up and down
density gives the magnetization.
30
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Magnetization
31
&control calculation = 'scf',pseudo_dir = ”
/
&systemibrav=3,celldm(1)=5.25,nat=1,ntyp=1,ecutwfc=25.0,occupations='smearing'smearing='gauss',degauss=0.05,nspin=1starting_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}4 4 4 1 1 1
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 anti-ferromagnetic
iron
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Current research ...
Images removed due to copyright restrictions. Please see: http://commons.wikimedia.org/wiki/Image:RBG-LED.jpg http://www.wallpaper.net.au/wallpapers/aviation/Airbus-A380-1024x768.jpg
http://www.sergiosilva.us/images/block4.jpg
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Literature
Charles Kittel, Introduction to Solid State Physics
Ashcraft and Mermin, Solid State Physics
wikipedia, “solid state physics”, “condensed matter physics”, ...
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Lesson outline
Review
structural properties
Calc. the band structure
Calc. the DOS
Metal/insulator
Magnetization
34
Image removed due to copyright restrictions. Please see:Fig. 3 in Thonhauser, T., and K. M. Rabe. "Fcc Breathing Instability inBaBiO3 From First Principles." Physical Review B 73 (2006): 212106.
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5. Quantum Modeling of Solids: Basic Properties
6.
Class outline
35
theory practice
1. It’s A Quantum World: The Theory of Quantum Mechanics
2. Quantum Mechanics: Practice Makes Perfect
4. From Atoms to Solids
3. From Many-Body to Single-Particle; Quantum Modeling of Molecules
Advanced Prop. of Materials; What else can we do?
7. Review session
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