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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
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
Motivation!
3
mechanical propertiespr
electrical propertiespr
optical propertiespr
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.
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.
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.
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
Review: Crystal symmetries
8
Image removed due to copyright restrictions.Please see: http://commons.wikimedia.org/wiki/Crystal_structure
Review: The inverse lattice
real space lattice (BCC) inverse lattice (FCC)
9
xy
a
z
a1a2
a3
Figure by MIT OpenCourseWare.
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
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
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.
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.
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.
The electron density
electrondensity
of silicon
15
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
Structural properties
Etot
finding the equilibrium
lattice constant
alat a
mass density
17
V0 V
Structural properties
Etotfinding the
stress/pressure and the bulk
modulus
∂E ∂p ∂2E p = − σbulk = −V = V
∂V ∂V ∂V 2
18
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.
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
Calculating the band structure
21
Image removed due to copyright restrictions. Please see the Genepattern MultiplePW module.
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.
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
Calculating the DOS
24
Image removed due to copyright restrictions. Please see the Genepattern MultiplePW module.
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)
Metal/insulator
26
diamond: insulator
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.
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
Simple optical properties
29
Image removed due to copyright restrictions.Please see: http://commons.wikimedia.org/wiki/Image:Srgbspectrum.png
Magnetizationspin-polarized calculation:
separate density for electrons with spin
up downIntegrated difference between up and down
density gives the magnetization.
30
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
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
32
Literature
Charles Kittel, Introduction to Solid State Physics
Ashcraft and Mermin, Solid State Physics
wikipedia, “solid state physics”, “condensed matter physics”, ...
33
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.
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