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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