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4 From Atoms to Solids

Part II Outline1. Its A Quantum World:The Theory of

Quantum Mechanics

2. Quantum Mechanics: Practice MakesPerfect3. From Many-Body to

Single-Particle; Quantum Modeling ofMolecules

7. Nanotechnology8. Solar Photovoltaics: Converting

Photons into Electrons9. Thermoelectrics: Converting

Heat into Electricity

10.Solar Fuels: Pushing Electrons up5. Quantum Modeling of Solids: Basic

a Hill

11.Hydrogen Storage: the Strengthof Weak InteractionsProperties

6. Advanced Prop. of Materials:What 12. Reviewelse can we do?

4. From Atoms to Solids

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Motivation

?

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Lesson outline

ReviewPeriodic potentialsBlochs theoremEnergy bands

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r1

Review: Next? Helium

H=Ee-

H1 +H2 +W(r1,r2) =E(r1,r2)

T1 +V1 +T2 +V2 +W (r1,r2) =E(r1,r2)

+

e-

r2

r12

2

2 2 2 2 2e e 2m1 40r1 2m2 40r2

+e2

40r12(r1,r2) =E(r1,r2)

cannot be solved analytically problem!

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Review:The Multi-Electron Hamiltonian2 e2Remember

the good old days of the

1-electron H-atom??

Theyre

over!

H =E 2m2 40r

(r) =E(r)

N N N n N n n nH= "% h2 #2Ri+ 1%%ZiZje2 "h2%#2ri "%% Zie2 + 1%% e2

i=12Mi 2i=1j=1Ri"Rj 2m i=1 i=1j=1Ri"rj 2i=1j=1 ri"rj

i$j i$jkinetic energy of ions kinetic energy of electrons electron-electron interaction

potential energy of ions electron-ion interaction

H R1,...,RN;r1,...,rn( )"R1,...,RN;r1,...,rn( ) = E"R1,...,RN;r1,...,rn( )Multi-Atom-Multi-Electron Schrdinger Equation

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Born-Oppenheimer Approximation

Electrons and nucleias separate systems

H= "2

h

m2$n #r2i "$N$n

R

Z

ii

"e2rj

+

2

1$n$nri

e

"2rji=1 i=1j=1 i=1j=1

i%j

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Born-Oppenheimer ApproximationElectrons and nuclei

as separate systemsH= "h2$n #2ri "$N$n Zie2 + 1$n$n e2

2mi=1 i=1j=1Ri"rj 2i=1

ij%=1jri"rj

... but this is anapproximation!

electrical resistivity

superconductivity

....

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Review: Solutions

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wavefunc

tion:

complicate

d!

electrondensieasy!

ty:

Review: DFT= r1,r2, . . . ,rN

Walter KohnDFT, 1964

n=n r

All aspects of the electronic structure of a systemof interacting electrons, in the ground state, inan external potential, are determined by n(r)

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Review: DFTion

source unknown. All rights reserved.This content is excluded from our CreativeCommons license. For more information,see http://ocw.mit.edu/fairuse.

electron density The ground-state energy is a

functional

of the electron density.

E n =T n +Vii+Vie n +Vee n kinetic ion-electron

ion-ion electron-electron

The functional is minimal at the exactground-state electron density n(r)The functional exists... but it is unknown!

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Review: DFTE n =T n +Vii+Vie n +Vee n

kinetic ion-ion ion-electron electron-electron

2electron density n(r) =|i(r)|i

groundstate =minE[n]

Find the wave functions that minimize theenergy using a functional derivative.

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Review: DFTFinding the minimum leads to

Kohn-Sham equations

ion potential Hartree potential exchange-correlation

potential

equations for non-interacting electrons

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Review: DFT

Only one problem: vxcnot known!approximations necessary

local density general gradient

approximation approximation

LDA GGA

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3.021jIntroductionto ModelingandSimulation N. Marzar MIT, May2003)

Iterations to selfconsistency

Review: Self-consistent cycleKohn-Sham equations

-

n(r)=i

|i(r)|2

s

cfloop

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Review: DFT calculationsscfloop

total energy = -84.80957141 Rytotal energy = -84.80938034 Rytotal energy = -84.81157880 Rytotal energy = -84.81278531 Rytotal energy = -84.81312816 Ry exiting loop;total energy = -84.81322862 Ry result precise enoughtotal energy = -84.81323129 Ry

At the end we get: 1) electronic charge density2) total energy

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Review: DFT calculationstotal energy = -84.80957141 Rytotal energy = -84.80938034 Rytotal energy = -84.81157880 Rytotal energy = -84.81278531 Rytotal energy = -84.81312816 Ry exiting loop;total energy = -84.81322862 Ry result precise enoughtotal energy = -84.81323129 Ry

scfloop

At the end we get:1) electronic charge density

2) total energy

Structure Elastic Vibrational ...

constants properties

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Review: Basis functionsMatrix eigenvalue equation:

=ciii

expansion in=

orthonormalized basisfunctions

cii=Ecii

i idrjH

icii=E

drj

icii

iHjici=EcjHc=Ec

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Review: Plane waves asbasis functionsplane wave expansion: (r) =cjeGjri

j plane waveCutoff for a maximum G is necessary and results in a finite basis set.

Plane waves are periodic,thus the wave function is periodic!

periodic crystals: atoms, molecules:Perfect!!! (next lecture) be careful!!!

Image by MIT OpenCourseWare.

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From atoms to solids

The ground state electron configuration of a system is constructed by putting the

available electrons, two at a time (Pauli principle), into the states of lowest energy

Solid EnergyAtom Molecule

Antibonding p

p

Antibonding s

Bonding ps

Bonding s

Conduction band

from antibonding

p orbitals

Conduction band

from antibonding

s orbitals

Valence band from

p bonding orbitals

Valence band from

s bonding orbitals

k

Image by MIT OpenCourseWare.

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nrgy

Energy bands

Metal Insulator Semiconductor

occupied

empty

energy

gap

NB: boxes = allowed energy regions

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Crystal symmetriesA crystal is built upof a unit cell and

periodic replicas thereof.

lattice unit cellImage of M. C. Escher's "Mobius with Birds" removed due to copyright restrictions.

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C l i

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Crystal symmetriesBravais

Lattice

Triclinic

Monoclinic

Orthorhombic

Tetragonal

Trigonal

Cubic

Hexagonal

ParametersSimple

(P)

Volume

Centered (I)

Base

Centered (C)

Face

Centered (F)

a1= a2= a3

12= 23= 31

a1= a2= a3

23= 31= 900

12= 900

a1= a2= a312= 23= 31 = 90

0

a1= a2= a312= 23= 31 = 90

0

a1= a2= a312= 120

0

23= 31= 900

a1= a2= a3

12= 23= 31 = 900

a1= a2= a312= 23= 31 < 120

0

a3

a1

a2

4 Lattice Types

7CrystalClasses

Image by MIT OpenCourseWare.

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Lattice and basissimplecubic

facecentered

cubic

basis

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The inverse latticeThe real space lattice is described by three basis vectors:

R =n1a1+n2a2+n3a3

The inverse lattice is described by three basis vectors:G =m1b1+m2b2+m3b3

i Gj(r) =cjei reGR = 1j

automatically periodic in R!

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The inverse latticereal space lattice (BCC) inverse lattice (FCC)

xy

a

z

a1a2

a3

Image by MIT OpenCourseWare.

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The Brillouin zone

inverse lattice

The Brillouin zone is a specialunit cell of the inverse lattice.

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The Brillouin zone

Brillouin zone of the FCC lattice

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Periodic potentialsmetallic sodium

V(r)

Dr. Helmut Foll. All rights reserved. This content is

2m2+V(r)

=E

excluded from our Creative Commons license.For more information, see http://ocw.mit.edu/fairuse.

R R R

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l

Periodic potentialsIt becomes much easier if you use the

periodicity of the potential!V(r) =V(r+R)

attice vector

Blochs theorem k r) =eikruk(r)(NEW quantum number k that

lives in the inverse lattice!

uk(r) =uk(r+R)

P d l

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Periodic potentialsBlochs theoremk(r) =eikruk(r)uk(r) =uk(r+R)

= 2/k

k(r) = eik.r

k = 0

u(r)

k = /a

a

Image by MIT OpenCourseWare.

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Periodic potentials

Results of the Bloch theorem:

k(r+R) =k(r)eikr2 2 charge density|k(r+R)| =|k(r)| is lattice periodic

k r

k+G r

Ek=Ek+Gif solution also solution

with

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Periodic potentialsSchrdinger certain quantumequation symmetry number

hydrogen spherical n,l,m(r)atom symmetry[H,L ] =HL L H = 0

[H,Lz] = 0

periodic translational n,k rsolid symmetry[H,T] = 0

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The band structureDifferent wave functions can satisfy the Bloch theorem for the same k:

eigenfunctions and eigenvalues labelled with kand the index n

energy bands

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Th b d

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The band structure

energy levels

in the Brillouin zone

Figure by MIT OpenCourseWareL

EC

EV

X1

-10

0

6

E(eV)

S1

X

k

U,KG G

G1

G'25

G15

D SL

Silicon

Image by MIT OpenCourseWare.

occupied

unoccupied

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The band structurefolding of the band structure

3 2

k

kx

a

3aa

2aa

a

k

kx

3 2 a

3aa

2aa

a

Image by MIT OpenCourseWare.

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The band structurereal band structure

k

kx

k

kx3 2 a

3aa

2aa

a

3 2 a

3aa

2aa

a

Image by MIT OpenCourseWare.

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Review

ReviewPeriodic potentialsBlochs theoremEnergy bands

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LiteratureCharles Kittel, Introduction to Solid State

Physics

Richard M. Martin, Electronic Structurewikipedia,solid state physics,condensed

matter physics, ...

Simple band structure simulations:http://phet.colorado.edu/simulations/sims.php?sim=Band_Structure

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MIT OpenCourseWarehttp://ocw.mit.edu

3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and SimulationSpring 2011

For information about citing these materials or our Terms of use, visit:http://ocw.mit.edu/terms.