1.021, 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2011

Part II – Quantum Mechanical Methods : Lecture 4

From atoms to solids

Jeffrey C. Grossman

Department of Materials Science and Engineering Massachusetts Institute of Technology

4. From Atoms to Solids

Part II Outlinetheory & practice example applications

1. It’s A Quantum World:The Theory of Quantum Mechanics

2. Quantum Mechanics: Practice Makes Perfect

3. From Many-Body to Single-Particle; Quantum Modeling of Molecules

7. Nanotechnology

8. Solar Photovoltaics: Converting Photons into Electrons

9. Thermoelectrics: Converting Heat into Electricity

10.Solar Fuels: Pushing Electrons up

5. Quantum Modeling of Solids: Basic

a Hill

11.Hydrogen Storage: the Strength of Weak Interactions

Properties

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

4. From Atoms to Solids

Motivation

?

Lesson outline

• Review

• Periodic potentials

• Bloch’s theorem

• Energy bands

r1

Review: Next? Helium

Hψ = Eψ

e- �H1 + H2 + W

�ψ(�r1, �r2) = Eψ(�r1, �r2)

�T1 + V1 + T2 + V2 + W

�ψ(�r1, �r2) = Eψ(�r1, �r2)

+

e -

r2

r12

� �22

2 �22

2e e−

2m ∇1 −

4π�0r1 −

2m ∇2 −

4π�0r2 +

e2

4π�0r12

�ψ(�r1, �r2) = Eψ(�r1, �r2)

cannot be solved analytically problem!

!

!

Review:The Multi-Electron Hamiltonian

�2 e2Rememberthe good old days of the

1-electron H-atom??

They’re over!

Hψ = Eψ �

− 2m

∇2 − 4π�0r

�ψ(�r) = Eψ(�r)

N N N n N n n n

H = "% h2

#2 Ri +

1 %% ZiZ je

2

" h2

%#2 ri "%% Zie

2

+ 1 %% e2

i=1 2Mi 2 i=1 j=1 Ri " R j 2m i=1 i=1 j=1 Ri " rj 2 i=1 j=1 ri " rji$ j i$ j

kinetic 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 Schrödinger Equation

!

Born-Oppenheimer Approximation

Electrons and nucleias “separate” systems

H = " 2hm

2

$ n

#r2 i "$

N

$ n

RZ

i

i

"

e2

rj + 21 $

n

$ n

ri

e"

2

rji=1 i=1 j=1 i=1 j=1 i% j

!

Born-Oppenheimer Approximation

Electrons and nucleias “separate” systems

H = " h2

$ n

#2 ri "$

N

$ n Zie

2

+ 1 $

n

$ n e2

2m i=1 i=1 j=1 Ri " rj 2 i=1 ij% =1 j ri " rj

... but this is anapproximation!

• electrical resistivity

• superconductivity

• ....

Review: Solutions

Moller-Plessetperturbation theory

MP2

coupled clustertheory CCSD(T)

quantum chemistrydensity

functionaltheory

density functional

theory

Review:Why DFT?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

Image by MIT OpenCourseWare.

wave function:

complicated!

electron densi easy!

ty:

Review: DFT

ψ = ψ(�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)

Review: DFT

ion

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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!

Review: DFT

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

2electron density n(�r) = �

|φi(�r)|i

Eground state = min E[n]φ

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

Review: DFT

Finding the minimum leads toKohn-Sham equations

ion potential Hartree potential exchange-correlation potential

equations for non-interacting electrons

Review: DFT

Only one problem: vxc not known!approximations necessary

local density general gradient approximation approximation

LDA GGA

3.021j Introduction to Modeling and Simulation N. Marzari (MIT, May 2003)

Iterations to selfconsistency

Review: Self-consistent cycleKohn-Sham equations

-

n(�r) = �

i

|φi(�r)|2

scf loop

Review: DFT calculationssc

f loo

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

At the end we get: 1) electronic charge density 2) total energy

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

f loo

p

At the end we get: 1) electronic charge density 2) total energy

Structure Elastic Vibrational ... constants properties

Review: Basis functionsMatrix eigenvalue equation: ψ =

� ciφi

i

expansion in Hψ = Eψ orthonormalized basis

functions H

� ciφi = E

� ciφi

i i � d�r φ∗

j H �

i

ciφi = E �

d�r φ∗ j

�

i

ciφi

�

i

Hjici = Ecj

H�c = E�c

Review: Plane waves asbasis functions

plane wave expansion: ψ(�r) = �

cj e Gj ·�ri �

j plane wave

Cutoff 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.

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 Energy Atom Molecule

Antibonding p

p Antibonding s

Bonding p s

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.

Ener

gy

Energy bands

Metal Insulator Semiconductor

occupied

empty

energy gap

NB: boxes = allowed energy regions

Crystal symmetries

A 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.

Crystal symmetries

1023 particlesper cm3

Since a crystal is periodic, maybe we can get away with modeling only the unit cell?

Crystal symmetries

BravaisLattice

Triclinic

Monoclinic

Orthorhombic

Tetragonal

Trigonal

Cubic

Hexagonal

Parameters Simple(P)

VolumeCentered (I)

BaseCentered (C)

FaceCentered (F)

a1 = a2 = a3α12 = α23 = α31

a1 = a2 = a3α23 = α31 = 900

α12 = 900

a1 = a2 = a3α12 = α23 = α31 = 900

a1 = a2 = a3α12 = α23 = α31 = 900

a1 = a2 = a3α12 = 1200

α23= α31= 900

a1 = a2 = a3α12 = α23 = α31 = 900

a1 = a2 = a3α12 = α23 = α31 < 1200

a3

a1

a2

4 Lattice Types

7 C

ryst

al C

lass

es

Image by MIT OpenCourseWare.

Lattice and basis

simplecubic

facecentered

cubic

basis

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

R� = n1�a1 + n2�a2 + n3�a3

The inverse lattice is described by three basis vectors:G� = m1

�b1 + m2�b2 + m3

�b3

i � � Gj ·ψ(�r) = �

cj e i � �r e G·R = 1

j automatically periodic in R!

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

x y

a

z

a1a2

a3

Image by MIT OpenCourseWare.

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

Periodic potentialsmetallic sodium

V (�r)

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

2�

m ∇2 + V (�r)

�ψ = Eψ

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

� −

�R �R �R

l

Periodic potentials

It becomes much easier if you use theperiodicity of the potential!

V (�r) = V (�r + R�)

attice vector

Bloch’s theorem ψ�k r) = e i�k·�r u�k(�r)(�

NEW quantum number k that lives in the inverse lattice!

u�k(�r) = u�k(�r + R�)

Periodic potentials

Bloch’s theorem

ψ�k(�r) = e i�k·�r u�k(�r)

u�k(�r) = u�k(�r + R�)

λ = 2π/k

ψk (r) = eik.r

k = 0

u(r)

k = π/a

a

Image by MIT OpenCourseWare.

Periodic potentials Results of the Bloch theorem:

ψ�k(�r + R�) = ψ�k(�r)e i�k·�r

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

ψ�k(�r) −→ ψ�k+ �G(�r)

E�k = E�k+ �G

if solution also solution

with

Periodic potentialsSchrödinger certain quantumequation symmetry number

hydrogen spherical ψn,l,m(�r)

atom symmetry [H, L2] = HL

2 − L2H = 0

[H, Lz] = 0

periodic translational ψ n, (�k �r)

solid symmetry [H, T ] = 0

The band structure

Different wave functions can satisfy the Bloch theorem for the same k: eigenfunctions and eigenvalues labelled with k and the index n

energy bands

The band structure

energy levelsin the Brillouin zone

Figure by MIT OpenCourseWare.

L

EC EV

X1

-10

0

6

E (e

V)

S1

X

k

U,K Γ Γ

Γ1

Γ' 25

Γ15

∆ ΣΛ

Silicon

Image by MIT OpenCourseWare.

The band structure

energy levels in the Brillouin zone

Figure by MIT OpenCourseWare.

L

EC EV

X1

-10

0

6

E (e

V)

S1

X k

U,K Γ Γ

Γ1

Γ' 25

Γ15

∆ ΣΛ

Silicon

Image by MIT OpenCourseWare.

occupied

unoccupied

Energy bandssimple example: infinitesimal small potential

solutions: plane waves with quadratic energies

E�k = E�k+ �G

g-1 g1 g2 g3

k

1. BZ

E

Image by MIT OpenCourseWare.

The band structure

folding of the band structure

3� 2� �

�k

kx

a 3� aa

2� aa

� a

�k

kx 3� 2� � a

3� aa

2� aa

� a

Image by MIT OpenCourseWare.

The band structure

real band structure

�k

kx

�k

kx 3� 2� � a

3� aa

2� aa

� a

3� 2� � a

3� aa

2� aa

� a

Image by MIT OpenCourseWare.

Review

• Review

• Periodic potentials

• Bloch’s theorem

• Energy bands

Literature

• Charles Kittel, Introduction to Solid State Physics

• Richard M. Martin, Electronic Structure

• wikipedia,“solid state physics”,“condensed matter physics”, ...

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

MIT OpenCourseWarehttp://ocw.mit.edu

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

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