1.021 , 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2012 Part II – Quantum Mechanical Methods : Lecture 9

Some Review & Introduction to Solar PV

Jeffrey C. Grossman

Department of Materials Science and Engineering Massac husetts Institute of T ec hnology

Par t II T opics

1. It ’ s a Quantum W orld: The Theor y of Quantum Mechanics

2. Quantum Mechanics: Practice Mak es P erf ect

3. Fr om Man y-Body to Single-Par ticle; Quantum Modeling of Molecules

4. Application of Quantum Modeling of Molecules: Solar Thermal Fuels

5. Application of Quantum Modeling of Molecules: Hydr ogen Storage

6. Fr om Atoms to Solids

7. Quantum Modeling of Solids: Basic Pr oper ties

8.

Advanced Pr op . of Materials: What else can w e do?

11. Application of Quantum Modeling of Solids: Nanotechnolog y

Lesson outline

• Discussion of PSET

• Re vie w f or the Quiz

• Int r oduction to Solar PV

?

mec hanical

pr oper ties

Motivation: ab-initio modeling!

?

electrical pr oper ties

?

optical

pr oper ties

Vision without Action is a Dream Action without Vision is a Nightmare

Japanese proverb

W h y quantum mechanics?

P r oblems in classical p h ysics that led to quantum mechanics:

• “classical atom”

• quantization of p r oper ties

• w a v e aspect of matter

• (black-body radiation), ...

W a v e aspect of matter

light matter

w a v e character par ticle character

_

_

_ _

_

_

_

_

_

_

_

_

_

Image by MIT OpenCourseWare.

Image in public domain. See Wikimedia Commons .

W a v e aspect of matter

par ticle : E w a v

and momentum p k

p k = n k k =

h k k

de B r oglie: fr ee par ticle can be described a as

plane w a v e with

ψ (

k r

.

, t ) = Ae

i ( k · � r − ωt )

�

λ = h mv

λ | k k |

Interp r etation of a w a v efunction

ψ ( k r, t ) w a v e function (complex)

| ψ | 2 = ψψ ∗ interpr etation as pr obability to find par ticle!

∞

ψψ ∗ dV = 1

−∞

Image by MIT OpenCourseWare.

Schrödinger equation

H time independent: ψ ( k r, t ) = ψ ( k r ) · f ( t )

f ˙ ( t ) H ψ ( k r )

i n = = const. = E f ( t ) ψ ( k r )

H ψ ( k r ) = Eψ ( k r )

ψ ( k r, t ) = ψ ( k r ) · e −

i Et

time independent Schrödinger equation stationar y Schrödinger equation

The h yd r ogen atom

stationar y

Schrödinger equation H ψ = E ψ

� n 2

� T + V � ψ = Eψ

2 �

— 2 m ∇

n 2 2

+ V ψ ( k r ) = Eψ ( k r )

e 2 �

The h yd r ogen atom

q u a n t u m n u m b e r s

Image by MIT OpenCourseWare.

The h yd r ogen atom

Energies:

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Atomic units

1 eV = 1.6021765 -19 J

1 Rydberg = 13.605692 eV = 2.1798719 -18 J

1 Har tr ee = 2 Rydberg

1 Bohr =5.2917721 -11 m

Atomic units (a.u.) :

Energies in Ry Distances in Bohr

Also in use: 1 Å =10 -10 m, nm= 10 -9 m

E v er ything is spinning ...

Stern–Gerlach experiment (1922)

F k = −∇ E

= ∇ m k · B k

Image courtesy of Teresa Knott.

E v er ything is spinning ...

ne w quantum n umber : spin quantum n umber f or elect r ons: spin quantum n umber can ON L Y be

up d o wn

Pauli ’ s exclusion p rinci p le

T w o elect r ons in a system cannot h a v e the same quantum n umbers!

quantum n umbers: main n: 1,2,3 ...

orbital l: 0,1,...,n-1

magnetic m: -l,...,l spin: up , do wn

h ydr ogen

... ... ... ...

3s 3p 3d

2s 2p

1s

P eriodic table of elements

This image is in the public domain. Source: Wikimedia Commons .

Next? Helium?

e ­

r 12

H ψ = E ψ

+ e ­

r 2 H 1 + H 2 + W ψ ( k r 1 , k r 2 ) = Eψ ( k r 1 , k r 2 )

T 1 + V 1 + T 2 + V 2 + W ⇥ ψ ( k r 1 , k r 2 ) = Eψ ( k r 1 , k r 2 )

n 2 2 e 2

n 2 2 e 2

e 2 ⇥

— 2 m ∇ 1 −

4 πs 0 r 1

— 2 m ∇ 2 −

4 πs 0 r 2

+ 4 πs

r 12

ψ ( k r 1 , k r 2 ) = Eψ ( k r 1 , k r 2 )

cannot be solv ed anal yticall y p r oblem!

Solutions

quantum chemistr y

density functional theor y

Molle r -Plesset per turbation theor y MP2

coupled cluster theor y CCSD(T)

The T w o Paths

Ψ is a w a v e function of all positions & time .

-

-

-

-

-

-

- -

n 2 2 ⇥ ∂

— 2 m ∇

+ V ( k r, t ) ψ ( k r, t ) = i n ψ ( k r, t )

∂t

Chemists (mostl y) P h ysicists (mostl y)

Ψ = something simpler

H = something simpler

“mean field” methods

Walter Kohn

W orking with the Density

E [ n ] = T [ n ] + V ii + V ie [ n ] + V ee [ n ]

kinetic ion-electr on

ion-ion electr on-electr on

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ion potential Har tr ee potential exchange-cor r elation

potential

W h y DFT?

Image by MIT OpenCourseWare.

Density functional theor y

E [ n ] = T [ n ] + V ii + V ie [ n ] + V ee [ n

kinetic ion-ion ion-electr on electr on-electr on

elect r on density n ( k r ) =

Σ | φ i ( k r ) | 2

i

E ground state = m i n E [ n

φ

Find the w a v e functions that minimize the energ y using a functional derivativ e .

Self-consistent cycle

K ohn-Sham equations

n ( k r ) =

Σ | φ i ( k r ) | 2

i

Density functional theor y

Onl y one p r oblem:

v xc not kno wn!!!

a pp r o ximations necessar y

local density general gradient a pp r o ximation a pp r o ximation

L D A GGA

• structur e

• bulk modulus

binding energies r eaction paths

• shear modulus f or ces

• elastic constants

pr essur e

str ess

• vibrational pr oper ties ...

• sound v elocity

?

Co n v ergence f or molecules

Cr ystal symmetries

S BC FC

= simple

= body center ed

= face center ed

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The i n v erse lattice

The r eal space lattice is described b y thr ee basis v ectors:

R k = n 1 k a 1 + n 2 k a 2 + n 3 k a 3

The in v erse lattice is described b y thr ee basis v ectors:

G k = m 1 k b 1 + m 2 k b 2 + m 3 k b 3

e iG · R = 1

ψ ( k r ) =

Σ c j e iG j · r

j

automaticall y periodic in R!

The Brillouin zone

in v erse lattice

The Brillouin zone is a special unit cell of the in v erse lattice .

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The Brillouin zone

Brillouin zone of the FCC lattice

P eriodic potentials

V ( k r )

R k

R k

R k

© Prof. Dr. Helmut Föll. A ll rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse .

V ( k r ) = V ( k r +

R k )

l attice v ector

NEW quantum n umber k that liv es in the in v erse lattice!

Bloch ’ s theor em ψ k ) = e ik · r u k ( k r )

u k ( k r ) = u k ( k r + R k )

I n v erse lattice

Results of Bloch ’ s theor em:

ψ ( k r + R k ) = ψ ( k r ) e ik · R

| ψ ( k r + R k ) | 2 = | ψ ( k r ) | 2

charge density

k k is lattice periodic

if solution

G ⇤ k ( → r ) — → G ⇤ k + G ⇤ ( → r )

also solution

with

E ⇤ k = E ⇤ k + G ⇤

Structural p r oper ties

finding the str ess/pr essur e and the bulk modulus

E tot

V 0 V

∂E ∂p ∂ 2 E

p = − σ bulk = − V = V

∂V ∂V ∂V 2

Calculating the band structu r e

1. Find the con v erged gr ound state

density and potential.

3- step p r ocedur e 2. F or the con v erged potential calculate

the energies at k-points along lines .

3. Use some softwar e to plot the band

6

 15

0

 '

25

X

E C

1

E

V

S 1

-10

L 

 





k

X U,K





structur e .

n ( k r ) = Σ | φ i ( k r ) | 2

i

K ohn-Sham equations

silicon

Metal/insulator

6

E C

0 E V

F ermi energ y

Ar e an y bands cr ossing the F ermi energ y?

YES: ME T AL NO: INSUL A T OR

-10

Number of electr on in unit cell: EVEN: M A YBE INSUL A T OR ODD: FOR SURE ME T AL

L   

k

X U,K  

Image by MIT OpenCourseWare.

E=hv

Simple optical p ro p er ties

6

u  noccupied

15

0

 '

25

X

1

E C E

V

S 1 occupied

-10

 

L







k

X

U,K





Image by MIT OpenCourseWare.

shor test wa v e length visible 400 nm ; cor r esponds to photon with E=3.1 eV

Image in the public domain.

Vibrational p r oper ties

lattice vibrations ar e called: phonons

What is the fr equency of this vibration?

i r on

Magnetization

&control

calculation = 'scf',

/

pseudo_dir = ''

&system

ibrav=3,

celldm(1)=5.25, nat=1,

ntyp=1,

ecutwfc=25.0,

occupations='smearing'

/

smearing='gauss', degauss=0.05,

starting_magnetization(1)=0.0

nspin=1

&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

star ting magnetization f or each atom

perf orm thr ee calculations and find lo w est energ y:

non spin-polarized spin-polarized

f er r omagnetic anti-f er r omagnetic

With the band structur e and DOS w e find:

• electrical conductivity (insulator/metal/semiconductor)

• thermal conductivity

• optical pr oper ties

• magnetization/polarization

• magnetic/electric pr oper ties

• ...

?

Co n v ergence f or solids

Summar y of p r oper ties

structural p r oper ties electrical p r oper ties optical p r oper ties magnetic p r oper ties vibrational p r oper ties

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It ’ s Not Only About W arming: Abundance of

Aff or dable Energ y Resou r ces Can Uplift the W orld

x4 x13

HUMAN WELL-BEING INCREASES WITH INCREASED PER-CAPI T A ENERGY USE

© AIP Publishing. All rights reserved. This content is excluded from our Creative

Commons license. For more information, see http://ocw.mit.edu/help/faq-fair- use/ . 44 Cour tesy: Vladimir Bulo vic

It ’ s Not Only About W arming: Abundance of Aff or dable Energ y Resou r ces Can Uplift the W orld

Nor th American Electrical Black out

08/14/2003

… August 15 …

just 24 hours into black out Air P ollution was Reduced

SO 2 >90% O 3 ~50%

Light Scattering Par ticles ~70%

“This clean air benefit was r ealized o v er m uch of eastern U .S . ”

Marufu et al., Geoph ysical Resear ch Letters 2004

b y 4:13 pm 256 po w er plants w er e off-line

Images courtesy of Vladimir Bulovic. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/ .

Cour tesy: Vladimir Bulo vic

45

Ma p of the W orld Scaled to Energ y

Consumption b y Countr y

Newman, U. Michigan, 2006

Courtesy of M. E. J. Newman .

• In 2002 the w orld burned energ y at a rate of 13.5 TW

• Ho w fast will w e burn energ y in 2050?

• (assume 9 billion people)

• If w e use energ y li k e in U .S. w e will need 102 TW

• Conser vativ e estimate: 28~35 TW

Energ y use estimates fr om Nocera, Daedalus 2006 & Le wis and Nocera, PNAS 2006

U .S. Energ y Consumption

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47

Energ y f r om the Sun

Courtesy of SOHO/EIT (ESA & NASA) consortium.

• Energ y r eleased b y an ear thqua k e of magnitude 8 (10 17 J):

• the sun deliv ers this in one second

• Energ y humans use ann uall y (10 20 J):

• sun deliv ers this in one hour

• Ear th ’ s total r esour ces of oil (3 trillion bar r els, 10 22 J):

• the sun deliv ers this in tw o d a ys

48

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Spring 20 1 2

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