1.021 , 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2012 Part II – Quantum Mechanical Methods : Lecture 8

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

Jeffrey C. Grossman

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

1

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

9.

Application of Quantum Modeling of Solids: Solar Cells Par t I

10. Application of Quantum Modeling of Solids: Solar Cells Par t II

11. Application of Quantum Modeling of Solids: Nanotechnolog y

2

Lesson outline

0.000 (A)

0. 005 (B)

0. 010 (C)

5

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3

-4

-4

-5

-5

• Brief Re vie w

• Optical p r oper ties

energy [eV]

• Magnetic p r oper ties

• T ranspor t p r oper ties

Vibrational p r oper ties

X W L K X’

0 2 4

DOS

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3

K eeping Rel e vant

© Bulletin of the Atomic Scientists. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use / .

Ma y/J une 2001, Bulletin of the Atomic Scientists 4

R e vi e w: i n v erse lattice

Schrödinger

cer tain

quantum

equation

symmetr y

n umber

h yd r ogen atom

spherical symmetr y

ψ n,l,m ( → r )

[ H, L 2 ] = H L 2 − L 2 H = 0

[ H, L z ] = 0

periodic solid

translational symmetr y

ψ n , → k ( → r )

[ H, T ] = 0

5

R e vi e w: i n v erse lattice

k z

some G

k y

k x

G ⇤ k ( → r ) — → G ⇤ k + G ⇤ ( → r )

E ⇤ k = E ⇤ k + G ⇤

6

R e vi e w: The band structu r e

E (eV)

k is a contin uous variable

6

15

0

'

25

X

1

E C

E

V

S 1

-10

L

k

X

U,K

Image by MIT OpenCourseWare.

7

The F ermi energ y

6

E (eV)

u noccupied

F ermi energ y

0

-10

15

X 1

' 25

E C E V

S 1 occu pie d

each band can hold:

2N electr ons and y ou h a v e (electr ons per unit cell)*N

or

tw o electr ons and y ou h a v e

L

k

X U,K

(electr ons per unit cell)

Image by MIT OpenCourseWare.

8

silicon

Electrical p r oper ties

6

E (eV)

15

E V

E C

' 25 X 1

0

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

S 1

-10

Number of electr ons 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.

9

Electrical p r oper ties

diamond: insulator

10

Elect r on T ranspor t

E

E- field

dv = eE − 1 v = 0

At equilibrium

v =

eτE

m

dt m τ

m, e

ne 2 τ

Electrical conductivity

j = nev = E ≡ σE

Electric cur r ent

m

ne 2 τ

σ =

m

11

Elect r on T ranspor t

F ermi function

Calculating σ f r om band structur e

4 π 3 ∂E

σ = e 2 τ

∫ d k

− ∂f v ( k ) v ( k )

v ( k ) =

1

Cur vatur e of band structur e

E (eV)

k ∇ k

E ( k )

6

15

0

'

25

X

1

E C

E

V

S 1

-10

L

k

X

U,K

12 Image by MIT OpenCourseWare.

Figure showing the silicon energy band diagram. Figure showing the silicon energy band diagram. Figure showing the silicon energy band diagram. Figure showing the silicon energy band diagram. Figure showing the silicon energy band diagram. Figure showing the silicon energy band diagram. Figure showing the silicon energy band diagram.

6

0

ie d

E C E V

ga p

S 1 occu pied

-10

U,K

X

k

L

1

25

X

'

1 5

u n occup

Simple optical p r oper ties

E (eV)

E=hv

photon has almost no momentum:

onl y v er tical transitions

possible

energ y con v ersation and momentum con v ersation a ppl y

13

Image by MIT OpenCourseWare.

Silicon Solar Cells H a v e to Be Thick ( $$$ )

It ’ s all in the band- structur e!

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Simple optical p ro p er ties

Image in the public domain .

15

Magnetism

S

N

Origin of magnetism: electr on spin

An electr on has a magnetic moment of μ B , Bohr magneton.

Spin up Spin do wn

n ↑ n ↓

μ = μ B ( n ↑ − n ↓ )

16

Magnetization

spin-polarized calculation: separate density f or elect r ons with spin

up do wn

Integrated diff er ence betw een up and do wn density giv es the magnetization.

17

Magnetism

In r eal systems, the density of states needs to be consider ed.

3

DOS (states/eV)

2

bcc F e

1

0

-1

-2

-3

E - E F (eV)

μ = μ B

∫

E F

dE [ g ↑ ( E ) − g ↓ ( E )]

-6 -4 -2 0 2

18

Quantum Molecular Dynamics

…and let us, as natur e dir ects, begin first with first principles.

Aristotle ( P oetics, I)

Vatican fresco "The School of Athens" by Raphael. Image is in the public domain. F=ma

Aristotle depicted by Raphael, holding his

Use Hellmann-F e ynman! Ethics. Image is in the public domain.

19

Carbon Nanotube G r o wth

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20

Silicon Nanocluster G r o wth

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21

W ater

Henr y C a v endish was the first to describe cor r ectl y the composition of water (2 H + 1 O), in 1781.

He r epor ted his findings in terms of phlogiston (later the gas he made was pr o v en to be h ydr ogen) and dephlogisticated air (later this was pr o v en to be o xygen).

C a v endish was a pr etty neat gu y .

Image is in the public domain.

A Univ ersity dr opout, he also compar ed the conductivities of electr ol ytes and expr essed a v ersion of Ohm's l a w .

His last major w ork was the first measur ement of Ne wton's gr a vitational constant, with the mass and density of the Ear th. The accuracy of this experiment was not impr o v ed f or a centur y .

22

W ater

Which of the f ollo wing is the cor r ect pictur e f or H 2 O?

Courtesy of Martin Chaplin, London South Bank University. Used with permission.

Cool water site: http://ww w .lsbu.ac .uk/water /

23

Classical or Quantum?

Mor e than 50 classical potentials in use tod a y f or wate r .

Which one is best?

Courtesy of Martin Chaplin, London South Bank University. Used with permission . 24

Mg++ in W ater

Courtesy of Martin Chaplin, London South Bank University. Used with permission .

Impor tant Diff er ences!

25

fo r c e

Vibrational p r oper ties

lattice vibrations ar e called: phonons

What is the fr equency of this vibration?

26

Vibrational p r oper ties

animated phonons on the w eb

http://dept. k ent.edu/p r ojects/ksuviz / lee viz/phonon/phonon.html

• sound in solids determined b y acoustical phonons (shock w a v es)

• some optical p r oper ties r elated to optical phonons

• heat ca pacity and transpor t r elated to phonons

27

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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Aerothermodynamic image of shuttle, courtesy NASA .

28

Literatu r e

•

Charles Kittel, Int r oduction to Solid State P h ysics

• Ashc r oft and Mermin, Solid State P h ysics

•

• wikipedia, “phonons”, “lattice vibrations”, ...

solar PV: tons of w eb sites, e .g.: http:// pv education.org/pvcd r om

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