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 Massachusetts Institute of Technology

1

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

Part II Topics 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

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

5. Application of Quantum Modeling of Molecules: Hydrogen Storage

6. From Atoms to Solids

7. Quantum Modeling of Solids: Basic Properties

9. Application of Quantum Modeling of Solids: Solar Cells Part I

10. Application of Quantum Modeling of Solids: Solar Cells Part II

11. Application of Quantum Modeling of Solids: Nanotechnology

2

Lesson outline

0. 0.

005 (B) 010 (C)

0.000 (A)

000

111

222

333

-5-5-5

-4-4-4

-3-3-3

-2-2-2

-1-1-1

5

4

3

2

1

0

-1

-2

-3

-4

-5

• Brief Review

• Optical properties

• Magnetic properties

• Transport properties

• Vibrational properties

0 2 4ΓΓΓ XXX WWW LLL ΓΓΓ KKK X’X’X’ DOS

ener

gyen

ergy

[eV

][e

V]

ener

gy [

eV]

3

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Keeping Relevant

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

May/June 2001, Bulletin of the Atomic Scientists 4

Review: inverse lattice Schrödinger certain quantum

equation symmetry number

hydrogen atom

spherical symmetry

n,l,m r)ψ (

[H, L2] = HL2 − L2H = 0

[H, Lz] = 0

periodic solid

translational symmetry

ψ n,�k( r)

[H, T ] = 0

5

Review: inverse lattice

kx

ky

kz

some G

6

�⇤k(⇤r) �⇥ �⇤k+ ⇤G(⇤r)

E⇤k = E⇤k+ ⇤G

Review:The band structure

k is a continuous variable

7

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

Image by MIT OpenCourseWare.

The Fermi energy

Fermi energy

each band can hold: 2N electrons and you have (electrons per unit cell)*N

or two electrons and you have

(electrons per unit cell)

8

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

occupied

unoccupied

Image by MIT OpenCourseWare.

Electrical properties

Are any bands crossing the Fermi energy?

YES: METAL NO: INSULATOR

Number of electrons in unit cell: EVEN: MAYBE INSULATOR ODD: FOR SURE METAL

9

silicon

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

Fermi energy

Image by MIT OpenCourseWare.

Electrical properties

diamond: insulator

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Electron Transport E dv eE 1E-field = − v = 0

dt m τ

At equilibrium

m, e ne2τ

j = nev = E ≡ σE

v = eτE m

Electric current Electrical conductivity

σ = ne2τ m

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m

�

Electron Transport Calculating σ from band structure

dk2τσ = e −4π3 ∂E

∂f

v(k)v(k)

Fermi function

1 v(k) = ∇kE(k)

Curvature of band structure

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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

Image by MIT OpenCourseWare.

Simple optical properties

E=hv

photon has almost no momentum:

only vertical transitions possible

energy conversation and momentum conversation apply

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

occupied

unoccupied gap

13

Image by MIT OpenCourseWare.

Silicon Solar Cells Have to Be Thick ($$$)

It’s all in the band-structure!

© Helmut Föll. All rights reserved. This content is excluded from our Creative14Commons license. For more information, see http://ocw.mit.edu/fairuse.

Simple optical properties

Image in the public domain.

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Magnetism S N

Origin of magnetism: electron spin An electron has a magnetic moment of μB, Bohr magneton.

μ = μB (n↑ − n↓)

Spin up Spin down n↑ n↓

16

Magnetization spin-polarized calculation:

separate density for electrons with spin

up down Integrated difference between up and down

density gives the magnetization.

17

Magnetism In real systems, the density of states needs to be considered.

3

2

1

0bcc Fe -1

-2

-3-6 -4 -2 0 2 E - E (eV)

μ = μB dE[g↑ (E) − g↓(E)] EF

F

DO

S (

stat

es/e

V)

18

�

Quantum Molecular Dynamics

…and let us, as nature directs, begin first with first principles. Aristotle (Poetics, I)

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

F=ma Aristotle depicted by Raphael, holding hisEthics. Image is in the public domain.

Use Hellmann-Feynman!

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Carbon Nanotube Growth

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Silicon Nanocluster Growth

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

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Water Henry Cavendish was the first to describe correctly the composition of water (2 H + 1 O), in 1781.

He reported his findings in terms of phlogiston (later the gas he made was proven to be hydrogen) and dephlogisticated air (later this was proven to be oxygen).

Cavendish was a pretty neat guy.

Image is in the public domain.

A University dropout, he also compared the conductivities of electrolytes and expressed a version of Ohm's law.

His last major work was the first measurement of Newton's gravitational constant, with the mass and density of the Earth.The accuracy of this experiment was not improved for a century. 22

Water Which of the following is the correct picture for H2O?

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

Cool water site: http://www.lsbu.ac.uk/water/ 23

Classical or Quantum?

More than 50 classical potentials in use today for water.

Which one is best?

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

Mg++ in Water

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

Important Differences! 25

Vibrational properties

lattice vibrations are called: phonons

force

What is the frequency of this vibration?

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Vibrational properties animated phonons on the web

http://dept.kent.edu/projects/ksuviz/ leeviz/phonon/phonon.html

• sound in solids determined by acoustical phonons (shock waves)

• some optical properties related to optical phonons

• heat capacity and transport related to phonons 27

Summary of properties

structural properties

electrical properties

optical properties

magnetic properties

vibrational properties

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Image of Airbus A380 on Wikimedia Commons. License: CC-BY-SA. Images of circuit board,phone, hard drive © sources unknown. All rights reserved. This content is excluded from ourCreative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.Aerothermodynamic image of shuttle, courtesy NASA .

Literature

• Charles Kittel, Introduction to Solid State Physics

• Ashcroft and Mermin, Solid State Physics

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

• solar PV: tons of web sites, e.g.: http:// pveducation.org/pvcdrom

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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modelling and SimulationSpring 2012 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.