L6_P02

1.021 , 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2012 Part II – Quantum Mechanical Methods : Lecture 6

From Atoms to Solids

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?

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

Lesson outline

• Briefly h yd r ogen storage

• P eriodic potentials

• Bloch ’ s theor em

• Energ y bands

Hyd r ogen Storage

Image s in the public domain.

P r esident Bush Launches the Hyd r ogen Fuel Initiati v e

" T onight I am pr oposing $1.2 billion in r esear ch funding so that America can lead the w orld in de v eloping clean, h ydr ogen- po w er ed automobiles.

"A simple chemical r eaction betw een h ydr ogen and o xygen generates energ y , which can be used to po w er a car pr oducing onl y wate r , not exhaust fumes.

"With a ne w nationa l commitment , our scientists and engineers will o v er come obstacles to taking these cars fr om laborator y to sho wr oom so that the first car driv en b y a child born toda y could be po w er ed b y h ydr ogen, and pollution-fr ee .

"J oin me in this impor tant inno vation to mak e our air significantl y cleane r , and our countr y m uch less dependent on f or eign sour ces of energ y ."

2003 Sta te of the Union Addr ess

Jan uary 28, 2003

Fr om Patr o vic & Millik en (2003) Images are in the public domain.

A History of Hydrogen as a Fuel

Chemical Re vie ws, 2004, V ol. 104, No . 3, Gr ochala and Edwar ds

© ACS Publications. From: Grochala, W., and Peter P. Edwards. Chemical Reviews 104 (2004): 1283-1315. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use / .

The Hyd r ogen Fuel Challenge

• The lo w v olumetric density of gaseous fuels r equir es a storage method which densifies the fuel.

• This is par ticularl y true f or h ydr ogen because of its lo w er energ y density r elativ e to h ydr ocarbon fuels

• 3 MJ/l (5000 psi H 2 ), 8 MJ/l (LH 2 ) vs. 32 MJ/l (gasoline)

• Storing enough h ydr ogen on v ehicles to achie v e gr eater than 300 miles driving range is difficult.

• Storage system ad ds an ad ditional w eight and v olume abo v e that of the fuel.

Ho w do w e achie v e adequate stor ed energ y in an efficient, saf e and cost-eff ectiv e system?

Ho w large of a gas tank do y ou want?

V olume Comparisons f or 4 kg V ehicular H 2 Storage

Schla pbach & Züttel, Natur e , 15 No v . 2001

Comp r essed/Liquid Hyd r ogen Storage

P ac kaging v olume and safety ar e k ey issues

Chemical H-Stor a ge

• Onl y the light elements.

• Shor t list: Li, Be , B, C , N, O , F , Na,

Mg, Al, Si, and P .

• No to xicity!

• List becomes onl y eight elements.

• Not a lot of r oom to do chemistr y!

Chemical Re vie ws, 2004, V ol. 104, No . 3, Gr ochala and Edwar ds

© ACS Publications. From: Grochala, W., and Peter P. Edwards. Chemical Reviews 104 (2004): 1283-1315. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/ .

Lots of Materials Choices

Cr ystalline Nanopor ous Materials P ol ymer Micr ospher es

Self-Assembled Nanocomposites Advanced Hydrides

Inorganic – Organic Compounds BN Nanotubes

Hydr ogenated Amorphous Carbon Mesopor ous Materials

Bulk Amorphous Materials (B AMs)

Ir on Hydr ol ysis Nanosize P o wders Metallic Hydr ogen

Hydride Alcohol ysis

Fr om Patr o vic & Millik en (2003)

Lots of Materials Choices


Formula	Formula wt.% Hydrogen
CH 4	25
H 3 BNH 3	19.5
LiBH 4	18.3
(CH 3 ) 4 NBH 4	18
NH 3	17.7
Al(BH 4 ) 3	16.8
Mg(BH 4 ) 2	14.8
LiH	12.6
CH 3 OH	12.5
H 2 O	11 . 2
LiAlH 4	10.6
NaBH 4	10.6
AlH 3	10.0
MgH 2	7.6
NaAlH 4	7.4

Example: BN Nanotubes

Figure 1 The morphologies of BN nanotubes: (a) multiwall nanotubes and (b) bamboo-like nanotubes. Scale bar: 100 nm.

Figure 2 The hydrogen adsorption as a function of pressure in multiwall BN nanotube s and bamboo nanotubes at 10 MPa is 1.8 and 2.6 wt %, respectively, in sharp contras t to the

0.2 wt % in bulk BN powder. The values reported here have an error of <0.3 wt %.

Figures removed due to copyright restrictions.

R. Ma, Y . Bando, H.Zhu, T . Sato, C. Xu, and D. W u, “Hydrogen Uptake in Boron Nitride Nanotubes at Room T emperature”, J. Am. Chem. Soc., 124 , 7672-7673 (2002).

Example: NaAl

NaAlH 4

Images of sodium alanate © Sandia/U.S. Dept. of Energy. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use / .

Sodium alanate doped with T i is a r eversible material hyd r ogen storage app r oach.

3NaAlH 4 Na 3 AlH 6 + 2Al + 3H 2 3NaH + Al + 3/2H 2

3.7 wt% 1.8 wt%

Low hydrogen capacity and slow kinetics are issues

Metal Hydrides

• Some metals absorb h ydr ogen to f orm metal h ydrides

• These r elease the h ydr ogen gas when heated at lo w pr essur e and r elativ el y high temperatur e

• Thus the metals soak up and r elease h ydr ogen li k e a sponge

• Hydr ogen becomes par t of the chemical structur e of the metal itself and ther ef or e does not r equir e high pr essur es or cr y ogenic temperatur es f or operation

BU T : ir r e v ersibility pr oblem!

Image removed due to copyright restrictions. Table 1 From: Grochala, W. , and Peter P. Edwards. Chemical Reviews 104 (2004) : 1283-315.

Chemical Re vie ws, 2004, V ol. 104, No . 3, Gr ochala and Edwar ds

The r e is n o ON E material yet.

• Ther e is, as y et, no material kno wn to meet sim ultaneousl y all of the k e y r equir ements and criteria.

• Palladium metal has long been vie w ed as an attractiv e h ydr ogen- storage medium, exhibiting r e v ersible beh a vior at quite lo w temperatur e . Ho w e v e r , its poor storage efficiency (less than 1 wt %) and the high cost of palladium ($1000 per ounce) eliminate it fr om an y r ealistic consideration

• On the other hand, the composite material “Li3Be2H7” is a highl y efficient storage medium (ca. 8.7 wt % of r e v ersibl y stor ed H), but it is highl y to xic and operates onl y at temperatur es as high as 300 °C .

The r e is no ONE material y et.

• Or ta k e AlH3: the compound is a r elativ el y lo w temperatur e (150

°C), highl y efficient (10.0 wt %) storage material and contains chea p Al metal ($1300 per tonne), but, unf or tunatel y , its h ydr ogen upta k e is almost completel y ir r e v ersible .

• Similarl y , an alkaline solution of NaBH4 in H2O constitutes a supe r - efficient storage system (9.2 wt % h ydr ogen), and full contr ol m a y be gained o v er H2 e v olution b y use of a pr oper catal yst, but the star ting material cannot be simpl y (economicall y) r egenerated.

• Finall y , pur e water contains 11.1 wt % of H, but its decomposition r equir es m uch thermal, electric , or chemical energ y .

• Recentl y advanced technolog y of h ydr ogen storage in nitrides and imides allo ws f or eff ectiv e (6.5-7.0 wt % H) but high- temperatur e (ar ound 300 °C) storage .

PERFECT Problem for Computational Quantum Mechanics!

• Hydr ogen storage: stor e h ydr ogen in a lightw eight and compact manner f or mobile a pplications.

• Bulk materials ar e often too stable .

d d

• E.g. MgH 2 : 7.7wt%, Δ H 0 = 75 kJ/mol, T ~ 300 o C

d

• Desirable Δ H 0 = 20 − 50 kJ/mol

d

• Δ H 0 can be tuned b y the size of nanopar ticles.

Image is in the public domain.

MgH 2

bulk

80

HF (Wagesmans, et al., 2005)

DFT-B97 (Wagemans, et al., 2005) DFT-PW91 (Wang & Johnson, 2008) DFT-PBE

QMC-DMC

Expt.: bulk

60

 H (kJ/mol H 2 )

40

20

0

-20

0 10 20 30 40 (MgH 2 ) N

Bulk

Wu, Allendorf, and JCG, JACS (2009)

All o ying and Nanostructuring M a y be

the K e y , but Phase Space is Enormous

© RSC Publishing . From Physical Chemistry Chemical Physics 14 (2012): 6611–16. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use / .

Wagner, Allendorf, and JCG, PCCP (2012)

F r om Atoms to Solids

?

F r om atoms to solids

Atom

Molecule

Solid

Ener gy

Antibonding p

Conduction band from antibonding p orbitals

p

Antibonding s

Conduction band from antibonding s orbitals

s

Bonding p

V alence band from p bonding orbitals

Bonding s

V alence band from s bonding orbitals

k

Image by MIT OpenCourseWare.

The gr ound state electr on configuration of a system is constructed b y putting the

a vailable electr ons, tw o at a time (Pauli principle), into the states of lo w est energ y

Energ y bands

empty

energ y ga p

occupied

Metal Insulator Semiconductor NB: bo x es = allo w ed energ y r egions

Cr ystal symmetries

A cr ystal is built up of a unit cell and

periodic r eplicas ther eof.

lattice unit cell

Image of Sketch 96 (Swans) by M.C. Escher removed due to copyright restrictions.

Cr ystal symmetries

cr ystal/solid

10 23 par ticles per cm 3

Since a cr ystal is periodic , m a ybe w e can get a w a y with modeling onl y the unit cell?

Cr ystal symmetries

S = simple

B C = body center ed F C = face center ed

Lattice a nd basis

simple cubic

basis

face center ed cubic

Image s by MIT OpenCourseWare.

The in v e r se

(or “ r ecip r ocal”) lattice

Associated with each r eal space lattice , ther e exists something w e call a r ecip r ocal lattice .

The r ecip r ocal lattice is the set of w a v e-v ectors which ar e commensurate with the r eal space lattice .

It is defined b y a set of v ectors a*, b*, and c* such that a* is perpendicular to b and c of the Br a vais lattice , and the p r oduct a* x a is 1.

The i n v erse lattice

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

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

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

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

e i G ⌅ · R ⌅ = 1

G ( → r ) = Σ c e i G ⇧ j · ⇧ r

j

automaticall y periodic in R!

The i n v erse lattice

r eal space lattice (BCC) in v erse lattice (FCC)

z

a

a 2

a 1

x

a 3

y

Image by MIT OpenCourseWare.

The Brillouin zone

in v erse lattice

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

Image by Gang65 on Wikimedia Commons. License: CC-BY-SA. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/ .

Re c i p r o c a l L a t t i c e & B r il l ouin Zone

It is defined b y a set of v ectors a*, b*, and c* such that a* is perpendicular to b and c of the Br a vais lattice , and the p r oduct a* x a is 1.

In par ticular :

a * 

b  c a  b  c

Image is in the public domain.

a * 

Brillouin

b  c

a  b  c

from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/ .

Surfaces of the first, second, and third Brillouin zones for body-centered cubic and face-centered cubic crystals. Images are in the public domain.

The Brillouin zone

Brillouin zone of the FCC lattice

P eriodic potentials

metallic sodium

V ( ⇤ r )

R ⇥ R ⇥ R ⇥

.

⇥

— n D 2 + V ( ⌃ r ) 1 = E 1

2 m

P eriodic potentials

It becomes m uch easier if y ou use the periodicity of the potential!

V ( ⌅ r ) = V ( ⌅ r + R ⌅ )

lattice v ector

Results in a VE R Y impor tant ne w concept.

Bloch ’ s Theor em

Bloch ’ s Theo r em

Recip r ocal lattice v ectors h a v e special p r oper ties of par ticular value f or calculations of solids.

W e write the r ecip r ocal lattice v ector :

G  2  n a *  2  m b *  2  o c *

W e ad ded the 2 simpl y f or con v enience , and the n, m, o , ar e integers.

No w consider the beh a vior of the function exp(iGr).

Bloch ’ s Theo r em

exp( i G  r )  exp  i ( 2  n a *  2  m b *  2  o c * )  (  a   b   c ) 

 exp  i ( 2  n   2  m   2  o  ) 

 cos( 2  n   2  m   2  o  )  i sin( 2  n   2  m   2  o  )

As r is varied, lattice v ector coefficients ( α , β , γ ) change betw een 0 and 1 and the function exp(i G · r ) changes to o .

Ho w e v e r , since n, m, and o ar e integral, exp(i G · r ) will al w a ys var y with the periodicity of the r eal-space lattice .

e i G ⌅ · R ⌅ = 1

G ( → r ) = Σ c e i G ⇧ j · ⇧ r

j

automaticall y periodic in R!

Bloch ’ s Theo r em

The periodicity of the lattice in a solid means that the values of a function (e .g., density) will be identical at equivalent points on the lattice .

The w a v efunction, on the other hand, is periodic but onl y when m ultiplied b y a phase facto r .

This is kno wn as Bloch ’ s theor em.

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

G ⌅ k

( → r ) = e i ⌅ k · ⌅ r u

( → r )

⌅ k

u ⇥ k ( ⌅ r ) = u ⇥ k ( ⌅ r + R ⌅ )

P eriodic potentials

Results of the Bloch theor em:

G ⌅ k

( → r + R → ) =

G ⌅ k

( → r ) e i ⌅ k · R ⌅

Courtesy Stanford News Service .

License CC-BY.

| G ⇥ ( → r + R → ) | 2 = | G ⇥ ( → r ) | 2

k k charge density

is lattice periodic

if solution

l ⇤ k ( ⇤ r ) — → l ⇤ k + G ⇤ ( ⇤ r )

also solution

with

E ⇤ k = E ⇤ k + G ⇤

P eriodic potentials


Schrödinger	certain	quantum
equation	symmetry	number

hydrogen spherical G

atom symmetry

[ H , L 2 ] = H L 2 — L 2 H = 0

[ H , L z ] = 0

n,l,m

( → r )

periodic translational solid symmetry

[ H , T ] = 0

G n , ⌅ k ( → r )

 k (r) = e ik . r

 = 2  /k

u(r)

a

k = 0

k =  /a

P eriodic potentials

Bloch ’ s theor em

G ⌅ k

( → r ) = e i ⌅ k · ⌅ r u

( → r )

⌅ k

u ⇥ k ( ⌅ r ) = u ⇥ k ( ⌅ r + R ⌅ )

Image by MIT OpenCourseWare.

The band structu r e

Diff er ent w a v e functions can satisfy the Bloch theor em f or the same k : eigenfunctions and eigen values labelled with k and the index n

energ y bands

The band structu r e

6

 15

0

 '

25

X

1

E C

E

V

S 1

-10

L



 





k

X

U,K





E (eV)

Silicon

energ y le v els

in the Brillouin zone

Image by MIT OpenCourseWare.

The band structu r e

E (eV)

Si l i co n

6

u  noccupied

15

0

 '

25

X

1

E C

E

V

S 1 occupied

-10

 

L







k

X

U,K





energ y le v els

in the Brillouin zone

Image by MIT OpenCourseWare.

Literatu r e

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

• Richar d M. Mar tin, Elect r onic Structur e

• wikipedia, “solid state p h ysics”, “condensed matter p h ysics”, ...

• Simple band structur e sim ulations: http:// phet.colorad o .edu/sim ulations/sims.php?

sim=Band_Structur e

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3 . 021 J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modelling and Simulation

Spring 20 1 2

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