Molecules to solid state materials

Today:

Bonding in molecules

Quantum theory of solids

FINAL EXAM is Monday, Dec. 15 10:30A-1P HERE

Duane G1B20.

EXTRA CREDIT HWK 14: Practice questions for the

Final Exam. Available on the website. Due Mon. 10AM

Please fill out the online participation survey. Worth

10points on HWK 13.

Electronic structure of atom determines its form

(metal, semi-metal, non-metal):

- related to electrons in outermost shell

- how these atoms bond to each other

Semiconductors

Chemical Bonding- Main ideas:

1. involves outermost electrons and their wave

functions

2. interference of wave functions

(one wave function from each atom) that produces

situation where atoms want to stick together.

3. degree of sharing of an electron across 2 or more

atoms determines the type of bond

Ionic or Inert MetallicCovalent

electron completely

transferred from one atom to

the other, or not at all.

electron equally shared

between adjacent atoms

electron shared between

all atoms

in solid

Degree of sharing of electron

Li+ F- or Helium H2 Solid Copper

Ionic Bond (NaCl)Na (outer shell 3s1) Cl (outer shell 3s23p5) Has one weakly bound

electron

Low ionization energy

Needs one electron to fill

shell

Strong electron affinity

Na+ Cl-

Attracted by coulomb

attractionCoulomb attraction

Energ

y

Separation

of ions

V(r)

Na+Cl-

Repulsion of electrons

Na+ Cl-

Covalent BondSharing of an electron… look at example H2

+

(2 protons (H nuclei), 1 electron)

Proton 1 Proton 2

1

Wave function if electron

bound to proton 1

Protons far apart …

Potential energy curve

Covalent BondSharing of an electron… look at example H2

+

(2 protons (H nuclei), 1 electron)

Proton 1 Proton 2

Proton 1 Proton 2

1

2

Wave function if electron

bound to proton 1

Protons far apart …

Wave function if electron

bound to proton 2

Covalent BondSharing of an electron… look at example H2

+

(2 protons (H nuclei), 1 electron)

1 2

If 1 and 2 are both valid solutions,

then any combination is also valid solution.

Subtract solutions

(antisymmetric):

- = 1-2

(molecular orbitals)

Add solutions

(symmetric):

+ = 1 + 2 and

-2

+ = 1 + 2

- = 1-2

Look at what happens to these wave functions as you

bring the protons closer…

Visualize how electron cloud is distributed… for

which wave function would this cloud distribution

tend to keep protons together? (bind atoms?) …

what is your reasoning?

a. S or +

b. A or -

Look at what happens to these wave functions as you

bring the protons closer…

+ puts electron density

between protons .. glues

together protons.

- … no electron density

between protons … protons

repel (less / not stable)

Bonding Orbital Antibonding Orbital

Energ

y (

mo

lecu

le)

Separation of protons

V(r)

Energy of + as distance decreases

(more of electron cloud between them)

Energy of - as distance decreases

1 2 (molecular orbitals)

-2

+ = 1 + 2

- = 1-2

Smaller proton-proton

repulsion.

Smaller electron KE.

Larger proton-proton

repulsion.

Larger electron KE.

Quantum Bound State Sim

What would you expect for

two square wells?For two atoms?

Now FIX the protons: what does the electron energy look like

“Degenerate”

energy levels

Energy levels

split apart

If protons far away, symmetric and antisymmetric state both have same energy as

ground state of electron bound to single proton:

Eatom

As protons get closer together, symmetric and antisymmetric state become more

distinct and energy levels split:

Eatom +

Eatom –

As separation decreases, energy

splitting increases

Same idea with p-orbital bonding … need constructive

interference of wave functions between 2 nuclei.

Sign of wave function matters!

Determines how wave functions interfere.

Why doesn’t He-He molecule form?

Not exact same molecular orbitals as H2+, but similar.

With He2, have 4 electrons …

fill both bonding and anti-bonding orbitals. Not stable.

So doesn’t form.

Now almost infinite power!

Know how to predict everything about behavior of atoms and electrons or

anything made out of them:

1. Write down all contributions to potential energy,

includes e-e, nuc.-nuc., nuc.-e for all electrons and nuclei.

q1q2/r1-2 + q2q3/r1-3 + qnuc1qnuc2/rqnuc1-qnuc2 +q1qnuc1/r1-nuc1 +

one spin up and one down electron per state req....

(plus little terms involving spin, magnetism, applied voltage)

2. Plug potential energy into Schrod. eq., add boundary. cond.

3. Solve for wave function elec1,(r1, r2, rnuc1, ...)elec2,

nuc1,

nuc2, ...

get energy levels

for system

calculate/predict everything there is to know!!

why "almost"...one little problem...

Big Picture.

almost

• With three objects (1 nuclei + 2 electrons) solving

eq. very hard.

• Gets much harder with each increment in number of

electrons and nuclei !!

Don’t need to always solve S. E. exactly--

Use various models and approximations.

Not perfect but very useful, tell a lot.

(lots of room for cleverness, creativity, intuition)

Limitations of Schrodinger

1. Energy levels and spacings in atoms molecules solids

2. How energy levels determine how electrons move.

Insulators, conductors, semiconductors.

3. Using this physics for nifty stuff like copying machines,

diodes and transistors (all electronics), light-emitting diodes.

What happens to energy levels as we

put a bunch of atoms together?

How does atom-atom interaction lead to band structure?

Quantum Bound State Sim

What would you expect

for two square wells?For two atoms?

Now FIX the protons: what does the electron energy look like

Bound State Sim.. Many Wells

countless levels smeared together, individual levels

indistinguishable. "bands" of levels. Each level filled with 2

electrons until run out.

1

2

3

In solid, `1022 atoms/cm3, many!! electrons, and levels

En

erg

y

atom level bands

more atoms

“band gap” ~ few eV

“valence

band”

“conduction

band”empty

filled with electrons

filled with electrons

empty

Which band structure goes with

which material?empty

full

1. Diamond 2. copper 3. germanium (poor conductor)

Energ

y

only top 2 filled and lowest 2 empty bands shown

x y zelement w

a. 1=w, 2=x, 3=y b. 1=z, 2=w, 3=y c. 1=z, 2=y, 3=x

d. 1=y, 2= w, 3=y. e. 1=w, 2=x, 3=y

25 eV

0

And so much more…

• Quantum 1, 2, and electives!

• Quantum theory of statistical mechanics!

• Relativistic quantum field theory!

Thanks for a great semester!

Lecture ended here.

QM of electrical conductionenergy levels of atoms molecules solids

at 1

Energ

y

at 2at 3

at 4

many levels!

top energy wave functions spread waaaay out

QM of electrical conduction

energy levels of atoms molecules solids

at1-at2 molec

inner electrons stick close

to nuclei. Outer e’s get

shared.

at 1

Energ

y

at 2

multielectron atoms

Quantum Mechanics to understand (predict, control, etc.)

flow of electricity through materials.

insulators, conductors,

QM control current flow in semiconductors

results: transistors, cell phones, iPods,…

The foundation of modern technology

Where to start in understanding flow of electrons in object

at QM level?

V

V

What is important for flow of current from QM perspective?

a. electrons move through material as classical particles, so QM effects are only a

minor effect.

b. spacing of electron energy levels is important because big spacing between levels

means electrons can move easily.

c. spacing of electron energy levels is important because small spacing between

levels means electrons can move easily.

d. QM is important because the shape of the wave function determines the direction

in which electron can move.

e. some other QM effectsmall compared to what?

Nanotechnology: how small does a wire have to be

before movement of electrons starts to depend on size

and shape due to quantum effects?

How to start?

Need to look at

Energy level spacing compared to thermal energy, kT.

Almost always focus on energies in QM.

Electrons, atoms, etc. hopping around with random energy kT.

Larger than spacing, spacing irrelevant. Smaller, spacing big deal.

So need to calculate energy levels.

pit depth compared

to kT?

from class 20 months ago : )

Energ

y Separation of protons

V(r)

Energy of + as distance decreases

(more of electron cloud between them)

Energy of - as distance decreases

1 2(molecular orbitals)

-2

+ = 1 + 2

- = 1-2

Potential energy of electron due to two protons:

Potential energy of electron due to single proton: V = -ke2/r

Ground state wave function of

electron in this potential:

(r) ~ e-r

Eatom

+ =

Ground state wave function of electron (symmetric/bonding):

1st excited state wave function (antisymmetric/antibonding):

+

+ =

=

For every energy level for 1 proton, 2 energy levels for 2 protons.

Look at what happens to these wave functions as bring protons closer…

+ puts electron density between protons ..

glues together protons.- … no electron density between protons

… protons repel (not stable)

Bonding Orbital Antibonding Orbital