5/2/2014
Solid-State Band Theory
The application of quantum mechanics to extended crystals
Atoms
Solid-State
Band
Theory
Atomic Orbitals
Molecules
Molecular Orbitals
Crystals
Band States
Crystal – an extended periodic array of atoms or molecules
Lattice – the coordinate system describing the positions of
the atoms in the crystal
Chem 332
Gentry, 2013
5s
Relative Size of Atomic Orbitals
Silver (Ag): 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 4d10 5s1
4s
4p
Overlap of Atomic Orbitals in Crystal Lattice
Lower orbitals too small in size to interact
But 5s orbitals big enough that overlap with neighboring atoms
5s
4d
Size of Orbitals
3s
5s
3d
3p
1s
4
2s
1s
Ag
1s
Ag
Ag
Lattice Spacing
3
2p
1s
Ag
Size of Orbitals
2
Localized Atomic
Orbitals
1s
1s
5s
Ag+
Ag+
Ag+
Ag+
Molecular Orbital Model
5s
Use atomic orbitals as basis set to build crystal states
Ag+
Ag+
Ag+
Ag+
ψ crystal =
Therefore can get mixing from one atom to next
S =
∫φ
*
5 s, atom1
∑
ci ⋅ φ 5 s,i
i = atoms
Overlap exists between neighboring 5s orbitals
ψ4
Ag+
Ag+
Ag+
Ag+
ψ3
Ag+
Ag+
Ag+
Ag+
ψ2
Ag+
Ag+
Ag+
Ag+
ψ1
Ag+
Ag+
Ag+
Ag+
⋅ φ5 s, atom 2dτ ≠ 0
Approach #1: Molecular Orbital Model
Use atomic orbitals as basis set to build crystal states
ψ crystal =
∑
ci ⋅ φ5s,i
i = atoms
Approach #2: Free Electron Gas Model
Use particle-in-a-box model as basis set for crystal states
ψ crystal =
1
 nπ 
⋅ sin 
⋅x
2π
 L

The Problem: very cumbersome since so
many atoms in a sizeable crystal
1
5/2/2014
Free Electron Gas
Use particle-in-a-box model to build states that span crystal
ψ crystal =
1
 nπ 
⋅ sin 
⋅ x
2π
 L

Free Electron Gas
Use particle-in-a-box model to build states that span crystal
ψ4
ψ4
Ag+
Ag+
Ag+
Ag+
ψ3
Ag+
Ag+
Ag+
Ag+
Ag+
Ag+
Ag+
Ag+
Standard particle in a box
ψ2
Ag+
Ag+
Ag+
 nπ 
ψ crystal = sin 
⋅x
 L

En =
ψ1
Ag+
k =
nπ
L
Ag+
2
Ag+
ψ crystal = sin ( k x )
… or …
Ag+
Ag+
•
•
- Not exact since ignores local Ag+ attractions
- But better starting point for describing states
that span large crystal
2
L = length of crystal
h n
8me L2
n = 1→ ∞
Ignores the atomic interactions that form the crystal
But is a good starting point to form new band states
Not-So-Free Electron Gas
MO Energy Calculations
Combine Free Electron Gas with Molecular Orbital Theory
Extended Overlap and Mixing of 5s1 orbitals
ψ4
Ag+
Ag+
Ag+
5s
Ag+
• Shape of wave functions still described by free e‒ states
k = nπ
L
ψ crystal = sin ( k x )
1s
1s
Ag
1s
Ag
1s
Ag
Ag
… But only N number of band states allowed
N = number of atomic orbitals being mixed together
(i.e. number of atoms)
… And energies of crystal states based on MO theory
Form a “band” of possible energy states
Equal energy gaps rather than free p.-in-a-box (Ep.box ~ n2)
α = coulomb energy
energy of electron staying on one atom but close to another
β = resonance energy
energy of electron swapping between neighboring atoms
S = overlap integral
Hückle Model for Conjugated π System
2
atoms
∆E = 2 β
∆E = 4 β
∆E = 3.34 β
4
atoms
Silver Crystal Lattice
∆E = 4 β
N
atoms
N
atoms
•
Core electrons localized on atoms
•
Mix 5s1 atomic orbitals across crystal …
•
Generates band of lattice states
- approximated by sine waves
- have increasing # of nodes
- extend across the lattice
•
Due to very large N, energy gaps are
very small relative to thermal energy
2 MO’s
4 MO’s
ethene
1,3-butadiene
N MO’s
conjugated polyene
N MO’s
As number of atoms increases,
- get more molecular orbitals ( = # of atoms)
- energy gap between each level gets smaller
- total spread of energies increases, but reaches 4β limit
* energies assume linear chain
2
5/2/2014
Other Atomic Bands
Fermi Level and Fermi Energy
Silver (Ag): [Kr] 5s1 4d10
Band States
→
5p
Energy
→
5p
Conduction
Band
Band Gap
Region with no states
5s
5s
Band Overlap
4d
Overlapping bands
create one large set
of continuous states
4d
Energy
Atomic Orbitals
(first empty band)
“Fermi Level”
= highest occupied
band state
Valence
Band
“Fermi Energy”
(uppermost
occupied band)
= energy of electrons
at Fermi Level
[Kr]
Can get additional splitting of bands due
to atomic term states and lattice scattering
Electrical Conductivity
Electrical Conductivity
Conductor
Semiconductor
Insulator
Why are Fermi Level and Band Gap Important?
‒
← e‒ flow
e‒ →
Ag+
e‒ Ag+
e‒
e‒
+
Ag+
← e‒ flow
e‒ →
+
e‒ Ag
Fermi
Must ADD electron at one end, REMOVE an electron at other end
Adding electron requires putting new electron in empty state
Level
Fermi level in
middle of
valence band
Fermi level at top of
valence band
Fermi level at top of
valence band
Band gap “small”
vs. thermal energy
Band gap large vs.
thermal energy
Can easily
add electron
Fermi
level
Gap = 40 * kT at 298K
for silicon
Electrical Conductivity of Sodium
Na
Unable to add
electron since
no state within
thermal energy
Conductor
Insulator
Electrical Conductivity of Magnesium
Mg
3p
3p
Fermi
3s
atomic states
Fermi
Level
Level
3s
3s
band state
Sodium (Na) is a metallic conductor
e‒
‒ Na contributes one atomic 3s per atom
‒ and each band state can take two e‒
‒ therefore only half fill the 3s band
atomic states
3p
band states
3s
Magnesium (Mg) would be electrical insulator if only
considered 3s band
‒ Mg contributes two atomic 3s e‒ per atom
‒ and each band state can only take two e‒
‒ this would completely fill the 3s band
BUT because of band overlap, 3p also contributes to
overall valence band
3
5/2/2014
Diode Junction
Doped Silicon Semiconductors
undoped
material
“p-type”
material
“n-type”
material
(Si)n
(Si)n-1 + B
(Si)n-1 + P
A conducting device that allows e‒ ‘s to flow only in 1 direction
‒
e‒
→
n-type
Fermi
n-type
Level
1) Inject
electrons on
n side
Fermi level at top of
valence band
Si bands arise from
P1/2 and P3/2 term
states in atom
Boron has 1 less e‒
Creates a positive (p)
“hole” in the valence
band
Phosphorous has 1
extra e‒
Places a negative (n)
electron in the
conduction band
+
e‒ →
p-type
p-type
(Si)n-1 + P (Si)n-1 + B
2) Electrons
“fall downhill”
from n to p
3) Remove
electrons from
p side
e‒ flow →
NPN Transistor
Collector
Device controls main flow of e‒ ‘s by
use of small control voltage
‒
Base
Quantum Dots
Emitter
Band widths and band gap
depend on
+
size of solid
e‒
→
n-type
p-type
emitter
base
n-type
e‒ →
(fewer atoms = fewer states)
collector
control bias voltage
e‒ ‘s normally allowed to flow from n→p, but not p→n
- Electrons will flow from emitter to base
- BUT current will not flow from base to collector, UNLESS…
Can switch on the base / collector junction by applying
bias voltage to the base
relaxation
UV
excitation
Optical
emission
CdSe
red
10+ nm
yellow
6 nm
blue
3 nm
← Size of Solid Particle
4