establishing common language crystal structure and binding
TRANSCRIPT
Organic ElectronicsStephen R. Forrest
Establishing Common LanguageCrystal Structure and Binding
Chapter 1.4, 2.1-2.4
Week 1-2
Organic ElectronicsStephen R. Forrest
Objectives: Structure of Organic Solids
• Introduce basic terminology of organic materials• Discuss relationship of crystal structure to properties• Introduce the basic terminology and concepts
of crystals and crystal lattices- Fill in the gaps
• Discuss crystal binding- Physical properties and constants
• Learn about organic lattices and their equilibrium structures- Energy minimization
• Growth and epitaxy
Organic ElectronicsStephen R. Forrest
First: Establishing a vocabulary
(Without a common language, there can be no common understanding)
• Illustration of molecular structure• Basic molecular units and structures• Standard terminologies
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Organic ElectronicsStephen R. Forrest
Benzene, phenyl, aryl
• Benzene: C6H6
• Phenyl: the phenyl group or phenyl ring is a cyclic group of atoms with the formula C6H5. Phenyl groups are closely related to benzene.
• Aryl: A functional group of the form C6H5 attached to a molecule
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Organic ElectronicsStephen R. Forrest
• Alkanes are the simplest organic molecules, consisting of only carbon and hydrogen and with only single bonds between carbon atoms. Alkanes are the basis for naming the majority of organic compounds (their nomenclature). Alkanes have the general formula CnH2n+2.
• Polyacenes: The acenes or polyacenes are a class of organic compounds, and polycyclic aromatic hydrocarbons made up of linearly fused benzene rings.
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Organic ElectronicsStephen R. Forrest
• Aromaticity: materials consisting of closed carbon rings where the C bonds are in resonance.
- Double-single bond structure of the molecule allows the bond arrangement to alternate between C atom pairs in the ring.
- To be aromatic the molecule must be planar and have an ODD number of π electron pairs. (see anaromatic)
• Polycyclic Aromatic Hydrocarbon (PAH): organic compounds containing only carbon and hydrogen—that are composed of multiple aromatic rings (organic rings in which the electrons are delocalized)
• Radical: A charged molecule that is either anionic or cationic. A cation is a positively charged molecule, and an anion is negatively charged. An excess electron is typically denoted by a solid dot.
Pentacene anion
Aromatics and Radicals
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Organic ElectronicsStephen R. Forrest
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JP¤
π-system
Electron density
Anthracene: A classic aromatic molecule
• C14H10• It is a PAH with form C4n+2H2n+4• n=number of rings
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Organic ElectronicsStephen R. Forrest
Conjugation
• a conjugated system is a system of connected p-orbitals with delocalized electrons in a molecule
• Conjugation lowers the overall energy of the molecule and increases stability.
• Conjugation is conventionally represented as having alternating single and multiple bonds.
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Organic ElectronicsStephen R. Forrest
A Few Polyacenes
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Benzene
Naphthalene
Anthracene
Tetracene
Pentacene
Hexacene
Pyrene
OvaleneAnthanthrene
BenzoperylenePerylene
Coronene
Organic ElectronicsStephen R. Forrest
Monomer, Oligomer• Monomer: a molecular complex with a well-defined molecular weight that
consists of a single irreducible unit
• Oligomer: a molecular complex that consists of a few monomer units, where the number of monomers is, in principle, not limited but is always well defined. (e.g. dimer, trimer, tetramer, etc.)
Pt N N
N N
n
b.#a.# c.#
Pt N N
N N
n
b.#a.# c.#
mer (Greek: meros)-part
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Organic ElectronicsStephen R. Forrest
thiophene
dibenzofuranindolizine
oxazole thiazole
imidazole isoindole
Heterocycles
Metalorganics
Ir(ppy)3 Ir(ppy)2acac
Ligand
PtOEP CuPc
Porphyrins Phthalocyanines
ligand: Latin: can be tied
Organic ElectronicsStephen R. Forrest
Dendrimer, Polymer
(b)$(a)$
Pt N N
N N
n
b.#a.# c.#
• Dendrimer: An oligomer that is built with repeat units radiating (like dendrites) from a central core. Each concentric repeat unit is a generation. (a) 54 ferrocene (so named due to the 54 ferrocene groups on its periphery), and (b) a 4th generation dendrimer.
• Polymer: a molecular complex consisting of an undefined number of monomeric repeat units and hence undefined molecular weight.
12
Organic ElectronicsStephen R. Forrest
At what point does the molecular picture give way to the solid?
Eg#
BW# Conduc,on##Band#
Valence#Band#
DOS#
Energy#
(a)#
Eg#HOMO#
Energy#
HOMO:1#HOMO:2#
• ##• ##• ##• ##
• ##• ##• ##• ##
• ##• ##• ##• ##
LUMO+2#LUMO+1#LUMO#
• ##• ##• ##• ##
(b)#
DOS#Eg#
BW# Conduc,on##Band#
Valence#Band#
DOS#
Energy#
(a)#
Eg#HOMO#
Energy#
HOMO:1#HOMO:2#
• ##• ##• ##• ##
• ##• ##• ##• ##
• ##• ##• ##• ##
LUMO+2#LUMO+1#LUMO#
• ##• ##• ##• ##
(b)#
DOS#Eg#
BW# Conduc,on##Band#
Valence#Band#
DOS#
Energy#
(a)#
Eg#HOMO#
Energy#
HOMO:1#HOMO:2#
• ##• ##• ##• ##
• ##• ##• ##• ##
• ##• ##• ##• ##
LUMO+2#LUMO+1#LUMO#
• ##• ##• ##• ##
(b)#
DOS# ? ?
Organic ElectronicsStephen R. Forrest
Crystal Morphologies
14
• Electrical conduction– Range from conducting to insulating
• Solid state: not liquids or gases• Organics found and exploited in all morphological types
Structure determines electronic properties
Types of solids
CrystallineAmorphous Polycrystalline
Organic ElectronicsStephen R. Forrest
Crystal Structure
15
Lattice
Unit cell
Periodic arrangement of atoms in a crystal
Small volume of crystal that may be used to reproduce the entire crystal—space filling
Primitive unit cell
Smallest unit cell that describes the crystal Body centered
cubic (BCC)
Simple cubic
lattice constant
a
Lattice constant is a material parameter
Organic ElectronicsStephen R. Forrest
Lattices
a"b" R=la+mb$
Translation vector: R=la+mb+nc
A translation vector moves a R point to an equivalent point in the lattice
Volume: VCell = a • (b× c)
Organic ElectronicsStephen R. Forrest
Common Semiconductor Crystal Structures
17
Diamond Zinc Blende
÷øö
çèæ 0,
2,2aa
÷øö
çèæ
4,4,4
aaa
(Si, Ge) (GaAs, InP, AlSb, …)
Ga
As
Organic ElectronicsStephen R. Forrest
Bravais Lattices• These lattices define the crystal structure
!
Lattice System Bravais Lattice
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Rhombohedral
Hexagonal
(pcc) (bcc) (fcc)
Cubic
!
Almost all organics fallInto these lattice types
Can you think of a moleculeThat fits this structure?
coordination #=12 (fcc)
Organic ElectronicsStephen R. Forrest
Reciprocal Lattice
• Condition of self transformation: ψ(r)=ψ(r+R) ~
• Then G is the reciprocal lattice vector:
• The reciprocal lattice defined by G, has an identical symmetry to the physical lattice defined by R.
• It is then straightforward to show that the primitive reciprocal lattice vectors are defined by the following relationships:
• What is the relationship between the unit cell volume in reciprocal to real space?
eiGir = eiGi(R+r )
G iR = 2π
a = 2π b× c
VCell b = 2π c × a
VCell c = 2π a × b
VCell
Organic ElectronicsStephen R. Forrest
Miller Indices
20
Need a method to describe crystalline direction/planes
plane ( h k l )
direction [ h k l ]
h k l are Miller indices defining the plane or direction
Organic ElectronicsStephen R. Forrest
Defining Crystal Directions and Planes
• Miller indicies
• Miller indices: h, k, l, are the lowest integers that are the inverse of the intercepts between the plane and the axes
• (a,b,c) define plane• {a,b,c} define set of equivalent planes (e.g. (100), (010), (001), etc. for cubic lattice)• [a,b,c] for lattice direction• <a,b,c> for set of equivalent lattice directions
(100)% (101)% (212)%
Organic ElectronicsStephen R. Forrest
Determining Miller Indices
22
1. Find intercepts (as multiple of a lattice constant)
2. Take reciprocal
3. Multiply by lowest common denominator
x
z
y
intercepts at (1, 2, ∞)
reciprocal (1, 1/2, 0)
LCM (2, 1, 0)
(210) plane
Vector perpendicular to plane is [210] direction
Organic ElectronicsStephen R. Forrest
Example: Miller Index
23
Determine the representation of the plane below
a
a
a2/3 a
x
y
z
Organic ElectronicsStephen R. Forrest
Ionic Bonds
For a solid, pairwise ionic interactions must be summed over all N ions, which leads to a net attractive energy of:
Uij (r) = ± q2
4πε0 ri − rj
Uattract (r) =q2
4πε01Rij
− 1Rij − a
⎛
⎝⎜
⎞
⎠⎟
i, j
N
∑
Similar atoms
Oppositely charged atoms
δ- δ+
NaCl (fcc)TTF-TCNQ: charge transfer complex
α fcc = 6 − 122+ 8
3− 62+ ...⎛
⎝⎜⎞⎠⎟= 1.7476
Utot (r) =σrm
− αq2
4πε01r
Uattract
Madelung Constant ⇒
Organic ElectronicsStephen R. Forrest
Equilibrium Crystal Structure
∂Utot
∂r r=r0
= 0
!1#
!0.5#
0#
0.5#
1#
0# 1# 2# 3# 4# 5# 6# 7#
Poten&al)of)an)Ionic)Solid)
r/r0$Utot(r)/Utot(r
0)$
Repulsion##A6rac:on#
Utot (r0 ) = − 1σ
αq2
4πε0m⎛⎝⎜
⎞⎠⎟
m⎡
⎣⎢⎢
⎤
⎦⎥⎥
1m−1
(1−m) = αq2
4πε0m⎛⎝⎜
⎞⎠⎟1r0(1−m)
k = ∂2U(r)∂r2 r=r0
= αq2
2πε0m⎛⎝⎜
⎞⎠⎟1r03 (1−m)
Force constant
r0 =mσαq2
4πε0⎡⎣⎢
⎤⎦⎥
1/(m−1)
≈ mσαq2
4πε0⎡⎣⎢
⎤⎦⎥
1/m
Utot (r) =σrm
− αq2
4πε01r
Equilibrium condition
Organic ElectronicsStephen R. Forrest
Covalent BondingShared electron systems between ionic cores
H2, Si, Ge, C….
Organic Molecules with C-C bondse.g. Anthracene
H = −
2
2m∇r2 − 2
2Mi
∇Ri
2 +V (r,Ri )i∑
HΨ(r,R) = ΕΨ(r,R)
V (r,R1,R2 ) = − q2
4πε01
r −R1
+ 1r −R2
− 1R1 −R2
⎛
⎝⎜⎞
⎠⎟
It’s complicated as N increases!
H2+
Organic ElectronicsStephen R. Forrest
Solutions to the H2+ Molecule
anti-bonding orbital
bonding orbital
- Two solutions exist
electron density concentrated between ionic cores
electron density concentrated repelled
Organic ElectronicsStephen R. Forrest
van der Waals bonding• Purely electrostatic instantaneous induced dipole-induced dipole interaction
between π-systems of nearby molecules.
A"
B"
A"
B"
t=t0" t>t0"
Φ(r) = 1
4πε0ε r
qr+ p i r
r3 + 12
Qij
rirj
r5i, j∑ + ⋅⋅⋅
⎡
⎣⎢⎢
⎤
⎦⎥⎥
Multipole potential is from expansion
Monopole dipole quadropole
Medium around the dipole is polarized
p = rρ(r)d 3r∫ : Dipole moment
E(r) = 3n̂(p i n̂)− p
4πε0ε rr3
: Field from dipole moment
Er =2pcosθ4πε0ε rr
3
Eθ =psinθ4πε0ε rr
3
Eφ = 0
Organic ElectronicsStephen R. Forrest
Dipole interactions
U = −p i E(r)r1"
p2" r12n̂12
p1"
r2"
O$
Fixed dipoles
Applying Boltzmann statistics to the energy as a function of angle, it can be shown:
For p1=p2
Keesom interaction:
Important relationships:
U = − 12kx2 (in one dimension for an SHO)
⇒ ′U = F = −kx⇒ ′′U = −k⇒Bulk modulus = B =Vol ′′U( ) = −Vol k( )
= force constant = compressibility of the solid
U(r12 ) = − 2p4
3 4πε0ε r( )kTr6 = − ADD
r6
E(r) = 3n̂(p i n̂)− p
4πε0ε rr3
U r12( ) = p1 i p2 − 3 n̂12 i p1( ) n̂12 i p2( )
4πε0ε rr123 = p2
4πε0ε r
p̂1 i p̂2
r123 −
3 r12 i p̂1( ) r12 i p̂2( )r12
5
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
Organic ElectronicsStephen R. Forrest
Van der Waals interaction
Induced dipoles depend on the polarizability (α) of the molecule (which may different from
the medium, εr: pind (r) =αE(r)
θ" n̂12
r12$
U (r12 ) = −
Adisp
r126
: Dispersion interaction
U(r) = 4ε σr
⎛⎝⎜
⎞⎠⎟12
− σr
⎛⎝⎜
⎞⎠⎟6⎡
⎣⎢
⎤
⎦⎥ : Lennard-Jones 6-12 potential
For van der Waals, the induced dipoles are parallel (p̂1 = p̂2 )
Then our energy equation reduces to:U == p1p24πε0ε rr12
3 1− 3cosθ{ } ≈ − p1p22πε0ε rr12
3
The field from one dipole at the other: E r( ) = 3n̂ p i n̂( )− p
4πε0ε rr123 ≈ p1
2πε0ε rr123
⇒ p2 =2α p1
4πε0ε rr123
From which we get the “London interaction energy”: UvdW = − α p12
πε0ε r( )r126
Organic ElectronicsStephen R. Forrest
van der Waals Coefficients Between atoms
-1
-0.5
0
0.5
1
0 1 2 3 4 5 6 7
U(r)/U(r0)
r/r0
1/r
1/r6
Atoms i-j r0,ij
Å
ε ij
kcal mol-1 C-C 4.00 0.150 C-N 3.75 0.155 C-O 3.60 0.173 C-S 4.00 0.173 C-H 3.00 0.055 N-C 3.75 0.155 N-N 3.50 0.160 N-O 3.35 0.179 N-S 3.75 0.179 N-H 2.75 0.057 O-C 3.60 0.173 O-N 3.35 0.179 O-O 3.20 0.200 O-S 3.60 0.200 O-H 2.60 0.063 S-C 4.00 0.173 S-N 3.75 0.179 S-O 3.60 0.200 S-S 4.00 0.200 S-H 3.00 0.063 H-C 3.00 0.055 H-N 2.75 0.057 H-O 2.60 0.063 H-S 3.00 0.063 H-H 2.00 0.020
!
Ucrystal =12
U(Rij )i≠ j∑ : Equilibrium crystal structure found by calculating and then minimizing all
atom-atom potentials over N atoms in molecules in solid- Local vs. global minima?- Huge numbers of degrees of freedom (6 per molecule!)- Thermodynamics important (different structures with different kBT)
PkT
= NV
+ B2 T( ) NV
⎛⎝⎜
⎞⎠⎟2
+ B3 T( ) NV
⎛⎝⎜
⎞⎠⎟3
+ .....
Virial coeff’ts from vdWinteractions
• Deviations from ideal gas law
Coulomb=long range
vdW=short range
Organic ElectronicsStephen R. Forrest
Hydrogen bonds
• Directional• Coulombic
UH (r) = −δ + pcosθr2
O-H bond: 1ÅO…H bond: 1.6 -1.8 Å
O − H ⋅⋅⋅O : Must be linear otherwise O-O repulsion dominates
H⋅⋅⋅O and H⋅⋅⋅N are usually only important
Precise form of potential (Coulombic, exponential) usually not critical