Bonding and valence electron density
Cox Chapter 3
Bonding and crystal form
Si
NaCl
Ionic bonding
Ibach +Lueth
Covalent bonding
Metallic bonding
B. Dam
Vaste Stof fysica
kamer U044
telefoon 47917
mail [email protected]
C M S M a s t e r s : Electronic Structure of Solids.
Charge localized at atoms
Charge localized between
atoms along the bond direction
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Bonding and crystal form
C M S M a s t e r s : Electronic Structure of Solids.
Silver
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Rutile TiO2
Crystal Order
Crystalline
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Amorphous
Poly-crystalline
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
1
Quartz SiO2
Perovskite CaTiO3
C M S M a s t e r s : Electronic Structure of Solids.
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Crystal form & Crystal structure
Pyrite FeS2
18th century:
A crystal is a 3D-repitition of
identical units
Crystal form reflects the
symmetry and the shape of
its basic units
Now:
Crystal form is related to a
minimization of the surface
free energy
Crystal form minimizes the
number of unsaturated bonds
C M S M a s t e r s : Electronic Structure of Solids.
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Filling 2D space with repetitive units
Which is not a
unit cell?
What type?
2
The primitive unit-cells of BCC and FCC are non-cubic!
Simple cubic
Body centered cubic
Abundance of Space Groups / Crystal lattice
–
–
–
–
–
–
–
Face centred cubic
•Choose the highest symmetry
P21/c
P212121
P -1
P21
C2/c
Pnma
P21212
29.2%
18.8
11.1
10.9
5.4
1.5
1.0
monoclinic
orthorhombic
triclinic
monoclinic
monoclinic
orthorhombic
orthorhombic
Simple-minded phycisists prefer hexagonal and cubic structures
•Choose the smallest unit
C M S M a s t e r s : Electronic Structure of Solids.
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Three common structures/stackings
structures/stackings
FCC
BCC
The complicated crystal structure of boron
HCP
C M S M a s t e r s : Electronic Structure of Solids.
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Cubic close packed: abcabcabcabc-stacking
Chapter3: bonding
Crystal structure: abababab-stacking in hcp
c
b
a
Hexagonal lattice with two independent positions: 0,0,0 and 1/3, 2/3, 1/2
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
3
3-mm diamond in eclogite
Lattice planes
b
a
Crystal morphology
is characterized
C M S M a s t e r s : Electronic
Structure of Solids. by a bipyramid. Why??
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Lattice planes
Lattice planes
Set of planes with identical lattice geometry
(01) lattice planes
b
Set of planes with identical lattice geometry
(02) lattice planes?
b
a
a
C M S M a s t e r s : Electronic Structure of Solids.
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Lattice planes
Lattice planes
Set of planes with identical lattice geometry
Only in a centered
lattice the (02) planes
exist as lattice planes!
b
a
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Set of planes with identical lattice geometry
(11) lattice planes
b
a
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
4
Lattice planes
Crystal morphology is determined by dhkl
Set of planes with identical lattice geometry
For orthogonal crystal systems
d
hkl
1
=
h
a
2
+
2
k
b
2
2
+
l
c
2
2
The larger dhkl the more prominent the crystal orientation
(21) lattice planes
b
(21) lattice planes
b
a
a
C M S M a s t e r s : Electronic Structure of Solids.
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Diamond Structure (C, Si Ge)
Ge)
Diamond Structure (C, Si Ge)
Ge)
A
A
B
Diamond crystal structure:
Tetrahedral bonding of C , Si and Ge
Diamond crystal structure:
Tetrahedral bonding of C , Si and Ge
B
FCC with atoms on (0,0,0) and (¼,¼,¼)
FCC with atoms on A=(0,0,0) and B=(¼,¼,¼)
(111)-plane densely packed, 1 unsaturated bond per atom
(111)-plane densely packed, 1 unsaturated bond per atom
(100)-plane, 2 unsaturated bonds per atom
C M S M a s t e r s : Electronic Structure of Solids.
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
3-mm diamond in eclogite
Crystal form & Crystal structure
Simple cubic
d100 > dhkl
Body centered cubic
d110 > d200
Face centred cubic
d111> d200 > d220
The crystal morphology depends on:
•the type of Bravais lattice
•the bonding between the atoms
M S M a s t e r s : Electronic
Structure of Solids.
CrystalCmorphology
is characterized
by a (111)-bipyramid
Chapter3: bonding
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
5
Crystal morphology is determined by dhkl
Ionic bonding
For orthogonal crystal systems
d
hkl
=
What is the charge distribution
1
2
h
a
2
+
k
b
2
2
l
c
+
2
2
The larger dhkl the more prominent the crystal orientation
Where are the energy levels
What determines the gap
– More chance of bonding network within dhkl
– Less chance of unsaturated bonds
– Lower surface free energy
This very phenomenological picture holds for all types of
bonding!
C M S M a s t e r s : Electronic Structure of Solids.
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Ionisation/affinity
Ionisation/affinity
The ionic levels of Na+ and Cl- in the gas phase
Gas phase ionisation costs energy: E=I-A
Electron
affinity
Ionisation
energy
On site e-e
repulsion
Na
Na + 5 eV--> Na++e
Cl+e --> Cl- + 3.5 eV
Filled p-level ClEmpty s-level Na+
Eg = A - I = -1.5 eV
Cl
C M S M a s t e r s : Electronic Structure of Solids.
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Radial distribution function
1
H
3 4
Li Be
11 12
Na Mg
19 20 21 22
K Ca Sc Ti
37 38 39 40
Rb Sr Y Zr
55 56 57 72
Cs Ba La Hf
87 88 89
Fr Ra Ac
57 58
La Ce
5 6 7
B C N
13 14 15
Al Si P
23 24 25 26 27 28 29 30 31 32 33
V Cr Mn Fe Co Ni Cu Zn Ga Ge As
41 42 43 44 45 46 47 48 49 50 51
Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb
73 74 75 76 77 78 79 80 81 82 83
Ta W Re Os Ir Pt Au Hg Tl Pb Bi
8
O
16
S
34
Se
52
Te
84
Po
9
F
17
Cl
35
Br
53
I
85
At
59 60 61 62 63 64 65 66 67 68 69 70 71
Pr Nd Pm SmEu Gd Tb Dy Ho Er Tm Yb Lu
2
He
10
Ne
18
Ar
36
Kr
54
Xe
86
Rn
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
6
1
H
3 4
Li Be
11 12
Na Mg
19 20 21 22
K Ca Sc Ti
37 38 39 40
Rb Sr Y Zr
55 56 57 72
Cs Ba La Hf
87 88 89
Fr Ra Ac
57 58
La Ce
23 24 25 26 27 28
V Cr Mn Fe Co Ni
41 42 43 44 45 46
Nb Mo Tc Ru Rh Pd
73 74 75 76 77 78
Ta W Re Os Ir Pt
5 6 7
B C N
13 14 15
Al Si P
29 30 31 32 33
Cu Zn Ga Ge As
47 48 49 50 51
Ag Cd In Sn Sb
79 80 81 82 83
Au Hg Tl Pb Bi
8
O
16
S
34
Se
52
Te
84
Po
9
F
17
Cl
35
Br
53
I
85
At
2
He
10
Ne
18
Ar
36
Kr
54
Xe
86
Rn
59 60 61 62 63 64 65 66 67 68 69 70 71
Pr Nd PmSmEu Gd Tb Dy Ho Er Tm Yb Lu
The ionic levels of Na+ and Cl- in the gas phase
1
H
3 4
Li Be
11 12
Na Mg
19 20 21 22
K Ca Sc Ti
37 38 39 40
Rb Sr Y Zr
55 56 57 72
Cs Ba La Hf
87 88 89
Fr Ra Ac
57 58
La Ce
5 6 7
B C N
13 14 15
Al Si P
23 24 25 26 27 28 29 30 31 32 33
V Cr Mn Fe Co Ni Cu Zn Ga Ge As
41 42 43 44 45 46 47 48 49 50 51
Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb
73 74 75 76 77 78 79 80 81 82 83
Ta W Re Os Ir Pt Au Hg Tl Pb Bi
Eg = A - I+Ei = 3 eV
2
He
10
Ne
18
Ar
36
Kr
54
Xe
86
Rn
http://www.webelements.com/webel
ements/scholar/properties/imageline/
The ionic levels of NaCl:
NaCl: the Madelung contribution
Bringing the ions closer
Na+ 3s together in the lattice
results in a further splitting
of the Na+ and Cl- levels
Na+ 3s
Eg = A - I = -1.5 eV
9
F
17
Cl
35
Br
53
I
85
At
59 60 61 62 63 64 65 66 67 68 69 70 71
Pr Nd Pm SmEu Gd Tb Dy Ho Er Tm Yb Lu
Gas phase ionisation costs energy: E=I-A
Filled p-level ClEmpty s-level Na+
8
O
16
S
34
Se
52
Te
84
Po
Filled
Filled
p-level
level Cl
Cl-Empty
Emptys-level
level Na
Na++
Cl- 3p
Cl- 3p
Energy gain due to gas phase ionic interaction: 4.5 eV
Madelung energy
Sum the interaction of ion i with all
other ions j
Sum over all i
E=
±1
e2
4πε 0 a0
∑r
i, j
ij
Divide by 2
Divide by the number of ion pairs
Em = − Am
Madelung potential
Each lattice has a different
Madelung constant Am:
NaCl= 1.7475
CsCl= 1.7629
What is the effect of the
Madelung potential on the
position of the energy levels
of Na and Cl?
Eg=A-I+Em = 17 eV
e2
4πε 0 a0
Electrostatic energy per ion-pair
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
7
The ionic levels of NaCl:
NaCl: the Madelung contribution
The ionic levels of NaCl:
NaCl: the polarization contribution
Na+ 3s Madelung term:
Na+ 3s
Positive binding energy
for ion pairs!
Eg=A-I+Em = 17 eV
Filled
Filled
p-level
level Cl
Cl-Empty
Emptys-level
level Na
Na++
The ions are not in a vacuum.
Each charge polarizes its
surroundings.
Filled
Filled
p-level
level Cl
Cl-Empty
Emptys-level
level Na
Na++
Cl- 3p
Cl- 3p
The ionic levels of NaCl:
NaCl: orbital overlap
Understanding energy gaps
Madelung term largest
Na+ 3s
Ionisation/Affinity energy much smaller term
Eg = 9 eV
Filled
Filled
p-level
level Cl
Cl-Empty
Emptys-level
level Na
Na++
Polarisation and bandwidth decrease with distance
Due to overlap the levels
eV
g= 9
broaden E
into
bands
Cl- 3p
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
The gap roughly decreases with the lattice spacing
Ionisation energy difference is a measure for the gap
Ev
Correct trend when cation same
(especially for F)
With heavy anions a trend is
hardly seen
Why?
5 eV
Na
gap ~ 10 eV
15 eV Cl
Ag: Ionization potential ~2eV
larger than in Sodium
The difference in ionisation energy
appears to scale with the gap?!
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
ClDifference in
ionisation
energy is
close to the
gap energy!
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
8
Ionisation energy difference is a measure for the gap
Ev
5 eV
Na
gap ~ 10 eV
15 eV Cl
Difference in
ionisation
energy is
close to the
gap energy!
Cl- vacuum
~ The ionic affinity
in bulk appears to be
the same as the
ionisation energy in
vacuum!
Cl- bulk
Ionisation energy difference is a measure for the gap
Ev
5 eV
Na
gap ~ 10 eV
15 eV Cl
Cl- vacuum
~ The ionic affinity
Difference in
ionisation
energy is
close to the
gap energy!
in bulk appears to be
the same as the
ionisation energy in
vacuum!
Cl- bulk
e
Cl+
Cl
+
e
+
+ e
Cl (gas) <=>
C M S Cl
M a s+
ters:
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Ionisation energy difference is a measure for the gap
Ev
5 eV
Na
gap ~ 10 eV
15 eV Cl
Difference in
ionisation
energy is
close to the
gap energy!
Cl+ +
Cl (gas) <=>
C M S M a s t ee
rs:
Understanding energy gaps
in bulk appears to be
the same as the
ionisation energy in
vacuum!
Cl- bulk
Electronic Structure of Solids.
Chapter3: bonding
<=> Cl + e
Cl- vacuum
~ The ionic affinity
~ The on-site
repulsion is of the
same order as the
attraction by cations
in bulk
e
Cl+
Electronic Structure of Solids.
Chapter3: bonding
Cl-(bulk)
Cl
+
e
+
Cl-(bulk) <=> Cl + e
Understanding energy gaps: bandwidth
Ag: Ionization potential ~2eV
larger than in Sodium
The difference in ionisation energy
scales with the gap, comparing
halides
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Understanding energy gaps: Top of the valence band
Na+ 3s
Cation Conduction band
Filled level ClEmpty level Na+
Fluorine 2p valence bandwidth correlates nicely with the
F-F distance, which is a measure for the overlap of the
bonding orbitals
Anionic valence band
Cl-
3p
Cationic valence band
Na 2p
9
Understanding energy gaps: Top of the valence band
Covalent solids: from bonds to bands
Strongly directional bonds
Cu+
4s
Electron pile-up between
atoms
Cation Conduction band
How to check
this?
Filled level ClEmpty level Cu+
In group IV hybridisation:
s2p2 -> sp3
Cationic valence band Cu 3d
‘Excitation’ energy must be
compensated by bonding
energy
Anionic valence band
Cl- 3p
Covalent solids: from bonds to bands
Covalent solids: from bonds to bands
p
Top is p-like
p33
p
s
sp3
sp3
Heteropolar bonding: ionic vs covalent
s
sp3
Heteropolar bonding: ionic vs covalent
∆E = Ei2 + Ec2
Ei
∆E = Ei2 + Ec2
Ei = E A − EB
Ei
Ec = 2VAB
For solids ∆E is not the gap, but represents the energy
difference between the band midpoints
Depending on Ei the bonding electrons have ‘B’-character
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
sp3
Ei = E A − EB
Ec = 2VAB
In homo-nuclear compounds ∆E ~ Ec
The value for ∆E can be measured by PES
We would like to have a scale for Ei
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
10
Heteropolar bonding: ionic vs covalent
Heteropolar bonding
Ei = Ea − Eb
∆E = Ei2 + Ec2
Ei = X A − X B
Ei = E A − EB
Ei
Ei
X A = electroneg ativity
Ec = 2VAB
For isoelectronic solids (Ge, GaAs, ZnSe and CuBr) the
bondlength is constant and hence Ec too. The gaps differ.
For isoelectronic solids (Ge, GaAs, ZnSe and CuBr) the
bondlength is constant and hence Ec too. The gaps differ.
From the measured gap (∆E) and the extrapolated Ec (Ge=5.6)
From the measured gap (∆E) and the extrapolated Ec (Ge=5.6)
EiGe= 0.0eV
EiGe= 0.0eV
EiGaAs =1.9eV EiZnSe =3.8eV EiCuBr =5.6eV
C M S M a s t e r s : Electronic Structure of Solids.
EiGaAs =1.9eV EiZnSe =3.8eV EiCuBr =5.6eV
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Heteropolar bonding
Ei = Ea − Eb
Ei = X A − X B
Ei
X A = electroneg ativity
1
H
3 4
Li Be
11 12
Na Mg
19 20 21 22
K Ca Sc Ti
37 38 39 40
Rb Sr Y Zr
55 56 57 72
Cs Ba La Hf
87 88 89
Fr Ra Ac
57 58
La Ce
Actually XA is related to the electron affinity.
According to Mullliken X=0.185(I+A)
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
The Ei/Ec ratio predicts the crystal structure of binary
compounds
23 24 25 26 27 28
V Cr Mn Fe Co Ni
41 42 43 44 45 46
Nb Mo Tc Ru Rh Pd
73 74 75 76 77 78
Ta W Re Os Ir Pt
5 6 7
B C N
13 14 15
Al Si P
29 30 31 32 33
Cu Zn Ga Ge As
47 48 49 50 51
Ag Cd In Sn Sb
79 80 81 82 83
Au Hg Tl Pb Bi
2
He
10
Ne
18
Ar
36
Kr
54
Xe
86
Rn
59 60 61 62 63 64 65 66 67 68 69 70 71
Pr Nd PmSmEu Gd Tb Dy Ho Er Tm Yb Lu
c
c
z
y x
Ionic
b
Fi=Ei/∆E=0.785
a
RockSalt
Salt
Rock
Rock Salt
Ei
9
F
17
Cl
35
Br
53
I
85
At
The Ei/Ec ratio predicts the crystal structure of binary
compounds
Ionic
Fi=Ei/∆E=0.785
8
O
16
S
34
Se
52
Te
84
Po
Wurtzite
Ei
b
a
Covalent
Covalent
Diamond structure
Ec
Ec
11
Question 1: Mg2NiH4
Question 1: Structure of Mg2NiH4
This is the simplified Mg2NiH4 strukture
Blue balls / tetraedra = NiH4
White/Grey balls =Mg
a
a
x
z
z
y
by
x
a
a
a
x
y
z
c
b
x
c
z
y
c
b
b
c
C M S M a s t e r s : Electronic Structure of Solids.
a = 0.649
The monoclinic low temperature structure derives
from a high temperature fluorite structure.
Derive the angle between a and c and the length
of a, for the ideal case
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
Chapter3: bonding
Q3: explain
the behaviour
of the affinity
for the first
three rows of
the PS
Question 2: The hcp lattice
What c lattice constant is needed to make this a
really close packed lattice for atoms with radius ½a
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
http://www.webelements.com/webel
ements/scholar/properties/imageline/
1
H
3 4
Li Be
11 12
Na Mg
19 20 21 22
K Ca Sc Ti
37 38 39 40
Rb Sr Y Zr
55 56 57 72
Cs Ba La Hf
87 88 89
Fr Ra Ac
57 58
La Ce
23 24 25 26 27 28
V Cr Mn Fe Co Ni
41 42 43 44 45 46
Nb Mo Tc Ru Rh Pd
73 74 75 76 77 78
Ta W Re Os Ir Pt
5 6 7
B C N
13 14 15
Al Si P
29 30 31 32 33
Cu Zn Ga Ge As
47 48 49 50 51
Ag Cd In Sn Sb
79 80 81 82 83
Au Hg Tl Pb Bi
8
O
16
S
34
Se
52
Te
84
Po
9
F
17
Cl
35
Br
53
I
85
At
2
He
10
Ne
18
Ar
36
Kr
54
Xe
86
Rn
59 60 61 62 63 64 65 66 67 68 69 70 71
Pr Nd PmSmEu Gd Tb Dy Ho Er Tm Yb Lu
Radial distribution function
C M S M a s t e r s : Electronic Structure of Solids.
Chapter3: bonding
12