lectures 21-22 solid state materials. electronic structure and conductivity 1) band theory the...
Post on 18-Dec-2015
219 views
TRANSCRIPT
![Page 1: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/1.jpg)
Lectures 21-22
Solid state materials. Electronic structure and conductivity 1) Band theory
• The electronic structure of solids can also be described by MO theory. • A solid can be considered as a supermolecule.
• One mole of atoms (NA), each with X orbitals in the valence shell contributes X moles of atomic orbitals producing X moles of MO’s.
Consider qualitatively bonding between N metal atoms of ns1 configuration (Li, Na etc) arranged in a chain; N = 2, 4, NA. Assume that X=1 for simplicity.
• In the case of N ~ NA atoms they form not bonds but bands.
• The band appearing in the bonding region is called valence band. The antibonding region is called conduction band.
• In the case of metals the valence and conduction bands are immediately adjacent.
N = 2 Li atoms NA Li atoms
F
valence band
conduction band
Fermi level
4 Li atoms
![Page 2: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/2.jpg)
2) Band theory. Insulators, semiconductors, conductors
• If we apply now an electrostatic potential to a conductor, the population of the energy levels will tend to change and electrons will be able to flow using empty adjacent conduction band.
• In the case of insulators and semiconductors, the energy gap between the valence and conduction bands is more or less significant; electrons cannot easily get into the conduction band and cannot move along the sample; thermal or photo-energy is needed to bring some electrons to the conduction band.
Bandgap,
eV
Conductivity, -1cm-1
C 6.02 < 10-18
Si 1.09 5·10-6
Ge 0.72 0.02
Sn 0.07 104
F
no potentialnegative potential positive potential
F
Bandgap
Insulators
Semiconductors
F
Bandgap
T (intrinsic conductivity)
For h (photoconductivity)
![Page 3: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/3.jpg)
3) Crystal Orbital theory
• The band structure of a crystalline material of virtually any complexity can be found through the application of the MO theory for solid state materials (Crystal Orbital theory).
• One of the ways to model a real (finite size) crystal is by using cyclic boundary conditions assuming that a chain of bound atoms forms a very large ring.
• It turns out that the energy levels in a cyclic molecule composed of N hydrogen atoms look as shown below.
N = 3 N = 4 N = 5 N = 6 N = 7
energy level of isolated s-orbitals
EEnergy levels of resulting MO's are indicated with
![Page 4: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/4.jpg)
4) Crystal orbitals (Bloch functions)
• If we have N hydrogen atoms with atomic wave functions m (m = 1 … N) related by symmetry and spaced at distance a, we can get N MO’s n (n = -N/2, …, 0, …, N/2) which are called Bloch functions.
• For the n-th crystal orbital, n, we will have:•
• When n changes from 0 to N/2, variable k = 2n/(aN)) (wave vector) changes from 0 to /a and the type of the MO changes from the completely bonding 0 to the completely antibonding N/2:
• Energy levels of the resulting set of MO’s (band structure) can be described with help of continuous functions E and density of states dn/dE (DOS)
N
mmn m
N
nim
N
n
1
)2
sin()2
cos(
N/2 = -1+2-3+...
k=/a
E, eV
0
-10
-20
a = 3 A a = 2 A a = 1 A
k
0 0 0/a /a /a
k k DOS
E, eV
a = 1 A
a k=0
0 = 1+2+3+...
![Page 5: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/5.jpg)
5) Bonding in solids: Crystal Orbital Overlap Population
• A common way to analyze bonding in solids is by calculating and analyzing the crystal orbital overlap population (COOP).
• COOP is defined in the same way as the bond order is defined in MO theory of molecules.
• For any two atoms i and j COOP(i-j) = 2cicjSij (Sij is the overlap integral for two atomic wavefunctions; summation should be performed for all pairs of overlapping orbitals of atoms i and j). A negative value of COOP means antibonding situation while a positive value is characteristic for bonding.
• For the chain of hydrogen atoms the lower half of the band is bonding while the upper half is antibonding (see diagram on the right).
COOP
E, eV
0
10
20
-10
-20
0 /a
k
E
0
a k=0
k=/a
Sij < 0
Sij > 0
![Page 6: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/6.jpg)
6) Simplified picture of bonding in crystalline metals
• Using crystal orbital theory we can rationalize the well-known fact that the metals with highest melting points are those belonging to 6th and 7th groups (see diagram below). COOP
E
ps
d
s, p - band
d - band
+0-
E
d - band
s, p - band
14 e's
12-13 e's
6-7 e's
-50
450
950
1450
1950
2450
2950
3450
3950
Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In La Hf Ta W Re Os Ir Pt Au Hg Tl
Me
ltin
g p
oin
t, o
C
![Page 7: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/7.jpg)
7) The Peierls distortion
• When working with highly symmetrical structures one has to be cautious. • Highly symmetrical structures with not completely filled degenerate or near-degenerate levels are a
subject to distortions which lower the symmetry and the energy of the system (Peierls distortion). • Diagrams on the left and in the center show how we can form bands for polymeric dihydrogen (-
MO) with twice larger four-atomic unit 2a and then distort the polymer to produce an array of dihydrogen molecules (the diagram on the right).
• Similarly an infinite polyene -HC=HC-HC=HC-… polyacetylene will have alternating HC-HC and HC=HC bonds due to the Peierls distortion. Because of the large band gap it will behave not as a conductor but as an semiconductor.
H2 or HCE E
F
2a
0 /2ak
E
F
2a
0 /2ak
a
CH
![Page 8: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/8.jpg)
8) Band structure of one dimensional polymers: a stack of PtII square planar complexes
• In some cases one dimensional consideration is sufficient for a satisfactory analysis of band structure of solids – one dimensional polymers. For example, we can get a satisfactory description of bonding and conductivity of K2[Pt(CN)4Clx] (x = 0 … 0.3) using just one-dimensional model of crystal.
• The complexes K2[Pt(CN)4Clx] (x = 0 … 0.3) have Pt(CN)4 – squares stacked one above another with Pt-Pt separation of 3.3 (x = 0) or 2.7-3.3 Ǻ (0 < x < 0.3).
• Purely PtII complex (x = 0 in the formula above) is an insulator while oxidized cyanoplatinates are low-dimensional conductors.
a
arrangement of anions in K2[Pt(CN)4]
arrangement of anions in K2[Pt(CN)4Clx]
2a
z
![Page 9: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/9.jpg)
9) Forming bands: Principles
• To predict a qualitative band structure of stacked [Pt(CN)4]2-, we will consider [PtH4]2- as a model.
• We will need for this analysis a MO diagram of PtL4.
• Each of the monomer’s MOs generates a band when we form a polymer. We can analyze all MO’s one by one and then combine all bands together.
• To get an idea about bands width use the rule which states that better orbital overlap will produce a wider band (>>):
Pt 4 L -GO's
d
b1g
s
p
eg
a1g
b2g
a1g
eu
a2u
b1g
a1g
eu
pz
dx2-y2
dxy
dz2
dyzdxzpz
dx2-y2
dxy
dz2
dyzdxz
E
z
x
y
a
z
![Page 10: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/10.jpg)
10) How bands behave
• To learn, how the “frontier” bands will run (“up” or “down”) let’s write corresponding Bloch functions for frontier orbitals, pz, and all d-orbitals, for k = 0 and k = /a.
k=/apz
E
0 /a
k
pz
runs downa k=0
k=0 k=/adz2
E
0 /a
k
dz2
runs up
k=0 k=/adxz
E
E
E
k=0 k=/adxy
E
zx
y
![Page 11: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/11.jpg)
11) Band structure of a stacked [PtH4]2-
• The predicted band structure of a stacked [PtH4]2- in the center match well a calculated diagram on the right.
• With band structure or DOS diagram in hands we can answer the questions: 1) why oxidized K2[Pt(CN)4Clx] (x>0) is a conductor and 2) why Pt-Pt distance shortens as x increases.
pz
dx2-y2
dxy
dz2
dyzdxz
E
k
0 /a
pz
dx2-y2
dxy
dz2
dyzdxz
E
z
x
y
![Page 12: Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d255503460f949fbc43/html5/thumbnails/12.jpg)
12) Bonding and conductivity in stacked [PtH4]2-
• Conductivity. The Fermi level of stacked [PtH4]2- is on the top of the z2-band since the monomer HOMO is dz2 orbital. The conduction band is pz-band which is almost 3 eV higher in energy.
• When the z2-band is completely filled (case of PtIIL4), no conductivity is expected / observed. For partially oxidized materials z2–band is filled only partially and we expect and observe conductivity.
• Bonding. In solids like in molecules if bonding and antibonding MO’s are completely filled, the net bonding is zero.
• For partially oxidized materials K2[Pt(CN)4Clx] (x = 0 … 0.3) z2–band is partially empty and we observe (dz2-dz2) bonding between Pt atoms.