Lectures 21-22

Solid state materials. Electronic structure and conductivity 1) Band theory

• • • The electronic structure of

solids

A solid can be considered as a can also be described by MO theory.

supermolecule

. One mole of atoms (

N A

), each with

X

atomic orbitals producing

X

orbitals in the valence shell contributes moles of MO’s.

X

moles of • • • Consider qualitatively bonding between N metal atoms arranged in a chain; N = 2, 4, N A . Assume that

X

of ns 1 configuration (Li, Na etc) =1 for simplicity. In the case of

N

~

N A

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 4 Li atoms N A Li atoms conduction band   

F

Fermi level   valence band

•

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

. 

F

• negative potential no potential positive potential 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 

F Insulators

Bandgap 

F Semiconductors

T (intrinsic conductivity) or h  (photoconductivity) 

F

C Si Ge Sn Bandgap, eV 6.02

1.09

0.72

0.07

Conductivity, W -1 cm -1 < 10 -18 5 ·10 -6 0.02

10 4

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 as shown below. hydrogen atoms look

E

energy level of isolated s-orbitals Energy levels of resulting MO's are indicated with N = 3 N = 4 N = 5 N = 6 N = 7

• • • •

4) Crystal orbitals (Bloch functions)

If we have

N

functions f m hydrogen atoms with atomic wave (m = 1 …

N

) related by symmetry and spaced at distance

a

, we can get N MO’s y n (n = -

N

/2, …, 0, …,

N

/2) which are called Bloch functions. For the n-

th

crystal orbital, y n , we will have: E, eV y

n



m N

  1   cos( 2 p

n N m

) 

i

sin( 2 p

n N m

)   

m

a

= 3 A

a

= 2 A When n changes from 0 2 p n/(

aN

bonding y 0 to

N

/2 , variable k = )) (wave vector) changes from the type of the MO changes from the completely to the completely 0 antibonding to y p / N/2

a

: and

a

k=0 k= p /

a

0 -10

a

= 1 A E, eV

a

= 1 A -20 y 0 = f 1 + f 2 + f 3 +...

y N/2 = f 1 + f 2 f 3 +...

• 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

) 0 k p /

a

0 k p /

a

0 k p /

a

DOS

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). E, eV 20 • COOP is defined in the same way as the bond order is defined in MO theory of molecules. 10 • • For any two atoms

i

and

j

COOP(

i

-

j

) = S 2c i c j S ij (S ij 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 for bonding. positive value is characteristic 0 -10 -20 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). 0 k p /

a

E

a

k= p /

a

S ij < 0 k=0 0 COOP S ij > 0

•

6) Simplified picture of bonding in crystalline metals

E E Using crystal orbital theory we can rationalize the well known fact that the metals with highest melting points are those belonging to 6 th and 7 th groups (see diagram below).

p s d s, p - band d - band s, p - band 0 14 e's 11-12 e's 6-7 e's d - band + COOP 3950 3450 2950 2450 1950 1450 950 450 -50 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

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 2

a

and then distort the polymer to produce an array of dihydrogen molecules. 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.

E

2

a

E H

2

or HC

a

CH



E

 F  2

a

0 p /2

a

k

0  F p /2

a

k