1. Chapter 5
Metals: Energy Bands

2. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Free Electron Model? + + + + + + + + + + + + + + + + - + + + + + + + + + + + + + + + +
V = 0

3. Free Electron Model e
g(e) eF 1 2 nF 3 n e
g(e)
Energy gap
Energy band

4. Contents 1. Introduction
2. Energy spectra in atoms, molecules, and solids
3. Nearly Free Electron Model
4. Wave Equation of Electron in a Periodic Potential
5. Density of States and Fermi Surface
6. Velocity of Bloch Electron and Effective Mass
7. Electrical Conductivity

5. + - + - - 5.2. Energy spectra in atoms, molecules, and solids
- Energy spectrum of a Li atom
: 1S2 2S1
V(x) V + x 2P - + K L M 2S - 1S - - -
Atomic orbitals -

6. + + + + + + + + + + + + - Energy spectrum of a Li molecule (Li2)
Discrete doublets + 1S 2S 2P + 1S 2S 2P + 1S 2S 2P + 1S 2S 2P + 1S 2S 2P + 1S 2S 2P + + 1S 2S 2P + 1S 2S 2P + 1S 2S 2P + 1S 2S 2P + 1S 2S 2P -
Energy level splitting occurs in physics because of external fields or other effects.

7.    Molecular orbitals H2 molecule H2+ ion The more overlapped, V 
V(x) r x    x
The more overlapped, V 
But the more overlapped, Er 
Li2 molecule
s* antibonding or antisymmetric orbital
2s* r 2S
s bonding or symmetric orbital 2s  
Molecular orbitals

8. + + 2P As (orbital E) , splitting  2S 1S splitting: small 1S
Li[1S22S1] + Li[1S22S1]
→ Li2[(1s)2(1s*)2(2s)2]

9. - Energy spectrum of Li solid
Bonding molecular orbital
antibonding molecular orbital
atomic orbital 2P 2S
1Li
2Li
3Li
4Li
5Li
NLi
N closely spaced sublevels 2P
Energy band 2S

10. + + Atom Molecule (Li2) x 1S 2S 2P 1ss* 2Sss* 2Pss* Crystal orbitals
Solid
wavefunctions
Solid
delocalized orbitals 2P
gap 2S
energy gap 1S

11. + + Atom Molecule (Li2) x 1S 2S 2P 1S 2S 2P Li metal Solid 2P
No energy gap 2S 1S

12. + 5.3 Nearly Free Electron (NFE) Model Free electron model -
Free electron wavefunctions,
Cf: wave equation for
traveling wave
Band electrons →
Perturbed only weakly by the periodic potential of the ions cores

13. Free electron model e k
What happens at ?
Standing waves!
1st BZ boundaries
No solution of Schrödinger Eq.

14. What happened in phonons on the boundary of the 1st BZ? Standing wave!
k = p/a or – p/a t1 t2 t3 t4 t5
traveling wave x
standing wave x

15. Electron waves at traveling wave Standing wave In phase Incident wavex t2
Reflected wavelets
Standing wave

16. 2 different standing waves formed!
Standing waves at
Incident wave t1 t2 x
2 different standing waves formed!
Reflected wavelets

17. 2 different standing waves formed!
directions!
No sign change for x → -x
Sign change for x → -x

18. Origin of the energy gap
Probability density of an electron:
For pure traveling wave,
For standing waves, e
g(e)
Energy gap
Energy band - -
Traveling waves x
V(x)

19. Er(+) < E (traveling electron) Er(+) < Er(-)
: equal portions - -
Traveling waves x
V(x)
Er(+) < E (traveling electron)
Er(+) < Er(-)
Er(-) > E (traveling electron)
Eg = Er(-) - Er(+)

20. Free electron model
NFE model e e
1st BZ Eg k
-p/a
p/a k
Eg = Er(-) - Er(+) at k = ±p/a
Origin of the Eg
→2 standing waves: piling up electrons at different regions
NFE model
→rough approximation for simple metals (Na, K, Al, etc)

21. 5.4 Wave Equation of Electron in a Periodic Potential
5.4.1 Bloch Functions
Eigenvalue Eq.:
Eigenvalue
Eigenvector or Eigenfunction
Periodic zone scheme
Reduced zone scheme
Extended zone scheme
5.4.2 Wave Equation of Electron in a Periodic Potential

22. 5.4 Wave Equation of Electron in a Periodic Potential
5.4.1 Bloch Functions -
V(r)
Schrödinger Eq.:
periodic potential
Schrödinger eq.
(1.14)
crystal potential
: a lattice vector
Bloch theorem:
(Bloch function) 
One electron wavefunction
(translational symmetry)

23. a crystal orbital, delocalized throughout the solid
(Bloch function)
V(r)
Bloch function:
a crystal orbital, delocalized throughout the solid
: periodic in the crystal

24. 5.4.2 Wave Equation of Electron in a Periodic Potential
→ Energy bands in solids

25. Schrödinger Eq.: Li metal (Homework) Eigenvector or Eigenfunction2P
Li metal
(Homework)
Eigenvector
or Eigenfunction
Eigenvalue Eq.:
Eigenvalue
e(k)
E3,k
Many solutions
E gap
3rd band
→ Energy Eigenvalue
E2,k
2nd band
E1,k
(n: band index)
1st band k

26. How about eigenvalues, En(k)?
(Homework)
(Reciprocal lattice vector)
p/a
2p/a
-p/a
First Brillouin Zone k
w/(4C1/M)1/2
Ex) ka = 1.2p
ka = -0.8p
1.2p/a
-0.8p/a e
Ex) ka = 1.2p
ka = -0.8p
G(2p/a) k
-p/a
-0.8p/a
p/a
1.2p/a
2p/a
First Brillouin Zone

27. (for 1-D) Free electron model e(k) 4th Band 3rd Band 2nd Band 1st Band
E1,k
E2,k
e(k)
E3,k
1st band
2nd band
3rd band
E gap k
2nd BZ
1st BZ
2nd BZ
Periodic zone scheme k
Reduced zone scheme
1st BZ

28. : smoothing sharp corner in band intersections
NFE model
e(k)
e(k)
E2g
E1g k k
1st BZ
Reduced zone scheme
Extended zone scheme
potential
: smoothing sharp corner in band intersections

29. 5.5. Density of States and Fermi Surface
5.5.1 Number of Orbitals in a Band
5.5.2 Density of States and Fermi Surface

30. 5.5.1 Number of Orbitals in a Band
Linear crystal
→ N atoms in L (length) L 1 2 3 4 5 6 7 8 a N
1st BZ
Zone boundary
# of states inside the 1st zone:
(# of unit cells)
Each band has N states inside the 1st BZ.
Maximum # of electrons in a single band: 2N
(# of crystal orbitals inside the 1st BZ)
= (# of unit cells)

31. Questions?
1. For a single atom of valence one per unit cell,
the band is ( ) % filled with electrons.
2. For a single atom of valence two per unit cell,
the band is ( ) % filled with electrons.
3. For two atoms of valence one per unit cell,
the band is ( ) % filled with electrons.

32. 5.5.2 Density of States and Fermi Surface
NFE model k e
p/a
-p/a
Free electron model k e
1st BZ Eg ky
p/a
Energy contour kx
-p/a
p/a
-p/a

33. g(e)de: # of states per unit volume between e and e+de
Density of states ky
p/a
g(e)de: # of states per unit volume between e and e+de de
For low k, kx
-p/a
p/a
-p/a e
g(e)
N at 0ºK eF
For spin degeneracy,

34. ky
p/a
-p/a de e
g(e) et
Near the top of the band,
(Homework)

35. For overlapped bands, e
g(e) et e
g(e) et 3d 2p 2s 4s
divalent metals
transition metals
# of electrons actually occupying between e and e +d e, dn(e):

36. Fermi surface (FS) FS
: the surface in k-space inside which all the states are occupied by valence electrons.
Why FS?
Crystal potential → FS?
FS shape? ky
p/a
For small n,
→ free-electron
behavior
→ Spherical FS
-p/a
p/a ky
Ex) Li, Na, K,….
As n ,
FS shape distorted
-p/a

37. e
g(e)
Monovalent metals
→ partially filled
Ex) Na
one atom per cell
→ one electron per cell eF e
g(e)
→ 50% filled 2p 2s
Divalent metals
→2 bands overlapped eF
→2 electrons per atom e
g(e)
Insulators
→ completely filled
N at 0ºK
(semiconductor) eF

38. Q: eF in monovalent metal?
→ free-electron behavior
Q: me dependence of eF? e
g(e) eF

39. ky FS
Q: FS in polyvalent metals?
large n
FS → intersecting ZBs
For empty lattice model kx ky FS
[111]
1st zone ky kx
1st zone FS kx
2nd zone
[111]
2nd zone
Extended zone scheme
Reduced zone scheme

40. ky FS
Energy bands?
e(k)
ZB[111]
ZB[100] kx
[111]
1st zone eF ky FS kx
k[111]
k[100]
[111]
2nd zone
Reduced zone scheme

41. 5.6 Velocity of Bloch Electron and Effective Mass
For Bloch electron,
For free electron,
(wave packet) ky
p/a ky
p/a kx
-p/a
p/a kx
-p/a
p/a
-p/a
(m*: effective mass)
-p/a

42. Q: v(any given electron) → constant?
Yes!
Exceptions:
- Collision with phonons
e(k)
- Electric or magnetic field
For low k,
For high k, k
Zero v at ZB?
Standing waves ! k

43. Effective Mass
Q: (Bloch electron at low k) vs. (free electron): the same?
Difference → mass!
(free electron)
(Bloch electron)
Under electric field,
In momentum space
Curvature of E-band

44. →opposite to the applied force Lattice force!
e(k)
p/a
e(k)
Small mass k
Large mass k k
For low k,
m*→constant m*
As k , m* 
For k > kc,
m* < 0 k
p/a kc
Acceleration < 0
→opposite to the applied force
Lattice force!

45. e(k)
Q: How are electrons moved
under ?
p/a k
electron k

46. → vacant orbitals in a band → positive charge, +e
Holes
e(k)
→ vacant orbitals in a band
→ positive charge, +e
(+ve m*, -e) ke k
one hole → near ZB
electron removed
m*(ke) < 0,
e(k)
mh*를 –me*로 정의하면 편리 (me*가 negative이므로)
m*는 e vs k band의 curvature함수: VB의 top부근에서 slope이 점점 감소하므로 m*가 negative가 됨
반면, CB의 bottom에서는 slope이 zero에서 점점 증가하므로 m*가 positive.
(+ve mh*, +e) +e
hole k

47. e(k)
Under electric field,
→All electrons move except the hole
→one vacant site movement CB VB
a vacant site e je ve vh h ke jh
e(k) +e
VB: Valence band
hole
CB: Conduction band k

48. 5.7 Electrical conductivity
What happens for electrons under E ?
At t, ky
Fermi surface ky
At t = 0,
Fermi sphere kx kx
Net current

49. kx ky
At t,
t: collision time
(uncompensated electrons)
(energy increment by ) e
g(e) eF

50. For spherical FS,
s  g(eF)
(for free electron model)
Metal :
g(eF) ,
s 
Metal
Insulator e
g(e)
Insulator :
g(eF) ~ 0,
s ~ 0
vF(insulator) > vF(metal) eF eF

51. Limit of the Band Theory
For Na metal s
Metal
Insulator a ac T