Trends in III-V and II-VI Compounds
Larger atoms, weaker bonds, smaller U, smaller Eg, higher μ, more costly!
Energy Gap and Lattice Constants
2.6
AlP
0.517
GaP
2.2
0.563
AlAs
0.620
2.0
Energy gap (eV)
1.8
0.689
AlSb
1.6
0.775
GaAs (D)
1.4
1.2
0.885
InP(D)
1.033
Si
1.0
1.240
0.8
0.6
2.067
Indirect band gap
0.4
1.550
GaSb (D)
Ge
3.100
InAs (D)
Direct band gap (D)
0.2
0
0.54
Wavelength (microns)
2.4
0.477
6.200
InSb (D)
0.55
0.56
0.57
0.58
0.59
0.60
0.61
0.62
0.63
0.64
0.65
Lattice constant (nm)
Figure by MIT OpenCourseWare.
©1999 E.A. Fitzgerald
1
Properties of non-free e
• Electrons near the diffraction condition are not
approximated as free
• Their properties can still be viewed as free e- if an ‘effective mass’ m* is used
h2k 2
E=
2mec*
h2k 2
E=
2m
h2
m = 2
∂ E
∂k 2
*
ec
h2k 2
E=
2mev*
h2
m = 2
∂ E
∂k 2
*
ev
−π/a
Note: These
electrons have
negative mass!
π/a
2
©1999 E.A. Fitzgerald
1-D Crystal Metals and Insulators
•
•
•
•
•
How do band gaps affect properties of materials?
Only electrons near EF participate in properties
If EF is in the middle of the band, free e- and metallic behavior
If EF is near the band gap, changes in materials properties may occur
Need to find out where EF is!
Where kF=π/a if we
2k L 2L want to see how many
N= F =
π
a
electrons are in first
band
EF with 2e- per unit cell
EF with 1e- per unit cell
−π/a
©1999 E.A. Fitzgerald
π/2a π/a
Note: L/a is the number of unit
cells in the 1-D crystal; therefore,
the number of electrons per unit
cell, which depends on n and the
crystal structure, determine where
EF is with respect to the band gap
3
1-D Crystal Metals and Insulators
• 2e- per unit cell: EF at band edge: 2 possibilities
– Band gap >> kT: electrons at band max can not accept
energy from electric fields; no conduction, insulating
behavior
– Band gap near kT: some thermal fluctuations large
enough to allow population of second band; carriers are
there, but less than for free e-, semimetal
• 1e- per unit cell: EF in middle of band: free e-, metallic
Note: crystal structure (number of atoms per primitive cell) and
valence (number of conduction electrons per atom), combined
with band gap size, determine the electronic properties
4
©1999 E.A. Fitzgerald
Higher Dimensions (2 and 3-D)
•
1-D: E(kx); 2-D: E(kx,ky); 3-D: E(kx,ky,kz)
‘Fermi Surface’
E
Zone center
ky
First zone
π/a
First zone
0
kx
−π/a
−π/a
0
π/a
π/a
3-D
−π/a
π/a
2-D
1-D
Zone center
©1999 E.A. Fitzgerald
−π/a 0
5
Metals and Insulators
• Covalent bonds, weak U seen by e-, with EF
being in mid-band area: free e-, metallic
• Covalent or slightly ionic bonds, weak U to medium U, with EF near band edge
– EF in or near kT of band edge: semimetal
– EF in gap: semiconductor
• More ionic bonds, large U, EF in very large
gap, insulator
6
©1999 E.A. Fitzgerald
Insulators
• Very large band gaps=no conduction electrons at
reasonable temperatures
• All electrons are bound
• Optical properties of insulators are derived from
the electric field being able to temporarily move
electrons: polarization
• We will return to the interaction of E-field with
bound electrons in Dielectrics Section
7
©1999 E.A. Fitzgerald