Semiconductors
Institute of Solid State PhysicsTechnische Universität Graz
Semiconductors
32
exp300
cc
B
ETn Nk T
32
exp300
vv
B
ETp Nk T
density of electrons in the conduction band
density of holes in the valence band
effective density of states in the conduction band at 300 K
effective density of states in the valence band at 300 K
The electrical contribution to the thermodynamic properties of insulators depend on band edges
http://lamp.tu-graz.ac.at/~hadley/ss1/semiconductors/intrinsic.php
Narrow bandgap semiconductors
Ev Ec
Use the programs for metals for small bandgap semiconductors.
Large gap -> large effective mass
22
2
*
2
1
d E tadk
mt
2 cos( )kE t ka
narrow bands -> large effective mass
2*
2
2
md Edk
Measuring the effective mass
*ceBm
Cyclotron resonance
Resonant absorption occurs when rf waves with the cyclotron resonance frequency are applied. This can be used to experimentally determine the effective mass.
Knowing the effective mass, the scattering time can be calculated from the measured conductivity.
2
*scne
m
direct bandgap:k = 0
photons can be emitted
indirect bandgap:k = 0
phonons are emitted
Direct and indirect band gaps
Momentum must be conserved when photons are absorbed or emitted.
Light emitting diodes
Extrinsic semiconductors
The introduction of impurity atoms that can and electrons or holes is called doping.
n-type : donor atoms contribute electrons to the conduction band. Examples: P, As in Si.
p-type : acceptor atoms contribute holes to the valence band. Examples: B, Ga, Al in Si.
Ionization of dopants
Easier to ionize a P atom in Si than a free P atom
4
2 2 208nmeE
h n
2*0
0r
mm
Ionization energy is smaller by a factor:
Ionization energy ~ 25 meV
images from wikipedia
Czochralski Process
add dopants to the melt
Crystal growth
Neutron transmutation
30Si + n 31Si + 31Si 31P +
image from wikipedia
Float zone Process
Crystal growth
image from wikipedia
Chemical vapor deposition
Epitaxial silicon CVD SiH4 (silane) or SiH2Cl2 (dichlorosilane)PH3 (phosphine) for n-doping or B2H6 (diborane) for p-doping.
http://www.microfab.de/foundry/services/diffusion/index.html
Gas phase diffusion
AsH3 (Arsine) or PH3 (phosphine) for n-doping B2H6 (diborane) for p-doping.
Ion implantation
Implant at 7º to avoid channeling
Donors
Five valence electrons: P, As
States are added in the band gap just below the conduction band
Ec
Eg
Ev
ED
n-type: n ~ ND Many more electrons in the conduction band than holes in the valence band.
majority carriers: electrons; minority carriers: holes
D(E)
(T=0)
Acceptors
Three valence electrons: B, Al, Ga
States are added in the band gap just above the valence band
Ec
Eg
Ev
EA
p-type: p ~ NA Many more holes in the valence band than electrons in the conduction band.
majority carriers: holes; minority carriers: electrons
(T=0)
D(E)
Donor and Acceptor Energies
Energy below the conduction band Energy above the valence band
n-type
For n-type, n ~ density of donors, p = ni
2 /n
n-type ND > NA, p ~ 0
exp cD c
B
En N Nk T
ln cc B
D
NE k TN
p-type
For p-type, p ~ density of acceptors, n = ni
2/p
p-type NA > ND, n ~ 0
exp vA v
B
Ep N Nk T
ln vv B
A
NE k TN
exp2
gi v c
B
En N N
k T
Intrinsic semiconductors
intrinsicextrinsic
freeze-out
1/T
log 1
0(n i
) cm
-3
GaAs
Si
Ge
300 K
Extrinsic semiconductors
At high temperatures, extrinsic semiconductors have the same temperature dependence as intrincic semiconductors.
Energy spectra of donors in silicon
Ionized donors and acceptors
For Ev + 3kBT < < Ec- 3kBT Boltzmann approximation
1 4exp
AA
A
B
NNE
k T
1 2exp
DD
D
B
NNE
k T
4 for materials with light holes and heavy holes (Si)2 otherwise
Mostly, ND+ = ND and NA
- = NA
ND = donor density cm-3 ND+ = ionized donor density cm-3
NA = donor density cm-3 NA- = ionized donor density cm-3
Charge neutrality
n + NA- = p + ND
+
Carrier concentration vs. Fermi energy
Why dope with donors AND acceptors?
collector base emitter
n pn+
lightly doped p substrate
Bipolar transistor
MOSFET
Bipolar Junction Transistor