Transcript Semiconductors

Physics 355
Semiconductors
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Diamond
 In the diamond structure, the
carbon atoms are arranged on an
fcc-type lattice with a total of 16
electrons per primitive cell.
The valence band and 7 lower bands are full, leaving no
electrons in the conduction band.
Diamond
Electrons may be thermally
activated to jump a gap. At
room temperature, kBT is
only 0.026 eV. To jump the
energy gap, the electron
requires very high
temperatures. So, diamond
is an excellent insulator.
ρ = 1018 -m
Graphite
ρ = 9 -m
Silicon
 Silicon has the diamond structure.
 There are 14 electrons per primitive
cell.
 Gap is only 1.12 eV, however.
Now there is a small (but finite) chance for a few
electrons to be thermally excited from valence band to
conduction band.
Silicon
}1.12 eV
Effective Mass Revisited
 An electron moving in the solid under the
influence of the crystal potential is subjected
to an electric field.
 We expect an external field to accelerate the
electron, increasing E and k and change the
electron’s state.
Effective Mass Revisited
  eV
and
d
dV
dV dx
 e
 e
dt
dt
dx dt
d dk
dV
 e
vg
dk dt
dx
dk
dV
v g
 e
vg
dt
dx
d
dV
k  e
dt
dx
1 d
vg 
 dk
d
k  eE
dt
Effective Mass Revisited
d  1 d  1 d  d  dk
a
 

 
dt
dt   dk   dk  dk  dt
dvg
1 d 2  dk   1 d 2  d



k
   2 2 

2
 dk  dt    dk  dt 
eE = F
1
me
Effective Mass Revisited
 1 d 
me   2 2 
  dk 


2
1
 This relates the curvature of the band to the “effective
mass.”
 One can show that a free electron “band” gives an effective
mass equal to the rest mass of an electron.
 Electrons in a crystal are accelerated in response to an
external force just as though they were free electrons with
effective mass me.
 Usually , me < m0.
Effective Mass Revisited
Effective Mass Revisited
Material
Si (4.2 K)
Ge
GaAs
InSb
ZnO
ZnSe
Electron Effective Mass Hole Effective Mass
Group IV
1.08
0.56
0.555
0.37
Groups III-IV
0.067
0.45
0.013
0.60
Groups II-VI
0.,19
1.21
0.17
1.44
in multiples of the free electron mass
m0 = 9.11  1031 kg
Experimental Measurement
 Traditionally effective masses were measured using cyclotron
resonance, a method in which microwave absorption of a
semiconductor immersed in a magnetic field goes through a sharp peak
when the microwave frequency equals the cyclotron frequency.
 In recent years effective masses have more commonly been
determined through measurement of band structures using techniques
such as angle-resolved photoemission or, most directly, the de Haasvan Alphen effect.
 Effective masses can also be estimated using the coefficient g of the
linear term in the low-temperature electronic specific heat at constant
volume Cv. The specific heat depends on the effective mass through
the density of states at the Fermi level and as such is a measure of
degeneracy as well as band curvature.
Electrons & Holes
Electrons & Holes
For the electrons occupying the vacant
states,
 1 d 2 

0
  2 dk 2 


(Negative!) and the electrons will move in
same direction as electric field (wrong
way!)
In a semiconductor, there are two charge carriers:
• Electrons (conduction band)
• negative mass
• negative charge
• Holes (valence band)
• positive mass,
• positive charge
Carrier Concentration
To calculate the carrier concentrations in energy bands we
need to know the following parameters:
 The distribution of energy states or levels as a function of
energy within the energy band, D().
 The probability of each of these states being occupied by
an electron, f().

n   f ( ) D( ) d 
c
v
1  f ( ) D( ) d


p
A band is shown for a one-dimensional crystal. The
square represents an initially empty state in an otherwise
filled band. When an electric field is applied, the states
represented by arrows successsively become empty as
electrons make transitions.
The band is completely filled except for a state marked by
a square. Except for the electron represented as a circle,
each electron can be paired with another, so the sum of
their crystal momentum vanishes. The total crystal
momentum for the band and the crystal momentum of the
hole are both ħk.
The empty state and the unpaired electron for two times
are shown when an electric field is applied. The change in
momentum is in the direction of the field.
Conduction Band Carrier Concentration
n

 0
f ( ) D( ) d
dN ( )
1  2me 
D( ) 
 2 3 
dE
2 

3/ 2

For  >> F, the Boltzmann distribution approximates the F-D
distribution:
1
(  F ) / k BT
f ( ) 

e
1  e (  F ) / k BT
 e (  c ) / k BT e ( c  F ) / k BT
which is valid for the tail end of the distribution.
Conduction Band Carrier Concentration
ne
 ( c  F ) / kBT


 me k BT 
N C  2

2
 2  
0
  c  / kBT
D( ) e
d
3/ 2
n  NC e
 F  c  / k BT
Conduction Band
Carrier Concentration
Valence Band Carrier Concentration
 The hole distribution is related to the electron distribution,
since a hole is the absence of an electron.
1
1
f h  1  f e  1    / k T
   / k T
F
B
B 1
e
1 e F
 e  F / k BT
(as long as  F    k BT )
 The holes near the top of the valence band behave like
particles with effective mass mh; and the density of states is
1  2mh 
D( )  2  2 
2 

3/ 2
  v
 mh k BT 
p   1  f ( )  D( ) d   2 
2 

 2

v
3/ 2
e
  v  F  / k B T
Equilibrium Relation
 Multiply n and p together:
3
 k BT 
3/2  g / k BT
np  4
e
 mh me
2
 2 


 
 The product is constant at a given temperature.
 It is also independent of any impurity concentration at a
given temperature. This is because any impurity that adds
electrons, necessarily fills holes.
 This is important in practice, since we could reduce the
total carrier concentration n + p in an impure crystal via the
controlled introduction of suitable impurities – such
reduction is called compensation.
Intrinsic Semiconductors
n  p  ni
N c e  F  c  / k BT  N v e  v  F / k BT
 Nc 
   c   F   F   v
 k BT ln 
 Nv 
c
 Nc 
c  v 1
F

F 
 k BT ln 
v
2
2
 Nv 
Extrinsic Semiconductors
 Extrinsic semiconductors: we can add impurities
to make a material semiconducting (or to change
the properties of the gap).
 There are 2 types of extrinsic semiconductors:
p-type and n-type
 These are materials which have mostly hole
carriers (p) or electron carriers (n).
 These give you ways of modifying the band gap
energies (important for electronics, detectors, etc).
Extrinsic Semiconductors: n type
•
Add a small amount of phosporus (P:
3s23p3) to Silicon (Si: 3s23p2)
(generally, a group V element to a
group IV host) P replaces a Si atom
and it donates an electron to the
conduction band (P is called the donor
atom). The periodic potential is
disrupted and we get a localized energy
level, D.
• This is an n-type semiconductor – more electrons around that
can be mobile; and the Fermi energy is closer to the
conduction band.
Extrinsic Semiconductors: n type
Phosphorus provides an extra electron.
 C – D = 45 meV
So, its easy for the donor electrons to enter
the conduction band at room temperature.
This means that at room temperature n ND.
This is called complete ionization (only true if ni
<< ND). Therefore, by doping Si crystal with
phosphorus, we increase the free electron
concentration.
At low temperature, these extra electrons get
trapped at the donor sites (no longer very mobile)
- the dopant is frozen out.
Extrinsic Semiconductors: p type
 Next suppose Si atom is replaced with Boron (B: 2s22p) to
Silicon (Si: 3s23p2). Again, we have a perturbed lattice and
a localized E-level created.
 Boron is missing an electron and accepts an
electron from valence band, creating a hole.
 Therefore doping with B increases hole
concentration. We call this p-type doping,
the electron concentration n is reduced.
 F moves closer to  V.
c
F
A
v
Extrinsic Semiconductors
Extrinsic Semiconductors
Boron in Silicon
Mass Action Law
• Valid for both intrinsic and extrinsic semiconductors.
• It is important in devices to control n and p
concentrations and to suppress the influence of the
intrinsic concentration.
• These equations are important in establishing upper
limits in semiconductor operating temperature.
• We generally require ni << (minimum doping density)
and, practically, this means we need doping
concentrations above 1014 cm3.
Semiconductors