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8
10월 19일 중간고사 1: 6:00 – 8:00 pm
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8.1 Band Structure
Hybridization of s- and p-states Sp3 state
Two s+p bands, lower filled
higher empty for Ge, Si Group IV
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Calculated band structure for Si.
Eg, at T=0 K
C :
5.48
Si :
1.17
Ge :
0.74
Sn(gray) : 0.08
T 2
EgT  Eg0 
,
T  D
  5 104 eV / K
D: Debye Temperature
Dependence of band-gap on temperature
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8.2 Intrinsic Semiconductors
At elevated temperature, Semiconductor become conducting
For intrinsic semiconductors,
EF   E g / 2
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Assuming, me*=mh*
Z(E): density of states
N(E): number of electrons per unit energy
N*: number of electrons
dN *  N ( E )dE,
V  2m 
Z (E) 


4 2  2 
N (E)  2  Z (E)  F (E)
3/ 2
E1/ 2
  E  EF  
1
F (E) 
 exp   

k
T
 E  EF 
  B 
exp 
 1
 kBT 
V
N* 
2 2
V
N 
2 2
*
3/ 2
 2m 
 2 




0
3/2
E
1/ 2
  E  EF  
 exp   
  dE
  kBT  
  E 
 EF   1/2
exp 
 0 E  exp   
  dE
 kBT 
  kBT  
 EF  k BT
 2m 
1/2
  2  exp 
(

k
T
)

B
k
T
2


 B 
EF
V 2mk BT 3/2
 (
)
exp(
)
2
4 
k BT
V

2 2
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 2m 
 2 


3/2
Inserting that EF=-Eg/2, and effective mass ratio me*/m0,
1  2m k 
Ne   0 2 B 
4 

3/2
m 


m
 0
*
e
3/2
T
3/2
  Eg
exp   
  2kBT
m 
 Ne  4.84 10  
 m0 
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*
e
3/ 2
T
3/ 2

 .

  Eg 
exp  

2
k
T
  B 
Number of electrons in the conduction band per unit volume (cm3)
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Fermi Level in Semiconductors:
Emax

Emax n 
EC

EG
1
=
Ec
2 N v ( E ) f ( E )dE
2
0
EF
=
Ev
=
1
2
(
2
1
2 2
=
(
2
1
2
(
2me*
2
2me*
2
2me*
(
2
2
)
3

2
1
  E  ( EC  EF )  
2
E
exp
 
 dE
0

kT

 

1
  E 
 ( E  EF ) 
2
) 2 exp   C
E
exp
    dE

kT

0
  kT  
2me*
2
3
1
  E  ( EC  EF )  
) 2 E 2 exp  
  dE
kT

 
3
)
3
2
1
kT
 ( E  EF ) 
( kT ) 2 exp   C

2
kT


3
 2 m kT  2
 ( EC  EF ) 
=2 
exp



h
kT




*
e
2
 2 m kT 
 ( E  EF ) 
=N C exp   C
where,
N

2


C

kT


 h

*
e
2
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When m*e=m, Nc =2.5x1019 cm-3
3
2
Fermi Level in Semiconductors:
Emax
Ec
Eg
Ev
By the same token,
 ( E  EV ) 
p  NV exp   F

kT


where, NV is the effective density of states in the valence band
2 mh* kT 2 3
NV  2(
)
h2
By multiplying together,
np  N C NV exp(
EG
)
kT
The number should be constant for a given material for a given temperature.
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Intrinsic Semiconductor:
The electron density of free electrons equals the density of free holes.
Therefore, if ni is the intrinsic carrier density,
ni2  np  N C NV exp(
EG
)
kT
since, n=p,
 ( EC  EF ) 
 ( EF  EV ) 
N C exp 

N
exp
V



kT
kT




EG
( EC  EF )
)  N C exp(
)
2kT
kT
EG
NC
EG 3kT
me*
kT
( EC  EF ) 
 ( ) ln( ) 
(
) ln( * )
2
2
NV
2
4
mh
1
n  ( N C NV ) 2 exp(
combining,
me* 3 4
E
ni  N C ( * ) exp( G )
mh
2kT
10
me* 3 4
E
 i  N C q( * ) ( n   p ) exp( G )
mh
2kT
Mobility
Ohm’s law


,

j  ,
j  N e,
 N


e  N  e.
  N e e e  N h e h ,
*


m
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  4.84 10  
 m0 
Again,
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For intrinsic semiconductor,
3/2
  Eg  
T e( e  h ) exp   
 ,
  2kBT  
3/2
me* 3/4
E
  N C e( * ) ( n   p ) exp( G )
mh
2kT
Ne  N h
  N e e e  N h e h ,
m 
  4.84 10  
 m0 
*
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Due to lattice vibration
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3/ 2
T
3/ 2
  Eg
e( e  h ) exp   
  2kBT
Increasing the number of carriers

 ,

8.3 Extrinsic Semiconductors
8.3.1 Donors and Acceptors
For intrinsic semiconductor, 109 electrons per cubic centimeter
P. Binding E = 0.045 eV
Doping : adding small amounts of impurities (III or V) to intrinsic semiconductors
Dopant in substitutional manner
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8.3.2 Band Structure
Donor electrons & thermally excited electrons
CB
VB
Impurity states; donor or acceptor levels
n-type, major carrier: electrons p-type, major carrier: holes
B, Al, Ga, In
P, As, Sb
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For extrinsic semiconductor,
Charge neutrality is important,
n  nA  ( N D  nD )  p
n  N A  N D  p
Ec
Ed
Ef
Ea
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Ev
where,
( E  EF )
1
nD  N D {( ) exp[ D
 1]}1
g
kT
( E  EA )
1
( N A  nA )  N A{( ) exp[ F
 1]}1
g
kT
g  2 (spin orientation) (degeneracy factor)
or ,
( E  ED )
N D  ( N D  nD )  N D {2 exp[ F
 1]}1
kT
( E  EF )
N A  nA  N A{2 exp[ A
 1]}1
kT
( EC  EF )
n  N C exp[
]
kT
( EF  EV )
p  NV exp[
]
kT
8.3.3 Temperature Dependence of the Number of Carriers
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8.3.4 Conductivity
  Nde ee ,
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8.3.5 Fermi Energy
n-type semiconductor, Nd=1016 atoms per
cubic centimeter
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8.4 Effective Mass
1
2

d
E
*
2
m   2  .
 dk 
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In the presence of electric field, electrons at the bottom of
conduction band and holes at the top of the valence band
move in opposite directions in real space (same sign mass
but different sign charge), whereas electrons and holes both
at the top of the valence band move in the same direction
(different sign mass cancels different sign charge).
8.5 Hall Effect
jx   Nx e
Due to Ex
FL  e( v  B)   x Bz e,
 y   x Bz .
Hall field
FH  e y since FH  FL  0
jx Bz I x Bz Ly
N

e y
Ax eVy
Vy : Hall voltage measured in the y-direction
Hall constant
RH 
Ey
jx Bz
1
RH  
Ne
Negative value for electron
Positive value for hole
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8.6 Compound Semiconductors
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8.7 Semiconductor Devices
8.7.1 Metal-Semiconductor Contacts
(a) rectifying contact: convert AC to DC
(b) ohmic contact: electrons can easily flow in both directions
draw the I-V curve:
electrons like to roll downwhill
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(Space charge region)
Holes wan to drift upward
8.7.2 Rectifying Contacts (Schottky Barrier Contacts)
Work function: energy difference between the Fermi energy and the
ionization energy
(Vacuum level)
(electron affinity)
Contact potential
- Diffusion current: electrons from both sides cross the potential barrier at equilibrium state
- Drift current: the transport of thermally created electrons and holes
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The width of the depletion
region in the n-type
semiconductor:
Assuming full depletion model
(there are no free electrons in
the depletion region and that
the only charge there is the
charge on spatially uniform
ionized donors.
 2  N D q

x 2
 r 0

 0 and  =D at x  wd
x
and,  =0 at x  0
B.C.: E 
  N D qx

C
x
 r 0
C
N D qwd
 r 0
  N D q


( x  wd )
x
 r 0
By integration a second time between x  0 and x  wd
N D q 2
D  (
) wd
2 r  0
wd  (
wd  [
2 r  0D 1/2
)
N D q
2 r  0 (D  app )

D
N q
since, C 
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1

(C / A) 2
]1/2
 r 0 A
d
2(qD  qapp )
 r  0 q 2 N D
Reverse bias
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Forward bias
I MS
I SM
  M    
 ACT exp   
 ,
k
T
  B 
2
  M  S  eVappl .  
 ACT exp   
 ,
k
T
B

 
2
A: Area of the contact, C: constant
Thermionic emission:
J x  (4 qm / h3 )(kT ) 2 exp( q / kT )  CT 2 exp( q / kT )
where,
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C  (4 qmk 2 / h3 ) : Richardson const.=1.02x106 A  m 2  K 2
S  
For low enough temperatures, Fermi level lies close to the conduction band,
See Fig. 8.10
I net  ISM  I MS
Consists of saturation current and a voltage-dependent term
  M  S  
I S  ACT exp   

  kBT  
2
I net

 eV
 I S exp 
 kBT

 
  1 .
 
A few advantages over p-n diode
No annihilation of electrons and holes, charge carrier , electron
Better heat removal
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8.7.3 Ohmic Contacts (Metallizations)
M  S
The formation of highly doped region to make an Ohmic contact.
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8.7.4 p  n
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Rectifier (Diode)
Diffusion of electrons in p-type region
The saturation current in the case of
reverse bias is given by the Shockley
equation, which is also called ideal
diode law:
Ideal diode law :
 Cep Dep Chn Dhn
I S  Ae 

 L
Lhn
ep

Diffusion of holes in n-type region

 ,

The electrons in the p-type region and the holes
in the n-type region can diffuse to the junction
area and be swept away when the reverse bias
voltage is applied.
Chn : concentration of holes in the n-type region
Cep : concentration of electrons in the p-type region
Einstein relation:
Dep 
ep kBT
e
Lep  Dep  ep ,
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8.7.5 Breakdown Voltage and Zener Diode
When the reverse voltage of a p-n diode is increased above a critical
value, the high electric field strength caused some electrons to become
accelerated to a velocity at which impact ionization occurs.
The breakdown voltage, which is
the result of this avalanching
process, depends on the degree
of doping: the higher the doping
the lower the breakdown voltage.
Tunneling or Zener breakdown
occurs when the doping is heavy
and thus the barrier width becomes
very thin.
-takes place at low reverse voltages
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8.7.6 Solar Cell (Photodiode)
A photodiode consists of a p-n junction.
A Si PV device yields an inherent voltage of 0.6 V.
Diffusion length of carriers: 10 – 200 m depending on the quality of Si.
Quantum Efficiency:
Si: 20 – 28% efficiency
The goal is to produce for terrestrial applications inexpensive solar cells having 20%
efficiency or better and a lifetime of about 20 years.
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8.7.8 Tunnel Diode (OK)
Degenerated doping – high doping
Fermi level lies in the conduction and valence band
Depletion width is very narrow (~10 nm)
Show negative current-voltage characteristics.
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8.7.9 Transistors
n-p diode p-n diode
For signal amplification
Smaller and higher resistivity
+
Climb
Unbiased n-p-n bipolar junction
transistor
diffuse
acceleration
The E-B diode is forward biased,
whereas the B-C diode is strongly
reverse biased.
Electron flow from E to C can be controlled
by bias voltage on the Base
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Transistors : amplification of music or voice
electronic switch (on & off) for logic and memory
(1) Bipolar : current flow through n-type as well as through p-type
Heavily doped
# of holes kept to a minimum(light doping) or thin
doping level is not critical
The voltage applied between emitter and base modulates the transfer of
the electrons from the emitter into the base region.
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(2) Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET)
Can be controlled
Electric field
Can be controlled
Unipolar : current flow only through n-type
Two types: depletion-type MOSFET
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Normally on
* Enhancement-type
MOSFET
Normally off
This type is dominating in IC circuit industry
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N-MOSFET
P-MOSFET
If both are integrated on one chip and wired in series, this technology is labeled
CMOSFET (complementary MOSEFT)
For information processing, low operating voltage, low power short channel for
High speed.
MOSFET = MOST (metal-oxide-semiconductor transistor)
= MISFET (metal-insulator-semiconductor field-effect transistor)
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(2) Junction Field-Effect Transistor (JFET)
Normally on, depletion type
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Transistors
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(2) GaAs MESFET
Use of computer still demands higher switching speed device- GaAs
seems to be the answer with its higher electron mobility
Source, Drain – Ohmic contact
Gate – Schottky contact
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8.7.10 Quantum Semiconductor Devices
The energy level is separated due to the size quantization.
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46
8.7.11 Semiconductor Device Fabrication
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8.7.12 Digital Circuits and Memory Devices
8-39: AND device (A(G) and B(S) on makes On)
8-41: NAND (NOT-AND) device with one load
MOSFET and two input MOSFET transistors,
A and B on – OFF
50 A or B off - On
Either
8-40: inverter circuit (Gate on – OFF, Gate off – On)
8-42: OR device: Either A or B on - On
8-43: NOR device: Either A or B on - OFF
8-44: SRAM memory device called R-S flipflop with latch
8-46: DRAM memory device
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