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ENE 311
Lecture 10
Ohmic Contact
• For metal-semiconductor contacts with low doping
concentration, the thermionic-emission current dominants
the current transport.
• Rc can be written as
k
 eb / kT 
Rc  ** e
eA T
(1)
* As seen from equation (1), in order to have a small value
of Rc, a low barrier height should be used.
Ohmic Contact
• For metal-semiconductor contacts with high
doping concentration, the barrier width
becomes very narrow and the tunneling current
becomes dominant.
• The tunneling current can be found by
 4 m*   V  
e s
b

J  J 0 exp  
ND


(2)
Ohmic Contact
• The specific contact
resistance for high doping
is
 4 m*
1
e s
 J0 

RC
ND


 4 m*  
e s b
 exp  




N
D



Upper inset shows the tunneling process.
Lower inset shows thermionic emission over the low barrier.
Ohmic Contact
Ex. An ohmic contact has an area of 10-5 cm2 and a
specific contact resistance of 10-6 Ω-cm2. The
ohmic contact is formed in an n-type silicon. If
ND = 5 x 1019 cm-3 and b = 0.8 V, and the electron
effective mass is 0.26m0, find the voltage drop
across the contact when a forward current of 1
A flows through it.
Ohmic Contact
Soln The contact resistance for the ohmic contact
is RC/Area = 10-6/ 10-5 = 0.1 Ω.
1
J

RC V
Let
V 0
4 me* s
.cm 2 
1
 C2
4 0.26  9.1  1031  11.9  8.85  10 12
14
-3/2 -1
C2 

1.9

10
m
V
34
1.05  10
 C2 b  V  
I  I 0 exp  

N D 

 C 
 C 
I
A

 I 0  2  exp   2 b 
 N 

V V 0 RC
N D 
D 


Ohmic Contact
Soln
1
 C2b 
A  C2 
I0 

 exp 

RC  N D 
 ND 
 5  1019  106 
 1.9  1014  0.8 
 10  
 exp 

19
16
 1.9  1014 
 5  10  10 


 8.13  108 A
N D  I0 
At I = 1A, we have b  V 
ln    0.763 V
C2
 I 
V  0.8  0.763  0.037 V
Transistor
• Transistor (Transfer resistor) is a multijunction
semiconductor device.
• Generally, the transistor is used with other
circuit elements for current gain, voltage gain,
or even signal-power gain.
• There are many types of transistors, but all of
them are biased on 2 major kinds: bipolar
transistor and unipolar transistor.
Bipolar Junction Transistor (BJT)
• The BJT was invented by Bell laboratories in
1947. It is an active 3-terminal device that can
be used as an amplifier or switch.
• It is called bipolar since both majority and
minority carriers participate in the conduction
process.
• Its structure is basically that 2 diodes are
connected back to back in the form of p-n-p or
n-p-n.
Bipolar Junction Transistor (BJT)
Bipolar Junction Transistor (BJT)
• (a) A p-n-p transistor
with all leads grounded
(at thermal
equilibrium).
• (b) Doping profile of a
transistor with abrupt
impurity distributions.
• (c) Electric-field
profile.
• (d) Energy band
diagram at thermal
equilibrium.
Bipolar Junction Transistor (BJT)
Operational Mode
Emitter-base
junction
Collector-base
junction
Active (normal)
Forward
Reverse
Cutoff
Reverse
Reverse
Saturation
Forward
Forward
Inverse
Reverse
Forward
Bipolar Junction Transistor (BJT)
• When the transistor is
biased in the active
mode, holes are injected
from the p+ emitter into
the base and electrons
are emitted from the n
base into the emitter.
•
For the collector-base
reverse biased junction, a
small reverse saturation
current will flow across
the junction.
Bipolar Junction Transistor (BJT)
• However, if the base
width is very narrow, the
injected holes can diffuse
through the base to reach
the base-collector
depletion edge and then
float up into the
collector.
•
This is why we called
them “emitter” and
“collector” since they
emit or inject the carriers
and collect these injected
carriers, respectively.
Bipolar Junction Transistor (BJT)
• IEp is the injected hole
current. Most of these
injected holes survive the
recombination in the
base, they will reach the
collector giving ICp.
• There are three other
base current: IBB, IEn, and
ICn. IBB is the electrons
that must be supplied by
the base to replace
electrons recombined
with the injected holes.
IBB = IEp – ICp.
Bipolar Junction Transistor (BJT)
• IEn is the injected electron
current (electrons
injected from the base to
the emitter.).
• ICn corresponds to
thermally generated
electrons that are near
the base-collector
junction edge and drift
from the collector to the
base.
Bipolar Junction Transistor (BJT)
I E  I Ep  I En
(4)
I C  I Cp  I Cn
(5)
I B  I E  IC  I En   I Ep  ICp   ICn
(6)
Bipolar Junction Transistor (BJT)
• The crucial parameter called “common-base
current gain” α0 is defined by
0 
I Cp
(7)
IE
• Substituting (4) into (7) yields
0 
I Cp
I Ep  I En
 T
(8)
Bipolar Junction Transistor (BJT)
• γ is the emitter efficiency written as

I Ep
IE
(9)
• αT is the base transport factor written as
T 
I Cp
I Ep
(10)
Bipolar Junction Transistor (BJT)
• For a well-designed and fabricated transistor, IEn
is small compared to IEp and ICp is close to IEp.
• Therefore, γ and α are close to 1 and that
makes α0 is close to unity as well. Thus, the
collector current can be expressed by
I C  T I Ep  I Cn   0 I E  I Cn
(11)
Bipolar Junction Transistor (BJT)
• Normally, ICn is know as ICB0 or the leakage
current between the collector and the base with
the emitter-base junction open.
• Thus, the collector current can be written as
IC   0 I E  ICB 0
(12)
Bipolar Junction Transistor (BJT)
In order to derive the current-voltage expression
for an ideal transistor, we assume the following:
• The device has uniform doping in each region.
• The hole drift current in the base region and
the collector saturation current is negligible.
• There is low-level injection.
• There are no generation-combination currents
in the depletion regions.
• There are no series resistances in the device.
Bipolar Junction Transistor (BJT)
Minority carrier distribution in various regions of a p-n-p
transistor under the active mode of operation.
Bipolar Junction Transistor (BJT)
• The distributions of the minority carriers can be found by
pn ( x)  pn 0e
eVEB / kT
x

1 
 W
x


  pn (0) 1  

 W
nE ( x)  nE 0  nE 0  eeVEB / kT  1 e
nC ( x)  nC 0  nC 0e
 x  xC 


L
 C 
x  xE
LE
for x  - xE
for x  xC
pn0, nE0, and nC0 are the equilibrium minority-carrier
concentrations in the base, emitter, and collector, respectively.
LE and LC are emitter and collector diffusion lengths,
respectively.
Bipolar Junction Transistor (BJT)
• Now the minority-carrier distributions are
known, the current components can be
calculated. The emitter current can be found by
I Ep 
eADp pn0
W
e
eVEB / kT
I En 
I E  I Ep  I En  a11  eeVEB / kT  1  a12
eAD p pn 0
 D p pn 0 DE nE 0 
a11  eA 

 , a12 
W
L
W

E



eADE nE 0 eVEB / kT
e
1
LE
(16)
Bipolar Junction Transistor (BJT)
• The collector current is expressed by
I Cp 
eADp pn0
W
e
eVEB / kT
I Cn
eADC nC 0

LC
I C  I Cp  I Cn  a21  eeVEB / kT  1  a22
a21  a12 
eADp pn 0
W
 Dp pn 0 DC nE 0 
, a22  eA 


W
L
C


(17)
Bipolar Junction Transistor (BJT)
• The ideal base current is IE – IC or
I B   a11  a21   eeVEB / kT  1   a12  a22 
(18)
Bipolar Junction Transistor (BJT)
Ex. An ideal Si p+-n-p transistor has impurity
concentrations of 1019, 1017, and 5 x 1015 cm-3 in
the emitter, base, and collector regions,
respectively; the corresponding lifetimes are
10-8, 10-7, and 10-6 s. Assume that an effective
cross section area A is 0.05 mm2 and the
emitter-base junction is forward-biased to 0.6
V. Find the common-base current gain of the
transistor. Note: DE = 1 cm2/s, Dp = 10 cm2/s, DC
= 2 cm2/s, and W = 0.5 μm.
Bipolar Junction Transistor (BJT)
Soln
In the base region,
L p  D p p  10  107  103 cm
9.65  10

n
pn 0 

NB
1017
2
i

9 2
 9.31 102 cm -3
Bipolar Junction Transistor (BJT)
In the emitter region,
LE  DE E  1 108  104 cm
nE 0
I Ep
I Cp
2
i
n


NE
 9.65 109 
1019
2
 9.31 cm -3
1.6  1019  5  104  10  9.31  10 2 0.6 / 0.0259
4


e

1.7137

10
A
4
0.5  10
 I Ep  1.7137  104 A
1.6  1019  5  104  1  9.31 0.6 / 0.0259
8
I En 

e

1

8.5687

10
A


4
10
I Cp
0 
 0.9995
I Ep  I En
Bipolar Junction Transistor (BJT)
Bipolar Junction Transistor (BJT)
• The general expressions of currents for all
operational modes are

e
 
 1  a  e

 1
I E  a11 eeVEB / kT  1  a12 eeVCB / kT  1
I C  a21
eVEB / kT
22
eVCB / kT
(19)
Current-Voltage Characteristics of
Common-Base Configuration
• In this
configuration, VEB
and VCB are the
input and output
voltages and IE
and IC are the
input and output
currents,
respectively.
Current-Voltage Characteristics of
Common-Emitter Configuration
• In many circuit
applications, the
common-emitter
configuration is
mostly used
where VEB and IB
are the input
parameters and
VEC and IC are the
output
parameters.
Current-Voltage Characteristics of
Common-Emitter Configuration
• The collector current for this configuration can
be found by substituting (6) into (12)
IC  0  I B  IC   ICB 0
0
I CB 0
IC 
IB 
1  0
1  0
(20)
Current-Voltage Characteristics of
Common-Emitter Configuration
• We define β0 as the common-emitter current
gain as
0 
I C
0

I B 1   0
(21)
• Then, ICE0 can be written as
I CE 0
I CB 0

   0  1 I CB 0
1  0
(22)
Current-Voltage Characteristics of
Common-Emitter Configuration
• Therefore, (20) becomes
IC  0 I B  ICE 0
(23)
• Since α0 is generally close to unity, β0 is much
larger than 1.
• Therefore, a small change in the base current
can give rise to a much larger change in the
collector current.
Frequency response
• (a) Basic transistor
equivalent circuit (low
frequency).
• (b) Basic circuit with the
addition of depletion and
diffusion capacitances
(higher frequency).
• (c) Basic circuit with the
addition of resistance and
conductance (high
frequency).
Frequency response
• For a high frequency, we
expect to have these
following components:
• CEB = EB depletion
capacitance, Cd =
diffusion capacitance, CCB
= CB depletion
capacitance, gm =
transconductance = iC/vEB,
gEB = input conductance =
iB/vEB, gEC = iC/v = output
conductance, rB = base
resistance, and rC =
collector resistance.
Frequency response
• The current gain will decrease after the certain
frequency is reached. The common-base current
gain α can be expressed by

0
1  j  f / f 
(24)
• where α0 is the lowest frequency common-base
current gain and fα is the common-base cutoff
frequency.
Frequency response

0
1  j  f / f 
(25)
where fβ is the common-emitter cutoff frequency
given by (1-α0) fα.
• Whereas fT is the cutoff frequency when β = 1.
fT  0 1  0  f  0 f
• fT is pretty close to but smaller than fα.
(26)
Frequency response
• fT is pretty close to but smaller than fα.