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Semiconductor Devices - Hour 30
BJT Example: Part II
(MCD: Entire Bipolar Calculation)
Last Time:
Started with the physical structure of a bipolar transistor
micron
4 −
cm⋅
N
Emitter
P Base
N
-^
Collector
Picked the doping of each layer:
N
d_E
19
cm
3
⋅
N
a_B
17
cm
3
⋅
N
d_C
(^15)
cm
3
Picked the full ("metallurgical") thickness of each layer:
0.2micron
0.2micron
250micron
Immediately ran into a problem - what are the "effective" layer thicknesses?
x
E_eff
x
B_eff
x
C_eff
W
EB
W
BC
1
Effective thicknesses depend on depletion layers - which depend on voltages applied to junctionsSo next I had decide how I was going to USE the transistor!
Chose:
- "Common emitter" amplifier circuit- Power supply voltage of 5 volts- Emitter-base forward bias of 0.5 volts- Small enough load resistor that could ignore
its voltage drop => collector at ~ 5 volts
R
load
C
B
~ 5 volts
E
0 volts
Note, final choice (small load R) may not be good circuit sense but allowed me to simplify OUR problemV
applied_left
V
p_layer
V
n_layer
−
V
BE
V
applied_right
V
p_layer
V
n_layer
−
V
BC
V
BE
0.5volt :=
V
BC
volt
Signs on voltages are EXTREMELY important and follow from diagram and V
xy
= V
x^
- V
y
2
Choice 1: Complex hyperbolic equations with both V
BE
and V
BC
voltage factors (book/my earlier lectures):
Completely general
: Any layer thickness / Any junction bias
However:
A bear to use!
Choice 2: "Straight-line" approximations for emitter and base layers
Appropriate if
:^
Emitter & base layers are << thinner than their minority carrier's diffusion length
AND
E-B junction slightly forward biased / B-C junction strongly reverse biased
This is exactly what you will want anyway, if are trying to design an amplifying transistor!
So we WANT a transistor that will obey simpler equations - just need to check to see if we DO satisfy conditions:
Voltage bias conditions WERE chosen to satisfy requirements!Are our emitter and base layers thinner than diffusion lengths?
Had to fully evaluate parameters of MINORITY CARRIER in each layer:
4
Equilibrium minority carrier concentration is each layer is n
(^2) i / (majority carrier concentration):
p
no_C
n
(^2) i
n
no_C
=
p
no_E
n
(^2) i
n
no_E
=
n
po_B
n
(^2) i
p
po_B
=
p
no_E
cm
(^3)
n
po_B
(^3)
⋅^
cm
(^3)
p
no_C
(^5)
⋅^
cm
(^3)
Then to get minority carrier diffusion lengths ( L
D
τ⋅
) needed minority diffusivity and lifetime
Assumed for all minority carriers:
τ
6 −
sec⋅
To get the diffusivities, used Einstein's Relationship:
D μ
k T
⋅ q
=
For which we needed the minority carrier mobilities in each layer - based on ion concentration in each layer:
Emitter
Base
Collector
N
d_E
(^19)
×
cm
(^3)
N
a_B
(^17)
×
cm
(^3)
N
d_C
(^15)
×
cm
(^3)
Doping = Ions:
5
- Determine minority carrier TYPE:
P in emitter
N in base
P in collector
- Figure out ion concentration that causes scattering which controls mobility:
Ions = donor / acceptor that created MAJORITY CARRIERS
(^19)
⋅^
cm
(^3)
⋅^
emitter
(^17)
⋅^
cm
(^3)
⋅^
base
(^15)
⋅^
cm
(^3)
⋅^
collector
- Use correct curve in chart above (one for that TYPE carrier at that layer's ION concentration) Anonymous show of hands:
Does this minority stuff make sense yet?Should we go over more carefully or are you content w/ screwing up on final?
Used mobilities and Einstein Relationship to get minority carrier diffusivities for each layer:
D
p_E
μ
p_E
k T
⋅ q
D
n_B
μ
n_B
k T
⋅ q
D
p_C
μ
p_C
k T
⋅ q
D
p_E
cm
(^2) sec
D
n_B
cm
(^2) sec
D
p_C
cm
(^2) sec
Which finally yielded the minority carrier diffusion lengths:
7
L
p_E
D
p_E
τ⋅
L
n_B
D
n_B
τ⋅
L
p_C
D
p_C
τ⋅
L
p_E
13.461 micron
L
n_B
45.539 micron
L
p_C
34.435 micron
Which we compared with the layer's effective thicknesses:
x
E_eff
0.199 micron
x
B_eff
0.1 micron
x
C_eff
247.42 micron
For emitter and base:
Effective layer thickness ARE << minority diffusion lengths
For collector:
Effective thickness >> minority diffusion length
Use Straight lines
Use exponential / hyperbolic
For the minority carrier profiles in NPN bipolar transistor (from later part of lecture 27):
δ
p
nE
x( )
p
no_E
e
q V
BE ⋅^ k T
⋅^
x
E
x −
(^
x
E_eff ⋅
Emitter
Base
δ
n
pB
x( )
n
po_B
e
q V
BE ⋅^ k T
⋅^
x
B
x' −
(^
x
B_eff
⋅^
x' x
B_eff
−
8
- Emitter hole current:
Mathcad input:
P-Base
q
19 −
coul
N-Emitter
eV
q volt
N-Collector
k
5 −
⋅^
eVK
T
300K
p
n_E
x( )
p
no_E
e
q V
BE ⋅^ k T
⋅
x
E_eff
Diffusion current = - (charge) * Diffusivity * (concentration gradient)
J
pE
q D
p_E ⋅^
p
no_E ⋅
x
E_eff
e
q V
BE ⋅^ k T
⋅
J
pE
5 −
×
A
cm
(^2)
Holes flowing into thin emitter N-emitter
of an NPN bipolar transistor
- Electrons recombination inside the base:
10
P-Base
N-Emitter
N-Collector
n
po_B
e
q V
BE ⋅^ k T
⋅
n
p_B
x( )
x
B_eff
Recombination = Total stored minority charge / minority lifetime:
J
RB
q x
B_eff ⋅^
n
po_B ⋅ 2
τ⋅^
e
q V
BE ⋅^ k T
⋅
J
RB
7 −
×
A
cm
(^2)
Electrons recombining in thin base
of an NPN bipolar transistor
- Electrons flowing through the base:
11
Which in the thin base / low recombination / straight-line model this is ~ J
nC
J
nC
J
nE
:=
- Holes leaking in from collector:
P-Base
N-Emitter
N-Collector
δ
p
n_C
p
no_C
e
x − L p_C
⋅
Which plugging slope into diffusion current expression gives:
J
pC
q D
p_C ⋅^
p
no_C ⋅ L
p_C
J
pC
10 −
×
A
cm
(^2)
Holes flowing out of a thick collector
13
Summarizing our calculated currents for this transistor with our choices of operating voltages:
J
nE
A
cm
(^2)
J
RB
7 −
×
A
cm
(^2)
J
nC
A
cm
(^2)
J
pE
5 −
×
A
cm
(^2)
J
pC
10 −
×
A
cm
(^2)
Can now calculate this transistor's operating parameters (
with
our choices of voltages)
α
T
= "Base Transport Factor" = fraction of minority carriers making it across the base:
14
J
RB
J
nC
6 −
×
α
T
= 1 - 2.411 x 10
"Common-Emitter" Current Gain:
J
nE
J
nC
J
RB
J
E
J
nE
J
pE
J
C
J
nC
J
pC
J
pE
J
B
J
pE
J
RB
J
pC −
J
pC
β
J
C
J
B
D
p_E
N
a_B ⋅^
x
B_eff ⋅
D
n_B
N
d_E ⋅^
x
E_eff ⋅
x B_effLn_B
2
Calculating from the formula:
16
β
D
p_E
N
a_B ⋅^
x
B_eff ⋅
D
n_B
N
d_E ⋅^
x
E_eff ⋅
x
B_effLn_B
2
We have a pretty high gaintransistor!!
β
(^3)
×
Calculating from currents:
β
J
C
J
B
J
nC
J
pC
J
pE
J
RB
J
pC −
β
J
nC
J
pC
J
pE
J
RB
J
pC −
β
(^3)
×
γ^
"Emitter Injection Efficiency" = desirable E-B current / all E-B current
J
nE
Don't want J
pE
because adds to J
B
input I must provide
J
pE
17
