The Bipolar Transistor
Ambipolar Transport &
Fundamental Concepts
Nonequilibrium Excess Carriers
Whenever there is current, the
semiconductor is in nonequilibrium state
Excess carriers are generated by various
processes, but in pairs.
Transport of excess carriers determines
the current
Excess Carriers and Excess Carrier Life Time
At thermal equilibrium
n0 p0 = ni2
when in nonequilibrium state, excess carriers are (typically)
excited in pairs
p = p0 + δp
n = n0 + δn
recombination can be approximated as
(
dn
= α r ni2 − np
dt
)
to the first order, or for low-level injection
dδn
= −α r (n0 + p0 )δn
dt
δp = δn
Excess Carrier Life Time
dδn
δn
= −α r (n0 + p0 )δn = −
dt
τ ex
τex is called the excess carrier life time
for n-type material n0>>p0 and
for p-type material p0>>n0 and
1
τ ex ≈
α r n0
1
τ ex ≈
α r p0
Generation-Recombination Processes
Band-to-Band Generation and Recombination
recombination-generation centers: defects, surface states, etc.
Auger Recombination
Continuity Equations
or
∂p 1
p
+ ∇J p = g p −
∂t e
τp
∂n 1
p
+
∇J n = g n −
∂t − e
τn
J p = eµ p pE − eD p ∇p
J n = eµ n pE + eDn∇n
∂p
p
+ ∇(µ p pE − D p ∇p ) = g p −
∂t
τp
n = n0 + δn
∂n
n
− ∇(µ n nE + Dn∇n ) = g n −
τn
∂t
or
D p ∇ 2 p − µ p (E∇p + p∇E ) + g p −
p = p0 + δp
p
τp
=
∂p
∂t
∂n
Dn∇ n + µ n (E∇n + n∇E ) + g n − =
τ n ∂t
2
n
δp = δn
Time-dependent diffusion equations for
excess carriers
Assume equilibrium carrier concentrations n0 and p0 are
time and space invariant
For example, Homogeneous region
D p ∇ 2δp − µ p (E∇δp + p∇E ) + g p −
Dn∇ 2δn + µ n (E∇δn + n∇E ) + g n −
n = n0 + δn
p = p0 + δp
p
τp
n
τn
=
∂δp
∂t
=
∂δn
∂t
δp = δn
Ambipolar Transport of Excess Carriers
Excess carriers tend to transport together
Once they are separated, an additional internal field is
created and tends to pull them back together
Results in Ambipolar Transport especially in weak
external field region
The additional internal field itself may be small compare
to external field, but its gradient can be large
E = Eapp + Eint
Eint << Eapp
but
∇Eint ≈ ∇Eapp
Ambipolar transport of excess carriers
D p ∇ 2δp − µ p (E∇δp + p∇E ) + g p −
Dn∇ 2δn + µ n (E∇δn + n∇E ) + g n −
p
τp
n
τn
=
=
∂δp
∂t
∂δn
∂t
δp = δn
Cancel away the gradient of the field term, one has
ambipolar transport equation for excess carriers:
D′∇ 2δn + µ ′E∇δn + g − R =
∂δn
∂t
Where ambipolar transport drift constant and mobility are
µ n nD p + µ p pDn
′
D =
µn n + µ p p
µn µ p ( p − n)
µ′ =
µn n + µ p p
Limits of Extrinsic Doping and Low Injection
For extrinsic doping and low injection, ambipolar
parameters reduce to the minority-carrier values.
For p-type region
Dn∇ 2δn + µ n E∇δn + g ′ −
δn ∂δn
=
τ n0
∂t
For n-type region
D p ∇ δp − µ p E∇δp + g ′ −
2
δp ∂δp
=
τ p0
∂t
The steady state solution yields excess carrier distribution
that give rise to current expressions.
Bipolar Transistor: Basic Structure
Bipolar Transistor: The Modes of Operation
Cutoff: no current in the junctions
Forward active: Ic controlled by VBE
Saturation: Ic not controlled by VBE
Bipolar Transistor: The Modes of Operation
Load Line (from C-E loop)
VCE = VCC − I C RC
Minority-carrier distribution: Forward-active mode
Apply ambipolar transport
equation to each region:
• steady state
• g’=0 (Why?) E=0 (Why?)
Base Region: p-type
Dn∇ δn + µ n E∇δn + g ′ −
2
δn ∂δn
=
τ n0
∂t
d 2δnB δnB
DB
−
=0
2
dx
τ B0
δnB (x ) = nB (x ) − nB 0
nB (0 ) = nB 0 e eVBE
nB ( xB ) = nB 0 e eVBC
kT
kT
≈0
Band Diagram
Whenever Fermi levels split, excess carriers are generated.
The excess carrier concentration is proportional to
exp(eVsplit/KT)-1 or exp(eVbias/KT)-1
Minority-carrier distribution: Forward-active mode
Emitter Region: n-type
d 2δp E δpE
−
=0
DE
dx′2
τ E0
δpE (x′) = p E (x′) − pE 0
pE (0 ) = pE 0 e eVBE
kT
pE ( x′E ) = pB 0
Minority-carrier diffusion length:
LE ≡ DEτ E 0
Minority-carrier distribution: Forward-active mode
Collector Region: n-type
d 2δpC δpC
−
=0
DC
dx′′2 τ C 0
δpC (x′′) = pC (x′′) − pE 0
pC (0 ) = pC 0 e eVBC
kT
≈0
pC ( xC′′ ) = pC 0
Minority-carrier diffusion length:
LC ≡ DCτ C 0
Simplified Current Relations: Forward-Active
Collector current
• B-C is reverse bias, drift
current is negligible,
diffusion current
dominates
iC = eDn ABE
iC =
dnC ( x )
dx
− eDn ABE
nB 0 eVBE Vt
xB
iC = I S eVBE
Vt
Simplified Current Relations: Forward Active
Emitter current
• B-E is forward biased,
drift current is not
negligible.
• diffusion current:
iE1 = iC = I S eVBE
• drift current:
iE 2 = I S 2 eVBE
Vt
iE = iE1 + iE 2 = I SE eVBE
Vt
Vt
Common-Base Current Gain
Both collector and emitter current is proportional
to exp(VBE/Vt), the ratio between two is therefore
a constant, which is defined as common base
current gain
iC = I S eVBE
iE = iE1 + iE 2 = I SE eVBE
Vt
α≡
iC
<1
iE
Vt
Other Modes of Operation
Cutoff mode
Minority carrier
distribution in base
region is almost flat
and there is no
diffusion current
All pn junctions are
reverse biased, no
drift current either.
Other Modes of Operation
Saturation mode
Both pn junctions
are forward
biased. Drift
current dominates
VBE influence on
collector current is
much weaker
Common-Base Current Gain:
Contributing Factors
Emitter injection efficiency factor:
γ=
J nE
J nE + J pE
Base transport factor:
αT =
J nC
J nE
Recombination factor:
δ=
J nC
= γα T δ
α=
J nE + J pE + J R
J nE + J pE
J nE + J pE + J R
Emitter Injection Efficiency Factor
Emitter injection efficiency factor:
γ=
J nE
J nE + J pE
J pE = −eDE
dδpE
dx ′
J nE = −eDB
γ≈
1
N D x
1+ B E B
N E DB xE
dδnB
dx
x′=0
x =0
Base Transport Factor
Base transport factor:
αT =
J nC
J nE
J nC = −eDB
dδnB
dx
J nE = −eDB
1
1  xB 
≈ 1 −  
αT ≈
cosh xB LB
2  LB 
dδnB
dx
2
LB ≡ DBτ B 0
x = xB
x =0
Recombination Factor
Recombination factor:
δ=
J nE + J pE
J nE + J pE + J R
J R = J r 0 eVBE
2Vt
J E = J nE + J pE = J s 0eVBE Vt
δ=
1
J r 0 −VBE
1+
e
J s0
2Vt
Non-ideal Effects: Base Width Modulation
xB is a function of the B-C voltage
Early Voltage
Non-ideal Effects: High Injection
Reduction in emitter injection efficiency
J pE ↑, γ ↓, α ↓
Non-ideal Effects: Emitter Bandgap Narrowing
ni2 = N c N v e
niE2 = ni2 e
pE 0
Reduction in emitter injection efficiency
− E g kT
∆E g kT
niE2
ni2 ∆E g
=
=
e
NE NE
pE 0 ↑, γ ↓, α ↓
kT
Non-ideal Effects: Current Crowding
Lateral voltage drop affects carrier injection
Non-ideal Effects: Breakdown Voltage
Punch-through
Avalanche breakdown
Hybrid-PI Equivalent Circuit Model
Frequency Limitations
α=
α0
1 + j f fT
Cutoff Frequency fT:
fT =
1
2πτ ec
Common emiter current gain β
α
β=
1−α
β ≈
fT
f
Emitter-to-collector time delay
HW7. 7.3, 8.7, 8.16, 10.3,10.4, 10.9
Due Thursday, May 1st,