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1
Chapter 11-1 Detailed Quantitative Analysis
pnp transistor, steady state, low-level injection. Only drift and diffusion, no external generations One dimensional etc.
Assumptions:
General approach is to solve minority carrier
diffusion equations for each of the three regions:
The goal is to relate transistor performance parameters
( γ, αT , βdc etc. ) to doping, lifetimes, base-widths etc.
G L
p x
p D t
p
τ
(^2) p
2 p L (^2) n
2 n G
n x
n D t
n
τ
and ∂∆
2
General Quantitative Analysis
Under steady state and when G L= 0,
(^2) n
2 n (^) τ =
∂ ∆ n x
n 0 D (^2) p
2 p (^) τ =
∂ ∆ p x
p D and
For the base in pnp, we are interested only in holes.
p 2
2 p = τ
∂ ∆ p x
p D
The rigorous analysis is carried out in chapter 11, but we are going to take a more simplified approach.
3
Review: Operational Parameters
These parameters can be related to device parameters such as doping, lifetimes, diffusion lengths, etc.
Base transport factor : αT = I C / I EP
Collector to emitter current gain: αDC = αT γ
Collector to base current gain: βDC = αDC / (1 – αDC )
Injection Efficiency : γ= I EP /( I EP+ I EN)
I E I C
I B
−
I EP
I BR
4
Review: Indirect thermal recombination-generation
n 0 , p 0 – under thermal equilibrium n , p – under arbitrary conditions, functions of t
∆ n = n – n 0 ∆ p = p – p 0
Low-level injection condition is assumed.
- Change in the majority carrier concentration is negligible. For example, in n-type material, ∆ p << n 0 ; n ≈ n 0. in p-type material, ∆ n << p 0 ; p ≈ p 0.
∆ n and ∆ p are deviations in carrier concentrations from their equilibrium values. ∆ n and ∆ p can be both positive or negative. ∆ n and ∆ p are termed excess carriers – excess above the equilibrium concentration.
Majority carriers are electrons in n-type material and holes in p-type material.
7
Review of P-N Junction Under Forward Bias (cont.)
I n = q A D E d∆ n /d x E = – ( q A D E/ L E ) ∆ n E(0) I p = – q A D B d∆ p /d x B = ( q A D B/ L B) ∆ p B(0)
Total current I = I P + (– I N) (“–” because x E and x B point in opposite directions)
= ( q A D B/ L B) ∆ p B(0) + ( q A D E/ L E) ∆ n (^) E (0)
= ( q A D B/ L B) p B0 [ exp ( q V EB / kT ) – 1 ] +
- ( q A D E/ L E) n E0 [ exp ( q V EB / kT ) – 1 ]
≈ ( q A D B/ L B) p B0 exp ( q V EB/ kT ) + ( q A D E/ L E) n E0 exp ( q V EB/ kT )
Note : I p and I n can also be calculated based on the fact that Q p has to be replaced every τ B seconds Î I p = Q p / τ B and I n = Q n / τ E and I E = I P + I N
8
Simplified Analysis
Consider the carrier distribution in a forward active pnp transistor
n E0^ p B
n C
Emitter Base Collector
n C(0)
p B(0)
n E(0)
9
Simplified Analysis (cont.)
n E0 , p B0 and n C0 = equilibrium concentration of minority carriers in emitter, base and collector
n E(0), p B(0) and n C(0) = minority carrier concentration under forward active conditions at the edge of the respective depletion layers
∆ n E (0), ∆ p B(0) and ∆ n C(0) = excess carrier concentration at the edge of the depletion layers
E
(p+)
B
(n)
C
(p)
10
Simplified Analysis (cont.)
∆ n E (0) = n E (0) – n E0 = n E0 [exp ( q V EB / kT ) – 1]
∆ p B (0) = p B (0) – p B0 = p B0 [exp (q V EB / kT ) – 1]
By taking the slopes of these minority carrier distribution at the depletion layer edges and multiplying it by “ q A D n,p ”, we can get hole and electron currents.
Note that I n = q A D n (d n / d x ) and I p = – q A D p (d p / d x )
13
Calculation of Currents (cont.)
Emitter Current (cont.)
I EN corresponds to electron current injection from base to emitter since E-B junction is forward biased.
I EN = q A ( D (^) E / L E) n E0 [exp ( q V EB / kT ) – 1 ] ≈ q A ( D (^) E / L E) n E0 [exp ( q V EB / kT )] ---- (C)
14
Calculation of Currents (cont.)
Base Current, I B
-supplies electrons for recombination in base -supplies electrons for injection to emitter
I B =^ q A p B0 [ W B / (2^ τB )] [exp ( qV EB /^ kT ) ]
q A ( D (^) E / L E) n E0 exp ( qV EB / kT )
( recombination) + (electron injection to emitter)
Now we can find transistor parameter easily.
15
Calculation of Currents (cont.)
Base transport factor , α (^) T
(same as eq. 11.42 in text)
B B 0 B
E E 0 E /
/ 1
1
D p W
D n L
=
2
B
B 0 EB B B
EB B B 0 B
B
EB B 0 B
B
EP
T C
2
1 1
1 exp 2
exp
exp
⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛
=
τ ⎟+ ⎠
⎞ ⎜ ⎝
⎛
⎟ ⎠
⎞ ⎜ ⎝
⎛
α = =
L
W kT
qV p qAW kT
qV p W
qAD
kT
p qV W
qAD
I
I
Emitter injection efficiency, γ
γ = I EP / [ I EP + I EN ]
= 1 / [ 1 + I EN / I EP ]
= 1 / [ 1 + (C) / (B) ]
16
Calculation of Currents (cont.)
Î n E0 = n i^2 / N E … doping in emitter Î p B0 = n i^2 / N B … doping in base
B E E
E B B B E B
E B E 01
D L N
D W N
D L p
D W n
γ = (Eq 11.41 in textbook)
αdc = γ αT
βDC = αDC / (1– αDC )
