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MOS Transistors Yannis Tsividis
Small-Signal Modeling - Source-Drain and Output Conductances
Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011 1
These slides are based on Y. Tsividis and C. McAndrew, “Operation and Modeling of the MOS Transistor”, Copyright © Oxford University Press, 2011. They are meant to be part of a lecture, and may be incomplete or may not even make sense without the accompanying narration.
Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011
Source-drain conductance
In saturation, we have a major problem:
2
In non-saturation: 𝑔𝑠𝑑 =𝜕𝐼𝐷𝑆𝜕𝑉𝐷𝑆
=𝑊
𝐿𝜇𝐶′𝑜𝑥(𝑉𝐺𝑆 - 𝑉𝑇 -𝛼𝑉𝐷𝑆)
𝑔𝑠𝑑 =𝜕𝐼𝐷𝑆𝜕𝑉𝐷𝑆
0 𝑉𝐷𝑆
𝑔𝑠𝑑
Error: > 50%
𝐼𝐷𝑆
0 𝑉𝐷𝑆
Error: < 1%
Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011
1. Channel length modulation
Simplest model:
𝐼𝐷𝑆 = 𝐼𝐷𝑆′ 1 +
𝑉𝐷𝑆 − 𝑉𝐷𝑆′
𝑉𝐴
𝑔𝑠𝑑 =𝜕𝐼𝐷𝑆𝜕𝑉𝐷𝑆
=𝐼𝐷𝑆′
𝑉𝐴 can be very inaccurate.
3
𝐼𝐷𝑆
𝑉𝐷𝑆
𝐼𝐷𝑆′
0 𝑉𝐷𝑆′
Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011
Channel length modulation, cont’d:
𝐼𝐷𝑆 =𝐼𝐷𝑆′
1 −𝑙𝑝𝐿
(saturation)
𝑔𝑠𝑑 =𝜕𝐼𝐷𝑆𝜕𝑉𝐷𝑆
=𝜕𝐼𝐷𝑆𝜕𝑙𝑝
𝜕𝑙𝑝
𝜕𝑉𝐷𝑆≈ 𝐼𝐷𝑆
′1
𝐿
𝜕𝑙𝑝
𝜕𝑉𝐷𝑆
E.g. with 𝑙𝑝 = 𝑙𝑎ln 1 +𝑉𝐷𝑆 − 𝑉𝐷𝑆
′
𝑉𝐸
𝑔𝑠𝑑 =𝐼𝐷𝑆′
𝐿𝑙𝑎
𝑉𝐸 + 𝑉𝐷𝑆 − 𝑉𝐷𝑆′
=𝐼𝐷𝑆′
𝑉𝐴 𝑉𝐷𝑆
4
A more careful analysis:
Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011
2. DIBL
𝑔𝑠𝑑 =𝜕𝐼𝐷𝑆𝜕𝑉𝐷𝑆
=𝑊
𝐿
𝜇𝐶𝑜𝑥′
𝛼𝑉𝐺𝑆 − 𝑉 𝑇 𝑉𝐷𝑆 −
𝜕𝑉 𝑇𝜕𝑉𝐷𝑆
⇒ 𝑔𝑠𝑑 = 𝑔𝑚(−𝜕𝑉 𝑇𝜕𝑉𝐷𝑆
)
𝑔𝑚
5
𝐼𝐷𝑆 =𝑊
𝐿
𝜇𝐶𝑜𝑥′
2𝛼𝑉𝐺𝑆 − 𝑉 𝑇 𝑉𝐷𝑆
2 (saturation)
For a given DIBL model, use 𝑉 𝑇 𝑉𝐷𝑆 = 𝑉𝑇 + ∆𝑉𝑇,𝐷𝐼𝐵𝐿(𝑉𝐷𝑆) in the above to get 𝑔𝑠𝑑 (see book).
Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011 6
𝑔𝑠𝑑 =𝐼𝐷𝑆′
𝑉𝐴𝑊 , saturation
𝑉𝐴𝑊 can be smaller than the strong inversion 𝑉𝐴
In weak inversion saturation, DIBL can be dominant:
Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011
Body-drain conductance
𝑔𝑏𝑑 ≡𝜕𝐼𝐵𝜕𝑉𝐵𝐷
7
Can be important in analog circuits in some processes.
g bd2
g bd1
M1
M2
g bd2
M1
M2
g bd2
M1
M2
g bd1
Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011
Output conductance
𝑔𝑜 =𝜕𝐼𝐷𝜕𝑉𝐷𝑆
=𝜕 𝐼𝐷𝑆 + 𝐼𝐷𝐵 + 𝐼𝐷𝐺
𝜕𝑉𝐷𝑆
= 𝑔𝑠𝑑 + 𝑔𝑏𝑑 + 𝑔𝑔𝑑
8
𝑉𝐷𝑆
𝑉𝐷𝑆
𝐼𝐷
𝑔𝑜 0
0
(logscale)
Non − saturation
CLM and DIBL
Impact ionization
Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011 9
𝐿 = 5 μm
Large variation
𝑔𝑜(S)
𝑉𝐷𝑆(V)
𝐿 = 0.09 μm
𝑔𝑜(S)
𝑉𝐷𝑆(V)
In long channel case, CLM can be made very small, & impact ionization effects are revealed.
Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011 10
Effect of extrinsic resistances:
𝑔𝑚,𝑒𝑓𝑓 =𝑔𝑚
1 + 𝑔𝑚𝑅𝑠𝑒
𝑉 𝑆𝐵 = 𝑉𝑆𝐵 − 𝑅𝑏𝑒𝐼𝐷𝐵
𝐼𝐷𝐵 ↑ ⇒ 𝑉 𝑆𝐵 ↓ ⇒ 𝑉 𝑇 ↓ ⇒ 𝐼𝐷𝑆 ↑
𝑔𝑜,𝑒𝑓𝑓 ≈ 𝑔𝑠𝑑 + 𝑔𝑔𝑑 + 𝑔𝑏𝑑 1 + 𝑔𝑚𝑏𝑅𝑏𝑒
𝑅𝑠𝑒 𝑅𝑑𝑒
