1. Lecture 13: Part I: MOS Small-Signal Models
Prof. Niknejad

2. Lecture Outline MOS Small-Signal Model (4.6)
Diode Currents in forward and reverse bias ( )
Department of EECS
University of California, Berkeley

3. Total Small Signal Current
Transconductance
Conductance
Department of EECS
University of California, Berkeley

4. Role of the Substrate Potential
Need not be the source potential, but VB < VS
Effect: changes threshold voltage, which changes the drain current … substrate acts like a “backgate”
Q = (VGS, VDS, VBS)
Department of EECS
University of California, Berkeley

5. Backgate Transconductance
Result:
Department of EECS
University of California, Berkeley

6. Four-Terminal Small-Signal Model
Department of EECS
University of California, Berkeley

7. MOSFET Capacitances in Saturation
Gate-source capacitance: channel charge is not controlled by drain in saturation.
Department of EECS
University of California, Berkeley

8. Gate-Source Capacitance Cgs
Wedge-shaped charge in saturation  effective area is (2/3)WL
(see H&S for details)
Overlap capacitance along source edge of gate 
(Underestimate due to fringing fields)
Department of EECS
University of California, Berkeley

9. Gate-Drain Capacitance Cgd
Not due to change in inversion charge in channel
Overlap capacitance Cov between drain and source
is Cgd
Department of EECS
University of California, Berkeley

10. Junction Capacitances
Drain and source diffusions have (different) junction capacitances since VSB and VDB = VSB + VDS aren’t the same
Complete model (without interconnects)
Department of EECS
University of California, Berkeley

11. P-Channel MOSFET
Measurement of –IDp versus VSD, with VSG as a parameter:
Department of EECS
University of California, Berkeley

12. Square-Law PMOS Characteristics
Department of EECS
University of California, Berkeley

13. Small-Signal PMOS Model
Department of EECS
University of California, Berkeley

14. MOSFET SPICE Model
Many “levels” … we will use the square-law “Level 1” model
See H&S Spice refs. on reserve for details.
Department of EECS
University of California, Berkeley

15. Part II: Currents in PN Junctions

16. Diode under Thermal Equilibrium
Minority Carrier Close to Junction
Thermal
Generation
p-type
n-type - + − − −
Recombination +
Carrier with energy
below barrier height + −
Diffusion small since few carriers have enough energy to penetrate barrier
Drift current is small since minority carriers are few and far between: Only minority carriers generated within a diffusion length can contribute current
Important Point: Minority drift current independent of barrier!
Diffusion current strong (exponential) function of barrier
Department of EECS
University of California, Berkeley

17. Reverse Bias Reverse Bias causes an increases barrier to diffusion
Diffusion current is reduced exponentially
p-type
n-type - + −
Drift current does not change
Net result: Small reverse current
Department of EECS
University of California, Berkeley

18. Forward Bias
Forward bias causes an exponential increase in the number of carriers with sufficient energy to penetrate barrier
Diffusion current increases exponentially
p-type
n-type - + −
Drift current does not change
Net result: Large forward current
Department of EECS
University of California, Berkeley

19. Diode I-V Curve Diode IV relation is an exponential function
This exponential is due to the Boltzmann distribution of carriers versus energy
For reverse bias the current saturations to the drift current due to minority carriers
Department of EECS
University of California, Berkeley

20. Minority Carriers at Junction Edges
Minority carrier concentration at boundaries of depletion region increase as barrier lowers … the function is
(minority) hole conc. on n-side of barrier
(majority) hole conc. on p-side of barrier
(Boltzmann’s Law)
Department of EECS
University of California, Berkeley

21. “Law of the Junction”
Minority carrier concentrations at the edges of the
depletion region are given by:
Note 1: NA and ND are the majority carrier concentrations on
the other side of the junction
Note 2: we can reduce these equations further by substituting
VD = 0 V (thermal equilibrium)
Note 3: assumption that pn << ND and np << NA
Department of EECS
University of California, Berkeley

22. Minority Carrier Concentration
Diffusion Length
The minority carrier concentration in the bulk region for
forward bias is a decaying exponential due to recombination
Department of EECS
University of California, Berkeley

23. Steady-State Concentrations
Assume that none of the diffusing holes and electrons recombine  get straight lines …
This also happens if the minority carrier diffusion lengths are much larger than Wn,p
Department of EECS
University of California, Berkeley

24. Diode Current Densities
Department of EECS
University of California, Berkeley

25. Fabrication of IC Diodes
cathode
annode p+ p
p-type n+
n-well
p-type
Start with p-type substrate
Create n-well to house diode
p and n+ diffusion regions are the cathode and annode
N-well must be reverse biased from substrate
Parasitic resistance due to well resistance
Department of EECS
University of California, Berkeley

26. Diode Small Signal Model
The I-V relation of a diode can be linearized
Department of EECS
University of California, Berkeley

27. Diode Capacitance
We have already seen that a reverse biased diode acts like a capacitor since the depletion region grows and shrinks in response to the applied field. the capacitance in forward bias is given by
But another charge storage mechanism comes into play in forward bias
Minority carriers injected into p and n regions “stay” in each region for a while
On average additional charge is stored in diode
Department of EECS
University of California, Berkeley

28. Charge Storage
Increasing forward bias increases minority charge density
By charge neutrality, the source voltage must supply equal and opposite charge
A detailed analysis yields:
Time to cross junction
(or minority carrier lifetime)
Department of EECS
University of California, Berkeley

29. Diode Circuits Rectifier (AC to DC conversion) Average value circuit
Peak detector (AM demodulator)
DC restorer
Voltage doubler / quadrupler /…
Department of EECS
University of California, Berkeley