1. Spring 2007EE130 Lecture 30, Slide 1 Lecture #30 OUTLINE The MOS Capacitor Electrostatics Reading: Chapter 16.3

2. Spring 2007EE130 Lecture 30, Slide 2 Bulk Semiconductor Potential,  F p-type Si: n-type Si: EcEc EFEF EvEv EiEi qFqF EcEc EFEF EvEv EiEi |q  F |

3. Spring 2007EE130 Lecture 30, Slide 3 Voltage Drops in the MOS System In general, where qV FB =  MS =  M –  S V ox is the voltage dropped across the oxide (V ox = total amount of band bending in the oxide)  s  is the voltage dropped in the silicon (total amount of band bending in the silicon) For example: When V G = V FB, V ox =  s = 0 i.e. there is no band bending

4. Spring 2007EE130 Lecture 30, Slide 4 MOS Band Diagrams (n-type Si) Inversion –V G < V T –Surface becomes p-type Accumulation –V G > V FB –Electrons accumulate at surface Depletion –V G < V FB –Electrons repelled from surface Decrease V G (toward more negative values) -> move the gate energy-bands up, relative to the Si decrease V G

5. Spring 2007EE130 Lecture 30, Slide 5 Biasing Conditions for p-type Si V G = V FB V G < V FB V T > V G > V FB increase V G

6. Spring 2007EE130 Lecture 30, Slide 6 Accumulation (n+ poly-Si gate, p-type Si) EcEc E FS EvEv |q  S | is small,  0 E c = E FM EvEv MOS 3.1 eV 4.8 eV p-type Si + _ VGVG V G < V FB + ++ + + + - - - --- GATE |qV G | | qV ox | Mobile carriers (holes) accumulate at Si surface

7. Spring 2007EE130 Lecture 30, Slide 7 Accumulation Layer Charge Density p-type Si + _ VGVG V G < V FB + ++ + + + - - - --- GATE Q acc (C/cm 2 ) From Gauss’ Law: (units: F/cm 2 ) xoxo

8. Spring 2007EE130 Lecture 30, Slide 8 Depletion (n+ poly-Si gate, p-type Si) EcEc E FS EvEv E c = E FM EvEv 3.1 eV 4.8 eV p-type Si + _ VGVG V T > V G > V FB - - - - - - + + + +++ GATE qV G qV ox qSqS MOS W Si surface is depleted of mobile carriers (holes) => Surface charge is due to ionized dopants (acceptors)

9. Spring 2007EE130 Lecture 30, Slide 9 Depletion Width W (p-type Si) Depletion Approximation: The surface of the Si is depleted of mobile carriers to a depth W. The charge density within the depletion region is Poisson’s equation: Integrate twice, to obtain  S : To find  s for a given V G, we need to consider the voltage drops in the MOS system…

10. Spring 2007EE130 Lecture 30, Slide 10 Voltage Drops in Depletion (p-type Si) p-type Si + _ VGVG - - - - - - + + + +++ GATE Q dep (C/cm 2 ) From Gauss’ Law: Q dep is the integrated charge density in the Si:

11. Spring 2007EE130 Lecture 30, Slide 11 Surface Potential in Depletion (p-type Si) Solving for  S, we have

12. Spring 2007EE130 Lecture 30, Slide 12 Threshold Condition (V G = V T ) When V G is increased to the point where  s reaches 2  F, the surface is said to be strongly inverted. (The surface is n-type to the same degree as the bulk is p-type.) This is the threshold condition. V G = V T

13. Spring 2007EE130 Lecture 30, Slide 13 MOS Band Diagram at Threshold (p-type Si) EcEc E FS EvEv E c = E FM EvEv qV G qV ox qsqs MOS WTWT qFqF qFqF

14. Spring 2007EE130 Lecture 30, Slide 14 Threshold Voltage For p-type Si: For n-type Si:

15. Spring 2007EE130 Lecture 30, Slide 15 Strong Inversion (p-type Si) p-type Si + _ VGVG - - - - - - + + + +++ GATE Significant density of mobile electrons at surface (surface is n-type) As V G is increased above V T, the negative charge in the Si is increased by adding mobile electrons (rather than by depleting the Si more deeply), so the depletion width remains ~constant at W= W T  x  M O S x WTWT

16. Spring 2007EE130 Lecture 30, Slide 16 Inversion Layer Charge Density (p-type Si)