1. MOSFET Cross-Section

2. A MOSFET Transistor Gate Source Drain Source Substrate Gate Drain

3. MOSFET Schematic Symbols

4. Formation of the Channel for an Enhancement MOS Transistor

5. Water Analogy of a (subthreshold) MOSFET

6. Channel Current vs. Gate Voltage 00.511.522.533.544.55 0 50 100 150 200 250 300 350 400 450 Gate voltage (V) Channel Current (  A) 0.40.60.811.21.41.61.82 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 Gate voltage (V) Channel Current (A) Above-ThresholdSub-Threshold In linear scale, we have a quadratic dependence In log-scale, we have an exponential dependence VTVT VTVT I th

7. MOS Capacitor Picture

8. MOSFET Channel Picture

9. MOS Capacitor Picture

10. MOS Electrostatics Condition is called flatband --- the voltage when this occurs is called flatband This state is the baseline operating case --- a capacitive divider has one free parameter V fb

11. MOS Electrostatics Depletion Condition --- gate charge is terminated by charged ions in the depletion region Part of this region is often referred to as weak-inversion

12. MOS Electrostatics Inversion --- further gate charge is terminated by carriers at the silicon--silicon-dioxide interface

13. MOS Structure Electrostatics

14. MOS Capacitor Picture

15. MOS-Capacitor Regions Q s = ln( 1 + e ) (  (V g - V T ) - V s )/U T Q s = e (  - V s )/U T Depletion (  (V g - V T ) - V s < 0) Q s = e (  (V g - V T ) - V s )/U T Inversion (  (V g - V T ) - V s > 0) Q s = (  (V g - V T ) - V s )/U T

16. MOS Capacitor Picture

17. MOSFET Channel Picture

18. Calculation of Drain Current

19. No recombination 0 2 2  dx nd DnDn n dn qDJ  l nn drainsource n  0 l  varies as  V G  n = Ax + B RG dx nd D dt dn n    2 2 0 00   eeII UVVUVV TdgTSg   / / 0 

20. MOSFET Current-Voltage Curves   1 /)( 0 //)( 0 / // 0 TSG TdsTSG TD TSTG uV VV uVuV VV uV uVu VV DS eI eeI eeeII        Saturation 4 Tds UV    eeII uVVuVV TdgTSg   // 0     1 / / 0 TSd TSg uVV uVV eeI    

21. MOSFET Current-Voltage Curves   1 /)( 0 //)( 0 / // 0 TSG TdSTSG TD TSTG uVKV uVuV uV uVu DS eI eeI eeeII          eeII uVVuVV TdgTSg   // 0     1 / / 0 TSd TSg uVV uVV eeI    

22. Subthreshold MOSFETs In linear scale, we have a quadratic dependence In log-scale, we have an exponential dependence G S D nFET G S D B pFET

23. Determination of Threshold Voltage 0.40.50.60.70.80.91 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Gate voltage (V) Drain current / subthreshold fit V T = 0.86

24. Drain Current --- Source Voltage

25. Drain Characteristics

26. Origin of Drain Dependencies Increasing Vd effects the drain-to-channel region: increases depletion width increases barrier height

27. Current versus Drain Voltage Not flat due to Early effect (channel length modulation) I d = I d (sat) (1 + (V d /V A ) ) or I d = I d (sat) e Vd/VA R out I d (sat) GND I out

28. Early Voltage Length Dependence Width of depletion region depends on doping, not L Might expect Vo to linearly vary with L

29. MOSFET Operating Regions Band-diagram picture moving from subthreshold to above-threshold Surface potential moving from depletion to inversion

30. Qualitative Above-Threshold I = (K/2  ) ( (  (V g - V T ) - V s ) 2 - ( (  (V g - V T ) - V d ) 2 )

31. Above Threshold MOSFET Equations I = (K/2  ) ( (  (V g - V T ) - V s ) 2 - (  (V g - V T ) - V d ) 2 ) Saturation: Q d = 0 I = (K/2  ) ( (  (V g - V T ) - V s ) 2 If  = 1 (ignoring back-gate effects): I = (K/2) ( 2(V gs - V T ) V ds - V ds 2 )

32. Detailed MOSFET Derivation I =  Q(x) E(x) Current is constant through the channel (no loss) ( E(x) = Electric Field ) Q(x) = C T (  (V g - V T ) - V(x) ) C T = C D + C ox Q s = C T (  (V g - V T ) - V s ), Q d = C T (  (V g - V T ) - V d ) E(x) = - = (1 / C T ) d V(x) dx d Q(x) dx I = (  / C T ) Q(x) d Q(x) dx (  = C ox / C T )

33. Detailed MOSFET Derivation I = (K/2  ) ( (  (V g - V T ) - V s ) 2 - ( (  (V g - V T ) - V d ) 2 ) Integrate with respect to length: I = (  / C T ) Q(x) d Q(x) dx Q s = C T (  (V g - V T ) - V s ) Q d = C T (  (V g - V T ) - V d ) I L = (  / 2 C T ) Q(x) 2 | I = (  / 2 C T ) ( 1 / L) ( Q s 2 - Q d 2 ) K =  C ox (W/L) I = (  C T / 2 ) ( 1 / L) ( (  (V g - V T ) - V s ) 2 - (  (V g - V T ) - V d ) 2 )

34. MOSFET Equations I = (K/2) ( (V g - V T - V s ) 2 - (V g - V T - V d ) 2 ) Saturation: (Q d = 0) I = (K/2) (V gs - V T ) 2 When ignoring back-gate effects (we often do):  = 1 I = (K/2) ( 2(V gs - V T ) V ds - V ds 2 ) Above-Threshold: Subthreshold: I = I s e (1 – e ) V gs /U T -V ds /U T Saturation: ( V ds > 4 U T ) I = I s e V gs /U T (V d > V g - V T )

35. Output Characteristics of the Above-Threshold MOSFET Interpretation of large-signal model

36. MOSFETs G S D nFET G S D B pFET 00.511.522.533.544.55 0 50 100 150 200 250 300 350 Gate voltage (V) Current (  A) K = 37.861  A/V 2 V T = 0.806

37. Drain Current - Gate Voltage

38. Drain Current --- Source Voltage 00.10.20.30.40.50.60.70.80.91 0 0.5 1 1.5 2 2.5 3 3.5 4 Gate voltage (V) sqrt(Drain current (  A))  (V g - V T ) = 0.595 K/  = 74.585  A/V 2 (  = 0.54) (  = 0.7)

39. An Ohmic MOSFET I = (K/2  ) ( (  (V g - V T ) - V s ) 2 - (  (V g - V T ) - V d ) 2 ) If V d ~ V s, (small difference) (V d - V s = 50mV) I = K (V g - V T )(V d - V s )

40. A MOSFET in Saturation Saturation: Q d = 0 I = (K/2  ) ( (  (V g - V T ) - V s ) 2 V s = 0 I = (K  /2) (V g - V T ) 2

41. More Ohmic Region Data I = (K/2  ) ( (  (V g - V T ) - V s ) 2 - (  (V g - V T ) - V d ) 2 ) Take the derivative of I with respect to V d (V s = 0 ) dI / d V d = (K/2  )( 0 - (-2) (  (V g - V T ) - V d ) ) = (K/2  )(  (V g - V T ) - V d )

42. Influence of V DS on the Output Characteristics

43. Current versus Drain Voltage Not flat due to Early effect (channel length modulation) I d = I d (sat) (1 + (V d /V A ) ) or I d = I d (sat) e Vd/VA R out I d (sat) GND I out

44. Early Voltage Length Dependence Width of depletion region depends on doping, not L Might expect Vo to linearly vary with L

45. Small-Signal Modeling gmVgmVroro V3V3 V2V2 V2V2 V1V1 +V-+V- V3V3 V2V2 V1V1 V3V3 V2V2 V1V1 gmgm roro Above VT MOSFET Sub VT MOSFET AvAv I I I / U T 2I / ( V 1 -V 2 -V T )V A / I V A / U T 2V A / ( V 1 -V 2 -V T )

46. Small-Signal Modeling gmVgmVroro V3V3 V2V2 V2V2 rr V1V1 +V-+V- V3V3 V2V2 V1V1 V3V3 V2V2 V1V1 gmgm roro rr BJT Above VT MOSFET Sub VT MOSFET AvAv   (U T  ) / I I I I / U T 2I / ( V 1 -V 2 -V T ) V A / I V A / U T 2V A / ( V 1 -V 2 -V T )

47. Capacitances in a MOSFET

48. MOSFET Depletion Capacitors

49. Overlap Capacitances

50. Capacitance Modeling

52. Velocity Saturation Ideal Drift (Ohm’s Law) Si Crystal Velocity Limit

53. Effect of Velocity Saturation L = 76 nm MOSFET VTVT Square-law region

55. Small-Signal Modeling (with kappa) gmVgmVroro V3V3 V2V2 V2V2 V1V1 +V-+V- V3V3 V2V2 V1V1 V3V3 V2V2 V1V1 gmgm roro Above VT MOSFET Sub VT MOSFET AvAv I I  I / U T 2I / ( V 1 -V 2 -V T )V A / I  V A / U T 2V A / ( V 1 -V 2 -V T )

56. Small-Signal Modeling (with kappa) gmVgmVroro V3V3 V2V2 V2V2 rr V1V1 +V-+V- V3V3 V2V2 V1V1 V3V3 V2V2 V1V1 gmgm roro rr BJT Above VT MOSFET Sub VT MOSFET AvAv   (U T  ) / I I I I / U T  I / U T 2I / ( V 1 -V 2 -V T ) V A / I V A / U T  V A / U T 2V A / ( V 1 -V 2 -V T )