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MOSFETs

MOSFETs. A little bit of history. V SG. V SD. More electrons. Operation of a transistor. V SG > 0 n type operation. Gate. Insulator. Source. Drain. Channel. Substrate. Positive gate bias attracts electrons into channel Channel now becomes more conductive . V SG. V SD.

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MOSFETs

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  1. MOSFETs ECE 663

  2. A little bit of history.. ECE 663

  3. VSG VSD More electrons Operation of a transistor VSG > 0 n type operation Gate Insulator Source Drain Channel Substrate Positive gate bias attracts electrons into channel Channel now becomes more conductive

  4. VSG VSD Operation of a transistor Gate Insulator Source Drain Channel Substrate Transistor turns on at high gate voltage Transistor current saturates at high drain bias

  5. VSG VSD Start with a MOS capacitor Gate Insulator Source Drain Channel Substrate

  6. MIS Diode (MOS capacitor) – Ideal ECE 663

  7. Questions • W What is the MOS capacitance? QS(yS) What are the local conditions during inversion? yS,cr How does the potential vary with position? y(x) How much inversion charge is generated at the surface? Qinv(x,yS) Add in the oxide: how does the voltage divide? yS(VG), yox(VG) How much gate voltage do you need to invert the channel? VTH How much inversion charge is generated by the gate? Qinv(VG) What’s the overall C-V of the MOSFET? QS(VG)

  8. Ideal MIS Diode n-type, Vappl=0 Assume Flat-band at equilibrium qfS EC EF Ei EV ECE 663

  9. Ideal MIS Diode n-type, Vappl=0 ECE 663

  10. Ideal MIS Diode p-type, Vappl=0 ECE 663

  11. Ideal MIS Diode p-type, Vappl=0 ECE 663

  12. Accumulation Pulling in majority carriers at surface ECE 663

  13. But this increases the barrier for current flow !! n+ p n+ ECE 663

  14. Depletion ECE 663

  15. Inversion yB Need CB to dip below EF. Once below by yB, minority carrier density trumps the intrinsic density. Once below by 2yB, it trumps the major carrier density (doping) ! ECE 663

  16. Sometimes mathscan help… ECE 663

  17. P-type semiconductor Vappl0 Convention for p-type: y positive if bands bend down ECE 663

  18. Ideal MIS diode – p-type CB moves towards EF if y > 0  n increases VB moves away from EF if y > 0  p decreases ECE 663

  19. Ideal MIS diode – p-type At the semiconductor surface,  = s ECE 663

  20. Surface carrier concentration • s < 0 - accumulation of holes • s =0 - flat band • B>s >0 – depletion of holes • s =B - intrinsic concentration ns=ps=ni • s > B – Inversion (more electrons than holes) EC EF ECE 663

  21. Want to find , E-field, Capacitance • Solve Poisson’s equation to get E field, potential based on charge density distribution(one dimension) E E E ECE 663

  22. Away from the surface,  = 0 • and ECE 663

  23. Solve Poisson’s equation: E = -dy/dx d2y/dx2 = -dE/dx = (dE/dy).(-dy/dx) = EdE/dy EdE/dy ECE 663

  24. Solve Poisson’s equation: • Do the integral: • LHS: • RHS: • Get expression for E field (d/dx): ECE 663

  25. Define: Debye Length Then: E > 0 E < 0 y > 0 y < 0 + for  > 0 and – for  < 0 ECE 663

  26. Use Gauss’ Law to find surface charge per unit area ECE 663

  27. Accumulation to depletion to strong Inversion • For negative , first term in F dominates – exponential • For small positive , second term in F dominates -  • As  gets larger, second exponential gets big yB = (kT/q)ln(NA/ni) = (1/b)ln(pp0/√pp0np0) (np0/pp0) = e-2byB yS > 2yB ECE 663

  28. Questions • What is the MOS capacitance? QS(yS) • What are the local conditions during inversion? yS,cr • How does the potential vary with position? y(x) • How much inversion charge is generated at the surface? Qinv(x,yS) • Add in the oxide: how does the voltage divide? yS(VG), yox(VG) • How much gate voltage do you need to invert the channel? VTH • How much inversion charge is generated by the gate? Qinv(VG) • What’s the overall C-V of the MOSFET? QS(VG)

  29. Charges, fields, and potentials • Charge on metal = induced surface charge in semiconductor • No charge/current in insulator (ideal) metal insul semiconductor depletion inversion ECE 663

  30. Charges, fields, and potentials Electrostatic Potential Electric Field ECE 663

  31. Depletion Region Electrostatic Potential Electric Field ECE 663

  32. Depletion Region Electrostatic Potential Electric Field y = ys(1-x/W)2 Wmax = 2es(2yB)/qNA yB= (kT/q)ln(NA/ni) ECE 663

  33. Questions • What is the MOS capacitance? QS(yS) • What are the local conditions during inversion? yS,cr • How does the potential vary with position?y(x) • How much inversion charge is generated at the surface? Qinv(x,yS) • Add in the oxide: how does the voltage divide? yS(VG), yox(VG) • How much gate voltage do you need to invert the channel? VTH • How much inversion charge is generated by the gate? Qinv(VG) • What’s the overall C-V of the MOSFET? QS(VG)

  34. Couldn’t we just solve this exactly?

  35. Exact Solution U = by US = byS UB = byB dy/dx = -(2kT/qLD)F(yB,np0/pp0)  U  dU/F(U) =  x/LD US F(U) = [eUB(e-U-1+U)-e-UB (eU-1-U)]1/2

  36. Exact Solution r = qni[eUB(e-U-1) – e-UB(eU-1)]  US  dU’/F(U’,UB) =  x/LD U F(U,UB) = [eUB(e-U-1+U) + e-UB (eU-1-U)]1/2

  37. Exact Solution NA = 1.67 x 1015 Qinv ~ 1/(x+x0)a x0 ~ LD . factor

  38. Questions • What is the MOS capacitance? QS(yS) • What are the local conditions during inversion? yS,cr • How does the potential vary with position?y(x) • How much inversion charge is generated at the surface? Qinv(x,yS) • Add in the oxide: how does the voltage divide? yS(VG), yox(VG) • How much gate voltage do you need to invert the channel?VTH • How much inversion charge is generated by the gate? Qinv(VG) • What’s the overall C-V of the MOSFET? QS(VG)

  39. Threshold Voltage for Strong Inversion • Total voltage across MOS structure= voltage across dielectric plus s ECE 663

  40. Notice Boundary Condition !! eoxVi/tox = esys/(W/2) Before Inversion After inversion there is a discontinuity in D due to surface Qinv Vox (at threshold) = es(2yB)/(Wmax/2)Ci = ECE 663

  41. VG = Vfb + ys + (kstox/kox) √(2kTNA/e0ks)[bys + eb(ys-2yB)]1/2 Local Potential vs Gate voltage yox ys Initially, all voltage drops across channel (blue curve). Above threshold, channel potential stays pinned to 2yB, varying only logarithmically, so that most of the gate voltage drops across the oxide (red curve).

  42. Look at Effective charge width ~Wdm/2 ~tinv Initially, a fast increasing channel potential drops across increasing depletion width Eventually, a constant potential drops across a decreasing inversion layer width, so field keeps increasing and thus matches increasing field in oxide

  43. Questions • What is the MOS capacitance? QS(yS) • What are the local conditions during inversion? yS,cr • How does the potential vary with position?y(x) • How much inversion charge is generated at the surface? Qinv(x,yS) • Add in the oxide: how does the voltage divide? yS(VG), yox(VG) • How much gate voltage do you need to invert the channel? VTH • How much inversion charge is generated by the gate? Qinv(VG) • What’s the overall C-V of the MOSFET? QS(VG)

  44. Charge vs Local Potential Qs≈√(2e0kskTNA)[bys + eb(ys-2yB)]1/2 Beyond threshold, all charge goes to inversion layer

  45. How do we get the curvatures? NEW Add other terms and keep Leading term EXACT

  46. Inversion Charge vs Gate voltage Q ~ eb(ys-2yB), ys- 2yB ~ log(VG-VT) Exponent of a logarithm gives a linear variation of Qinv with VG Qinv = -Cox(VG-VT) Why Cox?

  47. Questions • What is the MOS capacitance? QS(yS) • What are the local conditions during inversion? yS,cr • How does the potential vary with position?y(x) • How much inversion charge is generated at the surface? Qinv(x,yS) • Add in the oxide: how does the voltage divide? yS(VG), yox(VG) • How much gate voltage do you need to invert the channel? VTH • How much inversion charge is generated by the gate? Qinv(VG) • What’s the overall C-V of the MOSFET?QS(VG)

  48. Capacitance For s=0 (Flat Band): Expand exponentials….. ECE 663

  49. Capacitance of whole structure • Two capacitors in series: Ci - insulator CD - Depletion OR ECE 663

  50. Capacitance vs Voltage ECE 663

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