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					         Lecture 16

         OUTLINE
• The MOS Capacitor (cont’d)
  – Electrostatics



   Reading: Pierret 16.3; Hu 5.2-5.5
  Bulk Semiconductor Potential, fF

• p-type Si:                                                Ec

                                                            Ei
                                                    qf F
                                               EF
                                                            Ev


• n-type Si:                                                Ec
                                               EF
                                                    |qfF|
                                                            Ei
                                                            Ev


EE130/230M Spring 2013   Lecture 16, Slide 2
 Voltage Drops in the MOS System
• In general,

    where
           qVFB = FMS = FM – FS
           Vox is the voltage dropped across the oxide
           (Vox = total amount of band bending in the oxide)

            fs is the voltage dropped in the silicon
               (total amount of band bending in the silicon)



• For example:
       When VG = VFB, Vox = fs = 0, i.e. there is no band bending
EE130/230M Spring 2013             Lecture 16, Slide 3
  MOS Band Diagrams for n-type Si
 Decrease VG toward more negative values
 à the gate electron energy increases relative to that in the Si

                  decrease VG                    decrease VG




• Accumulation                  • Depletion                    • Inversion
   – VG > VFB                      – VG < VFB                     – VG < VT
   – Electrons                     – Electrons                    – Surface
     accumulate at                   repelled from                  becomes p-
       surface                       surface                        type

EE130/230M Spring 2013             Lecture 16, Slide 4
  MOS Band Diagrams for p-type Si
                                    increase VG                 increase VG




              VG = VFB   VG < VFB               VT > VG > VFB




EE130/230M Spring 2013              Lecture 16, Slide 5
                     Accumulation
               (n+ poly-Si gate, p-type Si)
                                                        M          O        S
         VG < VFB
                                                          3.1 eV           | qVox |

                                          Ec= EFM

                           GATE                  Ev
                          - - - - - -                                  |qVG |
                                               xo                                |qfS| is small, » 0
                       + + + + + +
       +                                                                                      Ec
VG     _
                       p-type Si                          4.8 eV                             EFS
                                                                                             Ev



     Mobile carriers (holes) accumulate at Si surface


 EE130/230M Spring 2013                 Lecture 16, Slide 6
     Accumulation Layer Charge Density
           VG < VFB

                                                         From Gauss’ Law:


                              GATE
                         - - - - - -
                                                    xo
                         + + + + + +
       +
VG     _                       Qacc (C/cm2)
                        p-type Si                                           (units: F/cm2)




     EE130/230M Spring 2013                   Lecture 16, Slide 7
                        Depletion
                (n+ poly-Si gate, p-type Si)
                                                                 M         O       S
         VT > VG > VFB
                                                                 qVox
                                                                               W
                                                                                                     Ec

                              GATE                                                                   EFS
                         + + + + + +                              3.1 eV                qf S         Ev
                             - - - - - -                                                       qVG
       +
V      _                                           Ec= EFM
                         p-type Si
G
                                                           Ev                  4.8 eV




     Si surface is depleted of mobile carriers (holes)
     => Surface charge is due to ionized dopants (acceptors)

    EE130/230M Spring 2013                 Lecture 16, Slide 8
      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 fS:

                                                       To find fs for a given VG, we
                                                       need to consider the voltage
                                                       drops in the MOS system…


EE130/230M Spring 2013           Lecture 16, Slide 9
Voltage Drops in Depletion (p-type Si)
                                                     From Gauss’ Law:
                           GATE
                      + + + + + +
                          - - - - - -
     +
VG   _
                             Qdep (C/cm2)
                                                     Qdep is the integrated
                      p-type Si
                                                     charge density in the Si:




 EE130/230M Spring 2013                     Lecture 16, Slide 10
      Surface Potential in Depletion
               (p-type Si)


• Solving for fS, we have




EE130/230M Spring 2013      Lecture 16, Slide 11
       Threshold Condition (VG = VT)
• When VG is increased to the point where fs reaches
  2fF, the surface is said to be strongly inverted. This
  is the threshold condition.

     VG = VT




 (The surface is n-type to the same degree as the bulk is p-type.)

EE130/230M Spring 2013     Lecture 16, Slide 12
   MOS Band Diagram at Threshold
            (p-type Si)
                                           M      O        S

                                           qVox       WT
                                                                      qf F
                                                                             Ec

                                           qf F                              EFS
                                                               qf s          Ev
                                                                        qVG

                             Ec= EFM

                                    Ev




EE130/230M Spring 2013   Lecture 16, Slide 13
                         Threshold Voltage
• For p-type Si:




• For n-type Si:




EE130/230M Spring 2013        Lecture 16, Slide 14
              Strong Inversion (p-type Si)
    As VG is increased above VT, 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 = WT
                                                              WT   r(x)
                                                                   M O S
                              GATE
                         + + + + + +
                                                                           x
                             - - - - - -
        +
V       _
                         p-type Si
G




    Significant density of mobile electrons at surface
    (surface is n-type)
    EE130/230M Spring 2013                 Lecture 16, Slide 15
     Inversion Layer Charge Density
               (p-type Si)




EE130/230M Spring 2013   Lecture 16, Slide 16

				
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