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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 qfF Ei (bulk ) EF • p-type Si: kT fF ln( N A / ni ) 0 q Ec EF qfF Ev • n-type Si: kT fF ln( N D / ni ) 0 q EE130/230M Spring 2013 Lecture 16, Slide 2 EF Ei Ec |qfF| Ei Ev Voltage Drops in the MOS System • In general, VG VFB Vox fs 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) qfS Ei (bulk ) Ei ( surface) • 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 the gate electron energy increases relative to that in the Si decrease VG • Accumulation – VG > VFB – Electrons accumulate at surface EE130/230M Spring 2013 decrease VG • Depletion – VG < VFB – Electrons repelled from surface Lecture 16, Slide 4 • Inversion – VG < VT – Surface becomes ptype MOS Band Diagrams for p-type Si increase VG VG = VFB EE130/230M Spring 2013 VG < VFB VT > VG > VFB Lecture 16, Slide 5 increase VG Accumulation (n+ poly-Si gate, p-type Si) M VG < VFB 3.1 eV O S | qVox | Ec= EFM GATE - - - - - + + + + + + VG + _ Ev |qVG | xo Ec p-type Si 4.8 eV Mobile carriers (holes) accumulate at Si surface EE130/230M Spring 2013 |qfS| is small, 0 Lecture 16, Slide 6 EFS Ev VG VFB Vox Accumulation Layer Charge Density VG < VFB Vox VG VFB From Gauss’ Law: GATE - - - - - + + + + + + VG + _ Qacc (C/cm2) xo ox Qacc / ε SiO2 Vox x Qacc / Cox ox o where Cox ε SiO2 / xo p-type Si (units: F/cm2) Qacc Cox (VG VFB ) 0 EE130/230M Spring 2013 Lecture 16, Slide 7 Depletion (n+ poly-Si gate, p-type Si) M VT > VG > VFB qVox O S W Ec GATE + + + + + + VG + _ qfS 3.1 eV - - - - - - p-type Si Ec= EFM 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 4.8 eV qVG EFS Ev 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 qN A (0 x W ) d ρ qN A • Poisson’s equation: dx ε Si ε Si (0 x W ) • Integrate twice, to obtain fS: qN A 2 fS W 2 Si EE130/230M Spring 2013 2 SifS W qN A Lecture 16, Slide 9 To find fs for a given VG, we need to consider the voltage drops in the MOS system… Voltage Drops in Depletion (p-type Si) From Gauss’ Law: GATE + + + + + + VG - - - - - - + _ Qdep (C/cm2) p-type Si ox Qdep / ε SiO2 Vox ox xo Qdep / Cox Qdep is the integrated charge density in the Si: Qdep qN AW 2qN A SifS 2qN A sifS VG VFB fS Vox VFB fS Cox EE130/230M Spring 2013 Lecture 16, Slide 10 Surface Potential in Depletion (p-type Si) 2qN A sifS VG VFB fS Cox • Solving for fS, we have 2 qN A si 2Cox (VG VFB ) 1 fS 1 qN A si 2Cox qN A si fS 2 2Cox EE130/230M Spring 2013 2Cox (VG VFB ) 1 1 qN A si 2 Lecture 16, Slide 11 2 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 fS 2fF E i (bulk ) Ei ( surface) 2Ei (bulk ) EF Ei ( surface) EF Ei (bulk ) EF nsurface N A (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 kT N A fS 2fF 2 ln q ni W WT qVox 2 Si (2f F ) qN A qfF Ec= EFM Ev EE130/230M Spring 2013 Lecture 16, Slide 13 O S WT qfF qfs Ec EFS Ev qVG Threshold Voltage • For p-type Si: 2qN A sifS VG VFB fS Vox VFB fS Cox 2qN A Si (2fF ) VT VFB 2fF Cox • For n-type Si: VT VFB 2fF EE130/230M Spring 2013 2qN D Si 2fF Cox 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 (x) WT M O S GATE + + + + + + VG + _ x - - - - - - p-type Si fS 2fF Significant density of mobile electrons at surface (surface is n-type) EE130/230M Spring 2013 Lecture 16, Slide 15 2 si (2fF ) W WT qN A Inversion Layer Charge Density (p-type Si) VG VFB fS Vox VFB 2fF (Qdep Qinv ) Cox 2qN A s (2fF ) Qinv VFB 2fF Cox Cox Qinv VT Cox Qinv Cox (VG VT ) EE130/230M Spring 2013 Lecture 16, Slide 16