Lecture #21 OUTLINE The MOS Capacitor • Electrostatics Reading: Course Reader Spring 2003 EE130 Lecture 21, Slide 1 MOS Capacitor Structure • Typical MOS capacitors and MOS capacitor (cross-sectional view) transistors in ICs today employ – heavily doped polycrystalline Si (“poly-Si”) film as the gate- electrode material GATE •n+-type, for “n-channel” xox transistors (NMOS) + + •p-type, for “p-channel” VG _ transistors (PMOS) –SiOas the gate dielectric Si 2 • band gap = 9 eV ε • r,SiO2 = 3.9 – Si as the semiconductor material • p-type, for “n-channel” transistors (NMOS) • n-type, for “p-channel” transistors (PMOS) Spring 2003 EE130 Lecture 21, Slide 2 1 ψ Bulk Semiconductor Potential B ψ ≡ − q B Ei (bulk) EF • p-type Si: Ec kT E ψ = > ψ i B ln(N A / ni ) 0 q B EF q Ev • n-type Si: Ec E F |qψ | kT B ψ = − < Ei B ln(N D / ni ) 0 q Ev Spring 2003 EE130 Lecture 21, Slide 3 MOS Equilibrium Energy-Band Diagram Ec 3.1 eV 3.1 3.1 eV 3.1 E c E E c f E v 2 E v 9eV SiO polysilicon + N gate body P-Silicon P-Silicon body Ev (a) (b) How does one arrive at this energy-band diagram? Spring 2003 EE130 Lecture 21, Slide 4 2 Guidelines for Drawing MOS Band Diagrams •Fermi level EF is flat (constant with distance x) in the Si – Since no current flows in the x direction, we can assume that equilibrium conditions prevail • Band bending is linear in the oxide – No charge in the oxide => d /dx = 0 so is constant => dEc/dx is constant • From Gauss’ Law, we know that the electric field strength in the Si at the surface, Si, is related to the electric field strength in the oxide, ox: = ε Si ≅ dEc = × dEc ox Si 3 Si so 3 εox dx oxide dx Si (at the surface) Spring 2003 EE130 Lecture 21, Slide 5 MOS Band-Diagram Guidelines (cont.) • The barrier height for conduction-band electron flow from the Si into SiO2 is 3.1 eV χ χ – This is equal to the electron-affinity difference ( Si and SiO2) • The barrier height for valence-band hole flow from the Si into SiO2 is 4.8 eV • The vertical distance between the Fermi level in the metal, EFM, and the Fermi level in the Si, EFS, is equal to the applied gate voltage: = − qVG EFS EFM Spring 2003 EE130 Lecture 21, Slide 6 3 Voltage Drops in the MOS System • In general, = + +ψ VG VFB Vox s where φ φ φ qVFB = MS = M – S Vox is the voltage dropped across the oxide (Vox = total amount of band bending in the oxide) ψ s is the voltage dropped in the silicon (total amount of band bending in the silicon) ψ = − q s Ei (bulk) Ei (surface) ψ For example: When VG = VFB, Vox = s = 0 i.e. there is no band bending Spring 2003 EE130 Lecture 21, Slide 7 Special Case: Equal Work Functions Φ Φ M = S What happens when the work function is different? Spring 2003 EE130 Lecture 21, Slide 8 4 General Case: Different Work Functions Spring 2003 EE130 Lecture 21, Slide 9 Flat-Band Condition E0 χ =0.95 eV SiO2 Ec qΦ M χ qΦ = χ + (E –E ) 3.1 eV 3.1 eV Si s Si c F Ec, EF Ec VFB Ef Ev Ev 9eV P-type Si N+ poly-Si SiO E : Vacuum level 0 4.8 eV E0 – Ef : Work function E0 – Ec : Electron affinity Si/SiO energy barrier 2 Ev Spring 2003 EE130 Lecture2 21, Slide 10 5 MOS Band Diagrams (n-type Si) Decrease VG (toward more negative values) -> move the gate energy-bands up, relative to the Si decrease VG decrease VG • Accumulation • Depletion • Inversion – VG > VFB – VG < VFB – VG < VT – Electrons – Electrons – Surface accumulate at repelled becomes surface from surface p-type Spring 2003 EE130 Lecture 21, Slide 11 Biasing Conditions for p-type Si increase VG increase VG VG = VFB VG < VFB VT > VG > VFB Spring 2003 EE130 Lecture 21, Slide 12 6 Accumulation (n+ poly-Si gate, p-type Si) MO S VG < VFB 3.1 eV | qVox | Ec= EFM GATE E v |qV | - - - - - - G ψ ≈ |q s| is small, 0 + + + + + + + E VG _ c p-type Si 4.8 eV EFS Ev ≅ + Mobile carriers (holes) accumulate at Si surface VG VFB Vox Spring 2003 EE130 Lecture 21, Slide 13 Accumulation Layer Charge Density ≅ − Vox VG VFB VG < VFB From Gauss’ Law: = −Q / ε ox acc SiO2 GATE - - - - - - tox = = − + + + + + + Vox oxtox Qacc / Cox + VG _ 2 Qacc (C/cm ) where C ≡ ε / t ox SiO2 ox p-type Si (units: F/cm2) = − − > ⇒ Qacc Cox (VG VFB ) 0 Spring 2003 EE130 Lecture 21, Slide 14 7 Depletion (n+ poly-Si gate, p-type Si) MO S VT > VG > VFB qVox Wd Ec GATE EFS + + + + + + ψ 3.1 eV q s Ev - - - - - - qVG + V _ E = E G p-type Si c FM Ev 4.8 eV Si surface is depleted of mobile carriers (holes) => Surface charge is due to ionized dopants (acceptors) Spring 2003 EE130 Lecture 21, Slide 15 Depletion Width Wd (p-type Si) • Depletion Approximation: The surface of the Si is depleted of mobile carriers to a depth Wd. • The charge density within the depletion region is ρ ≅ − ≤ ≤ qN A (0 x Wd ) d = ρ ≅ − qN A ≤ ≤ • Poisson’s equation: (0 x Wd ) dx εSi εSi ψ • Integrate twice, to obtain S: ψ qN 2ε ψ To find s for a given VG, we ψ = A W 2 ⇒ W = Si s s ε d d need to consider the voltage 2 Si qN A drops in the MOS system… Spring 2003 EE130 Lecture 21, Slide 16 8 Voltage Drops in Depletion (p-type Si) From Gauss’ Law: = −Q / ε GATE ox dep SiO2 + + + + + + - - - = = − - - - Vox oxtox Qdep / Cox V + G _ 2 Qdep (C/cm ) Qdep is the integrated p-type Si charge density in the Si: = − = − ε ψ Qdep qN AWd 2qN A Si s 2qN ε ψ = +ψ + = +ψ + A si s VG VFB s Vox VFB s Cox Spring 2003 EE130 Lecture 21, Slide 17 Surface Potential in Depletion (p-type Si) 2qN ε ψ = +ψ + A si s VG VFB s Cox ψ • Solving for S, we have qN ε 2C 2 (V −V ) ψ = A si 1+ ox G FB −1 s 2C qN ε ox A si 2 qN ε 2C 2 (V −V ) ψ = A si 1+ ox G FB −1 s 2C 2 qN ε ox A si Spring 2003 EE130 Lecture 21, Slide 18 9 Threshold Condition (VG = VT) ψ • When VG is increased to the point where s reaches 2ψΒ, 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. ψ = ψ VG = VT ⇒ s 2 B − = []− E i(bulk) Ei (surface) 2 Ei (bulk) EF − = −[]− Ei (surface) EF Ei (bulk) EF = ⇒ nsurface N A Spring 2003 EE130 Lecture 21, Slide 19 MOS Band Diagram at Threshold (p-type Si) MO S ψ = ψ = kT N A s 2 B 2 ln qVox W q ni dm qψ B E ε ψ c 2 Si (2 B ) W = W = qψ EFS d dm F qψ E qN A s v qVG Ec= EFM Ev Spring 2003 EE130 Lecture 21, Slide 20 10 Threshold Voltage • For p-type Si: 2qN ε ψ = +ψ + = +ψ + A si s VG VFB s Vox VFB s Cox 2qN ε (2ψ ) = + ψ + A Si B VT VFB 2 B Cox • For n-type Si: 2qN ε 2ψ = + ψ − D Si B VT VFB 2 B Cox Spring 2003 EE130 Lecture 21, Slide 21 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 d= Wdm Wdm ρ(x) GATE M O S + + + + + + - - - - - - + V _ x G p-type Si ψ ≅ ψ s 2 B ε ψ Significant density of mobile electrons at surface ≅ = 2 si (2 B ) (surface is n-type) Wd Wdm qN A Spring 2003 EE130 Lecture 21, Slide 22 11 Inversion Layer Charge Density (p-type Si) = +ψ + VG VFB S Vox (Q + Q ) = + ψ − dep inv VFB 2 B Cox 2qN ε (2ψ ) = + ψ + A s B − Qinv VFB 2 B Cox Cox = − Qinv VT Cox ∴ = − − Qinv Cox (VG VT ) Spring 2003 EE130 Lecture 21, Slide 23 12.