Lecture 7 PN Junction and MOS Electrostatics(IV) Metal-Oxide-Semiconductor Structure (contd.) Outline 1. Overview of MOS electrostatics under bias 2. Depletion regime 3. Flatband 4. Accumulation regime 5. Threshold 6. Inversion regime Reading Assignment: Howe and Sodini, Chapter 3, Sections 3.8-3.9 6.012 Spring 2009 Lecture 7 11. Overview of MOS electrostatics under bias • Built-in potential across MOS structure increases from φB to φB + VGB • Oxide forbids current flow ⇒ – J=0 everywhere in semiconductor – Need drift = -diffusion in SCR • Must maintain boundary condition at Si/SiO2 interface – E ox / Es ≈ 3 Application of bias: How can this be accommodated simultaneously? ⇒ quasi-equilibrium situation with potential build-up across MOS equal to φB + VGB 6.012 Spring 2009 Lecture 7 2Important consequence of quasi-equilibrium: ⇒ Boltzmann relations apply in semiconductor [they were derived starting from Jn = J p =0] n(x) = nieqφφφφ(x) kT − qφφφφ(x) kT p(x) = nie and 2 np = ni at every x 6.012 Spring 2009 Lecture 7 32. Depletion regime For VGB>0, metal attracts electrons and repels holes ⇒ Depletion region widens For VGB<0, metal repels electrons and attracts holes ⇒ Depletion region shrinks 6.012 Spring 2009 Lecture 7 4In depletion regime, all results obtained for thermal equilibrium apply if φB → φB+VGB. For example: Depletion region thickness: 2C 2 (φφφφB + VGB) εεεεs ox xd (VGB ) = 1+ − 1 Cox εεεεsqNa Potential drop across semiconductor SCR: qN x2 VB (VGB ) = a d 2ε s Surface potential φφφφ(0) =φφφφp + VB(VGB) Potential drop across oxide: Vox (VGB ) = qN axdtox ε ox 6.012 Spring 2009 Lecture 7 5Flatband Voltage: VGB = VFB = −φφφφB =−(φφφφN+ −φφφφp ) 3. Flatband At a certain negative VGB, depletion region is wiped out ⇒ Flatband 6.012 Spring 2009 Lecture 7 64. Accumulation regime If VGB < VFB accumulation of holes at Si/SiO2 interface 6.012 Spring 2009 Lecture 7 75. Threshold Back to VGB>0. For sufficiently large VGB>0, electrostatics change when n(0)=Na ⇒ threshold. Beyond threshold, we cannot neglect contributions of electrons towards electrostatics. Let’s compute the gate voltage (threshold voltage) that leads to n(0)=N . 6.012 Spring 2009 Lecture 7 8 Key assumption: use electrostatics of depletion (neglect electron concentration at threshold)Computation of threshold voltage. Three step process: First, compute potential drop in semiconductor at threshold. Start from: n(0) = nieqφ(0) kT Solve for φ(0) at VGB = VT: kT n (0) kT Nφφφφ(0 ) = • ln = • ln a = −φφφφpVGB =VTq ni VGB = VTq ni Hence: VB (VT ) = −2φφφφp 6.012 Spring 2009 Lecture 7 9Computation of threshold voltage (contd.) Second, compute potential drop in oxide at threshold. Obtain xd(VT) using relationship between VB and xd in depletion: Solve for xd at VGB = VT: Then: Vox (VT ) = Eox (VT )tox = qNa xd (VT ) εεεεox tox = 1 Cox 2εεεεs qNa (−2φφφφp ) VB (VGB = VT ) = qNaxd 2 VT() 2εεεεs = −2φφφφp xd (VT ) = xd max = 2εεεεs (−2φφφφp ) qNa 6.012 Spring 2009 Lecture 7 10Computation of threshold voltage. (contd..) Finally, sum potential drops across structure. Solve for VT: VT + φφφφB = VB(VT ) + Vox (VT ) = −2φφφφP + 1 Cox 2εεεεsqNa (−2φφφφp ) VGB ==== VT ==== VFB −−−− 2φφφφP ++++1 C ox 2εεεεsqNa(−−−−2φφφφp) Key dependencies: • If N a ↑↑↑↑ ⇒⇒⇒⇒ VT ↑↑↑↑. The higher the doping, the more voltage required to produce n(0) = Na • If C ox ↑↑↑↑ (tox ↓↓↓↓) ⇒⇒⇒⇒ VT ↓↓↓↓. The thinner the oxide, the less voltage dropped across the oxide. 6.012 Spring 2009 Lecture 7 116. Inversion What happens for VGB > VT? More electrons at Si/SiO2 interface than acceptors ⇒⇒⇒⇒ inversion. Electron concentration at Si/SiO2 interface modulated by VGB ⇒ VGB ↑ → n(0) ↑ → |QN| ↑ : Field-effect control of mobile charge density! [essence of MOSFET] Want to compute QN vs. VGB [charge-control relation] Make sheet charge approximation: electron layer at Si/SiO2 is much thinner than any other dimension in problem (tox, xd). 6.012 Spring 2009 Lecture 7 12Charge-Control Relation To derive the charge-control relation, let’s look at the overall electrostatics: 6.012 Spring 2009 Lecture 7 13Charge-Control Relation (contd.) Key realization: n(0) ∝ eqφφφφ(0) kT qNaxd ∝ φφφφ(0) Hence, as VGB ↑ and φ(0) ↑ , n(0) will change a lot, but |Qd| will change very little. Several consequences: • xd does not increase much beyond threshold: 2εεεεs(−2φφφφp) xd (inv.) ≈ xd (VT) = = xd,maxqNa • VB does not increase much beyond VB(VT) =-2φP (a thin sheet of electrons does not contribute much to VB.): VB (inv.) ≈ VB(VT ) =−2φφφφP 6.012 Spring 2009 Lecture 7 14Charge-Control Relation (contd..) • All extra voltage beyond VT used to increase inversion charge Qn. Think of it as capacitor: – Top plate: metal gate – Bottom plate: inversion layer Q = CV ⇒ QN = −Cox (VGB − VT ) for VGB > VT Coul/cm2 6.012 Spring 2009 Lecture 7 15 Existence of QNand control over QNby VGB⇒ key to MOS electronicsWhat did we learn today? Summary of Key Concepts In inversion: = Cox (VGB − VT ) for VGB > VTQN 6.012 Spring 2009 Lecture 7 16MIT OpenCourseWarehttp://ocw.mit.edu 6.012 Microelectronic Devices and Circuits Spring 2009 For information about citing these materials or our Terms of Use, visit:
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