EECS 6.012 Spring 1998Lecture 6 I. The MOS Capacitor in Thermal EquilibriumA. Introduction • Oxide = SiO 2 ... a near-perfect insulator.• Assume zero charge in the oxide, electric field is constant and potential is linear in the oxide.• n + polysilicon has a potential which is the maximum possible in silicon: φ n+ = 550 mV• Example parameters tox =15nm=150A Na = 10 17 cm-3• φ p = -420 mV• The surface potential φ s is the potential at the Si/SiO2 interfaceg• Strategy: same as pn junction electrostatics: first thermal equilibrium, then with an applied bias.,, ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,metal interconnect to bulkmetal interconnect to gaten+ polysilicon gategate oxide εox = 3.9 εo p-typeεs = 11.7 εox0EECS 6.012 Spring 1998Lecture 6 B. Thermal Equilibrium MOS Electrostatics - Qualitative,, ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,metal interconnect to bulkn+ polysilicon gategate oxideεox = 3.9 εo p-typeεs = 11.7 εo x0++_+_+__φmn+φpmVox,oVBo++ +++++++++++++++++−−−−−−−−−−−−−−−−−−−Xdometal interconnect to gateρo(x)Eo(x)φo(x)−tox−tox−toxXdoXdoxxXdox000φn++QG500mVcontactpotential φpmcontactpotential φmn+ −500mV(a)(b)φpVox,oVBo_++_−qNaEECS 6.012 Spring 1998Lecture 6 II. MOS Capacitor under BiasA. Introduction • No steady-state current between the n + poly gate and the substrate. • J n = 0 and J p = 0• Absence of current implies that we can relate potential to carrier concentration in the silicon substrate.• Flatband condition: cancel built-in drop by applying VFB = -970 mV for Na = 10 17 cm -3 • For V GB = V FB , the internal potential is constant through the structure since both the gate and the p-type substrate have the same potential.,, ,,,,,,,,,,,,,,,,,,,,,,,,,,,,metal interconnect to bulkmetal interconnect to gaten+ polysilicon gategate oxideεox = 3.9 εo p-typeεs = 11.7 εo VGBx0++_+_+__φmn+φpmVoxVB+_VGBφn+φp–()– VFB==EECS 6.012 Spring 1998Lecture 6 C. Quantitative Electrostatics • E Field and Potential in OXIDE• E Field and Potential in Silicon• ResultE 0+()Q–Bεs--------------qNaXdoεs----------------------------==E0-()εsεox------------qNaXdoεs----------------------------qNaXdoεox---------------------------- Eox===φ0() φp– φn+φp–qNaXdotoxεox---------------------------------------–=Ex()qNaXdox–εs---------------------------------------------=φn +φp–qNaXdoCox----------------------------qNaXdo22εs----------------------------+=Xd otoxεsεox------------12Cox2φn+φp–qεsNa-------------------------------------------------------+ 1–=φsφx=0()φn+φp–qNaXdoCox----------------------------– φp+==EECS 6.012 Spring 1998Lecture 6B. MOS Capacitor in Flatband • When VGB = VFB, the gate is at the same potential as the p-type bulk -- there is no charge on the gate or in the silicon.. • When VGB < VFB, the gate is more negative than the p-type bulk and takes on a negative charge; holes are piled up (accumulated) under the oxide.ρ(x)−tox−toxx0E(x)0x−toxφ(x) 0(a)(b)φpmφmn+VGB = VFB 250mV−250mV−500mV−750mV−1.0Vx_+_+EECS 6.012 Spring 1998Lecture 6C. MOS Capacitor in Accumulation• Charge density, electric field, and potential in accumulation: VGB < VFB = - 0.97 V.• The MOS capacitor in accumulation is a parallel plate capacitor, with the (negative) gate charge given by QG = Cox (VGB - VFB) φmnρ(x)-toxx0QG- QG-toxE(x)0xEox QGεox-toxφ(x) 0x= φpmVGB-750mV-500mV-250mV-1.0V-1.25V(accumulated holes)+VGB - VFBchargedensityelectricfieldpotential, ,,,,,,,,,,,,,,,+_, ,,,,,,,,+_p-typeVGB= VFB VGB< VFB p-typexx0+ ++ ++ + +++ ++ + +− − − − − − − − − −− −−EECS 6.012 Spring 1998Lecture 6D. MOS Capacitor in Depletion• Now we make VGB > VFB. The p-type substrate depletes--in order to have negative charge available to terminate the electric field from the positive charge on the gate.• Analysis is the same as before except replace VGB - VFB for the quantity (φn+ - φp)• Surface potential at oxide/silicon interface is now positive --> n-type(slightly, ns = 1013 cm-3). ρ (x)-toxXdx-qNa-toxXdxE (x)00QG-toxXdxφ (x) 0φs = 180 mVφmnφpmVGB-500 mV500 mV1 V+chargedensityelectricfieldpotentialEECS 6.012 Spring 1998Lecture 6E. The Threshold Voltage• Keep increasing VGB --> surface potential keeps increasing. At some point, the surface is n-type (i.e., we say that it is inverted) and the electron charge is a significant contribution to the charge density.• We approximate that onset of inversion as the point where the electron concentration ns at the surface is the same as the hole concentration Na in the bulk. • The gate-bulk potential at the onset of inversion is called the threshold voltage, VTn. • To find the threshold voltage, we consider the electrostatics in depletion (no electrons at the surface at the onset of inversion) -- with the surface potential equal to the opposite of the bulk potential:φsφp–=-toxXd,maxxφ (x) 0φs = - φp = 420 mVφmnφpmVGB = VT-500 mV500 mV1 V+1.5 VVoxVB= 2|φp |EECS 6.012 Spring 1998Lecture 6F. Threshold Voltage Expression• We can solve for the threshold voltage:• The drop across the depletion region is• The drop across the oxide is• The bulk charge in inversion is• Adding up the terms, the threshold voltage is:VTnVFB– VoxVB+=VBφsφp– φp– φp–2φp–== =VoxEoxtoxQ–Bεox----------toxQB–Cox----------== =,, ,,,,,,,,,,,+_,, ,,,,,,,,,,,+_p-typeVFB < VGB < VTn VGB > VTn p-typexx+ ++ ++ + +++ ++++−−−−−−−−−−−−−++++++++−−−−−−−−Xd++−−Xd.maxdepletion regiondepletion regioninversionlayerQBqNaXd– qNa2φp–12⁄()qNa()εs⁄-------------------------------------------–2qεsNa2φp–()–== =VTnVFB2– φp1Cox--------2 q εsNa2–
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