6.012 - Microelectronic Devices and Circuits Lecture 9 - MOS Capacitors I - Outline • Announcements Problem set 5 - Posted on Stellar. Due next Wednesday. • Qualitative description - MOS in thermal equilibrium Definition of structure: metal/silicon dioxide/p-type Si (Example: n-MOS) Electrostatic potential of metal relative to silicon: φm Zero bias condition: Si surface depleted if φm> φp-Si (typical situation) Negative bias on metal: depletion to flat-band to accumulation Positive bias on metal: depletion to threshold to inversion • Quantitative modeling - MOS in thermal equilibrium, vBC = 0 Depletion approximation applied to the MOS capacitor: 1. Flat-band voltage, VFB 2. Accumulation layer sheet charge density, qA* 3. Maximum depletion region width, XDT 4. Threshold voltage, VT 5. Inversion layer sheet charge density, qN* • Quantitative modeling - vBC ≠ 0; impact of vBC < 0 Voltage between n+ region and p-substrate: |2φp-Si | → |2φp-Si| - vBC Clif Fonstad, 10/8/09 Lecture 9 - Slide 1n-Channel MOSFET: Connecting with the npn MOSFET A very similar behavior, and very similar uses. MOSET GSD+––+vGSvDSiGiDiBvBEvCEiC0.6 V0.2 VForward Active RegionFARCutoffCutoffSaturationiC ! !F iBvCE > 0.2 ViB ! IBSeqVBE/kTInput curveOutput familyBJT BEC+––+vBEvCEiBiCvDSiDSaturation (FAR)CutoffLinearorTriodeiD ! K [vGS - VT(vBS)]2/2!Clif Fonstad, 10/8/09 Lecture 9 - Slide 2p-SiBG+vGSn+Dn+S–vDSvBS+iGiBiDMOS structures An n-channel MOSFET In an n-channel MOSFET, we have two n-regions (the source and the drain), as in the npn BJT, with a p-region producing a potential barrier for electrons between them. In this device, however, it is the voltage on the gate, vGS, that modulates the potential barrier height. The heart of this device is the MOS capacitor, which we will study today. To analyze the MOS capacitor we will use the same depletion approximation that we introduced in conjunction with p-n junctions. Clif Fonstad, 10/8/09 Lecture 9 - Slide 3The n-MOS capacitor Right: Basic device with vBC = 0 p-Sin+BSGSiO2+–vGS(= vGB)C Below: One-dimensional structure for depletion approximation analysis* Clif Fonstad, 10/8/09 Lecture 9 - Slide 4 BG+–p-SiSiO2x-tox0vGB* Note: We can't forget the n+ region is there; we will need electrons, and they will come from there.Electrostatic potential and net charge profiles φ(x) Zero bias: vGB = 0 -tox xd φmx φp ρ(x) qNAxd xd x-tox qD* = -qNAxd−qNA Clif Fonstad, 10/8/09 Lecture 9 - Slide 5Electrostatic potential and net charge profiles φ(x) Depletion: VFB < vGB < 0 -tox xd φm x φp vGB < 0 ρ(x) qNAxd xd x-tox -qNAxd−qNA Clif Fonstad, 10/8/09 Lecture 9 - Slide 6Electrostatic potential and net charge profiles φ(x) Flat band : vGB = VFB -tox x vGB = VFB VFB = φp– φm φp ρ(x) x-tox φm VFB = φp – φm Clif Fonstad, 10/8/09 Lecture 9 - Slide 7Electrostatic potential and net charge profiles φ(x) Accumulation : vGB < VFB -tox x vGB < VFB ρ(x) φp φm - C*(vGB - VFB)ox -tox C*(vGB - VFB)ox Clif Fonstad, 10/8/09 Lecture 9 - Slide 8 xElectrostatic potential and net charge profiles φ(x) Flat band : vGB = VFB -tox φm x vGB = VFB VFB = φp– φm φp ρ(x) x-tox Clif Fonstad, 10/8/09 Lecture 9 - Slide 9Electrostatic potential and net charge profiles φ(x) Depletion: VFB < vGB < 0 -tox xd φm x φp VFB < vGB < 0 ρ(x) qNAxd xd x-tox -qNAxd−qNA Clif Fonstad, 10/8/09 Lecture 9 - Slide 10Electrostatic potential and net charge profiles φ(x) Depletion: vGB = 0 -tox xd φmx φp ρ(x) qNAxd xd x-tox qD* = -qNAxd−qNA Clif Fonstad, 10/8/09 Lecture 9 - Slide 11Electrostatic potential and net charge profiles φ(x) Depletion: 0 < vGB < VT Weak inversion: φ(0) > 0 xd x 0 < vGB -tox φm< VT φp J = 0 ⇒ n(x) = nie-qφ(x)/kT ρ(x) and p(x) = nieqφ(x)/kT qNAxd φ(0)↑ ⇒ n(0)↑ xd x qD* = -qNAxd-tox −qNA Weak inversion: φ(0) > 0 ⇒ n(0) > p(0) Clif Fonstad, 10/8/09 Lecture 9 - Slide 12Electrostatic potential and net charge profiles φ(x) Threshold: vGB = VT vGB = VT -φp -tox x φp φm XDT At threshold φ(0) = - φp ρ(x) qNAXDT φ(0) = -φp ⇒ n(0) = NA XDT x qD* = -qNAXDT -tox −qNA Clif Fonstad, 10/8/09 Lecture 9 - Slide 13VT = B + |2φp| + (2εSi|2φp|qNA)1/2/Cox* Electrostatic potential and net charge profiles -tox x φ(x) φp φm -tox x −qNA XDT XDT ρ(x)qNAXDT VT – VFB Threshold*: vGB = VT -φp qD * = -qNAXDT |2φp| qNAXDT/Cox * vGB = VT VT – VFB = |2φp| + qNAXDT/Cox * VFXDT = (2εSi|2φp|/qNA)1/2 qD * = -qNAXDT = -(2εSi|2φp|qNA)1/2 VT = VFB + |2φp| + (2εSi|2φp|qNA)1/2/Cox * Clif Fonstad, 10/8/09 * At threshold φ(0) = - φp Lecture 9 - Slide 14Electrostatic potential and net charge profiles Inversion: VT < vGB -tox x φ(x) φp φm XDT VT < vGB -φp |2φp | -tox x −qNA XDT ρ(x) qNAXDT + Cox *(vGB - VT) qN * = - Cox *(vGB - VT) qD * = -qNAXDT qD*, depletion regioncharge unchanged qN* = Inversion layer charge(sheet of mobile electrons inSi near the Si-oxide interface) Clif Fonstad, 10/8/09 Lecture 9 - Slide 15Electrostatic potential and net charge profiles - regions and boundaries φXDT p φ(x) -tox φφ(x) φ(x) φp φvGB -φ|2φp |-tox vGB -tox xd xxx mm φpm φpvGB qNAXDT + xd ox ox ρ(x) ρ(x) C*(vGB - VT)ρ(x) qNAxd ox C- C*(vGB - VFB) -t-tox XDT x-tox xx qD* = -qNAXDT −qNA * −qNAqD = -qNAxdox *(vGB - VFB) vGB * qN = - Cox *(vGB - VT) Acccumulation Depletion Inversion Flat Band VoltageThreshold Voltage vGB < VFB VFB< vGB < VT VT < vGB vGB |qNA)1/2/C– φVT = VFB+|2φp|+(2εSi|2φp ox *VFB = φp m φ(x) φm φφ(x) XDT φ-φpvGB |2φ |p-tox -tox xx mvGB φpp qNAXDT ρ(x) ρ(x) -tox -tox XDT xx −qNA qD* = -qNAXDT Clif Fonstad, 10/8/09 Lecture 9 - Slide 16Clif Fonstad, 10/8/09 Lecture 9 Slide 17 vGB Electrostatic potential and net charge profiles- the grand procession from accumulation to inversion -VT φ(x) Accumulation : vGB < VFB -φp -tox x0 vGB < VFBVFB ρ(x) φp φm - C*(vGB - VFB)ox-tox x Cox *(vGB - VFB) Clif Fonstad, 10/8/09 Lecture 9 -- Slide 17Clif Fonstad, 10/8/09 Lecture 9 Slide 18 vGB Electrostatic potential and net charge profiles- the grand procession from accumulation to inversion -VT φ(x) Flat band : vGB = VFB φ(0) = φp -φp -tox φm x0 vGB = VFB VFB φp= φ – φVFB p m ρ(x) x-tox VFB = φp– φm Clif Fonstad, 10/8/09 Lecture 9 -- Slide 18Clif Fonstad, 10/8/09 Lecture 9 Slide 19
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