# Microelectronic Devices and Circuits Lecture 9 - MOS Capacitors I - Outline Announcements Problem set 5 -

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1 Microelectronic Devices and Circuits Lecture 9 - MOS Capacitors I - Outline Announcements Problem set 5 - Posted on Stellar. Due net Wednesday. Qualitative description - MOS in thermal equilibrium Definition of structure: metal/silicon dioide/p-type Si Electrostatic potential of metal relative to silicon: Zero bias condition: Si surface depleted if > -Si Negative bias on metal: depletion to flat-band to accumulation Positive bias on metal: depletion to threshold to inversion (Eample: n-mos) (typical situation) Quantitative modeling - MOS in thermal equilibrium, v BC = 0 Depletion approimation applied to the MOS capacitor: 1. Flat-band voltage, 2. Accumulation layer sheet charge density, q A 3. Maimum depletion region width, 4. Threshold voltage, V T 5. Inversion layer sheet charge density, q N Quantitative modeling - v BC 0; impact of v BC < 0 Voltage between n+ region and p-substrate: 2-Si 2-Si - v BC Clif Fonstad, 10/8/09 Lecture 9 - Slide 1

2 n-channel MOSFET: Connecting with the npn MOSFET A very similar behavior, and very similar uses. MOSET G i G + i D D + v DS id Linear or Triode Saturation (FAR) i D! K [v GS - V T (v BS )]2/2! BJT B i B + v BE i C C + v CE E v GS ib! IBSe qvbe/kt vce > 0.2 V i B S FAR ic Saturation Cutoff Forward Active Region ic!!f ib vds Cutoff vbe 0.6 V 0.2 V Cutoff Input curve Output family Clif Fonstad, 10/8/09 Lecture 9 - Slide 2 vce

3 MOS structures G S v GS + v DS i G D i D n+ n+ p-si An n-channel MOSFET v BS + B 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, v GS, 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 approimation that we introduced in conjunction with p-n junctions. i B Clif Fonstad, 10/8/09 Lecture 9 - Slide 3

4 The n-mos capacitor C S n+ v GS (= ) + G SiO 2 Right: Basic device with v BC = 0 p-si B Below: One-dimensional structure for depletion approimation analysis + SiO 2 p-si G B -t o 0 Note: We can't forget the n+ region is there; we will need electrons, and they will come from there. Clif Fonstad, 10/8/09 Lecture 9 - Slide 4

5 Electrostatic potential and net charge profiles φ() Zero bias: = 0 -t o d ρ() d d -t o q D = - d Clif Fonstad, 10/8/09 Lecture 9 - Slide 5

6 Electrostatic potential and net charge profiles φ() Depletion: < < 0 -t o d < 0 ρ() d d -t o - d Clif Fonstad, 10/8/09 Lecture 9 - Slide 6

7 Electrostatic potential and net charge profiles φ() Flat band : = -t o = = ρ() = -t o Clif Fonstad, 10/8/09 Lecture 9 - Slide 7

8 Electrostatic potential and net charge profiles φ() Accumulation : < -t o < ρ() -t o C ( - ) o - C ( - ) o Clif Fonstad, 10/8/09 Lecture 9 - Slide 8

9 Electrostatic potential and net charge profiles φ() Flat band : = = -t o = ρ() -t o Clif Fonstad, 10/8/09 Lecture 9 - Slide 9

10 Electrostatic potential and net charge profiles φ() Depletion: < < 0 -t o d < < 0 ρ() d d -t o - d Clif Fonstad, 10/8/09 Lecture 9 - Slide 10

11 Electrostatic potential and net charge profiles φ() Depletion: = 0 -t o d ρ() d d -t o q D = - d Clif Fonstad, 10/8/09 Lecture 9 - Slide 11

12 Electrostatic potential and net charge profiles φ() Depletion: 0 < < V T Weak inversion: φ(0) > 0 -t o d 0 < < V T ρ() J = 0 n() = n i e -qφ()/kt and p() = n i e qφ()/kt d φ(0) n(0) -t o q D = - d d Weak inversion: φ(0) > 0 n(0) > p(0) Clif Fonstad, 10/8/09 Lecture 9 - Slide 12

13 Electrostatic potential and net charge profiles = V T φ() Threshold: = V T - At threshold φ(0) = - -t o ρ() φ(0) = - n(0) = N A -t o q D = - Clif Fonstad, 10/8/09 Lecture 9 - Slide 13

14 Electrostatic potential and net charge profiles φ() Threshold: = V T = V T V T -t o - /C o = (2ε Si 2 / ) 1/2 2 ρ() V T = 2 + /C o V B )1/2 T = (2ε Si 2 ) 1/2 /C o V F -t o q D = - q D = - = -(2ε Si 2 ) 1/2 Clif Fonstad, 10/8/09 At threshold φ(0) = - Lecture 9 - Slide 14

15 Electrostatic potential and net charge profiles V T < φ() Inversion: V T < -φp -t o 2 + C o ( - V T ) ρ() q N = Inversion layer charge (sheet of mobile electrons in Si near the Si-oide interface) -t o q D = - q D, depletion region q N = - C o ( - V T ) charge unchanged Clif Fonstad, 10/8/09 Lecture 9 - Slide 15

16 Electrostatic potential and net charge profiles - regions and boundaries φ() φ() -φ -t o -t o d -t o φ φ φ m m + p φ() m 2 ρ() C ( - V T ) ρ() d o ρ() - C ( - ) -t -t q D = - C q o ( - ) v D = - GB d q N = - C o ( - V T ) -t o o o d o X DT Acccumulation Depletion Inversion < < < V T V T < Flat Band Voltage Threshold Voltage φ V ) 1/2 T = + 2 +(2ε Si 2 /C = m o φ() φ() - -t -t o 2 o φ φ p ρ() m φp ρ() -t o -t o q D = - Clif Fonstad, 10/8/09 Lecture 9 - Slide 16

17 Electrostatic potential and net charge profiles - the grand procession from accumulation to inversion - V T φ() Accumulation : < - 0 -t o < ρ() - C ( - ) o -t o C o ( - ) Clif Fonstad, 10/8/09 Lecture 9 - Slide 17

18 Electrostatic potential and net charge profiles - the grand procession from accumulation to inversion - V T φ() Flat band : = φ(0) = 0 = - -t o = ρ() -t o = Clif Fonstad, 10/8/09 Lecture 9 - Slide 18

19 Electrostatic potential and net charge profiles - the grand procession from accumulation to inversion - V T φ() Depletion: < < 0 < φ(0) - -t o d 0 < < 0 ρ() d d -t o - d Clif Fonstad, 10/8/09 Lecture 9 - Slide 19

20 Electrostatic potential and net charge profiles - the grand procession from accumulation to inversion - V T φ() Depletion: = 0 < φ(0) < 0-0 -t o d ρ() d -t o d q D = - d Clif Fonstad, 10/8/09 Lecture 9 - Slide 20

21 Electrostatic potential and net charge profiles - the grand procession from accumulation to inversion - V T φ() - d Depletion: 0 < < V T Weak Inversion: φ(0) > < -t o < V T d ρ() -t o q D = - d d Clif Fonstad, 10/8/09 Lecture 9 - Slide 21

22 Electrostatic potential and net charge profiles - the grand procession from accumulation to inversion - V T = V T φ() Threshold: = V T φ(0) = t o ρ() -t o q D = - V T = (2ε Si 2 qna )1/2 /Co Clif Fonstad, 10/8/09 Lecture 9 - Slide 22

23 Electrostatic potential and net charge profiles - the grand procession from accumulation to inversion - V T V T < φ() Inversion: V T < - < φ(0) 0 - -t o 2 + C ( - V T ) o ρ() -t o qd = - q N = - C ( - V T ) o Clif Fonstad, 10/8/09 Lecture 9 - Slide 23

24 Bias between n+ region and substrate, cont. Reverse bias applied to substrate, I.e. v BC < 0 C v GC + G SiO 2 v BC < 0 v BC + n+ p-si Soon we will see how this will let us electronically adjust MOSFET threshold voltages when it is convenient for us to do so. B Clif Fonstad, 10/8/09 Lecture 9 - Slide 24

25 With voltage between substrate and channel, v BC < 0 V T (0) Flat band: = φ() No difference from when v BC = 0 0 = -t o = φp ρ() -t o Clif Fonstad, 10/8/09 Lecture 9 - Slide 25

26 φ() With voltage between substrate and channel, v BC < 0 V T (0) Depletion: 0 < < V T (v BC ) - v BC - d No difference from when v BC = 0 0 -t o VFB d ρ() -t o q D = - d d Clif Fonstad, 10/8/09 Lecture 9 - Slide 26

27 φ() With voltage between substrate and channel, v BC < 0 V T (0) Depletion: 0 < < V T (v BC ) No difference from when v BC = 0 - v BC - 0 -t o ρ() -t o qd = - Clif Fonstad, 10/8/09 Lecture 9 - Slide 27

28 V T (v BC ) φ() With voltage between substrate and channel, v BC < 0 V T (0) = V T (v BC ) At threshold: = V T (v BC ) Big difference from when v BC = 0 - v BC 2 v BC 0 - -t o (v CB = 0) (v BC < 0) ρ() (v BC < 0) -t o q N = - DT Clif Fonstad, 10/8/09 Lecture 9 - Slide 28

29 φ() With voltage between substrate and channel, v BC < 0 = V T (v BC ) - v BC - 2 v BC -t o X (v DT CB = 0) (v BC < 0) Threshold: v GC = V T (v BC ) with v BC < 0 -vbc )qna]1/2 /Co ρ() V T (v BC ) = [2ε Si ( 2 {This is v GC at threshold} (v BC < 0) = [2ε Si ( 2 -v BC )/ ] 1/2 -t o q N = - DT -v BC ) ]1/2 Clif Fonstad, 10/8/09 Lecture 9 - Slide 29 q N = -[2ε Si ( 2 (v BC < 0)

30 Microelectronic Devices and Circuits Lecture 9 - MOS Capacitors I - Summary Qualitative description Three surface conditions: accumulated, depleted, inverted Two key voltages: flat-band voltage, ; threshold voltage, V T The progression: accumulation through flat-band to depletion, then depletion through threshold to inversion Quantitative modeling Apply depletion approimation to the MOS capacitor, v BC = 0 Definitions: such that φ(0) = -Si V T such that φ(0) = -Si C o ε o /t o Results and epressions (For n-mos eample) 1. Flat-band voltage, = -Si 2. Accumulation layer sheet charge density, q A = C o ( ) 3. Maimum depletion region width, = [2ε Si ( 2-Si -v BC )/ ] 1/2 4. Threshold voltage, V T = 2-Si + [2ε Si ( 2-Si -v BC )] 1/2 /C o 5. Inversion layer sheet charge density, q N = C o ( V T ) Clif Fonstad, 10/8/09 Lecture 9 - Slide 30

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