CS 15 Fall 23, Lecture 8 Lecture 8: Capacitor ad PN Juctio Prof. Nikejad
Lecture Outlie Review of lectrotatic IC MIM Capacitor No-Liear Capacitor PN Juctio Thermal quilibrium
lectrotatic Review 1 lectric field go from poitive charge to egative charge by covetio +++++++++++++++++++++ lectric field lie diverge o charge ρ I word, if the electric field chage magitude, there ha to be charge ivolved! Reult: I a charge free regio, the electric field mut be cotat!
lectrotatic Review 2 Gau Law equivaletly ay that if there i a et electric field leavig a regio, there ha to be poitive charge i that regio: +++++++++++++++++++++ lectric Field are Leavig Thi Bo! d d Recall: Q / ρ ds Q d ds S Q
lectrotatic i 1D verythig implifie i 1-D d ρ ρ d ρ ' + ' Zero field boudary coditio Coider a uiform charge ditributio ρ ρ 1 ' ' ρ ρ 1 ρ 1
lectrotatic Potetial The electric field force i related to the potetial eergy: d Negative ig ay that field lie go from high potetial poit to lower potetial poit egative lope Note: A electro hould float to a high potetial poit: d F e q e 1 2 d F e e e
More Potetial Itegratig thi baic relatio, we have that the potetial i the itegral of the field: C dl d l I 1D, thi i a imple itegral: ' ' Goig the other way, we have Poio equatio i 1D: d ρ 2 2
Boudary Coditio Potetial mut be a cotiuou fuctio. If ot, the field force would be ifiite lectric field eed ot be cotiuou. We have already ee that the electric field diverge o charge. I fact, acro a iterface we have: 1 1 2 2 S ds 1 1 S + 2 2 S Q iide Q iide S 1 1 S + 2 2 1 2 2 1 Field dicotiuity implie charge deity at urface!
IC MIM Capacitor Bottom Plate Top Plate Bottom Plate Cotact Thi Oide Q C By formig a thi oide ad metal or polyilico plate, a capacitor i formed Cotact are made to top ad bottom plate Paraitic capacitace eit betwee bottom plate ad ubtrate
Review of Capacitor + +++++++++++++++++++++ dl t o Q Q ds A A t o t o ds Q ds Q Q C C A t o For a ideal metal, all charge mut be at urface Gau law: Surface itegral of electric field over cloed urface equal charge iide volume
Capacitor Q- Relatio Q y +++++++++++++++++++++ Qy y Q C Total charge i liearly related to voltage Charge deity i a delta fuctio at urface for perfect metal
A No-Liear Capacitor Q y +++++++++++++++++++++ Qy y Q f We ll oo meet capacitor that have a o-liear Q- relatiohip If plate are ot ideal metal, the charge deity ca peetrate ito urface
What the Capacitace? For a o-liear capacitor, we have Q f C We ca t idetify a capacitace Imagie we apply a mall igal o top of a bia voltage: Q f df + v f + v d Cotat charge The icremetal charge i therefore: Q Q + q f df d + v
Small Sigal Capacitace Break the equatio for total charge ito two term: Q Q + q Icremetal Charge f df d + v Cotat Charge q df d v C v df C d
ample of No-Liear Capacitor Net lecture we ll ee that for a PN juctio, the charge i a fuctio of the revere bia: Q j qn 1 a p b oltage Acro NP Juctio Charge At N Side of Juctio Cotat Small igal capacitace: C j dq d j qn 2 a b p 1 1 b C 1 j b
Carrier Cocetratio ad Potetial I thermal equilibrium, there are o eteral field ad we thu epect the electro ad hole curret deitie to be zero: d d o J q µ + qd µ D o q kt o o d kt q d d o th d
Carrier Cocetratio ad Potetial 2 We have a equatio relatig the potetial to the carrier cocetratio kt q d o th d d If we itegrate the above equatio we have th l We defie the potetial referece to be itriic Si: i
Carrier Cocetratio eru Potetial The carrier cocetratio i thu a fuctio of potetial i e / th Check that for zero potetial, we have itriic carrier cocetratio referece. If we do a imilar calculatio for hole, we arrive at a imilar equatio p i e / th Note that the law of ma actio i upheld th 2 / th / p i e e i 2
The Dopig Chage Potetial Due to the log ature of the potetial, the potetial chage liearly for epoetial icreae i dopig: th l 26m l i i 6m log 6m log 1 1 p 1 26m l1 log 1 1 1 Quick calculatio aid: For a p-type cocetratio of 1 16 cm -3, the potetial i -36 m N-type material have a poitive potetial with repect to itriic Si
PN Juctio: Overview The mot importat device i a juctio betwee a p-type regio ad a -type regio Whe the juctio i firt formed, due to the cocetratio gradiet, mobile charge trafer ear juctio p-type N A lectro leave -type regio ad hole leave p-type regio Thee mobile carrier become miority carrier i ew regio ca t peetrate far due to recombiatio + + + + + + + + + + + + + + + + Due to charge trafer, a voltage differece occur betwee regio Thi create a field at the juctio that caue drift curret to oppoe the diffuio curret N D -type I thermal equilibrium, drift curret ad diffuio mut balace
PN Juctio Curret Coider the PN juctio i thermal equilibrium Agai, the curret have to be zero, o we have d o J q µ + qd d o q µ qd D D p d µ dp µ p o o kt q kt q 1 1 p d dp
PN Juctio Field p-type -type N A N D p N a p J diff p i N 2 d p N d i N 2 a J diff + + Traitio Regio
Total Charge i Traitio Regio To olve for the electric field, we eed to write dow the charge deity i the traitio regio: ρ q p + N d N a I the p-ide of the juctio, there are very few electro ad oly acceptor: ρ q p N a p < < Sice the hole cocetratio i decreaig o the p- ide, the et charge i egative: N a > p ρ <
Charge o N-Side Aalogou to the p-ide, the charge o the -ide i give by: ρ q + N d < < The et charge here i poitive ice: N d > ρ > N d i N 2 diff a J + + Traitio Regio