ECE66: Sold State Devces Lecture 13 Solutos of the Cotuty Eqs. Aalytcal & Numercal Gerhard Klmeck gekco@purdue.edu Outle Aalytcal Solutos to the Cotuty Equatos 1) Example problems ) Summary Numercal Solutos to the Cotuty Equatos 1) Basc Trasport Equatos ) Grddg ad fte dffereces 3) Dscretzg equatos ad boudary codtos 4) Cocluso
Cosder a complcated real devce example x Metal cotact Upassvated surface 1 3 Acceptor doped Lght tured o the mddle secto. The rght rego s full of md-gap traps because of daglg bods due to u-passvated surface. Iterface traps at the ed of the rght rego (That s where the daglg bods are ) The left rego s trap free. The left/rght regos cotacted by metal electrode. 3 Recall: Aalytcal Soluto of Schrodger Equato 1) d ψ + k ψ N ukows for N regos ) ψ ( x ) ψ ( x + ) Reduces ukows 3) ψ dψ ψ + x xb x xb dψ + x xb x xb Set N- equatos for N- ukows (for cotuous U) 4) Det(coeffcet matx) 5) ψ ( x, E) 1 Ad fd E by graphcal or umercal soluto for wave fucto 4
Recall: Boud-levels Fte well E U(x) 1) ψ As kx + B cos kx ψ Me + Ce α x + α x ψ De + Ne α x + α x ) Boudary Codtos ψ ( x ) ψ ( x + ) a 5 Aalogously, we solve for our devce Solve the equatos dfferet regos depedetly. Brg them together by applyg boudary codtos. 6
Rego : Traset, Uform Illumato, Uform dopg (uform) 1 J N r + N g N t q ( + ) + G t τ (uform) p 1 r + g t q J p P p J qµ E + qd N N N Recall Shockley- Read-Hall qpµ E qd p J p p P Acceptor doped ( p + p) p + G Majorty carrer t τ p Electrc feld stll zero because ew carrers balace D q p + N N q p + N N ( + ) ( + D A p D A ) + 7 Example: Traset, Uform Illumato, Uform dopg, No appled electrc feld ( ) + G t τ ( x, t) A + Be t τ t, ( x,) A B t, ( x, ) Gτ A t ( ) ( x, t) Gτ 1 e τ Acceptor doped No carrers yet geerated Steady state, o chage carrers wth tme tme 8
t (steady-state) r N (trap free) (ogeerato) g N Rego 1: Oe sded Morty Dffuso at steady state Steady state Acceptor doped 1 Trap-free 1 dj rn + gn t q d D N d E J N qµ N E + qdn d D N (due to serto of electros from cetral rego) 9 Example: Oe sded Morty Dffuso d DN ( x, t) C + Dx ' x a, ( x' a) C Da (Metal has hgh electro desty as compared to semcoductor) x ', (x ' ') C Just substtute x above eq. ' (, ) ( ') x x t x 1 a J q υ m m x x Metal cotact a 1
t (steady-state) r N (ot trap free) (ogeerato) g N Rego 3: Steady state Morty Dffuso wth recombato E d D N (due to serto of electros from cetral rego) D N D d ( + ) τ N d τ Flux Steady state Acceptor doped 3 Trap-flled 11 Dffuso wth Recombato D N d τ Fuctoally smlar to Schrodger eq. b Metal cotact x L x L ( x, t) Ee + Fe 3 x b, ( x b) F Ee b L x, ( x ) E + F ( x ) X () x t e e e (1 e ) x L b L x L (, ) ( ) b L x 1
( x) Gτ ( ) (') 1 ( x ') ( x ) 1 x a ' (') x L b L x L ( ) ( ) b L x e e e (1 e ) x Combg them all. 1 3 Match boudary codto x' Gτ 1 a d Calculatg curret JN qµ N E + qdn b Gτ e e e (1 e ) x L b L x L ( ) b L 13 Aalytcal Solutos Summary 1) Cotuty Equatos form the bass of aalyss of all the devces we wll study ths course. ) Full umercal soluto of the equatos are possble ad may commercal software are avalable to do so. 3) Aalytcal solutos however provde a great deal of sght to the key physcal mechasm volved the operato of a devce. 14
Outle Aalytcal Solutos to the Cotuty Equatos 1) Example problems ) Summary Numercal Solutos to the Cotuty Equatos 1) Basc Trasport Equatos ) Grddg ad fte dffereces 3) Dscretzg equatos ad boudary codtos 4) Cocluso 15 The 5 equatos we derved the past few lectures have bee used for the logest tme the dustry ad academa to uderstad carrer trasport devces. Preface It s useful to kow the essetals of how these equatos are mplemeted o a moder computer so that oe uderstads some of the fer detals volved creatg tools that smulate thee pheomea. Uderstadg some of these detals helps oe become a power user of the smulato tools that mplemet the physcs. Oe also uderstads the lmtatos re. umercal ssues ad applcablty rages of results. 16
Equatos to be solved derved last tme + ( D A ) D q p + N N 1 J r + g t q p 1 r + g t q J N N N J qµ E + qd N N N J P P P qpµ E qd p P P P Bad-dagram Dffuso approxmato, Morty carrer trasport, Ambpolar trasport 17 1) The Semcoductor Equatos Coservato Laws: ot specfc to a partcular problem - Uversal D ρ J q q ( ) ( g N r N ) ( J p ) ( g P r P ) (steady-state) Costtutve relatos: specfc to problem at had reflect physcs of the problem D κε E κε V ( p + N + D N ) q A ρ J qµ E + qd J p pqµ p E qd p p g, f (, p) etc. N P 18
1) The Mathematcal Problem The Semcoductor Equatos D ρ J q q ( ) ( g N r N ) ( J p ) ( g N r N ) 3 coupled, olear, secod order PDE s for the 3 ukows: Why are these equatos coupled? Potetal Feld curret chages potetal chages feld ad so o V ( r ) ( r ) p( r ) Coservatos laws: exact Trasport eqs. (drft-dffuso): approxmate 19 Outle Aalytcal Solutos to the Cotuty Equatos 1) Example problems ) Summary Numercal Solutos to the Cotuty Equatos 1) Basc Trasport Equatos ) Grddg ad fte dffereces 3) Dscretzg equatos ad boudary codtos 4) Cocluso
() exact umercal solutos Grddg total legth dvded to N parts - equal (uform grddg), or - uequal (adaptve ad o-uform grddg) Varables descrbed at each pot. a ) The Grd N odes 3N ukows V p V o ad V +1 s kow because these are voltages at source ad dra. 1 Fte Dfferece Expresso for Dervatve a df f ( xo) f ( x o + a ) ( o ) ( a a df f x + a f x + ) + f(x) a df ( x+ 1/ ) f + 1 a f x -1 x x +1 x cetered dfferece
The Secod Dervatve f f ( x a) ( x ) df a d f + f + a + +... ( x a) ( x ) x a x a df a d f f a +... x a x a ( + ) ( x a) ( x ) f x a f d f 1 + 1 a d f + f f a f + f x a 3 pot formula, could be exteded to N pots depedg o the umber of dervatves we carry our expaso 3 Outle Aalytcal Solutos to the Cotuty Equatos 1) Example problems ) Summary Numercal Solutos to the Cotuty Equatos 1) Basc Trasport Equatos ) Grddg ad fte dffereces 3) Dscretzg equatos ad boudary codtos 4) Cocluso 4
) Cotrol Volume 3 ukows at each ode: V,, p x ( -1) () ( +1) Need 3 equatos at each ode cotrol volume 5 Dscretzg Posso s Equato ρ / εo V K s V ( 1) V ( ) +V ( + 1) a D ρ D ε ε V K s E-K s q (p +N + K s ε D, N A, ) Sce V o ad V -1 are kow, as are carrer cocetrato o dopg (or lack thereof) cotacts, we fd V 1 terate from ths pot to solve for potetal. F V ad ( V V, V,, p ) 1, + 1 ( -1) () ( +1) D L D R Oce ths potetal s foud, solve cotuty equato to obta ew carrer cocetratos 6
Dscretzg Cotuty Equatos J q ( g N r N ) dv d JL qµ + ktµ ( -1) () ( +1) J L The smplest approach.. JL ktµ + V V a( kt / q) a 1 1 1 + ( ) F V, V,,, p, p 1 1 1 7 Three Dscretzed Equatos F F F V p x ( -1) () ( +1, j) 3 ukows at each ode N odes 3N ukows ad 3N equatos (coupled to each other) 8
Numercal Soluto Posso Equato Oly Have a system of 3N olear equatos to solve Recall Posso s equato at ode (): [A]: F V ( V V, V,, p ) 1, + 1 lear f ad p are kow V [ A] V b V 1 V V N 9 p p Boudary codtos Cotacts are assumed large ad equlbrum detaled balace ad law of mass-acto apply!! N+ 1 N+ 1 Dopat desty a Oe could have uequal materals o the two cotact sdes, oe must be careful to V V A V use the rght trsc cocetrato <-> materal. 3
-cotuty p-cotuty Numercal Soluto Posso N11 R11 V11 ( x1 ) ( x) V>... ( xm) m+ 1 R11 P11 V 11 p( x1 ) p p( x ) V>p.. Pmm p( xm ) pm+ 1 V 11 V ( x1 ) Q 11 V ( x) >Q p>q... V mm V ( xm) Q mm Off-dagoal terms are Posso-Cotuty equatos talkg to each other. Recombato-geerato terms also feed to cotuty equatos. 31 3) Ucoupled Numercal Soluto The semcoductor equatos are olear! (but they are lear dvdually) Guess V,,p Ucoupled soluto procedure repeat utl satsfed Solve Posso for ew V Solve electro cot for ew Solve hole cot for ew p 3
Summary 1) Two methods to solve drft-dffuso equato cosstetly aalytcal ad umercal. ) Aalytcal soluto provdes great sght ad the soluto methodology s smlar to that of Schrodger equatos. 3) Numercal soluto s more versatle. Oe begs wth a set of equatos ad boudary codtos, dscretze the equatos o a grd wth N odes to obta 3N olear equatos 3N ukows, ad solve the system of olear equatos by terato. 33