ECE606: Solid State Devices Lecture 14 Electrostatics of p-n junctions

Transcription

1 ECE606: Solid State evices Lecture 14 Electrostatics of - juctios Gerhard Klimeck gekco@urdue.edu Outlie 1) Itroductio to - juctios ) rawig bad-diagrams 3) ccurate solutio i equilibrium 4) Bad-diagram with alied bias Ref. Semicoductor evice Fudametals, Chater 5

2 What is a iode good for. solar cells Gas lasers Orgaic LE valache Photodiode Ga lasers Image.google.com 3 - Juctio evices Schematic of a - iode Symbol Cotact P-doed ocket P -doed substrate Cotact Poit-cotact diode 4

3 Toic Ma Toic Ma (Today : iode i Equilibrium) Equilibrium C Small sigal Large Sigal Circuits iode Schottky iode i Equilibrium. (o eteral voltage alied) BJT/HBT MOS 5 rawig Bad iagram i Equilibrium Previously costat i homogeeous semicoductors. But for diode: f()!! + ( ) = q + 1 = J r + g t q 1 = r + g t q J J = qµ E + q J P P P = qµ E q P P P Equilibrium (Start here) I equilibrium J=0 (o curret flow). But, Electric fields or diffusio might still be reset. etailed balace o-equilibrium (refie later) C d/dt=0 Small sigal d/dt ~ jωt Trasiet --- full solutio 6

4 P ad doed Material Side by Side E -doed P-doed Vacuum level (determied by material) Ec Coductio Bad EF Coductio Bad EV Valece Bad Valece Bad EF -k k For P-doed E F close to E V 7 Formig a Juctio Sace charge (fied) Mobile carriers oor-side (-side) Squares are fied door atoms. Every door atom has give away oe electro (blue circle) ccetor-side (P-side) Squares are fied accetor atoms. Every accetor atom has catured oe electro (from the valece bad). Every accetor atom leaves behid oe hole i the valece bad. (red circle)

5 Formig a Juctio Sace charge (fied) Mobile carriers = Juctio = 0 = i Valid oly i equilibrium Before joiig the ad the side Formig a Juctio iffusio E-field Sace charge (fied) Mobile carriers = Juctio = 0 = i Before joiig the ad the side eletio aroimatio ctual Carrier Cocetratios fter joiig the ad the side l(), l() -side -side eletio aroimatio Bulk Regio eleted Regio Bulk Regio 10

6 Formatio of a Juctio + l( ) l() l() l( ) et charge o -side: q (subtract gree from blue lie) Sace charge regio: fied door atoms eglect electros i -side X : deletio regio i -side Q=-+ - q eleted Regio et charge o P-side: -q (subtract brow from red lie) Sace charge regio: fied accetor atoms X : deletio regio i P-side red ad blue are of equal size charge balace Bulk Regio Bulk Regio eletio Regio meas deleted of mobile charges -q 11 Sketch of Electrostatics Potetial Itegrate V bi ositio E-field Itegrate E ma q q = = K ε o K ε o s s ositio -+- q eleted Regio ositio Bulk Regio Bulk Regio -q 1

7 Sketch of Electrostatics Ivert to go from otetial to eergy scale Potetial ositio Bad iagram ositio I equilibrium Fermi-level must be flat 13 Outlie 1) Itroductio to - juctios ) rawig bad-diagrams 3) alytical solutio i equilibrium 4) Bad-diagram with alied bias 14

8 Short-cut to Bad-diagram eutral Sace Charge eutral Vacuum level χ rawig Recie 1) Start with EF ) Ec/Ev i bulk -side 3) Ec/Ev i bulk -side 4) Vacuum level i 5) Vacuum level i P 6) Joi vacuum levels 7) Trasfer vacuum sloes to joi Ec/Ev χ 1 E C E V E F is equivalet to solvig the Poisso equatio 15 Built-i Potetial: boudary lways true i equilibrium qv bi χ 1 χ 1 E g, + χ + qv i = χ + E 1 1 b g, delta 1, determied via doig cocetratios X 1, material arameters Built-i otetial V bi ukow! 1 χ χ1 qv bi = E g, + = E + k T l + k T l + = k T l + ( χ χ 1 ) e ( χ ) g, B B χ1 V, C, 1 B E g, / k B T V, C, 1 16

9 Iterface Boudary Coditios Homo - Juctio Hetero - Juctio E = (/kεo) ositio Field ot cotiuous across juctio ositio = K ε E(0 ) = K ε E(0 ) = E K = K (0 ) E(0 + ) islacemet is cotiuous across the iterface, field eed ot be Built-i voltage for Homo-juctios eutral Sace Charge eutral rawig Recie 1) Start with EF ) Ec/Ev i bulk -side 3) Ec/Ev i bulk -side 4) Vacuum level i 5) Vacuum level i P 6) Joi vacuum levels 7) Trasfer vacuum sloes to joi Ec/Ev qv bi χ 1 χ 1 Vacuum level E C E F E V Zero for homo-juctios qv = k T l + ( χ χ ) = k T l = k T l e bi B E 1 g, / kbt B E g / kbt B V, C, 1 V Ce i 18

10 alytical Solutio of Poisso Equatio eleted Regio Q= -+ - q -q ositio d V Sε 0 = + + ( ) K q d 19 alytical Solutio for Homojuctios (Charge) -+- Itegrate E-field Itegrate q -q ositio E E ( 0 ) + ( 0 ) E-field q = k ε 0 q = k ε = s s 0 Potetial Small! V bi ositio ositio Itegrate + ( 0 ) ( 0 ) E E qvbi = + q q = + k ε k ε s 0 s 0 Potetial 0

11 eletio Regios i Homojuctios eutral Sace Charge eutral Solve for, qv bi = q q = + k ε k ε s 0 s 0 = = ksε 0 q ksε 0 q bi V ( + ) bi V ( + ) Small Project: Solve the same roblem for a hetero-juctio 1 Comlete alytical Solutio ρ = 0 ρ = q( ) P ρ = 0 ρ Itegrate - 0 E Itegrate V - q q q... 0 Ksε o If you eed to calculate electric de q =... 0 field at secific Ksε o d oits 0..., 0 ( ) q E q dε = 0 d ε = ε ( ) d d K ε KSε S 0 0 q q E( ) = ( + )... 0 E( ) = ( ) 0 K ε... KSε 0 S 0

12 Outlie 1) Itroductio to - juctio trasistors ) rawig bad-diagrams 3) alytical solutio i equilibrium 4) Bad-diagram with alied bias 3 Toic Ma Equilibriu m C Small sigal Large Sigal Circuit s iode Schottk y BJT/HB T MOS iode i o-equilibrium (Eteral C voltage alied) 4

13 lyig Bias to - Juctio IV characteristics of a iode To be discussed i detail l(i) 3 1. iffusio limited. mbiolar trasort 1 3. High ijectio 6,7 4. R-G i deletio Reverse Bias Forward Bias V 5. Breakdow 6. Tra-assisted R-G 4 7. Esaki Tuelig 5 5 Forward ad Reverse Bias Curret flows easily Forward Bias Curret does ot flow easily Reverse Bias 6

14 Bad iagram with lied Bias + ( ) = q + 1 = J r + g t q 1 = J r + g t q J J = qµ E + q P P P = qµ E q P P P Bad diagram (this segmet) et segmet / lecture 7 lyig a Bias: Poisso Equatio o bias qv bi E C -E F E F -E V Questio: Ma value of V bi? swer: for degeerate s.c., if E C -E F =0, E F -E V =0 E g lied bias q(v bi -V) ( ) = e ( ) = e i i ( F ) β Ei ( F E ) β i -qv E C -F F -E V ( F F ) β i = e P-side grouded 8

15 eletio Widths -V G From revious lecture (homo-juctio) = q ( + ) k ε s 0 = Vbi ( V) q q = + k ε k ε ( V ) q V bi s 0 s 0 k ε ( V V) s 0 = bi q ( + ) lied bias What about heterojuctios? 9 Fields ad eletio at Forward/Reverse Biases Charge P V<0 V=0 V>0 Electric Field Potetial V<0 V<0 V=0 V>0 V=0 V>0 Positio Positio V<0 V=0 V>0 Barrier height is reduced at forward biases Sigificat icrease of eak field at reverse bias Positio

16 Summary P-Juctio Electrostatics 1) Learig to draw bad-diagrams is oe of the most imortat toics you lear i this course. Bad-diagrams are a grahical way of quickly solvig the Poisso equatio. ) If you cosistetly follow the rules of drawig bad-diagrams, you will always get correct results. Try to follow the rules, ot guess the fial result. 31 ECE606: Solid State evices - diode I-V characteristics Gerhard Klimeck gekco@urdue.edu

17 Outlie 1) erivatio of the forward bias formula ) Solutio i the oliear regime 3) I-V i the ambiolar regime 4) Coclusio Ref. SF, Chater 6 33 Toic Ma Equilibrium C Small sigal Large Sigal Circuits iode Schottky iode i o-equilibrium (Eteral C voltage alied) BJT/HBT MOSFET 34

18 Cotiuity Equatios for - juctio iode + ( ) E = q + 1 = J r + g t q 1 = J r + g t q J J = qµ E + q P P P = qµ E q P P P This sectio 35 lyig a Bias: Poisso Equatio Review qv bi o bias E C -E F E F -E V q(v bi -V) lied bias -qv E C -F F -E V 36

19 eletio Widths Review -V G = q ( + ) k ε s 0 = Vbi ( V) q q = + k ε k ε ( V ) q V bi s 0 s 0 k ε ( V V) s 0 = bi q ( + ) 37 Flat Quasi-Fermi Level u to Juctio E C Equilibrium: May electros - side. Few electros P-side. E V etailed balace of drift ad diffusio forces. o et curret E C J E V J Forward Bias: Electro J ad hole J currets across juctio. iffusio force uchaged. (because doig did ot chage) But, drift force icreased due to alied bias et curret 38

20 1 Various Regios of I-V Characteristics l(i) 6,7 q l( I) ~ V k T B 3 1. iffusio limited. mbiolar trasort 1 3. High ijectio V 4. R-G i deletio 5. Breakdow 4 6. Tra-assisted R-G 5 7. Esaki Tuelig 39 Recall: Oe Sided Miority iffusio If curret is cotiuous, oe ca calculate for the curret aywhere. Positio does t matter! Calculate at easiest ositio. Miority carrier curret o P-side. We kow the solutio from earlier ssume steady state, o r, g -V q(v bi -V) Steady state ccetor doed F -E V 1 dj = r + g t q d J = qµ E + q d d d 0 = d 40

21 Boudary Coditios Boudary Coditios + ( = 0 ) = e + ( = 0 ) = e i i ( F Ei ) β ( F Ei ) β ifferece of Quasi-Fermi-levels equals alied voltage ( F F ) β qvβ = i e = i e q(v bi -V ) (0 ) = (0 ) (0 ) VG i = qvβ ( e 1) Solutio rofile VG = 0 + (0 ) + (0 ) = i qv β = e -V F X=0 + F 41 Right Boudary Coditio ( = W) i ( = W ) = 0 E C V metal Ifiite surface recombiatio velocity at metal ecess carrier cocetratio = 0 E V 4

22 Eamle: Oe Sided Miority iffusio d 0 d = satz (, t) = C + V Plug i B.C. = W, ( = W ) = 0 C = W i qvβ = 0', ( = 0) = ( e 1) = C i qv (, ) ( β t = e 1) 1 W Fial result: Ecess electro carrier cocetratio (P-side) as fuctio of ositio 43 Electro & Hole Flues i qv ( ) ( β = e 1) 1 W J = qµ E + q Curret (electros) d q = = J i q d = 0 W e qv ( β 1) F F Curret (holes) d q i qv J = q = e d W = 0' β ( 1) 44

23 Total Curret Forward Bias l J qv k T + l( cost.) T B Reverse Bias JT cost. iffusio curret = + J q i i T W W e To get total curret, add electro ad hole curret values at idetical -oits. (assume o recombiatio) qvβ ( 1) l(i) (4) () (3) (1) V F F (5) 45 Outlie 1) erivatio of the forward bias formula ) Solutio i the oliear regime 3) I-V i the ambiolar regime 4) Coclusio 48

24 oliear Regime (3) i i ( qv ) ( F F β JT = q + e 1) W W q( V aj ) 0 ( bj β = I e 1) Today s lecture: oliear Regime (,3) l(i) 3 ssumtio of flat Quasi- Fermi levels ivalid here 1 6,7 V 49 Flat Quasi-Fermi Level u to Juctio? J d = qµ E + q d df J W J = µ F = Rewrite ito o-equilibrium form, re-arrage J equatio d µ β ( F Ei ) d df ( F Ei ) = ie q = β q β ie d E d ro of Quasi-Fermi level across the juctio roortioal to curret! ew diffusio comoet: Plug this ito origial J equatio d df q = q β d E d df k T = µ E = B q d µ q F F W 50

25 Forward Bias: oliear Regime i (0 + ) = e ( F F ) β juctio ( ) i ( qv F F ) β + i ( qv F F ) β = e (0 ) = e 1 i i ( qv ) ( F F β JT = q + e 1) W W J W F = µ J W F = µ V F F Still diffusio domiated trasort? Sice Quasi-Fermi levels are ot flat i oliear regime (drift), this aroimatio becomes worse. 51 Outlie 1) erivatio of the forward bias formula ) Solutio i the oliear regime 3) I-V i the ambiolar regime 4) Tuelig ad I-V characteristics 5) Coclusio 5

26 Regio (): mbiolar Trasort ( qv F F ) β / qv JT q + ie l( JT ) W W kbt Today s lecture: mbiolar Trasort regime () Questio: Where does the come from? l(i) 3 1 6,7 V 53 = i e i q( V ) ( )( ) ( 1) F F β + + = i e q( V F F ) β = i ( e 1) q( V F F ) β / e i ( F F ) β Ecess carrier cocetratios >> Currets qi ( qv F F ) J = q = e W W qi ( qv F F ) J = q = e W W oliear Regime: mbiolar Trasort Here ot egligibly small. mbiolar trasort! Thus β / β / ote: juctio ever disaears, eve for large forward bias! 54

27 Outlie 1) erivatio of the forward bias formula ) Solutio i the oliear regime 3) I-V i the ambiolar regime 4) Tuelig ad I-V characteristics 5) Coclusio 55 Forward Bias oliearity (7): Esaki iode Esaki-iode: Heavily doed diode l(i) 3 F F 1 F emty F 6,7 V F X o states! F I Heavy doig 1 Tuelig i diodes. obel Prize (Esaki) V

28 Reverse Bias (5): Zeer Tuelig F Tuelig (triagular barrier) emty +V F 4 Zeer tuelig occurs i every diode. (reverse bias) 5 emty Remember: Tuelig through a triagular barrier I = qt υ 4 T = α k 4cosh α d + sih αd k α (.49 F) 57 Coclusio 1) I-V characteristics of a - juctio is defied by may iterestig heomea icludig diffusio, ambiolar trasort, tuelig etc. ) The searate regios are idetified by secific features. Oce we lear to idetify them, we ca see if oe or the other mechaism is domiated for a give techology. 3) I the et class, we will discuss a few more o-ideal effects. 58