3.8. Other Unipolar Junctions

Transcription

1 3.8. Oher Uipolar ucios The meal-semicoducor jucio is he mos sudied uipolar jucio, be o he oly oe ha occurs i semicoducor devices. Two oher uipolar jucios are he - homojucio ad he - Heerojucio. The - homojucio frequely occurs i semicoducor devices are heavily doped regios are commoly added o reduce he overall resisace ad improve he coac resisiviy. Mos exbooks igore he effec of such jucios as he aalysis is more difficul ad he overall effec o he device is ypically small. We prese he elecrosaic aalysis of he - here i par for compleeess bu also o se he sage for he aalysis of he - heerojucio. The - heerojucio frequely occurs i heerojucio devices. Such occurrece is o always deliberae, bu heir aalysis is, albei complex, eed whe opimizig a Heerojucio device desig. I his secio we prese he elecrosaic aalysis of he - homojucio ad heerojucio as well as he aalysis of he - heerojucio curre The - homojucio Whe coacig semicoducor devices oe very ofe icludes highly doped semicoducor layers o lower he coac resisace bewee he semicoducor ad he meal coac. This added layer causes a - jucio wihi he device. Mos ofe hese jucios are igored i he aalysis of device i par because of he difficuly reaig hem correcly, i par because hey ca simply be igored. The build-i volage of a - jucio is give by: i 1 ( E E F F ) l q N d Nd (3.8.1) Which meas ha he buil-i volage is abou 59.4 me if he dopig coceraios differ by a facor 1. I is because of his small buil-i volage ha his jucio is ofe igored. However large variaios i dopig coceraio do cause sigifica poeial variaios. The ifluece of he - jucio mus be evaluaed i cojucio wih is curre volage characerisics: if he - jucio is i series wih a p- diode, he issue is wheher or o he - jucio affecs he operaio of he p- jucio i ay way. A low curre desiies oe ca expec he p- diode o domiae he curre flow, whereas a high curre desiies he - jucio could play a role if o desiged properly. For he aalysis of he - jucio we sar from a fla bad eergy bad diagram coecig he wo regios i absece of a elecric field. Oe ca visualize ha elecros will flow from he regio ad accumulae i he -ype regio. However, sice he carrier coceraio mus be coiuous (his is oly required i a homojucio), he carrier desiy i he -ype regio is smaller ha he dopig coceraio of he regio, ad he regio is o compleely depleed. The full depleio approximaio is herefore o applicable. Isead oe recogizes he siuaio o be similar o ha of a meal-semicoducor jucio: he regio is depleed bu has a small volage across he semicoducor as i a Schoky barrier wih small applied volage, whereas he -ype regio is accumulaed as i a Ohmic coac. A geeral soluio of his srucure requires he use of equaio (3.3.2).

2 A approximae soluio ca be obaied i he limi where he poeial across boh regios is smaller ha he hermal volage. The charge i he - srucure regio is he give by solvig he liearized Poisso equaio: ε s i a ρ ( x) exp for x < LD ( LD L ) L D D ε s i a x ρ ( x) exp for x > L ( L L ) L D D D x D (3.8.2) (3.8.3) where he - ierface is locaed a x, ad L D, ad L D, are he exrisic Debye leghs i he maerial, give by: LD ε skt 2 q Nd (3.8.4) L D ε skt (3.8.5) 2 q N d Applyig Poisso's equaio agai oe fids he poeials o be: i a ( x) LD exp for x < LD L L D D L D x ( x) ( i a )[1 exp ] for x < L L L D D x D (3.8.6) (3.8.7) The soluios for he charge desiy, elecric field, poeial ad eergy bad diagram are ploed i he Figure 3.8.1:

3 Disace (m) Charge Desiy (C/cm 2 ) Disace (m) , -1, -15, -2, -25, -3, Elecric Field (/cm) Disace (m) Poeial () E c E i E v Disace (m) Eergy (e) Figure Charge, elecric field, poeial ad eergy baddiagram i a silico - srucure wih N d 1 16 cm -3, N d 1 17 cm -3 ad a.

4 The - heerojucio Heerojucios ca be foud i a wide rage of heerojucio devices icludig laser diode high elecro mobiliy rasisors (HEMTs) ad heerojucio bipolar rasisors (HBTs). Of hose, he HEMT aurally coais such heerojucio, while he oher devices could coai a uieioal - heerojucio. As a sarig poi of he aalysi cosider a - heerojucio icludig a spacer layer wih hickess d as show i Figure Ec Ef qf i DE c Ec Ef Ev DEv Ev x -layer -d spacer layer -layer Figure Flabad eergy diagram of a - heerojucio icludig a spacer layer wih hickess d. The buil-i volage for a - heerojucio wih dopig coceraios N d ad N d is give by: i 1 ( E q F EF ) l N N d c, N d N c, Ec (3.8.8) Where N c, ad N c, are he effecive desiies of saes of he low ad high-doped regio respecively. Ulike a homojucio, he heerojucio ca have a buil-i volage, which is subsaially larger ha he hermal volage. This jusifies usig he full depleio approximaio for he depleed regio. For he accumulaed regio oe also has o cosider he ifluece of quaizaio of eergy levels as carriers are cofied by he elecric field ad he heeroierface. I he ex wo secios we aalyze he eergy bad diagram of he - heerojucio wih ad wihou he iclusio of quaizaio ad compare he wo soluios Aalysis wihou quaizaio For he classic case where he maerial does o become degeerae a he ierface oe ca use (3.3.22) o fid he oal charge i he accumulaio layer: Q acc ε E ε 2[exp 1] L D (3.8.9)

5 while he poeials ad he field ca be solved for a give applied volage usig: sp i a (3.8.1) 2 2 ε s, E 2qε N d ε s Ed sp ε sp (3.8.11) (3.8.12) The subscrip sp refers o he udoped spacer layer wih hickess d, which is locaed bewee he wo doped regios. These equaios ca be solved by sarig wih a cerai value of, which eables he calculaio of he elecric field, he oher poeials ad he correspodig volage, a Aalysis icludig quaizaio The aalysis of a - heerojucio icludig quaized levels is more complicaed because he eergy levels deped o he poeial, which ca oly be calculaed if he eergy levels are kow. A self-cosise calculaio is herefore required o obai a correc soluio. A approximae mehod, which also clarifies he seps eeded for a correc soluio is described below 1. Sarig from a cerai desiy of elecros per ui area, N s, which are prese i he accumulaio layer, oe fids he field a he ierface: E qn s ε (3.8.13) We assume ha oly he 1 eergy level is populaed wih elecros. The miimal eergy ca be expressed as a fucio of he elecric field 2 : E 2 h 1/ 3 9π 2 / 3 1 ( ) ( qe ) 2m 8 (3.8.14) The badgap discoiuiy E c ca he be relaed o he oher poeials of he jucio, yieldig: ε N E (3.8.15) c, Ec qa E1 ktl[exp( ) 1] ktl q q sp qnc, qw N d 1 A similar aalysis ca also be foud i Weisbuch ad ier, Quaum Semicoducor Srucure pp 4-41, Academic Pres See for isace F. Ser, Phys. Rev. B 5 p 4891, (1972).

6 where he poeial ad sp, i ur ca be expressed as a fucio of E : 2 2 ε s, E (3.8.16) 2qε N d sp ε ε E sp d (3.8.17) These equaios ca be combied io oe rascedeal equaio as a fucio of he elecric field, E. Ec qa E1 ε E ktl[exp( ) 1] ktl qnc, qw N 2 2 c, ε s, E (3.8.18) N 2ε N d d Oce E is kow all poeials ca be obaied Compariso of he wo soluios wih ad wihou quaizaio We ow compare boh approaches by preseig a umerical soluio obaied by implemeig he equaios above for he case wih as well as ha wihou quaizaio. Keep i mid ha eiher is exac as i would require a self-cosise umerical aalysis ha icludes he calculaio of he poeial based o he quaum mechaical disribuio of he charge i each of he quaized levels. The eergy bad diagram ad he shee charge versus applied volage are preseed i Figure ad Figure respecively. Eergy (e) E F, E c E F, Disace (m) Figure Eergy baddiagram of a Al.4 Ga.6 As/GaAs - heerosrucure wih N d 1 17 cm -3, N d 1 16 cm -3, d 1 m ad a.15. Compariso of aalysis wihou quaizaio (upper curve) o ha wih quaizaio (lower curve)

7 Shee Desiy (cm -2 ) 9E11 8E11 7E11 6E11 5E11 4E11 3E11 2E11 1E Applied olage () Figure Elecro desiy, N s, i he accumulaio regio versus applied volage, a, wih quaizaio (op curve) ad wihou quaizaio (boom curve). From he figures oe fids ha he aalysis wihou quaizaio predics a larger barrier o he lef of he ierface ad a larger shee charge for a give volage. The acual soluio is expeced o be somewhere i bewee he wo preseed here, especially for he case where more ha oe quaized level exiss ad is occupied.

8 Curres across a - heerojucio Curre raspor across a - heerojucio is similar o ha of a meal-semicoducor jucio: Diffusio, hermioic emissio as well as uelig of carriers across he barrier ca occur. However o ideify he curre compoes oe mus firs ideify he poeials ad by solvig he elecrosaic problem. From he bad diagram oe fids ha a barrier exiss for elecros goig from he o he -doped regio as well as for elecros goig i he opposie direcio. The aalysis i he firs secio discusses he hermioic emissio ad yields a closed form expressio based o a se of specific assumpios. The derivaio also illusraes how a more geeral expressio could be obaied. The ex secio describes he curre-volage characerisics of carriers raversig a depleio regio, while he las secio discusses how boh effecs ca be combied Thermioic emissio curre across a - heerojucio The oal curre due o hermioic emissio across he barrier is give by he differece of he curre flowig from lef o righ ad he curre flowig from righ o lef. Raher ha rederivig he expressio for hermioic emissio, we will apply equaio (3.4.6) o he - heerojucio. Oe complicaio arises from he fac ha he effecive mass of he carriers is differe o each side of he heero-jucio which would seem o idicae ha he Richardso cosa is differe for carrier flow from lef o righ compared o he flow from righ o lef. A more deailed aalysis reveals ha he differece i effecive mass causes a quaum mechaical reflecio a he ierface, causig carriers wih he higher effecive mass o be refleced back while carriers wih he smaller effecive mass are o firs order uaffeced 3. We herefore use equaio (3.4.6) for flow i boh direcios while usig he Richardso cosa correspodig o he smaller of he wo effecive masse yieldig: H E ( ) 2 c x E q F, Ec ( x ) EF, E (3.8.19) c A T {exp[ exp[ ]} kt kt where he poeials are relaed o he applied volage by 4 : i a (3.8.2) ad he buil-i volage is give by: i E c ( Ec ( x ) E ) ( Ec ( x ) E F F ) q (3.8.21) Combiig hese relaios yields: 3 A.A. Griberg, "Thermioic emissio i heerojucio sysems wih differe effecive elecroic masse" Phys. Rev. B, pp , No spacer layer is assumed i his derivaio, bu could be added if desired.

9 H A T 2 B (3.8.22) a exp[ ]exp[ ][exp 1] where he barrier heigh B is defied as: Ec ( Ec( x ) EF ) B q (3.8.23) Assumig full depleio i he depleio regio ad usig equaio (3.8.9) for he accumulaed regio, he charge balace bewee he depleio ad accumulaio layer akes he followig form: 2 ε qn ε d 2[exp 1] (3.8.24) Combiig equaios [3.2.2] wih [3.2.16] yields a soluio for ad. For he special case where ε s N d ε s N d ad >> hese equaios reduce o: i a exp (3.8.25) The curre (give by [3.2.18]) ca he be expressed as a fucio of he applied volage a H qa T i a a (1 )exp( B (3.8.26) )[exp 1] k i Whereas his expressio is similar o ha of a meal-semicoducor barrier, i differs i ha he emperaure depedece is somewha modified ad he reverse bias curre icreases almos liearly wih volage. Uder reverse bia he jucio ca be characerized as a cosa resisace, R H, which equals: R H A H a AA T 2 exp( B ) (3.8.27) where A is he area of he jucio. This shows ha he resisace chages expoeially wih he barrier heigh. Gradig of he heerojucio is ypically used o reduce he spike i he eergy bad diagram ad wih i he resisace across he ierface Calculaio of he Curre ad quasi-fermi level hroughou a Depleio Regio We ypically assume he quasi-fermi level o be cosa hroughou he depleio regio. This assumpio ca be jusified for a homojucio bu is o ecessarily correc for a heerojucio p- diode. For a homojucio p- diode we derived he followig expressio for he mioriy carrier desiy i he quasi-eural regio of a "log" diode

10 p p (exp a 1) exp( x L ) (3.8.28) so ha he maximum chage i he quasi-fermi level, which occurs a he edge of he depleio regio, equals: df dx d( kt l ) i dx kt L (3.8.29) so ha he chage of he quasi-fermi level ca be igored if he depleio regio widh is smaller ha he diffusio legh as is ypically he case i silico p- diodes. For a heero-jucio p- diode oe ca o assume ha he quasi-fermi level is coiuou especially whe he mioriy carriers eer a arrow badgap regio i which he recombiaio rae is so high ha he curre is limied by he drif/diffusio curre i he depleio regio locaed i he wide badgap semicoducor. The curre desiy ca be calculaed from: qµ E qd d dx (3.8.3) Assumig he field o be cosa hroughou he depleio regio oe fids for a cosa curre desiy he followig expressio for he carrier desiy a he ierface: While for zero curre oe fid for a arbirary field Ex (3.8.31) Nd exp( ) q µ E E ( x) dx Nd exp( ) Nd exp( ) (3.8.32) Combiig he wo expressios we posulae he followig expressio for he carrier desiy: qµ E max N d exp( ) (3.8.33) The carrier desiy ca also be expressed as a fucio of he oal chage i he quasi-fermi level across he depleio regio, E f ; N d EF, (3.8.34) exp( )exp( ) which yields he followig expressios for he curre desiy due o drif/diffusio:

11 qµ E max Nd EF, exp( )(exp( ) 1) kt (3.8.35) where is E max he field a he heerojucio ierface. If E f equals he applied volage, as is he case for a - heerosrucure, his expressio equals: qµ E max N d exp( )(exp( a ) 1) (3.8.36) which reduces for a M-S jucio o: dd hermioic µ E max v R (3.8.37) so ha hermioic emissio domiaes for v R << µ E max or whe he drif velociy is larger ha he Richardso velociy Calculaio of he curre due o hermioic emissio ad drif/diffusio The calculaio of he curre hrough a - jucio due o hermioic emissio ad drif/diffusio becomes sraighforward oce oe realizes ha he oal applied volage equals he sum of he quasi-fermi level variaio, E f, across each regio. For his aalysis we herefore rewrie he curre expressios as a fucio of E f, while applyig he expressio for he drif/diffusio curre o he maerial. hermioic A T 2 E (3.8.38) B F, 1 exp( ) exp( )[exp 1] kt drif B E F, 2 / diffusio qµ E max N exp( )[exp 1] c, kt (3.8.39) 5 I should be oed here ha he drif/diffusio model is o loger valid as he drif velociy of he carriers approaches he hermal velociy.