Transent Effects n Hgh ltage de Stack under everse Bas Papež zech Techncal Unversty n Prague, Techncká, Prague 6, B Kjecký, J Kžíšek POLOOIČE, as, Nvdvrská 8a, Prague 4, J Hejhal student zech Techncal Unversty n Prague, Techncká, Prague 6, bstract Ths artcle deals wth a descrptn and analyss f the fast transent prcesses whch can ccur durng a lcal nn-destructve breakdwn n a crcut arranged by seral cnnectn f reverse based hgh-vltage slcn ddes The estence f the nn-destructve breakdwn was bserved at sme measurements f reverse current-vltage characterstcs f ndvdual ddes Hwever, the study f ths phenmenn s very dffcult fr many seral cnnected ddes n stack That s why, a physcal mdel was created fr reflectn f ndvdual lcal breakdwn n ths case aldty f ths mdel was verfed by means f crcut smulatn f the nvestgated prcess Further, statstcal sgnfcance f ths prcess was cnsdered wth respect t relablty and lfetme f hgh-vltage dde stack (HS) Keywrds: slcn dde, lcal breakdwn, mcrplasma, physcal mdel, crcut smulatn Lst f symbls capacty f space charge regn (S) F a avalanche current capactve current ttal dde leakage current d dffusn current rg generatn current N number f seral cnnected ddes - N number f cycles - N densty f dnrs m - n ntrnsc densty f carrers m - P prbablty - Qa avalanche charge q electrn charge resstr S PN junctn area m t tme s appled vltage reverse vltage drp f dde n reverse vltage f dde n chan vltage drp f resstr vs saturatn velcty ms - crdnate m 0 S wde befre avalanche m S wde as a functn f tme m relatve permttvty Fm - 0 permttvty f vacuum Fm - parameter f the Pssn dstrbutn - S lfetme f carrers n S s Intrductn The presence f mcr-defects and nhmgenetes n semcnductr slcn ften affects the behavur f PN junctn under reverse bas Suffcently hgh lcal mamum f electrc feld due t an appearance f current flaments (mcrplasmas) r effects referred as secnd breakdwn (mesplasmas) Bth ths effects are knwn lng-tme and descrbed n detal [ - 4] The rgn f mcrplasma s bund tgether wth lcal avalanche breakdwn Hgh electrc feld causes an avalanche multplcatn f electrns and hles whch traverse thrugh the space charge regn (S) and transfrm a dstrbutn f rgnal electrc feld Smultaneusly, the temperature f breakdwn place ncreases Bth these prcesses (decreasng f the electrc feld, ncreasng f the temperature) evke swtchng ff f the avalanche and the dde reverse vltage s restred agan The ttal prcess cnssts f tw phases: - avalanche s rgn and swtchng ff (rder ns duratn) - restratn f the rgn state (rder ms duratn) If the reverse vltage des nt decrease r, n the cntrast, ncreases, the prcess descrbed abve s repeated after any tme
On the bass f physcal analyss t s pssble t study prcesses cnnected wth the mcrplasma ccurrence fr ne dde and smple electrc crcut slutn f mre cmplcated respnses n real crcut requres a means f standard crcut analyss The devce and ts mcrplasma - prcess are perfrmed by substtute crcut s that pssble naccuracy s generally acceptable Ths way allws t antcpate the behavur f the crcut cnssted f a number f devces Mrever, a measure f nfluence f the ther cmpnents n the crcut can be judged s that results f measurement are crrectly nterpreted Physcal mdel and crcut smulatn cmmnly gve a pssblty t create a pcture f avalanche breakdwn respnse n hgh vltage dde stacks (HS) These devces are ften used n electrstatc fly-ash separatrs at a thermal pwer plants r at a dagnstcs X-ray equpments everse vltage f HS eceeds 00 k usually and drect measurement f sme electrc parameters s very dffcult The results f the bth physcal analyss and crcut smulatn yeld mprtant nfrmatn as t the functn relablty and lfetme f HS Thery Physcal mdel t frst, a lcal avalanche prcess and ts respnse wll be nvestgated fr an ndvdual dde (see Fg ) The generatn f avalanche charge, ts mvng thrugh the S and shft f charge neutral bundary s shwn n Fg Fg : Smple crcut fr physcal analyss P v S Q a S Fg : Scheme f lcal avalanche breakdwn 0 N valanche charge Qa s transprted thrugh the S and ts velcty s vs (saturatn velcty s equal 0 5 ms - [5]) recmbnatn f the carrers s neglgble durng ther transprt Ttal reverse current f the dde durng the avalanche s cmpsed by avalanche current a, by generatn current rg and by dffusn current d T smplfy stuatn, we can put a» rg + d Then equatn descrbng avalanche prcess has the frm Qa s S d () slutn f the Pssn equatn fr abrupt asymmetrcal PN junctn can be wrtten lke () Further, fr any tme t must be vald r d, () d ( ) mbnng Eq and wth Eq we gve d a b, (4) where b a Qa s, b S The slutn f Eq 4 wll be substantally smplfed f we put (5) In ther wrds, the change f the dde vltage s relatvely small n cmparsn wth the state befre avalanche Then, we can wrte fr = - a t b e (6) wth respect t the fact that» fter fnsh f avalanche prcess, the dde cmes back t ts frmer state Ttal reverse current may be epressed as, (7) rg d where rg s gven by knwn relatn n q rg S (8) sc
and dffusn current d s cnstant Because the change f rg s als small, then r cnst (9) rg d Smlar cnsderatn s reasnable fr the dynamc capactance f PN junctn (gven by the S wh) S cnst (0) estred prcess can be descrbed by equatn d r () Slvng Eq we have O r e t, () where 0 s the vltage drp n the resstr mmedately after fnsh f the avalanche prcess Nw, the stuatn wll be dscussed when the lcal reversble breakdwn ccurs n any dde belngng t a chan f N devces In ther wrds, (N-) seral cnnected ddes are added n Fg valanche prcess can be epressed by equatn Qa s d d and fr vltage dstrbutn t s vald N n n r (), (4) where n s average vltage drp n any dde f the chan and, n» Then, Eq may be transfrmed as d Q a s (5) N and fr N» t s d Qa S (5 ) Fr t = 0 t s = /N and slutn f Eq 5 s n the frm N N r n smplfed relatn Qa s t (6) Bt, (6 ) N where Q a s / N and B Immedately after swtchng ff f the avalanche prcess, the ttal vltage s dvded between the dde after breakdwn and ther ddes n the chan Ths can be epressed by means f S whs (resultng frm Pssn equatn) N N, (7) n where s S wde f the dde after avalanche, n s S wde f any ther dde and 0 s the same parameter befre start f breakdwn Because d n n d (8) N and the current flwng thrugh all ddes s the same, we can wrte d nq sc If N», Eq 9 has the frm d nq where d nq N sc (9), (0) sc Slvng Eq 0 we get e t 0, () where 0 s S wde mmedately after end f avalanche Transfrmatn f the t the by means f Pssn equatn gves (after small crrectn) t 0 e, () where 0 s the vltage dfference befre start and after end f the avalanche prcess The cncrete pcture f physcal analyss can be made nly by means f characterstc parameters f the tested dde chps and relevant physcal magntudes: = 0 m -, S = 50-5 m, = 0 - Fm -, qn = m - (5 ), = 0 pf, S = 50-5 s, s = 0 5 ms - = 0 k N = 80 Q a = 0-0 Then, we can calculate the values and tme cnstants n Eq 6,, 6, Fr Eq 6 and t = ns s
7 9, 8, 0 s Fr Eq s b 7 0 9, 8, 0 s Fr Eq 6 and t = ns s 0,, B 6, 7 0 s Fr Eq and = 50 s 0 0, 0 s It s evdent, the vltage drp s the same fr bth cases (ndvdual dde r dde chan) The value f vltage drp s abut % f the dde reverse vltage, but the tme cnstants f restred prcess are very dfferent (50 ) rcut smulatn ccrdng t the results f physcal analyss perfrmed abve the breakdwn prcess generates nly small change f the dde reverse vltage Under these cndtns, the dde (generally nn-lnear element) may be apprmated by lnear substtute crcut created by a parallel cmbnatn f a resstr and capactr The magntude f resstr s gven by the rat f reverse vltage and crrespndng current n the dde wrk pnt, the magntude f capactr s near t PN - junctn capactance at chced reverse vltage Mcrplasma frmng prcess s smulated by current surce, whch generates trapezdal current pulse The charge f ths pulse - current ntegral alng the tme - s equal t mcrplasma dscharge and the duratn f dscharge s equal t length f pulse nvenence f the dde representatn by substtute crcut was judged by means f cmparsn wth results f physcal analyss The crcut n Fg s replaced by crcut shwn n Fg, the shape f ntatng current pulse s n Fg 4 The tme dependences f the breakdwn respnses receved by means f bth the physcal and crcut analyss fr parameters = 0 k, = 0 M and = 0 pf are shwn n Fg 5 Ther mamum mutual devatn s abut % d d d Fg : Indvdual dde n smulated crcut 00 m ns Fg 4: Smulated shape f the current pulse at breakdwn [] 5 0 5 0 5 0 0 00 400 600 800 000 Fg 5: alculated dependences f the vltage change after fnsh f breakdwn (physcal analyss vs crcut cmputer smulatn ) Epermental t [ns] ll presented measurements have been carred ut n dde chps used fr cmmercal prductn f HS, type 808 (made by Plvdče, as, ) 808 cntans 80 pcs f seral cnnected dde chps nsde ceramc clumn and ts mamum repettve reverse vltage s 60 k The dde chps are dpped n slcn l, wrkng temperature f PN-junctn s 5 The ndvdual dde chp s cnsttuted by slcn slce 70 m thck, wth dameter 5 mm Specfc resstvty f basc slcn s 87 cm, PN junctn depth s 85 m Slcn wafer s cntacted by M-electrdes n bth sdes Ths sandwch s sldered t u-saucer cvered by N-layer Perphery f PN-junctn s prtected by slcn rubber The measurement f reverse characterstcs was carred ut at 5, tested ddes were dpped n thermstatc bath flled by slcn l Pwer supply f dc vltage allwed an regulated rate f rse f ttal dc vltage n nterval -00 s, the value f dde seral resstr was 0 k The curse f reverse current-vltage characterstcs was mntred by scllscpe glent 546 Eamples f reverse current - vltage characterstcs are shwn n Fg 6 a, b, a detal f the reversble breakdwn s n Fg 7
a b Fg 6a, b: Eamples f epermental I - reverse characterstcs (wth breakdwn nse); repeated frequency s 0,0 Hz, tme measurement abut 00 s - ae: reverse vltage (500 /dv) y - ae: reverse current (00 /dv) befre measurement (standard peratn made by prducer) 4 scussn a) ccrdance between physcal mdel and crcut smulatn The results f physcal analyss ndcate that the reverse vltage drp durng reversble lcal breakdwn s the same fr bth ndvdual dde and any dde n seral chan It s especally evdent f the Eq 6 s transfrmed n frm d fr t 0 and B N () d t, (4) where t s avalanche charge transt tme thrugh S Fr suffcently small Qa (0-0 - 0-9 ) the value f s 0-00 and t s nt affected by ther crcut parameters The cmparsn f tme cnstants n Eq 6 and Eq shws the fundamental nfluence f crcut parameters n restred prcess It s many tmes slwer n seral chan f ddes n cmparsn wth ne ndvdual dde T eplan the shape f the pulse n Fg 5, t s necessary t cnsder sme nfluence f real crcut The respnse f mcrplasma dscharge reflects a parastc capactances n crcut, especally the capactance f cnnectng sheldng cables Substtute schema f measurng crcut s shwn n Fg 8 OS M d d d 50 50 Fg 8: Substtute scheme f measurng crcut Fg 7: etal recrd f breakdwn pulse (I - characterstcs n Fg 6a) - ae: ndcated vltage ( /dv) y - ae: tme (5 s/dv) epeated measurements shwed that lng tme average frequency f current peaks (Fg 6 a) s abut 0 s - at the reverse vltage 000 Further, t was cnfrmed a ttal dsappearance f these peaks f the chps were dred several hurs abve temperature 00 mmedately [] 4 0-5 0 5 t [s] Fg 9: alculated dependence f the ndcated vltage n measurng crcut (cmpare wth Fg 7)
The crcut s cmpleted by tw capactrs (50 pf) whch represent parastc capactances alculated tme dependence f the vltage ndcated by scllscpe nput durng mcrplasma breakdwn s shwn n Fg 9, the same real dependence recrded by scllscpe s n Fg 7 Bth dependences ehbt very gd accrdance mparsn between ndvdual dde and the dde n real measurng crcut shws smaller change f vltage and lnger tme f dscharge decay caused by bth crcut capactances Substtute schema f the dde chan s n Fg 0 Smple dde crcut was cmpleted by further ddes These ddes are represented by parallel cmbnatn f resstr and capactr wth the values crrespndng t ttal cmbnatn f substtute resstance and capactance f the dde chan esultng curse f smulatn fr ddes n chan s n Fg Fg 0: Substtute crcut fr HS [] 5 0 5 0 5 0 d s d d s 0 00 00 00 400 500 t [us] Fg : esultng curse f smulatn fr ddes n chan The tme respnse s relatvely lng n ths case It crrespnds t lng tme cnstant n the actve dde crcut, whch s nt nfluenced by hgh ladng mpedance created by ther seral ddes The vltage decreasng s relatvely small and t cannt cause any damage f ther ddes The respnse curse descrbed abve s vald fr chan cmbnng much mre ddes, because the ttal mpedance s determned by parallel cmbnatn f ndvdual dde mpedance ( + ) and resultng mpedance f remanng passve ddes n chan (S + S) The chan mpedance s much greater than the mpedance f any ndvdual dde (N-tmes) and ts tme cnstant has the same value lke tme cnstant f ne dde That s why, the tme cnstant f whle transent prcess s the same and desn t depend n number f ddes Only the value f mamum vltage drp can be varable, ts value s ncreased abut 0 % fr great number f seral ddes n cmparsn wth Fg b) Lfetme and relablty Many tmes repeated lcal breakdwn can cause a destructn f devce after lng tme Materal dlatatns evked by fast change f temperature ntate mechancal cracks number f temperature cycles N causng a damage f devce can be epressed n frm [6] 9 00 N (5) T where T s a lcal ncreasng f temperature ( ) durng cycle If a surface f the mcrplasma dscharge s typcally abut 50 m [], then fr = 000 and a = 00 m (stuatn dscussed abve), lss energy n slcn s abut 40-7 J and temperature ncreasng f the avalanche regn s abut 4,5 Usng Eq 5 we receve average lfetme f devce - 9,6 years fr average frequency f breakdwns 0 s - Hwever ths apprmate calculatn des nt nclude the stuatn when a net mcrplasma dscharge s ntated suffcently fast after prevus dscharge Ths s mplcated by the statstcal nature f prcess Then, the value T n Eq 9 can be ncreased and, n the ther hand, number f cycles N wll cme dwn T judge a sgnfcance f ths pssblty, t s useful t make a calculatn f the tme needful fr clng dwn f mcrplasma regn back t temperature f the surrundng S Let us cnsder a slcn wafer whch has bth ts sdes kept n a cnstant temperature Suddenly, very small lcal heat surce - mcrplasma - appears nsde the wafer (durng rder f ns) generated heat s cnducted ut gradually Let the mcrplasma regn has a cylnder frm wth a base radus a (see Fg )
Fg : Slcn wafer wth mcrplasma ht spt The clng dwn f mcrplasma regn s descrbed by equatn f heat cnductn The slutn fr ths case has fllwng frm [7] T T Tm a a 0 erf erf (6) T0 K / P t K / P t where: T s actual temperature n -pnt fr tme t Tm s mamum temperature durng mcrplasma rgn T0 s the temperature befre mcrplasma rgn K s the thermal cnductvty, f S P s the specfc heat f S s the densty f S Let s estmate the tme needful fr temperature decreasng t 0 % f mamum temperature value Tm n the centre f mcrplasma regn ( = 0) Then, there s erf and mcrplasma regn a 0, 00 K / P t (7) a 0, 089 (8) K / P t alculated tme t s equal t 6 s fr a =4 m If the mcrplasma wll appear durng ths tme agan (n the same place), the number f cycles N necessary fr devce destructn wll be lwer T estmate the prbablty P f 6 s cncdence f the tw dscharges, the Pssn dstrbutn can be used k P e (9) k! where = t, s average lng tme frequency f any phenmenn and k s number f ths phenmenn durng tme nterval t If = 0 Hz, t = 6 s and k =, then 9 8, 0 a S-wafer >>a P (0) It means, ths stuatn s realsed wth tme perd equal t 0,5 year That s why, an nfluence f repeated dscharge c-ncdence n devce lfetme may be neglected 5 nclusns The physcal analyss and cmplement crcut smulatn were carred ut fr dde ehbtng transent lcal breakdwn (rgn f mcrplasma) n stuatns as fllws: - ndvdual dde wth seral resstr (analyss and measurement) - seral chan f many ddes (prgnss) In the frst case, bth physcal analyss and crcut smulatn were cmpared wth epermental dependences The very gd accrdance was fund amng all these ways It was shwn that - decreasng f the reverse vltage drp durng lcal breakdwn s the same fr ndvdual dde r fr dde seral chan; the transent reductn f the reverse vltage s abut %; - duratn f lcal breakdwn s rder f ns, duratn f restred prcess s rder f ms fr dde stuated n seral chan; - repeated rgn f mcrplasma can lead t a reductn f devce lfetme; hwever, ths nfluence s nt sgnfcant under cndtns dscussed abve cknwledgement uthrs wsh t epress the thanks t cmpany Plvdce, as fr knd agreement wth publcatn f ths artcle eferences [] H F Jhn: Prceedngs f the IEEE, l 55, N 8, 967 [] S K Ghandh: Semcnductr Pwer evces, J Wlley & Sns, NY 977 [] K av: Imperfectns and Impurtes n Semcnductr Slcn, J Wlley & Sns, NY 98 [4] S T Hsu: Sld State Electrncs, l 4, pp 487-497 (97) [5] M Kubát: ýknvá plvdčvá technka, SNTL Praha 978 [6] Benda, J Gwar, Grant: Pwer Semcnductr evces Thery and pplcatns, J Wlley & Sns, NY 999 [7] W M hsenw, J P Hartnett: Handbk f Heat Transfer, McGraw-Hll, NY 97, p -6 ddresses f the authrs Papež, J Hejhal, zech Techncal Unversty n Prague, Techncká, Prague 6,, e-mal: papez@feldcvutcz B Kjecký, J Kžíšek, POLOOIČE, as, Nvdvrská 8a, Prague 4,, e-mal: bkjecky@plvdcecz