The pn Junction. Φ = n. 2.1 The pn junction under forward bias (steady-state)

Transcription

1 Pwer Electrnic. The pn unctin under frward ia (teady-tate) If the p-regin i externally pitively-iaed with repect t the n-regin a hwn in figure.3, the cl narrw and current flw freely. The emf pitive ptential upplie hle t the p-regin, while the negative emf ptential prvide electrn t the n-regin. The carrier th cmine, ut are cntinuuly replenihed frm the emf urce. A large emf urce current flw thrugh the dide, which i termed frward-iaed. The dide i the implet iplar emicnductr device. It cmprie p-type and n- type emicnductr material rught tgether, uually after diffuin, t frm a (tep r arupt) unctin a hwn in figure.la. A depletin layer, r alternatively a pace charge layer, cl, i uilt up at the unctin a a reult f diffuin caued y the large carrier cncentratin gradient. The hle diffue frm the p-ide int the n-ide while electrn diffue frm the n- ide t the p-ide, a hwn in figure.l. The n-ide, ling electrn, i charged pitively ecaue f the net dnr charge left ehind, while the p-ide cnverely ecme negatively charged. An electric ptential arrier, ξ, uild up, creating a drift current which ppe the diffuin flw, th f which alance at thermdynamic equilirium a hwn in figure.lc. There are n free carrier in the cl. The zer external ia, uilt-in, unctin ptential r cl ptential i given y I I DRIFT k T N N A D = n () q n i (.) IDIFFUSION Ф where q i the electrn charge,.6 x -9 C k i Bltzmann cntant,.38 x -3 J/K T i the unctin temperature, K. Thu ϕ = k T / q =.59 e at rm temperature, 3 K. One imprtant feature f the pn unctin i that current (hle) flw freely in the p t n directin when frward-iaed, that i, the p-regin i iaed pitive with repect t the n-regin. Only a mall leakage current flw in the revere vltage ia cae. Thi aymmetry make the pn unctin dide ueful a a rectifier, exhiiting tatic vltage-current characteritic a illutrated in figure.. Figure.. The tep unctin. (a) The unctin if carrier did nt diffue: + inied dnr,- inied acceptr, + hle and - electrn; () electrn and hle mvement: ---- diffuin flw, drift flw; (c) equilirium ditriutin f inied impuritie and free carrier; and (d) cl electric field and vltage.. The pn unctin under revere ia (teady-tate) If a ia vltage i applied acr the p and n regin a hwn in figure.3c, with the p-terminal negative with repect t the n-terminal, then the cl widen. Thi i ecaue electrn in the n-regin are attracted t the pitive external emf urce while hle in the p-regin are attracted t the negative emf ptential. A the cl widen, the peak electric field ξ m at the unctin increae a hwn in figure.3d.

2 Pwer Electrnic The nly current that flw i the mall leakage current which i due t carrier generated in the cl r minrity carrier which diffue t the unctin and are cllected. The unctin i termed revere-iaed. Increaing applied revere ia eventually lead t unctin revere vltage reakdwn,, a hwn in figure., and the dide current i cntrlled y the external circuit. Breakdwn i due t ne f three phenmena, depending n the dping level f the regin and, mt imprtantly, n the cncentratin f the lwer dped ide f the unctin. n regin. Minrity carrier aciated with the leakage current are accelerated t kinetic energie high enugh fr them t inie ilicn atm n clliin, therey creating a new hle-electrn pair. Thee are accelerated in ppite directin, ecaue f the high electric field trength, clliding and iniing repeatedly - hence the term avalanche, impact iniatin r carrier multiplicatin. If the lighter dped ilicn regin ha a cncentratin f 3 4 < N c < 5 ( /cc) then the avalanche vltage may e apprximated y 3-3/4 = 5.34 N () (.3) c The peak electric field at the unctin will e 5 /8 ξ = 3.9 N (/m) c (.4) and the width f the cl, mainly in the lighter dped regin, at reakdwn i given y W = / ξ (.5) N 7/8 = 4 (m) c ( ) ff q kt I = I e Figure.. Typical I- tatic characteritic f a ilicn pn unctin dide, and the effect f unctin temperature, T... Punch-thrugh vltage The revere vltage extend the cl t at leat ne f the hmic cntact and the device preent a hrt circuit t that vltage in exce f the punch-thrugh vltage, PT. Punch-thrugh tend t ccur at lw temperature with device which emply a lw cncentratin regin (uually the n-ide), a i uual with highvltage device. The punch-thrugh vltage fr ilicn can e apprximated y 7.67 N W c c () 6 PT = (.) where N c i the cncentratin in /cc f the lighter dped regin and W c i the width f that regin in µm... Avalanche reakdwn Avalanche reakdwn r multiplicatin reakdwn, i the mt cmmn mde f reakdwn and ccur when the peak electric field, ξ m, in the cl at the unctin exceed a certain level which i dependent n the dping level f the lighter dped (d) Figure.3. Diagrammatic repreentatin f a pn unctin dide hwing minrity carrier flw: (a) withut external applied vltage; () with frward applied vltage; (c) with revere applied vltage; and (d) electric field and cl change with increaed revere applied vltage.

3 3 Pwer Electrnic 4..3 Zener reakdwn Field r Zener reakdwn ccur with heavily dped unctin regin and at uually le than 5 revere ia. It ccur when the cl i t narrw fr avalanche yet the electric field grw very large and electrn tunnel directly frm the valence and n the p-ide t the cnductin and n the n-ide. Thi revere current i called the Zener effect. Thee three mde f revere vltage reakdwn are nt necearily detructive prvided the current i unifrmly ditriuted. If the current denity in a particular area i t high, a lcal ht pt may ccur, leading t device thermal detructin..4 Mdel fr the unctin dide Semicnductr device mdel are ued extenively fr pwer electrnic circuit imulatin. A aic piecewie-linear mdel i applicale t imple manual calculatin, where the terminal I- characteritic are empirically mdelled aed n ideal circuit element. A mre cmplex and accurate mdel i required fr cmputer tranient analyi imulatin. Such accurate mdel are aed n the emicnductr phyic f the device. Many pwer witching emicnductr device manufacturer prvide value fr the mdel parameter uitale fr circuit imulatin in the package PSpice and SABER..3 Thermal effect The pn unctin current, I, hwn in figure., i related t the cl vltage,, accrding t - q / k T I ( ) = I[e - ] (A) (.6) where I, i the revere leakage current in amp. The frward cnductin vltage decreae with increaed unctin temperature, T. That i, the n-tate vltage ha a negative temperature cefficient. In practical ilicn pn dide, at lw current, the temperature cefficient i typically -.4 m/k, ecming le negative with increaed current. At higher current, the cefficient ecme pitive ecaue f the reduced carrier mility at higher temperature, which caue nn-cl regin t increae in reitance. The effect f the change in temperature cefficient at higher current, in practical device, are hwn dtted in figure.. Neglecting the expnential ilicn ad gap temperature dependence, the temperature effect at high current, n the diffuin cntant cmpnent f the leakage current I in equatin (.6), called the aturatin current, i given y.8 T I ( T) = I ( 5 C) 3 (.7) Silicn caride dide have a higher temperature cefficient, typically +8m/K. The avalanche vltage increae with temperature, a de the revere leakage current. The effect f temperature n the revere ia characteritic are hwn in figure.. In the cae f ilicn caride, increaed temperature decreae the avalanche vltage and increae the leakage current. The ilicn temperature cefficient fr avalanche i pitive ince the mean ditance etween clliin i reduced ecaue f the increaed thermal energy, which increae the viratinal amplitude. Higher electric field are neceary fr the carrier t gain ufficient kinetic energy fr iniatin. Equatin (.6) al indicate that the revere ia current increae with increaed unctin temperature. Thi pitive temperature cefficient de nt generally reult in thermal intaility with ilicn device, prvided ufficient heat inking i emplyed n maller device. Figure.4. Piecewie-linear apprximatin f unctin dide characteritic: (a) ideal dide with an ffet vltage and reitance t accunt fr lpe in the frward characteritic and () mdel including revere ia characteritic.

4 5 Pwer Electrnic 6.4. Piecewie-linear unctin dide mdel The pn unctin dide i a unilateral device that, t a gd apprximatin, cnduct current in nly ne directin. Figure.4a hw a piecewie-linear (pwl) mdel f the dide that i uitale fr tatic mdelling in pwer electrnic circuit. It include a perfect dide, an n-tate vltage urce E, and a erie reitr f value R t accunt fr the lpe in the actual frward characteritic. The frward I- characteritic at a given temperature i given y ( I ) = E + I R fr > E () (.8) F F F F The mdel in figure.4a de nt incrprate the tatic revere characteritic f leakage and avalanche. Thee are hwn in figure.4, where frm equatin (.3) mdel the avalanche limit and R i ( = / I ) give linear leakage current prpertie fr a given unctin temperature. The three dide cmpnent are aumed perfect. The mdel given y equatin (.8) i adequate fr calculatin f tatic alancing requirement f parallel and erie cnnected dide and thyritr, a cnidered in ectin. and the aciated prlem,.4,.5, and.9 t.. Example.: Uing the pwl unctin dide mdel An apprximatin t the frward characteritic f the dide hwn in figure.4a, i given y =. +. I. Fr a cntant current f 45A fr ⅔ f a cycle, F F calculate the dide i. n-tate vltage; ii. mean pwer l; and iii. rm current. Slutin i. The n-tate vltage at 45A i given y ( i ) =. + i. = =.45 F F F ii. If the n-tate duty cycle i δ=⅔, the average pwer l i P = δ I =.45 45A = 43.W F F iii. The dide rm current i given y I = δ I = 45A = 36.7A rm dc Example.: Static dide mdel A Schttky dide i ued t half-wave rectify a quare wave ±5 urce in erie with a Ω reitr. If the dide mdel hwn in figure.4 i mdelled y R =. Ω, E =., R i = Ω, and = 3, then determine: i. the dide mdel frward and revere ia perating pint equatin fr the erie circuit ii. the lad current and dide vltage iii. the rectifier le (neglecting any recvery effect) and the lad pwer diipatin iv. Etimate the pwer diipated in the lad if the urce i ac with the ame fundamental cmpnent a the quare wave. v. What i the nn-fundamental pwer diipated with the quare wave urce. Slutin i. When the dide i frward iaed IF i = ( v E ) fr v. DF F DF DF R Kirchhff vltage law fr the erie circuit give R =.Ω = i R + v F L DF =±5 Eliminating the dide vltage v DF give erie circuit current E i = fr E E=. F R + R L i = fr < < E F The dide vltage i therefre given y RL=Ω R + ER L v = = E + i R fr i > DF F R + R L When the dide i revered iaed, elw the revere reakdwn vltage i = v fr v < R DR DR Ri = i R + v R L DR Eliminating the dide vltage v D give erie circuit current i = R R + R i L The dide vltage i thu given y R i v i R DR R i R + R ii. The circuit vltage and current are, when the dide i frward iaed, i L IR Ri =Ω 5 -. i = F = 4.65A.Ω + Ω = A.Ω =.35 DF If R L >>R, the dide current equatin can e implified uing R =. When the dide i revere iaed 5 i = = 5.mA R Ω + Ω = 5mA Ω = 5. DR If R i >>R L the dide current and vltage equatin can e implified uing R L =. DR L

5 7 Pwer Electrnic 8 iii. The rectifier le are, when frward iaed, PD = v if F DF = A = 5.7W and when revere iaed PD = v ir R DR =5 5mA =.5W Ttal dide le are therefre ½( ) =.68W. The pwer frm the quare wave upply i ½ ( A + 5 5mA ) = W with.68 = 7.3W diipated in the Ω reitr lad. iv. The magnitude f the fundamental f a quare wave i 4/π time the quare wave magnitude, that i, 5 4/π = 9. peak. The frward iaed dide de nt cnduct until the upply vltage exceed.. Thi i a mall percentage f the ine wave, hence can e neglected in the l etimate. The frward current flw i apprximately if = in ωt = 8.7 in ωt Ω +.Ω The rm f a ine i / it magnitude and / again fr a half wave rectified ine. That i 8.7A i = = Frm 9.35A rm The revere leakage current i given y ir 9. = inωt =.9 inωt Ω + Ω which give an rm current f i Rrm 9mA = = 9.5mA rm The pwer diipated in the Ω lad reitr i P = ( i L F + ir ) RL rm rm = ( ) Ω= 87.4W + 9µW = 87.4W Clearly, the revere leakage current related cmpnent i negligile. v. With the quare wave, frm part iii., 7.3W are diipated in the lad ut frm part ii., nly 87.4W are diipated fr a ine wave with the ame fundamental magnitude. The 9.9W difference i pwer prduced y the harmnic f the fundamental (3 rd, 5 th, ). Fr a reitive heating lad thi pwer prduce ueful heating, ut in a mtr the harmnic pwer wuld prduce unwanted trque pulatin and mtr heating..4. Semicnductr phyic aed unctin dide mdel The charge-carrier dide mdel hwn in figure.5 i neceary fr tranient circuit analyi invlving dide. The pn unctin dide i aumed t have an arupt r tep unctin. The mdel cmpnent are vltage dependant current urce, I n and I, vltage dependant capacitance C t and C, and erie acce reitance R. I() = I e I() ande R / γϕ ( ) ( + )/ ϕ = I e ϕ = k T / q I+I C +C t cathde m C ( ) fr η ele C() = ( ( ) m η + m + ) C ( + m ) ( η ) di C() = t t t d Figure.5. PSpice tranient analyi circuit mdel f the pn unctin dide. The ideal dide current I i given y equatin (.6). The dide current I mdel revere vltage reakdwn, where the reakdwn vltage i aumed due t avalanche and i given y equatin (.3). The vltage dependant tranit capacitance, C t, which i dminant under frward ia, i related t the minrity carrier lifetime t t. The vltage dependant cl (depletin layer) capacitance C, which i dminant under revere ia, invlve the zer ia unctin ptential vltage, given y equatin (.) and the zer ia unctin capacitance C. In the cae f the ilicn caride Schttky dide, C >> C t. The cl capacitance, C () can e evaluated frm the pn dide tructure and dping prfile, a fllw..4.i - Determinatin f C Pin equatin, in cnunctin with Gau law, fr the ne dimeninal tep unctin hwn in figure.6, give d dξ qn qn dx dx = = D A = (.9) The dielectric permittivity = r cmprie the free pace permittivity = 8.854x - F/m and the relative permittivity r =.8 fr ilicn and 9.7 fr SiC.

6 9 Pwer Electrnic q N fr D x n x d ( x) < < ξ = dx q N fr < x< x A Qn=qAxnND xn ξm qna pace charge layer Q (cm -3 ) qnd ξ (/cm) p + xp xp Qn= -Qp Qp=-qAxpNA (.) q q = N x+ dx + N x+ dx x p ( ξ ) ( ξ ) D m A m x n (.) = ½ξ W m Since the charge each ide f the metallurgical unctin mut alance, equatin (.) can e rearranged t give the cl width. W = + q N N A D (.3) Frm equatin (.), a zer ia vltage exit withut the preence f any external vltage. Therefre, t incrprate nn-equilirium cnditin, the electrtatic arrier ptential ecme -, where i the externally applied revere ia vltage. Cnequently the exprein fr the cl width ecme: W = ( ) - + q N N A D (.4) The cl width vltage dependence can e expreed in term f the zer ia cl width, W () W W( ) = + - q N N A D = W - W W x = x = n p + N / N + N / N D A A D x ( n ) = x - xp( ) = x - p n (.5) (.6) Figure.6. The charge Q, electric field ξ, and vltage ptential, in the pace charge layer f a tep pn unctin. Integrating th part f equatin (.) ver the hwn und, give ξ(x): q N fr D x m x n x d( x) + ξ < < = ξ( x) = (.) dx q Nx+ ξ fr < x< x A m p q q where the maximium field intenity (at x = ) i ξ = N x N x m = D n A p The piece-wie paralic vltage ptential acr the cl hwn in figure.6, i given y integratin f the electric field, that i The magnitude f the vltage dependant charge n each ide f the unctin i N N N N D A D A Q( ) = qa W = A q - N + N N N D A + D A (.7) = Q - The unctin capacitance i given y differentiatin f equatin (.7) with repect t - D A dq q N N A C = = A = d( -) ( -) N + N W D A (.8) Equatin (.8) can e rearranged t give the PSpice capacitance frm, in term f the zer ia unctin capacitance C.

7 Pwer Electrnic C C ( ) = q N N D A where C = A N + N D A (.9) The electric field at the metallurgical unctin, frm equatin (.) i given y ξ ( ) = ξ - where ξ = / W (.).4.ii - One-ided pn dide equatin When N A >>N D, which i the uual cae in high vltage pn dide, equatin (.) t (.) are apprximated y the fllwing ne-ided dide equatin. W = qnd x and x n p Q = A q N C qnd = A D (.) Thee equatin hw that the cl penetrate mtly int the n-ide, (hence the name ne-ided), which upprt mt f the vltage. Example.3: Space charge layer parameter value A µm thick p-type x 6 /cc ilicn epitaxial layer i grwn n an n - -type x 4 /cc ilicn utrate, f area cm, t frm an arupt pn unctin. Calculate the fllwing PSpice parameter value, at rm temperature: i. zer ia unctin ptential, ; ii. zer ia cl width, maximum electric field, charge, and unctin capacitance, W, ξ, Q, C ; and iii. avalanche reakdwn vltage,. If the utrate i 5µm thick, fr a revere ia, calculate: iv. cl width and penetratin depth each ide f the unctin, W, x n, x p ; v. charge each ide f the unctin, maximum electric field, and the capacitance, Q, ξ, C. Slutin i. Frm equatin (.), the zer ia uilt-in vltage i k T NN = = q 9 =.59 ln(8.89 ) = ln A ni.59ln.5 D ii. Frm equatin (.5), (.), (.7), and (.9) W = + q N N A D W =.65µm 9 + =.6 ξ = / W =.534 /.65µm ξ =.4 M/m N N D A Q = A q N + D N A Q = = 4. nc 4 9 q N N D A C = A N + D N A C = 3.95 F = 3.95 nf =.53. iii. Frm equatin (.), the etimated punch-thrugh vltage i = 7.67 N W c c = (5) = PT That i, punch thrugh ccur when the revere ia i greater than the perating vltage,. If the dide i t reakdwn due t avalanche then the avalanche reakdwn vltage given y (equatin (.3)) mut e le than PT, = 5.34 Nc 3 = ( ) = 689 -¾ -¾ iv. Frm equatin (.5) the cl width at - revere ia i = = W W - = 4.6µm.65µ +.533

8 3 Pwer Electrnic 4 Frm equatin (.6) the cl penetratin int each ide f the unctin at - i W 4.6 W 4.6 x = = x = = n p + N / N N / N + D A A D x = 4.µm x =.57µm n p Nte that when N A >>N D, x n W, thu the lwer the relative cncentratin, the deeper the cl penetratin and the higher the prtin f upprted in N D. The unctin cl can under thee circumtance e analyed aed n implified equatin called ne-ided unctin equatin. v. The charge magnitude each ide f the unctin, hwn in figure.6, i given y equatin (.7). The electric field at the unctin i given y equatin (.), while the unctin capacitance at - i given y equatin (.9): = = Reading lit See chapter reading lit. Q Q nC ξ = ξ - = ξ = 7.5M/m = 4.n M ( < ξ = 5M/m ) C = C n + =.533 = 9pF Fraer, D. A., The Phyic f Semicnductr Device, OUP, 977. Wlf, H. F., Semicnductr, Jhn Wiley-Intercience, New Yrk, 977. Yang, E. S., Fundamental f Semicnductr Device, McGraw-Hill, 978. Prlem.. A ilicn dide i t have a reakdwn vltage f. If reakdwn i due t the avalanche mechanim calculate i. the cncentratin f the n - regin ii. the width f the n - regin iii. iv. the maximum electric field the expected punch-thrugh vltage aed n part i. and ii. [ x 4 /cc, 83µm,.4x 5 /cm, 57 ].. What i the punch-thrugh vltage fr a ilicn tep unctin with an n - dping level f 5x 3 /cc and a width f µm? Calculate the dping level and cl width fr a imilarly vltage rated ilicn avalanche dide auming equatin (.3) and (.5) are valid. [5.3, 5.7x 6 /cc,.63µm].3. An arupt ilicn pn unctin cnit f a p-type regin cntaining 6 cm -3 acceptr and an n-type regin cntaining 5x 4 cm -3 dnr. Calculate i. the uilt-in ptential f thi p-n unctin. ii. the ttal width f the cl regin if the applied vltage a equal,.5 and -. iii. maximum electric field in the cl regin at,.5 and -. iv. the ptential acr the cl regin in the n-type emicnductr at,.5 and Cnider an arupt pn dide with N A = 8 cm -3 and N D = 6 cm -3. Calculate the unctin capacitance at zer ia if the dide area i -4 cm. Repeat the prlem while treating the dide a a ne-ided dide and calculate the relative errr..5. Repeat example. uing the ingle-ided dide equatin in equatin (.), where N A >>N D. Calculate the percentage errr in uing the aumptin..6. A ilicn pn dide with N A = 8 cm 3 ha a capacitance f -7 F/cm at an applied revere vltage f. Calculate the dnr denity N D..7. A ilicn pn dide ha a maximum electric field magnitude f 7 /cm and a cl width f µm. The acceptr cncentratin i time the dnr denity. Calculate each dping denity..8. Repeat example. fr the equivalent 4H ilicn caride unctin dide having the ame electrical perating cnditin. Ue the ilicn caride data given elw. See prlem.4,.5, and.9 t.. Ueful SI data fr ilicn and ilicn caride: q =-.6x -9 C ξ =8.85x - F/m ξ rsi =.8 ξ rsic =9.7 kt/q =.59 e at 3ºC n isi =.5x 6 m -3 n isic =.5x -3 m -3