Chapter 3: Energy Band and Charge Carriers Semiconductor

Transcription

1 Chaptr 3: nrgy Band and Charg Carrrs Smondutor 3. Bondng Fors and nrgy bands n Sold Th bas dffrn btwn th as of an ltron n a sold and that of an ltron n an solatd atom s that n th sold th ltron has a rang, or band, of aalabl nrgs. Th dsrt nrgy lls of th solatd atom sprad nto bands of nrgs n th sold baus n th sold th wa funtons of ltrons n nghborng atoms orlap, and an ltron s not nssarly loalzd at a partular atom. For xampl, an ltron n th outr orbt of on atom fls th nflun of nghborng atoms, and ts orall wa funton s altrd. Naturally, ths nflun affts th potntal nrgy trm and th boundary ondtons n th Shrödngr quaton, and w would xpt to obtan dffrnt nrgs n th soluton. 3.. Bondng fors n Solds Th ntraton of ltrons n nghborng atoms s alld bond. Th bondng of a sold srs th ry mportant funton of holdng th rystal togthr. Ion bond: Ion bonds ar a typ of hmal bond basd on ltrostat fors btwn two oppostly-hargd ons. In on bond formaton, a mtal donats an ltron, du to a low ltrongatty to form a post on. In ordnary tabl salt, th bonds btwn th sodum (Na) and hlorn (Cl) ons ar on bonds. In a NaCl latt, ah Na atom s surroundd by sx nars nghbor Cl atoms, and rsa. Th ltron strutur of Na (Z) s [N]3s, and Cl (Z7) has th strutur [N]3s 3p 5. In th latt ah Na atom gs up ts outr 3s ltron to a Cl atom, so that th rystal s mad up of ons wth ltron struturs of th nrt atoms N and Ar. Howr, th ons ha nt ltr hargs aftr th ltron xhang. Th Na + on has a nt post harg, hang lost an ltron, and th Cl - on has a nt ngat harg, hang gand an ltron. On th ltron xhangs ha bn mad btwn th Na and Cl atoms to form Na + and Cl - ons, th outr orbts of all atoms ar ompltly flld. Sn th ons ha th losd-shll onfguratons of th nrt atoms N and Ar, thr ar no loosly bound ltrons to partpat n urrnt flow; as a rsult, NaCl s a good nsulator. Mtall bond: Mtall bondng s th bondng wthn mtals. It nols th dloalzd sharng of fr ltrons among a latt of mtal atoms. In a mtal atom th outr ltron shll s only partally flld, usually by no mor than thr ltrons. In Na has only on ltron n th outr orbt. Ths ltron s loosly bound and s gn up asly n on formaton. In th mtal th outr ltron of ah alkal atom s ontrbutd to th rystal as a whol, so that th sold s mad up of ons wth losd shlls mmrsd n a sa of fr ltrons. Th fors holdng th latt togthr ars from an ntraton btwn th post on ors and th surroundng fr ltrons. Ths s on typ of mtall bondng. Obously, thr ar omplatd dffrns n th holdng fors for arous mtals, as dnd by th wd rang of mltng tmpraturs [34K for Hg (Mrury), 3643K for W (tungstn)]. Howr, th mtals ha th sa of ltrons n ommon, and ths ltrons ar fr to mo about th rystal undr th nflun of an ltr fld. Coalnt bond: Coalnt bondng s an ntramolular form of hmal bondng haratrzd by th sharng of on or mor pars of ltrons btwn two omponnts, produng a mutual attraton that holds th rsultant molul togthr. Atoms tnd to shar ltrons n suh a way that thr outr ltron shlls ar flld. Atom n th G, S, or C damond latt s surroundd by four narst nghbors, ah wth four ltrons n th outr orbt. In ths rystals ah atom shars ts aln ltrons wth ts four nghbors. Th bondng fors ars from a quantum mhanal ntraton btwn th shard ltrons. Ths s known as oalnt bondng; ah ltron par onsttuts a oalnt bond. Th two ltrons ar ndstngushabl; xpt that thy must ha oppost spn to satsfy th Paul xluson Prnpl [No two ltrons n an ltron systm an ha th sam st of four quantum numbrs, s (Sharp), p (Prnpl), d (Dffus), f()fundamntal. Ths statmnt that no two ltrons may oupy th sam quantum stat s known as Paul xluson Prnpl]. 3.. nrgy Bands As solatd atoms ar brought togthr to form a sold, arous ntratons our btwn nghborng atoms. Th fors of attraton and rpulson btwn atoms wll fnd a balan at th propr ntratom spang for th rystal. In th pross, mportant hangs our n th ltron nrgy ll onfguraton and ths hangs rsult n th ard ltral proprts of sold. It has bn sn [Fg. -8] th orbtal modl of a S atom, along wth th nrgy lls of th arous ltrons n th oulomb potntal wll of th nulus. Lt us fous on th outrmost shll or aln shll, n3, whr two 3s and two 3p ltrons ntrat to form th four hybrdzd sp 3 ltrons whn th atoms ar brought los togthr. In Fg. 3-, w shmatally show th oulomb potntal wlls of two atoms los to ah othr, along wth th wa funtons of two-ltron ntrd on th two nul. By solng th Shrodngr for suh an ntratng systm, w fnd that th ompost two ltron wa funtons ar lnar ombnatons of th nddual atom orbts (LCAO). Th odd or ant-symmtr ombnaton s alld th ant-bondng orbtal, whl th n or symmtr ombnaton s th bondng orbtal. It s sn that th bondng orbtal has a hghr alu of th wa funton (and thrfor th ltron probablty dnsty) than th ant-bondng stat n th rgon btwn th two nul. Ths orrsponds to th oalnt bond btwn th atoms. To dtrmn th nrgy lls of th bondng and ant-bondng stats, t s mportant to rognz that n th rgon btwn th two nul th oulomb potntal nrgy V(r) s lowrd (sold ln n Fg. 3) ompard to solatd atoms (dashd ln n Fg. 3-).

2 Ψ Atom orbtals Ψ Bondng orbtal + + 4N Stats ltrons 6N Stats N ltrons V(r) + _ Ant-bondng Ant-bondng nrgy ll Rlat nrgy of ltrons Band gap g 4N Stats 4N ltrons N Stats N ltrons 8N Stats 4N ltrons 6N Stats N ltrons Outr shll N Stats N ltrons Bondng nrgy ll Mddl shll Innr shll Rlat spang of atoms Fgur 3-3 nrgy lls n S as a funton of ntr-atom spang. Fgur 3- Lnar ombnaton of atom orbtals (LCAO): Th LCAO whn atoms ar brought togthr lads to dstnt normal modls-a hghr nrgy ant-bondng orbtal, and a lowr nrgy bondng orbtal. [Not that th ltron probablty dnsty s hgh n th rgon btwn on ors (oalnt band), ladng to lowrng of th bondng nrgy ll and th ohson of th rystal. If nstad of atoms, on brngs togthr N atoms, thr wll b N dstnt LCAO, and N losly-spad nrgy lls n a bond.] Th potntal nrgy n th rgon s lowrd baus an ltron hr would b attratd by two nul; rathr than just on. For th bondng stat th ltron probablty dnsty s hghr n ths rgon of lowrd potntal nrgy than for th ant-bondng stat. 3 As a rsult, th orgnal solatd atom nrgy ll would b splt nto two, a lowr bondng nrgy ll and a hghr ant-bondng ll. It s th lowrng of th nrgy of th bondng stat that gs rs to ohson of th rystal. For n smallr ntr-atom spangs, th nrgy of th rystal gos up baus of rpulson btwn to nul and othr ltron ntratons. Sn th probablty dnsty s gn by th squar of th wa funton, f th ntr wa funton s multpld by-, t dos not lad to a dffrnt LCAO. Th mportant pont to not n ths dsusson s that th numbr of dstnt LCAO, and th numbr of dstnt nrgy lls dpnds on th numbr of atoms that ar brought togthr. Th lowst nrgy ll orrsponds to th totally symmtr LCAO, th hght orrsponds to th totally antsymmtr as and th othr ombnatons lad to th nrgy lls nn btwn 4

3 Whn two atoms ar ompltly solatd from ah othr so that thr s no ntraton of ltron wa funtons btwn thm, thy ha dntal ltron struturs. As th spang btwn two atoms boms smallr, howr, ltron wa funtons bgn to orlap. Th xluson prnpl dtats that no two ltrons n gn ntratng systm may ha th sam quantum stat; thus thr must b at most on ltron pr ll aftr thr s a splttng of th dsrt nrgy lls of th solatd atoms nto nw lls blongng to th par rathr than no nddual atoms. In a sold, many atoms ar brought togthr, so that th splt nrgy lls form ssntally ontnuous bands of nrgs. As an xampl, Fg. 3.3 llustrats th magnary formaton of a slon rystal from solatd slon atoms. ah solatd slon atom has an ltron strutur s s p 6 3s 3p n th ground stat. ah atom has aalabl two s stats, two s stats, two s stats, sx p stats, two 3s stats, sx 3p stats, and hghr stats. If w onsdr N atoms, thr wll b N, N, 6N, N, and 6N stats of typ s, s, p, 3s, and 3p, rsptly. As th ntratom spang drass, ths nrgy lls splt nto bands, bgnnng wth th outr (n3) shll. As th 3s and 3p bands grow, thy mrg nto a sngl band omposd of mxtur of nrgy lls. Ths band of 3s-3p lls ontans 8N aalabl stats. As th dstan btwn atoms approahs th qulbrum ntratom spang of slon, ths band splts nto two bands sparatd by an nrgy gap g. Th uppr band (alld th onduton band) ontans 4N stats, as dos th lowr (aln) band. Thus, apart from th low-lyng and tghtly bound or lls, th slon rystal has two bands of aalabl nrgy lls sparatd by an nrgy gap g wd, whh ontan no allowd nrgy lls for ltrons to oupy. Ths gap s somtms alld a forbddn band, sn n a prft rystal t ontans no ltron stats. W should paus at ths pont and ount ltrons. Th lowr s band s flld wth th N ltrons whh orgnally rasd n th ollt s stats of th solatd atoms. Smlarly, th s band and p wll ha N and 6N ltrons n thm, rsptly. Howr, thr wr 4N ltrons n th orgnal solatd n3 shlls (N n 3s stats and N n 3p stats). Ths 4N ltrons must oupy stats n th aln band or th onduton band n th rystal. At K th ltrons wll oupy th lowst nrgy stats aalabl to thm. In th as of S rystal, thr ar xatly 4N stats n th aln band aalabl to th 4N ltrons. Thus at K, ry stat n th aln band wll b flld, whl th onduton band wll b ompltly mpty of th ltrons. As w shall s, ths arrangmnt of ompltly flld and mpty nrgy bands has an mportant fft on th ltral ondutty of th sold Mtals, Smondutors, and Insulators ry sold has ts own haratrst nrgy band strutur. Ths araton n band strutur s rsponsbl for th wd rang of ltral haratrsts obsrd n arous matrals. Th slon band strutur of Fg. 3-3, for xampl, an g a good ptur of why slon n th damond latt s a good nsulator. To rah suh a onluson, w must onsdr th proprts of ompltly flld and ompltly mpty nrgy bands n th urrnt onduton pross. Bfor dsussng th mhansms of urrnt flow n solds furthr, w an obsr hr that for ltrons to xprn alraton n an appld ltr fld, thy must b abl to mo nto nw nrgy stats. Ths mpls thr must b mpty stats (allowd nrgy stats whh ar not alrady oupd by ltrons) aalabl to th ltrons. mpty g Flld Insulator mpty Flld Fgur 3-4 Typal band struturs at K. For xampl, f rlatly fw ltrons rsd n an othrws mpty band, ampl unoupd stats ar aalabl nto whh th ltrons an mo. On th othr hand, th slon band strutur s suh that th aln band s ompltly flld wth ltrons at K and th onduton band s mpty. Thr an b no harg transport wthn th aln band, sn no mpty stats ar aalabl nto whh ltron an mo. Thr ar no ltrons n th onduton band, so no harg transport an tak pla thr thr. Thus slon has a hghr rsstty typal of nsulator. Smondutor matrals at K ha basally th sam strutur as nsulators- a flld aln band sparatd from an mpty onduton band by a band gap ontanng no allowd nrgy stats (Fg. 3-4). Th dffrn ls n th sz of th band gap g, whh s muh smallr n smondutors than n nsulators. For xampl, th smondutor S has a band gap of about.v ompard wth 5V for damond. Th rlatly small band gaps of smondutors allow for xtaton of ltrons from th lowr (aln) band to uppr (onduton) band by rasonabl amounts of thrmal or optal nrgy. For xampl, at room tmpratur a smondutor wth -V band gap wll ha a sgnfant numbr of ltrons xtd thrmally aross th nrgy gap nto th onduton band, whras an nsulator wth g-v wll ha a nglgbl numbr of suh xtatons. Thus an mportant dffrn btwn smondutors and nsulators s that th numbr of ltrons aalabl for onduton an b nrasd gratly n smondutors by thrmal or optal nrgy. g Smondutor Partally flld Flld Mtal 5 6

4 In mtals th band thr orlap or ar only partally flld. Thus ltrons and mpty nrgy stats ar ntrmxd wthn th bands so that ltrons an mo frly undr th nflun of an ltr fld. As xptd from th mtall ban struturs of Fg. 3-4, mtals ha a hgh ltral ondutty Drt and Indrt Smondutors Th thought xprmnt of ston 3.. n whh solatd atoms wr brought togthr to form a sold, s usful n pontng out th xstn of nrgy bands and som of thr proprts. Othr thnqus ar gnrally usd, howr, whn quanttat alulatons ar mad of bond struturs. In a typal alulaton, a sngl ltron s assumd to tral through a prftly prod latt. Th wa funton of th ltron s assumd to b n th form of a plan wa mong. For xampl, n th x-drton wth propagaton onstant k, also alld wa tor. Th spa-dpndnt wa funton for th ltron s jk xx ψ k ( x) U ( k x, x) (3-) whr th funton U(k x,x) modulats th wa funton aordng to th prodty of th latt. Th band strutur of GaAs has a mnmum n th onduton band and a maxmum n th aln band for th sam k alu (k). On th othr hand, S has ts aln band maxmum at a dffrnt alu of k than ts onduton band mnmum. Thus an ltron makng a smallst-nrgy transton from th onduton band to th aln band n GaAs an do so wthout a hang n k alu: on th othr hand, a transton from th mnmum pont n th S onduton band to th maxmum pont of th aln band rqurs som hang n k. Thus thr ar two lasss of smondutor nrgy bands; drt and ndrt (Fg. 3-5). W an show that an ndrt transton, nolng a hang n k, rqurs a hang of momntum for th ltron. Th drt and ndrt smondutors ar dntfd n Appndx III. In a drt smondutor suh as GaAs, an ltron n th onduton band an fall to an mpty stat n th aln band, gng off th nrgy dffrn g as a photon of lght. On th othr hand, an ltron n th onduton band mnmum of an ndrt smondutor suh as S annot fall drtly to th aln band maxmum but must undrgo a momntum hang as wll as hangng ts nrgy. For xampl, t may go through som dft stat ( t) wthn th band gap. In an ndrt transton whh nols a hang n k, th nrgy s gnrally gn up as hat to th latt rathr than as mttd photon. Ths dffrn btwn drt and ndrt band strutur s ry mportant for ddng whh smondutors an b usd n ds rqurng lght output. g (a) Drt hh/λ.4/λ g k (b) Indrt Fgur 3-5 Drt and ndrt ltron transtons n smondutors: (a) drt transton wth aompanyng photon msson; (b) ndrt transton a a dft ll. In suh a alulaton, allowd alus of nrgy an b plottd s. th propagaton onstant k. Sn th prodty of most latts s dffrnt n arous drtons and th full rlatonshp btwn and k s a omplx surfa whh should b sualzd n thr dmnsons. t g k hplank s Constant J-s Frquny of th radaton. λ Wa-lngth Lght loty 3 8 m/s 3. Charg Carrrs n Smondutors Th mhansm of urrnt onduton s rlatly asy to sualz n th as of a mtal; th mtal atoms ar mbddd n a sa of th rlat fr ltrons, and ths ltrons an mo as a group undr th nflun of an ltr fld. Th smondutor has a flld aln band and an mpty onduton band at K, w must onsdr th nras n onduton band ltrons by thrmal xtatons aross th band gap as th tmpratur s rasd. In addton, aftr ltrons ar xtd to th onduton band, th mpty stats lst n th aln band ontrbut to th onduton pross. Th ntroduton of mpurts has an mportant fft on th nrgy band strutur and on th aalabl of harg arrrs. Thus thr s onsdrabl flxblty n ontrollng th ltral proprts of smondutor. 3.. ltrons and Hols As th tmpratur of a smondutor s rasd from K, som ltrons n th aln band r nough thrmal nrgy to b xtd aross th band gap to th onduton band. Hol: An mpty stat n th aln band s rfrrd to as a hol. ltron-hol Par: If th onduton band ltron and th hol ar ratd by th xtaton of th aln band ltron to th onduton band, thy ar alld an ltron-hol par (abbratd HP). If a flld band, all aalabl nrgy stats ar oupd. For ry ltron mong wth gn loty, thr s an qual and oppost ltron moton lswhr n th band. 7 8

5 If w apply an ltr fld, th nt urrnt s zro baus for ry ltron j mong loty j thr s a orrspondng ltron j wth th loty j. Fg. 3-8 llustrats ths fft n trms of th ltron s. wa tor plot for th alan band. Sn k s proportonal to ltron momntum, t s lar th two ltrons ha oppost drtd lots. g N J ( q) ( q) j ( jth ltron mssn g) (3-b) Sn th frst trm of (3-b) s zro n aordan (3-a), th nt urrnt s J q j ( Nt urrnt du to th mssn g of jth ltron) In othr words, th urrnt ontrbuton of th hol s qualnt to that of a postly hargd partl wth th loty j that of th mssng ltron. Th harg transport s atually du to th moton of th nw unompnsatd ltron j. Its urrnt ontrbuton (-q)(- j) s qualnt to that of a postly hargd partl wth th loty j. From th abo dsusson t s ralzd that th urrnt flow n th smondutor an b aount by th moton of ltrons and hols of harg arrrs. In th aln band, hol nrgy nrass oppostly to ltron nrgy, baus th two arrrs ha oppost harg. Thus hol nrgy nrass downward n Fg. 3-8 and hols, skng th lowst nrgy stat aalabl, ar gnrally found at th top of th alan band. In ontrast, onduton band ltrons ar found at th bottom of th onduton band. Fgur 3-7 ltron-hol pars n a smondutor. Wth N ltron/m 3 n th band w xprss th urrnt dnsty usng a sum or all of th ltron lots, and nludng th harg q on ah ltron. In a unt olum, N J ( q) ( flld band) (3-a) 3.. fft Mass Th ltrons n a rystal ar not ompltly fr, but nstad ntrat wth prod potntal of th latt. As a rsult, thr wa partl moton annot b xptd to b th sam as for ltrons n fr spa. Thus, n applyng th usual quatons of ltrodynams to hang arrrs n a sold, w must us altrd rsults of partl mass. Th alulaton of fft mass must tak nto aount th shap of th nrgy bands n thrdmnsonal k-spa, takng approprat arags or th arous nrgy bands. k j k j k Th ltron momntum an b wrttn as Pmħk. Thn P h m k m m Thus th ltron nrgy s parabol wth wa tor k. Th ltron mass s nrsly rlatd to th uratur (sond drat) of th (, k) rlaton, sn d h dk m Although ltrons n solds ar not fr, most nrgy bands ar los to parabol at thr mnma (for onduton band) or maxma (for aln band). W an also approxmat fft mass nar thos band xtrma from th uratur of th band. Fgur 3-8 A aln band wth all stats flld, nludng stats j and j, markd for dsusson. Th jth ltron wth wa tor k j s mathd by an ltron at j wth th oppost wa tor k j. thr s no nt urrnt n th band unlss an ltron s rmod. For xampl, f th jth ltron s rmod, th moton of th ltron at j s no longr ompnsaton. Now f w rat a hol by rmong th jth ltron, th nt urrnt dnsty n th alan band nols th sum or all lots, mnus th ontrbuton of th ltron w ha rmod. 9 Th fft mass of an ltron n a band wth a gn (, k) rlatonshp s found as follows: * h m (3-3) d / dk Thus th uratur of th band dtrmns th ltron fft mass Intrns Matral A prft smondutor wth no mpurts or latt dft s alld an ntrns matral.

6 In suh matral thr ar no harg arrr at K, sn th aln band s flld wth ltrons and th onduton band s mpty. At hgh tmpratur ltron-hol pars ar gnratd as aln band ltrons ar xtd thrmally aross th band gap to th onduton band. Ths HPs ar th only harg arrrs n ntrns matral. - h + - h + - : ltron h + : Hol 3..4 xtrns Matral In addton to th ntrns arrrs gnratd, t s possbl to rat arrrs n smondutors by purposly ntrodung mpurts nto th rystal. Ths pross, alld dopng, s th most ommon thnqu for aryng ondutty of smondutor. Thr ar two typs of dopd smondutors, n-typ (mostly ltrons) and p-typ (mostly hols). Whn a rystal s dopd suh that th qulbrum arrr onntratons n and p ar dffrnt from arrr onntraton n, th matral s sad to b xtrns. Whn mpurts or latt dfts ar ntrodud nto an othrws prft rystal, addtonal lls ar ratd n th nrgy band strutur, usually wthn th band gap. An mpurty from olumn V of th prod tabl (P, As and Sb) ntrodus an nrgy ll ry nar th onduton band n G or S. Ths ll s flld wth ltrons at K, and ry lttl thrmal nrgy s rqurd to xt ths ltrons to th onduton band (Fg. 3-a). Thus at 5-K rtually all of th ltrons n th mpurty ll ar, donatd to th onduton band. Suh an mpurty ll s alld a donor ll and th olumn V mpurts n G or S ar alld donor mpurts. Smondutors dopd wth a sgnfant numbr of donor atoms wll ha n>>(n,p) at room tmpratur. Ths s n-typ matral. Fgur 3- ltron-hol pars n th oalnt bondng modl of th S rystal. If on of th S aln ltrons s brokn away from ts poston n th bondng strutur suh that t boms fr to mo about n th latt, a onduton ltron s ratd and a brokn bond (hol) s lft bhnd. Th nrgy s rqurd to brak th bond s th band gap nrgy g. d d Sn th ltrons and hols ar ratd n pars, th onduton band ltron onntraton n (ltron/m 3 ) s qual to th onntraton of hols n th aln band p (hols/m 3 ). ah of ths ntrns arrr onntratons s ommonly rfrrd to as n. Thus for ntrns matral npn (3-6) At a gn tmpratur thr s a arrr onntraton of HPs n. Obously, f a stady stat arrr onntraton s mantand, thr must b rombnaton of HPs at th sam rat at whh thy ar gnratd. Rombnaton s ours whn an ltron n th onduton band maks transton (drt or ndrt) to an mpty stat (hol) n th aln band, thus annhlatng th par. If w dnot th gnraton rat of HPs as g (HP/m -3 -s) and th rombnaton rat r, qulbrum rqurs that r g (3-7a) ah of ths rats s tmpratur dpndnt. g (T) nrass whn th tmpratur s rasd, and a nw arrr onntraton n s stablshd suh that th hghr rombnaton rat r (T) just balans gnraton. At any tmpratur, w an prdt that th rat of rombnaton of ltrons and hols r s proportonal to th qulbrum onntraton of ltrons n and th onntraton of hols p : r α rn p α rn g (3-7b) Th fator α r s a onstant of proportonalty whh dpnds on th partular mhansm taks pla. TK a Fg. 3- nrgy band modl and hmal bond modl of dopants n smondutors. Smlarly, an mpurty from olumn III of th prod tabl (B, Al, Ga and In) ntrodus an nrgy ll ry nar th aln band n G or S. Ths lls ar mpty of ltrons at K (Fg. 3- T5K (a) Donaton of ltrons from donor ll to onduton band T5K TK (b) Aptan of aln band ltrons by an aptor ll and th rsultng raton of hols a

7 b). At low tmpraturs, nough thrmal nrgy s aalabl to xt ltrons from th aln nto th mpurty ll, lang bhnd hols n th aln band. Sn ths typ of mpurty ll apts ltrons from th aln band, t s alld an aptor ll, and th olumn III mpurts ar aptor mpurts n th G and S. Dopng wth aptor mpurts an rat a smondutor wth a hol onntraton p muh gratr that th onduton band ltron onntraton n. Ths typ s p-typ matral. f() TK Whn a smondutor s dopd n-typ or p-typ on typ of arrr domnats. For n-typ matral ltrons ar majorty (domnat) arrrs and hols ar mnorty arrrs. For p-typ matral hols ar majorty (domnat) arrrs and ltrons ar mnorty arrrs. 3.3 Carrr onntraton Th alulatng smondutor proprts and analyzng d bhaor, t s oftn nssary to know th numbr of harg arrrs pr m 3 n th matral. Th majorty arrr onntraton s usually obous n haly dopd matral, sn on majorty arrr s obtand for ah mpurty atom. In that as th mnorty arrr s not obous. To obtan quaton for th arrr onntraton, w shall apt th dstrbuton funton as gn Th Frm Ll Th dstrbuton of ltrons or a rang of allowd nrgy lls at thrmal qulbrum s f ( ) ( F + (3-) whr, k s Boltzmann s onstant (k8.6-5 V/K.38-3 J/K). Th funton f(), th Frm-Dra dstrbuton funton, gs th probablty that an aalabl nrgy stat at wll b oupd by an ltron at absolut tmpratur T. Th quantty F s alld th Frm ll, and t rprsnts an mportant quantty n th analyss of smondutor bhaor. For an nrgy qual to th Frm ll nrgy F, th oupaton probablty s f ( ) F (3-) ( ) / + F F kt Thus an nrgy stat at th Frm ll has th probablty (/) of bng oupd by an ltron. At K th dstrbuton taks th smpl rtangular form shown n Fg Wth TK n th dnomnator of th xponnt, f() s /(+) whn th xponnt s ngat (< F), and s /(+ ) whn th xponnt s post (> F). Ths rtangular dstrbuton mpls that at K ry aalabl nrgy stat up to F s flld wth ltrons, and all stats abo F ar mpty. / T > T Fgur 3-4 Th Frm-Dra dstrbuton funton Th Frm funton s symmtral about F for all tmpratur; that s, th probablty f( F + ) that a stat abo F s flld s th sam as probablty [-f( F- )] that a stat blow F s mpty. Th symmtry of th dstrbuton of mpty and flld stats about F maks th Frm ll a natural rfrn pont n alulatons of ltron and hol onntraton n smondutors. Th rlaton btwn f() and th band strutur an b sualzd by turnng th f() s. dagram on ts sd so that sal orrsponds to th nrgs of th band dagram (Fg. 3-5). For ntrns matral, w know that th onntraton of hols n th aln band s qual to th onntraton of ltrons n th onduton band. Thrfor, th Frm ll F must ls at th mddl of th band gap. Sn f() s symmtral about F, th ltron probablty tal f f() xtndng nto th onduton band of Fg. 3-5a s symmtral wth th hol probablty tal [-f()] n th aln band. In n-typ matral th Frm ll ls nar th onduton band (Fg. 3-5b) suh that th alu of f() for ah nrgy ll n th onduton band nrass as F mos losr to. Thus th nrgy dffrn (- F) gs masur of n. In p-typ matral th Frm ll ls nar th aln band (Fg. 3-5) suh that th [- f()] tal alu s largr than th f() tal abo. Th alu of ( F-) ndats how strongly p-typ th matral s. T T At tmpratur hghr than K, som probablty xsts for stats abo th Frm ll to b flld. For xampl, at TT n Fg. 3-4 thr s som probablty f() that stats abo F ar flld, and thr s a orrspondng probablty [- f()] that stats blow F ar mpty. 3 4

8 xampl: Th Frm ll n a S sampl at qulbrum s loatd at. V blow th onduton band. At T3K, dtrmn th probablty of oupany of th aptor stats f th aptor stats rloatd at.3 V abo th aln band. f() Soluton:. V (a) Intrns matral F F. V (b) n-typ matral f() f() / a.3 V From abo fgur, a- F{.3-(.-.)} V -.87 V kt V/K V w know that, f ( a ). ( ) / /(758.4 a F kt ) + F f() / 3.3. ltron and Hol Conntratons at qulbrum Th Frm dstrbuton funton an b usd to alulat th onntratons of ltrons and hols n a smondutor, f th dnsts of aalabl stats n th aln and onduton bands ar known. [-f() ] Th onntraton of ltron n th onduton band s n f ( ) N( ) d (3-) whr N()d s th dnsty of stats (m -3 ) n th nrgy rang d. Th subsrpt usd wth th ltron and hol onntraton symbols (n, p ) ndats qulbrum ondtons. () p-typ matral F Th numbr of ltrons pr unt olum n th nrgy rang d s th produt of th dnsty of stats and th probablty of oupany f(). Thus th total ltron onntraton s th ntgral or th ntr onduton band as n q. (3-). f() / Fgur 3-5 Th Frm dstrbuton funton appld to smondutors. Th funton N() [Appndx IV] s proportonal to (/), so th dnsty of stats n th onduton band nrass wth ltron nrgy. On th othr hand, th Frm funton boms xtrmly small for larg nrgs. Th rsult s that th produt of f()n() drass rapdly abo, and ry fw ltrons oupy nrgy stats far abo th onduton band dg. Smlarly, th probablty of fndng an mpty stat (hol) n th aln band [-f()] drass rapdly blow, and most hol oupy stats nar th top of th aln band. Ths fft s dmonstratd for ntrns, n-typ and p-typ matrals n Fg

9 Th ltron onntraton (n trms of fft dnsty of stats N whh s loatd at th onduton band dg ) n thrmal qulbrum an also b wrttn as follows: n N f ( ) (3-3) 3/ * πm nkt whr, N [Appndx IV] h It s assumd that th Frm ll F ls at last sral kt blow th onduton band. Thn ( F >> and th Frm funton f() an b smplfd as ( F f ( ) ( F + (3-4) For ths ondton th onntraton of ltrons n th onduton band s F n N (3-5) F ltrons Hols By smlar argumnt, th onntraton of hols n th aln band s p N [ f ( )] (3-7) * 3/ πm pkt whr, N s th fft dnsty of stats n th aln band. h For F largr than by sral kt, th probablty of fndng an mpty stat at s F f ( ) (3-8) ( F + From ths quatons, th onntraton of hols n th aln band s F p N (3-9) (a) Intrns matral N() f() Carrr onntraton F Th ltron and hol onntratons prdtd by qs. (3-5) and (3-9) ar ald whthr th matral s ntrns or dopd, prodd thrmal qulbrum s mantand. Thus for ntrns matral, F ls at som ntrns ll nar th mddl of th band gap, and th ntrns ltron and hol onntratons ar n N, p N (3-) Th produt of n and p at qulbrum s a onstant for a partular matral and tmpratur, n f th dopng s ard: F F g / kt n p N N N N N N (3-a) s g / kt N N N N N n p N (3-b) Th ntrns ltron and hol onntratons ar qual, n p ; thus th ntrns onntratons g / kt n N N (3-3) Th onstant produt of ltron and hol onntratons n q. (3-) an b wrttn onnntly as n p n (3-4) Th ntrns onntraton for S at room tmpratur s approxmatly n.5 m (b) n-typ () p-typ N() f() Carrr onntraton F N() f() Carrr onntraton Fgur 3-6 Shmat band dagram, dnsty, Frm-Dra dstrbuton, and th arrr onntraton. 8 N()f() N()[-f()]

10 Usng q. (3-), th qs. (3-5) and (3-9) an b wrttn as ( F n n p n ( F (3-5) In ths xampl, S s dopd n-typ wth donor onntraton N d of 5 m -3. At ry low tmpratur (larg /T) nglgbl ntrns HPs xst, and th donor ltrons ar bound to th donor atoms. It sn from th quaton (3-5) that th ltron onntratons s n whn F s a th ntrns ll, and that n nrass xponntally as th Frm ll mos away from toward th onduton band. Smlarly, th hol onntratons p ars from n to largr alus as F mos from toward th aln band. xampl 3-5 A S sampl s dopd wth 7 As atoms/m 3. What s th qulbrum hol onntraton p at 3K? Whr s F rlat to? Soluton: N d 7 atoms/m 3 ; kt.59 V; n.5 m -3 ; Sn N d>>n, w an approxmat n N d n (.5 ) 3 p.5 m -3 n 7 ( F W know that n n 7 n F kt ln.59 ln.47 V n.5 Th rsultng band dagram s Tmpratur Dpndn of Carrr Conntratons Th araton of arrr onntraton wth tmpratur s ndatd by q. (3-5) ( F ( F n n ; p n (3 5) Th ntrns arrr n has a strong tmpratur dpndn (q. 3-3) and that F an ary wth tmpratur. / kt n N N g (3 3) F.47 V Th tmpratur dpndn of ltron onntraton n a dopd smondutor an b sualzd as shown n Fg V n (m-3) 7 Intrns 5 3 xtrns /T (K) - Fgur 3-8 Carrr onntraton s. nrs tmpratur for S dopd wth 5 donors/m 3. As th tmpratur s rasd, ths ltrons ar donatd to th onduton band, and at about K (/T) all th donor atoms ar onzd. Ths tmpratur rang s alld onzaton rgon. On th donor atoms ar onzd, th onduton band ltron onntraton s n N d 5 m-3, sn on ltron s obtand for ah donor atom. Whn ry aalabl xtrns ltron has bn transfrrd to th onduton band, n o s rtually onstant wth tmpratur untl th onntraton of ntrns arrrs n boms omparabl to th xtrns onntraton N d. Fnally, at hghr tmpratur n s muh gratr than N d, and th ntrns arrrs domnat. In most ds t s dsrabl to ontrol th arrr onntraton by dopng rathr than by thrmal HP gnraton. Thus on usually dops th matral suh that th xtrns rang xtnds byond th hghst tmpratur at whh th d to b usd Compnsaton and Spa Charg Nutralty Whn th onpt of dopng was ntrodud, w assumd th matral ontand thr N d donors or N a aptors, so that th xtrns majorty arrr onntratons wr n N d or p N a, rsptly, for th n-typ or p-typ matral. If oftn happns, howr, that a smondutor ontans both donors and aptors. For xampl, Fg. 3-9 llustrats a smondutor for whh both donor and aptors ar prsnt, but N d>n a. Ionzaton

11 Th prdomnan of donors maks th matral n-typ, and th Frm ll s thrfor n th uppr part of th band gap. Sn F s wll abo aptor ll a, ths ll s ssntally flld wth ltrons. Howr, Wth F abo, t annot b xptd that a hol onntraton n aln band ommnsurat wth th aptor onntraton. In fat, th fllng of th a stats ours at th xpns of th donatd onduton band ltrons. Ths mhansm an b sualzd as follows: Assum an aptor stat s flld wth a aln band ltrons wth a hol rsultng n th aln band. Ths hol s thn flld by rombnaton wth on of th onduton band ltrons. xtndng ths log to all th aptor atoms, t s xptd that th rsultant onntraton of ltrons n th onduton band to b N d-n a nstad of th total N d. Ths pross s alld ompnsaton. xampl: A S s dopd wth 9 5 m -3 donors and 3 5 aptors. Fnd th poston of th Frm ll at 4 C. Soluton: N d9 5 m -3 ; N a3 5 m -3 ; n.5 m -3 ; Sn N d>n a, th Frm ll nar to th donor stats. W obtan, N rdn d-n a6 5 m -3 donor atoms; Thus th Frm ll dpnds on th donor atoms N rd. Now w onsdr n N rd baus N rd>>n. 5 n 5 6 So w obtan, F kt ln (8.6 V / K)(73 + 4K) ln. 348 V. n.5 Th rsultng band dagram s d F F.348 V. V a Fgur 3-9 Compnsaton n an n-typ smondutor (N d>n a) Rfrn [] Bn. G. Strtman and Sanjay Banrj: Sold Stat ltron Ds, Prnt-Hall of Inda Prat Lmtd. By ths pross t s possbl to bgn wth an n-typ smondutor and add aptors untl N an d and no donatd ltrons rman n th onduton band. In suh ompnsatd matral n n p o, and ntrns onduton s obtand. Wth furthr aptor dopng th smondutor boms p-typ wth a hol onntraton of ssntally N a-n d. Th xat rlatonshp among th ltron, hol, donor and aptor onntratons an b obtand by onsdrng th rqurmnts for spa harg nutralty. If th matral s to rman ltrostatally nutral, th sum of th post hargs (hols and onzd donor atoms) must balan th sum of th ngat hargs (ltrons and onzd aptor atoms): + p + N d n + N a (3-7) Thus n Fg 3-9 th nt ltron onntraton n th onduton band s n p + ( N + d + N a ) (3-8) If th matral dopd n-typ (n >>p ) and all th mpurts ar onzd, w an approxmat q. (3-8) by n N d + N a.

12 Insulator: A ry poor ondutor of ltrty s alld an nsulator. In an nsulator matral th alan band s flld whl th onduton band s mpty. Th onduton band and alan band n th nsulator ar sparatd by a larg forbddn band or nrgy gap (almost V). In an nsulator matral, th nrgy whh an b suppld to an ltron from a appld fld s too small to arry th partl from th fld alan band nto th mpty onduton band. Sn th ltron annot aqur suffnt appld nrgy, onduton s mpossbl n an nsulator. Smondutor: A substan whos ondutty ls btwn nsulator and ondutor s a smondutor. A substan for whh th wdth of th forbddn nrgy rgon s rlatly small (almost V) s alld smondutor. In a smondutor matral, th nrgy whh an b suppld to an ltron from a appld fld s too small to arry th partl from th fld alan band nto th mpty onduton band at K. As th tmpratur s nrasd, som of th alan band ltrons aqur thrmal nrgy. Thus, th smondutors allow for xtaton of ltrons from th alan band to onduton band. Ths ar now fr ltrons n th sns that thy an mo about undr th nflun of n a small-appld fld. Mtal: A mtal s an xllnt ondutor. In mtals th band thr orlap or ar only partally flld. Thus ltrons and mpty nrgy stats ar ntrmxd wthn th bands so that ltrons an mo frly undr th nflun of an ltr fld. Drt Matral: Th matral (suh as GaAs) n whh a transton of an ltron from th mnmum pont of onduton band to th maxmum pont of aln band taks pla wth th sam alu of K (propagaton onstant or wa tor) s alld drt matral. Indrt Matral: Th matral (suh as S) n whh a transton of an ltron from th mnmum pont of onduton band to th maxmum pont of aln band taks pla wth th dffrnt alus of K (propagaton onstant or wa tor) s alld drt matral. Stady-stat: Stady-stat rfrs to a nonqulbrum ondton n whh all pross ar onstant and ar baland by opposng pross (.g., a sampl wth a onstant urrnt or a onstant optal gnraton of HPs just baland by rombnaton. Voabulary: Commnsurat [adj.]: - (wth sth) mathng sth n sz, mportan, qualty t. Conton: Th pross n whh hat mos through a gas or a lqud as th hottr part rss and th oolr, har part snk Sattr: [rb] to throw or drop thngs n dffrnt drtons so that thy or an ara of ground; to mo or to mak popl or anmals mo ry qukly n dffrnt drtons [noun] a small amount or numbr of thngs sprad or an ara. Quas-: That appars to b sth but s not rally so; Partly, almost. Ampl: nough or mor than nough. ltrongatty: ltrongatty s a masur of th ablty of an atom or molul to attrat ltrons n th ontxt of a hmal bond. Th typ of bond formd s largly dtrmnd by th dffrn n ltrongatty btwn th atoms nold. Atoms wth smlar ltrongatts wll shar an ltron wth ah othr and form a oalnt bond. Howr, f th dffrn s too grat, th ltron wll b prmanntly transfrrd to on atom and an on bond wll form. Furthrmor, n a oalnt bond f on atom pulls slghtly hardr than th othr, a polar oalnt bond wll form. ltropostty: Th rrs of ltrongatty, th ablty of an atom to los ltrons, s known as ltropostty. Dloalzd ltrons: In physs dloalzd ltrons ar ltrons n a molul that ar not assoatd wth a sngl atom or a oalnt bond. Dloalzd ltrons ar part of a p ltron systm that xtnds or sral adjant atoms. ltrostats: ltrostats s th branh of physs that dals wth th fors xrtd by a stat (.. unhangng) ltr fld upon hargd objts qulbrum: Th trm rfrs to a ondton of no xtrnal xtaton xpt fo tmpratur, and no nt moton of harg (.g., a sampl at a onstant tmpratur, n th dark, wth no flds appld). 3 4

13 xampl: An unknown smondutor has g. and N N. It s dopd wth 5 m -3 donors whr th donors ll s. blow. Gn that F s.5 blow, alulat n and th onntraton of ltrons and hols n th smondutor at 3K. Soluton: Gn g.; N N ; N d 5 m -3 ; kt(3k).6 V. F- (.-.5).85 V. - F ( F-) V. g / kt g / kt g / kt n N N ; n N ; N n ; n p n ; n N F )/ kt ; p N F F kt ; p N ) / F kt ; n n ( ) / F kt ; p n ( ) / d F. V.5 V g. V 5