ECE606: Solid State Devices Lecture 7
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1 C606: Sold Stat vcs Lctur 7 Grhard Klmck gkco@purdu.du Rfrc: Vol. 6, Ch. 3 & 4 Prstato Outl Itrsc carrr coctrato Pottal, fld, ad charg -k dagram vs. bad-dagram Basc cocpts of doors ad accptors Law of mass-acto & trsc coctrato Statstcs of doors ad accptor lvls Itrsc carrr coctrato Tmpratur dpdc of carrr coctrato Multpl dopg, co-dopg, ad havy-dopg Coclusos
2 Carrr strbuto OS F- coctrato f ( ) = top c c ( ) ( ) g f d c υ gc g υ ( ) ( ) f ( ) F c ( ) ( ) g f ( ) ( ) gυ f υ bot ( ) ( ) p = gυ f d 3 lctro Coctrato 3 solds ( ) ( ) top = g f d c c ( ) * * top m m C = 3 β c π ħ + ( ) * m m * C 3 β c π ( ) ( ) β ( ) ħ c F c + = C F c c F π ( η ) η β ( ) C F d d Iclud sp factor of Assum wd bads C * 3 m F ( η ) = h + 0 π β ξ dξ ξ η 4
3 Boltzma vs. Frm-rac Statstcs η = C F ( ηc ) C η β ( ) c f c C F > 3 π η c F ( η c ) F gc ( ) f ( ) ( ) ( ) gυ f 5 ffctv sty of Stats β ( c F ) = C F ( ηc ) C f c F > 3β π F gc ( ) f ( ) ( ) ( ) gυ f C F V As f all stats ar at a sgl lvl C 6
4 Law of Mass-Acto = C β ( ) c F p = V ( ) + β v F F p = C V = C V β ( ) β g c Product s dpdt of th Frm lvl! Vry usful balac quato! Wll us t oft v 7 Frm-Lvl for Itrsc Smcoductors = p = β C V = g F = C V β g F G = + l β ( ) β ( ) β + V = p = C c v V C 3 k 8
5 Summary OS ad Frm Fuctos W dscussd how lctros ar dstrbutd lctroc stats dfd by th soluto of Schrodgr quato. Sc lctros ar dstrbutd accordg to thr rgy, rrspctv of thr momtum stats, th prvously dvlopd cocpts of costat rgy surfacs, dsty of stats tc. tur out to b vry usful. => wll ot dscuss Schrodgr q. aymor => vrythg s capturd baddgs ad ffctv masss W stll do ot kow whr F s for gral smcoductors If w dd, w could calculat lctro coctrato. 9 Rfrc: Vol. 6, Ch. 3 & 4 Prstato Outl Itrsc carrr coctrato Pottal, fld, ad charg -k dagram vs. bad-dagram Basc cocpts of doors ad accptors Law of mass-acto & trsc coctrato Statstcs of doors ad accptor lvls Itrsc carrr coctrato Tmpratur dpdc of carrr coctrato Multpl dopg, co-dopg, ad havy-dopg Coclusos 0
6 -k dagram vs. bad-dagram Pottal rgy c υ Ktc rgy P.. = = qv c rf Posto Rsolvd -k agram c c c υ υ υ
7 -k agram vs. Bad-dagram V=0 V=V c rf ( ) P.. = = qv x rf 3 Pottal, Fld ad Charg P.. = c rf rf qv = c rf dv = = dx dc q dx d ρ = = dx dx d V I most practcal cass start from charg ad drv pottals! => Usful to lar graphcal tgrato 4
8 Rfrc: Vol. 6, Ch. 3 & 4 Prstato Outl Itrsc carrr coctrato Pottal, fld, ad charg -k dagram vs. bad-dagram Basc cocpts of doors ad accptors Law of mass-acto & trsc coctrato Statstcs of doors ad accptor lvls Itrsc carrr coctrato Tmpratur dpdc of carrr coctrato Multpl dopg, co-dopg, ad havy-dopg Coclusos 5 Carrr strbuto OS F- coctrato f ( ) = top ( ) ( ) g f d c c c υ gc g υ ( ) ( ) f ( ) F ( ) ( ) g f => Rplac Klmck C606 by dlta Fall 0 fuctos ots adoptd from at Alam bad dg c ( ) ( ) gυ f υ bot ( ) ( ) p = gυ f d Crtcal tms hr; Itrsc smcoductor has VRY fw actv lctros s of th ordr of 0 0 /cm 3. I 0 /cm 3 atoms! o ot clud th coulomb tractos of dvdual fr lctros Oft good ough to forgt about th strbuto of carrrs rgy 6 C F V
9 ffctv sty of Stats F gc ( ) f ( ) ( ) ( ) gυ f C F V As f all stats ar at a sgl lvl C β ( c F ) = C F ( ηc ) C f β ( c F ) > 3 π Oft good ough to forgt about th strbuto of carrrs rgy => Rplac Klmck C606 by dlta Fall 0 fuctos ots adoptd from at Alam bad dg 7 -k agram vs. Bad-dagram V=0 V=V C V All quatum mchacs s ow hdd a sgl pot pr bad! 8
10 Pottal, Fld ad Charg P.. = rf c rf qv = c rf dv = = dx q dx dc d ρ = = dx dx d V I most practcal cass start from charg ad drv pottals! => Usful to lar graphcal tgrato 9 Rfrc: Vol. 6, Ch. 3 & 4 Prstato Outl Itrsc carrr coctrato Pottal, fld, ad charg -k dagram vs. bad-dagram Basc cocpts of doors ad accptors Law of mass-acto & trsc coctrato Statstcs of doors ad accptor lvls Itrsc carrr coctrato Tmpratur dpdc of carrr coctrato Multpl dopg, co-dopg, ad havy-dopg Coclusos 0
11 Isulator, Smcoductor, Mtal Mtal:» Coducts lctros v at vry low tmpraturs» Frm Lvl crosss multpl bads v at vry low tmpraturs Smcoductor» Vry wakly coductg»s: g=.v ~0 0 /cm /cm 3 0. a trllo»gaas: g=.4v ~0 6 /cm /cm 3»G: g=0.8v ~0 3 /cm /cm 3 0. a bllo Isulator» ot coductg»so, g= 9V, ~0-68 /cm3 Th whol arch has about 0 50 atoms! If you mad th whol worl out of glass thr would b ot o lcto coductv at room tmpratur! β = C V F g Smplfd Plaar Vw of Atoms
12 oor Atoms v wth doors, matral s charg utral 3 oor Atoms H-aalogy = + r 0 = r 0 4
13 oor Atoms Ral ad rgy Spac r 0 ~0s mv m * 4 host = ( Ks,hostħ) 4πε 0 m0q = 4 4 ( πεħ) 0 q * m = 3. 6 m 0 host m m * host 0 K s,host K s,host T = /β~k B T~5mV at T=300K 5 Assumpto of Larg Radus r 0 r 4πε K = 0 s,host,p * mhostq 4πε = m = r ħ. 9 r,p = A =. 9 A m K 0ħ 0 s,host * 0q mhost K s,host, H * mhost / m0 a~0.5m=5a => hudrds of S atoms 6
14 Charactrstcs of oor Atoms Th umbr of door atoms s much smallr compard to host atoms. Thrfor, th lctros from o door atom ca go to th othr door atoms oly va th coducto /valc bads of th host crystal. Just lk a Hydrog atom, t s possbl to hav multpl localzd lvl for a gv atom (s th blu lvls). () Good doors lv clos to th coducto bad, so that thy ca offr lctros asly. Howvr, f thy ar blow th mdgap, th door lvls ar markd wth () to dffrtat thm from accptor atoms (whch lv clos to th valc bad). 7 Accptor Atoms v wth accptor, matral s charg utral 8
15 Charactrstcs of Accptor Atoms r 0 A T 9 Amphotrc opats oor-typ accptor-typ 30
16 How to Rad th Tabl 3 Cocluso Itrsc carrr coctrato s so small that smcoductor must b dopd to mak t usful. A dopg atom bhavs lk a H-atom, xcpt that th dlctrc costat ad ffctv masss ar gv by by thos of th host atom. 3
17 Rfrc: Vol. 6, Ch. 3 & 4 Prstato Outl Itrsc carrr coctrato Pottal, fld, ad charg -k dagram vs. bad-dagram Basc cocpts of doors ad accptors Law of mass-acto & trsc coctrato Statstcs of doors ad accptor lvls Itrsc carrr coctrato Tmpratur dpdc of carrr coctrato Multpl dopg, co-dopg, ad havy-dopg Coclusos 33 Lookg ahad: Carrr-sty w/opg A bulk matral must b charg utral ovr all A + p + dv = 0 Furthr f th matral s spatally homogous + p + = 0 A ( F V ) / kbt ( C F )/ kbt V C A + = 0 ( F ) / kbt ( A F )/ kbt Lt us s how th formula com about 34
18 Charactrstcs of oor Atoms r 0 () T Localzd vs. localzd Stats stats/pr-bad (wth sp) - stats/pr-bad (wth sp)
19 Statstcs of oor Lvls P = ( F )/ kbt ( ) / k T ( )/ k T F B F B Z u / d P 0 / / Z ( F ) kbt / 0 Z ( F ) kbt 0 / Z / x x x Coulomb tracto forbds ths cofgurato Statstcs of oor Lvls u / d P 0 / / Z ( F ) kbt 0 / Z ( F ) kbt / 0 Z Prob. that th door s mpty (chargd) f P / Z = = = ( F ) / kbt ( F ) / kbt P00 + P0 + P 0 / Z + / Z + Prob. that th door s flld wth at last o lctro (utral) f00 = = ( F )/ kbt + + ( F ) / kbt ot th xtra factor.
20 Coulomb xcluso for Bad lctros? =4 = =0 P = ( F ) / kbt ( ) / k T ( ) / k T F B F B Z stat P / Z ( k T B F ) Z Localzd vs. Bad lctros 6 π/l x 5 0π/L x 4 8π/L x 3 6π/L x 4π/L x π/l x Two lctros (v wth oppost sp) ca ot b at th sam posto ad sam rgy bcaus of lctrostatc rpulso Lx 3 6π/(L x /) 4π/(L x /) p/(l x /) Bad lctros (wth oppost sp) d ot b at th sam posto, so thy ca shar occupy sam rgy lvl. L x / L x / Wh w dvd spac by a factor of, th umbr of stats (.g. 6 hr) dos ot chag.
21 Accptor Atoms Stat []. Hol prst - chargs Stat [0] Hol flld. chargs Statstcs of Accptor Lvls S ad G C V from lh from hh. ach atom cotrbuts stats (up & dow sp) to a bad, thrfor a bad has stats.. vry tm a host atom s rplacd by a mpurty atom, stats ar dsappar pr a bad ad appar as localzd stats (sort of). 3. Thrfor a accptor atom clos to hh ad lh bads rmovs four stats from thos bads. 4. Bcaus of Coulomb tracto oly hol ca sat ths 4 stats: th stats ar 0000, 000, 000, 000, 000.
22 umbr ad rgy Cosdratos Stats I HH/LH bads ) [0000] s th chargd stat as t has lctros, but - protos. ) Sgl hol cofgurato [000] s uchargd, as w hav - lctros, ad - protos sam s tru for [000], [000], [000] stats. 3) Gog from [0000] to [000] stats, th umbr of lctros gos dow by ( =-). 4) Gog from [0000] to [000] stats rgy gos dow by A, bcaus o lctro s o logr occupyg th hgh rgy lvl at A. Statstcs of Accptor Lvls P 0000 ( 0 0x F ) / kbt = ( F ) / kbt Z Stps 3 & P000 = P000 = P000 = P000 = ( F ) / kbt ( ( ) )/ k T ( )/ k T A F B A F B Z f 0000 P P + P + P + P + P + 4 A F B 0000 = = ( )/ k T
23 Flld ad mpty oor/accptor Lvls - stats mpty + = f = 00 ( F ) / kbt Stats I HH/LH bads (Two hols ca ot sat togthr) flld A A = A f [ ] 0000 = A A F B + 4 ( )/ k T strbutos ar physcal. f f = + F ) kb g ( / T = = k T + + gracy factor ε ( ) / k T ( ' ) / k T B F B F B ffctv door lvl
24 Summary A bulk matral must b charg utral ovr all p A dv = 0 Furthr f th matral s spatally homogous + p + + A = 0 A + = 0 ( F ) / kbt ( A F ) / kbt ( F V ) / kbt ( C F ) / kbt A V Rfrc: Vol. 6, Ch. 3 & 4 Prstato Outl Itrsc carrr coctrato Pottal, fld, ad charg -k dagram vs. bad-dagram Basc cocpts of doors ad accptors Law of mass-acto & trsc coctrato Statstcs of doors ad accptor lvls Itrsc carrr coctrato Tmpratur dpdc of carrr coctrato Multpl dopg, co-dopg, ad havy-dopg Coclusos 48
25 Carrr-dsty wth Uform opg A bulk matral must b charg utral ovr all + p + + A dv = 0 Furthr f th dopg s spatally homogous + p + + = 0 A F tgral vs. F fucto? ( ) A V F β F V A F β ( C F ) + = 0 β ( F ) β ( A F ) π π F A + F B A F )/ kbt = 0 ( F V )/ kbt ( C )/ kbt V A ( )/ k T ( (approx.) Oc you kow F, you ca calculat, p, +, A-. Itrsc Coctrato + p + + = 0 A A C + = 0 ( F )/ kbt ( A F ) / kbt ( F V ) / kbt ( C F )/ kbt V β ( ) β ( ) c F + v F C V p = 0 = G V F = + l β C
26 Rfrc: Vol. 6, Ch. 3 & 4 Prstato Outl Itrsc carrr coctrato Pottal, fld, ad charg -k dagram vs. bad-dagram Basc cocpts of doors ad accptors Law of mass-acto & trsc coctrato Statstcs of doors ad accptor lvls Itrsc carrr coctrato Tmpratur dpdc of carrr coctrato Multpl dopg, co-dopg, ad havy-dopg Coclusos 5 Carrr sty wth oors I spatally homogous fld-fr rgo + p + + = 0 A Assum -typ dopg C + ( F ) / kbt ( A F ) / kbt = 0 ( F V ) / kbt ( C F ) / kbt V A (wll plot xt sld)
27 Tmpratur-dpdt Coctrato Frz out xtrsc Itrsc Tmpratur Physcal Itrprtato Frz out xtrsc Itrsc Tmpratur
28 lctro Coctrato wth oors C F C = C = β ( ) β β C F + = + β F ( ) = β ( c ) + + C ξ lctro coctrato wth oors + p + = 0 + = 0 F B + ( F V ) / kbt ( C F ) / kbt V C ( ) / k T p = + = 0 + o approxmato so far. ξ
29 Hgh oor dsty/frz out T 0 ξ + = ξ ξ ξ = 4 ξ ξ = + Tmpratur Frz out xtrsc Itrsc ( ) ξ β C C xtrsc T ( ) C / kt C ξ 4 ξ ξ = + lctro coctrato quals door dsty hol coctrato by xp= Tmpratur Frz out xtrsc Itrsc 4 ξ ξ +
30 Itrsc T = for < ( F )/ kbt + + F + = 0 + ξ Frz out xtrsc Tmpratur Itrsc + = 0 = xtrsc/itrsc T For = Frz out xtrsc Itrsc For = Tmpratur What wll happ f you us slco crcuts at vry hgh tmpraturs? Badgap dtrms th trsc carrr dsty.
31 trmato of Frm-lvl β ( ) = c F C F = C + l β C p + + = 0 V F C + = 0 ( F )/ kbt + ( )/ k T ( )/ k T F V B C B Rfrc: Vol. 6, Ch. 3 & 4 Prstato Outl Itrsc carrr coctrato Pottal, fld, ad charg -k dagram vs. bad-dagram Basc cocpts of doors ad accptors Law of mass-acto & trsc coctrato Statstcs of doors ad accptor lvls Itrsc carrr coctrato Tmpratur dpdc of carrr coctrato Multpl dopg, co-dopg, ad havy-dopg Coclusos 6
32 Multpl oor Lvls Multpl lvls of sam door p + + = 0 A ( F )/ kbt + ( F + )/ kbt ( A + 4 F )/ kbt Codopg p + + = A 0 ( F )/ kbt ( F )/ kbt ( A F )/ kbt Havy opg ffcts: Badtal Stats k k?
33 Havy opg ffcts: Hoppg Coducto Badgap arrowg k K? p = C V β.g. Bas of HBTs * G Bad trasport vs. hoppg-trasport.g. a-slco, OL Arragmt of Atoms Poly-crystall Th Flm Trasstors Amorphous Oxds Crystall
34 Poly-crystall matral Isotropc badgap ad cras scattrg Bad-structur ad Prodcty PRB, 4, 508, 97 dagawa, PRL, 00,0390, 008 Prodcty s suffct, but ot cssary for badgap. May amorphous matral show full sotropc badgap
35 Coclusos. Charg utralty codto ad law of mass-acto allows calculato of Frm-lvl ad all carrr coctrato.. For smcoductors wth fld, charg utralty wll ot hold ad w wll d to us Posso quato. 3. Havg dopg ffcts play a mportat rol carrr trasport.
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