Dept. of Materials Science and Engineering. Problem Set 8 Solutions

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1 MSE 30/ECE 30 Elctrical Prortis Of Matrials Dt. of Matrials Scinc and Enginring Fall 0/Bill Knowlton Problm St 8 Solutions. Using th rlationshi of n i n i i that is a function of E g, rcrat th lot shown in Kasa figur 5.6 which is also shown in your class nots. ot that th quation adjacnt to figur 5.6 is incomlt so do not us it. Fully labl your lot including, but not limitd to, labling ach lot rlativ to th smiconductor (.g., lgnd) and th room tmratur intrinsic carrir concntrations of ach sminconductor (.g., vrtical lin and valus). Do not forgt to includ figur cations and figur cation labls. Plas commnt on th findings rvald by your lot and also comar it to figur 5.6. Hint: valus for ffctiv mass can b found in tabl 5. of Kasa. Solution: E g Using th quation from your slids: ni cv, and valus for c and v or m ff for lctrons and hols from Tabl 5. in Kasa, w can thn crat th lots: Intrinsic Carrir Concntration 0 8 Room Tm x0 3 cm -3 ni cm x0 0 cm -3 Si G 0 6.x0 8 cm -3 Gas T K - Fig. log of intrinsic lctron carrir concntration vrsus invrs tmratur (rrhnius lot) is lottd as a function of th invrs of tmratur for Gas, Si, and G. Th dashd vrtical lin indicats room tmratur and th labld concntration ar intrinsic carrir concntrations at room tmratur.

2 MSE 30/ECE 30 Elctrical Prortis Of Matrials Dt. of Matrials Scinc and Enginring Fall 0/Bill Knowlton ot that th stst slo is Gas and th shallowst slo is G indicating that E g,g < E g,si < E g,gas. Bcaus of this rlation, not that th intrinsic carrir concntration, n i, at room tmratur follows: n i,g > n i,si > n i,gas. Th lot abov is lottd with both c and v ar a function of tmratur and ar givn as: c 3 3 * * b m h b v m and h h. W us th ffctiv masss that ar scifically for th dnsity of stats calculations and thy can b found in tabl 5. in Kasa 3 rd dition on ag 386. lso not that th lins ar not quit linar. This is du to th T 3/ dndnc in th r-xonntial.. Using th fctiv Mass roximation, dscrib th conct of th Hydrognic Modl and shallow doants. Plas us mathmatical rlationshis in your dscrition for E doant and r doant. Th nrgy and bond lngth rlations for th fctiv Mass roximation ar givn by: * * m m Edoant EH 3.6 (V) () r r r r rdoant rh 0.5 (Å) () m * m * whr m * and ξ r ar th ffctiv mass and th rlativ dilctric constant, rsctivly. Th comlimntary nam for th fctiv Mass roximation is th Hydrognic modl which stms from th fact that th aroximation incororats th Bohr radius and nrgy. Th addition of th ffctiv mass and th rlativ dilctric constant into th Bohr radius and nrgy rflcts th natur of th lctronic nvironmnt of th matrix matrial on th singl carrir (sinc it is singl, it is hydrogn-lik) of th doant atom. Th aroximation rdicts vry long bond lngths of th carrir-doant atom rlativ to th lattic constant of th matrix and vry low bond nrgis rlativ to th nrgy band ga. Th lattr commnt suggsts that th doant nrgy is vry clos to th band dg, thus it is shallow into th nrgy band ga from th band dg hnc th nam shallow doant.

3 MSE 30/ECE 30 Elctrical Prortis Of Matrials Dt. of Matrials Scinc and Enginring Fall 0/Bill Knowlton Rfrncs: [] M. McClusky & E. Hallr, Doants & Dfcts in Smiconductors (CRC Prss, 0) Ch.. [] P. Y. Yu and M. Cardona, Fundamntals of Smiconductors Physics and Matrial Prortis. ( Sringr-Vrlag, 996), [3]. Hays and. M. Stonham, Dfcts and Dfct Procsss in onmtallic Solids (w York: Wily, 985). P. 59 []. K. Jonschr, Elctronic rortis of amorhous dilctric films, Thin Solid Films, vol., no. 3,. 3 3, In ordr to driv i for an intrinsic smiconductor and for an xtrinsic smiconductor, on must first rov quation. Prov th following and show all sts. f ( E) f ( E) FD, h FD, Solution: FD, h FD, E f ( E) f ( E) [] E E E E E E [] E 3

4 MSE 30/ECE 30 Elctrical Prortis Of Matrials Dt. of Matrials Scinc and Enginring Fall 0/Bill Knowlton. In th cours nots, w driv that: n D D Using th sam aroach, driv for a -ty smiconductor using th aroach rsntd in class (i.. using g(e) and f(e)). B sur to show all mathmatical sts and rovid short xlanations to rciv full crdit. Th answr you should obtain is th following: Hint: Start with th following: ffd, E E h E g [] * [] * f E E [3] Solution: FD, h Substitut quations [] and [3] into []: E g E g E g E g E f E g E f E g [] i for intrinsic smiconductors is givn by: Ev V Solv for E f and assum that E V = 0.

5 MSE 30/ECE 30 Elctrical Prortis Of Matrials Dt. of Matrials Scinc and Enginring Fall 0/Bill Knowlton V ln V E f E V ln [5] Substitut [5] into [] V ln [5] f V ln E g gv E [5] [] Lt: V [6] Substitut [6] into [5] [7] Cross-multily and collct trms: g E 0 [8] W hav a quadratic quation that can b solvd with th quadratic formula which is givn by: b b ac a a, b, c Substituting into th quadratic quation: 5

6 MSE 30/ECE 30 Dt. of Matrials Scinc and Enginring Elctrical Prortis Of Matrials Fall 0/Bill Knowlton 6 [9] Sinc is gratr than or qual to zro, w only tak ositiv valus. Thus: [0]

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