NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics Enrgy Bands, Basics of Transports and Optical Procsss in Smiconductors R. John Bosco Balaguru Profssor School of Elctrical & Elctronics Enginring SASTRA Univrsity B. G. Jyaprakash Assistant Profssor School of Elctrical & Elctronics Enginring SASTRA Univrsity Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics Tabl of Contnt 1. ENERGY BANDS IN SEMICONDUCTORS...3. BASICS OF TRANSPORTS IN SEMICONDUCTORS...6.1 MOBILITY 6. CONDUCTION BY DIFFUSION...7.3 CONTINUITY EQUATIONS...8.4 DEGENERATE SEMICONDUCTOR.9 3. OPTICAL PROCESSES IN SEMICONDUCTORS.10 3.1 EXCITONIC EFFECTS.11 4. QUIZ AND ASSIGNMENT...13 4.1 SOLUTIONS 14 5. REFERENCES. 16 Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics 1 Enrgy Bands in Smiconductor This lctur provids th basis of band gap and charg transitions btwn thm in smiconducting matrials Bohr's thory says that th lctrons in an isolatd atom hav crtain dfinit discrt amount of nrgy. If larg numbrs of atoms ar brought clos to on anothr to form a crystal, thy bgin to influnc ach othr. Th valnc lctrons ar attractd by th nuclus of th othr atoms. This maks modifications in th nrgy lvls of outr shlls lctrons and starts splitting whn brought narr to ach othr. Th nrgy gap btwn th valnc and conduction band is calld as forbiddn gap. Th basics of band gap ar as follows: Th smiconducting proprty which includs lctrical and optical proprtis dpnds on th nrgy gap Bands in k-spac that dtrmins th ffctiv mass of lctrons and hols In most of th solids, th nrgy bands around th maxima and minima in k-spac ar considrd as parabolic Th dirct and indirct gap compound smiconductor has a strong influnc in th proprtis which ar rlatd to optolctronic applications. Th lmntal smiconductors such as Si, G and group III-V compound smiconductors hav cubic symmtry and th maximum of valnc band is also locatd at k0 and th minimum of conduction may or may li on k0. InGaAs smiconductor th minimum of conduction band is locatd at k0 or clos to th bordr of first Brillouin zon. In such cas, th mission or absorption of photons taks plac dirctly by th rcombination of lctrons and hols without th chang in wav vctor. Usually th transition is indicatd by a vrtical lin in k-spac as shown in Fig. 1 (a). Th wav numbr of th lctron aftr th transition rmains sam as th initial wav vctor whr th wav numbr of th photons (k p/ħ) is ngligibl in comparison with thos of lctrons. Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics If an lctron maks a transition btwn th maximum of valnc band to th minimum of conduction band in indirct gap smiconductors such as Si or G, th lctrons has to changs its wav vctor and th nrgy diffrnc has to b givn out as thrmal nrgy. This is shown in Fig. 1(b). For this procss to tak plac a third particl calld phonon may b mittd or absorbd with th diffrnc in crystal momntum. Sinc th third particl is ndd for this transition, th probability of photon mission is lowr for indirct nrgy gap than with dirct nrgy gap smiconductors. Th optolctronic dvics such as smiconductor diod lasr or light mitting diods wr fabricatd with th hlp of dirct nrgy gap smiconductors. Fig. 1. Elctronic transition across th gap for (a) dirct smiconductor and (b) indirct smiconductor Th nrgy bands of Gallium arsnid and Silicon wr shown in Fig. rspctivly. GaAs has dirct nrgy of 1.43 V whras Silicon is indirct gap smiconductor with a gap of 1.1 V. Th ffctiv mass is givn by th quation as, 1 m * 1 d E dk (1) Th quation for thr dimnsions is givn by 1 * m ij 1 E k k i j Th ffctiv mass can b calculatd along any pr-dtrmind dirction using this way. Th dpndncy of E E(k) is sam along th thr main dirctions for GaAs and th following approximation will b valid () Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics k E( k) ( 3) * m From quation (3), th surfac of constant nrgy, E(k) constant ar sphrical for GaAs. Howvr, for silicon E E(k) is mor complicatd, by slcting th axs proprly w can writ th quation as k l kt E( k) + (4) * m * l mt whr k l and k t ar th longitudinal and transvrs componnts of k and th longitudinal and transvrsal lctron ffctiv masss ar givn by m l and m t. Th lctron ffctiv masss for silicon ar givn by m l 0.98m 0 and m t * 0.066m 0. Eqn. 4 indicats th surfac constant nrgy is of llipsoids as shown in Fig. 3. Fig.. Enrgy band of typical smiconductors: Gallium arsnid (lft) and silicon (right) Fig. 3. Constant nrgy surfacs in silicon around th minima of th conduction band Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics Basics of Transports in Smiconductors This lctur provids you th fundamntal paramtrs of charg carrir transports in smiconducting matrials Th carrir transport in smiconductors is similar to that of mtals if th lctric fild is of low magnitud. Th lctrical conductivity of smiconductors is much lowr than mtals bcaus of low carrir concntration. Th carrir concntration in th givn rgion of smiconductor can b changd just with th illumination of light with nough nrgy in ordr to crat lctron-hol pairs thn th lctrons and hols mov by diffusion du to thir concntration gradints. Th following sction xplains th various paramtrs rlatd to transport proprtis in smiconductors..1 Mobility Th lctrical conduction mchanism of smiconductors was xplaind with th hlp of smi classical modl. Undr th influnc of xtrnal lctric fild th charg carrirs xprinc scattring or collisions du to th prsnc of phonons and doping impuritis. Th scattring rsmbls th kind of frictional forc which acts in th opposit dirction to th motion. According to Nwton s scond law in on dimnsion, * m * dυ qf υ m τ ( 5) dt In Eqn.(5) (5) th assumption is mad that th frictional forc is proportional to th drift vlocity of th lctron v and τ is th rlaxation tim. Whn th fild is switchd off. t / τ [ υ ] υ ( 6) t 0 Th lctron vlocity xponntially dcays to zro with th charactristic tim qual to τ. In th stady stat, will b zro, thn from quation (5) th drift vlocity acquirs a constant valu givn by qτ υ F ( 7) * m Th proportionality constant is rlatd with th drift vlocity and lctric fild is calld as lctron mobility givn as F υ ( 8) Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics From quations (7) and (8) th lctron mobility is also givn as qτ m * (9) In th similar way th hol mobility is givn as Th currnt dnsity of th lctron is writtn as qτ h (10) m h * h J qnυ (11) whr q is th lctronic charg and n is th dnsity of charg carrirs and from quation (8) in trms of mobility J qn F ( 1) Using th ohm s law in point form th rlation btwn J and E is givn as J σ F (13) whr σ is th lctrical conductivity. Th currnt in th smiconductor is du to both by lctrons and hols, from quations (11) and (1) th xprssion for th lctrical conductivity in th smiconductor is givn as σ q n + p ) ( 14) ( h whr n and p ar lctron and hol concntration rspctivly.. Conduction by diffusion Diffusion currnts in th smiconductor aris du to th gradint in carrir concntration. Thr ar svral factors for diffusion: random thrmal motion, scattring producd by lattic imprfctions and concntration gradint. Th gnral quation of diffusion can b usd to discuss th diffusion currnt in smiconductors. Th diffusion currnts for lctrons and hols in on dimnsion is givn by dn J qd dx (15) Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics J h dp qdh (16) dx whr D and D h ar known as th diffusion cofficints for lctron and hol rspctivly. Th ngativ sign in th quation (16) indicats th lctrons movs in th ngativ x-dirction and hols mov in th positiv x-dirction du to th action of th concntration gradint that maks th currnt positiv. Th carrirs may gt scattrd whn thy mov ithr by th concntration gradint or by th lctric fild. Einstin rlations ar usd to rlat lctrons and hols as D D h h kt q kt q (17) (18).3 Continuity quations Dpnding upon th biasing, an lctron in conduction band may rcombin with th hol in th valnc band. Lt g and r b th gnration and rcombination rats rspctivly, thn th continuity quation can b writtn for th rat of chang of lctron dnsity n. Considring th on dimnsional modl th concntration gradint is a function of position and tim. Th chang of th lctron dnsity in a small rgion btwn x and x + dx is givn by n t ( n F) n g r + + D (19) x x Th last two trms ar th divrgnc from quation (1) and (15). Th xcss of carrir concntration n is dfind as ( x, t) n0 n n (0) whr th lctron concntration is n(x,t). Th carrir concntration for hols is givn by ( x, t) p0 p p ( 1) From th charg balanc or nutrality condition th xcss lctron concntration is balancd by th corrsponding hols which is givn as 0 n n n0 p p p () Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics Th minority carrirs i.. th xcss lctron concntration n in a p-typ smiconductor is writtn as ( n) ( n) ( n) t g r + F + D (3) x x Similarly, th continuity quation for th n-typ smiconductor is givn by this quation ( p) ( p) ( p) t whr p is th xcss hol concntration..4 Dgnrat Smiconductor g r + hf + Dh (4) x x Th non-dgnrat smiconductors ar thos for which th Frmi nrgy lvl E F is situatd in th gap whr th nrgy is about 3kT or mor away from band dgs. Th classical statistics can b applid for ths smiconductors, and whn th dopant concntration incrass th Frmi lvl approachs th band dgs and whn n or p xcds N c or N v, th Frmi lvl movs to conduction band in th cas of n-typ smiconductors or towards valnc band in th cas of p-typ smiconductor. For dgnrat smiconductors, th wav functions of lctrons in adjacnt impurity atoms ovrlap as in th cas of lctrons in crystals and impurity lvls form narrow impurity bands. Th impurity bands of original donor and accptor lvls ovrlaps with th conduction and valnc bands rspctivly. Th addd conduction and valnc bands ar calld band tail stats. Th rsult of band tailing lads to th phnomnon of bandgap narrowing. Bandgap narrowing plays important rol in th opration of lasr diods and in th absorption spctrum of havily dopd smiconductors. 3 Optical procsss in smiconductors This lctur provids you th fundamntal of optical procss and xcitonic ffct, in optical proprtis of smiconductor Whn th light is incidnt on a smiconductor, optical phnomna such as absorption, transmission and rflction taks plac. Th absorption spctrum of smiconductor occurs whn th nrgy of th photon which is incidnt on th smiconductor is qual Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics or largr than th nrgy gap of smiconductor and lctronic transitions taks plac from valnc bands to mpty conduction bands. Ths transitions may b ithr dirct or indirct and th absorption cofficint can b calculatd with th hlp of tim dpndnt prturbation thory. Th othr absorption procss in smiconductor corrsponds to lctronic transition btwn donor lvls and th conduction bands and from th valnc band to accptor lvls, ths absorption paks ar sn in infrard rang. In th cas of havily dopd smiconductors th optical absorption by fr carrirs plays an important rol sinc th absorption cofficint is dirctly proportional to th carrir concntration. Howvr, in ionic crystals, optical phonons can b dirctly xcitd by lctromagntic wavs bcaus of strong lctric dipol coupling btwn phonons and transvrs optical phonons. Th absorption paks lis in th infrard rang du to lattic vibrations. If th light intnsity incidnt on th surfac of solid b I o, thn th transmittd intnsity b I(z) at a distanc z from th surfac which is givn by th xprssion as I α z ( z) I 0 whr α is th absorption cofficint of th matrial which dpnds on th wavlngth of light and is givn in cm -1. Lt th pntration dpth is rprsntd as 1/ α. If th absorption cofficint is highr, smallr will b th pntration dpth. For compound smiconductors such as GaAs, α incrass vry sharply whn th nrgy of th photon is gratr than E g, du to dirct transition. In contrast for indirct smiconductors such as Si, G th incras of α ar slowr du to indirct rcombination of lctron hols with th participation of phonon, rsulting in th incras of α is not as sharp as in th cas of indirct smiconductor. 3.1 Excitonic ffcts An xciton is bound lctron-hol pair whr th lctron and hol ar hld by th Coulomb intraction. Typically it is found in smiconductor spcially in quantum dots. Whn on lctron lavs th valnc Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics band and jump up into conduction band, a hol is gnratd at valnc band. Extrnal prturbation such as photon is ndd to xcit th lctron into conduction band. In smiconductor, lctron-hols pair (or xciton) bhavs diffrntly than individual particls (lctron and hols). Hnc it nds to b tratd as a nwly ntir systm (quasi particl). Th xistnc of priodic potntial in smiconductor, lctrons intraction from nighboring lctrons and surfac traps giv ris to th xciton nrgy lvl slightly blow th band gap as shown in Fig. 4, though classically band gap is known as forbiddn rgion. Fig. 4. Exciton nrgy lvls Exciton formation plays an important rol in quantum wll structurs du to th confinmnt ffcts which nlarg th ovrlapping of th lctron and hol wav functions. Thr ar two diffrnt typs of xcitons (i). xcitons for which th wav functions of th lctrons and hol has only a slight ovrlap which is calld as Wannir Mott xcitons and ar usually sn in smiconductors. (ii) Othr typ of xciton ar obsrvd in insulators hav a small radius of th ordr of th lattic constant and ar calld Frnkl xcitons. Th lctron and hols which ar within a mdium of dilctric constant ε r ε 0, whr ε r is th high frquncy rlativ dilctric constant of th mdium. Th bound stats of th xcitons ar givn by E n RH 1 13.6V Rx, n 1,,3,... m ε n m ε n n 0 r 0 r whr R H is th Rydbrg constant for th hydrogn atom and R x is calld th xciton Rydbrg constant. Fig. 5 shows th absorption of GaAs for th photon nrgis nar to th gap. Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics 5. Excitation absorption spctrum of GaAs at 1.K Fig. Excitons ar much mor obsrvd in intrinsic smiconductors than in dopd smiconductors, whr th fr charg carrirs partially hold th Coulombic intraction btwn th lctron and th hol. In th cas of high dnsity scn, bixcitons consists of two xcitons. Bixcitons ar dtctd in bulk smiconductors, quantum wll and dots. In addition trions consists of an xciton plus ithr a hol or an lctron has bn noticd in nanostructurd III-IV quantum wlls and supr lattics. 4 Quiz And Assignmnt 1. Th particl involvd in indirct transition is. Enrgy of a particl is invrsly proportional to its mass 3. GaAs smiconductor maks transition btwn bands for mission or absorption of photon 4. Wav numbr of photons is givn as 5. For dirction transition typ matrials, th minimum of conduction lis at spac. 6. A sction of a Silicon dvic is kpt at room tmpratur (T300K). This sction is dopd with 10 15 cm -3 of accptor atoms. A stram of minority carrirs is injctd at x0 and thir distribution in th sampl is assumd to b linar dcrasing from a valu of 10 11 cm -3 at x0 to quilibrium valu at xw. whr W is 10m. Dtrmin th diffusion currnt dnsity of lctrons. (Givn th lctron mobility in th p-rgion, 1331cm /Vs.) Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics 7. Calculat th mobility and rlaxation tim of th lctron in silvr. Givn: th rsistivity 1.54x10-8 Ωm at room tmpratur. Elctron dnsity5.8x10 8 /m 3. 8. Th mobilitis of fr lctrons and hols in pur G ar 0.38 and 0.18 m /Vs. For pur Si it is 0.13 and 0.05 m /V.s. Find th valus of intrinsic conductivity for both th matrials. Assum n i.5 X 10 19 m -3 for G and n i 1.5 X 10 16 m -3 for Si at room tmpratur. 9. Find th intrinsic carrir concntration of G, if its intrinsic rsistivity at 300 K is 0.47 Ωm. It is givn that th lctronic charg is 1.6 X 10-19 C and th lctron and hol mobilitis at 300K ar 0.39 and 0.19 m /V.s. 10. Find th rsistanc of an intrinsic G rod 1 cm long, 1 mm wid and 1mm thick at 300 K. For G, n i.5 X 10 19 /m 3, μ n 0.39 m V -1 s -1 and μ p 0.19 m V -1 s -1. 11. In intrinsic Ga As, th lctron and hol mobilitis ar 0.85 m/vsand 0.04 m/vsrspctivly. Find th conductivity (Givn n i 1.98 X 10 1 m -3 ). 1. Intr band absorption in th smiconductor dpnds upon a) band structur b) Vibrational structur c) Elctronic structur d) absorption 13. Excitons ar obsrvd in smiconductors. 14. Which smiconductors will mit light nrgy? Why? 4.1 Solutions 1. Phonon. Effctiv 3. Dirct 4. k p/ħ 5. k0 6. Th thrmal quilibrium dnsitis of th majority and minority carrirs ar calculatd to b 15 3 5 3 po 10 cm and no 1x10 cm 1331cm / Vs Th lctron diffusion currnt dnsity is givn by, J n qdn dn dx Th diffusion constant is obtaind from Einstin s rlationship at T300K: D KT n q 0.059x1331 34.48cm / s Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics Th slop of th lctron dnsity 4, so that 1.6x10 J n 0.55mA cm x34.48x( 0 14 ) dn dx is -(10 11-1x10 5 )/(10x10-4 )-10 14 cm -14 cm - Th lctron mov in th positiv x-dirction, causing a currnt in th ngativ x- dirction. 7. Rlaxation tim; Mobility; m τ ρn 9.1x10 8 1.54x10 x5.8x10 14 4.8x10 scond 31 8 x(1.6x10 ) τ 1 m ρn 1 8 8 1.54x10 x1.6x10 x5.8x10 8. Intrinsic 3 7x10 m volt. s conductivity for G σ qn ( + ) i i n p 1.16 10.5 10 19 ( 0.38 + 0.18).4Ω m 9. Intrinsic conductivity for Si ( + ) σ i qni n p 16 1.16 10 1.5 10 1 1 σ i.13ω m ρ 0.47 σ i qni ( n + p ).13 1.6 10 ni 0.39 + 0.19.13 0.93 10 n n i.13 0.93 ( 10 ) i 4 ( 0.13+ 0.05) 4.3 10 Ω m ( ).3 10 19 m 3 Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics 10. σ qn i 1.6 10 ρl R A 1 σa.3 i ( + ) n p.5 10 1 10 19 3 3 ( 1 10 1 10 ) ( 0.39 + 0.19) 4310Ω.3Ω m σ qn ( + ) i i n p 11. 1.6 10 1.98 10 1. a) band structur 13. Intrinsic smiconductors 1 7 ( 0.85 + 0.04).8 10 Ω m 14. A dirct nrgy gap smiconductor mits light nrgy du to dirct rcombination of lctron and hols whn forward biasd. 5 Rfrncs [1] Jan G. Korvink, Andras Grinr, Smiconductors for Micro-and Nanotchnology- An introduction for Enginrs, Wily-VCH, 00 [] J.M. Martinz- Duart, R.J.Martin-Palma, F. Agullo-Ruda, Nanotchnology for Microlctronics and Optolctronics, Elsvir, 006. [3] David K. Frry, Stphn M. Goodnick and Jonathan Bird, Transport in Nanostructurs, Cambridg Univrsity Prss, 009. [4] B Kramr, T Brands W Hauslr, K Jaurgui, W Pfaff and D. Winmann, Intractions and transport in nanostructurs, Smicond. Sci. Tchnol. 9 (1994) 1871-1878. Joint Initiativ of IITs and IISc Fundd by MHRD
NPTEL Elctrical & Elctronics Enginring Smiconductor Nanodvics Joint Initiativ of IITs and IISc Fundd by MHRD