Solid State Theory Physics 545 Band Theory III

Transcription

1 Solid Stat Thory Physics 545 Band Thory III

2 ach atomic orbital lads to a band of allowd stats in th solid Band of allowd stats Gap: no allowd stats Band of allowd stats Gap: no allowd stats Band of allowd stats

3 Indpndnt Bloch stats F1 Solution of th tight binding modl is priodic in. Apparntly hav an infinit numbr of -stats for ach allowd nrgy stat. In fact th diffrnt -stats all quivalnt. () α= 10 γ = 1 Bloch stats ir.r ψ ( r R) ) ψ ( r ) Lt = + G whr is in th first Brillouin zon and G is a rciprocal lattic vctor. ψ( r π/a 0 π/a R) i.r ig.r ψ( r) But G.R = 2πn, n-intgr. Dfinition of th rciprocal lattic. So ig.r i.r = 1 and ψ( r + R) ψ( r) i is xactly quivalnt to. i.r.r [111] dirction Th only indpndnt valus of ar thos in th first Brillouin zon.

4 Rducd Brillouin zon schm Th only indpndnt valus of ar thos in th first Brillouin zon. Discard for > π/a Rsults of tight binding calculation 2π/a -2π/a Displac into 1 st B. Z. Rsults of narly fr lctron calculation Rducd Brillouin zon schm

5 xtndd, rducd and priodic Brillouin i zon schms Priodic Zon Rducd d Zon xtndd d Zon All allowd stats corrspond to -vctors in th first Brillouin Zon. Can draw () in 3 diffrnt ways

6 Th numbr of stats in a band Indpndnt -stats in th first Brillouin zon, i.. x < π/a tc. 2πn x Finit crystal: only discrt -stats allowd x = ±, n x = 0,1,2,...,,... tc. L Monatomic simpl cubic crystal, lattic constant a, and volum V. On allowd stat pr volum (2π) 3 /V in -spac. Volum of first BZ is (2π/a) 3 Total numbr of allowd -stats in a band is thrfor 3 2π ( 2π ) V N 3 a V = a = Prcisly N allowd -stats i.. 2N lctron stats (Pauli) pr band This rsult is tru for any lattic: ach primitiv unit cll contributs xactly on -stat to ach band.

7 Mtals and insulators In full band containing 2N lctrons all stats within th first B. Z. ar occupid. Th sum of all th -vctors in th band = 0. A partially filld band can carry currnt, a filld band cannot Insulators hav an vn intgr numbr of lctrons pr primitiv unit cll. With an vn numbr of lctrons pr unit cll can still hav mtallic bhaviour du to ban ovrlap. Ovrlap in nrgy nd not occur in th sam dirction F 0 π a Mtal du to ovrlapping bands

8 F 0 π 0 π a a 0 π a F mpty Band nrgy Gap Full Band Partially Filld Band nrgy Gap Part Filld Band Part Filld Band Full Band INSULATOR MTAL MTAL or SMICONDUCTOR or SMI-MTAL

9 Bands in 3D Grmanium In 3D th band structur is much mor complicatd than in 1D bcaus crystals do not hav sphrical symmtry. Figur rmovd to rduc fil siz Th form of () is dpndnt upon th dirction as wll as th magnitud of.

10 Chmical bonds and lctron bands. a) Numbr of lctrons in any band is finit bcaus th dnsity of stats is finit. ρ ( ) = 3 D ) = 8π 2m( ) 3 h 3/ 2 1/ 2 N = top 8π 2 m h ( ) 3 bottom 3/ 2 1/ 2 d b) Bands ar formd from molcular orbitals.

11 Filling of nrgy Bands An important proprty of a full band is that it is UNABL to carry a nt currnt sinc for ach stat in th band w can idntify a corrsponding stat with qual and OPPOSIT momntum that is filld by an lctron. To driv a nt currnt through th crystal it is ncssary to induc an IMBALANC in th filling of momntum stats For an nrgy band that is filld compltly howvr this rquirs that w xcit lctrons ACROSS th forbiddn gap. Situation in which th lowst nrgy band is filld compltly with lctrons NRGY GAP th only way in which a nt currnt can flow is to xcit lctrons across th nrgy gap if th nrgy gap is larg howvr xcitation ti cannot b achivd and so no nt currnt is allowd to flow π/a π/a

12 By th sam argumnts if th nrgy band is PARTIALLY filld thn it should b vry ASY to gnrat a nt currnt flow in th crystal In this situation th forbiddn gap lis FAR away from th highst filld lctron stats and so it is asy to us an lctric fild to gnrat an imbalanc in th filling of momntum stats A small applid voltag will thrfor gnrat a LARG currnt as w discussd prviously for fr lctrons NRGY GAP NRGY GAP π/a π/a π/a π/a NO APPLID LCTRIC FILD SMALL LCTRIC FILD APPLID

13 lctronic band thory prsnts us a natural schm for CLASSIFYING diffrnt typs of matrials MTALS should b matrials whos upprmost nrgy band is only PARTIALLY filld with lctrons. This xplains why ths matrials ar GOOD conductors of lctricity W xpct that insulators on th othr hand should b matrials whos nrgy bands ar ithr COMPLTLY full or mpty so that an nrgy gap BLOCKS currnt flow in ths matrials FORBIDDN GAP FORBIDDN GAP FILLD STATS FILLD STATS MTAL INSULATOR

14 Band structur of mtals monovalnt mtals multivalnt mtals, smimtals

15 What typs of lmnts produc partial or complt filling of nrgy bands? Th GROUP I lmnts should b good MTALS sinc ths lmnts hav only ON valnc lctron, whras coordination numbr is If w hav a crystal composd of N atoms thr will thrfor b N valnc lctrons which will HALF-FILL a singl nrgy band Th GROUP IV lmnts should b INSULATORS sinc ths lmnts hav FOUR valnc lctrons and so in an N-atom crystal thr will 4N valnc lctrons that FILL two nrgy bands compltly FILLD STATS FILLD STATS GROUP I FILLD STATS GROUP IV FILLING OF NRGY LVLS BY TH VALNC LCTRONS OF GROUP I & IV LMNTS

16 Smiconductors In crtain matrials nown as SMICONDUCTORS howvr th nrgy gap that sparats th highst filld band in th ground stat from th lowst mpty band is SMALL * Such matrials ar INSULATORS at zro tmpratur sinc thir ground stat is on in which th nrgy bands ar ithr compltly full or mpty * Sinc th forbiddn gap is small howvr lctrons can b XCITD across it at highr tmpraturs to PARTIALLY fill th nxt band Th matrial will no longr b an insulator at this tmpratur but will CONDUCT lctricity FORBIDDN GAP FILLD STATS FILLD STATS FILLD STATS FILLD STATS INSULATOR T = 0 INSULATOR T > 0 SMICONDUCTOR T = 0 SMICONDUCTOR T > 0

17 Som gnral COMMNTS on smiconductors Th nrgy band that holds th valnc lctrons in th ground stat is nown as th VALNC BAND. It is usually formd by Bonding Orbitals. Th lowst mpty band is nown as th CONDUCTION BAND. It is usually formd by antibonding orbitals. Th nrgy gap that sparats ths bands is usually dnotd as g * Room tmpratur t smiconductors ar gnrally matrials in which h g is a FW V ( 3 V) This should b compard to a thrmal nrgy of approximatly 40 mv that is availabl to lctrons at room tmpratur (300 K) SMICONDUCTOR g (V) Si G InSb InAs InP GaP GaAs GaSb AlSb 0 K 300 K CONDUCTION BAND VALNC BAND g SMICONDUCTOR T = 0

18 Concpt of a hol At highr tmpraturs lctrons in smiconductors may b xcitd into th conduction band whr thy ar abl carry an lctrical currnt * ach lctron lavs bhind an MPTY stat in th valnc band and to account for currnt flow in smiconductors w must ALSO considr th rol of ths HOL stats * If th valnc band is COMPLTLY filld, thn th total crystal momntum of this band is qual to ZRO sinc for any occupid -stat w can idntify if an corrsponding filld stat with OPPOSIT momntum 2 NRGY GAP 1 Th total crystal momntum in a filld nrgy band is xactly qual to zro to illustrat this considr th total momntum du to occupation of stats 1 & 2 stat 1 corrsponds to an lctron with positiv momntum whil stat 2 corrsponds to on with qual and opposit momntum th nt crystal momntum of lctrons occupying stats 1 & 2 is zro and this pairing can b rpatd for all othr stats in th band π/a π/a

19 Whn th valnc band is compltly filld with lctrons w can writ i = 0 (17) * if w xcit AN lctron from th stat with wavnumbr in th valnc band into th conduction band quation (17) for th valnc band may now b RWRITTN as i = (18) i * Th mpty stat in th valnc band may thrfor b viwd as a HOL which has OPPOSIT momntum to th lctron that was xcitd out of that stat = (19) h

20 Sinc th hol corrsponds to a missing lctron its nrgy may b writtn as h( h ) = ( ) = ( ) (20) * quation 10.4 shows that lctrons and hols hav OPPOSIT nrgy scals sinc moving DOWNWARD in th valnc band implis INCRASING hol nrgy NRGY GAP HOL 2 Moving downwards in th valnc band corrsponds to incrasingi hol nrgy in th pictur shown hr hol 1 thrfor has mor nrgy than hol 2 this is not too difficult to undrstand if w thin of th total nrgy of th lctrons lft in th band HOL 1 sinc hol 1 corrsponds to a missing lctron from a lowr nrgy lctron stat than hol 2 th total nrgy of lctrons in th band is highr for hol 1 than for hol 2 π/a π/a gy f g f

21 Whil th nrgy scals ar oppositly dirctd for lctrons and hols w can show that th hol VLOCITY is th SAM as that of th stat from which th lctron is missing * To do this w simply rplac h by and h ( h ) by ( ) ) ( 1 h h h h d d v = v d d d d = = = ) ( 1 )) ( ( 1 * Using th sam approach w can also show that th ffctiv mass of th hol is a NGATIV quantity ))) ( ( ( ) ( 1 ) ( * h h h h h d d d d m = = * 2 1 )) ( ( 1 m d d = =

22 It is important to apprciat that w do NOT actually hav positivly-chargd carrirs in th smiconductor but that th hols hl simply bhav bh AS IF thy had positiv i charg * Th basic ida is that ACH of th OCCUPID lctron stats in th valnc band rsponds to xtrnally-applid filds as w would xpct for a ngativly-chargd carrir. r Th NT rspons of th band howvr LOOKS li th rspons of a singl particl with a POSITIV charg! Whn w discuss conduction band proprtis of smiconductors or insulators w rfr to lctrons, but whn w discuss th valnc proprtis, w rfr to hols. This is bcaus in th valnc band only th missing lctrons or hols lad to currnt flow. LCTRIC FILD NRGY GAP NRGY GAP π/a π/a AN LCTRIC FILD APPLID IN TH +x DIRCTION ACCLRATS LCTRONS IN TH x DIRCTION AND SO TH HOL STAT APPARS TO B ACCLRATD IN TH +x DIRCTION π/a π/a

23 Gnral commnts Thrmal vibrations or nrgy can b usd to crat a hol by xciting an lctron from th valnc band to th conduction band In anintrinsic i i (undopd) d) smiconductor, th numbr of hols in th valnc band quals th lctrons in th conduction band Hols can mov about th valnc band and rcombin with lctrons in th conduction band (to disappar)

24 tatistics of lctrons and hols in smiconductors On can us Boltzmann statistics for lctrons and hols if thir nrgy is small in comparison with F 1 f ( ) = = xp F, if > 3. T T 5 F xp(( F ) / B ) + 1 B lctron xcitation on smiconductors B T n = N c * πmbt, 2 h ( ) / 2 F c BT Nc = 2 3/ 2 p = N v * ( ) / 2 π m T v F BT h B, Nv = 2 2 h 3/ 2 For pur smiconductor F g/2 n=p C T 2/3 xp(- g /2T) C is a matrial constant np n p C CT 2/3 xp(- g /T)=n i n i is concntration ti of intrinsic i i carrirs

25 Impuritis in smiconductors: Doping n-typ Band Diagram p-typ Band Diagram Impuritis crat lvls in th forbiddn gap of smiconductors.

26 Impuritis in smiconductors: concntration of carrirs Ionizd Dopant Concntration: n= 1 1 (N N ) xp p= d 2 (N N ) xp a C d V a 2 2T 2 2T N d and N a is th concntration of donors and accptors rspctivly d and a arinv V

27 Commnts on lctron and hol concntration In pur smiconductor concntration of hols and of lctrons ar qual. Concntration of hols and of lctrons dpnds on g.=> Insulator ar smiconductor with vry larg g Mobility in smiconductors has xactly th sam maning as in mtals. Total 2conductivity 2 n τ p τ can b p xprssd as: σ = n μ + pμp = + * * m m p C t ti f l t d h l i Concntration of lctrons and hols in smiconductors can b tailord by introducing impuritis (doping) In this cas concntration of lctrons and

28 Intrsting to now Band gap incrass with incrasing th strngth of th chmical bonds in th smiconductor. xampl: Diamond > Silicon > Grmanium In many smiconductor alloys th band gap changs almost linarly l with composition. i xampl: 1) GaAs AlAs 2) HgT CdT

29 Chmistry of doping Doping activity of impuritis (dopants) dpnd on th charg stat of th impurity rlativ to th charg in unprturbd lattic. Doping activity dpnds on sit, at which dopant is incorporatd. Som dopants may b both donors and accptors (amphotric). Impuritis may b hav charg mor than on. Som dopants ar natural dfcts that ar inhrnt to th matrial.

30 Chmistry of doping: xampls. 1. lmntary smiconductors: C (diamond), Si, G C (diamond) Si G Band gap K conductivity < (Ω cm) (Ω cm) (Ω cm) -1 Si Si Si Si Si As 1. Activity of substitutionally incorporatd dopants is dfind by th diffrnc in numbr of valnc lctrons. 2. Intrstitial dopants (mostly mtals) ar donors. B h + Li 3. Som intrstitial dopants (Au, Ni) may b amphotric. 4. Influnc of natural dfcts is ngligibl.

31 Chmistry of doping: xampls. 2. III-V smiconductors. Structur: cubic Band gaps K (sphalrit) or hxagonal AlN 6.2 w AlAs w GaN 3.44 w GaP 2.27 s GaAs 1.42 s InP 1.34 s InAs InSb s s Sphalrit Wurtzit 1. Most of th dfcts ar substitutional. i 2. Th substitution sit is largly dfind by ionic radii. xampl: B substituts Ga, T substitut As in GaAs. Homwor: Dtrmin th doping action of B and T in GaAs. Intrsting to now: both structurs ar polar (hav a spcific dirctio

32 ZnS ZnT CdS CdT Chmistry of doping: xampls. 3. II-VI smiconductors w w w s Structur: cubic (sphalrit) or hxagonal 1. Most of th dfcts ar substitutional. 2. Th substitution sit is largly dfind by ionic radii. xampl: In substituts Cd, As substitut T in CdT. 3. lctrical proprtis of II-VI crystals ar stronly affctd by natural dfcts. xampl of natural dfcts: 1) Cadmium vacancis in CdT may wor as a singl or doubl chargd accptor. Thrfor, undopd CdT is always p- typ. In 2) smiconductors Anion vacancis with ar a vry donors larg in band all II-VI gap smiconductors. accptor-li dfcts ar not thrmodynamically stabl if th nrgy of vacancy formation, V, is smallr than th rcombination Conduction nrgy, band. r Conduction band Donor lvl Donor lvl Rcombination Valnc band nrgy, r Accptor lvl Valnc band