Semiconductor crystals

Transcription

1 Dpt of Phys M.C. Chang Smiconductor crystals Ovrviw Th concpt of hol and ffctiv mass Intrinsic smiconductor Dopd smiconductor

2 Familis of smiconductors Elmnts Compounds Bonding bcoms mor ionic

3 Basic proprtis (at 300K) G Si GaAs GaN nrgy gap (V) 0.67 (i) 1.11 (i) 1.43 (d) 3.39 (d) lattic typ Diamond Diamond Zincblnd Wurtzit Smiconductor is insulator at 0 K, but bcaus of its smallr nrgy gap (diamond = 5.4 V), lctrons can b thrmally xcitd to th conduction band (and transport currnt) asily. 2 ovrlapping hcp lattics Si-basd dvic can ndur highr working tmpratur than G-basd (75 o C) dvic ( Si has a largr band gap). For som intrsting history of smiconductor industry, s 矽晶之火, by M.Riordan and L.Hoddson.

4 Dirct band gap (GaAs, GaN ) ω=e g Indirct band gap (Si, G ) ω=e g + Ω Dirct band gap smiconductor can mit light fficintly (1μm=1.24 V)

5 A filld band dos not carry currnt (Pirls, 1929) Elctric currnt dnsity j = ( v) = V 3 1 dk 1 filld k (2 π ) 3 ε n( k ) k For crystals with invrsion symmtry, ε n (k)= ε n (-k) lctrons with momnta ħk and -ħk hav opposit vlocitis no nt currnt in quilibrium A narly-filld band j = v V filld k = v V k 1st BZ =+ v V unfilld k unfilld k v unoccupid stats bhav as + charg carrirs

6 Th concpt of hol (Pirls, 1929) If an lctron of wavvctor k is missing, thn k= -k. Altrnativly spaking, a hol with wavvctor k h is producd (and k h = -k ). Th lowr in nrgy th missing lctron lis, th highr th nrgy of th whol systm. If th nrgy of a filld valnc band is st to zro, thn ε h E = ε ( k) ε ( k ) on band 0 filld band = ε ( k ) ε ( k ) 0 h 0 Th missing lctron k

7 Effctiv mass Nar th bottom of a conduction band, th nrgy disprsion is approximatly parabolic, ε( k) = ε + kk + O( k ) ε + p p 2 1 ε ( k ) i j 0 i j 2 i, j ki kj 2 i, j m* ij important Effctiv mass matrix m k k ε ( k ) * 2 ij i j Th lctron nar band bottom is lik a fr lctron (with m*). For a sphrical FS, m * ij =m* δ ij, only on m* is nough. In gnral, lctron in a flattr band has a largr m*. Ngativ ffctiv mass: If ε(k) is (.g. top of valnc band), thn m * <0 a i or d d dω d = dt dt dk dk dk k m k gi = = 2 ω υ * = 1 ( ) * ma= k = E i j i j j ij j i lctron (-) with ngativ m* = hol (+) with positiv m*.

8 For llipsoidal FS, thr can b at most thr diffrnt m * s Eg. th FS of Si is mad of six idntical llipsoidal pockts. k ε( k ) = εg m 2m 2m kx y kz L T T T L For Si, ε g = 1.1 V, m L = 0.9 m, m T = 0.2 m (It s mor difficult for th lctron to mov along th L dirction bcaus th band is flattr along that dirction.)

9 Mor band structurs and Frmi surfacs Common faturs Som usful paramtrs m L/ m T m HH/ m LH Δ Si 0.91/ / V GaAs / V

10 Ovrviw Th concpt of hol and ffctiv mass Intrinsic smiconductor (no doping) Dopd smiconductor

11 DOS and carrir dnsity DOS for fr lctron (ch 6) D( ε ) V 2π 2m = 2 2 3/2 ε DOS (pr V) for smicond g h ( ε ) Frmi distribution 1 f ( ε ) = ( ε μ) kt B + 1 ε g ( μ = if m = m 2 carrir dnsity n = f( ε ) g( ε) dε 0 1 2π 2m h = 2 2 3/2 h ) ε Top of valnc band is st as ε=0 3/2 1 2m g( ε ) = ε ε 2 2 g 2π

12 Considr th non-dgnrat cas: For lctrons,ε-μ >> k B T f ε μ ( )/ kt ( ε ) 1 For hols, μ- ε>> k B T 1 fh() ε 1 f( ε) = ( μ ε ) k B T + 1 ( ε μ)/ kt ( ε) 1 f h lctron dnsity in conduction band: hol dnsity in valnc band: n = f( ε) g ( ε) dε i ε g 2 2 3/2 3/2 1 2m = 2 2 N c g ( ) 1/2 ( μ ε)/ kt B ε εg dε ( μ ε )/ kt g 1/2 ( ε εg)/ kt B ( ε εg ) d( ε εg ) 0 ( μ ε )/ k T g 1 2m 2π = 2π B ε = B π /2 0 p = [1 f( ε)] g ( ε) dε i 1 2m 2π 1 2π h 2 2 2m h = 2 2 = N N V V μ / k T B h 3/2 0 3/2 mkt h = 2 2 B2 π ( ε ) 1/2 μ / kt B 1/2 ε h 0 3/2 ( ε μ)/ kt B ε / kt h dε B dε h N c mkt = 2 2 B2 π 3/2 np g = N N ε i i C V / k T B Valid vn with doping

13 Carrir dnsity and nrgy gap In intrinsic smiconductor, n i = p i = ( N C N V 1/ 2 ) E Th dnsity of intrinsic carrirs dpnds only on th nrgy gap. G / 2 k B T Chmical potntial (2μ E G ) / kbt = N / V 1 1 μ = EG + kbtln( NV / NC) = EG + kbtln( mh/ m) 2 4 Th 2nd trm is vry small bcaus k B T<<E G N C (For Si, th atom dnsity is cm -3.)

14 Extrinsic carrirs by doping

15 Impurity lvl: Bohr atom modl Th ionizd impurity atom has a hydrogn-lik potntial, (m m, ε 0 ε) E d 4 m 13.6 m = = V ε ε m Dilctric constant of Si = 11.7 (G=15.8, GaAs=13.13), Effctiv mass for Si = 0.2 m. Thrfor, th donor ionization nrgy = 20 mv. Bohr radius of th donor lctron: a d 2 ε 0.53ε = A 2 = m m/ m For Si, it's about 50 A (justifis th us of a constant ε).

16 Impurity lvls in Si Enrgy-band point of viw Conduction band Valnc band

17 Light mitting diod

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19 Solid stat lighting From S. Nakamura s slid

20 Th invntion of blu-light LED GaN can mit blu light bcaus of its larg band gap (3.4 V). First blu-light LED: Shuji Nakamura First blu-light lasr: Shuji Nakamura 中村修二 Bfor 2006 Aftr S a nic Intrviw with Nakamura : Scintific Amrican, July, 2000

21 UV Watr Purifir Krosn lighting and firwood ar usd by 1/3 of th world; thy caus countlss firs and ar vry infficint. LED-backlit LCD TV Blad runnr (1982)