Electrons in Solids. Bosons. Stimulated Emission. Probability: Boltzmann Factor. He-Ne Laser. Laser Construction. Today:

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1 Today: lctrons in Solids. Rviw Chaptrs 40, 4 Frmions and bosons Boltzmann factor nrgy bands in solids Frmi-Dirac distribution function. Som solid-stat applications Bosons Lasrs work bcaus photons ar bosons. Atom mits a nw photon into xactly th sam quantum stat as th original photon. This kps happning until thr is a strong bam of many photons all in th sam quantum stat. Produc intns, strongly collimatd lasr bam. Probability: Boltzmann Factor Givn any systm in thrmal quilibrium at tmpratur T. Th rlativ probability of any two stats is ( )/ kt kt / > kt thn P ) << ) So if ( For xampl in a gas th numbr of atoms in an xcitd stat is smallr than th numbr in th ground stat. Stimulatd mission hf Without pumping, at tmpratur T: n n < Nd pumping of a mtastabl lvl to produc population invrsion (n > n 0 ). 0 H-N Lasr Ky to any lasr is finding a mtastabl stat which can b pumpd (populatd) somhow. Gt population invrsion (chat th Boltzmann factor). Lasr Construction

2 Frmions Q.40- Pauli xclusion principl: No two lctrons in th sam stat. That is, on lctron for ach st of quantum numbrs: (n, l, m l, m s ). This givs th priodic tabl! Carbon ground stat: s s p Which of th following dscribs th ground stat of th nitrogn atom (Z7)? () s (4) s s s s p () s (5) s s s s p s p d () s s (6) s p p 5 Q.40- () s Which of th following dscribs th ground stat of th nitrogn atom (Z7)? (4) s Carbon ground stat: s s p Bgin with carbon and plac on mor lctron in th lowst availabl subshll, which is p. Not that p orbitals can hold 6 lctrons. s s s p () s (5) s s s s p s p d () s s (6) s p p 5 Ch.40 Sctions 5, 6, 0 (Rad ths sctions but don t worry about th dtails.) 40-5: Strn-Grlach xprimnt History Showd quantization of angular momntum 40-6: Magntic Rsonanc Proton spins ar flippd by oscillating magntic fild. nrgy of flip is diffrnt for diffrnt molculs: NMR. MRI givs maps of diffrnt molculs insid th body. 40-0: X-rays (Rad this sction but don t worry about th dtails.) lctron bam in an X ray tub collids with a havy mtal targt,.g. Mo. Rsulting x-ray photon nrgis form continuous and charactristic spctra. + A + A A A + hf X-rays (continud) (Rad this sction but don t worry about th dtails.) Charactristic X rays ar from collisions in which a vacancy is cratd in an innr shll of a havy atom. For xampl in cobalt (Z7) if thr is a vacancy in th n shll (K shll), thn an lctron from th n shll (L shll) can jump down to fill th vacancy. Th nrgy of th mittd photon in that cas is about 7 kv (wavlngth about 0.8 nm). Th jump from n to n is calld K α ; from n to n is K β tc. Transitions down to an n vacancy ar calld L α L β tc.

3 Ch. 4: lctrons in a mtal Atomic cor stats ar full. Conduction band is half full. Frmi nrgy F shows highst filld stat. asy for lctron to jump to slightly highr nrgy stat, and thn mov through solid. Good conductor. Frmi-Dirac Distribution Function At T0, stats ar filld just up to Frmi nrgy. At T>0, som lctrons hav gottn a thrmal boost. ( F ) + Boltzmann s constant: k V / K xampl: (Basd on Problm 4-5) Considr a stat in th conduction band of a mtal, with nrgy 0.05 V abov th Frmi nrgy. What is th probability that this stat is occupid at room tmpratur? kt (8.6 0 V / K )(00 K ).058 V V 40 Solution :.05 / Solution : Q.4- Considr a mtal at tmpratur T 500 K. Find th probability for an lctron to occupy a stat in th conduction band with nrgy 0.0 V abov th Frmi nrgy: F 0.0 V...5 %. 7 %. % 4. 8 % 5. 5 % Q.4- Considr a mtal at tmpratur T 500 K. Find th probability for an lctron to occupy a stat in th conduction band with nrgy 0.0 V abov th Frmi nrgy: F 0.0 V. Solution: us th Frmi-Dirac distribution function:..5 %. 7 %. % 4. 8 % 5. 5 % kt (8.6 0 V / K )(500 K ) 0. 5 V ( F )/ kt % 0.0 / Dopd smiconductors p-n junctions LDs Photoclls Additional topics

4 Dopd smiconductors n-typ: add atom with on mor valnc lctron..g. add P (Z5) to Si (Z4) xtra s li in donor lvls. p-typ: add atom with on lss valnc lctron..g. add Al (Z) to Si (Z4) xtra mpty accptor lvls. p-n Junction Junction btwn p-typ and n- typ matrials. Diffusion across th boundary forms a dpltion layr, and potntial diffrnc V 0. Usd to form a diod. Forward voltag allows currnt to flow LtoR by rducing V 0 and d 0. LD Currnt passing through p-n junction; lctrons and hols mt in th dpltion layr; many rcombin. This rlass an nrgy approximatly qual to th band gap. Light-mitting diod. Photo-diod Rvrs of LD. Incoming photon crats an lctronhol pair in th dpltion layr of a p-n junction. This allows flow of currnt; light nrgy is convrtd into lctrical nrgy. mission of photons! hf hf Photon from sun xcits lctron from valnc band to conduction band. Silicon Photocll Tandm solar cll. V. 8V contacts UT/CPH Two thin-film smiconductor junctions. On band gap tund to rd light, on to blu. TCO

5 lctrons in Solids Today: Back to Solids: Chaptr 4. Rviw Frmions and bosons Boltzmann factor nrgy bands in solids Frmi-Dirac distribution function. Som solid-stat applications Dnsity of stats and lctrons A crucial quantity for lctronic proprtis of a solid is th dnsity of stats. W will not try to larn how to calculat it. Its maning is straightworward: N( d Numbr of quantum stats with nrgy in th rang to +d. Multiply by probability of finding an lctron in a stat with nrgy to gt numbr of lctrons with nrgy in rang d. If s wr not frmions, this would b But bcaus s ar frmions, thy oby th Frmi-Dirac probability distribution function instad. Charg carrirs in smiconductor lctrons in conduction band conduction ) Multiply by Avagadro s numbr to gt rough stimat of charg carrirs pr mol: F Hols in valnc band For xampl ( g F )/ kt (.5)/ g V 0.5 V F Diamond lattic Insulators All bands ithr full or mpty. Band gap is larg. Diamond: 5. 5 V xcllnt insulator Band gaps Why is diamond an insulator? Why is 5.5 V a larg band gap? lctrons in filld bands cannot mov bcaus of Pauli xclusion principl. Must mov up to conduction band. Boltzmann factor dosn t lt this happn: g >> kt Room tmpratur kt is about.05 V. 5.5 / Also 5.5V is largr than nrgy of photon of visibl light. So diamond is transparnt, photons ar not absorbd. Shining light on a diamond dosn t mak it a conductor. 96