λ = 2L n Electronic structure of metals = 3 = 2a Free electron model Many metals have an unpaired s-electron that is largely free

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1 5.6 4 Lctur #4-6 pag Elctronic structur of mtals r lctron modl Many mtals av an unpaird s-lctron tat is largly fr Simplst modl: Particl in a box! or a cubic box of lngt L, ψ ( xyz) 8 xπ ny L L L n x π y nzπ z,, sin sinsin L π ( n, n, n ) ( n + n + n ) n, n, n x y z,,, ml x y z x y z PIB wavfunctions ar sinusoidal, wit intgral numbr of wavlngts in t box λ 4 a ( π x L) ( ) ψ 4 Bsin 4 nods λ a ( π x L) ( ) ψ Bsin nods λ L n n λ a λ a x ( π x L) ( ) ψ Bsin nod Rwrit wavfunction in trms of wavvctor componnts k π π n k π x n k x y y z π λ L L L ψ x 8 L ( x, yz, ) sin( kx) sin( ky) sin( kz) n z x y z ( π x L) ( ) ψ Bsin nods Wavvctor magnitud givn by kx + ky + k z k ( x + y + z ) L π L n n n # nods n -

2 5.6 4 Lctur #4-6 pag So nrgy is Or can writ k m ( ka) ma to xprss as function of unitlss quantity (ka) a L ( ka) π ( nx + ny + nz ) sinc a/l is xtrmly small, t nrgis ar ssntially continuous. Quadratic dpndnc of vs. k drivs from PIB nrgis ~ n. vry diffrnt from ponon disprsion rlation ~ k Wavvctor can go to valus largr tan π/a, sinc t lctrons go vrywr in t crystal, not just t lattic points. k Dnsity of stats k z k y k x

3 5.6 4 Lctur #4-6 pag How many stats ar tr in a rang dk about som valu k? Imagin radius k and a tin sll of ticknss dk. Volum in t sll is ind total numbr of stats witin tis volum 4π kdk π k x L n x so if n x incrass by, k x incrass by π/l. D volum lmnt is (π/l) π /V volum in k-spac occupid by on stat (V volum of crystal) # stats in t rang from k to k + dk is 4 kdk L dn π k dk 8 π L π # stats pr unit wavvctor ( ) ( 8 to includ only positiv octant) Dnsity of stats: w nd # stats pr unit nrgy k m m k dk m d dn L L m m k dk π π d V m dn 4π d Dnsity of stats - # stats wit nrgy btwn and d rmi nrgy of a mtal Eac wavvctor stat accommodats two lctrons, spin up and down. Maximum wavvctor k max k rmi wavvctor k Maximum nrgy rmi nrgy m N total numbr of lctrons V k V N N ( ) k dk k k π π π rmi wavvctor V lctrons pr orbital Unfilld lvls ~ -8 V illd lvls

4 5.6 4 Lctur #4-6 pag 4 k N ( ) π rmi nrgy m m V N /V numbr dnsity of carrir lctrons Dfin rmi tmpratur : k k Boltzmann s constant, not wavvctor Dfin rmi surfac: surfac of a spr in k-spac, wit radius k MEAL K Na Li Au Ag Cu N /V (cm - ).4x.5x 4.6x 5.9x 5.88x 8.55x E (V) (K) 4,4 6,4 54,5 64, 64, 8,6 k (A - ) ypically ~ -8 V, : ~,-8, K. mpratur dpndnc of lctron distribution K., i.. at K,.5 V.. So only lctrons vry clos to t rmi lvl migt b promotd to igr lvls abov it. rmi-dirac population distribution fd ( ) n i,d ( ) ( i µ ) ( ) + + Enrgy lvls ar vry narly continuous, so w don t nd subscript i. f D () ~ K K K Only a vry small fraction of t lctrons ar xcitd abov at ordinary.

5 5.6 4 Lctur #4-6 pag 5 Elctronic at capacity of a mtal Now tat w know t nrgy lvls, t dnsitis of stats, and t - dpndnt populations, w can calculat t lctronic contribution to t at capacity and otr trmodynamic quantitis. E ( ) ( ) ( ) f ( ) l Cl ifd i i fd i D i NV, i + i i intgral ovr t numbr of stats sum of nrgy x avrag bcaus t total intgratd population of ac lvl ara of f is -indpndnt dn dn ( ) ( ) ( ) ( ) fd d fd d intgral ovr t nrgy rang of all t stats D d dn V m dn V m 4π de 4π C V m l d ( ) fd ( ) 4π NV, All t -dpndnc is at sinc t drivativ trm lswr, so C l m ( ) V f D d( ) π ( ) NV, ( ) ( ) f D + ( ) NV, + Dfin x x + dx d C l x ( ) V m x V m x dx k dx π x ( ) x π + ( + ) can xtnd lowr limit to - sinc x is vry small for low nrgis π x

6 5.6 4 Lctur #4-6 pag 6 C l V m 6 N k ( m) ( m) V ( π ) C l Nπ otal lctronic + vibrational at capacity of a mtal Low tmpraturs: Dby modl for vibrational at capacity π C C + C + γ + A V 4 π N N l vib 5 θ D Linar -dpndnc dominats at vry low Low- rsult can b plottd as C V γ + A Potassium C V mj mol K C V mj.8 mol K mj mol K slop A intrcpt γ ( K )

7 5.6 4 Lctur #4-6 pag 7 Mtal mj γ xpt ( ) mol K γ γ xpt modl Mtal mj γ xpt ( ) mol K Li.7. Cu.69.4 Na.7.5 Ag.66. K.. 5 Ni 7 5 γ γ xpt modl or good mtals, t fr-lctron modl works rasonably wll. Hig tmpraturs: classical quipartition rsult for vibrational at capacity π N CV Cl + Cvib + Nk Elctronic contribution is ngligibl at K. Would bcom significant at. but bfor tn t crystal mlts! Classical quipartition rsult would av ac lctron contribut k to C V du to fr translation. Wy is it rally muc lss? Bcaus only t lctrons nar contribut! fraction nar is rougly, so Cl N k. Smiconductors ratmnt of t lctronic nrgy lvls including t priodic nuclar potntial yilds s tat can b sparatd by gaps. MEAL Mor unfilld lvls SEMICONDUCOR Mor unfilld lvls & gaps INSULAOR Conduction or localizd stats Enrgy gap Unfilld lvls illd lvls Conduction Bandgap Valnc Wid gap Valnc

8 5.6 4 Lctur #4-6 pag 8 Smiconductors gap can b narrow, ~.5 V trmally inducd conductivity du to lctrons xcitd into conduction Calculat t -dpndnt carrir population dnsity troug statistical mcanics Application of quilibrium statistical mcanics to smiconductors Calculat n, # of lctrons pr unit volum in conduction n, # of ols pr unit volum in valnc Excitd lctron lavs + ol in t valnc rally just t absnc of an lctron. Otr lctrons can fill t vacancy in ffct, ol transport, vn toug it s lctrons tat ar moving. g Conduction - + lctron > g ol < Valnc g Assum narrow gap, i.. << g or >> or an lctron in t conduction, > g >> ( ) fd ( ) ( ) + Boltzmann statistics! Conduction of a smiconductor as lots mor stats tan particls to fill tm Boltzmann statistics dn d # stats wit nrgy btwn and + d d dn d f D ( ) # lctrons in stats wit nrgy btwn and + d d dn m ( ) n d fd ( ) d ( g ) g V d g π PIB rsult, starting at nrgy g Rcall from fr-lctron modl of a mtal

9 5.6 4 Lctur #4-6 pag 9 dn V m π d now xtndd to t smiconductor conduction r lctron modl for conduction k g as in mtal m sam dnsity of wavvctor & nrgy stats as in mtal sam form for dn as in mtal Rwrit m ( g) ( g) n d ( g) π g m ( g) x m ( g) dxx π π π x g ( ) n π m ( g ) Not g < - biggr gap smallr n Sam form as translational partition function! - also from PIB modl & Boltzmann statistics! Now considr ols A ol is rally t absnc of an lctron for any nrgy f f D If If f D f (no ols) f D <, som lctrons ar promotd and tr ar som ols. f ( ) f D ( ) ( ) ( ) + + (rcall < ) >> Probability incrass as nrgy incrass! Wy? Bcaus t ol is t absnc of an lctron t lctrons sould fill t lowst availabl lvls, laving t ol at ig nrgy in t valnc.

10 5.6 4 Lctur #4-6 pag Hol k-dpndnc Valnc lctrons intract strongly wit parnt ions wavfunctions wit igr dnsity on nucli av lowr nrgis sort wavlngt (ig k) stats av lowr nrgis Curvatur of vs. k is ngativ, not positiv. k (rcall < ) m So bot valnc and conduction s av k nrgy dpndnc nar k. k nrgy dpndnc valnc as t sam form for dnsity of stats & population -dpndnc Conduction Valnc ~ k ~ k k dn m n d f( ) d ( ) V d π m x π m dxx π ( ) Using xprssions for n and n, trat - and + formation just lik cmical raction, & calculat t quilibrium constant. SOLID SOLID m g q 4 π + K n n Lik K q xprssion for ractions tratd by Boltzmann statistics!

11 5.6 4 Lctur #4-6 pag π m π m g D Rcall: K q q q, q translational partition functions Eg D Rsult is Boltzmann-lik bcaus is in t gap. n >>, >> so D distribution rducd to Boltzmann distribution. and ffctiv masss In gnral t mass m m m sinc lctrons intract wit t ions. Effctiv mass valus can account approximatly for t intractions. Witout t assumption tat m m : n n m m g π π n n n m π π 4 ( m ) n But n So π ( g ) m (or n ( mm ) g m π g ( g) ( g ) m m m 4 m g m g ln + ln 4 m 4 m 4 ) If m g m tn is in t middl of t gap If m m tn is skwd toward t conduction or valnc.

12 5.6 4 Lctur #4-6 pag Impurity lvls in smiconductors Just atom of boron pr, Si incrass conductivity by x at room! Doping wit impuritis ( dopants ) is usd to control smiconductor proprtis. B as on lss - tan Si in its valnc sll. Similar cangs in conductivity occur upon addition of P, As, or Sb. s av on mor valnc - tan silicon. Wat s appning? As substituts for Si in ttradrally bondd ntwork, maks 4 covalnt bonds. Extra - is only wakly bound As is an lctron donor. Extra - binding nrgy ~. V K, so t conductivity riss a lot. Ionization of t dopant is vry asy. n-dopd smiconductor p-dopd smiconductor g Conduction unpaird lctron As donor lvl g Conduction B accptor lvl Valnc Valnc B also substituts into ttradral Si sit. o mak 4 covalnt bonds, it borrows an - from t valnc it s an lctron accptor. is lavs a ol. In ts cass n n # conduction lctrons # ols