Solids - types. correlates with bonding energy

Transcription

1 Solids - types MOLCULAR. Set of sigle atoms or molecules boud to adjacet due to weak electric force betwee eutral objects (va der Waals). Stregth depeds o electric dipole momet No free electros poor coductors easily deformed, low freezig temperature freezig boilig bodig eergy He 1 K 4 K H 14 K 0 K Ar 84 K 87 K.08 ev/molecule H O 73 K 373 K 0.5 ev/mol CH 4 90 K 111 K 0.1 ev/mol correlates with bodig eergy 1

2 Ioic Solids Positive ad egative ios. Strog bod ad high meltig poit. o free electros poor coductor Potetial vs sep distace R R similar potetial as molecule. ~5 ev molecules ad ~6 ev solid (NaCl) each Cl- has 6 adjacet Na+, 1 ext Cl-, etc V (6Na 6e ) 4 R 1.7e all 4 R V (1Cl 1e ) 4 R eergy levels similar to molecules except o rotatios.electroic i UV ad vibratioal i IR. Ofte trasparet i visible 0 0 0

3 COVALNT. Share valece electros (C, H, etc) strog bods (5-10 ev), rigid solids, high meltig poit o free electros isulators usually absorb i both visible ad UV MTALLIC. s-p shell covalet bods. But d shell electros leftover (smaller value of lower eergy but larger <r>) ca also be metallic eve if o d shell if there is a ufilled bad 1-3 ev bods, so weaker, more ductile, medium meltig temp free electros ot associated with a specific uclei. Wavelegth large eough so wavefuctios overlap ad obey Fermi-Dirac statistics coductors M field of photo iteracts with free electros ad so absorb photos at all l 3

4 Bads i Solids lowest eergy levels very similar to free atoms large kietic eergy large p, small l 0 Z h re atoms K 13.6eV l Zeff ( ) p little overlap with electros i other atoms ad so arrow eergy bad higher eergy levels: larger l wavefuctios of electros from differet atoms overlap eed to use Fermi-Dirac statistics may differet closely spaced levels: Bad: valece vs coductio depeds o whether bad is filled or ot a 4s,4p,3d 3s,3p s,p 1s 4

5 Multielectro eergy levels the eed for totally atisymmetric wave fuctios causes the eergies to split whe the separatio distace R < wavelegth if far apart N degeerate(equal) states overlap still N states but differet eergy (differet spis ad differet spatial states, mixed symmetries) N based o how may electros overlap small D betwee differet levels ( 1,,3,4... N ) N! large for the outer shell a almost cotiuous eergy bad ature of the eergy bads determies properties of solid -- filled bads -- empty bads -- partially filled bads -- eergy width of bad -- eergy gaps betwee bads -- desity of states i bads terms 6 electros R 5

6 Coductio vs valece eergy levels i 4s/4p/3d bads overlap ad will have coductio as log as there is t a large D to available eergy states (ad so ca readily chage states) x00000x0 xxxxxxxx T=0 xxx0xx0x T>0 xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx x=electro 0=empty state ( hole ) sometime curret is due to holes ad ot electros good coductors have 1 or more coductio/free electros/holes per atom coductio valece 6

7 Li ad Be Bads Atoms: Li Z=3 1s s 1 ufilled coductor Be Z=4 1s s filled isulator But solids have eergy bads which ca overlap p s Atom 1s solid there is the just a sigle ps bad Be fills the bad more tha Li but the top (the Fermi ergy) is still i the middle of the bad. So ufilled bad ad both are metals 7

8 Magesium Bads Atoms: Z=1 1s s p 6 3s filled isulator like Be 8N Atomic separatio R the 3p level becomes a bad with 6N eergies. The 3s becomes a bad with N eergies They overlap becomig 1 bad with 8N eergy levels ad o gaps ad is a metal as ufilled levels above Fermi eergy BUT, if R becomes smaller, the bads split (bods) givig a eergy gap for C, Si, Ge 6N N 3p 3s 8

9 C,Si,Ge Bads similar valece C:s p Si:3s 3p Ge:4s 4p 4N 6N p,3p,4p 4N N s,3s4s Atomic separatio R 8N overlappig eergy levels for larger R R becomes smaller, the bads split ito 4N bod ad 4N atibod. a eergy gap for C (7eV) ad Si, Ge (~1 ev) filled T=0 (gap) empty T=0 9

10 Quatum Stats: Fermi Gas Model 1 probability / eergy ( ) ( F e 3 8V (m ) desity of states D( ) 3 h F ( T 0) 3 What are the umber of coductio electros excited to > F for give T? ad so available for coductio N total at T DN ( DN N 0 1 F D( 3kT F D( ) d D( ) ) D( F F 0 F ) D )( 3) kt.05ev 4eV 3N 3/ V 8m ( ) 3 h F 1/ h 8m 3/ *D 3/ F / 3 T=0 F T>0 ) / kt 1/ 1 1/ 10

11 Semicoductors Filled valece bad but small gap (~1 ev) to a empty (at T=0) coductio bad look at desity of states D ad distributio fuctio D valece coductio D* If T>0 F Fermi eergy o ceter of gap for udoped. Always where ()=0.5 D() typically goes as sqrt() at top of valece bad ad at bottom of coductio bad 11

12 Semicoductors II Distributio fuctio is ( ) e if ( F F 1 ) / kt gap 1 e / 1 ( kt F ) / T 300 ( g ) e g / kt so probability factor depeds o gap eergy g 1eV Si g 6eV C estimate #electros i coductio bad of semicoductor. Itegrate over *D factors at bottom of coductio bad 1

13 Coductio i semicoductors INTRINSIC. Thermally excited electros move from valece bad to coductio bad. Grows with T. PHOTOLCTRIC. If photo or charged particle iteracts with electros i valece bad. Causes them to acquire eergy ad move to coductio bad. Curret proportioal to umber of iteractios (solar cells, digital cameras, particle detectio.) XTRINSIC. Dope the material replacig some of the basic atoms (Si, Ge) i the lattice with oes of similar size but a differet umber (+- 1) of valece electros 13

14 Doped semicoductors Si(14) 3s 3p P(15) 3s 3p 3 Al(13) 3s 3p 1 Si 4 covalet bods. Fill all valece Si= Si =Si eergy levels (use all electros) 1 ev gap Si Si sigle electro loosely boud to P Si= P =Si (~looks like Na) 0.05 ev coductio bad Si e Si = Si 0.06 ev ca break oe of the Si=Si Si= Al -Si bods. That electro Al. The hole moves to the Si atom Si=Si=Si hole 14

15 Doped semicoductors II coductio bad door electros.05 ev.06 ev acceptor holes valece bad P-doped -type extra e.05 ev to move from door to coductio bad Al-doped p-type missig e= (hole).06 ev to move from valece to coductio bad The Fermi ergy is still where ( F ) = ½. dopig moves F 15

16 Doped Semicoductors III Addig P (-type). Sice oly.05 ev gap some electros will be raised to coductio bad where ()= ½ is i door bad D -type valece coductio D() F p-type F addig Al (p-type). some electros move from valece to acceptor bad. ()= ½ ow i that bad 16

17 Supercoductivity Resistace goes to 0 below a critical temperature T c elemet T c resistivity (T=300) Ag mohms/m Cu mohms/m Ga 1.1 K 1.7 mo/m Al 1..8 S Pb 7.. Nb may compouds (Nb-Ti, Cu-O-Y mixtures) have T c up to 90 K. Some are ceramics at room temp Res. T 17

18 Supercoductors observatios For differet isotopes, the critical temperature depeds o mass. ISOTOP FFCT M cos t ( S ) agai shows supercoductivity due to iteractios with the lattice. If M ifiity, o vibratios, ad T c 0 spike i specific heat at T c idicates phase trasitio; eergy gap betwee coductig ad supercoductig phases. Ad what the eergy differece is. B field adds eergy, ca cause quech if abiove critical value plasma gas liquid solid supercoductor 0.5 T c vibratios K M ta 115,117,

19 What causes supercoductivity? Bardee-Cooper-Schrieffer (BCS) model paired electros (cooper pairs) coupled via iteractios with the lattice gives et attractive potetial betwee two electros if electros iteract with each other ca move from the top of the Fermi sea (where there are t iteractios betwee electros) to a slightly lower eergy level Cooper pairs are very far apart (~5,000 atoms) but ca move coheretly through lattice if electric field resistivity = 0 (uless kt oise overwhelms breaks lattice couplig) atoms electro electro 19

20 Coditios for supercoductivity Temperature low eough so the umber of radom thermal phoos is small iteractios betwee electros ad phoos large ( large resistivity at room T) umber of electros at = Fermi eergy or just below be large. Phoo eergy is small (vibratios) ad so oly electros ear F participate i makig Cooper pairs (all actio happes at Fermi eergy) electros i Cooper pair have atiparallel spi space wave fuctio is symmetric ad so electros are a little closer together. Still 10,000 Agstroms apart ad oly some wavefuctios overlap (low large wavelegth) 0

21 Coditios for supercoductivity electros i pair have equal but opposite mometum. Maximizes the umber of pairs as weak bods costatly breakig ad reformig. All pairs will the be i phase (other mometum are allowed but will be out of phase ad also less probability to form) e ip r differet times differet pairs P pair p 1 p 0 if electric field applied, as wave fuctios of pairs are i phase - maximizes probability -- allows collective motio uimpeded by lattice (which is much smaller tha pair size) total

22 ergy levels i S.C. electros i Cooper pair have eergy as part of the Fermi sea ( 1 ad = F D) plus from their bidig eergy ito a Cooper pair (V 1 ) 1 1 V1 1 F 1 ad are just above F (where the actio is). If the coditio is met the have trasitio to the lower eergy supercoductig state F 1 gap s.c. ormal Temperature T C ca oly happe for T less tha critical temperature. Lower T gives larger eergy gap. At T=0 (from BCS theory) gap 3kT C

23 Magetic Properties of Materials H = magetic field stregth from macroscopic currets M = field due to charge movemet ad spi i atoms - microscopic B 0( H M ) M ch c magetic susceptibility ca be : c( T), c( H ), scalar, vector ca have residual magetism: M ot equal 0 whe H=0 diamagetic c < 0. Currets are iduced which couter applied field. Usually Supercoductig c = -1 ( perfect diamagetic) 3

24 Paramagetism Atoms ca have permaet magetic momet which ted to lie up with exteral fields if J=0 (Helium, filled shells, molecular solids with covalet S=0 bods ) c = c 10 most, c 10 Fe assume ufilled levels ad J>0 = # upaired magetic momets/volume + = umber parallel to B - = umber atiparallel to B = momets wat to be parallel as B B( atiparallel) B( parallel ) 4

25 Paramagetism II Use Boltzma distributio to get umber parallel ad atiparallel ( B / kt B / kt where M = et magetic dipole momet per uit volume ca use this to calculate susceptibility(curie Law) M H M M average if B kt Ce Ce e e B / kt B / kt ) e e B / kt B / kt ( 1 B / kt) (1 B / kt) B (1 B / kt) (1 B / kt) kt B 0 H 0M 0H ( c small) c H B kth 0 kt 5

26 Paramagetism III if electros are i a Fermi Gas (like i a metal) the eed to use Fermi-Dirac statistics C e C e ( B ( B 1 1 reduces umber of electros which ca flip, reduces iduced magetism, c smaller atiparallel parallel 1 F ) / kt 1 F ) / kt B 0 kt F B F tur o B field. shifts by B atiparallel states drop to lower eergy parallel 6

27 Ferromagetism Certai materials have very large c (1000) ad a o-zero B whe H=0 (permaet maget). c will go to 0 at critical temperature of about 1000 K ( o ferromagetic) 4s: Fe6 3d6 Co7 3d7 Ni8 3d8 6s: Gd64 4f8 Dy66 4f10 All have ufilled ier (lower ) shells. BUT lots of elemets have ufilled shells. Why are a few ferromagetic? Sigle atoms. Fe,Co,Ni D subshell L=. Use Hud s rules maximize S (symmetric spi) spatial is atisymmetric ad electros further apart. So S= for the 4 upaired electros i Fe Solids. Overlap betwee electros bads but less overlap i ier shell. overlappig chages spi couplig (same atom or to adjacet atom) ad which S has lower eergy. Adjacet atoms may prefer havig spis parallel. depeds o geometry iteruclear separatio R 7

28 Ferromagetism II R small. lots of overlap broad bad, may possible eergy states ad magetic effects diluted F R large. ot much overlap, eergy vs differece small R medium. broadeig of eergy bad similar to magetic shift almost all i state P P A A F F (umagetized)- Fe Co Ni vs (magetized) R M 8