Two ways of understanding electronic structure in solids:

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1 Bnd structure from the top down Two wys of understndng electronc structure n solds: Top-down - strt from free electron gs nd mpose correctons for ctully hvng on cores nd crystl lttce: nerly-free electron pcture. Bottom-up - consder electrons n ndvdul toms nd smll molecules, nd wtch wht hppens when lrger solds re ult up. In ths lecture we ll strt quc overvew of the top-down pproch; we ll loo t the ottom-up vew lter. Free electron gs fetures: Sngle-prtcle sttes leled y wvevector prmeter,, wth p =, =. m Boundry condtons only permt certn dscrete vlues of. Ignore nterctons: llowed mny-prtcle sttes re (totlly ntsymmetrc lner comntons of products of sngle-prtcle sttes. All three of these ponts re preserved n the nerly-free electron pcture. 1

2 Our strtegy: enerl propertes of electrons n perodc potentl Specfc emple: sngle-prtcle prolem n 1d n we perodc potentl - llustrtve. Set stge for lter dscussons of other nondeltes. Perodc potentl - Bloch Theorem Consder potentl V(r tht s perodc n spce wth some sptl perod R, V ( r + R = V ( r where R s n nteger lner comnton of some ss vectors: R = n + n + n Trnslton opertor: T f ( r = f ( r R R + Schroednger equton: H ψ ( r = + V ( r ψ ( r = ψ ( r m Snce H nd T R commute, ψ cn e smultneously n egenstte of oth: Hψ ( r = ψ ( r, T ψ ( r = c( R ψ ( r Snce two dfferent trnsltons commute, c ( R + R' = c( R c( R' R

3 Perodc potentl - Bloch Theorem II To preserve normlzton, t must e possle to wrte c s unt mgntude comple numer. c( = ep( π Ths mples c( R = ep( R where = 1 j 1 + = πδ j + The llowed vlues of re set y our choce of oundry condtons. We now see: T ψ ( r = ψ ( r + R = ep( R ψ ( r R 3 3, Bloch s Theorem. recprocl lttce vectors, defned here Bloch functons Suppose: u ( r = ep( r ψ ( r u ( r + R = ep( ( r + R ψ ( r + R = ep( r ψ ( r = u ( r So, f u (r s strctly perodc n R, sngle-prtcle wvefuncton tht utomtclly stsfes the Bloch relton s: ψ ( r = ep( r u ( r Sngle-prtcle egensttes of perodc potentls loo le plne wves modulted y functons tht re strctly perodc wth the sme perodcty s the potentl. 3

4 Bloch functons stll lels sttes, ut no longer drectly corresponds wth momentum. Sttes loo most le plne wves when s fr from recprocl lttce vector. Bloch theorem true for perodc potentls - don t hve to e we! Born-von Krmn oundry condtons Now mpose perodc oundry condtons, ssumng the sze of the smple n the drecton s N : ψ ( r = ψ ( r + N ( ψ ( r N = ep N ψ ( r + Bloch s thm. ep ( N = ep( πn j =, j = 1,,3 N = 3 = 1 j N N = 1 So, for lttce wth N=N 1 N N 3 stes, there re N llowed vlues of tht re consstent wth perodc oundry condtons. 4

5 Born-von Krmn oundry condtons 3 = = 1 j, j = 1,,3 N N So, for lttce wth N=N 1 N N 3 stes, there re N llowed vlues of tht re consstent wth perodc oundry condtons. Allowed volume n -spce per sngle-prtcle stte = ((π/l d, just s n free electron gs w/ perodc.c. cse! Cn mp out ll llowed vlues of even when restrctng components to mgntudes less thn the (nsde the frst Brlloun zone. For ech, there re multple solutons to the Schroednger equton (energy egenvlues; we numer these wth nd nde, usully leled n. Wht out lrger? Add recprocl lttce vector to : ψ n + ( r = ep( ( + r u = ep( r[ u = ep( r u ~ ( r = ψ n' ( r n + n + ( r ( rep( r] Any stte comptle wth oundry condtons nd wth outsde the frst zone s dentcl to stte nsde the frst zone wth dfferent nd nde! If we wnt, we cn lmt to just the frst Brlloun zone (mgntude of ech component smller thn the respectve nd stll e le to descre ll the sngle prtcle sttes. 5

6 nergy egenvlues Alterntely, we cn llow to rnge over ll -spce, nd lel the sttes such tht ( r = ( r n+ n Snce the energy s perodc n -spce wth the recprocl lttce, t hs mm nd mnm: For ech n, the energes of ll levels leled y fll n nd. nergy nd Fgure from Mrder. 6

7 7 Krong-Penney: specfc 1d toy model V( -V Two regmes: -V < < > Cse 1 -V < < We now tht: (, ( 1/ < < + + = V m Be Ae α ψ α α m De Ce < < + =, ( 1/ β ψ β β V( -V Then for Bloch s theorem:, ( ( ( < < + = + Be Ae u α α, ( ( ( < < + = + De Ce u β β c

8 Cse 1 contnued Need wvefuncton nd frst dervtve (nd hence u ( nd u ( to e contnuous t the edges of the wells. A + B = C + D ( α A ( α + B = ( β C ( β + D ( α Ae Ae ( α c ( α c + Be ( α + Be + ( α + c + ( α + c = Ce ( β = ( β Ce + De ( β ( β + ( β + De ( β + Four equtons, four unnowns. Resultng condton: α β cos( αc cosh( β sn( αcsnh( β = cos( αβ Cse Follow sme procedure. α + cos( αc cos( β sn( αcsn( = cos( α Both cses cn e wrtten s: F( F ( = cos( l Clerly, f F( > 1, ths cnnot e stsfed! Not ll re comptle wth oundry condtons. 8

9 9 We perodc potentls We cn do the generl prolem of sngle electron n we perodc potentl y perturton theory. = r r ( ( e c ψ Fourer decompose potentl V(r usng recprocl lttce vectors, nd wrte Schroednger equton: = + ' ' ( c V c m Free electron cse=unpertured cse: m q = q We perodc potentls When s fr wy from ny, frst order result: + + V Clerly somethng nterestng hppens when denomntor vnshes. Ths hppens t the edge of Brlloun zone, nd corresponds to the prtcle strongly dffrctng off the lttce potentl. The result ner tht pont: 4 ( V + ± V ± At tht pont:

10 Result: Sctterng off lttce potentl cuses gps to open t edges of Brlloun zone: From Kel To summrze: Perodc potentl mens sngle-electron egensttes re Bloch sttes. Perodc oundry condtons set the llowed vlues of. For gven, there re multple dscrete vlues of llowed. Sttes wth ner recprocl lttce vector (hvng perodcty tht s hrmonc of the lttce hve energes strongly ffected y lttce, even when lttce potentl s we. Result: energy gps open up t prtculr vlues of : not ll energes re llowed nymore. Note tht ths s ll for lrge ul crystllne solds. Wht s est wy to thn n nno cse? Wht hppens wth nterfces & defects? 1

11 Net tme Bsc nds n solds Dffrcton & crystl structure - very ref gude. 11

12 lectrons n we perod potentls II Wht we sw lst tme loong t the sngle-prtcle prolem: Perodc potentl mens sngle-electron egensttes re Bloch sttes. Perodc oundry condtons set the llowed vlues of. For gven, there re multple dscrete vlues of llowed. Sttes wth ner recprocl lttce vector (hvng perodcty tht s hrmonc of the lttce hve energes strongly ffected y lttce, even when lttce potentl s we. Result: energy gps open up t prtculr vlues of : not ll energes re llowed nymore. Crystl structure I In ul, mny solds re crystllne. Hve dscrete trnsltonl nd rottonl symmetres. Rel-spce structure s perodc - repettons of sngle unt cell. Smllest unt cell tht gves full structure: prmtve unt cell Cn descre structure y lttce nd ss. r 31 1 lttce ss 1

13 Crystl structure II Wgner-Setz prmtve cell: ll ponts closer to sngle lttce pont thn ny other. Type of stcng depends on energetcs of ondng. Surfces hve dfferent energes per tom thn ul, so nnoscle crystls (hgh surfce to volume rto cn hve dfferent structure thn ul mterls! Crystl structure III - Mller Indces Crystllogrpher s wy of lelng plnes of toms: Determne the ntercepts of the plne long the crystllogrphc es, n terms of unt cell dmensons. Te recprocls. Wrte s ntegers rther thn frctons. Negtves re wrtten usng overlnes: (-1 = (1 Trplets: (hl; qudruplets: (hjl

Common crystl structures http://cst-www.nrl.nvy.ml/lttce/ http://www.chem.o.c.u/cl/heyes/structure_of_solds/strucsol.

com/~chn/solidstat/crystal/ Smple cuc (1,, (,1, (,,1 Body-centered cuc

Sr, Rh, Pd, Ag, Ce, T, Ir, Pt, Au, P, Th Hegonl close-pced /(1,-3 1/, /(1,

Co, Zn, Y, Zr, Tc, Ru, d, T, Py, Ho, r, Tm, Lu, Hf, Re, Os, Tl Common

nterpenetrtng fcc lttces dsplced y 1/4. Result of ll sp 3 covlent onds.

dsplced y 1/4, ech lttce dfferent speces.

14 Common crystl structures Smple cuc (1,, (,1, (,,1 Body-centered cuc /(-1,1,1 /(1,-1,1 /(1,1,-1 Fce-centered cuc /(,1,1 /(1,,1 /(1,1, Al, Cu, N, Sr, Rh, Pd, Ag, Ce, T, Ir, Pt, Au, P, Th Hegonl close-pced /(1,-3 1/, /(1, 3 1/, c(,,1 W, L, N, K, V, Cr, Fe, R, N, Mo, Cs, B, u, T Mg, Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, d, T, Py, Ho, r, Tm, Lu, Hf, Re, Os, Tl Common crystl structures II - semconductors Dmond /(,1,1 /(1,,1 /(1,1, Two nterpenetrtng fcc lttces dsplced y 1/4. Result of ll sp 3 covlent onds. C, S, e, Sn Znc lende /(,1,1 /(1,,1 /(1,1, Two nterpenetrtng fcc lttces dsplced y 1/4, ech lttce dfferent speces. ZnS, AgI, AlAs, AlP, AlS, BAs, BN, BP, BeS, BeSe, BeTe, CdS, CuBr, CuCl, CuF, CuI, As, P, S, HgS, HgSe, HgTe, InAs, InP, MnS, MnSe, SC, ZnSe, ZnTe 3

15 Recprocl ss vectors We sw lst tme tht there re specl vectors n -spce (lso clled recprocl spce tht ehve le: j = πδ The defne lttce n recprocl spce just s the do n rel spce. In 3d, 1 = π 3 1 = π = π 3 j 3 ( 3 1 ( 3 1 ( Recprocl lttce vectors Usng the s, we cn uld up lttce n recprocl ( spce. The recprocl lttce s the set of ponts n recprocl spce gven y nteger lner comntons of the recprocl lttce vectors: = c + where c 1, c, c 3 re ntegers c c3 3 Wht s specl out the s? Any functon n rel spce wth the perodcty of the (rel spce lttce cn e wrtten ectly s sum le: 1 ρ ( r = ρ e r 4

16 Recprocl lttces Rel spce Recprocl spce SC π/ FCC 4π/ BCC 4π/ Brlloun zones I ch pont n the recprocl lttce s recprocl lttce vector. Rememer: when s close to such vector, the electronc sttes re strongly ffected y the lttce potentl (gps!. All unque vlues comptle wth.c. my e wrtten wthn the frst BZ - tht s, wthn the frst Wgner-Setz unt cell of recprocl spce. Wht do these Brlloun zones loo le? 5

Brlloun zone - FCC Γ = ( Χ = (1 1 1 1 L = (

17 Brlloun zone - FCC Γ = ( Χ = ( L = ( 1 W = ( K = ( U = ( Imge from Mrder. Brlloun zone - BCC Γ = ( H = (1 1 1 N = ( P = ( Imge from Mrder. 6

18 Why should we cre out Brlloun zones nd recprocl spce? Reson #1. Fllng of sttes n recprocl spce determnes electronc propertes of ul solds. Recll our free-electron gs procedure: Fnd llowed sngle prtcle sttes, leled y. Usng (, fgure out the energy levels of those sttes. For nonnterctng electrons, fnd mny prtcle ground stte y fllng those levels from the ottom up, two electrons per sngle-prtcle stte (spn. Cn do sme thng here, ut ( no longer smple! Bnd dgrms Imges from Blemore. Free prtcle We perodc potentl 7

19 Bnd dgrms Strt fllng sngle-prtcle sttes from the ottom. Where do we end up? F n mddle of nd: metl g F such tht nteger numer of nds ectly full: nd nsultor Specl cse: g s smll = ntrnsc semconductor. Complcton: Imges from Blemore. Lttce spcng depends on drecton. Result: nds cn overlp n energy. 8

20 Rel nd structures ermnum Dmond BZ Imges from Hrrson More out ths on Mondy. Why should we cre out Brlloun zones nd recprocl spce? Reson #. Plnes n recprocl spce leled y Mller ndces me dffrcton eperments possle! Brgg plnes: the set of ll equspced prllel plnes contnng ll the stes n lttce. π 1 1 π π The spcng etween (hl plnes s gven y where hl = h 1 + +l hl 3 9

21 Dffrcton Bsc de: Constructve nterference from perodc plnes leds to pes n dffrcted ntensty long drectons dependent on λ of ncomng wve. d θ θ θ θ Totl etr dstnce trveled y ottom ry = d sn θ. Constructve nterference requres Brgg condton d snθ = nλ Dffrcton nd ntenne In mny respects, dffrcton prolems for smll numers of sctterers re very smlr to prolems out ntenn rrys. Bsc methodology: Defne coordntes ncely. Assume ech sctterer s source of sphercl wves of wvenumer. Fnd mpltude t poston of nterest. A tot = A1 + A = A ep( r + A ep( r' y r r = r - 1

22 Dffrcton nd ntenne Me legtmte ppromtons. r >> r' r cosθ y θ r r = r - Fnd epresson for scttered ntensty, proportonl to mpltude : I ~ = = = A A A A tot ep( r + ep( ( r cosθ ep( r(1 + ep( ( cosθ 1+ ep( ( cosθ Dffrcton technques Sctterng of some wve from smple tells us out the structure of the smple. -ry dffrcton: electronc densty dstruton neutron dffrcton: mss densty dstruton, mgnetc orderng θ θ Send n, get out, wth =. Brgg condton ends up eng = hl. Intensty ~ Fourer component of lttce potentl. 11

23 Dervton of Brgg condton tr phse for lower pr of rys = ( d cosθ d cosθ ' + Assume = d = closest spcng = d ( ' Snce d s n emple of rel spce lttce vector R, nd ths phse must e n nteger multple of π to get constructve nterference, we fnd from defnton of s tht θ θ ( ' R = πj ( ' = The dfference n ncdent nd outgong must e recprocl lttce vector to constructve nterference ( dffrcton mmum. Dffrcted ntensty Incomng plne wves ch sctterng ste = source of outgong sphercl wves Scttered mpltude ~ For perodc lttce, ρ ( r = Intensty ~ Amp ~ ρ( r ρ e r e ( dr Integrl over smple volume r ( r e ρ For =, ntegrnd ~ 1; I ~ V Otherwse, I ~. dr 1

Types of dffrcton θ θ Sngle-crystl dffrcton: For fed λ, nowng smple orentton, dffrcton pes

Cn deduce structure from spot postons. Spcng of spots s nversely prop. to lttce spcngs.

For ny θ, some of the grns re gong to hve n hl meetng the Brgg condton.

24 Types of dffrcton θ θ Sngle-crystl dffrcton: For fed λ, nowng smple orentton, dffrcton pes only t certn specfc ngles, when Brgg condton s stsfed -- Lue spots. Cn deduce structure from spot postons. Spcng of spots s nversely prop. to lttce spcngs. Powder dffrcton: Smple s rndomly orented grns. For ny θ, some of the grns re gong to hve n hl meetng the Brgg condton. ch grn produces spots t prtculr θ,φ, so tht ddng the spot ptterns ncoherently produces set of pes t prtculr vlues of θ. Powder dffrcton 13

25 Dffrcton tdts Fnte T cuts ntensty, ut does not ffect wdth of pes (! Fnte sze does ffect pe wdths - possle troule for nnopowders. Amorphous mterls / lquds show or 3 very rod rngs, ndctve of very short-rnge order (ond lengths / nterprtcle spcngs. To summrze: Defne crystl y lttce + ss n rel spce, unt cell. Rel spce lttce cn e used to defne recprocl spce lttce. Recprocl spce lttce unt cell = Brlloun zone Rel spce plnes + recprocl lttce vectors leled y Mller ndces. Becuse of perodc lttce potentl, gps open up n free electron energy for ner edge of Brlloun zone. Detls of Brlloun zone & fllng determne electronc stte of mterl (more on ths net tme. Lttce vectors n recprocl spce determne loctons of dffrcton pes. Dffrcton s powerful method for structure determnton. 14

26 Net tme: More electronc propertes + nds Dopng of semconductors Boundres & surfce sttes Impurtes 15

Electrochemical Thermodynamics. Interfaces and Energy Conversion

Electrochemical Thermodynamics. Interfaces and Energy Conversion CHE465/865, 2006-3, Lecture 6, 18 th Sep., 2006 Electrochemcl Thermodynmcs Interfces nd Energy Converson Where does the energy contrbuton F zϕ dn come from? Frst lw of thermodynmcs (conservton of energy):