Chapter Three Crystal binding

Transcription

1 Chate Thee Cystal binding Why d atms fm cystals slids? Answe : Inteatmic fces that bind atms. Atms bind due t the Culmb attactive fces between electns and neighbing atmic ins. Cntents: Tyes and stengths f binding fces easn f cystal stuctue fmatin Mechanical eties f cystals [ stess ] [ elastic cnstant ] x [ stain ]

2 Chesive enegy U the enegy that must be added t the cystal t seaate its cmnents int neutal fee atms at est Enegy f fee atms Cystal enegy Hence, U>0 t fm a stable slid Magnitude ~ -0eV excet f the inet gas cystals (0.0-0.eV) U E in (Inizatin enegy Binding enegy f valence electns) U cntls the melting temeatue and bulk mdulus

3 Tyes f bnds (a) Van de Waals (Mlecula) (b) Cvalent (c) Metallic (d) Inic Electns lcalized amng atms Electns shaed by the neighbing atms Electns fee t mve thugh samle Electns tansfeed t adjacent atms (a) (b) (c) (d) -

4 (a) Mlecula bnding Inet gas cystals : He, Ne, A, K, Xe, n Tansaent Insulats cmletely filled ute electn shells Weakly bnding van de Waals bnding FCC stuctues excet f He and He 4 high inizatin enegies lw melting temeatues Nen Agn Kytn Xenn Chesive enegy (ev/atm) Melting temeatue (K) Inizatin enegy (ev) What hlds an inet gas cystal? 4

5 Phase diagams f (a) 4 He and (b) He. 5

6 Van de Waals Lndn Inteactin Cnside tw identical inet gas atms Neutal: sitive nucleus sheically symmetic distibutin f electn chage N inteactin between atms N chesin (NO slid)? Fluctuating dile-dile inteactin eviews: q d -q P qd n() Attactive inteactin between the atms P d n() P E ~ P as >>d P P F ~ P P U ~ 4 Electic fields Attactive fce eulsive fce 6

7 Inet gas slids On aveage sheically symmetic distibutin f electn chage with the sitive nucleus in the cente P But themal fluctuatins (finite T) cause instantaneus electic dile mment P (t) 0 P 0 0 v P fluctuatins v P 0 On adjacent atms if the diles ae andm thee culd be n net fce (time aveage) But dile induces a dile in neighbing atms that always gives an attactive fce 7

8 Mdel f inet gas slid tw identical linea hamnic scillats e x -e e x -e and ae the mmenta f these tw scillats C is the fce cnstant Hamiltnian f the unetubed system n Culmb inteactin H Cx m m Cx Hamiltnian f Culmb inteactin enegy f the system H e e x x e x e x x, x << e x x 8

9 9 Nmal mde tansfmatin -- symmetic (s) and anti-symmetic (a) ; x x x ; x x x a s a s Ttal Hamiltnian afte the tansfmatin a a s s x e C m x e C m H Cx m Cx m H m C ω, H m / e C m C ω m / e C m C ω H, a a s s Tw fequencies f the culed scillats symmetic (s) and anti-symmetic (a)

10 0 K K / a / s C e ) ( C e ω C e m C m / e C ω C e ) ( C e ω C e m C m / C - e ω Theefe, the ze int enegy f the culed scillats is lweed fm the unculed scillats by ω hω h a s ω hω h The ze int enegy The unculed scillats The culed scillats 6 A C e 8 ω U h Attactive inteactin

11 The van de Waals inteactin, the Lndn inteactin, the induced dile-dile inteactin A U 6 < 0 A 4 e hω hω α C wheeα is the electnic laizability P P fluctuatin induced Plaizability f the atm P P α E α P P P U ~ ~ α < 0 6 P α E

12 What limits attactin? -- eulsive fce (Pauli exclusin incile) Tw electns cannt have all thei quantum numbe s the same. A B A B Chage distibutins vela When chage distibutins f tw atms vela, thee is a tendency f electns fm atm B t ccuy in at states f atm A ccuied by electns f atm A, and vice vesus. Pauli exclusin incile events multile ccuancy, and electn distibutin f atms with clsed shells can vela nly if accmanies by the atial mtin f electns t unccuied high enegy state f the atm. The electn vela inceases the ttal enegy f the system and gives a eulsive cntibutin t the inteactin.

13 Emiical fmula f such eulsive tential B U 0 > The ttal tential f inet gas system 8 U() U Pauli U vdw B A 6 σ 4ε σ 6 U()/4ε U U Pauli the Lennad-Jnes tential - U vdw /σ whee emiical aametes A4εσ 6 and B4εσ ae detemined fm indeendent measuements made in the gas hase. Values f ε[enegy] and σ[length] ae shwn in Table 4.

14 N atms in the cystal U ttal ( ) N 4ε i j σ nn σ nn 6 whee nn is neaest neighb distance and nn is the distance between atm i and atm j U ttal ( ) N 4ε i j σ nn i j 6 σ nn 6 dimensinless Bth lattice sums can be dne f any stuctue. Sum f / n cnveges aidly f lage n. Me distant neighbs have me influence n the latte tem than the fme tem. 4

15 FCC stuctue,.88 and i j i j HCP stuctue,.9 and i j i j Bth stuctues have neaest neighbs. BCC stuctue, i j 9.48 and i j 6.50 BCC stuctue has 8 neaest neighbs. 5

16 Chesive enegy f inet gas cystals at 0K -- minimum U ttal (Equilibium) at du d ttal.09σ, σ Nε (.)( ) σ Nε (4.45)(6) U ttal ( ) σ 6 σ (4.45)( 6) 7 6 (.)() (4.45)(6) 0 (.5)(4Nε ) is a minimum Nen Agn Kytn Xenn FCC stuctue (Å) σ (Å) /σ Deviatin Quantum cectins 6

17 Exect stuctue t fm cystals which have lwest enegy, lagest chesive enegy Gibbs fee enegy : GU TS PV Assuming T0, P0, and n kinetic enegy f atmic mtin SC BCC HCP FCC β α i j 6 i j /Nε U tt FCC is faved. 7

18 (d) Inic bnding Alkali halides Electn tansfes between atms t fm tw sitely chaged ins. Stng electstatic fces dminate. Electn cnfiguatin : clsed electnic shells F examles, LiF : Li (S ) instead f Li (S S) F - (S S 6 ) instead f F (S S 5 ) Like inet gas atms but Chage distibutin is sheically symmetic. w/. sme disttin f chage distibutin nea the egin f cntact with neighbing atms Electn density distibutin in the base lane f NaCl elative electn cncentatin 8

19 Need t cnside the inizatin enegies and electn affinities f atms Inizatin enegy I Electn affinity A enegy that must be sulied in de t emve an electn fm a neutal atm enegy that is gained when an additinal electn is added t a neutal atm Inic bnding is duced wheneve an element w/. a elatively lw inizatin enegy is cmbined with an element w/. a high electn affinity. 9

20 e.g. NaCl cystal What is its chesive enegy? Valence electn lsely bund t in Na : e - Inizatin enegy 5.4eV (enegy t emve electn fm Na) Na Na 5.4eV Na e - Cl : Cl e - Seven valence electns tightly bund desie a filled ute shell Electn affinity enegy.6ev e - Cl - Cl.6eV NaCl : Na Cl -.8Å In bind by electstatic attactin U~-e /4πε ~ -7.9eV Na Cl - NaCl 7.9eV Inic bund Na Cl NaCl 6.7eV chesive enegy 0

21 N ins in the cystal and U is the inteactin enegy between ins i and j U λ ex( ) ρ sht ange Pauli eulsive U λ ex( ) ρ q q i i q q j j ( i j ) CGS lng ange electstatic whee neaest neighb distance U tt qiq j U Nz λ ex( ) ρ i j N zλ ex( ) ρ i j αq whee z numbe f neaest neighbs f any in α ' ± j Madelung cnstant

22 minimum U ttal (Equilibium) du d tt Nzλ ρ At equilibium, ex( ) ρ ex( Nα q ρ ) 0 ραq zλ U tt Nα q ρ Sht ange eulsive ρ 0. Madelung enegy Madelung cnstant α : gemetic sum deends n elative distance, numbe, and sign f neighbing atms cystal stuctues and basis

23 One dimensin : line f ins f altenating signs α ' ± α and ln ' ± α x 4 ( x) x 4 x 4 x 4 α ln().86 f ne dimensinal chain

24 In thee dimensins, it is me cmlicated t calculate α. vey slwly cnvegent vey lng ange electstatic fces Secial mathematical ticks ae used t calculate Madelung cnstant. stuctue NaCl (FCC) CsCl (BCC) GaAs (Zinc blende) ZnS (Wutzite) Cdinate N α Highe cdinatin numbe gives lage Madelung cnstant. It deends n the stuctue f the cystal but nt unit cell dimensins. 4

25 Madelung enegy E M Nα q 4πε in a binay inic cystal w/. N ins in the cystal Smetimes, the ce-ce eulsive enegy in the geneal fm is cnsideed. E Nα q ttal 4πε NA n Equilibium de PdV TdS The fist law f themdynamics At T0, the equilibium samle vlume is detemined by de dv de d 0 0 5

26 The equilibium neaest-neighb distance eq 4πε na α q Then, the ttal enegy ( n ) E ttal Nα q eq 4πε n cystal eq n (Å) A(J m n ) LiF LiCl The isthemal cmessibility V κ d E dv T eq κ NaF NaCl KCl dp κ VNa V dv NC T α q ( n ) 6πε C ( eq ) 4 n 6πε C α q κ CsCl ( eq ) 6 4

27 (b) Cvalent bnding C, Si, Ge Tetahedal bnd C C Oganic chemisty / diamnd Si Si Ge Ge } Semicnduct 7.eV/atm 4.6eV/atm.9eV/atm 4 atms in the valence band bnd t 4 neighbing atms diamnd Tetahedal bnding Natue f chemical bnds in a diamnd zinc blende stuctue 7

28 (/4,/4,/4) (0,/,/) (/,0,/) Tetahedal s bnd Fu lbes emanate fm an atm at the cente f a cube. Othe atms ae at the ends f the dtted lines and lbes int fm them twad the cube cente. (0,0,0) (/,/,0) High electn cncentatin The bnd is usually fmed fm tw electns, ne fm each atm aticiating in the bnd. Electn fming the bnd tend t be atially lcalized in the egin between tw atms jined by the bnd. The sins f tw electns in the bnd ae antiaallel. sin deendent culmb enegy disttin f electn clud aund atms 8

29 Calculated valence electn cncentatin in Ge. 9

30 Cnside simle cvalent bnd : H - H Bth hydgen atms wuld like t fm a filled ute shell -- shae electns Tw cases : (same sins n electns) (site sins n electns) Pauli exclusin incile fbids tw electns with the same states. same sins: electns must stay aat 0 site sins: electns can ccuy the same lace

31 Pauli exclusin incile mdifies the distibutin f chage accding t sin ientatin. Enegy is lwe when electns send time between nuclei -- attactive Culmb inteactin fm bth Sin-deendent Culmb enegy Exchange inteactin

32 Neutal H has nly ne electn cvalent bnding with ne the atm But, thee wuld be a hydgen bnd between them unde cetain cnditins. ~ 0.eV being fmed nly between the mst electnegative atms, such as F, O, and N. F - F - H HF - is stabilized by a hydgen bnd. In the exteme inic fm f the hydgen bnd, the hydgen atm lses its electn t anthe atm in the mlecule; the bae tn fms the hydgen bnd. The hydgen bnd cnnects nly tw atms.

33 (c) metallic bnding mst metals High electical cnductivity : a lage numbe f electns in a metal ae fee t mve. cnductin electns Oute electns f atms that fm metals ae lsely bund. The tential enegy baie between atms is educed, the electn enegy may be well abve the tential enegy maximum and thei wave functins ae then nealy lane waves in egins between atms. Weak binding, ~5eV/atm Metals tend t cystallize in elatively clsed acked stuctues : hc, fcc, bcc,

34 Mechanical eties f slid Bnd --- hamnic scillatin F k ( ) Cystal --- A cllectin f hamnic scillats a hmgeneus cntinuus medium athe than a eidic aay f atms Aly fces dislacements f atms stess stain D Elastic egime F A ku A stess [Nt/m ] kl u A l dimensinless elastic cnstant [Nt/m ] σ C Elastic behavi is the fundamental distinctin between slids and fluids. Elasticity descibes the dimensinal change unde extenal stesses. 4