Chap 8 Nearly free and tightly bound electrons

Transcription

1 Chap 8 Nearly free ad tightly boud electros (<<) Nearly-free electro lattice perturbatio empty lattice approximatio Fermi surface (>>) Tightly boud electro Liear combiatio of atomic orbitals Waier fuctio Special topic: Geometric phase i crystallie solid Dept of Phys M.C. Chag

2 Nearly-free electro

3 Lattice perturbatio to plae wave ε ε ε ( ) ( ) ( ') 0 C G U C G G G' G U + = >> G G ' 2m O-th order: By iteratio, 1 whe (0) 0 (0) G = G ε = ε ; C ( G) G1 = 0 whe G G ε ε ( ) ( ) 0 ( ) C G + U C G G G G 1 = 1 1 Let G=G 1, Let G G 1, ε = ( ) (1) 0 2 ε U O U G = (1) 1 C ( G) ε 0 ε U G G G G 1 st order eergy correctio 1 st order state correctio ε ε 0 0 G G If, for G=G 2 G 1, the the perturbatio above fails. 2 1,

4 Nearly-free electro (for illustratio, cosider 1D) The Bloch state ψ ( ) ( ) ( ) i G x x C G e G = is a superpositio of exp[i(-g)x], exp[ix], exp[i(+g)x] Free electro: ε exp[i(+g)x] exp[ix] exp[i(-g)x] -g 0 g Uder a wea perturbatio: If ~0, the the most sigificat compoet of ψ 1 (x) (at low eergy) is exp[ix] (little superpositio from other plae waves) If ~g/2, the the most sigificat compoets of ψ 1 (x) ad ψ 2 (x) (at low eergy) are exp[i(-g)x] ad exp[ix], others ca be eglected. previous page ext page

5 Degeerate perturbatio If {G 1, G 2, G m } give similar eergy ε 0 -G (ad are away from other eergy levels), the for G {G 1, G 2, G m }, oe has m 0 U C( Gi) Gi G (1) i 1 C = ( G) =, G G 0 0 ε ε for G= {G 1, G 2, G m }, oe has 0 ( ε ε ) or ε G G i m C ( G ) + U C ( G ) + U C ( G') = 0 G i i G j j G i G ' G i j= 1 G' G ε U U C ( G ) 0 G G G G G1 2 1 m 1 1 U G1 G2 U = 0 Gm G m 1 0 U U ε ε C ( Gm ) G 1 Gm Gm 1 Gm G m 1 st order eige-eergy ad 0-th order eige-states 1 i

6 For example, m=2 ear = -G, 0 ε ε U (0) G C 0 U ε ε C ( G) G 2 G 21 = 0 Eergy eigevalues ε + ε ε ε (1) G G ε ± = ± whe ˆ G ε = ε G = G 2 2 U G 2 for a ear a Bragg plae, eed to use degeerate perturbatio ad the eergy correctio is of order U

7 Bac to the example with m=2, Bloch states with q o the Bragg plae ε = ε ± ± U (1) 0 G U U (0) G G C * 0 U U = G ( ) 2 G C 2 G From iversio symmetry, U G is real, the C (0) 1 1 = C ( ) 2 1 G ± ψ (0) i r i( G) r ( r) = C (0) e + C ( G) e ± ± ± 2 G r 2cos (0) 2 2 ψ ( r ), ± = 2 G r 2si 2 Bragg reflectio at BZB forms two stadig wave with a fiite eergy differece (eergy gap)

8 Higher Brilloui zoes Reduced zoe scheme Same area At zoe boudary, poits to the plae bi-sectig the G vector, thus satisfyig the Laue coditio Gˆ = G 2 G Bragg reflectio at zoe boudaries produce eergy gaps (Peierls, 1930)

9 Beyod the 1 st Brilloui zoe BCC crystal FCC crystal

10 Empty lattice i 2D 2D square lattice Free electro i vacuum: ε = 2m 2 2 Free electro i empty lattice: ( G) ε = ε = = + G 1 st BZ 2m How to fold a parabolic surface bac to the first BZ? 2π/a M Γ X

11 Folded parabola alog ΓX (reduced zoe scheme) For U 0, there are eergy gaps at BZ boudaries M Γ X 2π/a

12 Empty FCC lattice Eergy bads for empty FCC lattice alog the Γ-X directio.

13 Compariso with real bad structure The eergy bads for empty FCC lattice Actual bad structure for copper (FCC, 3d 10 4s 1 ) d bads From Dr. J. Yates s ppt

14 Fermi surface for (2D) empty lattice For a moovalet elemet, the Fermi wave vector F = 2π a 3 2 For a divalet elemet F = For a trivalet elemet 4π a 1 F = 6π a Distortio due to lattice potetial

15 Fermi surface of alali metals (moovalet, BCC lattice) F = (3π 2 ) 1/3 = 2/a 3 F = (3/4π) 1/3 (2π/a) ΓN=(2π/a)[(1/2) 2 +(1/2) 2 ] 1/2 F = ΓN

16 Fermi spheres of alali metals Percet deviatio of from the free electro value

17 Fermi surface of oble metals (moovalet, FCC lattice) Bad structure (empty lattice) F = (3π 2 ) 1/3, = 4/a 3 F = (3/2π) 1/3 (2π/a) Fermi surface (a cross-sectio) ΓL= F = ΓL

18 Fermi surfaces of oble metals Periodic zoe scheme

19 Fermi surface of Al (trivalet, FCC lattice) 1 st BZ 2 d BZ Empty lattice approximatio Actual Fermi surface

20 Tightly boud electro

21 Tight bidig model: Eergy bads as a extesio of atomic orbitals

22 Covalet solid d-electros i trasitio metals Alali metal oble metal "We have the rather curious result that ot oly is it possible to obtai coductio with boud electros, but it is also possible to obtai ocoductio with free electros. A. Wilso

23 Tight bidig method (Bloch, 1928) importat Let a m (r) be the eigestate of a electro i the potetial U at (r) of a isolated atom. at atomic H atam() r = ε m am() r orbital Cosider a crystal with N atoms at lattice sites R, A wave fuctio with traslatio symmetry (but still ot a eergy eigestate) ϕ m ( r) = d( R) a ( r R) R 1 i R = e am( r R) N R m Liear combiatio of atomic orbitals (LCAO) Chec: ϕ 1 i R' ( r + R) = e am( r ( R' R)) m N R' 1 i R i R' i R = e e am( r R') = e ϕ ( r) m N R'

24 A eergy eigestate (Bloch state) ψ ( r) = C mϕ ( r) m m Schrödiger equatio 2 p H = + U( r) 2m H ψ = ε ψ ( H ) ϕ ε ψ = 0 m' m ( H ) C ' ϕ ε ϕ = 0 m m' m defie H = ϕ H S m' ' ( ) ( ) mm m m' mm' the = ϕ m ϕ m' ( ε ) H S C ϕ mm' mm' m' m' R' m' = 0 1 i Hmm' = a H a e mr m' R' N RR, ' where r a a ( r R '). 1 i S = a a e mm' mr m' R' N RR, ' = δ mm + a a ' R 0 mr α mm' m'0 ( R) e i i R i ( R R' ) ( R R' ), Overlap itegral importat

25 importat 2 p H = + Uat + U U 2m = H +ΔU at ( ) at U(r) U(r) a m (r) 1 H = a H a e mm' mr m' R' N RR, ' = i ( R R' ) a H a + a H a e m0 m'0 mr m'0 R 0 i R i i a H a = a H +ΔU a m0 m'0 m0 at m'0 = δ ε at mm' m m0 m'0 mm' at a H am'0 = ε m a a mr mr m'0 + a mr = ε α ( R) + γ ( R) + a β ΔU a at m mm' mm' ΔU a at at mm' mm' m mm' m mm' mm' m'0 H = δ ε + β + ε α ( ) + γ ( ) eergy shift due to the potetial of eighborig atoms. (U i Marder s) iter atomic matrix elemet betwee earby atoms. (t i Marder s)

26 m' m' ( ε ) H S C = 0 same status as the cetral eq. i NFE model ε mm' mm' m' ( δ ' + α '( )) + βm m' + γm m' ( ) Cm ' = ε ( δmm' + αm m' ( )) at m mm mm i. e. AC = ε BC C = ε C 1 ( B A) i so far o approximatio has bee used! m' : a eigevalue problem C m' Approximatio 1: The ras of A, B deped o the umber of atomic orbitals a m s-orbital, m=1 p-orbital, m=1 3 d-orbital, m=1 5 s-p mixig, m=1 4 etc Approximatio 2: Keep oly a few overlap itegrals (e.g. for NN ad NNN) α mm' ( R) = a a mr m'0 γ ( R) = a ΔU a mm' mr m'0 β = a ΔU a mm' m0 m'0 (o R-depedece)

27 Example: s-bad from the s-orbital (m=1) at εs ( 1 + α( ) ) + β + γ( ) C = ε ( 1 + α( )) C at β + γ( ) ε = ε s α( ) α( ) = α α R 0 R 0 i R 3 * ( Re ) ; ( R) = dras( r Ra ) s( r) β = dra( r) ΔUa( r) γ ( ) = γ Δ 3 * s s i R 3 * ( Re ) ; γ ( R) = dras( r R) Uas( r ) If we eep oly the NN itegrals, the ( R ) ( R ); ( R ) ( R α = α γ = γ ) have bee used ( ) α( ) = 2 α cos, α = α( R ) half of NN half of NN ( R) 0 0 γ ( ) = 2 γ cos, γ = γ ( R ( R) 0 0 NN NN )

28 Square lattice at 3D ε εs + β + γ( ) = 2 γ cos +cost. half of NN ( R) =2γ 0 ( cos a x + cos a y + cos a z ) 2D ε=2γ 0 ( cos a x + cos a y ) 0 Eergy cotours 1D ε =2γ cos a 0 Desity of states From Dr. P. Youg s at UCSC

29 Waier fuctio (1937) 1 i R i ψ ( r) = e ( ) Ca m m r R N R m let w ( r R) = C a ( r R) m m m i.e. ψ ( r ) = w ( r R) = 1 N 1 N R e i R 1 stbz w ( r R) e i R ψ ( r ) (localized) ψ ψ δ δ δ δ = ' ' ' w w =, ' R ' R' ' R, R' A orthoormal set Compariso Bloch state Waier fuctio Eergy eigestate ot a eergy eigestate i T ( P rp ) w = Rw i R ψ e = ψ R Kivelso, PRB, 82 R R (for 1D) Exteded fuctio orthoormal basis localized fuctio orthoormal basis

30 Waier fuctio for the Kroig-Pey model Pederse et al, PRB 1991

31 Tight-bidig model (TBM) As a basis, Waier fuctios are better tha atomic orbitals: w a w R ' R' ' R, R' a = δ δ δ δ mr m' R' mm' R, R' H p = + 2 w, R wr H wr ' w H U r ' R ' = ' R, ' R' 2m ( ) Oe-bad approx. (omit ) H = w H w, H w H w TB R RR, ' R' RR, ' R R' RR, ' R U w w + t w w, U H ; t = H R R R R, δ R R+ δ R R, R R, δ R, R+ δ R, δ 1 st BZ H = ε ψ ψ, ε = U + t e TB For a uiform system, U, t are idep of R, the by 1 i R w = e ψ R N δ i δ O-site eergy Hoppig amplitude (usually are treated as parameters) Cf: spectrum from LCAO

32 Geometric phase (aa Berry phase)

33 Brief itroductio of the Berry phase Adiabatic evolutio of a quatum system Eergy spectrum: H( r, p; λ ) After a cyclic evolutio E(λ(t)) x x 0 λ(t) +1-1 λ( T ) = λ(0) ψ i T dt ' E ( t ') 0 e = ψ, λ( T), λ( 0) Dyamical phase Phases of the sapshot states at differet λ s are idepedet ad ca be arbitrarily assiged ψ iγ ( λ) e ψ, λ( t), λ( t) Do we eed to worry about this phase?

34 No! Foc, Z. Phys 1928 Schiff, Quatum Mechaics (3rd ed.) p.290 Pf : Cosider the -th level, Ψ λ t dt' E ( ') ( ) t iγ λ 0 t = e e ψ, λ () HΨ () t = i Ψ () t λ λ t γ ψ ψ λ λ = i 0, λ, λ ψ ' i ( ) e φ = λ ψ, λ, λ A φ = (λ) A (λ) λ Choose a φ (λ) such that, i A (λ) Redefie the phase, (gauge trasformatio) Statioary, sapshot state Hψ = Eψ, λ, λ A (λ)=0 Thus removig the extra phase.

35 Oe problem: = φ λ A( λ ) does ot always have a well-defied (global) solutio Vector flow A Vector flow A φ is ot defied here Cotour of φ Cotour of φ C C γ ( T ) γ (0) = A dλ = 0 A dλ 0 C C M. Berry, 1984 : Parameter-depedet phase NOT always removable!

36 ψ i T dt ' E( t ') iγ C 0 e e = ψ λ( T ) λ(0) Berry phase (gauge idepedet, path depedet) γ = ( λ ) d λ C C A Idex eglected Berry coectio (or Berry potetial) A( λ ) i ψ ψ λ ( R i Marder s) λ Berry curvature (or Berry field) Ω( ) A( ) = i γ C λ λ ψ ψ C λ λ λ λ λ Stoes theorem (3-dim here, ca be higher) 2 = A dλ = Ω da S λ λ 1 λ() t λ 3 S C λ

37 Berry phase i crystallie solid (for the -th bad) 2 ε where H ( u ) = u H ( ) = + + Ur ( ) 2m i Berry phase u i u d γ = C Berry coectio A ( ) = u i u 2 Berry curvature Ω ( ) = A ( ) = i u u Stoes theorem γ = = A d A d 2 C S x z S C y

38 Symmetry ad Berry curvature For o-degeerate bad: Space iversio symmetry Time reversal symmetry Ω ( ) = Ω ( ) Ω ( ) = Ω ( ) both symmetries Ω ( ) = 0, Whe could we see ozero Berry curvature? Ω ( ) 0 SI symmetry is broe TR symmetry is broe bad crossig electric polarizatio QHE moopole For more, see Xiao D. et al, Rev Mod Phys 2010

39 Berry phase (crossig the BZ) i oe dimesio g 1 =5 g 2 =4 Dirac comb model 0 b a Lowest eergy bad: γ 1 For a 1D lattice with iversio symm, Berry phase ca oly be 0 or π g 2 =0 γ 1 =π r =b/a Rave ad Kerr, EPJ B 2005

40 Realistic Berry curvature for BCC Fe (itrisic) aomalous Hall effect (Karplus ad Luttiger, 1954) From Dr. J. Yates s ppt I additio to ε (), there is a 2 d fudametal quatity Ω ()