Green s func+on in LMTO. What they are. What they re good for.

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1 Green s func+on in LMTO What they are. What they re good for.

2 Dual nature of LMTO: basis set method and mul+ple Varia+onal approach: ψ nk = RL nk k C RL χ RL Hψ nk = E nk ψ nk R'L' k χ RL k H E nk χ R'L' nk C R'L' = 0 LMTO s χ RL are op+mal minimal basis set. Matrix elements easy to compute because constructed from solu+ons of Schrödinger eq. inside each region at energy ε υ in range Small basis set Errors in orbitals lead to second order errors in energy: φ and φdot errors ε- ε υ è errors in Enk of order (ε- ε υ ) 3

3 Tail cancella+on point of view HEAD TAIL Solu+on if all tails cancel Usual view: disadvantage non- linear in energy root searching No longer a disadvantage when using GF methods.

4 Green s func+ons for H (E H )ψ = 0 (E H )G(E) =1 H = H 0 +V (E H 0 )ψ = Vψ (E H 0 )G 0 (E) =1 ψ = ψ 0 + G 0 (E)Vψ ψ(r) = ψ 0 (r)+ dr' G 0 (E, r, r')v(r')ψ(r') Basis of sca@ering theory Ψ 0 (r) Born approxima+on

5 G(E) = Dealing with poles n n E n are poles of G(E) n E E n G + (E) = G(E + iδ) G + (t) = 1 2π = Θ(t) ie ie t n n n n retarded eigenstates What if E=E n? n n e iet de n E + iδ En If t>0 If t<0 Similarly G (E) = G(E iδ) advanced

6 DOS etc. 1 δ 0+ 1 ### P iπδ(e E n ) E E n + iδ E E n D(E) = 1 π Tr ImG(E + i0 +) = 1 d π de Im lndetg(e + i0 +) D RL (E) = 1 π ImG RL,RL(E + i0 + ) Phase formula ReG + (E) = 1 π P ImG+ (E ') de ' Hilbert transform E E '

7 Example D(E)=step func+on 4 D(E) 2 ReG(E) D(E) = Θ(E E 1 ) Θ(E E 2 ) ReG(E) =P E 2 E 1 de ' E E ' = ln E 2 E E 1 E

8 Applica+on to point defects Assume localized basis set and perturba+on restricted to orbitals on defect U A = P A UP A P A = i i P B = i i A i A i (E H )G(E) =1 (E H 0 U)G(E) =1 G(E) = G 0 (E)+ G 0 (E)UG(E) G(E) = [1 G 0 (E)U] 1 G 0 (E) Dyson equa+on

9 Block form of matrices A B A B

10 Only need to invert finite matrix within subspace spanned by the defect! Solving for localized gap states Zeros of det of same finite matrix give defect localized states in gap

11 Advantages /disadvantages Region where poten+al perturbed much smaller than where wave func+on perturbed Can give wave func+on or GF on many atoms outside impurity region Gives symmetry resolved total changes in DOS in band con+nua as well as defect levels No problems with band alignment etc. Difficult to go beyond ASA and include relaxa+ons etc.

12 Self- Consistent Method for Point Defects in Semiconductors: Applica+on to the Vacancy in Silicon J. Bernholc, Nunzio O. Lipari, and Sokrates T. Pantelides Phys. Rev. 41,

13 Examples Ab Ini)o Calcula+on of Hyperfine and Superhyperfine Interac+ons for Shallow Donors in Semiconductors Harald Overhof and Uwe Gerstmann,Phys. Rev. 92,

14 Rela+on to linear response δn i δv j = P ij δn i = π 1 Tr ImδG ii (z) dz δg ii (z) = G ii (z) G ii 0 (z) = P ij = π 1 G ij 0 (z) G ji 0 (z)dz j G 0 ij (z)δv j G 0 ji (z) Useful to accelerate self- consistency convergence

Interfaces and surfaces Key concept: principal

$matrices, e.g. decima+on, con+nued frac+on etc.$

15 Interfaces and surfaces Key concept: principal layer Layer thick enough so that layer n interacts With n- 1 and n+1 but not further x x x x x x x x x x x x x x x x Hamiltonian becomes tridiagonal Special methods can be used to invert such matrices, e.g. decima+on, con+nued frac+on etc. See PGF program

16 Principal layer method Bulk Green s func+on can be obtained from difference equa+on whose eigenvalues give complex band structure A.B. Chen et al. PRB39,923(1989) The semi- infinite GF is given by s b G mn = G mn G b m0 [G b 00 ] 1 b G 0n The GF in central layer is obtained adding one layer at a +me star+ng from leu semi- infinite, then connec+ng it to semi- infinite on right,and retracing one s steps

0 SL 00 = G 00 [E1 11 H 11 H 10 G (SL) 0,0 H 01 ]G (1) 11 =1 11 Last step

17 General step (n 1) [E1 nn H nn H nn 1 G n 1,n 1 H n 1n ]G (n) nn =1 nn 0 1 n- 1 n N- 1 N N+1 Surface layer of semi- infinite leu First step H n- 1,n G 0 SL 00 = G 00 [E1 11 H 11 H 10 G (SL) 0,0 H 01 ]G (1) 11 =1 11 Last step (N 1) [E1 NN H NN H NN 1 G N 1N 1 SR H N 1N H NN+1 G N+1N+1 H N+1N ]G NN =1 NN

18 Retracing steps (N 1) G NN 1 = G NN H NN 1 G N 1N 1 (N 1) G N 1N 1 = G N 1N 1 (N 1) + G N 1N 1 H N 1N G NN 1

19 Faleev et al., Phys. Rev. B 71, (2005) Transport problems Interface region a l r b Leu semi- infinite C Right semi- infinite [z1 CC H CC H la G Ls aa (z)h al H rb G Rs bb (z)h br ]G CC (z) =1 CC Σ L (z) Γ L = i[σ L + Σ L ] = 2 Im Σ L + I = e π Σ R (z) de[ f L (E) f R (E)]Tr[Γ R G + Γ L G ] Fermi distribu+on for μ L G rl G lr

20 g and big G [P(z) S]g(z) =1 [z H ]G(z) =1 Par+al wave expansion of real space GF [z + 2 v(r)]g(r, r,', z) = δ(r r') G(r, r', z) = δ RR' LL' * + φ RL φ RL L * (z *, r R < (z * ', r R )G RL,R'L' (z)φ R'L' (z, r R' ) )W { φ RL (z),φ irr RL (z)} 1 φ irr RL (z, r > R ) O. Gunnarsson, O. Jepsen, and O. K. Andersen Phys. Rev. B 27, 7144 (1983)

So, if we know g(z) we can find G(z) and charge

1 2 d ln P(z) dz if choosing φ irr to match φ γ +[

21 So, if we know g(z) we can find G(z) and charge density, PDOS,etc. ρ(ε, r) = 1 π Im(G(r, r, i0 +) ε F ρ(r) = ρ(ε, r) dε = 1 2πi = 1 π G(r, r, z)dz Im G(r, r, z)dz C E F G(z) = 1 2 d ln P(z) dz if choosing φ irr to match φ γ +[ P(z)] 1/2 g(z)[ P(z)] 1/2 " D RL (z) = π 1 Im g RL,RL (z) P RL (z) 1 # $ 2 C d dz ln P % RL (z) & '

22 Advantages and disadvantages of g/g More localized purturba+on δp(z) or δs than δh Only works in ASA or difficult to make FP (FP- KKR uses cells) g(z) has extra poles from Pdot(z) 1 d ln P(z) = 1 2 dz z V P α 1 " C z (z) = (γ α) # $ z V α V α = C Δ γ α, z z + (z ε ν ) 3 p p = ( φ γ ) 2 % & ' In 2 nd order we know them exactly Third order correc+ons

23 Applica+on of GF s to magne+c exchange interac+ons θ/2 i j θ/2 FM star+ng point δe ij δe i δe j = J ij θ 2 / 4 Liechtenstein et al. JMMM 67, 65 (1987) Andersen force theorem ε F [ ] εδd(ε)dε = εδn(ε) ε F ε F δ N(ε)dε Because rota+on of spin does not change Charge density in sphere

24 Lloyd s formula Inverse Bloch- sum (IFT) gives us real space exchange interac+ons Spin waves J i, j+t = e ik T k J i, j (k) ) 2 det ω(k)δ ij 2 # & + % J ij (k) δ ij J ik (0)( 2 * + µ i µ i $ k ' µ j,. -. = 0

25 Example: ScN:Mn Exchange interac+ons from supercell between near neighbors Exchange interac+ons From Liechtenstein approach

26 Without gap correc+on LDA metallic RKKY like With gap correc+on: semiconduc+ng

27 Doping effects Changing E F allows to study doping effect

28 Importance for magne+c exchange Supercell approach of examining E AF - E F relies on short- range assump+on, so interac+ons with images outside cell are negligible This is ouen NOT TRUE What is calculated in SC approach is sum over man small long- range interac+ons Long- range and fluctua+ng exchange interac+ons Can easily examine Fermi level or doping dependence. Gain more insight.

29 Coherent Poten+al Approxima+on H = H 0 + (U m Σ) m m + Σ m H Σ 0 = H 0 + Σ, H Σ 1 = (U m Σ) m m = V m m m G 0 Σ (z) = G 0 (z Σ) G = G 0 Σ + G 0 Σ T Σ G 0 Σ T Σ m T m Σ T Σ m = (1 G 0 mm (z)v m ) 1 V m m m G = G 0 Σ T m Σ = 0 & dup(u) ( 1 (U Σ) 0 G Σ 0 0 ) 1 ) (U Σ) = 0 '( *+ c A T A Σ + c B T B Σ = 0 m Dyson eq Σ represents average medium self- energy Approxima+on : each sca@erer is independent Alloy m T- matrix for one sca@erer in medium Choose medium Σ so on average no sca@ering Random impuri+es

30 Disordered local moments Random poten+als correspond to random orienta+ons of spins Average in CPA is over all direc+ons of spins

31 Conclusions/summary Green s func+ons are useful for point defects, linear response, interfaces, transport, magne+c exchange interac+ons Allow self- consistent calcula+on via local G RR But also give long- range G ij containing useful informa+on on how defects spread, transport occurs etc. Mul+ple sca@ering g vs. Hamiltonian G At present: only some of it implemented, see lmgf, lmpg

ELECTRONIC STRUCTURE OF DISORDERED ALLOYS, SURFACES AND INTERFACES

ELECTRONIC STRUCTURE OF DISORDERED ALLOYS, SURFACES AND INTERFACES llja TUREK Institute of Physics of Materials, Brno Academy of Sciences of the Czech Republic Vaclav DRCHAL, Josef KUDRNOVSKY Institute