Optical Properties with Wien2k

Transcription

1 Optical Properties with Wien2k Elias Assmann Vienna University of Technology, Institute for Solid State Physics Aug 13 Menu 1 Theory Screening in a solid Calculating ϵ: Random-Phase Approximation 2 Practical Calculations optic: Momentum Matrix Elements joint: Imaginary Part of Dielectric Tensor kram: Derived Quantities 3 Examples Ambrosch-Draxl and Sofo, Comp. Phys. Commun. 175, 1 (2006)

2 Appetizer optical conductivity Re σ ij = ω 4π Im ϵ ij refractive index n ii = extinction coefficient k ii = ( ϵ ii + Re ϵ ii )/2 ( ϵ ii Re ϵ ii )/2 Im ϵ ij (ω) absorption coefficient energy loss function α ii = 2ωk/c L ij = Im(ϵ 1 ) ij reflectivity R ii = (n 1)2 + k 2 (n + 1) 2 + k 2 ω sum rules N eff = dω Im ϵ(ω ) 0 Screening Consider a test charge Q in a solid: V(r r ) = Q r r V(q) = 4πQ q 2 e will move to screen the charge effective potential W; dielectric function V = ϵ W Simplest model: Thomas-Fermi W(r) = e k TFr r 4π W(q) = k 2 TF + q 2 k 2 TF = 4πN (E F)

3 Ansatz for W d d W dr dt ϵ 1 (r ; t) V(r r ; t t ) R r = R + d Bare V(r, r ; t, t ) = V(r r )δ(t t ) is translation invariant and instantaneous Response depends on position in unit cell, is retarded W R (d, d ; t) = dd 1 dd 2 ϵ 1 R(d 1, d 2 ; t) R V(R + d d [d 1 d 2 R]) The Dielectric Function q W G (q, ω) = ϵ 1 GG (q, ω) V G (q, ω) G light is long-wavelength: W G (q, ω) ϵ 1 G0(q, ω) V 0 (q, ω) G k = G + q G = 0, q 0 macroscopic ϵ (u.c. average): W(q, ω) = ϵ 1 00(q, ω) V 0 (q, ω) ϵ M (q, ω) = 1 ϵ 1 00(q,ω) neglect local-field effects: ϵ M (q, ω) ϵ 00 (q, ω)

4 Calculating ϵ: The RPA V(q) = ϵ(q, ω) W(q, ω) ˆ Poisson: q 2 W = 4π ( Q + δn) W = V + 4π q 2 δn ˆ linear response: δn = χ V P W V = (1 4π q 2 P) W ˆ random-phase approximation: P to lowest order P = G 0 (1, 2) G 0 (2, 1) Intra- and Interband transitions free e : Lindhard formula P = 4π f (ε k+q ) f (ε k ) q 2 Ω ε k k+q ε k ω intraband interband Bloch e : P = 4π q 2 Ω knn A nn kq f (ε k+q ) f (ε k ) ε k+q ε k ω

5 Intra- and Interband transitions intraband: Drude model, (ω p : plasma frequency) Im ϵ intra = ω p 2 ω (ω ) interband: joint density of states: ρ(ω) = dk δ ε c (k) ε v (k) ω c,v v-c transition probability ( selection rules ) given by momentum matrix elements Im ϵ ij (ω, 0) 1 dk δ ε ω 2 c (k) ε v (k) ω c,v ck p i vk vk p j ck Symmetry Constraints ϵ ij = ϵ ji is always symmetric. Additional constraints from crystal symmetry: ϵ = U 1 ϵ U cubic tetragonal, trigonal, hexagonal monoclinic orthorhombic triclinic

6 Program Flow lapw1 Kohn-Sham eigenstates optic momentum matrix elements (case.symmat) joint imaginary part of dielectric tensor (case.joint) kram derived quantities ˆ Kramers-Kronig Re ϵ ij = δ ij + 2 π P dω all optical constants 0 Ω Ω 2 ω 2 Im ϵ ij optic: Momentum Matrix Elements 0 normal SCF run converged density 1 x kgen dense k-mesh (check convergence!) 2 x lapw1 -options eigenvectors on dense mesh 3 x lapw2 -fermi -options case.weight ˆ metals: TETRA in case.in2 4 x optic -options momentum matrix elements case.symmat: ck p i vk vk p j ck core-level spectra: Kevin Jorissen s lecture tomorrow 10:30

7 case.inop optic: Input and Output #k-points, 1st k-point Emin Emax [Ry], NBvalMAX 2 #indep. elements (symmetry/soc) 1 Re xx 3 Re zz OFF 3 write mommat2?, #spheres spheres to sum over symmetry 1: Re xx 4: Re xy 2: Re yy 5: Re xz 3: Re zz 6: Re yz spin-orbit 7: Im xy 8: Im xz 9: Im yz case.symmat vk p i ck ck p j vk case.mommat2 (if ON) vk p i ck joint: Im(ϵ), (Joint) Density of States case.injoint lower, upper, upper-val bandindex Emin ( 0), de, Emax [Ry] ev units [ev / ryd / cm-1] 4 mode 2 #indep. elements broadenings for Drude (mode=6,7) case.joint Im ϵ ij dk δ ε c (k) ε v (k) ω ck p i vk vk p j ck ρ 1 c,v

8 joint: Modes of Operation physical (all bands) band analysis 1 joint DOS 0 joint DOS 3 regular DOS 2 DOS 4 Im ϵ interband 5 interband 6 Im ϵ intraband (Drude) 7 intraband Im ϵ ij c,v,k δ ε c (k) ε v (k) ω ck p i vk vk p j ck sphere analysis ck = MT,I α ck α NB: cross-terms are missed! case.inop OFF 3 mommat2?, #spheres spheres to sum over kram: Kramers-Kronig Analysis case.inkram 0.1 interband broadening 0.0 energy shift (scissors operator) 1 add intraband contributions? 1/ plasma frequencies (joint, mode=6) broadenings for Drude models output ˆ case.epsilon Re ϵ, Im ϵ ˆ case.sigmak Re σ, Im σ ˆ case.sumrules ˆ case.absorp Re σ, α ˆ case.eloss loss function ˆ case.reflectivity R ˆ case.refraction n, k

9 More Stuff You May Need to Know spin-polarized calculations Kramers-Kronig is not additive. 1 x joint -up && x joint -dn 2 addjoint-updn 3 x kram procedure for metals 1 x joint (mode=6) plasma frequencies ω pij 2 x joint (mode=4) interband Im ϵ 3 x kram (intra=1, insert ω p ) Im ϵ intra = ω p 2 ω (ω ), Re ϵintra = 1 ω p ω Kramers-Kronig needs Im ϵ in a large energy range Some Limitations ˆ linear optical properties only ˆ W = ϵ 1(1) V + ϵ 1(2) V 2 + ˆ Kohn-Sham eigenstates interpreted as excited states scissors operator: ε c (k) ε LDA (k) + Δ c ˆ independent-particle approx. (no e h + interaction) Bethe-Salpeter (BSE) Peter Blaha s lecture (13:00) ˆ LDA/GGA are not exact hybrid DFT, effective potentials Peter Blaha DFT+U, LDA+DMFT my lecture (tomorrow 9:00)

10 Example: Al, k-mesh Convergence Comparison to Experiment REELS = reflection electron energy loss spectroscopy Optical Constants for 17 Elemental Metals Werner et al., Phys. Chem. Ref. Data 38, 1013 (2009)

11 Comparison to Experiment REELS = reflection electron energy loss spectroscopy Optical Constants for 17 Elemental Metals Werner et al., Phys. Chem. Ref. Data 38, 1013 (2009)