Optical properties by wien2k

Transcription

1 Optical properties by wienk Robert Laskowski star.edu.sg Institute of High Performance Computing Singapore

2 outline Theory: independent particle approximation optic, joint, tetra. inputs / outputs, examples

3 Dielectric function V ext (r,t ) V ext (r,t )+V s (r,t ) Electrons respond, screening potential (Vs) V (r,t )=V ext (r,t )+V s (r,t ) Photon electric field V ext G Screening potential = ϵgg' V G ' G' dielectric constant

4 Independent particles single particle eigenstates (IPA) H 0 nk =ε nk Time dependence in the linear regime: V ext,v s, n e general form of the potential V (r )= 1 iωt i (q+g)r VGe Ω q,g irreducible of polarizability: P= n V 0 ng q, = P G G' q, V G' q, G' P 0 GG ' f m,k q f l,k 1 q, = [MGlm k,q ] MG' lm k,q lmk m,k q l, k G i q G r M lm k,q = lk e m, k q

5 Random phase approximation The screening potential Vs connects to the induced charge density n by Poisson equation Definition of ϵgg: GG'= GG' q G P V sg=ν(q+b)ng (q) V ext G 0 GG' = ϵg G' V G ' G' q, random phase approximation (RPA) S. L. Adler, Phys. Rev. 16, 413 (196) N. Wiser, Phys. Rev. 19, (1963)

6 Macroscopic dielectric constant Macroscopic external field has no G dependence ext V G (q)=δg, 0 V ext (q) ext ext V G = ϵg G ' V G ' V 0=ε 1 V 00 0 G' with local field effects: 1 M q, = q, neglecting local field effects: ϵm (q, ω)=ϵ00 (q, ω)=1 v (q)p 0 (q, ω)

7 ϵm (q, ω) in periodic systems ϵm (q, ω)=1 v (q) P 0 (q, ω) 4πe ε(q,ω)=1 lim η 0 q Ω A A i, j k,q k,i, j f (ε i, j' k,q i k +q ε intra-band transitions j k ) f (ε ) j k ε ω i η q =δ i, j +(1 δ i, j ) P i, j m ωi, j intra-band inter-band transitions i k +q inter-band

8 Long wave limit Find q 0 limit of P, use k p method 0 P (q 0,ω)=4 π vck v k pi c k c k p j v k (εck εvk ω)(εck εvk ) The expression for calculations of single particle excitation spectra 16 π ℑ(ϵij )= Ω ω vk pi ck ck p j vk δ(εkc εvk ω) vck Key quantity is the momentum matrix (optic program): vk pi ck

9 Interpretation joint density of states δ(εkc εvk ω) hν vck transition probability 16 π ℑ(ϵij )= Ωω vk pi ck ck p j vk δ(εkc εvk ω) vck

10 Momentum matrix elements vk pi ck Ψ v k Ψc k xi l,m determines character of the state Derivative of the wave function in z direction

11 Interpretation L vk pi ck L W W L±1 ± L character of the valence state couples to L-1 or L+1 character of the conduction band 16 π ℑ(ϵij )= Ωω vk pi ck ck p j vk δ(εkc εvk ω) vck

12 Symmetry triclinic monoclinic (α, β = 90 ) orthorhombic tetragonal, hexagonal cubic

13 Optical functions Dielectric tensor 16 ℑ ij = vk pi ck ck p j vk kc vk vck ω ' ℑ ϵi j (ω ') ℜ ϵi j =δi j π P d ω' ω ' ω 0 Optical conductivity Refractive index ℜ i j = (ω) ni i = ϵi i (ω) +ℜϵi i (mii 1) +k ii Reflectivity Ri i (ω)= Absorption Aii (ω)= Loss function Lii (ω)= ℑ( ℑ 4 i j (n ii +1)+k ii ωk ii (ω) c 1 ) ϵii (ω) i i ℜ i i k i i =

14 Magneto-optics Cubic, no SOC KK Cubic, with SOC and magnetism along z KK KK

15 How to do calculations? Run SCF calculations Increase k-mesh!!! compute momentum matrix elements with optic program (x optic) compute imaginary part of the dielectric function with joint program (x joint) use kram program for computing other optical constants (x kram)

16 Computing momentum matrix elements After running SCF, or restoring old run update potential (x lapw0) use better k-mesh (x kgen) generate eigenvectors (x lapw1 -???) Calculate momentum matrix elements (x optic -???) vk pi ck vk p j ck

17 optic program input, output input OFF output case.inop number of k-points, first k-point energy window for matrix elements number of cases (see choices) Re <x><x> Re <z><z> Im <x><y> write not-squared matrix elements to file? case.symmat vk pi ck vk p j ck case.mommat (ON) vk p j ck Choices: 1...Re<x><x>...Re<y><y> 3...Re<z><z> 4...Re<x><y> 5...Re<x><z> 6...Re<y><z> 7...Im<x><y> 8...Im<x><z> 9...Im<y><z>

18 Integrate joint DOS, joint program x joint, computes imaginary part of the dielectric tensor components, and more get Fermi level (x lapw -fermi -???) do the BZ integration (x joint -???) 16 π ℑ(ϵij )= Ωω vk pi ck ck p j vk δ(εkc εvk ω) vck

19 joint program input, output output input case.injoint ev case.joint lower and upper band index Emin, de, Emax [Ry] output units ev / Ry switch number of columns broadening for Drude terms choose gamma for each case! Switch: 0...JOINT DOS for each band combination 1...JOINT DOS sum over all band combinations...dos for each band 3...DOS sum over all bands 4...Im(EPSILON) total 5...Im(EPSILON) for each band combination 6...intraband contributions 7...intraband contributions including band analysis

20 kram program input, output input case.inkram (metal) broadening gamma energy shift (scissors operator) add intraband contributions 1/0 plasma frequency broadening for intraband part case.inkram (semiconductor) broadening gamma energy shift (scissors operator) add intraband contributions 1/0 Intra-band contribution output case.epsilon case.sigmak case refraction case.absorp case.eloss ω ' ℑ ϵ i j (ω ') ℜ ϵ i j =δ i j π P dω' ω ' ω 0

21 Example: Al, k-point convergence always check k-point convergence (use dense k-mesh!!!)

22 Example: Al, sumrules for KK transformation you need Im(ε) in a wide energy range ω ' ℑ ϵ i j (ω ') ℜ ϵ i j =δ i j π P dω' ω ' ω 0 be aware that LAPW linearization breaks down for high conduction states!!!

23 Theory vs experiment W. Werner, et al J. Phys. Chem. Ref. Data 38, 1013 (009)

24 Core level spectroscopy absorption (XAS) emission (XES) core-hole

25 Core level spectroscopy (XAS) X-ray absorption (XAS) core electrons are excited in to a conduction band each core shell introduces an absorption edge, (indexed by the principal number of a core level) hv L3 L p1/ L1 K s 1s p3/ SOC core levels

26 Core level spectroscopy (XES) X-ray emission (XES) Knock out core electron, valence electron fills core level and hν is emitted hv L3 L p1/ L1 K s 1s p3/ SOC core levels

27 XAS, XES, final state rule Final state determines the spectrum XAS - final state has a hole in core state, and additional e- in conduction band. Core-hole has large effect on the spectrum XES - final state has filled core, but valence hole, this is usually well screened, thus one sees the groundstate.

28 Core hole in wienk No core hole (ground state) Z+1 approximation (eg., replace C by N) usually not a good approximation (maybe in metals?) also not very good Core-hole (supercell) calculations: remove 1 core electron on ONE atom in the supercell, add 1 electron to conduction band remove 1 core electron, add 1 electron as uniform background charge, considers statically screened e-h coulomb correlation fractional core hole (consider different screening)

29 XAS, Mg K-edge in MgO xx supercell calculation, with core hole in one of the Mg atoms. This allows the conduction state to relax (adjust to the larger effective nuclear charge), but also to have static screening from the environment. core hole, no supercell: Z+1 (AlO) groundstate

30 Core-hole approach, limitation absorption is proportional to the projected DOS (empty band) ℑ M = vk p ck E ℏ ck vk branching ratio (eg. L/L3) is proportional to multiplicity of involved core states (p1/, p3/) L,3 edge of Ca in CaF L3 L plain DFT core hole DFT (supercell with core hole on excited atom)

31 L/L3 edges of 3d metals L3 L

32 Core-hole calculation in wienk generate super-cell (x supercell) initialize SCF, define core hole/add extra valence electron run SCF remove extra valence electron execute xspec task in wweb calculate eigenstates (x lapw1 -up/dn) calculate partial charges (x lapw -qtl -up/dn) execute x xspec Dipole approximation

33 Core-hole calculation in wienk generate super-cell (x supercell) initialize SCF, define core hole/add extra valence electron remove extra valence electron run SCF execute xspec task in wweb calculate eigenstates (x lapw1 -up/dn) calculate partial charges (x lapw -qtl -up/dn) execute x xspec Dipole approximation

34 xspec input file XES NbC: C K 1 0 0,0.5,0.5-0,0.1,3 EMIS AUTO XAS (Title) ( atom) (n core) (l core) (split, int1, int) (EMIN,DE,EMAX in ev) (type of spectrum) (S) (gamma0) (W) (band ranges AUTO or MAN (E0 in ev) (E1 in ev) (E in ev) NbC: C K (Title) (atom) 1 (n core) 0 (l core) 0,0.5,0.5 (split, int1, int) -,0.1,30 (EMIN,DE,EMAX in ev) ABS (type of spectrum) 0.5 (S) 0.5 (gamma0) Details in user guide

35 Core level spectroscopy (XMCD) X-ray magnetic circular dichroism (optic program) Independent particles approximation L Pardini, et al Computer Physics Communications 183 (01)

36 Core level spectroscopy (XMCD) X-ray magnetic circular dichroism (x optic) case.inop XMCD 1 L OFF : NKMAX, NKFIRST : EMIN, EMAX, NBvalMAX : optional line: for XMCD of 1st atom and L3 spectrum : number of choices (columns in *symmat) Re<x><x> Re<y><y> Re<z><z> Re<x><y> Re<x><z> Re<y><z> : ON/OFF writes MME to unit 4 conduction states are calculated with SOC Core states are calculated with lcore program (atomic Dirac solver)

37 Core level spectroscopy (XMCD) X-ray magnetic circular dichroism (x joint) ev XMCD : LOWER,UPPER,upper-valence BANDINDEX : EMIN DE EMAX FOR ENERGYGRID IN ryd : output units ev / ryd : omitt these 4 lines for non-xmcd : core energies in Ry (grep :P case.scfc) : core-hole broadening (ev) for both core states : spectrometer broadening (ev) : SWITCH : NUMBER OF COLUMNS : BROADENING (FOR DRUDE MODEL - switch 6,7)

38 Core level spectroscopy (EELS) ELNES vs XAS

39 EELS in wienk (telnes program) Within the dipole approximation the momentum transfer vector in non-relativistic EELS plays the same role as polarization vector in XAS telnes3 program also handles non-dipole transitions and relativistic corrections See details in users guide

40 Theory vs experiment details of the band structure matter band gap problem independent particles approximation, RPA