Linear response to an electric field: absorption and energy-loss Independent particle, Local fields effects, and Time-Dependent DFT

Transcription

1 Linear response to an electric field: absorption and energy-loss Independent particle, Local fields effects, and Time-Dependent DFT D. Sangalli

2 Motivations: probe your system Scattering of Transmission Beer-Lambert Law I=I0e-ax +- An instrument electrons or X-Ray Plasmons and Excitons (see also next talk ) to probe the energy levels of a material Many Body Effects!!!

3 Motivations: linear response Reaction is Stong perturbation hard to? predict? P = E + 2E2 + 3E3 + 4E4 +...

4 Motivations: linear response Reaction is Stong hard to perturbation predict?? P = E + 2E2 + 3E3 + 4E Reaction is Small perturbation Excuse me much easier to Yes.. predict P ~ E tip?

5 Motivations: theoretical spectroscopy By comparing ultraviolet-visible reflectance spectra of ancient and artificially aged modern papers with ab-initio TD-DFT estimate acting calculations, the as it abundance was of chromophores possible and oxidized functional groups and yellowing. yellowing A. Mosca Conte et al., Phys. Rev. Lett. 108, (2012) to identify responsible of paper Treasure map

6 Outline EXPERIMENS: Electron Energy Loss, Inelastic X-ray scattering and Absorption Connection to the microscopic dielectric function in linear response Correction to the non-interacting system: macroscopic induced field, local fields and correlation Final remarks, advanced topics and conclusions

7 EELS, IXSS and absorption Electron Energy Loss Spectroscopy Fast electrons used to probe VIS - UV Here the number of electrons is conserved q can be different from zero

8 EELS, IXSS and absorption Inelastic X-ray scattering spectroscopy X ray source used to probe NUV - EUV Here the number of electrons is conserved q can be different from zero

9 EELS, IXSS and absorption Here the number of electrons is conserved the Electric field in the VIS-NUV range is used to In the VIS / NUV range only q ~ 0 probe the VIS-NUV range

10 EELS, IXSS and absorption The number of electrons is conserved! VIS UV Energy: range valence conduction transitions are involved Dielectric function in the linear regime How can we compute the macroscopic dielectric function?

11 Poles of and -1 Transverse excitations Absorption has a peak if the imaginary part of, 2, has a peak For inter-band / intra-band transitions Longitudinal excitations Electron energy loss has a peak if: * the imaginary part of, 2, is small and the real part, 1, is zero For plasmons transitions * the real part, 1, is big and 2 has a peak small peaks for inter-band / intra-band transitions is a tensor, we just compute its longitudinal component thanks to the relation in the limit q 0 : L(q, )= T(q, )

12 Electric fields

13 From the fields to the potentials This is true only for longitudinal fields This is the classical induced field Which is the correct equation to use?

14 Micro Macro connection (1) (2) This implies that the macroscopic can be obtained via equation (1)

15 Micro Macro connection macro < < < > > < > < Average over the super-cell > macro > This is the dipole approximation: The external field is uniform in the supercell This is wrong We can compute the microscopic dielectric function and then do the average of its inverse How can we compute the variation of the density against the external potential?

16 The independent particle response δ ρip χ0= δ V ext χ 0 (r,r ',ω)= ij ψ j (r) ψ i (r) ψi (r ') ψ j (r ' ) ω Δ ϵij +i η It is easy to compute ϵ 0 (q, ω)=1 4π iqr iqr ' d r d r ' e e χ 0 (r, r ', ω) 2 q Contains informations on the eh transitions It is often not enough to describe the absorption or the electron energy +- loss of a system We need a theory for interacting particles

17 Interacting system

18 Interacting system δρi χ= δ V ext NI 0= V eff E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984)) Interacting System Non Interacting KS System Petersilka et al. Int. J. Quantum Chem. 80, 584 (1996) 1 [ 2 V eff r, t ] i r,t =i i r,t 2 t V eff (r,t )=V H (r, t )+V xc (r, t)+v ext (r, t ) V H V xc = 1 V ext V ext 0... by using... I = NI 0 V ext = V ext V H V xc v q, = 0 q, 0 q, v f xc q, q, f xc TDDFT is an exact theory for neutral excitations!

19 A two points equation in G-space χ (r + R, r ' + R ',t t ' ) χ G, G ' (q, ω) Repeated indexes are summed (G'',G''') χ G, G ' (q,ω)=χ G0, G ' (q, ω)+ χ 0G,G ' ' (q, ω)(δg ' ', G ' ' ' v G ' ' +f Gxc' ', G ' ' ' (q, ω)) χ G ' ' ',G ' (q,ω) We need the macroscopic component χ 0,0 (q, ω)=χ 00,0 ( q, ω)+ χ 00,G ' ' (q, ω)(δg ' ', G ' ' ' v G ' ' +f Gxc' ', G ' ' ' (q, ω))χ G ' ' ', 0 (q, ω) With a compact notation χ (q, ω)=χ 0 ( q, ω)+ χ 0 (q, ω)(v G=0 + vg >0 + f xc (q,ω)) χ (q,ω) Where we have matrix products in G space v G=0 G, G ' =δg, G ' δ G, 0 v 0 v GG >0, G ' =δg, G ' v G δ G, G ' δg, 0 v 0 It is convenient to split the induced field in a macroscopic term plus the microscopic components

20 The macroscopic induced field χ (q, ω)=χ 0 ( q, ω)+ χ 0 (q, ω)(v G=0 + vg >0 + f xc (q,ω)) χ (q,ω) For isolated system the is always zero, when other components are neglected the result is the IP approximation For extended system the is always important and so it is always included. When other components are neglected the result is the RPA without Local Fields approximation χ ( q, ω)=χ 0 (q, ω)+ χ 0 ( q, ω)(v G=0 + vg >0 + f xc (q,ω)) χ (q,ω) ϵ=(1+ v χ) 1 ϵ 1=1+v χ Francesco Sottile, PhD thesis χ (q, ω)=χ 0 (q, ω)+ χ 0 (q, ω)(v G >0 + f xc ( q, ω))χ ( q, ω) ϵ=1 v χ 1 1 ϵ =(1 v χ)

21 Difference between EELS and Abs χ (q, ω)=χ 0 ( q, ω)+ χ 0 (q, ω)(v G=0 + vg >0 + f xc (q,ω)) χ (q,ω) Francesco Sottile, PhD thesis χ=χ + χ v G=0 χ ℑ[ϵ]= ℑ[v χ ] 1 ℑ[ϵ ]= ℑ[v χ ]

22 The macroscopic induced field χ ( q, ω)=χ 0 (q, ω)+ χ 0 ( q, ω)(v G=0 + vg >0 + f xc (q,ω)) χ (q,ω)

23 The local fields effect χ (q, ω)=χ 0 ( q, ω)+ χ 0 (q, ω)(v G=0 + vg >0 + f xc (q,ω)) χ (q,ω) Francesco Sottile, PhD thesis The effects is usually small in solids and big in isolated systems / molecules It is important when the system is not uniform in the super-cell

24 The local fields effect χ (q, ω)=χ 0 ( q, ω)+ χ 0 (q, ω)(v G=0 + vg >0 + f xc (q,ω)) χ (q,ω) Ge-NWs 1D systems Parall dir Perp dir Phys. Rev. B 72, (2005) 1D systems For the same reason in 1D Perp dir (or also in 2D systems) the effect can be very dependent on the field polarization / hbn probed direction

25 TDDFT in G-space χ (q, ω)=χ 0 ( q, ω)+ χ 0 (q, ω)(v G=0 + vg >0 + f xc (q,ω)) χ (q,ω) The Adiabatic Local Density Approximation A xc [n] δ A xc [n] v xc [n](r, t)= δ n(r,t ) f adiab xc ALDA A xc [n]= dt d r e xc (n(r,t )) v ALDA xc e xc (n(r, t )) [n](r,t )= n δ v xc [n](r, t) δ v xc [n(t )](r, t) [n](r, t, r ',t ' )= δ(t t ' ) δ n(r ', t ' ) δ n(r ',t ) f ALDA (r, t, r ', t ' )=δ(t t ' )δ (r r ') xc 2 e xc (n(r, t)) 2 n

26 TDDFT in G-space χ (q, ω)=χ 0 ( q, ω)+ χ 0 (q, ω)(v G=0 + vg >0 + f xc (q,ω)) χ (q,ω) Small improvements over RPA with LF in solids Usually RPA/TDDFT very good in the description of EELS With the adiabatic LDA functional Small improvements over RPA in absorption. Phys. Rev. Lett. 86, 5962 (2001)

27 TDDFT in G-space χ (q, ω)=χ 0 ( q, ω)+ χ 0 (q, ω)(v G=0 + vg >0 + f xc (q,ω)) χ (q,ω) Small improvements over RPA with LF in solids Usually RPA/TDDFT very good in the description of IXSS Silicon, IXXS With the adiabatic LDA functional H. Weissker et al., Phys. Rev. Lett. 97 (2006)

28 Bad results on extended systems With the adiabatic LDA functional Silicon Solid Argon

29 TDDFT in the eh-space A theory for an electron-hole couple: we open the bubble χ ( q, ω)=χ 0 (q, ω)+ χ 0 (q, ω)(v G >0 + f xc ( q, ω)) χ ( q, ω) χ 1 (ω)=χ 1 0 (ω) v G>0 f xc (ω) L 1 (ω)=l 1 0 (ω) v G >0 f xc (ω) The equation is projected in the eh-space (KS) and rewritten As an eigenvalue problem 1 G>0 xc χ 1 (ω)=(χ ) (ω) v f ij, hk 0 ij, hk ij, hk ij, hk (ω) H 2p ij, hk =(ϵi ϵ j ) δi, h δ j, k (f j f i ) K ij, hk ' xc K ij, hk ' =v G>0 +f ij, hk ij, hk This can be diagonalized to obtain the eigenvalues (the poles of epsilon) and the eigenvectors, which describe the wave-function of the excitation

30 Good results for molecules With the adiabatic LDA functional Benzene K. Yabana, and G.F. Bertsch, J. Mod. Phys. 75 (1999)

31 Questions up to now? otherwise we go on with the advanced topics

32 Advanced topics: spin Spin is aways there Spin polarized systems Non polarized systems ( xc xc Δ ϵ v f v f xc xc v f Δ ϵ v f Δ ϵij, =Δ ϵij, 1 1 FT = 2 1 T S ψi, =ψi, 1 1 F S= 2 1 ( ) () xc +f xc ) a Δ ϵ 2v 2f TDDFT is an approx to TD-spin-DFT ij, hk ( Δ ϵij, Δ ϵij, ) ij, hk ψi, ψi, Spin symmetry is broken, double excitations needed M.E. Casida et al.: Linear-Response Time-Dependent Density Functional Theory for Open-Shell Molecules, Lect. Notes Phys. 706, (2006) xc xc Δ ϵ (f f ) Δ ϵ Δ ϵ 2v (f ) xc xc Δ ϵ v f v f xc xc v f Δ ϵ v f xc This will be also true in BSE P. Romaniello et al.: J. Chem. Phys. 130, (2009) D. Sangalli et at.: J. Chem. Phys (2011) Important for molecules, less (?) important for solids

33 Advanced topics: spin-simmetry In molecules for example the energy dissiociation of the H2 system is not correctly described To be explored in solids

34 Advanced topics: drude term The intraband transitions are hard to computed and require an ultra-fine sampling of the k-space A. Marini, PhD Thesis Thus in practice a drude model with the complex D parameter is implemented.

35 Advanced topics: better fxc Libcx library in yambo but one has to understand the physics: see next Lecture by M. Palummo

36 Advanced topics: surface spectr. C. Hogan, R. Del Sole, and G. Onida, Phys. Rev. B (2003)

37 Advanced topics: surface spectr. High-Resolution EELS Anisotropy Reflectance anisotropy spectra Si Si: RAS... missing surf excitons... GaAs: HREELS GaAs: RAS M. Palummo et al., Phys. Rev. Lett. 94, (2005)

38 Advanced topics: gauges The longitudinal part of the electro-magnetic field can be described in two gauges which are Often referred as length and velocity gauge J. E. Sipe et al., R. Del Sole et al., Phys. Rev. B 48, (1993) Phys. Rev. B 29, 4631 (1984) The length gauge is particularly appealing in the, dipole approximation, i.e. spatially uniform field. Beyond it requires an infinite multi-poles expansion and misseses the transverse terms. What about the transverse part??

39 Advanced topics: the Kerr effect The dielectric tensor (velocity gauge), the optical conductivity tensor and the Kerr paramenters Mz Magneto optical Kerr effect An example where the transverse part shows up hard (q 0 limit) in the longitudinal gauge

40 Advanced topics: the Kerr effect Fe D. Sangalli et al., Phys. Rev. B 86, (2012) In Metals IP-RPA is usually very good

41 Advanced topics: dichroism D. Varsano et al. Chem. Phys. Chem. 11, 4481 (2009) We need to consider the terms proportional to the magnetic field / the the induced magnetization

42 Advanced topics: dichroism Optical Rotatory Dispersion (ORD) Electric Circular Dichroism (ECD) S. Nielsen et al. Chem. Phys. Chem. 6, 2619 (2005) TDDFT: good results for molecules

43 Acknowledgments To prepare this lecture I used slides and pictures from many other lectures, papers, thesis, etc so thanks to: Matteo Gatti, Francesco Sottile, Claudio Attaccalite, Andrea Marini, Valerio Olevano, Conor Hogan, Daniele Varsano, Maurizia Palummo and the ETSF and...

44 Thank you for your attention!