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TDDFT Excitons TDDFT for extended systems II: Excitons Carsten A. Ullrich University of Missouri Rutgers University, August 9, 2019 2 Outline ● Introduction to excitons ● TDDFT for periodic systems ● Optical spectra and exciton binding energies ● xc functionals for excitons ● TDDFT vs. BSE: derivation of Wannier equation https://etsf.polytechnique.fr/Research/Excitons 3 Absorption of light in a solid Matter Light Absorption Transmission Let us consider the absorption of light in a solid with a gap. 4 Absorption of light across the band gap CB GaAs hν VB ► Light comes in with photon energy at least as large as the band gap ► Photon gets absorbed, promotes electron across the gap, leaving a hole behind 5 Absorption spectra of insulators/semiconductors CB hν VB will produce an absorption spectrum like this: Absorption photon energy Eg 6 Absorption spectra of insulators/semiconductors ZnO GaAs Eg R.G. Ulbrich, Adv. Solid State Phys. 25, P. Gori et al., Phys. Rev. B 81, 125207 (2010) 299 (1985) In the experiment, one finds sharp peaks at the absorption threshold… In fact, there are peaks below the band gap energy: Excitons. 7 What is an exciton? ► After their creation, the electron - and the hole are not completely free, but experience Coulomb attraction. + ►This gain in electrostatic energy can lower the onset of absorption, and change the spectral strength. Excitons are bound electron-hole pairs. 8 Elementary view of Excitons Mott-Wannier exciton: Frenkel exciton: weakly bound, delocalized tightly bound, localized on over many lattice constants a single (or a few) atoms ►In semiconductors with small ►In large-gap insulators, or band gap and large ε in low-ε organic materials 9 Excitonic features in the absorption spectrum H. Haug & S.W. Koch Quantum Theory of the Optical and Electronic Properties of Semiconductors H-atom: Elliott formula optical gap ● Sharp peaks below the onset of the single-particle gap ● Redistribution of oscillator strength: enhanced absorption close to the onset of the continuum 10 Wannier equation and excitonic Rydberg Series ● ϕ()r is exciton wave function 22 2 e ● includes dielectric screening r ()rr E () ● derived from Bethe-Salpeter eq. 2mrr Sham and Rice, Phys. Rev. 144, 708 (1966) Cu2O GaAs R.G. Ulbrich, Adv. Solid State Phys. 25, R.J. Uihlein, D. Frohlich, and R. Kenklies, 299 (1985) PRB 23, 2731 (1981) 11 Excitons in nanoscale systems G. D. Scholes and G. Rumbles, Nature Mater. 5, 683 (2006) Jang & Mennucci, Rev. Mod. Phys. 90, 035003 (2018) Frenkel excitons in light-harvesting systems: purple bacteria 12 Optical excitations in bulk Insulators Optical transitions in insulators are challenging for TDDFT: ● band gap opening ● excitons 13 Insulators: three different gaps The Kohn-Sham gap approximates the optical gap (neutral excitation), not the band gap! Band gap: Eg = Eg,KS + ∆ xc optical exciton Optical gap: Eg = Eg − E0 14 Hybrid functionals for the band gap Matsushita, Nakamura and Oshiyama, PRB 84, 075205 (2011) see also Skone, Govoni and Galli, PRB 93, 235106 (2016) 15 Excitons: comparison of first-principles methods* L. J. Sham and T. M. Rice, Phys. Rev. 144, 708 (1966) M. Rohlfing and S. Louie, PRB 62, 4927 (2000) G. Onida, L. Reining, R. Rubio, RMP 74, 601 (2002) S. Sharifzadeh, J. Phys.: Cond. Mat. 30, 153002 (2018) Many-body perturbation theory: Based on Green’s functions ● moves (quasi)particles around ● one-particle G: electron addition and removal – GW ground state ● two-particle L: electron-hole excitation – Bethe-Salpeter equation ● intuitive: contains the right physics (screened e-h interaction) by direct construction Time-dependent DFT: Based on the electron density ● moves the density around ● Ground state: Kohn-Sham DFT ● response function χ: neutral excitations of the KS system ● efficient (all interactions are local), but less intuitive how the right physics is built in * Matteo Gatti, TDDFT School 2010, Benasque 16 Excitons with TDDFT: equivalent approaches 1. Calculate the dielectric function via Dyson equation (computationally more efficient, gives optical spectrum) 2. Solve Casida equation (more expensive, gives precise exciton binding energies) 3. Real-time TDDFT (potentially even more efficient) C.A. Ullrich and Z.-H. Yang, Topics in Current Chem. 368 (2015) Y. -M. Byun and C.A. Ullrich, Phys. Rev. B 95, 205136 (2017) T. Sander and G. Kresse, JCP 146, 064110 (2017) 17 TDDFT Linear response in periodic systems χ ′ ω = χ ′ ω + 3 3 ′χ ω (r,r , ) s (r,r , ) ∫ d x∫ d x s (r,x, ) 1 × + f xc (x,x′,ω)χ(x′,r′,ω) | x − x′ | Periodic systems: χ(r,r′,ω) = χ(r + R,r′ + R,ω) Fourier transform: χω(,,)rr′′=∑∑ee−ii(qGr +⋅ )( qG +′′ ) ⋅ rχ(qGqG++ , ,) ω q∈BZ G,G′ χ(,qq ωχ )= (, ω ) + χ (, q ω ) GG′′ssGG ∑ GG1 GG12 ×+{VfG()qδ GGxc GG (, qqωχ )} GG′ (, ω ) 1 12 12 2 18 The dielectric tensor ∂B ∇ ⋅D = n ∇×E = − free ∂t Maxwell equations ∂D ∇ ⋅B = 0 ∇× H = j + free ∂t 3 Def. of dielectric tensor: D(r,ω) = ∫ d r′ε (r,r′,ω)E(r′,ω) In periodic solids: Dq(,ωεω )= (, q ) E (, q ω ) GG∑ GG′ ′ G′ This is the microscopic dielectric tensor. But for comparison with spectroscopy, we would like the macroscopic dielectric function: () () () DEmac mac mac Problem: we cannot calculate the macroscopic dielectric function directly! This would ignore the local-field effects (microscopic fluctuations). 19 Homogeneous systems In a homogeneous, isotropic system, things would be easy: ε hom (ω) = limε hom (q,ω) mac q→0 hom hom T hom T and ε (q,ω) = ε L (q,ω)qˆqˆ + εT (1− qˆqˆ ) hom hom and ε L (0,ω) = εT (0,ω) The connection to optics is via the refractive index: ~ 2 ε mac (ω) = n 2 2 Reε mac = n +κ Imε mac = 2nκ 20 The macroscopic dielectric function −1 For cubic symmetry, −1 εω( )= lim ε′ (q , ω ) =0 one can prove that mac → GG G q 0 G′=0 εωGG′ (q , ): longitudinal component of dielectric tensor (a.k.a. dielectric matrix) To make progress, we need a connection with response theory: scalar n (,)r′′ ω ω= 33′′ εω ω+ ′′ 1 dielectric V11(,r )∫∫ dr (,rr , ) V (, r ) dr function: |r′- r ′′ | − χω(,,)rr′′ ′ so that ε13(,rr′ , ωδ )= ( r −+ r′ ) dr ′′ ∫ |-rr′′ | −1 and for a periodic system, εGG′′′(,q ωδ )=GG +V G () qq χ GG (, ω ) 21 The macroscopic dielectric function From this, one obtains εωmac ( )= 1 − limV0 (qq ) χ 00 ( , ω ) q→0 There is a subtle, but very important point to be noted. Here we use a modified response functionχωGG′ (q , ): χ(,qq ωχ )= (, ω ) + χ (, q ω ) GG′′ssGG ∑ GG1 GG12 ×+{VfG()qδ GGxc GG (, qqωχ )} GG′ (, ω ) 1 12 12 2 0 forG = 0 where the long-range part of the Coulomb V ()q = 4π interaction has been G ≠ 2 forG 0 removed: ||qG+ G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002) 22 Density response of periodic systems ext Hxc δnG (,q ω )= ∑∑ χsGG′ (, q ω ) δVG′ (, q ω )+ fnGG′ ′′ (, qqωδ )G′′ (, ω ) GG′ ′′ Loss function: Optical absorption: response to a response to total microscopic macroscopic external scalar potential. classical perturbation. Loss spectrum Optical spectrum includes plasmons. includes excitons. Density eigenmode: Density eigenmode: set set ext ext H δωVG′ (,q )= 0 δVG′ (,qq ω )+= fn00 δω 0 (, ) 0 23 Excitation energies from TDDFT Excitation energies follow A B X − 10 X from eigenvalue problem ** = Ωn (Casida 1995): B A Y 01Y Hxc Avckk,, v′′ c ′ =−+() EE c k v kδδδ vv′ cc ′ kk ′ Fvc kk v ′′ c ′ Hxc BFvckk,, v′′ c ′= vc kk v ′′ c ′ 24π ⋅ −⋅ H= iiGr ′′Gr ′′ Fvckk, v′′ c ′ ∑ 2 cek v kk v e c k V G0≠ G xc 2 i((q+ G)r′ ⋅ ′′−i q +⋅ G)r ′′ Fvck,, v′′ c k ′ = limq→ 0∑ fxc GG′()qk ce v k v k e c k V GG′ 24 Direct calculation of exciton binding energies δδω+ Hxc + Hxc = −Ω ∑∑vkk,, v′′ c kk c ′′ cv kF vc k , v′′′ c k X v ′′′ c k FYvck, v′′′ c k v ′′′ c k n X vck vc′′kk ′ vc′′ ′ Hxc +δδω +=Hxc Ω ∑∑FXvck, v′′ c k v ′′′ c k vkk,, v′′ c kk c ′′ cv kFYY vc k , v′′′ c k v ′′′ c k n vc k vc′′kk ′ vc′′ ′ Tamm-Dancoff Approximation (TDA) Using time-reversal symmetry, Full Casida eq. can be transformed into δδω22+= ωω Hxc Ω ∑ vkk ,v′′ c kk ,c ′′ vc k2 cv k c′′′ v kFZ vc k, v ′′′ c k v ′′′ c k n Z vck vc′′k ′ T. Sander, E. Maggio, and G. Kresse, PRB 92, 045209 (2015) More expensive than calculating Im ε(ω), but more precise (no artificial line broadening) 25 Optical absorption in Insulators: TDDFT Silicon RPA and ALDA both bad! ►absorption edge red shifted (electron self-interaction) ►first excitonic peak missing (electron-hole interaction) Why does the LDA fail?? ►lacks long spatial range ►need new classes of xc functionals G. Onida, L. Reining, A. Rubio, RMP 74, 601 (2002) S. Botti, A. Schindlmayr, R. Del Sole, L. Reining, Rep. Prog. Phys. 70, 357 (2007) 26 The xc kernel for periodic systems ′ i(q+G)r −i(q+G′)r f xc (r,r ,ω) = ∑ ∑e f xc,GG′ (q,ω)e q∈FBZ G,G′ TDDFT requires the following matrix elements as input: xc i((q+⋅ G) r ′′−i q +⋅ G) r ′′ Fvck,, v′′ c k ′ = lim q→ 0∑ fxc GG′(,)qω ce k v kk v e c k GG′ Most important: long-range ( q → 0 ) limit of “head” (G = G′ = 0) : 1 cekkiqr v → q f exact (q,ω) → q0→ xc,00 q→0 q2 ALDA → Therefore, no excitons but fxc,00 (q )q→0 const.
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