297 K 297 K 83 K PRB 36, 4821 (1987) C. Tarrio, PRB 40, 7852 (1989)

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1 Finite temperature calculations of the electronic and optical properties of solids and nanostructures: the role of electron-phonon coupling from Initio perspective Andrea Marini National Reserach Council (CNR), Italy June 29th 2012, Tokyo Zero-Point Motion an Ab-

2 Real life is at finite temperature (I) 297 K PRB 36, 4821 (1987) C. Tarrio, PRB 40, 7852 (1989) 8K 297 K 83 K

3 Real life is at finite temperature (II)... unfortunately theorists do not even bother to compare their calculations with low-temperature measurements, using more easily accessible room temperature spectra." 793 K = 68 mev QP gap is 1200 mev YES! We do need phonons. At the absorption threshold the QP lifetime is EXACTLY infinite

4 Real life is at finite temperature (III) phonon sidebands G. Baldini, A. Bosacchi, and B. Bosacchi, PRL, 23, 846 (1969) Phonons are responsible for the low-temperature saturation of the excitonic lifetime PRL 95, (2005) PRL 95, (2005) There are many different flavours of excitons: polaritons, polaronic excitons... whose physical properties are influenced by the excitonphonon interaction

5 Electron-phonon today An ultra-short laser pulse pumps electrons in the conduction The non-thermal electronic distribution relaxes via e-e and e-ph scatterings The electronic population is probed after varying delays with photoemission See Monday's talk!

6 ℏ u T 1 2N Bose T 4M 2 The quantistic zero-point motion effect

7 THE Motivation Electron-Phonon Are purely electronic theories reliable???

8 THE limitations... In this talk... I will not extensively discuss formal aspects (vertex, conserving props., ) I will (try) not do extensive and heavy math (see references for that) I will not talk about superconductivity, Fröhlich or Holstein Hamiltonians I will use LDA or GGA, nothing more complicated (including SC-GW)

9 Outline (I) Ab-Initio Polarons Finite temperature excitons Spectral functions and the QP-approximation

10 Outline (II) Ab-Initio Polarons The Heine-Allen-Cardona Approach (static) The Hedin-Lundqvist approach (dynamical) The Diagrammatic approach (dynamical) Density Functional Perturbation Theory Finite temperature excitons The Bethe-Salpeter equation in the polaronic basis A phonon induced kernel for the Bethe-Salpeter equation Finite temperature optical spectra of solids Spectral functions and the QP-approximation q Quasiparticles and spectral functions Polarons as entangled electron-phonon states k k q

11 = Independent QPs Indirect exciton-phonon scattering Ab-Initio Polarons

12 The quasi-harmonic approximation Phonon population Thermal expansion Grüneisen parameters It can be easily calculated using DFT and finite-differences

13 The Heine-Allen-Cardona Approach (I) For a review see M. Cardona, Solid State Commun. 133, 3 (2005). Is I st Using standard 1 and theory and the fact that 2nd order perturbation Debye-Waller Now we can rewrite the displacement operator using the canonical operators Fan

14 The Heine-Allen-Cardona Approach (II) v By imposing an acoustic sum rule Non-Diagonal DW corrections can be important in isolated systems like molecules X. Gonze et al, Annalen der Physik 523, 168 (2011) Clear dependence on the temperature Polaron damping neglected

15 The Heine-Allen-Cardona Approach (III) R. B. Capaz et al. PRL. 94, (2005) F. Giustino, et al. PRL, 105, (2010) E. Cannuccia PRL 107, (2011) Electronic Gap: ev Renormalization: 615 mev S. Zollner et al. PRB 45, 3376 (1992) A. Marini PRL 101, (2008)

16 Green's functions: an (over)simplified picture (I) G. Strinati, Nuovo Cimento 11, 1 (1988) SF e icl t r a lp a Re E Ik =ϵnk icle t r a ip s a Qu Spectral Functions SF E Ik =ℜ( E nk )+i ℑ (E nk ) ρ( x ',t ) ψ H ( x,t ) ρ( x ',t ) Electronic & Nuclear density ψ H ( x,t )

17 Diagrammatic vs Hedin-Lundqvist approach (I) LH and SL, Solid. State Phys. 23, 1 (1969); RvL, PRB 69, (2004) Feynmann ρ( x ',t ) ψ H ( x,t ) diagrams ψ H ( x,t ) Equation of motion H H + ϕ( x)ρ( x, t ) [i t h(1)]g (1,2)=δ (1,2)+ dr Σ(1,3)G (3,2) ψ H ( x, t ) ψ H, ϕ (x,t ) 1 δ ψ H ( x, t) ρ( x ', t) ψ H ( x, t) δϕ δ G (4,5) Σ(1,2) d3 d4 v (1,3)G (1,4) δ ϕ(3)

18 Green's functions: an (over)simplified picture (II) [i t h(1)]g (1,2)=δ (1,2)+ dr Σ(1,3)G (3,2) icl t r pa l a Re Time Fourier transformation E Ik =ϵnk Linear expansion E nk =ϵnk +Σ(ϵnk )+Σ ' (ϵnk )( E nk ϵnk ) QP approximation Z nk Gnk (ω)= (ω E nk ) Z nk = 1 (1 Σ ' (ϵnk )) F S e art p si a Qu S e l ic E Ik =ℜ( E nk )+i ℑ (E nk ) F

19 Hedin-Lundqvist approach (II) RvL, PRB 69, (2004) W ph (r 1, r 2, w) δ n( x, t) Pe ( x,t ; x ', t) δ ϕ( x ',t ' ) δ N ( x, t) D ( x, t ; x ',t ) δ ϕ( x ',t ' ) & = + 1/ 2 = m Γ ( electron ) M atom Migdal's Fan self-energy Fan theorem

20 Hedin-Lundqvist approach (III): the FAN self-energy (ω=0) 1 dr ϵ atoms 1 (r, r 1) ϵ. V ion (r 1) Fan V SCF (r) Nqλ Nqλ (LH) (HAC) (LH) (HAC) STATIC & ADIABATIC limit Fan term in Heine-AllenCardona Theory ω ϵnk ϵnk ϵn ' k q ωq λ Fan

21 Density Functional Perturbation Theory [S. Baroni, REVIEWS OF MODERN PHYSICS, 2001, 73, 515] DFPT is composed by a self-consistent linear system El-Ph Dynamical matrix Matrix ωq λ elements g qλ nn ' k d V KS n ' k q ( M ω q λ ) n k d uq λ 1 By working out the DFPT definition of the KS potential it is possible to link DFPT to the static limit of the electron-phonon potential defined in MBPT

22 = Independent QPs Indirect exciton-phonon scattering Direct exciton-phonon scattering Finite temperature excitons

23 Excitons: the polaronic picture Quasihole and quasielectron = K L K2 1, el 0 = L [ K K 2,3 K 1, K3 2, 2 The excitons are the poles of and eigenstates of the Bethe-Salpeter Hamiltonian el L el H K, K '= e h K, K ' K Quasiparticle energies are real in the optical range The BS Hamiltonian is Hermitian 1, K2 K =0 ] L 3, K2

24 The dynamical BSE = + = + + Polarons + Coulomb The dynamical BSE [AM, R. Del Sole, PRL 91, (2003)] was introduced to sum the frequency dependent Coulomb interaction Phononic?

25 The dynamical BSE II: the phonon term Partial summation of the electronic part The phononic part of the Bethe-Salpeter kernel reduces to a self-energy-like operator (without introducing bosonic coordinates) wit h Excitonic Self-Energy The momenta conservation makes the polaronic (indirect) term dominant q k k k k p k p k Q There is no Ab-Initio justification for simple bosonic scattering pictures Q p

26 Excitons: the polaronic picture = Quasihole and quasielectron polarons The BS Hamiltonian is NOT Hermitian 2, T S T E T 1 T =[2 ℑ E T ] 1

27 Finite T excitons Bright to dark (and AM, Phys. Rev. Lett. 101, (2008) vice versa) transitions CHANGE in the excitonic state...gradual worsening of optical efficiency INCOHERENT contribution

28 Giant polaronic effects in nanostructures

29 Spectral Functions and QP picture SF e icl t r a lp a Re icle t r a ip s a Qu Spectral Functions Expanding in eigenstates of the total Hamiltonian the QP picture holds when there is a dominant (and sharp!) pole that collects most of electronic charge SF

30 Spectral Functions in the HEG (I) Debye Model in the HEG ω q λ ω At zero temperature g qn λn' k g k2 ϵn k =ϵk 2 Σ Fan (ω)=i g 2 d 4 k (2 π) 4 D ( p k )G (k ) D ( p k )=(( p0 k 0 ) ω +i η) 1 G (k )=(k 0 ϵk ±i η)

31 Spectral Functions in the HEG (II) ℑ( Σ Fan ω ϵk (ω)) E k =ϵk +Σ k (E k ) QP Ek= ϵk (1+g2 N ω 2 ) Z nk Gnk (ω)= (ω E nk ) 1 Z nk = (1 Σ ' (ϵnk )) ℜ( Σ Fan (ω))

32 Spectral Functions in the HEG (III) 2 2 g N ω =.5

33 Dynamical effects in Diamond! E. Cannuccia, Phys. Rev. Lett. 107, 107, (2011) ϵ2 (ω) d ω ' ℑ [G Γ (ω ω' )] ℑ [G Γ ' (ω ' )] 15c 25c On-Mass shell (HAC) 620 mev Δ E g (T 0) (615 mev in F. Giustino, S.G. Louie and M.L. Cohen, PRL 105, (2010)) Quasiparticle approximation 670 mev

34 C-based nanostructures: polymers Transpolyacetylene Zero-Point Motion u 0.1 a.u. 2 2 u 2C 0.2 a.u. u H 0.3 a.u. u 0.4 a.u. 2 Science (1998) H Polyethylene C C H

35 Breakdown of the QP picture E. Cannuccia, Phys. Rev. Lett. 107, (2011)

36 Polarons in an Hamiltonian representation (I) q k q Phonon population: k k + q k + q Basis set T = 0 K H Spectral Function

37 Polarons in an Hamiltonian representation (II) Ik u² Ik The polaronic wave-function The atomic indetermination IN the polaronic state

38 Are polymers an isolated case?

39 Spectral Functions the CBM and VBM Silicon Diamond hexagonal-bn MgO

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43 Optimal MPI (and, shortly, OpenMP) Paralellization Improved and efficient memory distribution Interfaced to common and well-known ground-state GPL codes: abinit, QEspresso, CPMD Development work-in-progress within the PRACE (Partnership for Advanced Computing in Europe) in collaboration with the Italian and Portughese Super-Computing centers Yambo: an ab-initio tool for excited state calculations, Andrea Marini, Conor Hogan, Myrta Grüning, Daniele Varsano, Comp. Phys. Comm. 180, 1392 (2009)

44 Quasi Particles Electron-Phonon Coupling Electronic Correlation Real Particles Multi-component particles Conclusions Polaronic-induced effect can be HUGE. They can even lead to the breakdown of the electronic picture The weak correlation is counterbalanced by the lowenergy activation process (~Debye energy) Potential ground-breaking consequences on mobility, optical properties,transport... Critical re-examination of ALL purely electronic band- structure calculations on Chttp:// based nanostructures?

45 People and references... PhD Thesis (2011) Giant polaronic effects in polymers: breakdown of the Elena Cannuccia quasiparticle picture E. Cannuccia and AM Ab initio study of entangled electron-phonon states in polymers, submitted to Phys. Rev. B (2012) E. Cannuccia and AM Effect of the quantistic zero-point atomic motion on the opto- electronic properties of diamond and trans-polyacetylene, 107, (2011) AM Ab-Initio Finite Temperature Excitons, Phys. Rev. Lett. 101, (2008) M. Cardona, Solid State Commun. 133, 3 (2005). L. Hedin and S. Lundqvist, Solid. State Phys. 23, 1 (1969) ;R. von Leeuwen, PRB 69, (2004)