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Ab-initio real-time spectroscopy: application to non-linear optics

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Ab-initio real-time spectroscopy: application to non-linear optics
C. Attaccalite and M. Gruning

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Ab-initio real-time spectroscopy: application to non-linear optics

  1. 1. Ab-initio real-time spectroscopy: application to non-linear optics C. Attaccalite, Institut Néel Grenoble M. Grüning, Queen's University, Belfast
  2. 2. What is it non­linear optics? Materials equations: ϵ 0 E (r , t)=D (r , t)−P (r , t) Electric Displacement Electric Field Polarization In general: P(r ,t )=P 0 + χ(1) E+ χ(2) E 2 +O( E3 ) First experiments on linear­optics  by P. Franken 1961 References: Nonlinear Optics and Spectroscopy The Nobel Prize in Physics 1981 Nicolaas Bloembergen
  3. 3. The first motivation to study non-linear optics is in your (my) hands This is a red laser This is not a green laser!!
  4. 4. How it works  a green laser pointer
  5. 5. To see “invisible” excitations The Optical Resonances in Carbon Nanotubes Arise from Excitons Feng Wang, et al. Science 308, 838 (2005);
  6. 6. Probing symmetries and crystal structures Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical SecondHarmonic Generation Nano Lett. 13, 3329 (2013) Second harmonic microscopy of MoS2 PRB 87, 161403 (2013)
  7. 7. … and even more ….. Photon entanglement Second harmonic generating (SHG) nanoprobes for in vivo imaging PNAS 107, 14535 (2007)
  8. 8. Spectroscopy:   experimental point of view
  9. 9. Spectroscopy:   theoretical point of view Linear response theory: + fast approach + analysis of the results - difficult for non-linear response - limited to equilibrium phenomena
  10. 10. Real­time approach External perturbation a uniform electric field −1 dA (t ) E( t)= c dt Time-dependent Schrodinger equations i ∂ Ψ ( x 1,. . x n)=[∑i ∂ t1 p2 i +r⋅E+V (r 1,. .r N )] Ψ ( x 1,. . , x n) 2m P (r ,t )=χ (1) E +χ(2) E 2 +O(E 3 )
  11. 11. Non­linear optics Non-linear optics can calculated in the same way of TDDFT as it is done in OCTOPUS or RT-TDDFT/SIESTA codes. Quasi-monocromatich-field p-nitroaniline Y.Takimoto, Phd thesis (2008)
  12. 12. The problem of bulk polarization ● Polarization for isolated systems is well defined 〈R〉 1 1 ̂ P= = ∫ d r n( r )= 〈 Ψ∣R∣ Ψ 〉 V V V ● How to define polarization as a bulk quantity?
  13. 13. Bulk polarization, the wrong way
  14. 14. The bulk polarization!! King-Smith-Vanderbilt formula n 2ie Pα = d k ∑ n=1 〈 un k∣ ∂ ∣un k 〉 3 ∫BZ ∂ kα (2 π) b King-Smith and Vanderbilt formula Phys. Rev. B 47, 1651 (1993) Berry's phase !! 1) it is a bulk quantity 2) time derivative gives the current 3) reproduces the polarizabilities at all orders 4) is not an Hermitian operator
  15. 15. Our computational setup
  16. 16. Let's add some correlation in 4 steps 1) We start from the DFT (Kohn-Sham) Hamiltonian: 2) Renormalization of the band structure due to correlation (GW) universal, parameter h k free approach 3) Charge fluctuations (time-dependent Hartree) Δ ρ→Δ V H 4) Electron-hole interaction
  17. 17. We reproduce results obtained from linear response theory: C. Attaccalite, M. Gruning, A. Marini, Phys. Rev. B 84, 245110 (2011)
  18. 18. SHG in bulk semiconductors: SiC, AlAs, CdTe SiC AlAs E. Luppi, H. Hübener, and V. Véniard Phys. Rev. B 82, 235201 (2010) CdTe E. Ghahramani, D. J. Moss, and J. E. Sipe, Phys. Rev. B 43, 9700 (1991) I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, J. Opt. Soc. Am. B 14, 2268 (1997) J. I. Jang, et al. J. Opt. Soc. Am. B 30, 2292 (2013)
  19. 19. THG in silicon D. J. Moss, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 41, 1542 (1990) D. Moss, H. M. van Driel, and J. E. Sipe, Optics letters 14, 57 (1989)
  20. 20. Local-fields and excitonic effects in h-BN monolayer IPA IPA + GW + TDSHF independent particles +quasi-particle corrections +time-dependent Hartree (RPA) +screend Hartree-Fock (excitonic effects)
  21. 21. MoS2 single-layer Second harmonic microscopy of monolayer MoS2 N. Kumar et al. Phys. Rev. B 87, 161403(R) (2013) Observation of intense second harmonic generation from MoS2 atomic crystals L. Malard et al. Phys. Rev. B 87, 201401(R) (2013) Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation Y. Li et al. NanoLetters, 13, 3329 (2013)
  22. 22. What next? … SFG, DFG, optical rectification, four-wave mixing, electron-optical effect, Fourier spectroscopy, etc.... SHG in liquid-liquid interfaces, nanostructures Pump and probe experiments Dissipation, coupling with phonons..... time resolved luminescence....
  23. 23. Acknowledgement  Myrta Grüning, Queen's University Belfast Reference: 1) Real-time approach to the optical properties of solids and nanostructures: Time-dependent Bethe-Salpeter equation C. Attaccalite, M. Gruning and A. Marini PRB 84, 245110 (2011) 2) Nonlinear optics from ab-initio by means of the dynamical Berry-phase C. Attaccalite and M. Grüning, Phys. Rev. B 88, 235113 (2013) 3) Second Harmonic Generation in h-BN and MoS2 monolayers: the role of electron-hole interaction M. Grüning and C. Attaccalite, Phys. Rev. B 89, 081102(R) (2014)
  24. 24. The King-Smith and Vanderbilt formula We introduce the Wannier functions Blount, 1962 We express the density in terms of Wannier functions Polarization in terms of Wannier functions [Blount 62]
  25. 25. How to perform k-derivatives? M (k )v k =λ (k )v k Solutions: 1) In mathematics the problem has been solved by using second, third,... etc derivatives SIAM, J. on Matrix. Anal. and Appl. 20, 78 (1998) 2) Global-gauge transformation Phys. Rev. B 76, 035213 (2007) 3) Phase optimization Phys. Rev. B 77, 045102 (2008) 4) Covariant derivative Phys. Rev. B 69, 085106 (2004)
  26. 26. Wrong ideas on velocity gauge In recent years different wrong papers using velocity gauge have been published (that I will not cite here) on: 1) real-time TD-DFT 2) Kadanoff-Baym equations + GW self-energy 3) Kadanoff-Baym equations + DMFT self-energy Length gauge: p2 H= + r E +V (r ) 2m Velocity gauge: 1 H= ( p−e A) 2 +V (r ) 2m Analitic demostration: K. Rzazewski and R. W. Boyd, Journal of modern optics 51, 1137 (2004) W. E. Lamb, et al. Phys. Rev. A 36, 2763 (1987) Ψ (r ,t ) e −r⋅A (t ) Ψ (r , t) Well done velocity gauge: M. Springborg, and B. Kirtman Phys. Rev. B 77, 045102 (2008) V. N. Genkin and P. M. Mednis Sov. Phys. JETP 27, 609 (1968)
  27. 27. Post-processing real-time data P(t) is a periodic function of period TL=2π/ωL pn is proportional to χn by the n-th order of the external field Performing a discrete-time signal sampling we reduce the problem to the solution of a systems of linear equations
  28. 28. SHG in frequency domain
  29. 29. Berry's phase and Green's functions n 2ie Pα = d k ∑ n=1 〈 un k∣ ∂ ∣un k 〉 3 ∫BZ ∂ kα (2 π) b King-Smith and Vanderbilt formula Phys. Rev. B 47, 1651 (1993) The idea of Chen, Lee, Resta..... Z. Wang et al. PRL 105, 256803 (2010) Chen, K. T., & Lee, P. A. Phys. Rev. B, 84, 205137 (2011) R. Resta, www-dft.ts.infn.it/~resta/sissa/draft.pdf
  30. 30. A bit of theory Which is the link between Berry's phase and SHG?
  31. 31. The Berry phase A generic quantum Hamiltonian with a  parametric dependence IgNobel Prize (2000) together with A.K. Geim for flying frogs … phase difference between two ground  eigenstates at two different ξ cannot have any physical meaning Berry, M. V. . Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 392(1802), 45-57 (1984).
  32. 32. ...connecting the dots... the phase difference of a closed-path is gauge-invariant therefore is a potential physical observable γ is an “exotic” observable which cannot be expressed in terms of any Hermitian operator
  33. 33. Berry's geometric phase −i Δ ϕ≃〈 ψ( ξ)∣∇ ξ ψ(ξ)〉⋅Δ ξ Berry's connection M γ=∑ s=1 Δ ϕs , s +1 →∫C i〈 ψ(ξ)∣∇ ξ ψ( ξ)〉 d ξ ● Berry's phase exists because the system is not isolated ξ is a kind of coupling with the “rest of the Universe” ● In a truly isolated system, there can be no manifestation of a Berry's phase Berry's Phase and Geometric Quantum Distance: Macroscopic Polarization and Electron Localization R. Resta, http://www.freescience.info/go.php?pagename=books&id=1437
  34. 34. Examples of Berry's phases Aharonov-Bohm effect Molecular AB effect Ph. Dugourd et al. Chem. Phys. Lett. 225, 28 (1994) Correction to the Wannier-Stark ladder spectra of semiclassical electrons R.G. Sadygov and D.R. Yarkony J. Chem. Phys. 110, 3639 (1999) J. Zak, Phys. Rev. Lett. 20, 1477 (1968) J. Zak, Phys. Rev. Lett. 62, 2747 (1989)
  35. 35. Let's add some correlation in 4 steps 1) We start from the Kohn-Sham Hamiltonian: hk universal, parameter free approach 2) Single-particle levels are renormalized within the G0W0 approx. hk+ Δ hk 3) Local-field effects are included in the response function h k + Δ h k +V H [Δ ρ] Time-Dependent Hartree 4) Excitonic effects included by means of the Screened-Exchange h k + Δ h k +V H [Δ ρ]+ Σ sex [Δ γ ]
  36. 36. Bulk polarization, the wrong way 3 3) P∝ ∑n , m k 〈 ψn k∣r∣ψm k 〉 ● 〈 ψ n k∣r∣ ψm k 〉 intra-bands terms undefined ● diverges close to the bands crossing ● ill-defined for degenerates states
  37. 37. Electrons in a periodic system ϕ n k (r + R)=e i k R ϕn k (r ) [ ] 1 2 p +V ( r ) ϕn k ( r)=ϵ n ( k)ϕn k ( r ) Bloch orbitals solution of 2m a mean-field Schrödinger eq. ϕn k ( r + R)=e [ Born-von-Karman boundary conditions ik r unk( r ) Bloch functions u obeys to periodic boundary conditions ] 1 (p +ℏ k )2 +V (r ) un k (r )=ϵ n (k)u n k (r ) 2m We map the problem in k-dependent Hamiltonian and k-independent boundary conditions k plays the role of an external parameter
  38. 38. What is the Berry's phase related to k? n 2ie Pα = d k ∑ n=1 〈 un k∣ ∂ ∣un k 〉 3 ∫BZ ∂ kα (2 π) b King-Smith and Vanderbilt formula Phys. Rev. B 47, 1651 (1993) Berry's connection again!!

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