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Non-linear optics by means of dynamical Berry phase


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We present an ab-initio real-time based computational approach to nonlinear optical properties in Condensed Matter systems. The equation of mot ions, and in particular the coupling of the electrons with the external electric field, are derived from the Berry phase formulation of the dynamical polarization. The zero-field Hamiltonian includes crystal local field effects, the renormalization of the independent particle energy levels by correlation and excitonic effects within the screened Hartree- Fock self-energy operator. The approach is validated by calculating the second-harmonic generation of SiC and AlAs bulk semiconductors : an excellent agreement is obtained with existing ab-initio calculations from response theory in frequency domain . We finally show applications to the second-harmonic generation of CdTe the third-harmonic generation of Si.

Reference :
Real-time approach to the optical properties of solids and nanostructures : Time-dependent Bethe-alpeter equation Phys. Rev. B 84, 245110 (2011)

Nonlinear optics from ab-initio by means of the dynamical Berry-phase
C. Attaccalite and M. Gruning Phys. Rev. B 88 (23), 235113 (2013)

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Non-linear optics by means of dynamical Berry phase

  1. 1. Non-linear optics by means of dynamical Berry phase C. Attaccalite, Institut Néel Grenoble M. Grüning, Queen's University, Belfast
  2. 2. What is it non­linear optics? References: Nonlinear Optics and Spectroscopy The Nobel Prize in Physics 1981 Nicolaas Bloembergen First experiments on linear­optics  by P. Franken 1961 D(r ,t)=ϵ0 E(r ,t)+P(r ,t) ∇⋅E(r ,t)=4 πρtot (r ,t) ∇⋅D(r ,t)=4 πρext (r ,t) From Gauss's law:Materials equations: Electric Displacement Electric Field Polarization P(r ,t)=χ(1) E+χ(2) E2 +O(E3 ) In general:
  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 Second-Harmonic Generation Nano Lett. 13, 3329 (2013)
  7. 7. … and even more ….. Second harmonic generating (SHG) nanoprobes for in vivo imaging PNAS 107, 14535 (2007) Second harmonic microscopy of MoS2 PRB 87, 161403 (2013)
  8. 8. A bit of theory Which is the link between Berry's phase and SHG?
  9. 9. The Berry phase IgNobel Prize (2000) together with A.K. Geim for flying frogs A generic quantum Hamiltonian with a  parametric dependence … 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).
  10. 10. ...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
  11. 11. Berry's geometric phase Berry's Phase and Geometric Quantum Distance: Macroscopic Polarization and Electron Localization R. Resta, −i Δ ϕ≃〈ψ(ξ)∣∇ξ ψ(ξ)〉⋅Δ ξ γ=∑s=1 M Δ ϕs, s+1→∫C i〈 ψ(ξ)∣∇ξ ψ(ξ)〉 d ξ Berry's connection ● 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
  12. 12. Examples of Berry's phases Molecular AB effectAharonov-Bohm effect Correction to the Wannier-Stark ladder spectra of semiclassical electrons Ph. Dugourd et al. Chem. Phys. Lett. 225, 28 (1994) 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)
  13. 13. The problem of bulk polarization ● How to define polarization as a bulk quantity? ● Polarization for isolated systems is well defined P= 〈 R〉 V = 1 V ∫d r n(r)= 1 V 〈 Ψ∣̂R∣Ψ 〉 1) P= 〈R〉sample V sample 2) P= 〈 R〉cell Vcell 3) P∝∑nm k 〈 ψnk∣r∣ψmk 〉
  14. 14. Bulk polarization, the wrong way 1 1) P= 〈R〉sample V sample
  15. 15. Bulk polarization, the wrong way 2 2) P= 〈 R〉cell Vcell Unfortunately Clausius-Mossotti does not work for solids because WF are delocalized
  16. 16. Bulk polarization, the wrong way 3 3) P∝∑n, mk 〈ψn k∣r∣ψm k〉 〈ψnk∣r∣ψm k〉 ● intra-bands terms undefined ● diverges close to the bands crossing ● ill-defined for degenerates states
  17. 17. Electrons in a periodic system ϕnk(r+ R)=eik R ϕn k(r) Born-von-Karman boundary conditions [ 1 2m p 2 +V (r) ]ϕn k(r)=ϵn(k)ϕn k(r) Bloch orbitals solution of a mean-field Schrödinger eq. ϕn k(r+R)=e ik r unk(r) Bloch functions u obeys to periodic boundary conditions [ 1 2m (p+ℏk)2 +V (r)]un k(r)=ϵn(k)unk (r) We map the problem in k-dependent Hamiltonian and k-independent boundary conditions k plays the role of an external parameter
  18. 18. What is the Berry's phase related to k? King-Smith and Vanderbilt formula Phys. Rev. B 47, 1651 (1993) Pα= 2ie (2π) 3 ∫BZ d k∑n=1 nb 〈un k∣ ∂ ∂ kα ∣unk 〉 Berry's connection again!!
  19. 19. King-Smith and Vanderbilt formula Pα= −ef 2πv aα Nkα ⊥ ∑kα ⊥ ℑ∑i Nk α −1 tr ln S(ki , ki+qα ) .. discretized King-Smith and Vanderbilt formula.... Phys. Rev. B 47, 1651 (1993) An exact formulation exists also for correlated wave-functions R. Resta., Phys. Rev. Lett. 80, 1800 (1998)
  20. 20. From Polarization to the Equations of Motion L= i ℏ N ∑n=1 M ∑k 〈 vkn∣˙vkn〉−E 0 −v Ε⋅P i ℏ ∂ ∂t ∣vk n〉=Hk 0 ∣vk n〉+i e Ε⋅∣∂k vk n〉 ∂k It is an object difficult to calculate numerically due to the gauge freedom of the Bloch functions ∣vk m〉→∑n occ Uk ,nm∣vkn 〉 I. Souza, J. Iniguez and D. Vanderbilt, Phys. Rev. B 69, 085106 (2004)
  21. 21. Our computational setup
  22. 22. As expected we reproduce results obtained from linear response theory: C. Attaccalite, M. Gruning, A. Marini, Phys. Rev. B 84, 245110 (2011)
  23. 23. Let's add some correlation in 4 steps hk 1) We start from the Kohn-Sham Hamiltonian: hk+Δhk universal, parameter free approach 2) Single-particle levels are renormalized within the G0 W0 approx. hk+Δhk+V H [Δρ] 3) Local-field effects are included in the response function hk+Δhk+V H [Δρ]+Σsex [Δ γ] Time-Dependent Hartree 4) Excitonic effects included by means of the Screened-Exchange
  24. 24. SHG in bulk semiconductors: SiC, AlAs, CdTe AlAs SiC 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) E. Luppi, H. Hübener, and V. Véniard Phys. Rev. B 82, 235201 (2010)
  25. 25. 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)
  26. 26. Nonlinear optics in semiconductors from first-principles real-time simulations TDSE
  27. 27. What next? … SFG, DFG, optical rectification, four-wave mixing, electron-optical effect, Fourier spectroscopy, etc.... SHG in liquid-liquid interfaces, nanostructures Dissipation, coupling with phonons..... luminescence, light emission,strong fields... Open questions? ●  Dissipative effects? How?   ●  Coupling dynamical Berry phase with Green's functions?  ●  Coupling dynamical Berry phase with density matrix hierarchy  equations, BBGKY? Z. Wang et al. PRL 105, 256803 (2010) Chen, K. T., & Lee, P. A. Phys. Rev. B, 84, 205137 (2011) R. Resta,
  28. 28. Acknowledgement  Myrta Grüning, Queen's University Belfast Reference: 1) Real-time approach to the optical properties of solids and nanostructures:Real-time approach to the optical properties of solids and nanostructures: Time-dependent Bethe-Salpeter equationTime-dependent Bethe-Salpeter equation, PRB 84, 245110 (2011) 2) Nonlinear optics from ab-initio by means of the dynamical Berry-phaseNonlinear optics from ab-initio by means of the dynamical Berry-phase C. Attaccalite and M. Grüning. 3) Second Harmonic Generation in h-BN and MoSSecond Harmonic Generation in h-BN and MoS22 monolayers: the role of electron-hole interactionmonolayers: the role of electron-hole interaction M. Grüning and C. Attaccalite submitted to NanoLetters
  29. 29. 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]
  30. 30. How to perform k-derivatives? 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) M (k )vk =λ(k)vk
  31. 31. 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: H = p2 2 m +r E+V (r) Ψ(r ,t ) Velocity gauge: H = 1 2 m ( p−e A)2 +V (r) e−r⋅A(t) Ψ(r ,t) 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) 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)
  32. 32. 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
  33. 33. SHG in frequency domain
  34. 34. King-Smith and Vanderbilt formula Phys. Rev. B 47, 1651 (1993) Pα= 2ie (2π) 3 ∫BZ d k∑n=1 nb 〈un k∣ ∂ ∂ kα ∣unk 〉 The idea of Chen, Lee, Resta..... Berry's phase and Green's functions Z. Wang et al. PRL 105, 256803 (2010) Chen, K. T., & Lee, P. A. Phys. Rev. B, 84, 205137 (2011) R. Resta,