Expos de soutenance pour le titre de Docteur de lcole Polytechnique - PowerPoint PPT Presentation

Exposé de soutenance pour le titre de Docteur de l’École Polytechnique Spécialité: Physique Iurii Timrov 27 March 2013, École Polytechnique 1/57

Outline 1. Introduction 1.1 Motivation 1.2 Material: Bismuth 1.3 State of the art methods 2. Results 2.1 High-energy response: new approach for EELS 2.2 Low-energy response: free-carrier response 3. Conclusions 2/57

Outline 1. Introduction 1.1 Motivation 1.2 Material: Bismuth 1.3 State of the art methods 2. Results 2.1 High-energy response: new approach for EELS 2.2 Low-energy response: free-carrier response 3. Conclusions 3/57

Motivation How to understand the nature of materials? Perturb them and see what happens! 4/57

Motivation Optics: q → 0, ω → 0 EELS: q � = 0, ω � = 0 ω 2 Drude model: ǫ ( ω ) = 1 − p Loss function − Im [ ǫ − 1 ( q , ω )] ω ( ω + i γ ) 5/57

Motivation Ab initio description of the full charge-carrier response of bismuth to external perturbations: low-energy and high-energy response. 6/57

Why do we need a new method for EELS? 1. Bridging the valence-loss and the core-loss EELS. It is computationally ex- pensive for state-of-the- art methods to describe EEL spectra of complex systems in the energy range up to 100 eV. C. Wehenkel et al., Solid State Comm. 15 , 555 (1974) 7/57

Why do we need a new method for EELS? 2. Calculation of EEL spectra of large systems (hundreds of atoms). Example: Calculation of surface plasmons ⇒ Simulation of the surface is needed = Figure: View of a 5-layer slab model of a surface, as used in periodic calculation. ⇓ Large number of atoms ⇓ Computationally demanding task for state-of-the-art methods D. Scholl and J. Steckel, “DFT: A practical introduction” (2009). 8/57

Low-energy response: photoexcited bismuth Photoexcitation of Bi ⇐ ⇒ Pump-probe THz expt. (L. Perfetti, J. Faure.) Theoretical model is needed in order to explain the evolution of the Drude plasma frequency ω p after the photoexcitation of Bi. 9/57

Outline 1. Introduction 1.1 Motivation 1.2 Material: Bismuth 1.3 State of the art methods 2. Results 2.1 High-energy response: new approach for EELS 2.2 Low-energy response: free-carrier response 3. Conclusions 10/57

Material: Semimetal Bismuth . Issi, Aus. J. Phys. 32 , 585 (1979) J.-P M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82 , 3045 (2010). 11/57

Crystal and Electronic Structure A7 rhombohedral structure: Peierls distortion of sc lattice Semimetallicity is due to the Peierls distor- tion: Overlap between valence and conduc- tion bands. The Fermi surface consists of 1 hole pocket and 3 electron pockets. Y. Liu et al., Phys. Rev. B 52 , 1566 (1995). . Issi, Aus. J. Phys. 32 , 585 (1979) J.-P 12/57

Crystal and Electronic Structure A7 rhombohedral structure: Peierls distortion of sc lattice Semimetallicity is due to the Peierls distor- tion: Overlap between valence and conduc- tion bands. The Fermi surface consists of 1 hole pocket and 3 electron pockets. Y. Liu et al., Phys. Rev. B 52 , 1566 (1995). . Issi, Aus. J. Phys. 32 , 585 (1979) J.-P 12/57

Crystal and Electronic Structure A7 rhombohedral structure: Peierls distortion of sc lattice Semimetallicity is due to the Peierls distor- tion: Overlap between valence and conduc- tion bands. The Fermi surface consists of 1 hole pocket and 3 electron pockets. Y. Liu et al., Phys. Rev. B 52 , 1566 (1995). . Issi, Aus. J. Phys. 32 , 585 (1979) J.-P 13/57

Spin-orbit coupling (SOC) Spin-orbit coupling is a coupling of electron’s spin S with its orbital motion L . The SOC Hamiltonian reads: H SOC ∝ ∇ V ( L · σ ) , � σ 0 � where V is the potential, and σ are Pauli spin-matrices: S = � . 2 0 σ material SOC-assisted split- ting of levels at Γ (eV) 0.04 Si 0.3 GaAs 0.8 InSb 0.3 As 0.6 Sb 1.0 Pb Bi 1.5 In bismuth the spin-orbit coupling is very strong! A. Dal Corso, J. Phys. Condens. Matter 20 , 445202 (2008). 14/57

Spin-orbit coupling (SOC) Spin-orbit coupling is a coupling of electron’s spin S with its orbital motion L . The SOC Hamiltonian reads: H SOC ∝ ∇ V ( L · σ ) , � σ 0 � where V is the potential, and σ are Pauli spin-matrices: S = � . 2 0 σ material SOC-assisted split- ting of levels at Γ (eV) 0.04 Si 0.3 GaAs 0.8 InSb 0.3 As 0.6 Sb 1.0 Pb Bi 1.5 In bismuth the spin-orbit coupling is very strong! A. Dal Corso, J. Phys. Condens. Matter 20 , 445202 (2008). 14/57

Kohn-Sham band structure of bismuth X. Gonze et al., Phys. Rev. B 41 , 11827 (1990) A. B. Shick et al., Phys. Rev. B 60 , 15484 (1999) I. Timrov, J. Faure, N. Vast, L. Perfetti et al., Phys. Rev. B 85 , 155139 (2012) 15/57

Kohn-Sham band structure of bismuth X. Gonze et al., Phys. Rev. B 41 , 11827 (1990) A. B. Shick et al., Phys. Rev. B 60 , 15484 (1999) I. Timrov, J. Faure, N. Vast, L. Perfetti et al., Phys. Rev. B 85 , 155139 (2012) 16/57

Outline 1. Introduction 1.1 Motivation 1.2 Material: Bismuth 1.3 State of the art methods 2. Results 2.1 High-energy response: new approach for EELS 2.2 Low-energy response: free-carrier response 3. Conclusions 17/57

Density Functional Theory Ground-state: DFT The Kohn-Sham equation: − 1 � � 2 ∇ 2 + V KS ( r ) ϕ i ( r ) = ε i ϕ i ( r ) . The Kohn-Sham potential V KS ( r ) : | r − r ′ | d r ′ + δ E xc [ ρ ( r )] ρ ( r ′ ) � + V ext ( r ) . δρ ( r ) The charge-density: occ | ϕ i ( r ) | 2 . ρ ( r ) = � i The quantum Liouville equation: [ ˆ ρ ] = 0 . H KS , ˆ Hohenberg and Kohn, Phys. Rev. (1964) Kohn and Sham, Phys. Rev. (1965) 18/57

Historical note 19/57

Time-Dependent Density Functional Theory Ground-state: DFT Excited-state: TDDFT The Kohn-Sham equation: The TD Kohn-Sham equation: − 1 − 1 � � � � ϕ i ( r , t ) = i ∂ 2 ∇ 2 + V KS ( r ) 2 ∇ 2 + V KS ( r , t ) ϕ i ( r ) = ε i ϕ i ( r ) . ∂ t ϕ i ( r , t ) . The TD Kohn-Sham potential V KS ( r , t ) : The Kohn-Sham potential V KS ( r ) : | r − r ′ | d r ′ + δ E xc [ ρ ( r )] ρ ( r ′ ) | r − r ′ | d r ′ + δ E xc [ ρ ( r , t )] ρ ( r ′ , t ) � � + V ext ( r ) . + V ext ( r , t ) , δρ ( r ) δρ ( r , t ) The charge-density: The TD charge-density: occ occ | ϕ i ( r , t ) | 2 . | ϕ i ( r ) | 2 . ρ ( r ) = � ρ ( r , t ) = � i i The quantum Liouville equation: The TD quantum Liouville equation: ρ ( t ) ] = i ∂ [ ˆ ρ ] = 0 . [ ˆ H KS , ˆ H KS ( t ) , ˆ ∂ t ˆ ρ ( t ) . Hohenberg and Kohn, Phys. Rev. (1964) Runge and Gross, PRL (1984) Kohn and Sham, Phys. Rev. (1965) Onida, Reining, Rubio, RMP (2002) 20/57

Time-Dependent Density Functional Theory Ground-state: DFT Excited-state: TDDFT The Kohn-Sham equation: The TD Kohn-Sham equation: − 1 − 1 � � � � ϕ i ( r , t ) = i ∂ 2 ∇ 2 + V KS ( r ) 2 ∇ 2 + V KS ( r , t ) ϕ i ( r ) = ε i ϕ i ( r ) . ∂ t ϕ i ( r , t ) . The TD Kohn-Sham potential V KS ( r , t ) : The Kohn-Sham potential V KS ( r ) : | r − r ′ | d r ′ + δ E xc [ ρ ( r )] ρ ( r ′ ) | r − r ′ | d r ′ + δ E xc [ ρ ( r , t )] ρ ( r ′ , t ) � � + V ext ( r ) . + V ext ( r , t ) , δρ ( r ) δρ ( r , t ) The charge-density: The TD charge-density: occ occ | ϕ i ( r , t ) | 2 . | ϕ i ( r ) | 2 . ρ ( r ) = � ρ ( r , t ) = � i i The quantum Liouville equation: The TD quantum Liouville equation: ρ ( t ) ] = i ∂ [ ˆ ρ ] = 0 . [ ˆ H KS , ˆ H KS ( t ) , ˆ ∂ t ˆ ρ ( t ) . Hohenberg and Kohn, Phys. Rev. (1964) Runge and Gross, PRL (1984) Kohn and Sham, Phys. Rev. (1965) Onida, Reining, Rubio, RMP (2002) 20/57

Fluctuation-dissipation theorem Optical absorption Perturbation: electric field ⇓ Polarization of the dipole: d ( ω ) = χ ( ω ) E ext ( ω ) χ is the polarization-polarization correlation function Im ǫ ( ω ) ∝ S ( ω ) S ( ω ) = 2 π ω Im χ ( ω ) S is the oscillator strength ◮ Im ǫ : Measured experimentally ◮ S : Fluctuation of polarization ◮ Im χ : Dissipation of energy 21/57

Two implementations of linear-response TDDFPT Optical absorption spectra of finite systems Liouville-Lanczos approach Conventional TDDFT approach Definition: Independent-transition polarizability χ 0 � � ˜ ext ( r , ω ) ˆ V ′ ρ ′ ( ω ) χ ( ω ) ≡ Tr ( f v − f c ) ϕ c ( r ) ϕ ∗ v ( r ) ϕ v ( r ′ ) ϕ c ( r ′ ) χ 0 ( ω ) = � ω − ( ε c − ε v ) + i η v , c ˆ ρ ′ ( ω ) =? Quantum Liouville equation: ρ ( t ) ] = i ∂ [ ˆ H KS ( t ) , ˆ ∂ t ˆ ρ ( t ) Linearization + Fourier transform: ρ 0 ] ( ω − ˆ ρ ′ ( ω ) = [˜ V ′ L ) · ˆ ext ( ω ) , ˆ ρ ′ ≡ [ˆ H 0 ρ ′ ] + [ˆ ρ 0 ] ˆ L · ˆ KS , ˆ V HXC , ˆ χ ( ω ) = � ˜ L ) − 1 [˜ ρ 0 ] � Dyson-like equation: ext ( ω ) | ( ω − ˆ V ′ V ′ ext ( ω ) , ˆ χ = χ 0 + χ 0 ( v Coul + f xc ) χ ⇓ Use of Lanczos recursion method Onida, Reining, Rubio, RMP (2002) Rocca, Gebauer, Saad, Baroni, JCP (2008) 22/57