Temperature dependence and zero-point renormalization of the electronic structure of solids

Transcription

1 Temperature dependence and zero-point renormalization of the electronic structure of solids X. Gonze, Université catholique de Louvain, Belgium Collaborators : S. Poncé, Y. Gillet, J. Laflamme, U.C. Louvain, Belgium G. Antonius, M. Côté, U. de Montréal, Canada A. Marini, CNR Italy P. Boulanger, CEA Grenoble 1

2 Temperature dependence of electronic/optical properties Motivation - peaks shift in energy - peaks broaden with increasing temperature : decreased electron lifetime L. Viña, S. Logothetidis and M. Cardona, Phys. Rev. B 30, 1979 (1984) - even at 0K, vibrational effects are important, due to Zero-Point Motion Usually, not taken into account in First-principles calculations M. Cardona, Solid State Comm. 133, 3 (2005) 2

3 Allen-Heine-Cardona theory + first-principles Review Optical absorption of Silicon. Excellent agremeent with Exp. Mostly broadening effect, imaginary part of the Fan term (not discussed in this talk) Diamond Zero-point motion in DFT : 0.4 ev for the direct gap Diamond Zero-point motion in DFT+GW : 0.63 ev for the direct gap, in agreement with experiments G. Antonius, S. Poncé, P. Boulanger, M. Côté & XG, Phys. Rev. Lett. 112, (2014) 3

4 Overview Motivation Thermal expansion and phonon population effects Ab initio Allen-Heine-Cardona (AHC) theory Temperature effects with GW electronic structure Breakdown of the adiabatic quadratic approximation for infra-red active materials References : X. Gonze, P. Boulanger and M. Côté, Ann. Phys 523, 168 (2011) S. Poncé et al, Comput. Materials Science 83, 341 (2014) G. Antonius, S. Poncé, P. Boulanger, M. Côté and X. Gonze, Phys. Rev. Lett.112, (2014) S. Poncé et al, Phys. Rev. B. 90, (2014) S. Poncé et al, J. Phys. Chem 143, (2015) G. Antonius et al, Phys. Rev. B 92, (2015) Also : Many-body perturbation theory approach to the electron-phonon interaction with density-functional theory as a starting point, A. Marini, S. Poncé and X. Gonze, Phys. Rev. B 91, (2015) 4

5 Thermal expansion and phonon population effects 5

6 Quasi-harmonic approximation: Divide and conquer a refresher Constant-pressure temperature dependence of the electronic eigenenergies : two contributions ε nk ε nk ε nk lnv T = T + lnv T P P V T Constant volume Constant temperature = α P (T ) Thermal expansion coefficient Contribution of the phonon population, i.e. the vibrations of the atomic nuclei, at constant volume + Contribution of the thermal expansion, i.e. the change in volume of the sample, at constant temperature 6

7 Ab initio thermal expansion n(ωq,m ) V 1 α (T ) = γ q,m 3B q, m ω q, m T Mode-Grüneisen parameters γ m,q = (ln ωm,q ) (ln V ) Alternative path : minimisation of free energy 7

8 The thermal expansion Ab initio thermal expansion contribution Linear thermal expansion coefficient of bulk silicon G.-M. Rignanese, J.-P. Michenaud and XG Phys. Rev. B 53, 4488 (1996) 8

9 Thermal expansion contribution to the gap of Si - - Calculation * Exp But total exper. change between 0K and 300K = 0.06 ev...thermal expansion contribution is negligible (for Si) NOT always the case, can be of same size : black phosphorus (Villegas, Rocha, Marini, arxiv: ), Bi2Se3 family (Monserrat & Vanderbilt, arxiv: ). 9

10 Phonon population effects Different levels of approximation : -dynamics of the nuclei classical quantum? -harmonic treatment of vibrations or anharmonicities? -adiabatic decoupling of nuclei and electronic dynamic, or non-adiabatic corrections? -independent electronic quasi-particles (DFT or GW), or many-body approach with spectral functions? At least 5 first-principle methodologies : (1) Time-average (2) Thermal average (3) Harmonic quadratic approximation + thermal average (4) Diagrammatic approach (Allen-Heine-Cardona) (5) Exact factorization (H. Gross and co-workers) 10

11 The phonon population Nuclear dynamics : dimerscontribution: Diatomic molecules Diatomic molecules = simplest system to study temperature dependence of eigenvalues. - discrete levels, well described with the theory of the molecular orbitals - only one relevant vibration mode. (6 modes decouple as 3 translations, 2 rotations + the stretch.) 11

12 The phonon population contribution: Average eigenenergies in the BO approx. Diatomic molecules Electronic eigenenergies, function of the bond length ε n (ΔR) => (1) Time-average of eigenenergies from Molecular Dynamics trajectories, ΔR(t) at average T, with 1 τ ε n (T ) = lim τ τ 0 ε n (ΔR(t)) dt Pros : well-defined procedure ; compatible with current implementations and computing capabilities ; ε n (ΔR(t)) from DFT or GW ; anharmonicities Cons : if classical dynamics => no zero-point motion ; adiabatic (vibrations, but no exchange of energy ) ; hard for solids (supercell) 12

13 The phonon population contribution: Average eigenenergies in the BO approx. Diatomic molecules Electronic eigenenergies function of the bond length ε n (ΔR) (2) Thermal average with accurate quantum vibrational states, 1 ε n (T ) = e Z m E ph (m) k BT ( * χ m (ΔR)ε n (ΔR) χ m (ΔR)dΔR ) Z = e E ph (m) k BT m Pros : zero-point motion ; ε n (ΔR(t)) from DFT or GW ; anharmonicities Cons : hard to sample more than a few vibrational degrees of freedom ; adiabatic (vibrations, but no exchange of energy ) 13

14 The phonon population Average eigenenergies : BO andcontribution: harmonic approx. Diatomic molecules (3) Thermal average with quantum vibrational states in the harmonic approximation, and expansion of ε n (ΔR) to second order (quadratic approximation) 2 ε 1 εn 0 2 n εn = εn + ΔR + ΔR R 2 R 2 1 E ph (m) = ω (m + ) 2 1 nvib (T ) = ω e k BT 1 T-dependent phonon occupation number (Bose-Einstein) δε n (T ) = ε n 1 n (T ) + vib nvib 2 ε n (ΔR) from DFT or GW ; Pros : zero-point motion ; tractable for molecules Cons : hard for solids (supercells) ; no anharmonicities ; quadratic ; adiabatic (vibrations, but no exchange of energy ) 14

15 Ab initio Allen-Heine-Cardona theory 15

16 Long history of the theory of T-dependent effects In a semi-empirical context (empirical pseudopotential, tight-binding) Work from the 50 : H. Y. Fan. Phys. Rev. 78, 808 (1950) ; 82, 900 (1951) E. Antoncik. Czechosl. Journ. Phys. 5, 449 (1955). Debye-Waller contribution. H. Brooks. Adv. Electron 7, 85 (1955) + Yu (PhD thesis, unpubl., Brooks supervisor) Fan Debye-Waller Within 2nd order perturbation theory treatment of electron-phonon effect, both contributions are needed (of course ). Unification by : Allen + Heine, J. Phys. C 9, 2305 (1976). Allen + Cardona, Phys. Rev. B 24, 7479 (1981) ; 27, 4760 (1983). => the Allen-Heine-Cardona (AHC) theory 16

17 Allen-Heine-Cardona Review (AHC) formalism Second-order (time-dependent) perturbation theory (no average contribution from first order) For solids (phonons have cristalline momentum) If adiabatic BO... neglect the phonon frequencies with respect to the electronic gap, no transfer of energy : + occupation number ε kn 1 1 (T ) + from Bose-Einstein δε kn (T,V = const) = n qj N q qj nqj 2 statistics ε kn 2ε kn ξκ a (qj)ξκ ' b ( qj) iq.( Rκ ' b Rκ a ) 1 = e nqj 2ω qj κ aκ ' b Rκ a Rκ ' b Mκ Mκ ' Electron-phonon coupling energy (EPCE) Phonon mode factor ξκ a (qj) phonon eigenmodes, κ = atom label, a=x, y, or z 17

18 2ε kn Eigenvalue changes R R κa κ 'b ˆ V + ρ ( r' )dr'+ de xc ε kn = φkn H k φkn H =T+ nucl r-r' dρ (r ) (0) (1) (0) = φ φ Hellman-Feynman theorem : ε kn(1) H kn k kn Review? One more derivative : ε (2) kn = φ (0) kn H (2) k φ (0) kn ( 1 (0) (1) + φkn H k +q φk(1)qn + (c.c) 2 ) Fan self-energy Debye-Waller Antoncik (1) In AHC, φn obtained by sum-over-states φn(1) = m n φ m(0) φ m(0) H (1) φn(0) εn εm 18

19 Derivatives ofreview the Hamiltonian? ρ ( r' ) de xc ˆ H =T+V nucl + dr'+ r-r' dρ (r ) V nucl = Vκ (r-rκ ) κ In AHC, use of semi-empirical pseudopotential => rigid-ion approximation Upon infinitesimal displacements of the nuclei, the rearrangement of electrons due to the perturbation is ignored H (2) pure site-diagonal 2V nucl = 0 for κ κ ' Rκ a Rκ ' b Debye-Waller contribution pure site-diagonal Moreover, invariance under pure translations 0=ε (2) n = φ (0) n H (2) transl φ (0) n ( 1 (0) (1) + φn H transl φn(1) + (c.c) 2 Reformulation of the Debye-Waller term. ) 19

20 Ad. AHC = Ad. Fan + rigid-ion Debye-Waller Review ε kn ε kn (Fan) ε kn (DW RIA ) = + nqj nqj nqj κ 'b Hκ ' φ φkn κ a Hκ φk +qn' φk +qn' ε kn (Fan) 1 kn ξκ a (qj)ξκ 'b ( qj) iq.( Rκ 'b Rκ a ) = ℜ e nqj ω qj κ aκ 'bn' ε kn ε k +qn' Mκ Mκ ' ε kn (DW RIA ) 1 = ℜ nqj ω qj κ aκ 'bn' φkn κ a Hκ φkn' φkn' κ 'b Hκ ' φ kn ε kn ε kn' 1 ξκ a (qj)ξκ b ( qj) ξκ 'a (qj)ξκ 'b ( qj) + 2 Mκ Mκ ' Good : only first-order electron-phonon matrix elements are needed (+ standard ingredients from first-principles phonon/band structure calculations) ; no supercell calculations Bad : (1) summation over a large number of unoccupied states n (2) is the rigid-ion approx. valid for first-principles calculations? (3) If first-principles calculations : DFT electron-phonon matrix elements, as well as eigenenergies, while MBPT should be used (4) Adiabatic approx. : phonon frequencies neglected in denominator 20

21 ImplementationReview Sum over state present in the AHC formalism, replaced by the use of Density-Functional Perturbation Theory quantities (Sternheimer equation) => large gain in speed. φn(1) = φ m n (0) m φ m(0) H (1) φn(0) εn εm For converged calculations for silicon : sum over states 125 h DFPT 17h 21

22 Numerical study diamond Re :: ZPR ZPMinview -Implementation in ABINIT ( -Plane wave + pseudopotential methodology -Converged number of plane waves ( Hartree) -k point sampling : 6x6x6 is sufficient for the generation of the first-order Hamiltonian -Sampling on the q phonon wavevectors for the Fan term is a big issue ZPM δε Γn = 1 ε Γn 1 N q qj nqj 2 φγn κ a Hκ φqn' φqn' ε Γn (Fan) 1 κ 'b Hκ ' φ Γn ξκ a (qj)ξκ 'b ( qj) iq.( Rκ 'b Rκ a ) = ℜ e nqj ω qj κ aκ 'bn' ε Γn ε qn' Mκ Mκ ' Indeed (1) interband contributions have strong variations (2) intraband contributions diverge due to the denominator 22

23 Intraband divergence small q Re : ZPMforview 15 ε Γn (Fan) 1 f (qjn) = lim lim ε nqj q 0 q 0 ω qj Γn ε qn ε Γn (Fan) 1 2 Optic modes : lim nqj q q 0 However, for acoustic modes, Fan/DDW contribs cancel each other 23

24 Intraband divergence on isoenergetic surface Re : ZPM view 15 Set of isoenergetic wavevectors ε Γn (Fan) 1 f (qjn) = lim lim nqj q ε qn q qiso q qiso ω qj ε Γn ε qn 1 qiso.(q qiso ) Problem only for the ZPR of the conduction state 24

25 Phonon wavevector integration Review Bottom of conduction Band at Gamma δε kn (T,V = const) = ε kn 1 1 n qj (T ) + N q qj nqj 2 G. Antonius, S. Poncé, P. Boulanger, M. Côté & XG, Phys. Rev. Lett. 112, (2014) Top of valence band 25

26 Rate of convergence Re : ZPM view S. Poncé et al, Comput. Materials Science 83, 341 (2014) 26

27 Smoothing the Redenominator : ZPM view φγn κ a Hκ φqn' φqn' ε Γn (Fan) 1 κ 'b Hκ ' φ Γn ξκ a (qj)ξκ 'b ( qj) iq.( Rκ 'b Rκ a ) = ℜ e + iδ nqj ω qj κ aκ 'bn' ε Γn ε qn' Mκ Mκ '... dramatically helps the convergence to a (slightly) different value If imaginary part = 100 mev : q grid #q in IBZ ZPM HOMO ZPM LUMO ZPM gap (mev) (mev) (mev) 8x8x8 x4s x12x12 x4s x16x16 x4s x20x20 x4s x24x24 x4s x28x28 x4s x32x32 x4s

28 Changing the Re : ZPM view imaginary delta f (qjn) + iδ ε Γn ε qn For very large q-wavevector sampling, rate of convergence understood, + correspond to expectations 28

29 Cross-checking Review Independent implementation in Quantum/Espresso + Yambo (without the Sternheimer trick, though) => excellent agreement with ABINIT 0.4 ev ZPR Band gap mev mev S. Poncé et al, Comput. Materials Science 83, 341 (2014) 29

30 DFT+AHC T-dependent bandgap : diamond Not bad, but still too small effect? S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete & XG, J. Chem. Phys. 143, (2015) 30

31 DFT T-dependent band structure Diamond 0 Kelvin (incl. Zero-point motion) Note the widening of the bands = lifetime S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete & XG, J. Chem. Phys. 143, (2015) 31

32 DFT T-dependent band structure Diamond 300 Kelvin Note the widening of the bands = lifetime S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete & XG, J. Chem. Phys. 143, (2015) 32

33 DFT T-dependent band structure Diamond 900 Kelvin Note the widening of the bands = lifetime S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete & XG, J. Chem. Phys. 143, (2015) 33

34 DFT T-dependent band structure Diamond 1500 Kelvin Note the widening of the bands = lifetime S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete & XG, J. Chem. Phys. 143, (2015) 34

35 Temperature effects with GW electronic structure 35

36 GW energies + frozen-phonon in supercells + occupation number ε kn 1 1 from Bose-Einstein δε kn (T,V = const) = n qj (T ) + N q qj nqj 2 statistics 2 ε kn ε 1 ξκ a (qj)ξκ ' b ( qj) iq.( Rκ ' b Rκ a ) kn = e nqj 2ω qj κ aκ ' b Rκ a Rκ ' b Mκ Mκ ' Finite-difference evaluation of the derivatives of the GW electronic energies wrt phonons, using supercells G. Antonius, S. Poncé, P. Boulanger, M. Côté & XG, Phys. Rev. Lett. 112, (2014) 36

37 Electron-phonon coupling energies EPCE ε kn nqj from DFT, G0W0 and scgw Significantly larger decrease of the gap within G0W0 and scgw compared to DFT G0W0 and scgw very close to each other Bottom of conduction Band at Gamma Top of valence band G. Antonius, S. Poncé, P. Boulanger, M. Côté & XG, Phys. Rev. Lett. 112, (2014) 37

38 DFT + perturbative phonons + GW + frozen-phonon in supercells Zero-point motion in DFT : 0.4 ev for the direct gap Zero-point motion in DFT+GW : 0.63 ev for the direct gap, in agreement with experiments G. Antonius, S. Poncé, P. Boulanger, M. Côté & XG, Phys. Rev. Lett. 112, (2014) 38

39 Breakdown of the adiabatic quadratic approximation for infra-red active materials 39

40 Boron nitride renormalization of gap when the imaginary delta tends to zero, the ZPR diverges such a divergence is confirmed by a «post-mortem» analysis 40

41 Electric field with IR-active optic modes Collective displacement with wavevector q 0 (1) H q(1) = Vext(1),q + VH(1),q + Vxc,q 4π Zκ vκ (q 0) = 2 + Cκ + O(q 2 ) q i (G + q)α e i(g+q).τ vκ (G + q) Ω0 nq(1) (1) VH,q (G) = 4π 2 G+q Vext(1),q (G) = nq(1) q when q 0 Both the external and Hartree potentials can diverge like 1/q. Definition of the polarization of a phonon mode : Pα(1) (qj) = Zκ,αβ ξκβ (qj) Zκ,αβ Pα = Ω0 uκ, β κβ δ E=0 Born effective charge tensor for atom Associated electric field Eα = 4π Ω0 (1) P δ (qj)qδ δ q ε γδ γ q κ = ih q(1) (G = 0) γδ δ 41

42 Quadratic approx. with IR-active optic modes ε Γn (Fan) 1 = ℜ nqj ω qj κ aκ 'bn' φγn κ a Hκ φqn' φqn' κ 'b Hκ ' φ Γn ξκ a (qj)ξκ 'b ( qj) iq.( Rκ 'b Rκ a ) e ε Γn ε qn' Mκ Mκ ' H q(1) = κ a Hκ ξκ a (qj) diverges like 1/q for polar optic modes. κa 2 At band extrema, the denominator diverges like 1/q. 2 For non-polar modes, divergence like 1/q, can be integrated 4 For polar optic modes, divergence like 1/q, cannot be integrated The adiabatic quadratic approximation breaks down for materials with IR-active optic modes. [ Note : In gapped systems, only elemental solids do not have IR-active modes] S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete & XG, J. Chem. Phys. 143, (2015) 42

43 Non-adiabatic AHC theory Beyond Rayleigh-Schrödinger perturbation theory MBPT Fan self-energy : H v(1) H v(1)* This yields a renormalizable theory Different levels : 0 On-the-mass shell approximation ε λ = ε λ0 + Σ ep ( ε λ λ) 0 Quasi-particle approximation ε = ε 0 + Σ ep (ε ) ε λ = ε λ0 + Z λ Σ ep λ (ε λ ) λ λ λ λ Or even spectral functions S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete & XG, J. Chem. Phys. 143, (2015) 43

44 T-dependent bandgaps for several insulators Zero-temperature limit and High-temperature linear slope 0 ε λ = ε λ0 + Σ ep λ (ε λ ) S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete & XG, J. Chem. Phys. 143, (2015) 44

45 LiF self-energy and spectral function Example for the top of the valence band ε λ = ε λ0 + Σ ep λ (ε λ ) 0 ε λ = ε λ0 + Z λ Σ ep λ (ε λ ) G. Antonius, S. Poncé, E. Lantagne-Hurtebise, G. Auclair, XG & M. Côté, Phys. Rev. B 92, (2015) 45

46 Spectral function in the BZ (0 Kelvin) Continuous green line : electronic structure without el-phonon coupling Satellite at VB Gamma VB diffuse spectral function except at the top G. Antonius, S. Poncé, E. Lantagne-Hurtebise, G. Auclair, XG & M. Côté, Phys. Rev. B 92, (2015) 46

47 Dynamical ep renormalisation for 4 solids On top of DFT : from static AHC (delta=0.1 ev), to position of peak maximum G. Antonius, S. Poncé, E. Lantagne-Hurtebise, G. Auclair, XG & M. Côté, Phys. Rev. B 92, (2015) From peak max of spectral function 47

48 A few more data for ZPM - all within DFT - Not of equivalent quality Different approximations... Big shifts (>1eV) : CH4 crystal 1.7 ev (Montserrat et al, 2015) NH3 crystal 1.0 ev (Montserrat et al, 2015) Ice 1.5 ev (Montserrat et al, 2015) HF crystal 1.6 ev (Montserrat et al, 2015) Helium (at 25 TPa) 2.0 ev (Montserrat et al, 2014) Medium shifts (>0.2 ev, like C-diam, LiF, MgO, BN, AlN) : Helium (at 0 GPa) 0.40 ev (Montserrat, Conduit, Needs, 2013) LiNbO ev (Friedrich et al, 2015) Polyethylene 0.28 ev (Canuccia & Marini, 2012) Small shifts (<0.2 ev, like Si) : LiH 0.04 ev (Montserrat, Drummond, Needs, 2013) LiD 0.03 ev (Montserrat, Drummond, Needs, 2013) GaN 0.13 ev (Kawai et al, 2013) GaN 0.15 ev (Nery & Allen, arxiv: , 2016) Trans-polyacetylene 0.04 ev (Canuccia & Marini, 2012) Black phosphorus 0.02 ev (Villegas, Rocha & Marini, 2016) Bi2Se3 family <0.02 ev (Montserrat & Vanderbilt, 2016) 48

49 Summary - Many techniques to get the T-dependent electronic structure and zero-point motion effect -Many effects to be taken into account : thermal expansion, Fan, Debye-Waller, dynamical self-energy, anharmonicities, non-rigid ion behaviour, accurate starting electronic structure (GW), accurate electron-phonon coupling (GW) - Sampling phonon wavevector (= supercell size) is is a challenge -The static quadratic approximation breaks down for infra-red active solids (both for AHC and supercell case) -Still lot of work to do to improve our tools 49