Temperature and zero-point motion effects on the electronic band structure

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Boulanger (CEA-Grenoble), S. Poncé (UCL) M. Côté, G. Antonius (U. Montréal) + help from A.

1 Lausanne 2012 Temperature and zero-point motion effects on the electronic band structure X. Gonze Université Catholique de Louvain, Louvain-la-neuve, Belgium Collaborators : P. Boulanger (CEA-Grenoble), S. Poncé (UCL) M. Côté, G. Antonius (U. Montréal) + help from A. Marini, E. Canuccia for the QE+YAMBO/ABINIT comparison + discussions with F. Giustino Lausanne, Switzerland, 8 November

2 Temperature dependence of electronic/optical properties Motivation The electrical and optical properties clearly depend on temperature... - 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) Lausanne, Switzerland, 8 November

3 Even at 0 Kelvin... Motivation... vibrational effects are important, because of Zero Point motion (ZPM). From M. Cardona, Solid State Comm. 133, 3 (2005) The ZPM renormalization is important! We must also calculate it. Lausanne, Switzerland, 8 November

4 Motivation Vertex correction... And beyond? For materials with lighter atoms, remaining discrepancy 0.1 ev ev Due to phonons, at least partly! From Shishkin, Marsman, Kresse, PRL 99, (2007) Lausanne, Switzerland, 8 November

5 Outline (1) Motivation (2) Brief review of first-principles phonon methodology (3) The Allen-Heine-Cardona approach (AHC) (4) Corrections beyond the rigid-ion approximation (5) Application to diamond... Work in progress... X. G., P. Boulanger and M. Côté. Ann. Phys. (Berlin) 523, (2011). G. Antonius, S. Poncé, X. G. and M. Côté, in preparation Lausanne, Switzerland, 8 November

6 The present theoretical approach : keywords Density Functional Theory (from QM and EM to total energies of solids, molecules, nanosystems...) Plane Waves / Pseudopotentials (representation on a computer) ABINIT (the actual software) Density Functional Perturbation Theory /Linear Response (derivatives of total energy - dynamical matrices, phonon band structures,... ) Lausanne, Switzerland, 8 November

7 Brief review of phonon methodology

8 Density functional theory... The electronic energy can be obtained by the minimisation of occ E el { ψ α } = ψ α α under constraints of orthonormalization ψ α ψ β = δ αβ for the occupied orbitals (independent-particle system). The density is given by ρ( r occ ) = ψ * α ( r )ψ α ( r ) Approximation needed for The eigenfunctions also obey where ˆT + ˆVnucl ψ α + 1 ρ( r)ρ( r ') dr dr '+E xc ρ 2 r r ' Ĥ=ˆT + ˆV nucl + ρ( r ') r r ' E xc [ ρ] Lausanne, Switzerland, 8 November α Ĥ ψ α =ε α ψ α dr '+ δe xc δρ r ( ) [ ] (Kohn-Sham equation) ˆV nucl = V κ (r R κ ) κ

9 Interatomic Force Constants - IFC (I) Expansion of the total energy of a (periodic) crystal with respect to small deviations of atomic positions from the equilibrium ones (second-order): E tot ({ ΔR }) = E (0) tot + ΔR a κα 1 2 E tot 2 aκα R a κα Rκ'α' a' a'κ'α' a' ΔR κ'α' a ΔR kα where is the displacement along direction α of the atom k in the cell labeled a, from its equilibrium position R k + R a The matrix of IFC s is defined as C κα,κ'α' ( a,a' ) = 2 E tot R a κα Rκ'α' a' Lausanne, Switzerland, 8 November

10 Interatomic Force Constants (II) Its Fourier Transform (using translational invariance) C κα,κ'α' q ( ) = Cκα,κ'α' a' (0,a') e i q. R a' allows one to compute phonon frequencies and eigenvectors as solution of the following generalized eigenvalue problem: κ'α' C κα,κ'α' ( q).ξ κ'α' (mq) = M κ. ω 2 mq. ξ κα (mq) phonon displacement pattern masses square of phonon frequencies Note : If the eigenvalue is negative (imaginary frequency), the system is unstable against spontaneous deformation How to get second derivatives of the energy? Density Functional Perturbation Theory! Lausanne, Switzerland, 8 November

11 General framework of perturbation theory * * A ( λ)= A (0) + λa (1) + λ 2 A (2) + λ 3 A (3)... E { ψ; V nucl } Hypothesis : we know (0) V nucl ( λ)= V nucl (1) + λv nucl + λ 2 (2) V nucl +... through all orders, as well as ψ (0), n (0), E (0) We would like to calculate E (1), E (2), E (3)... ρ α (1), ρ α (2), ρ α (3)... ψ α (1), ψ α (2), ψ α (3)... ε α (1), ε α (2), ε α (3)... Lausanne, Switzerland, 8 November

12 Linear-response The unperturbed quantities ψ (0), Ĥ (0) (0), ε α must be precomputed (1) Also, ˆV nucl must be available. Then, ψ (1) can be found by solving self-consistently Ĥ (0) -ε α (0) ψ α (1) = P unocc Ĥ (1) ψ α (0) with Ĥ (1) = (1) ˆV nucl + 1 r-r' + occ δ2 E xc δρ( r )δρ( r') ρ (1) ( r') dr' ρ (1) ( r ) = ψ (1)* α ( r )ψ (0) α ( r ) + ψ (0)* α ( r )ψ (1) α ( r ) α under constraint ψ α (0) ψ β (1) = 0 Late 80 + early 90 Baroni, Gianozzi, Testa, XG, Savrasov... Alternative approach : frozen-phonon... But DFPT is much more efficient : allows to treat efficiently all wavevectors. See the Review of Modern Physics 73, 515 (2001) by S. Baroni, S. De Gironcoli, A. Dal Corso, and P. Giannozzi. Lausanne, Switzerland, 8 November

13 The Allen-Heine-Cardona approach

14 Long history of the theory of T-dependent effects In a context : Work from the 50, based on 2nd order perturbation theory treatment of the electron-phonon effect Fan Debye-Waller + empirical psps/tight-binding : 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)... 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 Lausanne, Switzerland, 8 November

15 Long history of the theory of T-dependent effects First-principles context : R. D. King-Smith, R. J. Needs, V. Heine and M. J. Hodgson. Europhysics Letters 10, 569 (1989) A. Marini, Phys. Rev. Lett. 101, (2008) F. Giustino, S. Louie, M. Cohen, Phys. Rev. Lett. 105, (2010) E. Canuccia and A. Marini, Phys. Rev. Lett. 107, (2011) E. Canuccia and A. Marini, Eur. Phys. J. B 85, 320 (2012) Lausanne, Switzerland, 8 November

16 Applications of Review AHC theory Silicon. Marini PRL (2008) Excellent agremeent with exp. Mostly broadening effect, imaginary part of the Fan term (not discussed in this talk) Giustino, Louie, Cohen, PRL105, (2010) Diamond. Direct gap at Gamma. ZPM 615 mev Excellent agremeent with exp. T-dep To be compared with Cardona estimation of 370 mev for the indirect exciton gap...?!? ZPM confirmed by Canuccia & Marini, Eur. Phys. J. B 85, 320 (2012) Lausanne, Switzerland, 8 November

17 Allen-Heine-Cardona theory Review Second-order (time-dependent) perturbation theory (no average contribution from first order). If one neglects the phonon frequencies with respect to the electronic gap : δε kn (T,V = const) = 1 ˆn qj (T ) + 1 N q 2 ε kn n qj = 1 2ω qj κ aκ 'b qj 2 ε kn R κ a R κ 'b ε kn n qj ξ κ a ( qj)ξ κ 'b ( qj) M κ M κ ' e iq.(r κ 'b R κa ) Phonon mode factor ξ κ a ( qj) phonon eigenmodes, κ = atom label, a=x, y, or z + occupation number from BE statistics Lausanne, Switzerland, 8 November

18 2 εkn R κ a R κ 'b Ĥ= T+ ˆ ˆV nucl + ρ( r' ) dr'+ de xc r-r' dρ r Eigenvalue changes? ε n = φ n Ĥ φ n Review Hellman-Feynman theorem : ε (1) (0) n = φ n Ĥ (1) (0) φ n One more derivative : ( ) ε n (2) = φ n (0) Ĥ (2) φ n (0) + 1 ( 2 φ (0) n Ĥ (1) (1) φ n + (c.c)) Debye-Waller Antoncik Fan self-energy In AHC, φ n (1) obtained by sum-over-states φ n (1) = φ m (0) m n φ m (0) Ĥ (1) φ n (0) ε n ε m Lausanne, Switzerland, 8 November

19 Derivatives of Review the Hamiltonian? ( ) Ĥ= T+ ˆ ˆV nucl + ρ r' r-r' dr'+ de xc dρ r ( ) ˆV nucl = V κ (r-r 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 κ Ĥ (2) pure site-diagonal! 2 ˆVnucl R κ a R κ 'b = 0 for κ κ ' Debye-Waller contribution pure site-diagonal! Moreover, invariance under pure translations 0 = ε (2) (0) (2) n = φ n Ĥ transl φ n (0) ( (0) (1) (1) Ĥ transl φ n + (c.c)) φ n Reformulation of the Debye-Waller term. Lausanne, Switzerland, 8 November

20 The AHC theory : Fan term and site-diagonal Review Debye-Waller term ε kn = ε (Fan) kn n qj n qj + ε kn (DW diag ) n qj ε kn (Fan) = 1 φ V kn κ a κ φk + φ qn ' k + qn ' κ 'b V κ ' φkn n qj ω qj Rκ aκ 'bn ' εkn ε k + qn ' ξ κ a ( qj)ξ κ 'b ( qj) M κ M κ ' e iq.(r κ 'b R κa ) ε kn (DW diag ) = 1 φ kn κ av κ φ φ kn ' kn ' κ 'b V κ ' φkn n qj ω qj Rκ aκ 'bn ' εkn ε kn ' 1 2 ξ κ a ( qj)ξ κ b ( qj) M κ + ξ κ ' a ( qj)ξ κ 'b ( qj) Good : only first-order electron-phonon matrix elements are needed (+ standard ingredients from first-principles phonon/band structure calculations) Bad : (1) summation over a large number of unoccupied states n (2) neglect of non-site-diagonal Debye-Waller term, while the rigid-ion approx. is not valid for first-principles calculations (3) DFT first-principles calculations, while MBPT should be used Lausanne, Switzerland, 8 November M κ '

21 Implementation Review Sum over state present in the AHC formalism, replaced by the use of Density-Functional Perturbation Theory quantities => large gain in speed. φ n (1) = φ m (0) m n φ m (0) Ĥ (1) φ n (0) ε n ε m For converged calculations for silicon : sum over states 125 h DFPT 17h X. G., P. Boulanger and M. Côté. Ann. Phys. (Berlin) 523, (2011). Lausanne, Switzerland, 8 November

22 Beyond the rigid-ion approximation

23 ... + non-site-diagonal Review Debye-Waller term? ε kn = ε (Fan) kn n qj n qj + ε kn (DW diag ) n qj + ε kn (DW non diag ) n qj ε kn (DW non diag ) = 1 φ n qj 2ω kn κ a κ 'b H φkn qj κ aκ 'b ξ κ a ( qj)ξ κ 'b ( qj) M κ M κ ' e iq.(r κ 'b R κa ) 1 2 ξ κ a ( qj)ξ κ b ( qj) M κ + ξ κ ' a( qj)ξ κ 'b ( qj) M κ ' This term vanishes indeed for κ = κ ' Is it an important contribution to the temperature effect? Quite difficult to compute from first principles for a solid... not present in the DFPT for phonons! Lausanne, Switzerland, 8 November

The phonon population Phonon population effect: dimers contribution: Diatomic molecules Diatomic molecules are the simplest system study the temperature dependence of their eigenvalues.

24 The phonon population Phonon population effect: dimers contribution: Diatomic molecules Diatomic molecules are the simplest system study the temperature dependence of their eigenvalues. This simplicity stems from the fact that they have discrete levels, well described with the theory of the molecular orbitals, and, especially, only one relevant vibration mode. (The 6 modes decouple as 3 translations, 2 rotations and the stretch mode.) We can write the eigenenergies of the electronic states as a Taylor series on the bond length: ε n = ε n 0 + ε n R ΔR ε n R 2 ΔR2 => A frozen-phonon approach can deliver 2 ε n R 2 for reference Lausanne, Switzerland, 8 November

25 The non-diagonal Debye-Waller Beyond the rigid-ion approximation The case of diatomic molecules term is simple enough, that we can also evaluate directly the importance of the non-diagonal Debye-Waller term. ε n (Fan + DDW ) n str = ω str M 1 M 2 R φ H n R φ n ' φ H n ' 1 R2 φ n n ' ε n ε n ' ε n (NDDW ) = n str 2ω str M 1 M 2 φ n 2 H R1 R 2 φ n Does not cancel, unlike with rigid-ion hypothesis! Only Hartree + xc contribution For the hydrogen dimer : 2 E HOMO R 2 Fan+diagDW = Ha bohr 2 A large difference : a factor of 2! 2 E HOMO R 2 Lausanne, Switzerland, 8 November all contribs = Ha bohr 2

Small molecules The non-waller term AHC Excellent agreement The Non-Diagonal DW term is a sizeable contribution to the total : equal in size but opposite

26 Small molecules The non-waller term AHC Excellent agreement The Non-Diagonal DW term is a sizeable contribution to the total : equal in size but opposite for H2, 10-15% for N2 and CO, 50% for LiF. X. G., P. Boulanger and M. Côté. Ann. Phys. (Berlin) 523, (2011). Lausanne, Switzerland, 8 November

27 Application to diamond

28 The non-diagonal Debye-Waller Solids term - Large number of vibration modes (full Brillouin Zone) - Still one can focus on the contribution of selected, commensurate vibration modes - e.g. Zero-point motion correction to the HOMO eigenenergy (mev) : DDW+Fan (=AHC) NDDW Gamma L X Finitedifferences 2/3 L So, the NDDW can be as much as 20%, but as small as 1%. Apparently, always the same order of magnitude (< 5 mev), while the DDW+Fan can vary widely, and especially diverge near critical points (see next slides). G. Antonius, S. Poncé, X. G. and M. Côté, in preparation Lausanne, Switzerland, 8 November

29 Numerical study Re : ZPM in view diamond -Target numerical accuracy 10 mev -Direct band gap at Gamma (like Giustino, Marini) - Converged number of plane waves ( Hartree) - k point sampling : 6x6x6 is sufficient for the generation of the first-order H - Sampling on the q phonon wavevectors for the Fan term is a big issue! δε ZPM Γn = 1 N q qj ε Γn n qj 1 2 ε Γn (Fan) = 1 φ Γn κ a V κ φ qn ' φ qn ' κ 'b V κ ' φ Γn ξ κ a( qj)ξ κ 'b ( qj) e n qj ω qj Rκ iq.(r κ 'b R κa ) aκ 'bn ' ε Γn ε qn ' M κ M κ ' Intraband contributions diverge due to the denominator! Lausanne, Switzerland, 8 November

30 The strong divergence Re : ZPM for view small q 15 lim ε Γn(Fan) q 0 n qj = lim 1 q 0 f ( ω qj qjn) ε Γn ε qn Acoustic modes : Fan/DDW contribs cancel each other Optic modes : lim ε Γn(Fan) 1 q 0 n qj q 2 f ( qjn) 0 Lausanne, Switzerland, 8 November

31 Divergences on Re isoenergetic : ZPM view surface 15 Set of isoenergetic wavevectors lim ε Γn(Fan) q q iso n qj = q lim 1 qiso f ( ω qj qjn) ε Γn ε qn 1 q ε qn qiso.( q q iso ) Only for the ZPM of the conduction state Lausanne, Switzerland, 8 November

32 q grid convergence Re : ZPM : slow view / fluctuations q grid #q in IBZ ZPM HOMO (mev) ZPM LUMO (mev) ZPM gap (mev) 6x6x6 x4s x8x8 x4s Non sense* 10x10x10 x4s x12x12 x4s x14x14 x4s Non sense* 16x16x16 x4s x18x18 x4s x20x20 x4s *Variations become wider... Lausanne, Switzerland, 8 November

33 Convergence Re wrt : to ZPM q sampling view Convergence like 1/Nq for q point grids Nq*Nq*Nq Confirm theoretical understanding Lausanne, Switzerland, 8 November

34 Smoothing the Re denominator : ZPM view ε Γn (Fan) = 1 φ V Γn κ a κ φ qn ' φ qn ' κ 'b V κ ' φ Γn n qj ω qj Rκ aκ 'bn ' ε Γn ε qn ' + iδ ξ κ a ( qj)ξ κ 'b ( qj) M κ M κ ' e iq.(r κ 'b R κa )... dramatically helps the convergence Take imaginary part to be 100 mev (as done by Giustino) : q grid #q in IBZ ZPM HOMO (mev) ZPM LUMO (mev) ZPM gap (mev) 8x8x8 x4s x12x12 x4s x16x16 x4s x20x20 x4s x24x24 x4s x28x28 x4s x32x32 x4s Converged within 10 mev! But... not to the value of Giustino et al (-615 mev) Lausanne, Switzerland, 8 November

35 More tests... Re : ZPM view -Three different pseudopotentials (TM, two UPF) including the one from Giustino. At 28x28x28 x4s grid. Pseudo ZPM HOMO (mev) ZPM LUMO (mev) ZPM gap (mev) Troullier-Martins (own psp) Berkeley 2005.UPF (from FG) pz-vbc.upf (from AM) Computed the ZPM shift from 2-atom primitive cell, 8-atom conventional cell, 16-atom supercell : perfect consistency. -Compared with finite difference calculations for selected phonon wavevectors (shown previously) Lausanne, Switzerland, 8 November

36 External checks Re : QE+YAMBO : ZPM view vs ABINIT Lausanne, Switzerland, 8 November

37 External checks Re : el-ph : ZPM matrix view element Lausanne, Switzerland, 8 November

External checks Re : ZPM : ZPM (not fully view converged) With a 10x10x10 q-point grid and 100 bands, near perfect agreement between ABINIT and YAMBO : 0.465 ev for the ZPM of the HOMO-LUMO gap?

38 External checks Re : ZPM : ZPM (not fully view converged) With a 10x10x10 q-point grid and 100 bands, near perfect agreement between ABINIT and YAMBO : ev for the ZPM of the HOMO-LUMO gap?!? Sublety : published results with YAMBO do not use a regular grid, but a random sampling. But results with EPW (F. Giustino) use a regular grid!????????? Lausanne, Switzerland, 8 November

39 External checks Re : temperature : ZPM view dependence Lausanne, Switzerland, 8 November

40 Summary - Electronic levels and optical properties depends on vibrational effects... Allen, Heine, Cardona formalism. - The thermal expansion contribution is easily calculated using DFT + finite differences, but only a very small contribution - The non-diagonal Debye-Waller term was shown to be non-negligible for several dimers (50%!). Apparently, its relative effect on the temperature dependence of the electronic bands in diamond is much smaller (1-2%). - The calculation of the phonon population contribution for systems with many vibration modes can be done efficiently within DFPT + the rigid-ion approximation. Very difficult q point convergence! Lausanne, Switzerland, 8 November

41 Perspectives - Code comparison is important! But the discrepancy is not yet understood... - Of course : DFT (DFPT) calculation => One should try GW! - Need broadening parameter (100 mev), of course non ab initio. - Treatment of AHC : static phonons... The later work by E. Canuccia & A. Marini includes dynamical effects... Spectral function exhibit peaks, but the weight can be much smaller than Link with the approach by Eiguren & Draxl? Lausanne, Switzerland, 8 November

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

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,