First-principles calculations of insulators in a. finite electric field

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1 Université de Liège First-principles calculations of insulators in a finite electric field M. Veithen, I. Souza, J. Íñiguez, D. Vanderbilt, K. M. Rabe and h. Ghosez Supported by: FNRS Belgium, VW Stiftung, Communauté Française & Rutgers University Many thanks: Donald R. Hamann

2 Outline Introduction: Difficulties related to the electric field perturbation 1 st part: Structural response to macroscopic electric fields Formalism Berry phase calculation of and the ddk Implementation In practice: how to optimize the structure for a fixed polarization Applications nd 2 part: First-principles approach to insulators in finite electric fields Formalism Definition of an electric field dependent energy functional For < c : minimization of the energy functional Computation of forces and stresses Implementation in ABINIT In practice: how to do a finite electric field calculation Applications Conclusions

3 Nonperiodic lectric field perturbation: difficulties The perturbation - r is nonperiodic and unbound from below Methods based on Bloch's theorem and eigenstates Unbound from below nk do not apply nergy can always be lowered by transferring charge from valence states in one region to conduction states in a distant region c v e - L g lectric field bends the energy bands Dielectric breakdown by interband (Zener) tunneling An infinite crystal in the presence of an electric field does not have a ground-state

4 1 st Method: Structural response to macroscopic electric fields lectric field derivatives of arbitrary order can be computed from DFT R. W. Nunes and X. Gonze, RB 63, (2001) Low-order Taylor expansion of the energy with respect to the electric field Dependence of the energy on the structural degrees of freedom is preserved to all orders Two step approach: 1) Map out the energy as a function of the polarization 2) Use this energy surface & field-coupling term to compute the ground state structure in the presence of the electric field More informations Na Sai, K. M. Rabe and D. Vanderbilt, RB 66, (2002) H. Fu and R.. Cohen, Nature 403, 281 (2000)

5 " #! Constrained polarization approach: formalism Structural response at constant electric field F min R, F R,, olarization: thermodynamic conjugate of : Legendre transformation ( ): R eq eq Strain and atomic positions R at fixed electric field 1 F F R, F, min R, min F R, F R,,, Strain and atomic positions at fixed polarization: R eq & eq Inverse Legendre transformation F min F

6 $ & % A 4 < 0 = 1, + ) * ; / > 2 B 3 9 ' J L H F G K I ( Q U M T O N Constrained polarization approach: formalism Taylor expansion of F(R, ) around = 0 F R,, F R,, 0 -. F R,, , 7 8 : 2 F R,?, 0 CD Truncation of the Taylor expansion at the lowest order F 1 R,, F R,, 0 R,, 0 Approximation is supposed to be valid in systems where the ionic contribution to the polarization dominates (f. ex. ferroelectrics) Resulting expression F min, R, F R,, 0 RS R,, 0

7 V ^ Z _ [ ] ` \ f d j e i o p W Y X ~ ƒ u } v s q r t Ž ˆ š Constrained polarization approach: formalism System of equations to be solved F R,, 0 R F R,, 0 fhg R,, 0 R R,, 0 khl 0 acb 0 mcn R,, 0 In practice: Taylor expansion of F(R,,0) and (R,,0) F R, R,, 0, 0 F R 0, R 0, 0, 0 0, 0 f R 1 ŠZ Œ wcx R y{z e v1 2 K { R 2 2 C { 2 R { R 0, 0 : trial guess of the initial coordinates and strains ( f, : Hellmann-Feynmann forces and stresses K : interatomic force constants C : rigid atom elastic constants : coupling parameters between R and Z * : Born effective charges e : clamped ion piezoelectric tensor R = R R 0 & = - 0 )

8 Ÿ ª ž œ µ ¹ º ½ ¼» Á À ¾ Constrained polarization approach: formalism Linear systems of equations 1 K Z 1 C e 0 Z e R «f ± ² ³ = difference between initial and target values of Iterative solution Choose R 0 & 0 Compute f, (Hellmann-Feynmann theorem) (Berry phase) K, C,, e & Z * (linear response) Solve linear system of equations to obtain new atomic positions and strains R 0 + R R ÂÄÃ repeat until converged

9 Ñ Ð Å Ò Æ Ö Ô Ø Ù Ó Ç ã á Ý ß à ä â È É å è ë ê æ é ì Ê Ì Berry phase calculation of and the ddk Definition of the Born effective charges Z Computation of : string averaged Berry phase ÍÏÎ R k-b k k+b 2 2 e Õ3 Computation of Z * (linear response, non-stationarry expression) Z ÚÜÛ 2 i e ROBLM: If the ddk is computed from linear response, the relation between Z * and is only satisfied in a the limit of a dense k-point mesh Finite k-point grid: finite difference formula of the ddk 2 dk Þ3 dk ln occ m det S k, k R k b Comments: Ë No violoation of the charge neutrality k ç1 2b b b b b Same arguments in case of the piezoelectric tensor

10 í î ï ñ ò ó ô õ ö ø Implementation ABINIT: new Berry phase routine (berryphase_new.f) asier to use olarization in cartesian coordinates ð MI parallelization over k-points ANADDB: Structure of the DDB Header Total energy 1 st -order energy derivatives 2 nd -order energy derivatives New routine relaxpol.f Called from anaddb.f Compute in cartesian coordinates Solve linear system of equations Compute residual forces and stresses Update atomic positions and lattice constants

11 ù ú û ü ý þ ÿ In practice (v.4.3.x) 1 st step: Ground state calculation of the forces and the stress tensor nd 2 step: Berryphase calculation of and the ddk kptopt = 3 berryopt = -3: use berryphase_new.f routine rfdir = 1 1 1: compute projection of & ddk along x, y and z nband = number of occupied bands th 3 step: Linear response calculation th 4 step: Use ANADDB to compute new atomic positions and lattice constants polflag = 1 targetpol = target value of the polarization (cartesian coordinates & C/m 2 ) relaxat, relaxstr natfix, iatfix, istrfix specify which degrees of freedom are allowed to relax Repeat steps 1, 2, 3 & 4 until convergence is reached

12 Application: double well in BaTiO3

13 Application: c/a of btio 3 Motivation: btio 3 thin films grown on a SrTiO 3 substrate as a function of SrTiO 3 btio 3 xperiment: c/a decreases with decreasing film thickness Theory (effective hamiltonian): depolarizing electric field reduction of the polarization reduction of c/a Result a has been fixed at the theoretical lattice constant of SrTiO 3 C. Lichtensteiger et al., cond-mat/

14 Application: non-linear dielectric response of tetragonal btio 3 z as a function of an electric field along z Ionic dielectric constant: = (linear response = 17.37) Tunability: d zz d z pm V

15 2 nd method: First-principles approach to insulators in finite electric fields For a supercell of size L = N k a (a = unit cell size, N k = number of k- points), use only fields smaller than For < c, we can minimize the energy functional c g N k a F F KS a 3 FKS = Kohn-Sham energy at zero electric field = macroscopic polarization that is computed as a Berry phase of field polarized Bloch functions Long-lived metastable state For > c, F has no minimum More informations I. Souza, J. ĺñiguez and D. Vanderbilt, hys. Rev. Lett. 89, (2002)

16 ; 8 9 : = < # ' & % $! " 5. ( 7 3, - 4 / 6 Minimization of the energy functional F Strategy: use the preconditione cg minimization implemented in ABINIT to minimiez F KS Gradient of the energy functional F G nk F > G nk H KS k W nk usual zero-field term electric field term lectric field term k-b k k+b W nk )+* C m 1 S mn k, k b u mk b S mn k, k b u mk b Transform G nk into a preconditioned cg search direction D nk

17 @ A? I R Q N M L J O K C B H G Minimization of the energy functional F Update of a state new cos old sin D nk Minimize F( ) D = 0: analytic formula for min M. C. ayne et al., Rev. Mod. hys. 64, 1045 (1992) 0: no analytic formula for performed numerically F, line minimization must be min

18 S Y WX UV Z ^ [ ] \ a c _` b T Computation of forces and stresses Field polarized Bloch functions are stationary points of F, the Hellmann- Feynmann theorem yields for the force on atom j f j df dr j F r j (no implicit dependence on r j via the wavefunctions) F F KS a 3 el ion no explicit dependence on r j f j F KS r j ez j Similar arguments show that the expression used to compute the stress under zero electric field remains valid

19 d e f g h gstate.f initberry.f scfcv.f Implementation in ABINIT (4.3.x) Initialization of Berry phase calculation: olarization, ddk & electric field store informations in the efield_type structured datatype berryphase_new.f Initialize polarization pel_cg(1:3) & Initialize overlap matrices Start SCF optimization of wavefunctions vtorho.f vtowfk.f cgwf.f Non-SCF line minimizations. For each iteration update overlap matrices add the electric field contribution to the gradient perform line minimization numerically (linemin.f) Update polarization: add change in Berry phase to pel_cg(1:3) nd SCF optimization of wavefunctions berryphase_new.f Recompute polarization, check that it is consistent with the value obtained from the update of the wavefuntions

20 i j k l m n o p q s u In practice (4.3.x) 1 st step: erform a calculation under zero electric field nd 2 step: lectric fiel calculation (use wavefunctions computed during the first step to initialize the calculation) berryopt = 4 efield(1:3) = cartesian coordinates of the electric field in atomic units nsym = 1 kptopt = 3 nband = number of valence bands COMMNTS: insulators only r MI parallelization not yet implemented no spin polarization (spin polarization probably available in ABINITv4.4.x) t You should increase the amplitude of the electric field in small steps Suggestion: use multiple datasets. For each dataset, use a slightly larger value of the electric field and take wavefunctions of the previous dataset to initialize the SCF cycle.

21 ~ y v x } w Applications: AlAs, electric field along x Force on Al along the x direction Born effective charges: Z f z { Finit electric field calculation: Z * = Linear response calculation: Z * =

22 Š ˆ Applications: AlAs, electric field along x Applications: AlAs, electric field along x Optical dielectric constant F ƒ Ž ŒŽ Ÿž œ š Ÿž œ š «Ÿž ª Ž

23 ± Applications: AlAs, electric field along (1,1,1) olarziation as a function of = *(1,1,1) Non-linear optical susceptibilities Finite electric field calculation: (2) = pm/v Non-linear response calculation: (2) = pm/v

24 I. Souza et al., RL 89, (2002) Application to cubic semiconductors

25 ²» Conclusions First-principles calculations of insulators in finite electric fields 1 st method: structural response to macroscopic electric fields ³ Derivatives of the energy with respect to the electric field can be accurately computed from DFT Low order Taylor expansion of the energy with respect to the electric field µ Map out the energy as a function of the polarization Iterative optimization of the structure under the constrained of a fixed polarization nd 2 method: first-principles approach to insulators in finite electric fields ractical scheme for computing the electronic structure of insulators under a finite bias ¹ For < c: minimization of a modified energy functional º Accurate computation of the total energy, the polarization, the forces and stresses Finite difference calculation of electric field derivatives

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