Homogeneous Electric and Magnetic Fields in Periodic Systems

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1 Electric and Magnetic Fields in Periodic Systems Josef W. idepartment of Chemistry and Institute for Research in Materials Dalhousie University Halifax, Nova Scotia June /24

2 Acknowledgments NSERC, Canada Research Chairs for funding 2/24

3 Outline /24

4 Density functional theory Minimize occ occ E el {ψ} = ψ α T+v ext ψ α +E Hxc [n] ǫ βα ( ψ α ψ β δ αβ ) α αβ where occ n(r) = ψα(r)ψ β (r) and gradient is δe/δ ψ α α 4/24

5 Planewaves and pseudopotentials Periodicity of the solid leads to Bloch theorem: ψ nk (r) e ik r u nk (r) and the cell periodic part is expanded in planewaves: u nk (r) = u nk (G)e ig r G This is efficient if the core electrons are replaced by pseudopotentials. 5/24

6 Projector Augmented Wave Method The PAW method (Blöchl) projects from pseudofunctions back to all-electron valence space functions. ψ = T ψ T = 1+ i,r [ ] φ ir φ ir p ir ψ A ψ = ψ A ψ + ij,r ψ p ir p jr ψ ( ) φ ir A φ jr φ ir A φ jr 6/24

7 electric field V(R+r) = V(r) V(r)+eE r Obvious coupling between external electric field E and electric charge leads to energy term ee r This term is OK for finite systems but not for infinite systems! Appear to have lost all bound states! 7/24

8 Modern Theory of Polarization King-Smith and Vanderbilt showed that polarization does not suffer from unboundedness: P = ie (2π) 3 dk u nk k u nk n BZ Nunes and Gonze showed how polarization enters into a well-posed minimization scheme with finite electric field: E[ψ,E] = E[ψ] ΩE P(ψ) 8/24

9 Inclusion of a Finite Electric Field Minimize E = E 0 P E, where: P is computed via PAW transform and discretization 1 Generalized norm constraint is imposed: ψ n S ψ m = δ nm On-site dipole contribution from T is included: Form gradient: ũ nk T k i kt k ũ nk, ϕ I q,r,k = e ik (r R) ϕ I q,k δe/δ u mk = δe 0 /δ u mk E δp/δ u mk Implemented in Abinit, including spin polarized systems, spinors, spin-orbit coupling 1 King-Smith and Vanderbilt, cond-matt; et al., Comp. Mater. Sci. 58, 113 (2012) 9/24

10 Convergence with k-mesh DFPT ε Finite field 11.5 experiment Inverse k mesh/bohr Convergence with mesh size for Si 10/24

11 Applications Polarization is computed as a function of applied field and fit to the form (SI units for polarization and field): Polarization, C m -2 (a) x10 8 3x10 8 4x10 8 5x10 8 Electric Field, V m -1 (b) 6x10-5 P i = ǫ 0 χ (1) ij E j +2ǫ 0 d ijk E j E k, Polarization, C m -2 5x10-5 4x10-5 3x10-5 2x x x10 8 3x10 8 4x10 8 5x10 8 Electric field, V m -1 11/24

12 Applications High and low frequency susceptibility: χ αβ = dp α /de β Second order susceptibilities Compound ǫ 0 ǫ d 123 pm/v AlP (LDA) (PBE) (expt) AlAs (LDA) (PBE) (expt) AlSb (LDA) (PBE) (PBE + SO) 9.76 (expt) /24

13 Application: MgO Dielectric Method PAW E-field, PBE PAW DFPT, LDA NCPP DFPT, LDA Expt ǫ N.B. in DFPT, 2 E E i E j 0 is computed directly, without presence of a field. Polarization x x x MgO in Finite Electric Field 2.0x x10-5 slope = χ = πχ = x x10-5 Electric field (a.u.) 1.0x /24

14 Photoelasticity Inverse of dielectric tensor changed by stress or strain: B ij = p ijkl ǫ kl = π ijkl σ kl Compound ǫ p 11 p 21 p 44 Si (LDA) (PBE) (expt) C (LDA) (PBE) (expt) /24

Photoelasticity in oxides Quantity MgO BaO SnO C 11 325.8 158.3 111.7 C 33 43.4 C 12 98.8 46.8 95.0 C 13 18.9 C 44 162.5 35.7 30.

15 Photoelasticity in oxides Quantity MgO BaO SnO C C C C C C π π π π π π ǫ ǫ /24

16 magnetic in insulators One approach to magnetic in periodic insulators is the long wavelength approach of Louie and co-workers: B Bcos(q r) with q 0. Problematic: cannot always find k u such that k u u 0 k = 0 AND u0 k+g = eig r u 0 k. In 2005 and 2006, Ceresoli, Thonhauser, Resta, and Vanderbilt established: 1 M = 2c(2π) 3Im dk k u n k (H k δ nn +E nn k) k u nk nn BZ C = i 2π n BZ dk k u nk k u nk 16/24

17 Magnetic Translation Symmetry Recall gauge-dependent Hamiltonian: H = 1 2 (p+ 1 c A)2 +V In 2010, Essin et al. 2 (see also Brown, Zak) discussed magnetic translation symmetry: O r1,r 2 = Ōr 1,r 2 e ib r 1 r 2 /2c where Ō has lattice symmetry. 2 PRB (2010) 17/24

18 Density operator perturbation theory Rather than perturbing the wave function, can work with the density operator: 3 : ρ = ρρ ρ 1 = ρ 1 ρ 0 +ρ 0 ρ 1 +O(2) Using magnetic translation symmetry operation, all field dependence has been transferred FROM the Hamiltonian TO the density operator and we must perturb ρ r1,r 2 = ρ r1,r 2 e ib r 1 r 2 /2c 3 Lazzeri and Mauri, PRB (R) (2003) 18/24

19 A New Theory of Orbital Magnetic Susceptibility Based on the the previous ideas Xavier Gonze and I have developed a complete treatment of magnetic field response in a periodic insulator. Key new ingredient: 4 T k = Ṽk W k + m=1 1 m! ( ) ( i m m ) ε αnβ 2c nγ n B αn n=1 ( β1 βm Ṽ k )( γ1 γm Wk ), E (n) dk = (2π) 3Tr[( ρ(n) kvv + ρ(n) kcc ) H k ]. BZ 4 X. Gonze and, PRB 84, (2011) 19/24

20 Development of ρ Density operator perturbation theory is used: 5 ρ (1) kd = i 2c ε αβγb α ( β ρ (0) k )( γ ρ (0) k ), ρ (2) kd = ρ (1) k ρ(1) k quadratic + i 2c ε αβγb α [( β ρ (0) k )( γ ρ (1) k )+( β ρ (1) k )( γ ρ (0) k )] linear ( 2 ) 1 8c 2 ε αnβ nγ n B αn β1 β2 ρ (0) k. γ 1 γ2 ρ (0) k frozen n=1 Full ρ (n), including CV and VC parts, may be subsequently recovered if needed. 5 McWeeny Phys Rev 126, 1028 (1962); Lazzeri and Mauri, PRB (R) (2003) 20/24

21 Checking the Theory The theory recovers the result for magnetization of Essin et al., essentially E (1) Second order, E (2), is a new, rigorous result for orbital susceptibility We then checked it with a tight binding model: analytical versus numerical 21/24

22 Checking the theory A sites at corners, initially occupied B sites at centers, initially empty on-site energies: E A < E B A A couplings s; A B couplings t B field applied perpendicular to plane: H r1,r 2 = H r1,r 2 e ib r 1 r 2 E (2) B t A s s Example with t = 2.0. E (2) computed from theory, and by direct diagonalization 22/24

23 Mixed Perturbations In addition to the results for ρ (n) and E (n) due to magnetic, we have also established the response to mixed magnetic and other perturbations µ: 2 E µ B = BZ dk (2π) 3Tr [ ρ (1) H k µ ] 23/24

24 Summary Modern theory of polarization and finite in PAW formalism Applications to linear and nonlinear electric susceptibility Works with spin-orbit, spin-polarized, etc. New theory of orbital magnetic susceptibility, extension to mixed perturbations Implementation in Abinit underway MANY thanks to Xavier Gonze, Marc Torrent, Abinit development and theory community 24/24

Berry Phases and Curvatures in Electronic-Structure Theory. David Vanderbilt Rutgers University

Berry Phases and Curvatures in Electronic-Structure Theory David Vanderbilt Rutgers University Rahman prize for: Theory of polarization (King-Smith & Vanderbilt) Ultrasoft pseudopotentials Three quick