First-Principles Calculation of Electric Polarization Fumiyuki - PowerPoint PPT Presentation

First-Principles Calculation of Electric Polarization Fumiyuki Ishii Kanazawa University The International Summer workShop 2018 on First-Principles Electronic Structure Calculations (ISS2018), July 5, 2018

Electric polarization • Fundamental physical quantity of insulators • Characterize dielectric properties of insulators • Piezoelectricity, Ferroelectricity, Magnetoelectric effect • Many applications • Capacitor, Piezoelectric device, Ferroelectric memory • Momentum dependence: Characterize topological insulators

Perturbations and Responses 1. Mecanical 2. Thermal 3. Electric 4. Magnetic 5. Chemical Perturbations Responses 1. Mecanical Elasticity Thermal Electromechanical Magnetostriction Osmotic pressure expansion 2. Thermal Thermal Thermal Pyroelectric/ Thermomagnetic Heat diffusion insulating conductivity Thermoelectric (Peltier) Piezoelectric Pyroelectric/ Electric Polarization Magnetoelectric Battery 3. Electric Thermoelectric Electric Conductivity (Seebeck) Magnetostriction Thermomagnetic Magnetoelectric Magnetization ? 4. Magnetic 5. Chemical Osmotic pressure Heat diffusion Battery ? diffusion Based on the table of Hidetoshi Takahashi

In the textbook … The polarization P is defined as the dipole moment per unit volume, averaged over the volume of a cell. s s P = 1 = 1 ∑ ∑ p i r n q n - - - - - - - V V ⬇ ⬇ ⬇ ⬇ ⬇ ⬇ ⬇ i n s s + + + + + + + c = 1 ∑ r n q n 1 cell V n c Periodic boundary condition S:sample ,C:cell Dipole sum of discrete charges

Problems in electric polarization • Resta (1992): Contrary to common textbook statements , the dipole of a periodic charge distribution is ill defined, except the case in which the total charge is unambiguously decomposed into an assembly of localized and neutral charge distributions. P is not a bulk property, while the variations of P are indeed measurable .

Ca Can we we comp mpute e P from m ch charge de dens nsity ty ? ? Charge distribution is continuous in real materials. R. M. Martin, PRB 9, 1998(1974). Local polarization field P el ( r ) ∇⋅ P el ( r ) = − ρ ( r ) P el = 1 ∫ P ( r ) d r Ω cell = 1 + 1 [ ] ds d r ρ ( r ) r r n ⋅ P ( r ) ∫ ∫ Ω Ω cell surface cell to cell term (current) Conclusion � •Absolute value of polarization is not bulk property •Dipole moment divided by unit cell volume ≠ Polarization

Observation of electric polarization ⬇ • Current induced by perturbation ( ) = ∂ P A J λ ∂λ • Change in polarization by perturbation ⬆ ∂ P ( ) d λ ∫ ∫ Δ P = J λ = ∂λ d λ In classical way: j = − ne v Δ t ∫ [ ] − − ne r (0) [ ] Δ P = − ne v dt = − ne r ( Δ t ) 0 = P ( Δ t ) − P (0)

Electric polarization expressed by wave function occ P = e X X h k n j r j k n i V n =1 k H j k n i = E k n j k n i occ d P e d X X dt h k n j r j k = n i dt V n =1 k occ e X X “ h @ t k n j r j k n i + h k n j r j @ t k ” = n i V n =1 k 1 occ e X X X ( h @ t k n j k m ih k m j r j k = n i V n =1 m =1 k h k n j r j k m ih k m j @ t k + n i )

Electric polarization expressed by wave function occ 1 d P e X X X ( h @ t k n j k m ih k m j r j k = n i dt V n =1 m =1 k h k n j r j k m ih k m j @ t k + n i ) Velocity operator h k m j v j k i � h k m j [ r ; H ] j k n i = i � ( E k n ` E k m ) h k m j r j k n i = n i h k m j v j k n i h k m j r j k n i = i � ( E k n ` E k m ) n i ) ˜ h k n j r j k ( h k m j r j k m i =

Electric polarization expressed by wave function occ @ h @ t k 0 n j k m ih k m j v j k 1 n i ` ie d P X X X = ` c:c A ( E k n ` E k V � dt m ) n =1 m 6 = n k Bloch wavefunction and its periodic part e i k ´ r j u k j k n i = n i H j k E k n j k n i = n i e ` i k ´ r He i k ´ r j u k E k n j u k n i = n i H j u k ~ E k n j u k n i n i = h k m j v j k h u k v j u k n i = m j ~ n i

Heisenberg Equation of Motion i � d r = [ r ; H ] dt i � v = [ r ; H ] Bloch wavefunction and its periodic part e ` i k ´ r He i k ´ r ~ = H i � d r ! e i k ´ r = i � ~ e ` i k ´ r [ r ; H ] e i k ´ r e ` i k ´ r = v dt if [ r k ; H ] = 0 , ` i r e ` i k ´ r He i k ´ r + e ` i k ´ r He i k ´ r i r r k ~ = H r k ~ ` i [ r ; ~ H ] = � ~ = H v m jr k ~ H h k m j v j k h u k v j u k n i = h u k j u k n i = m j ~ n i �

Electric polarization expressed by wave function occ 0 1 @ h @ t k n j k m ih k m j v j k n i d P ` ie Z X X = ` c:c BZ d k A 8 ı 3 � ( E k n ` E k dt m ) n =1 m 6 = n occ 0 @ h @ t u k n j u k m ih u k v j u k 1 ` ie m j ~ n i Z X X = ` c:c BZ d k A 8 ı 3 � ( E k n ` E k m ) n =1 m 6 = n occ m jr k ~ 0 1 @ h @ t u k n j u k m ih u k H j u k n i ` ie Z X X = ` c:c BZ d k A 8 ı 3 ( E k n ` E k m ) n =1 m 6 = n occ ` ie Z X “ ” h @ t u k n jr k u k n i ` hr k u k n j @ t u k = n i BZ d k 8 ı 3 n =1

First-order perturbation theory ‹ ~ H ( k + ´ k ) ` ~ ~ = H ( k ) H j u k + ∆ k j u k i = n i n m j ‹ ~ m ih u k H j u k n i H 2 ) X j u k + O ( ‹ ~ + E k n ` E k m m 6 = n m jr k ~ m ih u k H j u k n i jr k u k X j u k n i ’ E k n ` E k m m 6 = n Ordinary derivative to partial derivative dt j u k ¸ ;t i = @ k ¸ j u k ¸ ;t i dk ¸ d + @ t j u k ¸ ;t i = @ t j u k ¸ ;t i dt

Electric polarization expressed by wave function Z ´ t dtd P dt = P (´ t ) ` P (0) 0 Z ´ t occ ` ie Z X “ h @ t u k n jr k u k n i ` hr k u k n j @ t u k ” = n i dt BZ d k 8 ı 3 0 n =1 Z ´ t occ ` ie Z X “ ” @ t h u k n jr k u k n i ` r k h u k n j @ t u k = dt BZ d k n i 8 ı 3 0 n =1 For k ¸ direction ; P ¸ (´ t ) ` P ¸ (0) ie Z = dk ˛ dk ‚ ˆ 8 ı 3 Z ´ t Z G ¸ occ X “ @ k ¸ h u k n j @ t u k n i ` @ t h u k n j @ k ¸ u k ” n i dt dk ¸ 0 0 n =1

Electric polarization expressed by Berry phase (King-Smith & Vanderbilt 1993) Z G ¸ occ P ¸ ( t ) = ` ie Z X dk ¸ h u k n ( t ) j @ k ¸ j u k n ( t ) i dk ˛ dk ‚ 8 ı 3 0 n =1 Z G ¸ occ e Z X dk ¸ h u k n ( t ) j @ k ¸ j u k = Im n ( t ) i dk ˛ dk ‚ 8 ı 3 0 n =1

Example: Orthorhombic unitcell Case: ( k ˛ ; k ‚ ) = (0 ; 0) sampling , G ˛ = 2 ı b , G ‚ = 2 ı c Z G ¸ occ e Z X dk ¸ h u k n ( t ) j @ k ¸ j u k P ¸ ( t ) = Im n ( t ) i dk ˛ dk ‚ 8 ı 3 0 n =1 e Z = dk ˛ dk ‚ ffi ( t ) 8 ı 3 2 ı 2 ı e e ea = c ffi ( t ) = 2 ıbc ffi ( t ) = 2 ıabc ffi ( t ) 8 ı 3 b ffi ( t ) ea ea ! = ffi ( t ) = 2 ı ˙ cell ˙ cell 2 ı

Numerical calculation of Berry Phase Z G ¸ occ X dk ¸ h u k n ( t ) j @ k ¸ j u k ffi ( t ) = Im n ( t ) i 0 n =1 We difine overlap matrix S ( k ; k 0 ; t ) , where S nm ( k ; k 0 ; t ) ” h u k n ( t ) j @ k ¸ j u k n ( t ) i . We use well-known matrix identity, det exp A = exp tr A , when A = log S $ exp A = S . log det S = tr log S . Z G ¸ n ( t ) j u k 0 ¸ h u k ffi ( t ) = Im dk ¸ tr @ k 0 m ( t ) ij k 0 = k 0 Z G ¸ ¸ S ( k ; k 0 ; t ) j k = k 0 = Im dk ¸ tr @ k 0 0

Numerical calculation of Berry Phase A = log S $ exp A = S det exp A = exp tr A , log det S = tr log S S nm ( k ; k 0 ; t ) j k = k 0 = ‹ mn Z G ¸ ¸ S ( k ; k 0 ; t ) j k = k 0 ffi ( t ) = Im dk ¸ tr @ k 0 0 Z G ¸ ¸ S ( k ; k 0 ; t ) 2 4 @ k 0 3 = Im dk ¸ tr 5 S ( k ; k 0 ; t ) 0 j k = k 0 Z G ¸ ¸ log S ( k ; k 0 ; t ) j k = k 0 = Im dk ¸ tr @ k 0 0 Z G ¸ ¸ log det S ( k ; k 0 ; t ) j k = k 0 = Im dk ¸ @ k 0 0

Numerical calculation of Berry Phase Z G ¸ ¸ log det S ( k ; k 0 ; t ) j k = k 0 ffi ( t ) = Im dk ¸ @ k 0 0 If we use k-point sampling mesh J along k ¸ direction, k ¸;s = sG ¸ =J and ´ k ¸ = G ¸ /J. J ` 1 X ffi ( t ) = Im lim ´ k ¸ ˆ ´ k ¸ ! 0 s =0 log det S nm ( k ¸;s ; k ¸;s + ´ k ¸ ; t ) ` log det S nm ( k ¸;s ; k ¸;s ; t ) ´ k ¸ J ` 1 X ffi ( t ) = Im lim log det S nm ( k ¸;s k ¸;s + ´ k ¸ ; t ) ´ k ¸ ! 0 s =0

Numerical calculation of Berry Phase J ` 1 X ffi ( t ) = Im lim log det S nm ( k ¸;s k ¸;s + ´ k ¸ ; t ) ´ k ¸ ! 0 s =0 J ` 1 Y = Im ´ k ¸ ! 0 log lim det S nm ( k ¸;s k ¸;s + ´ k ¸ ; t ) s =0 ◯ Ｇ ◯ ◯ S+1 S Ｇ ● ０ ● ●

Overlap matrix S in OpenMX ψ ( k ) e i k · r u ( k ) σ µ ( r ) = σ µ ( r ) , N 1 c ( k ) e i R n · k � � √ = σ µ,i α φ i α ( r − τ i − R n ) , N n ⟨ | ⟩ i α ⟨ u ( k ) σ µ | u ( k + ∆ k ) ⟨ ψ ( k ) σ µ | e i k · r e − i k · r e − i ∆ k · r | ψ ( k + ∆ k ) ⟩ = ⟩ , σν σν ⟨ ψ ( k ) σ µ | e − i ∆ k · r | ψ ( k + ∆ k ) = ⟩ , σν 1 e − i k · ( R n − R n ′ ) × c ( k ) ∗ σ µ,i α c ( k + ∆ k ) � � = σν ,j β N n , n ′ i α ,j β ⟨ φ i α ( r − τ i − R n ) | e − i ∆ k · ( r − R n ′ ) | φ j β ( r − τ j − R n ′ ) ⟩ . r ′ = r − τ i − R n ,