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
The International Summer workShop 2018
Elasticity Thermal expansion Electromechanical Magnetostriction Osmotic pressure
Thermal insulating Thermal conductivity Pyroelectric/ Thermoelectric (Peltier) Thermomagnetic Heat diffusion
Piezoelectric Pyroelectric/ Thermoelectric (Seebeck) Electric Polarization Electric Conductivity Magnetoelectric Battery
Magnetostriction Thermomagnetic Magnetoelectric Magnetization ?
Osmotic pressure Heat diffusion Battery ? diffusion
Perturbations Responses
Based on the table of Hidetoshi Takahashi
s
i s
s
n s
c
nqn n c
S:sample ,C:cell
+
+
+
+
+
+
+
1 cell
Dipole sum of discrete charges
Periodic boundary condition
The polarization P is defined as the dipole moment per unit volume, averaged over the volume of a cell.
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.
∇⋅Pel(r) = −ρ(r)
Local polarization field Pel(r)
Charge distribution is continuous in real materials.
Pel = 1 Ω P(r
cell
∫ )dr = 1 Ω drρ(r)r
cell
∫ + 1 Ω r n ⋅P(r)
[ ]ds
surface
∫
Conclusion
cell to cell term (current)
J λ
( ) = ∂P
∂λ ΔP = J λ
( )dλ
= ∂P ∂λ dλ
A
j = −nev ΔP = −nev
Δt
dt = −ner(Δt )
[ ] − −ner(0) [ ]
= P(Δt) − P(0)
In classical way:
Electric polarization expressed by wave function
P = e V
X k
X n=1
h k
njrj k ni
Hj k
ni = Ek nj k ni
dP dt = e V
X k
X n=1
d dth k
njrj k ni
= e V
X k
X n=1 “
h@t k
njrj k ni + h k njrj@t k ni ”
= e V
X k
X n=1 1 X m=1
(h@t k
nj k mih k mjrj k ni
+ h k
njrj k mih k mj@t k ni)
Electric polarization expressed by wave function
dP dt = e V
X k
X n=1 1 X m=1
(h@t k
nj k mih k mjrj k ni
+ h k
njrj k mih k mj@t k ni)
Velocity operator
h k
mjvj k ni
= ih k
mj[r; H]j k ni = i(Ek n ` Ek m)h k mjrj k ni
h k
mjrj k ni
= h k
mjvj k ni
i(Ek
n ` Ek m)
h k
njrj k mi
= (h k
mjrj k ni)˜
Electric polarization expressed by wave function
dP dt = `ie V
X k
X n=1 X m6=n @h@t k nj k mih k mjvj k ni
(Ek
n ` Ek m)
` c:c
1 A
Bloch wavefunction and its periodic part
j k
ni
= eik´rjuk
ni
Hj k
ni
= Ek
nj k ni
e`ik´rHeik´rjuk
ni
= Ek
njuk ni
~ Hjuk
ni
= Ek
njuk ni
h k
mjvj k ni
= huk
mj~
vjuk
ni
Heisenberg Equation of Motion
idr dt = [r; H] iv = [r; H]
Bloch wavefunction and its periodic part
~ H = e`ik´rHeik´r e`ik´r[r; H]eik´r = e`ik´r idr dt
!
eik´r = i~ v if [rk; H] = 0, rk ~ H = `ire`ik´rHeik´r + e`ik´rHeik´rir rk ~ H = `i[r; ~ H] = ~ v h k
mjvj k ni
= huk
mj~
vjuk
ni = huk mjrk ~
H
ni
Electric polarization expressed by wave function
dP dt = `ie 8ı3
Z BZ dk
X n=1 X m6=n @h@t k nj k mih k mjvj k ni
(Ek
n ` Ek m)
` c:c
1 A
= `ie 8ı3
Z BZ dk
X n=1 X m6=n @h@tuk njuk mihuk mj~
vjuk
ni
(Ek
n ` Ek m)
` c:c
1 A
= `ie 8ı3
Z BZ dk
X n=1 X m6=n @h@tuk njuk mihuk mjrk ~
Hjuk
ni
(Ek
n ` Ek m)
` c:c
1 A
= `ie 8ı3
Z BZ dk
X n=1 “
h@tuk
njrkuk ni ` hrkuk nj@tuk ni ”
First-order perturbation theory
‹ ~ H = ~ H(k + ´k) ` ~ H(k) juk+∆k
n
i = juk
ni
+
X m6=n
juk
mihuk mj‹ ~
Hjuk
ni
Ek
n ` Ek m
+ O(‹ ~ H2) jrkuk
ni
’
X m6=n
juk
mihuk mjrk ~
Hjuk
ni
Ek
n ` Ek m
Ordinary derivative to partial derivative
d dtjuk¸;ti = @k¸juk¸;tidk¸ dt + @tjuk¸;ti = @tjuk¸;ti
Electric polarization expressed by wave function
Z ´t
dtdP dt = P (´t) ` P (0) = `ie 8ı3
Z ´t
dt
Z BZ dk
X n=1 “
h@tuk
njrkuk ni ` hrkuk nj@tuk ni ”
= `ie 8ı3
Z ´t
dt
Z BZ dk
X n=1 “
@thuk
njrkuk ni ` rkhuk nj@tuk ni ”
For k¸ direction; P¸(´t) ` P¸(0) = ie 8ı3
Z
dk˛dk‚ ˆ
Z ´t
dt
Z G¸
dk¸
X n=1 “
@k¸huk
nj@tuk ni ` @thuk nj@k¸uk ni ”
(King-Smith & Vanderbilt 1993)
P¸(t) = `ie 8ı3
Z
dk˛dk‚
X n=1 Z G¸
dk¸huk
n(t)j@k¸juk n(t)i
= e 8ı3
Z
dk˛dk‚
X n=1
Im
Z G¸
dk¸huk
n(t)j@k¸juk n(t)i
Example: Orthorhombic unitcell
Case: (k˛; k‚) = (0; 0) sampling , G˛ = 2ı
b , G‚ = 2ı c
P¸(t) = e 8ı3
Z
dk˛dk‚
X n=1
Im
Z G¸
dk¸huk
n(t)j@k¸juk n(t)i
= e 8ı3
Z
dk˛dk‚ffi(t) = e 8ı3 2ı b 2ı c ffi(t) = e 2ıbcffi(t) = ea 2ıabcffi(t) = ea 2ı˙cell ffi(t) = ea ˙cell
ffi(t)
2ı
!
Numerical calculation of Berry Phase
ffi(t) =
X n=1
Im
Z G¸
dk¸huk
n(t)j@k¸juk n(t)i
We difine overlap matrix S(k; k0; t), where Snm(k; k0; t) ” huk
n(t)j@k¸juk n(t)i.
We use well-known matrix identity, det exp A = exp tr A, when A = logS $ exp A = S. log detS = tr logS. ffi(t) = Im
Z G¸
dk¸tr@k0
¸huk
n(t)juk0 m(t)ijk0=k
= Im
Z G¸
dk¸tr@k0
¸S(k; k0; t)jk=k0
Numerical calculation of Berry Phase
A = logS $ exp A = S det exp A = exp tr A, log detS = tr logS Snm(k; k0; t)jk=k0 = ‹mn ffi(t) = Im
Z G¸
dk¸tr@k0
¸S(k; k0; t)jk=k0
= Im
Z G¸
dk¸tr
2 4@k0
¸S(k; k0; t)
S(k; k0; t)
3 5 jk=k0
= Im
Z G¸
dk¸tr@k0
¸log S(k; k0; t)jk=k0
= Im
Z G¸
dk¸@k0
¸log detS(k; k0; t)jk=k0
Numerical calculation of Berry Phase
ffi(t) = Im
Z G¸
dk¸@k0
¸log detS(k; k0; t)jk=k0
If we use k-point sampling mesh J along k¸ direction, k¸;s = sG¸=J and ´k¸ = G¸/J. ffi(t) = Im lim
´k¸!0 J`1 X s=0
´k¸ˆ log det Snm(k¸;s; k¸;s + ´k¸; t) ` log det Snm(k¸;s; k¸;s; t) ´k¸ ffi(t) = Im lim
´k¸!0 J`1 X s=0
log det Snm(k¸;sk¸;s + ´k¸; t)
Numerical calculation of Berry Phase
ffi(t) = Im lim
´k¸!0 J`1 X s=0
log det Snm(k¸;sk¸;s + ´k¸; t) = Im lim
´k¸!0 log J`1 Y s=0
det Snm(k¸;sk¸;s + ´k¸; t)
０
S
Ｇ Ｇ◯
◯
ψ(k)
σµ (r)
= eik·ru(k)
σµ (r),
= 1 √ N
N
eiRn·k
iα
c(k)
σµ,iαφiα(r − τi − Rn),
⟨ | ⟩ ⟨u(k)
σµ |u(k+∆k) σν
⟩ = ⟨ψ(k)
σµ |eik·re−ik·re−i∆k·r|ψ(k+∆k) σν
⟩, = ⟨ψ(k)
σµ |e−i∆k·r|ψ(k+∆k) σν
⟩, = 1 N
c(k)∗
σµ,iαc(k+∆k) σν,jβ
e−ik·(Rn−Rn′) × ⟨φiα(r − τi − Rn)|e−i∆k·(r−Rn′)|φjβ(r − τj − Rn′)⟩.
r′ = r − τi − Rn,
⟨u(k)
σµ |u(k+∆k) σν
⟩ = 1 N
c(k)∗
σµ,iαc(k+∆k) σν,jβ
e−ik·(Rn−Rn′) × ⟨φiα(r′)|e−i∆k·(r′+τi+Rn−Rn′)|φjβ(r′ + τi − τj + Rn − Rn′)⟩.
− ⟨u(k)
σµ |u(k+∆k) σν
⟩ =
c(k)∗
σµ,iαc(k+∆k) σν,jβ
eik·Rn × ⟨φiα(r′)|e−i∆k·(r′+τi−Rn)|φjβ(r′ + τi − τj − Rn)⟩, =
c(k)∗
σµ,iαc(k+∆k) σν,jβ
eik·Rne−i∆k·(τi−Rn)⟨φiα(r′)|e−i∆k·r′|φjβ(r′ + τi − τj − Rn)⟩,
e−i∆k·r′ ≈ 1 − i∆k · r′.
⟨u(k)
σµ |u(k+∆k) σν
⟩ =
c(k)∗
σµ,iαc(k+∆k) σν,jβ
eik·Rne−i∆k·(τi−Rn) ×
⟨φiα(r′)|φjβ(r′ + τi − τj − Rn)⟩ − i∆k · ⟨φiα(r′)|r′|φjβ(r′ + τi − τj − Rn)⟩
First-principles study of spontaneous polarisation and water dipole moment in ferroelectric ice XI
Fumiyuki Ishiia*, Kei Teradab and Shinichi Miuraa
aFaculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kanazawa 920-1192, Japan; bDepartment of Computational Science, Faculty of Science, Kanazawa University, Kanazawa 920-1192, Japan
(Received 3 August 2010; final version received 16 October 2010) Using density functional calculations, spontaneous polarisation of proton-ordered ferroelectric ice XI phase is calculated for the first time. Spontaneous polarisation along the c-axis of orthorhombic Cmc21 structure is calculated to be 21 mC=cm2, which corresponds to water dipole moment 3.3 D. We have performed systematic calculation of the water dipole moment in proton-ordered ice without ambiguity. Keywords: water molecule; ice; density functional theory; electric polarisation; electric dipole moment, electronic structure Molecular Simulation
charge density in a unitcell is not sufficient to determine the polarization.
cell
cell
surface
E.R. Batista, S.S. Xantheas, H. Jonsson (J. Chem. Phys. 111, 6011(1999))
E.R. Batista, S.S. Xantheas, H. Jonsson (J. Chem. Phys. 111, 6011(1999))
plane of the cluster. The figure displays the charge density partitioned ac- cording to the Voronoi I dotted line and Voronoi II solid line schemes see text. In the Voronoi I scheme, the Voronoi cell is constructed around
II, the Voronoi cells are around three ‘‘atomic’’ centers per molecule: one at the oxygen atom and the other two shown with crosses on the O–H bonds, at 40% of the displacement from the oxygen atom to the hydrogen nucleus. Although both surfaces are very similar, the latter passes closer through the minimum of the charge density between the molecules.
Figure 2. The charge density of ice XI phase viewed from a-axis perpendicular to the polarisation direction. Contours are drawn on a logarithmic scale (from 1.0e-4 to 1.0 e/bohr3).
3 2.5 1.5 2 2.5 3 Water dipole moment (D) 3.5 4 4.5 3.5 4.5 4 5 5.5 ROO (Å)
Figure 4. Water dipole moment of model ice with a perspective view of the structure. The triangle indicates water dipole moment versus oxygen–oxygen distance ROO. The lines are a guide to the eye.