J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Electric Polarization and Homogeneous
Electric Fields in Periodic Systems
Josef W. Zwanziger
iDepartment of Chemistry and Institute for Research in Materials
Dalhousie University
Halifax, Nova Scotia
May 2014
1 / 26
Outline
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
1 Computational Framework
2 Homogeneous electric fields
2 / 26
Density functional theory
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Minimize
occ
occ
X
X
hψα |T +vext |ψα i+EHxc [n]−
ǫβα (hψα |ψβ i−δαβ )
Eel {ψ} =
α
αβ
where
n(r) =
occ
X
ψα∗ (r)ψβ (r)
α
and gradient is δE /δhψα |
3 / 26
Planewaves and pseudopotentials
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Periodicity of the solid leads to Bloch theorem:
ψnk (r) ∝ e i k·r unk (r)
and the cell periodic part is expanded in planewaves:
X
unk (r) =
unk (G)e i G·r
G
This is efficient if the core electrons are replaced by
pseudopotentials.
4 / 26
Projector Augmented Wave Method
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
The PAW method (Blöchl) projects from pseudofunctions back
to all-electron valence space functions.
|ψi = T |ψ̃i
i
Xh
T = 1+
|φi R i − |φ̃i R i hp̃i R |
i ,R
hψ|A|ψi = hψ̃|A|ψ̃i +
X
hψ̃|p̃i R ihp̃jR |ψ̃i ×
ij,R
hφi R |A|φjR i − hφ̃i R |A|φ̃jR i
5 / 26
Polarization
J. W.
Zwanziger
Computational
Framework
Polarization refers to electric dipole moment per volume.
Simple in finite systems, not simple to compute in extended
systems (thermodynamic limit).
Homogeneous
electric fields
6 / 26
Homogeneous electric field
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
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 / 26
Modern Theory of Polarization
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
This approach begins with work of Resta:
Z
1
P(λ) =
drρλ (r)r
V
e X
fi hφi |r|φi i
=
V
i
e X
′
P (λ) =
fi hφ′i |r|φi i + hφi |r|φ′i i
V
i
Shift of emphasis to derivative of P is crucial step.1
1
Resta, Ferroelectrics 136, 51 (1992)
8 / 26
To first order:
J. W.
Zwanziger
−ie X X hφi |p|φj ihφj |V ′ |φi i
fi
+ c.c
P (λ) =
me V
(Ei − Ej )2
′
Computational
Framework
Homogeneous
electric fields
i
j6=i
V′
where
is the perturbed KS potential and Ei are the KS
eigenenergies.
Working in the parallel transport gauge where hφ′i |φj i = 0 for
all occupied states j, can write2
occ unocc −ief X X X hφkn |p|φkm ihφkm |V ′ |φkn i
P (λ) =
+ c.c
me NΩ
(Ei − Ej )2
m
n
′
k
2
Gonze PRA 52, 1096 (1995); King-Smith and Vanderbilt, PRB 47, 1651
(1993)
9 / 26
Further development:
J. W.
Zwanziger
By exploiting commutators such as [∂/∂k, Hk ] and parallel
transport gauge, formula in k-space becomes
Computational
Framework
Homogeneous
electric fields
Pα′ (λ) =
occ
−ief X X
[h∂ukn /∂kα |∂ukn /∂λi + c.c]
NΩ
n
k
Note that only ground state wavefunctions appear here!
Going from summation to integration, and integrating by parts,
the key result is obtained:
∆P = P1 − P0
occ I
ife X
Pα =
dkhukn |∂/∂kα |ukn i
8π 3 n
10 / 26
Finite difference formula for line integral
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
The integration along the loop in k space is done via
discretization:3
Y
XZ
dethunkj |umkj +b i
dkα hunk |∇kα |unk i → Im ln
n
j
This form is invariant to phase differences between the states
at different k points. In case of PAW:
hunkj |umkj +b i → hũnkj Tkj |Tkj +b ũmkj +b i
3
Resta, PRL 80, 1800 (1998)
11 / 26
Total polarization
J. W.
Zwanziger
Computational
Framework
The total electric polarization is the sum of the electronic part
above and the ionic part:
Homogeneous
electric fields
P = Pion + Pelec ,
with
Pion =
X
Z i ri
i
just the sum over the charges and positions of the ions in the
unit cell, then folded into unit interval (-1,1).
12 / 26
Executing the calculation in Abinit
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
To compute the polarization, just do the following:
Insulating system
nband is number of valence bands, no empty bands
normal ground state calculation
berryopt -1
rfdir 1 1 1
PAW, nsppol, nspinor 2 all ok
13 / 26
berryopt output
J. W.
Zwanziger
In output file for AlAs in zero field:
Computational
Framework
Homogeneous
electric fields
Summary of the results
Electronic Berry phase
Ionic phase
Total phase
Remapping in [-1,1]
Polarization
Polarization
7.682370411E-03
-7.500000000E-01
-7.423176296E-01
-7.423176296E-01
-1.435570235E-02 (a.u. of charge)/bohr^2
-8.213580860E-01 C/m^2
Polarization in cartesian coordinates (a.u.):
(the sum of the electronic and ionic Berry phase has been folded into [-1, 1])
Electronic berry phase:
0.257330072E-03
0.257330072E-03
0.257330072E-03
...includes PAW on-site term: 0.000000000E+00
0.000000000E+00
0.000000000E+00
Ionic:
-0.251221359E-01 -0.251221359E-01 -0.251221359E-01
Total:
-0.248648059E-01 -0.248648059E-01 -0.248648059E-01
14 / 26
Inclusion of a Finite Electric Field
J. W.
Zwanziger
Minimize E = E0 − P · E,
4
where:
P is computed as above
Computational
Framework
Norm constraint is imposed: hψn |S|ψm i = δnm (S is
identity if NCPP)
Homogeneous
electric fields
Form gradient:
δE /δhumk | = δE0 /δhumk | − E·δP/δhumk |
Implemented in Abinit, including PAW,5 spin polarized
systems, spinors, spin-orbit coupling
4
Nunes and Gonze PRB 63, 155107 (2001); Stengel, Spaldin, Vanderbilt
Nature Physics 5, 304 (2009)
5
Zwanziger et al., Comp. Mater. Sci. 58, 113 (2012)
15 / 26
Finite field in Abinit
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
ndtset 4 (say)
getwfk -1
rfdir 1 1 1
efield1 3*0.0
berryopt1 -1 (Start with zero field, build up field slowly)
efield2 0.0001 0.0 0.0
berryopt2 4
Look at polarization in output file
16 / 26
Forces and Stress in finite field
J. W.
Zwanziger
In NCPP, no additional force terms arise (because
additional polarization term does not involve ion positions
so Hellman-Feynman force (derivative w.r.t position is
zero)
Computational
Framework
Homogeneous
electric fields
There is NCPP stress6
In PAW, due to ion position dependence of projectors,
there are additional force and stress terms in finite field.
In Abinit, NCPP includes force and stress but PAW is in
development.
6
Souza, Íñiguez, Vanderbilt, PRL 89, 117602 (2002)
17 / 26
Forces for E-field with PAW
J. W.
Zwanziger
The problem is that in Im ln Π det M, the M matrix elements
Computational
Framework
Homogeneous
electric fields
Mn,m = hũnkj Tkj |Tkj +b ũmkj +b i
depend on ionic position through the projectors in T . They
must be included in the Hellmann-Feynman force through7
X ∂
∂
Im ln Π det M = Im
ln det M
∂Ri
∂Ri
X
−1 ∂M
= Im
Tr M
∂Rj
7
Nunes and Gonze PRB 63, 155107 (2001); Audouze, Jollet, Torrent, Gonz
PRB 73, 235101 (2006)
18 / 26
Effect of PAW force
J. W.
Zwanziger
Forces included in jzwanzig/7.7.3-private. Example: AlAs
with PAW and finite electric field.
0.1
Homogeneous
electric fields
Force error: (F − Ffd)/Ffd
Computational
Framework
Ionic force
Ionic+PAW force
0.05
0
−0.05
−0.1
0
25
50
75
100
125
150
kptrlen
19 / 26
Convergence of polarization with k-mesh
J. W.
Zwanziger
Computational
Framework
14
Convergence with
mesh size for Si.
13.5
DFPT
13
12.5
P = χE
ǫ∞ = 1 + 4πχ
ε∞
Homogeneous
electric fields
Finite field
12
experiment
11.5
11
10.5
−50
0
50
100
150
200
250
300
350
400
Inverse k mesh/Bohr
20 / 26
Applications
(a)
J. W.
Zwanziger
Homogeneous
electric fields
Polarization is computed as a
function of applied field and fit
to the form (SI units for
polarization and field):
Pi =
(1)
ǫ0 χij Ej
+ 2ǫ0 dijk Ej Ek ,
-1.615
-1.620
-1.625
-1.630
-1.635
0
10
8
8
2x10
8
3x10
8
5x10
8
5x10
4x10
8
Electric Field, V m-1
(b)
6x10-5
Polarization, C m-2
Computational
Framework
Polarization, C m-2
-1.610
5x10-5
4x10-5
3x10-5
2x10-5
10-5
0
-1x10-5
0
10
8
8
2x10
8
3x10
4x10
8
Electric field, V m-1
21 / 26
Applications
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
High and low frequency susceptibility: χαβ = dPα /dEβ
Second order susceptibilities
Compound
AlP (LDA)
(PBE)
(expt)
AlAs (LDA)
(PBE)
(expt)
AlSb (LDA)
(PBE)
(PBE + SO)
(expt)
ǫ0
10.26
10.09
9.8
11.05
10.89
10.16
12.54
12.83
11.68
ǫ∞
8.01
7.84
7.5
8.75
8.80
8.16
11.17
11.45
9.76
9.88
d123 pm/V
21.5
23.2
32.7
38.8
32
98.3
103
98
22 / 26
Application: MgO Dielectric
J. W.
Zwanziger
-6.2090x10-2
1.0x10-4
8.0x10-5
6.0x10-5
slope = χ = 0.1662
1+4πχ = 3.089
-6.2095x10-2
4.0x10-5
2E N.B. in DFPT, ∂E∂i ∂E
is
j 0
computed directly, without
presence of a field.
MgO in Finite Electric Field
-6.2085x10-2
2.0x10-5
ǫ∞
3.089
3.057
3.063
3.014
0
Homogeneous
electric fields
Method
PAW E-field, PBE
PAW DFPT, LDA
NCPP DFPT, LDA
Expt
Polarization
Computational
Framework
Electric field (a.u.)
23 / 26
Photoelasticity
J. W.
Zwanziger
Computational
Framework
Inverse of dielectric tensor changed by stress or strain:
Homogeneous
electric fields
∆Bij = pijkl ǫkl = πijkl σkl
Compound
Si (LDA)
(PBE)
(expt)
C (LDA)
(PBE)
(expt)
ǫ
12.4
12.2
11.7
5.71
5.79
5.65–5.7
p11
-0.106
-0.112
-0.094
-0.263
-0.268
-0.244 – -0.42
p21
0.015
0.010
0.017
0.0673
0.0643
0.042–0.27
p44
-0.052
-0.061
-0.051
-0.160
-0.171
-0.172 – -0.162
24 / 26
Photoelasticity in oxides
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Quantity
C11
C33
C12
C13
C44
C66
π11
π33
π12
π13
π44
π66
ǫ∞
11
ǫ∞
33
MgO
325.8
BaO
158.3
98.8
46.8
162.5
35.7
-0.980
0.990
0.172
-0.176
-0.446
-1.26
3.04
4.27
SnO
111.7
43.4
95.0
18.9
30.4
85.2
-1.70
0.91
2.19
6.20
2.31
0.97
8.67
7.04
25 / 26
Summary
J. W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Modern theory of polarization and finite electric fields
NCPP, PAW, spinors, spin polarized systems
Applications to linear and nonlinear electric susceptibility
MANY thanks to Xavier Gonze, Marc Torrent, Abinit
development and theory community
26 / 26