Homogeneous Electric and Magnetic Fields in Periodic Systems

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Oct 18, 2013 (3 years and 9 months ago)

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J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Homogeneous Electric and Magnetic Fields in
Periodic Systems
Josef W.Zwanziger
iDepartment of Chemistry and Institute for Research in Materials
Dalhousie University
Halifax,Nova Scotia
June 2012
1/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Acknowledgments
NSERC,Canada Research Chairs for funding
2/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Outline
1
Computational Framework
2
Homogeneous electric fields
3
Homogeneous finite magnetic fields
3/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Density functional theory
Minimize
E
el
{ψ} =
occ
￿
α

α
|T+v
ext

α
i+E
Hxc
[n]−
occ
￿
αβ
ǫ
βα
(hψ
α

β
i−δ
αβ
)
where
n(r) =
occ
￿
α
ψ

α
(r)ψ
β
(r)
and gradient is δE/δhψ
α
|
4/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Planewaves and pseudopotentials
Periodicity of the solid leads to Bloch theorem:
ψ
nk
(r) ∝ e
i k∙r
u
nk
(r)
and the cell periodic part is expanded in planewaves:
u
nk
(r) =
￿
G
u
nk
(G)e
i G∙r
This is efficient if the core electrons are replaced by
pseudopotentials.
5/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Projector Augmented Wave Method
The PAW method (Bl¨ochl) projects from pseudofunctions back
to all-electron valence space functions.
|ψi = T|
˜
ψi
T = 1 +
￿
i,R
￿

i R
i −|
˜
φ
i R
i
￿
h˜p
i R
|
hψ|A|ψi = h
˜
ψ|A|
˜
ψi +
￿
ij,R
h
˜
ψ|˜p
i R
ih˜p
j R
|
˜
ψi ×
￿

i R
|A|φ
j R
i −h
˜
φ
i R
|A|
˜
φ
j R
i
￿
6/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Homogeneous 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
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Modern Theory of Polarization
King-Smith and Vanderbilt showed that polarization does
not suffer from unboundedness:
P = −
ie
(2π)
3
￿
n
￿
BZ
dkhu
nk
|∇
k
|u
nk
i
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
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
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:hψ
n
|S|ψ
m
i = δ
nm
On-site dipole contribution from T is included:
h˜u
nk
|T

k
i ∇
k
T
k
|˜u
nk
i,

I
q,R,k
i = e
−i k∙(r−R)

I
q,k
i
Form gradient:
δE/δhu
mk
| = δE
0
/δhu
mk
| −E·δP/δhu
mk
|
Implemented in Abinit,including spin polarized systems,
spinors,spin-orbit coupling
1
King-Smith and Vanderbilt,cond-matt;Zwanziger et al.,Comp.Mater.
Sci.58,113 (2012)
9/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Convergence with k-mesh

ε∞
10.5
11
11.5
12
12.5
13
13.5
14
Inverse k mesh/Bohr
−50 0 50 100 150 200 250 300 350 400
DFPT
Finite
eld
experiment
Convergence with mesh size for Si
10/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Applications
Polarization is computed as a
function of applied field and fit
to the form (SI units for
polarization and field):
P
i
= ǫ
0
χ
(1)
ij
E
j
+2ǫ
0
d
ijk
E
j
E
k
,

Polarization, C m
-2
-1x10
-5
0
10
-5
2x10
-5
3x10
-5
4x10
-5
5x10
-5
6x10
-5
Electric field, V m
-1
0
10
8
2x10
8
3x10
8
4x10
8
5x10
8
(b)

Polarization, C m
-2
-1.635
-1.630
-1.625
-1.620
-1.615
-1.610
Electric Field, V m
-1
0
10
8
2x10
8
3x10
8
4x10
8
5x10
8
(a)
11/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
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) 10.26 8.01 21.5
(PBE) 10.09 7.84 23.2
(expt) 9.8 7.5
AlAs (LDA) 11.05 8.75 32.7
(PBE) 10.89 8.80 38.8
(expt) 10.16 8.16 32
AlSb (LDA) 12.54 11.17 98.3
(PBE) 12.83 11.45 103
(PBE + SO) 9.76
(expt) 11.68 9.88 98
12/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Application:MgO Dielectric
Method ǫ

PAW E-field,PBE 3.089
PAW DFPT,LDA 3.057
NCPP DFPT,LDA 3.063
Expt 3.014
N.B.in DFPT,

2
E
∂E
i
∂E
j
￿
￿
￿
0
is
computed directly,without
presence of a field.
MgO in Finite Electric Field
Polarization
-6.2095x10
-2
-6.2090x10
-2
-6.2085x10
-2
Electric field (a.u.)
0
2.0x10
-5
4.0x10
-5
6.0x10
-5
8.0x10
-5
1.0x10
-4
slope = χ = 0.1662
1+4πχ = 3.089
13/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Photoelasticity
Inverse of dielectric tensor changed by stress or strain:
ΔB
ij
= p
ijkl
ǫ
kl
= π
ijkl
σ
kl
Compound ǫ p
11
p
21
p
44
Si (LDA) 12.4 -0.106 0.015 -0.052
(PBE) 12.2 -0.112 0.010 -0.061
(expt) 11.7 -0.094 0.017 -0.051
C (LDA) 5.71 -0.263 0.0673 -0.160
(PBE) 5.79 -0.268 0.0643 -0.171
(expt) 5.65–5.7 -0.244 – -0.42 0.042–0.27 -0.172 – -0.162
14/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
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 30.4
C
66
85.2
π
11
-0.980 0.990 -1.70
π
33
0.91
π
12
0.172 -0.176 2.19
π
13
6 6.20
π
44
-0.446 -1.26 2.31
π
66
0.97
ǫ

11
3.04 4.27 8.67
ǫ

33
7.04
15/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Homogeneous magnetic fields in insulators
One approach to magnetic fields in periodic insulators is
the long wavelength approach of Louie and co-workers:
B →B cos(q · r) with q →0.
Problematic:cannot always find |∂
k
ui such that
h∂
k
u|u
0
k
i = 0 AND |u
0
k+G
i = e
i G∙r
|u
0
k
i.
In 2005 and 2006,Ceresoli,Thonhauser,Resta,and
Vanderbilt established:
M=
1
2c(2π)
3
Im
￿
nn

￿
BZ
dkh∂
k
u
n

k
|×(H
k
δ
nn
′ +E
nn

k
)|∂
k
u
nk
i
C =
i

￿
n
￿
BZ
dkh∂
k
u
nk
| ×|∂
k
u
nk
i
16/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
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
r
1
,r
2
=
¯
O
r
1
,r
2
e
−i B∙r
1
×r
2
/2c
where
¯
O has lattice symmetry.
2
PRB 81 205104 (2010)
17/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
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

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
r
1
,r
2
= ¯ρ
r
1
,r
2
e
−i B∙r
1
×r
2
/2c
3
Lazzeri and Mauri,PRB 68 161101(R) (2003)
18/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
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
=
˜
V
k
˜
W
k
+

￿
m=1
1
m!
￿
i
2c
￿
m
￿
m
￿
n=1
ε
α
n
β
n
γ
n
B
α
n
￿
×(∂
β
1
· · · ∂
β
m
˜
V
k
)(∂
γ
1
· · · ∂
γ
m
˜
W
k
),
E
(n)
=
￿
BZ
dk
(2π)
3
Tr[(˜ρ
(n)
kVV
+ ˜ρ
(n)
kCC
)
¯
H
k
].
4
X.Gonze and J.W.Zwanziger,PRB 84,64445 (2011)
19/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Development of ρ
Density operator perturbation theory is used:
5
˜ρ
(1)
kD
=
i
2c
ε
αβγ
B
α
(∂
β
˜ρ
(0)
k
)(∂
γ
˜ρ
(0)
k
),
˜ρ
(2)
kD
= ˜ρ
(1)
k
˜ρ
(1)
k
quadratic
+
i
2c
ε
αβγ
B
α
[(∂
β
˜ρ
(0)
k
)(∂
γ
˜ρ
(1)
k
) +(∂
β
˜ρ
(1)
k
)(∂
γ
˜ρ
(0)
k
)] linear

1
8c
2
￿
2
￿
n=1
ε
αnβnγn
B
αn
￿

β1

β2
˜ρ
(0)
k
.∂
γ1

γ2
˜ρ
(0)
k
frozen
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 68
161101(R) (2003)
20/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
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
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
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
r
1
,r
2
=
¯
H
r
1
,r
2
e
−i B∙r
1
×r
2

E(2)
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
s
0 0.05 0.1 0.15 0.2
s
t
A
B
Example with t = 2.0.E
(2)
computed from theory,and by
direct diagonalization
22/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Mixed Perturbations
In addition to the results for ρ
(n)
and E
(n)
due to
magnetic fields,we have also established the response to
mixed magnetic and other perturbations µ:

2
E
∂µ∂B
=
￿
BZ
dk
(2π)
3
Tr
￿
˜ρ
(1)
∂H
k
∂µ
￿
23/24
J.W.
Zwanziger
Computational
Framework
Homogeneous
electric fields
Homogeneous
finite magnetic
fields
Summary
Modern theory of polarization and finite electric fields 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