Berry Phases and Curvatures
in Electronic-Structure Theory
David Vanderbilt
Rutgers University
March APS Meeting, Baltimore, March 13 2006
Rahman prize for:
– Theory of polarization (King-Smith & Vanderbilt)
– Ultrasoft pseudopotentials
Three quick preliminaries:
• Who was Aneesur Rahman?
• Who is Dominic King-Smith?
• A parable about referee reports…
March APS Meeting, Baltimore, March 13 2006
Who was Aneesur Rahman?
“Father of Molecular Dynamics”
•B
! orn Hyderbad, India
• Educ. Cambridge, Louvain
• Argonne Natl. Labs 1960-85
• U. Minnesota 1985-87
• Died 1987
• Rahman Prize established in
1992 with funds from IBM
Photo courtesy Sam Bader via Marie-Louise Saboungi
March APS Meeting, Baltimore, March 13 2006
Who is Dominic King-Smith?
“Father of Bettina”
• PhD, Cambridge, UK
• Postdoc at Rutgers `91-`93
• Biosym/MSI/Accelrys `93-`01
• Presently at:
Accelrys Job title:
“Product Manager, Quantum Mechanics”
March APS Meeting, Baltimore, March 13 2006
Ultrasoft Pseudopotentials
March APS Meeting, Baltimore, March 13 2006
Berry Phases and Curvatures
in Electronic-Structure Theory
David Vanderbilt
Rutgers University
March APS Meeting, Baltimore, March 13 2006
Introduction
• By mid-1990s, density-functional perturbation theory
allowed calculation of linear response to E-field
• However, it was not known how to:
– Calculate polarization itself
– Treat finite E-fields
• Analogous problem of calculating orbital magnetization
also unsolved
March APS Meeting, Baltimore, March 13 2006
Introduction
• Solutions of these problems are now in hand
– Modern theory of polarization (1993)
– Treatment of finite E-fields (2002)
– Orbital magnetization (2005)
• Solutions rely heavily on two crucial ingredients:
– Wannier functions
– Berry phases and related quantities
This talk:
Brief survey of methods!
Almost nothing on applications
March APS Meeting, Baltimore, March 13 2006
Outline of Talk
• Introduction
• Berry phases, potentials, and curvatures
• Realizations:
– Electric polarization
– Wannier functions
– Electric fields
– Anomalous Hall conductivity
– Orbital magnetization
• Summary and prospects
March APS Meeting, Baltimore, March 13 2006
Berry phases
u4Ò
u3Ò
u2Ò
unÒ =u1Ò
…
Now take limit
that density of
points Æ∞
un-1Ò
March APS Meeting, Baltimore, March 13 2006
Berry phases
ulÒ
l=1
l=0
Continuum
limit
March APS Meeting, Baltimore, March 13 2006
(Context: Molecular coordinates)
z2
ulÒ
l=1
l=0
Na3
(z1, z2)
z1
March APS Meeting, Baltimore, March 13 2006
Context: k-space in Brillouin zone
ukÒ
ky
l=1
l=0
Bloch function
0
kx
2p/a
March APS Meeting, Baltimore, March 13 2006
Stokes theorem: Berry curvature
ukÒ
ky
0
W
kx
2p/a
March APS Meeting, Baltimore, March 13 2006
Context: k-space in Brillouin zone
ukÒ
ky
l=1
l=0
Bloch function
0
kx
2p/a
March APS Meeting, Baltimore, March 13 2006
Spanning the BZ
l=0
l=1
ukÒ
ky
Bloch function
0
kx
2p/a
March APS Meeting, Baltimore, March 13 2006
Does any of this
have any connection
to real physics
of materials?
March APS Meeting, Baltimore, March 13 2006
Outline of Talk
• Introduction
• Berry phases, potentials, and curvatures
• Realizations:
– Electric polarization
– Wannier functions
– Electric fields
– Anomalous Hall conductivity
– Orbital magnetization
• Summary and prospects
March APS Meeting, Baltimore, March 13 2006
P = dcell / Vcell ?
+
–
+
–
+
–
+
–
+
–
+
–
• Textbook picture
(Claussius-Mossotti)
• But does not correspond
to reality!
March APS Meeting, Baltimore, March 13 2006
Ferroelectric PbTiO3
(Courtesy N. Marzari)
P = dcell / Vcell ?
dcell =
March APS Meeting, Baltimore, March 13 2006
P = dcell / Vcell ?
dcell =
March APS Meeting, Baltimore, March 13 2006
Berry-phase theory of electric polarization
March APS Meeting, Baltimore, March 13 2006
Berry-phase theory of electric polarization
Berry potential!
March APS Meeting, Baltimore, March 13 2006
Simplify: 1 band, 1D
l=0
l=1
ky
ukÒ
0
kx
2p/a
March APS Meeting, Baltimore, March 13 2006
Discrete sampling of k-space
March APS Meeting, Baltimore, March 13 2006
Discretized formula in 3D
where
March APS Meeting, Baltimore, March 13 2006
Sample Application: Born Z*
+2 e ?
+4 e ?
–2e ?
Paraelectric
Ferroelectric
–2e ?
March APS Meeting, Baltimore, March 13 2006
Outline of Talk
• Introduction
• Berry phases, potentials, and curvatures
• Realizations:
– Electric polarization
– Wannier functions
– Electric fields
– Anomalous Hall conductivity
– Orbital magnetization
• Summary and prospects
March APS Meeting, Baltimore, March 13 2006
Wannier function representation
(Marzari and Vanderbilt,
1997)
“Wannier center”
March APS Meeting, Baltimore, March 13 2006
Mapping to Wannier centers
Wannier
center
rn
March APS Meeting, Baltimore, March 13 2006
Mapping to Wannier centers
Wannier dipole theorem
DP =
Sion (Zione) Drion
+ Swf (– 2e) Drwf
• Exact!
• Gives local description of
dielectric response!
March APS Meeting, Baltimore, March 13 2006
Ferroelectric BaTiO3
(Courtesy N. Marzari)
Wannier functions
in a-Si
Wannier functions
in l-H2O
Fornari et al.
Silvestrelli et al.
Wannier analysis of PVDF polymers and copolymers
Courtesy S. Nakhmanson
Note upcoming release of public max-loc Wannier code…
(Organized by Nicola Marzari)
March APS Meeting, Baltimore, March 13 2006
Outline of Talk
• Introduction
• Berry phases, potentials, and curvatures
• Realizations:
– Electric polarization
– Wannier functions
– Electric fields
– Anomalous Hall conductivity
– Orbital magnetization
• Summary and prospects
March APS Meeting, Baltimore, March 13 2006
Electric Fields: The Problem
Easy to do in practice:
But ill-defined in principle:
Zener
tunneling
For small E-fields, tZener >> tUniverse ; is it OK?
March APS Meeting, Baltimore, March 13 2006
Electric Fields: The Problem
y(x) is very
messy
•
is not periodic
• Bloch’s theorem does not apply
March APS Meeting, Baltimore, March 13 2006
Electric Fields: The Solution
• Seek long-lived resonance
• Described by Bloch functions
• Minimizing the electric enthalpy functional
(Nunes and Gonze, 2001)
Usual EKS
Berry phase polarization
Souza, Iniguez, and Vanderbilt, PRL 89, 117602 (2002);
P. Umari and A. Pasquarello, PRL 89, 157602 (2002).
March APS Meeting, Baltimore, March 13 2006
Electric Fields: Implementation
As long as Dk is not too small:
• Can use standard methods to find minimum
• The
e · P term introduces coupling between k-points
–p/a
0
p/a
k
March APS Meeting, Baltimore, March 13 2006
Sample Application: Born Z*
Can check that previous results
for BaTiO3 are reproduced
March APS Meeting, Baltimore, March 13 2006
Outline of Talk
• Introduction
• Berry phases, potentials, and curvatures
• Realizations:
– Electric polarization
– Wannier functions
– Electric fields
– Anomalous Hall conductivity
– Orbital magnetization
• Summary and prospects
March APS Meeting, Baltimore, March 13 2006
Anomalous Hall effect
Ferromagnetic Material
March APS Meeting, Baltimore, March 13 2006
Anomalous Hall effect
• Karplus-Luttinger theory (1954)
– Scattering-free, intrinsic
Semiclassical equations
of motion:
• Skew-scattering mechanism (1955)
– Impurity scattering
• Side-jump mechanism (1970)
– Impurity or phonon scattering
• Berry-phase theory (1999)
– Restatement of Karplus-Luttinger
Sundaram and Niu, PRB 59,
14925 (1999).
March APS Meeting, Baltimore, March 13 2006
Stokes theorem: Berry curvature
ukÒ
ky
0
W
kx
2p/a
March APS Meeting, Baltimore, March 13 2006
Anomalous Hall conductivity of SrRuO3
Wz for kz=0
Z. Fang et al, Science 302,
92 (2003).
March APS Meeting, Baltimore, March 13 2006
X. Wang, J. Yates, I. Souza, and D. Vanderbilt, G23.00001
(Tuesday 8am).
Wz(kx,kz)
in
bcc Fe
See also Y.G. Yao et
al., PRL 92, 037204
(2004).
March APS Meeting, Baltimore, March 13 2006
Outline of Talk
• Introduction
• Berry phases, potentials, and curvatures
• Realizations:
– Electric polarization
– Wannier functions
– Electric fields
– Anomalous Hall conductivity
– Orbital magnetization
• Summary and prospects
March APS Meeting, Baltimore, March 13 2006
Orbital Magnetization
M is a bulk property?
K
fl K = M x n
K is only apparently a
surface property?
P is a bulk property
-s
+s
fl s = P ⋅ n
s is only apparently a
surface property
March APS Meeting, Baltimore, March 13 2006
Theory of orbital magnetization
T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta,
Phys. Rev. Lett. 95, 137205 (2005).
• Context:
– Ferromagnetic insulators
– Single-particle approximation
– Vanishing magnetic field
• Used Wannier representation to derive a
formula for the orbital magnetization
March APS Meeting, Baltimore, March 13 2006
Orbital currents in Wannier representation
ÔwsÒ
=
ÔwsÒ
r
+
Local Circulation
(LC)
·vÒ
ÔwsÒ
r
Itinerant Circulation
(IC)
March APS Meeting, Baltimore, March 13 2006
T. Thonhauser, H6.00001 (Tuesday 11:15am)
(invited talk)
Something new
Berry curvature
See also D. Xiao, J. Shi and Q.
Niu, PRL 95, 137204 (2005).
March APS Meeting, Baltimore, March 13 2006
Summary and Prospects
• Berry phases are everywhere!
• We discussed:
– Electric polarization
– Electric fields
– Anomalous Hall coefficient
– Orbital magnetization
• Other “hot topics”:
– Multiferroics and magnetoelectric effects
– Single graphene sheets
– Spin Hall effect and spin injection
• More Berry phases lurking around the corner?
March APS Meeting, Baltimore, March 13 2006
Extras
March APS Meeting, Baltimore, March 13 2006
Electric Fields: Justification
Seek
long-lived
metastable
periodic
solution
March APS Meeting, Baltimore, March 13 2006
Electric Fields: The Hitch
•
•
•
•
There is a hitch!
For given E-field, there is a limit on k-point sampling
Length scale LC = 1/Dk
Meaning: LC = supercell dimension
Nk = 8
Lc = 8 a
• Solution: Keep Dk > 1/Lt =
e/Eg
March APS Meeting, Baltimore, March 13 2006
X. Wang, J. Yates, I. Souza, and D. Vanderbilt, G23.00001
(Tuesday 8am).
Anomalous Hall conductivity of bcc Fe
See also Y.G. Yao et al., PRL
92, 037204 (2004).
March APS Meeting, Baltimore, March 13 2006
Orbital Magnetization
K=Mxn
K
Is M a bulk property?
Is K only apparently a
surface property?
Definition:
If K is predetermined at all surfaces in such a
way that K = M x n for some vector M, then
M is the bulk magnetization.
March APS Meeting, Baltimore, March 13 2006
Orbital Magnetization
Clarification:
• Microscopic M(r) defined via — x M(r) = J(r)
• M(r) ill-defined: M(r) fi M(r) + M0 + —h
• Therefore, cannot define M as cell average of M(r)
Conclusion: M is not, even in principle, a functional
of the bulk current distribution J(r)
(Hirst, RMP, 1997)
Just as: P is not, even in principle, a functional
of the bulk charge density distribution r(r)
March APS Meeting, Baltimore, March 13 2006
Strong reasons to expect bulk M
• Nearsightedness:
Edge of
type A
Surface current depends only
on local environment
Iy(A)
• Stationary quantum state:
dr/dt = 0
• Conservation of charge:
—⋅J = 0
So:
Mz
Iy(B)
Edge of
type B
Iy(A) = Iy(B) = Mz
March APS Meeting, Baltimore, March 13 2006
Comparison: P vs. M
Electric Polarization
Orbital Magnetization
Defined for insulators only
Insulators and metals
with broken TR symmetry
r(r) insufficient in principle;
need access to Berry physics
J(r) insufficient in principle;
need access to Berry physics
r operator
r ¥ v operator
“Quantum of polarization”
No quantum (no monopoles)
Derivable from adiabatic theory
No obvious adiabatic theory
Derivable from Wannier rep.
Derivable from Wannier rep.?
March APS Meeting, Baltimore, March 13 2006
Ultrasoft Pseudopotentials
Then, the good news:
*
Sidney Redner, Physics Today, June 2005.
* (A hot paper is…) “defined as a nonreview paper with 350 or more citations, an average ratio of
citation age to publication age greater than two-thirds, and a citation rate increasing with time.”
March APS Meeting, Baltimore, March 13 2006
Ultrasoft Pseudopotentials
Then, the good news:
Sidney Redner, APS talk,
March, 2004; Physics
Today, June 2005.
March APS Meeting, Baltimore, March 13 2006