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March APS Meeting, Baltimore, March 13 2006
Berry Phases and Curvaturesin Electronic-Structure Theory
David VanderbiltRutgers 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?
Photo courtesy Sam Bader via Marie-Louise Saboungi
• !Born 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
“Father of Molecular Dynamics”
March APS Meeting, Baltimore, March 13 2006
Who is Dominic King-Smith?
“Father of Bettina”
Accelrys Job title:“Product Manager, Quantum Mechanics”
• PhD, Cambridge, UK
• Postdoc at Rutgers `91-`93
• Biosym/MSI/Accelrys `93-`01
• Presently at:
March APS Meeting, Baltimore, March 13 2006
Ultrasoft Pseudopotentials
March APS Meeting, Baltimore, March 13 2006
Berry Phases and Curvaturesin Electronic-Structure Theory
David VanderbiltRutgers University
March APS Meeting, Baltimore, March 13 2006
Introduction
• By mid-1990s, density-functional perturbation theoryallowed 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 magnetizationalso 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
u3Ò
u2Ò
unÒ =u1Ò
u4Ò
…un-1Ò
Now take limitthat density of
points Æ∞
March APS Meeting, Baltimore, March 13 2006
Berry phases
ulÒ
l=0l=1
Continuumlimit
March APS Meeting, Baltimore, March 13 2006
(Context: Molecular coordinates)
ulÒ
l=0l=1
(z1, z2)
z1
z2
Na3
March APS Meeting, Baltimore, March 13 2006
Context: k-space in Brillouin zone
ukÒ
l=0l=1
kx
ky
0 2p/a
Bloch function
March APS Meeting, Baltimore, March 13 2006
Stokes theorem: Berry curvature
ukÒ
kx
ky
0 2p/a
W
March APS Meeting, Baltimore, March 13 2006
Context: k-space in Brillouin zone
ukÒ
l=0l=1
kx
ky
0 2p/a
Bloch function
March APS Meeting, Baltimore, March 13 2006
Spanning the BZ
Bloch function
ukÒ
l=0 l=1
kx
ky
0 2p/a
March APS Meeting, Baltimore, March 13 2006
Does any of thishave any connection
to real physicsof 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 correspondto reality!
Ferroelectric PbTiO3 (Courtesy N. Marzari)
March APS Meeting, Baltimore, March 13 2006
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
ukÒ
l=0 l=1
kx
ky
0 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*
Paraelectric Ferroelectric
+2 e ?
+4 e ?
– 2 e ?
– 2 e ?
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
Wanniercenter
rn
March APS Meeting, Baltimore, March 13 2006
Wannier dipole theorem
DP = Sion (Zione) Drion
+ Swf (– 2e) Drwf
• Exact!• Gives local description of
dielectric response!
Mapping to Wannier centers
Ferroelectric BaTiO3 (Courtesy N. Marzari)
Wannier functionsin a-Si
Fornari et al.
Wannier functionsin l-H2O
Silvestrelli et al.
Courtesy S. Nakhmanson
Wannier analysis of PVDF polymers and copolymers
March APS Meeting, Baltimore, March 13 2006
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:
For small E-fields, tZener >> tUniverse ; is it OK?
But ill-defined in principle:Zener
tunneling
March APS Meeting, Baltimore, March 13 2006
Electric Fields: The Problem
• is not periodic• Bloch’s theorem does not apply
y(x) is verymessy
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–p/a 0k
March APS Meeting, Baltimore, March 13 2006
Sample Application: Born Z*
Can check that previous resultsfor 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
• Skew-scattering mechanism (1955)
– Impurity scattering
• Side-jump mechanism (1970)
– Impurity or phonon scattering
• Berry-phase theory (1999)
– Restatement of Karplus-Luttinger
Semiclassical equationsof motion:
Sundaram and Niu, PRB 59,14925 (1999).
March APS Meeting, Baltimore, March 13 2006
Stokes theorem: Berry curvature
ukÒ
kx
ky
0 2p/a
W
March APS Meeting, Baltimore, March 13 2006
Z. Fang et al, Science 302,92 (2003).
Wz for kz=0
Anomalous Hall conductivity of SrRuO3
March APS Meeting, Baltimore, March 13 2006
X. Wang, J. Yates, I. Souza, and D. Vanderbilt, G23.00001(Tuesday 8am).
See also Y.G. Yao etal., PRL 92, 037204
(2004).
Wz(kx,kz)
in
bcc Fe
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?
fl K = M x n
K is only apparently asurface property?
K
+s-s
P is a bulk property
fl s = P ⋅ n
s is only apparently asurface 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
rr= +
ÔwsÒ ÔwsÒ ÔwsÒ
Local Circulation(LC)
Itinerant Circulation(IC)
·vÒ
March APS Meeting, Baltimore, March 13 2006
Berry curvature
Something new
T. Thonhauser, H6.00001 (Tuesday 11:15am)(invited talk)
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
Seeklong-livedmetastable
periodicsolution
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 = 8a
• 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., PRL92, 037204 (2004).
March APS Meeting, Baltimore, March 13 2006
Orbital Magnetization
K = M x n
Is M a bulk property?
Is K only apparently asurface property?
Definition:If K is predetermined at all surfaces in such away that K = M x n for some vector M, thenM is the bulk magnetization.
K
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)
Just as: P is not, even in principle, a functionalof the bulk charge density distribution r(r)
Conclusion: M is not, even in principle, a functionalof the bulk current distribution J(r)
(Hirst, RMP, 1997)
March APS Meeting, Baltimore, March 13 2006
Strong reasons to expect bulk M
• Nearsightedness:Surface current depends onlyon local environment
• Stationary quantum state:dr/dt = 0
• Conservation of charge:—⋅J = 0
Edge oftype A
Edge oftype B
Iy(A)
Iy(B)
So: Iy(A) = Iy(B) = Mz
Mz
March APS Meeting, Baltimore, March 13 2006
Comparison: P vs. M
Defined for insulators only
r(r) insufficient in principle;need access to Berry physics
r operator
“Quantum of polarization”
Derivable from adiabatic theory
Derivable from Wannier rep.
Insulators and metalswith broken TR symmetry
J(r) insufficient in principle;need access to Berry physics
r ¥ v operator
No quantum (no monopoles)
No obvious adiabatic theory
Derivable from Wannier rep.?
Electric Polarization Orbital Magnetization
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 ofcitation 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; PhysicsToday, June 2005.