Wannier Functions: ab-initio tight-bindingjry20/esdg2.pdf · Wannier Interpolation •Map low...
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Wannier Functions: ab-initio tight-binding
Jonathan Yates
Cavendish Laboratory, Cambridge University
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Representing a Fermi Surface
Lead Fermi surface
Accurate description of Fermi surface properties requires a detailed sampling of the Brillouin Zone
Al Fermi surface
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Representing a Fermi Surface
Lead Fermi surface
•Accurate description of Fermi surface properties requires a detailed sampling of the Brillouin Zone
• Full first-principles calculation expensive
Al Fermi surface
Scalar Relativistic with spin-orbit coupling Fe fermi surface
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Outline
•Wannier Functions •One band
•Isolated Set of Bands
•Entangled Bands
•Wannier Interpolation•Accurate and Efficient approach to Fermi surface and spectral properties
•Examples•Anomalous Hall Effect
•Electron-Phonon Coupling
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Wannier Functions
-9-8.5
-8-7.5
-7-6.5
-6-5.5
-5
L G X K G
wnR(r) =V
(2!)3
!
BZ"nk(r)e!ik.Rdk !nk(r)! e!i!(k)!nk(r)
Bloch stateQuantum # kWannier function
Quantum # R
GaAs
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Wannier Functions
-9-8.5
-8-7.5
-7-6.5
-6-5.5
-5
L G X K G
-6-4-2 0 2 4 6
L G X K G
wnR(r) =V
(2!)3
!
BZ"nk(r)e!ik.Rdk !nk(r)! e!i!(k)!nk(r)
!nk(r)!!
n
U (k)mn!mk(r)
Multiband - Generalized WF
Bloch stateQuantum # kWannier function
Quantum # R
GaAs
Si
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Maximally Localised Wannier Functions
wnR(r) =V
(2!)3
!
BZ
"#
m
U (k)mn"mk(r)
$e!k.Rdk
Choose Uk to minimise quadratic spread (Marzari, Vanderbilt)
|uk!
!uk|uk+b"U (k)Ab-initio code Wannier code
Wannier localisation in R gives Bloch smoothness in k !rot
nk (r) =!
m
U (k)mn!mk(r)
WF defined in basis of bloch states
-6-4-2 0 2 4 6
L G X K G
Si valence bands
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Wannier Functions: A local picture
Si GaAs
1- Intuitive picture of local bonding
2- Wannier centres give bulk polarisation in a ferroelectric
Paraelectric Ferroelectric
BaTiO3
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MLWF - Entangled Bands
Disentanglement procedure(Souza, Marzari and Vanderbilt)
Obtain “partially occupied” Wannier functions
Essentially perfect agreement within given energy window
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Copper
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Wannier InterpolationAccurate and Efficient approach to Fermi surface and spectral properties
•Exploit localisation of Wannier functions.•Require only matrix elements between close
neighbours.
Lead Fermi surface
1st principles accuracy at tight-binding costInterpolation of any one-electron operator
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Wannier Interpolation
1- Obtain operator in Wannier basisOmn(R) = !Om|O|Rn" =
!
k
e!ik·R UO(k)U†
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Wannier Interpolation
1- Obtain operator in Wannier basis
2- Fourier transform to an arbitrary k-point, k’Omn(k!) =
!
R
eik.·R Omn(R)
Omn(R) = !Om|O|Rn" =!
k
e!ik·R UO(k)U†
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Wannier Interpolation
1- Obtain operator in Wannier basis
2- Fourier transform to an arbitrary k-point, k’
3- Un-rotate matrix at k’
Omn(k!) =!
R
eik.·R Omn(R)
!!m,k! |O|!n,k!" = [U†k!O(k!)Uk! ]
What we need diagonalises H
Omn(R) = !Om|O|Rn" =!
k
e!ik·R UO(k)U†
All operations involve small matrices (NwanxNwan) => FAST!
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ConvergenceWannier interpolated object converges to ab-initio value, exponentially with k-grid sampling.
Lead
4x4x4
10x10x10 K-mesh size
Band interpolation error
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Wannier Interpolation - Summary
Bloch Stateseg planewave basis
• Each k-point expensive• Calculation of MLWF
converges exponentially
Wannier basis
•Each k-point cheap•Introduce occupancies and compute properties (only polynomial convergence)
Sample on coarse grid Sample on fine grid
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Wannier Interpolation - Fe•18 spinor Wannier functions•Keep up to 4th neighbour overlaps•Cost 1/2000 of full calculation
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Wannier Interpolation
H !-2
-1.5
-1
-0.5
0
0.5
1
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Wannier Interpolation pt2
1- k-derivatives can be taken analytically
eg band gradient
2- Position operator matrix elements well defined
Hk =!
R
eik·RH(R)
!Hk
!k!= i
!
R
R!eik·RH(R)
repeat for higher derivatives
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Hall Effect!!"#$%&'#(!)'**!+,,+-.!/)'**!01234
!!5&67'*689!)'**!+,,+-.!/)'**!011:4!
!!;6!!!,%+*$!&++$+$<
!!=+##67'>&+.%97!?#+'@9!.%7+!#+A+#9'*
!
!!B#699+$!!"!'&$!!!!,%+*$9
!!!!?#+'@9!.%7+C#+A+#9'*
!!;6!7'>&+.%-!6#$+#!&++$+$
!
!!"#$%&'#(!)'**!+,,+-.!/)'**!01234
!!5&67'*689!)'**!+,,+-.!/)'**!011:4!
!!;6!!!,%+*$!&++$+$<
!!=+##67'>&+.%97!?#+'@9!.%7+!#+A+#9'*
!
!!B#699+$!!"!'&$!!!!,%+*$9
!!!!?#+'@9!.%7+C#+A+#9'*
!!;6!7'>&+.%-!6#$+#!&++$+$
!
Ordinary Hall Effect (Hall 1879)
Anomalous Hall Effect (Hall 1880)
Crossed E and B fields
B breaks time-reversal
No B field needed!
Ferromagnetism breaks time-reversal
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Mechanisms for the AHEIntrinsic
1954 Karplus and Luttinger property of electron motion in perfect lattice “dissipationless” current (spin-orbit effect)
Extrinsic1955 Smit “skew scattering”1970 Berger “side-jump”
Intrinsic (again)1996- Rexamination of KL in terms of Berry’s phases Niu (Texas), Nagaosa (Tokyo)
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Electron Dynamics
r =1!
!"nk
!k! k"!n(k)
!k = !eE! er"B
Textbook wavepacket dynamics
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Electron Dynamics
r =1!
!"nk
!k! k"!n(k)
!k = !eE! er"B
Textbook wavepacket dynamics missing a term: “anomalous velocity”
“k-space magnetic field” caused by lattice potential“k-space Lorentz force” even when B=0
!xy =!e2
(2")2h
!
n
"
BZdk fn(k) !n,z(k)
Fermi function
Anomalous Hall Conductivity
Berry Curvature
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Jonathan Yates Tyndall Institute 27th March 2007
Finite-differences difficult: phases, band-crossings
bcc iron (LAPW) Yao et al (PRL 2004)
Kubo formula
Extremely computationally expensive!final number within 20% of experiment
Berry Curvature
!n(k) = !Im"#kun,k|$ |#kun,k%
!n(k) = !!An(k)An(k) = i!unk|!k|unk"
!n,z(k) = !2Im!
m!=n
vnm,x(k) vmn,y(k)(!m(k)! !n(k))2
Berry Curvature
Berry Potential
Ab-initio Calculation
cell periodic Bloch state
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Band-structure bcc Fe
N P! N H! N P H!
-8
-6
-4
-2
0
2
4
Spin up Spin downDFT (PBE) + spin-orbit
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Berry curvature bcc Fe
-1e3
1e3
-1e5
1e5
1e1 -1e1
N P! N H! N P H!
-8
-6
-4
-2
0
2
4
Berr
y Cu
rvat
ure
(ato
mic
uni
ts)
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Berry curvature bcc Fe
-1e3
1e3
-1e5
1e5
1e1 -1e1
N P! N H! N P H!
-8
-6
-4
-2
0
2
4
Berr
y Cu
rvat
ure
(ato
mic
uni
ts)
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Berry Curvature
-1e3
1e3
-1e5
1e5
1e1 -1e1
P H!
-0.4
-0.2
0
0.2
0.4
Berr
y Cu
rvat
ure
(ato
mic
uni
ts)
!tot,z =!
n
fn(k) !n,z(k)
Total Berry Curvature
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Our CalculationAb-initio k-grid: 8x8x8Interpolation k-grid: 320x320x320 + 7x7x7 “adaptive refinement”
(when Ω> 100 a.u.) Wannier Interpolation
planewaves + pseudopotentials
758 (Ω cm)-1
Existing Calculation*Kubo formula +
LAPW
751 (Ω cm)-1
*Yao et al PRL (2004)
Experiment ~1000 (Ω cm)-1
Differences: Scattering, Approximate DFT, Experimental uncertainty
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Berry curvature in ky=0 plane
Total Berry curvature (atomic units)
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Berry curvature in ky=0 plane
26% opposite spin
21% like spin
53% smooth background
Contributions
Total Berry curvature (atomic units)
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Wannier Interpolation
•Map low energy electronic structure onto “exact tight binding model”
•Compute quantities with ab-initio accuracy and tight-binding cost
Basis set independent (planewaves, LAPW, guassians etc)Any single particle level of theory (LDA, LDA+U, GW)
•Many applications:Accurate DOS, joint-DOS, Fermi SurfacesSpin-relaxationMagneto-crystalline anisotropyNMR in metals (Knight shift)Electron-phonon interaction
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AcknowledgmentsIvo Souza : UC Berkeley
Wannier Interpolation: Phys Rev B 75 195121 (2007)
Anomalous Hall EffectXinjie Wang, David Vanderbilt : Rutgers UniversityPhys Rev B 74 195118 (2006)
Wannier90 codeArash Mostofi, Nicola Marzari MITReleased under the GPL at www.wannier.org
Electron-PhononFeliciano Guistino, Marvin Cohen, Steven Louie : UCBPhys Rev Lett 94 047005 (2007)
Funding - Lawrence Berkeley National Laboratory Corpus Christi College, Cambridge