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
Representing a Fermi Surface
Lead Fermi surface
Accurate description of Fermi surface properties requires a detailed sampling of the Brillouin Zone
Al Fermi surface
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
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
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
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
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
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
MLWF - Entangled Bands
Disentanglement procedure(Souza, Marzari and Vanderbilt)
Obtain “partially occupied” Wannier functions
Essentially perfect agreement within given energy window
Copper
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
Wannier Interpolation
1- Obtain operator in Wannier basisOmn(R) = !Om|O|Rn" =
!
k
e!ik·R UO(k)U†
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†
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!
ConvergenceWannier interpolated object converges to ab-initio value, exponentially with k-grid sampling.
Lead
4x4x4
10x10x10 K-mesh size
Band interpolation error
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
Wannier Interpolation - Fe•18 spinor Wannier functions•Keep up to 4th neighbour overlaps•Cost 1/2000 of full calculation
Wannier Interpolation
H !-2
-1.5
-1
-0.5
0
0.5
1
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
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
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)
Electron Dynamics
r =1!
!"nk
!k! k"!n(k)
!k = !eE! er"B
Textbook wavepacket dynamics
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
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
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
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)
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)
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
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
Berry curvature in ky=0 plane
Total Berry curvature (atomic units)
Berry curvature in ky=0 plane
26% opposite spin
21% like spin
53% smooth background
Contributions
Total Berry curvature (atomic units)
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
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