ab initio pseudopotential calculations of the orbital...
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1ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Ab initio pseudopotential calculations of the orbital
magnetizationDavide Ceresoli
Present address:Department of Materials Science and Engineering
MIT
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2ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Acknowledgments
Francesco MauriAri SeitsonenInstitut de Minéralogie et Physique des Milieux Condensés (IMPMC)Université Pierre et Marie Curie, Paris, France
Uwe GerstmannIMPMC and University of Paderborn, Paderborn, Germany
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3ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Outline● Modern Theory of the orbital magnetization
● Application: EPR g-tensor in molecules and solids
● Orbital magnetization of Fe, Co and Ni
● Conclusions
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4ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Orbital magnetization
● ... but unquenched in nanostructures
LSJ
● Two contributions to the total magnetization
● Orbital magnetization usually small in solids ...
☞ Spin-Orbit interaction
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5ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Spin-orbit
spin-other-orbit (SOO)+
relativistic masscorrections
+...
Zeeman term
SO term
Schrekenbach and Ziegler, J. Phys. Chem. A 101, 3388 (1997)
Origin of Morb in solid and molecules
Atomic units,● molecule radicals● paramagnetic defects in solids● ferromagnetic metals
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6ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Definition
Isurf
● Well defined in finite systems!● Problems with periodic systems
● position operator r incompatible with PBCs● surface currents
1990's: Modern Theory of Polarization
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7ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Periodic systemsPass to Wannier functions
Evaluate “Local Circulation” of the Wannier orbital
What about surfaces?
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8ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Surface WFs
Thonhauser, Ceresoli, Vanderbilt, Resta, PRL 95, 137205 (2005)Ceresoli, Thohauser, Vanderbilt, Resta, PRB 74, 024408 (2006)
Each surface WF carries a net
current!
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9ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Periodic systems
● Derived independently by two groups (2005-2006)Ceresoli, Resta, Thonhauser, VanderbiltXiao, Yao, Fang, Shi, Vignale, Niu
● Valid for insulators and metals● Easy to implement in all-electron (AE) electronic structure
codes● Extra terms for pseudopotentials (PS)
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10ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
GIPAW
[1] Ismail-Beigi, Chang and Louie, PRL 87, 087402 (2001)[2] Pickard and Mauri, PRL 91, 196401 (2003)[3] Pickard and Mauri, PRB 63, 245101 (2001)[4] Blöchl, PRB 50, 17953 (1994)
Origin of extra terms: NLPP's coupling to EM fields [1,2]
Correct recipe: Gauge Including Projector Augmented Wave [3]● Gauge invariant, AE and PS eigenvalues coincide● Reconstruct the AE wvfcs from PS wvfcs● Based on the PAW method [4]
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11ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
GIPAW transformation
Inneroperator
projector
projector
AE atomic
PS atomic
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12ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
GIPAW reconstructionaugmentation region1.
2. 3.
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13ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Orbital magnetization
All quantities calculated with PS hamiltonian and wavefunctions!
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14ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Reconstruction termsParamagnetic term
Diamagnetic term
same as in: Pickard and Mauri, PRL 88, 086403 (2002)
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15ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
EPR spectroscopyEPR = Electron Paramagnetic Resonance
Tipical fields ~0.5 TResonance ~14 GHz
● Paramagnetic defects in solids
● g-tensor and hyperfine couplings very sensitive to chemical environment
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16ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Connection to the orbital magn.For a spin 1/2
The g factor is
∆gSO2.002319...
“chemical shift”
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17ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Calculation of the g-tensor● non perturbative method, SO interactions to all orders● 3 SCF calculations (j = 1..3) including SO● ∆gSO directly from Morb
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18ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Linear response (LR) method
● SCF calculation (no SO included)● LR with respect to uniform B (3 pertubations)● ∆gSO from induced current
Pickard and Mauri, PRL 88, 086403 (2002)
Linearizing ∆gSO with respect to SO coupling strenght
Current induced by uniform magnetic field
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19ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Technical details● 8000 Å3 Cubic supercell● 100 Ry PW cutoff● PBE functional● 2x2x2 k-points● Norm conserving PPs● 2 GIPAW projectors x angular momentum channel● du/dk computed as a covariant derivate
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20ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Linear response● GIPAW linear response (LR) recently implemented in quantum-
Espresso by Ceresoli, Seitsonen and Gerstmann● Available for production in Espresso-4.0● Capabilities
Magnetic susceptibility NMR shielding tensors Electric Field Gradients (EFGs) EPR g-tensor Hyperfine couplings XAS (under development, S. Fabris and Y. Yao) XANES (under development, G. Gougoussis and M. Calandra)
References and codes:● www.gipaw.net● www.quantum-espresso.org
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21ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Results for diatomic radicals
● expt. data: solid matrix● values in ppm● SOO not included
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22ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Results for molecule radicals
● expt. data: solid matrix● values in ppm● SOO not included
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23ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
GIPAW corrections
1. Relativistic Mass Corrections
CN H2O+
∆Mbare ~ 5-10 %
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24ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Advantages over LR● need only SCF calculations → LDA+U, EXX, OEP, B3LYP, ...● no magnetic field● no symmetry restrictions● SO interaction to all orders
Work in progress● benchmark against paramagnetic defects in solids● speedup● convergence w.r.t. k-points
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25ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Orbital magnetization in ferromagnets
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26ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Einstein only experiment!http://www.ptb.de/en/publikationen/jahresberichte/jb2005/nachrdjahres/s23e.html
Einstein-de Haas effectThe effect corresponds to the mechanical rotation that is induced in a ferromagnetic material (of cylindrical shape and originally at rest), suspended with the aid of a thin string inside a coil, on driving an impulse of electric current through the coil. To this mechanical rotation of the ferromagnetic material (say, iron) is associated a mechanical angular momentum, which, by the law of conservation of angular momentum, must be compensated by an equally large and oppositely directed angular momentum inside the ferromagnetic material.
By measuring Mtot and Jtot you canextract S and L!
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27ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Orbital magnetization in metals
Previous calculations● LDA and GGA underestimate Morb [1,5]● better agreement with orbital polarization (OP) [2,3,4]● ... and CDFT [3,4]● [6] provides a link between CDFT and OP
Which DFT is better for the orbital magnetization?
[1] Singh, Callaway, Wang, PRB 14, 1214 (1976)[2] Eriksson, Johanson, Albers, Boring, Brooks, PRB 42, 2707 (1990)[3] Ebert, Battocletti, Solid State Commun. 98, 785 (1996)[4] Ebert, Battocletti, Gross, Europhys. Lett. 40, 525 (1997)[5] Sharma, Pittalis, Kurth, Shallcross, Dewhurst, Gross, PRB 76, 100401 (2007)[6] Morbec, Capelle, Int. J. Quantum Chem., in press (2008)
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28ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Results for Fe, Co and Ni
[7] Meyer and Asch, J. Appl. Phys. 32, 330S (1961)
all values in µBThis work: PBE, 90 Ry, up to 32x32x32 k-points
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29ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
k-points convergence
Fe
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30ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Contributions to the magnetization
all values in µB
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31ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Conclusions● Derived orbital magnetization formula for ab initio
pseudopotential calculations● Non perturbative method to compute EPR g-tensor
tested against small molecule radicals
● We computed the orbital magnetization of Fe, Co and Ni● Work in progress
- evaluate speedup with respect to linear response method- combine non perturbative EPR method and LDA+U
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32ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Extra slides● Coupling to NLPPs (ICL)● Coupling to NLPPs (Pickard-Mauri)● Effective spin hamiltonian● Spin-orbit
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33ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Non local pseudopotential
•R
r
r '
augmentationregion
Ismail-Beigi, Chang and Louie, PRL 87, 087402 (2001)
● gauge invariant
Magnetic field coupling to non local potentials
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34ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
● gauge invariant
● same AE and PS eigenvalues
Non local pseudopotentialsMagnetic field coupling to non local potentials
•R
r
r '
augmentationregion
Pickard and Mauri, PRL 91, 196401 (2003)
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35ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Effective spin hamiltonian
g tensor hyperfinecouplings
Zero fieldsplitting
SOO + other relativistic
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36ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
GIPAWGauge Including Projector Augmented Wave [1]
● Correct treatment of magnetic field coupling● Gauge invariant● AE and PS eigenvalues coincide● Based on the PAW formalism [2]● Yields accurate AE properties from PS wavefunctions
[1] Pickard and Mauri, PRB 63, 245101 (2001)[2] Blöchl, PRB 50, 17953 (1994)
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37ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Technical details● Norm conserving PPs● 90 Ry PW cutoff, 0.001 Ry cold smearing● PBE functional● 2 GIPAW projectors x angular momentum channel● du/dk computed via k·p method● up to 32x32x32 k-points● spin constrained along easy axis (Fe [100], Ni [111], Co [001])
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38ES08 - 20th Annual Workshop on Recent Developments in Electronic Structure Methods
Periodic systems