Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems...
Transcript of Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems...
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Orbital Magnetization in Periodic Systems
Raffaele Resta
Dipartimento di Fisica Teorica, Università di Trieste,and CNR-INFM DEMOCRITOS National Simulation Center, Trieste
MSSC2009, Torino, September 2009
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Modern theory of electrical polarization
Theory developed since ' 1992, nowadays mature.
The theory is based on a Berry phase.Several first-principle calculations have been performed inmany nonmetallic materials: spontaneous polarization inferroelectrics, piezoelectricity, infrared spectra in solids andliquids....Most electronic-structure computer codes on the marketimplement the Berry phase as a standard option:CRYSTAL, PWSCF, ABINIT, VASP, SIESTA,CPMD...
![Page 3: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/3.jpg)
Modern theory of electrical polarization
Theory developed since ' 1992, nowadays mature.
The theory is based on a Berry phase.Several first-principle calculations have been performed inmany nonmetallic materials: spontaneous polarization inferroelectrics, piezoelectricity, infrared spectra in solids andliquids....Most electronic-structure computer codes on the marketimplement the Berry phase as a standard option:CRYSTAL, PWSCF, ABINIT, VASP, SIESTA,CPMD...
![Page 4: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/4.jpg)
Modern theory of electrical polarization
Theory developed since ' 1992, nowadays mature.
The theory is based on a Berry phase.Several first-principle calculations have been performed inmany nonmetallic materials: spontaneous polarization inferroelectrics, piezoelectricity, infrared spectra in solids andliquids....Most electronic-structure computer codes on the marketimplement the Berry phase as a standard option:CRYSTAL, PWSCF, ABINIT, VASP, SIESTA,CPMD...
![Page 5: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/5.jpg)
Modern theory of orbital magnetization
Theory started in 2005, and is still work in progress.
Only model-Hamiltonian results published so far.The very first ab-initio calculations are in press by now:
NMR shielding tensors in several materials:T. Thonhauser, D. Ceresoli, A.A. Mostofi, N. Marzari, R.Resta, and D. Vanderbilt, J. Chem. Phys. 131, 101101(2009).Spontaneous magnetization in Fe, Co, and Ni:�D. Ceresoli, U. Gertsmann, A.P. Seitsonen, & F. Mauri,http://arxiv.org/abs/0904.1988.
These are pseudopotential implementations;there are good reasons which make all-electronimplementations desirable.
![Page 6: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/6.jpg)
Modern theory of orbital magnetization
Theory started in 2005, and is still work in progress.
Only model-Hamiltonian results published so far.The very first ab-initio calculations are in press by now:
NMR shielding tensors in several materials:T. Thonhauser, D. Ceresoli, A.A. Mostofi, N. Marzari, R.Resta, and D. Vanderbilt, J. Chem. Phys. 131, 101101(2009).Spontaneous magnetization in Fe, Co, and Ni:�D. Ceresoli, U. Gertsmann, A.P. Seitsonen, & F. Mauri,http://arxiv.org/abs/0904.1988.
These are pseudopotential implementations;there are good reasons which make all-electronimplementations desirable.
![Page 7: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/7.jpg)
Modern theory of orbital magnetization
Theory started in 2005, and is still work in progress.
Only model-Hamiltonian results published so far.The very first ab-initio calculations are in press by now:
NMR shielding tensors in several materials:T. Thonhauser, D. Ceresoli, A.A. Mostofi, N. Marzari, R.Resta, and D. Vanderbilt, J. Chem. Phys. 131, 101101(2009).Spontaneous magnetization in Fe, Co, and Ni:�D. Ceresoli, U. Gertsmann, A.P. Seitsonen, & F. Mauri,http://arxiv.org/abs/0904.1988.
These are pseudopotential implementations;there are good reasons which make all-electronimplementations desirable.
![Page 8: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/8.jpg)
Modern theory of orbital magnetization
Theory started in 2005, and is still work in progress.
Only model-Hamiltonian results published so far.The very first ab-initio calculations are in press by now:
NMR shielding tensors in several materials:T. Thonhauser, D. Ceresoli, A.A. Mostofi, N. Marzari, R.Resta, and D. Vanderbilt, J. Chem. Phys. 131, 101101(2009).Spontaneous magnetization in Fe, Co, and Ni:�D. Ceresoli, U. Gertsmann, A.P. Seitsonen, & F. Mauri,http://arxiv.org/abs/0904.1988.
These are pseudopotential implementations;there are good reasons which make all-electronimplementations desirable.
![Page 9: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/9.jpg)
Outline
1 Why do we need a kind of “exotic” theory?The dipole of a finite moleculeThe “dipole” of a solid
2 Role of macroscopic fields
3 Modern theory of polarization: main features
4 Modern theory of orbital magnetization
5 Application: NMR shielding tensor
6 Results
![Page 10: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/10.jpg)
Outline
1 Why do we need a kind of “exotic” theory?The dipole of a finite moleculeThe “dipole” of a solid
2 Role of macroscopic fields
3 Modern theory of polarization: main features
4 Modern theory of orbital magnetization
5 Application: NMR shielding tensor
6 Results
![Page 11: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/11.jpg)
From Feynman Lectures in Physics, Vol. 2
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From Feynman Lectures in Physics, Vol. 2
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From Feynman Lectures in Physics, Vol. 2
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TEXTBOOK DEFINITIONSARE NONCOMPUTABLE!
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Main message of this talk
MACROSCOPIC POLARIZATION P HASNOTHING TO DO WITH THE PERIODICCHARGE DENSITY OF THE POLARIZEDSOLIDBy analogous reasonement:ORBITAL MAGNETIZATION HAS NOTHINGTO DO WITH THE PERIODIC CURRENTDISTRIBUTION OF THE MAGNETIZEDSOLID
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Main message of this talk
MACROSCOPIC POLARIZATION P HASNOTHING TO DO WITH THE PERIODICCHARGE DENSITY OF THE POLARIZEDSOLIDBy analogous reasonement:ORBITAL MAGNETIZATION HAS NOTHINGTO DO WITH THE PERIODIC CURRENTDISTRIBUTION OF THE MAGNETIZEDSOLID
![Page 17: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/17.jpg)
Outline
1 Why do we need a kind of “exotic” theory?The dipole of a finite moleculeThe “dipole” of a solid
2 Role of macroscopic fields
3 Modern theory of polarization: main features
4 Modern theory of orbital magnetization
5 Application: NMR shielding tensor
6 Results
![Page 18: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/18.jpg)
Trivial definitions(atomic units througout)
m =12c
∫dr r× j(r) + mspin
d = −∫
dr n(r) + dnuclear
d is nonzero only in absence of inversion symmetry.m is nonzero only in absence of time-reversal symmetry.
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Trivial definitions(atomic units througout)
m =12c
∫dr r× j(r) + mspin
d = −∫
dr n(r) + dnuclear
d is nonzero only in absence of inversion symmetry.m is nonzero only in absence of time-reversal symmetry.
![Page 20: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/20.jpg)
Trivial definitions (cont’d)
m =12c
∫dr r× j(r) + mspin
d = −∫
dr rn(r) + dnuclear
Orbital and spin magnetization are well separated,either experimentally or in (nonrelativistic) QM.We are going to neglect mspin in the following(nonzero only in ferromagnets)dnuclear is trivial(though charge neutrality is essential)
![Page 21: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/21.jpg)
Trivial definitions (cont’d)
m =12c
∫dr r× j(r) + mspin
d = −∫
dr rn(r) + dnuclear
Orbital and spin magnetization are well separated,either experimentally or in (nonrelativistic) QM.We are going to neglect mspin in the following(nonzero only in ferromagnets)dnuclear is trivial(though charge neutrality is essential)
![Page 22: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/22.jpg)
Trivial definitions (cont’d)
m =12c
∫dr r× j(r) + mspin
d = −∫
dr rn(r) + dnuclear
Orbital and spin magnetization are well separated,either experimentally or in (nonrelativistic) QM.We are going to neglect mspin in the following(nonzero only in ferromagnets)dnuclear is trivial(though charge neutrality is essential)
![Page 23: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/23.jpg)
Trivial definitions (cont’d)
m =12c
∫dr r× j(r) + mspin
d = −∫
dr rn(r) + dnuclear
Orbital and spin magnetization are well separated,either experimentally or in (nonrelativistic) QM.We are going to neglect mspin in the following(nonzero only in ferromagnets)dnuclear is trivial(though charge neutrality is essential)
![Page 24: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/24.jpg)
Trivial definitions (cont’d)
m =12c
∫dr r× j(r) + mspin
d = −∫
dr rn(r) + dnuclear
Orbital and spin magnetization are well separated,either experimentally or in (nonrelativistic) QM.We are going to neglect mspin in the following(nonzero only in ferromagnets)dnuclear is trivial(though charge neutrality is essential)
![Page 25: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/25.jpg)
Independent electrons(either Hartree-Fock or Kohn-Sham, double occupancy)
m = 212c
∑n∈ occupied
〈ϕn|r× v|ϕn〉
d = −2∑
n∈ occupied
〈ϕn|r|ϕn〉+∑
i ∈ nuclei
ZiRi
Velocity v = i[H, r]Invariant by transformation to localized (e.g. Boys) orbitals.
m = 212c
∑n∈ occupied
〈wn|r× v|wn〉
d = −2∑
n∈ occupied
〈wn|r|wn〉+∑
i ∈ nuclei
ZiRi
![Page 26: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/26.jpg)
Independent electrons(either Hartree-Fock or Kohn-Sham, double occupancy)
m = 212c
∑n∈ occupied
〈ϕn|r× v|ϕn〉
d = −2∑
n∈ occupied
〈ϕn|r|ϕn〉+∑
i ∈ nuclei
ZiRi
Velocity v = i[H, r]Invariant by transformation to localized (e.g. Boys) orbitals.
m = 212c
∑n∈ occupied
〈wn|r× v|wn〉
d = −2∑
n∈ occupied
〈wn|r|wn〉+∑
i ∈ nuclei
ZiRi
![Page 27: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/27.jpg)
Independent electrons(either Hartree-Fock or Kohn-Sham, double occupancy)
m = 212c
∑n∈ occupied
〈ϕn|r× v|ϕn〉
d = −2∑
n∈ occupied
〈ϕn|r|ϕn〉+∑
i ∈ nuclei
ZiRi
Velocity v = i[H, r]Invariant by transformation to localized (e.g. Boys) orbitals.
m = 212c
∑n∈ occupied
〈wn|r× v|wn〉
d = −2∑
n∈ occupied
〈wn|r|wn〉+∑
i ∈ nuclei
ZiRi
![Page 28: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/28.jpg)
Outline
1 Why do we need a kind of “exotic” theory?The dipole of a finite moleculeThe “dipole” of a solid
2 Role of macroscopic fields
3 Modern theory of polarization: main features
4 Modern theory of orbital magnetization
5 Application: NMR shielding tensor
6 Results
![Page 29: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/29.jpg)
Macroscopic magnetization and polarization(naive textbook view)
M =mV
=1
2cV
∫dr r× j(r)
P =dV
=1V
(−∫
dr r n(r) +∑
i ∈ nuclei
ZiRi
)
![Page 30: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/30.jpg)
Macroscopic magnetization and polarization(naive textbook view)
M =mV
=1
2cV
∫dr r× j(r)
P =dV
=1V
(−∫
dr r n(r) +∑
i ∈ nuclei
ZiRi
)
![Page 31: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/31.jpg)
Where is the problem?
M =mV
=1
2cV
∫dr r× j(r)
Pelectronic =dV
= − 1V
∫dr r n(r)
Common drawback:Surface terms contribute extensively to the dipole:so M and P are apparently surface properties:not bulk ones!This is due to the unbound nature of the operator r.
![Page 32: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/32.jpg)
Where is the problem?
M =mV
=1
2cV
∫dr r× j(r)
Pelectronic =dV
= − 1V
∫dr r n(r)
Common drawback:Surface terms contribute extensively to the dipole:so M and P are apparently surface properties:not bulk ones!This is due to the unbound nature of the operator r.
![Page 33: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/33.jpg)
Where is the problem?
M =mV
=1
2cV
∫dr r× j(r)
Pelectronic =dV
= − 1V
∫dr r n(r)
Common drawback:Surface terms contribute extensively to the dipole:so M and P are apparently surface properties:not bulk ones!This is due to the unbound nature of the operator r.
![Page 34: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/34.jpg)
The theoretical framework of CM physics:periodic (Born-von Kármán) boundary conditions(for both crystalline and disordered systems)
The system has no surface by construction.Any quantity defined or computed within PBC is bydefinition “bulk”.However... The position operator r is incompatible withBorn-von Kármán PBCs.The matrix elements of r over Bloch orbitals are ill defined.
![Page 35: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/35.jpg)
The theoretical framework of CM physics:periodic (Born-von Kármán) boundary conditions(for both crystalline and disordered systems)
The system has no surface by construction.Any quantity defined or computed within PBC is bydefinition “bulk”.However... The position operator r is incompatible withBorn-von Kármán PBCs.The matrix elements of r over Bloch orbitals are ill defined.
![Page 36: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/36.jpg)
The theoretical framework of CM physics:periodic (Born-von Kármán) boundary conditions(for both crystalline and disordered systems)
The system has no surface by construction.Any quantity defined or computed within PBC is bydefinition “bulk”.However... The position operator r is incompatible withBorn-von Kármán PBCs.The matrix elements of r over Bloch orbitals are ill defined.
![Page 37: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/37.jpg)
The theoretical framework of CM physics:periodic (Born-von Kármán) boundary conditions(for both crystalline and disordered systems)
The system has no surface by construction.Any quantity defined or computed within PBC is bydefinition “bulk”.However... The position operator r is incompatible withBorn-von Kármán PBCs.The matrix elements of r over Bloch orbitals are ill defined.
![Page 38: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/38.jpg)
The theoretical framework of CM physics:periodic (Born-von Kármán) boundary conditions(for both crystalline and disordered systems)
The system has no surface by construction.Any quantity defined or computed within PBC is bydefinition “bulk”.However... The position operator r is incompatible withBorn-von Kármán PBCs.The matrix elements of r over Bloch orbitals are ill defined.
![Page 39: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/39.jpg)
The theoretical framework of CM physics:periodic (Born-von Kármán) boundary conditions(for both crystalline and disordered systems)
The system has no surface by construction.Any quantity defined or computed within PBC is bydefinition “bulk”.However... The position operator r is incompatible withBorn-von Kármán PBCs.The matrix elements of r over Bloch orbitals are ill defined.
![Page 40: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/40.jpg)
Outline
1 Why do we need a kind of “exotic” theory?The dipole of a finite moleculeThe “dipole” of a solid
2 Role of macroscopic fields
3 Modern theory of polarization: main features
4 Modern theory of orbital magnetization
5 Application: NMR shielding tensor
6 Results
![Page 41: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/41.jpg)
Molecules
For a molecule, one can speak of the external fieldsEext and Bext.
By definition, they are the fields in the vacuum region faraway from the molecule.They can also be regarded as a boundary condition for theintegration of Poisson equation.If Eext = 0, then d is the spontaneous (equilibrium)electrical dipole.If Bext = 0, then m is the spontaneous (equilibrium)magnetic dipole.The linear responses to nonvanishing Eext and Bext are bydefinition the linear electric and magnetic polarizability.
![Page 42: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/42.jpg)
Molecules
For a molecule, one can speak of the external fieldsEext and Bext.
By definition, they are the fields in the vacuum region faraway from the molecule.They can also be regarded as a boundary condition for theintegration of Poisson equation.If Eext = 0, then d is the spontaneous (equilibrium)electrical dipole.If Bext = 0, then m is the spontaneous (equilibrium)magnetic dipole.The linear responses to nonvanishing Eext and Bext are bydefinition the linear electric and magnetic polarizability.
![Page 43: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/43.jpg)
Molecules
For a molecule, one can speak of the external fieldsEext and Bext.
By definition, they are the fields in the vacuum region faraway from the molecule.They can also be regarded as a boundary condition for theintegration of Poisson equation.If Eext = 0, then d is the spontaneous (equilibrium)electrical dipole.If Bext = 0, then m is the spontaneous (equilibrium)magnetic dipole.The linear responses to nonvanishing Eext and Bext are bydefinition the linear electric and magnetic polarizability.
![Page 44: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/44.jpg)
Molecules
For a molecule, one can speak of the external fieldsEext and Bext.
By definition, they are the fields in the vacuum region faraway from the molecule.They can also be regarded as a boundary condition for theintegration of Poisson equation.If Eext = 0, then d is the spontaneous (equilibrium)electrical dipole.If Bext = 0, then m is the spontaneous (equilibrium)magnetic dipole.The linear responses to nonvanishing Eext and Bext are bydefinition the linear electric and magnetic polarizability.
![Page 45: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/45.jpg)
Molecules
For a molecule, one can speak of the external fieldsEext and Bext.
By definition, they are the fields in the vacuum region faraway from the molecule.They can also be regarded as a boundary condition for theintegration of Poisson equation.If Eext = 0, then d is the spontaneous (equilibrium)electrical dipole.If Bext = 0, then m is the spontaneous (equilibrium)magnetic dipole.The linear responses to nonvanishing Eext and Bext are bydefinition the linear electric and magnetic polarizability.
![Page 46: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/46.jpg)
Solids
For an infinite periodic solid, one cannot speak of externalfields Eext and Bext.
The measurable microscopic fields inside the material areE(micro)(r) and B(micro)(r)Average over a macroscopic scale −→ E and B(average over a cell in crystalline materials).The value of the E and B is not a bulk property;instead, it is an arbitrary boundary condition.Electronic-structure codes require a lattice-periodicalpotential, hence they impose E = 0 and B = 0(otherwise Bloch states would not exist!)The modern theories address P and M in zero fields only.
![Page 47: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/47.jpg)
Solids
For an infinite periodic solid, one cannot speak of externalfields Eext and Bext.
The measurable microscopic fields inside the material areE(micro)(r) and B(micro)(r)Average over a macroscopic scale −→ E and B(average over a cell in crystalline materials).The value of the E and B is not a bulk property;instead, it is an arbitrary boundary condition.Electronic-structure codes require a lattice-periodicalpotential, hence they impose E = 0 and B = 0(otherwise Bloch states would not exist!)The modern theories address P and M in zero fields only.
![Page 48: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/48.jpg)
Solids
For an infinite periodic solid, one cannot speak of externalfields Eext and Bext.
The measurable microscopic fields inside the material areE(micro)(r) and B(micro)(r)Average over a macroscopic scale −→ E and B(average over a cell in crystalline materials).The value of the E and B is not a bulk property;instead, it is an arbitrary boundary condition.Electronic-structure codes require a lattice-periodicalpotential, hence they impose E = 0 and B = 0(otherwise Bloch states would not exist!)The modern theories address P and M in zero fields only.
![Page 49: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/49.jpg)
Solids
For an infinite periodic solid, one cannot speak of externalfields Eext and Bext.
The measurable microscopic fields inside the material areE(micro)(r) and B(micro)(r)Average over a macroscopic scale −→ E and B(average over a cell in crystalline materials).The value of the E and B is not a bulk property;instead, it is an arbitrary boundary condition.Electronic-structure codes require a lattice-periodicalpotential, hence they impose E = 0 and B = 0(otherwise Bloch states would not exist!)The modern theories address P and M in zero fields only.
![Page 50: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/50.jpg)
Solids
For an infinite periodic solid, one cannot speak of externalfields Eext and Bext.
The measurable microscopic fields inside the material areE(micro)(r) and B(micro)(r)Average over a macroscopic scale −→ E and B(average over a cell in crystalline materials).The value of the E and B is not a bulk property;instead, it is an arbitrary boundary condition.Electronic-structure codes require a lattice-periodicalpotential, hence they impose E = 0 and B = 0(otherwise Bloch states would not exist!)The modern theories address P and M in zero fields only.
![Page 51: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/51.jpg)
A finite solid as very large cluster
One can again speak of the external fields Eext and Bext for afinite cluster cut from the bulk.
The microscopic fields E(micro)(r) and B(micro)(r) tend to Eextand Bext in the vacuum region far away from the cluster.Inside the material, the macroscopic averages of E(micro)(r)and B(micro)(r) are different from Eext and Bext because ofscreening.The relationship between unscreened (a.k.a. external) andscreened (a.k.a. internal) fields depends on shape.Hence, a given choice of the screened fields E and Bclosely corresponds to a choice of a sample shape.
![Page 52: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/52.jpg)
A finite solid as very large cluster
One can again speak of the external fields Eext and Bext for afinite cluster cut from the bulk.
The microscopic fields E(micro)(r) and B(micro)(r) tend to Eextand Bext in the vacuum region far away from the cluster.Inside the material, the macroscopic averages of E(micro)(r)and B(micro)(r) are different from Eext and Bext because ofscreening.The relationship between unscreened (a.k.a. external) andscreened (a.k.a. internal) fields depends on shape.Hence, a given choice of the screened fields E and Bclosely corresponds to a choice of a sample shape.
![Page 53: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/53.jpg)
A finite solid as very large cluster
One can again speak of the external fields Eext and Bext for afinite cluster cut from the bulk.
The microscopic fields E(micro)(r) and B(micro)(r) tend to Eextand Bext in the vacuum region far away from the cluster.Inside the material, the macroscopic averages of E(micro)(r)and B(micro)(r) are different from Eext and Bext because ofscreening.The relationship between unscreened (a.k.a. external) andscreened (a.k.a. internal) fields depends on shape.Hence, a given choice of the screened fields E and Bclosely corresponds to a choice of a sample shape.
![Page 54: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/54.jpg)
A finite solid as very large cluster
One can again speak of the external fields Eext and Bext for afinite cluster cut from the bulk.
The microscopic fields E(micro)(r) and B(micro)(r) tend to Eextand Bext in the vacuum region far away from the cluster.Inside the material, the macroscopic averages of E(micro)(r)and B(micro)(r) are different from Eext and Bext because ofscreening.The relationship between unscreened (a.k.a. external) andscreened (a.k.a. internal) fields depends on shape.Hence, a given choice of the screened fields E and Bclosely corresponds to a choice of a sample shape.
![Page 55: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/55.jpg)
Outline
1 Why do we need a kind of “exotic” theory?The dipole of a finite moleculeThe “dipole” of a solid
2 Role of macroscopic fields
3 Modern theory of polarization: main features
4 Modern theory of orbital magnetization
5 Application: NMR shielding tensor
6 Results
![Page 56: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/56.jpg)
P in a crystalline solid
P = Pionic + Pelectronic ψnk(r) = eik·runk(r)
The—by now famous—Berry-phase formula(King-Smith & Vanderbilt, 1993):
Pelectronic = − 2ie(2π)3
∑n∈ occupied
∫BZ
dk 〈unk|∂kunk〉
Equivalently, and perhaps more intuitively, in terms ofWannier functions:
Pelectronic = − 2eVcell
∑n∈ occupied
〈wn|r|wn〉
![Page 57: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/57.jpg)
P in a crystalline solid
P = Pionic + Pelectronic ψnk(r) = eik·runk(r)
The—by now famous—Berry-phase formula(King-Smith & Vanderbilt, 1993):
Pelectronic = − 2ie(2π)3
∑n∈ occupied
∫BZ
dk 〈unk|∂kunk〉
Equivalently, and perhaps more intuitively, in terms ofWannier functions:
Pelectronic = − 2eVcell
∑n∈ occupied
〈wn|r|wn〉
![Page 58: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/58.jpg)
P in a crystalline solid
P = Pionic + Pelectronic ψnk(r) = eik·runk(r)
The—by now famous—Berry-phase formula(King-Smith & Vanderbilt, 1993):
Pelectronic = − 2ie(2π)3
∑n∈ occupied
∫BZ
dk 〈unk|∂kunk〉
Equivalently, and perhaps more intuitively, in terms ofWannier functions:
Pelectronic = − 2eVcell
∑n∈ occupied
〈wn|r|wn〉
![Page 59: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/59.jpg)
P in a crystalline solid
P = Pionic + Pelectronic ψnk(r) = eik·runk(r)
The—by now famous—Berry-phase formula(King-Smith & Vanderbilt, 1993):
Pelectronic = − 2ie(2π)3
∑n∈ occupied
∫BZ
dk 〈unk|∂kunk〉
Equivalently, and perhaps more intuitively, in terms ofWannier functions:
Pelectronic = − 2eVcell
∑n∈ occupied
〈wn|r|wn〉
![Page 60: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/60.jpg)
Key points
The theory is formulated in terms of either Bloch orbitals orWannier functions.This requires a lattice-periodical potential, i.e. E = 0.The modern theory addresses the macroscopicpolarization induced by something other than amacroscopic field:ferroelectricity, piezoelectricity, lattice dynamics in polarcrystals (Born effective charges).It does not address permittivity (in its original formulation).Polarization is nonzero only in absence of inversionsymmetry.
![Page 61: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/61.jpg)
Key points
The theory is formulated in terms of either Bloch orbitals orWannier functions.This requires a lattice-periodical potential, i.e. E = 0.The modern theory addresses the macroscopicpolarization induced by something other than amacroscopic field:ferroelectricity, piezoelectricity, lattice dynamics in polarcrystals (Born effective charges).It does not address permittivity (in its original formulation).Polarization is nonzero only in absence of inversionsymmetry.
![Page 62: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/62.jpg)
Key points
The theory is formulated in terms of either Bloch orbitals orWannier functions.This requires a lattice-periodical potential, i.e. E = 0.The modern theory addresses the macroscopicpolarization induced by something other than amacroscopic field:ferroelectricity, piezoelectricity, lattice dynamics in polarcrystals (Born effective charges).It does not address permittivity (in its original formulation).Polarization is nonzero only in absence of inversionsymmetry.
![Page 63: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/63.jpg)
Key points
The theory is formulated in terms of either Bloch orbitals orWannier functions.This requires a lattice-periodical potential, i.e. E = 0.The modern theory addresses the macroscopicpolarization induced by something other than amacroscopic field:ferroelectricity, piezoelectricity, lattice dynamics in polarcrystals (Born effective charges).It does not address permittivity (in its original formulation).Polarization is nonzero only in absence of inversionsymmetry.
![Page 64: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/64.jpg)
Key points
The theory is formulated in terms of either Bloch orbitals orWannier functions.This requires a lattice-periodical potential, i.e. E = 0.The modern theory addresses the macroscopicpolarization induced by something other than amacroscopic field:ferroelectricity, piezoelectricity, lattice dynamics in polarcrystals (Born effective charges).It does not address permittivity (in its original formulation).Polarization is nonzero only in absence of inversionsymmetry.
![Page 65: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/65.jpg)
Key points
The theory is formulated in terms of either Bloch orbitals orWannier functions.This requires a lattice-periodical potential, i.e. E = 0.The modern theory addresses the macroscopicpolarization induced by something other than amacroscopic field:ferroelectricity, piezoelectricity, lattice dynamics in polarcrystals (Born effective charges).It does not address permittivity (in its original formulation).Polarization is nonzero only in absence of inversionsymmetry.
![Page 66: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/66.jpg)
Key points
The theory is formulated in terms of either Bloch orbitals orWannier functions.This requires a lattice-periodical potential, i.e. E = 0.The modern theory addresses the macroscopicpolarization induced by something other than amacroscopic field:ferroelectricity, piezoelectricity, lattice dynamics in polarcrystals (Born effective charges).It does not address permittivity (in its original formulation).Polarization is nonzero only in absence of inversionsymmetry.
![Page 67: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/67.jpg)
Key points
The theory is formulated in terms of either Bloch orbitals orWannier functions.This requires a lattice-periodical potential, i.e. E = 0.The modern theory addresses the macroscopicpolarization induced by something other than amacroscopic field:ferroelectricity, piezoelectricity, lattice dynamics in polarcrystals (Born effective charges).It does not address permittivity (in its original formulation).Polarization is nonzero only in absence of inversionsymmetry.
![Page 68: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/68.jpg)
Outline
1 Why do we need a kind of “exotic” theory?The dipole of a finite moleculeThe “dipole” of a solid
2 Role of macroscopic fields
3 Modern theory of polarization: main features
4 Modern theory of orbital magnetization
5 Application: NMR shielding tensor
6 Results
![Page 69: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/69.jpg)
Heuristically, by analogy with the electrical case
For an insulator, in absence of inversion symmetry, in zeroE field, we have
Pelectronic = − 2Vcell
∑n∈ occupied
〈wn|r|wn〉
By analogy, in absence of time-reversal symmetry, in zeroB field, it is tempting to write:
M = − 22cVcell
∑n∈ occupied
〈wn|r× v|wn〉
Question: Is this the correct formula for the bulkmagnetization?Answer: No!There is an additional term, having no electrical analogue.
![Page 70: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/70.jpg)
Heuristically, by analogy with the electrical case
For an insulator, in absence of inversion symmetry, in zeroE field, we have
Pelectronic = − 2Vcell
∑n∈ occupied
〈wn|r|wn〉
By analogy, in absence of time-reversal symmetry, in zeroB field, it is tempting to write:
M = − 22cVcell
∑n∈ occupied
〈wn|r× v|wn〉
Question: Is this the correct formula for the bulkmagnetization?Answer: No!There is an additional term, having no electrical analogue.
![Page 71: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/71.jpg)
Heuristically, by analogy with the electrical case
For an insulator, in absence of inversion symmetry, in zeroE field, we have
Pelectronic = − 2Vcell
∑n∈ occupied
〈wn|r|wn〉
By analogy, in absence of time-reversal symmetry, in zeroB field, it is tempting to write:
M = − 22cVcell
∑n∈ occupied
〈wn|r× v|wn〉
Question: Is this the correct formula for the bulkmagnetization?Answer: No!There is an additional term, having no electrical analogue.
![Page 72: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/72.jpg)
Heuristically, by analogy with the electrical case
For an insulator, in absence of inversion symmetry, in zeroE field, we have
Pelectronic = − 2Vcell
∑n∈ occupied
〈wn|r|wn〉
By analogy, in absence of time-reversal symmetry, in zeroB field, it is tempting to write:
M = − 22cVcell
∑n∈ occupied
〈wn|r× v|wn〉
Question: Is this the correct formula for the bulkmagnetization?Answer: No!There is an additional term, having no electrical analogue.
![Page 73: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/73.jpg)
Heuristically, by analogy with the electrical case
For an insulator, in absence of inversion symmetry, in zeroE field, we have
Pelectronic = − 2Vcell
∑n∈ occupied
〈wn|r|wn〉
By analogy, in absence of time-reversal symmetry, in zeroB field, it is tempting to write:
M = − 22cVcell
∑n∈ occupied
〈wn|r× v|wn〉
Question: Is this the correct formula for the bulkmagnetization?Answer: No!There is an additional term, having no electrical analogue.
![Page 74: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/74.jpg)
Heuristically, by analogy with the electrical case
For an insulator, in absence of inversion symmetry, in zeroE field, we have
Pelectronic = − 2Vcell
∑n∈ occupied
〈wn|r|wn〉
By analogy, in absence of time-reversal symmetry, in zeroB field, it is tempting to write:
M = − 22cVcell
∑n∈ occupied
〈wn|r× v|wn〉
Question: Is this the correct formula for the bulkmagnetization?Answer: No!There is an additional term, having no electrical analogue.
![Page 75: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/75.jpg)
From Wannier back to Bloch
v =i~[H, r] ; ψnk(r) = eik·runk(r) ; H(k) = e−ik·rHeik·r
M = − 1cVcell
∑n∈ occupied
〈wn|r× v|wn〉
= − ie~cVcell
∑n∈ occupied
〈wn|r× Hr|wn〉
= − ie~c(2π)3
∑n∈ occupied
∫BZ
dk 〈∂kunk| × H(k)|∂kunk〉,
Electrical analogy once more:
Pelectronic = − 2i(2π)3
∑n∈ occupied
∫BZ
dk 〈unk|∂kunk〉
![Page 76: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/76.jpg)
From Wannier back to Bloch
v =i~[H, r] ; ψnk(r) = eik·runk(r) ; H(k) = e−ik·rHeik·r
M = − 1cVcell
∑n∈ occupied
〈wn|r× v|wn〉
= − ie~cVcell
∑n∈ occupied
〈wn|r× Hr|wn〉
= − ie~c(2π)3
∑n∈ occupied
∫BZ
dk 〈∂kunk| × H(k)|∂kunk〉,
Electrical analogy once more:
Pelectronic = − 2i(2π)3
∑n∈ occupied
∫BZ
dk 〈unk|∂kunk〉
![Page 77: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/77.jpg)
From Wannier back to Bloch
v =i~[H, r] ; ψnk(r) = eik·runk(r) ; H(k) = e−ik·rHeik·r
M = − 1cVcell
∑n∈ occupied
〈wn|r× v|wn〉
= − ie~cVcell
∑n∈ occupied
〈wn|r× Hr|wn〉
= − ie~c(2π)3
∑n∈ occupied
∫BZ
dk 〈∂kunk| × H(k)|∂kunk〉,
Electrical analogy once more:
Pelectronic = − 2i(2π)3
∑n∈ occupied
∫BZ
dk 〈unk|∂kunk〉
![Page 78: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/78.jpg)
Model Hückel-like Hamiltonian in 2d(Haldane, PRL 1988)
Honeycomb lattice. Tight-binding with first- andsecond-neighbor hopping: t1 real, t2 complex.It breaks time-reversal symmetry.Two nonequivalent sites per cell. Insulator at half-filling(only the lowest band occupied).Zero flux through the unit cell:The macroscopic B field is zero;The Hamiltonian is lattice-periodical;Bloch & Wannier orbitals exist.
![Page 79: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/79.jpg)
Model Hückel-like Hamiltonian in 2d(Haldane, PRL 1988)
Honeycomb lattice. Tight-binding with first- andsecond-neighbor hopping: t1 real, t2 complex.It breaks time-reversal symmetry.Two nonequivalent sites per cell. Insulator at half-filling(only the lowest band occupied).Zero flux through the unit cell:The macroscopic B field is zero;The Hamiltonian is lattice-periodical;Bloch & Wannier orbitals exist.
![Page 80: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/80.jpg)
Model Hückel-like Hamiltonian in 2d(Haldane, PRL 1988)
Honeycomb lattice. Tight-binding with first- andsecond-neighbor hopping: t1 real, t2 complex.It breaks time-reversal symmetry.Two nonequivalent sites per cell. Insulator at half-filling(only the lowest band occupied).Zero flux through the unit cell:The macroscopic B field is zero;The Hamiltonian is lattice-periodical;Bloch & Wannier orbitals exist.
![Page 81: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/81.jpg)
Model Hückel-like Hamiltonian in 2d(Haldane, PRL 1988)
Honeycomb lattice. Tight-binding with first- andsecond-neighbor hopping: t1 real, t2 complex.It breaks time-reversal symmetry.Two nonequivalent sites per cell. Insulator at half-filling(only the lowest band occupied).Zero flux through the unit cell:The macroscopic B field is zero;The Hamiltonian is lattice-periodical;Bloch & Wannier orbitals exist.
![Page 82: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/82.jpg)
Model Hückel-like Hamiltonian in 2d(Haldane, PRL 1988)
Honeycomb lattice. Tight-binding with first- andsecond-neighbor hopping: t1 real, t2 complex.It breaks time-reversal symmetry.Two nonequivalent sites per cell. Insulator at half-filling(only the lowest band occupied).Zero flux through the unit cell:The macroscopic B field is zero;The Hamiltonian is lattice-periodical;Bloch & Wannier orbitals exist.
![Page 83: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/83.jpg)
Model Hückel-like Hamiltonian in 2d(Haldane, PRL 1988)
Honeycomb lattice. Tight-binding with first- andsecond-neighbor hopping: t1 real, t2 complex.It breaks time-reversal symmetry.Two nonequivalent sites per cell. Insulator at half-filling(only the lowest band occupied).Zero flux through the unit cell:The macroscopic B field is zero;The Hamiltonian is lattice-periodical;Bloch & Wannier orbitals exist.
![Page 84: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/84.jpg)
Model Hückel-like Hamiltonian in 2d(Haldane, PRL 1988)
Honeycomb lattice. Tight-binding with first- andsecond-neighbor hopping: t1 real, t2 complex.It breaks time-reversal symmetry.Two nonequivalent sites per cell. Insulator at half-filling(only the lowest band occupied).Zero flux through the unit cell:The macroscopic B field is zero;The Hamiltonian is lattice-periodical;Bloch & Wannier orbitals exist.
![Page 85: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/85.jpg)
Model Hückel-like Hamiltonian in 2d(Haldane, PRL 1988)
Honeycomb lattice. Tight-binding with first- andsecond-neighbor hopping: t1 real, t2 complex.It breaks time-reversal symmetry.Two nonequivalent sites per cell. Insulator at half-filling(only the lowest band occupied).Zero flux through the unit cell:The macroscopic B field is zero;The Hamiltonian is lattice-periodical;Bloch & Wannier orbitals exist.
![Page 86: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/86.jpg)
Formula assessed via computer experiments(2d, single occupancy, single band, atomic units)
(A) Periodic “bulk” system:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉
(B) Finite system of area L2 cut from the bulk(so-called “open” boundary conditions)
M =1
2cL2
∫dr r× j(r) =
12cL2
∑n∈ occupied
〈ϕn|r× v|ϕn〉
(A) numerically evaluated on a dense k-point mesh;(B) evaluated for large L values (up to 2048 sites).Do they converge to the same limit?
![Page 87: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/87.jpg)
Formula assessed via computer experiments(2d, single occupancy, single band, atomic units)
(A) Periodic “bulk” system:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉
(B) Finite system of area L2 cut from the bulk(so-called “open” boundary conditions)
M =1
2cL2
∫dr r× j(r) =
12cL2
∑n∈ occupied
〈ϕn|r× v|ϕn〉
(A) numerically evaluated on a dense k-point mesh;(B) evaluated for large L values (up to 2048 sites).Do they converge to the same limit?
![Page 88: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/88.jpg)
Formula assessed via computer experiments(2d, single occupancy, single band, atomic units)
(A) Periodic “bulk” system:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉
(B) Finite system of area L2 cut from the bulk(so-called “open” boundary conditions)
M =1
2cL2
∫dr r× j(r) =
12cL2
∑n∈ occupied
〈ϕn|r× v|ϕn〉
(A) numerically evaluated on a dense k-point mesh;(B) evaluated for large L values (up to 2048 sites).Do they converge to the same limit?
![Page 89: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/89.jpg)
Formula assessed via computer experiments(2d, single occupancy, single band, atomic units)
(A) Periodic “bulk” system:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉
(B) Finite system of area L2 cut from the bulk(so-called “open” boundary conditions)
M =1
2cL2
∫dr r× j(r) =
12cL2
∑n∈ occupied
〈ϕn|r× v|ϕn〉
(A) numerically evaluated on a dense k-point mesh;(B) evaluated for large L values (up to 2048 sites).Do they converge to the same limit?
![Page 90: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/90.jpg)
Formula assessed via computer experiments(2d, single occupancy, single band, atomic units)
(A) Periodic “bulk” system:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉
(B) Finite system of area L2 cut from the bulk(so-called “open” boundary conditions)
M =1
2cL2
∫dr r× j(r) =
12cL2
∑n∈ occupied
〈ϕn|r× v|ϕn〉
(A) numerically evaluated on a dense k-point mesh;(B) evaluated for large L values (up to 2048 sites).Do they converge to the same limit?
![Page 91: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/91.jpg)
Formula assessed via computer experiments(2d, single occupancy, single band, atomic units)
(A) Periodic “bulk” system:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉
(B) Finite system of area L2 cut from the bulk(so-called “open” boundary conditions)
M =1
2cL2
∫dr r× j(r) =
12cL2
∑n∈ occupied
〈ϕn|r× v|ϕn〉
(A) numerically evaluated on a dense k-point mesh;(B) evaluated for large L values (up to 2048 sites).Do they converge to the same limit?
![Page 92: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/92.jpg)
Formula assessed via computer experiments(2d, single occupancy, single band, atomic units)
(A) Periodic “bulk” system:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉
(B) Finite system of area L2 cut from the bulk(so-called “open” boundary conditions)
M =1
2cL2
∫dr r× j(r) =
12cL2
∑n∈ occupied
〈ϕn|r× v|ϕn〉
(A) numerically evaluated on a dense k-point mesh;(B) evaluated for large L values (up to 2048 sites).Do they converge to the same limit?
![Page 93: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/93.jpg)
Chasing the missing term:Localized-orbital (Boys/Wannier) analysis
The Boys/Wannier localized orbitals at the sample edgecarry a net current and contribute to M.However, even the additional edge contribution can becomputed using Bloch states and PBCs, where the systemhas no edge.
![Page 94: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/94.jpg)
Chasing the missing term:Localized-orbital (Boys/Wannier) analysis
The Boys/Wannier localized orbitals at the sample edgecarry a net current and contribute to M.However, even the additional edge contribution can becomputed using Bloch states and PBCs, where the systemhas no edge.
![Page 95: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/95.jpg)
Chasing the missing term:Localized-orbital (Boys/Wannier) analysis
The Boys/Wannier localized orbitals at the sample edgecarry a net current and contribute to M.However, even the additional edge contribution can becomputed using Bloch states and PBCs, where the systemhas no edge.
![Page 96: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/96.jpg)
Chasing the missing term:Localized-orbital (Boys/Wannier) analysis
The Boys/Wannier localized orbitals at the sample edgecarry a net current and contribute to M.However, even the additional edge contribution can becomputed using Bloch states and PBCs, where the systemhas no edge.
![Page 97: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/97.jpg)
Local circulation & itinerant circulation (2d)
Magnetization that originates from the circulation of theBoys/Wannier orbitals (same formula as before):
MLC = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉.
Magnetization that originates from the net current carriedby the Boys/Wannier orbitals (nonvanishing only for edgeorbitals):
MIC = − i2c(2π)2
∫BZ
dk ε(k)〈∂kuk| × |∂kuk〉.
The final magnetization formula:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.
Proof: both numerical and analytical
![Page 98: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/98.jpg)
Local circulation & itinerant circulation (2d)
Magnetization that originates from the circulation of theBoys/Wannier orbitals (same formula as before):
MLC = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉.
Magnetization that originates from the net current carriedby the Boys/Wannier orbitals (nonvanishing only for edgeorbitals):
MIC = − i2c(2π)2
∫BZ
dk ε(k)〈∂kuk| × |∂kuk〉.
The final magnetization formula:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.
Proof: both numerical and analytical
![Page 99: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/99.jpg)
Local circulation & itinerant circulation (2d)
Magnetization that originates from the circulation of theBoys/Wannier orbitals (same formula as before):
MLC = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉.
Magnetization that originates from the net current carriedby the Boys/Wannier orbitals (nonvanishing only for edgeorbitals):
MIC = − i2c(2π)2
∫BZ
dk ε(k)〈∂kuk| × |∂kuk〉.
The final magnetization formula:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.
Proof: both numerical and analytical
![Page 100: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/100.jpg)
Local circulation & itinerant circulation (2d)
Magnetization that originates from the circulation of theBoys/Wannier orbitals (same formula as before):
MLC = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉.
Magnetization that originates from the net current carriedby the Boys/Wannier orbitals (nonvanishing only for edgeorbitals):
MIC = − i2c(2π)2
∫BZ
dk ε(k)〈∂kuk| × |∂kuk〉.
The final magnetization formula:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.
Proof: both numerical and analytical
![Page 101: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/101.jpg)
Local circulation & itinerant circulation (2d)
Magnetization that originates from the circulation of theBoys/Wannier orbitals (same formula as before):
MLC = − i2c(2π)2
∫BZ
dk 〈∂kuk| × H(k)|∂kuk〉.
Magnetization that originates from the net current carriedby the Boys/Wannier orbitals (nonvanishing only for edgeorbitals):
MIC = − i2c(2π)2
∫BZ
dk ε(k)〈∂kuk| × |∂kuk〉.
The final magnetization formula:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.
Proof: both numerical and analytical
![Page 102: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/102.jpg)
An apparent drawback
The final magnetization formula:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.
If we change the energy zero by ∆ε, the magnetizationapparently changes by
∆M = − i2∆ε
2c(2π)2
∫BZ
dk 〈∂kuk| × |∂kuk〉.
ButC1 =
i(2π)2
∫BZ
dk 〈∂kuk| × |∂kuk〉
is the Chern number, a topological integer.C1 = 0 for “normal” insulators. C1 6= 0 only in rather exoticcases (such as the quantum Hall regime).
![Page 103: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/103.jpg)
An apparent drawback
The final magnetization formula:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.
If we change the energy zero by ∆ε, the magnetizationapparently changes by
∆M = − i2∆ε
2c(2π)2
∫BZ
dk 〈∂kuk| × |∂kuk〉.
ButC1 =
i(2π)2
∫BZ
dk 〈∂kuk| × |∂kuk〉
is the Chern number, a topological integer.C1 = 0 for “normal” insulators. C1 6= 0 only in rather exoticcases (such as the quantum Hall regime).
![Page 104: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/104.jpg)
An apparent drawback
The final magnetization formula:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.
If we change the energy zero by ∆ε, the magnetizationapparently changes by
∆M = − i2∆ε
2c(2π)2
∫BZ
dk 〈∂kuk| × |∂kuk〉.
ButC1 =
i(2π)2
∫BZ
dk 〈∂kuk| × |∂kuk〉
is the Chern number, a topological integer.C1 = 0 for “normal” insulators. C1 6= 0 only in rather exoticcases (such as the quantum Hall regime).
![Page 105: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/105.jpg)
An apparent drawback
The final magnetization formula:
M = − i2c(2π)2
∫BZ
dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.
If we change the energy zero by ∆ε, the magnetizationapparently changes by
∆M = − i2∆ε
2c(2π)2
∫BZ
dk 〈∂kuk| × |∂kuk〉.
ButC1 =
i(2π)2
∫BZ
dk 〈∂kuk| × |∂kuk〉
is the Chern number, a topological integer.C1 = 0 for “normal” insulators. C1 6= 0 only in rather exoticcases (such as the quantum Hall regime).
![Page 106: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/106.jpg)
Insulators, metals, and more(Back to many bands, 3d, double occupancy)
Magnetization in a “normal” insulator:
M = − ie~c(2π)3
∑n∈ occupied
∫BZ
dk 〈∂kunk|×[H(k)+ε(k) ]|∂kunk〉
Magnetization in a metal:
M = − ie~c(2π)3
∑n
∫εn(k)<µ
dk 〈∂kunk|×[H(k)+ε(k)−2µ] |∂kunk〉
The same formula holds in insulators with C1 6= 0.
![Page 107: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/107.jpg)
Insulators, metals, and more(Back to many bands, 3d, double occupancy)
Magnetization in a “normal” insulator:
M = − ie~c(2π)3
∑n∈ occupied
∫BZ
dk 〈∂kunk|×[H(k)+ε(k) ]|∂kunk〉
Magnetization in a metal:
M = − ie~c(2π)3
∑n
∫εn(k)<µ
dk 〈∂kunk|×[H(k)+ε(k)−2µ] |∂kunk〉
The same formula holds in insulators with C1 6= 0.
![Page 108: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/108.jpg)
Insulators, metals, and more(Back to many bands, 3d, double occupancy)
Magnetization in a “normal” insulator:
M = − ie~c(2π)3
∑n∈ occupied
∫BZ
dk 〈∂kunk|×[H(k)+ε(k) ]|∂kunk〉
Magnetization in a metal:
M = − ie~c(2π)3
∑n
∫εn(k)<µ
dk 〈∂kunk|×[H(k)+ε(k)−2µ] |∂kunk〉
The same formula holds in insulators with C1 6= 0.
![Page 109: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/109.jpg)
Discretization of the BZ integral
The BZ integral is discretized on a regular grid in k-space.
A smooth phase choice is irrelevant for evaluating |∂kunk〉.Band crossings are similarly irrelevant.(Same as in discretizing the Berry-phase polarization).
We also have the single k-point formula for noncrystallinesystems in a supercell framework (e.g. for Car-Parrinellosimulations).
![Page 110: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/110.jpg)
Discretization of the BZ integral
The BZ integral is discretized on a regular grid in k-space.
A smooth phase choice is irrelevant for evaluating |∂kunk〉.Band crossings are similarly irrelevant.(Same as in discretizing the Berry-phase polarization).
We also have the single k-point formula for noncrystallinesystems in a supercell framework (e.g. for Car-Parrinellosimulations).
![Page 111: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/111.jpg)
Discretization of the BZ integral
The BZ integral is discretized on a regular grid in k-space.
A smooth phase choice is irrelevant for evaluating |∂kunk〉.Band crossings are similarly irrelevant.(Same as in discretizing the Berry-phase polarization).
We also have the single k-point formula for noncrystallinesystems in a supercell framework (e.g. for Car-Parrinellosimulations).
![Page 112: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/112.jpg)
Discretization of the BZ integral
The BZ integral is discretized on a regular grid in k-space.
A smooth phase choice is irrelevant for evaluating |∂kunk〉.Band crossings are similarly irrelevant.(Same as in discretizing the Berry-phase polarization).
We also have the single k-point formula for noncrystallinesystems in a supercell framework (e.g. for Car-Parrinellosimulations).
![Page 113: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/113.jpg)
Outline
1 Why do we need a kind of “exotic” theory?The dipole of a finite moleculeThe “dipole” of a solid
2 Role of macroscopic fields
3 Modern theory of polarization: main features
4 Modern theory of orbital magnetization
5 Application: NMR shielding tensor
6 Results
![Page 114: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/114.jpg)
Definition: The NMR shielding tensor
An external magnetic field Bext is applied to a finitesample.The field induces an orbital current: the total (shielded)field inside the sample is B(r) = Bext + Bind(r).Notice: B(r) depends on sample shape.At nuclear site r = rs (to linear order):
Binds = −←→σ s · Bext , Bs = (1−←→σ s) · Bext
1−←→σ s =∂Bs
∂Bext
The tensor←→σ s is the quantity actually measured.
![Page 115: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/115.jpg)
Definition: The NMR shielding tensor
An external magnetic field Bext is applied to a finitesample.The field induces an orbital current: the total (shielded)field inside the sample is B(r) = Bext + Bind(r).Notice: B(r) depends on sample shape.At nuclear site r = rs (to linear order):
Binds = −←→σ s · Bext , Bs = (1−←→σ s) · Bext
1−←→σ s =∂Bs
∂Bext
The tensor←→σ s is the quantity actually measured.
![Page 116: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/116.jpg)
Definition: The NMR shielding tensor
An external magnetic field Bext is applied to a finitesample.The field induces an orbital current: the total (shielded)field inside the sample is B(r) = Bext + Bind(r).Notice: B(r) depends on sample shape.At nuclear site r = rs (to linear order):
Binds = −←→σ s · Bext , Bs = (1−←→σ s) · Bext
1−←→σ s =∂Bs
∂Bext
The tensor←→σ s is the quantity actually measured.
![Page 117: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/117.jpg)
Definition: The NMR shielding tensor
An external magnetic field Bext is applied to a finitesample.The field induces an orbital current: the total (shielded)field inside the sample is B(r) = Bext + Bind(r).Notice: B(r) depends on sample shape.At nuclear site r = rs (to linear order):
Binds = −←→σ s · Bext , Bs = (1−←→σ s) · Bext
1−←→σ s =∂Bs
∂Bext
The tensor←→σ s is the quantity actually measured.
![Page 118: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/118.jpg)
Definition: The NMR shielding tensor
An external magnetic field Bext is applied to a finitesample.The field induces an orbital current: the total (shielded)field inside the sample is B(r) = Bext + Bind(r).Notice: B(r) depends on sample shape.At nuclear site r = rs (to linear order):
Binds = −←→σ s · Bext , Bs = (1−←→σ s) · Bext
1−←→σ s =∂Bs
∂Bext
The tensor←→σ s is the quantity actually measured.
![Page 119: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/119.jpg)
Definition: The NMR shielding tensor
An external magnetic field Bext is applied to a finitesample.The field induces an orbital current: the total (shielded)field inside the sample is B(r) = Bext + Bind(r).Notice: B(r) depends on sample shape.At nuclear site r = rs (to linear order):
Binds = −←→σ s · Bext , Bs = (1−←→σ s) · Bext
1−←→σ s =∂Bs
∂Bext
The tensor←→σ s is the quantity actually measured.
![Page 120: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/120.jpg)
Definition: The NMR shielding tensor
An external magnetic field Bext is applied to a finitesample.The field induces an orbital current: the total (shielded)field inside the sample is B(r) = Bext + Bind(r).Notice: B(r) depends on sample shape.At nuclear site r = rs (to linear order):
Binds = −←→σ s · Bext , Bs = (1−←→σ s) · Bext
1−←→σ s =∂Bs
∂Bext
The tensor←→σ s is the quantity actually measured.
![Page 121: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/121.jpg)
Shape dependence
1−←→σ s =∂Bs
∂Bext
Suppose we neglect the macroscopic induced field, thusidentifying the macroscopic total B field inside the materialwith the external one Bext. Then
1−←→σ s =∂Bs
∂B
This is exact for a sample in the form of a slab, and Bext
normal to the slab.For other sample shapes, there is a (small) correction.For a spherical sample←→σ sphere
s ' ←→σ s − (8π/3)χ.
![Page 122: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/122.jpg)
Shape dependence
1−←→σ s =∂Bs
∂Bext
Suppose we neglect the macroscopic induced field, thusidentifying the macroscopic total B field inside the materialwith the external one Bext. Then
1−←→σ s =∂Bs
∂B
This is exact for a sample in the form of a slab, and Bext
normal to the slab.For other sample shapes, there is a (small) correction.For a spherical sample←→σ sphere
s ' ←→σ s − (8π/3)χ.
![Page 123: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/123.jpg)
Shape dependence
1−←→σ s =∂Bs
∂Bext
Suppose we neglect the macroscopic induced field, thusidentifying the macroscopic total B field inside the materialwith the external one Bext. Then
1−←→σ s =∂Bs
∂B
This is exact for a sample in the form of a slab, and Bext
normal to the slab.For other sample shapes, there is a (small) correction.For a spherical sample←→σ sphere
s ' ←→σ s − (8π/3)χ.
![Page 124: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/124.jpg)
Shape dependence
1−←→σ s =∂Bs
∂Bext
Suppose we neglect the macroscopic induced field, thusidentifying the macroscopic total B field inside the materialwith the external one Bext. Then
1−←→σ s =∂Bs
∂B
This is exact for a sample in the form of a slab, and Bext
normal to the slab.For other sample shapes, there is a (small) correction.For a spherical sample←→σ sphere
s ' ←→σ s − (8π/3)χ.
![Page 125: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/125.jpg)
Shape dependence
1−←→σ s =∂Bs
∂Bext
Suppose we neglect the macroscopic induced field, thusidentifying the macroscopic total B field inside the materialwith the external one Bext. Then
1−←→σ s =∂Bs
∂B
This is exact for a sample in the form of a slab, and Bext
normal to the slab.For other sample shapes, there is a (small) correction.For a spherical sample←→σ sphere
s ' ←→σ s − (8π/3)χ.
![Page 126: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/126.jpg)
Shape dependence
1−←→σ s =∂Bs
∂Bext
Suppose we neglect the macroscopic induced field, thusidentifying the macroscopic total B field inside the materialwith the external one Bext. Then
1−←→σ s =∂Bs
∂B
This is exact for a sample in the form of a slab, and Bext
normal to the slab.For other sample shapes, there is a (small) correction.For a spherical sample←→σ sphere
s ' ←→σ s − (8π/3)χ.
![Page 127: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/127.jpg)
Computations: (1) The “direct” approach(F. Mauri & coworkers)
The only viable approach so far for crystalline systems.
1−←→σ s =∂Bs
∂BEvaluated via linear-response theory.Finite-difference approach impossible: the crystallineeigenfunctions in presence of a finite B field cannot beevaluated.
![Page 128: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/128.jpg)
Computations: (1) The “direct” approach(F. Mauri & coworkers)
The only viable approach so far for crystalline systems.
1−←→σ s =∂Bs
∂BEvaluated via linear-response theory.Finite-difference approach impossible: the crystallineeigenfunctions in presence of a finite B field cannot beevaluated.
![Page 129: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/129.jpg)
Computations: (1) The “direct” approach(F. Mauri & coworkers)
The only viable approach so far for crystalline systems.
1−←→σ s =∂Bs
∂BEvaluated via linear-response theory.Finite-difference approach impossible: the crystallineeigenfunctions in presence of a finite B field cannot beevaluated.
![Page 130: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/130.jpg)
Computations: (1) The “direct” approach(F. Mauri & coworkers)
The only viable approach so far for crystalline systems.
1−←→σ s =∂Bs
∂BEvaluated via linear-response theory.Finite-difference approach impossible: the crystallineeigenfunctions in presence of a finite B field cannot beevaluated.
![Page 131: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/131.jpg)
Computations: (2) Our “converse” approach(Thonhauser, Mostofi, Marzari, Resta, & Vanderbilt, arXiv.org)
Exploits the modern theory of orbital magnetization.
It has an exact electrical analogue, routinely used tocompute Born effective charges (for lattice dynamics) byexploiting the modern theory of polarization (Berry phase).
![Page 132: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/132.jpg)
Computations: (2) Our “converse” approach(Thonhauser, Mostofi, Marzari, Resta, & Vanderbilt, arXiv.org)
Exploits the modern theory of orbital magnetization.
It has an exact electrical analogue, routinely used tocompute Born effective charges (for lattice dynamics) byexploiting the modern theory of polarization (Berry phase).
![Page 133: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/133.jpg)
The “converse” approach: main concept
1−←→σ s =∂Bs
∂B
Bs can be ideally measured via the torque acting on aclassical magnetic (point) dipole at site rs:
Bs = − ∂E∂ms
E is the energy per cell of a periodic lattice of such dipoles(one per cell) in a macroscopic field B.
1−←→σ s = − ∂
∂B∂E∂ms
= − ∂
∂ms
∂E∂B
![Page 134: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/134.jpg)
The “converse” approach: main concept
1−←→σ s =∂Bs
∂B
Bs can be ideally measured via the torque acting on aclassical magnetic (point) dipole at site rs:
Bs = − ∂E∂ms
E is the energy per cell of a periodic lattice of such dipoles(one per cell) in a macroscopic field B.
1−←→σ s = − ∂
∂B∂E∂ms
= − ∂
∂ms
∂E∂B
![Page 135: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/135.jpg)
The “converse” approach: main concept
1−←→σ s =∂Bs
∂B
Bs can be ideally measured via the torque acting on aclassical magnetic (point) dipole at site rs:
Bs = − ∂E∂ms
E is the energy per cell of a periodic lattice of such dipoles(one per cell) in a macroscopic field B.
1−←→σ s = − ∂
∂B∂E∂ms
= − ∂
∂ms
∂E∂B
![Page 136: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/136.jpg)
The “converse” approach: main concept
1−←→σ s =∂Bs
∂B
Bs can be ideally measured via the torque acting on aclassical magnetic (point) dipole at site rs:
Bs = − ∂E∂ms
E is the energy per cell of a periodic lattice of such dipoles(one per cell) in a macroscopic field B.
1−←→σ s = − ∂
∂B∂E∂ms
= − ∂
∂ms
∂E∂B
![Page 137: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/137.jpg)
The “converse” approach: main concept
1−←→σ s =∂Bs
∂B
Bs can be ideally measured via the torque acting on aclassical magnetic (point) dipole at site rs:
Bs = − ∂E∂ms
E is the energy per cell of a periodic lattice of such dipoles(one per cell) in a macroscopic field B.
1−←→σ s = − ∂
∂B∂E∂ms
= − ∂
∂ms
∂E∂B
![Page 138: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/138.jpg)
The “converse” approach: main concept
1−←→σ s =∂Bs
∂B
Bs can be ideally measured via the torque acting on aclassical magnetic (point) dipole at site rs:
Bs = − ∂E∂ms
E is the energy per cell of a periodic lattice of such dipoles(one per cell) in a macroscopic field B.
1−←→σ s = − ∂
∂B∂E∂ms
= − ∂
∂ms
∂E∂B
![Page 139: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/139.jpg)
The “converse” approach, cont’d
M = − 1Vcell
∂E∂B
1−←→σ s = − ∂
∂ms
∂E∂B
= Vcell∂M∂ms
.
In words:1−←→σ s is the macroscopic orbital magnetization linearlyinduced by a classical point dipole at rs and its periodicreplicas.Computations by finite differences, switching on the msperturbation and evaluating the induced macroscopicmagnetization M.If we “switch off” the electronic response, then∂M/∂ms = 1/Vcell, as it must be.
![Page 140: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/140.jpg)
The “converse” approach, cont’d
M = − 1Vcell
∂E∂B
1−←→σ s = − ∂
∂ms
∂E∂B
= Vcell∂M∂ms
.
In words:1−←→σ s is the macroscopic orbital magnetization linearlyinduced by a classical point dipole at rs and its periodicreplicas.Computations by finite differences, switching on the msperturbation and evaluating the induced macroscopicmagnetization M.If we “switch off” the electronic response, then∂M/∂ms = 1/Vcell, as it must be.
![Page 141: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/141.jpg)
The “converse” approach, cont’d
M = − 1Vcell
∂E∂B
1−←→σ s = − ∂
∂ms
∂E∂B
= Vcell∂M∂ms
.
In words:1−←→σ s is the macroscopic orbital magnetization linearlyinduced by a classical point dipole at rs and its periodicreplicas.Computations by finite differences, switching on the msperturbation and evaluating the induced macroscopicmagnetization M.If we “switch off” the electronic response, then∂M/∂ms = 1/Vcell, as it must be.
![Page 142: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/142.jpg)
The “converse” approach, cont’d
M = − 1Vcell
∂E∂B
1−←→σ s = − ∂
∂ms
∂E∂B
= Vcell∂M∂ms
.
In words:1−←→σ s is the macroscopic orbital magnetization linearlyinduced by a classical point dipole at rs and its periodicreplicas.Computations by finite differences, switching on the msperturbation and evaluating the induced macroscopicmagnetization M.If we “switch off” the electronic response, then∂M/∂ms = 1/Vcell, as it must be.
![Page 143: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/143.jpg)
The “converse” approach, cont’d
M = − 1Vcell
∂E∂B
1−←→σ s = − ∂
∂ms
∂E∂B
= Vcell∂M∂ms
.
In words:1−←→σ s is the macroscopic orbital magnetization linearlyinduced by a classical point dipole at rs and its periodicreplicas.Computations by finite differences, switching on the msperturbation and evaluating the induced macroscopicmagnetization M.If we “switch off” the electronic response, then∂M/∂ms = 1/Vcell, as it must be.
![Page 144: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/144.jpg)
The “converse” approach, cont’d
M = − 1Vcell
∂E∂B
1−←→σ s = − ∂
∂ms
∂E∂B
= Vcell∂M∂ms
.
In words:1−←→σ s is the macroscopic orbital magnetization linearlyinduced by a classical point dipole at rs and its periodicreplicas.Computations by finite differences, switching on the msperturbation and evaluating the induced macroscopicmagnetization M.If we “switch off” the electronic response, then∂M/∂ms = 1/Vcell, as it must be.
![Page 145: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/145.jpg)
Outline
1 Why do we need a kind of “exotic” theory?The dipole of a finite moleculeThe “dipole” of a solid
2 Role of macroscopic fields
3 Modern theory of polarization: main features
4 Modern theory of orbital magnetization
5 Application: NMR shielding tensor
6 Results
![Page 146: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/146.jpg)
NMR shielding tensor for H in selected molecules
experiment direct converseH2 26.26 26.2 26.2HF 28.51 28.4 28.5CH4 30.61 30.8 31.0C2H2 29.26 28.8 28.9C2H4 25.43 24.7 24.8C2H6 29.86 30.2 30.4
Hydrogen NMR chemical shielding σ, in ppm, for severaldifferent molecules.
Pseudopotential PW calculations in a large supercell.
Core contribution added according to the theory of Pickard &Mauri (2003).
![Page 147: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/147.jpg)
NMR shielding tensor for H in selected molecules
experiment direct converseH2 26.26 26.2 26.2HF 28.51 28.4 28.5CH4 30.61 30.8 31.0C2H2 29.26 28.8 28.9C2H4 25.43 24.7 24.8C2H6 29.86 30.2 30.4
Hydrogen NMR chemical shielding σ, in ppm, for severaldifferent molecules.
Pseudopotential PW calculations in a large supercell.
Core contribution added according to the theory of Pickard &Mauri (2003).
![Page 148: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/148.jpg)
NMR shielding tensor for H in liquid water
Five snapshots, 64 molecule-supercell:average over 640 H atoms.
Average and spread very similar to what previously found withthe direct method (and smaller supercells).
![Page 149: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/149.jpg)
NMR shielding tensor for H in liquid water
Five snapshots, 64 molecule-supercell:average over 640 H atoms.
Average and spread very similar to what previously found withthe direct method (and smaller supercells).
![Page 150: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/150.jpg)
Last but not least:Orbital Magnetization in Ferromagnetic MetalsD. Ceresoli, U. Gertsmann, A.P. Seitsonen, & F. Mauri
Still unpublished: http://arXiv:0904.1988
![Page 151: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/151.jpg)
Conclusions
We have a formula for the macroscopic magnetization Min a crystalline solid (either insulating or metallic).It is the magnetic analogue of the well establishedBerry-phase formula for the electric polarization P (ininsulators).Both formulæ apply to the case where the macroscopicfield (E or B) is zero.The very first ab-initio implementations are appearingthese days.I’m not satisfied with the existing analytical proof for themetallic case (but computer simulations on modelHamiltonians are a robust numerical proof).
![Page 152: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/152.jpg)
Conclusions
We have a formula for the macroscopic magnetization Min a crystalline solid (either insulating or metallic).It is the magnetic analogue of the well establishedBerry-phase formula for the electric polarization P (ininsulators).Both formulæ apply to the case where the macroscopicfield (E or B) is zero.The very first ab-initio implementations are appearingthese days.I’m not satisfied with the existing analytical proof for themetallic case (but computer simulations on modelHamiltonians are a robust numerical proof).
![Page 153: Orbital Magnetization in Periodic Systems - Crystal · Orbital Magnetization in Periodic Systems Raffaele Resta Dipartimento di Fisica Teorica, Università di Trieste, and CNR-INFM](https://reader030.fdocuments.in/reader030/viewer/2022021611/5c666c4109d3f2d12a8c481e/html5/thumbnails/153.jpg)
Conclusions
We have a formula for the macroscopic magnetization Min a crystalline solid (either insulating or metallic).It is the magnetic analogue of the well establishedBerry-phase formula for the electric polarization P (ininsulators).Both formulæ apply to the case where the macroscopicfield (E or B) is zero.The very first ab-initio implementations are appearingthese days.I’m not satisfied with the existing analytical proof for themetallic case (but computer simulations on modelHamiltonians are a robust numerical proof).