orbital magnetization in periodic systems · modern theory of orbital magnetization theory started...

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

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...

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.

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

Outline

6 Results

From Feynman Lectures in Physics, Vol. 2

TEXTBOOK DEFINITIONSARE NONCOMPUTABLE!

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

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

Outline

6 Results

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.

Trivial definitions(atomic units througout)

m =12c

d = −∫

dr n(r) + dnuclear

d is nonzero only in absence of inversion symmetry.m is nonzero only in absence of time-reversal symmetry.

Trivial definitions (cont’d)

m =12c

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)

m =12c

d = −∫

dr rn(r) + dnuclear

m =12c

d = −∫

dr rn(r) + dnuclear

m =12c

d = −∫

dr rn(r) + dnuclear

m =12c

d = −∫

dr rn(r) + dnuclear

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

m = 212c

∑n∈ occupied

〈ϕn|r× v|ϕn〉

d = −2∑

∑i ∈ nuclei

ZiRi

m = 212c

∑n∈ occupied

〈wn|r× v|wn〉

d = −2∑

∑i ∈ nuclei

ZiRi

m = 212c

∑n∈ occupied

〈ϕn|r× v|ϕn〉

d = −2∑

∑i ∈ nuclei

ZiRi

m = 212c

∑n∈ occupied

〈wn|r× v|wn〉

d = −2∑

∑i ∈ nuclei

ZiRi

Outline

6 Results

Macroscopic magnetization and polarization(naive textbook view)

M =mV

=1

2cV

∫dr r× j(r)

P =dV

=1V

(−∫

dr r n(r) +∑

i ∈ nucleiZiRi

)

Macroscopic magnetization and polarization(naive textbook view)

M =mV

=1

2cV

∫dr r× j(r)

P =dV

=1V

(−∫

dr r n(r) +∑

i ∈ nucleiZiRi

)

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.

M =mV

=1

2cV

∫dr r× j(r)

Pelectronic =dV

= − 1V

∫dr r n(r)

M =mV

=1

2cV

∫dr r× j(r)

Pelectronic =dV

= − 1V

∫dr r n(r)

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.

Outline

6 Results

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.

Molecules

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.

Solids

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.

Outline

6 Results

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 = −2e

Vcell

∑n∈ occupied

〈wn|r|wn〉

(2π)3∑

n∈ occupied

∫BZ

Pelectronic = −2e

Vcell

∑n∈ occupied

〈wn|r|wn〉

(2π)3∑

n∈ occupied

∫BZ

Pelectronic = −2e

Vcell

∑n∈ occupied

〈wn|r|wn〉

(2π)3∑

n∈ occupied

∫BZ

Pelectronic = −2e

Vcell

∑n∈ occupied

〈wn|r|wn〉

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.

Key points

Outline

6 Results

Heuristically, by analogy with the electrical case

For an insulator, in absence of inversion symmetry, in zeroE field, we have

Pelectronic = −2

Vcell

∑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.

Pelectronic = −2

Vcell

∑n∈ occupied

〈wn|r|wn〉

M = − 22cVcell

∑n∈ occupied

〈wn|r× v|wn〉

Pelectronic = −2

Vcell

∑n∈ occupied

〈wn|r|wn〉

M = − 22cVcell

∑n∈ occupied

〈wn|r× v|wn〉

Pelectronic = −2

Vcell

∑n∈ occupied

〈wn|r|wn〉

M = − 22cVcell

∑n∈ occupied

〈wn|r× v|wn〉

Pelectronic = −2

Vcell

∑n∈ occupied

〈wn|r|wn〉

M = − 22cVcell

∑n∈ occupied

〈wn|r× v|wn〉

Pelectronic = −2

Vcell

∑n∈ occupied

〈wn|r|wn〉

M = − 22cVcell

∑n∈ occupied

〈wn|r× v|wn〉

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

M = − 1cVcell

∑n∈ occupied

〈wn|r× v|wn〉

= − ie~cVcell

∑n∈ occupied

〈wn|r× Hr|wn〉

= − ie~c(2π)3

∑n∈ occupied

∫BZ

Pelectronic = −2i

(2π)3∑

n∈ occupied

∫BZ

M = − 1cVcell

∑n∈ occupied

〈wn|r× v|wn〉

= − ie~cVcell

∑n∈ occupied

〈wn|r× Hr|wn〉

= − ie~c(2π)3

∑n∈ occupied

∫BZ

Pelectronic = −2i

(2π)3∑

n∈ occupied

∫BZ

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.

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) = 1

2cL2∑

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?

M = − i2c(2π)2

∫BZ

M =1

2cL2

∫dr r× j(r) = 1

2cL2∑

M = − i2c(2π)2

∫BZ

M =1

2cL2

∫dr r× j(r) = 1

2cL2∑

M = − i2c(2π)2

∫BZ

M =1

2cL2

∫dr r× j(r) = 1

2cL2∑

M = − i2c(2π)2

∫BZ

M =1

2cL2

∫dr r× j(r) = 1

2cL2∑

M = − i2c(2π)2

∫BZ

M =1

2cL2

∫dr r× j(r) = 1

2cL2∑

M = − i2c(2π)2

∫BZ

M =1

2cL2

∫dr r× j(r) = 1

2cL2∑

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.

Local circulation & itinerant circulation (2d)

Magnetization that originates from the circulation of theBoys/Wannier orbitals (same formula as before):

MLC = −i

2c(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 = −i

2c(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

MLC = −i

2c(2π)2

∫BZ

MIC = −i

2c(2π)2

∫BZ

M = − i2c(2π)2

∫BZ

dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.

MLC = −i

2c(2π)2

∫BZ

MIC = −i

2c(2π)2

∫BZ

M = − i2c(2π)2

∫BZ

dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.

MLC = −i

2c(2π)2

∫BZ

MIC = −i

2c(2π)2

∫BZ

M = − i2c(2π)2

∫BZ

dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.

MLC = −i

2c(2π)2

∫BZ

MIC = −i

2c(2π)2

∫BZ

M = − i2c(2π)2

∫BZ

dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.

An apparent drawback

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∆ε2c(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).

M = − i2c(2π)2

∫BZ

dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.

∆M = − i2∆ε2c(2π)2

∫BZ

ButC1 =

i(2π)2

∫BZ

M = − i2c(2π)2

∫BZ

dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.

∆M = − i2∆ε2c(2π)2

∫BZ

ButC1 =

i(2π)2

∫BZ

M = − i2c(2π)2

∫BZ

dk 〈∂kuk| × [H(k) + ε(k)] |∂kuk〉.

∆M = − i2∆ε2c(2π)2

∫BZ

ButC1 =

i(2π)2

∫BZ

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)

M = − ie~c(2π)3

∑n∈ occupied

∫BZ

M = − ie~c(2π)3

∑n

∫εn(k)

M = − ie~c(2π)3

∑n∈ occupied

∫BZ

M = − ie~c(2π)3

∑n

∫εn(k)

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).

Outline

6 Results

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.

Shape dependence

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 spheres ' ←→σ s − (8π/3)χ.

Shape dependence

1−←→σ s =∂Bs∂B

Shape dependence

1−←→σ s =∂Bs∂B

Shape dependence

1−←→σ s =∂Bs∂B

Shape dependence

1−←→σ s =∂Bs∂B

Shape dependence

1−←→σ s =∂Bs∂B

Computations: (1) The “direct” approach(F. Mauri & coworkers)

The only viable approach so far for crystalline systems.

1−←→σ s =∂Bs∂B

Evaluated via linear-response theory.Finite-difference approach impossible: the crystallineeigenfunctions in presence of a finite B field cannot beevaluated.

1−←→σ s =∂Bs∂B

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).

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).

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

1−←→σ s =∂Bs∂B

Bs = −∂E∂ms

1−←→σ s = −∂

∂B∂E∂ms

= − ∂∂ms

∂E∂B

1−←→σ s =∂Bs∂B

Bs = −∂E∂ms

1−←→σ s = −∂

∂B∂E∂ms

= − ∂∂ms

∂E∂B

1−←→σ s =∂Bs∂B

Bs = −∂E∂ms

1−←→σ s = −∂

∂B∂E∂ms

= − ∂∂ms

∂E∂B

1−←→σ s =∂Bs∂B

Bs = −∂E∂ms

1−←→σ s = −∂

∂B∂E∂ms

= − ∂∂ms

∂E∂B

1−←→σ s =∂Bs∂B

Bs = −∂E∂ms

1−←→σ s = −∂

∂B∂E∂ms

= − ∂∂ms

∂E∂B

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.

M = − 1Vcell

∂E∂B

1−←→σ s = −∂

∂ms∂E∂B

= Vcell∂M∂ms

.

M = − 1Vcell

∂E∂B

1−←→σ s = −∂

∂ms∂E∂B

= Vcell∂M∂ms

.

M = − 1Vcell

∂E∂B

1−←→σ s = −∂

∂ms∂E∂B

= Vcell∂M∂ms

.

M = − 1Vcell

∂E∂B

1−←→σ s = −∂

∂ms∂E∂B

= Vcell∂M∂ms

.

M = − 1Vcell

∂E∂B

1−←→σ s = −∂

∂ms∂E∂B

= Vcell∂M∂ms

.

Outline

6 Results

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).

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).

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).

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).

Last but not least:Orbital Magnetization in Ferromagnetic MetalsD. Ceresoli, U. Gertsmann, A.P. Seitsonen, & F. Mauri

Still unpublished: http://arXiv:0904.1988

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).

Conclusions

Why do we need a kind of ``exotic'' theory?The dipole of a finite moleculeThe ``dipole'' of a solid

Role of macroscopic fieldsModern theory of polarization: main featuresModern theory of orbital magnetizationApplication: NMR shielding tensorResults

orbital magnetization in periodic systems · modern theory of orbital magnetization theory started...

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