Simulations of magnetic systems : ferro / antiferromagnetic ......ferro / antiferromagnetic ordering and other issues SIESTA tutorial, Santander, June 9, 2010 Andrei Postnikov Outline

Simulations of magnetic systems :ferro / antiferromagnetic ordering and other issues

SIESTA tutorial, Santander, June 9, 2010

Andrei Postnikov

Outline

1 Basics: spin and orbital magnetism

2 Isolated and periodic systems in the context of magnetism

3 Different cases of magnetism in condensed matter

4 Magnetic calculation; how to initialize

5 Magnetic calculation; how to analyze results

6 Use of Fixed Spin Moment

7 Concluding remarks

A.Postnikov (Universite Metz) magnetic systems June 2010 2 / 22

Basics: spin and orbital magnetism

An isolated atom or ionI Orbital magnetic moment of an electron:

~µ =−e

2me

~L ; µ = − e~2me

√l(l + 1) ; µ(orb)

z = −ml µB ; µB =e~

2me.

I Spin magnetic moment:

s = 12 ; ms = ± 1

2 ; µ(spin)z = ± 1

2gµB (g≈ 2), and not 12µB as expected.

Many atoms (condensed matter)I Spin moment, with respect to global quantization axis, ... easy:

µ(spin)z [in µB] = N

ms= 12

el −Nms=− 12

el ≡ N↑ −N↓ .

I Orbital moment, with respect to site Rj ... kompliziert!

µ(orb; j)z =

∑n

∫dk

(2π)3∑i

eik(Rj−Ri)〈Wni |(r−Rj)×p|Wnj〉


Basics: spin and orbital magnetism

An isolated atom or ionI Orbital magnetic moment: µ

(orb)z =

∑occ.α

∑↑,↓

(−mα) ;

I Spin magnetic moment: µ(spin)z =

∑occ.α

(∑↑

1−∑↓

1).

Condensed matter:I Spin moment: µ

(spin)z [in µB] = N↑ −N↓.

I Orbital moment, with respect to site Rj , for example as:

µ(orb; j)z =

∑n

∫dk

(2π)3∑i

eik(Rj−Ri)〈Wni |(r−Rj)×p|Wnj〉 ,

with Ψnk(r)︸︷︷︸(Bloch function)

=1√Nsites

∑i

eikRi Wn(r−Ri)︸︷︷︸(Wannier function)

.


Starting with an isolated atom ...

... e.g., calculation of atom for pseudopotgeneration. The atomic configuration is ourchoice.

In practice of generating (transferrable) pseu-dopotentials, the configuration is rarely cho-sen magnetic. However, a good generatedpseudopotential should work within reasons-ble ranges of magnetic solutions as well.

In reality, the Hund’s rules usually enforcemagnetism on an isolated atom!

(sometimes forgotten for H, C, O, ...)


Spin and orbital moments of an atom: Hund’s rules

Apply to the ground state of a multi-electron atom:

1 For a given electron configuration, the term with maximummultiplicity (2S + 1) – or, equivalently, the term with maximum S,the spin angular momentum, – has the lowest energy.

2 For a given multiplicity, the term with the largest value of L(the orbital angular momentum) has the lowest energy.

3 For a given term, in an atom with outermost subshell half-filled orless, the level with the lowest value of J (total angular momentum,J = S + L) lies lowest in energy. If the outermost shell is more thanhalf-filled, the level with highest value of J is lowest in energy.

These rules apply in the LS-coupling regime: J = L + S; total spinangular moment S =

∑i

si; total orbital angular moment L =∑i

li.

si, li: spin and orbital momenta of electrons.


From atom to molecule or small cluster

atom molecule, cluster

N↑, N↓ are integer;Q = N↑ +N↓ and M = N↑ −N↓ as well.

Levels broadening

physical: band dispersion, lifetime of states

technical: ElectronicTemperature

Solving Kohn-Sham equationsfor non-periodic systems ⇒discrete energy eigenvalues,molecular orbitalsas eigenfunctions;eventuallymultiple magnetic solutions.


Magnetism in the Density Functional Theory

The Hohenberg–Kohn theorems do not immediately care about ↑, ↓; ρ(r)does not need to be spin-resolved.Kohn–Sham equations, with a sophisticated enough vxc(r), should be(hopefully?) able to describe magnetism somehow.[

− ~2

2m∇2 + vext(r) +

∫ρ(r′)|r− r′|

dr′ + vxc(r)]ψi(r) = εi ψi(r) ;

ρ(r) =∑i occ.

|ψi(r)|2 ; vxc(r) ≡ vxc[ρ(r)] .


Magnetism in the Density Functional Theory

Kohn–Sham equations + spin: allow explicit dependence on spin-resolvedorbitals / densities, for the sake of convenience and efficiency. Gives rise toLSDA etc.[

− ~2

2m∇2 + vext(r) +

∫ρ(r′)|r− r′|

dr′ + v↑xc(r)]ψ↑i (r) = ε↑i ψ

↑i (r) ;[

− ~2

2m∇2 + vext(r) +

∫ρ(r′)|r− r′|

dr′ + v↓xc(r)]ψ↓i (r) = ε↓i ψ

↓i (r) ;

ρ↑(r) =∑i occ.

∣∣∣ψ↑i (r)∣∣∣2; ρ↓(r) =

∑i occ.

∣∣∣ψ↓i (r)∣∣∣2; ρ(r) = ρ↑(r) + ρ↓(r) ;

vxc(r) ≡ vxc[ρ↑(r), ρ↓(r)] .

Computational effort doubles. The ↑, ↓ solutions are coupled via theircontribution to vxc and the total density → Coulomb potential.


Spin and orbital magnetism in practice

A formal evaluation of orbital moment, as discussed at the beginning,technically would not present a problem; however this would not makemuch sense without tracing its back effect onto spin variables,Kohn–Sham equations, etc.

To do everything right, the calculation must be relativistic, or at leastto take spin-orbit interaction into account. Situation in Siesta: anapproximated “private” version exists (J. Ferrer, S. Sanvito, ...) andwaits incorporation into the main trunk.

Beyond spin-orbit, the orbital moments (lifting of orbital degeneracy,etc.) make an issue for orbital-dependent potentials. Situation inSiesta: LDA+U is a (working) example thereof.

In many high-symmetry crystalline systems, the orbital moment isquenched, and can be safely neglected. In other cases, see above.


Atom, molecule, condensed matter...

atom molecule, cluster

N↑, N↓ are integer;Q = N↑ +N↓ and M = N↑ −N↓ as well.

Levels broadening

physical: band dispersion, lifetime of states

technical: ElectronicTemperature

condensed matter(solid, amorphous, liquid)

Partially occupied bandsin both spin channels:Q (per unit cell) is integer;a priori no such constrainton M .


Condensed matter, varieties of magnetism:

Ferromagnetic metal:no special constraintson Mtot

An antiferromagnet: Mtot = 0

local moments (in general, not integer anddefinition-dependent): M1 = −M2

N↑1 = N↓2 ; N↓1 = N↑2



Dielectric (or semiconductor):

N↑ and N↓(number of occupied bands)are integer ⇒Mtot: integer;Mloc: definition dependent

gap in one spinchannel, e.g.:N↑, number ofoccupied spin-up bands,is integer.

Semimetal:

There are bands crossingEF, as in a metal, BUTsinceN↑ +N↓ = Q is integer,N↓ MUST be integer,hence:M = N↑ −N↓ is integer.



gap in one spinchannel, e.g.:N↑, number ofoccupied spin-up bands,is integer.

Semimetal:

There are bands crossingEF, as in a metal, BUTsinceN↑ +N↓ = Q is integer,N↓ MUST be integer,hence:M = N↑ −N↓ is integer.

All this is true only in the conventional (collinear) spin-polarized calculation,with ↑ and ↓ channels treated distinctly.

Non-collinear magnetism (allowed in Siesta) and/or

Spin-orbit coupling (included in “private” modifications)

break these rules about M↑, M↓.


A word on non-collinearity (NC)

A way to go beyond limitations of single-determinant approachcombined with unique spin quantization axis. In practical terms, itallows mixing of spin-↑ and spin-↓ states. With this, spin-resolvedband structures merge into a unique combined one. [One can tracethe fraction of ↑ or ↓ in each E(k), though].

This paradigm is quite “condensed matter-like” and has some of itsroots in the spin-wave theory. Quantum chemistry (“molecular-like”)has a different paradigm (mixing determinants).

The NC option exists in Siesta from its early days. Practicaldifficulties: memory and time demanding (as the matric size doubles),difficult to converge. Many mestastable solutions possible.

In principle, NC must be treated at the same time and on the samefooting with spin-orbit interaction: they both mix spin-↑ and spin-↓.However, this is not always the case (also in other codes).


Magnetic calculation: how to initialize

(Always needed):

SpinPolarized T

If nothing else defined, ALL atoms will be inilialized to maximal spin,that is not always desired. For maintaining full control, use e.g.

%block DM.InitSpin2 + # Atom Nr.2 initialized with max.spin UP5 - # Atom Nr.5 initialized with max.spin DOWN

%endblock DM.InitSpin– include here only those atoms which have to be magnetic; the restwill be initialized with zero spin

Yet another option: no block DM.InitSpin, but adding

DM.InitSpinAFsets atom 1 spin-UP, atom 2 spin-DOWN, etc. in alternation. Mayproduce not what you expected following a small error in atoms list!


Magnetic calculation: results; what to look at

Total and local magnetic moments. To get them, setWriteMullikenPop 1

I Think about: is it integer (or nearly) if it should? (→ magneticmolecules, magnetic dielectrics, half-metallic compounds)

I Is it reasonable, judging from the expected magnetic configuration?

Local magnetic moments, as extracted from Mulliken analysis.ATTENTION: these numbers are VERY dependent on the choice ofbasis and are difficult to compare between different calculationmethods (i.e., Siesta vs. abinit or WIEN2k).

Magnetic (spin) density RHO(↑)−RHO(↓), generated whenSaveRho Tand further on extracted using denchar or rho2xsf. Comparablebetween different methods (if calculated on the same level of DFT),also with experiment (i.e. as extracted from neutron scattering).


Magnetic calculation: results; spin moments

This example (for a “Mn3Cr” magnetic molecule) shows:Total moment175.012 el.↑ − 164.988 el.↓ → 10.024 µB per unit cell(whole molecule). Deviation from integer number (10 µB)is due to, e.g., “electronic temperature” level broadening.

Local moment at the Cr(1) site:mCr = 4.223 − 7.158 → −2.935 µB

(set opposite to the total moment). The “as calculated”local moment does not have to be integer, and may becomedifferent with another basis definition. Use with care!


Magnetic calculation: results; spin moments

This example (for a “Mn3Cr” magnetic molecule) shows:Total moment175.012 el.↑ − 164.988 el.↓ → 10.024 µB per unit cell(whole molecule). Deviation from integer number (10 µB)is due to, e.g., “electronic temperature” level broadening.

Local moment at the Cr(1) site:mCr = 4.223 − 7.158 → −2.935 µB

(set opposite to the total moment). The “as calculated”local moment does not have to be integer, and may becomedifferent with another basis definition. Use with care!


Magnetic calculation: spin density

Spatial spin density

σ(r) = ρ↑(r)− ρ↓(r)

visualized with XCrySDen, fortwo ±0.0025 e/A3 contour levels(shown by different colours).

What it is good for:get an idea about the localization ofmagnetic density; compare with dif-ferent calculations (as such resultsare not much method-dependent)and with experiment.


Magnetic calculation: “transferable” local spin moments

Mulliken analysis and corresponding issues (e.g., local spin moments)depend on the choice of basis functions and hence are not unique. Spatialcharge/spin density (RHO) “lives” on the spatial grid and “does not knowanything” about basis functions. Hence it can yield a “transferable”(albeit obviously not unique) “definition” of local magnetic moments:

matom α =∫

Ωaround α

dr[ρ↑(r)− ρ↓(r)

],

integrating, e.g., over “muffin-tin spheres” of a given radius, or over theBader’s bassins. The simplest “integration” (a summation over grid pointswhich fall within a given atom-centered sphere) is yielded by grdint, seehttp://www.uni-osnabrueck.de/apostnik/Download/grdint.f


Magnetic calculation: “transferable” local spin moments

matom α =∫

Ωaround α

dr[ρ↑(r)− ρ↓(r)

],

An an example, take a Fe13 cluster calculated with VASP∗ andSiesta+grdint. In both VASP and Siesta calculations, the local momentrefers to the atom-cenetered sphere of the radius 1.302 A.

m from Mulliken m integrated m integratedanalysis (SIESTA) (SIESTA) (VASP)

(central Fe atom) 2.75 2.61 2.70(peripheric Fe atom) 3.43 3.14 3.06

∗by Sanjubala Sahoo, University Duisburg-Essen


Magnetic calculation: an example of Fe (phase diagrams)

Energy(Volume) and M(V ) curvesfor bcc and fcc Fe from Siestacalculation (top panels), comparedwith all-electron FLAPW results(bottom panels). All calcula-tions are initialized for ferromag-netic phase. Note a sharp low-spin/ high-spin transition as function ofvolume in the fcc. The calculationsare done in the LDA which pro-duces a wrong ground-state struc-ture (fcc instead of bcc; see all-electron result).


Magnetic calculation: an example of Fe (phase diagrams)

bcc and fcc Fe, Siesta vs. FLAPW(all-electron) calculation. Notethat GGA fixes the problem of thewrong ground state of Fe.

Open circles in the Siesta panel in-

dicate results obtained with Fe pseu-

dopotential which treats the 3p states

as valence ones. Note better agree-

ment with all-electron calculation in

energy differences, even if the band

structure and magnetism are not af-

fected in a noticeable way by an inclu-

sion of Fe3p semicore states.


Fixed Spin Moment

FSM allows a conditioned minimisationof, say, Etot(Volume), subject to a con-straint N↑ − N↓ = M = const. Theaim is to explore the energy surfacesEtot(Volume,Moment) and to identifymetastable states, barriers, etc.


Fixed Spin Moment


Using FixSpin: an example of Ni4 magnetic molecule

The Fixed Spin Moment trick comes back to Schwarz and Mohn, J.Phys.F 14,L129 (1984). Fixes N↑, N↓ separately (from given Q and M), introducing different

“Fermi energies” E↑F, E↓F, that amounts to imposing effective magnetic field.

Stabilizing different magnetic configura-tions of the magnetic molecule

... that allows to compare their totalenergies and map the results onto theHeisenberg model.


Concluding remarks:

A given system may have multiplemagnetic solutions: the same N↑ +N↓, different N↑ − N↓. A typical ex-ample: HighSpin / LowSpin states ofFe. Such metastable states can be dif-ferently initialized and stabilized, e.g.,using FixSpin Tand TotalSpin 〈N↑−N↓〉For magnetic systems, GGA usually performs better than LDA.

In order to push the system into a different magnetic solution, notnecessarily initialize anew from atoms. Use DMtune.

Large broadenings kill magnetism, narrow peaks favour it.

With the electron number per unit cell is odd and if the system notfor sure a metal (or, if you don’t know), don’t forget to setSpinPolarized T.



Documents

Simulations of magnetic systems : ferro / antiferromagnetic ......ferro / antiferromagnetic ordering and other issues SIESTA tutorial, Santander, June 9, 2010 Andrei Postnikov Outline