Electronic structure and magnetic properties of solids

Electronic structure and magnetic properties of solids

Sergej Y. Savrasov*, I II III, Alexander I. LichtensteinIV, Vladimir AntropovV andGabriel KotliarII

I Department of Physics, New Jersey Institute of Technology, Newark, NJ 07102, USAII Department of Physics and Center for Material Theory, Rutgers University, Pscataway, NJ 08854, USAIII Department of Physics, University of Nijmegen, Nijmegen, The NetherlandsIV Department of Physics, Hamburg University, Hamburg, GermanyV Condensed Matter Physics Department, Ames Laboratory, Ames, IA 50011, USA

Received July 1, 2004; accepted October 3, 2004

Magnetism / Electronic structure / Spin waves /Spin susceptibility / Strong correlations /Computational crystallography

Abstract. We review basic computational techniques forsimulations of various magnetic properties of solids. Sev-eral applications to compute magnetic anisotropy energy,spin wave spectra, magnetic susceptibilities and tempera-ture dependent magnetisations for a number of real sys-tems are presented for illustrative purposes.

1. Introduction

This review covers main techniques and their applicationsdeveloped in the past to calculate properties of magneticsystems by the methods of electronic structure theory ofsolids. The basic tool which was used in connections withthese developments is a spin dependent version of densityfunctional theory [1] which will be reviewed in Section 2in its most general non-collinear form and including socalled LDAþU method [2]. One of the oldest applicationof this method is the problem of magnetic anisotropy en-ergy (MAE). Due to its technological relevance as well asthe smallness of MAE, such calculations represent a realchallenge to the theory and we review these efforts in Sec-tion 3. In Section 4 we address the problem of computingnon-collinear spin alignments which is solved using anelegant spin spiral approach [3]. This allows us to simu-late adiabatic spin dynamics of real magnets with rela-tively cheap computational effort, and some of the calcula-tions will be discussed and compared to experiments.Another development which has been undertaken in thepast is the direct calculation of exchange integrals by uti-lizing a formula of linear response theory [4]. This is re-viewed in Section 5. The spin waves, their dispersions andlifetimes as well as other spin fluctuations formally appearin the structure of dynamical spin susceptibility. The de-velopments of electronic structure based on linear re-sponse approaches which access this quantity is reviewed

in Section 6. The effect of electronic correlations are notsmall in the real materials and static mean field treatmentdone with local spin density functional or LSDAþU meth-od may not be adequate for a whole range of solids.Bringing the effects of dynamical correlations to computemagnetic properties of real systems has been recently un-dertaken by utilizing the dynamical mean field theory [5].This approach will be reviewed in Section 7 together withsome of its most recent applications. The reader is alsorefereed to several other reviews [6, 6] on a similar sub-ject as well as a few excellent books [8, 9].

2. Spin density functionals innon-collinear form

Here we describe a basic tool for computational studies ofmagnetic phenomena, a spin dependent version of the den-sity functional theory [1]. We use general non-collinearnotations and also treat the effects of spin-orbit coupling.We utilize local spin density approximation (LSDA) to theexchange-correlation functional and also show how thecorrections due to strong electronic correlation effects ap-pear in the functional in its simplest Hartree-Fock formknown as the LDAþU method [2]. In section 5 we willalso describe a more general first-principle method basedon dynamical mean field theory [5] to include the correla-tion effects in a more rigorous manner.

To describe a magnetic solid we consider a system offermions under an external potential Vext and an externalmagnetic field Bext. It is useful to introduce the notion ofthe Kohn-Sham potential VeffðrÞ and the Kohn-Sham mag-netic field BeffðrÞ. When spin-orbit coupling is present,the intra-atomic magnetization mðrÞ is not collinear, andthe solid may choose different atom dependent quantiza-tion axes which makes magnetic moments pointing in dif-ferent directions. Therefore, the magnetization must betreated as a general vector field, which also realizes non-collinear intra-atomic nature of this quantity. Such generalmagnetization scheme has been recently discussed [10]. Inthe non-collinear version of LSDA approach the total en-ergy functional E is considered as a functional of twovariables the charge density qðrÞ and magnetization den-

Z. Kristallogr. 220 (2005) 473–488 473# by Oldenbourg Wissenschaftsverlag, Munchen

* Correspondence author (e-mail: [email protected])

Toropova , Mikhail I. Katsnelson, Antoni an

sity mðrÞqðrÞ ¼

Pkj

fkj f~yykjðrÞ jIIj ~yykjðrÞg

¼Pkj

fkjP

s¼"#yðsÞkj

*ðrÞ yðsÞkj ðrÞ ; ð1Þ

mðrÞ ¼ gmB

Pkj

fkjf~yykjðrÞ jssj ~yykjðrÞg

¼ gmB

Pkj

fkjP

ss0 ¼ "#yðsÞkj

*ðrÞ ssss0yðs0Þkj ðrÞ ð2Þ

where jjf g denotes averaging over spin degrees of free-dom only, the spin angular momentum operator is ex-pressed in terms of Pauli matrices ss ¼ ss=2; II is 2� 2 unitmatrix, g is the gyromagnetic ratio which is for electronsequal to 2, and ~yykjðrÞ are the non-interacting Kohn-Shamparticles are formally described by the two-component spi-nor wave functions

~yykjðrÞ ¼yð"Þkj ðrÞ

yð#Þkj ðrÞ

0@

1A ð3Þ

which define the charge and non-collinear magnetizationdensities of the electrons.

LSDAþU method [2] introduces additional variable“occupancy spin density matrix” nnab

nnab ¼n""ab n"#ab

n#"ab n##ab

!ð4Þ

which represents the correlated part of electron density. Tobuild up occupancy matrix one introduces a set of loca-lized orbitals faðrÞ, associated with correlated electrons.Then

nss0

ab ¼Pkj

f ðEkjÞ hyðsÞkj jfai hfb jyðs0Þkj i : ð5Þ

The occupancy matrix becomes non-diagonal in spinspace when spin-orbit coupling is taken into account. Weinclude spin-orbit coupling in a variational way as sug-gested by Andersen [11].

One introduces a Lagrange multipliers matrix DVVab toenforce (5). The LSDAþU total energy functional is givenby:

E½qðrÞ;mðrÞ; nnab�

¼Pkj

f ðEkjÞ Ekj �ð

VeffðrÞ qðrÞ dr

þð

BeffðrÞmðrÞ dr�Pab

Pss0

DVss0

ab nss0

ab

þð

VextðrÞ qðrÞ dr�ð

BextðrÞmðrÞ dr

þ 1

2

ðdr dr0

qðrÞ qðr0Þjr� r0j

þ ELSDAxc ½q;m� þ EModel½nnab� � EModel

DC ½nnab� : ð6Þ

The energies Ekj are determined by Pauli-like Kohn-Shammatrix equation:

½ð�r2 þ VeffÞ II þ gmBBeff ssþ xlssþPab

DVVabjfai hfbj�

� ~yykj ¼ Ekj~yykj : ð7Þ

Here ll and ss are one-electron orbital and spin angular mo-mentum operator, respectively. xðrÞ determines the strengthof spin-orbit coupling and in practice is determined har-mon by radial derivative of the l ¼ 0 component of theKohn-Sham potential inside an atomic sphere:

xðrÞ ¼ 2

c2

dVeffðrÞdr

: ð8Þ

ELSDAxc ½q;m� is the LSDA exchange correlation energy.

When magnetization is present, the exchange-correlationenergy functional is assumed to be dependent on densityand absolute value of the magnetization:

ELSDAxc ½q;m� ¼

Ðdr Exc½qðrÞ; mðrÞ� qðrÞþÐ

dr fxc½qðrÞ; mðrÞ� mðrÞ ; ð9Þwhere mðrÞ ¼ jmðrÞj.

EModel½nnab� is a contribution from the Coulomb energyin the shell of correlated electrons

EModel½nnab� ¼ 12

Ps

Pabcdhacj vC jbdi nss

ab n�s�scd

þ 12

Ps

Pabcdðhacj vC jbdi � hacj vC jdbiÞ nss

ab nsscd

� 12

Ps

Pabcdhacj vC jdbi ns�s

ab n�sscd : ð10Þ

This expression is nothing else as a Hartree-Fock averageof the original expression for the Coulomb interaction

12

Pss0

Pabcdhabj vC jcdi cyascybs0cds0ccs ð11Þ

where the Coulomb interaction vCðr� r0Þ has to take intoaccount the effects of screening by conduction electrons. Thematrix hacj vC jbdi ¼ Uabcd is standardly expressed via theSlater integrals which for d-electrons are three constantsFð0Þ;Fð2Þ; and Fð4Þ considered as the external parameters ofthe method. Its determination can for example be done usingatomic spectral data, constrained density functional theorycalculations or taken from spectroscopic experiments.

Since part of this Coulomb energy is already taken intoaccount in LDA functional, the double-counting part de-noted EModel

DC ½nnab� must be subtracted. Frequently usedform [2] for EModel

DC ½nnab� is

EModelDC ½nnab� ¼ 1

2�UU�nnð�nn� 1Þ � 1

2�JJ½�nn"ð�nn" � 1Þ

þ �nn#ð�nn# � 1Þ� ð12Þwhere

�UU ¼ 1

ð2lþ 1Þ2Pabhabj 1

rjabi ; ð13Þ

�JJ ¼ �UU � 1

2lð2lþ 1ÞPabhabj 1

rjabi � habj 1

rjbai

� �ð14Þ

and where �nns ¼P

anss

aa and �nn ¼ �nn" þ �nn#.

The Kohn-Sham potential VeffðrÞ and Kohn-Sham mag-netic field BeffðrÞ are obtained by extremizing the func-tional with respect to qðrÞ and mðrÞ:

VeffðrÞ ¼ VextðrÞ þð

dr0qðr0Þr� r0j j þ

dELSDAxc ½q;m�dqðrÞ ; ð15Þ

BeffðrÞ ¼ BextðrÞ þdELSDA

xc ½q;m�dmðrÞ : ð16Þ

474 S. Y. Savrasov, A. Toropova, M. I. Katsnelson et al.

Extremizing with respect to nss0ab yield the correction to the

potential DVss0ab

DVss0

ab ¼dEModel

dnss0ab

� dEModelDC

dnss0ab

: ð17Þ

Then the spin diagonal elements for potential correctionare given by:

DVssab ¼

Pcd

Uabcdn�s�scd þ

PcdðUabcd � UadcbÞ nss

cd

� dab�UUð�nn� 1

2 Þ þ dab�JJð�nnss � 1

2 Þ; ð18Þ

and spin off-diagonal elements are given by

DVs�sab ¼ �

Pcd

Uadcbn�sscd : ð19Þ

The off-diagonal elements of the potential correction onlypresent when spin-orbit coupling is included, hence a rela-tivistic effect.

This completes the description of the method and wenow turn out to several applications of it.

3. Magnetic anisotropy of ferromagnets

One of the oldest problems which has been addressed inthe past is the calculation of the magneto-crystalline aniso-tropy energy (MAE) [13–17] of magnetic materials. TheMAE is defined as the difference of total energies with theorientations of magnetization pointing in different, e.g.,(001) and (111), crystalline axis. The difference is notzero because of spin-orbit effect, which couples the mag-netization to the lattice, and determines the direction ofmagnetization, called the easy axis.

Being a ground state property, the MAE should be ac-cessible in principle via spin density functional theory de-scribed above, and, despite the primary difficulty relatedto the smallness of MAE (�1 meV/atom), great efforts tocompute the quantity with advanced total energy methodscombined with the development of faster computers, haveseen success in predicting its correct orders of magnitudes[18–22]. However, the correct easy axis of Ni has notbeen predicted by the LSDA and a great amount of workhas been done to understand what is the difficulty. Theseinclude (i) scaling spin-orbit coupling in order to enlargeits effect on the MAE [19, 20], (ii) calculating torque toavoid comparing large numbers of energy [20], (iii) study-ing the effects of the second Hund’s rule in the orbitalpolarization theory [21], (iv) analyzing possible changes inthe position of the Fermi level by changing the number ofvalence electrons [22], (v) using the state tracking method[23], and (iv) real space approach [24].

It was recently suggested that the deficiency of the cal-culations is lying in improper treatment of the correlationeffects and the intra-atomic repulsion U and exchange Jshould be taken into account. It is important to performthe calculations for fixed values of magnetic momentswhich themselves show some dependency on U and J asstudied previously Kudrnovsky. Since the pure LSDA re-sult (U ¼ 0, and J ¼ 0) reproduces the experimental va-lues for magnetic moments in both Fe and Ni fairly well,the U � J parameter space should be scanned and the path

of U and J values which hold the theoretical momentsclose to the experiment was extracted.

The effect of correlations was found to be crucial inpredicting the correct axis of Ni. Fig. 1 shows the resultsof this calculated MAE as a function of Coulomb para-meter U. Walking along the path of parameters U and Jwhich hold the magnetic moment to 0.6mB the MAE firstincreases to 60 meV (U ¼ 0:5 eV, J ¼ 0:3 eV) and thendecreases. While decreasing it makes a rather flat areafrom U ¼ 1:4 eV, J ¼ 0:9 eV to U ¼ 1:7 eV, J ¼ 1:1 eVwhere MAE is positive and around 10 meV. After the flatarea, the MAE changes from the wrong easy axis to thecorrect easy axis. The correct magnetic anisotropy is pre-dicted at U ¼ 1:9 eV and J ¼ 1:2 eV. The change fromthe wrong easy axis to the correct easy axis occurs overthe range of dU � 0:2 eV, which is of the order of spin-orbit coupling constant (�0:1 eV).

For Fe (see Fig. 2), the MAE was calculated along thepath of U and J values which fixes the magnetic momentto 2:2mB. At U ¼ 0 eV and J ¼ 0 eV, the MAE is 0:5 eV.The correct MAE with the correct direction of magneticmoment is predicted at U ¼ 1:2 eV and J ¼ 0:8 eV. It isremarkable that the values of U and J necessary to repro-duce the correct magnetic anisotropy energy are very closeto the values which are needed to describe photoemissionspectra of these materials [26].

Recently, there has been a lot of experimental and the-oretical studies devoted to CrO2 [27–31]. This compoundhas unusual half-metallic nature, it is a metal in one spinchannel and an insulator in the other. CrO2 demonstrates

Electronic structure and magnetic properties of solids 475

Fig. 1. The magneto-crystalline anisotropy energy MAE ¼ Eð111Þ� Eð001Þ for Ni as function of U. The experimental MAE is markedby arrow (�2:8 meV). The values of exchange parameter J for everyvalue of U are chosen to hold the magnetic moment of 0:61mB.

Fig. 2. The magneto-crystalline anisotropy energy MAE ¼ Eð111Þ� Eð001Þ for Fe as function of U. The experimental MAE is markedby arrow (1:4 meV). The values of exchange parameter J for everyvalue of U are chosen to hold the magnetic moment of 2:2mB.

ferromagnetic ordering with magnetic moment 2mb per Cratom and Curie Temperature TC ’ 390 K. A number ofpublished works [27–31] addressed its band structure.

The results of the MAE computations within LSDAhave been studied recently. Total energy calculations forthree different directions [001], [010] and [102]. [001]axis indicate that easy magnetization axis within LSDA isconsistent with latest thin film experiments [33–35]. Thenumerical values of MAE however exceed the experimen-tal one approximately two times.

In order to figure out the influence of intra-atomic re-pulsion U on the magnetic anisotropy, calculations havebeen also performed for finite value of U changing it from0 to 6 eV (J ¼ 0:87 eV has been kept constant). Theseresults are presented in Fig. 3. MAE is decreasing rapidlystarting from LDA value �68 meV per cell and changes itssign around U � 0:8 eV. This leads to switching of correcteasy magnetization axis [001] to the wrong one, namely[102]. The biggest experimental value of MAE reported inthe literature is 15:6 meV per cell [35]. The calculatedMAE approaches this value around point U ¼ 0:6 eV. Tosummarize, the LSDAþU approach with U � 0:6 eV andJ ¼ 0:87 eV adequately describes the magneto-crystallineanisotropy of CrO2.

We can conclude that the calculations of magnetic an-isotropy energies in materials are only in its beginningstage of development. It is by now clear that the effects ofCoulomb correlations must be taken seriously into accountbut the smallness of MAE and its sensitivity to the valuesof U used in the calculations prompts us that a predictivepower of the method is yet to be achieved. More work isclearly required to develop robust algorithms calculatingthis highly important property of ferromagnets.

4. Spin spiral method and frozen magnoncalculations

While ferromagnetic spin alignments are relatively simpleto explore by the methods of spin density functional theo-ry, the situation gets formally much more complicatedwhen non-collinearity occurs. The simplest example of

non-collinearity is the antiferromagnetism which is the or-dering with wave-vector q corresponding to a zone-bound-ary point of the Brillouin zone. This requires a choice ofdoubled unit cell so that in the new lattice this vector qbecomes one of the reciprocal lattice vectors. In a moregeneral sense we can consider an appearance of a spinspiral which gives orientation of different magnetic mo-ments MtþR for atoms of the sublattice t þ R (t is thebasis vector, R is the primitive translation) in the form

MtþR ¼Mx

tþR

MytþR

MztþR

8><>:

9>=>; ¼ Mt

cos ðqRþ ftÞ sin ðqtÞsin ðqRþ ftÞ sin ðqtÞ

cos ðqtÞ

8><>:

9>=>; ;

ð20Þ

where Mt is the length of the atomic moment, and qt;ft

are the angles describing its orientation in the unit cellwith R ¼ 0: As we see, in the systems with non-collinearordering the situation gets immediately very complicatedas for arbitrary q the size of the unit cell becomes prohibi-tively large. Note that here we do not assume that the sizeof the moment is varied when going from one cell to an-other. The latter is another type of so called incommensu-rate magnetism, the most known example of which is Cr.

Here we will briefly review a very elegant way to solvethe problem of non-collinear magnetism where the size ofthe moment is kept constant which is known as the spinspiral method developed by Sandratski [7]. The problemwhen the size itself varies is more difficult one and theuse of linear response theory will be discussed later of thisreview. The formal point is to note that non-collinear mag-netic state destroys the periodicity of the original latticeand therefore the original Bloch theorem applied to theKohn-Sham states ~yykjðrÞ is no longer available. However,the Bloch theorem in the present form reflects the transla-tional symmetry properties of the lattice only it does nottake into account the symmetry properties of the spin sub-system. A generalized symmetry treatment of the Hamilto-nian for the non-collinear magnet can be developed. Letus define group operations acting on a spinor state ~yyðrÞ:These are (i) the translational operator TTR so thatTTRr ¼ rþ R, (ii) the generalized rotation gg which is apure rotation aa ¼ ðag; bg; ggÞ described by the 3� 3 j ¼ 1Wigner matrix Uj¼1

mm0 ðag; bg; ggÞ (where ag; bg; gg are theEuler angles) and possible shift by vector b for non-sym-morphic group so that ggr ¼ aarþ b, (iii) the rotationxx ¼ ðax; bx; gxÞ in the spin-1

2 subspace described by the

2� 2 j ¼ 12 Wigner matrix U j¼1=2

ss0 ðax; bx; gxÞ.The usefulness of these operations becomes immedi-

ately transparent if we explore symmetry properties ofnon-relativistic Hamiltonians Hss0 ðrÞ ¼ ð�r2 þ VeffÞ dss0

þ gmBBeffsss0 of the type (7) with non-collinear spin align-ments which assumes that the effective magnetic field Beff

inside each atom centered at t þ R transforms similar to(20). The translations R combined with the spin rotations

xxfqRtg ¼ ðqt; qRþ ft; 0Þ described by the unitary ma-trices UUj¼1=2 leave the spin spiral structure invariant. Wethus arrive to a generalized Bloch theorem:

~yykjðrþ RÞ ¼ eikR UUj¼1=2ðqt; qRþ ft; 0Þ ~yykjðrÞ : ð21Þ


-600

-500

-400

-300

-200

-100

0

100

0 1 2 3 4 5 6

Experimental Value

MA

E(µ

eV/c

ell)

U (eV)

E(010) - E(001)E(102) - E(001)E(111) - E(001)

Fig. 3. The magneto-crystalline anisotropy energies MAE for CrO2

as functions of U. The experimental value of MAE E½010� � E½001�¼ 15:6 meV per cell is shown by arrow.

Remarkably that this property allows us to restrict theconsideration to a chemical unit cell and not to the super-cell. Hence, instead of introducing the supercell whose sizedepends on the symmetry of wave vector q, the non-colli-near spin spiral states can still be treated with the Hamilto-nians whose size does not depend on the wavevector q:

A variety of different calculations has been done with thespin spiral approach [6, 6]. Halilov et al. [36, 37] have stud-ied adiabatic spin waves computed using a frozen magnonmethod. This is analogous to the frozen phonon methodwhere atoms are displaced from their equilibrium positionsand the total energy is restored as a function of these posi-tions. Here the calculation is simplified since there is nonecessity to introduce supercells, and can be performed forany general spin wave-vector q. Fig. 4 reproduces the com-parison between calculated using frozen magnon LSDA ap-proach and experimental [38] spin wave spectra for Fe.

As we see the obtained agreement in these calculationsis very satisfactory despite the fact that LSDA theory doesnot take into account important effects of dynamical corre-lations. For systems like NiO, the correlation effects be-come more important and the frozen magnon calculationsusing LSDAþU method have been carried out. Fig. 5shows the comparison between calculated using LSDAþUmethod and experimental spin wave spectra for NiO. Theoverall agreement is good but important discrepancies re-main which point out on the necessity to treat correlationseffects among d-electrons beyond its simplest Hartree-Fock treatment as it is done within LSDAþU.

5. Calculations of exchange constants

While the density functional theory provide a rigorous de-scription of the ground state properties of magnetic materi-als, the finite-temperature magnetism is estimated follow-ing a simple suggestion [4], whereby constrained DFT atT ¼ 0 is used to extract exchange constants for a classi-cal Heisenberg model, which in turn is solved using ap-proximate methods (e.g. RPA, mean field) from a classicalstatistical mechanics of spin systems [4, 36, 39, 40]. Therecent implementation of this approach gives reasonablevalue of transition temperature for iron but not for nickel[41]. The analysis of exchange parameters for differentclasses of magnetic materials such as dilute magneticsemiconductors [42], molecular magnets [43], colossalmagnetoresistance perovskites [44], transition metal alloys[45], hard magnetic materials such as PtCo [46] gives auseful information for magnetic simulations.

In this section we outline the general strategy for esti-mations of exchange interactions in solids. The most reli-able way to consider spin excitations of itinerant electronmagnets in the framework of the spin density functionalapproach is the use of frequency dependent magnetic sus-ceptibility within the linear response theory [47–49].

dm ¼ cc dBext ¼ cc0 dBeff ; ð22Þ

where c0 is a “bare” DFT and c is an enhanced suscept-ibility. In collinear magnetic structures there are no cou-pling between the longitudinal and transverse componentsand for the transverse spin susceptibility we have the fol-lowing equation:

c�ðr; r0;wÞ ¼ c�0 ðr; r0;wÞ þÐ

dr00 c�0 ðr; r00;wÞ� Iðr00Þ c�ðr00; r0;wÞ ð23Þ

where I ¼ Bxc=m is an exchange-correlation “Hund’s rule”interaction. This random-phase-approximation (RPA) likeequation is formally exact in the adiabatic time dependentversion of density functional theory (TD-DFT) [50]. Thebare susceptibility has the following form:

c�0 ðr; r0;wÞ ¼Pmn

fm" � fn#w� em" þ en#

� y*m"ðrÞ yn#ðrÞ w*n#ðr0Þ ym"ðr0Þ ð24Þ

where yms and ems are eigenstates and eigenvalues for theKohn-Sham quasiparticles. We can rewrite the equation forthe transverse susceptibility as

cc� ¼ ðmþ LLÞ ðw� IxcLLÞ�1 ð25Þwhere

Lðr; r0;wÞ ¼Pmn


y*m"ðrÞ yn#ðrÞ

� r½ym"ðr0Þ ry*n#ðr0Þ � y*n#ðr0Þ� rym"ðr0Þ� : ð26Þ

Spin wave excitations can be separated from the Stonercontinuum (e.g., paramagnons) only in the adiabatic ap-proximation, which means the replacement Lðr; r0;wÞ byLðr; r0; 0Þ in Eq. (25). Otherwise one should just find the


Fig. 4. Comparison between calculated using frozen magnon methodand experimental [38] spin wave spectrum for Fe.

Fig. 5. Comparison between calculated using LDAþU method andexperimental spin wave spectrum for NiO.

poles of the total susceptibility, and the whole concept of‘‘exchange interactions” is not uniquely defined. Neverthe-less, formally we can introduce the effective exchange in-teractions via the quantities

Wðr; r0;wÞ ¼ IxcLðr; r0;wÞ : ð27Þ

In the static limit one can show that

Wðr; r0; 0Þ ¼ 1

mðrÞ Jðr; r0; 0Þ � BxcðrÞ dðr� r0Þ ð28Þ

where an expression for frequency dependent exchangeinteractions has the following form

Jðr; r0;wÞ ¼Pmn


� y*m"ðrÞ BxcðrÞ yn#ðrÞ y*

n#ðr0Þ� Bxcðr0Þ ym"ðr0Þ : ð29Þ

The later coincides with the exchange integrals [4, 6] ifwe neglect the w –– dependence. Since Bxc � m we haveJ � m2 and the expression (27) vanishes in non-magneticcase, as it should be. Note that the static susceptibilitycc� w ¼ 0ð Þ can be represented in the following form

cc� 0ð Þ ¼ mðWW�1 � B�1xc Þ � m ~WW~WW ð30Þ

which is equivalent to the result of Ref. [51]

~WW~WW ¼ WWð1� B�1xc WWÞ�1 ð31Þ

for the renormalized exchange interaction if one definethem in terms of inverse static susceptibility [52]. Note,that the magnon frequencies are just eigenstates of the op-erator WW 0ð Þ which exactly corresponds to the expressionfrom the unrenormalized exchange interactions [4, 6].Note that for the long-wavelength limit q! 0 this resultturns out to be exact which proves the above statementabout the stiffness constant D: in the framework of thelocal approximation it is rigorous.

Let us compare the Eq. (31) with the Heisenberg-likedispersion law in the spherical approximation for the spin-wave spectrum in terms of static transverse susceptibility[6]:

wq ¼ m½c�1q � c�1

0 � : ð32Þ

Now we will use an expression JJ ¼ cc�1 which gives thewell-known connection between the inverse static suscept-ibility and the parameter F2 in the Landau theory ofphase transitions. To analyze the relationship between thedispersion law (32) and that commonly used in the DFT,wl

q ¼ mI½c0 � cq� I, we assume that the ratio

Dq ¼ ðcq � c0Þ c�10 � wl

q=mI ð33Þ

is small in the long-wave approximation. Then, by ex-panding Eq. (32) over the parameter Dq; we obtain thedesired result:

wq ¼ mðc�1q � c�1

0 Þ ¼ mc�10 Dqð1� DqÞ�1

’ mc�10 ðc0 � cqÞ c�1

0 ¼ mðJl0 � Jl

qÞ ð34Þ

or

Jq ¼ J0½1þ Dqð1� DqÞ�1� ¼ J0½1þ Dq þ D2q þ : : :� ;

ð35Þwhere the susceptibility has a matrix structure and

Jlq ¼ J0ð1þ DqÞ ¼ c�1

0 cqc�10 ¼ IcqI ð36Þ

is the matrix of the exchange parameter in the local (long-wave) approximation. We take into account, that in theferromagnetic state c�1

0 ¼ I: A form of Eq. (36) is thewidely accepted definition of the exchange coupling para-meter. The usual assumption of weak enhancementJq � J0 ¼ J0

q � J00 is valid due to the well-known result

c�1q ¼ Jq ¼ J0

q � I. As a consequence, the spectrum of ele-mentary excitations is not affected by exchange-correlationenhancement effects in linear response regime. So, the de-finition (36) is directly related to several very strong ap-proximations: rigid spin, smallness of spin wave disper-sion compared to the effective exchange splitting (atomiclimit).

Below we will briefly mention calculations of the ex-change parameters and adiabatic spin wave spectra usingthe multiple scattering technique in the local density ap-proximation with a linear muffin-tin orbital technique(LMTO) in the atomic sphere approximation [11]. Threeferromagnetic systems with entirely different degree of lo-calization of the local moment were considered: Gd, Niand Fe [6]. An important observation is that in Gd (highlylocalized moments, small spin wave dispersion) the ap-proximation based on the assumption Dq � 1 is comple-tely fulfilled (D at the zone boundary is less then 0:01),whereas in Ni (moderately localized moments, large spinwave dispersion) it is not valid at all (D ’ 0:6 at q ¼ Y),so that the corresponding matrix estimates using Eq. (34)demonstrate a strong enhancement of spin wave spectra atlarger q. Fe is an intermediate case where the maximumof D is 0.30.

Practical expression for the exchange coupling hasbeen derived [52] using multiple scattering method:

J�q ¼1

4p

ðeF

de Im TrLðT" � T#Þ00

ðdk T"kT#kþq

� ��1

� ðT" � T#Þ00 : ð37Þ

where Ts is a path operator. This expression properlytakes into account both exchange effects related to theStoner splitting and the different energy dispersion for dif-ferent spins in band magnets.

Analysis of the effective exchange coupling betweenatoms after Eq. (37) indicated that in Fe and Ni the maincontribution came from the renormalization of the firstnearest neighbor exchange, so that in bcc Fe it is in-creased from Jl

01 ¼ 16.6 meV to J01 ¼ 19.4 meV, whereasin fcc Ni Jl

01 ¼ 2.7 meV to J01 ¼ 8.3 meV. Such a cleardifference in the results indicates that the removal of thelong-wave approximation can serve as an indicator of thedegree of localization (parameter ðcq � c0Þ c�1

0 above) indifferent metallic magnets. In Fe and Ni, for instance, it atleast partially explains why long wavelength mean fieldestimates predict such a small Tc in FM Ni: 300–350 K in


Ref. [6], 340 K in present calculations, while experimentalresult is 630 K. First of all, long-wave approximation issuitable for such a ‘localized’ system as Fe, and the corre-sponding change in the nearest neighbor J01 is relativelysmall (correspondingly the increase of Tc is expected to besmall). Ferromagnetic Ni represents a rather itinerant sys-tem and any local approach (long wave approximation inparticular) might produce a large error. Also in Ni at Tc

the energy associated with ‘creation’ of moment is com-parable with the energy of its rotation and longitudinalfluctuations should be taken into account.

A large increase in J01 for Ni may indicate that tradi-tional mean-field approaches (based on the absence ofshort-range order) are not applicable for the itinerant sys-tems in general, predicting very high Tc (above 1000 K).Such a number is not consistent with the local densityapproach because the latter does not allow to have Tc infcc Ni larger then 500–540 K (600–640 K using gradientcorrections) and this number can be considered only as anindicator of non-applicability of mean field approach.

6. Dynamical spin susceptibility calculations

Local spin density functional theory and the LDAþUmethod have been applied with success to study fullwave-vector and frequency dependent spin susceptibility cof solids: Its knowledge accessible directly via neutron-scattering measurements is important due to significant in-fluence of spin fluctuations to many physical propertiesand phenomena [8], such, e.g., as the electronic specificheat, electrical and thermal resistivity, suppression ofsuperconductivity for singlet spin pairing, etc. As we al-ready discussed, in magnetically ordered materials, trans-verse spin fluctuations are spin waves whose energies andlifetimes are seen in the structure of transverse susceptibil-ity.

Large efforts in the past were put on the developmentof methods for ab initio calculations of the dynamical spinsusceptibility based either on the random-phase-approxi-mation (RPA) decoupling of the Bethe-Salpeter equation[53], or within density functional formalism [54]. Quanti-tative estimates of c with realistic energy bands, wavefunctions, and self-consistently screened electron-electronmatrix elements appeared in the literature more than 3decades ago [48, 55, 56]. More recently, a computation-ally efficient time-dependent generalization of the Stern-heimer approach [57] originally developed for ab initiocalculations of phonon dispersions, electron-phonon inter-actions and transport properties of transition-metal materi-als has been proposed [49].

This method is advantageous since it allows us toavoid problems connected to the summations over higherenergy states and inversions of large matrices. It consid-ers the response of electrons to a small external magneticfield

dBextðrtÞ ¼ db eiðqþGÞ r eiwt e�hjtj þ c:c ð38Þapplied to a solid. Here db ¼

Pm

dbmem shows a polariza-

tion of the field (m runs over x; y; z or over �1; 0; 1), wave

vector q lies in the first Brillouin zone, G is a reciprocallattice vector, and h is an infinitesimal positive quantity. Ifthe unperturbed system is described by charge densityqðrÞ and by magnetization mðrÞ, the main problem is tofind self-consistently first-order changes dqðrtÞ anddmðrtÞ ¼

Pn

dmnðrtÞ en induced by the field dBextðrtÞ. If

the polarization db in (38) is fixed to a particular m-thdirection, and dmðrtÞ is calculated afterwards, a m-th col-umn of the spin susceptibility matrix cnmðr; qþG;wÞ willbe found.

Employment of time-dependent (TD) version of densityfunctional theory (DFT) [50] to find the quantities dqðrtÞand dmðrtÞ can be useful because in order to find thedynamical response within TD DFT, only the knowledgeof these unperturbed Kohn-Sham states (both occupiedand unoccupied) is required; no knowledge of real excita-tion spectra (both energies and lifetimes) is necessary.This is the main advantage of such approach. Unfortu-nately, within TD DFT, an accurate approximation to thekernel Ixcðr; r0;wÞ describing dynamical exchange-correla-tion effects is unknown while some progress is currentlybeen made [58]. In the past, the so called adiabatic localdensity approximation (ALDA) [50] and a generalized gra-dient approximation [60] (GGA) are adopted to treatIxcðr; r0;wÞ. The addition of the local U corrections hasbeen addressed in the work [59].

One can develop a variational linear-response formula-tion. It has already been known that static charge and spinsusceptibilities appeared as second-order changes in thetotal energy due to applied external fields can be calcu-lated in a variational way. This was demonstrated longtime ago [61] on the example of magnetic response, and,recently [57, 62], in the problem of lattice dynamicswhich is an example of charge response. The proof is di-rectly related to a powerful “2nþ 1” theorem of perturba-tion theory and stationarity property for the total energyitself [63]. Any (2nþ 1)-th change in the total energy Etot

involves finding only (n)-th order changes in one-electronwave functions yi, and corresponding changes in thecharge density as well as in the magnetization. Any (2n)-thchange in Etot is then variational with respect to the (n)-th-order changes in yi.

A time-dependent generalization of these results wasaddressed [49], where the action S as a functional of qðrtÞand mðrtÞ is considered within TD DFT [50, 64]. Thesefunctions are expressed via Kohn-Sham spinor orbitals~yykjðrtÞ satisfying TD Schrodinger’s equation. Therefore, Sas the stationary functional of ~yykjðrtÞ is considered inpractice: When the external field is small, the perturbedwave function is represented as ~yykjðrÞ e�iEkj t þ d~yyiðrtÞand the first-order changes d~yykjðrtÞ define the inducedcharge density as well as the magnetization:

dq ¼Pkjðfd~yykj jIj ~yykjg þ f~yykj jIj d~yykjgÞ ; ð39Þ

dm ¼ gmB

Piðfd~yykj jssj ~yykjg þ f~yykj jssj d~yykjgÞ : ð40Þ

In order to find d~yykjðrtÞ, a time-dependent analog ofthe “2nþ 1” theorem is now introduced. Any (2nþ 1)-thchange in the action functional S involves finding only


(n)-th order changes in the TD functions ~yykjðrtÞ, and cor-responding changes in charge density as well as in themagnetization. Any (2n)-th change in S is then variationalwith respect to the (n)-th-order changes in ~yykjðrtÞ. Theproof is the same as for the static case [63] if the statio-narity property of S and the standard TD perturbation the-ory are exploited. For important case n ¼ 2, this theoremmakes the second-order change Sð2Þ in the action varia-tional with respect to the first-order changes d~yyiðrtÞ: Ifthe perturbation has the form (38), Sð2Þ is directly relatedto the real diagonal part of the dynamical spin susceptibil-ity Re ½cnmðqþG0, qþG, wÞ�G0¼G, thus allowing its var-iational estimate [64].

The problem is now reduced to find Sð2Þ as a functionalof d~yykjðrtÞ and to minimize it. This will bring an equa-tion for d~yykjðrtÞ. Any change in the action functional canbe established by straightforward varying S of TD DFT[50, 64] with respect to the perturbation (38). This is ana-logous to what is done in the static DFT to derive, forexample, the dynamical matrix [57]. Sð2Þ is found to be

Sð2Þ½d~yykj� ¼Pkj

2hd~yykjj H � i@kjI jd~yykji

þÐ

dq dVeff �Ð

dmðdBeff þ dBextÞ ð41Þ

where the unperturbed 2� 2 Hamiltonian matrixHH ¼ ð�r2 þ VeffÞ II � gmBssBeff þ xlss. dVeff and dBeff aretheir first-order changes induced by the perturbation (38)which involve the Hartree (for dVeff ) and the exchange-correlation contributions expressed via dq and dm in thestandard manner [54].

The differential equation for d~yykjðrtÞ derived from thestationarity condition of (41) is given by

ðH � i @kjIIÞ d~yykj þ ðdVeff II � mBs dBeffÞ ~yykj ¼ 0 : ð42Þ

Applications to transverse spin fluctuations in Fe andNi as well as calculations of paramagnetic response in Crand Pd have demonstrated an efficiency of the approach.Experimental evidence of an “optical” spin-wave branchfor Ni [65] and its absence for Fe [38] was correctly de-scribed. The dynamical susceptibility was calculated abinitio for paramagnetic Cr, a highly interesting material

due to its incommensurate antiferromagnetism [66]. Thecalculation predicted a wave vector of the spin densitywave (SDW), and clarified the role of Fermi-surface nest-ing. Strong long-wavelength spin fluctuations of Pd wereevident from these and earlier [56] theoretical studies.

We reproduce some of the calculations at Fig. 6 and Fig. 7which show calculated Im ½cðq;wÞ� for ferromagnetic Ni andparamagnetic bcc Cr. In particular, for Cr, a remarkable struc-ture is clearly seen for the q0s near (0, 0, xSDW � 0.9) 2p=a,where the susceptibility is mostly enhanced at low fre-quencies. This predicts Cr to be an incommensurate anti-ferromagnet (experimentally, xSDW ¼ 0:95).

A scheme for making ab initio calculations of the dy-namic paramagnetic spin susceptibilities of solids at finitetemperatures was recently described [67] where incom-mensurate and commensurate antiferromagnetic spin fluc-tuations in paramagnetic Cr and compositionally disor-dered Cr95V5 and Cr95Re5 alloys were studied togetherwith the connection with the nesting of their Fermi sur-faces.

To conclude, the developed approach is able to de-scribe known spin-fluctuational spectra of some real mate-rials which demonstrated its efficiency for practical ab in-itio calculations. however, more elaborate approximationsto the dynamical exchange and correlation are clearly re-quired in order to account for the observed discrepancies.The next section discussed a more general framework todeal with the correlation effects.

7. Dynamical mean field approachand the regime of strong correlations

Magnetic materials range from very weak having a smallmagnetization to strong ones which exhibit a saturatedmagnetization close to the atomic value. As we have seenin the previous sections, frequently, weak or itinerant mag-nets are well described by spin density wave theory, wherespin fluctuations are localized in a small region of momen-tum space. Quantitatively they are well described byLSDA. The transition from ordered to paramagnetic stateis driven by amplitude fluctuations. In strong magnets,


Fig. 6. Calculated Im½c�ðq;wÞ� (arb.units) for Ni. Experimental data[65] are indicated by balls.

Fig. 7. Calculated Im½c�ðq;wÞ� (Ry�1) for Cr.

there is a separation of time scales, �h=t is the time scalefor an electron to hop from site to site with hopping inte-gral t, which is much shorter than �h=J; the time scale forthe moment to flip in the paramagnetic state. The spinfluctuations are localized in real space and the transitionto the paramagnetic state is driven by orientation fluctua-tions of the spin. The exchange splitting J is much largerthan the critical temperature.

Obtaining a quantitative theory of magnetic materialsvalid both in weak and in strong coupling, above and be-low the critical temperature has been a theoretical chal-lenge for many years. It has been particularly difficult todescribe the regime above Tc in strong magnets when themoments are well formed but their orientation fluctuates.This problem, for example, arises in magnetic insulatorsabove their ordering temperature, when this ordering tem-perature is small compared to electronic scales, a situationthat arises in transition metal monoxides (NiO and MnO)and led to the concept of a Mott insulator. In these materi-als the insulating gap is much larger than the Neel tem-perature. Above the ordering temperature, we have a col-lection of atoms with an open shell interacting viasuperexchange. This is a local moment regime which can-not be accessed easily with traditional electronic structuremethods.

Two important approaches were designed to access thedisordered local moment (DLM) regime. One approach[68] starts form a Hubbard like Hamiltonian and intro-duces spin fluctuations via the Hubbard-Stratonovichtransformation [69–72] which is then evaluated using astatic coherent potential approximation (CPA) and im-provements of this technique. A dynamical CPA [73] wasdeveloped by Kakehashi [74]. A second approach beginswith solutions of the Kohn-Sham equations of a con-strained LDA approximation in which the local momentspoint in random directions, and averages over their orien-tation using the KKR-CPA approach [75, 76]. The averageof the Kohn-Sham Green functions then can be taken asthe first approximation to the true Green functions, andinformation about angle resolved photoemission spectracan be extracted [77, 78]. There are approaches that arebased on a picture where there is no short range order tolarge degree. The opposite point of view, where the spinfluctuations far away form the critical temperature are stillrelatively long ranged was put forward in the fluctuationlocal band picture [79–81].

Description of the behavior near the critical point re-quires renormalization group methods, and the low tem-perature treatment of this problem is still a subject of in-tensive research [82]. There is also a large literature ondescribing magnetic metals using more standard many-body methods [83–87] .

Dynamical mean field theory (DMFT) can be used toimprove treatment to include dynamical fluctuations be-yond static approximations like LSDA or LSDAþU. Thisis needed to get access to disordered local moment re-gime and beyond. Notice that single site DMFT includessome degree of short range correlations. Cluster methodscan be used to go beyond the single site DMFT to im-prove the description of short range order on the quasipar-ticle spectrum. DMFT also allows us to incorporate the

effects of the electron-electron interaction on the electro-nic degrees of freedom. This is relatively important in me-tallic systems such as Fe and Ni and absolutely essentialto obtain the Mott-Hubbard gap in transition metal mon-oxides.

Realistic implementation of DMFT within electronicstructure framework [88, 89] can be viewed as a spectraldensity functional theory [90–93]. The central quantity ofthis method is the local Green function GGlocðr; r0;wÞ,which in the non-collinear version should be considered asthe matrix in spin variables

GGloc ¼G"" G"#

G#" G##

� �: ð43Þ

The local Green function is only a part of the full one-elec-tron Green function restricted which is restricted within acertain cluster area. Due to translational invariance of theGreen function on the original lattice given by primitivetranslations fRg, i.e. GGðrþ R, r0 þ R, wÞ ¼ GGðr; r0;wÞ; itis always sufficient to consider r lying within a primitiveunit cell Wc positioned at R ¼ 0. Thus, r0 travels withinsome area Wloc centered at R ¼ 0. We therefore define thelocal Green function to be the exact Green functionGðr; r0; zÞ within a given cluster Wloc and zero outside. Inother words,

GGlocðr; r0;wÞ ¼ GGðr; r0;wÞ qlocðr; r0Þ ð44Þ

where the theta function is a unity when vector r 2 Wc,r0 2 Wloc and zero otherwise. A functional theory wherethe total free energy of the system GSDF considers the lo-cal Green function as a variable was developed recently[90–93]. It is more powerful than the ordinary DFT sinceit accesses both the energetics and local excitational spec-tra of real materials. The range of locality is in principleour choice and the theory has a correct scaling in predict-ing full k- and w dependent spectrum of excitations whenthis range is expanded till infinity. If the range is col-lapsed to a single site, only the density of states is pre-dicted.

A numerically tractable approach was developed com-pletely similar to the density functional theory with theintroduction of energy-dependent analog of Kohn-Shamorbitals. They are very helpful to write down the kineticenergy portion of the functional similar to as DFT consid-ers the density functional as the functional of Kohn-Shamwave functions. It is first useful to introduce the notion ofan local self-energy operator

Mss0

eff ðr; r0;wÞ ¼ VextðrÞ dðr� r0Þ dss0 þMss0

int ðr; r0;wÞ¼ ½VextðrÞ þ VHðrÞ� dðr� r0Þ� dss0 þMss0

xc ðr; r0;wÞ

which has the same range of locality as the local Greenfunction. Second, an auxiliary Green functionGss0 ðr; r0; iwÞ connected to our new “interacting Kohn-Sham” particles so that it is defined in the entire space bythe relationship

½G�1�ss0 ðr; r0;wÞ ¼ ½G�10 �

ss0 ðr; r0;wÞ �Mss0

int ðr; r0;wÞ :


where Gss00 ðr; r0; iwÞ is the non-interacting Green function,

which is given by

½G�10 �

ss0 ðr; r0;wÞ¼ ½ðwþ mþr2�VextðrÞÞ dss0 � xl sss0 � gmBBextsss0 �� dðr� r0Þ : ð45Þ

Since G is a functional of Gloc, it is very useful to viewthe spectral density functional GSDF as a functional of G:

GSDF½G� ¼ Tr ln G � Tr ½G�10 � G�1� G þFSDF½Gloc�

ð46Þwhere the first two contribution represent the kinetic con-tribution and the contribution from the external potentialand magnetic field. The unknown interaction part of thefree energy FSDF½Gloc� is the functional of Gloc: If the Har-tree term is explicitly extracted, this functional can be re-presented as

FSDF½Gloc� ¼ EH ½q� þFxcSDF½Gloc� : ð47Þ

To facilitate the calculation, the auxiliary Green func-tion Gss0 ðr; r0; iwÞ can be represented in terms of general-ized energy-dependent one-particle states

Gss0 ðr; r0; iwÞ ¼Pkj

yRðsÞkjw ðrÞ y

Lðs0Þkjw ðr0Þ

iwþ m� Ekjwð48Þ

which obey the one-particle Dyson equation for the

“right” eigenvector yRðsÞkjw ðrÞP

s0f½�r2 þ VextðrÞ þ VHðrÞ�

� dss0 þ gmBBeffsss0 þ xl sss0 g yRðs0Þkjw ðrÞþ ð49ÞP

s0

Ðdr0 Mss0

xc ðr; r0; iwÞ yRðs0Þkjw ðr0Þ ¼ Ekjwy

RðsÞkjw ðrÞ ð50Þ

and similar equation for the “left” eigenvector yLðsÞkjw ðrÞ

when it appears on the left.Evaluation of the free energy requires writing down the

precise functional form for FSDF½Gloc�: Unfortunately, it isa problem because the free energy F ¼ E � TS; where Eis the total energy and S is the entropy, and the evaluationof the entropy requires the evaluation of the energy as afunction of temperature and an additional integration overit. However, the total energy formula can be written byutilizing the Galitzky-Migdal expression for the interactionenergy [5]. It is given by:

E ¼ TPiw

eiw0þ Pkj

gkjwEkjw � TPiw

Pss0

Ðdr dr0

�Mss0

eff ðr; r0; iwÞ Gs0sðr0; r; iwÞþÐ

dr VextðrÞ qðrÞ �Ð

dr BextðrÞmðrÞþ EH ½q� þ 1

2 TPiw

Pss0

Ðdr dr0

�Mss0

xc ðr; r0; iwÞ Gs0sðr0; r; iwÞ ð51Þwhere

gkjw ¼1

iwþ m� Ekjw: ð52Þ

We have cast the notation of spectral density theory in aform similar to DFT. The function gkjw is the Green func-

tion in the orthogonal left/right representation which playsa role of a “frequency dependent occupation number”.

Both the charge density and non-collinear magnetiza-tion can be found from the integral over the k-space:

qðrÞ ¼ TPiw

Pkj

Ps

gkjwyLðsÞkjw ðrÞ y

RðsÞkjw ðrÞ eiw0þ ; ð53Þ

mðrÞ ¼ gmBTPiw

Pkj

gkjwPss0

sss0yLðsÞkjw ðrÞ y

Rðs0Þkjw ðrÞ eiw0þ :

ð54Þ

The central procedure which is yet to be determined isthe construction of the exchange-correlation part of theself-energy. It is formally a variational derivative of theexchange correlation part of the free energy functional

Mss0

xc ðr; r0; iwÞ ¼dFxc

SDF½Gloc�dGs0sðr0; r; iwÞ

¼ dFxcSDF½Gloc�

dGs0sloc ðr0; r; iwÞ

qlocðr; r0Þ : ð55Þ

The spectral density functional theory, where an exactfunctional of certain local quantities is constructed in thespirit of Ref. [90] uses effective self-energies which arelocal by construction. This property can be exploited tofind good approximations to the interaction energy func-tional. For example, if it is a priori known that the realelectronic self-energy is local in a certain portion of theHilbert space, a good approximation is the correspondinglocal dynamical mean field theory obtained for exampleby a restriction or truncation of the full Baym-Kadanofffunctional to local quantities in the spirit of Ref. [91]. Thelocal DMFT approximates the functional FSDF½Gloc� bythe sum of all two-particle diagrams evaluated with Gloc

and the bare Coulomb interaction vC. In other words, thefunctional dependence of the interaction part FBK½G� inthe Baym-Kadanoff functional for which the diagrammaticrules exist is now restricted by Gloc and is used as anapproximation to FSDF½Gloc�, i.e. FSDF½Gloc� ¼ FBK½Gloc�:Obviously that the variational derivative of such restrictedfunctional will generate the self-energy confined in thesame area as the local Green function itself.

Remarkably the summation over all local diagrams canbe performed exactly via introduction of an auxiliaryquantum impurity model subjected to a self-consistencycondition [5]. If this impurity is considered as a cluster C,the cellular DMFT (C-DMFT) can be used which breaksthe translational invariance of the lattice to obtain accurateestimates of the self energies.

Various methods such as LDAþU [2], LDAþDMFT[94] and local GW [92, 95] which appeared recently forrealistic calculations of properties of strongly correlatedmaterials can be naturally understood within spectral den-sity functional theory. Let us, for example, explore theidea of expressing the energy as the density functional.Local density approximation prompts us that a large por-tion of the exchange-correlation part Fxc½q� can be foundeasily. Indeed, the charge density is known to be accu-rately obtained by the LDA. Why not think of LDA as themost primitive impurity solver, which generates manifestlylocal self-energy with localization radius collapsed to a


single r point? It is tempting to represent FSDF½Gloc�¼ EH ½q� þ ELDA

xc ½q� þ ~FF½Gloc� �FDC½Gloc�; where the new

functional ~FFSDF½Gloc� needs in fact to take care of thoseelectrons which are strongly correlated and heavy, thusbadly described by LDA. Conceptually, that means thatthe solution of the cluster impurity model for the lightelectrons is approximated by LDA and does not need afrequency resolution for their self-energies.

Such introduced LDAþDMFT approximation considersboth the density and the local Green function as the para-meters of the spectral density functional. A further approx-imation is made to accelerate the solution of a single-siteimpurity model: the functional dependence comes fromthe subblock of the correlated electrons only. If localizedorbital representation fcag is utilized, a subspace of theheavy electrons fcag can be identified. Thus, the approxi-mation can be written as ~FFSDF½Gloc; abðiwÞ�, whereGloc; abðiwÞ is the heavy block of the local Green function:

Unfortunately, the LDA has no diagrammatic represen-tation, and it is difficult to separate the contributions fromthe light and heavy electrons. The ELDA

xc ½q� is a non-linearfunctional and it already includes the contribution to theenergy from all orbitals in some average form. Thereforewe need to take care of a non-trivial double counting, en-coded in the functional FDC½Gloc�. The precise form of thedouble counting is related to the approximation imposedfor ~FF½Gloc� and can be borrowed from the LDAþU meth-od [2]. Note that both LDAþU and LDAþDMFT meth-ods require separations of the electrons onto light andheavy which makes them basis-set dependent.

Iron and nickel were studied in Refs. [96, 97]. The val-ues U ¼ 2:3 ð3:0Þ eV for Fe (Ni) and the same value of theinteratomic exchange, J ¼ 0:9 eV for both Fe and Ni wereused, as came out from the constrained LDA calculations[89, 98, 99]. These parameters are consistent with those ofmany earlier studies and resulted in a good description ofthe physical properties of Fe and Ni. In Ref. [97] the generalform of the double counting correction VDC

s ¼ 12 TrsMsð0Þ

was taken. Notice that because of the different self ener-gies in the eg and t2g blocks the DMFT Fermi surfacedoes not coincide with the LDA Fermi surface.

The impurity model was solved by QMC in Ref. [97]and by the FLEX scheme in Ref. [100]. It is clear thatnickel is more itinerant than iron (the spin-spin autocorre-lation decays faster), which has longer lived spin fluctua-tions. On the other hand, the one-particle density of statesof iron resembles very much the LSDA density of stateswhile the DOS of nickel below Tc, has additional featureswhich are not present in the LSDA spectra [101–103]: thepresence of a famous 6 eV satellite, the 30% narrowing ofthe occupied part of d-band and the 50% decrease of ex-change spittoons compared to the LDA results. Note thatthe satellite in Ni has substantially more spin-up contribu-tions in agreement with photoemission spectra [103]. Theexchange splitting of the d-band depends very weakly ontemperature from T ¼ 0:6TC to T ¼ 0:9TC. Correlation ef-fects in Fe are less pronounced than in Ni, due to its largespin splitting and the characteristic bcc structural dip inthe density of states for the spin-down states near the Fer-mi level, which reduces the density of states for particlehole excitations.

The uniform spin susceptibility in the paramagneticstate, cq¼ 0 ¼ dM=dH, was extracted from the QMC simu-lations by measuring the induced magnetic moment in asmall external magnetic field [5]. The dynamical meanfield results account for the Curie-Weiss law which is ob-served experimentally in Fe and Ni. As the temperatureincreases above Tc, the atomic character of the system ispartially restored resulting in an atomic like susceptibilitywith an effective moment:

cq¼0 ¼m2

eff

3ðT � TCÞ: ð56Þ

The temperature dependence of the ordered magnetic mo-ment below the Curie temperature and the inverse of theuniform susceptibility above the Curie point are plotted inFig. 8 together with the corresponding experimental datafor iron and nickel [104]. The LDAþDMFT calculationdescribes the magnetization curve and the slope of thehigh-temperature Curie-Weiss susceptibility remarkablywell. The calculated values of high-temperature magneticmoments extracted from the uniform spin susceptibilityare meff ¼ 3:09 ð1:50Þ mB for Fe (Ni), in good agreementwith the experimental data meff ¼ 3:13 ð1:62Þ mB for Fe(Ni) [104].

The Curie temperatures of Fe and Ni were estimatedfrom the disappearance of spin polarization in the self-consistent solution of the DMFT problem and from theCurie-Weiss law in Eq. (56). The estimates for TC ¼ 1900ð700Þ K are in reasonable agreement with experimentalvalues of 1043 ð631Þ K for Fe (Ni) respectively [104],considering the single-site nature of the DMFT approach,which is not able to capture the reduction of TC due tolong-wavelength spin waves. These effects are governedby the spin-wave stiffness. Since the ratio of the spin-wave stiffness D to TC, TC/a2D; is nearly a factor of 3larger for Fe than for Ni [104] (a is the lattice constant),we expect the TC in DMFT to be much higher than theobserved Curie temperature in Fe than in Ni. Quantitativecalculations demonstrating the sizeable reduction of TC

due to spin waves in Fe in the framework of a Heisenbergmodel were performed in Ref. [41]. This physics whereby


χ-1M

eff2 /

3Tc

M(T

)/M

(0)

Fig. 8. Temperature dependence of ordered moment and the inverseferromagnetic susceptibility for Fe (open square) and Ni (open cir-cle) compared with experimental results for Fe (square) and Ni(circle) (from Ref. 97). The calculated moments were normalizedto the LDA ground state magnetization (2.2mB for Fe and 0.6mBfor Ni).

the long wavelength fluctuations renormalize the criticaltemperature would be reintroduced in the DMFT by theuse E-DMFT. Alternatively, the reduction of the criticaltemperature due to spatial fluctuations can be investigatedwith cluster DMFT methods.

Within the dynamical mean field theory one can alsocompute the local spin susceptibility defined by

cloc ¼g2

3

ðb0

dt S tð Þ Sð0Þh i ð57Þ

where S ¼ 12

Pm; s;s0

cymssss0cms0 is single-site spin operator.

It differs from the q ¼ 0 susceptibility by the absence ofspin polarization in the Weiss field of the impurity model.Eq. (57) cannot be probed directly in experiments but it iseasily computed within the DMFT-QMC. Its behavior as afunction of temperature gives a very intuitive picture ofthe degree of correlations in the system. In a weakly cor-related regime we expect Eq. (57) to be nearly temperatureindependent, while in a strongly correlated regime we ex-pect a leading Curie-Weiss behavior at high temperaturesclocal ¼ m2

loc=ð3T þ const:Þ where mloc is an effective localmagnetic moment. In the Heisenberg model with spin S,m2

loc ¼ SðSþ 1Þ g2 and for the well-defined local magneticmoments (e.g., for rare-earth magnets) this quantity shouldbe temperature independent. For the itinerant electronmagnets, mloc is temperature dependent due to a variety ofcompeting many-body effects such as Kondo screening,the induction of local magnetic moment by temperature[8] and thermal fluctuations which disorder the moments[105]. All these effects are included in the DMFT calcula-tions.

The comparison of the values of the local and theq ¼ 0 susceptibility gives a crude measure of the degreeof short-range order which is present above TC. As ex-pected, the moments extracted from the local susceptibilityEq. (57) are a bit smaller (2.8mB for iron and 1.3mB fornickel) than those extracted from the uniform magneticsusceptibility. This reflects the small degree of the short-range correlations which remain well above TC [106]. Thehigh-temperature LDAþDMFT clearly shows the presenceof a local moment above TC. This moment is correlatedwith the presence of high-energy features (of the order ofthe Coulomb energies) in the photoemission. This is alsotrue below TC, where the spin dependence of the spectra ismore pronounced for the satellite region in Nickel than forthat of the quasiparticle bands near the Fermi level. Thiscan explain the apparent discrepancies between differentexperimental determinations of the high-temperature mag-netic splittings [107–110] as being the results of probingdifferent energy regions. The resonant photoemission ex-periments [109] reflect the presence of local-moment po-larization in the high-energy spectrum above the Curietemperature in Nickel, while the low-energy angle re-solved photoemission investigations [110] results in non-magnetic bands near the Fermi level. This is exactly theDMFT view on the electronic structure of transition met-als above TC. Fluctuating moments and atomic-like config-urations are large at short times, which results in correla-tion effects in the high-energy spectra such as spin-

multiplet splittings. The moment is reduced at longer timescales, corresponding to a more band-like, less correlatedelectronic structure near the Fermi level.

NiO and MnO represent two classical Mott-Hubbardsystems. Both materials are insulators with the energy gapof a few eV regardless whether they are antiferro- or para-magnetic. The spin dependent LSDA theory strongly un-derestimates the energy gap in the ordered phase. This canbe corrected by the use of the LDAþU method. Both the-ories however fail completely to describe the local mo-ment regime reflecting a general drawback of band theoryto reproduce atomic limit. Therefore the real challenge isto describe the paramagnetic insulating state where theself-energy effects are crucial both for the electronic struc-ture and for recovering the correct phonon dispersions inthese materials. The DMFT calculations have been per-formed [111] by taking into account correlations amongd-electrons. In the regime of large U adequate for both forNiO and MnO in the paramagnetic phase the correlationswere treated within the well-known Hubbard I approxima-tion.

The calculated densities of states using theLDAþDMFT method for the paramagnetic state of NiOand MnO [111] have revealed the presence of both lowerand upper Hubbard subbands. These were found in agree-ment with the LDAþU calculations of Anisimov [2]which have been performed for the ordered states of theseoxides. Clearly, spin integrated spectral functions do notshow an appreciable dependence with temperature andlook similar below and above phase transition point.

The progress in the electronic structure calculation withDMFT allowed to apply the method to more complicatedsystems such as plutonium [112, 113]. The temperaturedependence of atomic volume in Pu is anomalous [114,115]. It shows an enormous volume expansion between aand d phases which is about 25%. Within the d phase, themetal shows negative thermal expansion. Transition be-tween d and higher-temperature e phase occurs with a 5%volume collapse. Also, Pu shows anomalous resistivity be-havior [116] characteristic for the heavy fermion systems,but neither of its phases is magnetic. The susceptibility issmall and relatively temperature independent. The photo-emission [117] shows a strong narrow Kondo-like peak atthe Fermi level consistent with large values of the linearspecific heat coefficient.

Density functional based LDA and GGA calculationsdescribe the properties of Pu incorrectly. They predictmagnetic ordering [118]. They underestimate [119] equili-brium volume of the d and e phase by as much as 30%,the largest discrepancy known in LDA, which usually pre-dicts the volume of solids within a few % accuracy evenfor such correlated systems as high temperature supercon-ductors.

To address these questions several approaches havebeen developed. The LDAþU method was applied to d Pu[120]. It is able to produce the correct volume of the dphase for values of the parameter U~ 4 eV consistent withatomic spectral data and constrained density functionalcalculations. Similar calculation has been performed by aso-called orbitally ordered density functional method[121]. However, both methods predict Pu to be magnetic,


which is not observed experimentally. The LDAþU meth-od is unable to predict the correct excitation spectrum.Also, to recover the a phase within LDAþU the para-meter U has to be set to zero which is inconsistent withits transport properties and with microscopic calculationsof this parameter. Another approach proposed [122] in thepast is the constrained LDA approach in which some ofthe 5f electrons, are treated as core, while the remainingare allowed to participate in band formation. Results ofthe self-interaction-corrected LDA calculations have beenreported [123], as well as qualitative discussion of thebonding nature across the actinides series has been given[124].

Dynamical mean-field calculations have been recentlyapplied with success to this material. They were per-formed in paramagnetic state and do not impose magneticordering. DMFT was able to predict the appearance of astrong quasiparticle peak near the Fermi level which existsin the both phases. Also, the lower and upper Hubbardbands were clearly distinguished. The width of the quasi-particle peak in the a phase is found to be larger by 30per cent compared to the width in the d phase. This indi-cates that the low-temperature phase is more metallic, i.e.it has larger spectral weight in the quasiparticle peak andsmaller weight in the Hubbard bands. Recent advanceshave allowed the experimental determination of thesespectra, and these calculations were consistent with thesemeasurements [117].

8. Conclusion

In conclusion, electronic structure methods have been ap-plied with success to study magnetic properties of solids.Many good results have been obtained for weakly corre-lated itinerant systems. New algorithms for inclusion ofmany-body effects are beginning to attack magnetic phe-nomena in strongly correlated materials. These successeshave generated a new synergy between the electronicstructure and many-body communities, which are jointlydeveloping new and powerful electronic structure tools forvirtual material exploration.

Acknowledgments. The work on magnetic anisotropy was supportedby the Office of Naval Research grant No. 4-2650.

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