Quasiparticles in Iron Silicides: GW versus LDA

0.87 MB·English

by  Natalie Zamkova

Quasiparticles in Iron Silicides:
GW versus LDA
Natalia Zamkova,1,2 Vyacheslav Zhandun,1 Sergey Ovchinnikov,1,2 and Igor Sandalov1,3,∗
1L.V.Kirensky Institute of physics, Siberian Branch of Russian Academy Sciences, 660036 Krasnoyarsk, Russia
2Siberian Federal University, 660041 Krasnoyarsk, Russia
3Applied Materials Physics, Department of Materials Science and Engineering,
KTH Royal Institute of Technology, SE 100 44 Stockholm, Sweden
5 The angle-resolved photoemission spectroscopy (ARPES) is able to measure both the spectra
1 and spectral weights of the quasiparticles in solids. Although it is common to interpret the band
0 structure obtained within the density-functional-theory based methods as quasiparticle spectra,
2 these methods are not able to provide the changes in spectral weights of the electron excitations.
r WeuseViennaAbinitioSimulationPackage(VASP)forevaluationofthequasiparticlespectraand
a theirspectralweightswithinHedin’sGWapproximation(GWA)forFe Siandα−FeSi ,providing,
3 2
M
thus, a prediction for the ARPES experiments. Comparison of the GGA-to-DFT and GWA band
structures shows that both theories reflect peculiarities of the crystal structures in similar shape

3 of the bands in certain k-directions, however, in general the difference is quite noticeable. We find
thattheGWAspectralweightofquasiparticlesinthesecompoundsdeviatesfromunityeverywhere

] and shows non-monotonic behavior in those parts of bands where the delocalized states contribute
i
c to their formation. Both methods lead to the same conclusion: those of iron ions in Fe Si which
3
s occupy the positions Fe(1), where they are surrounded by only Fe ions ( Fe(2) positions), have
-
l the d-electrons localized and large magnetic moment whereas Fe ions in the Fe(2) positions with
r
t theSinearest neighborshaved-electronsdelocalized andthemagneticmomentsquenched. TheSi
m influenceontheFeionstateisevenmorepronouncedintheα−FeSi whereallironionshavethe
2
. Siionsas nearest neighbors: bothGWA andGGA calculations producezero moment oniron ions.
at The advantages and disadvantages of both approaches are discussed.
m
PACSnumbers: 71.20.Be,71.20.Eh,71.20.Gj,75.20.Hr,75.47.Np,
- 71.20.-b,75.10.Lp,71.15.Dx,71.70.Ch,71.45.Gm
d
n
o
I. INTRODUCTION information (not thermodynamics), require for their in-
c
[ terpretationa knowledgeof the single-particle-excitation

properties. For example, all photoeffect-based measure-
3 A hope to use the electron spin additionally to the
ments belong to this class. The photoemission spec-
v charge as an information carrier has led to a develop-
troscopy (PES) provides direct measurement of the en-
8
ment of the spintronics. One feasible way to exploit
9 ergyofelectronicquasiparticles. Itsextension,theangle-
the spin degrees of freedom is to synthesize such mag-
8 resolvedPES(ARPES),whichusesthe synchrotronra-
netic semiconductors, which, on the one hand, should
5 diationfacilities,allowsforextractingalsoperpendicular-
0 be magnetic at room temperature, and, on the another to surface momentum dependence of quasiparticle en-
. hand, should be easily integrated with existing semicon-
1 ergy. Further refinement, the laser-based ARPES pro-
ductor industry. Therefore, it is desirable that it should
0 vides even better accuracy and resolution. Recently de-
5 be Si based1. A magnetic moment can be added by veloped method, the time-resolved two-photon photoe-
1 a transition-metal constituents. The other way is to mission (TR-2PPE) spectroscopy11, can monitor the
: use a magnetic metal for injecting of spin-polarizedelec-
v stateofanexcitedelectronduringthecourseofitstrans-
trons into, say, Si-based semiconductor. The technolo-
i formation by laser-induced surface reaction. Probably,
X gies which create magnetic epitaxial multi-layer films on
this is the only method capable to directly measure the
r Si, produce an interface, which contains the compounds
quasi-particle’s life-time. The transport and tunneling
a
of TSi, where T is a transition metal. This makes iron
experiments are even more evident examples where the
silicides compounds highly perspective materials both in
qusiparticle concept is a necessary ingredient for under-
bulk andfilm formanda detailedunderstandingoftheir
standing the underlying physics. However, general theo-
physics is on demand. Recently the formation of sin-
ries sometimes are not sufficient for describing real ma-
gle crystallineFe Si phases in the Fe/Si interface has
3 terials. For example, the predictions of the life-times
beendemonstratedbyseveralgroups2-4. Thetheoretical
withintheLandautheoryofFermi-liquiddonotdescribe
understanding of the ground-state properties like cohe- theexperimentsevenonAl andnoblemetals12,contrary
sive and structural properties is achieved long ago via
to the expectation. Indeed, electrons in these metals are
first-principle calculations based on the various realiza-
welldelocalizedandexpectedtobehaveasagoodFermi-
tion of local-density approximationto density functional
liquid. Furthermore, TR-2PPE experiments show that
theory5-10.
the lifetime ofanexcitedelectronin Al atafixedenergy
However, the experiments, that measure differential E <E dependsonthefrequencyofthepumppulse,i.e.,
F
2
on the band from which the electron originated. These of electronstates calculations of ironsilicides Fe Si and
3
examples convince that ab initio calculations for real α−FeSi obtainedby both methods is givenin Sec. III
2
materials are required. The most developed method of and in Sec. IV we discuss results and make conclusions.
electronic structure calculations, which is based on den-
sity functional theory, is designed and applicable for the
ground state properties only12,19. This gives rise to cer-
II. THE METHODS: GW VERSUS GGA FOR
taindoubtsthattheelectronicbandsobtainedwithinthe
IRON SILICIDES
frameworkdensity functional theory (DFT) by means of
local-(spin)-density approximation with or without gra-
A. Band structure and excitation energies.
dient corrections, can be interpreted as energies of ex-
cited states. Nevertheless,the bandstructure, generated
by Kohn-Sham equations, is ubiquitously exploited as The question if the DFT band structure can be in-
quasiparticle spectrum.Here we will compare the results terpreted as single-electron excitations’ energies have
of calculations within the version of gradient-corrected been discussed repeatedly (see, e.g.,5, review18 and the
local-density approximationto density functional theory book19). The single-electron Green’s function by defi-
(GGA) and GW approximation (GWA)12. In the latter nition describes the transitions between n and (n±1)
GistheGreen’sfunctionandW isthescreenedCoulomb electron states, and the solutions of the Dyson’s equa-
interaction. Both approaches have their own advantages tion provide the excitation energies and their spectral
and disadvantages. The exchange-correlation potential weights. The potential in the Kohn-Sham equation does
used in all modern implementaions of the Kohn-Sham not depend on energy and automatically looses infor-
machinery contains the correlationeffects that are much mation on possible deviations of the spectral weight of
beyond the random phase approximation used in GWA. quasi-particles from unity. Although the DFT based
Nevertheless the DFT based approach has the essential methods have been designed for the investigation of
drawbacks that i) the exchange-correlation potential is the ground-state properties and, it seems, there are no
calculated for the homogeneous electron gas, and the groundsforinterpretingthebandstructure,generatedby
transfer of these results to the inhomogeneous electron the Kohn-Sham equations as the energies of the single-
gasofrealmateralsisnoteasytojustify, andii)itisdif- electron excitations. However, the question about such
ficulttoimpovethecalculationsbyaddinginacontrolled a possibility is not so trivial and cannot be ruled out
waysomecorrections. Theconvincingexampleofitisthe by this argument only. First, one can notice that the
widely used LDA+U approximation, where many differ- same statement holds for the Hartree-Fock-determinant
ent forms of double-counting corrections are in use: the wavefunctionofthegroundstate,forwhichKoopmans20
wayto make it in a controlledwayis not foundyet. The proved that in closed-shell Hartree - Fock theory, the
GWmethoddoesnotcontainthisproblem,itiswellcon- first ionization energy of a molecular system is equal
trolled approximation. The strong advantage of GWA is to the negative of the orbital energy of the highest oc-
that it results in realquasiparticle excitations. However, cupied molecular orbital. There is an analogue of the
it requires much more of computer resources than, say, Koopmans’ theorem for DFT21. Second, Hoenberg-and-
GGA. It is expected that they should produce approxi- Kohn’s proof22 that the total energy of a system with
mately the same band structure, since both approaches interaction is a unique functional of the electron den-
ignore the intraatomic (often strong) correlations. If the sity ρ(r) led Sham and Schlueter23 to the equation,
problem of interest does not require the knowledge of which, at least, in principle, allows calculation of the
the spectral weights and/or is intended to use it as a exchange-correlationpotentialinDFTforanychosenap-
first step for a better approximations, it is preferable proximationfortheelectron-Green’s-functionself-energy
to use GGA. The latter usually is used as a perturba- Σ′. Schematically, G−1 = (G−1 −v )+v −Σ′ gives
0 xc xc
tional input for GWA. For this reason it makes sense to G=G +G (v −Σ′)G. Here G is the Green’s
DFT DFT xc 0
compare both quasiparticle and Kohn-Sham band struc- function of electrons in the Coulomb field of nuclei and
turesandtoanalysethedifferencebetweentheexchange- Hartree potential and the prime in Σ′ means that the
correlation potential and the real part of the GW self Hartree term is subtracted, Σ′ =Σ−v . Express-
Hartree
energy. Since the eigenvalues of the Kohn-Sham (KS) ing then the charge densities in terms of G, ρ = hGi ,
E
equations cannot be interpreted as quasi-particle (QP) i.e.,takingappropriatecontourintegraloverenergyfrom
excitations whereas the band structure generated by the bothsidesandusingtheequalityρDFT (r)=ρ (r)
model genuine
GW method can, we calculate GW QP energies. They onefinds0=hG (v −Σ′)Gi .Thisequation23 es-
DFT xc E
canserveasapredictonofARPESmeasurementsoniron tablishesthecorrespondencebetweentheself-energyand
silicides. the exchange-correlation potential. For this reason one
Thepaperisorganizedasfollows. InSec. IIwepresent can hope that at least for the v found for certain Σ′
xc
the details of our ab initio DFT and GW calculations the self-energy-based and Kohn-Sham eigenvalues coin-
and give brief discussion of their advantages and draw- cide. The problem is, however, that the eigenvalues ε
kn
backs. The comparison of band structure and densitiy and wave functions ϕ (r) of Kohn-Sham equation,
kn
3
p2
+v (r)+v (r)+v (r) ϕ (r)=ε ϕ (r), (1)
e−n Hartree xc kn kn kn
(cid:20)2m (cid:21)
and its self-energy-basedcounterpart,
p2
+v (r)+v (r) Φ (r,E )+ dr ReΣ′(r,r ,E )Φ (r ,E )=E Φ (r,E ), (2)
e−n Hartree kn kn 1 1 kn kn 1 kn kn kn kn
(cid:20)2m (cid:21) Z
represent more differential information than the elec- the excitations. Contrary to that the self-energy does
tron densities and, to the best of our knowledge, a depend on energy giving birth to the spectral weight of
prove of a statement that ε = E does not ex- the GW-QPs Z < 1; the remaining part of the spectral
kn kn ε
ists ( here n labels the bands). To be more pre- weight is shifted to incoherent excitations. It is instruc-
cise, Eq.(2) is an approximate version of the equa- tive to represent the spectral weight in terms of magni-
tion for the functions Φ (r,ω), which diagonalize tudes, encoded in the VASP. Let us write the equations
kn
the Green’s function in GWA at arbitrary energy for the Green’s functions, corresponding to Eqs.(1),(2):
ω, hΦkn(ω)|G−1(ω)|Φk′n′(ω)i = δkk′δnn′(ω−∆kn(ω)),
i.e., it contains energy-dependent self-energy Σ′(r,r1,ω) G−1(E)=G−D1FT(E)−(Σ′(E)−vxc). (3)
and Φ (r ,ω); the eigenvalues, which are poles of the
kn 1
Green’s function, are obtained from the equation Ekn = Within the “one-shot” (G0W0) approximation E = εkn
Re∆ (E ). At last, both equations are highly non- and we obtain
kn kn
linear and uniqueness theorem does not exist for them.
E −ε
kn kn
Particularly, the problem of choice of the physical so- Zkn ≃ . (4)
lution from multiple solutions generated by GWA is dis- ReΣGW(εkn)− vx(GcGA)
cussedinref.31. Thus,atthemomentweareatthestage (cid:16) (cid:17)kn
wheneventheformoftheexchange-correlationpotential This relation is fulfilled by the magnitudes
foraself-energyinGWapproximationisnotderivedand E ,ε , v(GGA) ,Σ (ε ), which are gener-
kn kn xc GW kn
thequestionaboutrelationshipbetweenKohn-Shamand (cid:16) (cid:17)kn
ated by VASP. As seen from Eq.(4) in the form18,
GWA energies remains open. It is interesting, however,
thatmoreaccurateexchange-correlationfunctionalsthan
the widely-used ones can be obtained for evaluation of Ekn =εkn+Zkn ReΣ′GW(εkn)− vx(GcGA) , (5)
the excitation energies: the authors of ref.5 state that, h (cid:16) (cid:17)kni
on the one hand, a surprising degree of agreement be- the role of the coefficient in the first term of the
tweenthe exactground-stateKohn-Shameigenvaluedif- quasiparticle-energy expansion with respect to presum-
ferences andexcitation energies,for excitations from the ably small perturbation ReΣ(ε ) − v , is played by
kn xc
highest occupied orbital to the unoccupied orbitals has the spectral weight. If the perturbation theory works,
beenachievedfor smallsystems; onthe other hand,that E − ε < (v −Σ) everywhere. Our calcula-
kn kn xc kn
“thepopularLDAandGGAfunctionalsarenowherenear tions confirm that this inequality is valid for iron sili-
sufficiently accurate”5. cides. A comparison of behavior of these two differ-
ences, k-dependence of the spectralweightsandananal-
ysis of contributions from different electrons into these
B. Band structure and spectral weights. k-dependences are given in Sec.(IIIC).
The PES-basedexperiments provide also the informa-
III. IRON SILICIDES
tion on the spectral-weights of QPs, Z . As was dis-
ε
cussed above, the “self-energy” in Kohn-Sham equation
The calculations presented in this paper are per-
(1) does not depend on energy and, therefore, corre-
formed using the Vienna Ab initio Simulation Pack-
sponding spectral weight, defined in the perturbation
age (VASP)13 with Projector Augmented Wave (PAW)
theory as Z =[1−∂Σ(E)/∂E]−1 , is always
kλ pseudopotentials14. The valence electron configurations
(cid:16) (cid:17)E=Ekλ
equal to unity. Insufficiency of the DFT-based approach 3d64s2 are taken for Fe atoms and 3s23p2 for Si atoms.
is seen also from the Slu¨ter & Sham’s equation for the One part of calculations is based on the density func-
exchange-correlation potential: since it is based on the tional theory where the exchange-correlation functional
equalities of model and genuine electron densities of is chosen within the Perdew-Burke-Ernzerhoff (PBE)
many-electronsystem,whichareintegralcharacteristics, parametrization15 and the generalized gradient approxi-
it does not provide the equation for spectral weights of mation (GGA).
4
IntheGWpartofcalculationsimplementedinVASP16 the contributions of Fe-ions’ d-states into electron den-
we report only one-shot approximation,so-called G W . sityofstates. ThemagneticmomentM ofFe(1) atom
0 0 Fe1
The GGA Kohn-Sham band structure and eigenfunc- is higher than the free-atom moment, MGGA = 2.52µ
Fe1 B
tions were taken as the input for the GW calculations. and MGWA = 2.55µ what corresponds approximately
Fe1 B
The self-energy Σ is computed as Σ ≈ iGGGAWGGA. to the d5 configuration. The Fe(2)atom has much lower
Throughoutboth GGAandGWA calculationsthe plane Fe Si α−FeSi
3 2
wave cutoff energy is 500 eV, the Brillouin-zone integra- a=5.6˚A 5.65˚A a=2.70˚A 2.69˚A
tionsareperformedonthegridMonkhorst-Pack17special (cid:0) (cid:1) (cid:0) (cid:1)
R Fe(1)−Fe(2) =2.45˚A c=5.13˚A 5.134˚A
points10×10×10forFe Siand12×12×6forα−FeSi (cid:16) (cid:17) (cid:0) (cid:1)
3 2
andGaussbroaderingwithsmearing0.05eVisused. For R(cid:16)Fe(2)−Si(cid:17)=2.45˚A zSi=0.272(0.28)
allGWcalculationsthenumberoffrequenciesis500and R Fe(1)−Si =2.83˚A R(Fe−Si)=2.30˚A
(cid:16) (cid:17)
160 electronic bands are used. We use the complex shift
R(Si−Si)=2.56˚A
of 0.043 eV for the calculation of self-energy.
Table I: Relaxed lattice parameters and the equilibrium dis-
tances between nearest ions. The experimental values23 are
A. Iron Silicides structure
given in brackets.
The calculations are performed for two iron silicides,
Fe3Si and α− FeSi2. The compound Fe3Si belongs moment, MFGeG2A = 1.34µB and MFGeW2A = 1.40µB. As
to DO structural type with the space symmetry group
3 will be seen from the analysis of DOS, the latter mo-
Fm¯3m. The iron atoms have two nonequivalent crys-
ments are formed by the delocalized d-states. The ex-
tallographic positions in fcc lattice, namely, Fe(1)and perimental values reported in works25 and26 are slightly
Fe(2) have different nearest surroundings: Fe(1) has different: Mexp = 1.2µ , Mexp = 2.4µ in Ref.25 and
eightFe(2)nearestneighborswhichformacube,whereas Mexp =1.35Fµe2, Mexp =B2.2µFe1in Ref.26B.
theFe(2) isinthetetrahedralsurroundingofbothSiand Fe2 B Fe1 B
Fe(1) atoms.
The iron bi-silicides have several structural modifi-
cations. The most stable phases are α − FeSi and B. Comparison of GGA and GW densities of
2
β −FeSi phases23,24. The compound α−FeSi has electron states
2 2
tetragonal lattice with P4/mmn space symmetry group
with one molecula per unit cell. Each iron atom here Fig.(1)displayscomparisonofthe GGA andGWden-
is located in the center of cube, consisting of the sili- sities of electronstates (DOS) for Fe Si andα−FeSi .
3 2
con atoms. This structure contains the planes which are The GGA part of results coincides with previous calcu-
formed by only iron and by only silicon atoms. These lationsof Fe Si25,27,28 and α−FeSi 24,27,29. The gen-
3 2
planes are orthogonal to tetragonal axis. Two planes eralfeaturesoftheDOSinbothcompoundsandapprox-
formed by silicon atoms are separated by wide empty imations are that the bands in the interval(−5,+5) eV
cavity, which does not contain the iron atoms. Accord- aroundFermienergyareformedbythed-electronsofiron
ing to our calculations this cavity is expected to play with a slightadmixture ofs− and p−electronsofSi and
essential role in the transport properties of α−FeSi . Fe. The Si valent s- and p−electrons are delocalized in
2
The rhombohedral cell have been used for the Fe Si the wide energy region with smeared maximum around
3
calculations. The equilibrium parameters and the dis- −4eVinbothcompounds. GWAchangesthe intensities
tances between nearest Fe and Si atoms for the Fe Si of the peaks mainly in the energy region deeply under
3
and α−FeSi structures have been found from the full Fermi surface, but the changes in Fe Si and α−FeSi
2 3 2
optimizationofthestructuregeometrieswithinGGAand are different. If in the GGA DOS of the Fe Si the peak
3
are shown in Tab. (I). The GW calculation have been locatedatE ∼−3.5 eVis shifted by GWA for about0.5
performed with the same structural parameters. eVandmadesharper,theGGApeaksinα−FeSi DOS
2
Both spin-polarized GGA and GW result in metallic inapproximatelythe sameenergyregion(I,II inbottom
states, ferromagnetic for Fe Si and paramagnetic with panel of Fig.(1)) is washed out within the GW calcula-
3
zero-spin Fe atoms for α−FeSi . For this reason all tions. ThereisadifferenceintheGWAvsGGAchanges
2
further calculations for α−FeSi have been performed in DOS for spin “up”(majority) and “down”: while the
2
within non-spin-polarized version of VASP. The struc- GW spin-majoritystates remainalmost untouched com-
turalinequivalence of the Fe-atoms’surroundingsin the paredtoGGA,thespin-minorityGWApeaksareshifted
Fe Sireflectsitselfinbothmagneticmomentvaluesand towards the Fermi energy.
3
Aswasmentionedabove,differentchemicalsurroundings atomsandthetetrahedralonefortheFe(2) atomsbythe
oftheFeatompositions,thecubiconeforFe(1) byFe(2)
5
10 spin up GGA 666 I GGA
G0W0 II
G0W0
555
5
DOS (arb. unit) −05 DOS (arb. unit)DOS (arb. unit)DOS (arb. unit)234234234
spin down 111
−10
000
−4 −3 −2 −1 0 1 2 −−−555 −−−444 −−−333 −−−222 −−−111 000 111 222 333 444 555
Energy (eV) Energy (eV)
Figure 1: The left panel: The spin-polarised density of electron states for Fe Si in the interval of energies [-4eV, 2eV]. Since
3
the GGA and the GW approximations producedifferent Fermi energies, εGGA(Fe Si)=7.88 eV and εGW(Fe Si)=8.44 eV,
F 3 F 3
theplotsarealigned forcomparison byplacingthezerointheenergyaxisofbothplotsatFermienergy. Therightpanel: The
DOSforα−FeSi intheenergyinterval[-5,5]eVwiththesametypeofalignmentoftheenergyaxes;εGGA(α−FeSi )=9.34
2 F 2
eV and εGW(α−FeSi )=10.03 eV.
F 2
Fe(1) and Si atoms as nearest neighbors reflect them- is illustrated in Fig.(2).
selves in different behavior of partial d-electron DOS. It
The general shape of partial d-DOS is not changed netism supression would be a formation of the low-spin
by GWA. The d-states of E symmetry of Fe(1) as well state within the localized d-electron picture. This state
g
as Fe(2) do not contribute into DOS at the Fermi level couldbeformedifthecrystal-fieldsplittingofthed-shell
in both in LDA and GWA (panel (A), (C), Fig.(2)). As wasstrongerthanthe Hund-exchangeone. However,the
seenfromFig.(2),panels(A),(B),thecontributiontothe densityofd-electronstatesdoesnotcontainbrightpeaks
magnetic moment on Fe(1) from E -orbitals (positive which might be interpreted as former d-levels splitted in
g
DOS in (A)) compensatedby the contributionfromT - the crystal field. Thus, one can conclude that if GGA
2g
orbitals ( the negative middle peak in (B)). This means and GWA are good approximations for α−FeSi2, the
that the T -orbitalsare responsible for formationof the keymechanismofthemagnetismdestructioninthiscom-
2g
large quasi-localized magnetic moment at Fe(1) atoms. pound is the delocalization of d-electrons.
Itisalsointerestingthattheusualsplittingofthed-shell The most pronounced changes in GWA compared to
intoT2g-andEg-symmetriesisviolatedherebythecon- GGA are experienced by T2g electrons. It is illustrated
tribution from exchange interaction: as seen from (A) onFig.(2), panels (E,F)for α−FeSi2: twopeaks (I and
and(B)panelsthe E -peakisin-betweentwoT -peaks. II) seen in the GGA DOS which have been mentioned
g 2g
A different behavior is demonstrated by d-electrons of aboveandwashedoutintheGWAareformednamelyby
Fe(2): its d-DOS is spead in a wide region of energies. T2g electrons. The same is valid for the “down” spin T2g
The d-electrons of both T2g- and Eg- symmetries con- states in the vicinity of the Fermi level in Fe3Si (panel
tribute to formation of magnetic moment of Fe(2). The (D) in Fig.(2)). At the same time the well-expressed lo-
delocalization of Fe(2) d-electrons reflects itself in the calized peaks formed by Eg orbitals remain intact. This
smaller moment than the one on Fe(1). The d-DOS of isunderstandable: the GWAis the approximationwhich
ironinα−FeSi ,whereFeatomsalsohaveSiatomsas takes into account the effects of screening which is pro-
2
neighbors,displaysbehaviorsimilartoFe(2) partialDOS vided namely by the delocaized electrons. These effects
ford-electronsinFe Si. Howeveronlythesedelocalized aretakenintoaccountwithinthe GGA andGWAdiffer-
3
ently and, therefore, one can expect that the difference
electrons are not able to form magnetism in α−FeSi .
2
Indeed, the criterium for magnetism formation for the will be more pronounced for these types of electrons.
delocalizedelectronsis muchharderto fulfilthan forthe
case of localized electrons whose role is played by T -
2g
electrons of Fe(1) in Fe Si . The absence of magnetism C. Comparison of GGA and GW band structures
3
in α−FeSi is easy to understand on the basis of well- and analysis of the spectral weights
2
knownStoner’smodelforamagnetismofthedelocalized
electrons: the criterium Jg(εF) > 1 is not fulfilled since The Kohn-Sham band structure calculated within
the density of electron states g(ε ) at the Fermi energy GGA does not differ from the known results for
F
εF istoosmall(hereJ isexchangeintegralbetweendelo- Fe3Si25,28 and for α − FeSi224,29,30. Here we com-
calizedelectrons). Analternativemechanismofthemag- pare the GGA and GW bands in some of symmet-
6
A B
6 Fe1 Eg GGA 6 Fe1 T2g GGA
G0W0 G0W0
4 4
unit) 2 unit) 2
b. b.
ar ar
S ( 0 S ( 0
O O
D D
p p
−2 −2
−4 −4
−4 −3 −2 −1 0 1 2 −4 −3 −2 −1 0 1 2
Energy (eV) Energy (eV)
C D
4 4
GGA GGA
Fe2 Eg Fe2 T2g
3 G0W0 3 G0W0
2 2
pDOS (arb. unit)−−0121 pDOS (arb. unit)−−0121
−3 −3
−4 −4
−5 −5
−4 −3 −2 −1 0 1 2 −4 −3 −2 −1 0 1 2
Energy (eV) Energy (eV)
E F
4 4
GGA GGA
Fe Eg Fe T2g
G0W0 G0W0
3 3 II
nit) I nit)
u u
b. b.
S (ar2 II S (ar2 I
O O
D D
p p
1 1
0 0
−5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5
Energy (eV) Energy (eV)
Figure 2: The partial spin-polarised d-electron DOS for Fe Si (upper four panels, (A),(B),(C),(D)) and α−FeSi (lower two
3 2
panels, (E),(F)) . Left panels display the contribution to DOS from dz2− and dx2−y2− states (Eg), while the right ones show
thecontribution from dxy− , dxz− and dyz− states (T2g).
ric directions. Figs.(3) and (4) show the band struc- the Fe Si in Fig.(3, left) near Γ−point, the dublet E
3 g
ture for Fe Si in the directions ΓX and ΓL, and for and the triplet T , are formed by the d-electrons of Fe
3 2g
α−FeSi in the directions ΓX , ΓM and ΓZ, where atoms. First empty band (A ) near Γ−point is formed
2 1g
Γ = (0,0,0), X = (2π/a)(1,0,0), L = (π/a)(1,1,1), by the s-states of both Fe and Si atoms.
M =(2π/a)(1,1,0), Z =(2π/c)(0,0,1).
The GGA and GWA band structures for α−FeSi
2
We report here the results of the comparison only for are shown in Fig.(4). Here the closest to the Fermi
thepartoftheGGAandGWAbandstructureswhichare energy filled bands are formed by the d-orbitals of Fe
withinseveralelectron-voltsvicinityoftheFermienergies atoms near Γ−point (D4h group) are B1g( dx2−y2), A1g
(remindthatGGAandGWAgeneratedifferentFermien- (dz2), the dublet Eg (dxz,dyz) and B2g( dxy). The low-
ergies,see the captureto Fig.(1)). The bandsarenamed est shown band (A ) is formed by the s-electrons of Fe
2u
inaccordancewiththeirsymmetriesintheΓ−point. The and p-electrons of Si. Again we observe the same ten-
closestto the Fermi energythree filled spin-up bands for dency: namelythe delocalizedstates,inthis case,s-and
7
Figure4: Thebandstructureofα−FeSi ;zerointheenergy
2
axisofGGAandGWAplotsischosenatcorrespondingFermi
energies.
“up” shown in Fig.(3).
FirstgeneralfeatureseenfrombothFig.(5)andFig.(6)
is that the spectral weights Z , that within the Kohn-
kn
Sham approach are equal to one by construction, within
the GWA are strongly decreased. Second feature, some-
what surprising, is that the non-monotonous behaviour
of Z arises everywhere where an admixture of delocal-
kn
ized electron states is present. Indeed, one observesthat
near the X point of the Brillouin band it is the admix-
(1)
ture of s- and p-electrons to d-states for the band T
2g
of Fe Si causes the changes in the Z ; the difference
3 kn
ReΣ′−v ischangedfasterthanE −ε (seepanelA
xc kn kn
atFig.(5)). The picture forthe empty bandsis different:
the s- and p-states of Si and d-states of Fe are mixed
in the center of the band, whereas the contribution of
Figure 3: The GGA and GW spin-up (top) and spin-down
the d-states is increased around boundary points X and
(bottom) bands for Fe Si. Zero in the energy axis of GGA
3 L. The quasiparticle energies of the excited states are
and GWA plots is chosen at corresponding Fermi energies.
lower than their Kohn-Sham counterparts. Again, the
largestdifferenceisobservedinthethosepartsoftheen-
ergy spectrum where the contributions from s− and p−
p-states,showthelargestdifferenceinGGAandGWA.If
states becomes significant.
thebandformedbyd-electronsclosetotheFermienergy
As seen from Fig.(6), the spectral weights Z of the
remain almost untouched, the lowest sp band is shifted kn
quasiparticles in α − FeSi also show a strongly non-
inGWAby∼1eV.Firstemptyband(E )nearΓ−point 2
u
monotonousdependence onk. For example,the spectral
is formed by the p-states of both Fe and Si atoms. In
weight Z of the B band in the direction ΓX (sec-
generaltheGGAvsGWAshiftisaroundhalfofelectron- k,B1g 1g
ond from top panel in Fig.(6,B) shows sharp changes in
voltfor the excited states,while the bandshape remains
the interval 0.55 < Z < 0.8 due to much faster de-
the same. As seenatthe rightpanelofFig.(4)inthe ΓZ k,B1g
pendence on k of the difference (Σ′−v ) than of the
direction, the purely d-bands are completely flat, while xc
(E −ε ) one. The lower part of the panel B in
the dispersionwhicharisesnear the boundaries is due to QP GGA
Fig.(6) explains the reason: again, the closer to the X
the admixture of s and p states. Analogous admixture
point the higher contribution from the delocalized s-
of sp electrons is observed around the boundary points
electrons. This confirms, thus, the general tendency no-
X and M. This is easy to understand since the motion
ticed above.
alongΓX directioninthe k-spacecorrespondsto motion
fromtheFeatomsplanetotheplaneofSiatomsinreal
space; Si atoms do not have d-electrons and can admix
the s- and p-states only. IV. DISCUSSION AND CONCLUSIONS
Here we return to the question asked in Sec.(IIB),
whichelectronsmainly areresponsibleforthe deviations The comparison of the band structures obtained in
of GWA fromGGA results. Let us consider the example the ab initio calculations within the VASP for Fe Si
3
of two E and T filled bands of quasiparticles for spin and α−FeSi in GGA and G W A shows that in gen-
g 2g 2 0 0
8
Figure 6: Theα−FeSi bands. The notations are thesame
2
as in Fig.(5), but the filled bands are imaged at the panels
A,B,C, while the emptyone is shown at D panel.
Figure 5: Fe Si quasiparticle filled bands(top panels), their
3
spectral weights Z (second from the top), the k-dispersion
k The other astonishing moment following from GWA cal-
ofnumeratoranddenominatorofEq.(4)(thirdfromthetop)
culations is that in spite of the fact that the electrons
and the character coefficients C (bottom panels, see text).
k
in both systems are well delocalized the spectral weights
The letters label different characters: d and p stand for the
Z of the quasiparticlebands E are decreasedalmost
empty and filled circles correspondingly, the black squares kn kn
denote s character. The filled bands are shown in (A) and by a half. One could assume that the reason for that is
(B)panels,while(C),(D)displaytheemptyones. Thedashed that the basis set is not sufficiently large, however, the
andsolidlineson(A)paneldenotethenon-degeneratebands results are not changed with further increase of number
Eg. The dashed and solid lines on (B), (D) panel denote of bands which are taken into account. This means that
non-degenerate and doubledegenerate bandsT2g. the GWA indeed shifts the remaining part of the weight
to the incoherent part of excitations.
Both GGA and GWA band structures and, corre-
eral the bands shape is similar. The difference between spondingly, the density of electron states, show that the
GGA and GWA bands becomes more pronounced in d-electrons of those Fe atoms which have Si nearest
those parts of the Brillouin zone where the delocalized neighbors, namely, Fe(2) atoms for Fe3Si and all
states give noticible contribution into quasi-particle en- Fe atoms in α − FeSi2, are more delocalized than
ergies. This observation is somewhat unexpectable since the d-electrons of Fe(1) atoms in Fe3Si which have
both approximations are designed for description of well only the other Fe atoms as neighbors. The partial
delocalized (Fermi-liquid like) electrons. There are at density of states of Fe(1) d-electrons with Eg and T2g
leasttwo sourceswhich cancontribute to this difference. symmetry in the Γ point has well-expressed peaks,
First is the fact that the standard GGA is not free from the positions of which could be ascribed to a splitting
the self-interactionwhile GWA takes into accountFermi in the crystal field. However, this splitting does not
statistics by construction. Second source is that GGA correspond to the standard picture of the quasiatomic
andGWAarequitedifferentapproximations,aswasdis- levels ε0T2g,σd†tσdtσ +ε0Eg,σd†eσdeσ, from which the bands
cussed in Introduction. are formed, the interactions renormalize these “levels”
9
ε0 → ε in such a way that their sequence terms the vertex corrections should be applied remains
T2g,σ T2g,σ
becomes ε <ε ,ε <ε highly non-trivial. This have been demonstrated, e.g.,
T2g,↑ T2g,↓ Eg,↑ T2g,↑.
We performed the calculations within the one-shot in ref.35, where it was shown that the vertex corrections
GW approximation (G W ), and it is worth to notice into effective interaction W may improve the results
0 0
why this approximation is preferable compared to the while turning on the corrections in the self-energy may
fully self-consistent GWA. First, it often gives the lead to unphysical quaiparticle dispersion. At last, the
results which describe PES experiments better (see, calculations within GWA with the vertex corrections
e.g., refs.32,33,34). Second argument is connected with demand for much stronger computer resources than
the nature of the self-consistency. Each term in the even scGWA. The ARPES experiments on iron silicides
self-consistent perturbation theory (scPT) corresponds would be of great help in further understanding of these
to whole series in the non-scPT. If we consider these compoundsand,possibly,couldmotivatemoreadvanced
corrections, we find that a part of corrections in each theoretical approachto the problem.
order of the non-scPT changes the matrix elements of
the Coulomb interaction, actually, via corrections to the
wave functions. The self-consistency loop changes the
eigenfunctions which diagonalize the GW equation for
Acknowledgement
the Green’s functions, and, correspondingly, dresses the
matrix elements of Coulomb interaction by the RPA
(random phase approximation) graphs only. All other TheauthorsthankforsupportfromRFBRNo. 14002-
contributions to the wave functions, which arise in the 00186,thePresidentofRussiunFederationGrants(NSH-
same orders of non-scPT, and, correspondingly, to the 2886.2014.2 and NSH-924.2014.2) and Physics Depart-
matrix elements of Coulomb interaction, are not taken ment of RAS program ”Electron correlations in systems
into account. However, in the case of non-homogeneous with strong interaction”. The calculations were per-
electrongasinreal-material,wherethegellium-likeparts formedwiththe computer resourcesofNRC “Kurchatov
can coexist with strongly correlatedliquid, the situation Institute” (ui2.computing.kiae.ru). I.S. thanks A.Ruban
it not alway obvious. The question how and to which for useful discussion.
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