Ab-initio calculations of the optical and magneto-optical properties of moderately correlated systems: accounting for correlation effects

0.22 MB·English

by  A. Perlov

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AB INITIO CALCULATIONS OF THE OPTICAL
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2 AND MAGNETO-OPTICAL PROPERTIES OF

] MODERATELY CORRELATED SYSTEMS:
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ACCOUNTING FOR CORRELATION EFFECTS
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a A.Perlov1,S.Chadov1,H.Ebert1,L.Chioncel2,A.Lichtenstein2,M.Katsnelson3
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- 1UniversityofMunich,Butenandstrasse5-13,D-81377,Munich,Germany
d 2UniversityofNijmegen,NL-6526EDNijmegen,TheNetherlands
n 3UppsalaUniversity,P.O.Box530,S-75121Uppsala,Sweden
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Abstract Theinfluenceofdynamicalcorrelationeffectsonthemagneto-opticalproperties
v
5 offerromagneticFeandNihasbeeninvestigated. Inadditionthetemperature
4 dependenceoftheself-energyanditsinfluenceontheDOSandopticalconduc-
5 tivityisconsidered. Magneto-opticalpropertieswerecalculatedonthebasisof
1 the one-particle Green’s function, which was obtained from the DMFT-SPTF
0 procedure. Itisshownthatdynamicalcorrelationsplayaratherimportantrole
4 inweaklycorrelatedFeandsubstantiallychangethespectraformoderatelycor-
0 relatedNi. Magneto-optical propertiesobtained forbothsystemsarefoundin
/
t betteragreementwithexperimentthanbyconventionalLDAcalculations.
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1. Introduction
n
o
Much information on the electronic structure of magnetic solids is gained
c
: by optical and magneto-optical measurements, being useful tools for analyz-
v
ing the dispersion of (quasi-particle) bands. However, measured optical and
i
X
magneto-opticalspectracanhardlybeinterpretedwithoutaccompanyingtheo-
r reticalcalculations. Forthispurpose oneingeneralhastosolveacorrespond-
a
ing many-electron problem, which is impossible without the use of more or
lesssevereapproximations. Formaterialswherethekineticenergyoftheelec-
tronsismoreimportantthantheCoulombinteractions,themostsuccessfulfirst
principlesmethodistheLocal(Spin-)DensityApproximation(L(S)DA)tothe
DensityFunctionaltheory(DFT)[1],wherethemany-bodyproblemismapped
ontoanon-interacting systemwithaone-electron exchange-correlation poten-
tial approximated by that of the homogeneous electron gas. For the last two
2
decades abinitio calculations oftheoptical andmagneto-optical properties of
solidsbasedonthisapproximation yieldedagoodbasisforsuchaninterpreta-
tion,oftenleading toaquantitative agreement betweentheoretical andexperi-
mentalspectra. Thesituationisverydifferentwhenweconsidermorestrongly
correlated materials, (systems containing f and d electrons) since in all the
calculations the LDA eigen-energies are implicitly interpreted to be the one-
particle excitation energies of the system. It is well known that there are two
possible sources of error connected with that approach: Firstly, the LDA pro-
vides only anapproximate expression forthe(local) exchange-correlation po-
tential. Secondly, even with the exact exchange-correlation potential at hand,
one is left with the problem that there is no known correspondence between
theKohn-Shameigen-energies andtheone-particle excitationenergies [2–5].
Foranexactdescription oftheexcitation energies thenon-local self-energy
hastobeconsidered. This,however,constitutesamany-bodyproblem. There-
fore,DFT-LDAcalculations mustbesupplemented bymany-body methods to
arriveatarealisticdescription oftheone-particle excitationsincorrelatedsys-
tems. Togive anexample, let usmention the GWapproximation [6] which is
well suited for the case of insulators and semi-conductors and has also been
applied successfully to transition metals [6–9]. Another approach is to con-
sider the Hubbard-type models where those Coulomb-interaction terms are
included explicitly that are assumed to be treated insufficiently within DFT-
LDA. Already the simplest Hartree-Fock like realization of such an approach
calledLDA+U[10]schemeallowedtoimproveconsiderablythedescriptionof
theopticalandmagneto-optical spectraofstronglycorrelated systems(mostly
containing rare earths elements [11, 12]). Themainadvantage of theLDA+U
scheme is the energy independence of the self-energy which allows to use
only slightly modified standard band structure methods for calculating opti-
cal and magneto-optical spectra. On the other hand the scheme works rather
good only for extremely correlated systems, where Coulomb interactions (U)
prevail considerably over the kinetic energy (bandwidth W). For moderately
correlated systems (U W) which applies for most 3d and 5f elements and
≈
their compounds one has to take into account a non-Hermitian energy de-
pendent self-energy to get a reasonable description of the electronic struc-
ture. Nowadays there are several approaches available to deal with this sit-
uation. ThemostadvancedoneistheDynamicalMean-FieldTheory(DMFT)
[13]. DMFT is a successful approach to investigate strongly correlated sys-
temswithlocal Coulombinteractions. Itusestheband structure results calcu-
lated,forexample,withinLDAapproximation, asinputandthenmissingelec-
tronic correlations are introduced by mapping the lattice problem onto an ef-
fectivesingle-siteproblemwhichisequivalenttoanAndersonimpuritymodel
[14]. Duetothisequivalence avarietyofapproximative techniques havebeen
usedtosolvetheDMFTequations, suchasIteratedPerturbation Theory(IPT)
Opticalandmagneto-opticalpropertiesofmoderatelycorrelatedsystems 3
[13, 15], Non-Crossing Approximation (NCA)[16, 17], numerical techniques
like Quantum Monte Carlo simulations (QMC) [18], Exact Diagonalization
(ED)[15, 19], Numerical Renormalization Group (NRG)[20], or Fluctuation
Exchange (FLEX) [21–23]. The DMFT maps lattice models onto quantum
impurity models subject toaself-consistency condition insuch awaythat the
many-body problem for the crystal splits into a single-particle impurity prob-
lem and a many-body problem of an effective atom. In fact, the DMFT, due
to numerical and analytical techniques developed to solve the effective im-
purity problem [13], is a very efficient and extensively used approximation
for energy-dependent self energy Σ(ω). At present LDA+DMFT is the only
available ab initio computational technique which is able to treat correlated
electronic systemsclosetoaMott-Hubbard MIT(Metal-Insulator Transition),
heavyfermionsandf-electronsystems.
Concerning the calculation of the optical spectra we have to face the fol-
lowingproblem: oneparticlewavefunctions arenotdefinedanymoreandthe
formalismhastoappliedintheGreenfunctionrepresentation. Sucharepresen-
tation has already been derived [24] and successfully applied for calculations
in the framework of Korringa-Kohn-Rostoker (KKR) Green-function method
for LSDA calculations. The only drawback of such an approach is that it is
highlydemandingastobothcomputational resourcesandcomputational time.
Inthispaperweproposeasimplifiedwaytocalculate opticalandmagneto-
opticalproperties ofsolidsintheGreenfunction representation basedonvari-
ationalmethodsofbandstructurecalculations.
Thepaperisorganized asfollowing: insection 2theformalism forGreen’s
function calculations of optical and magneto-optical properties that account
formany-body effects through aneffectiveself-energy ispresented. Then,the
DMFT-SPTF method for the calculation of the self-energy is considered. In
section 3 the obtained results of our calculations for Fe and Ni are discussed
andcompared withexperimental ones. Thelastsection 4contains theconclu-
sionandanoutlook.
2. Green’s function calculations of the conductivity tensor
Optical properties of solids are conventionally described in terms of either
the dielectric function or the optical conductivity tensor which are connected
viathesimplerelationship:
iω
σ (ω) = (ε (ω) δ ). (1)
αβ αβ αβ
−4π −
The optical conductivity is connected directly to the other optical properties.
Forexample, theKerrrotation θ (ω)andso-called Kerrellipticity ε (ω)for
K K
4
smallanglesand ε ε canbecalculated usingtheexpression [25]:
xy xx
| |≪| |
σ (ω)
xy
θ (ω)+iε (ω) = − . (2)
K K 1/2
σ (ω) 1+ 4πσ (ω)
xx ω xx
h i
Thereflectivitycoefficient risgivenby
(n 1)2+k2
r = − (3)
(n+1)2+k2
with n and k being the components of the complex refractive index, namely
refractive and absorptive indices, respectively. They are connected to the di-
electricfunction via:
n+ik =(ε +iε )1/2 . (4)
xx xy
Microscopiccalculations oftheopticalconductivity tensorarebasedonthe
Kubolinearresponse formalism[26]:
1 0
σ (ω) = dτe−i(ω+iη)τ [J (τ),J (0)] (5)
αβ β α
−¯hωV h i
Z−∞
involving the expectation value of the correlator of the electric current opera-
tor J (τ). In the framework of the quasiparticle description of the excitation
α
spectra of solids the formula can be rewritten in the spirit of the Greenwood
approach andmakinguseoftheone-particle Greenfunction G(E):
i¯h ∞ ∞
σ (ω) = dE dE′f(E µ)f(µ E′)
αβ π2V − −
Z−∞ Z−∞
Tr ˆj G(E′)ˆj G(E)
α β
ℑ ℑ +
"(E′ En+iη)(h¯ω+E E′o+iη)
− −
Tr ˆj G(E′)ˆj G(E)
β α
ℑ ℑ , (6)
(E′ En+iη)(h¯ω+E′ Eo+iη) #
− −
where G(E)standsfortheanti-HermitianpartoftheGreen’sfunction,f(E)
ℑ
istheFermifunctionandV isthevolumeofasample. Takingthezerotemper-
aturelimitandmakinguseoftheanalytical properties oftheGreen’sfunction
onecangetasimplerexpressionfortheabsorptive(anti-Hermitian)partofthe
conductivity tensor:
σ(1)(ω)= 1 EF dE tr ˆj G(E)ˆj G(E +h¯ω) . (7)
αβ πω αℑ βℑ
ZEF−ω h i
Opticalandmagneto-opticalpropertiesofmoderatelycorrelatedsystems 5
The dispersive part of σ (ω) is connected to the absorptive one via a
αβ
Kramers-Kronigrelationship.
The central quantity entering expression Eq.(7) is the one-particle Green’s
functiondefinedasasolution oftheequation:
[Hˆ +Σˆ(E) E]Gˆ(E) = Iˆ, (8)
0
−
whereHˆ isaone-particleHamiltonianincludingthekineticenergy,theelectron-
0
ionCoulombinteractionandtheHartreepotential,whiletheself-energyΣˆ(E)
describesallstaticanddynamiceffectsofelectron-electron exchange andcor-
relations. The L(S)DA introduces the self-energy as a local, energy indepen-
dent exchange-correlation potential V (r). As the introduction of such an
xc
additional potential does not change the properties of Hˆ we will incorporate
0
this potential to Hˆ and subtract this term from the self-energy operator.
LDA
This means that the self energy Σ used in the following is meant to describe
exchangeandcorrelation effectsnotaccounted forwithinLSDA.
With a choice of the complete basis set i the Green’s function can be
{| i}
represented as:
G(E) = i G (E) j , (9)
ij
| i h |
ij
X
withtheGreen’smatrixG beingdefinedas
ij
−1
G (E) = iHˆ j E ij + iΣˆ(E)j . (10)
ij
h | | i− h | i h | | i
h i
Dealing with crystals one can make use of Bloch’s theorem when choosing
basicfunctions ik . Thisleadstothek-dependent Green’sfunctionmatrix
| i
Gk(E) = [Hk EOk +Σk(E)]−1 . (11)
ij ij − ij ij
Introducing theanti-Hermitian partoftheGreen’sfunction matrixas
i
k k k
(E) = [G (E) G (E)] (12)
Gij 2 ij − ji
andtakingintoaccounttheabovementionedtranslationalsymmetryweobtain
thefollowingexpression fortheabsorptive partoftheoptical conductivity:
σabs = 1 EF dE d3k α(k,E) β(k,E +h¯ω) (13)
αβ πω Jij Jji
ZEF−¯hω Z ij
X
with
Jiαj(k,E) = Gikn(E)hnk|ˆjα|jki (14)
n
X
6
The efficiency and accuracy of the approach is determined by the choice
of ik . One of the computationally most efficient variational methods is the
| i
Linear Muffin-Tin Orbitals method [27] which allows one to get a rather ac-
curate description of the valence/conduction band in the range of about 1 Ry,
which is enough for the calculations of the optical spectra (¯hω < 6 8 eV).
−
This method has been used in the present work. A detailed description of the
application oftheabovesketchedapproachintheframeworkofLMTOcanbe
foundelsewhere[28].
Calculation of the self-energy
The key point for accounting of many-body correlations in the present ap-
proach isthechoice ofapproximation fortheself-energy. Asitwasdiscussed
intheIntroductiononeofthemostelaboratedmodernapproximationisDMFT.
For the present work we have chosen one of the most computationally ef-
ficient variants of DMFT: Spin polarized T-matrix plus fluctuation exchange
(SPTF)approximation [23], whichisbased onthegeneral many-body Hamil-
tonianintheLDA+Uscheme:
H = H +H
t U
Ht = tλλ′c+λσcλ′σ
λλ′σ
X
1
HU = 2 λ1λ2|v|λ′1λ′2 c+λ1σc+λ2σ′cλ′2σ′cλ′1σ, (15)
{λXi}σσ′(cid:10) (cid:11)
where λ = im are the site number (i) and orbital (m) quantum numbers,
σ = , is the spin projection, c+,c are the Fermion creation and annihila-
↑ ↓
tion operators, H is the effective single-particle Hamiltonian from the LDA,
t
corrected for the double-counting of average interactions among correlated
electrons as it will be described below. The matrix elements of the screened
Coulombpotential aredefinedinthestandard way
12 v 34 = drdr′ψ∗(r)ψ∗(r′)v r r′ ψ (r)ψ (r′), (16)
h | | i 1 2 − 3 4
Z
(cid:0) (cid:1)
where we define for briefness λ 1 etc. A general SPTF scheme has been
1
≡
presented recently [23]. For d electrons in cubic structures where the one-
site Green function is diagonal in orbital indices the general formalism can
be simplified. First, the basic equation for the T-matrix which replaces the
effectivepotential intheSPTFapproach reads
13 Tσσ′(iΩ) 24 = 13 v 24 1 13 v 56
h | | i β h | | i×
D (cid:12) (cid:12) E Xω X56
(cid:12)(cid:12) G(cid:12)(cid:12)σ(iω)Gσ′(iΩ iω) 56 Tσσ′(iΩ) 24 , (17)
5 6 −
D (cid:12) (cid:12) E
(cid:12) (cid:12)
(cid:12) (cid:12)
Opticalandmagneto-opticalpropertiesofmoderatelycorrelatedsystems 7
where ω = (2n+1)πT are the Matsubara frequencies for temperature T
≡
β−1 (n = 0, 1,...).
±
At first, we should take into account the “Hartree” and “Fock” diagrams
withthereplacement ofthebareinteraction bytheT-matrix
Σ(TH)(iω) = 1 13 Tσσ′(iΩ) 23 Gσ′(iΩ iω)
12,σ β 3 −
XΩ X3σ′ D (cid:12) (cid:12) E
1 (cid:12) (cid:12)
Σ(TF)(iω) = 13(cid:12) Tσσ(iΩ)(cid:12) 32 Gσ(iΩ iω) .
12,σ −β h | | i 3 −
Ω 3
XX
(18)
NowwerewritetheeffectiveHamiltonian(15)withthereplacement 12 v 34
h | | i
by 12 Tσσ′ 34 inH . Toconsider thecorrelation effects described dueto
U
P-HDcha(cid:12)nnel (cid:12)weEhave to separate density (d) and magnetic (m)channels as in
(cid:12) (cid:12)
Ref.[21(cid:12)] (cid:12)
1
d = c+c +c+c
12 √2 1↑ 2↑ 1↓ 2↓
(cid:16) (cid:17)
1
m0 = c+c c+c
12 √2 1↑ 2↑− 1↓ 2↓
(cid:16) (cid:17)
m+ = c+c
12 1↑ 2↓
m− = c+c . (19)
12 1↓ 2↑
ThentheinteractionHamiltoniancanberewritteninthefollowingmatrixform
1
H = Tr D+ Vk D+m+ V⊥ m−+m− V⊥ m+ , (20)
U 2 ∗ ∗ ∗ m ∗ ∗ m ∗
(cid:16) (cid:17)
where means the matrix multiplication with respect to the pairs of orbital
∗
indices, e.g.
V⊥ m+ = V⊥ m+ .
m ∗ 11′ m 11′,22′ 22′
(cid:16) (cid:17) X34 (cid:16) (cid:17)
Thesupervector Disdefinedas
d+
D = d,m0 ,D+ = ,
m+
(cid:16) (cid:17) (cid:18) 0 (cid:19)
andtheeffectiveinteractions havethefollowingform:
V⊥ = 12 T↑↓ 2′1′
m 11′,22′ −
(cid:16) (cid:17) D (cid:12) (cid:12) E
Vdd Vdm (cid:12) (cid:12)
Vk = (cid:12) (cid:12)
Vmd Vdd
!
8
V1d1d′,22′ = 12 12 Tσσ′ 1′2′ − 21 12|Tσσ|2′1′
V1m1′m,22′ = 21Xσσ′ Dσσ′ (cid:12)(cid:12)(cid:12)12 T(cid:12)(cid:12)(cid:12)σσ′ E1′2′ −Xσ21(cid:10) 12|Tσσ|(cid:11)2′1′
Xσσ′ D (cid:12)(cid:12) (cid:12)(cid:12) E Xσ (cid:10) (cid:11)
(cid:12) (cid:12)
V1d1m′,22′ = V2m2′d,11′ =
1
[ 12 T↑↑ 1′2′ 12 T↓↓ 1′2′ 12 T↑↓ 1′2′
2 − −
D (cid:12) (cid:12) E D (cid:12) (cid:12) E D (cid:12) (cid:12) E
+ 12 T(cid:12)↓↑ 1(cid:12)′2′ 12 T(cid:12)↑↑ 2(cid:12)′1′ + 12 T(cid:12)↓↓ 2(cid:12)′1′ ]. (21)
(cid:12) (cid:12) − (cid:12) (cid:12) (cid:12) (cid:12)
D (cid:12) (cid:12) E D (cid:12) (cid:12) E D (cid:12) (cid:12) E
To calculate the part(cid:12)icle-(cid:12)hole (P-H) co(cid:12)ntrib(cid:12)ution to the (cid:12)elect(cid:12)ron self-energy
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
wefirsthave towritethe expressions forthe generalized susceptibilities, both
transverse χ⊥ andlongitudinal χk. Onehas
−1
χ+−(iω) = 1+V⊥ Γ↑↓(iω) Γ↑↓(iω), (22)
m ∗ ∗
h i
where
Γσσ′ (τ) = Gσ(τ)Gσ′( τ)δ δ (23)
12,34 − 2 1 − 23 14
is an “empty loop” susceptibility and Γ(iω) is its Fourier transform, τ is the
imaginary time. The corresponding longitudinal susceptibility matrix has a
morecomplicated form:
χk(iω) = 1+Vk χk(iω) −1 χk(iω), (24)
∗ 0 ∗ 0
h i
andthematrixofthebarelongitudinal susceptibility is
1 Γ↑↑+Γ↓↓ Γ↑↑ Γ↓↓
k
χ = − , (25)
0 2 Γ↑↑ Γ↓↓ Γ↑↑+Γ↓↓ !
−
inthedd-,dm0-,m0d-,andm0m0-channels(d,m0 = 1,2inthesupermatrix
indices). An important feature of these equations is the coupling of longitu-
dinal magnetic fluctuations and ofdensity fluctuations. Itisnot present inthe
one-bandHubbardmodelduetotheabsenceoftheinteractionofelectronswith
parallel spins. For this case Eqs. (22) and (24) coincide with the well-known
resultofIzuyamaet.al.[29].
Nowwecan write the particle-hole contribution to the self-energy. Similar
toRef.[22]onehas
Σ(ph)(τ)= Wσσ′ (τ)Gσ′ (τ), (26)
12,σ 13,42 34
34,σ′
X
withtheP-Hfluctuation potential matrix:
Opticalandmagneto-opticalpropertiesofmoderatelycorrelatedsystems 9
Wσσ′(iω) = W↑↑(iω) W⊥(iω) , (27)
W⊥(iω) W↓↓(iω)
" #
werethespin-dependent effectivepotentials aredefinedas
1
W↑↑ = Vk χk χk Vk
2 ∗ − 0 ∗
1 h i
W↓↓ = Vk χk χk Vk
2 ∗ − 0 ∗
h i
W↑↓ = V⊥ χ+− χ+− V⊥
m ∗ e −e 0 ∗ m
h i
W↓↑ = V⊥ χ−+ χ−+ V⊥ . (28)
m ∗ − 0 ∗ m
h i
Here χk,χk differ from χk,χk by the replacement of Γ↑↑ Γ↓↓ in Eq.(25).
0 0 ⇔
We have subtracted the second-order contributions since they have already
beenteakeneintoaccountinEq.(18).
Ourfinalexpression fortheselfenergyis
Σ = Σ(TH)+Σ(TF)+Σ(PH) . (29)
Thisformulation takesintoaccountaccurately spin-polaron effectsbecauseof
the interaction with magnetic fluctuations [30, 31], the energy dependence of
the T-matrix which is important for describing the satellite effects in Ni[32],
contains exact second-order terms in v and is rigorous (because of the first
term)foralmostfilledoralmostemptybands.
Since the LSDA Green’s function already contains the average electron-
electron interaction, in Eqs. (18) and (26) the static part of the self-energy
Σσ(0)isnotincluded, i.e. wehave
Σ˜σ(iω) = Σσ(iω) Σσ(0). (30)
−
3. Results and discussion
The matrix elements of v appearing in Eq.(16) can be calculated in terms
of two parameters - the averaged screened Coulomb interaction U and ex-
changeinteractionJ [23]. Thescreeningoftheexchangeinteractionisusually
small and the value of J can be calculated directly. Moreover numeric cal-
culations show that the value of J for all 3d elements is practically the same
and approximately equal to 0.9 eV. This value has been adopted for all our
calculations presented here. At the same time direct Coulomb interaction un-
dergoes substantional screening and one has to be extremely careful making
the choice for this parameter. There are some prescription how one can get it
10
a) t2g 2 b)
0.2
Σ-8e(eV)x10 00..01 DOS(st/eV) 01
R -1
-0.1
-2
0.2 t2g -8 -6 -4 E-n2ergy(e0V) 2 4 6
-80 80 c)
x1 0.0
Σ(eV) -14-110s60
Re-0.2 σω1(),xx40
-0.4
-8 -6 -4 -2 0 2 4 6 0 1 2 3 4 5 6
Energy(eV) Energy(eV)
Figure1. Theself-energy(a)ofFeforthreedifferenttemperaturesandcorrespondingdensi-
tiesofstates(b)andopticalconductivitiesspectra(c).Full,dashedanddottedlinescorrespond
toT =125K,T =300K andT =900K,respectively.
within constraint LDA calculation [2]. However, results obtained in this way
depend noticeably onthe choice of the basis functions, wayof accounting for
hybridization etc. Nevertheless the order of magnitude coming out from vari-
ous approaches is the same giving the value of U in the range 1–4 eV. In the
present paper wearediscussing the influence ofthechoice ofU onthecalcu-
latedoptical spectra.
Another parameter entering SPTFequations is temperature. For a moment
wearemoreinterestedinthelowtemperaturepropertieswhilecomputationally
the higher the tempreture is, less computationally demanding are the calcula-
tions. This is why we decided first to consider the dependence of the self-
energyonthetemperature.
In Fig. 1 we show the self-energy obtained for Fe for three different tem-
peraturesaswellascorrespondingdensitiesofstatesandopticalconductivities
spectra. Onecansee thatdespite the differences inΣarequite noticeable this
leads only tomoderate changes inthe density ofstates and doesnotaffect the
opticalconductivity.
Much more important for the results is the parameter U. Fig. 2 shows as
an example the real part t component of Σ for T = 300K in Fe for var-
2g
ious values of U. Despite the overall shape of the curve is practically the
same the magnitude of the self-energy increasing with increase of U as it is
expected from the analytical expressions. This change in self-energy leads to
corresponding changes in the densities of states especially noticeable for the