Spectrum of extended systems from Reduced Density Matrix Functional Theory

0.28 MB·English

by  S. Sharma

Spectrum of extended systems from Reduced Density Matrix Functional Theory
S. Sharma1,∗ S. Shallcross2, J. K. Dewhurst1, and E. K. U. Gross1
1 Max-Planck-Institut fu¨r Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany. and
2 Lehrstuhl fu¨r Theoretische Festk¨orperphysik, Staudstr. 7-B2, 91058 Erlangen, Germany.
(Dated: December 6, 2009)
Wepresentamethodforcalculatingthespectrumofextendedsolidswithinreduceddensitymatrix
functional theory. An application of this method to the strongly correlated transition metal oxide
series demonstrates that (i) an insulating state is found in the absence of magnetic order and, in
addition, (ii) the interplay between the change transfer and Mott-Hubbard correlation is correctly
9 described. In this respect we find that while NiO has a strong charge transfer character to the
0 electronicgap,withsubstantialhybridizationbetweent2g andoxygen-pstatesinthelowerHubbard
0 band, for MnO this is almost entirely absent. As a validation of our method we also calculate the
2 spectra for a variety of weakly correlated materials, finding good agreement with experiment and
othertechniques.
c
e
PACSnumbers: 71.10.-w,71.27.+a,71.45.Gm,71.20.Nr
D

6
A derivate of the ground-state density functional the- ment with experimental data as well as with the results

] ory (DFT) calculations are the Kohn-Sham (KS) eigen- ofGW and dynamicalmeanfield theory (DMFT) calcu-
l
e values, which lead to a non-interacting spectrum. Even lations. As an additional test we determine the spectra
- though the KS equations represent an auxiliary non- forasetofsystemswithoutstrongcorrelations;themet-
r
t interacting system whose states and eigenvalues may be als Cu and V and the band insulator BN. The spectra
s
. quite different from the true quasi-particle system, em- thusfoundshowgoodagreementwithexperimentaldata
t
a pirical evidence shows that in many cases this single as well as with data from other techniques, indicating a
m
particle KS spectrum is in agreement with the x-ray general applicability of the method proposed here.
- photo-emissionSpectroscopy(XPS)andBremsstrahlung Within RDMFT the one-body reduced density matrix
d
isochromat spectroscopy (BIS) experiments. However, (1-RDM) is the basic variable [4, 5]
n
o for strongly correlated materials, this KS spectrum is in
[c fundamental disagreement with experimental reality. In γ(r,r′)=NZ dr2...drNΦ∗(r′,r2...rN)Φ(r,r2...rN),
the absence of spin-ordering all modern exchange cor-

(1)
1 relation (xc) functionals within DFT fail to predict an
v insulating ground-statefor transitionmetalmono-oxides where Φ denotes the many-body wave function. Di-
8 (TMOs), the prototypicalMott insulators. On the other agonalization of this matrix produces a set of natural
1
1 hand, it is well known experimentally that these materi- orbitals[4],φjk, andoccupationnumbers,njk,leadingto
1 alsareinsulatinginnatureevenatelevatedtemperatures the spectral representation
. (muchabovetheN´eeltemperature)[1,2]. Thisindicates
12 thatmagneticorderisnotthedrivingmechanismforthe γ(r,r′)= njkφjk(r)φ∗jk(r′), (2)
9 gap, but is instead a co-occurring phenomenon. In fact Xj,k
0 not only DFT, but most modern many-body techniques
where the necessary and sufficient conditions for ensem-
: alsofailtocapturetheinsulatingbehaviorinTMOswith-
v ble N-representability of γ [6] require 0 ≤ njk ≤ 1 for
Xi out explicit long range spin ordering. all j,k, and j,knjk = N. Here j represents the band
Inthisregardreduceddensitymatrixfunctionaltheory index and k Pthe crystal momentum.
r
a (RDMFT) has proved to be valuable in that it not only In terms of γ, the total ground state energy [5] of the
improvesupontheKSbandgapsforinsulatorsingeneral, interacting systemis (atomic units are used throughout)
butalsopredictsTMOsasinsulators,evenintheabsence
1
of long range spin-order[3]. This clearly points towards E[γ]=− lim ∇2rγ(r,r′)d3r′+ ρ(r)Vext(r)d3r
2Z r→r′ Z
its ability to capture physics well beyond the reach of
msuocsctesms othdeerneffdecatyivgenroeussndo-fstRatDeMmFeTthoadssa. gDroeuspnidtesttahties + 21Z ρ|(rr)−ρ(rr′′|)d3rd3r′+Exc[γ], (3)
theory is seriously hampered by the absence of a tech-
where ρ(r) = γ(r,r), Vext is a given external potential,
nique for the determination of spectral information.
and Exc we call the xc energy functional. In principle,
In this work, we present a method for calculating the Gilbert’s [5] generalization of the Hohenberg-Kohn the-
spectrum of extended systems within the framework of orem to the 1-RDM guarantees the existence of a func-
RDMFT. Using this approach the spectra of a set of tionalE[γ]whoseminimumyieldstheexactγandtheex-
TMOs is determined. In all cases we find good agree- act ground-state energy of systems characterized by the
2
external potential Vext(r). In practice, however, the cor- 3 0.06
relation energy is an unknown functional of the 1-RDM
2.5 0.04
and needs to be approximated. Although there are sev-
0.02
eral known approximations for the xc energy functional, 2
V) 0
the most promising for extended systems is the power e 0 0.01 0.02 0.03
y (1.5
functional[3] where the xc energy reads g
er
n
1 |γα(r,r′)|2 E 1
Exc[γ]=Exc[{φik},{nik}]=−2Z Z d3r′d3r |r−r′| 0.5
=−21Z Z d3r′d3rikX,jk′(niknjk′)α φ∗ik(r)φjk′(|rr)−φ∗jrk′′|(r′)φik(r′) 00 0.2 0.4 nik 0.6 0.8 1
(4)
FIG. 1: Change in total energy upon changing a single oc-
In this functional we set α = 0.565, so that the corre-
cupation number nik, with all others held fixed. Inset shows
lation energy of the homogeneous electron gas (HEG) thezoom of thesame for small values of nik.
is well reproduced in the range of r appropriate for
s
solids[7]. The method then becomes truly ab-intio in
thesensethatthetheorydoesnotdependonparameters
E E
F F
that are adjusted to the system to be calculated.
Expt. NiO FeO
The main aim of the present work is to determine the DMFT
GW
ionization potentials and electron affinities of extended RDMFT
systems within the framework of RDMFT. To achieve s)
this the total energy functional is first minimized with unit
respect to {φik} and {nik} for the N-particle system to arb.
obtain the ground state. The electron removal energies S ( CoO MnO
O
D
and electron addition energies are then calculated as
∂E[{φik},{nik}]
ǫik = . (5)
∂nik (cid:12)(cid:12)nik=1/2 -10 -5 0 5 -10 -5 0 5
Numerical tests show that the de(cid:12)pendence of the to- Energy (eV)
tal energy upon varying a single occupation number nik
FIG. 2: (Color online) Density of states (DOS) for the tran-
is essentially linear, see Fig. 1, and thus Eq. 5, where
sition metal oxides NiO, FeO, CoO, and MnO. Shown are
the derivative is calculated at half filling, describes the
XPS and BIS spectra, in addition to calculations using the
change in energy when a particle with momentum k is GW, DMFT, and RDMFT methods. The GW and DMFT
either added or removed from the system with all other resultsarefromspin-polarizedcalculations,andarevertically
degrees of freedom held fixed. For ǫik > µ (µ being the shifted for clarity, while the RDMFT calculations are spin-
chemical potential) this derivative then corresponds to unpolarized.
theelectronadditionenergy(EN+1−EN)andforǫik <µ
to the electron removal energy (EN −EN−1).
We now deploy this scheme for the determination of
the spectra of several extended solids; the strongly cor- from Refs. 14, 15, 16, 17 for MnO, CoO, NiO and FeO
related Mott insulators NiO, CoO, FeO and MnO; the respectively.
bandinsulatorBN;thetransitionmetalV,andtheNobel It is immediately apparent from Fig. 2 that RDMFT
metal Cu. All calculations are performed using the full- captures the essence of Mott-Hubbard physics: all the
potential linearized augmented plane wave code Elk[8], TMOs considered are insulating in the absence of spin
with practical details of the calculations following the order. Furthermore, examination of the spectra for NiO
scheme described in Ref. 3. and CoO reveals an excellent agreement between the
PresentedinFig.2arethe spectrageneratedviaEq.5 RDMFT peak positions andthe correspondingXPSand
for the Mott insulators under consideration. Also shown BIS data. In fact, not only are the peak positions well
areGW datatakenfromRefs.9,10anddynamicalmean reproduced, but also their relative weights. Turning to
field theory (DMFT) results that for NiO taken from MnO, one notes a similarly good agreement between
Refs. 11, 12 and for MnO from Ref. 13. For details of experiment and the RDMFT results for peak positions
these calculations we refer the reader to the afore men- with, however,somewhatworseagreementregardingthe
tioned works, however we note that in all cases the cal- relative weights.
culations are spin polarized. The experimental data are Inthe case ofFeO itmust be recalledthat Fe segrega-
3
E E
F F
NiO FeO
t
2g
e
g
nits) O-p
u
b.
ar
S (
O CoO MnO
D
al
arti
P
-10 -5 0 5 -10 -5 0 5
Energy (eV)
FIG.3: (Coloronline)PartialdensityofstatesforTMOsNiO,
FeO,CoO,andMnO.Inallcasesthegapisbetweenstatesof
eg and t2g character. Oxygen-p states are also presented to
highlight thehybridization effects.
tion,unavoidableinthiscompound,precludestheexper-
imental realizationof pure FeO samples. For this reason
theonlyexistingexperimentaldataisratherold,andthe
presumablysubstantiallybroadeneddatapresentsnodis-
FIG. 4: (Color online) Difference between LSDA charge
tinct features thatmay be used forcomparison. Turning density and charge density calculated using LSDA+U and
to a comparison of the RDMFT spectra with the corre- RDMFT,(ρ(r)−ρLSDA(r))forCoO.Positivevaluesindicate
spondingGW anddynamicalmeanfieldtheory(DMFT) localization of charge as compared to LSDA.
results, one notes that while for NiO all three methods
are in close agreement, for MnO the peak positions are
morecloselymatchedbetweentheGW andRDMFTcal- subtle interplay with charge transfer effects.
culations.
Aconfirmationofthis chargelocalizationmay be seen
As is well known, while the insulating state of TMOs inthechargedensitydifferenceρ(r)−ρ (r),shownin
LSDA
is drivenby a chargelocalizationdue to strongCoulomb Fig.4forRDMFTandLSDA+U calculationsofCoO.A
repulsion(Mott-Hubbardcorrelation),animportantaux- comparison with LSDA+U is instructive as this method
iliary mechanism is charge transfer[18] due to hybridiza- (withanappropriatechoiceofU)isabletoaccuratelyre-
tion between ligand and transition metal (TM) states. produce the insulating gaps of the TMO series. As CoO
Amongstthe TMOseriesthislattermechanismis gener- has an odd number of electrons per unit cell, spin polar-
allybelievedtoplayanimportantroleinthecaseofNiO, izationisrequiredfortheLSDA+U calculationtofindan
but to be of decreasing importance as atomic number is insulating ground state. However, despite the fact that
lowered,with the insulating state of MnO thought to be both the RDMFT and LSDA+U calculations yield simi-
drivenentirelybyMott-Hubbardcorrelation. Clearly,an lar gaps, a significant charge localization is seen only for
outstanding challenge for any ab-initio theory is to cap- the RDMFT calculation. Interestingly, one observes an
ture both these aspects of TMO physics. almostsphericalchargeaccumulationatthe oxygensite,
In Fig. 3 we present the site and angular momentum a result in agreement with experiment[19], but different
projected DOS for the TMOs considered in this work. from that found in the corresponding LSDA+U result.
The electronic gap, as expected, always occurs between The actual values of the insulating gaps that may be
lower and upper Hubbard bands dominated by states of extractedfromFig.2are4.0eV(4.3eV),2.5eV(3.4eV),
t2g andeg characterrespectively. However,whileforNiO 2.4eV(3.4eV),and2.0eV(3.9eV),forNiO,CoO,FeO,
one finds a significant component of oxygen-p states in and MnO respectively. The corresponding experimental
the lower Hubbard band, for the other TMOs this hy- gapisgiveninparenthesis. Onenotesthattheagreement
bridization between oxygen-p and TM-d states is much is closestfor the NiO whichhasthe lowestmagnetic mo-
reduced, and is almost absent in the case of MnO, indi- mentof1.7µ amongsttheTMOsconsideredhere,with
B
catingthatforthismaterialtheinsulatingstateisdriven the worst agreement for MnO which has the largest mo-
mostlybyMott-Hubbardcorrelations. Thuswefindthat ment of 4.7 µ . As the RDMFT calculations presented
B
notonlydoesRDMFTcapturetheMott-Hubbardcharge herearenon-magnetic(i.e., spindegenerate)the trendis
localization in the TMO series, but in addition also the natural, and indicates that for the large moment TMOs
4
the co-occurring magnetic order does contribute signifi- counted in this approach. It should be stressed that all
cantly to the magnitude of the gap. materials are calculated using the same RDMFT func-
tional that is designed to to reproduce the correlation
energy of the electron gas at typical metallic densities,
i.e. the approachis universally applicable to solids.
Vanadium
s)-6 -5 -4 -3 -2 -1 0 1
nit ∗ Electronic address: [email protected]
u Copper
b. [1] O.Tjernberg, S.S¨oderholm,G.Chiaia, R.Girard, U.O.
S (ar Karlsson, H. Nylen, and I. Lindau, Phys. Rev. B 54,
O 10245 (1996).
D-6 -5 -4 -3 -2 -1 0 [2] W. Jauch and M. Reehuis, Phys. Rev. B 70, 195121
Cubic-BN Expt. (2004).
LDA
GW [3] S. Sharma, J. K. Dewhurst, N. N. Lathiotakis, and
RDMFT E. K.U. Gross, Phys. Rev.B 78, 201103 (2008).
[4] P. O. L¨odwin, Phys. Rev. 97, 1974 (1955).
-12 -8 -4 0 4 8 12
Energy (eV) [5] T. L. Gilbert, Phys. Rev.B 12, 2111 (1975).
[6] A. Coleman, Rev.Mod. Phys. 35, 668 (1963).
FIG. 5: (Color online) DOS for (a) Vanadium, (b) Copper [7] N.Lathiotakis,S.Sharma,J.Dewhurst,F.Eich,M.Mar-
and (c) cubic-BN in comparison with XPS and BIS spectra. ques, and E. Gross, Phys.Rev.A 79, 040501 (2009).
Note that the LDA data is scissors corrected for BN. The [8] (2004), URLhttp://elk.sourceforge.net.
experimental data for BN is from Ref. 20 and for Cu and V [9] C. R¨odl, F. Fuchs, J. Furthmu¨ller, and F. Bechstedt,
from Refs. 21, 22. The GW data for Cu and V is from Ref. Phys. Rev.B 79, 235114 (2009).
23 [10] S.Kobayashi,Y.Ohara,S.Yamamoto,andT.Fujiwara,
Phys. Rev.B 78, 155112 (2008).
[11] O. Miura and T. Fujiwara, Phys. Rev. B 77, 195124
An essential validation of the RDMFT approach
(2008).
demonstrated here consists of showing that the physics
[12] X.Ren,I.Leonov,G.Keller,M.Kollar,I.Nekrasov,and
of strongly correlated TMOs is not captured at the ex- D. Vollhardt, Phys.Rev. B 74, 195114 (2006).
pense of poorly treating weakly correlated systems. In [13] J.Kunes,A.V.Lukoyanov,V.I.Anisimov,R.T.Scalet-
Fig. 5 are shown the DOS for the band insulator BN, as tar, and W. E. Pickett, Nat. Mat. 7, 198 (2008).
well as the metals V and Cu. All these are systems with [14] J. van Elp, R. H. Potze, H. Eskes, R. Berger, and G. A.
weakcorrelationand,reassuringly,weseethatVandCu Sawatzky, Phys.Rev.B 44, 1530 (1991).
[15] J. van Elp, J. L. Wieland, H. Eskes, P. Kuiper, G. A.
are indeed found to be metallic in RDMFT, while the
Sawatzky, F. M. F. de Groot, and T. S. Turner, Phys.
spectrum ofBNis foundto be ingoodagreementwith a
Rev. B 44, 6090 (1991).
scissors corrected LDA calculation and experiment. The [16] G. A. Sawatzky and J. W. Allen, Phys. Rev. Lett. 53,
gap for BN in RDMFT is found to be 6.9 eV, close to 2339 (1984).
the experimental value of 6.39 eV. The metallic DOS in [17] P. S. Bagus, C. R. Brundle, T. J. Chuang, and K. Wan-
RDMFT also shows broad agreement with experiment delt, Phys. Rev.Lett. 39, 1229 (1977).
and other calculations. [18] J. Zaanen, G. A.Sawatzky,and J. W. Allen,Phys. Rev.
Lett. 99, 156404 (1985).
Toconcludewehavepresentedamethodforthecalcu-
[19] S. L. Dudarev, L.-M. Peng, S. Y. Savrasov, and J.-M.
lationofspectrawithintheframeworkofRDMFT based
Zuo, Phys. Rev.B 61, 2506 (2000).
on the derivative of the total energy with respect to oc-
[20] K. Lawniczak-Jablonska, T. Suski, I. Gorczyca, N. E.
cupation number at half filling. Owing to the linear be- Christensen, K. E. Attenkofer, R. C. C. Perera, E. M.
haviour of the total energy upon varying a single oc- Gullikson, J. H. Underwood, D. L. Ederer, and Z. L.
cupation number, this derivative represents the change Weber, Phys. Rev.B 61, 16623 (200).
in energy upon addition or removal of a particle. We [21] W. Speier, J. C. Fuggle, R. Zeller, B. Ackermann,
K.Szot,F.U.Hillebrecht,andM.Campagna,Phys.Rev.
have shown that the spectral information obtained in
B 30, 6921 (1984).
this way gives a detailed account of the strongly corre-
[22] A. G. Narmonev and A. I. Zakharov, Phys. Met. Metal-
latednatureoftheTMOs,includingthebalancebetween logr. 65, 315 (1988).
Mott-Hubbard correlation and charge-transfer character [23] K. D. Belashchenko, V. P. Antropov, and N. E. Zein,
in these materials. Furthermore, both metals as well as Phys. Rev.B 73, 073105 (2006).
weaklycorrelatedbandinsulatorsarefoundtobewellac-