SmO thin films: a flexible route to correlated flat bands with nontrivial topology

2.6 MB·English

by  Deepa Kasinathan

SmO thin films: a flexible route to correlated flat bands with nontrivial topology
Deepa Kasinathan,1 Klaus Koepernik,2 L. H. Tjeng,1 and Maurits W. Haverkort1
1Max-Planck-Institut für Chemische Physik fester Stoffe
Nöthnitzer Str. 40, 01187 Dresden, Germany
2IFW Dresden, P.O. Box 270116, D-01171 Dresden, Germany
Using density functional theory based calculations, we show that the correlated mixed-valent
compoundSmOisa3Dstronglytopologicalsemi-metalasaresultofa4f-5dbandinversionatthe
Xpoint. The[001]surfaceBlochspectraldensityrevealstwoweaklyinteractingDiracconesthatare
quasi-degenerateattheM¯-pointandanothersingleDiracconeattheΓ¯-point. Wealsoshowthatthe
topologicalnon-trivialityinSmOisveryrobustandprevailsforawiderangeoflatticeparameters,
making it an ideal candidate to investigate topological nontrivial correlated flat bands in thin-film
5 form. Moreover, the electron filling is tunable by strain. In addition, we find conditions for which
1 theinversionisofthe4f-6stype,makingSmOtobearatheruniquesystem. Thesimilaritiesofthe
0 crystalsymmetryandthelatticeconstantofSmOtothewellstudiedferromagneticsemiconductor
2 EuO, makes SmO/EuO thin film interfaces an excellent contender towards realizing the quantum
anomalous Hall effect in a strongly correlated electron system.
n
a
J
Topologicalinsulators(TI)arematerialswhichexhibit tionoftopologicalnontrivialcorrelatedflatbands. Inad-
8
a fundamentally new physical phenomena that was first dition, experimental realization of the theoretically pre-

predicted by theorists[1, 2, 4–7] and subsequently ascer- dicted quantum anomalous Hall effect (QAHE) (quan-
]
l tained in experiments[8–12]. Although most work con- tized Hall conductance in the absence of an external
e
- siders band semiconductors, the concept of topology can magnetic field) has been challenging due to difficulties
r
t be extended to correlated systems for which many excit- in obtaining insulating bulk TI’s concurrent with homo-
s
ingneweffectscanbeexpected. Inanalogytocorrelation geneousmagneticdoping[28]. Recently,notwithstanding
.
t
a driven fractional quantum Hall states one might expect the presence of bulk carrier density (i.e. not a nominally
m fractionalCherninsulatingstatestoemergeincorrelated insulating bulk), long range ferromagnetism and QAHE
- topological insulators[13–17]. The experimental realiza- was successfully observed for the first time in Cr-doped
d
tion of such states would open a whole new field of pos- (Bi,Sb) Te TI thin films[29, 30]. Other proposals to-
n 2 3
sibilities. wards realizing a QAHE are to either grow a TI on top
o
c of a ferromagnetic insulator[31, 32]; transition-metal ox-
SmB was recently reported to be a correlated mixed
[ 6 ideheterostructures,[33,34]ortodepositalayerofheavy
valent topological insulator[18–23]. The highly corre-
atoms with large spin-orbit coupling on a magnetic insu-
1 lated flat Sm-f derived bands hybridize with the disper-
v lator[35]. Fortuitously, a well studied substrate with the
sive and itinerant Sm-d derived bands to form a mixed
2 same symmetry as SmO is readily available in the form
valent insulating ground state. Although the insulat-
0 of the ferromagnetic semiconductor EuO[36–42]. Con-
9 ing state and topology of the bulk of SmB6 is well de- sequently, we propose growing SmO/EuO thin film in-
1 fined, [001] surface of SmB is polar and therefore in-
6 terfaces to test the feasibility of obtaining a QAHE in a
0 herently unstable[24]. This hinders the unique determi-
. strongly correlated electron system.
1 nation of topological surface states and in turn inhibits
0 the creation of robust technological devices. Here, us- SmO crystallizes in the rock-salt structure with
5 ing density functional theory (DFT) based calculations, a=4.9414–4.943Å[43, 44]. Considering the lattice pa-
1
we show that SmO is a mixed valent correlated com- rametersoftrivalentneodymiumchalcogenidesandinter-
:
v pound with a band structure similar to the topological polating between neodymium and terbium compounds,
Xi nontrivial high-pressure gold phase of SmS[25]. In con- Leger et. al. obtain a lattice constant of 4.917Å for
trast to SmS, SmO is predicted to have a topological Sm3+O[43]. Similarly, considering other divalent lan-
r
a nontrivial ground state at ambient pressure. Addition- thanide oxides, a lattice constant of 5.15Å is expected
ally,ourcalculationsshowthatthetopologicalnontrivial forSm2+O.Ergo,theexperimentalcellparameteroffcc
ground-state of SmO is stable for a large range of Sm- SmOliesbetweenthatofSm3+OandSm2+O,whichlead
O distances, including both positive and negative strain. the authors[43] to conclude that samarium is in an in-
The Sm-f band filling is thus tunable by strain, which termediate valence state in SmO. Using Vegard’s law,
opens up the possibility to create correlated topological a valence of 2.92 was assigned[43]. Electrical resistiv-
nontrivial bands at different filling. Subsequent to the ity measurements as a function of temperature reveals
enormous success in the design and fabrication of semi- a T2 like increase between 4.2 and 20K, followed by a
conductors, it is presently possible to grow high-quality rapid increase up to 32 K. The resistivity shows a linear
oxide thin films and heterostructures[26, 27]. SmO thus temperature dependence above 60 K[45]. Krill and co-
seems the ideal candidate for the experimental realiza- workers[44]measuredthemagneticsusceptibilityofSmO
2
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FIG. 1. (Color online) Phase diagram (d) of SmO as a function of the fcc lattice parameter within LDA+SO+U (U = 6 eV,
J =0eV)alongwithrepresentativeFPLObandstructuresforthe(a)nontriviald−f bandinversion,(b)trivialinsulatorand
H
(a) nontrivial s−f band inversion scenarios. The 4f, 5d and 6s orbital character derived bands are represented by red, blue
and green colored symbols respectively. The size of the symbols represent the weight of the various orbital contributions to
the underlying bands. The Brillouin zone of an fcc lattice is displayed along with the eight time reversal invariant momenta
(TRIM) points.
and observed the roughly constant magnetic susceptibil- dictions by varying U (5 to 7 eV) and J (0 to 0.7 eV)
H
ity and the non-divergence of the low-temperature sus- and by using different functionals (LDA and GGA us-
ceptibility. They drew parallels to the susceptibilities of ing FPLO and the modified Becke-Johnson approach[49]
other intermediate valence compounds SmB and gold- with U=3 eV and J =0 eV using Wien2k[50]). Impor-
6 H
SmSunderpressure,whereinthelowtemperaturebehav- tantly, as pointed out by Martin and Allen[2] for SmB
6
ior is not explainable by crystal field effects alone[44]. and related systems, the symmetry of the relevant Sm
Theelectronicstructurecalculationsareperformedus- oneparticleorbitalsandthemanybodyelectronremoval
ing the full-potential non-orthogonal local orbital code Green’s function is the same. This allows for a possible
(FPLO)[46]. Thelocaldensityapproximation(LDA)with adiabaticcontinuationfromtheLDAresultsaspresented
thePerdewandWangflavor[47]oftheexchangeandcor- here to the full interacting system without changing the
relation potential was chosen. To account for the strong topology of the system. See Supplemental Material[52]
spin-orbit (SO) coupling of the f electrons, we employ for further discussion on the Sm 4f many body state.
thefullfour-componentrelativisticscheme. Additionally, Collected in Fig.S1(a) is the FPLO non-spin polarized,
thestrongCoulombrepulsionbetweenthe4f electronsof full-relativisticbandstructureofSmOwiththeinclusion
samarium are included in a mean-field way by applying ofthestrongCoulombinteraction(LDA+SO+U)forthe
LDA+SO+U withthe"fullylocalizedlimit"(FLL)dou- experimental lattice constant. With samarium being in
blecountingterm[48]. Thisleveloftheorywillnotsuffice the 2+ configuration, the 4f states are split into lower
todescribequantitativelytheSm4f spectralweightand lyingandfilled4f statesthatcanaccommodate6elec-
5/2
its dispersion, but is sufficient to predict the Sm d−f tronsandhigherlying(above5.5eV),empty4f states
7/2
inversion. We have tested the robustness of these pre- (seeSupplementalMaterial). Wedonotobserveadirect
3
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FIG. 3. (Color online) Maximum possible f valence due to
FIG. 2. (Color online) Left: Surface Bloch spectral density f −→ d or f −→ s promotion in SmO while retaining the
(ABl(k)) of the first 12 SmO layers of a semi-infinite solid nontrivial topology as a function of lattice parameter. The
with [001]-surface. Note, that there are two Dirac cones at color map follows that of Fig.S1(d): light blue = nontriv-
the X¯-point, while the one at the Γ¯-point falls into the bulk ial topology from d−f inversion, light green = nontrivial
projectedbandstructureandhenceformsasurfaceresonance topology from s−f inversion, light pink = trivial topology.
(see right panel). Right: ABl(k) with downwards shifted 4f- Theequilibriumlatticeparametersforahigh-spin3+and2+
electron pocket around the Γ¯-point to reveal the third Dirac statearecomputedusingL(S)DA+SO+U scheme(dashedred
cone. line). The estimates from an empirical extrapolation for the
limiting cases are obtained from Ref.43 (dashed green line).
The filled green circle denotes the experimental bulk SmO.
SmOremainsatopologicalsemi-metalforawiderangeoften-
energy gap around the Fermi level (E ). The material is
F sile strain (≥ 1%), since the calculated maximum valence, is
semi-metallic with a “warped gap” such that everywhere
above both the empirical extrapolation and L(S)DA+U+SO
in the BZ (at EF), band number x is always below band estimates.
number x+1. There are small electron and hole pockets
at X and Γ respectively, but there are no band crossings
between the highest occupied 4f5/2 bands and the low- Γ point. The correct band sequence is obtained by using
est unoccupied bands. Besides, varying U (5 to 7 eV) or the MBJLDA exchange potential[53]. Literature on the
JH (0 to 0.7 eV) does not change the above mentioned efficiency of MBJLDA over the traditional LDA+U for
features,sincetheoxygen2p,samarium5dand6sbands f electron systems is still scarce. In a recent work on
experience a constant shift with respect to the localized SmS, the authors have employed MBJLDA+SO+U (U
4f states. There exists only one report[45] in literature = 3 eV) to open a band gap of 0.2 eV in the ambient
on the transport properties of SmO wherein the resistiv- pressureblackphasesuchthatthegapisconsistentwith
itydecreaseswithdecreasingtemperatureandthuscould the available experiments[25]. Using the same param-
hint to metallic character, consistent with the lack of an eters, the authors conclude that the high pressure gold
energy gap in our calculations. In contrast to the local- phase of SmS is a topological metal. We compared the
ized and not very dispersive 4f bands, the 5d bands of LDA+SO+U resultswiththatofMBJLDA+SOandob-
samarium are dispersive, broad and dip below the Fermi tain a consistent picture for the band inversion between
level,retaininga100%weightattheXpoint. Asaconse- the two approaches (see Supplemental Material). No di-
quence,weobservea4f-5dbandinversionattheXpoint, rect band gap is opened with MBJLDA+SO+U (U = 3
resulting in a nontrivial topology, similar in sense to the eV), in accordance with our LDA+SO+U results and as
bandinversionsreportedforSmB6 andthehighpressure well as with the experimental report[45].
goldphaseofSmS.Thisbandinversionandtheresulting
To provide additional confirmation of the topological
topological indices 1;(000) allows us to classify SmO as
non-triviality in SmO and to explicitly identify the pro-
a 3D strongly topological semi-metal (more details are
tected surface states, we have calculated the Bloch spec-
provided in Supplemental Material).
tral density (A (k)) of the 12 topmost surface layers of
Bl
AwellknownissuewithLDAwhendealingwithsemi- a semi infinite solid[54] with [001] surface. We obtain
conductors, is the underestimation of band gaps. Since two weakly interacting quasi-degenerate Dirac-cones at
nontrivial topology depends on band inversions, under- M¯, and a single Dirac-cone at Γ¯ which is hidden in the
estimation of band gaps could sometimes lead to wrong bulk and becomes a surface resonance due to the semi-
sequenceoftheunderlyingbandstructure. Forexample, metallic nature of SmO. To clearly identify the Dirac-
in HgTe, a s-p TI, LDA correctly predicts the band in- cone at Γ¯, we have repeated the slab calculations such
version, but results in an incorrect band sequence at the that the hole pockets at Γ for the bulk are pushed down.
4
This in turn, allows one to unambiguously identify all of experimental reports available in literature, the va-
three Dirac-cones in a [001] terminated SmO. More de- lence assigned to Sm in SmO is 2.92, i.e. a promotion
tails are provided in the Supplemental Material. of 0.92 electrons. To actually calculate the amount of
Having established the nontrivial topology in SmO for electron promotion is a difficult task, since DFT in the
the experimentalbulk latticeparameter, we considerthe Kohn-Sham scheme is based on a single Slater determi-
scenario of growing thin films of SmO. In general, the nantapproximationandpreventsacorrectdescriptionof
latticeparameter of thesubstrateplays adecisive role in amanybodyintermediatevalencestate. Inotherwords,
determiningthelatticeparameterofthethinfilm. Then, the position of the 4f bands relative to the 5d/6s as
the relevant question to answer is the robustness of the calculated in DFT is not a reliable quantity (see Sup-
topological semi-metal state as a function of lattice pa- plemental Material). Yet, we can address the effect of
rameter variation. To this end, we have investigated the mixed valency on the topological properties of SmO in
topological indices for various lattice parameters. We a rigorous quantitative way, by calculating the amount
have retained the cubic symmetry of the unit cell (i.e of electrons (integration of the density of states) that
isotropic volume change) in SmO, based on the experi- can be accommodated in the dispersive 5d or 6s band,
mentalstudiesonisostructuralsamariumsystems,which thereby mimicking various amounts of 4f −→ 5d or 4f
evidences a negative value of the Poisson ratio ν (elastic −→6spromotion. Forthelatticeconstantswhichhasthe
constant C < 0), characteristic of an isotropic volume band inversion at the X point, we calculate the amount
12
change[55, 56]. Since the nontrivial topology is obtained of electrons that can be contained in the 5d band be-
duetod−f bandinversionattheXpoint,SmOistopo- fore beginning to populate the 6s band. Note that the
logical as long as the 5d band bottom is below the E . nontrivial topology is maintained as long as only the 5d
F
The nontrivial topology is suppressed when the 5d band is occupied and the 6s remains unoccupied. Fig.3 dis-
bottom moves above E or when other bands dip below plays this electron amount as a function of the lattice
F
E (Fig.S1(b)). In the case of SmO, it is the 6s band constant. Analogously, on the other side of the phase
F
bottom at the Γ point that shifts to lower values as a diagram, wherein non-triviality is manifested due to a
function of increasing lattice parameter. The shift of the s − f inversion, we calculate the amount of electrons
5d and 6s band bottoms (at X and Γ respectively) with that can be contained in the 6s band before beginning
respect to the E as a function of lattice parameter are to populate the 5d band. Equilibrium lattice constants
F
collectedinFig.S1(d)forLDA+SO+U approach. Treat- for the 3+ and 2+ limiting cases are estimated using
ing the strong 4f correlations on a mean-field level, the spin-polarizedL(S)DA+SO+U schemes(seeSupplemen-
topological semi-metal state is quite robust and remains tal Material). We observe that the theoretical estimated
so, up to 5.38Å, a 9% increase in lattice parameter. On valence and lattice constants always yield a topological
anexperimentallevel, thisresultisverypromising, since nontrivial ground state. Though the estimate from an
itprovidesalargerangeofsubstratelatticeconstantsfor empirical extrapolation for the SmO valence places the
which SmO thin films reveal topological non-triviality. bulkinthetrivialregionofthephasediagram,nontrivial
During the evaluation of the topological indices using topologyisquicklyreinstatedforlatticeparameterswith
LDA+SO+U, we became aware of another interesting a small tensile strain (≥ 1%). Spectroscopy experiments
feature,anontrivialtopologyduetos−f bandinversion arenecessarytoconfirmthesamariumvalency,andposi-
at Γ for 11% and larger lattice constants (≥ 5.5Å). In tions of the samarium s and d bands with respect to the
LDA+SO+U, the 5d band bottom shifts above the E f states in bulk SmO and SmO thin films.
F
and becomes trivial before the 6s band bottom dips be- Byvirtueofthelargesurfacetovolumeratio,thinfilms
low EF. Further increase in the lattice parameter then of systems with nontrivial topology with dominant sur-
results in a s − f band inversion at Γ which is again face states are highly sought-after. Owing to the simple
topological (Fig.S1(c)). Although, the probability for a fcc symmetry and the flexibility in observing the topo-
successfulgrowthofSmOonasubstratewitha11%and logical ground state in SmO for a wide range of lattice
larger lattice constant may be low, it nonetheless offers parameters, we anticipate plenty of options for suitable
another exciting route towards realizing a topologically substrates. One particular substrate that invokes special
nontrivial state, only this time with a s−f band inver- interestisthefccferromagneticsemiconductorEuO[36–
sion. We verified the validity of our results for a range 42],whichhasalatticeconstantof5.14Å,onlya4%lat-
of U values ( 5 to 7 eV) and as well as for the MBJLDA tice mismatch with that of bulk SmO. EuO by itself is
exchange potential. an attractive material with many functionalities, includ-
In the calculations so far, we have allowed Sm to be in ingmetal-insulatortransition,magneticphasetransition,
the intermediate valence state and have treated this on colossal magneto-resistance, etc[39–42]. These function-
a DFT level. The mixed valent situation can be viewed alities have produced an abundance of research on grow-
as a 4f to 5d promotion of a certain fractional amount ing high quality thin films of EuO. So, a quick progress
of electrons due to the hybridization of the localized 4f in the engineering of SmO/EuO thin film interfaces can
stateswiththebandlike5dstates. Basedonthehandful be expected. Additionally, this would open up the possi-
5
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Supplemental Material
CALCULATION OF TOPOLOGICAL INDICES and4L(1,1,1)2π/a. Thesignoftheproductofthepari-
2 2 2
tiesofalloccupieddoublets(Kramersdegeneratebands)
TocalculatetheZ topologicalindexforasystemwith at these eight TRIM points refer to a trivial topology,
2
when positive and to a nontrivial topology, when neg-
inversion symmetry, we use the parity criteria as pro-
ative. Under spatial inversion, the d orbitals are even
posed by Fu and Kane[1], wherein the product of the
while the f orbitals are odd. From the calculated band
paritiesofalloccupiedKramersdoubletsateachtimere-
structures, it is clear that the occupied bands between
versal invariant momentum (TRIM) is determined. For
0 to -2 eV at Γ and the four L points possess f orbital
afcclattice, andhenceabccBrillouinzone, wehavethe
character, while due to the 4f-5d band inversion, one of
following 8 TRIM points : 1 Γ (0,0,0); 3 X (1,0,0)2π/a
the occupied band at the three X points possesses d or-
bital character. The product of the parities for all the
occupied bands will then be negative, giving rise to a
nontrivial topology in the band structure of SmO. This
resultisvalidforbothLDA+SOandLDA+SO+U calcu-
lations. This qualitative discussion was confirmed by ex-
plicitlycalculatingthespacegrouprepresentationsofthe
inversion operator of all bands at the TRIM points (Ta-
bleS1). This gives access to the four topological indices
ν ;(ν ν ν ). Theresultingindicesforallbandsuptothe
0 1 2 3
highest occupied 4f bands are 1;(000), which makes
5/2
SmO strongly topological. As a result of the particular
arrangement of the bands around the E in SmO, which
F
creates a semi-metallic “warped gap” (no band crossings
atE ),itispossibletoapplytheparitycountingscheme
F
at the TRIM points, since this “warped gap” ensures the
continuity condition for the phases of the wave functions
in the whole BZ which leads to the counting scheme.
In consequence one could classify SmO as a 3D strongly
topological semi-metal.
Thetopologicalindicesswitchfromtrivialtonontrivial
at the bottom of the six 4f derived bands and switch
5/2
back to trivial above the band, which forms the elec-
FIG. S1. (Color online) The calculated FPLO band struc-
tron pockets around the X point, which in consequence
ture of SmO (a = 4.941Å) within a non-spin polarized, full-
tells us that topological surface bands could bridge en-
relativistic approximation including strong Coulomb correla-
ergy window from the lowest 4f band to the lowest
tion(LDA+SO+U). AU andJH valueof6eVand0eVhave 5/2
beenusedrespectively. Thesizeofthesymbolsrepresentthe unoccupied 5d band. In Table S1, we provide the details
weight of the various orbital contributions to the underlying ofthetopologicalcharacteroftheimportantbands. The
bands. The numbers within parentheses are band indices to energy separation of about 2 eV between the O2p bands
assistinassigningthevarioustopologicalindicesinTable.S1. and the lowest 4f state ensures that the gap between
7
TABLE S1. Calculated topological indices of selected bands
in SmO within LDA+SO+U approximation (a = 4.941Å).
1.0
ε refers of the band energy at Γ; n refers to the band in-
n
dices, shown in Fig.S1. Γ, X and L are the TRIM points.
ν ;(ν ν ν ) are the four topological indices. The parity of Sm 5d
0 1 2 3
theKramersdoubletatthevariousTRIMpointsaredenoted Sm 4f
5/2
by O (odd) and E (even). Additionally, the dominating or-
)
bital character of the bands at the TRIM points are listed. V
e
(
εn(Γ) eV, (n) Γ 3×X 4×L ν0;(ν1ν2ν3) gy 0.0 E
-3.650 O (2p) O (2p) E (2p) 0;(000) er F
n
E
-0.493 (1,2) O (4f ) E (5d) O (4f ) 1;(000)
5/2 5/2
+0.030 (3,4) O (4f ) O (4f ) O (4f ) 1;(000)
5/2 5/2 5/2
+0.030 (5,6) O (4f ) O (4f ) O (4f ) 1;(000)
5/2 5/2 5/2
+1.457 (7,8) E (6s) O (5d) E (5d) 0;(000)
-1.0
Γ X W K Γ L W U X
O2p and Sm 4f must be trivial due to the fact that SO
coupling is too small to create band inversions bridging
this gap. Hence the highest O2p band forms the trivial
base line for discussing the topological properties of the 0.0
)
V
remaining bands. In the table, we list the energy of the e
( E
bands at the Γ point, the band indices (as indicated in gy -0.1 F
r
Fig.S1), the parity of the Kramers doublets (O: odd, E: e
n
even) together with the dominating orbital character at E-0.2
thethreesymmetrydistinguishedclassesofTRIMpoints
as well as the resulting topological indices obtained from Γ X W K Γ L W U X
all bands including the ones mentioned in the first col-
umn. Notethat,bandsnumbered3...6aretrivialbands, FIG. S2. (Color online) Band structure of SmO with or-
and do not change the topology. bital character using MBJLDA+SO approximation as imple-
mented in WIEN2K. The topology and order of the bands are
consistent to the ones obtained using LDA+SO+U scheme.
Theblow-upinthebottompanelclearlyshowstheopeningof
BAND STRUCTURE USING MODIFIED BECKE
thewarpedband-gapalongΓ-X,butsmallerinsizecompared
JOHNSON APPROACH
to FPLO. For clarity we have plotted the bands without any
band character.
To make sure that the topological characteristics are
not dependent on the choice of exchange and correlation
functional, we calculated the band structure of SmO us- bandsaroundtheFermilevelwithinafewmeVaccuracy.
ing the recently proposed modified Becke-Johnson func- Afterwards, the obtained bulk hopping parameters were
tional (MBJLDA) as implemented in WIEN2K. Fig.S2 mapped onto a semi infinite solid and a Greens function
shows the band structure obtained using MBJLDA+SO. techniquewasusedtocalculateA (k)forthesurfacelay-
Bl
The f−d band inversion at X point is consistent to our ers. Theresultingsurfacebandsclearlyshowtwoweakly
LDA+SO+U calculations, though the hybridization gap interacting Dirac cones around the surface projected M¯-
and the warped band-gap in MBJLDA+SO are smaller point. In order to be of topology induced nature, there
(but, of the same order). mustbeanoddnumberofDiraccones. ThesurfaceBril-
louin zone for a [001] surface ends up with two bulk X-
pointsbeingprojectedontothesurfaceM¯-pointandone
SURFACE STATE CALCULATIONS bulk X-point being projected onto the surface Γ¯-point
(see Fig.S3). Hence, one expects two Dirac cones at M¯
To asses the nature of the surface band structure we and one at Γ¯. The back-folding of the bulk band struc-
calculated the Bloch spectral density (A (k)) of the 12 tureduetoprojectiononto[001]leadstotheappearance
Bl
topmost surface layers of a semi infinite solid with [001] of electron pocket states from the bulk X-point at Γ¯,
surface. For this purpose the DFT band structure was which overlap with the bulk projected states of the bulk
fitted with atom centered maximally projected Wannier Γ hole pocket at Γ¯. Hence, the third Dirac cone will be
functions of Sm 5d, 6s and 4f and O 2p character. A immersed into projected bulk states and can only form a
soft energy cutoff was employed to achieve band disen- surface resonance. In order to prove the existence of this
tanglementat+6eV.Theresultingfitmodelsallrelevant thirdDiraccone,weperformedanadditionalcalculation,
8
where the 4f electron pocket around the bulk Γ-point rect. The spin polarized state found in L(S)DA does not
are lowered in energy by application of a k-dependent represent the local Sm f6 character correctly. Within
potential. This removes bulk projected states from the L(S)DA the Sm f6 configuration has a local moment
low energy region at the surface Γ¯ point and reveals the (M = 2S +Lz = 3) whereas the many body ground-
z z
third Dirac cone. Finally, to rule out drastic effects of state is a singlet ((cid:104)J2(cid:105) = J(J +1) = 0) without a local
the relaxation of the surface electronic structure we per- moment (L = 0, S = 0, M = 0). The many-body
z z z
formed a full DFT slab calculation for 21 SmO layers. ground-state without a local moment is consistent with
The resulting slab band structure shows the same two experimentalobservationsondivalentSmcompounds. In
weakly interacting Dirac cones around the M¯-point as ordertoreproducethelocalsymmetryofthemanybody
the semi-infinite calculation. Slight shifts of bands oc- state of an f6 configuration (7F term with J = 0, be-
0
cur, but the overall structure equals that of the mapped longingtotheA representation)weusenon-spinpolar-
1g
model calculation. ized DFT calculations. In this case the f6 configuration
isrepresentedbyastatewithsixelectronsinthej =5/2
bands. Although this state has an incorrect expectation
NOTES ON THE MANY BODY STATE OF SMO value of L ((cid:104)L2(cid:105) = 24/7) and S ((cid:104)S2(cid:105) = 24/7) com-
pared to the many body state (and thus a to high local
TheSm4f statesareknowntobestronglycorrelated. Coulombenergy), itdoeshavetherightsymmetry(A )
1g
Nonethelessonecanobtainusefulinformationaboutthis and total momentum (J = 0). The f5 configuration is
system using DFT within the local density approxima- represented by single hole excitations starting from the
tion. The large onsite Coulomb repulsion between the f f6 configuration. A peculiarity of starting from a many
electrons restricts the local valence occupations to fluc- bodyf6 statewithA symmetryisthatthemanybody
1g
tuate between the f5 and f6 configurations. The lattice one electron excitations have the same symmetry as the
constants for the limiting cases of a pure 2+ (f6) and independent electron bands found in LDA, as shown by
pure 3+ (f5) SmO compound are determined using spin Martin and Allen[2].
polarizedL(S)DA+SO+U.Thelocalatomicf6 (f5)con-
figuration has a lowest Hunds-rule multiplet state which
belongs to the 7F (6H) term. LDA in the Kohn-Sham
scheme does capture these local states as they are sin-
[1] L. Fu, and C. L. Kane, Phys. Rev. B 76, 045302 (2007).
gle Slater determinant representable (as any Hunds-rule
[2] R. Martin, and J. Allen, J. Appl. Phys. 50, 7561 (1979).
high spin ground-state).
In order to determine the topology of the bands and
in order to answer the question if hybridization between
the Sm d derived bands and the Sm f derived bands is
allowed, it is important to have the local symmetry cor-
FIG. S3. (Color online) Bulk and surface Brillouin zone of a
fcc lattice.