Correlated Topological Insulators with Mixed Valence

0.76 MB·English

by  Feng Lu

Correlated Topological Insulators with Mixed Valence
∗ ∗
Feng Lu , JianZhou Zhao , Hongming Weng, Zhong Fang and Xi Dai
Beijing National Laboratory for Condensed Matter Physics,
and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
(Dated: January 29, 2013)
We propose the local density approximation (LDA)+Gutzwiller method incorporating Green’s
function scheme to study the topological physics of correlated materials from the first-principles.
ApplyingthismethodtotypicalmixedvalencematerialsSmB6,wefounditsnon-trivialZ2topology,
indicating that SmB6 is a strongly correlated topological insulator (TI). The uniquefeature of this
3 compound is that its surface states contain threeDirac cones in contrast to most known TIs.
1
0 PACSnumbers:
2

n Most of Z2 topological insulators (TI)[1–5] discovered !"#! !$#! $’!
a
up to now are semiconductors which are free of strong
J Γ
correlation effects and their topological nature can thus
6
be predicted quite reliably by first principle calculations ! !
2
based on density functional theory (DFT)[6]. The Z2 #!

] classification of band insulators has been generalized to $
l Γ %
e interactingsystemsbylookingatitsresponsetoexternal
r- electric magnetic field, namely the topological magneto- $&! "! !!
t electric effect (TME)[7]. A correlated insulator is a TI
s
. if the θ-angle defines in TME is π. Given the TME the-
t
a ory and several simplifications[8], however, its applica-
m
tion to realistic materials is still absent due to: (1) the
FIG. 1: (A) The CsCl-type structure of SmB6 with Pm3m
- lackofsuitablecompounds;(2)thedifficulty tocompute
d space group. Sm ions and B6 octahedron are located at the
reliably correlated electronic structures from the first-
n conner and center of the cubic lattice respectively. (B) The
o principles. Inthisletter,westudyaspecialclassofmate- bulk and surface BZ for SmB6.
c rials,themixedvalence(MV)compounds,whichcontain
[ rare-earth elements with non-integer chemical valence.

2 By combining the Gutzwiller variational approach from to be among the possible candidates of the interesting
v the first-principles and the Green’s function method for topologicalKondo insulators [16–18]. Although the ana-
3 theTME,wefoundSmB6 isa3DcorrelatedTI.Interest- lyticalstudiesofMVcompoundshavebeenconductedby
6 ingly, ithas three Diraccones onthe surface,in contrast several groups, the reliable first principle studies, which
8
to most of the known TIs. is crucial to identify the possible topological phases, is
5
stilllacking,duetothestrongcorrelationnatureofthese
.
1 Classic MV compounds, such as SmB6 and YbB12[9], materials[19, 20]. Here we report that the local density
1
share the following common features. (i) The x-ray approximation (LDA)+Gutzwiller method, a newly de-
2
photoelectron spectroscopy (XPS) and X-ray absorption veloped first-principles tool for correlated electron sys-
1
: spectra (XAS) contain peaks from both divalent and tems, enables us to search for the topological phases in
v
i trivalent multiplets with comparable spectral weight[10– correlated matters. By taking the MV compound SmB6
X 12],indicatingthevalenceofSmorYbtobecloseto2.5. as an example, we will focus on two key issues: (i) how
r (ii) A small semiconducting gapopens at leastin partof to compute Z2 topological index with LDA+Gutzwiller;
a
theBrillouinzone(BZ)atlowtemperature[13]. Previous (ii)what’sthetopologicalnatureofSmB6withthestrong
electronic structure studies indicate that the electrons correlationeffects among the f electrons?
transferfromSm(orYb)4f orbitalsto5dorbitals,which The LDA+Gutzwiller method combines the DFT
leadstofractionaloccupationin4f orbitals[14,15]. From within LDA with the Gutzwiller type trial wave func-
thebandstructurepointofview,theelectron-transferbe- tion, which takes care of the strong atomic features in
tween4f and5dorbitalsindicatespossiblebandinversion the ground state. Here we just sketch the most impor-
between them, which is a crucialingredient to realize TI tantaspects,andleavingdetailstoreference[21–23]. We
(if it happens odd number of times in the entire BZ). start from the common Hamiltonian (used for most of
At the mean while, SmB6 has been recently suggested the LDA++ schemes),
H =H +H +H (1)
Total LDA int DC
∗theseauthorscontributedequallytothepaper where HLDA is the single particle Hamiltonian obtained
2
by LDA and Hint is the local interaction term for the (A) LDA (C) LDA+Gutzwiller
4f electrons describedby the Slater integralsF0, F2, F4, 2 2
F6. In the presentstudy, for Sm atomwe chooseF0=5.8 j=7/2
V) V)
eV and F2 = 9.89eV, F4 = 7.08eV,F6 = 4.99eV to be e 0 e 0
their atomic values[24, 25]. H is the double count- y ( j=5/2 y ( j=5/2
DC g g
ingtermrepresentingtheinteractionenergyalreadycon- ner-2 ner-2
E E
sidered at the LDA level. In the present paper, we
-4 -4
compute the double counting energy using the scheme
described in reference[26]. We then use the following Γ X M Γ R M X R Γ X M Γ R M X R
Gutzwiller trial wave function, |Gi = P |0i = Q P |0i, (B) (D)
G i i
where|0iisanon-interactingstate(obtainedfromLDA), 0.2 0.2
Pi-ith=sitPeΓwiitλhΓi||ΓΓiiiibhΓeiin|gisththeeatGomutizcweiilgleenrsptarotejescatnodr λatΓtbhee- y (eV) 0 y (eV) 0
ing the variationalparametersto be determined by min- g g
er-0.2 er-0.2
imizing the ground state total energy (under Gutzwiller n n
E E
approximation)[22, 27, 28], EG=h0|Heff|0i+PΓλΓEΓ. -0.4 -0.4
Here H = P H P is called renormalized effec-
eff G LDA G
tive single particle Hamiltonian for the quasi-particles Γ X M Γ R M X R Γ X M Γ R M X R
andEΓ isthe eigenenergyoftheΓ-thatomiceigenstate.
FIG. 2: The calculated band structure for SmB6 in different
Theschemepreservestheniceaspectofbeingvariational,
energyscalesbyLDA(A,B)andLDA+Gutzwiller(C,D).The
however, it is beyond LDA (and also LDA+U) because
BandDarejustthezoominofAandBrespectively,around
the total energy EG now relies on the balance between thefermi level. Compared to theLDA results, the 4f j=7/2
the renormalized kinetic energy of quasi-particle motion bands in LDA+Gutzwiller have been pushed up to be about
and the local interaction energy, which is configuration- 4.0 eV above thefermi level, leaving theband structurenear
dependent. We note that including the complete formof the fermi level being dominated by 4f j=5/2 and 5d states.
The blue, red and grey colors represent the weight of the
allthe Slater integrals (F0 to F6) in the localinteraction
orbital character for 5d, 4f and 2p respectively.
is essential to obtain the correct electronic structure for
these materials.
The linear response theory for the coefficient of TME
where the second term describes the incoherent part
has been developed and simplified by Z. Wang et al[8].
of the Green’s function, which is ignorable for low fre-
Forinteractingsystems,whentheself-energycontainsno
quency. Therefore by applying the green’s function
singularity along the imaginary axis, the formula for the
method described above, we reach the conclusion that
TME coefficient only requires the single particle Green’s
function at zero frequency, gˆ(k,0). Because the singu- the Z2 invariance in LDA+Gutzwiller method is deter-
mined by the occupied eigenstates of the Gutzwiller ef-
lar point for the self-energy along the imaginary axis
fective Hamiltonian H , which can be interpreted as
only appears for a Mott insulator with completely lo- eff
the band structure of the “quasi-particles”.
calized f-orbitals, which is not the case for SmB6, we
can safely apply the above method here. Therefore the We now focus on SmB6, which is crystallized in the
way to determine the Z2 invariance is just diagonalizing CsCl-type structure with Sm ions and B6 clusters be-
the Hermite matrix −gˆ(k,0)−1 = Hˆ0 −µf +Σˆ(0) and ing located at the conner and body center of the cubic
treating the eigenstates with the negative eigenvalues as lattice respectively (see Fig.1). The LDA part of the
the “occupied states”. If the system also has spacial in- calculations has been done by full potential linearized
version center, the Z2 invariance can be simply deter- augmentedplane wave(LAPW) methodimplemented in
mined by counting the parities of these “occupied state” the WIEN2k package[29]. BZ integrationwas performed
on the time reversal invariant momenta (TRIM) points, onaregularmeshof12×12×12k points. The muffin-tin
thesameasthatdoneforthenon-interactingtopological radii(RMT) of 2.50 and 1.65 bohr were chosen for Sm
band insulators[1, 2]. and B atoms, respectively. The largest plane-wave vec-
In the Gutzwiller approximation (or the equivalent tor Kmax was give by RMTKmax = 8.5. The spin-orbit
slave boson mean field approach), the low energy single coupling (SOC) is included in all calculations.
particleGreen’sfunctionofaninteractingsystemcanbe ThebandstructureobtainedbyLDAisshowninFig.2
expressedbythequasi-particleeffectiveHamiltonianand (A and B), where we can find three major features.
the quasi-particleweightzˆ(whichingeneralis amatrix) (i) The Sm-4f orbitals, which split into the j=5/2 and
as, j=7/2manifoldsdue to the SOC,formnarrowbandsre-
spectivelywithwidtharound0.5eVneartheFermilevel.
zˆ (ii) The low energy band structure is semiconductor-like
gˆ(k,iω)= +gˆ (k,iω) (2)
iω−Hˆ +µ ic with a minimum gapabout 15meV along the Γ-X direc-
eff f
3
tion. (iii) There is clear band inversion features at the
X-points, where one 5d band goes below the f bands, 0.2 Γd
which reduces the occupation number of the f-states to 7
be around 5.5.
0.1
The first two features of the LDA band structures Γf
are not consistent with the experimental observations. V) 7
e 0
Firstly, the XPS measurements find quite strong 4f y (
multiplet peaks, indicating strong atomic nature of 4f g
er -0.1 Γf
electrons in SmB6 (in other words, most of the 4f En 6
electrons are not envolved in the formation of energy
-0.2
bands)[10, 11]. Secondly, both transport [30–32] and
optical [33–36] measurements reveal the formation of a
-0.3
small gap only for temperature below 50 K. The above
two features imply that the f electrons in SmB6 have Γ X Γ
both localized and itinerant natures and the correct de-
scription of its electronic structure should include both
FIG. 3: Band hybridizations between 4f and 5d along Γ to
ofthem. Thequalitativephysicalpicturecanbeascribed X direction. Theoriginal 4f j=5/2orbitalssplitintotwoΓf
7
to Kondo physics, which involves both itinerant and lo- and one Γf bands. The Γf bands cross and hybridize with
6 7
calized electronic states[17]. At high temperature, these another Γd7 band formed by5d orbitals.
localized orbitals (4f states) are completely decoupled
from the itinerant energy bands, forming atomic multi-
plet states. While in the low temperature, the coherent among the three Γ7 bands are allowedand generates the
hybridization between localized and itinerant states is hybridizationgap,whichisaround10meVnow. Wenote
gradually developed leading to the formation of “heavy that in contrast to the LDA results, the semiconductor
quasi-particle” bands. The insulating behavior appears gap obtained by LDA+Gutzwiller is indirect, which is
when the chemicalpotential falls into the “hybridization quite consistent with the transport measurements[36].
gap” between the heavy quasi-particle and the conduc- WewouldemphasizethatbyLDA+Gutzwiller,wecan
tionbands,whichismainlyofthe 5dorbitalcharacterin only obtain the quasi-particle part in the spectral func-
SmB6. tion, but not the Hubbard bands containing the atomic
Such Kondo picture can be nicely captured by multiplet structure[15, 20]. While the Gutzwiller type
our LDA+Gutzwiller calculations, which provide equal- wave function can well capture the multiplet features
footing descriptions of both the itinerant 5d bands in the ground state[22, 27, 37]. The Gutzwiller varia-
and those heavy quasi-particle states formed by 4f or- tional parameter λΓ determines nothing but the proba-
bitals. The quasi-particle band structures obtained by bility of each atomic configurationΓ in the groundstate
LDA+GutzwillerisshowninFig.2(CandD).Compared (which is defined as hG|ΓihΓ|Gi). In Fig. 4, we plot
with the LDA results, there are three major differences the corresponding probabilities for SmB6 obtained by
induced by the strong correlation effects. (i) The 4f LDA+Gutzwiller together with that obtained by LDA
j=7/2 bands are pushed up to be around 4.0 eV above wave function(h0|ΓihΓ|0i). The LDA ground state is
the fermilevel,leavingthe bandstructurenearthefermi dominatedbytheatomicconfigurationswiththenumber
level being dominated by 4f j=5/2 and 5d bands. (ii) off-electronsNf=6,ontheotherhandhowever,thedis-
The band width of those “heavy quasi-particle” bands tribution of the probability of atomic states obtained by
formed by 4f orbitals are much reduced to be less than LDA+Gutzwiller is almost equally concentrated on two
0.1eV.Thequasi-particleweightzofthese“heavyquasi- atomic multiplet states with five and six f-electrons re-
particle” bands is about 0.18, indicating that the quasi- spectively, leading to approximately+2.5 valence of Sm.
particles are formed by less than 20% of the 4f spec- Moreover, the Gutzwiller wave function provides cor-
tralweightandthe remainingweightisattributedtothe rect description of the intermixing between j = 7/2 and
atomic multiplets (or the Hubbard bands). (iii) A hy- j = 5/2 orbitals, which can not be captured by LDA
bridization gap between 4f quasi-particle bands and the only calculation and manifest itself in the average occu-
itinerant 5d bands appears along the Γ to X direction. pancy of the f orbitals. The occupancy of j = 5/2 and
The detail hybridization process is illustrated in Fig. 3. j =7/2 orbitals are 5.31 and 0.22 respectively using the
Along the Γ to X direction, the point group symmetry LDA type wave function (|0i). While the f orbital oc-
is lowered to be C4v, whose double group contains two cupancy is modified to be 3.64 for j = 5/2 and 1.89 for
irreducible two-dimensional representations, Γ6 and Γ7. j = 7/2 orbitals using the Gutzwiller type wave func-
Along Γ to X, the original 4f j=5/2 orbitals split into tion (|Gi). The dramatic increment of the occupancy
two Γ7 and one Γ6 bands, which cross with another Γ7 for j =7/2 orbitals is the important consequence of the
band formed by 5d orbitals. The hybridization terms F2−F6 terms in the atomic interactions, which can not
4
tight binding Hamiltonian and the same rotational in-
N = 5 variant Gutzwiller approach on a 40-layers slab. The
0.6 a Nff = 6 obtained quasi-particle bands of the slab are plotted in
N = 7
y f Fig.5. It is clearly seen that the surface states contain
bilit 0.4 threeDiraccones: oneconeislocatedatΓ¯ andthe other
ba two are at two X¯ points of the surface BZ. The inter-
o
Pr 0.2 esting multi-Dirac-cones behavior is quite unique among
theexistingTIsandisanaturalconsequenceoftheband
0.0 inversionat the bulk X points, which are projectedonto
0.6 b NNf == 65 to Γ¯ and two X¯ points of the (001) surface BZ. The
Nf = 7 multiple Dirac cones on the surface of SmB6 may gener-
y f
bilit 0.4 aqtueasini-tpearretsitcilneginptheyrsfeicraenlcpehepnaottmerennaisn,sscuacnhnainsgthtuenunneilqinuge
a
ob microscope (STM), which will be studied in our further
r
P 0.2 publications.
0.0
!
-5.0 -3.0 -1.0 1.0 3.0 5.0
Energy (eV)
"
Γ
FIG. 4: The probability of atomic eigenstates in the ground
state obtained by (a) LDA alone and (b) LDA+Gutzwiller.
N is the total number of f electrons for the corresponding
f
atomic eigenstates. The horizontal axis denotes the corre-
sponding atomic eigen energy.
beexpressedbyapure”density-density”forminanysin-
gleparticlebasisandgeneratestrongmulti-configuration
nature for the ground state wave function.
TABLEI:Theproductsofparityeigenvaluesoftheoccupied
states for TRIM points, Γ, X, R and M in theBZ. FIG. 5: The surface states of SmB6 on the (001) surface.
The surface states are obtained by LDA+Gutzwiller with a
Γ 3X R 3M 40 layer slab on the basis of projected Wannier functions.
+ − + + (Inset) The fermi surfaces for SS on (001) surface.
ThebandinversionfeaturearoundtheX pointsiswell In summary, we have developed the LDA+Gutzwiller
persisted in the LDA+Gutzwiller quasi-particle bands. method incorporating the Green’s function scheme to
As discussed in previous paragraphs,the topological na- study the the topological phases of strongly correlated
ture of this interacting systemis fully determined by the materials from the first-principles (beyond LDA and
Gutzwiller effective HamiltonianH . Since the spacial LDA+U). This method is systematically applicable to
eff
inversionsymmetryispresentforSmB6,wearenowable all correlated compounds as long as the quasi-particle
to determine its topological nature by simply counting weight is not reaching zero. Both quasi-particle bands
the parities of those occupied quasi-particle states at 8 andatomic multiplete structurescanbe wellcapturedin
TRIMpoints. AslistedinTableI,theparitiesareallpos- thepresenttechnique. Applyingthismethodontotypical
itiveexcepttheXpoints. Becausetherearetotallythree mixed valence compound SmB6, we demonstrate that it
equivalent X points in the whole BZ, the Z2 topological is a stronglycorrelated3D TI with unique surface states
index for SmB6 has to be odd, resulting in a strongly containing three Dirac cones on the (001) surface. The
correlatedtopological insulators with topologicalindices strong interaction among the f electrons reduces both
(1;111). the band width and the quasiparticle weight for almost
To see the topological surface states, we construct one order, but the topological feature remains.
a tight binding model using the projected Wannier This work was supported by the National Science
functions[38–40],whichcanreproducetheLDAbanddis- Foundation of China and by the 973 program of China
persion quite precisely. The surface states (SS) of the (No. 2011CBA00108and 2013CBP21700). We acknowl-
(001) surface are then obtained by combining the above edgethehelpfuldiscussionwithprofessorP.Coleman,O.
5
Eriksson and Dr. P. Thunstrom. B 79, 075114 (2009).
[22] N.Lanat,H.U.R.Strand,X.Dai,andB.Hellsing,Phys-
ical Review B 85, 035133 (2012), pRB.
[23] Y. Yao, C. Wang, and K. Ho, Physical Review B 83,
245139 (2011).
[1] L.FuandC.Kane,PhysicalReviewB76,045302(2007). [24] R.D.Cowan,Thetheory ofatomicstructure and spectra
(1981).
[2] L. Fu, C. Kane, and E. Mele, Physical Review Letters
98, 106803 (2007). [25] W. T. Carnall, G. L. Goodman, K. Rajna, and R. S.
Rana, J. Chem. Phys. 90, 3443 (1989).
[3] M.Z.HasanandC.L.Kane,ReviewsofModernPhysics
82, 3045 (2010). [26] V. I.Anisimov, F. Aryasetiawan, and A. I.Lichtenstein,
J. Phys.: Condens. Matter 9, 767 (1997).
[4] X. L. Qi and S. C. Zhang, Reviews of Modern Physics
83, 1057 (2011). [27] J. Bunemann, W. Weber, and F. Gebhard, Physical Re-
view B 57, 6896 (1998).
[5] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science
314, 1757 (2006). [28] F.Lechermann,A.Georges,G.Kotliar,andO.Parcollet,
Physical Review B 76, 155102 (2007).
[6] H.Zhang,C.-X.Liu,X.-L.Qi,X.Dai,Z.Fang,andS.-C.
Zhang, NaturePhysics 5, 438 (2009). [29] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka,
[7] X. L. Qi, R. Li, J. Zang, and S. C. Zhang, Science 323, and J. Luitz, WIEN2k, An Augmented Plane Wave +
LocalOrbitalsProgramforCalculatingCrystalProperties
1184 (2009).
[8] Z.WangandS.-C.Zhang,Phys.Rev.X2,031008(2012). (2001), ISBN 3-9501031-1-2.
[9] C. M. Varma, Rev.Mod. Phys. 48 (1976). [30] A. Menth, E. Buehler, and T. H. Geballe, Physical Re-
[10] J.Allen,L.Johansson,I.Lindau,andS.Hagstrom,Phys- view Letters22, 295 (1969).
ical Review B 21, 1335 (1980). [31] J. C. Nickerson, R. M. White, K.N. Lee, R. Bachmann,
[11] J. N. Chazalviel, M. Campagna, G. K. Wertheim, and T. H.Geballe, and G. W. Hull Jr, Physical Review B 3,
P.H. Schmidt,Physical Review B 14, 4586 (1976). 2030 (1971).
[12] E. Beaurepaire, J. P. Kappler, and G. Krill, Phys. Rev. [32] J. C. Cooley, M. C. Aronson, A. Lacerda, Z. Fisk, P. C.
B 41, 6768 (1990). Canfield,andR.P.Guertin,PhysicalReviewB52,7322
[13] J.Cooley,M.Aronson,Z.Fisk,andP.Canfield,Physical (1995).
ReviewLetters 74, 1629 (1995). [33] S. von Molnar, T. Theis, A. Benoit, A. Briggs, J. Flo-
[14] V. Antonov, B. Harmon, and e. al, Physical Review B quet,J.Ravex,andZ.Fisk,ValenceInstabilities (North-
(2002). Holland, Amsterdam, 1982), p. 389.
[15] E. R. Ylvisaker, J. Kunes, A. K. McMahan, and W. E. [34] T.Nanba,H.Ohta,H.Motokawa,K.S.,K.S.,andK.T.,
Pickett,Physical Review Letters 102, 246401 (2009). Physica B: Condensed Matter 186-188, 440 (1993).
[16] M.Dzero,K.Sun,V.Galitski,andP.Coleman,Physical [35] G.TravagliniandP.Wachter,PhysicalReviewB29,893
ReviewLetters 104, 106408 (2010). (1984).
[17] M.Dzero,K.Sun,P.Coleman,andV.Galitski,Physical [36] J.Demsar,V.Thorsmlle,J.Sarrao,andA.Taylor,Phys.
ReviewB 85, 045130 (2012). Rev. Lett.96, 037401 (2006).
[18] H.Miyazaki,T.Hajiri,T.Ito,S.Kunii,andS.-i.Kimura, [37] M. C. Gutzwiller, Phys.Rev.Lett. 10, 159 (1963).
Physical Review B 86, 075105 (2012). [38] I.Solovyev,Z.Pchelkina,andV.Anisimov,PhysicalRe-
[19] M. Colarieti-Tosti, M. I. Katsnelson, M. Mattesini, S. I. view B 75, 045110 (2007).
Simak,R.Ahuja,B.Johansson,C.Dallera,andO.Eriks- [39] J. Kunes, R. Arita, P. Wissgott, and e. al, Computer
son, Physical Review Letters 93, 096403 (2004). Physics Communications 181, 1888 (2010).
[20] P. Thunstrom, I. D. Marco, A. Grechnev, S. Lebegue, [40] A.A.Mostofi,J.R.Yates,Y.-S.Lee,I.Souza,D.Vander-
M. I. Katsnelson, A. Svane, and O. Eriksson, Physical bilt,andN.Marzari,ComputerPhysicsCommunications
ReviewB 79, 165104 (2009). 178, 685 (2008).
[21] X.Deng,L.Wang,X.Dai,andZ.Fang,PhysicalReview