Table Of ContentA new route to relativistic insulators: Lifshitz transition driven by spin fluctuations
and spin-orbit renormalization in NaOsO
3
Bongjae Kim,1 Peitao Liu,1 Zeynep Ergo¨nenc,1 Alessandro Toschi,2 Sergii Khmelevskyi,1,3 and Cesare Franchini1,∗
1University of Vienna, Faculty of Physics and Center for Computational Materials Science, Vienna, Austria
2Institut fu¨r Festko¨rperphysik, Technische Universita¨t Wien, Vienna, Austria
3Institute for Applied Physics, Technische Universita¨t Wien, Vienna, Austria
(Dated: December 23, 2016)
Insystemswhereelectronsformbothdispersivebandsandsmalllocalspins,weshowthatchanges
6
ofthespinconfigurationcantunethebandsthroughaLifshitztransition, resultinginacontinuous
1
metal-insulator transition associated with a progressive change of the Fermi surface topology. In
0
contrast to a Mott-Hubbard and Slater pictures, this spin-driven Lifshitz transition appears in
2
systemswithsmallelectron-electroncorrelationandlargehybridization. Weshowthatthissituation
c isrealizedin5ddistortedperovskiteswithanhalf-filledt2g bandssuchasNaOsO3,wherethestrong
e p−dhybridizationreducesthelocalmoment,andspin-orbitcouplingcausesalargerenormalization
D
of the electronic mobility. This weakens the role of electronic correlations and drives the system
towards an itinerant magnetic regime which enables spin-fluctuations.
2
2
PACSnumbers: 71.30.+h,75.47.Lx,71.27.+a
]
l
e
Metal-to-insulator transitions (MITs) are one of the ousMITwithLifshitzcharacteristics,fundamentallydif-
-
r most important phenomena in solid-state physics and ferent from relativistic Mott or purely Slater insulating
t
s their fundamental understanding represents an enduring states. Thenecessaryconditionsfortheonsetofthistype
.
t challenge in solid state theory [1, 2]. Different mech- ofspin-drivenLifshitzMITarethecoexistenceofasmall
a
m anisms have been invoked to explain the formation of U, a small local moment and an high degree of orbital
an insulating regime. Classical examples are realized in hybridization. This places the system at the border of a
-
d 3d transition metal oxides (TMOs), where the noncon- magnetic andelectronic instability where a Lifshitz MIT
n ducting state is typically understood within the Mott- ispossible. Thissituationcanberealizedinstructurally-
o Hubbardmodel as arisingfrom the competitionbetween distorted, half-filled t3 (or close to half-filled) 5d oxides
c 2g
strong electron-electron interaction (U) and the elec- such as NaOsO and Cd Os O or iridates. By taking
[ 3 2 2 7
tronicmobility,associatedtothe(non-interacting)band- NaOsO as a prototypical example, we show that here
3
3
width(W)[3–5]. Whenmovingtothemorespatiallyex- the balance between U and W is controlledby the SOC:
v
tended 4d and 5d orbitals the W increases and the U is the SOC induces a surprisignly large reduction of elec-
0
1 expected to become smaller, leading to the tendency to- tronicmobilitywhichrenormalizesthe U andpushes the
3 wards metallicity as in the itinerant magnet SrRuO3 [6]. system into a weakly correlatedand magnetically itiner-
3 In contrast to these expectations, recent theory and ex- ant regime subjected to spin fluctuations. We find that
0
perimenthaverevealedthatin’heavy’5dTMOs,the en- at high-temperature NaOsO is a paramagnetic metal
3
.
1 hanced spin-orbit coupling (SOC) strength [7] can lead butastemperaturedecreasesthechangesoftheamplitue
0 to the formation of a variety of novel types of quantum and direction of the spins lead to the continuous vanish-
6
statesincludingunexpectedinsulatingregimes[8–11]. In ing of holes and electrons pockets in the Fermi surface
1
the most representative example, Sr IrO , the concerted (FS), that do not involve any substantial modification
: 2 4
v action of a strong SOC and a moderate U leads to the of the underlying band structure [20], consistent with a
Xi opening ofasmallspectralgap[11,12]denominatedrel- Lifshitz-type MIT.
ativistic Mott gap. Other and more rare types of MITs
r Experimental observations indicate that NaOsO3 un-
a have been recently proposed for magnetic relativistic os-
dergoes a second-order MIT concomitant with the on-
mium oxides based on the Slater mechanism [13–16],
setofanantiferromagneticlong-rangeorderingataN´eel
drivenbyantiferromagnetic(AFM)order,wherethegap temperature (T ) of ≈ 410 K. This behavior is appar-
N
is opened by exchange interactionsandnot by electronic
entlyadaptabletoa Slatermechanism[13–16,21]. How-
correlation,[17]orLifshitz-likeprocesses[18,19],involv-
ever, the bad-metal behavior observed in a wide inter-
ing a rigid change of the Fermi surface topology [20].
mediate temperature region (0.1T <T <T ) [13], the
N N
In this work we show that in weakly correlated (small
Curie-WeissbehaviorofthesusceptibilityaboveT ,and
N
U) itinerant magnets the combined effect of longitudi-
the need of a sizable U in Density Functional Theory
nal and rotationalspin-fluctuations can cause a continu-
(DFT) calculations to open the gap, is in conflict with
an authentic Slater mechanism [21–23].
The fundamental properties of NaOsO have been
3
∗ [email protected] clearly exposed by Jung et al. [21]. In particular, it
2
was shown that the apparent discrepancy between the (a) 0.5 D
measured Os ordered moment of only 1 µB[14] and the 0.4 Metal m Os Insulator 1.6 Os
lcnaaorlmgmeinopam−letnd32gthacynobndrfiidfgoiurzmraattiaoionnn(tehslhaeotcuterlffdoenbcicteivaaentldtyrimrbeuadtguancbeeltseictt)hoeittilhnoe-- D (eV) 00..23 11..24 ( momentm
erant background [21]. However, the authors concluded 0.1 Kexp/KDFT 1.0 B)
thatthe roleofSOCis modestinthis system,alsobased 0.0 0.5 U SOC U (eV1).0 1.5 U cnRoSPOAC
on the, reasonable, consideration that the t32g associated (b) 1.0 Cu (c) nwoi tSh OSCOC
mwsautieibtvnhsiths,ateilmclLsuee[ffcf2hfe8c,ts=2sm9c0a]a.llnseItnrabtettehhseeetnrxfoohtnlihlbgoeiwtesovinneagnenwioneefgalwpigaqilirulbteliseanhlcolohywrebfidtithlloaaerldtbmritte2oalg--l K/KexpDFT 000...468 VCOrO22 SrRuO3CSrr2RuNOaO4sO 3 ItinerantM E-EF- 011
moment scenario, and we will explain that the SOC is 0.2 BaFe2As2 V2O3 ott -2
the crucial effect in paving the way for the continuous La2CuO4 LaNiO3
MIT in NaOsO3. 0.00.0 0.2 SO0.C4 stre0n.g6th 0.8 1.0 -3 S G T R G
Spin-orbit induced renormalization. In the presence of
FIG. 1. (color online). Relativistic “renormalization” of the
a MIT in a TMO, the clarification of the role played by
electronic correlation. (a) Energy gap and Os magnetic mo-
the electronic correlation U is a central aspect for the ment. (b) Kexp/KDFT as a function of the SOC strength
theoretical understanding. We begin, hence, by study- forNaOsO3 andotherreferencesystems. Themeasureddata
ing the effects of U and its interplay with the SOC for of the electron mobility are taken from Ref. [15] (NaOsO3),
the case of our interest, NaOsO . All calculations were Ref. [25] (LaNiO3), and Ref. [5] (all other materials). (c)
3
performed using the Vienna ab initio simulation pack- Paramagnetic band structure of NaOsO3 with and without
SOC.
age(VASP)[2]usingthe DFT+U method andincluding
relativistic effects (see Supplement).
can rewrite the kinetic energy ratio as:
We start by computing the dependence of experimen-
tally accessible observables on U. As usual, the value K ω2
exp p,exp
of the spectral gap ∆ and the magnetic moment µ = (1)
Os K ω2
are highly sensitive on the choice of U, as visualized DFT p,DFT
in Figure 1(a). For U = 0 the system is metallic, but
Surprisingly, K /K in NaOsO is dramatically
for U ≥ 0.3 eV NaOsO exhibits a finite gap that in- exp DFT 3
3 dependent on the SOC strength as shown in Fig. 1(b).
creases linearly with U. The experimentally reported
Without SOC, one should have classified the system as
low-temperature optical gap, 0.1 eV [15] and the µ
Os intermediate-strongcorrelatedwith K /K = 0.33,
≈ 1 µ [13] clearly indicate that the optimal value of exp DFT
B whichisclosetoFe-pnictidesuperconductors. AsSOCis
U for NaOsO should be chosen around 0.5 eV. At the
3 consideredintheDFTcalculations,however,K gets
DFT
sametime, if we calculateU entirelyab initio within the
systematicallyreduced, asthe estimated degreeofcorre-
constrained random phase approximation (cRPA) [27],
lation. For a full inclusion of SOC (SOC=1) we obtain
neglecting all relativistic effects (SOC=0), we obtain
K /K =0.76, a value similar to those of conven-
UnoSOC = 1.58 eV, a value similar to the one used in exp DFT
tional metals like Cr. To our best knowledge, NaOsO
3
previous studies (i.e., U = 2 eV [21–23]). As we demon-
representsthefirstsystemwheresuchastrongrenormal-
strate here, the root of this apparent controversy lays in
ization of the balance between SOC and correlations is
the suprising“competition”betweenSOCandelectronic
reported.
correlation.
Supporting evidence for the importance of SOC in
We study the interplay between SOC and correlation NaOsO is provided by the computed SOC energy, 0.4
3
bycomparingourDFT+U resultswithavailableinfrared eV/Os, and by comparing the electronic structure with
opticalexperiments. Onerathergeneralway[5,30,31]to andwithoutSOC(Fig.1(b)). TheinclusionofSOCleads
estimatethedegreeofelectroniccorrelationofasystemis to a widening of the band width by about 0.3 eV, a lin-
to evaluate the reductionof the electronic kinetic energy earizationof the bandnear Γ whichyields the formation
(mass renormalization) due to Coulomb repulsion. This ofaDirac-likefeatureand,mostimportantlyfortheelec-
can be quantified by the ratio K /K between the tron mobility, a band flattening near E that increases
exp DFT F
experimentally measured kinetic energy (K ), deter- the effective masses and therefore decreases K .
exp DFT
minedbyintegratingoverfrequencythe Drudecontribu- Therefore, by turning on the SOC the estimated de-
tionintheopticalconductivityσ andthecorrepsonding greeofcorrelationmovesgraduallyfromthebordertothe
D
“non-correlated”kineticenergyK obtainedby DFT Mott-physicstotheoneofconventionalmetallicsystems.
DFT
atU =0. Consideringthatσ canbeexpressedinterms This significant relativistic renormalization clarifies the
D
ofthe plasmafrequency, ω , via dωσ (ω)=ω2/8,one originoftheabovementionedinconsistenciesonthevalue
p D p
R
3
(a) 120 total sis is supported by the experiments of Lo Vecchio et al.
Os-t
2g showingthattheopticalconductivitydoesneithervanish
T=0 80
atT [15]norshowsacleardownturnatlowfrequencies
V) 40 N
e in the intermediate temperature region. Such a pecu-
(b) ates/ 0 liar temperature dependence of the conductivity is very
es (st12800 unusualforTMOs[15],where-inthepresenceofaMott-
at Hubbard MIT- the opposite trend can be observed [35].
of st 40 Similar temperature dependence properties were re-
T(c>)T/TN ensity 12 00 ported, instead, for the narrow-gap semiconductor FeSi:
D at low temperature FeSi is a paramagnetic insulator but
80
itdevelopsapseudogapassociatedwithabad-metalstate
40
as temperature increases [36–38]. This anomalous be-
0
havior is explained well by spin fluctuations in the con-
-1 0 1
Energy (eV) text of the theory of itinerant magnetism elaborated by
Moriya [39]. This similarity is of course very inspiring
FIG.2. (coloronline). High-temperatureparamagneticstate. for the identification of the physical mechanisms at play
Totalandt2g-projecteddensityofstates(DOS)ofthe(a)low- in NaOsO .
temperature insulating AFM ground phase, (b) disordered 3
Tosubstantiatethisideaweexploretheeffectoftrans-
paramagnetic phase (metallic) and (c) nonmagnetic phase
(metallic). The arros indicate theOssins. verse (rotational) and longitudinal spin fluctuations by
means of non-collinear fixed spin moment calculations.
As a firststep, we have simulatedthe effectof rotational
ofU inthissystem. Morequantitatively,byrescalingthe spinfluctuationsinthehigh-temperaturespindisordered
cRPAvalueofUnoSOC bytheSOC-inducedrenormaliza- configuration by performing non-collinear magnetic cal-
tion factor [31] we obtain USOC = 0.68 eV (comparable culationsonlargesupercellscontaining32Ossitesstart-
to the SOC energy) which leads to a much better de- ingfromrandomlyrotatedOsspinmomentsinaparam-
scription of the bandgap (see arrow in Fig. 1(a)). This agnetic arrangement (i.e., the total magnetic moment is
smallervalueofU isalsoconsistentwiththeU proposed zero), by fixing the magnitude of the magnetic moments
for similar compounds like LiOsO , U < 1 eV [32] and to the ground state value. The results are summarized
3
the itinerant magnet SrRuO , U ≈0.6 eV [33]. in Fig. 2. The starting point is the density of states
3
Importantly, the analogy with SrRuO is not limited (DOS) of the reference collinear AFM insulating state
3
to the low degree of correlation but also involves the at T = 0, Fig. 2(a). The paramagnetic DOS, Fig. 2(b),
magnetic itinerancy. Although NaOsO exhibits a high clearly shows that above T the system is an ordinary
3 N
T =410 K, the effective moment extracted from the the metal independently on the value of the local magnetic
N
Curie-Weiss behavior of the high temperature suscep- moment. Byallowingafullrelaxationofthemomentsthe
tibility, 2.71 µ [13], is much higher than the ground non-collinearparamagneticstateconvergestoaunrealis-
B
statemagneticmomentmeasuredbyneutrondiffraction, tic metallic non-spin-polarized state (Fig. 2(c)) with en-
≈ 1 µ [14], indicating a large Rhodes-Wohlfarth ratio tirelyquenchedlocalspinmoments,similarlyto the case
B
and suggestive of an itinerant antiferromagnetic behav- ofothermagneticallyitinerantmetalslikeCrandNi[40].
ior [34]. This magnetic itinerancy would not be compat- However,longitudinal spin fluctuations induce local mo-
iblewithalargeU andisessentialtoexplainthe MITin ments forming the true realistic high-temperature para-
NaOsO , as discussed below. magnetic state [41]. Thus, to explain the double anoma-
3
SpinfluctuationsandLifshitzMIT.Afterclarifyingthe liesbehaviorintheresistivitycurveatfinitetemperature
actualcorrelationdegree in NaOsO andthe crucialrole (Fig.3(a)), the effectofbothrotationalandlongitudinal
3
played by SOC to determine it, we are ready to address spin fluctuations should be taken into the account. To
themostintriguingfeatureofthiscompound–theorigin quantifytheamplitudeofthemomentsinducedbylongi-
of the continuous MIT at finite temperatures. tudinalspinfluctuationsatfinitetemperatureweemploy
Our main results are summarized in Figs. 2 and 3. the phenomenologicaltheory of Moriya,as describedbe-
First, we recall that the resistivity (ρ) curve of NaOsO low [39].
3
(Fig. 3(a)) shows two anomalies: one at T = 410 K, Within spin-fluctuation theory there is a universal re-
N
corresponding to a sudden increase of ρ, and the second lation between the magnitude of the local mean square
onearoundT =30K(T/T ≈0.1)[13,14]characterized fluctuating magnetic moment at the transition temper-
A N
by a steeper increment. The regionbelow T has a clear ature (M = <M2(T )>) and the ground state
A N
insulating-likebehaviorwithalargeandrapidlygrowing local moment (pM0) expressed by the Moriya formula
ρ, whereas in the intermediate region, T < T < T , < M2(T ) >= 3/5M2 [34, 39]. By combining the
A N N 0
ρ is always smaller then 1 Ωcm and grows more slowly, Moriya formula with the Mohn and Wohlfarth approx-
with a bad-metal/pseudogap behavior. This hypothe- imation[34], which assumesa linear temperature depen-
4
(a) (b) (c) (d)
104 TA TN
W ( cm)11110000--0242 I PS rM 11 0505 mTO/Ts=N1 =.107.0 OsO-t-2pg -000...022 YT S
rV)0.2 DDd 15 m Os=1.10 0.2
(e0.1 i 10 T/TN =0.3 0.0
5
D 0.0 V) 0 -0.2
II0.0 0.2 0.4DDii0.6TDD((/eeddT0.N))8 1.0 1.2 1.4 Density of states (states/e1111 0505055 mTmTOO//TTss==NN10 ==..090027..68 Energy (eV) --000000......020222
0
15 m =0.90 0.2
Os
PPSS 10 T/TN =1.0 0.0
5
-0.2
0
15 m =0 0.2
Os
10 high T 0.0
MM 5 -0.2
0
-1 0 1 G X S Y G Z U R T Z
Energy (eV)
FIG. 3. (color online). Temperature-dependent MIT. (a) Indirect band gap (∆i) and direct pseudogap (∆d) as a function of
temperature compared with the experimental resistivity curve readapted from Ref. [13]. The two anomalies in the resistivity
curves at TA and TN set the transition from an insulating (I) to a pseudogap (PS) state and from the PS to a purely metal
(M) state, respectively. These two anomalies are correlated with the closing of the insulating (∆i) and pseudo (∆d) gaps
(indicated by arrows). (b) Partial DOS (c) band structure and (d) FS for different temperatures (T/TN) corresponding to
different Os magnetic moment µOs (Eq. 2). (e) Schematic diagram of the MIT. I: AFM insulator; PS: AFM pseudogap state
with longitudinal (and small rotational) spin fluctuation; M: magnetically itinerant metal.
dence of the local mean square moment amplitude, we the transition to a metallic behavior for T > T . The
N
arriveto the following relationbetweenthe amplitude of entire process is sketched schematically in Fig. 3(e).
the fluctuating Os moment (M(T)) and temperature:
By refining our analyis, and considering the cor-
2 T
M(T)=M 1− . (2) responding evolution of the electronic bandstructure
0r 5T
N (Fig. 3(c)), we finally gain the complete description of
Using this relation we have conducted a series of calcu- the MIT in NaOsO . The bands changes almost rigidly
3
lations with fixed magnetic moments from µ = M = withtemperatureandleadtoacontinuouschangeofthe
Os 0
1.2 µ to 0 µ to examine the changes of the electronic FS topology in terms of the appearance of progressively
B B
structure upon temperature. The results are collected larger holes and electrons pockets (Fig. 3(d)): this rep-
in Fig. 3. At T = 0 (top panels) the system is an resents a clear hallmark of a Lifshitz transition [19, 20].
antiferromagnetic insulator with a narrow bandgap Lifshitz transitions like the one described here are
and an Os moment µ =1.17µ . As the temperature likely to be relevant for other magnetic materials lay-
Os B
increases the resistivity curve shows its first anomaly ingatthe borderbetweenalocalizedMottpicture anda
at T = T/T ≈ 0.1 (Fig. 3(a)), corresponding to the metallicregimesuchasNiS[44],or5dTMOswithaclose
A N
closing of the indirect band gap ∆ due to longitudinal to half-filled t3 configuration like Cd Os O [18, 19]
i 2g 2 2 7
spin fluctuations that pushes down the conduction band and iridates. Nearly half-filling, in fact, appears to be
minima at the Y point (Fig. 3(c)). Right above T , the the optimum balance between a localized (insulating)
A
system starts to develop a FS (Fig. 3(d)) and enters and deloclaized (metallic) scenario: a reduced filling is
a bad-metal state characterized by several hole and generally associated to strong U picture [43] (i.e. d1
electron pockets with a pseudogap (PS, see Fig. 3(e)) Ba NaOsO and d2 Ba CaOsO are Mott-like insula-
2 6 2 6
separating the valence and conduction bands. This tor [28]), whereas larger occupation increases the degree
pseudogap (∆ ) decreases with temperature and finally of metallicity (i.e. d4 BaOsO and NaIrO [42]); more-
d 3 3
closes for M(T) = µ = 0.9µ at about T ≃ T , over,thenearlyL =0statereducestheSOC-induced
Os B N eff
second anomaly in the resistivity curve indicated by an splitting which weaken the tendency towards Mott-SOC
arrow in Fig. 3(a). This second anomaly, due to the states and is functional for a rigid-band Lifshitz MIT.
rotational magnetic disorder described previously, set Also, structural distortions in 5d oxides are an impor-
5
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ACKNOWLEDGEMENTS
We thank A. Perucchi, S. Lupi and L. Boeri for help-
ful discussions. This work was supported by the joint
1
Supplementary information: A new route to relativistic insulators: Lifshitz transition driven by spin
fluctuations and spin-orbit renormalization in NaOsO
3
COMPUTATIONAL DETAILS
All calculations were performed using the Vienna ab initio simulation package (VASP) [S1, S2] using the DFT+U
method [S3] within the generalized gradient approximation and including spin-orbit interaction (with quantization
axisalongthe (0,0,1)direction). The on-siteCoulombU wascomputedfully ab initio usingthe constrainedrandom
phase approximation(cRPA) [S4]. For the cRPA calculations we used a maximally-localizedWannier functions basis
including the t states that ensured an excellent match with the DFT band structure around the Fermi energy
2g
as shown in Fig. S1). We used U = 0.4 eV, slightly smaller than the value estimated by the cRPA, 0.68 eV (see
main text). We adopted the generalized gradient approximation, U=0.4 eV, plane-wave cutoff of 400 eV and a t
2g
basis set for cRPA. The atomic position were fully relaxedat the experimental volume. A 10×6×10Monkhorst-Pack
meshwasused(4×3×4and20×20×20fornon-collinearcalculationsandplasmafrequencycalculations,respectively).
The lattice parameters were fixed to the experimental ones and internal positions of all atoms were fully relaxed.
A 10×6×10 Monkhorst-Pack mesh was used (reduced to 4×3×4 for non-collinear magnetic calculations in the large
supercellcontaining32Ossites)withaplane-waveenergycutoffof400eV.Inordertoestimatetheroleofrelativistic
effects on the electron correlation (relativistic renormalization) we have followed the standard procedure described
in Ref.S5 which involves the calculation of the plasma frequency in the paramagnetic phase using U = 0. For these
calculations we increasedthe k-point grid up to 20×20×20and very stringentenergy convergencecriteria (10−8 eV).
To model the high-temperature paramagnetic phase we adopted a supercell containing 32 Os sites starting from
randomlydistributedmagneticmomentswithinanon-collinearset-upresultinginazerototalmagneticmoment[S6–
S8]. The degreeof disorderwas verifiedby computing the spin-spin correlationfunction S: for the randomsystem we
have obtain S=0.2, to be compared with the ordered AFM system for which S=1.
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3
2
1
0
-1
-2
-3
S G T R G
FIG. S1. Superimposition of theDFT and t2g wannier-projected band structure.
2
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