Electronic correlations, magnetism and Hund's rule coupling in the ruthenium perovskites SrRuO$_3$ and CaRuO$_3$

0.96 MB·English

by  Hung T. Dang

Electronic correlations, magnetism and Hund’s rule coupling
in the ruthenium perovskites SrRuO and CaRuO
3 3
Hung T. Dang,1 Jernej Mravlje,2 Antoine Georges,3,4,5 and Andrew J. Millis6
1Institute for Theoretical Solid State Physics, JARA-FIT and JARA-HPC,
RWTH Aachen University, 52056 Aachen, Germany
2Joˇzef Stefan Institute, Jamova 39, Ljubljana, Slovenia
3Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France
4Centre de Physique Th´eorique, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France
5DQMC, Universit´e de Gen`eve, 24 quai Ernest-Ansermet, 1211 Gen`eve 4, Switzerland
6Department of Physics, Columbia University, New York, New York 10027, USA
5
A comparative density functional plus dynamical mean field theory study of the pseudocubic
1
ruthenate materials CaRuO and SrRuO is presented. Phase diagrams are determined for both
0 3 3
materialsasafunctionofHubbardrepulsionU andHund’srulecouplingJ. Metallicandinsulating
2
phasesarefound,asareferromagneticandparamagneticstates. Thelocationsoftherelevantphase

y boundaries are determined. Based on the computed phase diagrams, Mott-dominated and Hund’s
a dominated regimes of strong correlation are distinguished. Comparison of calculated properties to
M experimentsindicatesthattheactualmaterialsareintheHund’scouplingdominatedregionofthe
phase diagram so can be characterized as Hund’s metals, in common with other members of the

1 ruthenate family. Comparison of the phase diagrams for the two materials reveals the role played
3 by rotational and tilt (GdFeO -type) distortions of the ideal perovskite structure. The presence of
3
magnetisminSrRuO anditsabsenceinCaRuO despitethelargermassandlargertilt/rotational
3 3
] distortion amplitude of CaRuO can be understood in terms of density of states effects in the
l 3
e presence of strong Hund’s coupling. Comparison of the calculated low-T properties of CaRuO to
3
- those of SrRuO provides insight into the effects of magnetic order on the properties of a Hund’s
r 3
t metal. Thestudyprovidesasimultaneousdescriptionofmagnetismandcorrelationsandexplicates
s
the roles played by band theory and Hubbard and Hund’s interactions.
.
t
a
m PACSnumbers: 71.27.+a,75.50.Cc,72.15.Eb
-
d
I. INTRODUCTION larlyimportantroleintheinvestigationofelectroniccor-
n
relations. InmanyTMOmaterialsthetransitionmetald
o
c shells are partially filled. Interactions between electrons
Thenotionthattheelectronicpropertiesofcrystalline
[ in the d orbitals of a transition metal ion are character-
materials can to a large degree be understood in terms
ized by a sizable effective Coulomb repulsion U that is
2 of the energy bands arising from the solution of the eff
v Schr¨odinger equation for a single electron in a periodic close in magnitude to the bandwidth W of the d-derived
4 bands and leads to the formation of spin and orbital de-
potentialisfundamentaltocondensedmatterphysicsand
6 greesoffreedom. AsaresultthephysicsofTMOmateri-
its applications. Electrons are charged and the inter-
9 alsoftendifferssharplyfromthepredictionsofDFTand
3 electron Coulomb interaction cannot be neglected. Den-
involves an intricate interplay of charge, spin and orbital
0 sity functional theory (DFT), in essence a sophisticated
degreesoffreedom, whichisfurthermorehighlysensitive
. mean field treatment of electron-electron interactions,
1 to details of the crystal structure.
provides a very good approximation to the interacting
0 These issues have been intensively studied in the con-
5 electron problem, enabling the theoretical description
text of TMOs which crystallize in variants of the ABO
1 from first principles of many properties of many com- 3
perovskitestructure. Intheidealperovskitestructurethe
: pounds. However DFT does not describe all electronic
v B site ions lie on the vertices of a simple cubic lattice;
properties of all materials, and the cases where it fails
Xi can be taken to define the “strong correlation problem”. each B site ion is octahedrally coordinated by oxygen.
Few members of the family of ABO transition metal
ar One dramatic example of strong correlations is the oxides crystallize in the ideal cubic s3tructure: in most
“Mott” insulator [1]: a material in which the correla- materials a mismatch between the size of the A and B
tions are so strong that they lead to insulating behavior siteions(“tolerancefactor”lessthanone)leadstoacom-
in situations where band theory predicts a metal. Less pressive strain on the BO network. This strain is typ-
3
extreme cases, where the interactions do not drive the ically accommodated by a rotational and tilt (GdFeO -
3
material insulating but do lead to strong renormaliza- type) distortion of the BO octahedra that diminishes
6
tion of electron velocity relative to band theory, to large the width of the d-derived bands and lowers the degen-
and strongly temperature and frequency dependent elec- eracy of the d multiplets.
tron lifetimes, and to the occurrence of magnetic order,
Particular attention has been given to materials in
have been extensively documented [2].
which the B-site ion is drawn from the first transition
Transition metal oxides (TMOs) [1, 2] play a particu- metalrowoftheperiodictablesothatthe3dshellofthe
2
transition metal ion is partially occupied. In these ma- crystallize in GdFeO -distorted versions of the ABO
3 3
terials the key physics is the correlation-induced metal- perovskite structure; with the distortion amplitude be-
insulator transition (often referred to as the ‘Mott’ tran- ing larger in CaRuO than in SrRuO . SrRuO is fer-
3 3 3
sition [1] although actual atomic-scale physics may be romagnetically ordered (a rather rare behaviour among
more involved [3]). The prevailing understanding [2, 4] 4d TMOs) below a Curie temperature T ∼ 160K while
c
is that in most of these materials the basic “correlation CaRuO remains paramagnetic to lowest temperatures.
3
strength” is related to the proximity of the material to On the applied side, SrRuO is a convenient electrode
3
theMotttransition(butseeRefs.5–8fortheexceptional material, widely used as a substrate and magnetic ingre-
case of the nickelates) while the rotational and tilting dient in heterostructures and spin-valves [29–32]. Basic
distortions play a key role in determining this proximity. scientificquestionsremainopen,includingtheirdegreeof
For example, SrVO (nominal valence d1) crystallizes in correlation,theoriginoftheapparentlynon-Fermi-liquid
3
thesimplecubicstructureandisamoderatelycorrelated properties evident in the optical spectra [33, 34] and the
Fermi liquid [9]. In CaVO , a small-amplitude GdFeO reason for the magnetism, in particular why the appar-
3 3
distortion occurs; the material is still metallic but more ently less strongly correlated material SrRuO is mag-
3
correlated than SrVO [9–11]. In LaTiO the nominal d netic while the apparently more correlated CaRuO is
3 3 3
valence is also d1, however a larger-amplitude GdFeO not. There is also fundamental interest in obtaining a
3
distortion is present and the material is a Mott insu- better understanding of ruthenates in general, because
lator. In isoelectronic YTiO the distortion amplitude insightsgainedinthestudyofthepseudocubicmaterials
3
and the insulating gap are larger than in LaTiO [2, 12]. may shed light on the unconventional superconductivity
3
The differences between Sr- and CaVO or between La- of Sr RuO [26] and the metamagnetism and other phe-
3 2 4
and YTiO may be attributed to different amplitudes nomena observed in Sr Ru O [27].
3 3 2 7
of the GdFeO distortion. Theoretical work [13] showed
3 The question of the correct physical picture of the
that the key physics is a lifting of the degeneracy of the
pseudocubic ruthenate perovskites (whether they should
transition metal t levels; this is important because the
2g be regarded as weakly correlated itinerant metals or
criticalinteractionstrengthneededtodriveaMotttran-
as strongly correlated systems) is the subject of con-
sition depends strongly on orbital degeneracy, see, e.g.
troversy. On the experimental side, photoemission ex-
Ref. 14 and references therein. (The differences between
periments [35] do not detect Hubbard sidebands, sug-
the V-based and Ti-based materials arise in part from
gesting that the materials are not in proximity to a
difference in GdFeO distortion amplitude and in part
3 Mott transition. However, an earlier interpretation of
from the difference in relative electronegativities of Ti
the photoemission spectroscopy [36] indicated that siz-
and V [15, 16].)
able renormalizations occur at low energies [37]. Optical
However, proximity to a Mott insulating state is not spectroscopy [33, 34, 38, 39] indicates strong deviations
the only cause of correlated electron behavior. In heavy- from Fermi-liquid behavior, while transport experiments
fermionmaterialsalatticeversionoftheKondoeffectcan reveal very low Fermi liquid coherence scales (7K for
lead to enormous mass enhancements and other exotic SrRuO [40] and 1.5K for CaRuO [39]) and large mass
3 3
physics [17]. In transition metal oxides with nominal enhancements.
valences from d2 to d8 the Hund’s coupling can play a
On the theory side, early analyses [41, 42] of the elec-
crucialroleinproducingverylargerenormalizationseven
tronic structure based on spin-dependent density func-
for materials far from a Mott transition [18–23].
tional theory [local spin density approximation (LSDA)
In this regard transition metal oxides where the tran- or spin-dependent generalized gradient approximation
sition metal is drawn from the 4d series are of particular (GGA)] correctly describe many of the magnetic prop-
interest. Because 4d orbitals have a much greater spa- erties. The ferromagnetism in SrRuO was interpreted
3
tial extent than 3d orbitals, the effective bandwidth is as the result of a Stoner instability, and the presence of
largerandtheUeff issmaller, suggestingthatthe4dma- magnetisminSrRuO3anditsabsenceinCaRuO3wasre-
terials are in general likely to be farther from the Mott lated to the Fermi-level density of states, which is higher
state than the 3d materials. Although many of the 4d and more sharply peaked in SrRuO than in CaRuO .
3 3
series TMO are indeed itinerant metals, signatures of However the DFT calculations do not account for the
strong correlations, such as enhancement of the specific low coherence scales and large mass renormalizations.
heat [24, 25], magnetic transitions [26] and evidence of A more recent comparative study of magnetism using a
other unusual electronic phases [27], are clearly present, range of band theoretic techniques including the density
in particular in the ruthenate family [23, 26]. Further, functionalplusU methodconcludedthatU =0givesthe
some members of the 4d series (for example Ca2RuO4) bestdescriptionoftheexperimentallyobservedtransition
have been identified as Mott insulators [28]. Thus in the temperatures [43]. Within LSDA, properties of SrRuO
3
4d-series transition metal oxides the issue of the relative andCaRuO understrainwerecalculated[44,45]andthe
3
importanceofMottandHund’scorrelationsremainsun- predicted occurrence of ferromagnetism in CaRuO un-
3
clear, as does the role of the GdFeO3 distortions. dertensilestrainwasrecentlyobserved[46]. Ontheother
Here, we explore these issues by focusing on the two hand, many theoretical papers including the LSDA+U
of pseudocubic ruthenates: SrRuO and CaRuO . Both work of Rondinelli and collaborators [47] and several dy-
3 3
3
namical mean-field theory (DMFT) investigations [48– pushes the e levels up in energy so the relevant near-
g
51] assert that correlations beyond LSDA/GGA are im- Fermi-surfacebandsarederivedfromRut symmetryd
2g
portant. states with some admixture of the O 2p states. Because
Theexistingliteraturethussuggeststhatthechallenge Ru is a second-row transition metal ion, the d states are
presented by the perovskite ruthenates is to develop a expectedtobemoreextendedandtheon-siteinteraction
theory that includes the electronic correlations that pro- U is expected to be weaker than for the first-row transi-
videtheexperimentallyindicatedmassenhancementand tionmetalions,indicating[23]thatthematerialsarenot
otherrenormalizationswithoutspoilingthegoodaccount in the charge transfer regime of the Zaanen-Sawatzky-
of the magnetic phase diagram found in density func- Allen phase diagram [3]. We therefore adopt the “fron-
tional calculations. In this paper we address this chal- tier orbital” approach in which the low energy electronic
lenge by performing a systematic density functional plus propertiesareobtainedfromamultibandHubbardmodel
dynamical mean field theory study that includes realis- withhybridizationsandlevelsplittingsobtainedfromthe
tic electronic structure and investigates a wide range of near-Fermi-surface transition metal d-derived bands.
potentially relevant interaction parameters. We calcu- The Hamiltonian takes the general form
late the phase diagram in the (U,J) plane and by com-
H =H +H , (1)
paring calculated and measured properties we locate the kin onsite
perovskite ruthenates in the correlated Hund’s metal re- where H describes the dispersion of the bands derived
kin
gion of the phase diagram. Effective masses are found from the frontier orbitals and H the additional in-
onsite
to be large and coherence scales small in the paramag- teractions. The chemical potential is set to ensure that
netic phase. The greater tendency to magnetic ordering these bands contain four electrons per Ru.
in SrRuO3than in CaRuO3is accounted for in a manner To define the near-Fermi-surface bands of Hkin pre-
similartothatfoundinearlierelectronicstructurecalcu- cisely we use the non-spin-polarized generalized gradient
lations. Our results also provide insight into the general approximation (GGA) as implemented in the Quantum
issue of the effect of GdFeO3 distortions on the effective ESPRESSO density functional code [54, 55] to obtain
correlation strength of Hund’s metal. The low frequency electronic band structures and then project the result-
properties are affected by the density of states (which in ing bands onto maximally-localized Wannier functions
the ruthenates is reduced by GdFeO3 distortions) while (MLWF)[56,57]usingthewannier90code[58](details
the global and higher frequency correlation strength is are given in Appendix A).
controlled by the inverse bandwidth (which is increased In most of this paper we construct H by projecting
kin
by GdFeO3 distortions). the Kohn-Sham Hamiltonian onto t2g-symmetry Wan-
Therestofthispaperisorganizedasfollows. SectionII nier functions centered on the Ru sites. This procedure
describes the methods we use. Section III presents our captures correctly all of the electronically active frontier
computed ferromagnetic/paramagnetic (FM/PM) and orbitals and provides a reliable description of the phase
metal-insulatorphasediagramsanddiscussesthephysics diagram and quasiparticle properties. However, as will
behind them. Section IV discusses the differences of two be discussed in detail in Sec. V, this procedure leads to
ruthenates in the mass enhancement and the self-energy an overestimate of the magnetic moment in the magnet-
and uses this information to locate the materials on the icallyorderedphase. Obtainingacorrectestimateofthe
phasediagramofSec.III.ThemagneticphaseofSrRuO3 orderedmomentrequiresinclusionofbandsderivedfrom
isanalyzedindetailinSec.V.SectionVIpresentsasum- Ru e states. In our analysis of the magnetic state the
g
maryandprospectsforfuturework. Appendicesprovide e -derived bands are therefore retained, but because the
g
details of the calculations e -derived states are far from the Fermi level, they are
g
treatedbythemean-fieldapproximationusedinRef. 59.
The t orbitals are treated dynamically. As usual in
2g
II. CRYSTAL STRUCTURE, ELECTRONIC studiesoftransitionmetaloxides, theinteractionHamil-
STRUCTURE AND MODEL tonian is taken to be site-local and to have the rotation-
allyinvariantSlater-Kanamoriform[2]. Weusetheform
appropriate [23] for intra-t orbitals, since these are the
CaRuO and SrRuO crystallize in a Pnma symme- 2g
3 3
primary focus of this work
trycrystalstructurerelatedtotheidealcubicperovskite
structure by a GdFeO distortion corresponding to a tilt (cid:88) (cid:88)
3 H =U n n +(U −2J) n n +
and rotation of each RuO octahedron. The tilts and onsite α↑ α↓ α↑ β↓
6
rotations alternate in a four-sublattice pattern. SrRuO α α(cid:54)=β
3 (cid:88)
has a Ru-O-Ru bond angle of about 163◦, in between +(U −3J) n n +
ασ βσ (2)
the ideal perovskite Ru-O-Ru bond angle of 180◦ and α>β,σ
the Ru-O-Ru bond angle of 150◦ observed in CaRuO (cid:88)
3 +J (c† c† c c +c† c† c c ),
[52, 53]. α↑ β↓ α↓ β↑ α↑ α↓ β↓ β↑
α(cid:54)=β
ValencecountingimpliesthatinCaRuO andSrRuO
3 3
theRuisinthed4+ electronconfigurationwithfourelec- whereα,βareorbitalindicesandσisthespinindex. Dif-
trons in the Ru 4d shell. The octahedral ligand field ferent values of U and J have been used for ruthenates
4
in the literature. For Sr RuO , constrained LDA [60] III. QUALITATIVE PHYSICS
2 4
gives (U,J) values of (3.1 eV,0.7 eV) while the values
(2.3eV,0.25eV)[20]and(2.6eV,0.26eV)[61]havebeen
A. Electronic structure: Density of states
obtained from constrained random phase approximation
(cRPA) method. As discussed above, for the perovskite
ruthenates U values ranging from zero to rather large
numbers have been employed. For this reason and be- 1.2 (a) xy CaRuO
cause the behavior of the model for general parameters 0.6 3
isoftheoreticalinterest,weconsiderarangeofvaluesfor
0.0
U and J in this paper. However we restrict attention to
0.6
theregimeU >3J wheretheeffectiveon-siteinteraction
is positive in all channels. 1.2 SrRuO3
s
e
ploTyostinregalet-stithee doynnsaitmeicianltemraecatnio-finel[dEqt.he(o2r)y], (DweMFemT)- stat 01..62 (b) yz CaRuO3
[62]. This method allows us to map the Hamiltonian of 0.0
[Eq. (1)] into a multiorbital impurity model embedded y
t 0.6
in a fermion bath. The impurity model is solved by us- si
ingthehybridizationexpansionversionofthecontinuous en 1.2 SrRuO3
D
timequantumMonteCarlo(CT-HYB)[63]implemented 1.2 (c) zx CaRuO
3
in the TRIQS code [64] for rotationally invariant inter- 0.6
action using conserved quantities [65] to speed up the 0.0
calculations.
0.6
Careisrequiredinthedefinitionoftheimpuritymodel. 1.2 SrRuO3
Each Ru ion is at the center of an octahedron defined by
2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0
six oxygen ions. The Pnma structure means that the
Energy (eV)
localsymmetryaxesofagivenRuO octahedronarenot
6
parallel to the axes that define the global crystal struc-
FIG. 1. Orbitally-resolved density of states generated from
ture. If the e and t combinations of the d-derived
g 2g MLWFfittingsoftheDFTcalculationsforCaRuO (SrRuO )
states in a single octahedron are defined with respect to 3 3
on the positive (negative) half plane. Each panel is corre-
theglobal,ratherthanthelocal,symmetryaxes,theim-
spondingtoeachofthet orbitals. The(xy,yz,zx)occupan-
2g
purity model will contain off-diagonal terms which mix cies are (1.40,1.36,1.23) for CaRuO , and (1.34,1.26,1.40)
3
the different orbitals at the single-particle level. This for SrRuO , respectively. In the same order, the DOS at
3
causes a severe sign problem for the CT-HYB solver the Fermi level ν (states/eV per Ru atom) for each of
F
[66]. It is preferable to avoid this complication, follow- the three t orbitals are (0.83,0.73,0.79) for CaRuO , and
2g 3
ing Ref. 67, by using a local basis with symmetry axes (1.00,0.77,1.19)forSrRuO3. Theverticaldashedlinemarks
aligned along the octahedral directions appropriate to a the Fermi level.
given Ru, instead of the global axes. When restricted to
thet manifoldonly,theMLWFapproachusedherepro- Figure 1 shows the orbitally-resolved near-Fermi-
2g
ducesorbitalsthatarealreadyalignedwithrespecttothe surface density of states (DOS). The DOS of the two
localaxesoftheappropriateoctahedronsothehybridiza- materials are similar, as expected from the essentially
tion function is essentially diagonal. Thus if only t identical quantum chemistry, but the difference in the
2g
orbitals are retained the DMFT calculation is straight- magnitudeoftheGdFeO distortionoccurringinthetwo
3
forward: theimpuritymodelwithhybridizationfunction materials leads to two important differences in the DOS.
defined directly from the projection of the Kohn-Sham First,thet -derivedbandsinSrRuO areapproximately
2g 3
Hamiltonian onto the Wannier states is solved for one 10% wider than those of CaRuO (SrRuO bandwidth
3 3
Ru site. The self-energies for the other Ru sites are then ≈ 3.0eV, as compared to ≈ 2.7eV for CaRuO ). To
3
constructed by applying appropriate rotation operators. the extent that correlation effects scale as the ratio of
If the MLWF procedure is applied to the full d manifold an interaction strength to a bandwidth, this would sug-
(including both t and e orbitals) then the resulting gestthatCaRuO wouldbethemorestronglycorrelated
2g g 3
orbitalsturnoutnottobealignedtothelocalsymmetry material.
axes and an additional change of basis is required before However, SrRuO has the larger density of states at
3
solving the impurity model (see Appendix A). From the theFermilevel. ThiscanbetracedbacktothevanHove
solution of the dynamical mean field equations we de- singularityoftheundistortedcubicstructure,whichhap-
termine the phase (metal versus insulator, paramagnetic pens to lie very close to the Fermi level. Because the
versusferromagnetic)andsomepropertiesofthephases, GdFeO distortion lifts the degeneracy of the t lev-
3 2g
inparticularthequasiparticlemassenhancementandthe els, it splits the van Hove peak into three features. In
magneticmoment. Detailsofourprocedurefordetermin- SrRuO the splitting is small and the density of states
3
ing different phases are given in Appendix B. remains large. In CaRuO the splitting is larger, leading
3
5
toasmallerFermi-levelDOS.Totheextentthatcorrela- 5 5
tioneffectsarerelatedtotheFermi-leveldensityofstates, (a) SrRuO3 (b) CaRuO3
4 4
this suggests that SrRuO would be the more strongly
correlated material. In pa3rticular, the Stoner model of eV)3 PM FM PM FM 3
(
ferromagnetism relates the presence of magnetic order U 2 FM metal 2
to the value of the product of an interaction constant 1 PM metal 1
and the Fermi-level density of states [68], so the density PM insulator U<3J region
0 0
of states difference would suggest (in agreement with ex-
(c)PM insulator (d)PM insulator
perimentandwiththeDFTworkofRefs.41,42,and69) 4 4
thatSrRuO3ismorelikelytobemagneticthanCaRuO3. V)3 FM metal FM metal 3
Further, particularly in CaRuO , the splitting creates a e
densityofstatespeakbelowthe3Fermilevel. Theconsid- U (2 PM PM 2
metal CRO PM/FM metal CRO MIT
erationsofRef.67buildingonpreviousworkofVollhardt 1 1
SRO PM/FM SRO MIT
and collaborators [70–74] suggest that this peak is unfa- 0 0
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0
vorable to magnetism.
J (eV) J (eV)
FIG. 2. (Color online) Ferromagnetic metal/paramagnetic
B. Phase diagrams metal (FM/PM) and metal/Mott insulator (MIT) phase di-
agrams. The dotted lines indicate the metal-insulator phase
Figure2displaysthephasediagramsintheplaneofthe boundary, with the region above the lines being insulating.
The heavy solid line (red on-line) indicates the boundary of
Hubbard U and the Hund’s coupling J, determined by
the ferromagnetic region, while the light line (black on-line)
the procedure described in Appendix B. We considered
separatesthephysicallyrelevantpositiveeffectiveinteraction
metallic and insulating phases, paramagnetic and ferro-
(U >3J) from the unphysical negative interaction (U <3J)
magnetic phases. Antiferromagnetism was not studied.
region. (a,b)PhasediagramsforSrRuO (a)andCaRuO (b)
3 3
Focusfirstontheupperrightpanel,whichpresentsre- obtained from the DFT+DMFT procedure described in the
sultsforCaRuO3. Weseethatastheinteractionstrength text. Open circles (black on-line) indicate (U,J) points for
U isincreasedatfixedJ,thereisaphasetransitionfrom whichpropertieswerecomputedandaparamagneticmetallic
a metal (which may be paramagnetic or ferromagnetic) state was found. Closed circles (black on-line) indicate para-
toaMottinsulator. AsJ isincreasedatfixedU, atran- magneticinsulating solutions. Opensquares (redon-line)in-
sition to a ferromagnetic metal phase occurs but no fer- dicate (U,J) points for which a ferromagnetic metallic state
wasfound. (c)ThephasediagramsforSrRuO andCaRuO
romagnetismwasfoundininsulatingphasesintheU and 3 3
plotted together. Ferromagnetic phase boundary of CaRuO
J ranges that we studied. From this phase diagram one 3
indicated by heavy dashed line (blue on-line). (d) Phase di-
can identify two regimes: at large values of U, near the
agrams for SrRuO and CaRuO plotted together, but with
3 3
metal-insulatorphaseboundary,propertiesaremostsen-
the phase boundary of SrRuO rescaled by the ratio of the
3
sitive to the value of the effective Hubbard interaction. SrRuO bandwidth to the CaRuO bandwidth. Note that
3 3
Note in particular that at large J the phase boundary there is no FM insulating phase found in these phase dia-
becomes a straight line with slope U−3J. The quantity grams.
U −3J is the effective Hubbard interaction (correlation
strength)relevanttotheMotttransitionbecauseitgives
the lowest energy cost for a valence change from 2d4 to berequiredtoaccountforthefactthatCaRuO isapara-
3
high spin d3d5. In contrast, far from the metal-insulator magnetic metal, while SrRuO is a ferromagnetic metal.
3
phaseboundary,theHund’scouplingJ isthekeyparam- A much more significant difference between phase
eter: by increasing J a ferromagnetic phase is induced boundariesforthetwostructuresisfoundatsmallerval-
and there is a range of U in which the location of the ues of U <∼3 eV. There, the CaRuO3 phase diagram has
critical boundary is only weakly dependent on U. a significantly smaller region of ferromagnetism than the
The upper left panel displays our results for SrRuO . SrRuO one. Thus, in this regime much less fine-tuning
3 3
ThesamephasesarefoundbutthedifferenceinGdFeO of the parameters is needed to correctly account for the
3
distortion amplitude causes the location of the phase difference in magnetic behavior of the two materials. As
boundaries to be different. To highlight the differences discussed in Sec. IV, there are also other experimental
betweenthetwocalculationswepresentinFig.2(c)asu- indications suggesting that these two materials should
perposition of the two phase diagrams. At larger U the be viewed as being in this Hund’s coupling dominated
phaseboundariesareparallel(andcontrolledbyU−3J), regime. WealsonotethatthecRPAvaluesofU,J found
withCaRuO requiringaslightlysmallervalueofU tobe for the related Sr RuO material [61] are in the mod-
3 2 4
drivenintotheMottphase,asexpectedfromthesmaller erate U, larger J region where CaRuO is paramagnetic
3
bandwidth and larger t -level splitting following from but SrRuO is ferromagnetic.
2g 3
thelargerdistortioninCaRuO . However,inthisregion We now discuss further the qualitative origin of the
3
of the phase diagram the magnetic phase boundaries are observed differences between the phase diagrams of the
very similar, and an extreme fine-tuning of (U,J) would twostructures,bypresentinginFig.2(d)asuperposition
6
of the phase diagrams of the two materials, but with the (a) U=2.3 eV
0.0
U andJ valuesforSrRuO3rescaledbytheratior =1.11 0.0
of the SrRuO to the CaRuO bandwidth. At larger U CaRuO
3 3 3
0.1
the phase diagrams for both the metal-insulator transi- 0.1 SrRuO
3
tion and magnetism coincide in the rescaled plot. This 0.2
indicates that in the Mott-dominated region the physics ω 0.42eV
is controlled by the ratio of the interaction strengths to 0.2 0.3 c≈
)
thebandwidthsanddependsonlyweaklyonforexample ωn 0.4
(
the Fermi-level DOS. However, in the smaller U regime, mΣ 0.3
the magnetic phase diagrams do not coincide, indicating I 0.5
that in this regime the physics is clearly not controlled 0.0 0.2 0.4 0.6 0.8
0.4
solely by the difference in bandwidths.
Instead, the substantial difference in the critical J re-
quiredtodrivetheferromagnetictransitionisassociated 0.5
to the DOS in the near-Fermi-level region. One impor-
tant property is the value of the DOS at the Fermi level 0.0 0.5 1.0 1.5 2.0 2.5
ω (eV)
ν . InthestandardStonertheory[68]magnetismisasso- n
F
ciated with a value greater than unity of a dimensionless (b) U=3.0 eV
0.0
interactionparameterI constructedastheproductofan 0.0
appropriate interaction energy and the Fermi level den- 0.1 CaRuO3
0.1
sity of states (I = UνF > 1). Clearly the larger DOS SrRuO3
in SrRuO3makes it easier for the Stoner parameter to 0.2 0.2
ω 0.11eV
exceed the critical value and as discussed by Mazin and c≈
0.3
collaborators [41, 42] spin dependent DFT calculations 0.3
)
indeed indicate a Stoner parameter slightly greater than ωn 0.4 0.4
(
unity for SrRuO and slightly less for CaRuO . It is Σ
3 3 m
also worth noting that the Stoner theory is in essence I 0.5 0.5
a Hartree approximation. When the correlation is fully 0.6 0.00 0.05 0.10 0.15 0.20
treated, other factors such as the energy derivative of
the DOS at the Fermi level and indeed the structure of 0.7
the DOS far from the Fermi level are also important and
0.8
provide significant corrections to the simple Stoner esti-
mate. Dynamical mean field studies of related models 0.0 0.5 1.0 1.5 2.0 2.5
ω (eV)
[67,70–72]indicatethatforsystemswithcarrierconcen- n
tration such that the d shells are less than half occupied
such as La Sr VO (d valence d2−x) ferromagnetism FIG. 3. (Color online) The imaginary part of the Matsub-
1−x x 3
is favored if the DOS peak is at or below the Fermi level araself-energyImΣ(iωn)averagedoverthethreet2g orbitals,
calculated at U = 2.3 (a) and 3eV (b) with J = 0.4eV,
[67]. A particle-hole transformation allows us to relate
T =0.0025eV for SrRuO and CaRuO . The inset provides
the results of Ref. 67 (derived for a system with valence 3 3
an expanded version of the low frequency regime, allowing
near d2) to the ruthenates (valence d4), concluding that
the crossing point ω to be distinguished. Calculations are
intheruthenatesaDOSpeakatorabovetheFermilevel c
restricted to the paramagnetic order.
favors ferromagnetism. Therefore, as seen in Fig. 1, the
larger distortion of CaRuO produces below-Fermi-level
3
densityofstatespeaks, thusdisfavoringferromagnetism,
tively. We fix J = 0.4 eV as a representative value for
whileinSrRuO theDOSpeaksconcentrateattheFermi
3 which the Ca material is paramagnetic and the Sr mate-
level and ferromagnetism is favored.
rial is ferromagnetic.
Figure 3 displays the imaginary parts of the com-
puted Matsubara-axis DMFT self-energies restricted to
IV. SELF-ENERGY, MASS ENHANCEMENT the paramagnetic phase over a wide range of Matsub-
AND QUASIPARTICLE LIFETIME ara frequencies. We see that at low frequencies the self-
energy of the Sr-compound is larger, indicating that for
In this section we study the electron self-energy and quantitiesdominatedbylowenergiestheSrmaterialmay
quasiparticleproperties,choosinginteractionparameters beviewedasmorecorrelated. Ontheotherhand,abovea
U =2.3 and 3 eV as representatives of the Hund’s metal frequency ω the self-energy of the Ca material is larger,
c
and Mott-dominated regimes respectively. These values reflecting the effect of the difference of bandwidths on
are similar to those obtained for the related material the effective correlation strength at higher energies. The
Sr RuO from ab initio estimations using constrained Hubbard U and Hund’s coupling J compete in this re-
2 4
DFT [60] and constrained RPA [20] approaches, respec- spect. AtsmallerU [Fig.3(a)],theHund’scouplingeffect
7
Figure 4 shows the estimated mass enhancement for
7
30 thetwomaterialsatthetwovaluesofU underconsidera-
χ(iΩ ))
10 6 m /nm=0 251)− tion. Thecalculationsarerestrictedtotheparamagnetic
V
m5 e state and show a strong temperature dependence, which
ncement 8 m/43 1250χ(iΩ)(n=0 ialsoscsaaomlcimaantoeimfdeeswntatittwhiointthhoeSftf(cid:29)ohrem1loa/wt2io(qniunaaosntiphdaesrrltowicwolerfldcuso,chwteuriatehtnicoHenusnocdafl’aes
nha 6 2 10 metal physics) [19, 23]. As will be seen below, in the fer-
e 0.1 0.2 0.3 0.4
ss J(eV) romagneticstateofSrRuO3 thetemperaturedependence
Ma is cut off because the ferromagnetic order quenches the
4
slowspinfluctuations. Toreinforcethispointweshowin
CaRuO3 theinsetthatthemassenhancementandzeroMatsubara
2 SrRuO3 frequency impurity model spin correlation function have
identicalJ dependence. ComparisontoFig.1showsthat
0 50 100 150 200 250 300
Temperature (K) the local spin fluctuations involve an energy scale that is
much lower than that of the bare density of states.
FIG. 4. (Color online) The mass enhancement Z−1 com- In our simulations we were unable to reach tempera-
puted as described in the text for SrRuO (squares, black tureslowenoughtoobservethesaturationofthemassto
3
on-line) and CaRuO (triangles, red on-line) at J = 0.4eV itsT →0limitintheparamagneticphase. Nevertheless,
3
and U = 2.3eV (solid lines) and 3eV (dashed lines) as a we find the quasiparticles are becoming well defined at
function of temperature. In these calculations magnetism thelowesttemperaturesreachedinoursimulations. Fig-
is suppressed: all results are for the paramagnetic phase of
ure5presentsthetemperaturedependenceofΓ/T calcu-
themodel. Inset: comparisonofmassenhancement(squares,
lated for CaRuO at U = 2.3 and 3eV with J = 0.4eV.
greenon-line,leftyaxis)andzeroMatsubarafrequencyspin- 3
Below T = 70K for U = 2.3eV and below T = 30K
spin correlation function χ(iΩ )=(cid:82)β(cid:104)S (τ)S (0)(cid:105)dτ with
n=0 0 z z for U = 3eV the scattering rate becomes smaller than
S = 1(Ntot−Ntot) (circles, blue on-line, right y axis) com-
z 2 ↑ ↓ temperature, which is indicative of coherence. We ex-
puted for CaRuO at temperature T =2.5 meV≈29K with
3 pectthatasthetemperatureisloweredfurtherbelowthe
U =2.3eV and plotted as a function of J.
coherence scale the mass will saturate. From these con-
siderationsweestimatetheT =0massenhancementsto
is stronger, resulting in a wider range of low frequency beabout7and11fortheU =2.3and3eV,respectively.
(largerω ≈0.4eV)inwhichSrRuO ismorecorrelated.
c 3
For the larger U close to the MIT phase boundary, Mott 5 5
(a) U=2.3eV (b) U=3.0eV
physics associated with U becomes stronger, as signaled
by the decrease of ω by a factor of ∼ 4 [ω ≈ 0.1 eV in 4 4
c c
Figure 3(b)].
We now turn to two key physical quantities charac- 3 3
T
terizing quasiparticles in the metallic state, namely the Γ/
effective mass enhancement m(cid:63)/m (directly related in 2 xy 2
thelocalDMFTapproachtothequasiparticleweightZ)
1 yz 1
and the quasiparticle scattering rate (inverse lifetime) Γ.
zx
These are defined from the real and imaginary part of
0 0
the retarded self-energy on the real-frequency axis by: 0 50 100 150 200 250 300 0 50 100 150 200 250 300
T(K) T(K)
m∗ 1 ∂
= =1− ReΣ(ω+i0+)| , (3)
m Z ∂ω ω=0 FIG.5. (Coloronline)Inverseofthequasiparticlelifetime,Γ
Γ = −ZImΣ(ω+i0+)|ω=0. (Eq. (3)) calculated for CaRuO3 at J =0.4eV and U-values
indicated, divided by temperature and plotted as a function
Inferring real-axis quantities from Matsubara-frequency of temperature, for each of the t orbitals. Linearity cor-
2g
data in general requires analytical continuation. If, how- responds to Fermi liquid behaviour Γ ∝ T2. The horizontal
ever, the low frequency properties are reasonably well dashedlinemarkstheboundary(Γ≤T)ofthecoherencere-
described by the Fermi liquid fixed point (as is the case gion. Thethreeorbitalsdifferslightlyinoccupancy;thecloser
for the parameters we study), the low frequency limit the orbital is to half filling the more strongly it is correlated.
[20]. The convention for the orbitals is given in Appendix A.
of the real frequency self-energy may be inferred with
reasonable accuracy from the data at small Matsubara
frequencies, with 1−Z−1 ≈ dImΣ(iω )/dω | and We remark that Fig. 5 implies that the quasiparticle
n n ωn→0
Z−1Γ = −ImΣ(iω → 0). In practice, we extract Z scattering rate Γ varies as T2 up to T ≈ 150K, even
n
andΓbyfittingafourth-orderpolynomialtothefirstsix thoughthecharacteristicFermiliquidsignaturesinphys-
Matsubara-axisdatapointsforImΣ(iω )andcomputing ical observables (temperature independent mass/specific
n
the needed quantities from the fitting function. heat coefficient and magnetic susceptibility along with
8
(a) SrRuO
7 3 0.9 (a) xy
CaRuO
T=0.0250eV 3
T=0.0100eV 0.6 CaVO
6 T=0.0067eV 3
nt T=0.0050eV 0.3
e T=0.0025eV
m5
ce s0.0
n e
ass enha4 of stat00..69 (b) yz
m3 y
t0.3
si
n
2 De0.0
0.15 0.20 0.25 0.30 0.35 0.40 0.9 (c) zx
J (eV)
0.6
(b) CaRuO
7 3
T=0.0250eV=290K 0.3
T=0.0100eV=116K
6 T=0.0050eV=58K 0.0
T=0.0025eV=29K 3 2 1 0 1 2 3
ent Energy (eV)
m
e5
c
n
ha FIG. 7. Predicted orbitally-resolved spectral function of
n
s e4 CaRuO3. WeusedU =2.3eV,J =0.35eV,andtemperature
as T = 2.5 meV ≈ 29 K. Dashed line is the orbitally-averaged
m
spectralfunctionforCaVO withdataextractedfromRef.78.
3
3
2
0.15 0.20 0.25 0.30 0.35 0.40
J (eV) while Fig. 6 shows that Z−1 depends even more strongly
on Hund’s coupling J. As the interaction parameters
FIG.6. (Coloronline)Themassenhancementof(a): SrRuO are not likely to change significantly between the two
3
and (b): CaRuO vs. the Hund’s coupling J calculated at compounds,weassumethatSrRuO andCaRuO arede-
3 3 3
U = 2.3eV for several different temperatures. For SrRuO3, scribed by the same (U,J) values. Requiring that the
thelargersquareindicatestheJ valueclosesttotheonsetof calculated mass enhancement for CaRuO is close to but
3
ferromagnetism at the corresponding temperature. not higher than the measured mass, and at the same
time that J be such that SrRuO is ferromagnetic and
3
CaRuO is paramagnetic allows us to locate the materi-
3
quadratic transport scattering rate) are only evident be- als on the phase diagram.
lowmuchlowertemperatures(lowerthanthelowesttem- At U = 3eV the phase diagram of Fig. 2 requires
perature accessible in our U = 2.3, J = 0.4eV calcu- that 0.3 eV <∼ J <∼ 0.4eV while for J in this range the
lations). This behavior qualifies CaRuO as a ‘hidden masses resulting from our calculations are clearly above
3
Fermi liquid’ [75, 76] in which although the temperature the experimental value (≈ 10 at T = 50K and clearly
dependenceofe.g. theresistivitydeviatesfromT2 above increasing as T is decreased; see also the lower panel of
a very low temperature, the quasiparticle scattering rate Fig.8). Thuswearguethatthecombinationofthemass
remains ∼ T2 up to much higher temperatures and the and phase diagram are inconsistent with the possibility
deviationfromtheFermiliquidtemperaturedependence thattheperovskiteruthenatesareintheMott-dominated
expected for the resistivity is attributable to a tempera- regime.
ture dependence of the quasiparticle weight. On the other hand, our results at smaller U indicate
We now use the experimentally measured low temper- that the Hund’s metal regime can provide a good de-
ature specific heat coefficient to help constrain the pa- scription of the basic physics. A relatively wide range of
rameter values. The measured low-T specific heat of J is found for which SrRuO3is magnetic and CaRuO3is
CaRuO is ≈ 74mJ/molK2 [77], implying a mass en- not,whilethesensitivedependenceofthemassenhance-
3
hancement m(cid:63)/m ≈ 7 with respect to the DFT value. mentonJ,withmassesrangingfrommuchsmallerthan,
We note that the experimental value has a contribution to rather larger than, the measured value, means that
(ofunknownmagnitude)fromtheelectron-phononinter- a reasonable J ∼ 0.3eV and U ∼ 2−2.5eV (close to
action, so should be regarded as an upper bound on the those found from constrained DFT and cRPA methods
electronic contribution to the mass enhancement. Fig- [20, 60, 61]) will account for the basic physics.
ure 4 shows that Z−1 has a marked dependence on U It is interesting to look also at the correlated DOS.
9
We obtained the spectra at real frequencies using the 5.0 (a) Magnetization M and m∗/m of SrRuO3 1.8
maximum entropy method [79, 80] to continue the self- 3
energies[81]andthenusingthecontinuedself-energiesto 4.5 2 1.6
cpornessetnrutecdt tinheFoirgb.i7ta(llsyo-lridesolilnveesd).spTehcterad.enTshiteiersesouflsttsaateres m/m∗4.0 2M1 11..24per Ru)
onfeathr-eFtehrmreie-loervbeiltfianlseasrteruqcutiuteresiimnitlhaer.DCOoSmtpoartihsaotnsohfotwhne ment 3.5 00 100 200 3001.0 ( MµB
yffpeozerraebkntahstnleysde.nseTononfhaitenthtne∼eerCga−aactt0Rii.vnu4egOefV3mreDoqaduFneedTln∼ciDnyO+pFS0eiga.a3.kr5e1iesrVrepenuvoienlrlaemtldshaeultihpzxeaymdtaudtnhcifdhe- Mass enhance32..05 T (K) 000...468Magnetization
2.0
closer to the Fermi level, appearing at about −0.05 eV Majority spin 0.2
Minority spin
in the correlated band structure. The positive frequency
1.5 0.0
peak in the correlated band structure occurs at approx- 0 50 100 150 200 250 300
temperature (K)
imately the +0.2 eV energy of the DFT band structure,
(b) Mass enhancement of CaRuO
thusisrenormalizedbyamuchsmalleramount. Thedif- 9 3
ference is a manifestation of a large particle-hole asym- U=2.0 eV
8 U=2.3 eV
metry in the self-energy, whose influence on the spec-
U=3.0 eV
troscopy was discussed in [82]. The effect is a fingerprint
nt7
ofHund’smetalphysics,asrecentlynotedinRef.83. See e
m
also a very recent preprint [84] for calculations of DOS ce6
n
in SrRuO within an approach similar to ours. a
3 h
Figure 7 also presents for comparison the many-body s en5
density of states calculated for the d1 material CaVO as
3 m4
[78], in which the correlations are believed to arise from
Mottphysics(Hund’smetalphysicsrequiresahigheroc- 3
cupancy of the d level). The spectrum of the Mott ma-
terial consists of a central peak corresponding to a uni- 20 50 100 150 200 250 300
formly renormalized DFT DOS and clearly visible Hub- temperature (K)
bardsatellitesat−1.5and+3.5eV.Incontrastthespec-
trum of the Hund’s material CaRuO lacks clear Hub- FIG. 8. (Color online) (a) Solid curves: temperature depen-
3
bard sidebands, but as noted above does exhibit a more dence of orbitally-averaged spin-resolved mass enhancement
of SrRuO calculated at U = 2.3eV and J = 0.35eV (left
strongly renormalized and more highly structured quasi- 3
y axis). Dashed curve: magnetic moment M = (cid:104)m (cid:105) calcu-
particle peak. Although the Hubbard bands are more z
lated for same parameters (right y axis). Inset: temperature
visible in the Mott material, the correlations are in an
dependence of squared magnetic moment M2 =((cid:104)m (cid:105))2. (b)
important sense weaker, with the mass enhancement of z
Orbitally-averagedmassenhancementvs. T forCaRuO cal-
3
CaVO only about 2-3 [78] in contrast to the ∼ 7 that
3 culatedatJ =0.35eVanddifferentU values;noteCaRuO3is
we find for CaRuO3. paramagnetic at all T.
V. MAGNETIC PHASE OF SrRuO importantroleinthecalculationofdynamicalquantities
3
in the paramagnetic phase, acts to stabilize the magne-
In this section we show that the correlated Hund’s tization at partial polarization.
metal picture provides an adequate description of the The two panels of Fig. 8 compare the mass enhance-
magneticphaseofSrRuO3, preservingthesuccessfulde- mentsofSrRuO3andCaRuO3computedwithinthefive-
scription of the magnetic properties obtained from DFT band model. The lower panel shows the T-dependence
calculations [35, 43] while simultaneously providing a of the CaRuO3mass enhancements for three U values.
good account of the quasiparticle mass. While a restric- The behavior is in agreement with the three-band com-
tion of the correlated subspace to the Ru t orbitals putations discussed above: the mass increases as T is
2g
is adequate for most purposes, calculations within this decreased, and below a U-dependent scale saturates at
scheme yield half-metals, with magnetic moments that a U-dependent value. At the J = 0.35eV in this figure,
saturate at 2µ per Ru site and with too-small values we see that the mass is too small at U = 2.0eV and ex-
B
of the specific heat coefficient. We therefore treat a five- trapolates to a too-large value at U = 3eV, suggesting
band model that includes both e and t -symmetry or- that a U ≈ 2.3eV provides a reasonable description of
g 2g
bitals, with the e states and e -t interactions treated the physics.
g g 2g
in a mean field approximation. [The intra-t interac- The upper panel shows the temperature dependence
2g
tions remain as specified in Eq. (2)]. Inclusion of the e ofthespin-resolvedmassenhancementofSrRuO forthe
g 3
orbitals, whichliewellabovetheFermilevelandplayno parametersU =2.3,J =0.35eVthatprovideagoodde-
10
scriptionofCaRuO3,alongwiththetemperaturedepen- 50
80
dence of the calculated magnetic moment. The moment
)
usaetsuorfatthesemtoaagnveatliucera≈ng1i.n6g4fµrBom. E0.x8pteori1m.6enµtBspreeproRrtuvsiatle- 2mol K45 60 CUa=R2u.3Oe3V
[77, 85, 86], with more recent experiments converging on mJ/40 40
valuesbetween1.4and1.6µB [87,88]. DFTcalculations (γ
[41, 42, 51, 69, 89] report magnetic moments consistent ent 35 U=2.0eV 200 100 200 300
with experiment. We see that the Hund’s metal picture ci
provides a similarly good level of agreement. The inset effi 30
revealsthatthemagnetizationhasamean-field-liketem- co U=2.3eV
perature dependence M2 linear in Tc−T near the tran- eat 25
h
sition; extrapolation to M2 = 0 indicates a Curie tem- c
Tpecra≈tu1r6e0≈K.20F0luKctsulaigthiotnlyehffiegchtesritnhaanththreeeexdpimereinmseionntaall Specifi20 U =3.0 eV
magnet typically reduce the transition temperature by
15
∼ 30% relative to the DMFT value (see, e.g. Refs. 90 0 50 100 150 200 250 300
temperature (K)
and 91) so this value also is very reasonable.
At temperatures above the magnetic phase transition
FIG.9. (Coloronline)Specificheatcoefficientγ (Sommerfeld
the mass enhancement of SrRuO is very close to that
3
constant)calculatedforSrRuO asafunctionoftemperature
of CaRuO . (The differences discussed above between 3
3 T atJ =0.35eVandU valuesindicated. Inset: γforCaRuO
the paramagnetic phase mass enhancements of the two 3
at U =2.3eV and J =0.35eV.
materialsbecomemanifestonlyatlowT wherethemass
enhancement is large and very sensitive to the spin dy-
namicscontrolledbyJ andthedensityofstates). AsT is
decreasedbelowTc weseethatthemassenhancementin tion to the free energy F = −TrlnG. Here the spin-
SrRuO3becomes spin dependent, taking different values dependent quantities Σσ,Gσ and Z−σ1 are 12×12 ma-
forthemajorityandminorityspinchannels. Thesmaller trices, accounting for the twelve bands (three for each of
valueofthemassintheminority-spinchannelmaybeun- the four Ru atoms in the unit cell.) Use of the quasi-
derstood as a phase space effect. The dominant part of particle approximation in Eq. (4) is strictly correct only
the interaction is between opposite spin species, embod- in the low-T Fermi liquid regime, but is expected to be
ied in the S+S− part of the local interaction. Because of the correct order of magnitude even at higher tem-
m m(cid:48)
we are dealing with a greater than half-filled band, the peratures. In the orbital basis that minimizes the hy-
phasespaceavailableforaminorityspinelectrontoscat- bridization function (see Sec. II), Z is diagonal in site
σ
terintoamajorityspinstateislessthanthephasespace and orbital indices while the four Ru atoms are equiva-
availableforamajority-spinelectrontoscatterintoami- lentuptoarotationinorbitalspace. Inthiscase,Eq.(4)
nority spin state. More importantly, as the amplitude of simplifies to
the magnetic moment increases we see that the increase
inmassiscutoff,sotheconcavityofthem(cid:63)/m(T)curve
changes and the T dependence of the mass saturates be- π2k2 (cid:88)ImGmm((cid:126)r =0,i0−)
γ = B σ Z−1, (5)
low T ∼ 50K. This behavior is a natural consequence 3 π σm
σm
of the Hund’s metal physics, in which the large mass en-
hancementarisesfromslowfluctuationsofspontaneously
generated local moments whose formation and dynamics where the sum is over spin and the d orbitals of one
is very sensitive to J and T [19, 23]. The quenching of particular Ru atom. Figure 9 is produced using Eq. (5)
these moments in the ordered phase then cuts off the in- withZ andΣ0 =ReΣ (iω →i0+)obtainedfrom
σm σm σm n
creaseofthemassenhancement. Thevaluesobtainedfor DMFT.
the majority-spin mass are in reasonable correspondence
We see from Fig. 9 that the value U =2.3eV selected
withexperiment[69]althoughthecontributionofthemi-
on the basis of the phase diagram and CaRuO mass
nority spin channel to the overall specific heat requires 3
reproduces well the low-T experimental results (γ ∼
further investigation.
30 mJ/molK2 for SrRuO and 75 mJ/molK2 for
In Fig. 9, we present an estimate for the specific heat 3
CaRuO [69, 77]). Larger (3.0eV) or smaller (2.0eV)
coefficient γ = lim C/T for both materials. We ob- 3
T→0 valueofU areinconsistentwithexperiment. Essentialin
tain the specific heat by using the quasiparticle approxi-
obtaining this level of agreement is obtaining a correct
mation Σ (ω)≈Σ0 +(1−Z−1)ω in the formula
σ σ σ estimateforthemagnitudeofthesaturationmoment. A
γ = π2kB2 (cid:88) 1 (cid:90) d3(cid:126)k Tr(cid:104)ImG ((cid:126)k,i0−)Z−1(cid:105), (4) too small moment (U =2.0eV) leaves active spin fluctu-
3 π (2π)3 σ σ ations which cause a further increase in the mass, while
σ if the moment is too near to saturation (U =3.0eV) the
which is obtained from the quasiparticle approxima- γ is too small.