Soft and anisotropic local moments in 4 d and 5 d mixed-valence M 2 O 9 dimers
Ying Li, Alexander A. Tsirlin, Tusharkanti Dey, Philipp Gegenwart, Roser Valenti, Stephen M. Winter
SSoft and anisotropic local moments in 4 d and 5 d mixed-valence M O dimers Ying Li ∗ ,
1, 2
Alexander A. Tsirlin, Tusharkanti Dey † , Philipp Gegenwart, Roser Valent´ı ‡ , and Stephen M. Winter § Department of Applied Physics and MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,School of Science, Xi’an Jiaotong University, Xi’an 710049, China Institut f¨ur Theoretische Physik, Goethe-Universit¨at Frankfurt,Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany Experimental Physics VI, Center for Electronic Correlations and Magnetism,University of Augsburg, 86159 Augsburg, Germany (Dated: April 29, 2020)We investigate via exact diagonalization of finite clusters the electronic structure and magnetismof M O dimers in the mixed-valence hexagonal perovskites A B’M O for various different fillingsof 4 d and 5 d transition-metal M ions. We find that the magnetic moments of such dimers are deter-mined by a subtle interplay of spin-orbit coupling, Hund’s coupling, and Coulomb repulsion, as wellas the electron filling of the M ions. Most importantly, the magnetic moments are anisotropic andtemperature-dependent. This behavior is a result of spin-orbit coupling, magnetic field effects, andthe existence of several nearly-degenerate electronic configurations whose proximity allows occupa-tion of excited states already at room temperature. This analysis is consistent with experimentalsusceptibility measurements for a variety of dimer-based materials. Furthermore, we perform asurvey of A B’M O materials and propose ground-state phase diagrams for the experimentally rel-evant M fillings of d . , d . and d . . Finally, our results show that the usually applied Curie-Weisslaw with a constant magnetic moment cannot be used in these spin-orbit-coupled materials. I. INTRODUCTION
Oxides of 4 d and 5 d transition metals feature strongspin-orbit coupling, moderate electronic correlations andsizable metal-oxygen hybridization. Together, these ef-fects may give rise to exotic spin-orbital states, includingquantum spin liquids relying on strongly anisotropic ef-fective magnetic Hamiltonians . An interesting classof such materials is the 6H perovskite family A B’M O (M = Ir, Os, Re, Rh, Ru), which features face-sharingM O dimers as the central magnetic and structuralunit. Whereas the A-cations are almost always diva-lent, the oxidation state of the B’-cations ranges from1+ to 4+, thus imposing different charge states to thetransition metal M. Compounds with integer oxidationstates of M usually behave as common spin dimers ,where two magnetic centers are coupled by a moderatelystrong exchange interaction leading to a non-magneticground state. However, in 5 d systems with the non-magnetic ( J eff = 0) state of individual M ions, unusualeffects like “excitonic” magnetism can be expected if in-terdimer magnetic couplings exceed the energy gap toexcited multiplets with J eff >
0. The Ba B’Ir O iri-dates with divalent B’ = Zn, Ca, Sr serve as possibleexperimental examples for this scenario .In contrast, if the total charge of [A B’] is odd, it in-duces a half-integer oxidation state in M, i.e., a mixed (orintermediate) valence of the transition-metal ion . Inthis case, having an odd number of electrons per dimerensures a finite spin moment in the ground state. Var-ious different valencies are possible including d . in iri-dates with trivalent B’ , d . in ruthenates with triva-lent B’ or iridates with monovalent B’ , and d . inruthenates and osmates with monovalent B’ . Suchmaterials have been less explored theoretically, despite the fact that their experimental magnetic response re-veals several peculiarities. At high temperatures, wherethe mixed-valence dimers can be seen as isolated, mag-netic susceptibility deviates from the conventional Curie-Weiss behavior suggesting a non-trivial tempera-ture evolution of the local magnetic moment. At lowtemperatures, interactions between the dimers becomeimportant, and signatures of frustrated magnetic behav-ior including possible formation of a spin-liquid groundstate have been reported.As we discuss in this work, even on the level of asingle dimer a variety of different local states canbe realized as a function of electronic filling, dimer ge-ometry, and spin-orbit coupling strength. Understand-ing the magnetic models describing interactions betweensuch dimers first requires an understanding of the localelectronic structures of an individual dimer. Here, we en-deavor to obtain a microscopic insight into the electronicstate and magnetism of the mixed-valence M O dimerswith different fillings d . , d . and d . encountered in4 d/ d hexagonal perovskites. In particular, we focuson experimentally relevant details, such as the relationbetween ground state and paramagnetic susceptibility.Performing exact diagonalization (ED) of dimer clusters,we find a sizable state-dependent uniaxial anisotropy ofthe magnetic moment and its temperature dependence,which can be employed to identify different ground statesfrom experiment.This paper is organized as follows. In Section II, wefirst provide a preliminary description of the local Hamil-tonian for each dimer, followed by the definition of theeffective magnetic moment in Sec. III. In Sections IV–VI,we consider the ground states and behavior of the effec-tive moment for dimers with d . , d . , and d . fillingand in Section VII we present our conclusions. a r X i v : . [ c ond - m a t . s t r- e l ] A p r II. DIMER MODELA. Electronic Hamiltonian per Dimer
To first approximation, the hexagonal perovskites,A B’M O crystallize in the space group P /mmc , inwhich each dimer has local D h symmetry. The zero-fieldHamiltonian for each dimer is given by: H tot = H hop + H CF + H SO + H U (1)which is the sum of, respectively, the intersite hop-ping, on-site crystal field, spin-orbit coupling, andCoulomb interactions. We consider only the t g orbitals( d xy, , d xz , d yz ) on each metal atom. In this case, the lat-ter term is given by: H U = U (cid:88) i,a n i,a, ↑ n i,a, ↓ + ( U (cid:48) − J H ) (cid:88) i,a
18 for simplicity.
B. Single-Particle Levels
We first consider the evolution of the single-particleenergy levels assuming each dimer has D h point groupsymmetry, as described above. In the absence of SOC,the local trigonal crystal field weakly splits the t g or-bitals into singly degenerate a g ( √ ( d xy + d xz + d yz )) (c) (e)(b) e g a e g a a antibonding (A *1g )e g antibonding (E *g )a bonding (A )e g bonding (E g ) (d) ~~ λ xy e g a * ) g )) A A * E (1) g E (2) g E *(1) g E *(2) g x y zxx y z (a) O M Z XY j antibondinge j bondinge j antibondinge e j bondinge e a antibondinge e g antibondinge ⊕ e e g bondinge ⊕ e a bondinge λ λ C o m po s i t i on o f j / e bondinge bondinge bonding e antibondinge antibondinge antibonding FIG. 1. (a) M O face-shared bioctahedral dimer. Shownare the local and global coordinates; (b) Energy levels of thedimer from ED in the absence of spin-orbit coupling; (c) En-ergy levels-change as a function of spin-orbit coupling strength˜ λ = λ/t ; and (d) Composition of j / state in terms of single-particle levels as a function of ˜ λ . and doubly degenerate e g ( √ ( d xy + e πi d xz + e − πi d yz ), √ ( d xy + e − πi d xz + e πi d yz )). By convention, we labelsuch atomic combinations according to their representa-tion in D h , which describes a single trigonally distortedMO octahedron. For ∆ >
0, the a g levels lie below the e g levels, as depicted in Fig. 1. The intradimer hopping ispurely diagonal in this atomic basis, with t a g = t + 2 t and t e g = t − t . In the absence of SOC, these hop-pings lead to the formation of bonding and anti-bondingcombinations of atomic a g and e g orbitals. Fig. 1(b)shows the ordering of such levels for physically relevantparameters, t > t (cid:29) | t | .The inclusion of spin-orbit coupling (SOC) leads to thesplitting of the single-particle levels. In the limit of strongSOC ( λ (cid:29) ∆ , t , t ), the local atomic states are moreconveniently described in terms of doublet j / (with thespin, orbital and total effective angular momentum as S = , L eff = 1 , J eff = , respectively) and quadruplet j / ( S = , L eff = 1 , J eff = ) levels. Intradimer hop- ping is not diagonal with respect to the j / /j / char-acter, and therefore leads to both the mixing of the j -states and the formation of bonding/anti-bonding com-binations. Formally, the mixed spin-orbitals at interme-diate ˜ λ can be labelled according to their double grouprepresentation within D h , which admits three Kramersdoublet representations: e / , e / and e / , as shown inFig. 1(c). A character table for these states is given inthe Appendix A. Of these single-particle levels, the e / states have pure j / character, while the e / and e / states are mixtures of atomic j / and j / functions. Asan example of the relation between the atomic relativistic j / / j / basis and the 3 Kramers doublets, in Fig. 1 (d)we display the weights (cid:104) φ | j / (cid:105) of j / states in termsof the three Kramers doublets e / , e / and e / ( | φ (cid:105) states) as a function of ˜ λ . In the non-relativistic limit,˜ λ = 0, the states are distributed to all relativistic states.With increasing ˜ λ , the contribution to j / of the low-est bonding states e / and e / arises and finally thesestates become pure j / while the highest antibondingstate e / is finally a j / state, as it is the antibonding e / . In contrast, bonding e / shows a non-monotonouscomposition of j / and j / with ˜ λ .As a function of SOC strength ˜ λ , the ordering of thesingle-particle levels changes, such that the ground stateof a single dimer is sensitive to both ˜ λ and filling. Forintermediate values of ˜ λ found in real materials, this alsohas the effect of confining the j / bonding and j / anti-bonding levels into a narrow energy range, which signifi-cantly reduces the energy scales relevant to the magneticresponse of the dimer. The consequences for specific fill-ings are considered in sections IV–VI with inclusion ofCoulomb interactions. III. DEFINITION OF EFFECTIVE MOMENTS
Given that we are interested in dimers with open-shell(e.g. doublet) ground states, it is useful to characterizethem according to their magnetic response. For isolatedmagnetic ions or dimers, it is conventional to describe themolar magnetic susceptibility in terms of temperature-dependent effective moments µ α eff ( T ), defined by: χ α ( T ) = N A k B T [ µ α eff ( T )] (10)where α ∈ { x, y, z } . In simple cases where there areno low-lying excited multiplets, and the ground stateis a pure spin multiplet (there is no orbital degener-acy, and spin-orbit coupling can be neglected), thenthe susceptibility should follow a pure Curie Law, with µ eff = g S (cid:112) S ( S + 1) being temperature-independent andisotropic. However, when any of these conditions are vio-lated, as it is common in real materials with strong SOC,the susceptibility of isolated magnetic species will gener-ally be a more complex function of temperature.Additional non-Curie contributions to χ ( T ) occur, es-sentially, from two sources. First, there can exist low-lying excited multiplets that become thermally populatedat relevant temperatures. Second, when spin-orbit cou-pling is relevant, it is important to note that the SOCoperator H SOC = λ L · S generally does not commutewith the Zeeman operator, given by H Z = µ B H · M ,with M = ( g S S + g L L ). As a result, the magnetic fieldinduces mixing between different multiplets, introducingadditional terms in the susceptibility analogous to thephenomenon of van Vleck paramagnetism. These effectsare well documented, and have been studied for isolatedions both theoretically and experimentally . While itis clear that exchange couplings ∼ J between magneticspecies further impact the paramagnetic susceptibility,provided the temperature is sufficiently large T (cid:29) J ,then χ ( T ) should generally be dominated by local single-species effects. As a result, the behavior of µ α eff ( T ) rep-resents a first clue regarding the nature of the local elec-tronic ground state. We focus on this quantity in thefollowing sections. IV. d . FILLINGA. Survey of Materials
We first consider the case of d . filling that corre-sponds to three holes per dimer. This filling can be foundin A B’M O with trivalent B’ and M = Rh or Ir, amongwhich mostly the iridates have been studied experimen-tally. All of them show strong deviations from the Curie-Weiss behavior and develop a characteristic bend in theinverse susceptibility at 50 −
100 K . The linear fit tothe high-temperature part returns the effective momentsof 1.53 µ B /f.u. (B’ = In) and 1.79 µ B /f.u. (B’ = Sc) thatwere interpreted as the S = state of the mixed-valencedimer , although very high Curie-Weiss temperaturesof several hundred Kelvin put into question the validityof such a fit. Below the bend, a much lower effectivemoment of 0.76 µ B /f.u. (B’ = In) is obtained , and amagnetic entropy on the order of R ln 2 is released ,suggesting the ground-state doublet of hitherto unknownnature.At even lower temperatures, interactions between thedimers come into play. Ba YIr O undergoes long-rangemagnetic ordering at 4.5 K , whereas Ba InIr O reveals persistent spin dynamics down to at least20 mK, with local probes suggesting a collective, possi-bly spin-liquid behavior of local moments below 1 K .Ba ScIr O does not show long-range magnetic orderdown to at least 2 K, but pends experimental character-ization at lower temperatures .Rh atoms can also be accommodated in the 6H per-ovskite structure, but show strong site mixing with the B’atoms . Therefore, no experimental information onthe magnetism of pure Rh O dimers is presently avail-able. B. Phase Diagram
The theoretical ground state of a d . dimer as a func-tion of ˜ U and ˜ λ is shown in Fig. 2(a) for t = ∆ = 0.The color shading indicates the expectation value of thetotal spin moment squared (cid:104) ˆ S (cid:105) per dimer. In the non-relativistic limit ˜ λ →
0, there are two possible groundstates depending on the relative strength of ˜ t and ˜ U (and J H /t since we fix the ratio J H /U = 0 . e g and a g levels ( J H (cid:29) t ), the groundstate is a high-spin S = 3 / e / , and e / orbitals(Fig. 2 (a) upper left corner). In this configuration, theorbital angular momentum is completely quenched, and (cid:104) ˆ S (cid:105) = S ( S + 1) = 15 /
4. In contrast, for J H (cid:28) t ,a low-spin S = 1 / a g anti-bondingstate, and one hole occupying the e g anti-bonding state(lower left corner in Fig. 2 (a)). This is indicated by Low-SHigh-S (a) (b) (c) ~ ~ ~ ~~ ~ FIG. 2. (a) Phase diagram of theoretical ground state of a d . dimer as a function of ˜ U = U/t and ˜ λ = λ/t for t = ∆ = 0.We fix J H /U = 0 .
18. (b,c) Evolution of the zero-temperatureeffective moment (Eq. 10) for ˜ λ values corresponding to 4 d and 5 d materials, respectively. − − − − − − (a) (c) (b) (d) ~~ ~~~ ~~ ~~ ~~ ~ FIG. 3. Evolution of the zero-temperature effective moment(Eq. 10) as a function of ˜∆ = ∆ /t and ˜ t = t /t . (cid:104) ˆ S (cid:105) = 3 /
4. Provided C rotational symmetry is pre-served, this latter configuration has unquenched orbitalangular momentum, as the e g hole may occupy either the e / or e / single-particle levels shown in Fig. 1(c). Thecombined spin and orbital degrees of freedom provide anoverall four-fold ground-state degeneracy for ˜ λ = 0. Asthe partially occupied single-particle levels are predomi-nantly of j / character, we refer to this ground state ashaving total J eff ≈ / S ≈ / J eff ≈ / λ leads to a splitting of the quar-tet states into pairs of Kramers doublets. In the high-spinlimit, this can be viewed as the introduction of an on-sitezero-field splitting term ( S x + S y + S z ) that energeticallyprefers the m S = ± / L and S moments, stabilizing the m J = ± / λ , these cases are smoothly connected, such that there isno sharp “phase transition” for small ˜ λ on increasing ˜ U .As indicated in Fig. 2(a), (cid:104) ˆ S (cid:105) evolves continuously be-tween the high-spin and low-spin limits, as S is no longera good quantum number with finite SOC.Further increasing ˜ λ leads to a change in the orderingof the single-particle e / bonding and e / anti-bondinglevels, as discussed in Section II B. This stabilizes a dif-ferent “low-spin” J eff ≈ / j / level ( e / ), and onehole in the bonding j / level ( e / ).Considering real materials, hypothetical Rh-baseddimers should fall well within the high-spin S ≈ / J eff ≈ / J eff ≈ / O dimers areused. Similar experiments on the mixed-valence com-pounds would be required to identify the experimentalground states of real materials. In this context, a use- ful observation is that the different electronic configu-rations of the dimers can also be distinguished by theanisotropy and magnitude of the effective magnetic mo-ments µ α eff ( T ), as defined in Section III.In Fig. 2(b) we show the evolution of the zero-temperature limit of the effective moment per dimer µ eff (0) as a function of ˜ U for ˜ λ = 0 .
5, correspondingto the expected range for 4 d dimers of Rh. In this case,SOC reduces the average value of µ eff (0) to well belowthe spin-only value of √
15 for pure S = 3 / U . The crossover from the low-spin to high-spin ground state with increasing ˜ U leads to a reversalof the anisotropy of the effective moment; for small ˜ U , µ ab eff < µ c eff , while for large ˜ U , µ ab eff > µ c eff . We expect thathypothetical Rh-based dimers should fall into the lattercategory.In the region of parameters applicable to Ir-baseddimers (˜ λ = 1 .
4; Fig. 2(c)), increasing ˜ U leads to a phasetransition from a J eff ≈ / J eff ≈ / J eff ≈ / λ, ˜ U , ˜∆, and ˜ t . The influence of the lattertwo parameters is shown in Fig. 3(a,b). Modifying ˜∆ and˜ t can lead to reversals of the anisotropy of the effectivemoment in the J eff ≈ / t also stabilizes the J eff ≈ / J eff ≈ / U and/or ˜ t is characterized by a muchlarger average moment, which is strongly anisotropicwith µ c eff > µ ab eff . As shown in Fig. 3(c,d), this anisotropyis only weakly affected by crystal field ˜∆, and remainsfor a wide range of values of ˜ t hopping. C. Comparison to Experiment
For temperatures large compared to the interactionsbetween dimers, the effective moment per dimer can beextracted from experimental susceptibility data χ α via: µ α eff ( T ) ≈ (cid:114) k B TN A χ α exp ( T ) (11)where α indicates the field direction, N A is the Avogadroconstant, k B is the Boltzmann constant, and T is thetemperature. Given that all Ir-based d . dimers dis-cussed above show features in the specific heat at lowtemperatures T (cid:46)
10 K that account for a significantfraction of R ln 2 entropy, we assume that the magneticcouplings between dimers are relatively weak. As a re-sult, the temperature dependence of χ ( T ) above roomtemperature should be largely dominated by the evo-lution of µ eff ( T ). In Fig. 4(a)–(c), we thus show thetheoretical temperature dependence of the effective mo-ment for selected parameters corresponding to regionsexpected for real materials. For real materials, | t | = ( )( ) (d) Exp. cb ~~ ~~~~ ~ H ⊥ cH || cAve. H ⊥ cH || cAve. ~ ~ (a) FIG. 4. Temperature dependence of the magnetic momentsfor d . dimers for various ˜ U ( U/t ) and ˜ λ ( λ/t ) values for t = ∆ = 0. (a) Parameters relevant for isolated 4 d dimers. Thegreen area displays the region where the first excited state isoccupied. (b,c) Parameters relevant for isolated 5 d dimers in J eff ≈ / J eff ≈ / t . (d) Exper-imental effective moment for Ba InIr O derived from bothsingle-crystal and powder measurements. The insert of fig-ure 4 (d) is the scanning electron microscopy image of theBa InIr O crystal. . − . T = T /t ∼ . √ µ B as temperature is raised. Neverthe-less, below room temperature such materials will displaystrong deviations from Curie behavior even in the ab-sence of interdimer couplings. The green area above 0.047is the region where the first excited state is occupied.For Ir-based dimers, we consider ˜ λ = 1.1 and 1.6 cor-responding to the J eff ≈ / J eff ≈ / µ eff ( T )is a monotonically increasing function. The energy ofthe first excited state is above 0.2 t . For the J eff ≈ / √ J eff = 1 / J eff = 1 / , and therefore is not unique to the dimer case. Incontrast, the anisotropy is expected to be weaker in the J eff = 3 / √ InIr O (see Appendix B for details of crystal growthand characterization) and measured its magnetic suscep-tibility along different directions. Data up to 200 K wereobtained, while at higher temperatures the signal dropsbelow the sensitivity threshold of the magnetometer, butalready these data indicate weak anisotropy of the para-magnetic response. The effective moment is quite lowand approaches √ InIr O is J eff = 3 /
2. A sim-ilar J eff = 3 / √ .An ideal J = 3 / λ (cid:54) = 0. This explains why at low tempera-tures magnetic entropy of not more than R ln 2 is recov-ered , corresponding to the lower Kramers doublet.Low-temperature magnetism associated with this dou-blet should be then treated as an effective spin- with astrongly renormalized g -tensor. The very low g -value elu-cidates the fact that magnetization of Ba InIr O doesnot reach saturation even at 14 T, even though exchangecouplings are on the order of several Kelvin and would beeasily saturated in a conventional spin- antiferromagnetwith g = 2 . . V. d . FILLINGA. Survey of Materials
Materials with d . filling are found for two differentcompositions A B’M O . The first case corresponds toM being a group 8 element (Os, Ru) and trivalent B’,whereas the second case corresponds to M being a group9 element (Ir, Rh) and monovalent B’.The mixed-valence ruthenates have been reported for awide range of trivalent B’ ions, including Y, In, and manyof the lanthanides. All these compounds show strongdeviations from the Curie-Weiss behavior . An ef-fective spin- state was conjectured for Ba B’Ru O with B’ = In, Y, Lu from the relatively low valuesof the magnetic susceptibility and from the local mo-ment of 1.0 µ B /dimer in the magnetically ordered stateof Ba YRu O . On the other hand, Ba LaRu O shows an overall higher susceptibility that approachesthe Curie-Weiss regime for spin- above room temper-ature . The local moment of 2 . − . µ B /dimer in themagnetically ordered state also supports the spin- scenario. These dissimilar trends indicate the presence ofat least two competing electronic states in the d . Ru O dimers. Indeed, local excitations of magnetic origin wereobserved by inelastic neutron scattering around 35 meVand assigned to a transition between these states tenta-tively identified as S = 1 / S = 3 / .Similar to the d . iridates, the d . ruthenates exhibita range of magnetic ground states, from long-range orderconfirmed in B’ = La, Nd, Y and anticipated forB’ = Lu , to a static disordered state in Ba InRu O .To our knowledge, Os-based dimers with d . filling haveyet to be explored.Examples of the second case of materials includeBa (Na/Li)Ir O . Susceptibility measurementsshow signs of magnetic order at 75 and 50 K for B’ = Liand Na, respectively . Curie-Weiss fits of the magneticsusceptibility between about 150 K and room tempera-ture return the effective moments of 3.93 µ B /f.u. (B’ =Li) and 3.60 µ B /f.u. (B’ = Na) . On the other hand,Curie-Weiss temperatures of, respectively, −
576 K and −
232 K suggest that a true paramagnetic regime maynot have been reached in this temperature range. To ourknowledge, Rh-based dimers with d . filling were notreported. B. Phase Diagram
The theoretical ground state of a d . dimer as a func-tion of ˜ U and ˜ λ is shown in Fig. 5(a) for t = ∆ = 0.Similar to the d . case, the non-relativistic limit ˜ λ → J H /t , respectively. LargeHund’s coupling produces a high-spin S = 5 / e / , e / and bonding e / , e / orbitals. In this state, theorbital angular momentum is essentially quenched. Incontrast, weak interactions lead to a low-spin S = 1 / a g anti-bonding level, and three holes occupying the e g anti-bonding levels. This low-spin ground state has or-bital degeneracy associated with the single electron inthe e g anti-bonding levels implying an unquenched or-bital moment. In the non-relativistic limit, the groundstate is four-fold degenerate, with unpaired electrons oc-cupying single-particle levels with predominant j / char-acter. As a result, we refer to this low-spin ground stateas J eff ≈ /
2. A very narrow intermediate-spin S ≈ / d . case, a distinct transition occursbetween the low-spin and high-spin states, as can be seenin µ eff (0) as a function of ˜ U , shown in Fig. 5 (b) for˜ λ = 0 .
5. The introduction of small ˜ λ leads to spin-orbitallocking in the low-spin phase, which ultimately quenchescompletely the effective moment. As a result, µ eff (0) ≈ U . From the perspective of the single-particlelevels, the unpaired electron occupies an e / level, whichhas pure j / character. As a result, the effective momentis precisely zero. For the high-spin case, small λ leads to asplitting of the six degenerate S = 5 / m S = ± / Low-SHigh-S (a) (b) (c) High-SLow-S ~ ~ ~ ~ ~ ~ FIG. 5. (a) Phase diagram of theoretical ground state of a d . dimer as a function of ˜ U = U/t and ˜ λ = λ/t for t =∆ = 0. We fix J H /U = 0 .
18. (b,c) Evolution of the zero-temperature effective moment as a function of ˜ U for ˜ λ valuescorresponding to 4 d and 5 d materials, respectively. Note thatthe jumps in the figures are due to changes in J eff . forms the ground state, which leads to µ ab eff > µ c eff ≈ √ λ . These can beunderstood in the single particle picture from Fig. 1(c).As noted above, for small ˜ λ , the J eff ≈ / e g anti-bonding levelof e / symmetry, with pure j / anti-bonding character.With increasing ˜ λ , this level crosses the j / bonding levelof e / symmetry, leading instead to a ground state with J eff ≈ / λ leads toanother crossing of the e / level with an e / level thatis nominally j / anti-bonding. As a result, the groundstate in the ˜ λ → ∞ limit consists of two holes in theanti-bonding j / level ( e / ), two holes in the bonding j / level ( e / ) and one hole in the antibonding j / level( e / ). This latter J eff ≈ / S = 5 / (f) Exp. Ba LiIr O (e) (c) Exp. Ba InRu O Ba YRu O Ba LuRu O ~ ~~~ ~ H ⊥ cH || cAve. (b) ~ (a) ~ ~ (d) ~ ~~~ FIG. 6. Temperature dependence of the magnetic momentsfor d . dimers with various ˜ U and ˜ λ values for t = ∆ = 0.(a,b) Theoretical results for isolated 4 d dimers in high-spinand low-spin ground states, respectively. The green areas dis-play the regions where the 6 lowest excited states are filledsequentially. (c) Experimental effective moment for variousRu-based dimers based on data from Ref. 16. (d,e) Theoreti-cal results for isolated 5 d dimers in different J eff ≈ / t . (f) Experimentaleffective moment for Ba LiIr O derived from Ref. 17. weak SOC.In Fig. 5(c), we show the evolution of the zero-temperature effective moment as a function of ˜ U for˜ λ = 1 .
4, corresponding to 5 d materials. In this case, µ eff (0) can be used to distinguish the three differentground states. For small ˜ U , the effective moment is 0, asthe low-energy degrees of freedom have pure J eff ≈ / J eff ≈ / U , the effective moment jumps to an aver-age value ∼ √ µ B . With further increasing U , the av-erage effective moment again drops when entering the J eff ≈ / µ eff tends to grow with increasing U, J H . Theanisotropy µ ab eff > µ c eff is maintained throughout this lat-ter J eff ≈ / C. Comparison to Experiment
In Fig. 6, we show the temperature dependence of theeffective moment for selected parameters correspondingto regions expected for real materials. For Ru/Rh sys-tems, we consider ˜ λ = 0 .
5, with ∆ = t = 0 for simplic-ity. Fig. 6(a) and (b) show the expected behavior for thehigh-spin and low-spin ground states, respectively. Thegreen areas display the regions where the lowest excitedstates are sequentially filled as a function of temperature.For the high-spin state, we find a strong temperaturedependence as thermal fluctuations rapidly overwhelmthe local “single-ion” magnetic anisotropies. Thus theanisotropy of the effective moment is suppressed with in-creasing T , and the average value becomes comparable tothe pure spin value of √ µ B for S = 5 /
2. For the low-spin state, µ α eff ( T ) is nearly isotropic at all temperatures,and slowly decreases to zero as the temperature is low-ered. Considering real materials, Fig. 6(b) closely resem-bles the response of the powder samples of Ba YRu O ,Ba LuRu O , and Ba InRu O reported in Refs. 14 and16. Moreover, magnetic susceptibility measured on a sin-gle crystal of Ba InRu O shows a nearly isotropic mag-netic response , in agreement with the low-spin scenario.In Fig. 6(c), we plot the (cid:112) (3 k B T /N A ) χ avg ( T ) for allthree compounds, derived from the experimental χ ( T ) inRef. 16. As above, we expect | t | = 0 . − . T = T /t ∼ . d . iridates, this J eff = 3 / µ eff ( T ) at low temperatures results fromSOC, which couples the S = 1 / , andwith the observed electronic excitation around 35 meV that should be assigned to the transition between the twodoublets.An interesting exception to this picture is the behav-ior of Ba LuRu O , which shows a larger average µ eff over an extended temperature range. It may be possiblethat this material adopts a high-spin or intermediate-spin ground state instead; a non-trivial test of this sce-nario would be demonstration of significant anisotropy of µ α eff ( T ), with µ ab eff > µ c eff , if single crystal samples becomeavailable.For Ir- and Os-based dimers, the two realistic groundstates are both described as J eff ≈ /
2. As a result, theyare less distinguishable on the basis of magnetic suscep-tibility. In Fig. 6 (d,e), we compare µ eff ( T ) for ˜ U = 6and different values of ˜ λ = 1 . , . t in (e). In both cases,we find µ ab eff > µ c eff , with average values falling in simi-lar ranges. In Fig. 6 (f), we plot (cid:112) (3 k B T /N A ) χ avg ( T )for Ba LiIr O , derived from the experimental suscep-tibility from Ref. 17. The experimental value of ∼ µ B around room temperature is compatible with the theoret-ical results on panel (d) at ˜ T = 0 .
1. On the other hand,the effective moments of 3.60 and 3.93 µ B /f.u. obtainedfrom the Curie-Weiss fits are comparable to the high-temperature limit on panel (e). A general problem here isthat relatively high magnetic ordering temperatures of 50and 75 K indicate sizable exchange interactions betweenthe dimers. Therefore, it may be difficult to directly com-pare the theoretical and experimental data within theavailable temperature range. VI. d . FILLINGA. Survey of Materials
Materials with d . filling have A B’M O composi-tions with monovalent B’ and transition metal M be-longing to group 8 (Os, Ru). Of these, Ba NaRu O shows a rather abrupt decrease in the magnetic sus-ceptibility at 210 K, which is accompanied by a struc-tural transition and interdimer charge order . Thelow-temperature phase is thus composed of distinct(Ru ) O and (Ru ) O dimers, which do not main-tain d . filling. Interestingly, the Li analogue also ex-hibits a drop in susceptibility around 150 K, althoughthis is not associated with a structural transition ; itis presently unclear whether charge-order occurs in thiscase. The Os analogues, Ba B’Os O (B’ = Na, Li),have also been reported . Both compounds show noevidence of charge order, whereas kinks in the suscepti-bility around 10 −
13 K likely indicate magnetic order-ing. Curie-Weiss fits yield larger effective moments ofabout 5 µ B for the ruthenates and lower effective mo-ments of about 3.3 µ B for the osmates . However, allvalues should be taken with caution, because, similarto the d . iridates, large antiferromagnetic Weiss con-stants suggest that the true paramagnetic regime has notbeen reached, and/or the effective moment is stronglytemperature-dependent. B. Phase Diagram
The theoretical phase diagram for d . filling and ∆ = t = 0 is shown in Fig. 7(a). The green areas display thelowest five excited states region. In the non-relativisticlimit ˜ λ →
0, these materials also exhibit high-spin andlow-spin states depending on the strength of interac-tions. For the high-spin S = 5 / e g ( e / , e / ) and bond-ing e g ( e / , e / ) and bonding a g ( e / ) orbitals. At Low-SHigh-S (a) (b) (c)
Low-S High-S H i gh - S ~~ ~ ~ ~~ FIG. 7. P(a) Phase diagram of theoretical ground state of a d . dimer as a function of ˜ U = U/t and ˜ λ = λ/t for t =∆ = 0. We fix J H /U = 0 .
18. (b,c) Evolution of the zero-temperature effective moment as a function of ˜ U for ˜ λ valuescorresponding to 4 d and 5 d materials, respectively. lowest order, the orbital momentum is quenched. In con-trast, the low-spin state consists of all five electrons inthe bonding orbitals, while the anti-bonding levels areempty. The degenerate bonding e g levels contain threeelectrons, leading to an unquenched orbital degree of free-dom. However, in this case, the unpaired electron nomi-nally occupies a level with j / character, identifying thisground state as J eff ≈ / µ eff (0) as a function of ˜ U is shown inFig. 7(b) for ˜ λ = 0 .
5, corresponding to a reasonable valuefor Ru-based dimers.The energy of the first excited stateis above 0.2 t . For the high-spin state, the introductionof a small λ leads to a splitting of the six degeneratestates into three pairs of Kramers doublets. The m s = ± / µ ab eff >µ c eff ≈ √
3. In contrast, the low-spin state has a reversedanisotropy of the effective moment. This finding can beunderstood from the single-particle levels. For small λ ,the single unpaired electron occupies an e / orbital withmostly j / character, leading to a J eff ≈ / ≈ √
3, withan anisotropy µ ab eff < µ c eff due to mixing with J eff ≈ / λ in the phase diagram. For small ˜ U and large ˜ λ , asingle unpaired electron occupies an e / anti-bondinglevel of pure j / character, suggesting a J eff ≈ / µ eff (0) for ˜ U = 1 .
4, shownin Fig. 7(c). For intermediate values of ˜ U , the J eff ≈ / µ ab eff = 0. This reflects the fact that this doublet is essen-tially composed of m J = ± / µ c eff due to Hund’s coupling,but the transverse moment is essentially zero. We ex-pect that Os-based dimers may lie near the border of J eff ≈ / J eff ≈ / C. Comparison to Experiment
In Fig. 8, we show the computed temperature depen-dence of the effective moment for selected parameters cor-responding to regions expected for real materials. In thestrongly relativistic region relevant for Os-based dimers,the anisotropy of µ eff is generally weak. In Fig. 8(d)and (e), we show the temperature dependence of µ eff for˜ λ = 1 . .
6, corresponding to the J eff ≈ / J eff ≈ / t . For the latter case, the suppressed µ ab eff discussed above is observed only at relatively low temper-atures, below room temperature ( ˜ T = T /t ∼ . (cid:112) (3 k B T /N A ) χ avg ( T ) for Ba BOs O (B = Na, Li)based on data from Ref. 19 shown in Fig. 8 (f). In prin-ciple, the experimental powder data may be consistentwith both possible ground states; single-crystal measure-ments may be more valuable in the future.For Ru-based dimers, there is a stark contrast betweenthe behavior of the high-spin S ≈ / J eff ≈ / √ µ B due to SOC. In contrast, inthe low-spin case, µ α eff ( T ) evolves monotonically, is moreweakly anisotropic, and has average value comparableto a pure J eff = 1 / √
3. These results canbe compared to the experimental µ eff ( T ) of Ba LiRu O ,shown in Fig. 8(c), which is based on measurements of thepowder susceptibility in Ref. 18. For the entire temper-ature range, the average experimental effective momentremains on the order of √
3, which suggests that eitherthe material is in a low-spin state, or has a strongly sup-pressed moment due to strong interdimer couplings orincipient charge order. (f) Exp. Ba NaOs O Ba LiOs O (e) (c) Exp. Ba LiRu O (b) ~ ~~ ~~ ~ (d) H ⊥ cH || cAve. (a) ~ ~ ~~ ~~ FIG. 8. Temperature dependence of the magnetic momentsfor d . dimers with various ˜ U ( U/t ) and ˜ λ ( λ/t ) values for t = ∆ = 0. (a,b) Theoretical results for isolated 4 d dimers inhigh-spin and low-spin ground states, respectively. The greenarea displays the region where excited states are filled. (c) Ex-perimental effective moment for Ba LiRu O based on datafrom Ref. 18. (d,e) Theoretical results for isolated 5 d dimersin J eff ≈ / J eff ≈ / t . (f) Experi-mental effective moment for Ba (Na/Li)Os O derived fromRef. 19. VII. SUMMARY AND CONCLUSIONS
In this work, we considered the theoretical phase dia-grams and magnetic response of face-sharing dimers of 4 d and 5 d elements with fractional fillings d . to d . . Onthe basis of these studies we draw various conclusions:(i) For these complex materials with competingCoulomb, hopping, and SOC terms, the magnetic re-sponse of individual dimers exhibits strong departuresfrom conventional Curie behavior. As a result, Curie-Weiss analysis of the magnetic susceptibility may notbe applicable or, if applied, will return strange resultsthat do not reflect true nature of the local magnetic mo-ment. Except for the most standard case of large ˜ U andsmall ˜ λ , essentially all interaction regimes considered inour study lead to effective moments that are stronglytemperature-dependent up to at least ˜ T = 0 . J H , U , spin-orbitcoupling strength λ , and intradimer hoppings t , and t .An important consideration is that increasing λ leadsto several crossings of single-particle levels of differentsymmetry, which manifests in phase transitions betweendifferent ground-state configurations depending on thespecific filling. From the experimental perspective, thislevel crossing leads to electronic excitations lying withinthe energy range of 50 −
100 meV and accessible for in-elastic neutron scattering, which can provide additionalvaluable information on the electronic structure. It alsojustifies the application of empirical models that wereused to describe magnetic susceptibility of mixed-valencedimers in terms of excitations between several electronicstates . A caveat here is that none of these states fea-ture purely spin moments, so their energies and magneticmoments should be both treated as fitting parameters,with the risk of making the model overparameterized.(iii) For all considered fillings, 4 d Ru- and Rh-baseddimers may, in principle, exhibit either low-spin or high-spin ground states. From a survey of available experi-mental susceptibility data, we find that the vast majorityof such materials likely fall into a low-spin ground state.Provided C symmetry of the dimers is maintained, alllow-spin states have unquenched orbital moments, whichstrongly modify the magnetic response via coupling tothe S = 1 / d materials, various ground states are pos-sible, having either pure J eff = 3 / J eff = 1 / J eff = 3 / InIr O with d . filling of the Ir sites, themagnetic response is suggestive of a “low-spin” J eff ≈ / d and 5 d dimers with fractional fillings, which may serve as a start-ing point for future experimental and theoretical studiesof these complex materials. ACKNOWLEDGMENTS
R.V., S.W. and Y.L. thank Daniel Khomskii, IgorMazin, Sergey V. Streltsov, Markus Gr¨uninger and AdamA. Aczel for discussions and acknowledge the supportby the Deutsche Forschungsgemeinschaft (DFG) throughgrant VA117/15-1. Computer time was allotted at theCentre for for Scientific Computing (CSC) in Frankfurt.R.V. and S.W. acknowledge support from the NationalScience Foundation under Grant No. NSF PHY-1748958and hospitality from KITP where part of this work wasperformed. Y.L. acknowledge support from the Fun-damental Research Funds for the Central Universities(Grant No. xxj032019006) and China Postdoctoral Sci-ence Foundation (Grant No. 2019M660249). The work inAugsburg was supported by DFG under TRR80 and bythe Federal Ministry for Education and Research throughthe Sofja Kovalevskaya Award of Alexander von Hum-boldt Foundation (A.A.T.) ∗ [email protected] † Current address: Department of Physics, Indian Insti-tute of Technology (Indian School of Mines), Dhanbad,Jharkhand, 826004, India ‡ [email protected] § [email protected] W. Witczak-Krempa, G. Chen, Y. Kim, and L. Balents,Ann. Rev. Condensed Matter Phys. , 57 (2014). J. Rau, E.-H. Lee, and H.-Y. Kee, Ann. Rev. CondensedMatter Phys. , 195 (2016). R. Schaffer, E. K.-H. Lee, B.-J. Yang, and Y. B. Kim,Rep. Prog. Phys. , 094504 (2016). S. Winter, A. Tsirlin, M. Daghofer, J. van den Brink,Y. Singh, P. Gegenwart, and R. Valent´ı, J. Phys.: Con-dens. Matter , 493002 (2017). G. Cao and P. Schlottmann, Rep. Prog. Phys. , 042502(2018). J. Darriet, M. Drillon, G. Villeneuve, and P. Hagenmuller,J. Solid State Chem. , 213 (1976). Q. Chen, S. Fan, K. Taddei, M. Stone, A. Kolesnikov, J. Cheng, J. Musfeldt, H. Zhou, and A. Aczel, J. Amer.Chem. Soc. , 9928 (2019). A. J. Kim, H. O. Jeschke, P. Werner, and R. Valent´ı,Physical review letters , 086401 (2017). A. Nag, S. Middey, S. Bhowal, S. K. Panda, R. Mathieu,J. C. Orain, F. Bert, P. Mendels, P. Freeman, M. Mansson,H. M. Ronnow, M. Telling, P. K. Biswas, D. Sheptyakov,S. D. Kaushik, V. Siruguri, C. Meneghini, D. D. Sarma,I. Dasgupta, and S. Ray, Phys. Rev. Lett. , 097205(2016). A. Nag, S. Bhowal, M. M. Sala, A. Efimenko, I. Dasgupta,and S. Ray, Phys. Rev. Lett. , 017201 (2019). R. C. Byrne and C. W. Moeller, J. Solid State Chem. ,228 (1970). Y. Doi and Y. Hinatsu, J. Phys.: Condens. Matter ,2849 (2004). T. Sakamoto, Y. Doi, and Y. Hinatsu, J. Solid State Chem. , 2595 (2006). Y. Doi, K. Matsuhira, and Y. Hinatsu, J. Solid StateChem. , 317 (2002). M. S. Senn, S. A. J. Kimber, A. M. A. Lopez, A. H. Hill,and J. P. Attfield, Phys. Rev. B , 134402 (2013). D. Ziat, A. A. Aczel, R. Sinclair, Q. Chen, H. D. Zhou,T. J. Williams, M. B. Stone, A. Verrier, and J. A. Quil-liam, Phys. Rev. B , 184424 (2017). S.-J. Kim, M. D. Smith, J. Darriet, and H.-C. zur Loye,J. Solid State Chem. , 1493 (2004). K. E. Stitzer, M. D. Smith, W. R. Gemmill, and H.-C. zurLoye, J. Amer. Chem. Soc. , 13877 (2002). K. E. Stitzer, A. E. Abed, M. D. Smith, M. J. Davis, S.-J.Kim, J. Darriet, and H.-C. zur Loye, Inorg. Chem. ,947 (2003). L. Shlyk, S. Kryukov, V. Durairaj, S. Parkin, G. Cao, andL. De Long, J. Magn. Magn. Mater. , 64 (2007). T. Dey, M. Majumder, J. C. Orain, A. Senyshyn, M. Prinz-Zwick, S. Bachus, Y. Tokiwa, F. Bert, P. Khuntia,N. B¨uttgen, A. A. Tsirlin, and P. Gegenwart, Phys. Rev.B , 174411 (2017). S. V. Streltsov and D. I. Khomskii, Proceedings of theNational Academy of Sciences , 10491 (2016). S. V. Streltsov and D. I. Khomskii, Phys. Rev. B ,161112 (2014). K. I. Kugel, D. I. Khomskii, A. O. Sboychakov, and S. V.Streltsov, Phys. Rev. B , 155125 (2015). D. I. Khomskii, K. I. Kugel, A. O. Sboychakov, and S. V.Streltsov, Journal of Experimental and Theoretical Physics , 484 (2016). K. Foyevtsova, H. O. Jeschke, I. I. Mazin, D. I. Khomskii,and R. Valent´ı, Phys. Rev. B , 035107 (2013). S. M. Winter, Y. Li, H. O. Jeschke, and R. Valent´ı, Phys-ical Review B , 214431 (2016). A. Banerjee, C. A. Bridges, J.-Q. Yan, A. A. Aczel, L. Li,M. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu,J. Knolle, S. Bhattacharjee, D. L. Kovrizhin, R. Moessner,D. A. Tennant, D. G. Mandrus, and S. E. Nagler, NatureMater. , 733 (2016). B. H. Kim, G. Khaliullin, and B. I. Min, Phys. Rev. B ,081109 (2014). M. Kotani, J. Phys. Soc. Japan , 293 (1949). T. Dey, R. Kumar, A. V. Mahajan, S. D. Kaushik, andV. Siruguri, Phys. Rev. B , 205101 (2014). S. Panda, S. Bhowal, Y. Li, S. Ganguly, R. Valent´ı,L. Nordstr¨om, and I. Dasgupta, Physical Review B ,180403 (2015). R. Kumar, D. Sheptyakov, P. Khuntia, K. Rolfs, P. G.Freeman, H. M. Rønnow, T. Dey, M. Baenitz, and A. V.Mahajan, Phys. Rev. B , 174410 (2016). A. Revelli, M. M. Sala, G. Monaco, P. Becker, and L. Bo-haty, Science Advances , eaav4020 (2019). Y. Doi, Y. Hinatsu, Y. Shimojo, and Y. Ishii, J. SolidState Chem. , 113 (2001). Q. Chen, A. Verrier, D. Ziat, A. J. Clune, R. Rouane, X. Bazier-Matte, G. Wang, S. Calder, K. M. Taddei, C. R.dela Cruz, A. I. Kolesnikov, J. Ma, J.-G. Cheng, Z. Liu,J. A. Quilliam, J. L. Musfeldt, H. D. Zhou, and A. A.Aczel, “Realization of the orbital-selective Mott state atthe molecular level in Ba LaRu O ,” arXiv:2004.00661. H.-C. zur Loye, S.-J. Kim, R. Macquart, M. D. Smith,Y. Lee, and T. Vogt, Solid State Sci. , 608 (2009). S. A. J. Kimber, M. S. Senn, S. Fratini, H. Wu, A. H. Hill,P. Manuel, J. P. Attfield, D. N. Argyriou, and P. F. Henry,Phys. Rev. Lett. , 217205 (2012).
Appendix A: Character Table for D h Double Group
TABLE I. Partial character table for D h Double Group
E C C (cid:48) σ h S σ v E C S E / √ − − −√ E / − − E / −√ − − √ Appendix B: Details of crystal growth andcharacterization for Ba InIr O Small single crystals of Ba InIr O were grown fromthe pre-reacted polycrystalline material in the BaCl flux taken in the 1:10 ratio. The mixture was heatedto 1200 ◦ C in 9 hours and kept at this temperature for20 hours followed by a slow cooling to 950 ◦ in 90 hoursand a standard furnace cooling to room temperature.The resulting crystals had platelet hexagonal form andlinear dimensions of less than 0.3 mm. They showed samelattice parameters and hexagonal symmetry as the pow-der samples reported previously . Energy-dispersive x-ray spectroscopy (EDXS) analysis revealed the elementratio of In:Ir:Ba = 1:2.08(6):3.17(9) in good agreementwith the Ba InIr O composition. The c -direction of thecrystals was determined by monitoring reflections on thex-ray powder diffractometer (Rigaku MiniFlex, CuK α ra-diation).Magnetic susceptibility (Fig. 9) was measured on a sin-gle crystal using the MPMS III SQUID from QuantumDesign in the temperature range of 1 . −
200 K in theapplied field of 7 T. Above 200 K, the signal was belowthe sensitivity limit, owing to the very small sample size.As the sample mass could not be determined with suf-ficient accuracy, the data measured for H (cid:107) c and H ⊥ c were averaged and scaled against the powder data. Thepowder data were measured up to 650 K, as explained inRef. 21.3 FIG. 9. Temperature-dependent magnetic susceptibilitymeasured on the Ba InIr O9