Cation spin and superexchange interaction in oxide materials below and above spin crossover under high pressure
Vladimir A. Gavrichkov, Semyon I. Polukeev, Sergey G. Ovchinnikov
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Cation spin and superexchange interaction in oxide materials below and above spincrossover under high pressure.
Vladimir A. Gavrichkov, ∗ Semyon I. Polukeev, and Sergey G. Ovchinnikov
Kirensky Institute of Physics, Akademgorodok 50, bld.38, Krasnoyarsk, 660036 Russia (Dated: February 13, 2020)We derived simple rules for the sign of 180 ◦ superexchange interaction based on the multielectroncalculations of the superexchange interaction in the transition metal oxides that are valid both belowand above spin crossover under high pressure. The superexchange interaction between two cationsin d n configurations is given by a sum of partial contributions related to the electron-hole virtualexcitations to the different states of the d n +1 and d n − configurations. Using these rules, we haveanalyzed the sign of the 180 ◦ superexchange interaction of a number of oxides with magnetic cationsin electron configurations from d till d : the iron, cobalt, chromium, nickel, copper and manganeseoxides with increasing pressure. The most interesting result concerns the magnetic state of cobaltand nickel oxides CoO, Ni O and also La CoO , LaNiO isostructural to well-known high-T C andcolossal magnetoresistance materials. These oxides have a spin at the high pressure. Change ofthe interaction from antiferromagnetic below spin crossover to ferromagnetic above spin crossoveris predicted for oxide materials with cations in d (FeBO ) and d (CoO) configurations, while formaterials with the other d n configurations spin crossover under high pressure does not change thesign of the 180 ◦ superexchange interaction. I. INTRODUCTION
The mechanism of superexhcange interaction is wellknown for a long time . The effective Heisenberg Hamil-tonian describes the exchange interaction J of the mag-netic cations in the ground state. It is well known thatthere are many excited states for multielectron cations. However these states are not involved by the superex-change interaction, and Heisenberg model is a based the-ory because typically excited states lies well above themagnetic scale J and Curie/Neel temperatures ( T C / T N ).A low energy description of magnetic insulators may beviolated in two situations. The first one is related withintensive optical pumping when one of magnetic cationsis excited into some high energy state and its exchangeinteraction with the neighbor cation in the ground statechanges resulting in many interesting effects of the fem-tosecond magnetism. The other situation occurs at thehigh pressure when the cation spin crossover in magneticinsulators from the high spin (HS) to the low spin (LS)state takes place.
The spin crossover occurs due tocompetition between the energy of the crystalline field10 Dq and the parameter of intratomic Hund exchange J H . Typically, the applied pressure increases the crys-tal field, but does not significantly change the exchangeparameter J H . The spin crossovers are known for manytransition metal oxides with d ÷ d cations, and for tran-sition metal complexes, like metalorganic molecules ormolecular assemblies. Near crossover the energies oftwo states ε HS and ε LS are close to each other and con-ventional scheme of the superexhcange interaction calcu-lation should be modified.The spin crossovers have been experimentally detectedand investigated in a number of transition metal ox-ides. Calculations also confirm a possibility of the spincrossovers in these materials and their role in the metal-insulator transition.
In real, a situation is complicated by the observed structural and chemical instabilities ofsome oxide materials at the high pressures, which de-stroy the possibility of a comparison between the cal-culation of superexchange interaction and experimentaldata. The results of the experimental studies containboth the examples of stable FeBO with isostructuralspin crossover at ∼ O hematite. Further we will restrict ourselves tothe stable oxide materials and assume that there areisostructural areas on the phase P/T diagram of the ox-ides, where the magnetic ordering is governed mainly bystrong superexchange AFM interactions in Me − O − Mewith a bond angle of about 180 ◦ .The aim of our work is to answer the question of howthe 180 ◦ superexchange interaction depends on the cationspin in transition metal oxides at the high pressure, andcan simple changes in the crystal field without a spincrossover lead to a change in its nature from AFM toFM? In terms of the realistic p-d model that included-electrons of cation and p-electrons of oxygen the su-perexchange interaction arises via cation-anion p-d hop-ping t pd in the fourth order perturbation theory over the t pd (see for example ). Eliminating the oxygen statesone can obtain the effective Hubbard model with cation-cation hopping t ∼ t pd . ( ε p − ε d ) and then the effectiveHeisenberg model may be obtained by the unitary trans-formation of the Hubbard model with the superex-change interaction of the kinematic origin J ∼ t (cid:14) U .The superexchange interaction appears in a second orderperturbation theory over interband hopping t from theoccupied low Hubbard band into the empty upper Hub-bard band and back. It may be considered as result ofthe virtual excitation of the electron-hole pair.We start discussing the properties of the transitionmetal oxides with a model of the periodic lattice ofcations in d n configuration in a center of oxygen octa-hedra with a set of states | n i with energy ε n . The elec-tron addition (extra electron) results in the d n +1 states | e i with energies ε e ( n + 1). Similar electron removal (orhole creation) involves the d n − states | h i with energies ε h ( n − | n i ( we call thisstates neutral) into some | e i state (we call these excita-tion by electronic) and at site 2 the hole creation excitesthe neutral state into the one of the states | h i (we callthem hole). These electon-hole excitations are virtual,after their annihilation back the final state is again twocations in initial d n configurations. This approach allowsus to consider all partial contributions to the superex-change interaction including both the ground states aswell as excited states in all three sectors of the Hilbertspace: neutral N ( d n ), electronic N + (cid:0) d n +1 (cid:1) and hole N − (cid:0) d n − (cid:1) . Here we show that the sign of the parti-tial contributions J F Mij and J AF Mij to the total superex-change interaction J ij = J AF Mij + J F Mij is directly inde-pendent on the cation spin S ( d n ), but is controlled bythe spin ratio S (cid:0) d n − (cid:1) = S (cid:0) d n +1 (cid:1) (AFM interaction)or S (cid:0) d n − (cid:1) = S (cid:0) d n +1 (cid:1) ± J F Mij and J AF Mij , however they can lead toa change in their relative magnitudes, as a result, to achange in the sign of superexchange parameter J ij . Amain factor for the comparison between the AFM andFM interactions is the type σ or π overlapping orbitalsinvolved by the partial contributions. These character-istics is comparable in simplicity with the well knownGoodenough-Kanamori-Anderson rules, which are usedmany years by scientists in the analysis of the magneticstates of dielectric materials. In the paper we alsogeneralize the previous results for the superexchange in-teraction in iron borate under high pressure and opti-cal pumping to the different transition metals oxideswith magnetic ions in the d - d configurations.For the readers convenience the theoretical details areplaced in the Appendix below and in the main text willdiscuss the physical ideas. II. ADDITIVITY PROPERTIES OFSUPEREXCHANGE INTERACTION
We will work within a framework of the cell perturba-tion approach to calculate a magnitude of the superex-change, that logically fits into the LDA + GTB methodto study both the electronic structure, and the 180 ◦ superexchange interaction in oxide materials under thepressure and optical pumping. The conclusion of ourstudy will be some simple rules which can help to es-timate the sign of the superexchange in the oxide ma-terials at high pressure without complicate calculations.At this point we will take the superexchange Hamilto-nian (1) (see Appendix) as a working tool, in structurewhich there is a summation over the independent con-tributions involving the ground | n i = (cid:12)(cid:12) ( N , M S ) n (cid:11) ,excited electronic | e ( h ) i = (cid:12)(cid:12)(cid:12) ( N ± , M S ) e ( h ) E ( e ) and hole( h ) states at energies ε e ( h ) of the configuration space sec-tors N ± = n ± H s = X i = j J ij (cid:18) ˆ S in ˆ S jn −
14 ˆ n ( e ) in ˆ n ( h ) jn (cid:19) ,J ij = X he J ij ( h, n , e )(2 S h + 1) (2 S n + 1) (1)where J ij ( h, n , e ) = 2 (cid:16) t n h,n eij (cid:17) (cid:30) ∆ n he and ∆ n he = ε e + ε h − ε n . All definitions of the multielectron spinˆ S in and number of quasiparticles ˆ n ( e ) in operators are inthe Appendix. The second contribution in Eq.(1) dif-fers from the generally accepted method of writing thesuperexchange interaction and coincides with the usualform ˆ n i ˆ n j for half-filled shells, where there is electron-hole symmetry. The superexchange interaction parame-ter J ij in Eq.(1) is additive for all electronic | e i and hole | h i states in sectors N ± in Fig.1 and one is obtained insecond order of cell perturbation theory over the inter-band contribution δ ˆ H to the total Hamiltonian ˆ H ofelectron interatomic hopping: δ ˆ H = X ij ˆ h outij = X ij X nhe " t el,hnij X σ α + iσ ( en ) β jσ ( hn ) + t nh,leij X σ β + iσ ( nh ) α jσ ( ne ) , (2)that describes the creation and annihilation of the vir-tual electron (denoted by the operator α + iσ ) and hole(operator β + iσ ) pairs. Exactly the virtual excitationsthrough the dielectric gap ∆ nhe to the conduction bandand vice versa in Eq.(2) contribute to the superexchange interaction. The total multielectron Hamiltonian in therepresentation of the Hubbard operators looks likeˆ H = ˆ H + ˆ H , where ˆ H contains all multielectron statesof the involved d n and d n ± configurations, and ˆ H de-scribed all interatomic single electron hoppings (kinetic e e ... ... h h n n ... ´ - N N + N he J eh J ( ) n d ( ) - n d ( ) + n d FIG. 1. The scheme of the superexchange interaction illus-trating property of its additivity over virtual electron excita-tions involving all ground states J h e (dotted line, we callthis contribution the main exchange loop) and the excitedelectronic d n +1 contribution J h e (solid line, called the ex-cited exchange loop). energy):ˆ H = X i (X h ( ε h − N − µ ) X hhi + X n ( ε n − N µ ) X nni ++ X e ( ε e − N + µ ) X eei ) (3)ˆ H = X ij X rr ′ t rr ′ ij X ri + X r ′ j (4)for the material with magnetic cations in arbitrary d n electron configuration. Any Hubbard operator X ri = | p i h q | constructed in the full and orthogonal set of eigen-states | p i is numerated by a pair of indexes which de-notes the initial state | q i and the final state | p i of theexcitation. It is more convenient to numerate eachexcitation by single vector index r = ( p, q ) (so calledroot vector that plays a role of the quasiparticle bandindex). Here, electronic creation operators for vector in-dexes r = ( n, h ) or r = ( e, n ) excitations in Eq.(2) aredenoted by β + iσ ( N − → N ) and α + iσ ( N → N + ) respec-tively. The hopping matrix element in Eq.(4) is t rr ′ fg = X λλ ′ t λλ ′ fg X σ [ γ ∗ λσ ( r ) γ λ ′ σ ( r ′ ) + γ ∗ λ ′ σ ( r ) γ λσ ( r ′ )](5)and γ λσ ( r ) = h e | c + iλσ | n i × δ S ie ,S in ±| σ | × δ M e ,M n + σ (6)where a root vectors r and r ′ run over on all possiblequasiparticle excitations ( e, n ) and ( n, h ) between many-electron states | n i and | e ( h ) i with the energies ε n and ε e ( h ) in the sectors N and N ± of configuration space(Fig.1). These quasiparticle excitations are describedby nondiagonal elements t rr ′ fg = t nh,neij . In the conven-tional Hubbard model there is only one such element corresponding to the excitations between lower and up-per Hubbard bands. Using the results of Appendix(seeEq.(A.12)), we can represent the exchange parameter fora pair of interacting spins S in = S jn in the form ofEq.(7): J ij = J AF Mij + J F Mij . (7)This equality and its relationship with the spin S h ( e ) at the states | h ( e ) i was obtained in the works foriron borate and also was firstly briefly mentioned in theworks. The virtual electron interband ( n , e ) and( n , h ) hoppings correspond to only one of contributionsin the sum J ij = P he J ij ( h, n , e ), and any contribution J he = P ij J ij ( h, n , e ) can be represented by a double loopor the so-called exchange loop, marked by the same line(solid or dashed). In Fig.1 the contributions J he is illus-trated by double exchange loops with the arrows whichconnect the ground state of the magnetic ions | n i withall ground | h ( e ) i and excited | h ( e ) i states. III. RULES FOR A SIGN OF DIFFERENTCONTRIBUTIONS TO THESUPEREXCHANGE
The new result of this paper is the classification ofdifferent contrubutions by the relation between spins S h and S e . If in exchange loop S h = S e ± S h = S e it isAFM contribution. These two relations exhaust allpossible interrelations between spins for all nonzerocontributions, i.e. in any other case, the contributionto superexchange from this pair of states | h i and | e i issimply not available. The sign of the total exchangeinteraction (FM or AFM) depends on a relationshipbetween relative magnitudes of the contributions. Themain difficulty is a great number of excited states in N ± sectors of the configuration space. Due to the smallestdenominator ∆ n h e in the superexchange (1), themain exchange loop involving ground | h ( e ) i statescan form a dominant J h e contribution. However, thecontributions J he from the excited states | h ( e ) i in N ± sectors can compete with the main exchange loop dueto the dominant nominator, if the excited exchangeloop occurs by overlapping of states with e g symmetry,and the main exchange loop is formed by π bondingdespite not the smallest denominator ∆ n he in Eq.(1).The problem is that without complicated numericcalculation taking into account all hopping integrals(4), it is difficult to obtain the final answer about themagnitude and sign of the superexchange interaction.For example, such numerical calculations have beencarried out for La CuO with a configuration d , wherea number of the contributions exceeds ten ones. Wewill give a qualitative criterion that takes into accountboth factors in the case both σ or π overlapping in theHamiltonian (2) (where t el,hnij hopping is obtained bythe mapping of the multiband p-d model, which includesintegral for σ or π overlapping), and the energy gap∆ nhe in the arbitrary exchange loop J he . The minimalgap ∆ n h e just coincide with a dielectric gap E g inthe oxide materials. After comparing calculated signof the superexchange constant for magnetic ions in theelectron configurations d - d with experimental data,we found that in most cases there is no need to sum overall possible virtual hoppings (or exchange loops), it isenough to establish the criterion in form:1. For the σ overlapping e g states corresponding to con-tribution J h e , the sign of superexchange is controlledby the virtual electron excitations with participate of theground | h ( e ) i states and minimal magnitude of theenergy gap ∆ n h e ∼ E g . These excitations involved tothe main exchange loop is pictured in Fig.1 by a dashedline.2. In the case of π overlapping t g states for the virtualelectron excitations involving only the ground | h ( e ) i states the sign of superexchange is controlled by not themain exchange loop, but the virtual electron excitations(exchange loop) involving the excited states with the σ overlapping e g states. These virtual excitations arepictured in Fig.1 by solid line. If such exchange loops isabsent, the sign of superexchange is controlled still bythe main loop with the π overlapping.Here, the σ overlapping have the priority. Indeed, thesuperexchange interaction is proportional to the fourthdegree of the overlapping integral I σ ( π ) = ρ ( | ∆ R | ) χ σ ( π ) between the electron states of the anion and the magneticcation, where the radial part ρ ( | ∆ R | ) depends only onthe anion-cation distance ∆ R , and the angular part χ σ ( π ) depends on the angular distribution of the anions. Thesquared ratio ( I π / I σ ) of the overlapping integrals for e g and t g states involved in the superexchange through σ and π coupling in the same octahedral complexesis the following relation: ( I π / I σ ) = ( χ τ / χ σ ) = 1/3.Thus the fourth degree gives ratio of matrix elements ∼ .
1, i.e., competition between the contributions with aparticipation of virtual t g electron hopping and the onethrough σ coupling is possible, when the denominatorenergy ∆ nhe for excited loop J he is no more than 9 timeshigher in energy than the main loop energy ∆ n h e .Otherwise the σ type contribution from exchange loopis dominant. In case of several competing contributionssimple calculations of the multielectron energies belowand above the spin crossover at the high pressure canbe used to compare energy denominators of the AFMand FM contributions given by Eq.(7). Some exampleswill be given in the next section for oxide materials withd and d cations. IV. SUPEREXCHANGE IN OXIDES WITHCATIONS IN d AND d CONFIGURATIONS
Let us show, using the example of oxide materials CoOand Ni O with Ni +, Co + cations in the d electronconfiguration under high pressure, how our rules work.The energy of the neutral | n i (d ) states and electronic | e i (d ) and hole | h i (d ) states at the ambient pressureare shown in Fig.2(a). From the main exchange loopwith π overlapping our rules results in the FM sign of thecontribution J T, A . Competing AFM contribution is theexchange loop J T, T with the excited states (cid:12)(cid:12) T , (cid:11) and σ overlapping. Below we will check our rules by directcalculation for the main exchange loop. To derive the FMcontribution J T, A using angular momentum additionrules, we introduce the creation operators β + iσ ( n , h ) for N − ↔ N hole quasiparticles by Eq.(7) and α + iσ ( e , n )for N ↔ N + electron quasiparticles by Eq.(8). − β + i ↑ = r X , i + r X , i + r X − , − i + r X − , − i , β + i ↓ = r X , i + r X , i + r X − , i + r X − , − i − α + i ↑ (cid:0) A , T (cid:1) = r X , i + r X , − i + r X − , − i , α + i ↓ (cid:0) A , T (cid:1) = r X , i + r X , i + r X − , − i (8)Working further in framework of the cell perturbationtheory, we obtain in the second order the FM contribu-tion J T, A from the main exchange loop in Fig.2 withthe π overlapping: J T, A = − X i = j J ij (cid:0) T, A (cid:1) (5) (3/2) (cid:18) ˆ S in ˆ S jn + 14 ˆ n ( e ) in ˆ n ( h ) jn (cid:19) (9)where S in = , ˆ S + in = − β + i ↑ β i ↓ = − α i ↓ α + i ↑ , ˆ S zin = − P σ η ( σ ) β + iσ β iσ = − P σ η ( σ ) α iσ α + iσ , and also ˆ n ( e ) in = 5 P σ β + iσ β iσ and ˆ n ( h ) jn = 4 P σ α jσ α + jσ are the number ofelectron and hole quasiparticles involved in the superex-change interaction. According to a second point of thecriterion the FM contribution competes with the AFM J T, T contribution: J T, T = X i = j J ij (cid:0) T, T (cid:1) (3) (3/2) (cid:18) ˆ S in ˆ S jn −
14 ˆ n ( e ) in ˆ n ( h ) jn (cid:19) (10)from the virtual hoppings of e g electrons with participa-tion of the states (cid:12)(cid:12) T , (cid:11) and σ overlapping (see Fig.2(a)).Similarly to Eqs.(7) and (8), new α ′ + iσ and β ′ + iσ quasipar- ticles involved in this superexchange are given by theexpression: β ′ + i ↑ (cid:0) T, T (cid:1) = X , i + r X , i + r X − , − i , β ′ + i ↓ (cid:0) T, T (cid:1) = r X , i + r X − , i + X − , − i ; − α ′ + i ↑ (cid:0) T, T (cid:1) = r X , i + r X , − i + r X − , − i , α ′ + i ↓ (cid:0) T, T (cid:1) = r X , i + r X , i + X − , − i (11) ´ - N N + N ( ) d ( ) d ( ) d , T T A,T J A T , T T,T J g e g e g t ´ - N N + N ( ) d ( ) d ( ) d E A,A J A A E E,A J g e g e g e a b g t FIG. 2. Scheme of the 180 ◦ superexchange interaction in CoO:(a) at the ambient pressure, where AFM interaction is con-trolled by the contribution from the exchange loop J T, T with the excited states (cid:12)(cid:12) T , (cid:11) and σ overlapping. The con-tribution J T, A from the main exchange loop J A, A with π overlapping is showed by a dotted line; (b) under high pres-sure, where FM character is controlled by the main exchangeloop J A, A with the σ overlapping. The AFM contributionfrom the exchange loop J A, E with participation of excitedstates (cid:12)(cid:12) E (cid:11) has a large denominator. Here: ˆ S + in = 3 β ′ + i ↑ β ′ i ↓ = − α ′ i ↓ α ′ + i ↑ , ˆ S zin =3 P σ η ( σ ) β ′ + iσ β ′ iσ = − P σ η ( σ ) α ′ iσ α ′ + iσ and ˆ n ( e ) in =3 P σ β ′ + iσ β ′ iσ , ˆ n ( h ) in = 4 P σ α ′ iσ α ′ + iσ Calculation of energiesof the different states below and above spin crossoverallows us to obtain the energy denominators for the dif-ferent contributions to superexchange interaction. Forthe main exchange loop J T, A in Fig.2(a) the value∆ n he = U − J H , where U is the intra-atomic Coulombmatrix element (Hubbard parameter) and J H is the Hundexchange coupling, both U and J H are positive. Forthe contribution from exchange loop J T, T , ∆ n he = ε e + ε h − ε n = U + J H . At the typical magnitudes U = 6 eV and J H = 1 eV the ration of denominatorsis 5 /
8, and the ratio of numerators is 9 /
1. It provesthe dominant AFM contribution below spin crossover.With increasing pressure there is the spin crossover inconfiguration d . The pressure enter in the crystal fieldparameter 10 Dq that linearly increases with the pres-sure: below spin crossover at the ambient pressure when10 Dq < J H the cation Co is at the HS state, and | n i = (cid:12)(cid:12) T (cid:11) , | h i = (cid:12)(cid:12) T (cid:11) , | e i = (cid:12)(cid:12) A (cid:11) (see Fig.2(a)). Above spin crossover at 10 Dq > J H the cation Co is at the LS state | n i = (cid:12)(cid:12) E (cid:11) , and | h i = (cid:12)(cid:12) A (cid:11) (seeFig.2(b)). Thus, the ground | n i and hole | h i statesthe superexchange interactions in the cobalt monoxideunder high pressure is changed. The main exchange loop J A, A with the σ overlapping should be FM accordingour rules. J A, A = − X i = j J ij (cid:0) A, A (cid:1) (cid:18) ˆ S in ˆ S jn + 14 ˆ n ( e ) in ˆ n ( h ) jn (cid:19) (12)The AFM contribution from the exchange loop with theexcited states has the large denominator than the FMone (Fig.2b). J A, E = X i = j J ij (cid:0) A, E (cid:1) (cid:18) ˆ S in ˆ S jn −
14 ˆ n ( e ) in ˆ n ( h ) jn (cid:19) (13)These conclusions can be obtained analogously to theprevious Eq.(9) and Eq.(10), starting from building oper-ators β + iσ , α + iσ and β ′ + iσ , α ′ + iσ of the quasiparticles and fin-ishing with derivation of the Eqs.(12) and (13). We haveto compare the energy denominators. For FM contribu-tion J A, A , the energy ∆ A A = ε (cid:0) A, d (cid:1) + ε (cid:0) A, d (cid:1) − ε (cid:0) E, d (cid:1) = U − J H and ∆ A E = U . Taking into ac-count that all contributions have the same σ bonding, wecame to conclusion that resulting interaction in the LSstate for materials with the cations in d configurationwill be FM.Let’s compare our conclusions with the results for ironborate FeBO at the high pressure. Under pressure P ∼ GP a in the iron borate with cations Fe in theconfiguration d the spin crossover (cid:12)(cid:12) A (cid:11) → (cid:12)(cid:12) T (cid:11) occursat 10 Dq = 3 J H . Given above criterion tells us that thesign of the exchange interaction in iron borate is changedfrom AFM to FM with increasing pressure in agreementwith direct calculations. . This conclusion is also validfor another oxide materials with cations in the configu-ration d and octahedral environment.At the ambient pressure FM contributions from the ex-change loops are missing (Fig.3(a)). The AFM superex-change interaction is caused by the contribution J E, E from the σ bonding exchange loop with the excited | e i states. The calculation of the energy denominator is TABLE I. The examples of transition metals oxides with calculated sign of 180 ◦ superexchange interactions (in 3 and 5 columns),and also the magnetic ordering below and above the spin crossover (in 4 and 6 columns). The notations (ex) and (gr) indicatesthe nature of the main contribution to the superexchange: (ex) is the exchange loop involving excited states, (gr) is the mainexchange loop. Cation and electron Oxides Superexchange Ambient pressure Superexchange High pressureconfiguration below spin crossover (experiment) above spin crossover (experiment)d , Cr , CrO J F M T, A (gr) FM, T C = 90 K no crossover, FM up to S n = 1 J F M T, A (gr), P=56GPa, S n = 1d , Cr , LaCrO J AF M T, T (ex) AFM, T N = 298 K no crossover AFM, T N increases S n = J AF M T, T (gr), with a pressure up to S n = d , Fe , Mn , LaMnO J F M A, A ( gr ) AFM, with FM planes crossover is expected AFM, T N = 152 KS n = 2 T N = 140 K , to the LS state, at the pressure J F M A, T (gr), P=2 GPa. FM above S n = 1 the spin crossoveris predicted.d , Fe , Mn FeBO , J AF M E, E (ex) AFM, T N = 348 K spin crossover, T N ( C ) = 50 K ∗ , S n = (Fe O , MnO J F M T, A (gr) at P=49 GPa, S n = FM above thespin crossover ispredicted.d , Fe , Co Mg − x Fe x O, J AF M T, T (ex) AFM, T N = 37 K spin crossover to non magnetic S n = 2 (LaCoO ) nonmagnetic state above P=55 GPa with S n = 0d , Co , Ni , CoO, J AF M T, T (ex) AFM, T N = 290 K spin crossover spin crossover S n = (La CoO , LaNiO ) is expected, observed at J F M A, A (gr), P=80-90 GPa S n = d , Ni , Cu NiO J AF M E, E (ex) AFM, T N = 525 K no spin crossover, no spin crossover S n = 1 J AF M E, E , observed up to S n = 1 P=220 GPa *The critical temperature T N ( C ) of magnetic ordering in iron borate FeBO at the higher pressure was measured by Mossbauerspectroscopy, however, this method cannot distinguish the nature (FM or AFM) of the magnetic ordering. Up to nowthere is no experimental data on the magnetic ordering in the LS state of FeBO or any other materials with d cations. ∆ E T = U − Dq + 4 J H . Thus, the AFM exchangeinteraction at the ambient pressure may be estimatedas J E T ≈ t σ (cid:14) ( U + J H ). Crystal field increases withpressure, and at the critical pressure 10 Dq ( P c ) = 3 J H there is spin crossover (cid:12)(cid:12) A (cid:11) → (cid:12)(cid:12) T (cid:11) . Above the spincrossover, the nature of the FM superexchange interac-tion is obtained from the competition of FM( J T, A ) andAFM ( J T, A ) loops with the same type of π overlapping,where the FM contribution prevails (see Fig.3(b) due tothe smaller magnitude of the energy gap ∆ n he . We canestimate the competing FM and AFM by calculation oftheir energy denominators. For the main FM exchangeloop (dotted lines in Fig.3(b) the energy ∆ T A ≈ U − J H , and for the excited AFM loop (solid lines in Fig.3(b) theenergy ∆ T A ≈ U . That is why the FM contributiondominates. Nevertheless the AFM one strongly reducedthe total FM superexchange interaction, that can be es-timated as J F M = J E T + J E T ≈ t π U − J H − t π U = t π U − J H J H U (14)Thus, spin crossover in oxide materials with d cationsnot only changes the sign of exchange interaction, butalso reduces its amplitude by the factor J H / U << ´ - N N + N E,E J ( ) d ( ) d ( ) d A T E E T,E J g e g e g t ´ - N N + N A,T J ( ) d ( ) d ( ) d T T A,T J A T g t g t g t a b FIG. 3. Scheme of the 180 ◦ superexchange interaction inFeBO : (a) at the ambient pressure, where the main exchangeloop J E, T has a zero contribution because of zero over-lapping, and the σ overlapping exchange loops J E, E resultin AFM contribution only; (b) under high pressure, whereboth contributions J T, A (FM) and J T, A (AFM) are pro-portional to π overlaping. The FM contribution J T, A dom-inates. V. SUPEREXCHANGE IN OXIDES WITHCATIONS IN OTHER ELECTRONCONFIGURATIONS
Now, we can obtain the nature(FM or AFM) of thesuperexchange interaction for oxide materials with d - d cations under pressure, below and above the spincrossover in Tab.1, and also compare one with experi-mental data, where it is possible. In the oxide materialswith another d n ions, where n = 2 ,
3, spin crossover isnot possible, and ground states (cid:12)(cid:12) T (cid:11) and (cid:12)(cid:12) A (cid:11) is stableunder high pressure.d . Chromium dioxide CrO , where chrome cationCr has configuration d with spin S n = 1, is the ex-ample of FM contribution J T, A from the main exchangeloop involving the ground states of t g cation with the π overlapping at an arbitrarily pressure. FM ordering inchromium dioxide is known experimentally and persistsin orthorhombic phase of the chromium dioxide up toP=56Gpa. d . For chromium oxide LaCrO with cations Cr at the ground state (cid:12)(cid:12) A (cid:11) is stable under pressure, andthe dominant AFM contribution is given by the ex-change loop with the ground state | h i = (cid:12)(cid:12) T (cid:11) in thehole configuration d and the excited state | n i = (cid:12)(cid:12) T (cid:11) in the electron d configuration. Under high pressurewhen 10 Dq ( P ) > J H the crossover stabilizes the triplet | n i = (cid:12)(cid:12) T (cid:11) . The AFM sign of the exchange interactiondoes not change, but the same interaction J T, T is de-scribed by the main exchange loop and its value becomeslarger.d . In manganite LaMnO at the ambient pressurewith cations Mn at the ground HS state | n i = (cid:12)(cid:12) E (cid:11) ,the σ overlapping main loop J A, A results in the FM in-teraction. Under high pressure 10 Dq ( P ) > J H whenthe cations M n are in the intermediate spin state | n i = (cid:12)(cid:12) T (cid:11) , and all superexchange interactions resultsfrom the π bonding. The main exchange loop provides the FM interaction J A, T , with the energy denominator∆ A T = U − J H , while exchange via excited states givesthe AFM contribution J T, T with ∆ T T = U + J H , andthe total superexchange interaction has the FM sign. Itshould be emphasized that in this study we consider thecrystals with cations in the octahedral oxygen environ-ment. When we compare our conclusions about the FMinteraction with the magnetic state of manganite, we findthe disagreement with its AFM ordering at the ambientpressure. Nevertheless this AFM ordering consists of theFM ab planes that are AFM coupled. This disagreementis probably related to the dependence of the magneticordering on the type of orbital ordering in the oxide ma-terial with Jahn - Teller cations Mn . With increas-ing pressure, the spin crossover in is accompanied by thetransition of the cation Mn from the HS Jahn - Tellerstate (cid:12)(cid:12) E (cid:11) to the state (cid:12)(cid:12) T (cid:11) . Therefore, the orbital or-dering with increasing pressure should disappear, and theFM nature of the superexchange will manifest itself (seeTab. 1).d . At the ambient pressure, in the wustiteMg x Fe − x O with cations Fe in the configuration d there is a competition of two different contributions J AF M T, T with σ overlapping and J F M A, T with π overlapping,and the AFM contribution dominates. At high pressures(P = 55 GPa), the magnetic moment in the wustite isabsent as well as in all other compounds with cations inconfiguration d . The large class of such materials with S n = 0 in the ground state is given by the perovskitebased rare earth cobaltite LaCoO , where La is the 4fion.d . For nickel monoxide NiO with cations Ni in theconfiguration d situation is similar to the configurationd . There is no spin crossover in the neutral configu-ration d and at the ambient pressure the AFM inter-action J AF M E, E involves the excited state | h i = (cid:12)(cid:12) E (cid:11) inthe hole configuration d . Above the spin crossover inthe hole configuration this state becomes the ground one | h i = (cid:12)(cid:12) E (cid:11) , and the same AFM interaction J AF M E, E isgiven now by the main exchange loop. Thus, its valueincreases due to the spin crossover in the hole configu-ration d . Summarizing our analysis we get together allour conclusions in Tab.1, and also compare them withexperimental data, where it is possible. VI. CONCLUSIONS
The sign of the partial contributions J he to the to-tal superexchange interaction is directly independent onthe cation spin S ( d n ), but is controlled by the spinrelation S (cid:0) d n − (cid:1) = S (cid:0) d n +1 (cid:1) (AFM interaction) or S (cid:0) d n − (cid:1) = S (cid:0) d n +1 (cid:1) ± S ( d n ) = S (cid:0) d n ± (cid:1) ± and nickel monox-ide NiO ( S n = 1), or manganites LaMnO and wustiteMg − x Fe x O ( S n = 2), can have FM and AFM interac-tions respectively. The main factor for the comparisonbetween the AFM and FM interactions is the type over-lapping states involved by the contributions.The nature of the superexchange interaction with in-creasing pressure changes (AFM → FM) only in oxide ma-terials with cations in d (e.g. FeBO ) and d (e.g.CoO) configurations. Indeed spin crossover (cid:12)(cid:12) T (cid:11) → (cid:12)(cid:12) E (cid:11) with generating Jahn-Teller cations Co (cid:0) E (cid:1) in cobalt monoxide at P >
43 Gpa is accompaniedby: (i) transformation of the cubic rock salt-typestructure to mixed rhombohedrally distorted rock salt-type structure without significant volume change struc-ture; (ii) a resistance drop by eight orders of mag-nitude at the room temperature (43Gpa < P < > We did not findany studies related to high pressure effects in oxidesLa CoO ( S n = 3 / T N = 275 K at the ambient pres-sure ) and LaNiO − x (paramagnetic metal and ultra-thin film AFM insulator at the ambient pressure).Unlike the cobalt oxides LaSrCoO and LaCoO , withCo , the layered oxide La CoO has not been studiedunder the high pressure. However, these oxide materi-als isostructural to well-known high-T C and colossalmagnetoresistance materials could have interesting phys-ical properties at the high pressure ( > C supercon-ductors: doped and nonstoichiometric cuprates withthe multimode Jahn-Teller (cid:0) a + b (cid:1) ⊗ ( b g + a g ) ef-fect and iron based superconductors, have also thespin S n = 1/2 and on the other hand, pseudogap ef-fects and colossal magnetoresistance are observed in thedoped manganite La(Sr,Ba)MnO also with the Jahn-Teller Mn ( E ) cations. However, the cobalt oxideLa CoO at the high pressure is very likely different from cuprate La CuO at the ambient pressure in a sign ofthe superexchange interaction, despite the same cationspin 1 /
2. Indeed the interaction in nickel monoxide doesnot undergo any critical changes with increasing pres-sure, either in theory or experiment, up to 220 GPa. Note, in the oxide materials: CrO , NiO, La CuO withthe cations in the electron configurations d , d , d thespin crossover under high pressure is impossible.The results partially disagree with experimental dataat the ambient pressure only for oxide materials withJahn-Teller d cations like LaMnO , where the FM ab planes have AFM ordering. With increasing pressure,the spin crossover in manganite LaMnO is accompa-nied by the transition of the magnetic Jahn-Teller Mn cation to the state (cid:12)(cid:12) T (cid:11) . In according to our conclusions,the effects of orbital ordering should disappear, and theFM nature of the superexchange will manifest itself (seeTab.1). Indeed, below pressure 29 GPa the manganiteis not metallic and consists of a dynamic mixture of dis-torted and undistorted MnO octahedral. Above pres-sure 32 Gpa, undoped manganite already shows metallicproperties.
ACKNOWLEDGMENTS
We acknowledge the support of the Russian ScienceFoundation through Grant 18-12-00022.
Appendix: AFM and FM contributions tosuperexchange interaction
To derive Eqs.(1) and (6), we start from the Hamilto-nian of the p-d model, whereˆ H = ˆ H d + ˆ H p + ˆ H pd + ˆ H pp ˆ H d = X iλσ ( ε λ − µ ) d + λiσ d λiσ + U d n σλi ˆ n − σλi + 12 X λ ′ = λ X σ ′ V λλ ′ ˆ n σλi ˆ n σ ′ λ ′ i − J H d + λiσ d λi ¯ σ d + λ ′ i ¯ σ d λ ′ iσ ! , ˆ H p = X mασ ( ε α − µ ) p + αmσ p αmσ + U p n σαm ˆ n − σαm + 12 X α ′ = α,σ ′ V αα ′ ˆ n σαm ˆ n σ ′ α ′ m , ˆ H pd = X mi X αλσ " t λαim (cid:0) p + αmσ d λiσ + h.c. (cid:1) + V pdim X σ ′ ˆ n σαm ˆ n σ ′ λi , ˆ H pp = X mn X αβσ t αβmn (cid:0) p + αmσ p βnσ + h.c. (cid:1) . (A.1)Here, n σλi = d + λiσ d λiσ , n σαm = p + αmσ p αmσ , where theindices i ( j ) and m ( n ) run over all positions d λ = d x − y , d r − z , d xy , d xz , d yz and p α = p x , p y , p z localizedone electron states with energies ε λ and ε α ; t λαim and t αβmn the hopping matrix elements; U d , U p and J H are onesite Coulomb interactions and the Hund exchange inter-action, V pdim is the energy of repulsion of cation and an- ion electrons. A correct transition from the Hamiltonian(A.1) of the p-d model to the Hamiltonian (3) in the mul-tielectron representation of the Hubbard operators is pos-sible when constructing well localized Wannier cell oxy-gen states (cid:12)(cid:12) p + λiσ (cid:11) . Although, there is no general deriva-tion of the canonical transformation (cid:12)(cid:12) p + λiσ (cid:11) ↔ | p + αmσ i forarbitrary lattice symmetry, we assume that the canoni-cal representation does exist and that the Wannier celloxygen functions are sufficiently localized. In themultielectron representation the one-electron p + λiσ and d + λiσ operators can be written as a superposition of theHubbard operators that describe one electron excitationsfrom the LS and HS partner states | h ( e ) i with spin S e ( h ) = S n ± | n i : c + λiσ = X n "X e γ λ ( ne ) α + iσ ( ne ) + X h γ λ ( nh ) β + iσ ( nh ) , (A.2)where the new operators α + iσ ( en ) and β + iσ ( nh ) are no-tations for the electron addition to the ground state N → N + , and to the hole state N − → N , respectively.Calculation of the matrix elements in Eq.(5) in agree-ment with the rules of addition of angular momentumsresults in the following relations: α + iσ ( en ) = η ( σ ) M ν P − M ν r S n − η ( σ ) M e + 122 S n +1 X M e ,M n = M e − σiM e P − M e r S n + η ( σ ) M e + 122 S n +1 X M e ,M n = M e − σi (A.3)and β + iσ ( nh ) = η ( σ ) M n P − M n r S h − η ( σ ) M n + 122 S h +1 X M n ,M h = M n − σiM n P − M n r S h + η ( σ ) M n + 122 S h +1 X M n ,M h = M n − σi (A.4)where top and below lines are for S e = S n − | σ | and S e = S n + | σ | respectively. The superexchange interac-tion appears in the second order of the cell perturba-tion theory with respect to the hopping processes ˆ H inEq.(3), which corresponds to virtual excitations throughthe dielectric gap into the conduction band and back to valence band. These quasiparticle excitations corre-spond to the electron-hole excitations and are describedby off-diagonal elements with root vectors r = ( h, n )and ( n, e ). To highlight these contributions, we use aset of projection operators P h and P e , that generalizedthe Hubbard model analysis to the Mott-Hubbard ap-proach with an arbitrary quasiparticle spectrum, where P h = (cid:18) X hhi + P n X nni (cid:19) (cid:18) X hhj + P n ′ X n ′ n ′ j (cid:19) and P e = X eei + X e ′ e ′ j − X eei P e ′ X e ′ e ′ j with 1 h N h , 1 n N n and 1 e ( e ′ ) N e . These operators satisfies the re-lations (cid:18) N h P h =1 P h + N e P e =1 P e (cid:19) = 1 and P h P e = 0, P h P h ′ = δ hh ′ P h , P e P e ′ = δ ee ′ P e . We introduce the Hamiltonianof the exchange coupled ( i, j ) -pairs: ˆ h ij = (cid:16) ˆ h + ˆ h in (cid:17) +ˆ h out , where (cid:16) ˆ h + ˆ h in (cid:17) = P hh ′ P h ˆ h ij P h ′ + P ee ′ P e ˆ h ij P e ′ and ˆ h out = (cid:18)P h P h (cid:19) ˆ h ij (cid:18)P e P e (cid:19) + (cid:18)P e P e (cid:19) ˆ h ij (cid:18)P h P h (cid:19) is the intra- and interband contributions for Hamilto-nian ˆ H = P ij ˆ h ij respectively. In the unitary trans-formation the Hamiltonian for ( i, j ) -pairs is equal to˜ h ij = e ˆ G ˆ h ij e − ˆ G , where ˆ G satisfies the equation X h P h ! ˆ h ij X e P e ! + X e P e ! ˆ h ij X h P h ! ++ " ˆ G, X hh ′ P h ˆ h ij P h ′ + X ee ′ P e ˆ h ij P e ′ ! = 0 , (A.5)and the transformed Hamiltonian ˜ h ij in the second orderof cell perturbation theory over interband hopping ˆ h out can be represented as˜ h ij ≈ X hh ′ P h ˆ h ij P h ′ + X ee ′ P e ˆ h ij P e ′ ! + 12 " ˆ G, ( X h P h ! ˆ h ij X e P e ! + X e P e ! ˆ h ij X h P h !) (A.6)where X h P h ! ˆ h ij X e P e ! = X nσ X he t en,hnij α + iσ ( en ) β jσ ( hn ) , X e P e ! ˆ h ij X h P h ! = X nσ X he t ne,nhij β + iσ ( nh ) α jσ ( ne )(A.7) and ˆ G = X nhe " t en,hnij ∆ nhe X σ α + iσ ( en ) β jσ ( hn ) −− t nh,neij ∆ n he X σ β + iσ ( nh ) α jσ ( ne ) (A.8)with the energy denominator is ∆ nhe = ( ε e + ε h ) − ε n .The effects of the ligand environment of magnetic ionsare taken into account, due to the Wannier oxygen cellfunctions, as well as the exact diagonalization procedurewhen constructing the configuration space of the cell | n i | h ( e ) i states with energies ε n and ε e ( h ) respectively. In agreement with the relations:ˆ n ( e ) inσ = (2 S h + 1) β ( t )+ iσ ( nh ) β ( t ) iσ ( hn ) ,n ( h ) inσ = (2 S n + 1) α ( s ) iσ ( ne ) α ( s )+ iσ ( en ) (A.9) S + in = (cid:26) (2 S h + 1) β + i ↑ ( nh ) β i ↓ ( hn ) = (2 S n + 1) α ↓ ( ne ) α + ↓ ( en ) , S n = S h + | σ | ; S e = S n + | σ |− (2 S h + 1) β + i ↑ ( nh ) β i ↓ ( hn ) = − (2 S n + 1) α i ↓ ( ne ) α + i ↓ ( en ) , S n = S h − | σ | ; S e = S n − | σ | (A.10)ˆ S zin = (2 S h + 1) P σ η ( σ ) β + iσ β iσ = (2 S n + 1) P σ η ( σ ) α iσ α + iσ , S n = S h + | σ | ; S e = S n + | σ |− (2 S h + 1) P σ η ( σ ) β + iσ β iσ = − (2 S n + 1) P σ η ( σ ) α iσ α + iσ , S n = S h − | σ | ; S e = S n − | σ | and assuming that the ground state | n i = | n i is occupiedat T = 0 K , and the superexchange Hamiltonian takes theform:ˆ H s = X i = j ˜ h ij = X i = j (cid:26) J − ij ˆ S in ˆ S jn − J + ij ˆ n ( e ) in ˆ n ( h ) jn (cid:27) (A.11)where J − ij = X he ′ J ij ( h, n , e )(2 S h + 1) (2 S n + 1) − X he ′′ J ij ( h, n , e )(2 S h + 1) (2 S n + 1) , (A.12)and J + ij = X he ′ J ij ( h, n , e )(2 S h + 1) (2 S n + 1) ++ X he ′′ J ij ( h, n , e )(2 S h + 1) (2 S n + 1) (A.13) and ˆ n ( e ) in = P σ ˆ n ( e ) in σ , ˆ n ( h ) in = P σ ˆ n ( h ) in σ . Since in the firstcontribution ( P he ′ ... ) the exchange loops are summed with S h = S e , and in the second one ( P he ′′ ... ), the exchangeloops are with S h = S e ±
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