Model for ferromagnetism in d 0 materials .

1. Model for ferromagnetism in d0 materials. Georges Bouzerar Laboratoire Louis Néel CNRS Grenoble Introduction. Model for «d0» ferromagnetism. Magnetic exchange couplings and treatment of the effective Heisenberg Hamiltonian. Conclusions. Conference « Self organized strongly correlated electron systems » in Seillac 2006 Collaboration: T.Ziman (ILL Grenoble)

2. Introduction. I-What is «d0»  ferromagnetism ? • « d0 »  ferromagnetism = (i) Unusual form of ferromagnetism which do not require the presence of magnetic ions or (ii) non magnetic defect induced ferromagnetism. In these materials d or f shells of anions/cations are either (i) empty or (ii) completely filled. • Magnetic Moments induced dynamically. • Observation or prediction (based on ab-initio calculations) of ferromagnetism with very high Curie temperature in several materails: • dielectric oxides:CaO, HfO2, TiO2, ZnO (??),... [1-5]. • irradiated graphite[6]. • fullerenes[7]. • hexaborideCaB6[8]. • …… [1] I.S. Elfimov, S. Yunoki and G.A. Sawatzky, Phys. Rev. Lett. 89 216403 (2002). [2] M. Venkatesan et al. Nature (London) 430, 630 (2004), J.M.D. Coey et al., Phys. Rev. B, 72 024450 (2005). [3] S.R. Shinde et al. , Phys. Rev. B, 67 115211 (2003), K.A. Griffin et al. , Phys. Rev. Lett., 94 157204 (2005). [4] C. D. Pemmaraju and S. Sanvito, Phys. Rev. Lett., 94 217205 (2005). [5] J. Osorio-Guillén, S. Lany, S. V. Barabash, and A. Zunger, Phys. Rev. Lett., 96 107203 (2006). [6] P. Esquinazi et al. Phys. Rev. Lett., 87 227201 (2003). [7] TL Makarova et al., Nature 413 718 (2002). [8] R. Monnier and B. Delley, Phys. Rev. Lett., 91 157204 (2001).

3. II-Which are the relevant defects? A-First principle calculations. • The ab-initio calculations done for HfO2[Pemmaraju et al. PRL 2005] and for CaO[Elfimov et al. PRL 2002] suggest the crucial role of cationic vacancies (Hf4+ et Ca2+ ) : appearance of an almost localized magnetic moment on the oxygen atoms neighboring the vacancy. • In contrast, the case of oxygen vacancy (anionic vacancy) leads to the absence of a magnetic moment. • Very recently Osorio-Guillén et al. (PRL 2006) extended the study ofElfimov et al.Using first principle approach (based on pseudo potential method + GGA)have shown (i) that at equilibrium the solubility of cationic vacancies in CaO can not exceed 0.030 % and (ii) that he calculated exchange couplings are very short range, thus vacancy induced ferromagnetism in CaO should not be possible. Density of states in HfO2 (Pemmaraju et al.) Exchange couplings in CaO for 3.5% of vacancies (Osorio-Guillen et al.) Density of states in CaO(Elfimov et al.)

4. B-Experimental observations. • No clear identification of the defects responsible for the feroomagnetism. • Strong sensitivity to preparation conditions: recent experiments (LLN-Grenoble by R. Galéra, L.Ranno)performed on HfO2 under similar conditions as those reported by Coey et al. have shown that the samples were non magnetic . • The studies performed by Coey et al. on HfO2indicate that the induced magnetic moments appear to be located close to the interface between the material and substrate All these remarks underline the difficulty to control the ferromagnetism induced by intrinsic defects. Open issues: • Which are the defects responsible for the intrinsic ferromagnetism? • Which mechanism leads to ferromagnetism (nature of the couplings)? • Why are the reported Curie temperature systematically above room • temperature? • How could we control the d0 ferromagnetism? Is there some alternative? • ………

5. Model for d0 Ferromagnetism. Hubbard model (strongly correlated electron systems) + disorder to simulate the presence of cationic vacancies (defects in general) G.B. and T. Ziman PRL 2006. 2D Projection Sites i correspond to oxygen orbitals Each defect Dj is surrounded by 8 oxygen atoms (sc lattice).

6. Many body problem in presence of «correlated disorder» : require an appropriate treatment of the disorder: Exact Diagonalisation approach is impossible (Huge Hilbert space). Approximations: Single band + Unrestricted Hartree-Fock (UHF). UHF treatment works well for groundstate properties in presence of disorder. • Remarks: • Spins up/down are coupled: simultaneous diagonalisation in each spin sectors. • Disorder is treated exactly. • Local potential depends both on (i) spin and (ii) density of the itinerant carrier. • For a chosen disorder configuration the problem is solved self-consistently

7. Summary of the self-consistent procedure Choice of a configuration for the positions of the vacancies (defects), and fixed concentration of carriers and defects. convergence Determination of transport properties and magnetic properties (exchange couplings,…)

8. Exchange couplings and treatment of the effective Heisenberg Hamiltonian I-Determination of the exchange couplings. Exchange between 2 spins belonging to 2 different « clusters » Liechtenstein et al., PRB (1995) The total spin around a defect (cluster) The total exchange coupling between two classical spins is,

9. II-Treatment of the effective Heisenberg Hamiltonian. The problem is now mapped to the Heisenberg model Note that the couplings Jij(x,nh,U,V) It is necessary to treat (i) the effect of disorder (dilute system) and (ii) thermal fluctuations accurately. SC-LRPA approach: (1) Disorder is treated exactly and (2) Thermal fluctuations beyond a meanfield treatment within local RPA approximation [1-5]. [1] G. B.,T. Ziman and J. Kudrnovsky, Europhys. Lett., 69 812 (2005). [2] G. B., T; Ziman and J. Kudrnovsky, APL 85, 4941 (2004). [3] G. B., T; Ziman and J. Kudrnovsky PRB 72, 125207 (2005) [4] R. Bouzerar, G.B. and T. Ziman , Phys. Rev. B 73, 024411 (2006 ) [5] G.B., R. Bouzerar,T. Ziman and J. Kudrnovsky, PRB submited (2006)

10. III-SC-LRPA Approach The SC-LRPA approach has several advantages: • Allows us to treat simultaneously and on equal footings the thermal fluctuations and the effects of disorder (localization of the magnetic excitations) • It leads to an semi analytical expression for TC. • Extremely fast and easy to implement. With respect to Monte Carlo simulations it is 3 orders of magnitude faster. • For a given temperature T, one can calculate: • The distribution of the local magnetizations mi (T) . • The spectral function A(q,w,T) and the magnetic excitation spectrum. • The spin-spin correlation functions < Si .Sj >. • etc. ………

11. IV-Results and discussions. A-Induced moments: U=0.4W x=0.04 and g=3. DOS Spin density Charge density V EF __ up __down Magnetic moments around the defect Hole rich region around the defect • For fixed U a local moment appear on oxygen sites for sufficiently large V. • We observe simultaneously a phase separation in charge space. • The critical value depends on U , x et nh : Vc(U,x,nh).

12. B- Exchange couplings: U=0.4W x=0.04 and g=3. • Couplings are relatively of short range, they are similar to the those obtained by TB-LMTO in the diluted magnetic semiconductor GaMnAs or calculated in the V-Jpd model see Poster R. Bouzerar. • For small V the nearest neighbor coupling J1 is antiferromagnetic and oscillations can be observed. • The exchange increase with V and become more and more ferromagnetic • After a maximum is reached for V=0.45 W the couplings decrease.. • For large V couplings become AF: superexchange mechanism become dominant :frustration, spin glass.

13. C- Curie Temperature: effect of V. We are interested here in the effect of the impurity band position. We choose our parameters to fit ab-initio calculations for the host band bandwith of HfO2 and consider a reasonnable value for the Hubbard parameter : W =7 eV and U=2.5 eV. • Below V=0.3 W no ferromagnetism. • We observe a maximum of TC=750 K for V=0.45 W • There is a window for V for which TC reaches values beyond room temperature (impurity band is preformed but not yet separated from the valence bande) • For larger values of V TCdisappear , the frustration effects become important. • Weak variation of TCwith U (not shown) Absence of magnetic moments Superexchange mechanism dominates (frustration)

14. D- Curie Temperature: effects of the carrier density . We now study the effect of changing the density of carrier for fixed density of defects x=0.04 and impurity band position (V= 0.40 W and U=0.40 W). • Below nh/x =2.5 and beyond nh/x =4we find no ferromagnetic phase. • We observe a maximum of TC=500 K fornh/x =3. • There is a window for the carrier density for which TC is finite and can exceed room temperature. • On the basis of our model, our calculation predicts that CaO exhibits effectively a localized moment but no ferromagnetism • HfO2 case is very close lto the zone boundary of stability of the ferromagnetic phase. 3 1 2 4

15. Conclusion. • This study shows that ferromagnetism is indeed possible in a priori non-magneticinsulating oxides, that the exchange couplings are of relatively short range • Curie temperature far beyond room temperature can be reached. • The Curie temperature TC is essentially controlled by two relevant parameters: V and nh/x Optimal situation: • The value of the parameter V is such that the impurity band remains sufficiently close to the valence band. •The density of « itinerant » carriers per defect should be of order nh/x =3.

16. Good candidates for d0 ferromagnetism: A4+1-xA’+xO2 Li A’ A Na K Ti Rb Zr Cs Hf • From the experimental point of vue a crucial point concerns the solubility of the non magnetic A’ cation in AO2 The support of first principle calculations are of great importance. • Remark: • Some very recent calculations performed by the ab-initio group in Prague ( Kudrnovsky,Drchal, Maca) have already confirmed the existence of a well defined moment in ZrKO2 and TiKO2

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