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Anisotropic Superexchange in low-dimensional systems: Electron Spin Resonance

Anisotropic Superexchange in low-dimensional systems: Electron Spin Resonance. Dmitry Zakharov Experimental Physics V Electronic Correlations and Magnetism University of Augsburg Germany. Motivation. Anisotropic Exchange Dominant source of anisotropy for S=1/2 systems

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Anisotropic Superexchange in low-dimensional systems: Electron Spin Resonance

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  1. Anisotropic Superexchangein low-dimensional systems:Electron Spin Resonance Dmitry Zakharov Experimental Physics V Electronic Correlations and Magnetism University of Augsburg Germany

  2. Motivation • Anisotropic Exchange • Dominant source of anisotropy for S=1/2 systems • Produces canted spin structures • Ising or XY model are limit cases • Can be estimated by Electron Spin Resonance (ESR) • Electron spin resonance • Microscopic probe for local electronic properties • Ideally suited for systems with intrinsic magnetic moments • Spin systems of low dimensions • Variety of ground states different from 3D order e.g. spin-Peierls, Kosterlitz-Thouless • Short-range order phenomena and fluctuations at temperatures far above magnetic phase transitions

  3. Outline • Basic theory of anisotropic exchange • Introduction to electron spin resonance (ESR) • Full microscopical picture of the symmetric anisotropic exchange: NaV2O5 • Temperature dependence of the ESR linewidth in low-dimensional systems: NaV2O5, LiCuVO4, CuGeO3, TiOCl Outline

  4. Isotropic superexchange Two magnetic ions can interact indirect via an intermediate diamagnetic ion (O2-, F-,..)  potential exchange: like direct exchange describes the self-energy of the charge distribution → ferromagnetic; • kinetic exchange: the delocalized electrons can hop, what leads to the stabilization of the singlet state over the triplet: → antiferromagnetic spin ordering can be described through the perturbation treatment: Basic theory of anisotropic exchange

  5. Mechanism of anisotropic exchange interaction The free spin couples to the lattice via the spin-lattice interaction HLS=l(l·s)  the excited orbital states are involved in the exchange process  can be described as virtual hoppings of electrons via the excited orbital states (the additional perturbation term – (LS)-coupling – acts on one site between the orbital levels) • This effect adds to the isotropic exchange interaction an anisotropic part (dominant source of anisotropy for S=½ systems!) Basic theory of anisotropic exchange

  6. Theoretical treatment Clear theoretical description can be carried out in the framework of the perturbation theory: • Fourth order: describes 4 virtual electrons hoppings  Isotropic superexchange • Fifth order: 4 hoppings + on-site (LS)-coupling  Antisymmetric part of anisotropic exchange = Dzyaloshinsky-Moriya interaction • Sixth order: 4 hoppings + 2 times on-site (LS)-coupling  Symmetric part of anisotropic exchange = Pseudo-dipol interaction Basic theory of anisotropic exchange

  7. ra rb sa sb Antisymmetric part of anisotropic exchange There is a simple geometric rule allowed to determine the anisotropy produced by Dzyaloshinsky-Moriya interaction: Spin variables are going into the Hamiltonian of the antisymmetric exchange in form of a cross-product: j = {x, y, z}, h, j– orbital levels, Dhj– energy splitting, lj – operator of the LS-coupling, J – exchange integral. The direction of D (Dzyaloshinsky-Moriya vector) can be determined from: It should be no center of inversion between the ions! Basic theory of anisotropic exchange

  8. Symmetric part of anisotropic exchange Exchange constant of the pseudo-dipol interaction is a tensor of second rankG(ab)and does not allow a simple graphical presentation. a, b = {x, y, z}; h, z, z’– orbital levels.  Nonzero elements of G(ab)can be determined by the nonnegligible product of the matrix elements of the (LS)-coupling and the hopping integrals. Basic theory of anisotropic exchange

  9. Outline • Basic theory of anisotropic exchange • Introduction to electron spin resonance (ESR) • Full microscopical picture of the symmetric anisotropic exchange: NaV2O5 • Temperature dependence of the ESR linewidth in low-dimensional systems: LiCuVO4, CuGeO3, TiOCl How to study all this?

  10. E SZ = +1/2   L SZ = -1/2 E SZ = +1/2 Hres H SZ = -1/2 H Zeeman effect Zeeman energy in magnetic field H: eigen energies of the spinSZ= 1/2 magnetic microwave field^HwithE = h induces dipolar transitions Introduction to electron spin resonance

  11. Experimental Set-Up microwave source 9 GHz diode ESR signal sample magnet 0...18 kOe resonator microwave field <1Oe Introduction to electron spin resonance

  12. ESR quantities: intensity: local spin susceptibility resonance field: ħ=gBHres g = g- 2.0023 local symmetry linewidth H: spin relaxation, anisotropic interactions ESR signal Introduction to electron spin resonance

  13. Theory of line broadening Hamiltonian for strongly correlated spin systems: Zeeman energy isotropic exchange additional couplings • Strong isotropic coupling • averages local fields like in the case of fast motion of the spins • Narrowing of the ESR signals • Local fluctuating fields • local, statistic resonance shift • inhomogeneous broadening of the ESR signal Introduction to electron spin resonance

  14. Possible mechanisms of the ESR-line broadening Only the following mechanisms are dominant in concentrated low-dimensional spin systems: • Crystal field is absent forS = ½ (topic of this work) • Anisotropic Zeeman interaction negligible in case of nearly equivalent g-tensors on all sites; characteristic value of DH ~ 1 Oe • Hyperfine structure & Dipol interaction characteristic broadening about DH~10 Oe as result of the large isotropic exchange • Relaxation to the lattice produces a divergent behavior of DH(T) • Anisotropic exchange interactionsare themain broadening sources of the ESR line[R. M. Eremina.., PRB 68, 014417 (2003)] [Krug von Nidda.., PRB 65, 134445(2002)] Introduction to electron spin resonance

  15. Theoretical approach Linewidth of the exchange narrowed ESR line in the high-temperature approximation (T ≥J ): Schematic representation of the „exchange narrowing“ [R.Kubo et al., JPSJ 9, 888 (1954)] Second moment of a line: Introduction to electron spin resonance

  16. Outline • Basic theory of anisotropic exchange • Introduction to electron spin resonance (ESR) • Full microscopical picture of the symmetric anisotropic exchange: NaV2O5 • Temperature dependence of the ESR linewidth in low-dimensional systems: LiCuVO4, CuGeO3, TiOCl Let‘s start at last!

  17. c a VO5 b V4.5+ O2- one electron S = 1/2 Na ladder 1 ladder 2 NaV2O5 structure Full microscopical picture of AE: NaV2O5

  18. NaV2O5 susceptibility / ESR linewidth • One-dimensional system at T > 200 K; • Charge-ordering fluctuations 34K<T<200K; • “Zigzag” charge ordering at TCO= 34 K; • ESR linewidth at T > 200 K is about 102 Oe Full microscopical picture of AE: NaV2O5

  19. ra rb sb sa Antisymmetric vs. symmetric exchange Dzyaloshinsky-Moriya interactionPseudo-dipol interaction Dzyaloshinsky-Moriya interaction is negligible because of two almost equal exchange paths which calcel each other Standard mechanism by Bleaney & Bowers is not effective due to the orthogonality of the orbital wave functions What is the broadening source of the ESR line?! Full microscopical picture of AE: NaV2O5

  20. Conventional anisotropic exchange processes [B. Bleaney and K. D. Bowers, Proc. R. Soc. A 214, 451 (1952)] Full microscopical picture of AE

  21. AE with the spin-orbit coupling on both sites are not so effective because of the larger energy in denominator Dhj [Eremin, Zakharov, Eremina…, PRL 96, 027209 (2006)] Full microscopical picture of AE

  22. AE with hoppings between the excited levels is of great importance in chain systems due to the big hopping integrals thx and tjz between the nonorthogonal orbital levels [Eremin, Zakharov, Eremina…, PRL 96, 027209 (2006)] Full microscopical picture of AE

  23. Schematic pathways of intra-ladder AE Only one type of the anisotropic exchange – pseudo-dipol interaction with electron hoppings between the excited orbital levels – is possible in the ladders of NaV2O5 G(zz) – dominant! ground states excited states Full microscopical picture of AE: NaV2O5

  24. Schematic pathways of inter-ladder AE Instead, the “conventional” exchange mechanisms are dominant for the exchange of the spins from the different ladders Full microscopical picture of AE: NaV2O5

  25. Estimation of the exchange parameters Theoretical description of the angular dependence of the ESR linewidth by the moments method allows to determine the parameter of the dominant exchange path at high temperatures G(zz)≈ 5 K in good agreement with the estimations based on the values of hopping integrals and crystal-field splittings Temperature dependence of DH clearly shows the development of the charge-ordering fluctuations at T < 200 K [Eremin.., PRL 96, 027209 (2006)] Full microscopical picture of AE: NaV2O5

  26. Temperature dependence of DH in NaV2O5 Are there other systems to corroborate these findings? Which temperature dependence of the ESR linewidth is characteristic for the symmetric and antisymmetric part of anisotropic exchange in low-dimensional systems? Open questions

  27. Outline • Basic theory of anisotropic exchange • Introduction to electron spin resonance (ESR) • Full microscopical picture of the symmetric anisotropic exchange: NaV2O5 • Temperature dependence of the ESR linewidth in low-dimensional systems: NaV2O5, LiCuVO4, CuGeO3, TiOCl → Empirical answer!

  28. LiCuVO4 CuGeO3 NaV2O5 Temperature dependence of the ESR linewidth DH(T) in low-dimensional systems

  29. Universal temperature law DH(T) in low-dimensional systems

  30. Theoretical predictions High-temperature approximation fails for T<J (!) Field theory (M. Oshikawa and I. Affleck, Phys. Rev. B 65, 134410, 2002): (1) if only one interaction determines the linewidth: H (T, , ) = f (T ) · H (T , , ) linewidth ratio independent of temperature (2) low temperatures T<<J : H (T ) ~ T for symmetric anisotropic exchange H (T ) ~ 1/T 2 for antisymmetric DM interaction  in LiCuVO4, CuGeO3 and NaV2O5 symmetric anisotropic exchange dominant DH(T) in low-dimensional systems

  31. LiCuVO4 spin fluctuations (T > TN= 2.1 K) Linewidth ratio: deviations from universality → (1): if only one interaction determines the linewidth: H (T, , ) = f (T ) · H (T , , )  linewidth ratio independent of temperature CuGeO3 • lattice fluctuations • (T > TSP= 14.3 K) NaV2O5 • charge fluctuations • (T > TCO= 34 K) DH(T) in low-dimensional systems

  32. Universal behavior of the linewidth →(2): low temperatures T << J : H (T) ~ T for symmetric anisotropic exchange H (T) ~ 1/T 2 for antisymmetric DM interaction Is it possible to find a system with a large antisymmetric interaction and a high isotropic exchange constant J to observe a low-temperature 1/T2 divergence due to this interaction? DH(T) in low-dimensional systems

  33. TiOCl • There is no center of inversion between the ions in the Ti-O layers  Strong antisymmetric anisotropic exchange • Isotropic exchange constant J = 660 K [A. Seidel et al., Phys. Rev. B 67, 020405(R) (2003)] DH(T) in low-dimensional systems: TiOCl

  34. Analysis of the anisotropic exchange mechanisms Dzyaloshinsky-Moriya interactionPseudo-dipol interaction • D is almost parallel to the b direction • Dominant component of the tensor of the pseudo-dipol interaction is G(aa) DH(T) in low-dimensional systems: TiOCl

  35. Temperature dependence of DH The temperature and angular dependence of DH can be described as a competition of the symmetric and the antisymmetric exchange interactions! [Oe]KAE(∞)KDM(∞) H|| a 1429 1.397 H || b 765 2.319 H ||c930 1.344 [Zakharov et al., PRB 73, 094452 (2006)] DH(T) in low-dimensional systems

  36. Summary • Anisotropic exchange dominates the ESR line broadening in low dimensional S=1/2 transition-metal oxides • Unconventional symmetric anisotropic superexchange in NaV2O5 • Universal temperature dependence of the ESR linewidth in spin chains with dominant symmetric anisotropic exchange • Interplay of antisymmetric Dzyaloshinsky-Moriya and symmetric anisotropic exchange in TiOCl Summary

  37. Acknowledgements • Crystal growth NaV2O5: G. Obermeier, S. Horn (C1, Augsburg) TiOCl: M. Hoinkis, M. Klemm, S. Horn, R. Claessen (B3, C1, Augsburg) LiCuVO4: A. Prokofiev, W. Assmus (Frankfurt) CuGeO3: L. I. Leonyuk (Moscow) • German-russian cooperation (DFG and RFBR) M. V. Eremin (Kazan State University) R. M. Eremina (Zavoisky Institute, Kazan) V. N. Glazkov (Kapitza Institute, Moscow) L. E. Svistov (Institute for Crystallography, Moscow) • ESR group, Experimental Physics V (Prof. A. Loidl) H.-A. Krug von Nidda, J. Deisenhofer Thanks for your attention!

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  41. Direct exchange Exchange interaction is a manifestation of the fact that, because of the Pauli principle, the Coulomb interaction can give rise to the energies dependent on the relative spin orientations of the different electrons in the system. In case of the non negligible direct overlap of the wave functions jiof two neighbouring atoms, they should be modified because of the Pauli principle • Modification of the Hamiltonian: J – „overlap integral“. Direct exchange always stabilizes the triplet over the singlet according to the Hund‘s rule, favoring a ferromagnetic pairing of the electrons. Basic theory of anisotropic exchange

  42. LiCuVO4 structure / susceptibility Cu2+S = 1/2 chains along b orthorhombically distorted inverse spinel DH(T) in low-dimensional systems: LiCuVO4

  43. x y b-axis +Dab px Cu + - O -Dab O - - + + + + - - ra rb dxy O dx2-y2 - Cu(i) ground state Cu(j) excited state sb sa + py Antisymmetric vs. symmetric exchange Dzyaloshinsky-Moriya interactionPseudo-dipol interaction Antisymmetric exchange is NOTpossible in LiCuVO4 (!) Ring-exchange geometry strongly intensifies the pseudo-dipol exchange! DH(T) in low-dimensional systems: LiCuVO4

  44. Angular dependence of DH ring-exchange geometry high symmetric anisotropic exchange theoretically expectedJcc  2K DH(T) in low-dimensional systems: LiCuVO4

  45. O2- J J12 Cu2+ CuGeO3 structure / susceptibility 2 Cu2+S = 1/2 chains along c J12 0.1J T > TSP: (T ) not like Bonner-Fisher T < TSP: (T ) ~ exp{-(T )/T } DH(T) in low-dimensional systems: CuGeO3

  46. Antisymmetric vs. symmetric exchange Dzyaloshinsky-Moriya interactionPseudo-dipol interaction • Intra-chain geometry is the same as with LiCuVO4 D≡ 0G(zz) - dominant • Inter-chain exchange: ? G(yy)(Fig.a) and G(xx) (Fig.b) are not negligible DH(T) in low-dimensional systems: CuGeO3

  47. ESR anisotropy in CuGeO3 intra chain contribution inter chain contribution DH(T) in low-dimensional systems: CuGeO3

  48. Empty DH(T) in low-dimensional systems

  49. LiCuVO4 Cu2+ S = 1/2 chain J = 40 K TN = 2.1 K antiferromagnetic order NaV2O5 S = 1/2 per 2 V4.5+ ¼-filled ladder J = 570 K TCO = 34 K dimerization via charge order Model systems CuGeO3 Cu2+ S = 1/2 chain J = 120 K TSP = 14 K dimerized, spin-Peierls S = 0 ground state Introduction to electron spin resonance

  50. LiCuVO4 ga= 2.07 gb= 2.10 gc= 2.31 Cu2+ 3d9: g-2 > 0 highest g-value for H || c longest Cu-O bond NaV2O5 ga= 1.979 gb= 1.977 gc= 1.938 V4.5+ 3d0.5: g-2 < 0 strongest g-shift for H || c c c Resonance field, g-values - local symmetry CuGeO3 ga= 2.16 gb= 2.26 gc= 2.07 sum of two tensors local symmetry like in LiCuVO4 Introduction to electron spin resonance

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