1. Blaubeuren 2006 Relaxation mechanisms in exchange coupled spin systems – I Line broadening and the Kubo-Tomito approach Joachim Deisenhofer Université de Genève Blaubeuren 6th September 2006

2. Blaubeuren 2006 Outline ESR in exchange-coupled system: definition of the problem linear response and ESR: the Kubo-Tomita approach the effective spin Hamiltonian and the broadening mechanisms summary

3. Blaubeuren 2006 Motivation - ESR in transition-metal compounds ESR locally probes the spin of interest low-dimensional spin systems –S = 1/2 chains: LiCuVO 4, CuGeO 3, TiOCl - and ladders: NaV 2 O 5 –chains with larger spin PbNi 2 V 2 O 8 (S = 1), (NH 4 ) 2 MnF 5 (S =2) –2D honeycomb-lattice BaNi 2 V 2 O 8 metal-insulator-transitions –heavy-fermion-properties in Gd 1-x Sr x TiO 3 –spin-state transitions in GdBaCo 2 O 5+  materials with colossal magnetoresistance effect –magnetic structure in thiospinels FeCr 2 S 4, MnCr 2 S 4 –orbital ordering in La 1-x Sr x MnO 3

4. Blaubeuren 2006 parameters: intensity: local spin susceptibility resonance field:  g = g - 2.0023 local symmetry linewidth  H: spin-relaxation, anisotropic interactions The absorption signal

5. Blaubeuren 2006 Why one Lorentzian only? “Exchange narrowing” isotropic exchange interaction between neighbouring spins causes local field fluctuations originally Gaussian lineshapes are therefore “narrowed“ in the middle and extendend in the wings phenomenon similar to “motional narrowing“ in liquids as seen by NMR

6. Blaubeuren 2006 What are we measuring? - Definition of the problem usual case: Faraday configuration; microwave field perpendicular to static magnetic field couples to the spin Microwave absorption in the cavity is given by Problem: What is the (linear) response function of the spin system to the microwave perturbation?

7. Blaubeuren 2006 ESR in Linear Response – ingredients observed quantity: time-dependent perturbation: general spin susceptibility: Heisenberg equation of motion: Hamiltonian of the spin system:

8. Blaubeuren 2006 The effective spin Hamiltonian Hamiltonian for strongly correlated spin systems: Zeeman energy isotropic exchange additional interactions strong isotropic coupling  averages local fields similar to fast movements of the spins  “ exchange narrowing“ of the ESR signal local fluctuating fields  local, static resonance shift  inhomogenous broadening of the ESR signal e.g. crystal field anisotropic exchange dipole-dipole interaction hyperfine interaction

9. Blaubeuren 2006 final general expression: with lineshape is completely determined by : resonance line infinitely sharp at ω=ω 0 (δ –peak) assuming a Lorentzian lineshape and gives linewidth: and resonance shift: Where does linear response theory take us? Oshikawa and Affleck, PRB 65, 134410 (2002) Nagata and Tatsuke, J. Phys. Soc. Jpn. 32, 337 (1972)

10. Blaubeuren 2006 The Kubo-Tomita formula for the linewidth T  ∞: high-temperature approximation correlation function decays exponentially: characteristic time is governed by isotropic exchange J: equivalent expression: R. Kubo and K. Tomita, J. Phys. Soc. Jpn. 9, 888 (1954)

11. Blaubeuren 2006 Characteristics of the Kubo-Tomita approach - I temperature dependence is governed by the static susceptibility describes many 3D magnetic systems, e.g. manganites Causa et al., PRB 58, 3233 (1998)

12. Blaubeuren 2006 Characteristics of the Kubo-Tomita approach - II anisotropies completely contained in the second Moment M 2 : remaining tasks: calculate the second moment for the different contributions to the spin Hamiltonian, find the dominating line-broadening mechanism (and check for the anisotropy)

13. Blaubeuren 2006 The effective spin Hamiltonian Hamiltonian for strongly correlated spin systems: Zeeman energy isotropic exchange additional interactions strong isotropic coupling  averages local fields similar to fast movements of the spins  “ exchange narrowing“ of the ESR signal local fluctuating fields  local, static resonance shift  inhomogenous broadening of the ESR signal e.g. crystal field anisotropic exchange dipole-dipole interaction hyperfine interaction

14. Blaubeuren 2006 Line broadening mechanisms: Zero-Field Splitting (crystal field) Hyperfine interaction Dipole-dipole interaction Anisotropic Zeeman interaction Anisotropic exchange interactions: Symmetric anisotropic exchange Antisymmetric anisotropic exchange (Dzyaloshinsky-Moriya interaction)

15. Blaubeuren 2006 Crystal field – zero-field splitting two independent elements for orthorhombic symmetry: perturbation theory (spin-orbit coupling perturbs the crystal-field levels): important source of line broadening and anisotropy (e.g. in manganites)

16. Blaubeuren 2006 Anisotropic Zeeman interaction Different Cu sites give rise to superposition of resonance lines, e.g. in CuGeO 3 (1D Heisenberg spin-chain) exchange coupling along the b-direction yields then only one broadened resonance line Second moment depends on the external field!  good candidate for field-dependent linewidth B. Pilawa, J. Phys.: Cond. Mat. 9, 3779 (1997)

17. Blaubeuren 2006 Line broadening mechanisms: Zero-Field Splitting (crystal field) Hyperfine interaction Dipole-dipole interaction Anisotropic Zeeman interaction Anisotropic exchange interactions: Symmetric anisotropic exchange Antisymmetric anisotropic exchange (Dzyaloshinsky-Moriya interaction)

18. Blaubeuren 2006 magnetic ions interact indirectly via an intermediate diamagnetic ion (O 2-, F 1-,..) potential exchange: describes the self-energy of the charge distribution → ferromagnetic kinetic exchange: electrons can hop, stabilization of the singlet over the triplet state : → antiferromagnetic perturbation treatment: Reminder: Isotropic superexchange

19. Blaubeuren 2006 Mechanism of anisotropic exchange interaction This effect adds to the isotropic exchange interaction an anisotropic part (dominant source of anisotropy for S=½ systems!) free spin couples to the lattice via the spin-orbit interactionH LS = (l·s)  excited orbital states are involved in the exchange process  described as virtual hoppings of electrons via the excited orbital states (additional perturbation term – (LS)-coupling – acts on one site between the orbital levels)

20. Blaubeuren 2006 Theoretical treatment - perturbation theory 4th order: describes 4 virtual electrons hoppings  Isotropic superexchange 5th order: 4 hoppings + on-site (LS)-coupling  Antisymmetric part of anisotropic exchange = Dzyaloshinsky-Moriya interaction 6th order: 4 hoppings + 2 times on-site (LS)-coupling  Symmetric part of anisotropic exchange = Pseudo-dipol interaction

21. Blaubeuren 2006 Effective spin Hamiltonian of the antisymmetric exchange in form of a cross-product: direction of D (Dzyaloshinsky-Moriya vector) Perturbative result: Antisymmetric part of anisotropic exchange sasa sbsb rara rbrb j = {x, y, z},  – orbital levels,   – energy splitting, l j – operator of the LS-coupling, J – exchange integral.

22. Blaubeuren 2006 Symmetric part of anisotropic exchange ,  = {x, y, z};  ’ – orbital levels. Effecti ve exchange constant of the pseudo-dipol interaction is a tensor of second rank    and does not allow a simple graphical presentation: Nonzero elements of    can be determined by the product of the matrix elements of the (LS)-coupling and the hopping integrals.

23. Blaubeuren 2006 Uffhh – How to sort out things? Things are not as bad as they might look like: Second moments of the broadening mechanisms are known Geometry and order of magnitude estimations often allow to single out the dominant interactions Within the KT-approach the temperature dependence is the same for all spin-spin interactions  anisotropies are “very helpful” But… keep in mind: lots of questions for one single broad absorption line justification of the KT- approach (dimension)? possibility of spin-lattice (phonon) relaxation

24. Blaubeuren 2006 summary and outlook High-temperature KT-approach describes well 3D magnets Lineshape completely determined by perturbation term Dominating relaxation mechanism has to explain order of magnitude and anisotropy Tomorrow: Success of the KT-approach: the case of manganites Limits of the KT approach - the case of linear spin chains: e.g. NaV 2 O 5, LiCuVO 4, CuGeO 3, TiOCl

25. Blaubeuren 2006 Relaxation mechanisms in exchange coupled spin systems – II Two case files: the manganite system La 1-x Sr x MnO 3 and ESR in spin chain compounds Joachim Deisenhofer Université de Genève Blaubeuren 7th September 2006

26. Blaubeuren 2006 outline Part 1: the manganite system La 1-x Sr x MnO 3 Phase diagram of La 1-x Sr x MnO 3 ESR results spin relaxation in La 0.95 Sr 0.05 MnO 3 : orbital ordering Part 2: ESR in 1D Heisenberg spin chains Magnetic properties of spin S = 1/2 chains temperature dependence of the ESR linewidth in the model systems NaV 2 O 5, LiCuVO 4, CuGeO 3 the case of TiOCl

27. Blaubeuren 2006 phase diagram of La 1-x Sr x MnO 3 Hemberger et al., PRB 66, 094410 (2002).

28. Blaubeuren 2006 Jahn-Teller distortion in LaMnO 3 cooperative JT-distortions of MnO 6 octahedra in the orthorhombic O'- phase up to T JT  750 K A-type AFM (T N = 140 K): FM coupling in ac-planes AFM coupling between ac-planes Huang et al., PRB 55, 14987 (1997)

29. Blaubeuren 2006 estimate for  JT ? (~0.1eV instead of ~1eV) orbital excitations (orbitons) in LaMnO 3 or multiphonon processes? OO due to electron-phonon and/or electron-electron interaction? What is the value of  ? Suggestions: 90°, 102°, 106°, 120° orbital physics in LaMnO 3 OO ? x y z x y z

30. Blaubeuren 2006 parameters: intensity: local spin susceptibility resonance field:  g = g - 2.0023 local symmetry linewidth  H: spin-relaxation, anisotropic interactions the ESR Signal 

31. Blaubeuren 2006 ESR in La 1-x Sr x MnO 3 (0  x  0.2) Ivanshin et al., PRB 61, 6213 (2000) phase diagram Paraskevopoulos et al., J. Phys.:Cond.Mat. 12, 3993 (2000). linewidth La 0.95 Sr 0.05 MnO 3 : still an AFM insulator (T N = 140 K) reduced JT transition temperature T JT = 600 K  effective treatment like LaMnO 3

32. Blaubeuren 2006 T-dependence of the linewidth

33. Blaubeuren 2006 Line broadening mechanisms: Observed linewidth: ~ 1-2 kOe Zero-Field Splitting (crystal field): ~1 kOe Anisotropic exchange interactions: Symmetric anisotropic exchange ~1 Oe Antisymmetric anisotropic exchange ~1 kOe (Dzyaloshinsky-Moriya interaction) Hyperfine interaction: ~10 Oe Dipole-dipole interaction ~1 Oe Anisotropic Zeeman interaction ~1 Oe

34. Blaubeuren 2006 Crystal field – zero-field splitting two independent elements for orthorhombic symmetry: perturbation theory (spin-orbit coupling perturbs the crystal-field levels):

35. Blaubeuren 2006 Effective spin Hamiltonian of the antisymmetric exchange in form of a cross-product: direction of D (Dzyaloshinsky-Moriya vector) Dzyaloshinsky-Moriya interaction

36. Blaubeuren 2006 fitting the T-dependence of the linewidth 3(+1) fit parameters:  ZFS = 0.57 kOe  DM = 1.0 kOe  = 0.16 (critical JT-exponent) + ZFS-ratio E/D=0.37 Kochelaev et al., Mod. Phys. Lett. B 17, 459 (2003). JD et al., PRB 68, 214427 (2003).

37. Blaubeuren 2006 consistent ESR fitparameters ZFS ratio E/D=0.37 was obtained from fitting the g-factor anisotropy Deisenhofer et al., PRB 68, 214427 (2003)..

38. Blaubeuren 2006 zero-field splitting and orbital order perturbation theory:  ESR  =106° ESR  ND  =106° neutron diffraction [Rodriguez-Carvajal et al., PRB 57, R3189 (1998)] excellent agreement!

39. Blaubeuren 2006 summary - part I KT-approach allows to describe the temperature dependence and the anisotropy of the linewidth Dominant contribution are the ZFS and the DM interaction in agreement with neutron diffraction data type of orbital ordering in La 0.95 Sr 0.05 MnO 3 has been derived from the analysis of the ESR g-factor and linewidth (ZFS)

40. Blaubeuren 2006 outline Part 1: the manganite system La 1-x Sr x MnO 3 Phase diagram of La 1-x Sr x MnO 3 ESR results spin relaxation in La 0.95 Sr 0.05 MnO 3 : orbital ordering Part 2: ESR in 1D Heisenberg spin chains Magnetic properties of spin S = 1/2 chains temperature dependence of the ESR linewidth in the model systems NaV 2 O 5, LiCuVO 4, CuGeO 3 the case of TiOCl

41. Blaubeuren 2006 Susceptibility of free spins

42. Blaubeuren 2006 Interacting spins (3D)

43. Blaubeuren 2006 Interacting spins (1D)

44. Blaubeuren 2006 Real spin chains Only in ideal 1D antiferromagnets no phase transition In real systems: –weak inter-chain coupling not negligible  3D antiferromagnetic order at T << J –electron-phonon interaction  Spin-Peierls transition into dimerized ground state

45. Blaubeuren 2006 Spin-Peierls transition

46. Blaubeuren 2006 Model systems LiCuVO 4 Cu 2+ S = 1/2 chain J = 40 K T N = 2.1 K antiferromagnetic order NaV 2 O 5 S = 1/2 per 2 V 4.5+ ¼-filled ladder J = 570 K T CO = 34 K dimerization via charge order CuGeO 3 Cu 2+ S = 1/2 chain J = 120 K T SP = 14 K dimerized, spin-Peierls S = 0 ground state

47. Blaubeuren 2006 Temperature dependence of the ESR linewidth LiCuVO 4 CuGeO 3 NaV 2 O 5

48. Blaubeuren 2006 Universal temperature law

49. Blaubeuren 2006 Limits of the KT-approach High-temperature approximation fails for T < J (!) Field theoretical approach (M. Oshikawa and I. Affleck, Phys. Rev. B 65, 134410, 2002) For temperatures T << J :  H (T ) ~ T for symmetric anisotropic exchange  H (T ) ~ 1/T 2 for antisymmetric DM interaction  in LiCuVO 4, CuGeO 3 and NaV 2 O 5 symmetric anisotropic exchange is the dominant relaxation process

50. Blaubeuren 2006 Universal behavior of the linewidth low temperatures T << J :  H (T) ~ T for symmetric anisotropic exchange  H (T) ~ 1/T 2 for antisymmetric DM interaction What about the antisymmetric interaction? Observation of a low-temperature 1/T 2 divergence due to this interaction?

51. Blaubeuren 2006 The system TiOCl There is no center of inversion between the ions in the Ti-O- layers  antisymmetric anisotropic exchange [A. Seidel et al., Phys. Rev. B 67, 020405(R) (2003)] Isotropic exchange constant J = 660 K

52. Blaubeuren 2006 Analysis of the anisotropic exchange mechanisms Dzyaloshinsky-Moriya interaction Pseudo-dipol interaction D is almost parallel to the b-direction Dominant component of the tensor of the pseudo- dipol interaction is  (aa)

53. Blaubeuren 2006 Temperature dependence of  H [Oe] K AE (∞) K DM (∞) H || a 1429 1.397 H || b 765 2.319 H || c 930 1.344 The temperature and angular dependence of  H can be described as a competition of the symmetric and the antisymmetric exchange interactions! [Zakharov et al., PRB 73, 094452 (2006)]

54. Blaubeuren 2006 Summary Anisotropic exchange dominates the ESR line broadening in low dimensional S=1/2 transition-metal oxides  in contrast to estimations based on the KT-approach 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