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 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

Basic theory of anisotropic exchange Introduction to electron spin resonance (ESR) Full microscopical picture of the symmetric anisotropic exchange: NaV 2 O 5 Temperature dependence of the ESR linewidth in low- dimensional systems: NaV 2 O 5, LiCuVO 4, CuGeO 3, TiOCl Outline

Two magnetic ions can interact indirect via an intermediate diamagnetic ion (O 2-, F -,..)  potential exchange: like direct exchange describes the self-energy of the charge distribution → ferromagnetic; Isotropic superexchange Basic theory of anisotropic exchange  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:

Mechanism of anisotropic exchange interaction Basic theory of anisotropic exchange The free spin couples to the lattice via the spin-lattice interaction H LS = (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!)

Theoretical treatment Basic theory of anisotropic exchange 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 Clear theoretical description can be carried out in the framework of the perturbation theory:

Antisymmetric part of anisotropic exchange Basic theory 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: The direction of D (Dzyaloshinsky- Moriya vector) can be determined from: sasa sbsb rara rbrb j = {x, y, z},  – orbital levels,    – energy splitting, l j – operator of the LS-coupling, J – exchange integral. It should be no center of inversion between the ions!

Symmetric part of anisotropic exchange Basic theory of anisotropic exchange 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 nonnegligible product of the matrix elements of the (LS)-coupling and the hopping integrals. ,  = {x, y, z};  ’ – orbital levels.

Basic theory of anisotropic exchange Introduction to electron spin resonance (ESR) Full microscopical picture of the symmetric anisotropic exchange: NaV 2 O 5 Temperature dependence of the ESR linewidth in low- dimensional systems: LiCuVO 4, CuGeO 3, TiOCl Outline How to study all this?

Zeeman energy in magnetic field H : eigen energies of the spin S Z =  1/2 magnetic microwave field  H with E = h induces dipolar transitions E H H res S Z = -1/2   L  L S Z = +1/2 Zeeman effect S Z = +1/2 E H S Z = -1/2 Introduction to electron spin resonance

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

ESR signal ESR quantities: intensity: local spin susceptibility resonance field: ħ  =g  B H res  g = g local symmetry linewidth  H: spin relaxation, anisotropic interactions Introduction to electron spin resonance

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

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

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

Outline Basic theory of anisotropic exchange Introduction to electron spin resonance (ESR) Full microscopical picture of the symmetric anisotropic exchange: NaV 2 O 5 Temperature dependence of the ESR linewidth in low- dimensional systems: LiCuVO 4, CuGeO 3, TiOCl Let‘s start at last!

NaV 2 O 5 structure Full microscopical picture of AE: NaV 2 O 5 one electron S = 1/2 V 4.5+ O 2- ladder 1ladder 2 a b c VO 5 Na

NaV 2 O 5 susceptibility / ESR linewidth Full microscopical picture of AE: NaV 2 O 5 One-dimensional system at T > 200 K; Charge-ordering fluctuations 34K<T<200K; “Zigzag” charge ordering at T CO = 34 K; ESR linewidth at T > 200 K is about 10 2 Oe

Antisymmetric vs. symmetric exchange Full microscopical picture of AE: NaV 2 O 5 sasa sbsb rara rbrb 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?! Dzyaloshinsky-Moriya interaction Pseudo-dipol interaction

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

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

AE with hoppings between the excited levels Full microscopical picture of AE [Eremin, Zakharov, Eremina…, PRL 96, (2006)] is of great importance in chain systems due to the big hopping integrals t  and t  between the nonorthogonal orbital levels

Schematic pathways of intra-ladder AE Full microscopical picture of AE: NaV 2 O 5 Only one type of the anisotropic exchange – pseudo-dipol interaction with electron hoppings between the excited orbital levels – is possible in the ladders of NaV 2 O 5 ground states excited states  (zz) – dominant!

Schematic pathways of inter-ladder AE Full microscopical picture of AE: NaV 2 O 5 Instead, the “conventional” exchange mechanisms are dominant for the exchange of the spins from the different ladders

Estimation of the exchange parameters Full microscopical picture of AE: NaV 2 O 5 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  (zz) ≈ 5 K in good agreement with the estimations based on the values of hopping integrals and crystal-field splittings Temperature dependence of  H clearly shows the development of the charge- ordering fluctuations at T < 200 K [Eremin.., PRL 96, (2006)]

Temperature dependence of  H in NaV 2 O 5 Open questions 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?

Outline → Empirical answer! Basic theory of anisotropic exchange Introduction to electron spin resonance (ESR) Full microscopical picture of the symmetric anisotropic exchange: NaV 2 O 5 Temperature dependence of the ESR linewidth in low- dimensional systems: NaV 2 O 5, LiCuVO 4, CuGeO 3, TiOCl

Temperature dependence of the ESR linewidth LiCuVO 4 CuGeO 3 NaV 2 O 5  H(T) in low-dimensional systems

Universal temperature law  H(T) in low-dimensional systems

Theoretical predictions High-temperature approximation fails for T < J (!) Field theory ( M. Oshikawa and I. Affleck, Phys. Rev. B 65, , 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 LiCuVO 4, CuGeO 3 and NaV 2 O 5 symmetric anisotropic exchange dominant  H(T) in low-dimensional systems

Linewidth ratio: deviations from universality CuGeO 3 lattice fluctuations (T > T SP = 14.3 K) NaV 2 O 5 charge fluctuations (T > T CO = 34 K) LiCuVO 4 spin fluctuations (T > T N = 2.1 K)  H(T) in low-dimensional systems → (1): if only one interaction determines the linewidth:  H (T, ,  ) = f (T ) ·  H (T , ,  )  linewidth ratio independent of temperature

Universal behavior of the linewidth  H(T) in low-dimensional systems →(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/T 2 divergence due to this interaction?

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

Analysis of the anisotropic exchange mechanisms  H(T) in low-dimensional systems: TiOCl 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)

Temperature dependence of  H  H(T) in low-dimensional systems [Oe] K AE (∞) K DM (∞) H || a H || b H || c 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, (2006)]

Summary Anisotropic exchange dominates the ESR line broadening in low dimensional S=1/2 transition-metal oxides Unconventional symmetric anisotropic superexchange in NaV 2 O 5 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

Acknowledgements Crystal growth NaV 2 O 5 : G. Obermeier, S. Horn (C1, Augsburg) TiOCl: M. Hoinkis, M. Klemm, S. Horn, R. Claessen (B3, C1, Augsburg) LiCuVO 4 : A. Prokofiev, W. Assmus (Frankfurt) CuGeO 3 : 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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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. Direct exchange Basic theory of anisotropic exchange In case of the non negligible direct overlap of the wave functions  i of 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.

LiCuVO 4 structure / susceptibility Cu 2+ S = 1/2 chains along b orthorhombically distorted inverse spinel  H(T) in low-dimensional systems: LiCuVO 4

Antisymmetric vs. symmetric exchange sasa sbsb rara rbrb Antisymmetric exchange is NOT possible in LiCuVO4 (!) Ring-exchange geometry strongly intensifies the pseudo-dipol exchange! Dzyaloshinsky-Moriya interaction Pseudo-dipol interaction Cu O +D ab -D ab b-axis x y d x 2 -y 2 d xy pxpx pypy O O Cu(i) ground state Cu(j) excited state  H(T) in low-dimensional systems: LiCuVO 4

Angular dependence of  H ring-exchange geometry high symmetric anisotropic exchange theoretically expected J cc  2K  H(T) in low-dimensional systems: LiCuVO 4

CuGeO 3 structure / susceptibility 2 Cu 2+ S = 1/2 chains along c J 12  0.1 J T > T SP :  (T ) not like Bonner-Fisher T < T SP :  (T ) ~ exp{-  (T )/T } Cu 2+ O 2- J 12 J  H(T) in low-dimensional systems: CuGeO 3

Antisymmetric vs. symmetric exchange ?  (yy) (Fig.a) and  (xx) (Fig.b) are not negligible Dzyaloshinsky-Moriya interaction Pseudo-dipol interaction  H(T) in low-dimensional systems: CuGeO 3 Intra-chain geometry is the same as with LiCuVO 4 D ≡ 0  (zz) - dominant Inter-chain exchange:

ESR anisotropy in CuGeO 3 intra chain contribution inter chain contribution  H(T) in low-dimensional systems: CuGeO 3

Empty  H(T) in low-dimensional systems

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 Introduction to electron spin resonance

Resonance field, g-values - local symmetry LiCuVO 4 g a = 2.07 g b = 2.10 g c = 2.31 Cu 2+ 3d 9 : g-2 > 0 highest g-value for H || c longest Cu-O bond NaV 2 O 5 g a = g b = g c = V d 0.5 : g-2 < 0 strongest g-shift for H || c CuGeO 3 g a = 2.16 g b = 2.26 g c = 2.07 sum of two tensors local symmetry like in LiCuVO 4 c c Introduction to electron spin resonance

Temperature dependence Field theory ( M. Oshikawa and I. Affleck, Phys. Rev. B 65, , 2002 ): T << J :  H (T ) ~ T for symmetric anisotropic exchange Introduction to electron spin resonance

Summary  Electron spin resonance spin susceptibility, local symmetry, spin relaxation  1D S = 1/2 systems LiCuVO 4, CuGeO 3, NaV 2 O 5  H (T, ,  )  symmetric anisotropic exchange  Charge order in Na 1/3 V 2 O 5 g-value: V1 sites occupied  H ( ,  ): CO not linear but blockwise  H (T ): charge gap consistent with resistivity

Outlook – TiOCl, VOCl Ti 3+ (3d 1, S = 1/2) spin-Peierls A. Seidel et al., Phys. Rev. B 67, (2003) V. Kataev et al., Phys. Rev. B 68, (2003) J. Deisenhofer unpublished (EPV) V 3+ (3d 2, S = 1) Haldane T. Saha-Dasgupta et al., Europhys. Lett. Preprint (2004)

ESR spectrometer microwave (9.4; 34 GHz) electromagnet (bis 18 kOe) resonator, cryostat (He, N 2 : 1.6 – 670 K) control unit lock-in temperature control

ESR in transition metal oxides ESR measures locally at spin of interest  materials with colossal magneto resistance orbital order in La 1-x Sr x MnO 3 magnetic structure in thio spinels FeCr 2 S 4, MnCr 2 S 4  metal-insulator-transition heavy-fermion properties in Gd 1-x Sr x Ti O 3 change of the spin state in GdBaCo 2 O 5+   Low-dimensional spin systems S = 1/2 chains: LiCuVO 4, CuGeO 3 - and ladders: NaV 2 O 5 chains of higher 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

Anisotropic exchange antisymmetric exchange possible in CuGeO 3 and NaV 2 O 5 but not in LiCuVO 4 (!) G ij ~ r i ×r j SiSi SjSj riri rjrj Cu O +G ij -G ij b-axis anisotropic antisymmetric (Dzyaloshinsky-Moriya) ~(  g/g) ·J 1. order anisotropic symmetric ~(  g/g) 2 ·J 2. order conventional estimate

Paths in CuGeO 3 and NaV 2 O 5 CuGeO 3 chains like in LiCuVO 4 large contribution within chains additional contribution between chains  fully describable by symmetric exchange NaV 2 O 5 ladder more complicated than chain high J cc expected from ring structure Up to now no theoretical estimate c

High-temperature linewidth  Symmetric anisotropic exchange well describes the large linewidth for T >> J in LiCuVO 4, CuGeO 3 und probably also in NaV 2 O 5  Good agreement with recent theoretical results on the linewidth in S = 1/2 chains: (J. Choukroun et al., Phys. Rev. Lett. 87, , 2001) Contribution of symmetric anisotropic exchange is always larger than that of Dzyaloshinsky-Moriya interaction

Neutron scattering in CuGeO 3 temperature dependence of low-lying phonon modes in CuGeO 3 M. Braden et al., Phys. Rev. Lett. 80, 3634 (1998) 296 K 1.6 K (THz) intensity

Electron diffraction in CuGeO 3 temperature dependence of diffusive scattering intensity C. H. Chen and S.-W. Cheong, Phys. Rev. B 51, 6777 (1995) diffraction pattern of CuGeO 3 at 15 K intensity T (K) CuGeO 3

Comparison CuGeO 3

Anisotropic-exchange parameter

Outlook  Open Questions anisotropic exchange in NaV 2 O 5 connections to charge fluctuations LiCuVO 4 : comparison to NMR  ESR in the ground state AFMR in LiCuVO 4 triplet-excitations AFMR in CuGeO 3 impurity doping