Quantum kagome spin liquids

Transcription

1 Quantum kagome spin liquids Philippe Mendels - Univ Paris-Sud Orsa The discovery of Herbertsmithite, ZnCu 3 (OH) 6 Cl 2, which features a perfect kagome geometry, has been coined as the "end to the drought of spin liquids" [1,2]. It has triggered an intense activity on new kagome materials and related theories for the ground state of the quantum kagome Heisenberg antiferromagnet which has eluded any definitive conclusion for the last twenty years. This is indeed the first experimental example of a kagome lattice where no order has been found at any temperature well below J through all experimental techniques, among which µsr [3] has been by far the most accurate. This illustrates the power of the association of an enhancement of quantum fluctuations for S=1/2 spins with the frustration of antiferromagnetic interactions on the loosely connected kagome lattice to stabilize novel ground states of magnetic matter, namely quantum spin liquids. I will review and discuss our contribution to this field through µsr, in connection with our NMR work. Through the various Cu 2+ S=1/2 materials which we have studied, I ll illustrate some of the research thrusts in tracking down quantum spin liquid physics. This will encompass Zn and Mg-paratacamites, (Zn,Mg) x Cu 4-x (OH) 6 Cl 2 [), their polymorphs kapellasite [4] and haydeite, vesignieite [5] as well as the recently discovered triangular-based lattice Ba 3 CuSb 2 O 9 [6] compound. [1] P.A.Lee, Science, Perspectives 321, 1306 (2008). [2] For a review, see P. Mendels and F.Bert, Special Topics Section on "Novel States of Matter Induced by Frustration", J. Phys. Soc. Jpn 1, (2010). [3] Phys. Rev. Lett. 98, (2010). [4] Phys. Rev. Lett. 109, (2012). [5] Phys. Rev. B 84, (2011). [6] Phys. Rev. Lett., in press.

2 Quantum Quantum KagomeSpin Antiferromagnets liquids F. Bert, E. Kermarrec, A. Olariu, J. Quilliam, M. Jeong, A. Zorko Laboratoire de Physique des Solides, Université Paris-Sud, Orsay, France

3 Antiferromagnetism, Classical state à la Néel H r r = J ijsi. S j, Jij < 0 Τ Ν J -excitations = magnons (S=1) - 1/S quantum corrections (fluctuations) -> moments reduction < r r S. S i j >= e r r i j / ξ ;ξ +

4 Néel state, any alternative? 1972 Geometrical Frustration of magnetic interactions? Class. A C S r S r + S r = A + B C B 0 r

5 Néel AF state, any alternative? RVB =

6 Corner sharing: classical kagomé lattice S r S r + S r = 0 r A + B C C A B C A B C A A A C C B B C A B C C A C B C C A B A C A A C B B A C B A A C A B Macroscopic degeneracy Soft modes

7 Corner sharing: classical vs quantum kagomé lattice Classical soft modes Triangular Kagome <J/20 Back to 90 s Δ<J/20 No gap in the singlet sector Lecheminant, PRB 56, 2521 (1997) Waldtmann et al., EPJB 2, 501 (1998).

8 Other stream: can one stabilize a spin liquid in dimension >1? Spinon continuum Spin waves branches «Fractional excitations fractionnaires» ( S 1)

9 The IDEAL Highly Frustrated Magnet - Heisenberg or Ising spins - S=1/2 (quantum spins) - Corner sharing geometry: Kagomé (2D), hyperkagome or pyrochlore (3D) lattice - No «perturbation» (anisotropy, dipolar interaction, n.n. interactions, dilution) - For kagome: stacking should keep the 2D kagome planes uncoupled

10 Kagome Heisenberg antiferromagnet, a model of geometrically "frustrated " magnetism Néel 2011 Anderson 1972 S. Yan, D. Huse and S.R. White, Science 332 (2011)

11 Kagome Heisenberg antiferromagnet, a model of geometrically "frustrated " magnetism Quantum Spin Liquid Chiral spin liquid Messio et al., PRL 2012 diamond-pattern valence bond crystal honeycomb valence bond crystal S. Yan, D. Huse and S.R. White, Science 332 (2011)

12 Kagome ground state? - Quantum spin liquid? A state without any spontaneously broken symmetry Z 2 QSL Gapped magnetic excitations (S=1/2) Gapped non-magnetic excitations C v ~e - /T ; χ~e - /T Algebraic/Critical/Dirac/U(1) QSL Gapless excitations C v ~T 2 ; χ~t Short range RVB Long range RVB = Yan et al, science (2011) Hastings, PRB 63, 2000 Ran et al, PRL 98, 2007 Ryu et al, PRB 75, 2007

13 Large size numerical study (DMRG) Quantum Spin Liquid The ground-state of QKHA would be a gapped spin-liquid (short-range RVB) triplet valence bond crystal singlet S. Yan et al, Science 332 (2011)

14 The Cu 2+ world (S=1/2, ~Heisenberg) b

15 Materials are all existing minerals! Cu 2+ S=1/2 Herbertsmithite Volborthite Haydeeite Kapellasite Vesignieite Ross H Colman, David Boldrin, Andrew S Wills, UCL, UK

16 2005 Sciences, perspectives sept 2008

17 Kagome stats From C. Broholm (2010)

18 FRUSTRATION: unsatisfied pairwise interactions Disorder : Spin glasses (e.g. CuMn) No disorder : Geometric Frustration? NETWORK GEOMETRY : Triangular based lattice AF coupling? J 2 J 1? INTERACTIONS GEOMETRY: 2nd n.n. interactions J 2 / J 1 =0.5, J 1 (AF) (Carretta; Geibel)

19 Outline ISIS, PSI NMR - Zn and Mg Herbertsmithite Cu 3 (Zn,Mg)(OH) 6 Cl 2 - a gapless quantum spin liquid (summary) - the fate of defects - field induced solidification of the QSL - Vesignieite Cu 3 Ba(VO 5 H) 2 - local susceptibility (NMR) - heterogeneous frozen ground state (µsr+nmr) Quantum criticality Dzyaloshinky-Moriya interactions J1-J2 model on kagome lattice Novel spin liquid? -Kapellasite Cu 3 Zn(OH) 6 Cl 2 - competing exchange interactions: Collaborations (Herbertsmithite, Vesignieite, Kapellasite) samples Ross H Colman, David Boldrin, Andrew S Wills, UCL, UK P. Strobel, Institut Neel, Grenoble, France A. Harrisson, M. de Vries, Edinburgh, UK F. Duc, Toulouse

20 M.I.T., 2005 Herbertsmithite: ZnCu 3 (OH) 6 Cl 2 Cu 2+, S=1/2 Cu Zn OH Cl

21 µsr : ZnCu 3 (OH) 6 Cl 2 : zero field Polarisation mk H Herbertsmithite Zero Field µsr Deuterated time (µs) OH-µ : 85% Cl - : 15% upper limit of a frozen moment for Cu 2+, if any : 6x10-4 µ B P. Mendels et al, PRL 98, (2007) No order or frozen disorder down to 50 mk despite J=190 K! Liquid down to J/4000 under ZF Also: ac-χ Helton et al, PRL (2007) µsr O. Ofer et al, cond-mat/

22 µsr : phase diagram of paratacamites Zn x Cu 4-x (OH) 6 Cl 2 Polarization 0.9 x = 1.0 x =0.5 x = x = x = 0.15 x = 0 T=1.5K, Zero Field time (µs) - Paramagnetic (x=1 type) component - Oscillations smeared out - x=0 : fully ordered below ~18K X.G. Zheng et al, PRL 95, (2005) Large dynamical domain 0.66<x<1 -> small influence of interlayer coupling P. Mendels et al, PRL 98, (2007)

23 Susceptibility: χmuons, D NMR vs 17O NMR (intrinsic, defects) µsr D NMR O. Ofer et al., ArXiv (2010) T. Imai, Phys. Rev. B (2011) A. Olariu et al., Phys. Rev. Lett (2008) 17 O lineshift (%) χ ac O NMR T (K) 0 χ SQUID (10-4 cm 3 /mol Cu) Cu Cl Zn O

24 Gapless spin liquid 17 O lineshift (%) 2 1 J/ χ SQUID (10-4 cm 3 /mol Cu) (ms -1 ) T T 1-1 ~ T 0.7+/ T -1 (K -1 ) T (K) Olariu et al, PRL 100, (2008) See also Imai et al, PRL 100, (2008) Inelastic Neutron scattering: Helton et al, PRL (2007) χ ~ω -0.7

25 Very Low T Spin Dynamics 6.8 T 0.1 M. Jeong et al, Phys. Rev. Lett 107 (2011) 1/T 1 (ms -1 ) E-3? 1E-4 1E T (K)

26 Very Low T Spin Dynamics: freezing under a field T 6.7 T T c M. Jeong et al, Phys. Rev. Lett 107 (2011) /T 1 (ms -1 ) 1E-3 1E-4 1E T (K)

27 Small frozen moment ~ 0.1µ B A Quantum Critical Point? T c ~ (H-H c ) 0.65 H c = 1.5 T ~ 2.3 k B T c µsr µ B H c ~ J/180 Algebraic critical spin liquid? No gap (H = 0) Instability ( H 0) M ~ H α but Tc ~ H DM interaction -> L.R.O. Ran et al, PRL (2007)

28 Ideally. Nobody is perfect

29 Magnetic defects : Zn/Cu intersite mixing Cu Cu on the Zn site Nearly free ½ spins (~ %) Zn defects Freedman et al, JACS, 132 (2010) J (AF), F ~ few K Zn in the kagome plane -> magnetic vacancy? J (AF) ~180 K (~ 1-7%) 97 Cu Zn/Cu OH For a review and discussion, see P. Mendels and F. Bert, J. Phys. Soc. Jpn (2010) J. Phys. Conf. Series (2011) 119 Cl

30 Polarisation G mk ZF 10G 80G time (µs) Out-of-plane (?) defects (2) Electronic spin signal Dynamical relaxation (T 1 process) P = exp( λt) λ = 2γ 2 H 2 µ µ / - Weak decrease of the electronic spin fluctuation rate when T->0 ν - H µ ~ 20 G very small ->relaxation involves few Cu 2+ (defects?) ZnCu 3 (OH) 6 Cl 2 : x=1 - Dynamical plateau vanishes when x Frozen fraction Static P2 1 /n R3m Dynamic Zn content Energy scale ~1K in the defect channel

31 Mg- Herbertsmithite: control of inter-site defects E. Kermarrec, Phys. Rev. B (R) (2011)

32 Dzyaloshinskii-Moriya interactions: quantum criticality Disordered For classical spins, DM stabilizes ordered phases (cf jarosites) M. Elhajal et al, PRB 66, (2002) In the quantum case, a moment free phase survives up to D/J~0.1 O. Cepas et al, PRB 78, (R) (2008) Y. Huh et al, PRB 81, (2010) L. Messio et al, PRB 81, (2010) Herbertsmithite

33 Dzyaloshinskii-Moriya interactions: quantum criticality Disordered For classical spins, DM stabilizes ordered phases (cf jarosites) M. Elhajal et al, PRB 66, (2002) In the quantum case, a moment free phase survives up to D/J~0.1 ESR: A. Zorko et al., PRL 101 (2008) O. Cepas et al, PRB 78, (R) (2008) Y. Huh et al, PRB 81, (2010) L. Messio et al, PRB 81, (2010) S. El Shawish et al, PRB 81, (2010) Herbertsmithite H c ~ D c - D herb

34 Outline - Zn and Mg Herbertsmithite Cu 3 (Zn,Mg)(OH) 6 Cl 2 - a gapless quantum spin liquid (summary) - field induced solidification of the QSL - Vesignieite Cu 3 Ba(VO 5 H) 2 - local susceptibility (NMR) - heterogeneous frozen ground state (µsr+nmr) Quantum criticality Dzyaloshinky-Moriya interactions -Kapellasite Cu 3 Zn(OH) 6 Cl 2 - competing exchange interactions Collaborations Ross H Colman, David Boldrin, Andrew S Wills, UCL, UK L. Messio, C. Lhuillier, B. Bernu, Paris, France B. Fak, CENG, Grenoble, France

35 Vesignieite Cu 3 Ba(VO 5 H) 2 Y. Okamoto et al, JPSJ 78, (2009) a weakly distorted kagome lattice 0.1 % (Volborthite : 3%) R.C. Colman et al., Phys Rev B, Rapid Com 2011 Coll. UCL London

36 V NMR (T>9K) Cu hexagon susceptibility χ int J = (K-σ)J/A (10-2 K µ B / koe) Vesignieite (J = 53 K) 1 Herbertsmithite (Olariu2008, J=170 K) Volborthite (Hiroi2001, J=84 K) Theory (Misguich2007, adjusted) T/J Shift Herbertsmithite J. Quilliam et al, arxiv: J ~ 50 K - Curie tail ~ 7% S=1/2 - Kink at T~9K R.C. Colman et al (2011)

37 Vesignieite µsr: two components One static disordered- one dynamical Not zero slope Dynamical component Not zero Static component! At least 4 muon sites P(t) = f f P f (t) + (1-f f )P para (t) T < 9K decoupling µ stat ~ 0.1 µ B R.C. Colman et al., Phys Rev B, Rapid Com (2011) Coll. UCL London

38 Vesignieite µsr: two components Frozen fraction Coupled slow dynamics non frozen fraction No macroscopic phase separation. Spin dynamics of all Cu 2+ suppressed down to spin freezing for 40%.

39 Vesignieite V NMR T<9K Static component below 9 K ~ 0.2 µ B J. Quilliam et al, PRB (R), 2011

40 Vesignieite NMR -> two components Static + dynamic (lost) Seen in NMR Lost in NMR NMR Good agreement Coll. Orsay / UCL London µsr J. Quilliam et al. R.C. Colman et al

41 Dzyaloshinskii-Moriya induced quantum criticality D z =0.08J, D p ~0.01J J vesi ~ J herb /3; H vesi ~ H volb ESR: H~D 2 /J D/J vesi ~ A. Zorko et al, PRL 101, (2008) W. Zhang, H. Ohta et al., JSPJ (2010) Herbertsmithite Vesignieite

42 Conclusions Herbertsmithite Vesignieite Perfect kagome No freezing at H=0 Close to perfect kagome Partial J/6 Field induced freezing 0.44 < D/J < 0.08 QCP D/J > 0.1

43 Outline - Zn and Mg Herbertsmithite Cu 3 (Zn,Mg)(OH) 6 Cl 2 - a gapless quantum spin liquid (summary) - field induced solidification of the QSL - Vesignieite Cu 3 Ba(VO 5 H) 2 - local susceptibility (NMR) - heterogeneous frozen ground state (µsr+nmr) Quantum criticality Dzyaloshinky-Moriya interactions -Kapellasite Cu 3 Zn(OH) 6 Cl 2 - competing exchange interactions Collaborations Ross H Colman, David Boldrin, Andrew S Wills, UCL, UK L. Messio, C. Lhuillier, B. Bernu, Paris, France B. Fak, CENG, Grenoble, France

44 Quantum Kagome Antiferromagnets ZnCu 3 (OH) 6 Cl 2 B. Fak, E. Kermarrec et al, PRL, 2012

45 Kapellasite : a polymorph of Herbertsmithite Herbertsmithite Cu 3 Zn(OH) 6 Cl 2 Kapellasite 20 µm

46 35 Cl NMR 1.0 NMR lines T=78 K.) ụ (a ity s n te in R M N Cl-1 Cl-2 Cl-3 Cl Cu Zn NMR shift 46 % Cl-1 37 % Cl-2 15 % Cl-3 2 % Cl-4 Chemical analysis (ICP) Cu 2.3 Zn 1.7 (OH) 6 Cl 2 Neutron diffraction (Cu 0.73 Zn 0.27 ) 3 (Zn 0.88 Cu 0.12 ) (OH) 6 Cl 2 Kagome site p c =0.65

47 High-Temperature series expansion analysis J 1 ferro -15 K ~ further neighbor J 2,d 13 K B. Fak, E. Kermarrec et al, PRL, 2012

48 Interactions scheme L. Messio, et. al, PRB 83, (2011)

49 Classical ordered states on the Kagome lattice L. Messio, C. Lhuillier, G. Misguich, LPTMC Paris, CEA 8 Classical long-range ordered states allowed by symmetries L. Messio, et. al, PRB 83, (2011) Non coplanar state 12 sublattice spins order : «cuboc2» Neutrons experiment shows correlations reminiscent of this peculiar state

50

51 µsr -no freezing (no decoupling) -Persistent slow fluctuations

52 Neutron scattering (B. Fak, ILL) Cuboc-2 correlations survive up to 100 K (also specific heat)

53 µsr vs NMR

54 Conclusions Experimentally: two quantum spin liquids on the kagome lattice: Heisenberg + anisotropy and J1-J2 Fragility of the QSL: Defects: no Anisotropy, field: yes Is the gapless behavior intrinsic or not? Can we achieve a better understanding of the relaxation plateaus? Other materials: S=1/2 Vanadates Thank you to ISIS team!