High Field NMR studies of Quantum Spin Systems. From Bose Einstein Condensation to Magnetization Plateaus C. BERTHIER

Transcription

1 Institut d Etudes Scientifique de Cargese International Summer School MAGNETIC FIELDS FOR SCIENCE High Field NMR studies of Quantum Spin Systems. From Bose Einstein Condensation to Magnetization Plateaus C. BERTHIER Grenoble High Magnetic Field Laboratory

2 Outline of the lecture Introduction to Quantum spin systems. GS and magnetization Bose-Enstein Condensation vs Magnetization plateaus SrCu 2 (BO 3 ) A 2D othogonal dimer system BaCuSi 2 O 6 (Han purple) A model for 2D Bose Einstein Condensation? Conclusion

3 Low Dimensional Antiferromagnetic Quantum Spin System Classical magnetism H = Σ i,j J i,j S i S j S i, the electronic spins, are vectors, not operators leads to AF Néel Order, Ferrimagnetism, Helicoidal structures Semi-classical magnetism: Introduction of magnons and spin waves Quantum magnetism Low values of S (S = 1/2, 1); low number of first neighbors (low dimensionality, or weakly coupled dimers Frustration also plays an important role Quantum fluctuations dominates and lead to Peculiar Ground States very different from Néel states

4 Isolated dimers of S = 1/2 electronic spins Let s consider a pair of Cu 2+ ions, coupled by AF exchange J. I = J S 1 S 2 s = - 2 M + t 0 = 2 gμ B H t ±1 = ( ) E Δ =Δ+gμ B H H c H t Δ = J 63,65 Cu NMR spectrum ν Q s t -1 Δ =Δ-gμ B H H c H frequency ν = γ(1+k orb )H 0 Zero shift due to the el. spin!

5 Increasing the field above H c I = J S 1 S 2 s = - 2 M + t 0 = 2 gμ B H t ±1 = ( ) E Δ =Δ+gμ B H H c H t Δ = J 63,65 Cu NMR spectrum ν Q s t -1 Δ =Δ-gμ B H H c H ν = γ(1+k orb )H 0 +A hf <S z > frequency About 250 MHz shift K S!

6 A new Ground State in Quantum Antiferromagnets: The Spin liquid J M Sat J // M H c2 E Single spin excitations E(k) = J + J // cos ka Δ =Δ+gμ B H H c1 H t -1 Δ = J Δ =Δ-gμ B H H Possibility of magnetization plateaus or Bose Einstein Condensation Strong effect of H on C S ppt e - Δ/T (variation of the gap) H c1 H c2

7 A few peculiar GS of frustrated S = 1/2 AF systems 1D systems VBC J 1 -J 2 HAFC (J 2 /J ), two-leg spin ladders, zig-zag HAFC, 2D systems VBC Shastry Sutherland (orthogonal dimers) SrCu 2 (BO 3 ) 2 Square plane of dimers BaCuSi 2 O 6 3D systems VBC Coupled dimers (TlCuCl 3 ), RVB Spin-Liquid Spin ½ Kagome lattice 籠目格子?

8 The pseudo-vector picture Zero field GS described by vacuum. Becomes populated by bosonic particules for which H acts as chemical potential Pairs of Cu 2+ ( 2 spins S = ½) coupled by antiferromagnetic interaction J J, J coupling between the dimers Close to H c, only the low energy degree of freedom are considered. Triplets can be described as hard core bosons on a lattice (triplons) J, J determine the kinetic energy and the repulsive interaction between triplons. Kinetic energy is strongly reduced in frustrated systems T. Giamarchi and A. Tsvelik PRB 59 (1999); Momoi and Totsuka, PRB (2000); T.M. Rice Science (2002)

9 If kinetics energy dominates, the triplons can acquire a common phase (<Φ> 0) Bose Einstein Condensation (Giamachi and Tsvelik, PRB 1999, Nikuni et al., JPSJ 2001 (TlCuCl3), Jaime et al., PRL 2004 [Ba 2 Cu 2 Si 2 O 6 ] Order Parameter: Staggered Transverse Magnetization H.Tanaka et al., JPSJ (2001) If repulsive interaction dominates, the triplons may form a commensurate superlattice giving rise to magnetic plateaus «Wigner crystallization» mK Ba 2 Cu 2 Si 2 O 6 M. Jaime, PRL 93 (2004) SrCu 2 (BO 3 ) 2 Onizuka et al.,jsps 69(2000)

10 SrCuBO 3 : 1st realization of the Shastry-Sutherland Hamiltonian B.S. Shastry, B. Sutherland, Physica B108, 1069 (1981) Kageyama et al., PRL 84 (2000) 5876 J /J is always an eigenstate J < 0.7 J is the G.S J=84 K, J =54 K, J =8 K S.Miyahara & Ueda, JPS.J, 69 Supp. (2000) 72 J / J = Cu 2+ : S = 1/2 spins SrCu 2 (BO 3 ) 2 is close to the quantum critical point

11 Magnetization pattern within a quantum magnetization plateau. Oshikawa criterion: N (S m) Z ( N number of magnetic sites in GS cell) m average magnetization per site Oshikawa PRL (2000) Determination of the superstructure in the hard core bosons approximation S. Miyahara and K. Ueda, PRB 61 (2000) 3417 K. Onizuka et al., J. Phys. Soc. Jpn. 69 (1999) 1016.

12 What can we learn from NMR? Plateaus of "fractional" magnetisation are supposed to be stable, commensurate, spin-textured states. Experimentally, due to high magnetic field involved (> 27 Tesla), their magnetic structure can only only be accessed by NMR! 63,65 Cu NMR spectrum ν Q ν = γ(1+k orb )H 0 +A hf <S z > K orb =1.6%; ν Q = 21.1 (20.5) MHz A hf = 23.8 T/µ B K. Kodama et al., J. Phys. Cond. Matter 14 (2002) L14 frequency 20 MW resistive magnet at the GHMFL Most simple picture (h.c. bosons): outside the plateau one single site: one group of 6 lines inside the plateau: 2 inequivalent sites:one triplet, 7 singlets in a supercell 2 groups of 6 lines separated by 260 MHz

13 63,65 Cu NMR spectrum of SrCu 2 (BO 3 ) 2 at 35 mk Fit: 11 sites S z =0.3 S z =0.2 S z < 0 Site <H n > (T) Intensity / / / / / / / / / / /16 K. Kodama et al., Science 298, 395 (2002)

14 Theoretical description of of the spin superstructure Exact diagonalization on N=16 and 24 sites clusters of the full S.-S. Hamiltonian K. Kodama et al., Science 298 (2002) 395; S. S. Miyahara et al. PRB 68 (2003) Theory+ on-site coupling Experiment A = T/µ B =A on.site + B + C Theory+ all couplings In SrCu 2 (BO) 3, Cu 40 mk and 28 T gave the first evidence for a breaking of translational symmetry of in a non-trivial (1/8) magnetization plateau. Experiments on B showed that the transition is 1st order vs temperature (T C = 480 mk ) or magnetic field. K. Kodama et al., Physica B (2004)

15 phase transition into the 1/8-state : temperature sweep field sweep 63 Cu NMR Int ensit y ordered phase uniform phase T (K) 11 B NMR Discontinuous development of spin profile (order parameter) T 550 mk 11 B NMR Mixed phase (Order parameter gets slightly reduced near T C ) Intensity 480 mk 150 mk 1 st order Freq uency ( MHz)

16 11 B spectra as a probe of Cu spins polarization The central line is convoluted by 0.4 δ(ν - ν Q )+ 0.6 δ(ν - ν Q )+ 0.4 δ(ν - ν Q ). One can obtained a deconvoluted spectrum Hyperfine field at Cu site (T) central line 11 B 26 T 11 B NMR intensity complete spectrum central transition 11 B 27.5 T Frequency (MHz) Frequency (MHz) We shall focus on the the lowest frequency part or deconvolute whole spectar.

17 What is expected above the 1/8 plateau, at T??? or T 26 T These boron lines are the signature of the «magnetic crystal» This one corresponds to a uniform gas of magnetic excitations How will this spectra evolve? Frequency (MHz) / local field

18 13-19 February 2006 Dr. Shin-ichi Matsubara, Prof. Masashi Takigawa 19 February 2006

19 Participants SrCu 2 (BO 3 ) 2 M. Takigawa ( 瀧川仁 ), ISSP, Japan K. Kodama 1, S. Matsubara, ISSP, Japan H. Kageyama 2 M. Horvatić,C.B. GHMFL, France F. Mila, EPFL, Lausanne, Switzerland F. Becca 3, EPFL, Lausanne, Switzerland S Miyahara 4, EPFL, Lausanne, Switzerland Present address 1 JAERI, Tokai, Japan ; 2 Department of Chemistry, Kyoto University, Japan; 3 ITP, Trieste, Italy 4 Department of Physics, Aoyama Gakuin University

20 The Han Purple BaCuSi 2 O 6 : a prototype for Bose-Einstein Condensation? J >> J >> J (2D) Han purple BaCuSi 2 O 6 Chinese warrior from the terracota army (Xi an, more than 2000 years old) Ba 2 Cu 2 Si 2 O 6 M. Jaime, P RL 93 (2004) 87203

21 NMR Steffen Krämer, Mladen Horvatić, C.B. GHMFL, Grenoble, France Raivo Stern NICPB, Tallinn, Estonia Participants. Samples Tsuyoshi Kimura NHMFL, Los Alamos, U.S.A. Suchitra Sebastian, Ian Fisher Dept. of Applied Physics Stanford University, U.S.A S. Krämer et al., cond-mat\ accepted in PRB Rapid Communication

22 Macroscopic Evidence for Bose-Einstein Condensation BEC of triplets A = T/µ B M. Jaime et al., PRL 93, (2004) S.E. Sebastian et al., Nature 4732 (2006) ν = 2/3 S.E. Sebastian et al., PRB 72, (2004) Cross over from 3D to 2D BEC (ν = 2/d ) What about the microscopic picture?

23 Bose condensation of hard core bosons? J E for each dimer: Φ,Θ = e i Φ/2 cos(θ/2) + e +i Φ/2 sin(θ/2) { }/ 2 [Momoi and Totsuka, PRB 62, (2000)] H local projections of S Z, S : Φ/2 staggered S moments Appear when <Φ> 0 Bose Einstein Condensate Θ/2 S 1 S 2 NMR should detect a line-splitting (into two lines) corresponding to a Néel type order!

24 BEC and NMR Outside the BEC No transverse staggered magnetisation; same longitudinal magnetization <S z > on all equivalent dimers. NMR should detect a line-splitting (into two lines) provided there exist nondiagonal elements of the hyperfine coupling tensor! δν = A zx <S x > Inside the BEC 2 δν 14 N spectra above and below T BEC I = 1, 4 sites, one doublet per sites

25 29 Si NMR BaCuSi 2 O 6 Normalized BaCuSispectra T=40 mk 2 O 6 T = 50 mk 29 Si Si (B- B crit ) [ T] Cu dimer 29 Si NMR (I = 1/2) Oriented single crystal (B c) spectra between 23 T and 25 T between 40 mk and 600 mk (dilution refrigerator) Δf [ MHz ] B crit = T Evidence for phase transition at 50 mk with B crit = T Complicated and incommensurate phase?

26 29 Si vs 63,65 Cu NMR BaCuSi 2 O 6 single crystal 29 Si NMR, B 0 = 14.8 T B 0 // c-axis (normalized intensity) Intensity/2 The amplitude of the modulation does not follow the average local field M 1 Intensity/ f [MHz] T [K] approximate spin polarization (full = 0.5) Even in the "normal state", NMR reveals below 90 K a phase transition to a IC state that breaks the "ideal" tetragonal structure! Echo Intensity (a. u.) H 0 // c-axis, T = 8.9 K, MHz sweep 65 Cu high-freq. sat. : 2 sites + IC lineshape (magnetic!) 65 Cu central line X rays, E.C. Samulon et al.,prb (2006) Magnetic field (T)

27 Si around the Quantum Critical point At 50 mk ~ the QCP, the boson population suddenly increases as H becomes > H c1. The first moment of the line gives the density of bosons n. The order parameter is pp to n, so that the second moment ~ n is also increases linearly with H - H c1 T = 720 mk B 0 [ T ] M 1 M 2 M 1 [ khz ] M 2 [ 10 2 s -2 ] ( f 0-29 γ B 0 ) [ MHz ] B 0 [T] Because of the temperature and the decrease of the gap when increasing H, the 1st moment (pp. to the number of triplet states = density of bosons) increases before the BEC. However, the 2 nd moment explodes at the transition.

28 Phase boundary 800 T [ mk ] "BEC" B 0 [ T ] T BEC determined using the previous analysis of second moment. Linear dependence of T BEC vs (H H c1 ) below 600 mkin agreement with Sebastian et al., Nature Argument for a 2D BEC T BEC (H H c1 ) 2/d d dimensionality of the condensate Origin of this two dimemnsionality at low temperature?

29 From 3D to 2D as T BEC 0? The BEC forms at [π, π] in the plane. At very low boson concentration, the probability of hopping to to the next layer is zero due to frustration, and grows as the square of the concentration, due to interaction within the planes But.. S.E. Sebastian et al., Nature 4732 (2006) C. Battista et al., PRB (2006)

30 Two types of dimers Normalized echo intensity [arb. units] T = 8.9 K 167 MHz 63,65 Cu NMR B 0 = T 65 Cu upper sat B ext [ T ] Δ 0 = 3.9 mev Δ 0 = 4.4 mev Frequency (MHz) Corresponding to different types of planes with different B crit Temperature [ K ] Two different groups of lines with different shifts and different T dependence Two different types of dimers with Δ 1 (0) =1.15 Δ 2 Incomensurate modulation affects both types of dimers See Ch. Ruegg et al., PRL 98, (2007)

31 Line separation at 50 mk B T = 50 mk 0 [ T ] Ix ( f 0-29 γ B 0 ) [ MHz ] Intensity narrow part B 0 = T wide part ( f 0-29 γ B 0 ) [ MHz ] Δ(wide part) Δ(narrow part) = 4.5 wide part Δ B Δ A M 1 [khz] total line gμ B H 50 0 narrow part B [T]

32 Conclusions Microscopic picture of the «BEC» in BaCuSi 2 O 6 is complex, due to: Two (at least) different types of dimers, with different J couplings J 1 / J 2 = 1.15 Each goups of dimers is affected by the incommensurate structural modulation, leading to an incommensurate modulation of the spin density. T C (H ) in agreement with 2D BEC in the temperature range 50 mk-600 mk as found by torque. NMR measurements indicate that at H c1 the BEC occurs only in one over two planes along the c-axis, although a small tranfert exists. The density of the BEC is incommensuratetly modulated in the plane, due to a structural distorsion which modulates J.

33 Equipment of the NMR group at the GHMFL Broad band MHz NMR pulsed spectrometer T sweepable superconducting magnet He3-He4 dilution fridge (40 mk at 17 T) + VTI Access to resistive magnet up to 30 T at T down to 40 mk 34 T at T down to 400 mk Both types of magnets can be used for proposals submitted to the GHMFL In addition, 9T and 11.5 T superconducting magnets at the LSP (Grenoble University, Saint-Martin d Hères)