Field-induced ordering in a random-bond quantum spin ladder compound with weak anisotropy
FField-induced ordering in a random-bond quantum spin ladder compound with weakanisotropy.
G. S. Perren, W. E. A. Lorenz, E. Ressouche, and A. Zheludev ∗ Laboratory for Solid State Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland. INAC, SPSMS, CEA Grenoble, 38054 Grenoble, France (Dated: October 14, 2018)The field induced quantum phase transitions in the disorder-free and disordered samples of the spinladder material (CH ) CHNH Cu(Cl − x Br x ) are studied using magnetic calorimetry and magneticneutron diffraction on single crystal samples. Drastically different critical indexes and correlationlengths in the high field phase are found for two different orientations of the applied field. It is arguedthat for a field applied along the crystallographic c axis, as in previous studies, the transition isbest described as an Ising transition in random field and anisotropy, rather than as a magnetic BoseGlass to Bose Condensate transition. PACS numbers: 75.10.Nr, 75.40.-s, 75.30.-m, 75.40.Cx, 75.50.Ee, 75.50.Lk
I. INTRODUCTION
Shortly after the realization that field-induced mag-netic ordering in gapped quantum paramagnets canbe viewed as a Bose-Einstein condensation (BEC) ofmagnons, a new research thrust was aimed at under-standing the corresponding transition in disordered spinsystems (for a review see Ref. 5). The idea is that inthe presence of disorder, the magnetic BEC state maybe preceded by the magnetic analog of the long soughtfor Bose Glass (BG) state.
The organic quantum spinladder compound (CH ) CHNH Cu(Cl − x Br x ) (IPAXfor short) was one of the first materials studied in thiscontext. Indeed, neutron scattering experiments un-der applied field revealed a BG-like gapless, magnetizable(“compressible”) yet disordered magnetic state. Most ofthe experimental work since was done on other disorderedquantum magnets, such as NiCl − x Br x · ) and (C H N )Cu (Cl − x Br x ) . The main effortwas aimed at understanding the critical exponents of thequantum phase transition between the magnetic BECand BG phases. Of particular interest is the experi-mentally accessible exponent φ , which defines the field-temperature phase boundary: T c = ( H c − H ) φ . Theoriginal work of Fisher et al. predicted φ >
2, as op-posed to φ = 2 / φ ≈ . but were in turn con-tradicted by a later study consistent with Fischer’s orig-inal arguments. So far, experiments on materials likeNiCl − x Br x · ) and Tl − x K x CuCl have only added to the controversy, and the issue remainsunresolved.To date, there have been no systematic studies of criti-cal exponents in the “original” BG-prototype compoundIPAX. Moreover, from the start it was clear that thefield-induced ordering transition in this material is notexactly a BG to BEC one. In IPAX the high field phasehas static, but only short range magnetic order, with ahistory dependent correlation length, in contrast to true long range order expected for a magnetic BEC. A con-vincing explanation for this behavior is lacking. In thepresent work we address these issues in a series of bulkmeasurements and additional neutron diffraction studieson the x = 0 .
05 and x = 0 systems. We show that thepreviously observed short-range order is directly linked toa weak magnetic anisotropy present already in the parentcompound IPA-CuCl . We further argue that in the con-text of IPAX, and perhaps in most other quantum para-magnets with chemical disorder, it is more appropriateto speak of spin freezing in a random field environment,rather than of magnon condensation in a magnetic BG.IPA-CuCl crystallizes in a triclinic space group P a = 7 .
78 ˚A, b = 9 .
71 ˚A, c = 6 .
08 ˚A, α = 97 . ◦ , β = 101 . ◦ , and γ = 67 . ◦ . The magnetic proper-ties are due to S = 1 / ions that form weakly cou-pled two-leg ladders directed along the crystallographic a axis. The ground state is a spin singlet. The mag-netic exitation spectrum features a gap of ∆ = 1 . Based on these neutron data, nu-merical modeling allowed to determine the relevant ex-change constants. The magnon triplet is split into asinglet and doublet by 0.05 meV due to a very weakeasy-axis anisotropy of exchange interactions.
Theeasy axis is roughly along the crystallographic b direc-tion. The anisotropy is also manifest in the gyromagnetictensor with g a = 2 . g b = 2 .
22, and g c = 2 . Thefield-induced ordering transition occurs in H c ∼ . The transition can to some extent be viewed as a BECof magnons. This picture is disrupted by the resid-ual anisotropy which leads to are-opening of a small gapin the ordered state. In the Br-substituted compoundIPAX the crystal structure is almost unchanged up to x = 0 . The halogen substitution does not directlyaffect the magnetic Cu ions, but instead randomizesthe strengths of magnetic interactions. As mentionedabove, the result is a Bose-glass like precursor to thefield-induced ordered phase. a r X i v : . [ c ond - m a t . s t r- e l ] M a r II. EXPERIMENTAL DETAILS
The present study was performed on the same fullydeuterated IPA-CuCl and x = 0 .
05 IPAX (IPAX-0.05for short) single crystal samples as used in Refs. 27 and1, respectively. Thermal-relaxation calorimetry was car-ried out on ∼ He- He dilution refrigerator insert.The data were collected in two field orientations. A closeto axial geometry was realized with the magnetic fields H applied along the b ∗ -axis, which is conveniently per-pendicular to a natural cleavage plane of the crystals. Inthis orientation the field is only ∼ ◦ misaligned withthe magnetic easy axis. Alternatively, the field was ap-plied along the crystallographic c -direction, making italmost perpendicular to the anisotropy axis (transversegeometry).A new series of neutron diffraction measurements werecarried out on the x = 0 .
05 IPAX compound using fully adeuterated 150 mg single crystal. The magnetic field wasapplied along the b direction. This setting almost exactlyrealizes the axial geometry, in contrast to previous studiesof Ref. 1 where the field was applied in the transversedirection. In our experiments, the sample environmentwas a 12 T cryomagnet with a dilution refrigerator insert.Diffraction was carried out using λ = 1 .
218 ˚A˜neutronsfrom a Cu-(200) monochromator.
III. RESULTS AND DATA ANALYSISA. Calorimetry
Typical magnetic C ( T ) and C ( H ) curves measured forIPAX and IPA-CuCl are shown in Figs. 1 and 2, re-spectively. In the constant-field data, the Debye latticecontribution was measured in zero applied field and sub-tracted from the data shown. In all cases, a peak inthe specific heat curve was taken as a signature of theonset of long range order. For the disorder-free samplethe lambda anomalies are perfectly sharp, but somewhatrounded in IPAX-0.05. In that case, the lambda anomalyis particularly broad in the transverse-field H (cid:107) c geome-try, especially below T ∼ T scans, shown in solidlines in Figs. 1 and 2, respectively. For the disorder-freesystem the transition was simply identified with the sharpmaximum in the measured data. The results are shown inFig. 3. The IPAX-0.05 phase boundary data measuredin the transverse geometry using neutron diffraction inRef. 1 are also shown. They appear to be consistent withthe present calorimetric measurements, despite the largerscattering of data points. Also shown in Fig. 3 are datafor the parent compound from Ref. 29 collected in the FIG. 1. Magnetic specific heat vs. temperature measuredin IPA-CuCl (a) and IPAX (b) for a field applied almostparallel to the anisotropy axis, and IPAX in a field almostperpendicular to it (c). Solid lines are empirical fits to locatethe maximum. transverse-field geometry H (cid:107) a .In order to estimate the exponents φ , the C ( H ) phase-boundary data were analyzed using power-law fits. Thelatter were performed in a progressively shrinking tem-perature range, following the procedure described inRef. 12. The fit parameters were φ , H c and an over-all scale factor. The dependencies of the fitted value of φ on the temperature range used is plotted in Fig. 4. Wesee that the results are very stable for all fitting rangesbelow T max = 0 . B. Neutron diffraction
One of the more spectacular observations of the previ-ous neutron study of IPAX-0.05 in the transverse field ge-ometry was the substantial and history-dependent broad-ening of the antiferromagnetic Bragg peaks in the highfield phase. A key result of the present work is that thiseffect, while still present, is drastically reduced in the
FIG. 2. Specific heat vs. applied field, as measured in IPA-CuCl (a) and IPAX (b,c). The field is almost parallel to theanisotropy axis in (a,b), and almost perpendicular to it n (c).Solid lines are empirical fits to locate the maximum. φ H c (T) H (cid:107) b ∗ :IPA-CuCl H ⊥ b ∗ :IPA-CuCl ( H (cid:107) c ) 0.50(1) 9.99(1)IPAX-0.05 ( H (cid:107) a ) 0.49(1) 10.48(1)TABLE I. Parameters of power-law fits to the calorimetricphase boundary data in a temperature range T < T max =0 . in H ⊥ b ∗ was based ondata from Ref. 29. almost axially symmetric configuration. FIG. 3. Magnetic phase diagrams measured for IPA-CuCl (top) and IPAX-0.05 (bottom) for magnetic fields appliedalmost parallel to the easy anisotropy axis ( b ∗ -direction),and almost transverse to it. Solid and open squares arepositions of specific heat maxima in constant-temperatureand constant-field scans, respectively. Diamonds are neutrondiffraction data from Ref. 1. The H ⊥ b ∗ data for the parentcompound are from Ref. 29. Solid lines are power law fits asdescribed in the text. Figure 5 shows scans across the (0 . , . ,
0) magneticreflections in the high-field phase of IPAX-0.05 for themagnetic field applied almost perpendicular (Ref. 1) andalmost parallel to the anisotropy axis (this work). Thesedata were collected following either a field-cooling (FC)or zero-field cooling (ZFC) protocols. A Gaussian fit toinstrumental resolution is in all cases shown in a dashedline. The resolution was determined by measuring theweak nuclear scattering contribution at (0 . , . ,
0) dueto λ/ numeri-cally convoluted with the instrument resolution. The fitsare show in Figure 5 in solid lines and allow us to extract FIG. 4. Shrinking-fit-window analysis of the measured phaseboundaries. The plots show the least-squares fitted values ofthe phase boundary exponent φ vs. the temperature rangeused for the fit. The other two fit parameters, H c and anoverall scaling factor, are not shown. the correlation length ξ , tabulated in Table II.For a transverse field, the history-dependent shortrange order in IPAX-0.05 is also manifest in a hystereticfield- and temperature-dependencies of the diffractionpeak height. Such a hysteresis measurement for the closeto axial geometry is shown in Fig. 6. The sample was firstcooled in zero field to T = 50 mK, then the magnetic fieldwas raised to H = 11 . Overall, in theobserved behavior in the close to axial geometry is simi-lar to that previously seen in (C H N )Cu (Cl − x Br x ) ,but with an even smaller hysteretic effect. FIG. 5. Magnetic (0 . , . ,
0) Bragg peaks measured in IPAX-0.05 in for a magnetic field H = 13 T applied almost per-pendicular (a, b) and and almost parallel (c, d) to the easyanisotropy axis. The data were collected using field-colling(FC, open symbols) and zero-field-cooling (ZFC, closed sym-bols) protocols. The dash-dot lines represent experimentalGaussian resolution that was measured as described in thetext. Solid lines are fits to the data of Lorentzian-squaredfunctions folded with the experimental resolution. The ex-perimental data in panels (a) and (b) are from Ref. 1. ξ b (˚A) ξ c (˚A) ⊥ -geometry (Ref. 1):FC 30(4) 94(7)ZFC 8.3(1) 50(5) (cid:107) -geometry:FC 97(6) 83(4)ZFC 97(6) 83(4)TABLE II. Magnetic correlation length measured in IPAX-0.05 for a magnetic field applied almost perpendicular andalmost parallel to the anisotropy axis, in field-cooled andzero-field-cooled samples. The values are obtained fromLorentzian-squared fits to the diffraction data, as describedin the text. IV. DISCUSSION
The first thing to note is the clear differences in behav-ior of the disorder-free system in the two orientations. Inagreement with Ref. 28, for the off-axial field, BEC ofmagnons is a poor description for the criticality of thetransition. The measured value of the phase boundaryexponent is in better agreement with that of the 3+1 di-mensional Ising model ( φ = 1 /
2, since z = 1 and ν = 1 / FIG. 6. Peak intensity of the (0 . , . ,
0) Bragg reflection mea-sured in IPAX-0.05 for a magnetic field applied almost par-allel to the easy anisotropy axis. The measurements are per-formed following zero-field-cooling the sample, and throughan isothermal field ramping at low temperature, warming toabove the transition tempereture, field-cooling and an isother-mal ramping down of the applied field. The observed his-tory dependence is much smaller than previously seen in thetransverse-field geometry. at the upper critical dimension) then with φ = 2 / That the material behaves as an Ising system for H ⊥ b at T (cid:46) a ∗ , b ∗ )-plane is itself of the or-der of 0.5 K. Similar residual anisotropy has beena complication for most examples of magnon BEC inquantum magnets, including TlCuCl . Moreover,in IPA-CuCl , due to a low crystal symmetry, a truly ax-ially symmetric geometry can not be perfectly restoredby an choice of field direction. Nevertheless, applying afield close to the principal axis of anisotropy seems toapproximate the BEC scenario reasonably well, at leastas far as the measured critical exponent is concerned.For the disordered material, the short range orderingand history dependence is clearly strongest in the trans-verse field geometry. Therefore, it must be interpretedas an effect of disorder not on a BEC of magnons, buton a transition that is in the Ising universality class.The chemical Cl/Br disorder in IPAX will have severalconsequences for the magnetic Hamiltonian. First, thestrength of antiferromagneic interactions will be affecteddue to variances in Cu-Cl and Cu-Br covalency strengthsand distortions of superexchange bond angles. This typeof disorder is exactly what translates into a random po-tential for bosons, as envisioned in the magnetic BoseGlass picture. In addition however, distortions of thelocal Cu crystallographic environments will lead ro arandom component to the magnetic ion’s gyromagnetictensor. In the presence of a uniform external magneticfield, this will effectively generate a random spin field in the sample. It’s component along the applied uni-form field couples to magnetization, i.e., magnon density.Thus it adds to the random potential for magnons. How-ever, the transverse component of the spin field is directly coupled to the order parameter of the phase transition,namely the transverse magnetization. Such random fieldsare known to have drastic effects on phase transitions. The third effect of chemical disorder is random two-ion(exchange) anisotropy. In the presence of a uniform ex-ternal field a random anisotropy becomes again equiva-lent to a random field, with all the same consequences. We conclude that for a field applied perpendicular to theprincipal anisotropy axis the phase transition in IPAXis to be viewed as that in a Random Field and RandomBond (RF+RB) Ising transition, and surely not as a BECto BG transition, as implied in Ref. 1.The ideal RF Ising model is known to order in threedimensions. The quantum and thermodynamic phasetransitions are, in fact, governed by the same fluctua-tionless fixed-point. Nevertheless, the actual behaviorobserved in prototype materials such as diluted AFs in auniform field is quite different.
Rather than showinglong range order, these systems, of which Mn x Zn − x F isperhaps the best known example, go through a freezingtransition. Below that point there is static but only shortrange order. The system breaks up into microscopic do-mains. The domain size is history dependent, but canonly increase as long as one doesn’t exit into the param-agnetic phase. There is also no temporal relaxation of do-main size. This anomalous behavior has been attributedto the the simultaneous presence of RF and RB. Thissituation exactly corresponds to IPAX. Indeed, in thetransverse field configuration we observe almost exactlythe type of behavior as seen in Mn x Zn − x F at a fixedrandom-field strength.The pinning of domains is governed by the widthof the domain walls, i.e., by the relative magni-tude of anisotropic vs. axially symmetric magneticinteractions. For IPAX, the off-axial component ofanisotropy is weakest in the close to axial geometries.In these configurations the pinning and resulting history-dependent behavior can thus expected to be much weakerthan in the transverse-field case. This is totally con-sistent with our observations. In IPAX-0.05 in a fieldapplied almost along the anisotropy axis, the small hys-teretic effects and internal line width aside, we observewhat almost is a long-range-ordered phase. Perhapsby coincidence, adding disorder has almost no effecton the critical field H c in this case. Nonetheless, thephase boundary significantly modified. The measuredcritical exponent is very similar to that observed undersimilar conditions in (C H N )Cu (Cl − x Br x ) andNiCl − x Br x · ) . Regardless of whether φ ∼ and whether that scenario is at all rel-evant for IPAX-0.05 in the close-to-axial configuration,experimentally it seems to be a rather common situationin disordered magnets. 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