A new collective phenomenon arising from spin anisotropic perturbations to a Heisenberg square lattice manifested in paramagnetic resonance experiments
S. Cox, R.D. McDonald, J. Singleton, S. Miller, P.A. Goddard, S. El Shawish, J. Bonca, J.A. Schlueter, J.L. Manson
AA new collective phenomenon arising from spin anisotropic perturbations to aHeisenberg square lattice manifested in paramagnetic resonance experiments
S. Cox, R.D. McDonald, ∗ J. Singleton, S. Miller, P.A. Goddard, S. El Shawish, J. Bonca,
4, 3
J.A. Schlueter, and J.L. Manson National High Magnetic Field Laboratory, Los Alamos National Laboratory, MS-E536, Los Alamos, NM 87545, USA Clarendon Laboratory, Department of Physics, Oxford University, Oxford, UK OX1 3PU J. Stefan Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia Faculty of Mathematics and Physics, University of Ljubljana, Jadranska, 1000 Ljubljana, Slovenia Materials Science Division, Argonne National Laboratory, Argonne, IL 60439 USA Department of Chemistry and Biochemistry, Eastern Washington University, Cheney, WA 99004, USA
We report unexpected behaviour in a family of Cu spin- systems, in which an apparent gap inthe low energy magneto-optical absorption spectrum opens at low temperature. This previously-unreported collective phenomenon arises at temperatures where the energy of the dominant exchangeinteraction exceeds the thermal energy. Simulations of the observed shifts in electron paramagneticresonance spectral weight, which include spin anisotropy, reproduce this behavior yielding the mag-nitude of the spin anisotropy in these compounds. PACS numbers: 76.30-v, 75.10.Pq
The spin ( S ) two-dimensional (2D) square-latticequantum Heisenberg antiferromagnet system has longbeen interesting to theoretical physicists due to the va-riety of transitions that can arise [1, 2, 3]. Moreover,the role of S = fluctuations on a square lattice inthe mechanism for cuprate superconductivity is hotly de-bated [3, 4, 5, 6]. The recently discovered family of H-bonded metal-organic magnets [7] offer the possibility toreadily control the exchange parameters in a 2D systemby changing chemical composition, thus creating spin ar-chitectures with desirable properties ‘to order’ [8]. For anidealized 2D system, long range magnetic order would notoccur at finite temperature [2, 9]. However, in the metal-organic systems, interlayer coupling gives rise to a finiteNeel temperature [10, 11, 12]. For these quasi-2D systemsthe ordering temperature is dominated by the weakest(the interlayer) exchange interaction, whereas the satu-ration magnetic field is dominated by the strongest ex-change interactions, thus providing a means of estimat-ing the spatial exchange anisotropy in the system [8]. Itshould be noted that the more 2D the system, the widerthe temperature ( T ) range, T N < T < J/k B , over whichmagnetic fluctuations dominate. Here we demonstratethat a spin anisotropy perturbation to the Heisenbergsquare lattice results in a new collective phenomenonwithin this regime, manifested as a shift in electron para-magnetic resonance (EPR) frequency at low temperature.Fig 1 illustrates the dramatic shift of the EPR mag-netic field and line width with temperature for a singlefrequency, f . Multi-frequency EPR measurements, fit-ted to f = f + gµ B B/h , where B is the resonance field,show that the shifts correspond to changes in both thelow-field intercept f and effective g-factor g . At high T , f = 0, but as the phenomenon develops with decreasing T , f becomes finite and positive, behavior that stronglyresembles an energy gap opening. Not only is this un- FIG. 1: a) The Q-factor of the 26 GHz cavity-mode as a func-tion of magnetic field, illustrating the evolution of the EPRline width and resonant magnetic field with temperature in[CuHF (pyz) ]ClO . The data is for magnetic field appliedperpendicular to the planes. (b) Low-temperature magneti-sation data in the same compound [8]. expected in an S = system; it is ruled out by the lowtemperature magnetization, which shows a monotonic in-crease between H = 0 and the onset of saturation (seeFig. 1). Any gap would be manifested as a region ofreduced or zero d M /d H [8].We shall show below that this dramatic change is notlinked to the antiferromagnetic transition temperaturein either of the compounds studied, but is rather relatedto the intralayer exchange energy J . The metal-organiccompounds [CuHF (pyz) ]X were therefore selected forthis study as the choice of anion molecule X can al-ter J controllably by a factor ∼ J a r X i v : . [ c ond - m a t . s t r- e l ] A ug values have been determined to a high accuracy usingmagnetometry [8]. In these compounds, Cu ions arearranged in square-lattice layers, separated by pyrazine(pyz) molecules. The layers are held apart by bifluoridebridges [7, 8, 12]. Single crystals of [CuHF (pyz) ]X withX = ClO or PF , were produced by an aqueous chemicalreaction between the appropriate CuX salt and stoichio-metric amounts of the ligands (see [7, 13, 14] for prepa-ration method details and X-ray data). Both materialsundergo antiferromagnetic ordering, with T N = 1 .
94 Kfor X = ClO and T N = 4 .
31 K for X = PF . Note thatowing to differing summing conventions the definition of J used in the current paper is a factor two smaller thanthat in Ref. [8], i.e. herein J plane (X = ClO ) = 3.6 Kand J plane (X = PF ) = 6.2 K.EPR spectra of single-crystal samples were obtained intwo ways. In the first method, the sample was placed ina cylindrical resonant cavity and the Q-factor and reso-nant frequency of the cavity were measured at each fieldpoint. The change in Q-factor is proportional to the mi-crowave absorption of the sample [15]. These measure-ments were carried out in the frequency range 11-40 GHzusing a Hewlett-Packard 8722ET network analyzer. Thecavity was placed in a He flow cryostat that was capa-ble of stabilizing temperatures down to 1.5 K. For higherfrequencies, the sample was placed in a confocal etalonresonator. The transmission, and hence change in mi-crowave absorption, was measured in the frequency range60-120 GHz using an ABmm [16] Millimetre wave VectorNetwork Analyser. The etalon was placed in a He sys-tem that was capable of stabilizing temperatures down to0.6 K. For fixed temperatures, the EPR field was foundto be a linear function of frequency, allowing the inter-cept and effective g-factor to be determined as describedabove; experimental values will be compared with theo-retical predictions below.Before demonstrating that the shift in intercept f andeffective g-factor g (Fig. 1) is a collective phenomenoninvolving a large number of spins, we show that sim-pler models such as dimerization cannot account forthe data. Fig. 2a) illustrates the Zeeman splitting fornon-interacting electrons and b) the corresponding lin-ear frequency-magnetic field relationship with a zero fre-quency intercept [15]. Fig. 2c) and d) includes the effectof an antiferromagnetic scalar exchange interaction J .The | ↑↑(cid:105) , √ ( | ↑↓(cid:105) + | ↓↑(cid:105) ) and | ↓↓(cid:105) states will be degen-erate at B = 0, with the √ ( | ↑↓(cid:105) − | ↓↑(cid:105) ) state separatedby an energy − J [15]. Although this singlet-triplet gapcauses the optically active ( δS z = ± , δS Tot = 0) EPRtransitions within the triplet states to become ‘frozen out’at
T < J , it does not introduce a gap across which opti-cally active EPR transitions occur, which would lead toa finite frequency intercept.Fig. 2e) and f) includes a spin anisotropy in the ex-change interaction, J zz (cid:54) = J xx = J yy , lifting the zero field Magnetic Field E n e r gy € ↑€ ↓ a) Magnetic Field F r e qu e n c y b) Magnetic Field E n e r gy € ↑↓ −↓↑€ ↑↓ + ↓↑€ ↓↓ € ↑↑ e) Magnetic Field F r e qu e n c y f) Magnetic Field E n e r gy € ↑↓ −↓↑€ ↑↓ + ↓↑€ ↓↓ € ↑↑ c) Magnetic Field F r e qu e n c y d) FIG. 2: Illustration of how a spin-anisotropic exchange in-teraction can give rise to a an EPR spectrum with a finitefrequency intercept in the case of an antiferromagneticallycoupled dimer. Respectively (a), (c) and (e) show the Zee-man splitting for a non-interacting, isotropically coupled andspin-anisotropically coupled dimer. (b), (d) and (f) illustratethe corresponding frequency-magnetic field relationships ofthe resonant absorption. degeneracy of the S Tot = 1 , S z = ± S Tot = 1 , S z =0 states (Fig. 2), which in turn lifts the field degeneracyof the two intratriplet transitions, separating EPR lines2 and 3. Although some of the possible EPR lines nowhave finite frequency B = 0 intercepts, this simple modeldoes not reproduce the systematics of the effect reportedhere. √ ( | ↑↓(cid:105) − | ↓↑(cid:105) ) is the groundstate so that thetwo transitions potentially observable at low temperaturewill be lines 1 and 4; however, for these to be optically-active, off-diagonal exchange terms (e.g. J xy ) such as theDzyaloshinsky-Moriya interaction must be present (sincethey mix the singlet and triplet states) [15]. We notethat despite a Dzyaloshinsky-Moriya term arising froma lower order spin-orbit perturbation than the diagonalspin anisotropy terms, that they are precluded by the in-version symmetry about the mid point of the dominantexchange interaction. At elevated temperatures T > J ,one would observe an EPR spectrum dominated by lines2 and 3 (each with a finite B = 0 intercept); as the sam-ple cooled to T < J , this would change to a spectrumdominated by the much weaker (or even absent) lines 1and 4, with an interecept differing from that of 2 and 3.Such behavior does not lead to a smooth thermal evolu-tion of the EPR line or the preservation of spectral weightto temperatures
T < J that we observe.Having shown that a simple local distortion cannotcause the effect shown in Fig. 1, we now turn to a finitecluster approach applied to an anisotropic 2D Heisenbergmodel with spin anisotropy for two field orientations: H = J (cid:88) S zi S zj + ∆(S xi S xj + S yi S yj ) − g T µ B B (cid:88) S z , yi , where ∆ is the spin anisotropy, with ∆ = 1 . < > xy -antiferromagnet. It should be noted that the input pa-rameter to the model g T is a temperature independent g-tensor that reproduces the high temperature ( T >> J ) g-factor anisotropy arising from the spin-orbit interaction.This g-anisotropy is consistent with the magnetic d x − y orbital lying in the 2D planes [8]. J perp is assumed to bezero, since J perp << J plane , as demonstrated in [8]. Thismodel was calculated using full diagonalization at finitetemperature for 12, 14 and 16 sites [17]. There was littledifference between the results with different numbers ofsites, and therefore only the 14 site data are displayedhere. The simulations were carried out for anisotropyvalues ∆=1.02, 1.04, 1.06 and 1.08. It was found that in-creased spin anisotropy leads to the parallel shift of thepeak away from its initial position while at the same timethe peak gains a finite width. Multiple smaller peaks inaddition to the main peak were observed in the simula-tions for ∆ (cid:54) = 1, due to finite-size effects.To compare the simulations to the results for[CuHF (pyz) ]ClO , the resonant magnetic field, Lorentzlinewidth and spectral weight of the resonance were cal-culated for a resonance at 26 GHz. The resonant mag-netic field was calculated as B R = 2 B − (cid:104) ω (cid:105) , where B = 0 . J (26 GHz in units of J ) and (cid:104) ω (cid:105) = (cid:82) ω S xx ( ω, B )d ω (cid:82) S xx ( ω, B )d ω The Lorentz linewidth∆ B ∝ (cid:115) (cid:104) ω (cid:105) (cid:104) ω (cid:105) where ω rel = ω − (cid:104) ω (cid:105) . The spectral weight was calculatedas: I = (cid:90) ξ (cid:48)(cid:48) ( ω , B )d B ∼ (1 − e − ω /T ) (cid:90) S xx ( ω, B )d ω with the dynamic spin structure factorS xx ( ω ) = 1 N Re (cid:90) ∞ e ( iωt ) (cid:104) S x ( t )S x (0) (cid:105) d t. A Lorentzian fit to the experimental data yielded thelinewidth and spectral weight. As can be seen fromFig. 3, the results of the simulations reproduce the salientfeatures of experimental results for [CuHF (pyz) ]ClO .The experimental values clearly lie between the simu-lated values of ∆ = 1 .
04 and 1.08. Taking into accountthe fact that above T = 0 . J the 1.04 simulation pro-vides a good fit for both the position of the resonance and its linewidth, the best match for the data is given bya spin anisotropy of 1.04.From Fig 3 the changes in the position and widthof the EPR line in [CuHF (pyz) ]ClO , start as thetemperature is lowered through 3.5 K; this correspondsto T /J =1 (as opposed to T N = 1 .
94 K). Turning to[CuHF (pyz) ]PF , Fig. 4 shows that both g and f undergo a dramatic change on cooling through 4.5 K.Coincidentally, this is close to the ordering temperature T N [8]. The greater relative thermal separation of T N and T = J in the ClO compound than in the PF salt re-flects the smaller interlayer exchange energy; one mightsay that the former is a closer approximation to two di-mensionality than the PF compound [8]. In spite of theproximity of the ordering temperature to the onset ofthe EPR shifts, the fit of the model to the data from thePF compound that we will now give shows that it is theexchange interaction J that determines the temperaturescale of the effect and not T N .To compare the simulations for the results for[CuHF (pyz) ]PF the experimental g-factors and fre-quency intercepts were calculated. The theoretical g-factors and intercepts are given by the slopes and in-tercepts of the ω, B ) plots, which were obtained fromthe extrapolation through five calculated points, ω/ J =1 . , . , . , . , . B perpendicular to and B parallel to the square Cu planebeing reproduced. For the B parallel data, the fact thatwe do not observe an upturn in the g-factor and the smallpositive intercept at low temperature is most likely due toa small angular misalignment (since the B perpendiculareffects are so much larger).Considering the B perpendicular data, the g-factor ex-perimental data suggests a spin anisotropy above 1.08,whereas the intercept data suggests an anisotropy closeto 1.04. If we consider the corresponding data for X= ClO (for which data was taken at only a few valuesdue to the extreme weakness of the signal at low tem-peratures) we find that at T /J = 0 .
27 the experimentalvalues are g = 0 . g and intercept = 0 . J . This givesa similar anisotropy, 1.06 - 1.04. For both materials itcan be clearly seen that for T /J < i.e. the phase tran-sition has the effect of contracting the entropic (theoreti-cal temperature) scale. The shape of the intercept curvesuggests that the experimental value of the intercept hasnearly stabilised by the lowest
T /J values and thereforethe intercept is taken to give the best indication of the R e s o n a n t m a g n e t i c f i e l d ( T ) T/J g s µ B B / J a) L i n e w i d t h ( T ) g s µ B B / J b) N o r m a li z e d s p e c t r a l w e i g h t T N c) FIG. 3: A comparison of the experimental EPR data, cir-cles, from [CuHF (pyz) ]ClO measured at 26 GHz with the-oretical simulations, ∆ =1.02, 1.04, 1.06, 1.08, 1.10; solid,dot, dash, dot-dash and dot-dot-dash respectively. The leftand bottom axes are absolute units, the right and top axesare renormalized by the exchange energy and spectroscopicg-factor. a) The variation of the magnetic field at which reso-nance occurs as a function of temperature. b) The linewidth(full width at half maximum of a Lorentzian fit) of the res-onance peak as a function of temperature. c) The spectralweight (area under Lorentzian fit) normalized to its 15 Kvalue. In all the cases error bars are smaller than data points. spin anisotropy (around 1.04 in both compounds). Theapproximate conservation of spectral weight in both ex-perimental and theoretical results also demonstrates therobustness of our model.In conclusion we have observed a collective phe-nomenon in two members of the organic magnet system(Cu(HF )(pyz) )X that produces a shift in the frequencyintercept of the EPR data which resembles, but doesnot correspond to, a gap opening in the system. Analo-gously to an anisotropic g-tensor, the spin anisotropy inthe exchange interaction responsible for this effect mostlikely originates from spin-orbit coupling. As a result,this collective phenomenon is expected to be strongestfor low-spin transition metal ions in relatively low sym-metry environments, like the octahedral copper site (3 d )in (Cu(HF )(pyz) )X. Although the organic systems in-vestigated provide an ideal ‘low exchange energy scale’ g - f a c t o r T/J I n t e r c e p t ( G H z ) Temperature (K) g - f a c t o r T/J Temperature (K) I n t e r c e p t ( J ) a) b)c) d) FIG. 4: Comparison of the experimental g-factor for X =PF to simulations for different values of the spin anisotropy,∆ =1.02, 1.04, 1.06, and 1.08, solid, dotted, dashed and dot-dashed lines respectively, for (a) B perpendicular to and (b) B parallel to the Cu square latttice. Also, comparison ofthe experimental frequency intercept of resonant frequencyvs magnetic field and simulated values for (c) B parallel and(d) B perpendicular. environment in which to characterize this effect, stronglycoupled copper octahedra are ubiquitous in correlatedelectron systems; for example, J ≈ (cid:48) s K in the parentphase of the high T C superconductors.Work at the NHMFL occurs under the auspices ofthe National Science Foundation, DoE and the State ofFlorida. Work at Argonne is supported by a U.S. De-partment of Energy Office of Science laboratory, operatedunder Contract No. DE-AC02-06CH11357. The authorswould like to thank Pinaki Sengupta and Cristian Batistafor valuable discussions. ∗ Electronic address: [email protected][1] M. Kastner, R. Birgenau, and G. S. amd Y. Endoh, Rev.Mod. Phys. , 897 (1998).[2] U. Schollw¨ock, D. Farnell, and R. Bishop, eds., Quantummagnetism (Springer, Berlin, 2004).[3] E. Manousakis, Rev. Mod. Phys. , 1 (1991).[4] J. Schrieffer and J. Brooks, eds., High temperature su-perconductivity theory and experiment (Springer, Berlin,2007).[5] S. Julian and M. Norman, Nature , 537 (2007).[6] R. McDonald, N.Harrison, and J. Singleton, J. Phys. CM , 012201 (2009).[7] J.L. Manson, M.M. Conner, J.A. Schlueter, T. Lan-caster, S.J. Blundell, M.L. Brooks, T. Papageorgiou,A.D. Bianchi, J. Wosnitza, M.H. Wangbo, Chem. Com-mum. p. 4894 (2006).[8] P.A. Goddard, J. Singleton, P. Sengupta, R.D. McDon-ald, T. Lancaster, S.J. Blundell, F.L. Pratt, S. Cox, N.Harrison, J.L. Manson, H.I. Southerland, J.A. Schlueter,New Journal of Physics , 083025 (2008). [9] N. Mermin and H. Wagner, Phys. Rev. Lett. , 1133(1966).[10] N. Christensen, H. Ronnow, D. McMorrow, A. Harri-son, T. Perring, M. Enderle, R. Coldea, L. Regnault,and G. Aeppli, Proc. Nat. Acad. Sci. , 15264 (2007).[11] J. Choi, J. Woodward, J. Musfeldt, C. Landee, andM. Turnbull, Chem. of Materials , 2797 (2003).[12] T. Lancaster, S. Blundell, M. Brooks, P. Baker, F. Pratt,J. Manson, M. Conner, F. Xiao, C. Landee, F. Chaves,et al., Phys. Rev. B , 094421 (2007).[13] J.L. Manson, H. Southerland, J. Schlueter and K. Funk,preprint (2008). [14] T. Lancaster, S.J. Blundell, P.J. Baker, M.L. Brooks,W. Hayes, F.L. Pratt, J.L. Manson M.M. Conner, J.A.Schlueter, Phys. Rev. Lett. , 267601 (2007).[15] A. Abragam and B. Bleaney, Electron Paramagnetic Res-onance of Transition Ions (Oxford University Press, NewYork, 1970).[16] R. D. McDonald et al., Rev. Sci. Instrum. , 084702(2006).[17] for B//y the maximum number of sites used was 14 dueto the lack of S zz