Spin dynamics and magnetic resonance in two-dimensional Heisenberg antiferromagnets

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    Spin dynamics and magnetic resonance in twoAimensional Heisenberg antiferromagnets

    Rajiv R. P. SinghDepartment of Physics. University of California, Davis, California 95616

    Martin P. GelfandDepartment of Physics, University of California, Los Angeles, California 90024

    (Received 20 February l990; revised manuscript received 5 April 1990)

    High-temperature expansions are calculated and extrapolated to yield estimates of low-ordertime derivatives of one- and two-point dynamic correlations in the S 2 Heisenberg antifer-romagnet. Taking this model to describe the spin correlations in Y-aa-Cu-O, spin-lattice relaxa-tion rates (T~ ') for Y and planar Cu and 0 nuclei are estimated by means of Moriya's Gaussianapproximation. For Y and 0, (T~T) exhibits a broad plateau in the range 0.7& T/J & 1.5,and in that interval their Korringa ratios are also nearly constant, being 0.25 and 1.1 times thefree-e1ectron value, respectively.

    The normal state of perovskite superconductors hasbeen the subject of many recent magnetic resonance stud-ies. ' Analyses of Knight shifts, particularly in YBa2-Cu306+ have yielded insight into the real-space distri-bution of spin degrees of freedom. '2 Measurements ofspin-lattice relaxation rates T~ have provided evidenceof antiferromagnetic correlations between spins residingon planar Cu sites, even in the oxygenated materials.Here we study the spin dynamics in the 5 -' square lat-tice Heisenberg model,

    &spn-JZSt'SJ, J &0 (i)&ij )

    (We take 5-' ,so that J is an energy. ) For the insulat-ing parent compounds, there is a preponderance of evi-dence' that (I ) well describes the spin degrees of freedom.Our calculations should thus be directly applicable tothese systems. The possible relevance of our results to ex-periments in doped materials will be discussed at the endof the paper. We focus attention specifically on Y-Ba-Cu-O.

    We obtain high-temperature expansions for low-orderterms in short-time expansions of one- and (various) two-point correlation functions for the model (I). These ex-pansions, as well as that for the uniform susceptibility gare extrapolated by changes of variables and Pade approx-irnants; the estimates appear to be accurate, within a fewpercent, down to T 0. The task of estimating the dy-namic correlations at the Zeeman frequency to-J/hfrom the short-time expansions is not trivial, but this is re-quired to determine T~ '. Here we adopt the Gaussian

    approximation of Moriya, which is discussed below. Wealso employ simple hyperfine couplings for the various nu-clei. Our principle results are displayed in Fig. 2. Forplanar Cu, (T~T) increases with decreasing T, whilefor Y and planar 0, (T~T) ' attains a broad maximumat T =J. Indeed, over the range 0.7 & T/J & 1.5,(T~T) for Y varies by only 3%, and for 0 it variessomewhat more, roughly 12%. In both cases, the Korrin-ga ratios (T~TK ) ' (with K the Knight shift) vary byless than 10% over that range, and the numerical valuesfor Y and 0 are 0.25 and 1.1 times the free-electron result(4~/r )(y./y, ) '.

    Consider first the Cu sites, where the spins S; are locat-ed. An analysis similar to the present one (differing onlyin its use of lower-order expansions) has been given byChakravarty and Orbach. We assume the hyperfine cou-pling to be isotropic and on-site, with strength Ac. (Inreality, c/ttt, t is quite anisotropic, but for a disorderedelectronic spin system this will not affect the temperaturedependence of Ti '. It will, however, affect the Korringaratio, which, therefore, should not be directly compar-ed with experiment. ) The nuclear relaxation rate is cT~

    4 (Ac/h )Cc(to ), where Cc(to) is the Fouriertransform of the dynamic correlation function Cc(t)

    -', (S~(t) S~(0)). Here, angle brackets denote thermalaverages, and co- is so small compared to all other energyscales that it may be set to zero.

    Let the short-time expansion of C(t ) be


    996 @199QThe American Physical Society


    C(()) [ (o)]2[~(2)/ ](/2

    & exp( [p ' '] '-/2M "') (3)The resultin~ cTi is displayed in Fig. 2(a); note thatcT| =z' Ac/4h J. Let us now discuss the reliabilityof this approximation.

    Consider first the situation at T . For this case, we

    TABLE l. CoeScients c, of the high-temperature seriesgc,(J/T)"' for the p'"'.



    0.50.750.031 25

    0.242 1880.077 995

    0.072 2220.050627


    1.50.3750.468 750.096 354

    0.1106770.083 398




    0.500.093 750.023 438

    0.027 7340.016 113

    0.007 849


    1 125

    0.046 8750.055 5990.004622

    where the p" are, im licitly, functions of temperature.Quite generally, the p

    " may be written as moments ofthe spectral function C(to); since C(to) is necessarily positive, so are the p ". In particular, pP' 1 and p$'= ,' J(Si.Sq), where 1 and 2 are nearest-neighborsites. Since p$' is a multiple of the energy density, onemay obtain its expansion in powers of J/T from the litera-ture. Similar expressions may be constructed for thehigher moments, but, instead, we calculate the expansionsby a cluster method. For a given cluster, matrices forS,~;and S; can be constructed, and then the expressionfor p

    " as a multiple commutator may be evaluated per-turbatively, in powers of J/T. This procedure, which iscarried out entirely by computer programs, leads to theexpansions for the p(", which are displayed in Table I.

    There are now two tasks to consider: First, the extrapo-lation of the series, to yield estimates of the p'"', and,second, to estimate C(c0 0) given its low-order moments.Somewhat surprisingly, accurate extrapolation requiresrelatively little effort. Direct Pade approximation is un-suitable, as all the nondiagonal approximants will eithervanish or diverge at T 0. However, after transformingthe series to the variable y tanhJ/T, all Pades are atleast superficially well behaved. The principle weaknessof this scheme is that the approximants vary as+Be / + near T 0, regardless of how the func-tions actually behave.

    In Fig. 1, extrapolations are presented for p$), pP, andg. (The series for g is given in Ref. 7.) We may checktheir accuracy for p$and g by comparing them withquantum Monte Carlo at finite T, and with a different ex-pansion' at T 0. Apparently, they are quite reliableeven for T =0. This gives us confidence that all the ex-trapolations are likely to be accurate, at least within thelimits of consistency of the various Pades.

    The second task is more challenging. The simplest ap-proximation is to assume that C(t) is Gaussian. Usingthe second cumulant, ' M ) p ip [p(')], one ob-tains

    T/JFIG. 1. Plots of pq. ', pq=', and g. Error bars indicate the vari-

    ation among the various Pade approximants, which is negligiblefor p&''. The diamonds and squares are from Monte Carlo cal-culations on large (at least 32&& 32) lattices (Ref. 9); the crossesat T 0 are from Ref. 10.

    have obtained the p$'' up to n 14. By applying atheorem of Chebyshev, ' one can bound Cc(t) above andbelow at finite time; so long as the bounds lie close to oneanother, the function is well determined. One findss thatCc(t) decreases monotonically [in contrast to the one-dimensional (1D) case ], at least up to t =2h/J, atwhich point Cc(t ) =0.1. If one assumes that thelong-time behavior of C(t) does not contribute signifi-cantly to its time integral, Cc(0) is indeed well approxi-mated by (3). In principle, that assumption is unjustified.At long times, normal spin diffusion in 2D implies thatCc(t)I/t, ' ' and Cc(co) is thus logarithmicallydivergent as to 0. In real systems, however, there aremany cutoffs besides the finite to which prevent C fromdiverging, including interplanar coupling and anisotropy.Whatever the physical mechanism, so long as the effectivecutoff time is 10 (('t/J or less, the long-time contribution toCc(0)yields only a modest (10-20%) correction.

    For 30 spin systems, several approximations to thecorrelation functions at high T were compared with exper-iments by Gulley etal. ' The Gaussian approximationconsistently underestimated T( ' by roughly 30%, reason-able agreement was obtained with a more complicatedcorrelation function, using p to fix an additional param-eter. However, until suitable data become available forlayered CuO systems at T J so that a similar analysismay be performed, it seems reasonable to consider onlythe simple Gaussian form.

    As T is reduced the diflusion constant should increase,and make the long-time contribution to C(0) less impor-tant. However, lowering T below J has other, moredramatic eA'ects. In particular, antiferromagnetic fluctua-tions begin to soften. %e expect any calculation based ona short-time expansion to break down in the vicinity of theT 0 critical point; calculations for T (J/2 can be done


    2 One then obtains oT~ ' -' (A$/h )Co(0), with

    Cp(E) (S~(r)'S[(0)+S~(r)'Sp(0)&. (4)





    (c) YI

    I I I II

    I I I I1

    I I I I

    0 I I I I

    It is straightforward to show that pg 1 pP/4J, andpg"' -p$', while for pg, the expansion is given inTable I. The relaxation rate, estimated by the proceduredescribed above, is displayed in Fig. 2(b), with oT~

    z'~ A$/3 '~ h J. The scaled Korringa ratio is alsodisplayed, the Knight shift being 2Aog/)', y, .

    We can attempt to match these high-temperature re-sults for the relaxation rate to other calculations valid atlow temperatures, similar to the matching done for Cu byChakravarty and Orbach. Bulut etal. 2 ) provide de-tailed estimates of oT~ ', calculated on the basis ofSchwinger boson mean-field theory for T/J &0.6. Theirresults depend on a microscopic cutoff frequency e. As-suming this cutoff lies in the interval 10 4& hrp, /J&10 '-, their estimates cross ours in the temperature

    range 0.2 & T/J &0.5, which suggests that a smooth in-terpolation may be possible. A more detailed matching,however, would depend on the specific cutoff chosen, andwill not be attempted here.

    The Y atoms lie between CuO planes, at the centers ofplaquettes which have Cu atoms at the vertices. The sim-plest (and, again, a realistic) hyperfine interaction wouldbe an isotropic coupling to the eight closest Cu spins. Thespins in different planes are nearly uncorrelated above theNeel transition, and hence v T~

    ' 2(A$/5 )Cv(0), with

    (t ) -" (S~(r)' S~(0)+2S](r)' S2(0)+S (r)' S3(0)),(5)



    I I I I I I I I I I I I I I I

    FIG. 2. Plots of scaled T~ ', (TiT) ', and Korringa ratio yIfor (a) Cu, (b) 0, and (c) Y. Error bars, showing the uncer-tainties in estimated rates based on the variation of the Pade ap-proximants for the p'"', are shown for only one curve in eachplot; the relative uncertainty is the same for all curves.

    by invoking the dynamic scaling hypothesis.Finally, for the Cu sites, the Knight shift is given in the

    present notation by KcAcg/), ). Combining thiswith the preceding calculations for T~, we obtain esti-mates for the Korringa ratio, R=(T~TK ) '. For freeelectrons, this takes the value RFE (4m/t't)(y/y) atlow T; in Fig. 2(a) the scaled ratio Rc,/RFE is plotted.

    Let us now treat the 0 and Y nuclei. A planar 0 atomsits between a pair of nearest-neighbor Cu sites, and thesimplest hyperfine interaction (which appears to be realis-tic, for Y-Ba-Cu-0) is pShr AoIp (S~+S2), where Ip isthe nuclear spin and 1 and 2 denote the nearest Cu sites.

    where 1 and 2 are nearest-neighbors and I and 3 aresecond-nearest (i.e., nearest-diagonal) neighbors. In thiscase, pvt ' ,' pP', while the expansions for pv

    ' and pv'

    appear in Table I. Extrapolation and application of (3),as described above, yields the estimate of v T~ ' presentedin Fig. 2(c), where vTt ' 2(2z) A(/hJ. The Korrin-ga ratio is also plotted; note that it is primarily the in-crease in the number of Cu spins S; to which the nuclearspin is coupled, as one goes from Cu to 0 to Y, that ac-counts for the corresponding decrease in the Korringa ra-tio.

    These calculations show that Shastrys suggestion2(b)for the differing T dependences of the 0 and Cu relaxa-tion rates in high-T, materials can hold even outside ofthe dynamic scaling regime. However, their relevance tothe normal state of doped (conducting) Y-Ba-Cu-0remains unclear, as we have not accounted for any mobilecarriers. What is intriguing is that the results for theHeisenberg model for 0.7 & T/J & 1.5 are qualitativelysimilar to the experiments on YBa2Cu307 at temperatures100 K & T & 300 K. ' In that range, (cT~T) ' increasesas the temperature is lowered, whereas the correspondingratios for 0 and Y are nearly constant. The similaritymight appear superficial, as the exchange constant in theundoped system is greater than 1000 K, but there are fur-ther congruences. The magnetic correlation length is afew lattice spacings in both cases. Furthermore, at loweroxygen content both the spin susceptibility and (T~ T)for 0 and Y decrease as the temperature is lowered from


    100 K, rather reminiscent of Figs. 2(b) and 2(c) forT &0.7J. Finally, the Korringa ratios for Y and 0 inYBa2Cu30~ between T, and room temperature are'0.19 and' ' 1.4 times RFE, respectively; not too differentfrom the results for the Heisenberg model.

    A simple interpretation' is that J decreases with in-creasing doping, and is on the order of a few hundred de-grees. However, (CT~ T) ' also decreases with T forT & 100 K for the 067o materials, ' ' suggesting a spingap, which cannot be accounted for by the Heisenbergmodel. This interpretation is also inconsistent with thedoping dependence of T~ . within the Heisenberg model,the scale is set by 1/J, so a decrease in J with increaseddoping would imply a higher rate, contrary to the observa-tion of Imai etal. "' Finally, phenomenological anal-yses td'~t' of NMR experiments in YBa&Cu307 demanda large exchange coupling, on the order of a thousand,rather than a hundred, degrees. We conclude that thedoped systems cannot be modeled by a spin Hamiltonianwith a single energy scale. One should note, in this con-nection, that the Gaussian approximation can be appliedto the t-J model. Although the relevant moments haveonly been calculated at T =~, one does find that cT~decreases with doping, in agreement with the experimen-

    tal findings. Further studies of the short-time dynamics ofmodels of mobile fermions may prove informative.

    To summarize, we have calculated nuclear relaxationrates and Knight shifts for a 5= .' Heisenberg antifer-romagnet, making the simplest choices for the hyperfinecouplings consistent with the structure of Y-Ba-Cu-O.The estimation of T~ relied on the assumption that theappropriate dynamic correlation functions had a Gaussianform; if there is no overwhelming long-time contributionto the co =0 correlation function, the approximation maybe reasonable even though its accuracy is difficult togauge. %e have discussed the possible relevance of thesecalculations to experiments on Y-Ba-Cu-O.

    We thank S. Chakravarty, R. Orbach, A. Jagannathan,G. Zimanyi, D. Cox, D. Scalapino, G. Bickers, M. E.Fisher, R. Scalettar, D. Hone, and N. Bulut for discus-sions; one of us (M.P.G.) is particularly grateful to Pro-fessor Chakravarty and Professor Orbach for introducinghim to the theory of NMR, and many enlightening con-versations thereafter. This work was supported in partthrough National Science Foundation Grant No. DMR89-07664.

    'For example: (a) T. Imai etal. (unpublished); (b) H. Alloul,T. Ohno, and P. Mendels, Phys. Rev. Lett. 63, 1700 (1989);(c) W. W. Warren etal. , ibid 62, 1193. (1989); (d) C. H.Pennington etal. , Phys. Rev. B 39, 274 (1989); 32, 2902(1989); (e) P. C. Hammel et al. , Phys. Rev. Lett. 63, 1992(1989); M. Takigawa etal. , ibid 63, 1865 (.1989); M. Taki-gawa et al. , Phys. Rev. B 39, 7371 (1989).

    2See also (a) F. Mila and T. M. Rice, Physica C 157, 561(1989); (b) B. S. Shastry, Phys. Rev. Lett. 63, 1288 (1989);(c) A. J. Millis, H. Monien, and D. Pines, this issue, Phys.Rev. B 42, 167 (1990); (d) N. Bulut etal. , ibid 41, 1797.(1990); (e) D. L. Cox and B. R. Trees, ibid 41, 11260.(1990).

    3S. Chakravarty, in Proceedings of the 1989 Los Alamos Symposium, High Temperature Superconductivity, edited by K.S. Bedell, D. Coffey, D. E. Meltzer, D. Pines, and J. R.SchrieA'er (Addison-Wesley, Redwood City, CA, 1990), p.136.

    4T. Moriya, Prog. Theor. Phys. 16, 23 (1956); 16, 641 (1956).5S. Chakravarty and R. Orbach, Phys. Rev. Lett. 64, 224


    D. Forster, Hydrodynamic Fluctuations, Broken Symmetry,and Correlation Functions (Benjamin-Cummings, Reading,MA, 1975), especially Chaps. 2 and 3.

    7G. S. Rushbrooke, G. A. Baker, Jr., and P. J. Wood, in PhaseTransitions and Critical Phenomena, edited by C. Domb andM. S. Green (Academic, New York, 1974), Vol. 3.

    "R. R. P. Singh (unpublished).M. S. Makivic (private communication); see also H. -Q. Ding

    and M. S. Makivic, Phys. Rev. Lett. 64, 1449 (1990).'oR. R. P. Singh, Phys. Rev. B 39, 9760 (1989).''For all sites, ~e found that M' ' was estimated most reliably

    by using the approximants for the p'"', rather than by con-structing and extrapolating the series for the cumulants them-selves.

    ' J. M. R. Roldan, B. M. McCoy, and J. H. H. Perk, Physica136A, 255 (1986).P. M. Richards and M. B. Salamon, Phys. Rev. B 9, 32(1974).

    ~ "T. Morita, Phys. Rev. B 6, 3385 (1972).'s J. E. Gulley et al. , Phys. Rev. B 1, 1020 (1970).'6D. C. Johnston, Phys. Rev. Lett. 62, 956 (1989).


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