Volume 61, number 3 CHEIZIICAL PHYSICS LETIERS 1 &larch i979

SPIN DIFFUSION IN ~V~~~ENSIONAL ~~nX~-~iFE~~~~AGNETS

-WI-H DIPOLAR BROADENING *

T-T-P. CHEUNG, K-T. MCGREGOR+ and 2.G. SOOS Deparirment of clremistry, Princeton biziversity, Princeton. New Jersey 08540, US-4

Received 6 September i978 Revised manuscript received 3 November 1978

Comparison of the exchange-narroaed EPR linewidths AHof single crystals of K2XInF4 and f.3-propanediammonium JInCi4, both of which are two-dimensional antiferromamets with comparable isotropic exchange, yields direct evidence for spin diffusion due to the 4 = 0 secular dipolar term, with its cbamcteristic (3 cos2e - l)* angular dependence. A genem,l sczkling argument is developed for AH in isostructural systems wivith dipoku broadening in order to identify diffusive costri- butions and cut offs even when the absorption is lorentzian and the requkd spin correlation functions are not explicitly linOWl.

Electron pammagnetic resonance (EPR) studies [x,2] of (CH3)@MnC13 (TMMC) demonstrate one- dimensional (1-D) Heisenberg exchange between S = S/2 M&I) sites in MnCQ chains. Intrachain dipoIar broadening accounts for the shape, width, angular,

temperature, and frequency dependence of the ex- change-narrowed EPR absorption [2]. Richards [3] has reviewed the application of I&.&o-Tomita (KT) theory [4] for EPR in various lo~v-dimensional crystals of transition-metal complexes, whose static magnetic properties are discussed by de Jongh and Miedema IS]. KT theory applies to 3-D networks at high tem- perature, but memory function I63 techniques and improved evaluation of thermal averages allow the gen- eralization [7] to fmite temperature, where the ther- mal energy kT and the dominant isotropic exchange J are comparable. The latter is found from static suscep- tibility or magnetic heat capacity studies. The total- spin-torque (TSI) correlation function [24] G/(t) which is found by perturbation theory in the KT merhod, is cut off at iC as

* Supported by NSF GPCHE76-07377. + Present address: GCA-Technology Division, Bedford,

Bf~ssachusetts 01730, USA.

+ct) = @KT@) exP(-t/~c) W in Io~v-d~ensional systems, whereas spin diffusion [3] leads to $~,&t) a r-19(” at long f. General arguments [8,6] indicate that zc must be at least as large as a+r since the observed linewidth AHsets the experimental time scale.

Dipolar interactions between Ma(H) sites also dom- inate the broadening [9] in the 2-D antiferromagnet K$nF4, with square-planar MinF,$-layers. In con- trast to the 1-D EPR profile in TMMC, 2-D systems show only slight deviations from a lorentzian absorp- tion. Furthermore, numerical methods [9] are re- quired to match short- and long-time contributions to z&,&t) based on an assumed time evolution for un- known 2-D four-spin correlation functions_ Both the experimental and the theoretical analysis of spin dif- fusion in 2-D systems has consequently been approxi- mate and no estimates of the cut off 5c are available_

We take advantage of other [s] 2-D MnX$(X=F. Cl) crystals with primarily dipolar broadening to dem- onstrate diffusive effects even for largely lorentzian EPR absorption and without approximating the four- spin correlation functions in $&+ 2-D MnB$sys- terns [lo] will be deferred, as they probably involve other broadening mechanisms. The two isostructurai

Volume 61, number 3 CHEMKAL PHYSICS LETTERS 1 Lfarch 1979

cry%& considered are K$frrF~ and PDAMnCI4, where PDA is I ~-prop~ed~oni~_ Our method applies whenever &His dominated by dipoIar interac- tions between spins in similar networks_

The basic idea is that when long-time effects are neggigibie (7:’ = 0) and dipoku broadening dominates, the exchange-narrowed EPR linewidth AH in isostruc- turd systems scales appro-ximately as

2 = (+@Jt/J~- (2)

Here rl, r- are the fattice constants and Jt ,J2 are comparable isotropic exchange constants that both ex- ceed the Zceman energy “~0 =m#g- The PD&lnC14 parameters t1 I ] are ri = 5-14 a for the &In-Mn separation in the f3yer and [ I21 -J, = 4.6 K; those for K,MnF4 are f9] ‘2 = 4.20 A and -Jz = 4-I 5 K and lead to z = 3-72 in eq. (2). Since J1 and J2 we comp3rabie, the thermal averages of spin correla- tions functions [2_7] are essentially cquaI_ Since both excred fiwo by an order of magnitude for Ho =t 3 000 G, both secuiar and nonsecufar terms [3,4] contribute to &i_ Defming 0 = 0 to be the normal to the MnXz’ tayer, we obt3in the purefy experimental quantity

A(@) = AHI (63) - AH&i) 0)

shown in fig- l_ together with the observed 300 K NY(6) data for [9] K2MnF4 and PDAMnCI,. The Iat- ter \\ere collected at X-band (ffo ?- 3300G) as a func- tion of orientation and temperature. The fact that A(@) does not vanish, but Follows the known 193 (3 cos% - l)l angular dependence of the iong wave- Iength (q = 0) secular dipohr term, is direct evidence that this term has a cut off sc in eq- (I)_ Furthermore, rc for PD_UinQ is tsrger than rc for K,hfnF,, since d(0) is positive.

The scaie factor z in eq. (2) is derived for simplicity in the case of a ne& 1orentz.k EPR absorption, but c3n re3dlIy he treated in the general case [7] - The hdf- width t‘ of a lorentzian AH = 3-r/& is approximately [3,4.7]

fil) X cos(mwOr) exp(-r&) dr,

where fi(t) is the TST correlation function in ec_ (I), which diverges for D = 1 or 2 in the KT approxima-

45s

Fig- I- Peak-t-peak derivative linewidths AH of K2MnI_b and PDAMnt& st 300 K as a function of the an& 6 between the applied field and the normal to the DlnX$-layer_ The secular diffusive term A@> from eq. (3) follows a (3 cos% - 1)’ aa- @ar dependence.

lion of 7;’ = 0. Zu eq. <4), RZ = 0 denotes the secular dip&u term, whiIe the nonsecular 222 f 0 terms 3re modulated as cos(~zzw& The coupling constants A, go as r6 for dipolar broadening_ The composite index p denotes alI lattice sites in four-spin time correlation functions such as

j&r) = hYs~~t)s,xt)s,s;,,

where N is a normaIiz3tion constant for t = 0- The cru- cial point is that isotropic exchange (forJ%- fiw,, Ap)

governs the time evolution off,,(r) and dominates the thermal average < k Without decoupkg or evaluating any f m(t), it follows for large J that 5 = tfi/Jcl and K = kTp/ cl are reduced v3riabIes for time and tempera- ture, respectively, with the classical exchange Jcl = 2S(S + 1)J chosen for convenience- Any reduced t/J or T/J suffkes to show that changing variables in eq. (4) and setting r: ’ = 0 gives a r that scales as rm6J-“, thus obtaining z in eq- (2).

The short-time and m + 0 contributions to dW(or to r) do not diverge in the KT approximation and can be adequatefy handIed [2,3] without a cut off- Further- more, thermal averages [2,7] bf, (t) will be sIowIy changing functions for .h - ’ - kT/JcIs I_ It foIIows that both short-time and nonsecular contributions cancel

Volume 61, number 3 CHEMICAL PHYSICS LETTERS 1 March 1979

05- /- ,C

-05

/ /

/ , 8=0

00.’ 0

00 I 2 3 4

K= kT/JcI

Fig- 1. Temperature dependence of the normalized, 0 = 0 line- widths AH(l-)/~H(300) for K2BInF4 (+) and PDX\InCl4 (0) as a function of K = X-T//cl. The sea&r diffusive term (0) 1I(T)jA(300) decreases rapidly with temperature.

approximately for A(0) in eq. (3) whenever Jl , J2 are comparable and the magnetic ions form isostructural Iattices with identical p-summations in eq. (4). The secuIzr (n2 = 0) contributions to A(e), on the other hand, do not cancel unIess the cut off Tc for the two lattices is accidentally the same. Thus fig_ I shows A(0) to follow the known [9] angular behavior of the long-time (4 = 0) secular broadening, while fig_ 7_ gives the temperature dependence of A(0) between Ir; = 1.7 and 3-7 (T% 130 K to 300 K)_ Diffusive effects be- come Iess important at low temperature, in agreement with previous work on both [2] 1-D and [9] 2-D sys- tems.

The scaling argument can be estended to systems with unlike spins, provided that the magnetic ions are isostructural and that AH is dominated by dipolar broadening_ The cancellation of short-time contribu- tions to A(0) at finite temperatures is only exact for J1 = J2, while the approximate cancellation of nz f 0 components for J, , J2 S fiwo can be examined by ex- plicitIy assuming a short- and Iong-time behavior for I,!@)_ The general case [7] of non-lorentzian EPR profiles in low-dimensional crystals can also be carried out for any assumed e(f), and this procedure works well [2] In 1-D where static correlation functions for classical spins are known exactly_ We emphasize that eq. (3) provides an experimental handle on long-time contributions and on 5c even when the time and/or temperature dependence of G(r) is not known.

If we assume that fpo(rj goes as 7-l in 2-D for long times 7 %- 1, and for simplicity approximate the cut off in eq. (4) by a step function at rc, then the inte-

gral goes as In ic_ The secular contribution to eq. (3) is simply

Ao(ej = (3 co98 - ij*iMo~ ln(7,r /5~2j. (6)

where 7,-r, rc_, are for cut offs for PDAMnCI, and K2MnF4, respectively, and n-l, represents the 0 = 0 secular ApO sum in eq. (4) fGr PDAMnCI,. As dis- cussed above, the secular contribution Ao(Oj domi- nates A(0), although the region near 0 = 54” in fig. 1 clearly points to residual nonsecular effects. The angu- lar dependence of A,(e) is known in advance, while the sign of A&3) depends on the relative magnitude of Tcl and ic., _ Exact cancellation for n,(e) occurs accidentally for ~~~ = r,, _

The uncertainty in the calculated fifO in eq. (6) pre- cludes a quantitative result for i,.r /T& based on the fit to A(B) in fig. 1, since the uncertainty is exponentiated. According to theory [3,7], J1, Involves four-spin cor- relation functions divided by the spin diffusion con- stant. Neither is adequately known in 2-D. Calculated diffusion constants [ 131 differ by factors of 2 even in the far better understood 1-D case_ Our analysis of the experimentai A(0) curves in isostructural systems thus demonstra:e cut offs in 2-D systems, identify the dom- inzmt secular term, and establish the system with the smaller cut off, all without explicitly knowzng either the time or temperature dependence of the TST corre- lation function $(tj. Additional work is needed to ob- tain ~~ quantitatively.

Since the initial gaussian and the asymptctic behav- ior of G(t) are understood, it is natural [3,9,2] to as- sume the intermediate behavior and to match short- and long-time contnbutions. Now all contributions to I’ in eq_ (4) can be estimated and rather more compli- cated angular dependences are possible. The short- time contribution isfif7/we, where Icl, is the second moment and c+. = J/h% the exchange frequency_ Us- ing standard high-temperature results [3,4], we esti- mate that the short-time contributions in both K2MnF4 and PDAMnCI, give lo-208 of the EPR Iinewidth at 8 = 0 and T= 300 K and probably over 50% at 150 K_

Some limitations of the present approach must be mentioned. First, the related hlnB&antiferromagnet- ic layers [lo] have larger AH and g values that deviate significantly from the free electron value. The latter is not unexpecfed in view of the larger spin-orbit COU-

pling of Br-, but deviations from S-state ions indicate

459

Volume 61, number 3 CHEMICAL PHYSICS LE’ITBRS 1 March I979

other broadening mechanisms, beyond the dipolar in- teraction. The rather different angular dependence of MinRr~- Iayers and the rob? of spin diffiion in these

systems remains crpcn_ The structuraily related ferro-

magnetic CuCl~- Iayers, based OR S = lf2,3d9 C&(R) ions* have AZ$ dominated ]14] by symmetric and anti- symmetric corrections to the isotropic exchange, rather than by dip&r broadening_ The r-6 scahng in eq- (2) is clearfy restricted to systems with largely di- pofar broadening,

Second, experimentai uncertainties in AH of at fest C? G impose the practicaI limitation of o&y com- pariRg systems with z 2 2 in eq. (2). Other MnCI~- sahs [S] ~‘an be compared to f(7hS~F~. but not with

each other. In MA2hfnCl~, where MA is methylamine, the parameters am [I51 r= 5.10 A and f16] -J= 5.0 K, so that = = 0.93 in terms of PDAMnCIG_ The most recent OH data [ 161 show 3 %ery similar angular de- pendence and a sIightly broader AH for MA2MnC14, whiie earlier rest&s were nzrower j17J for MA2MnCId and broader ]I I] for PDAMnCl4. For = = 1 and com- parable Af2 in eq. (31, A(0) is quite sensitive to small experiment errors. Fortunately, changing iigands from F- to CI- results in a significant change in I and Ieads to z = 3-72 for the systks considered_

In summary, we have shown tfmt diffusive contribu- tion to the eschange-narrowed EPR linewidth A& can be identified by comparing isostructuraf systems with dipoiar broadening and taking advantage of the known scaIing z in eq. (2). The angular dependence of the ex- perinrentahy determined L#?) in eq- (3) estabtishes di- rectiy the dominant Iong-time secuIar contributions and the occurrence of cut offs 7;’ 2 AH in the

Kubo-Tomita approximation of the TST correlation function 4(t), even when the four-spin time correla- tion functions are not known and the observed absorp- tions are lorentzian, as iIIustrated by the 2-D systems K2M’lnF4 and PDAhfnCf3_

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