Journal ArticleThe spin glass transition of a quasi-1D organic-based magnet ([MnTPP][TCNE]) is explored using both ac and dc measurements. A scaling analysis of the ac susceptibility shows a spin glass transition near 4 K, with a viscous decay of the thermoremanent magnetization recorded above 4 K.We propose an extension to a fractal cluster model of spin glasses that determines the dimension of the spin clusters (D) ranging from _x0001_o-0.8 to  over 1.5 as the glass transition is approached. Long-range dipolar interactions  are suggested as the origin of this low value for the apparent lower critical dimension

Etzkorn, S. J.

Miller, Joel Steven

English

The University of Utah: J. Willard Marriott Digital Library

V o l u m e  89 , N u m b e r  2 0 P H Y S I C A L  R E V I E W  L E T T E R S 11 N o v e m b e r  2 0 0 2Viscous Behavior in a Q uasi-ID  Fractal Cluster GlassS. J. Etzkorn,1 Wendy Hibbs,2 Joel S. M iller,2 and A. J. Epstein1,3 1 Department o f Physics, The Ohio State University, Columbus, Ohio 43210-1106 2Department o f Chemistry, University o f Utah, Salt Lake City, Utah 84112-0850 3Department o f Chemistry, The Ohio State University, Columbus, Ohio 43210-1173 (Received 1 May 2002; published 23‘October 2002)The spin glass transition of a quasi-ID organic-based magnet ([MnTPP][TCNE]) is explored using both ac and dc measurements. A scaling analysis of the ac susceptibility shows a spin glass transition near 4 K, with a viscous decay of the thermoremanent magnetization recorded above 4 K. We propose an extension to a fractal cluster model of spin glasses that determines the dimension of the spin clusters (D) ranging from —0.8 to over 1.5 as the glass transition is approached. Long-range dipolar interactions are suggested as the origin of this low value for the apparent lower critical dimension.DOI: 10.1103/PhysRevLett.89.207201Recent failures [1,2] of mean-field models for spin glass behavior have renewed interest in non-mean-field interpretations. One such non-mean-field interpretation is the fractal cluster model [3 -6 ] which provides a physical meaning for the results, the fractal dimension of the spin clusters. This cluster fractal dimension can be calculated from critical exponents obtained through a scaling analy­sis of the glass transition, and knowledge of the effective dimension of the system. Previous results based on this model have concentrated on traditional 3D spin glass materials, including GdAl and Fea25Zn0 75F2, generally obtaining a cluster dimension of —2.5 [5,7]. In this paper, we propose a method for obtaining the fractal dimension of the spin clusters in a quasi-ID spin glass without any prior knowledge of the effective dimensionality o f the system.The organic-based magnet [Mn'I'PP]' [TCNH] • 2(1, 3-C6H4Cl2) (TPP is tetraphenylporphyrin, TCNE is tetracyanoethylene) is an electron transfer salt which forms ID polymers of alternating donors (MnTPP) and acceptors (TCNE) with large interchain spacings [8,9]. Previous magnetic data show ID chain behavior above —50 K with a transition to more complex behavior, in ­cluding spin glass behavior at lower temperatures [9-11]. Because o f weak interchain interactions in this class of materials, dipolar interactions between neighboring chains is the suggested mechanism for bulk magnetic ordering [11-13]. However, the transition to the spin glass phase has yet to be fully investigated.We report here a dynamic scaling analysis o f the ac susceptibility of [MnTPP][TCNE] together with ther­moremanent magnetization (TRM) decay data. The ac data support a spin glass transition near 4 K. The TRM  decay was measured for temperatures above this transi­tion and shows unusually long relaxation times near the glass transition. We propose an extension of the fractal cluster model which allows a determination o f the fractal dimension of the spin clusters within this material at various temperatures. Calculated values range fromPACS numbers: 75.50.Lk, 64.60.Ak, 75.10.Nr, 75.40.Gb—0.8 to over 1.5 near the glass transition, consistent with quasi-ID behavior. This anomalously low value for the apparent lower critical dimension is suggested to result from long-range dipolar interactions between clus­ters beyond the nearest neighbor.Polycrystalline samples of [Mn'I'PP]' [TCNH] • 2(1, 3-C6H4Cl2) were synthesized according to Ref. [9]. The polycrystalline powder was constrained for study by dispersing in eicosane (melting point 36-38°C ) and heat­ing above 40°C and subsequently cooled. The sample was then sealed in a quartz tube (for ac susceptibility mea­surements) or an airtight Delrin® sample holder (for TRM measurements). The ac susceptibility was measured with a LakeShore 7225 ac susceptometer/dc magne­tometer. Phase-sensitive detection allowed both the in-phase (x ') and out-of-phase (x ") components o f the susceptibility to be extracted. The instrument was ca li­brated for both phase and relative amplitude at each frequency by the use of a paramagnetic standard. The dc measurements were recorded on a Quantum Design PPMS-9 with the AC MS option.The ac susceptibility data were taken from 4.5 to 50 K, at frequencies from 11.0 Hz to 11.0 kHz with an ac field of1.0 Oe and zero applied dc field. In order to probe the glass transition, a dynamic scaling analysis of the out-of-phase component was performed according to the fractal cluster model [14] following the linear method proposed by Geschwind, Huse, and Devlin [15]. Only data points above the peak temperature were included in the scaling plot in order to avoid an overestimation o f the critical exponents [16].The linear method [15] enables independent determi­nation o f the ratio o f the critical exponents f3 / z  v (where v,  and z have their usual definitions [17]) by exam in­ing the frequency dependence of the x"(T,  co) peak. As pointed out in Ref. [15], the amplitude (x ”) and tempera­ture (Tp) o f the peak should satisfy the relation Xp(co)Tp (co) oc a)l3/ zv. Figure 1(a) shows a plot of log]0[^'p(co)rp(co)] versus log10(&>). The slope of a linear207 2 0 1 -1  0 0 3 1 - 9 0 0 7 /0 2 /8 9 (2 0 ) /2 0 7 2 0 1 (4 )$ 2 0 .0 0  ©  2 0 0 2  T h e  A m erica n  P hysica l S o c ie ty  2 0 7201 -1V o l u m e  89 , N u m b e r  20 P H Y S I C A L  R E V I E W  L E T T E R S 11 N o v e m b e r  20 0 2(a) 1.55£  150 .  a.°  1.45 O) o1.40Linear scaling: slope of fit ^gives value of /5/zvyX  p / z v =  0 .0 4 1 5  ± 0 . 0 0 1 1log 10^Full sc a lin g  p lot of x '  'ac• 11 HzTg = 4.1 ± 0 .1 5  K▼■33 Hz 110 Hzz v =  8 .9  ± 0 .1 5 333 Hz* A 1100 Hz• 3330 Hz© 11.0 kHz> t o v . . o __ Jtl. aV s * ?0.8 1.0 1.2 1.4 1.6 1.8 2.0e/Ct)1/zvFIG. 1 (color online), (a) A plot of x " ( l°)T (io )  at the peak temperature vs co^z,‘; the slope of the fitted line gives the ratio P /zi'. (b) A full scaling plot using the scaling function of Ref. [15]. Good data collapse was observed for temperatures from the peak temperature to —30 K.fit to  this data gives a value for f i / z v  =  0.0415 ±  0.0011. The value of this ratio was used to  guide the selection of z v  and (3 in the full scaling plot.The scaling function [15] x " T /o ) plzv ~  f ( e / o ]/zP) [e =  (T /T ,, -  1)] was used to  plot the ac susceptibility data above the peak  tem perature. G ood data collapse occurs over the three m easured decades of frequency for the values of zv  =  8.9 ± 0 .1 5  and T „ = 4 . 1 ±  0.15 K giving /3 =  0.369 ±  0.012 using the value for the ratio (3 /zv  found from  Fig. 1(a). Scaling was observed over a large range of tem peratures, from  the peak tem ­perature ( ~  9 K) up to 30 K. The full scaling plot of x "T Io )P lzl' vs e /o } ''^ 1' with the values of the critical exponents and transition tem perature given above is shown in Fig. 1(b).Thcrm orcm ancnt m agnetization m easurem ents were made upon cooling in a field of 1000 Oc from  60 K to the desired m easurem ent tem perature. The sam ple was held at the desired tem perature in the field for 5 m in prior to  reducing the field to zero. Because of lim itations of the instrum ent, the first m easurem ent was taken approxi­m ately 30 sec a fte r reaching zero field. The m agnetization decay was then recorded for periods of 45 m in to over120 h depending on the tem perature of the sample. The m agnetization decay was fit to a stretched exponential of the formM (t) =  M 0 +  a 0e (1)where M 0, cr0, t ,  and n arc the fitting param eters. In particular, M 0 is a param eter required to  com pensate for a small nonzero remanent m agnetization (M0 ~  0.01 cr0) due to  the presence of cluster glass behavior [11]. Stretched exponential relaxation has been observed experim entally  in other spin glass and quasi-ID  systems [18], as well as predicted by theoretical models, including the fractal cluster model [19,20].Fits of the TRM  data to Eq. (1) were made for tem ­peratures from  4 to  6 K and were valid  over the entire m easured tim e range. Above 6 K, the m agnetization relaxes too quickly to observe relaxation with this method. Data were generally taken until the m agnetiza­tion appeared constant with tim e plo tted  on a logarithm ic scale. However, the unusually long relaxation tim es ob­served near T„ made this experim entally  unachievable at some tem peratures. Typical relaxation curves and the corresponding fits arc shown in Fig. 2. In particular, these fits allow a d irect estim ation of the characteristic response tim e ( t )  of [M nTPP][TCN E] at different tem peratures.The characteristic response tim e of the system  at v a ri­ous tem peratures is shown in Fig. 3. The response tim es obtained from  both the ac data and the TRM  m easure­ments lie on the sam e curve. For the ac data, rv a lu e s  were determ ined at the peak tem perature by t  ~  1/&>,  while, for TRM  m easurem ents, r  values were determ ined from  fits to Eq. (1).Besides the determ ination of r ,  fits o f the data to Eq. (1) also allow a determ ination of the stretched exponential power («). The m easured values for both n and t  increase60003000 -j) ,§  1000TR M  d a ta  a t  v a r io u s  T  • 4 .5  K5.0  K5.5  K>  \  \101 102 103 104 Tim e (s)105FIG. 2 (color online). Typical TRM decay curves at various temperatures. The symbols represent the data, while the solid lines are a fit to Eq. (1).2 0 7 2 0 1 -2 2 0 7 2 0 1 -2V o l u m e  89 , N u m b e r  20 P H Y S I C A L  R E V I E W  L E T T E R S 11 N o v e m b e r  20 0 2COH1 0 2-10"1 0 £W▼▼T vsT▼T▼▼▼. F ro m F ro mTRM d a ta a AC d a taTA ' ▲S ▲ .\ / ▲6 8 10 12 T e m p e ra tu re  (K)14 16FIG. 3 (color online). The characteristic response time of [MnTPP][TCNE] versus temperature. Data from both dc and ac measurements can be seen on the curve. The data cover large ranges of both temperature and time, suggesting a slow, viscous transition.w ith decreasing tem perature. We discuss below the in ­crease in n from  ~0 .35  to ~ 0 .5  near the glass transition in  term s of the fractal dim ension of the spin clusters.The fractal dim ension o f the spin clusters (D) is usually obtained from  the relation [4,51d  =  D + p / v . (2)However, this equation requires knowledge of the effec­tive spatial dim ension (d) o f the system. W hile d  =  3 for most systems, it may be different from  three, in fact, a noninteger (fractal) effective spatial dim ension is also possible" [4], The chain structure o f [M nTPPl[TC N El suggests quasi-ID  behavior or d ~  1 though the exact value is not explicitly known. We present here another approach for calculating the fractal dim ension o f the spin clusters which avoids placing assum ptions on the value of d.The stretched exponential power (n) can be used to obtain the fractal dim ension of the spin clusters. Follow ing Ref. [201, « can be w ritten in  term s of critical exponents as n =  z v /{ z v  +  where z, v, and </> are standard critical exponents. The critical exponents </> and v are related to the fractal dim ension of the clusters by </> =  i’D  [51. Therefore we can relate D  to n byDzv  \1 — n(3)E quation (3) gives the fractal dim ension in  term s of critical exponents, and the stretched exponential power with no reference to the effective dim ension of the system.The critical exponent z v  { =  8.9) was obtained from  the ac susceptibility m easurem ents directly, while n is determ ined at various tem peratures from  fits o f the TRM  data to Eq. (1). The crossover exponent </> can be obtainedfrom  /3 through the use of scaling relations. The lack of a specific heat anom aly at the spin glass transition (as is typical o f this m aterial [211 and other spin glasses [22D places certa in  restrictions on the scaling relations involv­ing the specific heat exponent, a .  In particular, C am pbell points out that, for the specific heat to be regular at T„, I’d  =  4 [23,241. Using this result, and Eq. (2), </> can be related to /3  by </> =  4 — /3. This relation allows a deter­m ination o f D  from  Eq. (3) using quantities determ ined experim entally.As the param eter n is evaluated at different tem pera­tures, Eq. (3) gives D  in a tem perature dependent form. This allows physical insight into the evolution o f the spin clusters that generally cannot be achieved using Eq. (2). A plot of D  and rt versus tem perature for [M nTPPlfTC N El is shown in Fig. 4. Calculated values for D  increase with decreasing tem perature and range from  ~ 0 .8  to over 1.5 near T„. In particular, D  shows a rapid increase as the tem perature approaches T„.These results can be understood in term s of an an iso­tropic cluster glass model. [M nTPPlfTC N El is a typical m ember of a broad class o f m aterials which form  linear chains w ith large ( >  1.2 11m) interchain spacing [91. This large interchain spacing results in  a very weak exchange between neighboring chains w ith d ipolar interactions having the dom inant role in interchain interactions [10-131. S tructural data, as well as the strong m agneto- crysta lline anisotropy observed in this m aterial [251, support that spin clusters lie predom inantly along a chain.The values obtained for D  confirm this type of quasi- ID  behavior. A t higher tem peratures (T >  5.0 K), the values for D  ( ~  0.8) indicate that spin clusters are gen­erally constrained to a single chain. As the tem perature is lowered, D  becomes greater than 1 as clusters now in ­clude portions o f neighboring chains. As the glass tran ­sition is approached from  above, the increasing size o f theTem perature (K)FIG. 4 (color online). Variation of the stretched exponential power n from Eq. (1) and the calculated values for the fractal cluster dimension D as a function of temperature. The arrows indicate the axis for a particular plot. The symbols represent the data, while the lines arc merely a guide to the eye.20 7 2 0 1 -3 2 0 7 2 0 1 -3V o l u m e  89 , N u m b e r  20 P H Y S I C A L  R E V I E W  L E T T E R S II N o v e m b e r  2 0 0 2spin cluster leads to an increase in the strength o f the dipolar interactions. This effective “dipolar exchange” with parallel chains beyond nearest neighbor, as w ell as the uniaxial anisotropy o f [MnTPP][TCNE], may lead to the observed spin glass transition occurring for d <  3 [26],The response times (r) o f the system shown in Fig. 3 provide insight into the nature o f this glass transition. The t  from the ac and TRM data cover a large temperature range including temperatures up to ~ 4 Tg showing a slow transition to the frozen state. The slowness o f the tran­sition is also evident in the strong frequency dependent shift in the peak temperature o f the ac susceptibility [&Tp/ ( T pA  lo g /)  ~  0.178)1.The TRM data show unusually long relaxation times ( t  105 sec) near Tg suggesting viscous behavior o f the spin clusters. The long relaxation seen near Tg can be understood in terms o f the dipolar interaction between spin clusters o f neighboring chains. The interaction o f the magnetic clusters w ill create some weak 3D order which w ill compete with and slow the glassy relaxation o f the system.In conclusion, we have reported a detailed scaling analysis o f the ac susceptibility and TRM decay data for [MnTPP][TCNE]. Results indicate the presence o f a glass transition near 4 K, Long characteristic times and strong frequency dependence o f ac measurements suggest a slow, viscous transition to the glassy state. An extension o f the fractal cluster model for spin glasses [3 -6 ] enables a calculation o f the fractal dimension o f spin clusters in a temperature dependent form. Using this method, the spin system for [MnTPP][TCNE] was found to evolve from near linear spin clusters to higher dimensional clusters. The results were discussed in terms o f the dipolar inter­action between spin clusters o f neighboring chains. Values for the fractal dimension obtained for [MnTPP][TCNE] are consistent with quasi-ID behavior, and are believed to be the first reported for a system o f d  <  3.This work was supported in part by DOE Grant No. DE-FG02-86ER45271 and NSF Grant No. CHE- 9730984.[1] C. M. Newman and D. L. Stein, Phys. Rev. Lett. 76, 515 (1996).[2] C.M. Newman and D.L. Stein, Phys. Rev. E 57, 1356 (1998).[3] A. P. Malozemoff, S. E. Barnes, and B. Barbara, Phys. Rev. Lett. 51, 1704 (1983).[4] B. Barbara and A. P. Malozemoff, J. Less-Common Met. 94, 45 (1983).[5] A. P. Malozemoff and B. Barbara, J. Appl. Phys. 57, 3410(1985).[6] S. E. Barnes, A. P. Malozemoff, and B. Barbara, Phys. Rev. B 30, 2765 (1984).[7] P. H. R. Barbosa, E. P. Raposo, and M. D. Coutinho-Filho, Physica (Amsterdam) 295A, 140 (2001).[8] J.S. Miller, J.C. Calbrese, R.S. Mclean, and A.J. Epstein, Adv. Mater. 4, 498 (1992).[9] W. Hibbs, D. K. Rittenberg, K.-I. Sugiura, B. M. Burkhart, B. G. Morin, A. M. Arif, L. Liable-Sands, A. L. Rheingold, M. Sundaralingam, A. J. Epstein, and J. S. Miller, Inorg. Chem. 40, 1915 (2001).[10] C.M. Wynn, M.A. Girin. K.-I. Sugiura, E.J. Brandon, J. L. Manson, J. S. Miller, and A. J. Epstein, Synth. Met. 85, 1695 (1997).[11] M. A. Girtu, C.M. Wynn, K.-I. Sugiura, J.S. Miller, and A. J. Epstein, J. Appl. Phys. 81, 4410 (1997).[12] C.M. Wynn, M. A. Girtu, W. B. Brinckerhoff, K.-I. Sugiura, J. S. Miller, and A. J. Epstein, Chem. Mater. 9, 2156 (1997).[13] S. Ostrovsky, W. Haase, M. Drillon, and P. Panissod, Phys. Rev. B 64, 134418 (2001).[14] M. Continentino and A. P. Malozemoff, Phys. Rev. B 34, 471 (1986).[15] S. Geschwind, D. A. Huse, and G. E. Devlin, Phys. Rev. B 41, 4854 (1990).[16] D. Bertrand, A. Mauger, J. Ferre, and P. Beauvillain, Phys. Rev. B 45, 507 (1992).[17] /3, v, and z are the order parameter, correlation length, and dynamical exponents, respectively. See, for instance, L. P. Kadanoff, W. Gotze, D. Hamblen, R. Hecht, E. A. S. Lewis, V.V. Palciauskas, M. Rayl, and J. Swift, Rev. Mod. Phys. 39, 395 (1967); N. Bontemps, J. Rajchenbach, R.V. Chamberlin, and R. Orbach, Phys. Rev. B 30, 6514 (1984).[18] K. Ravindran, G.V. Rubenacker, D. N. Haines, and J. E. Drum heller. Phys. Rev. B 40, 9431 (1989).[19] R. M. C. de Almeida, N. Lemke, P. Jund, R. Jullien, I. A. Campbell, and D. Bertrand, J. Non-Cryst. Solids 287, 201 (2001).[20] M. A. Continentino and A. P. Malozemoff, Phys. Rev. B 33, 3591 (1986).[21] M. Sorai (private communication).[22] K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801(1986).[23] LA. Campbell, Phys. Rev. B 37, 9800 (1988).[24] This is derived from the scaling relation 2 — a  = vd, with the restriction that a  is a negative even integer, e.g., a = — 2. Choosing values for a < — 2, e.g., —4, — 6, . . .  results in nonphysical values (D >  3) of the fractal cluster dimension.[25] K. Nagai, T. lyoda, A. Fujishima, and K. Hashimoto, Chem. Lett. 1996, 591 (1996).[26] The value for the effective dimension (d) may be slightly underestimated due to deviations from scaling causing variations of /3 with temperature in the crossover region near the glass transition. It is also noted that TRM data below 4.5 K do not fully relax to a constant value. Fits of this data to Eq. (1) may lead to an underestimation of n and likewise d  at the glass transition.2 0 7 2 0 1 -4 2 0 7 2 0 1 -4

Viscous behavior in a quasi-1D fractal cluster glass

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Viscous behavior in a quasi-1D fractal cluster glass

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