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A novel approach to modelling non-exponential spin glass relaxation
ARTICLE IN PRESS
Physica B 397 (2007) 99–101 www.elsevier.com/locate/physb
A novel approach to modelling non-exponential spin glass relaxation R.M. Pickupa,, R. Cywinskia, C. Pappasb a
School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK b Hahn-Meitner Institut, Glienicker StraX e 100, 14109 Berlin, Germany
Abstract A probabilistic cluster model, originally proposed by Weron to explain the universal power law of dielectric relaxation, is shown to account for the non-exponential relaxation in spin glasses above Tg. Neutron spin echo spectra measured for the cluster glass compound Co55Ga45 are well described by the Weron relaxation function, j(t) ¼ jo(1+k(t/t)b)1/k, with the interaction parameter k scaling linearly with the non-Curie–Weiss susceptibility. r 2007 Elsevier B.V. All rights reserved. PACS: 75.50.Lk; 78.70.Nx; 72.25.Rb Keywords: Spin glass; Neutron spin echo; Non-exponential relaxation
Structural glass transitions and their associated dynamics have been extensively studied, but nevertheless remain relatively poorly understood. In many cases the structural dynamics above the glass transition, Tg, are characterised not by a simple exponential (Debye) relaxation function, but by a stretched exponential form, exp(–(t/t)b). Such stretched exponential relaxation (i.e., Kohlrausch or KWW relaxation [1,2]) is now widely recognised as being almost ubiquitous in disordered and strongly interacting electronic and molecular systems. While the dynamics of structural glasses are difficult to model, much can be learned by studying their magnetic analogues, spin glasses. In such systems magnetic spins freeze with random orientations below Tg, usually because of random or frustrated exchange interactions. In common with structural glasses, spin glasses exhibit (magnetic) creep below Tg and anomalous non-exponential (spin) relaxation above Tg. Extensive Monte Carlo calculations by Ogielski [3] based upon a 3d7J Ising spin glass model have shown that close to and above the spin glass temperature the dynamical spin autocorrelation function should take the modified Kohlrausch form t b qðtÞ ¼ hS i ð0ÞS i ðtÞi / tx exp . t Corresponding author. Tel.: +44 113 3433841; fax: +44 113 3433846.
E-mail address: [email protected] (R.M. Pickup). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.02.081
Neutron spin echo (NSE) measurements have shown that this functional form adequately describes the spin relaxation in, for example, Au1xFex, spin glasses [4]. The experimentally determined values of x and b closely follow those predicted by the Ogielski simulations: b increases from 1/3 at Tg to 1 at 4Tg, and x increases from 0 below Tg to 0.5 at high temperatures. However, the origin of such non-exponential relaxation remains the subject of debate. The simplest, and most commonly invoked, origin of Kohlrausch-like relaxation is a statistical distribution of independent (parallel) relaxation channels. However, it is also argued that the Kohlrausch form can equally well be attributed to hierarchically constrained dynamics in complex strongly interacting materials [5]. The empirical Ogielski relaxation function does not allow parallel and hierarchical relaxation processes to be distinguished. Moreover, Ogielski’s power law prefactor can lead to unphysical values for the correlation function (i.e., q(t)41) at short times. For a more detailed physical insight, it is therefore necessary to turn to alternative models of glassy relaxation which intrinsically embody the interactions that may lead to hierarchical relaxation. Such models should, of course, also satisfy the strict criterion that q(t) ¼ 1 as t-0. Weron [6] has proposed just such a model to explain the universal power law for dielectric relaxation. Weron’s
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rigorous probabilistic approach is based upon a cluster model which assumes that individual dipoles and their environment do not remain independent during the relaxation process. Instead the time taken for polarization fluctuations to reach equilibrium is a random variable that for each relaxing dipole depends upon two other random variables, namely the waiting time and the dissipation rate. Through these variables Weron is able to account for the effects of both intercluster and intracluster interactions with the characteristic timescale of a relaxing entity being restricted by the structural reorganisation of the surrounding clusters. Weron derives a novel relaxation function fðtÞ ¼ fo 1 þ k
t b 1=k t
,
where k (40) is an effective interaction parameter, related to the waiting time. k also provides a measure of the relative contribution of hierarchical relaxation processes. It should be noted that that the Weron power law reduces to the Kohlrausch form in the limit k-0, in which case b (14 b40) has precisely the same meaning as before, with the limit b-1 implying simple Debye (exponential) relaxation. Despite its rigour and elegance, the Weron function has never been employed in the analysis of either spin glass relaxation or NSE spin echo spectra. We have therefore chosen to test its applicability in the case of a spin glass-like system in which the growth and subsequent interaction of magnetic clusters has been well characterised. This system is the binary (CsCl) compound Co50+xGa50x. At x ¼ 0 the compound is Pauli paramagnetic, As x increases the resulting magnetic ‘‘antistructural’’ Co atoms (i.e., the excess Co atoms on the Ga sites) polarise the surrounding Co structural atoms leading to extended magnetic clusters and cluster-glass behaviour. Percolation of the clusters then leads to ferromagnetism for x45 [7]. A Co55Ga45 sample was prepared by argon arc melting, and subsequently annealed at 830 1C (to ensure a random distribution of Co antistructure atoms) before quenching into water. AC susceptibility measurements show that the sample has an effective glass temperature of 17 K. The NSE experiments were performed on the IN15 spectrometer at ILL (Grenoble), using neutrons of wavelength 0.8 nm. A typical Co55Ga45 spin relaxation spectrum, taken at 38 K and Q ¼ 0.34 nm1, is shown in Fig. 1. Only the Ogielski function fits the data well at this temperature, though S(Q, t)/S(Q, 0) rises above 1 at short Fourier times and there are large errors associated with the fitted parameters. However, none of the conventional functions can adequately fit the Co55Ga45 relaxation spectra at all temperatures, and the temperature dependences of the fitted parameters, are neither consistent nor physically realistic whichever of the functions is used. In contrast, the Weron function provides an excellent description of the Co55Ga45 spectra at all temperatures, as
Fig. 1. NSE spectrum for Co55Ga45 at 38 K. The lines represent the leastsquares fits of the indicated functions to the data.
Fig. 2. NSE spectra from Co55Ga45 at several temperatures. The solid lines are fits of the Weron function to the data.
can be seen in Fig. 2. We find that b ¼ 1, representing nondistributed Debye relaxation at all temperatures, whilst the characteristic relaxation time to follows a critical slowing form, diverging at Tg ¼ 16.6 K (close to the measured Tg). The interaction parameter k increases rapidly as T approaches Tg, indicating that intercluster interactions begin to dominate the relaxation process with decreasing temperature Surprisingly, k appears to scale directly with the non-Curie–Weiss susceptibility of Co55Ga45 (Fig. 3). The reasons for such scaling are not yet understood as the Weron model does not explicitly provide the physical meaning of k. It should also be noted that our recent NSE measurements on other spin glass (e.g., Cu1xMn, EuSr1xSx) and related random anisotropy (e.g., amorphous Er7Fe3) systems, as well as our re-analysis of previously published NSE spectra of spin glasses suggest that the Weron function provides an almost universally appropriate func-
ARTICLE IN PRESS R.M. Pickup et al. / Physica B 397 (2007) 99–101
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scales linearly with the paramagnetic susceptibility of the respective system. These observations, and their implications, will be discussed in detail elsewhere. In conclusion, we have found that the probabilistic cluster model developed by Weron to explain the universal power law relaxation in dielectric relaxation can be equally well applied to spin glass and cluster glass relaxation. Correspondingly the Weron function provides a novel but useful functional form with which to parameterise the evolution of spin relaxation above Tg as measured by NSE. We wish to thank B. Farago and K. Weron for invaluable discussions. R.M.P. gratefully acknowledges receipt of an EPSRC postgraduate studentship. Fig. 3. Linear scaling of the interaction parameter, k, with the real component of the AC susceptibility for Co55Ga45. Inset: w(T) for Co55Ga45.
tional form to describe non-exponential spin relaxation above Tg. It also affords a useful and unique insight into the evolution with temperature of intercluster interactions and, possibly, hierarchical relaxation processes. Moreover we have found that for each system, the parameter k1/b
References [1] [2] [3] [4]
F. Kohlrausch, Pogg. Ann. Phys. 119 (1863) 352. G. Williams, D.C. Watts, Trans. Faraday Soc. 66 (1970) 80. A.T. Ogielski, Phys. Rev. B 32 (1985) 7384. C. Pappas, F. Mezei, G. Ehlers, P. Manuel, I.A. Campbell, Phys. Rev. B 68 (2003) 54431. [5] R.G. Palmer, D.L. Stein, E. Abrahams, P.W. Anderson, Phys. Rev. Lett. 53 (1984) 958. [6] K. Weron, J. Phys.: Condens. Matter 3 (1991) 9151. [7] R. Cywinski, J.G. Booth, B.D. Rainford, J. Phys. F 7 (1977) 2567.
Physica B 397 (2007) 99–101 www.elsevier.com/locate/physb
A novel approach to modelling non-exponential spin glass relaxation R.M. Pickupa,, R. Cywinskia, C. Pappasb a
School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK b Hahn-Meitner Institut, Glienicker StraX e 100, 14109 Berlin, Germany
Abstract A probabilistic cluster model, originally proposed by Weron to explain the universal power law of dielectric relaxation, is shown to account for the non-exponential relaxation in spin glasses above Tg. Neutron spin echo spectra measured for the cluster glass compound Co55Ga45 are well described by the Weron relaxation function, j(t) ¼ jo(1+k(t/t)b)1/k, with the interaction parameter k scaling linearly with the non-Curie–Weiss susceptibility. r 2007 Elsevier B.V. All rights reserved. PACS: 75.50.Lk; 78.70.Nx; 72.25.Rb Keywords: Spin glass; Neutron spin echo; Non-exponential relaxation
Structural glass transitions and their associated dynamics have been extensively studied, but nevertheless remain relatively poorly understood. In many cases the structural dynamics above the glass transition, Tg, are characterised not by a simple exponential (Debye) relaxation function, but by a stretched exponential form, exp(–(t/t)b). Such stretched exponential relaxation (i.e., Kohlrausch or KWW relaxation [1,2]) is now widely recognised as being almost ubiquitous in disordered and strongly interacting electronic and molecular systems. While the dynamics of structural glasses are difficult to model, much can be learned by studying their magnetic analogues, spin glasses. In such systems magnetic spins freeze with random orientations below Tg, usually because of random or frustrated exchange interactions. In common with structural glasses, spin glasses exhibit (magnetic) creep below Tg and anomalous non-exponential (spin) relaxation above Tg. Extensive Monte Carlo calculations by Ogielski [3] based upon a 3d7J Ising spin glass model have shown that close to and above the spin glass temperature the dynamical spin autocorrelation function should take the modified Kohlrausch form t b qðtÞ ¼ hS i ð0ÞS i ðtÞi / tx exp . t Corresponding author. Tel.: +44 113 3433841; fax: +44 113 3433846.
E-mail address: [email protected] (R.M. Pickup). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.02.081
Neutron spin echo (NSE) measurements have shown that this functional form adequately describes the spin relaxation in, for example, Au1xFex, spin glasses [4]. The experimentally determined values of x and b closely follow those predicted by the Ogielski simulations: b increases from 1/3 at Tg to 1 at 4Tg, and x increases from 0 below Tg to 0.5 at high temperatures. However, the origin of such non-exponential relaxation remains the subject of debate. The simplest, and most commonly invoked, origin of Kohlrausch-like relaxation is a statistical distribution of independent (parallel) relaxation channels. However, it is also argued that the Kohlrausch form can equally well be attributed to hierarchically constrained dynamics in complex strongly interacting materials [5]. The empirical Ogielski relaxation function does not allow parallel and hierarchical relaxation processes to be distinguished. Moreover, Ogielski’s power law prefactor can lead to unphysical values for the correlation function (i.e., q(t)41) at short times. For a more detailed physical insight, it is therefore necessary to turn to alternative models of glassy relaxation which intrinsically embody the interactions that may lead to hierarchical relaxation. Such models should, of course, also satisfy the strict criterion that q(t) ¼ 1 as t-0. Weron [6] has proposed just such a model to explain the universal power law for dielectric relaxation. Weron’s
ARTICLE IN PRESS R.M. Pickup et al. / Physica B 397 (2007) 99–101
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rigorous probabilistic approach is based upon a cluster model which assumes that individual dipoles and their environment do not remain independent during the relaxation process. Instead the time taken for polarization fluctuations to reach equilibrium is a random variable that for each relaxing dipole depends upon two other random variables, namely the waiting time and the dissipation rate. Through these variables Weron is able to account for the effects of both intercluster and intracluster interactions with the characteristic timescale of a relaxing entity being restricted by the structural reorganisation of the surrounding clusters. Weron derives a novel relaxation function fðtÞ ¼ fo 1 þ k
t b 1=k t
,
where k (40) is an effective interaction parameter, related to the waiting time. k also provides a measure of the relative contribution of hierarchical relaxation processes. It should be noted that that the Weron power law reduces to the Kohlrausch form in the limit k-0, in which case b (14 b40) has precisely the same meaning as before, with the limit b-1 implying simple Debye (exponential) relaxation. Despite its rigour and elegance, the Weron function has never been employed in the analysis of either spin glass relaxation or NSE spin echo spectra. We have therefore chosen to test its applicability in the case of a spin glass-like system in which the growth and subsequent interaction of magnetic clusters has been well characterised. This system is the binary (CsCl) compound Co50+xGa50x. At x ¼ 0 the compound is Pauli paramagnetic, As x increases the resulting magnetic ‘‘antistructural’’ Co atoms (i.e., the excess Co atoms on the Ga sites) polarise the surrounding Co structural atoms leading to extended magnetic clusters and cluster-glass behaviour. Percolation of the clusters then leads to ferromagnetism for x45 [7]. A Co55Ga45 sample was prepared by argon arc melting, and subsequently annealed at 830 1C (to ensure a random distribution of Co antistructure atoms) before quenching into water. AC susceptibility measurements show that the sample has an effective glass temperature of 17 K. The NSE experiments were performed on the IN15 spectrometer at ILL (Grenoble), using neutrons of wavelength 0.8 nm. A typical Co55Ga45 spin relaxation spectrum, taken at 38 K and Q ¼ 0.34 nm1, is shown in Fig. 1. Only the Ogielski function fits the data well at this temperature, though S(Q, t)/S(Q, 0) rises above 1 at short Fourier times and there are large errors associated with the fitted parameters. However, none of the conventional functions can adequately fit the Co55Ga45 relaxation spectra at all temperatures, and the temperature dependences of the fitted parameters, are neither consistent nor physically realistic whichever of the functions is used. In contrast, the Weron function provides an excellent description of the Co55Ga45 spectra at all temperatures, as
Fig. 1. NSE spectrum for Co55Ga45 at 38 K. The lines represent the leastsquares fits of the indicated functions to the data.
Fig. 2. NSE spectra from Co55Ga45 at several temperatures. The solid lines are fits of the Weron function to the data.
can be seen in Fig. 2. We find that b ¼ 1, representing nondistributed Debye relaxation at all temperatures, whilst the characteristic relaxation time to follows a critical slowing form, diverging at Tg ¼ 16.6 K (close to the measured Tg). The interaction parameter k increases rapidly as T approaches Tg, indicating that intercluster interactions begin to dominate the relaxation process with decreasing temperature Surprisingly, k appears to scale directly with the non-Curie–Weiss susceptibility of Co55Ga45 (Fig. 3). The reasons for such scaling are not yet understood as the Weron model does not explicitly provide the physical meaning of k. It should also be noted that our recent NSE measurements on other spin glass (e.g., Cu1xMn, EuSr1xSx) and related random anisotropy (e.g., amorphous Er7Fe3) systems, as well as our re-analysis of previously published NSE spectra of spin glasses suggest that the Weron function provides an almost universally appropriate func-
ARTICLE IN PRESS R.M. Pickup et al. / Physica B 397 (2007) 99–101
101
scales linearly with the paramagnetic susceptibility of the respective system. These observations, and their implications, will be discussed in detail elsewhere. In conclusion, we have found that the probabilistic cluster model developed by Weron to explain the universal power law relaxation in dielectric relaxation can be equally well applied to spin glass and cluster glass relaxation. Correspondingly the Weron function provides a novel but useful functional form with which to parameterise the evolution of spin relaxation above Tg as measured by NSE. We wish to thank B. Farago and K. Weron for invaluable discussions. R.M.P. gratefully acknowledges receipt of an EPSRC postgraduate studentship. Fig. 3. Linear scaling of the interaction parameter, k, with the real component of the AC susceptibility for Co55Ga45. Inset: w(T) for Co55Ga45.
tional form to describe non-exponential spin relaxation above Tg. It also affords a useful and unique insight into the evolution with temperature of intercluster interactions and, possibly, hierarchical relaxation processes. Moreover we have found that for each system, the parameter k1/b
References [1] [2] [3] [4]
F. Kohlrausch, Pogg. Ann. Phys. 119 (1863) 352. G. Williams, D.C. Watts, Trans. Faraday Soc. 66 (1970) 80. A.T. Ogielski, Phys. Rev. B 32 (1985) 7384. C. Pappas, F. Mezei, G. Ehlers, P. Manuel, I.A. Campbell, Phys. Rev. B 68 (2003) 54431. [5] R.G. Palmer, D.L. Stein, E. Abrahams, P.W. Anderson, Phys. Rev. Lett. 53 (1984) 958. [6] K. Weron, J. Phys.: Condens. Matter 3 (1991) 9151. [7] R. Cywinski, J.G. Booth, B.D. Rainford, J. Phys. F 7 (1977) 2567.