Abstract
Spin-glass thin films exhibit many features different from the bulk. The freezing temperatures of spin-glass films are suppressed for reduced thickness and follow the Kenning relation. The dynamics are altered near the vacuum interface. These phenomena are closely related to the lower critical dimension of spin glasses, the spin-glass correlation length, and the dimensional crossover from = 3 to = 2. In this article, we review the experimental facts and theoretical perspectives for spin-glass thin films. We focus on canonical spin-glass systems with the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction between magnetic impurities in a nonmagnetic host. Open questions to be addressed are emphasized.
1 Introduction
Spin glasses [] with random spin orientations yet strong correlations have motivated theoretical developments [] to understand their emergent complexities and continue to spur new findings in complex systems []. After more than half a century of intensive efforts to uncover the physics of spin-glass dynamics, controversies remain. A consensus regarding the density of ground states and the stability of the spin-glass phase [, ] in an external field is still lacking.
Finite-size effects of spin glasses, first reported by the pioneering work of Kenning, Slaughter, and Cowen [], offer a new route to uncover some of their mysteries. A surge of research interest in spin-glass thin films focuses on the dimensional crossover from to [–]. In this work, we review the experimental discoveries and theoretical developments for metallic canonical spin-glass thin films. The freezing temperature, , one of the most studied quantities of thin films, is first introduced. The correlation length offers a unique lens through which to understand dimensional crossover. Simulations to extract the growth laws governing correlation lengths are introduced. We also examine the impact of the interface on the spin-freezing process. We conclude the paper with open questions and remarks.
2 The freezing temperature and dimensional crossover: experiments
The freezing temperature [, ] of spin glasses is different from the critical temperature . Approaching from above in a bulk sample, a continuous symmetry breaking in the phase space commences and signifies a phase transition. In thin film magnetometry measurements, is defined as the temperature below which the zero-field-cooled (ZFC) magnetization, , differs from the reference field-cooled magnetization (in some literature, is defined as the peak of ; the difference between the two is small in a small enough magnetic field).
The time dependence of naturally leads to the time dependence of . When cooled from the paramagnetic phase to the working temperature , exhibits a sudden jump after the field is switched on, and gradually increases toward . is, therefore, the highest temperature below which nonequilibrium dynamics set in for a fixed observation time . In a domain growth model, the length scale sets the relaxation time. Consequently, the equality of relaxation time and observation time gives rise to the observed freezing temperature.
The technical difficulty of extracting is the very weak magnetic signal of a thin film with dilute magnetic spins dispersed in a nonmagnetic host. To circumvent this issue, Kenning et al. [] have used multilayers of CuMn thin films, separated by layers of either pure Cu or Si, to decouple the direct contact between the CuMn thin films. They found that decreases monotonically as the CuMn film thickness is reduced. The data were originally fitted to the finite-size scaling form of Equation 1as proposed in the real-space droplet (domain) model, where is the film thickness. The droplet model assumes the existence of two ground states of the spin-glass phase, related by time-reversal symmetry. The finite-temperature properties are governed by low-lying excitations of droplets of typical size . In response to Kenning’s results, a new scaling ansatz for the free energy of the droplet in was proposed, Equation 2,
Because , the spin-glass phase is unstable in , and long-range order is destroyed. Fisher and Huse [] predicted that in the critical region,
Equation 3 is difficult to test experimentally, as an accurate estimation of is difficult. In addition, the error bars of the many exponents in Equation 3 contribute to the uncertainty of .
An alternative interpretation of the freezing temperature was given in []:where is the average spacing between the magnetic impurities, is the experimental time scale, and is the freezing temperature for the bulk sample. Equation 4 is referred to as the Kenning relation.
Figure 1 exhibits data for plotted against film thickness from [,], and []. It is assumed that the equilibrium correlation length perpendicular to the film plane, , has saturated at the film thickness at the time , indicative of a dimensional crossover.
FIGURE 1
Because the lower critical dimension of spin glass is between 2 and 3 (exactly 2.5 for Edwards–Anderson spin glass []), the critical temperature is zero after a dimensional crossover from to . The energy barrier height that governs the relaxation time was found to only depend on in a temperature range of 1 K at up to film thickness of 20 nm []. Equation 4 indicates that shifts to lower with slower cooling rates, or longer , which is qualitatively consistent with experimental observations (e.g., Figure 1).
Additional support for the dimensional crossover of spin-glass thin films was obtained through polarized neutron reflectometry (PNR) [, ]. An asymmetry parameter , defined aswas measured for AuFe thin films, where and are the reflectivities for the spin-up neutrons and spin-down neutrons, respectively. By fitting Equation 5 with a Parratt recursion formalism, the average magnetic moment for Fe atoms was obtained, as reproduced in Figure 2. In the high-temperature regime, can be described by the Brillouin function, , where is the angular momentum, and with being the magnetic moment of impurities.
FIGURE 2
Below 50 K, deviates from , in a range of 0.5–1.3 for nm. Exceptions are for nm and nm, which follow a paramagnetic line down to 2 K. In particular, for the 1 nm film, the measured magnetic moment 4.2 is very close to the value for non-interacting Fe atoms.
These measurements were performed in a very high magnetic field of 6 T. In large fields, the measured moment should correspond to a quasi-equilibrium spin-glass phase. Therefore, at large observation times and for ultra-thin film, Equation 4 implies a , consistent with the measurements. In addition, the freezing of the magnetic moment for the thicker films implies the existence of a spin-glass phase up to 6 T.
According to calculations based on an Ising model, an A-T line [] exists for the phase diagram of spin glasses, while the droplet picture predicts that the spin-glass phase vanishes in a magnetic field. However, equilibrium states are difficult to access in laboratory experiments, leaving a lack of consensus on the nature of the spin-glass phase in the presence of a magnetic field [, ].
3 Correlation length growth and dimensional crossover: simulations
It has been shown through Monte Carlo (MC) simulations and finite-size scaling [–] that the correlation length diverges at a finite temperature for a 3D-Ising Edwards–Anderson (EA) model with either Gaussian or bimodal interactions. Similar conclusions were reached for the 3D-Heisenberg spin glass [, ] with the aid of a much larger sample size. The 2D-Ising (Heisenberg) spin glass only exhibits a phase transition at , as shown by free energy calculations [] and MC simulations []. These findings imply that there must be a crossover of the spin-glass dynamics when the size of the system is reduced gradually by one dimension. Simulations are usually limited by sample size and time scales, but compared to experiments, they enjoy enhanced spatial resolutions and gain immediate access to spin configurations. They thus provide a unique route to understand dimensional crossover. In particular, they have been able to directly extract .
Of particular interest is the aging dynamics of when the spin-glass phase evolves from nonequilibrium towards equilibrium. In practice, simulations suddenly quench the sample from the paramagnetic state (infinite temperature) to a temperature comparable to experiments. Rieger et al. [, ] performed MC simulations on an Ising EA model with Gaussian interactions. In order to understand the aging phenomena observed in experiments, the autocorrelation function is introduced, defined bywhere is the duration in which the sample remains at after quench, and the averages are taken over thermal fluctuation and quenched disorder. For , is stationary and only depends on in Equation 6. For and , a power law of Equation 7 is found, for different obeys the simple scaling form,where if . This is not consistent with activated dynamics,which leads to a logarithmic scaling. The spatial correlation is calculated through Equation 10,and the correlation length at is given by Equation 11,
The growth of can be fitted with the activated dynamics of Equation 9 or a power-law dynamics,
However, Equation 12 naturally leads to the scaling form of Equation 8 when assuming , where . The power-law growth of was later verified by Joh et al. [].
A twelve-time-decade MC simulation of the Ising EA model was carried out by Fernandez et al. []. They clarified growth dynamics for and reinforced the evidence for the scaling function of Equation 8. They found that, Equation 13,where is the corrections-to-scaling exponent, , and is a small number. This verified that the energy barrier height inferred from aging dynamics is physical [].
The first MC simulation for dimensional crossover in a thin film geometry, comparable to experiments, was carried out by Victor Martin-Mayor and his coworkers []. The protocol used in this simulation closely resembled that of experiments: The sample was quenched to a working temperature, and the complete growth process (from nonequilibrium to equilibrium) of the correlation length was monitored. The correlation length is related to the autocorrelation function of Equation 14,where is the overlap between spin configuration and in replicas (a) and (b).
The estimator for the correlation length is given by Equation 15
Again, they found a power-law growth of the correlation length .
Four time regimes were identified, with different for . In the first, grows with the aging rate . Upon saturation of close to the film thickness , the growth of gradually speeds up in the second time regime. The aging rate of in the third regime finally matches , smaller than (faster dynamics). Finally, in the fourth regime, reaches to its equilibrium value .
Further analysis leads to a scaling function,where is the correlation length of a 3D sample. The invariance of Equation 16 allows a Kananoff–Wilson block spin transformation of the simulation results. These, in turn, lead to the mapping of the temperature of the film to an effective temperature in (a true monolayer film),
The mapping given by Equation 17 is remarkable in that an effective temperature can allow for treating the spin-glass problem exactly in . For example, for , . As the correction is negligible, the analyses of CuMn thin films in [–] are adequate.
4 Interfacial effects on spin freezing
Much of our understanding of dimensional crossover in thin films arises from multilayers of spin-glass films separated by non-magnetic metallic or insulating layers. It is natural to ask whether the interface between the spin-glass layers and the decoupling layers leads to artificial or unwanted effects. For example, as illustrated in Figure 1, the of the same CuMn film decoupled by Si is lower than films decoupled by Cu []. The RKKY interaction, responsible for spin-glass behavior in metallic spin glasses, is mediated by the conduction electrons. This long-range interaction is sharply cut off at the CuMn/Si boundary but falls off slowly at the CuMn/Cu boundary [], perhaps accounting for the difference. A quantitative analysis is lacking.
The first systematic study to address the effects of the decoupling layers was conducted by Granberg et al. []. They varied the thickness of Cu layers, , to explore the freezing process of CuMn/Cu multilayers. We identify the from Figure 1 of [] in order to display the time dependence of in Figure 3. The reduction of follows a logarithmic time dependence given by Equation 4. The rates of reduction, for different are displayed in Figure 4. The rate decreases with decreasing thickness of the decoupling layer until it reaches its value.
FIGURE 3
FIGURE 4
An explanation for the behaviors exhibited in Figures 3, 4 was given in []. For a given observation time, the inequality of Equation 18,holds for in , where is the equilibrium correlation length obtained in the FC state. The growth of the correlation length for the ZFC protocol obeys the growth law on the left of Equation 18. For spin-glass films fully decoupled from one another, is bounded by regardless of . grows faster than []. Then, any films with should be much more sensitive to the variation of the observation time scale. Between the and the fully decoupled layer limit, crosstalk occurs among the neighboring layers, leading to an intermediate sensitivity to the variation of observation time .
A direct probe of the surface dynamics has been carried out by depth-dependent muon-spin-relaxation (SR) studies []. The SR is a powerful technique to probe local spin orientations [–]. In the experimental setup, the polarized beam is stopped by the sample, and the decay positron emitted from is counted. The backward (EB) and forward (EF) counting rates are given by Equation 19,where is the lifetime of , and is the muon-spin-relaxation function. for completely polarized spins, and for completely depolarized spins. The depolarization process can be inferred from the asymmetric time evolution of and .
The muons are assumed to take random interstitial sites in the sample and do not diffuse in the lattice. For a CuMn (1 at%) sample, the atomic dipolar field (100 G) from the Mn impurity dominates compared to the average RKKY field (10 G) and the Cu nuclear dipolar field (4 G). For an ordered translational invariant magnetic phase (e.g., a ferromagnet), the muon spin will precess with a single frequency in the local dipolar field. In the case of spin glasses, the randomness of the local dipolar field leads to a rapid depolarization of the polarized muons.
Taking into account the static random local fields and their fluctuation, a stochastic theory of muon-spin-relaxation for was formulated by Uemura et al. []:where and with are the average amplitude of random fields. It was assumed that, Equation 21,where the brackets represent thermal averages, and the bar indicates spatial averages. Each spin has a preferred static component below , and a dynamic component with a fluctuating rate .
Therefore, the order parameter [] can be obtained by fitting Equation 20 to the experimental data. Figure 5 reproduced the extracted values of at different depth of a 220-nm-thick AuFe film by []. At the low-temperature regime, attains a finite value, signifying the onset of spin freezing. The dynamical fluctuations of spins are significantly reduced with increasing distance from the surface. This suggests an inhomogeneous freezing gradient along the direction of the film thickness because of the vacuum interface. It is likely that the RKKY interaction between magnetic impurities is modified by the vacuum interface. However, again, a treatment to quantify this effect is lacking.
FIGURE 5
Recent noise measurements [, ], covering a much larger temperature window (inaccessible in magnetometry measurements) for a larger collection of film thicknesses, suggest that the maximum barrier height is temperature dependent for thicker films. Although this appears to conflict with the magnetometry measurements [] at first glance, the noise is sensitive to the length scales associated with the electronic mean free path, which are much shorter than the range of the RKKY interaction. This may be the reason behind the discrepancy between the two experimental processes.
5 Conclusion
We have examined the evidence for dimensional crossover from to of spin-glass thin films. The results from the magnetometry and the PNR measurements are consistent with the Kenning relation, Equation 4. The correlation length serves as a caliber to quantify . The dependence of originates from the power-law growth of the correlation length, as detailed in large-scale simulations.
Although much has been understood concerning the dynamics of thin film spin-glasses, in our opinion, a few questions remain to be addressed:
(1) In the low-temperature regime of the spin-glass phase, the dynamics become too slow to be probed by magnetometry and simulations. The validity of the Kenning relation remains to be tested with novel experimental protocols or data analyses.
(2) Both the vacuum and Si interfaces alter the spin-freezing process. It remains unknown how the interface affects the spin correlations. A theoretical treatment of the interfacial effects would not only contribute to a deeper understanding of spin-glass physics but also would benefit devices utilizing junctions between spin-glasses and other magnetic ordering materials [, ].
Statements
Author contributions
QZ: writing–original draft and writing–review and editing. RO: writing–original draft and writing–review and editing.
Funding
The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. QZ was supported by China Postdoc Fund Grant No. 2022M722548, Shaanxi NSF Grant No. 2023-JC-QN-0018, and Central University Basis Research Fund Grant No. xzy012023044. RO was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Division of Materials Science and Engineering, under Award No. DE-SC0013599.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
spin glasses, dimensional crossover, freezing temperature, correlation length, interfacial effects
Citation
Zhai Q and Orbach RL (2024) Toward understanding the dimensional crossover of canonical spin-glass thin films. Front. Phys. 12:1488275. doi: 10.3389/fphy.2024.1488275
Received
29 August 2024
Accepted
07 October 2024
Published
05 November 2024
Volume
12 - 2024
Edited by
Ralph Chamberlin, Arizona State University, United States
Reviewed by
Aakansha, Banasthali University, India
Updates
Copyright
© 2024 Zhai and Orbach.
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*Correspondence: Raymond L. Orbach, orbach@utexas.edu
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.