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# Switchable hardening of a ferromagnet at fixed temperature

Edited* by Sidney Nagel, University of Chicago, and approved December 14, 2009 (received for review September 16, 2009)

## Abstract

The intended use of a magnetic material, from information storage to power conversion, depends crucially on its domain structure, traditionally crafted during materials synthesis. By contrast, we show that an external magnetic field, applied transverse to the preferred magnetization of a model disordered uniaxial ferromagnet, is an isothermal regulator of domain pinning. At elevated temperatures, near the transition into the paramagnet, modest transverse fields increase the pinning, stabilize the domain structure, and harden the magnet, until a point where the field induces quantum tunneling of the domain walls and softens the magnet. At low temperatures, tunneling completely dominates the domain dynamics and provides an interpretation of the quantum phase transition in highly disordered magnets as a localization/delocalization transition for domain walls. While the energy scales of the rare earth ferromagnet studied here restrict the effects to cryogenic temperatures, the principles discovered are general and should be applicable to existing classes of highly anisotropic ferromagnets with ordering at room temperature or above.

It is no accident that techniques to mechanically and magnetically harden and soften materials coincide. The same impurities and grain boundaries that prevent dislocations from flowing prevent magnetic domains from switching. Strategies to pin magnetic domains typically are implemented during materials preparation by varying the composition, structure, and morphology (1–7), locking in a set of properties that cannot, subsequently, be modified. Of particular technological interest is ultrahigh-density magnetic storage, where bits are warmed close to or above their ferromagnetic transition temperature for writing and then cooled for long-term retention and reading (8). A more powerful approach would be to adapt material properties in situ as circumstances demand, continuously tuning the energetics of domain reversal without changing temperature.

We propose a unique avenue to regulate magnetization reversal in real materials by exploiting the random-field Ising model (RFIM) (9, 10), where a site-random magnetic field acts to orient magnetization locally in competition with the underlying exchange couplings that favor homogeneous magnetism. Random fields act on ferromagnets via their pinning of domain walls, creating barriers to motion that increase with random-field amplitude. The rising barrier height decreases the probability of reversal at a given applied field, and if the amplitude of the random-field contribution can be controlled continuously, then the pinning potential can be increased or decreased on demand. There is an extensive experimental literature on the properties of the RFIM (11). Until recently, however, the work has been confined to site-diluted antiferromagnets that lack a net magnetization and, hence, are unsuited for examining domain reversal dynamics. Instead, the reversal dynamics of the RFIM have been studied via simulations (12, 13), focusing largely on the properties of avalanches and related phenomena.

For a tuneable random-field in a *ferromagnet*, we turn to the LiHo_{x}Y_{1–4}F_{4} salts, which can be described by the S = 1/2 Ising model. The parent compound, LiHoF_{4}, is a dipolar-coupled ferromagnet with Curie temperature T_{C} = 1.53 K. Partially substituting nonmagnetic yttrium for the magnetic holmium atoms results in a suppression of T_{C} with a breakdown of the long-range ordered state for x < 0.3 (14). Experiment (15) and theory (16, 17) from the last few years demonstrate that the combination of the random dilution of the magnetic ions, the off-diagonal components of the dipole interaction, and the application of a magnetic field transverse to the Ising axis, produce a site-random-field along the Ising axis whose strength scales with the external transverse field. LiHo_{x}Y_{1-x}F_{4} for x > 0 is, thus, a ferromagnetic realization of the RFIM. Whereas this realization only appears for T < 1 K, similar effects could be designed to occur at or above room temperature by using conventional high-anisotropy ferromagnets mixed with a nonmagnetic material in a manner analogous to yttrium doping in LiHo_{x}Y_{1-x}F_{4}.

The transverse field also can affect the dynamics of LiHo_{x}Y_{1-x}F_{4} by adding a *σ*_{x} term to the Hamiltonian (18, 19), mixing the original eigenstates and providing a quantum-tunneling pathway for domain reversal. The effective Hamiltonian is then: [1]where *J*_{ij} is the interspin coupling along the Ising axis, *h*_{i} is the random-field term proportional to the external transverse field *H*_{t}, and sets the scale of the quantum-tunneling rate. The dynamics of domain growth and reversal depend on which of the latter two terms in Eq. **1** dominate, as illustrated in Fig. 1*A*. In a regime where the random-field effects control the behavior, domains nucleate at pinning sites corresponding to local maxima in the random-field and have sharp, well-defined walls whose shape is determined by the distribution of these pinning sites. When quantum-tunneling dominates, the Heisenberg uncertainty principle implies broadened domain walls, averaging out the effects of any local pinning centers. In this tunneling regime, propagation of a domain wall follows dynamics akin to a particle tunneling through a potential barrier, where application of a transverse field tunes both the barrier height and the effective mass of the particle (Fig. 1*B*).

Our work on disordered ferromagnets in the LiHo_{x}Y_{1-x}F_{4} series has focused primarily on samples with composition x ∼ 0.5, where long-range order survives even in the presence of considerable random site disorder. The phase diagram of Fig. 1*C* for x = 0.44 results from the competition between terms in Eq. **1** with a complex ground state as *T* → 0 that can be accessed differently via classical and quantum trajectories (20–22) as well as domain dynamics that can be controlled with Γ (18). The behavior of the phase boundary, and in particular the presence of a low-temperature transverse-field-driven transition into a paramagnet state (19), demonstrate that it is necessary to incorporate a quantum mechanical treatment to properly model the magnetism, rather than a fully classical model such as the Stoner–Wohlfarth model in which the spins uniformly rotate into the transverse plane. In the semiclassical regime near the Curie point, *T*_{C}(*x* = 0.44) = 0.67 K (15), the critical behavior is dominated by random-field effects due to the internal field *h*_{i}. The experiments reported here involve dc magnetization loops measured to large longitudinal (along the Ising axis) fields in both the classical (*T* ∼ *T*_{C}) and quantum regimes. The application of a large longitudinal field polarizes the spins and erases the memory of a particular trajectory through the complex free energy landscape.

## Results and Discussion

Fig. 2 captures the essential result of the paper, showing magnetic hysteresis loops for LiHo_{0.44}Y_{0.56}F_{4} taken to saturation at two temperatures. At low temperature (*T* ≪ *T*_{C}), where the dynamics are dominated by quantum mechanics (18), applying a transverse field results in a narrowing of the hysteresis loop due to the increase in domain wall tunneling (Fig. 1*B*). Near the Curie temperature, on the other hand, applying the same field results in the opposite behavior, with the loop broadening, indicating that the random fields increase the pinning. The transverse field thus can be used to either narrow or broaden the magnetic hysteresis, with temperature used to select which regime is in effect.

The dynamics of magnetization reversal can be explored quantitatively by examining how the area enclosed by the hysteresis loop changes as a function of temperature, field, and sweep rate. We focus in Fig. 3 on the quantum-fluctuation-dominated (low temperature) regime. The enclosed area depends strongly on the time taken to traverse the loop (Fig. 3*A*). As the rate of change of the longitudinal field is reduced, the area of the loop shrinks monotonically (Fig 3*B*), indicating that even over the 36-h time span of the slowest loops, the system is still approaching equilibrium, underscoring once again the need to explicitly account for finite frequency effects due to slowly relaxing spin clusters at low *T* (23). We estimate the infinite-time behavior by fitting the data to *A* = *A*_{0} + *A*_{1}|*dH*_{l}/*dt*|^{1/2}, where area *A* = ∮4*πMdH* is integrated around a complete hysteresis loop. This experimentally observed form follows the *t*^{-1/2} approach to equilibrium predicted for Ising systems based on domain coarsening models (24).

As seen in Fig. 3*C*, the behavior of the system for *T* = 200 mK remains essentially unchanged until *H*_{t} = 3.5 KOe. With increasing transverse field and, hence, an increasingly larger term proportional to *σ*_{x} in the Hamiltonian of Eq. **1**, the tunneling rate increases, and the macroscopic hysteresis loop narrows monotonically; previous studies (18) have shown that the reversal proceeds by groups of up to 10 spins tunneling coherently. The observed *H*_{t} = 3.5 kOe onset for the narrowing corresponds to a crossover seen in the nonlinear dynamics of strongly diluted LiHo_{0.045}Y_{0.955}F_{4} (25), suggesting that a transverse-field-induced level crossing in the full microscopic Hamiltonian, including nuclear hyperfine terms (19, 26), is required before there is a significant probability of domain reversal via quantum tunneling. Above 3.5 kOe, the domain walls are still localized to some extent by the pinning centers, until their tunneling delocalizes them sufficiently for the hysteresis loop to close, with the enclosed area going to zero at the phase transition from ferromagnet to paramagnet, independently established via ac susceptibility (18). We, thus, can think of the quantum phase transition in this disordered ferromagnet as an Anderson localization/delocalization transition where domain walls play the role of particles and form bands of extended states analogous to Bloch waves (27, 28).

In the high-temperature, classical regime, different dynamics dominate the magnetization reversal. The ratio of *k*_{B}*T* to the energy barrier is sufficiently large that the primary mechanism for domain motion is thermally activated hopping (18). As in the quantum regime, the hysteresis loops narrow as the sweep rate is reduced (Fig 4*A*). Here, we discover that loop areas exhibit a square-root dependence on the sweep rate at fast rates, but approach saturation at intermediate rates of order 25 Oe/ min. We show in Fig. 4*B* that the sweep rate determining the crossover to saturation is transverse field-independent within the ferromagnet, only changing at the transition into the paramagnet (Fig. 4*D*). Unlike the quantum-tunneling regime that showed *t*^{-1/2} behavior throughout the range of times investigated, it is possible at higher *T* to access a cut-off length scale for the coarsening of domains, consistent with predictions for the random-field problem (29, 30). We obtain an estimate for this length scale via the relationship (31) , where the ratio of temperature to the bare coupling constant *k*_{B}*T*/*J* = 0.36 for *T* = 0.55 K, the ratio of the strength of the random field to the bare coupling constant *h*/*J* ∼ 0.3 for LiHo_{0.44}Y_{0.56}F_{4} in a 3.5 kOe transverse field (16, 17), and *t* = 6 × 10^{13} is a dimensionless time set by the product of the 10^{10} Hz attempt frequency for single-spin reversal (18) and the 100 min required to sweep the longitudinal field from saturation to zero. These values yield an equilibrium domain size ∼0.14 μm (260 unit cells) on a side, a cut-off scale an order of magnitude smaller than the ∼5 μm domains observed in pure LiHoF_{4} (32).

We map the transverse field dependence of the loop area in Fig. 4*C*. Unlike what we see for the lower temperature data in Fig. 3*C*, the hysteresis is non-monotonic with field, actually increasing with increasing transverse field up to *H*_{t} = 3.5 kOe and then decreasing. As the strength of the random-field term in Eq. **1** is increased, the effective pinning of the domains increases as well. For *H*_{t} > 3.5 kOe, quantum fluctuations once again accelerate domain reversal and the loop begins to narrow, eventually vanishing at the *H*_{t} = 5 kOe paramagnetic phase boundary. At higher *T*, where the ferromagnetic order is quenched for *H*_{t} ≤ 3.5 kOe, the main effect of the transverse field is to increase the pinning forces, in sharp contrast to the lower temperature regime closer to the quantum critical point, where the dominant effect of the transverse field is to increase quantum fluctuations.

We have demonstrated the isothermal control of magnetic domain dynamics via two competing mechanisms, the first, associated with classical random fields and the second with quantum tunneling of magnetic domains, in a disordered, anisotropic ferromagnet. The laboratory control of tunneling probabilities via the simple application of a transverse field permits a clear delineation of quantum criticality as the result of the quantum tunneling-induced dissolution and depinning of domain walls. While we have done our experiments for a model magnet with a low Curie point, we envision tailoring mixed rare earth/transition metal alloys to produce anisotropic magnets where transverse external fields yield tunable internal random fields at room temperature, hence opening the door for technological application of these effects. The starting point for such an effort would be highly anisotropic ferromagnets such as SmCo, FePt, or NdFeB materials that have been extensively characterized for potential use as storage media (33). Preparing the materials in needle-like grain structures would combine the intrinsic crystalline anisotropy with a strong shape anisotropy to produce a system where each grain couples to its neighbors via a dipole-dipole interaction. This would result in a room temperature analog to LiHoF_{4}, the parent compound of the material discussed in this work. In a manner similar to the substitution of yttrium for holmium, randomness could then be introduced by sintering needles of the magnetic material with those of a compatible nonmagnetic material. Whether quantum tunneling of domains can be dialed in at elevated temperatures for other materials is a more difficult question. The high characteristic energy scales for transition metals (34) and their oxides might allow us to make some progress should these scales be switchable optically, piezoelectrically, or via an electrical gate.

## Methods

We attached a commercial GaAs magnetometer (Toshiba THS118) with a (0.6 × 0.6) mm^{2} active area to a (0.8 × 0.8 × 3) mm^{3} single crystal LiHo_{0.44}Y_{0.56}F_{4} needle (Fig. 1*D*), and mounted the assembly in a differential configuration on the cold finger of a helium dilution refrigerator equipped with a 5T/2T vector magnet. The sample was cooled from above *T*_{C} to the operating temperature in zero field and a series of transverse fields subsequently applied. Hysteresis loops were swept to ± 2.5 kOe along the Ising axis, sufficient to saturate the magnetization for all temperatures and transverse fields and the Hall response was measured by using standard lock-in amplifier techniques in the Ohmic and frequency-independent limits. The applied field was simultaneously measured by using a second magnetometer chip inside the cryostat to eliminate any spurious hysteresis effects from trapped flux in the solenoid. The range of times probed was bounded from below by the response time of the instrumentation and from above by the 36 h maximum interval between refills of the cryostat helium reservoir.

## Acknowledgments

The work at the University of Chicago was supported by DOE Basic Energy Sciences under Grant No. DEFG02-99ER45789, while work in London was supported via the United Kingdom Engineering and Physical Sciences Research Council. D.M.S. acknowledges use of the central facilities of the NSF Materials Research Science & Engineering Center.

## Footnotes

^{1}To whom correspondence should be addressed. E-mail: tfr{at}uchicago.edu.Author contributions: D.M.S., G.A., and T.F.R. designed research; D.M.S. and T.F.R. performed research; D.M.S. analyzed data; and D.M.S., G.A., and T.F.R. wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.

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