Switchable hardening of a ferromagnet at fixed temperature

Results and Discussion

Fig. 2 captures the essential result of the paper, showing magnetic hysteresis loops for LiHo_0.44Y_0.56F₄ 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. 1B). 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.

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Fig. 2.

Magnetic hysteresis loops for LiHo_0.44Y_0.56F₄ in the quantum and classical random-field regimes. In the low-temperature quantum-dominated regime (left), applying a transverse field increases the tunneling rate, monotonically narrowing the hysteresis loop. In the high-temperature classical regime (right), applying a 3.5 kOe transverse field increases the effective pinning energy, widening the hysteresis loop.

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. 3A). As the rate of change of the longitudinal field is reduced, the area of the loop shrinks monotonically (Fig 3B), 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 = A₀ + A₁|dH_l/dt|^1/2, where area A = ∮4πMdH 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).

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Fig. 3.

Rate-dependent dissipation for LiHo_0.44Y_0.56F₄ in the quantum regime. (A) Hysteresis loops at T = 200 mK and H_t = 9.0 kOe at longitudinal field sweep rates of 200, 70, and 15 Oe/ min. (B) Area enclosed by hysteresis loops at T = 200 mK and H_t = 0,3.25,3.75,5,6,7.5,9,10 kOe vs. the square root of the sweep rate of the longitudinal magnetic field. Area calculated by integrating the magnetization around the full hysteresis loop. Lines are fits to . (C) Long-time-limit enclosed area (A₀) vs. transverse field. The area enclosed by the loop decreases monotonically with increasing transverse field, vanishing at the paramagnetic phase boundary. Lines are guides to the eye. Arrow shows T = 200 mK ferromagnetic-paramagnetic phase boundary measured independently via ac magnetic susceptibility (18).

As seen in Fig. 3C, 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_0.045Y_0.955F₄ (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_BT 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 4A). 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. 4B 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. 4D). 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_BT/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_0.44Y_0.56F₄ in a 3.5 kOe transverse field (16, 17), and t = 6 × 10¹³ is a dimensionless time set by the product of the 10¹⁰ 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 LiHoF₄ (32).

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Fig. 4.

Rate-dependent dissipation for LiHo_0.44Y_0.56F₄ in the classical random-field regime. (A) Hysteresis loops at T = 550 mK and H_t = 3.25 kOe at longitudinal field sweep rates of 200, 100, and 20 Oe/ min. (B) Area enclosed by hysteresis loops at T = 550 mK and H_t = 0,2,3.25,5 kOe vs. the square root of the sweep rate of the longitudinal magnetic field. Curves are fits to a constant A₀ at low sweep rate and above the crossover point. (C) Long-time-limit enclosed area (A₀) vs. transverse field. Increases in random-field pinning results in a peak in the curve at H_t = 3.25 kOe. Curve is a guide to the eye. Arrow shows T = 550 mK ferromagnetic-paramagnetic phase boundary (18). (D) Crossover time between equilibrium and activated regimes, showing a constant time until the transition to paramagnetism is reached. Lines are guides to the eye.

We map the transverse field dependence of the loop area in Fig. 4C. Unlike what we see for the lower temperature data in Fig. 3C, 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 LiHoF₄, 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.