Interpretation of thermal dependence of magnetic aftereffect for magnetic nanocomposite with slow decay rates

Abstract

A new experimental characterization is presented of time-, field-, and temperature-dependent dynamic effects in magnetization of a nanocomposite which displays slow decay. Field and temperature variations of irreversible susceptibility, χ _irr , decay coefficient, S, fluctuation field, h_f , and activation volume, V, have been calculated for the nanocomposite sample (Co₈₀Ni₂₀) using a recently developed modified Preisach–Arrhenius (MPA) model. The sample is composed of non-interacting nanoparticles having negligible reversible magnetization. Non-Arrhenius behavior is observed in both the maximum decay coefficient, S _max, and the fluctuation field, h_f , as a function of temperature T. The peak of both temperature curves are identical and occur at a critical temperature T_k of ∼50 K, which agrees with our experimental results. Based on the effect of a temperature-dependent chemical potential on energy barrier, h_f is studied for T < T_k and T ≥ T_k , respectively. A more complete MPA model that can predict the magnetization as function of time, field and temperature for a magnetic material with slow decay rates is proposed. This model uses a multi-variable analytical formula, , which incorporates the characteristic parameters.

Keywords:

• magnetic aftereffect
• fluctuation field
• numerical modeling
• Co₈₀Ni₂₀ nanocomposite

1.Introduction

In magnetic information storage devices where a certain magnetization state must be permanently conserved, the decay of magnetization towards anhysteretic ground state with time is normally slow due to the presence of large energy barriers. However, continuous reduction of particle sizes is required in order to achieve ultra-high storage density and the reversal of magnetization has become easier as a result. Thus, magnetic aftereffects of mono-domain ferromagnetic nanoparticles with diameters in the range 10–100 nm, and exhibiting slow decay, have attracted considerable interest [1–5 5].

A modified Presiach–Arrhenius (MPA) model has recently been presented [6] which predicts time reliability of magnetization state in magnetic nanoparticles based on (i) quasi-equilibrium thermodynamics [7,8], (ii) the assumption that the decay rate is maximized at the field that maximizes the irreversible susceptibility [9], and (iii) experimental and artificial data on a Co₈₀Ni₂₀ nanocomposite.

The model can predict the shape of the entire magnetic aftereffect curves, for magnetic materials with slow decay rate, which cannot be observed experimentally within a reasonable time interval. It resolved problems that were attributed to the instability of the Preisach function by including a feedback parameter derived from the Moving model [10,11] and the variance–variable model [12] into the operative field.

The following analytical formula is derived from the MPA model [6],

(1)

where t is the relative decay time, h_ri is the holding field within the region of interest, T is temperature, t_k is the critical time and σ _t is the standard deviation of decay rate. Equation (1)(1) enables us to predict the magnetization at any time during the decay process at a given h_ri and a given temperature.

In this investigation, the joint presence of field-induced effects and thermal activation in magnetic aftereffect is interpreted. Both field and temperature dependence are shown in characteristic magnetic state parameters, such as coercivity, irreversible susceptibility, decay coefficient and fluctuation field, that are derived from hysteretic and aftereffect measurements. Extension of Equation (1)(1) to a multi-variable function is proposed so that the precise nature of time, field and temperature dependence in magnetic aftereffect can be rigorously described. The accuracy of the extended model has been validated by experimental data.

2. Experimental characterization

The nanocomposite sample used in this study consists of monodispersed quasi-spherical Co₈₀Ni₂₀ ferromagnetic nanoparticles with 50 nm average diameter (i.e., homogeneous in size). The nanoparticles, prepared by the polyol method, are embedded in a diamagnetic PVC polymer matrix with uniform distribution [13]. The nanocomposite contains the Co₈₀Ni₂₀ nanoparticles at 0.4 wt % concentration. Hence, each nanoparticle is an isolated single-domain particle, and according to Wohlfarth [9], the activation volume of each nanoparticle is the actual particle volume. The sample also satisfies the MPA model criteria: (i) negligible reversible magnetization and (ii) negligible interaction between the magnetic nanoparticles. As shown in Figure 1, physical properites (i.e. size, shape, and distribution) of the nanocomposite sample are obtained from TEM and SEM images.

Figure 1. (a) TEM image [1] of Co₈₀Ni₂₀ nanoparticles with 50 nm average diameter and (b) SEM image [13] of the Co₈₀Ni₂₀ nanoparticles prepared by polyol process.

Hysteresis loops and magnetic aftereffect curves of the Co₈₀Ni₂₀ nanocomposite were measured at temperatures ranging from 5 K to 300 K, using a SQUID magnetometer (Quantum Design, MPMS XL 5.0 Tesla). The diamagnetic contribution of the PVC matrix has been measured separately and substracted in the data presented.

Figure 2a shows descending hysteretic curves at selected temperatures from 5 K to 300 K over the field range –5 Tesla to 5 Tesla. As the temperature increases from 5 K to 300 K, coercivity H_c decreases monotonically from 0.095 Tesla to 0.035 Tesla (Figure 2b, shown by an arrow).

Figure 2. (a) Hysteresis loops of the Co₈₀Ni₂₀ nanocomposite at selected temperatures from 5 K to 300 K after correction for the diamagnetic PVC matrix. Main panel: highlight of the hysteresis loops over field range –0.5 to 0.5 Tesla. Inset: hysteresis loops over the field range –5 to 5 Tesla. (b) Highlight of hysteresis loops in (a) over the field range –0.2 to 0.0 Tesla. The arrow illustrates the shift of decreasing coercivity H _c (increasing –H _c) as the temperature increases from 5 K to 300 K.

The irrversible susceptibility χ _irr [14] is defined as

(2)

Since the reversible magnetization is negligible for the Co₈₀Ni₂₀ nanocomposite, χ _irr can be derived from the remanent magnetization curve. As shown in Figure 3, χ _irr is dependent on both applied field and temperature. As temperature increases from 5 K to 300 K, the value of the maximum irrversible susceptibility χ _irr,max (i.e. the peak of each χ _irr curve) increases monotonically and the location of the χ _irr,max shifts to lower field (shown by arrow). For any combination of temperature and field, the value of the correponding irrversible susceptibility can be found on the three-dimensional surface in Figure 4.

Figure 3. Applied field dependence of irrversible susceptibility for the Co₈₀Ni₂₀ nanocomposite at selected temperatures from 5 K to 300 K. The maximum irreversible susceptibility χ _irr,max of each curve (peak value) is represented by black dot. The arrow illustrates the shift of increasing χ _irr,max as the temperature increases from 5 K to 300 K.

Figure 4. Applied field and temperature dependence of irreversible susceptibility for the Co₈₀Ni₂₀ nanocomposite. The color varies from light green for small values of χ _irr to purple for χ _irr,max.

3. Implementation of modified Preisach-Arrhenius model

The recently developed MPA model can predict the shape of the entire magnetic aftereffect curves for magnetic materials with slow decay rate, and thus, enables us to accurately extract the maximum decay coefficient, S _max, at a given holding field, H, and temperature, T. Figure 5 shows the predicted sigmodal-shaped decay curve at a given temperature (T = 50 K) and holding field (–0.1 Tesla) using MPA model (solid blue). The location and value of the maximum decay slope S _max (dotted) can be accurately calculated.

Figure 5. Extraction of decay coefficient from decay curve using MPA model.

After obtaining the values of decay coefficient at any given temperature and field, the applied field dependence of decay coefficient is studied for selected temperatures within the range of 10 K to 300 K. As shown in Figure 6, the decay coefficient is dependent on both the applied field and the temperature. At each temperature, the applied field at which the maximum decay coefficient (black dot) occurs has been defined as the remanent coercivity h_rc [15]. As the temperature increases from 10 K to 300 K, the value of h_rc decreases monotonically, as shown by the arrow. The value of S _max, increases as temperature increases from 10 K to 50 K (dotted line) and then decreases as temperature increases further from 50 K to 300 K (dashed line). Note: S _max reaches the maximum value when the temperature is 50 K.

Figure 6. Holding field dependence of decay coefficient, S, for the Co₈₀Ni₂₀ nanocomposite at selected temperatures from 10 K to 300 K. As temperature increases from 10 K to 300 K, S _max increases first from 10 K to 50 K, reaches peak value at 50 K and then decreases from 50 K to 300 K.

Non-Arrhenius behavior [16] is observed in the maximum decay coefficient S _max as a function of the temperature T curve, with the peak, S _max, at the critical temperature T_k of ∼50 K (Figure 7). The non-Arrhenius behavior implies that the density of states in energy space is not uniform and the energy distribution of magnetic excitations obeys quantum statistics rather than Maxwell–Boltzmann statistics which are assumed in the Preisach–Arrhenius model. Bose-Einstein statistics have been proposed to explain similar non-Arrhenius behavior in nano-magnetic materials [17,18]. In Figure 8, the maximum decay coefficient is plotted, in three dimensions, as a function of both temperature and holding field. For any combination of temperature and field, the value of the correponding maximum decay rate can be found on the surface in Figure 8.

Figure 7. Temperature dependence of the maximum decay coefficient for the Co₈₀Ni₂₀ nanocomposite. MPA model evaluated S _max at selected temperatures are represented by red dots. The non-Arrhenius behavior is illustrated by the black fitting curve.

Figure 8. Temperature and field dependence of the maximum decay coefficient for the Co₈₀Ni₂₀ nanocomposite. The maximum decay coefficient varies from light green for small values of S _max to dark blue for peak value of S _max.

4. Extension to modified Preisach-Arrhenius model

The fluctuation field h_f , introduced by Néel [19] and expanded by Street and Brown [14] is defined as

(3)

where S is the decay coefficient and χ _irr is irreversible susceptibility. Hence, h_f can be rigorously determined from macroscopic measurements of S and χ _irr . As shown in Figures 4 and 8, both S and χ _irr are dependent on the variation of T and H, where H is the holding field in S(H, T) and the applied field in χ _irr (H, T). Note, the fluctuation field may also have time dependence, which can be attributed to the gradual variation in magnetic properties of the moments during the thermal relaxation process.

In Figure 9, the fluctuation field is plotted, in three dimensions, as a function of both temperature and field. Note: the generated surface (highlighted by red circle) is expected to have percent error of ±0.5%, due to the limited amount of data obtained from measurements of S and χ _irr .

Figure 9. Temperature and applied field dependence of the fluctuation field for the Co₈₀Ni₂₀ nanocomposite. h_f were rigorously determined from macroscopic measurements of S and χ _irr .

At each holding field in the region of interest, the non-Arrhenius behavior has been observed in the fluctuation field vs. temperature curves. Figure 10 shows the temperature dependence of the fluctuation field at a representative applied field of –0.1 Tesla, with a peak value at a critical temperature T_k of ∼50 K. Even though the location of the critical temperature varies at different holding fields in the range of negative saturation to positive saturation, the variation is negligible within the region of interest. Hence, for the special case of a monodispersed nanoparticulate system that exhibits slow decay, the fluctuation field, the driving force of the magnetic thermal aftereffect over the energy barrier, can be assumed to be independent of the holding field within the region of interest. Therefore,

(4)

Figure 10. Temperature dependence of fluctuation field at a representative applied field of –0.1 Tesla.

Wohlfarth [9] suggested that h_f can alternatively be evaluated by equating two definitions of an activation energy

(5)

Hence, the h_f can alternatively been given by

(6)

where μ₀ is the Bohr magneton, k is the Boltzmann constant, M_S is the saturation magnetization, T is the temperature and V is the activation volume [20–22 22].

Della Torre and co-workers [16] introduced a temperature-dependent chemical potential ζ to describe the magnetic aftereffect when the holding field is close to coercivity, i.e. the system has a substantial spin entropy s. The temperature dependence of the fluctuation field below and above the critical temperature can then be described separately. For T < T_k , the fluctuation field h_f (T) can be described by Equation (6)(6), which is directly proportional to temperature, except for weak temperature changes associated with variations in M_S and V. For T ≥ T_k , the effect of chemical potential on the energy barrier is not negligible and the expression of h_f (T) is modified, accordingly:

(7)

Figure 11 shows the joint field and temperature dependences of the fluctuation field that were evaluated using Equation (7)(7). The difference between Figures 9 and 11 in the peak region (highlighted by red circle in Figure 9) is attributed to the inaccuracy in the measured data (±0.5%).

Figure 11. Temperature and applied field dependence of the fluctuation field for the Co₈₀Ni₂₀ nanocomposite. h_f were evaluted using Equation (7)(7).

It is proposed to incorporate the interpretation of the temperature dependence of magnetic aftereffect into the previously developed MPA model and to extend Equation (1)(1) to a multi-variable function, based on the behavior of fluctuation field

(8)

where critical time t_k (T) is given by

(9)

the standard deviation of decay rate σ _t (T) is given by

(10)

and fluctuation field h_f (T) can be given by Equation (6)(6) and Equation (7)(7) for T < T_k and T ≥ T_k , respectively.

Note that expanding Equation (8)(8) up to the 4^th order term (n = 4) gives a computationally efficient approximation of the equation.

5. Conclusion

The characteristic parameters, such as irreversible susceptibility, χ _irr , maximum decay coefficient, S _max, and fluctuation field, h_f , have been derived on dimensional grounds, i.e. both field and temperature variation, from hysteretic and thermal aftereffect measurements and theoretical analysis based on the MPA model. As the temperature increases, while χ _irr,max increases monotonically, S _max and h_f within h_ri demonstrate non-Arrhenius behaviors, which are interpreted as follows: (i) for T < T_k , h_f , derived by equating two definitions of activation energy, is directly proportional to temperature, except for weak variations in saturation magnetization M_S and activation volume V with temperature, (ii) for T ≥ T_k , a temperature-dependent chemical potential reduces the effective energy barrier, and as a result, exerts an influence on the behavior of h_f . These interpretations have been incorporated into the extended multi-variable analytical MPA model, which is quantatively consistent with the presented data.

Acknowledgements

The authors thank Dr. Vázquez of CISC, Madrid, Spain for providing us with the nanocomposite sample. The authors are grateful to Prof. M. Wagner and Dr. C. Yan of The George Washington University for collaboration and to Drs. C. Dennis and V. Provenzano of the National Institute of Standards and Technology for fruitful discussions. The work is partially supported by National Science Foundation under Contract no. 0733526 and no. 1031619.