Abstract
Rate (master) equations are ubiquitous in statistical physics, yet, to the best of our knowledge, a rate equation with memory has previously never been considered. We write down an integrodifferential rate equation for the evolution of a thermally relaxing system with memory. For concreteness we adopt as a model a singledomain magnetic particle driven by a small ac field and derive the modified Debye formulas. For any memory time Θ the inphase component of the resultant ac susceptibility is positive at small probing frequencies ω, but becomes negative at large ω. The system thus exhibits frequency induced diamagnetism. For comparison we also consider particle pairs with dipolar coupling. The memory effect is found to be enhanced by ferromagnetic coupling and suppressed by antiferromagnetic coupling. Numerical calculations support the prediction of a negative susceptibility which arises from a phase shift induced by the memory effect. It is proposed that the onset of frequency induced diamagnetism represents a viable experimental signature of correlated noise.
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Introduction
The Debye formulas determine the initial ac response of a two level, equilibrium system to an applied ac perturbation. We address here the dynamic magnetic susceptibility discussed by e.g. Bermúdez et al^{1}. or Klik et al.^{2}. Motivating the research is the underlying assumption of uncorrelated (white) noise adopted in all models to date and the need to investigate the implications of colored noise arising from the temporal correlation of heat bath properties. This is an important generalization of the theoretical approach and is especially timely in relation to the current advances in ultrafast laser processes involving femtosecond laser processes. Following the pioneering work of Beaurepaire et al.^{3} demonstrating ultrafast demagnetization of Ni, more recent work has shown extremely complex and subtle behavior during laser driven processes including thermally driven magnetization switching^{4,5} in which the magnetization is reversed by the action of a thermal pulse in the absence of an applied field. Ultrafast experiments rely on the rapid thermally driven relaxation of the spin system, modelled in refs^{4,5}. using a white noise thermostat, however it has been shown^{6} that correlated noise can significantly slow down the relaxation processes. Ultrafast laser processes are extremely complex, for this reason we consider here the study of the effects of correlated noise on the wellunderstood Debye formulas. This provides an important generalization of the Debye formulas and also provides an experimentally realizable means of determine the correlation times.
For simplicity we adopt the well known uniaxial singledomain particle with magnetic field applied parallel to its easy axis. The corresponding free energy^{7} is
with K the anisotropy constant, V the particle volume and the reduced field h = H/H_{ K } where H = H(t) is the time dependent applied magnetic field and H_{ K } = 2K/M_{ s } is the anisotropy field. Further, M_{ s } is the saturation magnetization of the particle and x = cosϑ, where ϑ is the angle spanned by the magnetization vector and the applied field.
At \(h < 1\) the system has two local minima located at x = ±1 and thermally activated transitions between these two minima take place with rates \({\kappa }_{12}={\kappa }_{1\to 2}\) and \({\kappa }_{21}={\kappa }_{2\to 1}\). Neglecting all intrawell processes^{8,9,10,11}, the thermally activated dynamics of the system are then given by the simple master (rate) Eq.^{7,12} for the occupation probabilities n_{1} and n_{2} = 1 − n_{1},
\({\dot{n}}_{1}=d{n}_{1}/dt\) and we made use of the probability normalization. For the uniaxial particle (1) there is \({\kappa }_{12}={f}_{0}\exp [q{\mathrm{(1}+h)}^{2}],\) and \({\kappa }_{21}={f}_{0}\exp [q{\mathrm{(1}h)}^{2}]\) where f_{0} is the prefactor, usually taken to be a constant^{13,14} and \(q=KV/{k}_{B}T\) where k_{ B } is the Boltzmann constant and T the ambient temperature. Let now
where 0 < h_{0} ≪ 1 is a small field amplitude and ω is the field frequency. Solving Eq. (2) in the first order of h_{0} we obtain for the reduced magnetization m = M/M_{ s } the expression
with the Debye susceptibilities
Γ = 2f_{0}e^{−q}. In these expressions we introduced the quantity \({\tilde{n}}_{\uparrow \uparrow }=\mathrm{1/4}\) whose significance shall become apparent later, when we consider the response of interacting particle pairs. It should be noted that we use only O(h_{0}) terms of the relaxation rates whose precise form is thus irrelevant. For this reason Eqs (5) and (6) hold also for other physical quantities, such as electric susceptibility^{1}. At this point is is worth evaluating the generality of the master equation expressions for the susceptibility. To do this we have evaluated the temperature dependence of the susceptibility in Supplementary Information S1. It is shown that Eqs (5) and (6) predict results in good agreement with experiment, with correct limiting behaviour as \(q\to \infty \) and \(q\to 0\). In Supplementary Information S1 we also show that the numerical approach, to be described shortly, give behaviour in good qualitative agreement.
If the system has memory, then in place of Eq. (2) we write the integrodifferential equation
with the memory kernel K(t), which we take to have the exponential form^{15,16}
Θ is the memory time. This topic has been discussed in a different context by Sun et al.^{17,18}.
To first order in h_{0} Eq. (7) is composed of two convolution integrals and is readily solved. The long time, steady state solution then has the form (4), with
The two functions χ_{1}(ω) and χ_{2}(ω) are plotted in Fig. 1, where we note we have used a large value of the escape rate, Γ = 1 Hz, for visual clarity. The salient feature of the inphase component χ_{1} is the fact that it becomes negative at sufficiently large frequencies for any Θ > 0, passes through a local minimum and only then rapidly approaches its asymptotic zero value from below. This means that the particle is paramagnetic at low frequencies, but diamagnetic at high frequencies. In the diamagnetic phase of the response the particle is apparently slowed down by the medium in which it finds itself and cannot follow the rapidly oscillating field. One can therefore expect to find the frequency induced diamagnetism in particles suspended in viscous fluids. However, the effect has previously been observed experimentally in single crystals by Rhyee et al.^{19}. Theoretically the effect can be so strong that it would render the magnetic permeability negative^{20}. The outofphase component is interesting in that its peak grows significantly higher and narrower with increasing memory time, so that we conclude that memory enhances energy losses and heating of the sample in a narrow frequency interval. This behavior is required from particles used to treat cancer by magnetic particle hyperthermia^{21}. We note that, in the fully hysteretic regime with a conventional (white noise) thermostat, El Mrbati et al.^{22} have shown diamagnetic behaviour, here we demonstrate a very general effect arising from the nature of the noise itself.
So far we have considered only noninteracting media. Now we shall consider an exactly solvable model system of two identical superparamagnetic particles with parallel easy axes^{7,23}. The particles interact with each other via dipolar coupling and have three metastable states: \(\uparrow \uparrow \), \(\uparrow \downarrow +\downarrow \uparrow \) and \(\downarrow \downarrow \). The mutual coupling is ferromagnetic if the bond angle β between the easy axes is zero, β = 0 and antiferromagetic if β = π/2. The thermally driven dynamics of the particle pair, at the small fields of interest here, are described by the three level master equation^{7}
where again we made use of the probability conservation n_{1} + n_{2} + n_{3} = 1. The two particle occupation probabilities are \({n}_{1}={n}_{\uparrow \uparrow }\), \({n}_{2}={n}_{\uparrow \downarrow +\downarrow \uparrow }\) and \({n}_{3}={n}_{\downarrow \downarrow }\). Obviously, n_{2} corresponds to a demagnetized state and the reduced magnetization m = 2(n_{1} − n_{3}). The rates of the thermally activated transitions between the individual minima, \({\kappa }_{ij}={\kappa }_{i\to j}\), are given by the formula
with
The dipole strength \(\rho ={M}_{s}^{2}V\mathrm{/2}K{R}^{3}\) where R is the distance between the particles. If the bond angle β = 0, then the critical field \({h}_{c}^{2}\mathrm{=(1}\rho \mathrm{)(1}3\rho )\) and ε = 8. At β = π/2 there is \({h}_{c}^{2}\mathrm{=(1}\rho \mathrm{)(1}+3\rho )\) and ε = −4. Introducing now memory into Eqs (11) and (12) in the same manner as was done in Eq. (7) and linearizing with respect to h_{0}, we obtain again the formulas (9) and (10), but with \({\rm{\Gamma }}=2{f}_{0}{e}^{q({h}_{c}^{2}+\varepsilon \rho )}\), and
in zero field. The formal equivalence of the two results, for interacting and noninteracting particles, follows from the fact that in the first order of h_{0} Eqs (11) and (12) yield a single rate equation for the reduced magnetization m. This property follows from the symmetry of the interacting system and is preserved also if a mean field theory is employed. However, it could not hold for more complex, strongly coupled particle assemblies whose response is thus not of the Debye type^{24}. It should be noted that ferromagnetic coupling with ε > 0 reduces the rate Γ and facilitates the onset of the diamagnetic state, while antiferromagnetic coupling, which favors the creation of magnetically neutral demagnetized states, has the opposite effect.
The dynamics of superparamgnets are also amenable to investigation via direct numerical simulation of the corresponding Langevin equation, the LandauLifshitzGilbert (LLG) equation. The Markovian rate equations of the form Eq. 2 correspond to the high barrier limit of the LLG^{14,25}, in what is termed the discrete orientation approximation. The preceding analysis of the nonMarkovian rate equations may then be understood as the direct modification of said high barrier rate equations to include explicit memory effects. In what follows we will compare this approach to that of an explicitly nonMarkovian Langevin equation for the magnetic system.
The LLG equation for the timeevolution of a single spin is
where α is the phenomenological damping constant, \(\gamma =1.76\times {10}^{11}\,{T}^{1}\,{s}^{1}\) is the gyromagnetic ratio and S is a unit vector representing the direction of the magnetic moment. The local magnetic field, H, is obtained from the spin Hamiltonian, \( {\mathcal H} \), as \({\bf{H}}=\frac{1}{{\mu }_{s}}\frac{\partial {\mathcal H} }{\partial {\bf{S}}}\), where μ_{ s } is the spin moment. This may include anisotropy, exchange and other fields as appropriate. At nonzero temperature the local field is augmented by a thermal field, H_{ th }, whose components are assumed to be spatially and temporally uncorrelated, thus
where (i, j) refer to Cartesian coordinates.
The LandauLifshitzMiyazakiSeki (LLMS) Eq.^{6}, is the natural nonMarkovian extension of the LLG, via the introduction of timecorrelated, coloured noise, Under this approach, the spin no longer includes an explicit damping term; rather we consider a spin undergoing only precessional motion. Damping and correlations are included via a bath magnetic field with a characteristic correlation time and which dynamically evolves to align with the spin while also being driven by a white noise. The time evolution of the spin and the bath field are of the form^{6}
where Θ is the noise correlation time, λ is the coupling of the bath to the spins and η is the correlated thermal magnetic field. The fluctuationdissipation theorem for the white noise R is
The Langevin equation for the spinonly is
where \(\psi (t)={e}^{t/{\rm{\Theta }}}\), \(\bar{{\bf{R}}}(t)\) is exponentiallycorrelated thermal noise \(\langle {\bar{R}}_{i}(t){\bar{R}}_{j}(t^{\prime} )\rangle =\frac{\lambda {k}_{B}T}{{\mu }_{s}}\psi (tt^{\prime} ){\delta }_{ij}\) and the last term represents the damping which contains an effect of retardation due to the finite relaxation time of the fluctuating field. The spinonly LLMS is very similar to the LLG equation in the form \(\frac{d{\bf{S}}}{dt}=\gamma ({\bf{S}}\times {\bf{H}}\alpha {\bf{S}}\times \frac{d{\bf{S}}}{dt})\) and by taking the \({\rm{\Theta }}\to 0\) limit, we may recover the LLG equation and obtain the relation α = γλΘ. Both sets of equations are integrated using the Heun method which is consistent with the Stratonovich calculus.
To calculate the susceptibility, we simulate the time evolution of an ensemble of noninteracting spins using either LLG or LLMS as appropriate. The spin is first evolved in time until the system has thermalised. Following this an oscillating magnetic field h_{ z } = h_{0}cos(ωt) was applied to the system along the anisotropy direction, where h_{0} is the field magnitude and ω the frequency. The real and imaginary susceptibility, respectively χ_{1} and χ_{2} are calculated as the cosine and sine Fourier transform of 〈S_{ z }〉, the ensemble average of the zcomponent of the spin. Calculations as a function of ac field magnitude were carried out to ensure the model was applied in the linear region of the system response.
For the following simulations we consider an ensemble of noninteracting Co nanoparticles in the absence of external fields. The particles are assumed to have a volume of V = 8 × 10^{−27} m^{3}, anisotropy energy KV = 3.36 × 10^{−21} J and a magnetic moment of μ_{ s } = 1.12 × 10^{−20} J/T. The temperature and damping parameters are taken such that q = 2 and α = 0.5. An ensemble of 10,000 spins were used for the simulations. We are especially interested in the behaviour of the real part of the dynamic susceptibility, χ_{1} in the vicinity of the analyticallypredicted diamagnetic phase transition.
The variation of the real part of the susceptibility with frequency (normalised by the Larmor frequency, \(\omega ^{\prime} =\omega {(\gamma {H}_{k})}^{1}\)) for various values of τ = ΘγH_{ k } for high frequencies, on the order of the expected phase transition, is shown in Fig. 2. This shows that the susceptibility with correlations becomes negative in this region, supporting the analytical predictions of the master equation approach and confirming that the negative susceptibility arises from a phase change induced by the slowing down of the response of the particle by the medium. We also note a crossover in the variation of χ_{1} with frequency, in agreement with the analytical predictions. Overall, the numerical calculations represent an important counterpart to the analytical model. The presence of a diamagnetic phase transition under both approaches leads us to anticipate that such a transition may be the generic result of memory in magnetic systems, regardless of the precise form of the memory kernel.
Finally, We consider the critical frequency for the transition to diamagnetic behaviour. From Eq. 9 the transition to diamagnetic behavior occurs at the frequency \({\omega }_{c}^{2}={\rm{\Gamma }}/{\rm{\Theta }}\propto {\tau }^{1}\). Thus we estimate that
relating the measurable critical frequency to the correlation time Θ. In Fig. 3 we show the correlation time dependence as exhibited by the LLMS and see that the lowτ behaviour coincides with the master equation, varying as 1/√τ. The prefactor likewise compares well to \(\sqrt{2{f}_{0}{e}^{q}}\), as the Larmor frequency is a good first approximation to the attempt frequency, so we may take f_{0} = 1 in the reduced units and the anticipated prefactor is then simply \(\sqrt{2{e}^{q}}=\mathrm{0.668,}\,0.52\) for q = 1.5, 2 respectively.
Overall, the behavior predicted by the LLMS approach shows nonmonotonic behavior with an increase for large Θ. We anticipate that this departure is due to the assumption of a fixed transition rate, Γ, between the wells in the master equation expression, an assumption which is not generally valid for the LLMS model, where the transition rate will be some, potentially complicated, function of Θ and λ. Given that we have maintained a fixed phenomenological damping, α = γλΘ, an increase in the correlation time will cause a corresponding decrease in the coupling of the bath and the spin. The effect of the lowλ limit on the escape dynamics is much the same as the lowα behaviour of the LLG and will generally lead to a decrease in the escape rate in the limit of vanishing λ. Similarly, an increase in the correlation time of the thermal field will tend to decrease the transition rate between wells^{26}. This effect is similar to the increase in the longitudinal and transverse relaxation times for exchangecoupled spins undergoing LLMS evolution^{6}.
In summary, we have presented analytical and numerical studies of the frequency dependent susceptibility of magnetic nanoparticles. Remarkably, the calculations, consistently between the analytical and numerical models, show a frequencyinduced diamagnetic susceptibility above a critical frequency dependent on the correlation time. The effect is shown to be enhanced by the presence of interparticle interactions, leading to the intriguing suggestion that interaction effects themselves may be considered as a memory effect. It should be noted that further work is needed to link the LLMS approach to the exponentially correlated noise introduced into the master equation formalism. However, this is a nontrivial problem which can only be achieved via the FokkerPlanck equation and is beyond the scope of the current work. However, we stress that the negative susceptibility arises from a phase shift induced by the low frequency filtering property of the nonMarkovian system for both distinct models. Thus it is suggested that the negative susceptibility will be a common feature of systems with correlated noise. The effects of heatbath correlations are not easy to investigate experimentally. Our prediction of frequencyinduced diamagnetic behavior represents an important prediction of a clear experimentally accessible signature of heatbath correlations, with Eq. 26 giving a direct relationship between the correlation time and the critical frequency for diamagnetic behaviour. Measurements of the correlation time would generate new understanding of the properties of thermal baths and their interaction with spin systems and would be especially important in the understanding of ultrafast magnetization processes as pointed out in ref.^{6}.
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Acknowledgements
The authors acknowledge support by the National Science Council grant number NSC982112M002012MY3. The work also made use of the facilities of N8 HPC provided and funded by the N8 consortium and EPSRC (Grant No. EP/K000225/1) coordinated by the Universities of Leeds and Manchester and the EPSRC Small items of research equipment at the University of York ENERGY (Grant No. EP/K031589/1).
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I.K. conducted analytical investigation of rate equations. J.M. implemented Langevin simulations and compiled data. All authors contributed to and reviewed the manuscript.
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Klik, I., McHugh, J., Chantrell, R.W. et al. Debye formulas for a relaxing system with memory. Sci Rep 8, 3271 (2018). https://doi.org/10.1038/s41598018210282
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DOI: https://doi.org/10.1038/s41598018210282
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