where
D =
Dx+ Dy
2 0 0
0 Dx+ Dy
2 0
0 0 Dz
(2.80)
and
ha= hx+ hy
2 ˆex+ hDCˆez (2.81) The equation represents an autonomous dynamic system, hence the methods of nonlinear dynamics (used in the previous section) can now be applied.
2.6 Hysteresis in ferromagnetic resonance
An evident effect of the bifurcations on the dynamics is hysteresis. A neces-sary condition for having hysteresis is the existence of more than one equilib-rium point. Whenever the equilibequilib-rium is destroyed the system state changes abruptly. Even if the previous equilibrium is restored the system does not go back until the new one becomes unstable.
In this section numerical simulations are executed in order to show the ef-fect of a bifurcation and to check the approximation introduced by Krylov-Boguliobov method in systems with small asymmetries. Let us consider a nanomagnet with κeff = 0.5, α = 0.05, hDC = 0.3 and hAC = 0.03, the bifurcation map is shown in figure 2.12.
Starting from ω = 0.55, the frequency is increased up to ω = 0.75 and then decreased back to ω = 0.55 as highlighted in the figure. The ampli-tude m⊥ = p1 − m2z is shown in figure 2.13: the analytical fold over the curve is represented in blue, the interval in which m⊥oscillates is represented in red. Starting from the lower curve, when the saddle-node bifurcation line is crossed the m⊥ jumps to a new equilibrium for ω ≈ 0.69 and stays on the higher curve. When the frequency is decreased m⊥remains on the higher curve until the hopf bifurcation occurs at ω ≈ 0.66. Since the system is now in a stable limit cycle m⊥ oscillates and the cycle grows in amplitude until it collides with a saddle and gets destroyed by the homoclinic bifurcation at ω ≈ 0.63.
A similar simulation is executed for a slightly asymmetric system. Let us con-sider a nanomagnet with Dx = −0.05, Dy = 0.05 and Dz = −0.5, while all the remaining parameters are the same used for the previous simulation, so
Figure 2.12: Bifurcation map of the system considered in section 2.6 obtained via numerical simulation. In the region indicated by the black double arrow, three bifurcation curves are crossed.
that the symmetrized model obtained with Krylov-Boguliobov method is un-changed.
Results of the simulation are in figure 2.14. As expected the asymmetry causes a ripple around the equilibrium state, nonetheless the averaged system still gives a good approximation of m⊥as long as the asymmetry is small.
2.6. HYSTERESIS IN FERROMAGNETIC RESONANCE 67
Figure 2.13: Simulation proposed in section 2.6 for a symmetrical system. Hysteresis are observed in the region between the homoclinic bifurcation curve and the saddle node bifurcation curve. The values of the equilibria - analytically obtained - are illustrated in blue
Figure 2.14: Simulation proposed in section 2.6 for a asymmetrical system. The equilibrium for the equivalent symmetrized system - ob-tained with the Krylov Boguliobov method - are illustrated in blue.
Since the asymmetry is small, the magnetization amplitude m⊥is sim-ilar to the one of the symmetric system except for a ripple.
Chapter 3
Noise in Nanosystems
In small systems the thermal fluctuations randomly influence the magnetiza-tion. Thermal fluctuations may allow the system to pass through an high en-ergy barrier after a long enough time as depicted in the figure 3.1.
An example of the probability density function evolution is shown in figure 3.2. Starting from a potential well, the distribution quickly changes in the well.
After a longer time the probability starts to grow in the other well and the equi-librium is reached.
The dimensions of magnetized devices used in magnetic storage technologies and spintronics are usually rather small and thermal effects must be included in the analysis.
These effects are usually studied by introducing an appropriate stochastic term in the Landau Lifshitz Gilbert (LLG) equation (1.82). The stochastic term usually added has the form of a random magnetic torque m × νhN(t), where hN(t) is a vector whose components are independent gaussian white noise processes, and ν is a parameter which measures the intensity of thermal per-turbations
∂m
∂t = −m × (heff+ νhN) − αm × [m × (heff+ νhN)] . (3.1) The assumption that the thermal noise is gaussian is usually motivated by the central limit theorem: the random fluctuations are the result of a very large number of statistically independent random events, hence the sum of their ef-fects tends to have a gaussian distribution. Moreover the choice of the gaussian distribution leads to results which are consistent with statistical mechanics.
On the other hand, the assumption that the noise has negligible correlation time reflects the hypothesis that the random perturbations are expected to have
69
Figure 3.1: Qualitative representation a small bistable systems af-fected by thermal fluctuations
Figure 3.2: Evolution of the probability density function in a peri-odic potential. After a long enough time, the probability of finding the system state in potential well is the same regardless of the initial probability density function.
3.1. PROBABILITY DENSITY FUNCTION AT EQUILIBRIUM 71
a correlation time much shorter than any time constant of magnetization dy-namics.
Since equation (3.1) is stochastic, an analysis of the magnetization dynamics requires a large number of realizations of the process in order to have reliable statistics. Alternatively the time evolution of the probability density function of the magnetization could be directly determined. These two approaches are those conceptually proposed by Langevin in 1908 [51] and by Einstein in 1905 [50] respectively.
The study of the probability density function is generally preferable when the noise is rather big compared to the drift. The equation governing the probabil-ity densprobabil-ity function evolution is called Fokker-Planck equation
∂w
∂t = −∇Σ·
(m × ∇Σg − α∇Σg) w −ν2 2 ∇Σw
, (3.2)
where w is the probability density function defined on the unit sphere, g is the Gibbs-Landau free energy and “∇Σ” and “∇Σ·” are respectively the gradient and the divergence on the surface of the unit sphere.
In the following we consider the influence of the thermal fluctuations in two different scenarios: the switching times statistical distribution for magnetic memories [49] and the persistence of data in a magnetic grain.
3.1 Probability density function at equilibrium
In the second chapter we analyzed the equilibrium of a uniformly magnetized body. When there is noise the system can not have static equilibrium anymore since hN makes the system non-autonomous and non-periodic. The magneti-zation is expected to move, but at the same time it is expected to spend longer time in the regions with low free energy.
Let us consider the Fokker-Planck equation (3.2), at the equilibrium the prob-ability density function must satisfy
∇Σ·
(−m × ∇Σg + α∇Σg) w +ν2 2 ∇Σw
= 0 . (3.3)
In order to solve (3.3) we impose that the term in square bracket is null, this procedure is also called detailed balance [43]. The solution can be found by applying separation of variables if we assume that the probability distribution depends only on the free energy, i.e.
w = w(g) . (3.4)
This assumption implies that the precessional term gives no contribution. In-deed we have
∇Σ· [(m × ∇Σg) w] = ∇Σ· (m × ∇Σg) w + (m × ∇Σg) · ∇Σw = 0 . (3.5) The first term on the right hand side of (3.5) is zero because flows of Hamilto-nian systems are divergenceless and the second term is null because ∇Σg and
∇Σw are parallel according to (3.4). Eventually we get w = 1
Z exp
−2α ν2g
, (3.6)
where Z is a renormalization constant needed to ensure that the total proba-bility is unitary. This result is also expected by statistical mechanics, in fact Boltzmann distribution reads
w = 1 Z exp
−µ0MS2V kBT g
. (3.7)
This last relation allows to find the value of ν, which is ν2
2α = kBT
µ0MS2V =⇒ ν =
s2αkBT
µ0MS2V . (3.8) This relation is often referred to as the fluctuation-dissipation relation.