Orientation of the magnetic moments in a stack of three ferromagnetic layers (0, 1, 2); the magnetic moments in layers 0, 1, and 2 (shown with short arrows) are coupled by the exchange interaction thus building an exchange spring tri-layer; H (shown with the arrow outside the stack) is an external magnetic field directed antiparallel to the magnetization in layer 0. The current J (shown with the long arrow) flows in the layer plane (that is along the x axis).
Current-voltage characteristics of the magnetic stack of Fig. 1 in which the magnetization directions in the layers are homogeneously distributed along the x direction (that is along the stack). It was calculated for R(Θ) = R + − R −cosΘ, R −/R + = 0.2, D 0 = 0.2; Jc = Vc /R(π). The branches 0–a and b–b′ of the IVC correspond to parallel and antiparallel orientations of the stack magnetization, respectively (the parts a–a′ and 0–b are unstable); the branch a–b corresponds to 0 ≤ Θ[T(V)] ≤ π.
The coordinate dependence of the temperature, T(x), and the magnetization direction, Θ(x), in the magnetic stack with a magneto-thermal-electric domain inside it, L II is the length of the “hot” part of the MTED which is defined in such a way that the length of the “cold” part is L I = L − L II. Calculations are made for R(Θ) = R + − R −cosΘ, R −/R + = 0.2, D 0 = 0.2; J/Jc = 1.0265, .
Temporal evolution of the length of MTED L II(t) for the case . The unstable limiting cycle (shown with a thick line) separate the phase plane into two regions: any initial state inside the limitingcycle develops into the length of the steady stationary MTED (shown with a dot) while an initial state outside it results in oscillations of the MTED length with an increasing in time amplitude until the MTED disappears, thatis L II reaches either L II ≈ 0 or L II ≈ L. Calculations are made for R −/R + = 0.2, D 0 = 0.2 and τ T /τ L = 0.1 where τ T and τ L are the characteristic times of the temperature and current developments, respectively; J 0 is the stabilization current.
Temporal evolution of the current J(t) and voltage drop in a MTED inside it for the case . The steady IVC of a homogeneous stack IVC with a MTED are shown with a dashed and dashed-dotted lines, respectively. The stack is in a bistable state: depending on the initial conditions the system goes either to a stable steady MTED (which is shown with a dot) or goes to a stable limiting cycle (the largest closed curve) corresponding to spontaneous oscillations of the current J(t), voltage drop , temperature T(t), and the magnetization direction Θ(t), the stack being in a homogeneous state (in which the MTED has disappeared). There is an unstable limiting cycle that separates the initial states which develop either to the MTED or the oscillations as is shown by two black arrows. Calculations are made for the same parameters, as on the Fig. 4 .
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