1. Department of Physics ,,Fudan Unviversity, Shanghai, China
Locking states in one and two dimensional spin-cluster chain induced by spin-pumping and static coupling
Jie Lin and Jiang Xiao
Department of Physics ,,Fudan Unviversity, Shanghai, China
We consider an 1-dimensional chain or a 2-dimensional lattice of magnetic clusters that are embedded in a non-magnetic metal bulk. In this system, the neighboring clusters are coupled via static exchange coupling and dynamic spin-pumping/spin-transfer torque coupling, the latter is resulting from the emission/absorption of spin current due to the dynamics motion of the magnetic moment in each cluster. We find that although the individual cluster may have different intrinsic resonance frequency, different clusters can reach a collective resonance states when the dynamics coupling is strong
Firstly, we fix static coupling to see the locked modes under different spin-pumping parameters. The upper three lines are the imaginary parts and the lower three are the real parts of eigen-frequencies. Those systems with greater spin-pumping effects always show greater locked modes which is consistent with two-body cases .The imaginary part of each eigen-frequency is mainly determined by spin-pumping effects.
Each cluster’s dynamic equation:
γi is the gyromagnetic ratio, Heff is the effctive fildld in each cluster,Cij is the coefficients of static coupling coming from a Heisenberg-type contribution, βij is the coefficient of spin-pumping( dynamic coupling).
Finite spin-pumping effects can lokce the two modes when the difference of intrinsic frequencies is not too great. While finite staic coupling tends to break the locked mode.
Secondly, we fix spin-pumping parameters to see the locked modes under different static coupling effects. Static coupling breaks the locked modes obviously. The static coupling does not influence the imaginary parts which is consistent with its instantaneous property
References:
1. Nonlocal magnetization dynamics in ferromagnetic heterostructures, Yaroslav Tserkovnyak, Arne Brataas , Gerrit E. W. Bauer and Bertrand I. Halperin, Reviews of Modern Physics, 77, 2005