in Thiele's derivation for magnetic bubble/skyrmion motion, I see that he defined the force density as
f=-gradf(H.M), where gradf is the gradient with Feynmann subscript notation (see attached file)
However I do not really see the connection here. Can someone help me to derive the force term? or maybe can someone explain to me why only the M term is being derivated here?
Article Steady-State Motion of Magnetic Domains
The force arises as a term obtained by taking the derivative of energy density with respect to magnetization. I think there should be other energy terms like exchange, anisotropy and DMI other than Zeeman term inside the bracket of the expression you have attached. You can see it more clearly if you go through the derivation of Thiele's equation starting from LLG equation.
Thanks Rabindra for your answer, kindly please direct us to the publication that has explained everything in detail
The specified task occurred baseline in the analysis of static stability of cylindrical magnetic domain (magnetic bubble domain) in magnetically uniaxial plate with thickness h and determine its equilibrium radius r depending on the external magnetic field H. The equilibrium equation of circular magnetic bubble is derived from the condition of vanishing of the variation of the total energy E=Ew+EH+EM, where there are summarized the domain wall energy, Zeeman energy of the interaction domain with the external field and the magnetostatic energy, respectively. The greatest difficulty is the computation of the last member. The derivative with respect to the radius of the domain ¶EM/¶r has the dimension of force and it is usually called a generalized magnetostatic force, or force function F(x). A good approximation is F(x) ~ x(1+3x/4)-1, where x=2r/h.
[See: Thiele A. A. // Bell System Techn.J. (1969) v.48, 10, 328;
Eshenfelder A. Physics and Technology of Magnetic Bubble Domains (1983);
Malozemoff A. P., J. C. Slonczewski, Magnetic Domain Walls in Bubble Materials, Academic, New York, 1979].
The part of the force, that you mentioned, perhaps, refers to the interaction of the domain with a uniform external field, which may cause movement of the domain due to the inhomogeneous distribution of the magnetization in the domain wall (especially with complex structure, as in hard magnetic bubbles). A similar approach, apparently, is applicable to magnetic vortices, skyrmions, which are characterized by the twisted magnetization structure.
[See also the following publications that may help you: Khodenkov Н.Е. Nonstationary equations of motion for magnetic bubble domains // Phys.St.Sol.(a), 1981, v.63, p.461-472.
Khodenkov H.Ye. Linear equation of motion of Bloch domain wall in fer-romagnet//Phys. Lett. A, 1976, v. 58, No 2, p.135-136. link.springer.com›article/10.1134/1.1310726
O’Dell T.H. Ferromagnetodynamics: The Dynamics of Magnetic Bubbles, Domains, and Domain Walls (MacMillan, New York, 1981].
Best regards!
Many thanks for your answer Alexander! :)
As you have mentioned, the writing of the force appears to regard the field as constant, which is strange because in the Thiele equation for skyrmion, none of the 'field's (gyroscopic, dissipative, etc) are actually constant.
At present I cant really complain much as the final result for the skyrmion speed equation matches with the simulation as well as the experimental result. But I always welcome any discussion regarding this. :D
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