Chiral Instability of the Homogeneous State of a Ferromagnetic Film on a Magnetic Substrate

Abstract

The conditions of formation of chiral magnetization distributions in the systems ferromagnet/superconductor and ferromagnet/paramagnet are theoretically determined. The formation of chiral states is caused by the magnetostatic interaction in inhomogeneous magnetic systems. The estimates performed demonstrate that the predicted effects can be experimentally observed.

APPENDIX

After performing Fourier transform, we write the addition to energy (4) that is related to magnetization vector direction fluctuations as

$$\begin{gathered} \Delta E = \frac{V}{2}\sum\limits_{\mathbf{q}}^{} {\{ (\alpha {{q}^{2}} + H){\text{|}}{{M}_{y}}({\mathbf{q}}){{{\text{|}}}^{2}}} \\ + \,(\alpha {{q}^{2}} + H - K){\text{|}}{{M}_{z}}({\mathbf{q}}){{{\text{|}}}^{2}}\} \\ + \frac{V}{2}\sum\limits_{{\mathbf{q}},i,k = y,z}^{} {{{D}_{{ik}}}({\mathbf{q}}){{M}_{i}}({\mathbf{q}}){{M}_{k}}( - {\mathbf{q}}).} \\ \end{gathered} $$

(A.1)

The Fourier transforms of the magnetostatic tensor components are calculated as follows:

$$\begin{gathered} {{D}_{{yy}}}({\mathbf{q}}) = \frac{{q_{y}^{2}}}{h}\int {{{e}^{{ - i{\mathbf{q}} \cdot {\boldsymbol{\rho }}}}}\frac{1}{{\sqrt {{{\rho }^{2}} + {{{(z - z{\kern 1pt} ')}}^{2}}} }}d{\boldsymbol{\rho }}dzdz{\kern 1pt} '} , \\ {{D}_{{zz}}}({\mathbf{q}}) = \frac{2}{h}\int {{{e}^{{ - i{\mathbf{q}} \cdot {\boldsymbol{\rho }}}}}\left\{ {\frac{1}{\rho } - \frac{1}{{\sqrt {{{\rho }^{2}} + {{h}^{2}}} }}} \right\}d{\boldsymbol{\rho }}} . \\ \end{gathered} $$

(A.2)

The integral in Eq. (A.2) is taken over the ferromagnet volume. Note that D_yz(q) = 0 for the off-diagonal component. Using the Bessel function of the first kind

$$\int\limits_0^{2\pi } {{{e}^{{ - iq\rho \cos \phi }}}d\phi = 2\pi {{J}_{0}}(q\rho ),} $$

we can write Eqs. (A.2) as follows

$$\begin{gathered} {{D}_{{yy}}}({\mathbf{q}}) = 2\pi \frac{{q_{y}^{2}}}{h}\int\limits_0^{} {\int\limits_{}^h {\int\limits_0^\infty {\frac{{{{J}_{0}}(q\rho )}}{{\sqrt {{{\rho }^{2}} + {{{(z - z{\kern 1pt} ')}}^{2}}} }}\rho d\rho dzdz{\kern 1pt} '} ,} } \\ {{D}_{{zz}}}({\mathbf{q}}) = \frac{{4\pi }}{h}\int\limits_0^\infty {{{J}_{0}}(q\rho )\left\{ {1 - \frac{\rho }{{\sqrt {{{\rho }^{2}} + {{h}^{2}}} }}} \right\}d\rho .} \\ \end{gathered} $$

(A.3)

After integration with respect to variable ρ in Eq. (A.3), we obtain [39]

$$\begin{gathered} {{D}_{{yy}}}({\mathbf{q}}) = 2\pi \frac{{q_{y}^{2}}}{{qh}}\int\limits_{}^h {\int\limits_0^{} {{{e}^{{ - q|z - z'|}}}dzdz{\kern 1pt} ',} } \\ {{D}_{{zz}}}({\mathbf{q}}) = 4\pi \frac{{1 - {{e}^{{ - qh}}}}}{{qh}}. \\ \end{gathered} $$

(A.4)

The integral in the first equation of (A.4) is taken according to the rule

$$\begin{gathered} \int\limits_{}^h {\int\limits_0^{} {{{e}^{{ - q|z - z'|}}}dzdz{\kern 1pt} '} } \\ = \int\limits_0^h {\left\{ {\int\limits_0^z {{{e}^{{ - q(z - z')}}}dz{\kern 1pt} '\, + \int\limits_z^h {{{e}^{{q(z - z')}}}dz{\kern 1pt} '} } } \right\}} dz \\ = \frac{{2h}}{q}\left( {1 - \frac{{1 - {{e}^{{ - qh}}}}}{{qh}}} \right). \\ \end{gathered} $$