Maxwell Equations and Landau–Lifshitz Equations

Abstract

This chapter will present the theoretical, mathematical and computational fundamentals for micromagnetics. The target of micromagnetics is to clarify the motion of magnetic moments in ferromagnetic materials and devices, which is described by the nonlinear Landau–Lifshitz equations, or the equivalent Landau–Lifshitz–Gilbert (LLG) equations. In the LLG equations, the time derivative of the magnet moment in a micromagnetic cell is controlled by the local effective magnetic field. The effective magnetic field contains the terms determined by the fundamental and applied magnetism in a magnetic material, including the external field, the crystalline anisotropy field, the exchange field, the demagnetizing field, and the magneto-elastic field. Among these field terms, the most time-consuming one in computation is the demagnetizing field, which will be calculated by the Green’s function method following the Maxwell’s equations.

Appendix

Appendix A

If a $a^{\prime }\times b^{\prime }$ rectangle locates at $z^{\prime }=0$, and the observation position vector is located at ${\mathbf{ r} }=(x,y,z)=(r_1,r_2,r_3)$, as seen in Fig. 2.4a, the three nonzero elements $N^{\mathrm{tri}}_{13},\;N^{\mathrm{tri}}_{23}\; {\text {and}}\; N^{\mathrm{tri}}_{33}$ in the demagnetizing matrix of a rectangular surface are ($\alpha =1,3$):

$$\begin{aligned} N^\mathrm{ rec} _{\alpha 3} ({\mathbf{ r} })= -{\frac{1}{{4\pi }}} \int \limits ^{a^{\prime }/2}_{-a^{\prime }/2} \mathrm{ d} r^{\prime }_1 \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} r^{\prime }_2 {\frac{{r_\alpha -r^{\prime }_\alpha }}{{\ \left[(r_1-r^{\prime }_1)^2+(r_2-r^{\prime }_2)^2+r^2_3\right]^{3/2}\ }}} \end{aligned}$$

(2.79)

We just have to do two integrations for $N^\mathrm{ rec} _{33}$ and $N^\mathrm{ rec} _{13}$, and the integration of $N^\mathrm{ rec} _{23}$ is totally analogy to $N^\mathrm{ rec} _{13}$. Let’s start with $N^\mathrm{ rec} _{13}$:

$$\begin{aligned} N^\mathrm{ rec} _{13}&= -{\frac{1}{{4\pi }}} \int \limits ^{a^{\prime }/2}_{-a^{\prime }/2} \mathrm{ d} x^{\prime } \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } {\frac{{x-x^{\prime }}}{{\ \left[(x-x^{\prime })^2+(y-y^{\prime })^2+z^2\right]^{3/2}\ }}} \nonumber \\&= -{\frac{1}{{4\pi }}} \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } \left.{\frac{1}{{\ \left[(x-x^{\prime })^2+(y-y^{\prime })^2+z^2\right]^{1/2}\ }}} \right|^{a^{\prime }/2}_{x^{\prime }=-a^{\prime }/2} \nonumber \\&= -{\frac{1}{{4\pi }}}\sum _q (-q) \ln \left(y^{\prime }-y+ \sqrt{R^2_1+(y-y^{\prime })^2+R^2_3}\right)^{b^{\prime }/2}_{y^{\prime }=-b^{\prime }/2} \nonumber \\&= -{\frac{1}{{4\pi }}}\sum _q \sum _w qw \ln \left(R-wR_2\right) \end{aligned}$$

(2.80)

The variables are defined in Table 2.2. The integration for $N^\mathrm{ rec} _{33}$ can also be done:

$$\begin{aligned} N^\mathrm{ rec} _{33}&= -{\frac{1}{{4\pi }}} \int \limits ^{a^{\prime }/2}_{-a^{\prime }/2} \mathrm{ d} x^{\prime } \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } {\frac{{z}}{{\big [(x-x^{\prime })^2+(y-y^{\prime })^2+z^2\big ]^{3/2}\ }}} \nonumber \\&= -{\frac{1}{{4\pi }}} \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } {\frac{{z}}{{\big [(y-y^{\prime })^2+z^2\big ]\ }}} \left.{\frac{{x^{\prime }-x}}{{\big [(x-x^{\prime })^2+(y-y^{\prime })^2+z^2\big ]^{1/2}\ }}} \right|^{a^{\prime }/2}_{x^{\prime }=-a^{\prime }/2} \nonumber \\&= -{\frac{1}{{4\pi }}}\sum _q \left. \arctan \left( \frac{R_1}{R_3} \frac{y^{\prime }-y}{\sqrt{R^2_1+(y-y^{\prime })^2+R^2_3}} \right) \right|^{b^{\prime }/2}_{y^{\prime }=-b^{\prime }/2} \nonumber \\&= -{\frac{1}{{4\pi }}}\sum _q \sum _w \arctan \,\frac{R_1R_2}{R_3R} \end{aligned}$$

(2.81)

The $N^\mathrm{ rec} _{\alpha 3} ({\mathbf{ r} })$ of a rectangular surface have be listed in Table 2.2 respectively.

Appendix A

If the surface located at $z^{\prime }=0$ is a right-angle triangle with right-angle side lengths $(a^{\prime },b^{\prime })$, the origin at the midpoint of the hypotenuse, and the hypotenuse defined by equation $y^{\prime }=-sx^{\prime }$ or $x^{\prime }=-vy^{\prime }$, and the observation position vector is located at ${\mathbf{ r} }=(x,y,z)$, there are still three nonzero elements in the demagnetizing matrix of a triangular surface, as seen in Fig. 2.4b. Here we can first do the integral for element $N^\mathrm{ tri} _{13}$:

$$\begin{aligned} N^\mathrm{ tri} _{13}=&\; -{\frac{1}{{4\pi }}} \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } \int \limits ^{a^{\prime }/2}_{-vy^{\prime }} \mathrm{ d} x^{\prime } {\frac{{x-x^{\prime }}}{{\ \left[(x-x^{\prime })^2+(y-y^{\prime })^2+z^2\right]^{3/2}\ }}} \nonumber \\=&\; -{\frac{1}{{4\pi }}} \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } \left\{ \frac{1}{\ \left[(a^{\prime }/2-x)^2+(y-y^{\prime })^2+z^2\right]^{1/2}\ }\right. \nonumber \\&\;- \left.{ \frac{1}{\ \left[(x+vy^{\prime })^2+(y-y^{\prime })^2+z^2\right]^{1/2}\ }} \right\} \nonumber \\=&\; \left.-{\frac{1}{{4\pi }}} \ln \left(y^{\prime }-y+ \sqrt{(a^{\prime }/2-x)^2+(y-y^{\prime })^2+z^2}\,\right) \right|^{b^{\prime }/2}_{y^{\prime }=-b^{\prime }/2} \nonumber \\&\;+{\frac{1}{{4\pi }}}\int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } \frac{1}{\sqrt{1+v^2}} \frac{1}{\ \left[(y^{\prime }-c_1)^2+r^2/(1+v^2)-c^2_1\right]^{1/2}\ } \nonumber \\=&\; {\frac{1}{{4\pi }}} \sum _w w \ln \left(R_\mathrm{ I} -wR_2\right) \nonumber \\&\;\left.+{\frac{1}{{4\pi }}}{\frac{1}{\sqrt{1+v^2}}} \ln \left(y^{\prime }-c_1+ \sqrt{(y^{\prime }-c_1)^2+c^2_2}\,\right) \right|^{b^{\prime }/2}_{y^{\prime }=-b^{\prime }/2} \nonumber \\=&\; -{\frac{1}{{4\pi }}} \sum _w w \left\{ {\frac{1}{\sqrt{1+v^2}}}\ln (P-wP_2)- \ln (R_\mathrm{ I} -wR_2)\right\} \end{aligned}$$

(2.82)

The symbols $c_1$, $c^2_2$, $P$, $P_2$, $R_\mathrm{ I}$, and $R_2$ used in Eq. (2.82) have be defined in Tables 2.2 and 2.3 respectively. The integration for $N^\mathrm{ tri} _{23}$ is totally analogy to $N^\mathrm{ tri} _{13}$, just with a $x\leftrightarrow y$ symmetry, therefore the derivation of $N^\mathrm{ tri} _{23}$ will be omitted here.

The integral for the matrix element $N^\mathrm{ tri} _{33}$ is the most complicated one, which include three parts:

$$\begin{aligned} N^\mathrm{ tri} _{33}&= -{\frac{1}{{4\pi }}} \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } \int \limits ^{a^{\prime }/2}_{-vy^{\prime }} \mathrm{ d} x^{\prime } {\frac{{z}}{{\ \left[(x-x^{\prime })^2+(y-y^{\prime })^2+z^2\right]^{3/2}\ }}} \nonumber \\&= -{\frac{1}{{4\pi }}} \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } \left.{\frac{{z}}{{\ \left[(y-y^{\prime })^2+z^2\right]\ }}} {\frac{{x^{\prime }-x}}{{\ \left[(x-x^{\prime })^2+(y-y^{\prime })^2+z^2\right]^{1/2}\ }}} \right|^{a^{\prime }/2}_{x^{\prime }=-vy^{\prime }} \nonumber \\&= -{\frac{1}{{4\pi }}}\left\{ N^{(1)}_{33} + N^{(2)}_{33} + N^{(3)}_{33}\right\} \end{aligned}$$

(2.83)

In Eq. (2.83), the derivation of the first term $N^{(1)}_{33}$ is actually very similar to one of the two terms in Eq. (2.81) for rectangular surface; in the second term, the numerator of the integrand is $z(-vy^{\prime }-x)$, which can be disassembled into two parts $zv(-y^{\prime }+y)$ and $z(-vy-x)$, and these two part just corresponds to the $N^{(2)}_{33}$ and $N^{(3)}_{33}$ respectively.

The integration for $N^{(1)}_{33}$ is just straightforward:

$$\begin{aligned} N^{(1)}_{33}&= \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } {\frac{{z}}{{\ \left[(y-y^{\prime })^2+z^2\right]\ }}} {\frac{{a^{\prime }/2-x}}{{\ \left[(a^{\prime }/2-x)^2+(y-y^{\prime })^2+z^2\right]^{1/2}\ }}} \nonumber \\&= \left.\arctan \left( \frac{a^{\prime }/2-x}{z} \frac{y^{\prime }-y}{\sqrt{(a^{\prime }/2-x)^2+(y-y^{\prime })^2+z^2}}\right) \right|^{b^{\prime }/2}_{y^{\prime }=-b^{\prime }/2} \nonumber \\&= \sum _w \arctan [(a^{\prime }/2-x)R_2/(zR_\mathrm{ I} )] \end{aligned}$$

(2.84)

In the derivation for $N^{(2)}_{33}$, complicated variables such as $c_1$, $c_2$ and $c_5=y-\mathrm{ i} z$ in Table 2.3 have to be defined, and the respective integral is:

$$\begin{aligned} N^{(2)}_{33}&= \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } {\frac{{zv}}{{\ \left[(y-y^{\prime })^2+z^2\right]\ }}} {\frac{{y^{\prime }-y}}{{\ \left[(x+vy^{\prime })^2+(y-y^{\prime })^2+z^2\right]^{1/2}\ }}} \nonumber \\&= \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime }{\frac{{zv}}{2}} \left[{\frac{1}{{\ y^{\prime }-c_5\ }}}+{\frac{1}{{\ y^{\prime }-c^*_5\ }}}\right] {\frac{1}{\sqrt{1+v^2}}}{\frac{1 }{{\ \left[(y^{\prime }-c_1)^2+c^2_2\right]^{1/2}\ }}}\ \ \ \ \end{aligned}$$

(2.85)

The previous integration include two terms which are complex conjugates of one another. Now let’s make an integration variable change $y^{\prime }-c_1=c_2\sinh \theta$ with the two integration limits of the angle $\theta$ as $\theta _1=\sinh ^{-1}[(-b^{\prime }/2-c_1)/c_2]$ and $\theta _2=\sinh ^{-1}[(b^{\prime }/2-c_1)/c_2]$, the integral in Eq. (2.85) has the form:

$$\begin{aligned} N^{(2)}_{33}\;&= \;{\frac{{zv}}{{2\sqrt{1+v^2}}}} \int \limits ^{\theta _2}_{\theta _1} \left[{\frac{{\mathrm{ d}\theta }}{{c_1-c_5+c_2\sinh \theta }}} + c.c.\right]\ \ \ \end{aligned}$$

(2.86)

Then make another variable change $e^\theta =u$, and define a new constant $\sinh \eta =(c_1-c_5)/c_2$, the integral becomes:

$$\begin{aligned} N^{(2)}_{33}\;&= \;{\frac{zv}{\sqrt{1+v^2}}} \int \limits ^{u_2}_{u_1} \left[{\frac{1}{{c_2}}}{\frac{{\mathrm{ d}\left(e^\theta \right)}}{{\left(e^\theta \right)^2+2[(c_1-c_5)/c_2]e^\theta -1}}} + c.c.\right]\ \ \ \nonumber \\&= \;{\frac{zv}{\sqrt{1+v^2}}} \int \limits ^{u_2}_{u_1} \left[\frac{1}{c_2}{\frac{\mathrm{ d}u}{(u+e^\eta )(u-e^{-\eta })}} + c.c.\right]\ \ \ \nonumber \\&= \!\! {\frac{zv}{\sqrt{1+v^2}}} \left\{ \left. {\frac{1}{{2c_2\cosh \eta }}} \ln {\frac{u-e^{-\eta }}{u+e^\eta }} \right|^{u_2}_{u\,=\,u_1} + c.c. \right\} \ \ \ \nonumber \\&= \; {\frac{zv}{\sqrt{1+v^2}}} \sum _w \Re \left\{ \frac{w}{\sqrt{(c_1-c_5)^2+c^2_2}} \ln \frac{V_+}{V_-} \right\} \nonumber \\&= \; {\frac{zv}{\sqrt{1+v^2}}} \sum _w \Re \left\{ \frac{w}{A e^{\mathrm{ i} \theta ^{\prime }/2}} \ln \frac{|V_+|e^{\mathrm{ i} \phi _+}}{|V_-|e^{\mathrm{ i} \phi _-}} \right\} \end{aligned}$$

(2.87)

In the previous derivation, the difficult part is to find integration limits $u_1$ and $u_2$. Actually $\sinh ^{-1}x=\ln [x+\sqrt{x^2+1}]$, therefore $u_1$, $u_2$, $e^\eta$ and $e^{-\eta }$ are:

$$\begin{aligned} u_1=e^{\theta _1}=\frac{1}{c_2}\left[-\left(\frac{b^{\prime }}{2}+c_1\right)+\sqrt{\left(\frac{b^{\prime }}{2}+c_1\right)^2+c^2_2}\right] \nonumber \\ u_2=e^{\theta _2}=\frac{1}{c_2}\left[+\left(\frac{b^{\prime }}{2}-c_1\right)+\sqrt{\left(\frac{b^{\prime }}{2}-c_1\right)^2+c^2_2}\right] \nonumber \\ e^\eta =\frac{1}{c_2}\left[+(c_1-c_5)+\sqrt{(c_1-c_5)^2+c^2_2}\right] \nonumber \\ e^{-\eta }=\frac{1}{c_2}\left[-(c_1-c_5)+\sqrt{(c_1-c_5)^2+c^2_2}\right] \end{aligned}$$

(2.88)

By defining $P_2={\frac{b^{\prime }}{2}}+w\ \!\! c_1$ ($w=+1$ and $w=-1$ are for the two integral limits $u_1$ and $u_2$ respectively), $A e^{\mathrm{ i} \theta ^{\prime }/2}$, $V_+$ and $V_-$ will have the forms in Table 2.3:

$$\begin{aligned} \frac{u-e^{-\eta }}{u+e^\eta }= \frac{-wP_2+\sqrt{P^2_2+c^2_2}+(c_1-c_5)-\sqrt{(c_1-c_5)^2+c^2_2}}{-wP_2+\sqrt{P^2_2+c^2_2}+(c_1-c_5)+\sqrt{(c_1-c_5)^2+c^2_2}} =\frac{V_-}{V_+} \end{aligned}$$

(2.89)

The integration of $N^{(3)}_{33}$ is similar to $N^{(2)}_{33}$, which also includes imaginary numbers:

$$\begin{aligned} N^{(3)}_{33}&= \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } {\frac{{z}}{{\ \left[(y-y^{\prime })^2+z^2\right]\ }}} {\frac{{x+vy}}{{\ \left[(x+vy^{\prime })^2+(y-y^{\prime })^2+z^2\right]^{1/2}\ }}} \\ \nonumber&= \int \limits ^{b^{\prime }/2}_{-b^{\prime }/2} \mathrm{ d} y^{\prime } {\frac{{x+vy}}{{2\mathrm{ i} }}} \left[{\frac{{-1}}{{\ y^{\prime }-c_5\ }}} +{\frac{1}{{\ y^{\prime }-c^*_5\ }}}\right] {\frac{1}{\sqrt{1+v^2}}}{\frac{1 }{{\ \left[(y^{\prime }-c_1)^2+c^2_2\right]^{1/2}\ }}} \end{aligned}$$

(2.90)

The rest of derivations are similar to Eq. (2.87), but the result is an imaginary part:

$$\begin{aligned} N^{(3)}_{33}\;&= \; -{\frac{{x+vy}}{{2\mathrm{ i} \sqrt{1+v^2}}}} \int \limits ^{\theta _2}_{\theta _1} \left[{\frac{{\mathrm{ d} \theta }}{{c_1-c_5+c_2\sinh \theta }}} - c.c.\right]\ \ \ \nonumber \\&= \; -{\frac{{x+vy}}{{2\mathrm{ i} \sqrt{1+v^2}}}} \int \limits ^{u_2}_{u_1} \left[{\frac{1}{{c_2}}}{\frac{{\mathrm{ d} u}}{{(u+e^\eta )(u-e^{-\eta })}}} - c.c.\right]\ \ \ \nonumber \\&= \; -{\frac{{x+vy}}{\sqrt{1+v^2}}} \sum _w \Im \left\{ \frac{w}{\sqrt{(c_1-c_5)^2+c^2_2}} \ln \frac{V_+}{V_-} \right\} \nonumber \\&= \; -{\frac{{x+vy}}{\sqrt{1+v^2}}}\sum _w \Im \left\{ \frac{w}{A e^{\mathrm{ i} \theta ^{\prime }/2}} \ln \frac{|V_+|e^{\mathrm{ i} \phi _+}}{|V_-|e^{\mathrm{ i} \phi _-}} \right\} \end{aligned}$$

(2.91)

Finally, insert the results of the three parts in Eqs. (2.84), (2.87) and (2.91) into Eq. (2.81), the most difficult matrix element of a triangular surface $N^\mathrm{ tri} _{33}$ can be found.