3. Micromagnetism

For this chapter we follow [Abert, 2019]. The micromagnetic model is used at the micron scale, and the main assumption is that the magnetization \(\mathbf{M} (\mathbf{x})\) is a continuous field \(\Mbf: \Omega \to \Omega\) with constant magnitude and can be written as

\[ \mathbf{M} (\xbf) = \Ms \ \mbf (\xbf), \]

where \(\Omega \subset \Rbb^3\) is the magnetic domain, the field \(\mbf\) is the unit magnetization (also called slightly incorrectly magnetization) and satisfied \(\| \mbf (\xbf) \| = 1\) on \(\bar\Omega\), and \(\Ms \in \Rbb\) is the spontaneous magnetization. This varies with temperature and at temperature \(0 K\) it is called saturation magnetization.

Let us define \(\Mcal\) the set of all possible unit magnetization profiles, i.e.

\[ \Mcal := \{ \mbf = (m_1, m_2, m_3) \ : \ m_i \in H^1(\Omega) \ \text{for} \ i=1,2,3 \}. \]

In this notation, we can write the unit condition as

\[ m_1^2 (\xbf) + m_2^2 (\xbf) + m_3^2 (\xbf) = 1 \qquad \forall \xbf \in \bar\Omega = \Omega \cup \partial\Omega. \]

Let us furthermore observe that this set together with the norm

\[ \| \mbf \|_\Mcal = \left[ \| m_1 \|_{H^1(\Omega)}^2 + \| m_2 \|_{H^1(\Omega)}^2 + \| m_3 \|_{H^1(\Omega)}^2 \right]^{1/2} \]

is a Banach space.