### The magnetic domain wall (DW) (Video)

Images and sketches of domain walls Domains in a ferromagnetic film. The magnetization points "up" in dark-coloured regions and "down" in light-coloured regions.  Sketch of a domain pointing "up" (left on the sketch) next to a domain pointing "down" (right on the sketch). The magnetic configuration between domains turns from down to up gradually. (Here, the magnetisation turns by going out of the plane of the screen.)

Magnetic domains are regions where the magnetisation is almost uniform.

A transition layer

A domain wall is the magnetisation configuration between two uniform states with different magnetisation.

A typical case is when the magnetisation rotates by $\pi$ between two domains (a 180-wall).

The magnetisation variation is best visualised on the Bloch sphere.

For example, consider a vector moving from the north to the south pole. The spherical parametrisation of the magnetisation with the angles $\Theta, \Phi$.

$$m_1 = \sin\Theta \cos\Phi,\quad m_2 = \sin\Theta \sin\Phi,\quad m_3 = \cos\Theta.$$

In a model with easy-axis anisotropy, we have two ground states, $\mathbf{m}=(0,0,\pm 1)$, or $\Theta=0, \pi$, that is, the north and south pole of the sphere.

We look for a topological soliton connecting the north and the south pole.

• We confine ourselves to one space dimension $\mathbf{m} = \mathbf{m}(x)$.
• We try the simplest possibility of a meridian on the Bloch sphere