Theoretical study of magnetic domain walls through a cobalt nanocontact

To calculate the magnetic ground state of nanoparticles we present a self-consistent first-principles method in terms of a fully relativistic embedded cluster multiple scattering Green's function technique. Based on the derivatives of the band energy, a Newton-Raphson algorithm is used to find the ground-state configuration. The method is applied to a cobalt nanocontact that turned out to show a cycloidal domain wall configuration between oppositely magnetized leads. We found that a wall of cycloidal spin structure is about 30 meV lower in energy than the one of helical spin structure. A detailed analysis revealed that the uniaxial on-site anisotropy of the central atom is mainly responsible to this energy difference. This high uniaxial anisotropy energy is accompanied by a huge enhancement and anisotropy of the orbital magnetic moment of the central atom. By varying the magnetic orientation at the central atom, we identified the term related to exchange couplings (Weiss-field term), various on-site anisotropy terms, and also those due to higher order spin interactions.

Figure 1

(a) The geometry of the contact viewed from the $\left(1\overline{1}0\right)$ direction. The leads are depicted as dark (blue) rectangles, the cobalt atoms forming the contact are represented by gray (orange) circles, and $a$ denotes the nearest neighbor distance in the fcc structure. The length of the contact is tuned via $x=$ 0.85, 0.90, 0.95, 1.00, 1.05, 1.10, and 1.15. Note that only the marked distances were scaled. (b) Sketch of the embedded cluster. Dark (blue) circles: selected atoms of the cobalt leads; gray (orange) circles: cobalt atoms in the nanocontact; empty circles: empty spheres around the contact. The directions of magnetization in the leads are marked by dark (blue) arrows.

Figure 2

Sketch of the local frame of reference. The unit vector ${\mathit{\sigma }}_i$ is parallel to the magnetization at site $i$, while the unit vectors ${\mathbf{e}}_{i1}$ and ${\mathbf{e}}_{i2}$ point into the transverse directions. Rotations around these axes by ${\phi }_{i1}$ and ${\phi }_{i2}$ are also indicated.

Figure 3

The cycloidal spin configuration obtained for the unstretched contact ($x=1$). The lengths of the arrows, indicated also with color coding, are proportional to the size of the spin magnetic moments.

Figure 4

Polar angles averaged within a layer of cobalt atoms in the contact with $x=1$ as the function of the distance from the central atom (in units of the fcc nearest neighbor distance, $a$). The solid curve displays the fit, $\vartheta \left(z\right)=-\frac{\pi }{2}\tanh (2z/d_{\mathrm{w}})$.

Figure 5

Width of the domain walls through the point contact as a function of the stretching factor, $x$. Note that $d_{\mathrm{w}}\left(1.00\right)=2.34a$, where $a$ is the fcc nearest neighbor distance. The solid line stands for the identity function.

Figure 6

Calculated atomic magnetic moments (${\mu }_{\mathrm{B}}$) in half of the nanocontact for the stretching factor, $x=1$. In the upper and lower panels shown are the spin and orbital moments, ${\mu }_{\mathrm{spin}}$ and ${\mu }_{\mathrm{orb}}$, respectively. For comparison, the spin moments at the Co surface and in the bulk are $1.82{\mu }_{\mathrm{B}}$ and $1.67{\mu }_{\mathrm{B}}/\mathrm{atom}$, while the corresponding values of the orbital moments are $0.14{\mu }_{\mathrm{B}}$ and $0.08{\mu }_{\mathrm{B}}$.

Figure 7

The spin and orbital moments of the central atom as a function of the stretching. Spin moments are displayed by open symbols; orbital moments are displayed by filled symbols as calculated in the cycloidal wall (CW, squares) and in the helical wall (HW, triangles) configurations.

Figure 8

The band energy of the nanocontact with $x=1.00$ while rotating the exchange field at each atomic site simultaneously around the (110) axis. By rotating all the spins by ${90}^{\circ }$ the system goes over from the cycloidal wall (CW) into the helical wall (HW). The dashed line denotes the leading Fourier component of the band energy, $-15.2\left[\mathrm{meV}\right]\cos \left(2\theta \right)$; see Eq. (8). Note that we shifted the zero level of the energy to the constant term, $K_0$.

Figure 9

Diamonds: Calculated energy differences between the helical and cycloidal domain walls, $E_{\mathrm{HW}}-E_{\mathrm{CW}}$; circles: on-site uniaxial magnetic anisotropy energy of the central atom (see text) as a function of the stretching parameter, $x$. Thin lines serve as a guide for the eye.