Microscopic origin of magnetic anisotropy in martensitic Ni${}_2$MnGa

The microscopic origin of magnetic anisotropy in the shape memory alloy Ni${}_2$MnGa is investigated by means of x-ray magnetic circular dichroism in transmission mode. Field- and angle-dependent dichroism spectra of epitaxial Ni${}_2$MnGa(101)/MgO(001) films reveal pronounced differences for magnetization aligned parallel and perpendicular to the film plane. These differences are related to an anisotropy of the orbital magnetic moment in agreement with the observed out-of-plane magnetocrystalline anisotropy. The spectral variation of the x-ray absorption originates from changes in the spin-projected density of states when the magnetization vector is rotated from the easy to the hard magnetic axis. Minority Ni states with $d_{3z^2-r^2}$ symmetry close to the Fermi energy form a wide half filled band for easy axis magnetization. When the magnetization is rotated into the hard axis the band narrowing of these states causes an increase of the mean kinetic energy of the electronic system. The opposite behavior of mostly unoccupied Ni states with $d_{\mathit{xy}}$ symmetry leads to an increase of the minority orbital moment for hard-axis magnetization.

Figure 1

(a) Two circle x-ray diffraction in Bragg Brentano geometry of a Ni${}_2$MnGa(101)/MgO(001) film. (b) Structural model of the martensitic phase neglecting modulation viewed along the Ni${}_2$MnGa [010] axis. The position of the (101) twin planes is indicated by the horizontal lines. (c) Room-temperature $\varphi$ scans of the {022} and {004} Ni${}_2$MnGa film and MgO {111} reflections revealing four variants of the cubic phase indicated by different symbols. (d) Top view of the relative orientation of one of the four variants with respect to the substrate.

Figure 2

X-ray absorption $\mu (H)$ versus external field $H$ applied perpendicular to the film surface (dots). The zero-field value $\mu (0)$ was subtracted and the difference was normalized to the saturation value $\Delta {\mu }_{\mathit{sat}}$ measured at the maximum field. The cross section of the initial susceptibility (black line) with the saturation value indicates the saturation field $H_s$. The initial susceptibility if only the shape anisotropy field $M_s=J_s/{\mu }_0$ were present is drawn for comparison [light-blue (gray) line].

Figure 3

XAS (a,b) and corresponding XMCD (c,d) absorption coefficients calculated from the transmitted X-ray intensity measured at $T=200$ K (a,c) and $T=290$ K (b,d). The variation of the incidence angle between x-ray beam/magnetization and the surface normal is indicated in the figure. Peak A marks the satellite peak emerging in the austenitic state[22] and peaks B and C indicate angular dependent spectral changes caused by Ni $d$ states with $d_{\mathit{xy}}$ symmetry. Spectra and arrows are shifted with an identical offset for clarity.

Figure 4

Angular dependence of the effective spin moment (a) and of the orbital to spin moment ratio (b) resulting from the sum-rule analysis at the Ni $L_{2,3}$ edge. The effective spin moment includes the magnetic dipole term. We assumed a number of $d$ holes of $N_h=1.5$ for Ni. Horizontal lines in (a) indicate the corresponding mean value of the effective spin moment. Full lines in (b) result from a fit to the function ${\mu }_{\mathrm{orb}}={\mu }_{\mathrm{orb}}(0)-\Delta {\mu }_{\mathrm{orb}}\sin {}^2\theta$.

Figure 5

(a) Site and spin-projected $d$-electron state densities for Ni (in arbitrary units) resulting from ab initio calculations (Ref. 29) shifted along the energy axis for direct comparison to the experimental data. Positive (negative) values indicate the majority (minority) density of states. (b) Difference of the XMCD signal measured at $\theta ={60}^{°}$ and normal incidence for $T<T_m$ [dark (blue) squares]. Full (red) line shows for comparison the XMCD signal measured at normal incidence.

Figure 6

Sketch of the orbital symmetry of electron states with $d_{3z^2-r^2}$ ($d_{z^2}$) and $d_{\mathit{xy}}$ symmetry indicating the dominating contribution to the DOS maxima for out-of-plane and in-plane magnetization direction, respectively. The two lower panels illustrate the opposite behavior of states with $d_{z^2}$ and $d_{\mathit{xy}}$ symmetry for perpendicular (left) and in-plane (right) magnetization. Dark shaded areas indicate occupied states.