Determination of spin chirality using x-ray magnetic circular dichroism

A threefold symmetric kagome lattice that has negative spin chirality can give a nonzero x-ray magnetic circular dichroism (XMCD) signal, despite the total spin moment amounting to zero. In order to explain this, I present here a rule for the rotational symmetry invariance of the XMCD signal. A necessary condition is the existence of an anisotropic XMCD signal for the local magnetic atom, which can arise from a spin anisotropy either in the ground state or the final state. The angular dependence of the XMCD as a function of the beam direction has an unusual behavior. The maximum dichroism is not aligned along the spin direction, but depends on the relative orientation of the spin with respect to the atomic orientation. Therefore, different geometries can result in the same angular dependence, and the spin direction can only be determined if the atomic orientation is known. The consequences for the x-ray magneto-optical sum rules are given. The integrated XMCD signals are proportional to the anisotropy in the orbital moment and the magnetic dipole term, where the isotropic spin moment drops out.

Figure 1

Structure of two-dimensional kagome lattice with threefold symmetry axes and corner-sharing triangles. Only the magnetic atoms are shown. In the spin-oriented state the red, blue, and yellow atoms have different spin directions. The dashed box indicates a unit cell of the magnetic structure.

Figure 2

Definition of the angles in the reference frame. The local atom (gray disk) is oriented along $x$, which makes an angle $\alpha$ with the $x^{\prime }$ axis of the laboratory frame. The spin direction $\widehat{\mathbf{S}}$ and the helicity vector $\widehat{\mathbf{P}}$ of the circularly polarized x rays are at an angle $\mu$ and $\gamma$, respectively, with respect to $x^{\prime }$.

Figure 3

Magnetic unit cells oriented along ${\alpha }_0=0^{\circ }$ with positive chirality ($h=1$) for (a) ${\mu }_0=0^{\circ }$ ($A_2$ phase), (b) ${\mu }_0={20}^{\circ }$, and (c) ${\mu }_0={90}^{\circ }$ ($A_1$ phase), and with negative chirality ($h=-1)$ for (d) ${\mu }_0=0^{\circ }$, (e) ${\mu }_0={20}^{\circ }$, and (f) ${\mu }_0={90}^{\circ }$.

Figure 4

The magnetic unit cell, oriented at ${\alpha }_0=0^{\circ }$, is shown in the inset at the left. The main image shows the angular dependence of the XMCD as a function of $\gamma$. The red, blue, and green curves give the XMCD for the three magnetic atoms with their spin directions shown by the color-coded arrows. The total XMCD is given by the thick black curve. The plots have been obtained from Eq. (6) for $I_x=4$, $I_y=2.5$, and the solid and dashed curves correspond to positive and negative signals, respectively. (a) Positive helicity ($h=1$) with ${\mu }_0={20}^{\circ }$, which gives a total $\text{XMCD}=0$. (b) Negative helicity ($h=-1$) with ${\mu }_0=0^{\circ }$, which gives a total $\mathrm{XMCD}=\frac{3}{2}(I_x-I_y)cos\gamma$, with maximum at $\gamma =-{\mu }_0=0^{\circ }$, i.e., along $+x^{\prime }$. (c) Negative helicity with ${\mu }_0={90}^{\circ }$, which gives a total $\mathrm{XMCD}=-\frac{3}{2}(I_x-I_y)sin\gamma$, with maximum at $\gamma =-{\mu }_0=-{90}^{\circ }$, i.e., along $-y^{\prime }$.

Figure 5

(a) The angular dependence as a function of $\gamma$ for negative helicity with ${\mu }_0={20}^{\circ }$. The description of the plot is as in Fig. 4. The total XMCD = $\frac{3}{2}(I_x-I_y)cos(\gamma +{20}^{\circ })$, which has a maximum for $\gamma =-{\mu }_0=-{20}^{\circ }$, thus at the opposite site of the $x^{\prime }$ axis than $S_0$ (red arrow). The lower panel shows the decompostion of the XMCD in isotropic and anisotropic part. (b) The isotropic part of the XMCD, which shows no total signal. (c) The anisotropic part, where all three magnetic atoms have the same angular dependence, shown by the brown curve, and which together add up to the black curve.