Theory of X-ray Magnetic Dichroism and Altermagnetism
[Note: This page uses MathJax to render TeX equations]
1. Introduction to X-ray Absorption
X-ray absorption spectroscopy (XAS) is a powerful element-specific technique. When X-rays interact with a material, core electrons are excited into unoccupied valence states. For transition metals (like Fe, Co, Ni), the most intense transitions occur at the \( L_{2,3} \) edges, where \( 2p \) core electrons are excited into the \( 3d \) valence band.
The transition probability from an initial core state \( |i\rangle \) to a final valence state \( |f\rangle \) is governed by Fermi's Golden Rule in the electric dipole approximation: $$ W_{fi} \propto |\langle f | \boldsymbol{\epsilon} \cdot \mathbf{r} | i \rangle|^2 \delta(E_f - E_i - \hbar\omega) $$ where \( \boldsymbol{\epsilon} \) is the X-ray polarization vector.
2. Ferromagnets and the Origin of XMCD Sum Rules
The "Two-Step" Model
X-ray Magnetic Circular Dichroism (XMCD) measures the difference in absorption between left- (\( \sigma^+ \)) and right-circularly (\( \sigma^- \)) polarized light. To understand why this works, we rely on the phenomenological "Two-Step Model".
Step 1 (The Spin Source): Circularly polarized photons carry angular momentum (\( \pm \hbar \)). When they excite a \( 2p \) core electron, selection rules dictate that \( \Delta m_l = \pm 1 \). Because the \( 2p \) core level is split by spin-orbit coupling into \( 2p_{3/2} \) (the \( L_3 \) edge) and \( 2p_{1/2} \) (the \( L_2 \) edge), this orbital angular momentum is partially transferred to the electron's spin. Thus, the X-ray acts as a source of highly spin-polarized photoelectrons.
Step 2 (The Spin Detector): In a ferromagnet, the \( 3d \) valence band is exchange-split, meaning there is an imbalance between the number of empty spin-up (majority) and spin-down (minority) states. By the Pauli Exclusion Principle, the spin-polarized photoelectrons can only transition into empty states of the same spin. Therefore, the difference in absorption directly maps the difference in the empty density of states for spin-up versus spin-down.
The Wigner-Eckart Theorem and Sum Rules
Where do the equations actually come from? In the early 1990s, Thole and Carra [Thole et al., PRL 1992] proved that if you express the dipole operator in spherical harmonics and apply the Wigner-Eckart theorem, the transition matrix elements separate into a radial integral and an angular part (Clebsch-Gordan coefficients). By summing these coefficients over all initial core states and final unoccupied states, the complex algebra miraculously collapses down to the ground-state expectation values of the orbital (\( \langle L_z \rangle \)) and spin (\( \langle S_z \rangle \)) operators.
The orbital sum rule dictates that integrating the dichroic signal over both edges yields the orbital moment: $$ \frac{\int_{L_3+L_2} (\mu_+ - \mu_-) d\omega}{\int_{L_3+L_2} (\mu_+ + \mu_0 + \mu_-) d\omega} = \frac{\langle L_z \rangle}{2 n_h} $$ The spin sum rule states that a linear combination of the integrals isolates the effective spin moment: $$ \frac{\int_{L_3} (\mu_+ - \mu_-) d\omega - 2\int_{L_2} (\mu_+ - \mu_-) d\omega}{\int_{L_3+L_2} (\mu_+ + \mu_0 + \mu_-) d\omega} = \frac{2\langle S_z \rangle + 7\langle T_z \rangle}{3 n_h} $$ Here, \( n_h \) is the number of \( 3d \) holes, and \( \langle T_z \rangle \) is the magnetic dipole operator. Because this relies on net spin-polarization, XMCD is immensely strong in ferromagnets.
3. Deep Dive: Deriving the Origin of XMLD
In an antiferromagnet, the opposing magnetic sublattices cancel each other out, yielding a net macroscopic magnetization of zero (\( M = 0 \)). Consequently, the macroscopic XMCD signal strictly vanishes. To probe these materials, we turn to X-ray Magnetic Linear Dichroism (XMLD) [van der Laan et al., PRB 1986]. But where does the XMLD equation actually come from? Let's derive it.
Step 1: Spin-Orbit Coupling and Charge Anisotropy
The fundamental driver of XMLD is the Spin-Orbit Coupling (SOC) Hamiltonian, \( H_{SOC} = \xi \mathbf{L} \cdot \mathbf{S} \). While the strong exchange interaction aligns the spins \( \mathbf{S} \) along a preferred magnetic axis (the Néel vector \( \mathbf{n} \)), the SOC physically locks the orbital angular momentum \( \mathbf{L} \) to that spin direction.
Because the orbital states dictate the spatial distribution of the electrons, locking \( \mathbf{L} \) to a specific axis deforms the typically spherical \( 3d \) charge density into an aspherical shape (a quadrupole moment). This charge anisotropy is the physical observable that linearly polarized X-rays detect.
Step 2: Transition Matrix Elements
Recall Fermi's Golden Rule. For linearly polarized light, the electric field vector \( \boldsymbol{\epsilon} \) interacts with the spatial coordinate \( \mathbf{r} \). We can decompose this dipole operator into components parallel (\( z \)) and perpendicular (\( x, y \)) to the magnetic quantization axis \( \mathbf{n} \).
If the polarization vector \( \boldsymbol{\epsilon} \) is at an angle \( \theta \) relative to \( \mathbf{n} \), the transition probability \( I(\theta) \) is simply the geometric projection of the parallel absorption (\( I_\parallel \), where \( \Delta m = 0 \)) and the perpendicular absorption (\( I_\perp \), where \( \Delta m = \pm 1 \)): $$ I(\theta) = I_\parallel \cos^2\theta + I_\perp \sin^2\theta $$
Step 3: The \( 3\cos^2\theta - 1 \) Rule
Using the trigonometric identity \( \sin^2\theta = 1 - \cos^2\theta \), we can rearrange the absorption intensity: $$ I(\theta) = I_\perp + (I_\parallel - I_\perp) \cos^2\theta $$
Physicists usually define the isotropic (average) absorption as \( I_0 = \frac{1}{3}I_\parallel + \frac{2}{3}I_\perp \), and the maximum linear dichroism as \( \Delta I = I_\parallel - I_\perp \). Substituting these definitions, the equation elegantly transforms into: $$ I(\theta) = I_0 + \frac{1}{2} \Delta I (3\cos^2\theta - 1) $$
Why does XMLD survive in Antiferromagnets?
Notice that the XMLD intensity depends on \( \cos^2\theta \). In a ferromagnet, all spins point to angle \( \theta \). In an antiferromagnet, half the spins point to \( \theta \) and half point to \( \theta + 180^\circ \). Because \( \cos^2(\theta + 180^\circ) = \cos^2\theta \), the XMLD signals from the two opposing sublattices add constructively! In contrast, XMCD depends linearly on \( \cos\theta \), so \( \cos(\theta) + \cos(\theta + 180^\circ) = 0 \), rendering it completely blind to antiferromagnetic order.
4. Altermagnetism: The \( d \)-wave Regime
Recently, a new, third fundamental class of magnetism has been discovered: Altermagnetism [Šmejkal et al., PRX 2022]. Like antiferromagnets, altermagnets have collinear, antiparallel spin sublattices resulting in zero net magnetization. However, unlike traditional antiferromagnets, their opposite-spin sublattices are connected by specific crystal rotations rather than simple translations or inversion.
This unique symmetry (often lacking an internal parity or combined parity-time reversal symmetry \( \mathcal{PT} \)) allows for strong, non-relativistic spin splitting in the momentum space (\( \mathbf{k} \)-space) that alternates in sign depending on the direction—much like the lobes of a \( d \)-wave or \( g \)-wave orbital. For the simplest \( d \)-wave case (as modeled in our simulation below), the spin splitting takes the form: $$ \Delta E(\mathbf{k}) \propto k_x^2 - k_y^2 $$
X-ray response of Altermagnets: Because the macroscopic magnetization is zero, standard spatially-integrated XMCD is zero. However, if one uses momentum-resolved techniques (like ARPES) or localized probes, one can observe strong spin-polarization effects identical to ferromagnets. Furthermore, macroscopic XMLD remains robust in altermagnets due to the inherent \( d \)-wave crystal/magnetic symmetry anisotropies, bridging the gap between ferromagnetic and antiferromagnetic phenomenology.
Interactive Simulation
Below is an interactive simulation demonstrating these exact principles. Choose "Altermagnet" to see the \( d \)-wave split Fermi surfaces (\( k_x^2 - k_y^2 \)). You can adjust the Spin Angle \( \phi_M \) and the X-ray polarization to see how the XMLD intensity scales with the \( 3\cos^2\theta - 1 \) rule.