Spin-Resolved Band Structures

• Spin-resolved band structures are a framework that defines the energy, momentum, and spin polarization of states in quantum materials, key for understanding spin-dependent phenomena.
• SARPES and advanced techniques accurately capture spin splitting, Rashba effects, and topological surface states with high resolution and sensitivity.
• Ab initio modeling combined with symmetry analysis underpins the design of spintronic devices and the exploration of next-generation quantum applications.

Spin-resolved band structures describe the energy–momentum dispersion of electronic, magnonic, or quasiparticle states together with their spin polarization, and constitute a central framework for understanding and exploiting spin-dependent phenomena in quantum materials. These structures not only resolve the energetics and symmetry of bands as functions of crystal momentum $\mathbf{k}$ but also specify the vectorial spin expectation values at each point in the Brillouin zone, enabling direct connections to topological invariants, symmetry protection, and spintronic functionalities. Accurate mapping and theoretical modeling of spin-resolved band structures underpin modern research in topological insulators, Rashba and Dresselhaus materials, spin-orbit coupled transport, quantum spin liquids, and complex magnetic systems.

1. Principles and Measurement Techniques

Spin-resolved band structures are obtained by measuring or calculating the energy, momentum, and spin polarization of states in a material. The primary experimental method is spin- and angle-resolved photoemission spectroscopy (SARPES), which extends conventional ARPES by adding spin filtering, typically via Mott or very-low-energy-electron-diffraction (VLEED) polarimeters. SARPES enables direct mapping of $\mathbf{k}$-resolved energy bands with quantification of individual spin components with high angular ($<1^\circ$) and energy (sub-10 meV) resolution (Zha et al., 2023). Spin-polarized ARPES is critical in surface-sensitive studies (e.g., topological insulators), but use of low-energy or soft X-ray photons increases probe depth and allows detection of buried or bulk spin textures (Berntsen et al., 2022, Vasilyev et al., 2018).

The key spin-resolved quantities are often presented as

$$P(\mathbf{k},E) = \frac{I_\uparrow(\mathbf{k}, E) - I_\downarrow(\mathbf{k}, E)}{I_\uparrow(\mathbf{k}, E) + I_\downarrow(\mathbf{k}, E)}$$

where $I_\uparrow$ and $I_\downarrow$ are spin-resolved intensities for chosen spin quantization axes. The measurement geometry, photon polarization, and experimental configuration determine which spin components and selection rules can be accessed.

Recent advances comprise high-throughput momentum microscopy for 3D spin texture mapping (Chernov et al., 2019), low-energy SARPES for buried interfaces (Berntsen et al., 2022), and the use of multichannel VLEED polarimeters offering improved stability and sensitivity (Zha et al., 2023).

2. Spin-Orbit Coupling, Symmetry Breaking, and Rashba Effects

Spin-orbit coupling (SOC) is fundamental in inducing spin splitting and complex spin textures in band structures, especially in materials lacking inversion symmetry. In systems with strong SOC and broken inversion symmetry, spin splitting arises even in nonmagnetic states, as epitomized by the Rashba and Dresselhaus effects: $$H_\text{Rashba} = \alpha_R (\sigma_x k_y - \sigma_y k_x), \qquad H_\text{SO} = \alpha (\sigma \times \mathbf{k}) \cdot \hat{z}$$ This produces spin-momentum locking whereby spin orientation is locked perpendicular to the momentum, yielding characteristic helicity textures detected in SARPES as, for instance, concentric Fermi rings with opposite spin polarization (Landolt et al., 2012, Sakano et al., 2012).

Giant Rashba splitting ($>300$ meV) is observed in bulk polar semiconductors such as BiTeI due to both strong SOC from heavy atoms and large built-in electric fields arising from non-centrosymmetric crystal structures. The theoretical and experimental analysis in these systems relies on parameterizing the splitting in terms of measured $k_0$ offsets and Rashba coefficients ($\alpha_\mathrm{R}$), which can be extracted from energy- and momentum-resolved SARPES and SX-ARPES (Sakano et al., 2012).

The interplay of orbital character, local symmetry, and spin-orbit fields can generate highly anisotropic and unconventional Rashba-like splittings, as found in oxide two-dimensional electron gases (2DEGs) such as (111)-KTaO$_3$, where the spin texture becomes 3-fold symmetric, deviating from the simple textbook Rashba model (Bruno et al., 2019). These are described by momentum-dependent effective Rashba parameters $\alpha_\mathrm{R}(\mathbf{k})$.

In centrosymmetric materials, spin texture can still emerge at interfaces, surfaces, or via local inversion asymmetry within atomic layers—an effect termed spin-layer locking (Zha et al., 2023).

3. Topological Phases and Surface States

Spin-resolved band structures provide direct evidence for topological phases. In strong three-dimensional topological insulators such as Bi$_{1-x}$Sb$_x$ or Sb$_2$Te$_3$, SARPES reveals surface states with Dirac-cone-like linear dispersion and spin-momentum locking, protected by time-reversal symmetry (0902.2251, Pauly et al., 2012). Here, the number of spin-polarized Fermi-level crossings between time-reversal-invariant momenta is an indicator of topological character: an odd number signals a strong topological insulator.

Mirror chirality ($\eta$), extracted from the arrangement and crossing (or non-crossing) of spin branches, provides additional characterization of topological surface states. For example, in Bi$_{1-x}$Sb$_x$, the absence of a band crossing between $\Sigma_1$ and $\Sigma_2$ identifies $\eta = -1$ (0902.2251). The behavior of spin-polarized bands and the presence of robust, symmetry-protected Rashba-type or Dirac states can be rationalized via first-principles calculations, symmetry criteria (e.g., arguments due to Pendry and Gurman), and DFT-based topological invariants (Pauly et al., 2012).

Spin-resolved mapping also enables the identification of Fermi arcs and their connectivity to bulk and surface resonance bands in topological semimetals such as WTe$_2$, including distinctions between surface and bulk states via comparison of spin signals on opposite crystal terminations (Wan et al., 2021).

4. Ab Initio Modeling, Multipole Frameworks, and Theoretical Approaches

Quantitative calculation and understanding of spin-resolved band structures demand methods that capture SOC, broken symmetries, and correlations. Fully relativistic ab initio band structure methods—such as Dirac-based Korringa-Kohn-Rostoker (KKR) with Coherent Potential Approximation (CPA) for disorder—incorporate SOC and allow for decomposition of electronic transport or conductivity into spin channels based on relativistic spin projection operators (Lowitzer et al., 2010). The matrix-element induced spin polarization in photoemission, especially in centrosymmetric, nonmagnetic materials, is analyzed via models incorporating optical spin orientation (Fano effect) and final-state interference, which require careful separation of spin components (Vasilyev et al., 2018).

In materials with negligible intrinsic SOC but nontrivial magnetic order (e.g., noncollinear or noncoplanar antiferromagnets), a "bottom-up" multipole approach classifies spin splitting according to cluster and bond multipoles (electric and magnetic toroidal) and their symmetry. Symmetric (even-in-$\mathbf{k}$) spin splitting arises in collinear magnets via electric multipole couplings, while antisymmetric (odd-in-$\mathbf{k}$) splitting is generically a consequence of magnetic toroidal multipoles present in noncollinear magnets (Hayami et al., 2020). This multipole decomposition formalism allows systematic prediction of candidate materials supporting tunable spin splitting absent SOC.

For accurate electronic structure with strong SOC, Spin-Current Density Functional Theory (SCDFT) and its generalized Kohn-Sham (GKS) extensions provide self-consistent inclusion of both spin densities and spin currents, employing nonlocal exchange potentials (a fraction $\alpha$ of Fock exchange). These approaches reliably reproduce SOC-induced band splittings and band gaps in both inversion-asymmetric monolayers (Rashba-I systems) and inversion-symmetric systems with hidden spin textures (Rashba-II) (Desmarais et al., 2023). In contrast, traditional SDFT with perturbative SOC inclusion underestimates these splittings due to lack of spin current feedback.

Quantum magnets and spin liquids are modeled using Abrikosov (fermionic) parton representations, with band structures for emergent spinon quasiparticles obtained by combining pseudofermion functional renormalization group (PFFRG)-derived effective interactions with PSG-classified mean-field ansätze (Hering et al., 2018). This scheme captures both pairing and hopping contributions to the spinon spectrum beyond simple mean-field.

5. Material-Specific Cases and Novel Phenomena

Spin-resolved band structure studies have elucidated the physics of diverse quantum materials:

• Ferromagnetic semiconductors (e.g., EuO): Spin-ARPES demonstrates indirect exchange splitting in O $2p$ bands caused by interaction with localized Eu $4f$ moments; the splitting is temperature dependent and vanishes above the Curie point (Heider et al., 2018).
• Heusler alloys: Spin- and momentum-resolved momentum microscopy reveals that experimental band filling and exchange splitting deviate from a rigid-band model, exhibiting spin gaps or near half-metallic behavior contingent on composition and valence electron count (Chernov et al., 2019).
• Chiral surfaces: Broken mirror symmetry in chiral two-dimensional lattices leads to three nonzero spin texture components. Notably, the longitudinal spin polarization is absent in achiral lattices and reverses sign between enantiomorphs, as demonstrated by explicit theoretical modeling and heuristic Lorentz-transformation-based analysis (Lewis et al., 2018).
• Quantum spin liquids: Emergent spinon band structures with Dirac nodes or small Fermi surfaces are theoretically derived using combined PFFRG and projective symmetry group classification, enabling systematic identification of spin-liquid ground states (Hering et al., 2018).
• Rectangular antidot magnonic lattices: Anisotropic magnonic band structures with tunable, direction-dependent bandgaps are measured and predicted, supporting directional spin-wave filtering via engineered spin-resolved bands (Lenk et al., 2012).

6. Applications, Implications, and Future Directions

Precise mapping and understanding of spin-resolved band structures have direct implications for:

• Spintronic devices: Robust, symmetry-protected spin currents on topological insulator surfaces, giant Rashba-split or Dirac bands, and enhanced spin polarization in tailored 2DEGs facilitate efficient spin injection, manipulation, and conversion.
• Quantum information: Tunable spin textures and interface engineering (e.g., in buried Dirac states) enable integration of quantum materials into spin-based logic and memory architectures, particularly when SARPES techniques are applied to buried layers (Berntsen et al., 2022).
• Nonreciprocal and magnetoelectric transport: Multipole-guided design of AFM materials with targeted spin splittings may provide giant nonreciprocal responses, magnetoelectric effects, and elastic tuning without the need for heavy elements or SOC.
• Photocurrent control: Band-resolved imaging of photocurrent generation in topological insulators confirms that only resonant optical transitions coupling to spin-orbital textured states provide net photocurrent, allowing for optical manipulation of spin currents (Soifer et al., 2017).
• Fundamental symmetry studies: Comprehensive spin structure mapping enables the identification of symmetry-protected surface states, extraction of mirror Chern numbers, and discrimination between bulk and surface contributions in topological and Weyl materials (0902.2251, Wan et al., 2021).

Advances in SARPES instrumentation, high-throughput and high-resolution measurement, bulk and interface sensitivity, and ab initio computational formalisms incorporating spin currents and SOC, will continue to expand the ability to resolve, control, and exploit spin-resolved band structures in quantum materials for both fundamental discovery and technological innovation.