Spin-Dependent Thermoelectrics

• Spin-dependent thermoelectric properties are defined as the interplay between spin and charge responses to thermal gradients, enabling the generation of both charge and spin currents.
• Key mechanisms such as band structure engineering, quantum interference, spin-orbit coupling, and magnetic anisotropy allow precise tuning of thermopower and spin figures of merit.
• Experimental prototypes, from quantum dot hybrids to graphene nanoribbons, demonstrate tunable energy conversion, paving the way for advanced spin-caloritronic applications.

Spin-dependent thermoelectric properties refer to the interplay between spin and charge degrees of freedom in thermoelectric transport, where thermal gradients not only generate charge currents (Seebeck effect) but also lead to spin currents or spin accumulations (spin Seebeck effect). Unlike conventional thermoelectric materials, spin-dependent thermoelectrics exploit the spin selectivity of quantum transport channels, band structure engineering, magnetic textures, or quantum interference to achieve or enhance thermoelectric energy conversion, energy filtering, or heat–to–spin current transduction.

1. Theoretical Formulation and Spin-resolved Transport Coefficients

Spin-dependent thermoelectric phenomena are fundamentally described using Onsager relations generalized to include spin, charge, and heat currents and their conjugate affinities. In a multi-terminal (e.g., quantum dot, nanoribbon, or tunnel junction) geometry, the electrical, spin, and heat current densities can be expressed as:

$$\begin{pmatrix} j_q \ j_e \ j_s \end{pmatrix} = \begin{pmatrix} L_{qq}& L_{qe}& L_{qs}\ L_{eq}& L_{ee}& L_{es}\ L_{sq}& L_{se}& L_{ss} \end{pmatrix} \begin{pmatrix} X_q \ X_e \ X_s \end{pmatrix}$$

where $j_e = j_\uparrow + j_\downarrow$ is the charge current, $j_s = j_\uparrow - j_\downarrow$ is the spin current, and $j_q$ is the heat current. The forces $X_q$, $X_e$, and $X_s$ correspond to gradients in temperature, electrochemical potential, and spin-dependent chemical potential, respectively. In each spin channel, the linear-response coefficients (e.g., conductance, Seebeck coefficient) are defined via Landauer–Büttiker or Green's-function integrals over the spin-resolved transmission $T_\sigma(E)$:

$$L_{n\sigma} = \frac{1}{h} \!\int\! dE\, (E-\mu)^n T_\sigma(E)\left[-\frac{\partial f}{\partial E}\right]$$

This leads to spin-resolved electrical conductance $G_\sigma = e^2 L_{0\sigma}$, thermopower (Seebeck) coefficient $$S_\sigma = -\frac{1}{eT} \frac{L_{1\sigma}}{L_{0\sigma}}$$, and electronic thermal conductance $$\kappa_\sigma = \frac{1}{T}[L_{2\sigma} - (L_{1\sigma})^2/L_{0\sigma}]$$. The charge and spin thermopower are, respectively:

$$S = \frac{G_\uparrow S_\uparrow + G_\downarrow S_\downarrow}{G_\uparrow + G_\downarrow}, \qquad S_s = S_\uparrow - S_\downarrow$$

The figure of merit for spin thermoelectricity generalizes the classic electronic form to include spin conductance and spin thermopower, $ZT_s = S_s^2 G_s T/\kappa_s$ (Trocha et al., 2017, Zberecki et al., 2014, Bennemann, 2011).

2. Key Physical Mechanisms: Role of Magnetism, Band Structure, and Quantum Effects

Spin-dependent thermoelectric properties arise from several physical mechanisms:

• Spin-polarized density of states (DOS): Ferromagnetic materials or leads introduce spin asymmetry in the DOS, which is crucial for generating a spin Seebeck effect. For example, a quantum dot coupled to a ferromagnet and a superconductor demonstrates that as spin polarization $p \to 1$, Andreev conductance is suppressed and spin thermopower $S_s$ is maximized (Trocha et al., 2017).
• Particle–hole symmetry breaking: Single-particle states at the edges of a superconducting gap or in defective 2D materials exhibit strong energy dependence in transmission, leading to large derivatives $\partial T_\sigma/\partial E$ and enhanced thermopower (Trocha et al., 2017, Zambrano et al., 12 Jan 2026).
• Impurity- and vacancy-induced states: Doping or random defects can introduce narrow in-gap states and Fano-type antiresonances, strongly enhancing the energy dependence of $T_\sigma(E)$ and thus spin thermopower, especially in low-dimensional nanoribbons and heterostructures (Zberecki et al., 2014, Gholami et al., 2021, Zambrano et al., 12 Jan 2026).
• Quantum interference, Kondo and Majorana correlations: Multi-level systems, such as coupled dots or magnetic molecules, show quantum interference (Dicke, Fano effects), Kondo resonance splitting, and signatures due to coupling to Majorana zero modes, all of which can be exploited to achieve or reverse spin thermopower and optimize $ZT_s$ (Karwacki et al., 2017, Wang et al., 2012, Majek et al., 17 Sep 2025).
• Spin-orbit interaction: Rashba spin–orbit coupling in dots or graphene heterostructures can strongly enhance spin thermopower and enable gate or field-tunability of critical points where the Seebeck coefficient vanishes or changes sign (Karwacki et al., 2014, Beiranvand et al., 2016).
• Magnetic anisotropy and single-spin impurities: Molecular anisotropy (uniaxial, transverse $D, E$) and large-spin scattering in tunnel barriers allow energy and angular momentum exchange, generating pure spin and heat currents without net charge transport (Misiorny et al., 2014, Misiorny et al., 2014, Manaparambil et al., 2021).

3. Material Realizations and Device Prototypes

Spin-dependent thermoelectric phenomena have been explored in several systems:

• Quantum dot hybrids with ferromagnetic and superconducting contacts enable electrically tunable large charge and spin thermopower via tuning of Coulomb interaction, spin polarization, and gate voltages. The divergent BCS DOS at gap edges and suppression of Andreev reflection yield strong peaks in $S$ and $ZT$ (Trocha et al., 2017).
• Silicene and graphene nanoribbons, particularly with edge doping (Al, P) or random vacancies, provide a tunable platform for achieving large $S_s$ and $ZT_s$. Defect engineering, edge magnetism, or ferromagnetic contacts can produce room-temperature performance exceeding bulk Wiedemann–Franz limits (Zberecki et al., 2014, Zambrano et al., 12 Jan 2026, Gholami et al., 2021).
• Molecular junctions and magnetic impurities displaying strong electronic correlations and magnetic anisotropy enable observation and control of spin thermopower, with Kondo or exchange-induced sign changes in $S_s$ (Manaparambil et al., 2021, Misiorny et al., 2014).
• Superconductor/ferromagnet bilayers under Zeeman splitting and spin-flip scattering exhibit a giant thermopower (up to $5 k_B/e$), attributed to a complex spin-dependent softening of superconducting gaps (Rezaei et al., 2017).
• Magnetic tunnel junctions and YIG/metal bilayers measured in the longitudinal spin Seebeck configuration convert magnonic spin currents, generated by a temperature gradient, to measurable voltages via the inverse spin Hall effect, with $S_{SSE} \sim \pm 1–2~\mu\mathrm{V/K}$ depending on the spin Hall angle and resistivity of the transition-metal overlayer (Ishida et al., 2013).
• Magnetic graphene: Both in ballistic and diffusive regimes, spin splitting and chemical potential tuning enable pure spin thermopower generation; the maximal spin Seebeck coefficient can be a few $10^2~\mu\mathrm{V/K}$ (Rameshti et al., 2014).

4. Optimization Strategies and Parameter Control

Maximizing spin-dependent thermoelectric response involves careful tuning of multiple material and device parameters:

• Spin polarization of leads/materials: Increasing ferromagnetic polarization enhances $S_s$ and suppresses unwanted channels (e.g., Andreev reflection) (Trocha et al., 2017).
• Band gap and energy scale alignment: Adjusting superconducting gap $\Delta$, chemical potential, or Fermi level to coincide with sharp DOS features optimally amplifies thermopower (Trocha et al., 2017, Zambrano et al., 12 Jan 2026).
• Coulomb interaction and quantum dot levels: Tuning interaction strengths can create multi-peak thermopower structures and additional sign reversals by aligning resonant levels with spectral singularities (Trocha et al., 2017, Karwacki et al., 2017).
• Quantum confinement and vacancy concentration: In nanoribbons, low to moderate defect concentration results in strong phonon suppression (phonon glass, electron crystal regime) while maintaining finite spin conductance and boosting $ZT_s$ (Zambrano et al., 12 Jan 2026).
• Geometry and symmetry breaking: Frustrated or ladder geometries, on-site asymmetry, light-induced Floquet modulations, and long-range hopping support tailored spin selectivity and maximize $ZT_s$ (Bhattacharya et al., 16 Dec 2025, Ganguly et al., 3 Jul 2025).
• Magnetic texture and anisotropy: Magnetic anisotropy enables scattering and quantum tunneling of angular momentum, which can mediate pure spin and heat current even in the absence of charge transport (Misiorny et al., 2014, Misiorny et al., 2014).

5. Experimental Signatures and Applications

Spin-dependent thermoelectric effects are detected via a variety of experimental protocols:

• Voltage measurement under thermal gradients (Seebeck effect): In YIG/metal LSSE experiments, a temperature gradient produces a measurable voltage in the metallic layer proportional to the magnon spin current via the inverse spin Hall effect (Ishida et al., 2013).
• Spin thermopower and spin voltage: Spin-resolved voltage can be extracted using spin-polarized contacts or ferromagnetic resonance.
• Negative differential thermoelectric resistance (NDTR): In silicene/graphene nanoribbons, NDTR regimes with decreasing thermocurrent as $\Delta T$ increases enable thermal switching functionality (Gholami et al., 2021).
• High $ZT_s$ or sign reversals: Observation of sign changes and enhancements in $S_s$ and $ZT_s$ as a function of gate, temperature, or bias indicate underlying spin-dependent mechanisms (e.g., Kondo–Majorana interplay, Rashba effects) (Majek et al., 17 Sep 2025, Karwacki et al., 2014).
• Pure spin current generation: Systems where $S_c=0$, $S_s \neq 0$ can serve as pure spin injectors or heat-driven spin batteries, a core concept for spin caloritronic devices (Trocha et al., 2017, Rameshti et al., 2014).

6. Challenges, Open Questions, and Outlook

While significant advances have been made in understanding and engineering spin-dependent thermoelectricity, several open challenges remain:

• Phonon thermal conductance: Accurate modeling and suppression of phonon contributions are essential for maximizing $ZT_s$, especially at room temperature (Zberecki et al., 2014, Zambrano et al., 12 Jan 2026).
• Scalability and material control: Reproducible synthesis of well-defined defects, controlled doping, or magnetic proximity effects is crucial for device fabrication.
• Spin accumulation and relaxation: Engineering slow spin relaxation in experimental platforms is necessary to observe maximal spin thermopower and non-equilibrium spin Seebeck effects, as emphasized in correlated dot and molecular junction settings (Manaparambil et al., 2023, Manaparambil et al., 2021).
• Nonlinear and far-from-equilibrium effects: Far-from-equilibrium transport regimes (large $\Delta T$, bias) exhibit nonlinearities and extra sign changes in thermopower not captured by linear response, requiring advanced methods (NRG + nonequilibrium Green's function techniques) for analysis (Manaparambil et al., 2023).
• Room-temperature operation: While sub-Kelvin spin-caloritronic devices have been demonstrated, extension of giant or high-efficiency spin-dependent thermoelectricity to technologically relevant temperatures motivates ongoing research (Zambrano et al., 12 Jan 2026).
• Integration with spin and quantum devices: The ability to tune spin thermoelectric coefficients via gating, magnetic field, structural asymmetry, or light opens avenues for integrating spin-caloritronic components with other quantum functional devices, including spin batteries, diodes, and heat-driven logic circuits (Gholami et al., 2021, Ganguly et al., 3 Jul 2025).

In summary, spin-dependent thermoelectric properties emerge from spin-selective band structure and scattering mechanisms, quantum coherence, and magnetic symmetry breaking in low-dimensional or engineered nanoscale systems. Advances in theory and device fabrication offer viable routes to pure spin current generation, efficient energy conversion, and multifunctional spin-caloritronic applications.