Half-and-Half Metal-Superconductor

• Half-and-Half Metal-Superconductor systems are nanoscale hybrids that integrate a half-metal with full spin polarization and a conventional superconductor, enabling unconventional pairing through interface-induced triplet correlations.
• Interface engineering, including spin mixing, spin–orbit coupling, and non-collinear magnetization, converts blocked singlet pairs into equal-spin triplets while tuning critical temperatures.
• Such structures exhibit topological superconducting phases, anomalous Andreev reflections, and potential Majorana modes, offering promising pathways for quantum device applications.

A half-metal/superconductor, or “half-and-half,” structure is a nanoscale hybrid in which a half-metallic ferromagnet is electronically coupled to a superconductor. In the half-metal, one spin band is metallic while the other has no states at the Fermi level; in the superconductor, the condensate is ordinarily spin-singlet. This combination blocks the usual singlet proximity effect in its simplest form, yet it does not eliminate superconducting phenomena. Across atomically thin spin valves, Josephson junctions, tunnel interfaces, and two-dimensional heterostructures, the interplay of full spin polarization, spin-active scattering, non-collinear magnetization, and spin–orbit coupling produces tunable critical temperatures, equal-spin triplet transport, odd-frequency condensates, and topological superconducting phases (Devizorova et al., 2019, Duckheim et al., 2010).

1. Physical principle and defining constraints

A half-metal is a ferromagnet with full spin polarization at the Fermi level: one spin species is metallic and the other is absent or gapped. In the atomically thin models of half-metal/superconductor spin valves, this is encoded by keeping only one finite spin band and sending the other to infinity, for example

$$\hat W= \begin{pmatrix} \xi(\mathbf p) & 0\ 0 & \infty \end{pmatrix},$$

while the superconducting layer carries a conventional spin-singlet $s$-wave pairing term of the form

$$\hat H_S=\sum_{\mathbf p}\left( \Delta^* \phi_{\mathbf p,2}\phi_{-\mathbf p,1} +\Delta\,\phi^\dag_{\mathbf p,1}\phi^\dag_{-\mathbf p,2} \right).$$

Because a spin-singlet Cooper pair requires both spin species, such a pair cannot directly propagate coherently into a half-metal. In a clean interface this suppresses ordinary Andreev reflection and the usual induced singlet pair amplitude in the half-metal (Devizorova et al., 2019, Kupferschmidt et al., 2010).

That obstruction is not absolute. Virtual tunneling processes, spin mixing at interfaces, spatially non-uniform magnetization, and spin–orbit coupling can convert singlet correlations into equal-spin triplet correlations compatible with the fully spin-polarized medium. In the diffusive limit the constraint is even sharper: only spin-polarized Cooper pairs which are non-locally and antisymmetrically correlated in time may exist in a half-metal, so the half-metal acts as an odd-frequency superconducting condensate. A common misconception is therefore that half-metals and spin-singlet superconductors are simply incompatible. The more precise statement is that conventional singlet propagation is blocked, whereas triplet conversion channels remain available and can dominate the low-energy physics (Fyhn et al., 2019, Wilken et al., 2012).

2. Microscopic descriptions and interface conversion mechanisms

The central theoretical distinction in this subject is between quasiclassical treatments and microscopic descriptions that retain band-structure details. For atomically thin multilayers, the superconducting critical temperature $T_c$ can depend on the relative shift $\varepsilon$ between the occupied spin band of a half-metal and the superconducting band, a dependence that the tight-binding Gor’kov approach captures explicitly. This is one reason the half-metal limit cannot be treated adequately by standard quasiclassical assumptions such as $h\ll E_F$ and weak band mismatch (Devizorova et al., 2019).

In ballistic junctions, the microscopic language is typically Bogoliubov–de Gennes scattering. For a non-centrosymmetric or Ising superconductor coupled to a half-metal, spin–orbit coupling at or near the interface supplies the missing spin rotation. In one formulation the spin–orbit term in the superconductor is

$$\hat H_{\rm SO} = \frac{\hbar}{2}\,[p\,\Theta(z)+\Theta(z)p]\cdot \sum_{i=1}^{3}\boldsymbol{\Omega}_i \sigma_i,$$

and the resulting Andreev process converts a majority electron into a majority hole, yielding triplet Andreev reflection rather than the conventional opposite-spin process. In a complementary scattering formulation for a half-metal with non-uniform magnetization, the local spin rotation enters through a perturbation proportional to the magnetization gradient, and the Andreev amplitude changes qualitatively depending on whether that gradient is perpendicular or parallel to the interface (Duckheim et al., 2010, Kupferschmidt et al., 2010).

Spin-active interfaces provide an additional control layer. In the Ising-superconductor/half-metal/Ising-superconductor junction, the interface potential contains both spin-independent and spin-dependent pieces,

$$\hat{\mathcal V}_{\text{int}} = (\mathcal V_0 \hat I + \boldsymbol{\sigma}\cdot \mathbf{\mathcal V_m})[\delta(x)+\delta(x-L)],$$

with spin mixing and spin flip controlled by the barrier magnetic moment. In half-metal/conical-magnet/superconductor structures, the conical magnet plays the same role dynamically: it rotates the injected spin adiabatically or non-adiabatically depending on the period of the exchange-field modulation, and the subgap conductance becomes governed by anomalous Andreev reflection (Acharjee et al., 2023, Wójcik et al., 2015).

3. Critical-temperature control, spin valves, and absolute switching

One of the most developed “half-and-half” phenomena is the spin-valve effect in structures such as $\mathrm{S/HM_1/HM_2}$ and $\mathrm{HM_1/S/HM_2}$. In the atomically thin tight-binding treatment, the leading angular dependence of the critical temperature is

$s$0

so the effect is monotonic in the angle $s$1 between the spin quantization axes of the two half-metals. The sign of $s$2 determines whether the device shows the standard spin-valve effect, $s$3, or the inverse effect, $s$4. For $s$5 with the central half-metal band shifted by $s$6, the crossover occurs at $s$7; for $s$8 with the adjacent half-metal band shifted, it occurs at $s$9. By contrast, shifting the side half-metal band in $$\hat H_S=\sum_{\mathbf p}\left( \Delta^* \phi_{\mathbf p,2}\phi_{-\mathbf p,1} +\Delta\,\phi^\dag_{\mathbf p,1}\phi^\dag_{-\mathbf p,2} \right).$$0 leaves $$\hat H_S=\sum_{\mathbf p}\left( \Delta^* \phi_{\mathbf p,2}\phi_{-\mathbf p,1} +\Delta\,\phi^\dag_{\mathbf p,1}\phi^\dag_{-\mathbf p,2} \right).$$1 for all $$\hat H_S=\sum_{\mathbf p}\left( \Delta^* \phi_{\mathbf p,2}\phi_{-\mathbf p,1} +\Delta\,\phi^\dag_{\mathbf p,1}\phi^\dag_{-\mathbf p,2} \right).$$2, so only the central band shift can invert the effect in that geometry (Devizorova et al., 2019).

The same theme appears in triplet spin valves built from nearly half-metallic ferromagnets. In the $$\hat H_S=\sum_{\mathbf p}\left( \Delta^* \phi_{\mathbf p,2}\phi_{-\mathbf p,1} +\Delta\,\phi^\dag_{\mathbf p,1}\phi^\dag_{-\mathbf p,2} \right).$$3 multilayer, the superconducting transition temperature as a function of the angle $$\hat H_S=\sum_{\mathbf p}\left( \Delta^* \phi_{\mathbf p,2}\phi_{-\mathbf p,1} +\Delta\,\phi^\dag_{\mathbf p,1}\phi^\dag_{-\mathbf p,2} \right).$$4 between the Heusler alloy and Ni magnetizations exhibits a deep minimum in the vicinity of the perpendicular configuration. The separation between the superconducting transition curves for $$\hat H_S=\sum_{\mathbf p}\left( \Delta^* \phi_{\mathbf p,2}\phi_{-\mathbf p,1} +\Delta\,\phi^\dag_{\mathbf p,1}\phi^\dag_{-\mathbf p,2} \right).$$5 and $$\hat H_S=\sum_{\mathbf p}\left( \Delta^* \phi_{\mathbf p,2}\phi_{-\mathbf p,1} +\Delta\,\phi^\dag_{\mathbf p,1}\phi^\dag_{-\mathbf p,2} \right).$$6 reaches up to $$\hat H_S=\sum_{\mathbf p}\left( \Delta^* \phi_{\mathbf p,2}\phi_{-\mathbf p,1} +\Delta\,\phi^\dag_{\mathbf p,1}\phi^\dag_{-\mathbf p,2} \right).$$7 K, and for the $$\hat H_S=\sum_{\mathbf p}\left( \Delta^* \phi_{\mathbf p,2}\phi_{-\mathbf p,1} +\Delta\,\phi^\dag_{\mathbf p,1}\phi^\dag_{-\mathbf p,2} \right).$$8 nm sample the reported value is $$\hat H_S=\sum_{\mathbf p}\left( \Delta^* \phi_{\mathbf p,2}\phi_{-\mathbf p,1} +\Delta\,\phi^\dag_{\mathbf p,1}\phi^\dag_{-\mathbf p,2} \right).$$9 K at $T_c$0 kOe. This is interpreted as a long-range triplet component efficiently drawn off by the half-metallic nature of the Heusler layer (Kamashev et al., 2019).

A distinct but related limit is the de Gennes absolute superconducting switch. In the $T_c$1 structure, EuS is an insulating ferromagnet, Au is a heavy metal, and Nb is an ultrathin $T_c$2-wave superconductor. The decisive interface parameter is the spin-mixing conductance at EuS/Au, modeled by

$T_c$3

With $T_c$4 and $T_c$5, the P-state superconductivity is completely quenched while the AP state remains superconducting. Experimentally,

$T_c$6

so

$T_c$7

This realizes de Gennes’ absolute switch: the same wire is superconducting in the antiparallel configuration and effectively metallic in the parallel configuration (Matsuki et al., 2024).

4. Josephson transport, Andreev spectra, and tunneling conductance

In half-metal/superconductor transport, the decisive low-energy question is not whether Andreev reflection exists in the conventional singlet form, but which unconventional version replaces it. At a clean spin-active half-metal/superconductor interface with a magnetization gradient perpendicular to the interface, the Andreev reflection amplitude is proportional to the excitation energy and vanishes at $T_c$8. Impurities at or in the immediate vicinity of the interface change this qualitatively: they generate a finite Andreev reflection amplitude at zero energy, and impurity-assisted Andreev reflection then dominates the low-bias conductance of an HS junction and the Josephson current of a long SHS junction (Wilken et al., 2012).

Josephson junctions with explicitly spin-active interfaces show a broader phenomenology. In the clean Ising-superconductor/half-metal/Ising-superconductor system, Bogoliubov–de Gennes solutions show that spin mixing, spin flipping, and Ising spin–orbit coupling split the Andreev bound states, modify the current–phase relation, and produce $T_c$9 transitions and $\varepsilon$0-junction behavior. The paper reports an additional splitting of the Andreev levels due to the superconductor’s spin–orbit coupling, different $\varepsilon$1-junctions obtained by tuning the barrier magnetic moment and the spin mismatch angle, finite sub-gap conductance, and anomalous Andreev levels for different half-metal lengths. It also states that the interplay of spin mixing and spin flipping processes with SOC can host Majorana modes in the proposed system (Acharjee et al., 2023).

In half-metal/conical-magnet/superconductor junctions, the conductance in the subgap region is mainly determined by anomalous Andreev reflection, and the distinction between adiabatic and non-adiabatic spin transport through the conical magnet becomes measurable. The conductance oscillates as a function of conical-magnet thickness; the oscillations are irregular in the non-adiabatic regime and regular in the adiabatic regime. In the non-adiabatic regime, decreasing the exchange-field amplitude produces a conductance peak for one particular conical-magnet thickness, in agreement with the experiment cited in the paper (Wójcik et al., 2015).

Tunnel spectroscopy provides a complementary limit. In $\varepsilon$2-insulator-Pb junctions, the observed increase in normalized differential conductivity is attributed to spin-polarized electron accumulation in the superconductor. The method yields a spin polarization $\varepsilon$3 of L2$\varepsilon$4-type ordered $\varepsilon$5 films at $\varepsilon$6 K close to $\varepsilon$7, directly tying half-metallicity to the nonequilibrium superconducting response (Linder et al., 2010).

5. Topological, nodal, and gapless superconducting phases

A central development in the field is the realization that half-metals can host topological superconductivity once spin–orbit coupling supplies the missing spin rotation. In a half-metal/superconductor heterostructure with interface spin–orbit coupling, the induced pairing on the spin-polarized band is chiral $\varepsilon$8. The effective gap on the half-metal side is described in the strong-hopping limit by

$\varepsilon$9

and the resulting state is a spinless chiral topological superconductor with a single chiral Majorana edge state. Band calculations in that work identify two atomic layers of VTe or CrO$h\ll E_F$0 as single-band half-metals over a wide Fermi-energy range, making them candidate materials for the platform (Chung et al., 2010).

A related route starts from a non-centrosymmetric superconductor with spin–orbit coupling. There the Andreev reflection amplitude for a majority electron incident on the half-metal is odd in momentum and even in energy, and in a lateral thin-film or wire geometry repeated coherent reflections open a proximity-induced minigap. For energies below that gap the lateral half-metal–superconductor contact becomes a perfect triplet Andreev reflector, and the low-energy theory maps onto a spinless $h\ll E_F$1-wave Hamiltonian. The paper then shows that a half-infinite wire supports a localized Majorana end state satisfying the Majorana condition $h\ll E_F$2 (Duckheim et al., 2010).

Two-dimensional half-metal/superconductor heterostructures with Rashba spin–orbit coupling exhibit a richer topological phase diagram when the magnetization direction is varied. In the lattice model studied for a half-metal film on an $h\ll E_F$3-wave superconductor, tuning the magnetization angle produces transitions among a fully gapped topological superconducting phase with counter-propagating Majorana edge modes, a globally gapless but locally gapped phase with unidirectional Majorana edge modes, and a nodal phase identified as a two-dimensional Weyl superconductor with Fermi-line edge states. The Weyl nodes appear when the magnetization lies in plane and are protected by mirror symmetry (Hao et al., 2016).

The CrI$h\ll E_F$4/Pb interface shows that such topological behavior can emerge even when the magnetic layer is originally an insulator. First-principles calculations show charge transfer from Pb into CrI$h\ll E_F$5, moving the Fermi energy to approximately $h\ll E_F$6 meV above the bottom of the lowest conduction band and thereby doping the monolayer into an effective half-metal. The same work argues that gapless topological superconductivity is generic in two-dimensional half-metal–superconductor heterostructures lacking two-fold in-plane rotational symmetry, because $h\ll E_F$7 prevents a full pairing gap on the entire Fermi surface. A sufficiently large proximity-induced pairing amplitude can, however, drive the system into a fully gapped topological superconducting phase (Margalit et al., 2020).

6. Odd-frequency condensates, vortices, and wider perspectives

In the diffusive regime, the half-metal/superconductor proximity effect acquires a particularly distinctive form. Because only one spin species conducts, the allowed condensate is an odd-frequency equal-spin triplet state. The quasiclassical treatment of a two-dimensional SHS junction in a perpendicular magnetic field shows that this odd-frequency condensate can support bona fide superconducting vortices inside the half-metal. At the vortex cores, the local density of states returns to the normal-state value, while the circulating supercurrent is necessarily spin-polarized because only one spin band is present. The paper further shows that magnetic disorder at the interfaces influences the position at which the vortices nucleate, suggesting an experimental route to infer effective interfacial misalignment angles from STM imaging of vortex positions (Fyhn et al., 2019).

The notion of a half-metal superconductor is not limited to proximity structures. A controlled microscopic theory of a two-dimensional near-ferromagnetic electron system shows that the ordered ferromagnetic phase instantly polarizes low-energy fermionic excitations and yields a half-metal. In that phase, two transverse Goldstone modes mediate an attractive interaction in a spatially odd channel, with the likely leading instability in the $h\ll E_F$8-wave channel. The pairing temperature in the ferromagnetic half-metal is found to be a fraction of the Fermi energy and significantly larger than in the paramagnetic phase near the transition, making the half-metallic superconducting state itself a legitimate many-body phase rather than only an interfacial effect (Raines et al., 30 Jun 2025).

Taken together, these results define “half-and-half metal-superconductor” as a family of hybrids and correlated phases in which full spin polarization does not extinguish superconductivity but redirects it into highly constrained channels: monotonic or inverted spin valves controlled by band alignment, equal-spin Andreev transport enabled by spin-active interfaces, odd-frequency triplet condensates in diffusive media, and topological superconductivity with Majorana or Weyl excitations when spin–orbit coupling and symmetry conditions are favorable (Devizorova et al., 2019, Chung et al., 2010).