Non-Local Transport Spectroscopy in Hybrids

• Non-local transport spectroscopy is a set of methods used to probe quantum coherence, spin dynamics, and scattering phenomena in nanoscale systems.
• It leverages mechanisms like crossed Andreev reflection and elastic cotunneling along with spin-dependent interface scattering to elucidate superconductivity and magnetism interplay.
• Temperature and energy-dependent measurements, framed by quasiclassical approaches, enable extraction of non-local resistance and conductance across spatially separated terminals.

Non-local transport spectroscopy is a suite of experimental and theoretical techniques designed to probe correlation, coherence, spatial information, and complex scattering phenomena in mesoscopic and nanoscale systems by measuring current or voltage responses at locations distinct from where the driving perturbation is applied. Unlike conventional (local) transport measurements that relate the response at a probe to the driving at the same location, non-local transport accesses the subtler, often quantum-coherent propagation of excitations, spins, charges, or quasiparticles across spatially separated terminals. The non-local signal is acutely sensitive to underlying symmetries, coherence, interface properties, and the presence of spin or other internal degrees of freedom, offering a powerful tool for the investigation of superconductivity, magnetism, topological phases, non-equilibrium dynamics, and even non-Hermitian physics.

1. Theoretical Frameworks for Non-Local Transport

Central to the quantitative theory of non-local transport in diffusive heterostructures is the quasiclassical Green's function approach, specifically the Usadel equation for diffusive superconducting and magnetic systems: $$iD\nabla(\mathcal{G} \nabla \mathcal{G}) = [\Omega + eV, \mathcal{G}],\quad \mathcal{G}^2 = 1,$$ where $\mathcal{G}$ is the Green–Keldysh matrix, $D$ the diffusion constant, and $\Omega$ contains the energy, superconducting order parameter $\Delta$, and—in ferromagnets—the exchange field $h$ (Kalenkov et al., 2010).

At interfaces, spin-active scattering is incorporated in the boundary conditions, with interface conductance parameters $G_T$ (spin independent), $G_m$ (spin dependent), and $G_\varphi$ (spin-mixing/phase-shift). The magnetic term $G_m$ in particular encodes the degree of spin selectivity at the interface, governing the emergence of non-local, spin-sensitive effects.

For three-terminal FSF (ferromagnet–superconductor–ferromagnet) structures in the tunneling and weak spin-dependence regime ($|G_m|, |G_\varphi| \ll G_T$), the non-local spectral conductance takes the form: $$g_{12}(\varepsilon) = [r_{\xi_S}(\varepsilon)/2]\, e^{-d/\xi_S(\varepsilon)}\, \left\{ \frac{\Delta^2 - \varepsilon^2}{\Delta^2} g_{T1}(\varepsilon)g_{T2}(\varepsilon) + (\mathbf{m}_1 \cdot \mathbf{m}_2) G_{m1}G_{m2} \frac{\Delta^2}{\Delta^2 - \varepsilon^2} \right\},$$ where $r_{\xi_S}$ is the Drude resistance of a segment of length $\xi_S$, $d$ is the SF contact separation, $g_{Ti}$ are the spin-independent interface conductances, and $\mathbf{m}_1,\mathbf{m}_2$ are the interface magnetization directions.

The energy and temperature-dependent non-local conductances can be integrated to yield the linear conductance matrix $G_{ij}(T)$ and, upon inversion of the current–voltage relations, the non-local resistance $R_{12}$, which can change sign depending on the mutual orientation of the magnetizations (i.e., $(\mathbf{m}_1 \cdot \mathbf{m}_2)$) (Kalenkov et al., 2010, Kalenkov et al., 2011).

2. Mechanisms of Non-Local Transport: CAR, EC, and Spin-Dependent Terms

Non-local transport in three-terminal hybrid structures arises fundamentally from two microscopic processes:

1. Crossed Andreev Reflection (CAR): An electron in one ferromagnetic electrode Andreev-reflects in the superconductor by pairing with an electron in the distant electrode, resulting in a nonlocal conversion of two electrons (from spatially separated regions) into a Cooper pair, and a simultaneous emission of a hole into the second electrode. CAR is highly sensitive to spin projection and the symmetry of the underlying superconducting order; for example, spin singlet Cooper pairing means that only pairs of electrons with opposite spins can participate (Kalenkov et al., 2011). CAR is suppressed for perfectly transparent interfaces.
2. Elastic Cotunneling (EC): An alternative process where an electron tunnels coherently from one electrode to the other, via a virtual intermediate state in the superconductor, without any net Cooper pair formation.

In FSF junctions, two main contributions to non-local conductance are observed:

• A conventional (spin-independent) higher-order (fourth-order in interface transmissions) term.
• A magnetic (spin-dependent) second-order term, proportional to $(\mathbf{m}_1 \cdot \mathbf{m}_2) G_{m1}G_{m2}$, which can dominate and drive the non-local conductance negative for antiparallel magnetizations at subgap energies.

Spin-active interface scattering crucially couples spin and transport, introducing cross-current terms and magnetoconductance even for weak spin polarization at the interface ($|G_m|\ll G_T$).

3. Exchange Fields, Disorder, and Suppression of Quantum Interference

In normal-superconductor-normal (NSN) structures, disorder-induced electron interference leads to phenomena such as the zero-bias anomaly in the tunneling density of states ($\sim 1/\sqrt{\epsilon}$). In FSF heterostructures with strong exchange fields $h$ in the ferromagnetic contacts ($\Delta \ll h_{1,2} \ll \varepsilon_F$), quantum interference is strongly suppressed in the ferromagnetic leads. The proximity effect in the F’s is consequently weak, and the disorder-induced zero-bias anomaly is largely eliminated, leaving the non-local conductance determined mainly by interface spin-dependent processes (Kalenkov et al., 2010). In contrast, interference effects persist in the superconducting wire (e.g., via Cooperon modes), affecting the local density of states and propagators relevant for conductance.

4. Energy and Temperature Dependence of Non-Local Conductance and Resistance

Non-local signals display marked dependence on energy (bias, temperature) relative to the superconducting gap $\Delta$:

• Subgap energies ($|\varepsilon| < \Delta$): Non-local conductance decays exponentially with contact separation ($\propto \exp[-d/\xi_S(\varepsilon)]$), with strong energy dependence due to the prefactors $(\Delta^2 - \varepsilon^2)/\Delta^2$ (conventional) and its reciprocal (magnetic term). For antiparallel magnetizations, sign reversal of $g_{12}$ and $R_{12}$ occurs.
• Above-gap energies ($|\varepsilon| > \Delta$): Magnetic effects and the proximity-induced modifications to the density of states become less relevant; non-local conductance approaches the behavior seen in non-magnetic systems.

Temperature enters via Fermi functions in the integrated conductances, producing distinctive temperature dependencies and correlating the sign and magnitude of the non-local resistance with interface alignment and thermal energy.

5. Experimental Proposals and Measurement Strategies

The theoretical framework provides clear guidance for experimental verification:

• Fabricate FSF devices with well-controlled and independently tunable interface magnetizations.
• Measure both local and non-local current–voltage characteristics, specifically extracting $G_{ij}(T)$ and $R_{ij}(T)$ from multi-terminal configurations over a range of temperatures and bias voltages.
• Detect sign reversals in $R_{12}$ at subgap energies and low temperatures as a function of the relative orientation of $\mathbf{m}_1$ and $\mathbf{m}_2$, a haLLMark of interface-driven magnetoconductance.
• Suppress or tune the exchange field to explore the regime where disorder-induced quantum interference either dominates (for small $h$) or is quenched (for large $h$).

6. Comparison with Related Systems and Broader Implications

The FSF non-local transport scenario generalizes to broader classes of hybrid structures, including NSN and more complex multi-terminal geometries (Kalenkov et al., 2011). In ballistic (clean) systems, similar energy- and spin-dependent effects arise, but the relative importance of CAR and EC and the influence of interface transparency are distinct: CAR vanishes for perfectly transparent barriers, while for tunneling barriers non-local signals are dominated by higher-order processes.

The sensitivity of non-local transport to spin-dependent interface scattering and the capability to induce sign changes in non-local resistance underpins applications in superconducting spintronics, non-local entanglement generation, and fundamental probes of quantum coherence and many-body correlations in nanostructures.

7. Summary

Non-local transport spectroscopy in diffusive FSF heterostructures is governed by a combination of spin-independent and spin-sensitive contributions to non-local conductance, with interface magnetizations playing a pivotal role in the magnitude and even the sign of the signal. Strong exchange fields suppress disorder-induced interference, isolating spin-dependent interface effects as the dominant mechanism for non-locality in the subgap regime. Theoretical predictions, mapped out by quantitative expressions for spectral and integrated conductances, provide a direct experimental blueprint for probing non-local magnetoelectric effects and enable discrimination of underlying quantum processes such as crossed Andreev reflection, elastic cotunneling, and their interplay with disorder and magnetism. Experimental validation would establish non-local transport spectroscopy as a stringent probe of spin-coherent and phase-coherent phenomena in hybrid superconducting systems (Kalenkov et al., 2010, Kalenkov et al., 2011).