Nonlocal Differential-Conductance Spectroscopy

Updated 15 October 2025

Nonlocal differential-conductance spectroscopy is a technique that measures the voltage-induced current response between spatially separated contacts to reveal spatial correlations and quantum coherence.
It employs diverse device architectures—such as multiterminal hybrid nanostructures, quantum dots, and planar Josephson junctions—to dissect local and nonlocal transport phenomena.
Key applications include distinguishing topological phases, probing Majorana zero modes and Andreev bound states, and enabling quantum control in nanoscale devices.

Nonlocal differential-conductance spectroscopy is a broad class of experimental and theoretical techniques in which the variation of the current with respect to an applied voltage (dI/dV) is measured under nonlocal conditions—i.e., where current injection and detection occur at spatially separated contacts or through distinct subsystems. Unlike conventional local tunneling spectroscopy, nonlocal differential-conductance measurements probe spatial correlations, coherence effects, and collective electronic phenomena, accessing information about quasiparticle propagation, electron–electron interactions, unconventional superconductivity, topological phases, and quantum interference in a wide spectrum of quantum materials and nanoscale systems.

1. Foundational Principles

The essence of nonlocal differential-conductance spectroscopy is the separation of injection and detection points: a bias voltage is applied at one terminal or subsystem, while the resultant current is measured at a physically distinct location. Technically, the measured quantity is a nonlocal conductance matrix element, such as $G_{ij} = dI_i/dV_j$ for current $I_i$ in terminal $i$ as a response to voltage $V_j$ applied at terminal $j$ . In two- or multi-terminal devices, this methodology directly interrogates electron transport processes that involve coherent traversals over finite distances, quasiparticle interference, crossed Andreev reflection, elastic cotunneling, and the interplay of local and extended quantum states.

Key features:

Nonlocality: The current response in one region contains information about electronic events or quantum states spatially separated from the biasing terminal.
Differential Spectroscopy: Measuring $dI/dV$ as a function of bias voltage provides energy-resolved spectral information, analogous to local tunneling spectroscopy, but with an added spatial dimension.
Selectivity for Nonlocal Processes: Distinct transport phenomena, such as crossed Andreev reflection, elastic cotunneling, quantum interference, or the presence of spatially extended bound states (e.g., Majorana zero modes, Andreev bound states), produce unique signatures in the nonlocal conductance that cannot be captured by local probes.

2. Methodologies and Device Architectures

Experimental realizations of nonlocal differential-conductance spectroscopy span a range of device architectures:

Multiterminal Hybrid Nanostructures: Semiconducting nanowires proximitized by superconductors with multiple tunnel probes, as in InAs/Al nanowires with side or end contacts. Local and nonlocal conductances are measured via lock-in detection by driving AC biases at separate terminals (Pöschl et al., 2022).
Quantum Dot Platforms: Double quantum dot devices in which tunneling through one quantum dot is influenced by the quantum state or bias conditions of a distant dot, enabling measurement of both local and nonlocal differential conductances (1311.0659).
Planar Josephson Junctions: Epitaxial hybrid junctions with quantum point contacts at each end to form a 2×2 conductance matrix, systematically mapping out gap closing/reopening events and the emergence of correlated subgap features (Banerjee et al., 2022).
Near-Field Microscopy: Optical or microwave probes scan over a two-dimensional electron system (2DES), with the height-dependent modulation of the probe's induced dipole capturing the nonlocal response of the 2DES. The probe elevation $z_0$ selects Fourier components of the nonlocal conductivity kernel $\sigma(q, \omega)$ , effectively sampling electron dynamics at various length scales (Khavronin et al., 2022).
Molecular Architectures: STM-based assembly and measurement of molecular clusters, where nonlocal dI/dV mapping reveals spatially extended electron correlations, Coulomb blockade, and negative differential conductance—crucially dependent on inter-molecular capacitance and local environment (Li et al., 7 Aug 2025).

Theoretical analysis of these setups is typically carried out using combinations of the scattering matrix formalism, nonequilibrium Green's functions, rate-equation master equations, and exact diagonalization of effective Hamiltonians (e.g., Kondo, Anderson impurity, or Bogoliubov–de Gennes frameworks).

3. Probing Quantum Coherence, Bound-State Nonlocality, and Topological Phases

Nonlocal differential-conductance spectroscopy is uniquely sensitive to the spatial character and quantum correlations of subgap and in-gap states:

Majorana Zero Modes (MZMs): In hybrid superconducting nanostructures, spatially separated MZMs produce correlated zero-bias peaks (ZBCPs) in end-to-end conductance measurements. Crucially, nonlocal measurements can distinguish true topological MZMs (showing correlated, symmetric conductance at both ends) from trivial Andreev bound states (ABSs), which lack such nonlocal correlations (Lai et al., 2019, Dourado et al., 2023).
Distinguishing Trivial and Topological States: Nonlocal conductance, being sensitive to the bulk properties and spatial extension of subgap states, readily discerns between inhomogeneous, localized zero-energy states, and states associated with gap closure across the whole system (topological phase transition) (Rosdahl et al., 2017, Kurilovich et al., 14 Sep 2024).
Fano Interference and Quantum Interference in Kondo Systems: In Kondo lattice systems, the quantum interference between tunneling into conduction and localized $f$ -states produces Fano lineshapes in the differential conductance, the analysis of which enables extraction of the Kondo exchange coupling and tunneling amplitude ratios (Figgins et al., 2010).
Andreev Bound State Nonlocality and BCS Charge: Local and nonlocal tunneling measurements in multiterminal geometries can quantitatively extract the electron–hole character ("BCS charge") of ABSs. Oscillations or sign changes in nonlocal conductance as a function of gate or bias track the crossover between electron-like and hole-like states, and expose the internal charge evolution of ABSs (Pöschl et al., 2022, Danon et al., 2019).
Negative Differential Conductance via Correlation Effects: In molecule-scale devices, NDC emerges nonlocally due to strong inter-site Coulomb blockade and capacitive interaction, as visualized by spatial conductance mapping and explained using tri-impurity Anderson models (Li et al., 7 Aug 2025).
Nonlocal Conductance as a Fingerprint of Andreev and Crossed Processes: In FM/SC heterostructures, overlapping ABSs at spatially separated interfaces mediate both crossed Andreev reflection (CAR) and elastic cotunneling (EC), leading to asymmetries and peak splittings in the nonlocal conductance that depend on interface separation and magnetization (Metalidis et al., 2010).

4. Symmetry Relations and Theoretical Constraints

Symmetry considerations play a central role in interpreting nonlocal differential-conductance measurements:

Particle–Hole Symmetry and Antisymmetry in Bias: For many superconducting and topological systems, nonlocal conductance matrix elements are odd functions of bias, $G_{RL}(V) = -G_{RL}(-V)$ , a direct reflection of the underlying particle–hole symmetry (Rosdahl et al., 2017, Kurilovich et al., 14 Sep 2024).
Microscopic and Geometric Symmetries: Conductance matrices in multiterminal setups obey constraints from microreversibility, antiunitary symmetries, and device geometry (such as mirror symmetry), which can be exploited to isolate transmission coefficients (normal, Andreev, and crossed) and to diagnose the relative strength and orientation of spin–orbit interactions (Maiani et al., 2022).
Conductance Matrix Decomposition: The antisymmetric and symmetric components of local and nonlocal conductance elements reveal detailed information about the charge admixture in bound states and the underlying scattering processes (Danon et al., 2019).
Topological Transition and Critical Behavior: Near the topological transition, the localization length of propagating quasiparticles in a disordered Majorana nanowire diverges logarithmically, $l(E) \propto \ln(1/E)$ , leading to a distinctive scaling in the suppression of nonlocal conductance with system length and permitting an operational definition for the width of the critical region (Kurilovich et al., 14 Sep 2024).

5. Applications: Spectroscopic Diagnosis and Quantum Device Control

Nonlocal differential-conductance spectroscopy serves as a key experimental and diagnostic tool in quantum materials research:

Discriminating Topological Phases: By providing bulk- and edge-sensitive spectroscopic data, nonlocal conductance spectroscopy can definitively distinguish Majorana modes from trivial states, map out the phase diagram of hybrid superconductors, and directly observe gap closings and reopening events associated with topological transitions (Banerjee et al., 2022, Lai et al., 2019).
Extracting Bulk Superconducting Parameters: Systematic measurement of the bias and length dependence of nonlocal conductance offers quantitative access to the induced gap, coherence length, and Thouless energy in proximitized nanostructures (Rosdahl et al., 2017).
Characterizing Electron Correlations and Quantum Interference: In both Kondo systems and molecular assemblies, analysis of nonlocal dI/dV lineshapes quantifies the role of quantum interference, electron–electron interaction, capacitive coupling, and cluster topology in shaping the device's electronic response (Figgins et al., 2010, Li et al., 7 Aug 2025).
Identifying Charge and Spin Textures: The local BCS charge of bound states, accessed through the ratio of symmetric and antisymmetric conductance components, pinpoints the electron–hole distribution at different spatial locations, crucial for understanding ABS and MBS physics (Danon et al., 2019, Pöschl et al., 2022).
Quantum Device Engineering: Nonlocal manipulation of conductance properties, e.g., tuning barrier asymmetry to modulate both ends of a Majorana wire or gate-induced switching of current through a quantum dot coupled to a Majorana, provides a route toward programmable topological transistors and single-electron control in nanoscale devices (Dourado et al., 2023).

6. Advanced Topics: Nonlocality Beyond Superconductors and High-Resolution Probes

The scope of nonlocal differential-conductance spectroscopy extends beyond hybrid superconductors:

Optical Probes of Nonlocal Response: Near-field microscopy, leveraging the variable height of a probe above a 2DES, provides access to the Laplace transform of the nonlocal conductivity kernel. Analysis of the height dependence disentangles Drude, hydrodynamic, and ballistic transport regimes via their distinct scaling laws and dependence on carrier density (Khavronin et al., 2022).
High-Temperature and Highly Disordered Systems: Nonlocal differential-conductance methodologies enable probing of Andreev reflection and gap extraction in granular or inhomogeneous superconductors—regimes where conventional local spectroscopies are ineffective or ambiguous (Gheorghiu, 2022).
Correlation-Controlled NDC and Nanoelectronics: Assembly of molecule-scale clusters produces nontrivial nonlocal conductance features, including NDC and programmable switching, opening new directions in molecular nanoelectronics (Li et al., 7 Aug 2025).

7. Limitations, Challenges, and Future Directions

Key limitations and ongoing research directions include:

Finite-Size and Coherence Effects: In one-dimensional or molecular systems, finite-size effects and quantum coherence between zero-mode and bulk states can lead to deviations from quantized conductance plateaus, overshoots, and detailed lineshape modifications—features essential to correct interpretation (Yao et al., 2021).
Breakdown of Symmetry in Nonideal Regimes: Quasiparticle leakage, dissipation, or voltage-bias–dependent deformation of the electrostatic potential can break symmetry relations and generate backgrounds or systematic errors in conductance matrices (Maiani et al., 2022).
Disorder and Criticality: The detailed energy and length scaling of nonlocal conductance under strong disorder, including infinite-randomness critical points and transitions between trivial and topological regimes, demand advanced theoretical modeling and careful experimental design (Kurilovich et al., 14 Sep 2024).
Integration with Quantum Information Architectures: The nonlocal control and detection of topological modes position nonlocal differential-conductance spectroscopy at the heart of experimental roadmaps for topological quantum computation, where definitive characterization of qubits requires the ability to unambiguously discern nonlocality and topological protection (Béri, 2011, Pöschl et al., 2022).

Nonlocal differential-conductance spectroscopy thus constitutes a versatile and rigorous probe in condensed matter physics, with the unique capacity to resolve spatial correlations, topological features, electron–hole quantum coherence, and the fundamental symmetry-encoded nature of low-energy excitations in quantum materials and artificial nanostructures.