Topologically Charged Nodal Surface Semimetal

• Topologically charged nodal surface semimetals are 3D materials where energy bands cross along a two-dimensional momentum surface, carrying quantized topological charges.
• Theoretical models using low-energy Hamiltonians and BdG formalism detail how interface matching and non-symmorphic symmetries produce unique scattering, Andreev reflection, and conductance oscillations.
• Experimental signatures, such as π-period oscillations in electrical and thermal transport, distinguish NSSMs from conventional Dirac and nodal-line semimetals in topological junctions.

A topologically charged nodal surface semimetal (NSSM) is a class of three-dimensional materials characterized by energy bands that cross along a two-dimensional surface in momentum space, rather than at discrete points (as in Weyl semimetals) or along one-dimensional lines (as in nodal-line semimetals). These nodal surfaces are protected by crystalline (generally non-symmorphic) symmetries and carry quantized topological charges, endowing NSSMs with unique band topology and transport phenomena. The interplay of topology and band geometry in such systems directly influences electronic, thermal, and proximity-induced superconducting properties, making NSSMs central in the study of topological matter and their junctions with insulators and superconductors (Pandit et al., 25 Jan 2026).

1. Band Structure and Topological Characterization

The minimal continuum low-energy Hamiltonian for a three-dimensional NSSM centered at a band-crossing point $\mathbf{k}_0 = (0,0,\pi)$ is

$$H_{\rm NSSM}(\mathbf{q}) = q_z(q_x \sigma_x + q_y \sigma_y) + q_z \sigma_z,$$

where $\mathbf{q}$ is the momentum measured from $\mathbf{k}_0$, and $\sigma_{i}$ ($i = x,y,z$) are Pauli matrices in orbital pseudospin space. Eigenvalues are

$$E_{\pm}(\mathbf{q}) = \pm q_z \sqrt{1 + q_\rho^2}, \qquad q_\rho^2 = q_x^2 + q_y^2.$$

The “nodal surface” is defined by $q_z = 0$, forming a two-dimensional manifold in the Brillouin zone where the conduction and valence bands touch. Crucially, this band crossing exhibits linear $q_z$ and quadratic $q_{x,y}$ dispersions. The stability of the nodal surface reflects underlying non-symmorphic symmetry and supports a $\mathbb{Z}$-valued topological charge corresponding to the winding of the pseudospin texture along loops that link the nodal surface (Pandit et al., 25 Jan 2026).

2. Junction Architectures and Theoretical Modeling

Transport in NSSM-based heterostructures is typically probed using planar junctions, for example, in a NSSM–Insulator–Superconductor (NSSM–I–SC) geometry. The theoretical description employs a Bogoliubov–de Gennes (BdG) formalism: $$H_{\rm BdG}(x,\mathbf{q}) = [H_{\rm NSSM}(\mathbf{q}) + U(x) - \mu_N]\tau_z + \Delta(x) \tau_x,$$ with spatially piecewise-defined potentials and superconducting pairing $\Delta(x)$. The insulator region is modeled as a thin barrier of width $d$ and height $V_0$, characterized in the limit $V_0 \to \infty$, $d \to 0$ with barrier strength $\chi = V_0 d / \hbar v_z$ held fixed. The electron and hole-like excitations in the NSSM, their interface matching, and the unique scattering amplitudes are central for analyzing transport across the junction (Pandit et al., 25 Jan 2026).

3. Electrical and Thermal Transport Phenomena

Distinctive features arise in electrical and thermal conductance in NSSM-based heterostructures:

• Scattering and Reflection Amplitudes: The structure of the NSSM spectrum, along with the thin-barrier limit, yields closed-form Andreev and normal reflection amplitudes that are periodic in $\chi$. For instance,

$$r(\theta) = \frac{e^{i\chi} \alpha_e - e^{-i\chi} \alpha_h \cos \gamma - 2i u v \sin\chi \cos\theta} {e^{i\chi} \alpha_e + e^{-i\chi} \alpha_h \cos \gamma + 2i u v \sin\chi \cos\theta},$$

and

$$r_A(\theta) = \frac{2 u v \cos \theta} {e^{i\chi} \alpha_e + e^{-i\chi} \alpha_h \cos \gamma + 2i u v \sin\chi \cos\theta},$$

where coherence factors $u,v$ and angular parameters are derived from the band structure and interface geometry.

• Conductance Oscillations: The differential tunneling conductance $G(eV)/G_N$ calculated via a generalized Blonder–Tinkham–Klapwijk (BTK) formula exhibits a universal $\pi$-periodic oscillation in $\chi$,

$\frac{G(0)}{G_N} \sim \frac{1}{1 + \sin^2 \chi},$

a qualitative distinction from graphene or silicene NIS junctions, which exhibit $\pi/2$ periodicity, and conventional NIS junctions where such oscillations are absent.

• Thermal Conductance: The thermal current, given by

$$\kappa(T) = \int_0^\infty dE \int_{-\pi/2}^{\pi/2} [1 - |r|^2 - (\cos\theta_A/\cos\theta)|r_A|^2] \frac{E^2}{4T^2 \cosh^2(E/2T)} \cos\theta d\theta,$$

also displays $\pi$-periodic oscillations as a function of $\chi$, robust across doping regimes.

These behaviors have no analog in standard NIS systems and result from the unique topology and dispersion anisotropy of the NSSM band structure (Pandit et al., 25 Jan 2026).

4. Comparison with Other Topological Junctions

The NSSM–I–SC junction's transport responses exhibit sharply different features compared to those based on graphene and silicene:

System	Conductance Oscillation Period	Maximal Conductance at $\chi=0$	Dispersion
NSSM–I–SC	$\pi$	Maximal	Linear–Quadratic
Graphene–NIS	$\pi/2$	Minimal	Dirac (linear)
Silicene–NIS	$\pi/2$	Minimal	Dirac (linear)

The $\pi$-periodicity in NSSM contrasts with the $\pi/2$ oscillations found in Dirac material-based junctions, highlighting the effect of the nodal-surface topology on quasiparticle interference and Andreev reflection.

5. Physical Mechanisms and Implications

The oscillatory transport and the angle/energy dependence of Andreev processes in NSSM-based junctions arise from the interplay of linear–quadratic band dispersion and the topological charge of the nodal surface. The matching conditions at interfaces, band anisotropy, and the role of Fermi surface mismatch (tuned via potentials $U_0$) impact both normal and Andreev reflection probabilities, enabling novel control over thermal and electrical transport. These findings suggest experimental routes to manipulate transport in topological materials distinct from conventional or Dirac systems (Pandit et al., 25 Jan 2026).

6. Experimental Considerations and Prospects

While the presented models and predictions are primarily theoretical, the identified universal oscillations and anisotropic responses provide experimental benchmarks for discriminating NSSMs from other nodal and Dirac systems. The NSSM–I–SC geometry, in particular, enables systematic studies of how topological nodal surfaces modify proximity-induced superconductivity and heat flow at the mesoscopic scale. Application directions include tunable nanoscale junctions for topological quantum devices and spectroscopy of novel quasiparticle phenomena accessible via electrical and thermal probes (Pandit et al., 25 Jan 2026).