Predictive surface-bulk transport in confined topological semimetals

Problem statement

Develop a quantitatively predictive theory of longitudinal transport in confined topological semimetals when mobile surface states exchange carriers with the bulk. The target is a film or wire whose dimensions, interfaces, temperature and measured transport observable are specified, with finite intervalley relaxation and boundary scattering. A useful theory must predict the thickness dependence of resistivity and identify which surface and bulk currents produce it. For a long diffusive film of thickness $d$, distinguish its sheet conductance $G(d)$ and resistivity $\rho(d)=d/G(d)$. The condition for a resistivity extremum, $G-dG'=0$, differs from equality of the separately defined surface and bulk currents. For a wire, retain its actual cross-sectional area and perimeter instead of replacing them by an unspecified film thickness. Contacts, ballistic transmission, band reconstruction, surface overlap and amorphous or crystalline structure must be treated in their applicable regimes. The difficult step is independent prediction rather than an unconstrained parallel-channel fit. Boundary capture and reemission must conserve charge and respect the applicable equilibrium reciprocity. Surface density of states, velocities, bulk mean free paths and scattering parameters need independently defensible constraints. If available observations cannot identify a requested crossover or distinguish mechanisms, characterize that nonidentification and specify an additional observable that can test it. The broader motivation is a transport theory that identifies material and geometry regimes capable of avoiding the resistivity penalty of conventional interconnect scaling. This challenge addresses the coupled semimetal part of that objective. Restricted derivations and conditional comparisons are useful contributions; they do not by themselves establish predictive validity across materials.

Current state

Classical size-effect theory is established, as reviewed by Sondheimer. Durand and colleagues solve a finite-thickness Boltzmann problem for band-geometric nonlinear transport while omitting mobile topological surface states and nonnegligible intervalley scattering from their stated treatment. Their linear transport framework supplies one starting point for a coupled extension. Coupled surface and bulk transport is already prior work. Gorbar and colleagues analyze dissipative Fermi-arc transport; Breitkreiz and Brouwer derive surface contributions with exchange and relaxation between sectors; Buccheri and colleagues formulate coupled bulk/arc phonon transport with explicit scattering processes. The unresolved target is therefore not the generic existence of surface conduction, a two-channel sum, or a thickness crossover. Lien and colleagues provide a theoretical CoSi thickness benchmark combining realistic band structure and Green-function transport, with a surface/bulk partition whose domain must be respected. Kumar and colleagues study surface-dominated conductance in NbAs slabs; their ballistic resistance-area quantities and unequal surface terminations require an explicit bridge before comparison with diffusive resistivity. Lim and colleagues report ultrathin amorphous TaP transport, and Roy and colleagues report TaSiAs nanowires and belts. These are different material classes and observables, not interchangeable realizations of one ideal Weyl model. A predictive solution must connect conserving boundary kinetics to specified material and geometric inputs, preserve the distinction between measurements and theoretical benchmarks, and test parameters independently of the same thickness curve used to demonstrate agreement. Missing supplements, unresolved source normalization, unmeasured geometry and unknown microscopic rates remain explicit gaps. A numerically consistent restricted law or a fit with as many adjustable coefficients as observations does not close those gaps.

Resolution criteria

1. Derive a charge-conserving coupled bulk/surface linear response for a specified Weyl film or wire, including finite intervalley and boundary scattering. State the collision operators, surface normalization, probability constraints, reciprocal gain/loss terms and physical validity domain. Recover the applicable classical and decoupled limits and independently check the unreduced governing equations. 2. Obtain $\rho(d)$ and its thickness derivative in closed form or through explicitly evaluable one-dimensional integrals in the declared geometry. Define the requested crossover precisely. Treat current equality separately from a resistivity extremum, and state existence, multiplicity and parameter-identifiability conditions. If a crossover is not identifiable, provide a constructive ambiguity or rigorous bound and a discriminating observable. 3. Compare against source-verified NbAs, TaP, TaSiAs and CoSi evidence, distinguishing experimental measurements from theoretical benchmarks and respecting actual geometry, temperature, interfaces, source versions and observable normalization. Show compatible predictions or explain quantitative incompatibility within the model's stated domain. Keep unavailable comparisons open until the required evidence is obtained. A different theoretical source does not replace a missing experimental dataset. 4. Establish at least one quantitative material prediction with parameters constrained independently of the observations used to test it. Report uncertainty, calibration choices, residual degrees of freedom, rejected parameter sets and out-of-domain extrapolations. Validate a prediction or a proposed discriminating observable against additional measured evidence with sufficient source detail to reproduce the comparison. 5. Provide the derivation, source identities, extraction procedure, executable calculation and complete positive and negative results for independent reproduction. A paper may make a partial contribution while these criteria remain open; a claim of resolution requires evidence addressing every criterion above.

Citations

1. Robin Durand, Louis-Thomas Gendron, Théo Nathaniel Dionne, Ion Garate. Nonlinear longitudinal current of band-geometric origin in wires of finite thickness. 2024.
2. Maxim Breitkreiz, Piet W. Brouwer. Large contribution of Fermi arcs to the conductivity of topological metals. 2019.
3. Francesco Buccheri, Alessandro De Martino, Rodrigo G. Pereira, Piet W. Brouwer, Reinhold Egger. Phonon-limited Transport and Fermi Arc Lifetime in Weyl Semimetals. 2021.
4. Shang-Wei Lien, Ion Garate, Utkarsh Bajpai, Cheng-Yi Huang, Chuang-Han Hsu, Yi-Hsin Tu, Nicholas A. Lanzillo, Arun Bansil, Tay-Rong Chang, Gengchiau Liang, Hsin Lin, Ching-Tzu Chen. Unconventional Resistivity Scaling in Topological Semimetal CoSi. 2022.
5. Sushant Kumar, Yi-Hsin Tu, Luo Sheng, Nicholas A. Lanzillo, Tay-Rong Chang, Gengchiau Liang, Ravishankar Sundararaman, Hsin Lin, Ching-Tzu Chen. Surface-dominated conductance scaling in Weyl semimetal NbAs. 2022.
6. E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, P. O. Sukhachov. Origin of dissipative Fermi arc transport in Weyl semimetals. 2016.
7. Dong-Hyun Lim, Young-Min Song, Yeji Kim, Ae Rim Choi, Hyun-Mi Kim, Hyeongkeun Kim, Sujin Kwon, Bonggeun Shong, Justin Shih, Asir Intisar Khan, Il-Kwon Oh. Scale-robust Low Resistance Transport in Atomic Layer Deposited Topological Semimetal Wafers on Amorphous Substrate. 2025.
8. Anand Roy, Ofek Goldreich, Guy Ohad, Barun Barick, Olga Brontvein, Ora Bitton, Katya Rechav, Yishay Feldman, Leeor Kronik, Ernesto Joselevich. Square Net TaSiAs Nanowires with Topological Surface Conduction and Linear Magnetoresistance. 2026.
9. E.H. Sondheimer. The mean free path of electrons in metals. 1952.

Linked papers

• Nonlinear longitudinal current of band-geometric origin in wires of finite thicknessrelated21 Sept 2026