Nonlinear longitudinal current of band-geometric origin in wires of finite thickness

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• SoundquroreVerified by submitter+0 (0 / 0)+

  The paper solves the Boltzmann equation with Fuchs-Sondheimer boundary conditions to second order in the field for a wire of thickness $a$, and shows that the odd-in-$p_x$ part of the second-order distribution, which alone carries the longitudinal current, can be obtained without the relaxation-time approximation (SM Sec. A). That decoupling is what makes the central result, a surface-localized nonlinear current proportional to the Berry curvature, credible. I re-derived the thick-film coefficient of eq. (23) analytically ($4/3$ to first order in the tilt) and reproduced it numerically from SM eq. (S19) to within 2.4 %, reproduced the 3D thin-film law $\tfrac{3}{2}(a/l)^2\ln(l/a)$, the position of the 2D maximum at $a \approx l$, and both order-of-magnitude estimates ($\bar\sigma_2/\bar\sigma_1 \approx 6.6\times10^{-4}\,\mu$m/V and $\bar\sigma_2 \approx 1.4\times10^{-6}\, l[\mathrm{nm}]\,\Omega^{-1}\mathrm{V}^{-1}$). All eleven references I checked exist. What moved the stance is that the claims are scoped to what the derivation supports: bulk states only, lowest Born approximation, no skew scattering, and the End Matter states plainly that the predicted current coexists with bulk nonlinear currents and is hard to isolate. I found three minor internal inconsistencies (the coefficient of the 2D thin-film law in eq. (19), the sign of eq. (23)/(S20) relative to (S19) and Figs. S2-S3, and the stated value of the 3D maximum), none of which changes a physical conclusion.

  What the derivation rests on, and whether it holds

  The result depends on one structural fact: with $\varepsilon^\chi_p$ even in $p_x$, the second-order Boltzmann equation splits into an even-in-$p_x$ part with an intractable collision term and an odd-in-$p_x$ part whose collision term reduces to $-f^{\chi,o}_{2,p}/\tau$ exactly (SM eqs. (S5)-(S6)). Only the odd part enters $j_x$. The authors do not hide that the even part cannot be computed consistently; Sec. E of the SM shows that the relaxation-time approximation for it violates particle conservation, and they drop every quantity that depends on it (valley polarization, edge charge, the Son-Yamamoto inhomogeneity term). This is the right call and it is stated. The solution (11) for the odd part follows from (10) with the Fuchs-Sondheimer boundary condition by the same integration that gives the classical (8). I checked (8) and (11) by substitution.

  Numerical re-derivation of the limits and estimates

  From the LaTeX source I evaluated $I(a/l)$ of eq. (17) and the valley-resolved average of SM eq. (S19). Results: (i) $I(0^+) = 1$, so $\bar j_{2,x}\to 0$ as $a\to 0$, and $I(10\,l) = 1.5\times10^{-4}$, so $\bar j_{2,x}\to j_{2D}$ for $a\gg l$; the maximum of $(1-I)/a$ sits at $a = 0.96\,l$, as stated. (ii) The large-$a$ braces of (S19) tend to $\tfrac{4}{3}\tilde\alpha_z$ to first order in the tilt (analytically: $\int_0^1 u(1-u^2)^2\,du = 1/6$ gives $-\tfrac{2}{3}\tilde\alpha_z$, plus $2\tilde\alpha_z$ from the shifted integration limits), which reproduces the prefactor $e^3 l^2/(12 a h^2 c)$ of eq. (23) exactly; numerically the ratio is $1.334$ at $\tilde\alpha_z = 0.05$ and $1.365$ at $0.3$. (iii) The 3D thin-film law $\tfrac{3}{2}(a/l)^2\ln(l/a)$ is reproduced to 1-4 % at $a/l = 10^{-2}, 10^{-3}$. (iv) Footnote 26 with $h$ the Planck constant, $m = 100$ meV and $\alpha_z = 0.3\times10^5$ m/s gives $6.6\times10^{-4}\,\mu$m/V, consistent with the quoted $10^{-3}\,\mu$m/V; with $\hbar$ instead it would be $1.0\times10^{-4}$, so $h$ is the intended constant. (v) Eq. (23) at $a = 0.5\,l$ with $\tilde\alpha_y = 0.3$, $\tilde\alpha_z^\chi = 0.3\chi$ gives $\bar\sigma_2 \approx 1.4\times10^{-6}\, l[\mathrm{nm}]\,\Omega^{-1}\mathrm{V}^{-1}$, consistent with the quoted $10^{-6}$.

  Three minor inconsistencies

  First, the $a\ll l$ branch of eq. (19) reads $\tfrac{9}{4}(a/l)^2\ln(l/a)\,j_{2D}$. Expanding the integrand of (17) for $\sin\theta \gg a/l$ gives $1 - I \approx \tfrac{3}{8}(a/l)^2\int\cos^2\theta/\sin\theta\,d\theta \approx \tfrac{3}{4}(a/l)^2\ln(l/a)$; numerically the ratio of $1-I$ to the printed law is $0.339$, $0.342$, $0.349$ at $a/l = 10^{-3}, 10^{-2}, 5\times10^{-2}$. The coefficient is $3/4$, not $9/4$. The scaling and the conclusion $\bar\sigma_2\to0$ are unaffected. Second, evaluating (S19) as printed gives a negative $\bar j^\chi_{2,x}$ for $\tilde\alpha_y, \chi\tilde\alpha_z^\chi > 0$ (ratio to eq. (23): $-1.024$ for $a/l \ge 20$), while (S20) and eq. (23) carry a plus sign; Figs. S2 and S3 plot $-\bar j_{2,x}$ as a positive quantity, so the figures agree with (S19) and the sign of (S20)/(23) is the odd one out. Only the sign convention is affected. Third, the text states a maximum of $j_{3D}/2$ at $a \sim 0.5\,l$; from (S19) I find $|\bar j_{2,x}| = 0.30\,j_{3D}$ at $a = 0.5\,l$ and the maximum of $|\bar j_{2,x}|$ at $a \approx 0.65\,l$. Both are within the stated $\sim$, and the $\bar\sigma_2$ estimate keeps its order of magnitude.

  Scope, novelty, and uptake

  The generalization of Fuchs-Sondheimer to Berry-curved bands is new as far as I can find; earlier nonlinear-current work (Sodemann-Fu, Isobe-Xu-Fu, Tsirkin-Souza, Das et al.) treats infinite systems, and the SM Sec. F compares the result against the Son-Yamamoto inhomogeneity term and against Xiao-Yao-Fang-Niu's orbital-moment current and explains why neither cancels it. The limitations matter for the paper's own motivation: the title problem is interconnect resistivity, but the paper computes only bulk-state transport in the lowest Born approximation, with no topological surface states, no size quantization, no intervalley scattering and no skew scattering, all of which the authors list. The result therefore says nothing yet about the linear resistivity scaling of topological-semimetal wires, which is the quantity the interconnect experiments measure; the authors name that extension as future work. Uptake is low: Semantic Scholar lists 3 citing works on 2026-09-20, none building on the result. Published in Phys. Rev. Lett. 133, 226302 (2024).

  • derivationminor

    The text states that $\bar j_{2,x}$ reaches a maximum of $j_{3D}/2$ at $a \sim 0.5 l$; evaluating SM eq. (S19) gives $|\bar j_{2,x}| = 0.30\, j_{3D}$ at $a = 0.5 l$ and the maximum of $|\bar j_{2,x}|$ at $a \approx 0.65 l$.

    • https://arxiv.org/abs/2402.14112v3· text after eq. (23); SM Fig. S3
  • internal_consistencyminor

    SM eq. (S19) as printed yields $\sum_\chi \bar j^\chi_{2,x} / j_{3D} = -1.024$ for $a/l \ge 20$, $\tilde\alpha_y = 0.3$, $\tilde\alpha^\chi_z = 0.3\chi$, whereas (S20) and eq. (23) carry the opposite sign; Figs. S2 and S3 plot $-\bar j_{2,x}$ as positive and so agree with (S19). The magnitude $e^3 l^2/(12 a h^2 c)$ is confirmed.

    • https://arxiv.org/abs/2402.14112v3· eq. (23); SM eqs. (S19), (S20); Figs. S2, S3
  • referencescitation check: upheld

    Reference [28], the chiral tellurium nonlinear conductivity measurement used for the comparison after eq. (23), exists.

    Evidence · citation_lookup/2.0.0

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  • referencescitation check: upheld

    Reference [4] exists.

    Evidence · citation_lookup/2.0.0

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  • referencescitation check: upheld

    Reference [22] exists.

    Evidence · citation_lookup/2.0.0

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  • derivationminor

    The $a \ll l$ branch of eq. (19) has coefficient $9/4$; the asymptotic expansion of eq. (17) gives $1 - I \approx \tfrac{3}{4}(a/l)^2 \ln(l/a)$, and numerical evaluation of (17) gives a ratio of $0.34 \pm 0.01$ between $1-I$ and the printed law at $a/l = 10^{-3}, 10^{-2}, 5\times10^{-2}$.

    • https://arxiv.org/abs/2402.14112v3· eqs. (17), (19); SM eq. (S16)
  • referencescitation check: upheld

    Reference [7] exists; it is the CoSi resistivity-scaling paper, npj Quantum Materials 8, 3 (2023).

    Evidence · citation_lookup/2.0.0

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  Impact prediction: top 40% of 1,624 Physics, Mesoscale and Nanoscale Physics papers, 2023-08-01 to 2024-01-31

  top 1%

  What to do next

  Next step on this line

  Add a Fermi-arc surface channel to the same Boltzmann problem and compute the linear resistivity versus thickness

  Ground

  The paper's motivation is interconnect resistivity scaling, but the calculation is bulk-only; the authors name the coexistence of bulk and surface states with intervalley scattering as the open direction.

  Action

  Couple a 2D surface-state distribution to the bulk Fuchs-Sondheimer problem through a surface-to-bulk scattering rate, solve at first order in $E_x$, and derive $\rho(a)$ with the crossover thickness where surface conduction dominates.

  Expected outcome

  A closed-form or one-integral thickness law that can be fitted to the NbAs, CoSi and TaP thin-wire data with a physically bounded specularity and surface-state density.

  A different direction

  Test the prediction numerically in a lattice model instead of waiting for a transport experiment

  Ground

  The End Matter concedes that isolating the surface-localized nonlinear current from bulk skew-scattering currents is experimentally hard.

  Action

  Compute the second-order longitudinal response of a gapped Dirac lattice model with disordered edges and opposite time-reversal-breaking boundaries (the configuration proposed in the End Matter) with a Kwant or tight-binding Kubo calculation.

  Expected outcome

  A thickness-independent nonlinear conductance that appears only when the boundaries break time reversal oppositely would confirm eqs. (11)-(12) beyond the semiclassical model.

  Would change this verdict: A demonstration that the even-in-$p_x$ part of $f_2$ feeds back into the odd part through the exact collision integral beyond the lowest Born approximation, so that the decoupling in SM eqs. (S5)-(S6) fails and eq. (11) is not the full odd part. Or a calculation showing that the neglected Son-Yamamoto term at second order, or the internal field $E_z$ of SM Sec. E, cancels the surface-localized current rather than adding to it. Either would remove the central result.