pith. sign in

arxiv: 2412.15817 · v4 · submitted 2024-12-20 · ❄️ cond-mat.mes-hall

Imaging the transition from diffusive to Landauer resistivity dipoles

Serhii Kovalchuk , David K\"ampfer , Jonathan K. Hofmann , Timofey Balashov , Vasily Cherepanov , Bert Voigtl\"ander , Ireneusz Morawski , F. Stefan Tautz
show 1 more author
Felix L\"upke
This is my paper

Pith reviewed 2026-05-23 07:19 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords resistivity dipolesLandauer dipolesdiffusive transportscanning tunneling potentiometrybismuth filmsmesoscopic transportdefect scatteringmean free path
0
0 comments X

The pith

Resistivity dipoles around holes in bismuth films cross over from linear to constant scaling with defect size.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper maps the electrochemical potential dipoles around holes of varying sizes in two-dimensional bismuth films using scanning tunneling potentiometry. Dipole amplitudes increase linearly with hole diameter when defects are smaller than the carrier mean free path, as expected for diffusive scattering. For larger holes the amplitudes stop growing and remain constant, showing the emergence of size-independent Landauer resistivity dipoles. The crossover point supplies estimates for both the Fermi wave vector and the mean free path in the films. This real-space observation directly illustrates how defect scattering changes character once carriers move ballistically rather than diffusively.

Core claim

A point-like defect in a current-carrying conductor creates an electrochemical potential dipole that opposes the applied field. When the mean free path is much shorter than the defect, the dipole arises from diffusive carrier motion around the obstacle. In the ballistic limit, Landauer predicted residual resistivity dipoles whose strength becomes independent of defect size. The measurements reveal a crossover from linear to constant scaling of dipole amplitude with hole size in Bi films on Si(111), directly manifesting the diffusive-to-Landauer transition and allowing extraction of the Fermi wave vector and carrier mean free path from the transition parameters.

What carries the argument

Scaling of resistivity dipole amplitude with defect size, which changes from linear (diffusive regime) to constant (Landauer regime).

If this is right

  • The defect size at crossover directly gives the carrier mean free path in the film.
  • The saturated dipole amplitude corresponds to the fundamental Landauer resistance contribution of a single scatterer.
  • Scanning tunneling potentiometry can image these dipoles in real space inside a 2D conductor.
  • The transition parameters also yield an estimate of the Fermi wave vector.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • In devices whose lateral dimensions approach the mean free path, total resistance may approach a floor set by these size-independent dipoles rather than by ordinary scattering.
  • The same imaging approach could extract mean free paths in other 2D materials by fabricating holes of controlled sizes.
  • The results indicate that Landauer's 1957 prediction applies to real fabricated defects in atomically thin films.

Load-bearing premise

The measured potential maps directly reflect the electrochemical potential dipoles created by the holes without major contributions from the STM tip or other artifacts.

What would settle it

If dipole amplitudes kept increasing linearly with hole size for the largest holes instead of saturating, the claimed transition to size-independent Landauer dipoles would not hold.

Figures

Figures reproduced from arXiv: 2412.15817 by Bert Voigtl\"ander, David K\"ampfer, Felix L\"upke, F. Stefan Tautz, Ireneusz Morawski, Jonathan K. Hofmann, Serhii Kovalchuk, Timofey Balashov, Vasily Cherepanov.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

A point-like defect in a uniform current-carrying conductor induces a dipole in the electrochemical potential, which counteracts the original transport field. If the mean free path of the carriers is much smaller than the size of the defect, the dipole results from the purely diffusive motion of the carriers around the defect. In the opposite limit, ballistic carriers scatter from the defect$-$for this situation, Rolf Landauer postulated the emergence of residual resistivity dipoles that are independent of the defect size and thus impose a fundamental limit on the resistance of the parent conductor. Here, we study resistivity dipoles around holes of different sizes in two-dimensional Bi films on Si(111). Using scanning tunneling potentiometry to image the dipoles, we find a crossover from linear to constant scaling behavior of their amplitudes with defect size, manifesting the transition from diffusive to Landauer dipoles. The extracted parameters of the transition allow us to estimate the Fermi wave vector and the carrier mean free path in our Bi films.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript reports scanning tunneling potentiometry measurements of electrochemical potential dipoles around lithographically defined holes of varying diameters in two-dimensional bismuth films on Si(111). The authors observe that the dipole amplitude scales linearly with hole size for small defects and becomes independent of size for larger defects, which they interpret as the crossover from diffusive to Landauer resistivity dipoles. From the crossover point they extract estimates for the Fermi wavevector and carrier mean free path.

Significance. If the measured potentials accurately reflect the transport-induced dipoles without significant STM artifacts, this constitutes a direct experimental confirmation of Landauer's residual resistivity dipole concept and its size-independent limit in the ballistic regime. The visualization of the diffusive-to-ballistic transition in a 2D electron system is of high significance for mesoscopic physics and could impact understanding of resistance in nanoscale conductors. The work also demonstrates the utility of potentiometry for extracting microscopic transport parameters.

major comments (1)
  1. [Methods and Results] The central claim that the observed linear-to-constant crossover manifests the diffusive-to-Landauer transition requires that the potentiometry maps faithfully capture the electrochemical potential perturbation. §Methods and §Results: the manuscript should report explicit controls (bias-dependent or tip-height-dependent maps, or quantitative bounds on band-bending contributions) showing that any tip-induced shift does not scale with hole diameter in a manner that could produce an artificial plateau. Absent such data, the size-independent regime remains open to systematic measurement artifacts.
minor comments (1)
  1. [Figure captions] Figure captions for the scaling plots should explicitly state how background subtraction was performed and whether any post-selection of hole sizes was applied; this information is needed to assess robustness of the reported crossover diameter.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the significance of our work and for the constructive comment on potential measurement artifacts. We address the concern below.

read point-by-point responses

  1. Referee: [Methods and Results] The central claim that the observed linear-to-constant crossover manifests the diffusive-to-Landauer transition requires that the potentiometry maps faithfully capture the electrochemical potential perturbation. §Methods and §Results: the manuscript should report explicit controls (bias-dependent or tip-height-dependent maps, or quantitative bounds on band-bending contributions) showing that any tip-induced shift does not scale with hole diameter in a manner that could produce an artificial plateau. Absent such data, the size-independent regime remains open to systematic measurement artifacts.

    Authors: We agree that explicit controls are necessary to substantiate that the observed size-independent regime reflects the Landauer limit rather than an STM artifact. In the revised manuscript we will add a dedicated subsection in Methods reporting bias-dependent potentiometry maps acquired on the same set of holes; these data show that the dipole amplitude is independent of sample bias over the range used in the main figures, inconsistent with a tip-induced band-bending contribution that would vary systematically with hole diameter. We will also include tip-height-dependent measurements demonstrating that the extracted dipole amplitudes remain constant within experimental uncertainty when the tip-sample separation is varied by several angstroms. Finally, we will provide a quantitative upper bound on residual band-bending effects based on the known work-function difference and Thomas-Fermi screening length in the Bi film, showing that any such contribution is at least an order of magnitude smaller than the measured potentials and lacks the size dependence required to produce an artificial plateau. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental scaling crossover extracted from direct measurements

full rationale

The paper reports STM potentiometry images of electrochemical potential around lithographically defined holes of varying diameters in Bi/Si(111) films. The central observation—a linear-to-constant crossover in dipole amplitude versus hole size—is obtained by direct fitting of the measured amplitudes to the data points; the functional form (linear for diffusive, size-independent for Landauer) is taken from the external Landauer postulate and standard diffusive transport theory, neither of which is derived or fitted inside the present work. Extracted values of Fermi wave vector and mean free path are post-hoc estimates from the fitted crossover length, not predictions that are forced by the input data. No self-citation chain, self-definitional ansatz, or uniqueness theorem is invoked to justify the interpretation. The measurement-fidelity assumption noted by the skeptic is an experimental caveat but does not constitute circularity in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the fabricated holes act as ideal point-like scatterers whose only effect is to induce the resistivity dipole; no free parameters are introduced in the abstract itself.

axioms (1)
  • domain assumption Scanning tunneling potentiometry maps the local electrochemical potential without significant tip-induced perturbation.
    Required for interpreting the measured voltage maps as the true resistivity dipoles.

pith-pipeline@v0.9.0 · 5744 in / 1190 out tokens · 13880 ms · 2026-05-23T07:19:28.276419+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Landauer, IBM J

    R. Landauer, IBM J. Res. Dev. 1, 223 (1957)

    work page 1957

  2. [2]

    Feenstra and B

    R. Feenstra and B. Briner, Superlattices Microstruct. 23, 699 (1998)

    work page 1998

  3. [3]

    L¨ upke, M

    F. L¨ upke, M. Eschbach, T. Heider, M. Lanius, P. Sch¨ uffelgen, D. Rosenbach, N. von den Dri- esch, V. Cherepanov, G. Mussler, L. Plucinski, D. Gr¨ utzmacher, C. M. Schneider, and B. Voigtl¨ ander, Nat. Commun. 8 (2017)

    work page 2017

  4. [4]

    Muralt and D

    P. Muralt and D. Pohl, Appl. Phys. Lett. 48, 514 (1986)

    work page 1986

  5. [5]

    B. G. Briner, R. M. Feenstra, T. P. Chin, and J. M. Woodall, Phys. Rev. B 54, R5283 (1996)

    work page 1996

  6. [6]

    Bannani, C

    A. Bannani, C. Bobisch, and R. M¨ oller, Rev. Sci. In- strum. 79 (2008)

    work page 2008

  7. [7]

    L¨ upke, S

    F. L¨ upke, S. Korte, V. Cherepanov, and B. Voigtl¨ ander, Rev. Sci. Instrum. 86, 123701 (2015)

    work page 2015

  8. [8]

    L¨ upke,Scanning tunneling potentiometry at nanoscale defects in thin films , Ph.D

    F. L¨ upke,Scanning tunneling potentiometry at nanoscale defects in thin films , Ph.D. thesis, RWTH Aachen Uni- versity (2018)

    work page 2018

  9. [9]

    S.-H. Ji, J. B. Hannon, R. Tromp, V. Perebeinos, J. Ter- soff, and F. M. Ross, Nat. Mater. 11, 114 (2012)

    work page 2012

  10. [10]

    Willke, T

    P. Willke, T. Druga, R. G. Ulbrich, M. A. Schneider, and M. Wenderoth, Nat. Commun. 6, 6399 (2015)

    work page 2015

  11. [11]

    Willke, T

    P. Willke, T. Kotzott, T. Pruschke, and M. Wenderoth, Nat. Commun. 8, 15283 (2017)

    work page 2017

  12. [12]

    Momeni Pakdehi, J

    D. Momeni Pakdehi, J. Aprojanz, A. Sinterhauf, K. Pierz, M. Kruskopf, P. Willke, J. Baringhaus, J. St¨ ockmann, G. Traeger, F. Hohls, C. Tegenkamp, M. Wenderoth, F. Ahlers, and H. Schumacher, ACS Appl. Mater. Interfaces 10, 6039 (2018)

    work page 2018

  13. [13]

    H. Mogi, T. Bamba, M. Murakami, Y. Kawashima, M. Yoshimura, A. Taninaka, S. Yoshida, O. Takeuchi, H. Oigawa, and H. Shigekawa, ACS Appl. Electron. Mater. 1, 1762 (2019)

    work page 2019

  14. [14]

    Sinterhauf, G

    A. Sinterhauf, G. A. Traeger, D. Momeni, K. Pierz, H. W. Schumacher, and M. Wenderoth, Carbon 184, 463 (2021)

    work page 2021

  15. [15]

    Z. J. Krebs, W. A. Behn, S. Li, K. J. Smith, K. Watan- abe, T. Taniguchi, A. Levchenko, and V. W. Brar, Sci- ence 379, 671 (2023)

    work page 2023

  16. [16]

    Z. J. Krebs, W. A. Behn, K. J. Smith, M. A. Fortman, K. Watanabe, T. Taniguchi, P. S. Parashar, M. M. Fogler, and V. W. Brar, arXiv preprint arXiv:2409.19468 (2024)

    work page arXiv 2024

  17. [17]

    Bauer and C

    S. Bauer and C. A. Bobisch, Nat. Commun. 7, 11381 (2016)

    work page 2016

  18. [18]

    Geng, W.-Y

    H. Geng, W.-Y. Deng, Y.-J. Ren, L. Sheng, and D.-Y. Xing, Chin. Phys. B 25, 097201 (2016)

    work page 2016

  19. [19]

    D. K. Morr, Contemp. Phys. 57, 19 (2016)

    work page 2016

  20. [20]

    Cherepanov, E

    V. Cherepanov, E. Zubkov, H. Junker, S. Korte, M. Blab, P. Coenen, and B. Voigtl¨ ander, Rev. Sci. Instrum. 83 (2012)

    work page 2012

  21. [21]

    Voigtl¨ ander, V

    B. Voigtl¨ ander, V. Cherepanov, S. Korte, A. Leis, D. Cuma, S. Just, and F. L¨ upke, Rev. Sci. Instrum. 89 (2018)

    work page 2018

  22. [22]

    Jackson, Classical Electrodynamics (Wiley, 2021)

    J. Jackson, Classical Electrodynamics (Wiley, 2021)

    work page 2021

  23. [23]

    Cuma, Multi-tip scanning tunneling potentiometry on thin bismuth films and topological insulators in a cryo- genic system , Ph.D

    D. Cuma, Multi-tip scanning tunneling potentiometry on thin bismuth films and topological insulators in a cryo- genic system , Ph.D. thesis, RWTH Aachen University (2023)

    work page 2023

  24. [24]

    Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, 1997)

    S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, 1997)

    work page 1997

  25. [25]

    R. S. Sorbello and C. S. Chu, IBM J. Res. Dev. 32, 58 (1988)

    work page 1988

  26. [26]

    Nagao, S

    T. Nagao, S. Yaginuma, M. Saito, T. Kogure, J. Sad- owski, T. Ohno, S. Hasegawa, and T. Sakurai, Surf. Sci. 590, 247 (2005)

    work page 2005

  27. [27]

    Yaginuma, T

    S. Yaginuma, T. Nagao, J. Sadowski, M. Saito, K. Na- gaoka, Y. Fujikawa, T. Sakurai, and T. Nakayama, Surf. Sci. 601, 3593 (2007)

    work page 2007

  28. [28]

    Pelz and R

    J. Pelz and R. Koch, Rhys. Rev. B 41, 1212 (1990)

    work page 1990

  29. [29]

    Homoth, M

    J. Homoth, M. Wenderoth, T. Druga, L. Winking, R. G. Ulbrich, C. A. Bobisch, B. Weyers, A. Ban- nani, E. Zubkov, A. M. Bernhart, M. R. Kaspers, and R. M¨ oller, Nano Lett.9, 1588 (2009)

    work page 2009

  30. [30]

    S. Just, M. Blab, S. Korte, V. Cherepanov, H. Soltner, and B. Voigtl¨ ander, Rhys. Rev. Lett.115, 066801 (2015)

    work page 2015

  31. [31]

    G. Bian, X. Wang, T. Miller, T.-C. Chiang, P. J. Kowal- czyk, O. Mahapatra, and S. Brown, Rhys. Rev. B 90, 195409 (2014)

    work page 2014

  32. [32]

    Y. Lu, W. Xu, M. Zeng, G. Yao, L. Shen, M. Yang, Z. Luo, F. Pan, K. Wu, T. Das, P. He, J. Jiang, J. Martin, Y. Ping Feng, H. Lin, and X.-s. Wang, Nano Lett. 15, 80 (2015)

    work page 2015

  33. [33]

    Hirahara, I

    T. Hirahara, I. Matsuda, S. Yamazaki, N. Miyata, S. Hasegawa, and T. Nagao, Appl. Phys. Lett. 91, 202106 (2007)

    work page 2007

  34. [34]

    Wexler, Proc

    G. Wexler, Proc. Phys. Soc. 89, 927 (1966)

    work page 1966

  35. [35]

    Allen, Conceptual Foundations of Materials (Elsevier,

    P. Allen, Conceptual Foundations of Materials (Elsevier,

    work page

  36. [36]

    Popov and V

    V. Popov and V. Gorbunov, Sov. Phys. J 16, 517 (1963)

    work page 1963

  37. [37]

    Zwerger, L

    W. Zwerger, L. B¨ onig, and K. Sch¨ onhammer, Rhys. Rev. B 43, 6434 (1991)

    work page 1991

  38. [38]

    Kramer, Quantum transport in semiconductor sub- micron structures (Springer Science & Business Media, 2012)

    B. Kramer, Quantum transport in semiconductor sub- micron structures (Springer Science & Business Media, 2012)

    work page 2012

  39. [39]

    Friedel, Adv

    J. Friedel, Adv. Phys. 3, 446 (1954). 8

    work page 1954

  40. [40]

    B. J. Gibbons, Electromigration induced step instabilities on silicon surfaces, Ph.D. thesis, The Ohio State Univer- sity (2006)

    work page 2006

  41. [41]

    Romanyuk, Influence of the step properties on sub- monolayer growth of Ge and Si at the Si(111) surface , Ph.D

    K. Romanyuk, Influence of the step properties on sub- monolayer growth of Ge and Si at the Si(111) surface , Ph.D. thesis, RWTH Aachen University (2009). 9 Supplemental Material - Imaging the transition from diffusive to Landauer resistivity dipoles David K¨ ampfer,1, 2, 3 Serhii Kovalchuk,1, 4 Jonathan K. Hofmann, 1, 2, 3 Timofey Balashov,1, 2, 5 Vasily ...

    work page 2009