新型拓樸外爾半金屬由於其由塊材能帶結構的拓撲結構引起的特殊量子特性而在表面上具有強大的載子傳輸性質。費米弧態對電導率有很大貢獻。我們使用手性拓樸半金屬 CoSi 作為模型系統來展示表面雜質效應下的電阻率縮放行為。由於第一原理計算電阻率的成本很高，我們開發了一種數值方法來計算費米黃金法則(Fermi golden rule)和久保公式(Kubo formula)中的 T 矩陣。我們的結果表明存在一個臨界厚度 ?? 劃分縮放趨勢。在 ?? 之上，我們證明了在相當低的表面雜質密度下，CoSi中的表面的載子主導傳輸導致電阻率隨著厚度的減小而降低。這與銅等傳統金屬形成對比，其中由於載流子從表面、晶界等散射，電阻率隨著尺寸的減小而增加。然而，在 ?? 以上，在高表面雜質密度下，由於強大的表面缺陷散射效應，CoS的電阻率隨著厚度的減小而增加。在這種情況下，CoSi的電阻率縮放行為變成了傳統金屬(如金屬銅)。低於??時，由於費米弧態的殘餘成為主要載流子通道，而與缺陷密度無關，電阻率隨著厚度的減小而降低。

Novel topological Weyl semimetals possess robust transport properties on surfaces due to their exotic quantum properties induced by the topology of the bulk band structure. Fermi-arc states provide a significant contribution to conductivity. We use CoSi, the chiral topological semimetal, as a model system to demonstrate resistivity scaling behaviors under the surface impurity effect. Because it is expensive to calculate the resistivity on the first principle method, we develop a numerical method to calculate the T-matrix from Fermi’s golden rule and Kubo's formula. Our results reveal that there exists a critical thickness dc dividing the scaling trends. Above dc, we demonstrate that the surface-dominated transport in CoSi in a quite low surface impurity density gives decreasing resistivity with reduced thickness. This contrasts with conventional metals such as copper, where resistivity increases with decreasing size due to carrier scattering from surfaces, grain boundaries, etc. However, above dc, at high surface impurity densities, the resistivity of CoSi increases with decreasing thickness due to the strong surface defect scattering effect. In this case, The resistivity scaling behavior of CoSi is the same as that of conventional metals. Below dc, the resistivity decreases with decreasing thickness because the electrons on the remnants Fermi arc become the dominant carriers regardless of defect density.

1. Kane, C. L. & Mele, E. J. Z2 Topological Order and the Quantum Spin Hall Effect. Phys. Rev. Lett. 95, 146802 (2005).
2. Hu, C. et al. Realization of an intrinsic ferromagnetic topological state in MnBi8Te13. Science Advances 6, eaba4275 (2020).
3. Yang, L. X. et al. Weyl semimetal phase in the non-centrosymmetric compound TaAs. Nature Phys 11, 728–732 (2015).
4. Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
5. Xu, S.-Y. et al. Experimental discovery of a topological Weyl semimetal state in TaP. Science Advances 1, e1501092 (2015).
6. Xu, S.-Y. et al. Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide. Nature Phys 11, 748–754 (2015).
7. Chang, C.-Z. et al. Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator. Science 340, 167–170 (2013).
8. Zhang, C. et al. Ultrahigh conductivity in Weyl semimetal NbAs nanobelts. Nat. Mater. 18, 482–488 (2019).
9. Takane, D. et al. Observation of Chiral Fermions with a Large Topological Charge and Associated Fermi-Arc Surface States in CoSi. Phys. Rev. Lett. 122, 076402 (2019).
10. Chang, G. et al. Unconventional Chiral Fermions and Large Topological Fermi Arcs in RhSi. Phys. Rev. Lett. 119, 206401 (2017).
11. Tang, P., Zhou, Q. & Zhang, S.-C. Multiple Types of Topological Fermions in Transition Metal Silicides. Phys. Rev. Lett. 119, 206402 (2017).
12. Chang, G. et al. Topological quantum properties of chiral crystals. Nature Mater 17, 978–985 (2018).
13. Souma, S. et al. Direct observation of nonequivalent Fermi-arc states of opposite surfaces in the noncentrosymmetric Weyl semimetal NbP. Phys. Rev. B 93, 161112 (2016).
14. Gall, D. The search for the most conductive metal for narrow interconnect lines. Journal of Applied Physics 127, 050901 (2020).
15. Chen, C.-T. et al. Topological Semimetals for Scaled Back-End-Of-Line Interconnect Beyond Cu. in 2020 IEEE International Electron Devices Meeting (IEDM) 32.4.1-32.4.4 (2020). doi:10.1109/IEDM13553.2020.9371996.
16. Tsai, C.-I. et al. Cobalt Silicide Nanostructures: Synthesis, Electron Transport, and Field Emission Properties. Crystal Growth & Design 9, 4514–4518 (2009).
17. Atomistic Simulation Software | QuantumATK - Synopsys. https://www.synopsys.com/silicon/quantumatk.html.
18. Yu, R., Qi, X. L., Bernevig, A., Fang, Z. & Dai, X. Equivalent expression of Z2 topological invariant for band insulators using the non-Abelian Berry connection. Phys. Rev. B 84, 075119 (2011).
19. Garnett, W. B. semi-infinite Green’s function. Surface states of ternary semiconductor alloys: Effect of alloy fluctuations in one-dimensional models with realistic atoms 31, 8 (1995).
20. G., P. Wannier. Phys. Cond. Matt. 32, (2020).
21. E, F. Fermi Golden rule. Nuclear Physics ISBN 978-0226243658, (1950).
22. Bruus, H. & Flensberg, K. Many-Body Quantum Theory in Condensed Matter Physics: An Introduction. (OUP Oxford, 2004).
23. Kaasbjerg, K. Atomistic $T$-matrix theory of disordered two-dimensional materials: Bound states, spectral properties, quasiparticle scattering, and transport. Phys. Rev. B 101, 045433 (2020).
24. B.A., L. & Julian, S. Lippmann-Schwinger. Phys. Rev. Lett. 79(3), 469–480 (1950).
25. Kubo, R. Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems. J. Phys. Soc. Jpn. 12, 570–586 (1957).
26. Jian-Hao, C. & W.G., C. Graphene. PRL 102, (2009).
27. VASP - Vienna Ab initio Simulation Package. https://www.vasp.at/.
28. YU, P. & Cardona, M. Fundamentals of Semiconductors: Physics and Materials Properties. (Springer, 2010).
29. Sondheimer, E. H. The mean free path of electrons in metals. Advances in Physics 1, 1–42 (1952).
30. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, 3865–3868 (1996).