Aharonov-Bohm effect and Aharonov-Bohm oscillations: from theories to experiments

Aharonov-Bohm effect and Aharonov-Bohm oscillations:from theories to experiments Zhongpeng Sun 2018/04/23

Aharonov-Bohm effect and Aharonov-Bohm oscillations:from theories to experiments • Theories: • Path integral theory • A-B effect • Weak localization and weak anti-localization • Altshuler–Aronov–Spivak effect • Experiments: • Aharonov-Bohm oscillations in carbon nanotubes • Aharonov-Bohm oscillations in topological insulator (Bi1.33Sb0.67)Se3 nanowires • Aharonov-Bohm oscillations in Dirac semimetal Cd3As2 nanowires

Path integral theory Propagator（传播子）： From Schrodinger Equation , In coordinate image , it can be represented as if we define Propagator . 曽谨言，量子力学.

Path integral theory The physical meaning of propagator: If we let , = If we symbolize the initial coordinate eigenvalue with in stead of ,

Path integral theory Supposing a particle is emitted from A, consider the probability P(B,A) that we find the particle at B. In the classical situation, In the quantum situation, The probability amplitude , then =

Path integral theory Under the hypothesis of infinite screens with infinite holes, The probability amplitude ), The basic assumption of Feynman is constructing the propagator as below: Here, is the action of the particle andis the lagrangian of the particle Therefore,

A-B effect According to Path Integral Theory, if there is no magnetic field. If there is a solenoid as shown in the figure, Here, Therefore,

A-B effect Considering , Here, is the magnetic flux through enclosed path. If we define magnetic flux quantum , The periodic interference fringes have been observed in the experiments.

Weak localization Under the quasi-classical condition , we can neglect the interference of scattered electron waves propagating along different paths and having approximately random phases. , =.

Weak localization However, there is aspecific class of trajectories, namely, self-crossing trajectories for which the wave interference turns out to be essential. Because of time reversal symmetry, two waves propagating along such trajectories in two opposite directions (conjugated waves) accumulate the same phase difference and amplitude. Therefore, at point O, which is twice the sum of the squared amplitude moduli. Thus the interference of conjugated waves favors particle localization and, hence, results in an increase of resistivity.(weak localization) Bergmann, G. (1984). Physics Reports, 107(1), 1-58

Weak localization and weak anti-localization If there are inelastic processes in trajectories, the weak localization will be broken. Conductance Take spins into consideration, Spin-flipping: similar with inelastic scattering. Spin-orbit interaction: lead to which is half the sum of the squared amplitude moduli.(weak anti-localization) Abrahams, E., Anderson, et al.(1979). Physical Review Letters, 42(10), 673. Van Haesendonck, C. et al. (1981) Physical Review Letters, 46(8), 565. Bergmann, G. (1984). Physics Reports, 107(1), 1-58

Altshuler–Aronov–Spivakeffect Conjugated waves propagating along the circumference of the cylinder acquire additional phase factors: magnetic flux quantum . This means that the cylinder resistivity will oscillate with the period .(AAS effect)

Altshuler–Aronov–Spivakeffect Aronov, A. G., & Sharvin, Y. V. (1987). Magnetic flux effects in disordered conductors. Reviews of modern physics, 59(3), 755. Al’tshuler, B. L., Aronov, A. G., & Spivak, B. Z. (1981). The Aaronov-Bohm effect in disordered conductors. Jetp Lett, 33(2), 94.

Aharonov-Bohm oscillations in carbon nanotubes

Aharonov-Bohm oscillations in carbon nanotubes Thus Comparing with the outer radius

Aharonov-Bohm oscillations in carbon nanotubes The outer radius . Long period: Minimum ~9T Second maximum ~18T Short period:

Aharonov-Bohm oscillations in carbon nanotubes Where may the “fast” magneto-oscillation come from? Winding number n>1. Why is particular winding number selected? Chiral currents in strained nanotube? Further experimental and theoretical studies are needed for its detailed understanding.

Aharonov-Bohm oscillations in topological insulator (Bi1.33Sb0.67)Se3 nanowires Cho, S., ... & Mason, N. (2015). Nature communications, 6, 7634. Cho, S., ... & Mason, N. (2013). Symmetry protected Josephson supercurrents in three-dimensional topological insulators. Nature communications, 4, 1689.

Aharonov-Bohm oscillations in topological insulator (Bi1.33Sb0.67)Se3 nanowires Rosenberg, G., Guo, H. M., & Franz, M. (2010). Physical Review B, 82(4), 041104. Hong, S. S….& Cui, Y. (2014). One-dimensional helical transport in topological insulator nanowire interferometers. Nano letters, 14(5), 2815-2821.

Aharonov-Bohm oscillations in topological insulator (Bi1.33Sb0.67)Se3 nanowires

Aharonov-Bohm oscillations in topological insulator (Bi1.33Sb0.67)Se3 nanowires Hong, S. S….& Cui, Y. (2014). One-dimensional helical transport in topological insulator nanowire interferometers. Nano letters, 14(5), 2815-2821.

Aharonov-Bohm oscillations in topological insulator (Bi1.33Sb0.67)Se3 nanowires

Aharonov-Bohm oscillations in Dirac semimetal Cd3As2 nanowires Wang, L. X., Li, C. Z., Yu, D. P., & Liao, Z. M. (2016). Aharonov–Bohm oscillations in Dirac semimetal Cd 3 As 2 nanowires. Nature communications, 7, 10769.

Aharonov-Bohm oscillations in Dirac semimetal Cd3As2 nanowires

Conclusion • A-B effect is a quantum effect that reflects the significance of the electromagneticpotential. • A-B oscillation is a robust transport phenomenon in cylinder-like nano-structures, like carbon nanotubes and topological insulator nanowires. • A-B oscillations can be tuned by the angle of the magnetic field, gate voltage, etc. Thank you for your listening!

Aharonov-Bohm effect and Aharonov-Bohm oscillations: from theories to experiments