Quantum Coherent Transport of 1D ballistic states in second order topological insulator Bi$_4$Br$_4$

Abstract

We investigate quantum transport in micrometer-sized single crystals of Bi$_4$Br$_4$, a material predicted to be a second-order topological insulator. 1D topological states with long phase coherence times are revealed via the modulation of quantum interference with magnetic field and gate voltage. In particular, we demonstrate the existence of Aharonov-Bohm interference between 1D ballistic states several micrometers long, that we identify as phase-coherent hinge modes on neighboring step edges at the crystal surface. These Aharonov-Bohm oscillations are made possible by a disordered phase-coherent contact region, the existence of which is confirmed by scanning transmission electron microscopy combined with energy-dispersive X-ray spectroscopy (STEM-EDX) of FIB lamellae. Their coherent nature modulates the transmission of the 1D edge states, leading to weak antilocalization and universal conductance fluctuations with surprisingly large characteristic fields and a strongly anisotropic behavior. These complementary experimental results provide a comprehensive, coherent description of quantum transport in Bi$_4$Br$_4$, and establish the material as a second-order topological insulator with topologically protected 1D ballistic states.

Figures (9)

• Figure 1: Structure of Bi$_4$Br$_4$ crystals far from and in the contact region. (a) Crystalline structure of Bi$_4$Br$_4$ and crystal orientation. Bi atoms are figured in red, and Br in blue. (b) Structure of Bi$_4$Br$_4$ along the $a$ and $b$ axis. The Z direction, perpendicular to the (a,b) plane of the flakes, is indicated. (c) High-resolution high-angle-annular dark-field (HAADF) STEM images of FIB lamellae of contacted Bi$_4$Br$_4$ flakes obtained along the $b$ direction, away from the contact area. The expected crystalline structure is superimposed on the picture. The agreement between theoretical and real atoms positions is excellent. (d) Similar image along the $a$ direction. (e) Contact region of Au/Pd (100 nm/7 nm) top-contacted electrodes, showing crystal damage both under and away from the electrode. The EDX analysis (right image) shows a diffusion of Bi in the Pd and, to a lesser extent, in the Au layers. The spurious Si signal in the gold electrode is due to the proximity of a fluorescence band of Au with the Si K$\alpha$. (f) Contact region for 7-nm-thick Pd electrodes placed below the Bi$_4$Br$_4$ flakes. Diffusion of Pd in the crystal is also visible. (g) Enlargement of the area denoted by the red dashed rectangle in (f), which evidence disordered contact regions of size around 100 nm.
• Figure 2: Anisotropic transport in Bi$_4$Br$_4$ sample $H_1$ at 10 mK and under a 500 nA dc bias. (a) Optical picture of sample $H_1$ with labeling of the electrodes. (b) Conductance matrix of sample $H_1$. The conductances are small in the transverse direction ($a$ axis of the crystal), highlighted by red hatching, and much higher, by a factor 10, in the longitudinal direction ($b$ axis of the crystal).
• Figure 3: Temperature dependence of electronic transport in Bi$_4$Br$_4$. (a) Temperature dependence of a representative Bi$_4$Br$_4$ sample, measured in a physical property measurement system. A Savitzky-Golay filter was applied to the data to get rid of the high-frequency noise. Right : optical picture of the sample used for temperature dependence. (b) Longitudinal ($R_{xx}$) and transverse ($R_{xy}$) magnetoresistance at temperature between 4 (purple) and 20 K (red) for sample $\mathcal{H}_1$. $R_{xy}^{\mathrm{antisym}}$ corresponds to the antisymmetric part of the transverse resistance and shows no Hall effect.
• Figure 4: Coherent quantum transport measurement in sample $\mathcal{L}$. (a) Optical image of sample $\mathcal{L}$. (b) Sketch of the different sample regions contributing to the quantum transport. (c) Magnetoconductance at 10 mK of all segments for an out-of-plane magnetic field. Correspondence between the segment and curve is indicated in the legend. The dashed line is a fit of the WAL component using formula (\ref{['eq:HLN']}). (d) Dependence of UCF amplitudes (in units of $e^2/h$) as a function of temperature. The amplitude is deduced from the gate dependence (round symbols) or the magnetic field dependence (triangle symbols) of the UCFs. The color code is the same as (c). The horizontal dashed lines are the expected value of the UCF for a diffusive conductor of the same length and same $\alpha$ as the considered section if self-averaging is assumed. (e) $l_\varphi$ extracted from both the UCFs (round symbols) and WAL (triangle symbols), showing a power-law dependence in temperature with exponent $-0.45$. $l_\varphi$ saturates for temperatures below 100 mK. The part of the curve in gray indicates the range of temperature where UCFs cannot be reliably extracted due to experimental noise.
• Figure 5: (a)-(c) : Magnetoconductance anisotropy at 3.8 K for the 4-$\mu$m segment of sample $\mathcal{L}$. (a) Top : schematic image of the flake and the $XY$ plane in which the magnetic field is applied. Bottom : variation of the conductance by sweeping the magnetic field angle in the $XY$ plane, for a magnetic field modulus of 1 T. The angle 0° corresponds to a field aligned along the $Y$ direction. Magnetoconductance is highest in the Y direction, defining an extended lobe in the perpendicular $X$ direction, as indicated by the red line. (b) Similar data for a magnetic field in the $XZ$ plane, showing a maximum magnetoconductance along the $Z$ direction. (c) Similar data for a magnetic field in the $Y$ plane, showing a maximum magnetoconductance along the $Z$ direction and side lobes, indicated by gray dashed lines, attributed to the geometry of the contacts (see the text). (d) Top : optical image of the sample $\mathcal{H}_2$. Bottom : sketch of the cross section of the flake $\mathcal{H}_2$. (e) Magnetoconductance with a magnetic field of 1 T in the $YZ$ plane for lateral contacts indicated in (d) in green (left part) or red (right part). The direction of maximum magnetoconductance is not related to the crystalline orientation but rather to the geometry of the lateral contacts.