Figure 1

Atom-chip trapping principle (Refs. 34, 35, 36). A cylindrically symmetric quadrupole magnetic field is created by superimposing a weak, homogeneous bias field $B_{\mathrm{bias}}$ with that from a wire with current $I$. Weak-field seeking atoms, i.e., atoms in a Zeeman state whose energy increases for increasing magnetic field magnitude, are trapped in a small region around the zero of the magnetic quadrupole field. If the microwire is attached to a surface via standard photolithography, then the trap may be brought to an arbitrarily close distance $r\propto I/B_{\mathrm{bias}}$ from the surface by adjusting the ratio of $I$ to $B_{\mathrm{bias}}$. (Surface potentials limit $r>200$ nm.) Trapped atoms may be translated perpendicular to the axis of the wire by rotating with an offset field $B_{\mathrm{offset}}$ the angle that $B_{\mathrm{bias}}$ subtends with the substrate surface, thus enabling the precise positioning of atoms above adjacent materials.Reuse & Permissions

Figure 2

Atom-chip microscopy principle (Refs. 31 and 32). (a) A Bose-Einstein condensate (BEC, shown in red) is confined above the atom chip by the $Z$ trap formed by the fields from the $Z$-shaped microwire (orange) and $B_{\mathrm{bias}}$ produced by an external (not shown) Helmholtz coil pair, the axis of which is aligned with $\widehat{x}$. Adjusting current $I$ in the trapping wire and $B_{\mathrm{bias}}$ controls, with submicron precision, the position of the BEC above the surface of the substrate. (b) Current running through the sample wire may not flow parallel to the wire's axis due to scattering centers (exaggerated in figure), which result in 1-ppm variations of the magnetic field above the sample (Ref. 33). This inhomogeneous field deforms the trap, imprinting density modulations onto the otherwise smooth, cigar-shaped BEC cloud. The absorption of a near-infrared laser casts a shadow onto a CCD camera, providing $\mu$m-scale resolution of density perturbations in the submicron-wide, $\sim$1-mm-long cloud (Ref. 37). The BEC can be recreated and repositioned every few seconds, thus in a few minutes providing a wide-area map of the inhomogeneous current flow. (c) To image transport in a TI, the atoms can be cantilevered over the sample, and held away from the $Z$ wire, by rotating $B_{\mathrm{bias}}$ in the $xz$ plane using $B_{\mathrm{offset}}$. The TI sample (shown in blue) may be mounted on the cryogenically cooled atom chip.Reuse & Permissions

Figure 3

Schematic of device under consideration. Contacts induce a longitudinal current across the channel, which is backgated to be within the bulk gap for the case of Bi${}_2$Se${}_3$. The density of a BEC held in a cigar-shaped atom-chip magnetic microtrap is distorted depending on the direction of current flow underneath. The atom chip supports a $Z$-shaped gold wire that, with current $I_{\text{dc}}$ and homogeneous bias magnetic field $B_{\text{bias}}$ along $-\widehat{x}$, creates a magnetic trap above the substrates (see Sec. 2). A $\widehat{z}$-oriented magnetic field $B_{\text{offset}}$ allows the BEC to be shifted laterally in the $\widehat{x}$ direction. $h$ is the height of the BEC above the material.Reuse & Permissions

Figure 4

Current profile, where red/bright (blue/dark) denote high (low) current density in (a) and (d) undoped Bi${}_2$Se${}_3$; (b) and (e) doped Bi${}_2$Se${}_3$; and (c) and (f) metal thin films. Positions of contact leads shown as gold rectangles in (a) and (d); leads in the other panels not shown. (a) Current is closely tied to within 3 nm of the top and bottom surfaces of the undoped TI Bi${}_2$Se${}_3$. (b) Doping by raising the chemical potential to 0.2 eV opens a parallel bulk conducting path as carriers reach the CB. Backscattering in the CB marginally decreases total current by a few percent. The structure in the current density is due to finite simulation size effects. (c) Carriers in the metal immediately diffuse and flow through the bulk of the channel. The current profile of the metal is normalized so that total contact current is equivalent to the undoped Bi${}_2$Se${}_3$ profile. (d)–(f) Current flow around two trenches in the top surface of the material. Trenches in these plots are 1 nm wide, 5 nm deep, and separated by 7 nm. (d) Current hugs the upper surface contour in the undoped Bi${}_2$Se${}_3$ sample, but (e) fails to do so in the doped Bi${}_2$Se${}_3$. (f) As in the doped TI, the trenches in a metal serve to spatially low-pass filter the $\widehat{z}$ current modulation seen in the undoped TI. Atom-chip microscopy can image depth of the current flow between the trenches, thus providing a measure of surface-to-bulk conductivity.Reuse & Permissions

Figure 5

Current flow around trenches in the material differs depending on whether the material is an ideal TI, a metal, or a TI with nonzero bulk conductivity like Bi${}_2$Se${}_3$. (a) Raster-scan method for observing the field along $\widehat{x}$. BECs are sequentially created and imaged at positions along $\widehat{x}$ to form the field profile in panel (c). (b) Single-shot detection method in which a single BEC can image the field profile in panel (d). (c) Field $B_y$ from the raster-scan method depicted in panel (a) is plotted 2 $\mu$m from surface. The Bi${}_2$Se${}_3$ sample is 150 $\mu$m long and 10 $\mu$m wide. The two trenches are 30 $\mu$m wide and 5 $\mu$m deep and are separated by $d=30$ $\mu$m. (d) Field $B_y$ from the single-shot method depicted in panel (b) and plotted 2 $\mu$m from surface. The Bi${}_2$Se${}_3$ sample is 500 $\mu$m long and 500 $\mu$m wide. The two trenches along $\widehat{x}$ are 100 $\mu$m wide and 5 $\mu$m deep and are separated by 30 $\mu$m. The two trenches along $\widehat{y}$ are $w=100$ $\mu$m wide, 20 $\mu$m long, and separated by $L=100$ $\mu$m. Panels (c) and (d) show the magnetic field response for a system with surface-to-bulk current ratio of 100$\%$ (green line), 50$\%$ (dashed-dotted light blue line), and 0$\%$ (dashed dark blue line). The magnetic fields are equivalent away from the midpoint between the trenches, while the magnetic field for the system with a 50$\%$ or 0$\%$ current ratio is less than for a 100$\%$ ratio. The magnetic field response follows this behavior for all current ratios, smoothly and monotonically decreasing as a function of doping level until the Fermi level reaches the conduction or valence bands.Reuse & Permissions

Figure 6

Transverse $\widehat{y}$ component of the dc magnetic field $B_y$ produced by longitudinal transport of surface current density 5 $\mu$A/nm though the material with two trenches as in Fig. 3. Field is shown for undoped (green line) and doped (dashed-dotted light blue line) Bi${}_2$Se${}_3$ and the metal model (dashed dark blue line). The magnetic field is calculated $z=-1$ nm from the top surface, neglecting surface currents along the left and right sides of the system, and is plotted along the length of the material. The surface current between the trenches in the undoped system creates a signature of topological current flow with an observable change in $B_y$ that is directly related to doping strength. Field magnitudes decrease in the doped system as total current is mitigated by conduction-band backscattering.Reuse & Permissions

Figure 7

(a) Plot of the surface current ratio between the trenches relative to the ideal surface current as a function of bulk doping strength for Bi${}_2$Se${}_3$. The highest energy carriers reach the first CB as soon as doping deviates from zero, causing a linear decay as more injected carriers have access to bulk states. The highest energy carriers have access to the second CB at a doping strength $\gtrsim \left|0.1\right|$ eV, and as a result, surface current exhibits. (b) Plot of the Bi${}_2$Se${}_3$ band structure used for the simulation in panel (a).Reuse & Permissions

Figure 8

Plot of angle dependence as electrons flow from left to right around a circular impurity in the metal model. Carriers flow around the single impurity primarily at 45${}^{\circ }$ angles, diffusing back to a uniform longitudinal current profile afterward.Reuse & Permissions

Figure 9

(a) Plot of the ratio of surface current between the trenches relative to an idealized surface current as a function of bulk doping strength for an idealized TI. The highest energy injected carriers begin to reach the CB minimum at doping strengths around $\pm 0.5$ eV. (b) Plot of the idealized TI band structure used for the simulation in panel (a).Reuse & Permissions

Figure 10

Same as Fig. 9, but with particle-hole asymmetry.Reuse & Permissions