Electric-field control of zero-dimensional topological states in ultranarrow germanene nanoribbons

Abstract

Reversible, all-electric control of symmetry-protected zero-dimensional modes has been a long-standing goal. In buckled honeycomb lattices, a perpendicular field couples to the staggered sublattice potential providing the required handle. We combine scanning tunneling microscopy and tight-binding theory to switch zero-dimensional topological end states reversibly on and off in ultranarrow germanene nanoribbons by tuning the electric field in the tunnel junction. Increasing the field switches off the end modes of topological two-hexagon wide ribbons, while the same field switches on zero-dimensional states in initially trivial three- and four-hexagon wide ribbons. This atomic scale platform realizes a proof-of-principle for a zero-dimensional topological field effect device, opening a path for ultrasmall memory, controllable qubits, and neuromorphic architectures.

Figures (3)

• Figure 1: (a) Large area STM topograph showing germanene nanoribbons (black dashed outlines) embedded in disordered nanowires on the Pt/Ge(110) surface. The ribbons run along the [-110] direction. (b) High resolution image of a two-hexagon wide nanoribbon. (c) Atomic resolution image of the two-hexagon nanoribbon overlaid on the tentative structural model. (d) and (e) Line profiles taken along the white and teal lines in panel (c), respectively, showing the lattice periodicity and buckling in the hexagonal lattice. (f) Schematic of the buckled honeycomb lattice. (g) STM image of a narrow nanoribbon; the cyan line indicates the spatial path of the line spectroscopy. (h) $dI(V)/dV$ spectra recorded at the end (red) and bulk (black) of the nanoribbon in (g). (i) $dI(V)/dV$ line spectroscopy recorded along the cyan line in (g), showing the localized end state. (j,k) $(dI(V)/dV)/I_0$ spectra recorded for increasing current setpoints ($I_0$) from I=0.2 nA to I=2.5 nA, for the bulk (j) and end (k) of the nanoribbon in (g). The images have been displaced for clarity; zero is represented by a dashed line. (l,m) Theoretically calculated LDOS, for several values of the staggered mass $M_S$, see inset in (m) for the color code. The results for the bulk are shown in (l) and for the edge in (m). The plots obtained for the lowest values of staggered mass, black curves in (l) and (m), should not be compared to the experiments because germanene always has a finite mass due to the buckling. The dashed lines in (j) and (l) indicate the closing of the bulk gap. In the corresponding regime of parameters, a (slightly displaced) zero-bias peak is observed at the ends, see (k) and (m).
• Figure 2: (a) Topological phase diagram in the $\lambda_{\text{SO}}-M_{\text{S}}$ plane for two-hexagon wide nanoribbons. The topological regime is depicted in red and the trivial regime in white. (b) Experimental $(dI(V)/dV)/I_0$ (solid lines) and calculated LDOS (dashed lines of the end state at low (red) and high (black) electric fields. (c) STM topography of an ultranarrow nanoribbon (two-hexagon wide), on which the spectra in (b) are recorded. (d) $dI(V)/dV$ point spectra acquired moving from the end ribbon (red) to the bulk (black) of the ribbon (see colored points in c for the location, the color of the marker corresponds to the color of the spectrum), showing the sharp decay of the localized end state. (e) dI/dV maps recorded at bias voltages of -150, 50, 100 and 250 mV under low (0.3 nA) and high (2.5 nA) current setpoints, illustrating the localization of the end state (50-100 mV) at the ribbon end and its disappearance at high electric fields. In contrast, the dangling bonds are still visible at high fields.
• Figure 3: (a) Repeated cycling (seven cycles) between low ($0.3\,$nA) and high ($2.5\,$nA) applied perpendicular electric fields at the end of a two-hexagon ribbon, demonstrating reproducible and reversible on/off switching of the end states. The inset shows the STM topography of the nanoribbon used to perform the cycling experiments. (b) Zoom-in of the dashed-box region in (a), highlighting the discrete switching of the 0D end state. (c) $(dI(V)/dV)/I_0$ and (d) the normalized intensity of the end mode as a function of a gradual change of the setpoint current, sweeping it from low to high and back. (e) Topological phase diagrams calculated for different values of $M_\text{S}$ and $\lambda_{\text{SO}}$ for the three- (top) and four- (bottom) hexagon wide nanoribbons. The topological regime is depicted in red and the trivial regime in white. (f) Calculated LDOS of the three- and four-hexagon wide nanoribbons, showing the predicted end states (localized at the end of the nanoribbons, see insets) at large perpendicular electric fields and their absence at small electric fields. (g) Experimental $(dI(V)/dV)/I_0$ spectra for three-hexagon (top) and four-hexagon (bottom) ribbons at low and high setpoints, showing reversible end-state appearance around $50$-$100\,$mV at high electric fields. Two cycles of low-high-low-high electric fields are depicted for both ribbons.