Velocity and confinement of edge plasmons in HgTe-based two-dimensional topological insulators

High-frequency transport in the edge states of the quantum spin Hall (QSH) effect has rarely been explored, though it could cast light on the scattering mechanisms taking place therein. Here we report on the measurement of the plasmon velocity in topological HgTe quantum wells both in the QSH and quantum Hall (QH) regimes, using harmonic GHz excitations and phase-resolved detection. We observe low plasmon velocities in both regimes, with, in particular, large transverse widths in the QH regime despite a sharp edge confinement profile. We ascribe these observations to the prominent influence of charge puddles forming in the vicinity of edge channels. Together with other recent works, it suggests that puddles play an essential role in the edge state physics and probably constitute a main hurdle on the way to clean and robust edge transport.

Figure 1

Sample geometry. (a) Sketch of the device: The light blue part is the HgTe mesa while the yellow parts (finger-gate electrodes RFg 1, 2 and ohmic contacts A, B) are made of gold. The red dashed region corresponds to the space covered by the top-gate DCg. A DC voltage $V_g$ is applied to RFg 1, 2 and DCg to uniformly tune the electron density $n$. (b) Image taken with an optical microscope of Sample A, showing the different gates and contacts of the sample as sketched in (a).

Figure 2

DC transport properties of the sample. (a) Two-terminal resistance $R_{2T}$ as a function of the gate voltage $V_g$ exhibiting a peak signaling the gap (indicated by the dashed lines), and the conduction and valence bands on either sides of this peak. (b) 2D color map of the ratio $R_K/R_{2T}$ as a function of gate voltage $V_g$ and magnetic field $B$. The different filling factors $\nu$ are labeled, and the white dotted lines are the lines $B_{\nu }$ used to fit the carrier density $n$ as a function of gate voltage $V_g$ (see main text). The contours between the different QH plateaus are highlighted as dashed black lines. The color scale is intentionally saturated at a maximum value $R_K/R_{2T}=10$ in order to distinguish more clearly the first QH plateaus.

Figure 3

Calibrated amplitude, phase, and velocity. Colormaps of the amplitude $M$ (a), phase $\varphi$ (b), and velocity $v$ (c) as function of the gate voltage $V_g$ applied on ${\mathrm{DC}}_g$ and the magnetic field $B$, obtained after calibration. The white shadings indicate regions where the signal amplitude $M$ is too small, so that $\varphi$ and $v$ are not reliably computed.

Figure 4

Velocity and plasmon transverse width in the Hall regime. (a) Linecuts of the velocity $v$ as function of magnetic field $B$ for three values of the density $n$. The gray dashed line shows fits to the law $v\propto B^{-1}$, valid at low fields. In the high field region, $v$ exhibits strong oscillations which become more pronounced as the density $n$ increases. (b) Transverse width $w$ as a function of filling factor $\nu$. For large $\nu$, i.e., low fields, $w$ is approximately constant and independent of $n$. For low $\nu$, the width $w$ oscillates, showing minimum for integer filling factors $\nu \in \mathbb{Z}$. (c) Sketch of the edge density profile $n_e\left(x\right)$ as a function of the distance $x$ from the edge: $n_e\left(x\right)$ saturates at $n_e\left(x\right)=n$ in the bulk of the material, and decreases to $n\left(x\right)=0$ at the edge. The blue shades indicate the compressible stripes while the white stripes are the incompressible ones. The bare plasmon width is given by the position of the innermost Landau level $x_{\mathrm{QH}}$, and is further increased by $w_p/2$ due to puddles. (d) Normalized reconstructed edge profile $n_e\left(x\right)$ obtained by plotting $1-1/2\nu$ as a function of $w$ for all data triplets $(n,B,w)$. The obtained profiles are shown as colored dots for various values of the bulk density $n$. The dashed lines represent the heuristic edge profile $f\left(x\right)$ for two extreme admissible values of the depletion depth $l=60$ and $150\mathrm{n}\mathrm{m}$.

Figure 5

Velocity measured in the QSH regime at $B\simeq 0$. Velocity $v$ as function of the gate voltage $V_g$ applied to DCg for three values of the magnetic field $B$ close to $B=0$. The gap region estimated from the resistance $R_{2T}$ is indicated by vertical dashed lines.