Resistance of helical edges formed in a semiconductor heterostructure

Time-reversal symmetry prohibits elastic backscattering of electrons propagating within a helical edge of a two-dimensional topological insulator. However, small band gaps in these systems make them sensitive to doping disorder, which may lead to the formation of electron and hole puddles. Such a puddle—a quantum dot—tunnel-coupled to the edge may significantly enhance the inelastic backscattering rate, due to the long dwelling time of an electron in the dot. The added resistance is especially strong for dots carrying an odd number of electrons, due to the Kondo effect. For the same reason, the temperature dependence of the added resistance becomes rather weak. We present a detailed theory of the quantum dot effect on the helical edge resistance. It allows us to make specific predictions for possible future experiments with artificially prepared dots in topological insulators. It also provides a qualitative explanation of the resistance fluctuations observed in short HgTe quantum wells. In addition to the single-dot theory, we develop a statistical description of the helical edge resistivity introduced by random charge puddles in a long heterostructure carrying helical edge states. The presence of charge puddles in long samples may explain the observed coexistence of a high sample resistance with the propagation of electrons along the sample edges.

Figure 1

A quantum dot of linear size $w$ (bottom) tunnel-coupled to a helical edge (top). The mean level spacing of the puddle is denoted by $\delta$ and its charging energy by $E_C$. The typical tunneling-induced level width is $\Gamma$.

Figure 2

(a) Helical edge conductance correction $\Delta G$ caused by a Coulomb blockaded quantum dot is plotted as a function of dimensionless gate voltage $N_g=C_g{V}_g/e$ and temperature $T$. (Here, $C_g$ is the dot-gate capacitance.) Half-integer values of $N_g$ correspond to points of charge degeneracy. For convenience, we measure $\Delta G$ in units of its peak value $\Delta G^{\text{peak}}=G_0{\Gamma }^2/g^2{\delta }^2$ at “high” temperature, $\delta >T\gg \Gamma$, see Eq. (2.7). Temperature decreases from dot level spacing $T\sim \delta \gg \Gamma$ (red, topmost curve) down to a very low value $T\sim \delta e^{-\pi \delta /\Gamma }$ (blue, second lowest curve), corresponding to odd-valley Kondo temperature at distance $\delta /E_C$ from a charge degeneracy point. Ground-state degeneracy in the odd valleys leads to a weak temperature dependence $\Delta G\propto {ln}^2(T/T_K)$ resulting in the conspicuous plateau seen in the graph. The plateau persists down to temperature $\delta e^{-\pi \delta /\Gamma }$ below which it starts to narrow from the sides (not shown in the figure). Once temperature is further lowered below $T\sim \delta e^{-\pi E_C/2\Gamma }$ (Kondo temperature in the bottom of the valley), the odd valley contribution starts to decrease (purple, lowest curve) as $\Delta G\propto T^4$, see Eq. (2.15). In the even valley, the power-law $\Delta G\propto T^4$ holds throughout the wide interval from level spacing to zero temperature. The same law applies to the peaks at temperatures below the level width, $T\ll \Gamma$. However, at temperatures above $\Gamma$ the peak $\Delta G\left(T\right)$ saturates to a constant value because of off-resonant scattering, see Eq. (2.7) and the discussion leading to it. (b) For comparison: two-terminal conductance $G(N_g,T)$ of a conventional quantum dot in the regime of the Kondo effect. Notice that $dG/dT<0$ in the odd valley because of Kondo effect, while in the even one $dG/dT>0$. This results in trace crossings in the graph, contrary to Fig. 2. Elastic tunneling makes $G$ finite even in the limit $T\to 0$ (blue curve). This is also in contrast with $\Delta G\left(T\right)$ in Fig. 2, which is monotonically decreasing at any $N_g$, and eventually vanishes at $T\to 0$ (purple, lowest curve) due to the inelastic nature of scattering.

Figure 3

An example of a virtual process contributing to the amplitude (2.2). The initial, intermediate, and final states of the dot are shown (from left to right). The dashed line marks the Fermi level. Here the upwards and downwards arrows correspond to the labels $L$ and $R$, respectively.

Figure 4

An example of a virtual process contributing to the anisotropic part of exchange couplings $J_{ij}$. The amplitude for this process is given in Eq. (2.12). The figure depicts the initial, intermediate, and final states of the dot. The oddly occupied dot starts from a “spin-up” Kramers state (left). A left mover from the edge tunnels into an empty dot level 1 (dashed line marks the Fermi level). It flips its spin by interacting, via $U$, with the electron on level 0, and then returns back to the edge, leaving the dot state unchanged (right).

Figure 5

At high temperatures, the correction to the conductance is given by $\Delta G={(2R_{\Gamma }+R_{{\tau }_{\text{e}\text{−}\text{e}}^{-1}})}^{-1}$, corresponding to three resistors in series: two resistors correspond to the weak tunneling between the edge (on the left) and the dot, while the third one describes the inelastic process in the dot, needed to convert right movers into left movers, and vice versa.

Figure 6

The temperature and gate-voltage dependence of the single-dot correction $\Delta G$ to the helical edge conductance. At very low temperatures, $T\ll T_K$, the low-temperature asymptote of $\Delta G$ for all gate voltages is a power-law with a universal exponent, $\Delta G\propto T^4$. The Kondo temperature $T_K$ depends on the gate voltage, attaining its highest value $\sim \Gamma$ upon approaching the Coulomb blockade peaks. Above $T_K$, the correction $\Delta G$ at the peaks and in the odd valleys becomes weakly-dependent on $T$, as also illustrated in Fig. 2; for even valleys, the $\Delta G\propto T^4$ law remains intact. Upon increasing the temperature above the level spacing $\delta$, the distinction between even and odd valleys disappears because of spinful thermal excitations in the dot; at the same time, thermal fluctuations of the dot electron number remain suppressed as long as $T\ll E_C$. In this limit $\Delta G\propto T^2$—a standard consequence of interaction in the Fermi-liquid theory, see Eq. (2.7). Finally, at temperatures $T\gg E_C$, direct tunneling of electrons into and out of the dot becomes possible, resulting in a temperature- and gate-voltage-independent asymptote of $\Delta G$, see Eq. (2.16) and the explanatory Fig. 5. The crossover leading to Eq. (2.16) depends on specific dot parameters, see Fig. 7.

Figure 7

The crossover from two-level dominant contribution in Eq. (3.25) to the high-temperature asymptote of Eq. (3.49). The order of the characteristic temperatures $T_1\sim {\delta }^2/\Gamma$ and $T_2\sim g\sqrt{\Gamma \delta }$ depends on the parameter $g^{2/3}\Gamma /\delta$.

Figure 8

Graph of the Kondo temperature as a function of gate voltage, $T_K\left(E_{+}\right)$. Here, $\Gamma >T>T_{\delta }$. At these temperatures, the main contribution to the gate-voltage-averaged $\Delta G$ comes from the regime around $E_{+}\approx \delta$. In region I, the dot spin is strongly-screened since $T<T_K\left(E_{+}\right)$. There, $\Delta G\propto T^4$ from the first term in Eq. (4.35). In region II, we have $T>T_K\left(E_{+}\right)$, and the second term of Eq. (4.35) is applicable.

Figure 9

Schematic drawing of the heterostructure. The dark grey top layer depicts the gate electrode, below which is the doping layer (light grey) with a random distribution of dopants (blue dots) with density $n_d$. The dopants induce a random potential $V\left(\mathbf{r}\right)$ on the quantum well (white) and charge puddles are formed (blue).

Figure 10

A helical edge with left- and right-propagating modes (top) and a charge puddle (bottom). In the absence of Coulomb blockade, puddles with an optimal level width $\Gamma \sim T$ yield the largest correction to the conductance. The level width is optimal in puddles at a distance $d_T$ from the edge. Those puddles whose distance from the edge is within $\Lambda$ of $d_T$ give the main contribution to resistivity $\varrho$ in Eq. (5.10)

Figure 11

The integration domain in Eq. (5.10) for Coulomb blockaded puddles. Equation (5.10) is valid also in the presence of large charging energy $E_C\gg \delta$. The domain of integration over $\Gamma$ in Eq. (5.10) can be divided into different regions determined by the temperature. The integrand, $\propto \Delta G^{\text{av}}$, is given by Eqs. (4.48), (4.51), and (4.54) in the regions I, II, and III, respectively. The dominant contribution to Eq. (5.10) at any temperature $T\ll \delta$ comes from the shaded part in region II, where $\Delta G^{\text{av}}$ is given by Eq. (4.52). Contribution from region III is never dominant in Eq. (5.10).