Shot noise in the edge states of two-dimensional topological insulators

We calculate the resistance and shot noise in the edge states of a two-dimensional topological insulator that result from the exchange of electrons between these states and conducting puddles in the bulk of the insulator. The two limiting cases where the energy relaxation is either absent or very strong are considered. A finite time of spin relaxation in the puddles is introduced phenomenologically. Depending on this time and on the strength of coupling between the edge states and the puddles, the Fano factor $\mathcal{F}=S_I/2eI$ ranges from 0 to 1/3, which is in an agreement with the available experimental data.

Figure 1

Contour plot of the conductance for one puddle given by Eq. (14) in coordinates effective coupling strength ${\varphi }_L$—normalized spin-flip time ${\eta }_1$. As the coupling and the spin-flip rate increase, it decreases from $e^2/2\pi \hslash$ to $e^2/4\pi \hslash$.

Figure 2

Contour plot of Fano factor for one puddle in the absence of energy relaxation given by Eq. (19) in coordinates effective coupling strength ${\varphi }_L$—normalized spin-flip time ${\eta }_1$. The maximum Fano factor corresponds to the maximum resistance of the edge states.

Figure 3

Contour plot of the conductance for a continuous distribution of puddles given by Eq. (25) in coordinates effective coupling strength ${\phi }_L$—normalized spin-flip time $\eta$. As the coupling and the spin-flip rate increase, it decreases from $e^2/2\pi \hslash$ to zero.

Figure 4

Contour plot of Fano factor for a continuous distribution of puddles in the absence of energy relaxation in coordinates effective coupling strength ${\phi }_L$—normalized spin-flip time $\eta$ according to Eq. (30). The Fano factor vanishes in the ballistic regime and tends to its maximum value 1/3 regardless of $\eta$ if the conductance of the edge states tends to zero.

Figure 5

Contour plot of Fano factor for a single puddle with a strong energy relaxation in coordinates effective coupling strength ${\varphi }_L$—normalized spin-flip time ${\eta }_1$ according to Eq. (38). The maximum of Fano factor at ${\eta }_1=0$ results from the randomness of tunneling between the puddle and the edge states. The maximum at ${\eta }_1\approx 1.6$ stems from the randomness of spin-flip scattering in the puddle.

Figure 6

Approximate coordinate dependence of the chemical potentials for spin-up and spin-down electrons in the puddles for the strong energy relaxation. There are finite jumps between the spin-dependent chemical potentials of the left and right reservoirs and potentials of the puddles.

Figure 7

Contour plot of Fano factor for a continuous distribution of puddles for a strong energy relaxation given by Eq. (43) in coordinates effective coupling strength ${\phi }_L$—normalized spin-flip time $\eta$. The smaller maximum of $\mathcal{F}$ is located at $\eta =0$, and the larger is reached at $\eta \to \infty$. $\mathcal{F}$ varies from zero to 1/4.