Spin-resolved spectroscopy of helical Andreev bound states

We propose a versatile setup that allows performing spin-resolved spectroscopy of helical Andreev bound states, proving their existence and therefore the topological nature of the Josephson junction that hosts them. The latter is realized on the helical edge of a 2D topological insulator, proximitized with two superconducting electrodes. The spectroscopic analysis is enabled by a quantum point contact, which couples the junction with another helical edge acting as a spin-sensitive probe. By means of straightforward transport measurements, it is possible to detect the particular spin structure of the helical Andreev bound state and to shine light on the mechanism responsible for their existence, i.e., the presence of protected Andreev reflection within the Josephson junction. We discuss the robustness of this helical Andreev spectrometer with respect to processes that weaken spin to charge conversion.

Figure 1

(a) Schematic of the setup. Green (yellow) rectangles represent superconducting (normal) leads. The 2DTI (gray) features helical edge states, shown with blue (spin-up) and red (spin-down) line/arrows. (b) The two classes of hABSs: Solid (dotted) lines represent electrons (holes). (c) Labeling of the electronic amplitudes, incoming ($\alpha$) or outgoing ($\beta$) with respect to the QPC.

Figure 2

[(a)–(c)] Nonlocal differential conductances $G_{12/21}(E,\chi )$ for the parameters: ${\lambda }_p=0.25$, $\overline{x}=1{\xi }_L$, ${\mathrm{\Delta }}_L=\mathrm{\Delta }$, ${\mathrm{\Delta }}_R=1.2\mathrm{\Delta }$, $D=6{\xi }_L$, with the coherence length ${\xi }_L=v/\left(\pi {\mathrm{\Delta }}_L\right)$. In (a) we plot $G_{12}$ for ${\lambda }_f=0$. In (b) and (c) we plot $G_{12}$ and $G_{21}$, respectively, with ${\lambda }_f=0.15$. In (d) we plot $|\delta G^{\nu }\left|(\pi /2)\right|$ [see Eq. (8)] as a function of ${\lambda }_p$ and ${\lambda }_f/{\lambda }_p$. All plots share the same color bar, in units of $e^2/h$.

Figure 3

Differential conductances in presence of electronic backscattering on the upper edge, induced by a magnetic impurity with $m=0.05$ and located at $x_{\mathrm{M}}=2{\xi }_L$. The other parameters are the same as in Figs. 2 and 2. The conductances are in units of $e^2/h$. Plots (a) and (b) share the same color bar with Fig. 2.

Figure 4

Sketch of the processes that contribute to the reflection at the QPC of electrons on the lower edge. As in Fig. 1, blue (red) lines indicate spin-up (spin-down) channels, while solid (dotted) line indicate electron (hole) channels. Moreover, green (yellow) rectangles indicate superconducting (normal) electrodes. Spin-preserving (spin-flipping) tunneling events at the QPC are denoted by orange circles (pink diamonds). In (a)–(d) we consider the presence of a single hABS with $\nu =-1$, i.e., consisting of left-moving spin-up electrons and right-moving spin-down holes. In (e)–(h) we consider a bound state resulting from the presence of a strong magnetic region (dark gray) to the left of the QPC, which induces perfect electronic backscattering. In (i), according to Eq. (B5), we show the amplitudes associated with each tunneling event.

Figure 5

Values of ${\mathrm{\Sigma }}^{\max }$ (in units of $e^2/h$), as a function of different parameters, in presence of a single magnetic impurity with strength $m$ and located at $x_M=2{\xi }_L$ [with ${\xi }_L=v/\left(\pi {\mathrm{\Delta }}_L\right)$]. All the panels share the same color bar. In panel (a), we study the dependence of ${\mathrm{\Sigma }}^{\max }$ on the tunneling amplitude ${\lambda }_p$ and on the ratio ${\lambda }_f/{\lambda }_p$ between spin-flipping and spin-preserving tunneling amplitudes at the QPC. Panel (b) highlights the very weak dependence of ${\mathrm{\Sigma }}^{\max }$ on $D$ and $\overline{x}$ (both in units of ${\xi }_L$). In panel (c), we consider different impurity strengths, by varying both the magnitude $\left|m\right|$ and the phase ${\phi }_m=arg\left(m\right)$. Each panel shares its fixed parameters with Fig. 3, that is $D=6{\xi }_L$, ${\mathrm{\Delta }}_L=\mathrm{\Delta },{\mathrm{\Delta }}_R=1.2\mathrm{\Delta },\overline{x}={\xi }_L$, $m=0.05$, $x_{\mathrm{M}}=2{\xi }_L$, ${\lambda }_p=0.25$, ${\lambda }_f=0.15$.

Figure 6

Values of ${\mathrm{\Sigma }}^{\max }$ (in units of $e^2/h$), as a function of different parameters, in presence of two magnetic impurities, with strengths $m_1$ and $m_2$, located on both sides of the QPC (i.e., $x_{M_1}<\overline{x}<x_{M_2}$). The two panels share the same color bar. In panel (a), we study the dependence of ${\mathrm{\Sigma }}^{\max }$ on the strength of impurity 1, by varying both its magnitude $|m_1|$ and phase ${\phi }_{m_1}=arg\left(m_1\right)$, while keeping $m_2=0.05$ fixed. The QPC is at $\overline{x}=0$ and the impurities are located at $x_{{\mathrm{M}}_2}=-x_{{\mathrm{M}}_1}=2{\xi }_L$. In panel (b), we plot ${\mathrm{\Sigma }}^{\max }$ as a function of the positions of the QPC ($\overline{x}$) and the impurity 2 ($x_{{\mathrm{M}}_2}$), both in units of ${\xi }_L$, for $m_1=m_2=0.05$ and $x_{{\mathrm{M}}_1}=-2{\xi }_L$. Both panels share the remaining parameters, which read $D=6{\xi }_L$, ${\mathrm{\Delta }}_L={\mathrm{\Delta }}_R=\mathrm{\Delta }$, ${\lambda }_p=0.25$, ${\lambda }_f=0.15$.

Figure 7

Differential conductances (in units of $e^2/h$) in presence of electronic backscattering on the upper edge, induced by a magnetic impurity with $m=0.05i$ and located at $x_{\mathrm{M}}=2{\xi }_L$ [with ${\xi }_L=v/\left(\pi {\mathrm{\Delta }}_L\right)$]. The remaining parameters are the same as in Figs. 2, 2, and 3. They read $D=6{\xi }_L$, ${\lambda }_p=0.25$, ${\lambda }_f=0.15$, $\overline{x}={\xi }_L$, ${\mathrm{\Delta }}_L=\mathrm{\Delta }$, ${\mathrm{\Delta }}_R=1.2\mathrm{\Delta }$.