Crossed Andreev reflection in spin-polarized chiral edge states due to the Meissner effect

We consider a hybrid quantum Hall-superconductor system, where a superconducting finger with oblique profile is wedged into a two-dimensional electron gas in the presence of a perpendicular magnetic field, as considered by Lee et al., Nat. Phys. 13, 693 (2017). The electron gas is in the quantum Hall regime at filling factor $\nu =1$. Due to the Meissner effect, the perpendicular magnetic field close to the quantum Hall-superconductor boundary is distorted and gives rise to an in-plane component of the magnetic field. This component enables nonlocal crossed Andreev reflection between the spin-polarized chiral edge states running on opposite sides of the superconducting finger, thus opening a gap in the spectrum of the edge states without the need of spin-orbit interaction or nontrivial magnetic textures. We compute numerically the transport properties of this setup and show that a negative resistance exists as a consequence of nonlocal Andreev processes. We also obtain numerically the zero-energy local density of states, which systematically shows peaks stable to disorder. The latter result is compatible with the emergence of Majorana bound states.

Figure 1

(a) Sketch of the considered setup: A narrow SC finger wedged into a 2DEG tuned into the QHE regime at filling factor $\nu =1$. A single chiral spin-polarized edge state is localized along the system boundaries and around the SC finger. (b) Side-view schematic of the oblique cross-section of the SC finger with width $W_s$ at the level of the 2DEG: The locally distorted magnetic field distribution $\mathit{B}\left(\mathit{r}\right)$ due to the Meissner effect. Owing to the oppositely directed in-plane component of the field at the opposite sides of the finger with the counter-propagating chiral (otherwise fully spin-polarized) edge states, the superconducting gap is induced by proximity in the spectrum of the edge states due to nonlocal CAR processes.

Figure 2

The proximity-induced CAR gap ${\mathrm{\Delta }}_c$ relative to SC parent gap ${\mathrm{\Delta }}_s$ as function of SC finger width $W_s$ in units of parent SC coherence length ${\xi }_s$, for different ratios of in-plane and out-of-plane components of Zeeman energy, ${\mathrm{\Delta }}_Z^{\parallel }/{\mathrm{\Delta }}_Z^{\perp }=\{1/8,1/4,1/2,1\}$. Numerical results are obtained for an infinite QHE-SC-QHE system with ${\mathrm{\Delta }}_Z^{\perp }=2{\mathrm{\Delta }}_s, {\mu }_s=5{\mathrm{\Delta }}_s, {\mu }_n=3.125{\mathrm{\Delta }}_s, m_s=m_n$, a Landau gap in the QHE part of ${\mathrm{\Delta }}_{LL}=\hslash e{B}_0/m_n=6.28{\mathrm{\Delta }}_s$, along with a perfect interface coupling ${\tilde{t}}_c$. (Inset) Low-energy spectrum of the infinite system at $W_s\approx 1.14{\xi }_s$ and ${\mathrm{\Delta }}_Z^{\parallel }/{\mathrm{\Delta }}_Z^{\perp }=1/2$, as indicated by yellow asterisk in main panel. Red (green) lines correspond to SC (edge states) in the absence of tunneling. Full lines stand for the electronlike while dash-dotted ones for the holelike parts of the spectra. Blue circles show the spectrum of a coupled QHE-SC-QHE junction, where the edge states are gapped due to CAR. This gap ${\mathrm{\Delta }}_c$ is shown as a function of $W_s$ in the main panel.

Figure 3

The absolute value of the proximity-induced CAR gap ${\mathrm{\Delta }}_c$ relative to SC parent gap ${\mathrm{\Delta }}_s$ as a function of the angle $\theta$, obtained numerically in an infinite QHE-SC-QHE system (see Appendix pp4 and, in particular, Fig. 10 for further details). The blue and red dotted curves are analytical fits with a sine function and a linear function, respectively, of $\theta$.

Figure 4

(a) Simulated transport measurement arrangement based on those in experiments [56, 60, 68]. (b) Andreev transmission coefficient between the two leads, $T^A$, as a function of the finger width $W_s$ at different temperatures $T$, is used to calculate (c) downstream ($R_D$) and (d) upstream ($R_U$) resistances. In both figures, dashed lines indicate the theoretically achievable minimum resistances, $R_{D/U}=\mp R_Q$ for $\nu =1$ filling. Results were obtained for the same parameters as the yellow curve in Fig. 2 and $L_s\approx 2.85{\xi }_s$. For details of numerical simulations, we refer also to Appendix pp3.

Figure 5

(a) LDOS at zero energy, $\mathrm{LDOS}(E=0$), in the finite-geometry finger setup for the same set of parameters as used in Fig. 4. Note that edge states along the SC finger gap out and a localized zero-energy MBS forms on the tip of the finger, as well as its delocalized partner in the lower edge states (away from the finger), invisible at the scale adapted to the MBS density. (b) A negative downstream resistance $R_D$ for smaller values of ${\mathrm{\Delta }}_Z^{\parallel }$ can be achieved by increasing the SC finger length $L_s$. The latter reduces the MBS wave-function overlap (localization characterized by induced CAR gap ${\mathrm{\Delta }}_c$), and thus also the energy-splitting away from $E=0$, rendering electron-to-hole conversion through the lower MBS (hybridized with the edge state) more efficient in the transport. All parameters are kept unchanged except for $W_s=0.82{\xi }_s$ and $T=0.006{\mathrm{\Delta }}_s$. For details of numerical simulations, we refer also to Appendix pp3.

Figure 6

(a) SC finger (red) and the in-plane component of the expelled magnetic field due to the Meissner effect. (b) Cross section of the wedge-shaped SC finger and the corresponding dimensions chosen in the simulations. The dashed line, where the width of the SC is $W_s=125\mathrm{nm}$, indicates the plane of the graphene sheet that lies $75\mathrm{nm}$ away from the bottom of the wedge.

Figure 7

(a) The in-plane magnetic field $B^{\parallel }/B_0$ as a function of the $x$ coordinate along the line determined by the dashed line in Fig. 6 for six different magnetic susceptibilities ranging from the perfect diamagnet ${\chi }_{\text{SC}}^{}=-1$ to zero magnetic response ${\chi }_{\text{SC}}^{}=0$. The in-plane component $B^{\parallel }$ is maximum for the perfect diamagnet and is absent if the Meissner effect is neglected. (b) The out-of-plane component of the magnetic field $B^{\perp }/B_0$ along the same line. The values of $B^{\perp }$ are only slightly modified close to the interface between the SC and the 2DEG. Thus one can argue that the spatial distribution of the edge state is only slightly affected. (c) The ratio of the in-plane and out-of-plane components as a function of magnetic susceptibility at $x=65$ nm. For a perfect diamagnet, the ratio could be as large as $B^{\parallel }/B_0\approx 1/2$ in this geometry, giving a tilt angle $\theta \approx \pi /4$.

Figure 8

(a) Quasi-1D discretization scheme for the infinite QHE-SC-QHE structure with periodic boundary conditions in $x$ direction, corresponding to Eqs. (B7, B8, B9). [(b)–(d)] Sketches of the QHE-SC-QHE lateral junction structure with the different possible CAR gap opening mechanisms: (b) The case of the main text, where oppositely directed in-plane magnetic field components in the QHE parts enable the otherwise spin-polarized chiral edges of the QHE at $\nu =1$ filling to be proximitized by the singlet-type SC. (c) The mechanism based on SOI considered in earlier works, where the spin-polarized $\nu =1$ edges gain overlap with the singlet structure of CPs by the spin-rotating effect of SOI in the SC. (d) Just for comparison purposes, we show the case of $\nu =2$ filling, where the chiral edges are unpolarized and CAR can take place without any additional ingredient. [(e)–(g)] Low-energy spectra with the CAR opened gaps of the respective systems in (b)–(d), obtained in the discretized model of (a). Color coding is consistent with the rest of the figure and the main text, green and purple lines show the lowest Landau levels and edge states in the separate QHE subsystem spectrum (with PBC only having the set of chiral edges present next to the SC that are affected by CAR). Red curves represent states in the unconnected SC region with width $W_s$. Full lines mean electronic and dashed ones hole states. Blue circles are the lowest eigenenergies of the hybrid QHE-SC-QHE system at different $k$, showing a significant induced cross-gap ${\mathrm{\Delta }}_c$. Parameters used for all (e)–(g): $N_n=50, N_s=26, t_s=t_c=t, {\mu }_s=0.1t, {\mathrm{\Delta }}_s=0.02t, \varphi =0.01$, and $\zeta =10a$. Only in (e), ${\mu }_n=0.06t, {\mathrm{\Delta }}_Z^{\perp }=0.04t, {\mathrm{\Delta }}_Z^{\parallel }=0.04t$, and ${\alpha }_s=0$. Only in (f), ${\mu }_n=0.06t, {\mathrm{\Delta }}_Z^{\perp }=0.04t, {\mathrm{\Delta }}_Z^{\parallel }=0$, and ${\alpha }_s=0.05t$. And only in (g), ${\mu }_n=0.1025t, {\mathrm{\Delta }}_Z^{\perp }=0.0025t, {\mathrm{\Delta }}_Z^{\parallel }=0$, and ${\alpha }_s=0$.

Figure 9

Schematic of the discretized finger geometry model with leads indicated that are used for transport calculations. The principal region described by Eq. (B10) is marked with the dashed black line and contains $N_x\times N_y$ sites, with $N_x=2N_n+N_s$. The SC finger is $W_s=(N_s-1)a$ wide and $L_s=(S-1)a$ long, its sites are marked with red and the red shaded area marks the region of zero magnetic flux through the plaquettes. All gray plaquettes bear a flux, even the region under the blue dashed lines that indicate the interface coupling between SC and QHE, quantified by $t_c$. ${\mathrm{\Sigma }}_{L/R}$ denote the self-energies associated with the left- and right semi-infinite leads, respectively.

Figure 10

The absolute value of the proximity-induced CAR gap ${\mathrm{\Delta }}_c$ relative to SC parent gap ${\mathrm{\Delta }}_s$ as a function of the ratio of in-plane and out-of-plane components of Zeeman energy, ${\mathrm{\Delta }}_Z^{\parallel }/{\mathrm{\Delta }}_Z^{\perp }$. We used the following parameters: $N_n=50, N_s=36, t_s=t_c=t, {\mu }_s=0.1t, {\mathrm{\Delta }}_s=0.02t, \varphi =0.01, {\mu }_n=0.0625t, {\alpha }_s=0$, and $\zeta =10a$. In this figure, we fix ${\mathrm{\Delta }}_Z^{\perp }=0.04t$, whereas in Fig. 3, we used with a Zeeman term of constant magnitude ${\mathrm{\Delta }}_Z=0.04t$ that is rotated homogeneously within the QHE regions (oppositely for the two QHE parts) from out-of-plane to in-plane directions with angle $\theta$.

Figure 11

(a) Oscillation of the absolute value of the crossed gap, ${\mathrm{\Delta }}_c$, induced in the $\nu =1$ QHE edge states, normalized by the superconducting gap in the SC region, ${\mathrm{\Delta }}_s$, as a function of the SC region width, $W_s$, measured in units of the discretized lattice spacing $a$, for fixed in-plane magnetic field ${\mathrm{\Delta }}_Z^{\parallel }/{\mathrm{\Delta }}_Z^{\perp }=1$ and different values of chemical potential in the SC, ${\mu }_s$. (b) Fourier transformation of the data displayed in (a). The theoretically estimated Fermi momentum values are indicated by the solid colored lines. As expected, the absolute value of the gap oscillates with the period set by $2k_{F,s}$. Parameters of the numerics not indicated in the figure: $N_n=50, t_s=t_c=t, {\mu }_n=0.0625t, {\mathrm{\Delta }}_Z^{\perp }=0.04t, \zeta =10a, {\mathrm{\Delta }}_s=0.02t, \varphi =0.01$, and ${\alpha }_s=0$. For a better understanding, we also indicate the chemical potential dependent superconducting coherence length ${\xi }_s$ for all curves, ${\xi }_s/a\approx \{31,47,63,79,95\}$, which characterize the scale of decay for the CAR gap. We observe irregular resonances for $W_s\lesssim {\xi }_s$, while a smooth decaying cosine-like behavior for $W_s>{\xi }_s$, in line with Eqs. (D1) and (D5).

Figure 12

(a) Double oscillation of the absolute value of the crossed gap, ${\mathrm{\Delta }}_c$, induced in the $\nu =1$ QHE edge states, normalized by the superconducting gap in the SC region, ${\mathrm{\Delta }}_s$, as a function of the SC region width, $W_s$, measured in units of the discretized lattice spacing $a$, for fixed SOI strength ${\alpha }_s=0.1t$ and different values of chemical potential in the SC, ${\mu }_s$. (b) Fourier transformation of the data displayed in (a). The theoretically expected Fermi-momentum values are indicated by the solid colored lines. (c) Double oscillation of the absolute value of the crossed gap, ${\mathrm{\Delta }}_c$, induced in the $\nu =1$ QHE edge states, normalized by the superconducting gap in the SC region, ${\mathrm{\Delta }}_s$, as a function of the SC region width, $W_s$, measured in units of the discretized lattice spacing $a$, for different values of SOI strength in the SC, ${\alpha }_s$, at constant SC chemical potential ${\mu }_s/t=0.9$. (d) Fourier transformation of the data displayed in (c). The theoretically expected fast oscillation period value is indicated by the solid black line, while dashed lines show the theoretically expected side lobe periods due to the slow envelope oscillations. Parameters of the numerics not indicated in the figure: $N_n=50, t_s=t_c=t, {\mu }_n=0.0625t, {\mathrm{\Delta }}_Z^{\perp }=0.04t, {\mathrm{\Delta }}_Z^{\parallel }=0, \zeta =10a, {\mathrm{\Delta }}_s=0.02t$, and $\varphi =0.01$. The superconducting coherence length ${\xi }_s$ for all curves in (a) takes values ${\xi }_s/a\approx \{31,47,63,79,95\}$, whereas in (c), it is fixed as ${\xi }_s/a\approx 95$. We observe irregular resonances for $W_s\lesssim {\xi }_s$, while a smooth, decaying double oscillatory behavior for $W_s>{\xi }_s$, in line with Eqs. (D2) and (D5).

Figure 13

(a) Oscillation of the absolute value of the crossed gap, ${\mathrm{\Delta }}_c$, induced in the $\nu =2$ QHE edge states, normalized by the superconducting gap in the SC region, ${\mathrm{\Delta }}_s$, as a function of the SC region width, $W_s$, measured in units of the discretized lattice spacing $a$, for different values of chemical potential in the SC, ${\mu }_s$. (b) Fourier transformation of the data displayed in the panel (a). The theoretically expected Fermi-momentum values are indicated by the solid colored lines. Parameters of the numerics not indicated in the figure: $N_n=50, t_s=t_c=t, {\mu }_n=0.1025t, {\mathrm{\Delta }}_Z^{\parallel }={\mathrm{\Delta }}_Z^{\perp }=0, \zeta =10a, {\alpha }_s=0, {\mathrm{\Delta }}_s=0.02t$, and $\varphi =0.01$. Superconducting coherence length ${\xi }_s$ for all curves reads ${\xi }_s/a\approx \{31,47,63,79,95\}$. We observe irregular resonances for $W_s\lesssim {\xi }_s$, while a smooth, decaying cosine-like behavior for $W_s>{\xi }_s$, in line with Eqs. (D1) and (D5).

Figure 14

Plot of the absolute value of the proximity-induced CAR gap ${\mathrm{\Delta }}_c$ relative to SC parent gap ${\mathrm{\Delta }}_s$, averaged over different values of $W_s$ as a function of $t_c/t$, normalized with respect to the same quantity at $t_c=t$. Curves are obtained numerically in an infinite QHE-SC-QHE system with $N_n=50, N_s=2,...,120, t_s=t, {\mu }_s=0.4t, {\mathrm{\Delta }}_s=0.02t, \varphi =0.01, \zeta =10a$, and for $\nu =1, {\mathrm{\Delta }}_Z^{\parallel }\ne 0$ with ${\mu }_n=0.0625t, {\mathrm{\Delta }}_Z^{\parallel }={\mathrm{\Delta }}_Z^{\perp }=0.04t, {\alpha }_s=0$, for $\nu =1, {\alpha }_s\ne 0$ with ${\mu }_n=0.0625t, {\mathrm{\Delta }}_Z^{\parallel }=0, {\mathrm{\Delta }}_Z^{\perp }=0.04t, {\alpha }_s=0.1t$, and finally for $\nu =2$ with ${\mu }_n=0.1025t, {\mathrm{\Delta }}_Z^{\parallel }={\mathrm{\Delta }}_Z^{\perp }=0$, and ${\alpha }_s=0$ (no SOI).

Figure 15

(a) The absolute value of the proximity-induced CAR gap ${\mathrm{\Delta }}_c$ relative to SC parent gap ${\mathrm{\Delta }}_s$ averaged over different values of $W_s$ with respect to the disorder (a) in the SC chemical potential ${\mu }_s$ and (b) in the interface coupling $t_c$, normalized with respect to the same quantity without disorder. Curves are obtained with the same respective parameters as in Fig. 14, by numerical exact diagonalization of a long but finite QHE-SC-QHE planar junction.

Figure 16

(a) Andreev transmission coefficient between the two leads, $T^A$, as a function of the finger width $W_s$ at different temperatures $T$ for the in-plane magnetic field case. The dash-dotted line at 0.5 indicates the threshold above which values of anomalous transmission negative downstream resistances are observed. (b) Calculated width dependence of downstream ($R_D$) resistances. The dashed line indicates the theoretically achievable minimum resistance, $R_D=-R_Q$ for $\nu =1$ filling. (c) Stability of the Andreev transmission coefficient $T^A$ as a function of the magnetic flux variation $\delta \varphi$. (d) Stability of the downstream resistance $R_D$ as a function of the magnetic flux variation $\delta \varphi$. This magnetic flux controls the orbital effect in the QHE region. The stability is checked for different widths associated with peaks in the transmission. In correspondence with the chosen values of $W_s$, lines are drawn in (a) and (b) with the same color code as in (c) and (d). Parameters: $N_x=150, N_y=120, S=90, t_s=t_c=t, {\mu }_s=0.1t, {\mathrm{\Delta }}_s=0.02t, \varphi =0.01, {\mu }_n=0.0625t, {\mathrm{\Delta }}_Z^{\perp }=0.04t, {\mathrm{\Delta }}_Z^{\parallel }=0.02t, \zeta =10a$, and ${\alpha }_s=0$.

Figure 17

(a) Andreev transmission coefficient between the two leads, $T^A$, as a function of the finger width $W_s$ at different temperatures $T$ for the SOI case. The dash-dotted line at 0.5 indicates the threshold above which values of anomalous transmission negative downstream resistances are observed. (b) Calculated width dependence of downstream ($R_D$) resistances. The dashed line indicates the theoretically achievable minimum resistance, $R_D=-R_Q$ for $\nu =1$ filling. (c) Stability of the Andreev transmission coefficient $T^A$ as a function of the magnetic flux variation $\delta \varphi$. (d) Stability of the downstream resistance $R_D$ as a function of the magnetic flux variation $\delta \varphi$. This magnetic flux controls the orbital effect in the QHE region. The stability is checked for different widths associated with peaks in the transmission. In correspondence with the chosen values of $W_s$, lines are drawn in (a) and (b) with the same color code as in (c) and (d).Note how the double oscillations of $T^A$ in (a) are reminiscent of the double oscillations of the induced CAR gap displayed for the $\nu =1$ SOI case in Fig. 12. Parameters: $N_x=150, N_y=120, S=90, t_s=t_c=t, {\mu }_s=0.1t, {\mathrm{\Delta }}_s=0.02t, \varphi =0.01, {\mu }_n=0.0625t, {\mathrm{\Delta }}_Z^{\perp }=0.04t, {\mathrm{\Delta }}_Z^{\parallel }=0, \zeta =10a$, and ${\alpha }_s=0.05t$.

Figure 18

(a) Andreev transmission coefficient between the two leads, $T^A$, as a function of the finger width $W_s$ at different temperatures $T$ for the $\nu =2$ case. The dash-dotted line at 1 indicates the threshold above which values of anomalous transmission negative downstream resistances are observed. (b) Calculated width dependence of downstream ($R_D$) resistances. The dashed line indicates the theoretically achievable minimum resistance, $R_D=-R_Q/2$ for $\nu =2$ filling. (c) Stability of the Andreev transmission coefficient $T^A$ as a function of the magnetic flux variation $\delta \varphi$. (d) Stability of the downstream resistance $R_D$ as a function of the magnetic flux variation $\delta \varphi$. This magnetic flux controls the orbital effect in the QHE region. The stability is checked for different widths associated with peaks in the transmission. In correspondence with the chosen values of $W_s$, lines are drawn in (a) and (b) with the same color code as in (c) and (d). Parameters: $N_x=150, N_y=120, S=90, t_s=t_c=t, {\mu }_s=0.1t, {\mathrm{\Delta }}_s=0.02t, \varphi =0.01, {\mu }_n=0.1025t, {\mathrm{\Delta }}_Z^{\perp }={\mathrm{\Delta }}_Z^{\parallel }=0, \zeta =10a$, and ${\alpha }_s=0$.

Figure 19

(a) LDOS at zero energy ($E=0$), in the finite-geometry SC finger setup. The green dash-dotted lines indicate the SC finger. Note that edge states along the SC finger gap out and a localized zero-energy MBS forms on the tip of the finger, as well as its delocalized partner in the lower edge states (away from the finger), invisible at the scale adapted to the MBS density. The strength of disorder is (a) $\delta {\mu }_s=0$, (b) $\delta {\mu }_s=0.5{\mathrm{\Delta }}_s$, (c) $\delta {\mu }_s={\mathrm{\Delta }}_s$, (d) $\delta {\mu }_s=2{\mathrm{\Delta }}_s$, (e) $\delta {\mu }_s=5{\mathrm{\Delta }}_s$, (f) $\delta {\mu }_s=10{\mathrm{\Delta }}_s$, (g) $\delta {\mu }_s=20{\mathrm{\Delta }}_s$, and (h) $\delta {\mu }_s=40{\mathrm{\Delta }}_s$. Apart from the presence of disorder, densities are obtained numerically with the same parameters as in Fig. 4 in the main text.