Intermediate chiral edge states in quantum Hall Josephson junctions

Abstract

A transfer-matrix-based theoretical framework is developed to study transport in superconductor-quantum Hall-Superconductor (SQHS) Josephson junctions modulated by local potential barriers in the quantum-Hall regime. The method allows one to evaluate the change in the conductivity of such SQHS Josephson junctions contributed by the intermediate chiral edge states (ICES) induced by these local potential barriers at their electrostatic boundaries at specific electron filling-fractions. It is particularly demonstrated how these ICES created at different Landau levels (LL) overlap with each other through intra- and inter-LL ICES mixing with the change in strength and width of the potential barriers. This results in different mechanisms for forming Landau bands when an array of such potential barriers are present. It is also demonstrated that our theoretical framework can be extended to study the lattice effect in a bounded domain in such SQHS Josephson junctions by simultaneously submitting the normal region to a transverse magnetic field and periodic potential.

Figures (5)

• Figure 1: We show the dispersion plots for the SNS junctions with barriers in the N region for $\nu = 5.5$ using Eq. \ref{['nbarrier']}. To show the presence of Landau levels, we set $\Delta_0 = 2.0 \hbar \omega_C$. In (a), we have one barrier with width $d=2$. In (b), (c) we have taken $4$ barriers in the N region with (b) $V_0 = 0.2$, separation $D = 2$ and $d= 1$, (c) $V_0 = 0.9$, $D = 0.5$ and $d=3$ and in (d) we have taken $40$ barriers in the N region with $d = 0.3$, $D = 0.3$. The red semicircles denote the classical electron orbit, and the blue semicircles denote the classical hole orbits. In (e) and (f), we show the dispersion for the monolayer graphene-based SQHS junction when the electrostatic barrier of height $V_0= 0.2$ and $0.8$ is present in the QH region respectively. In (g), we compare the conductivity of the SNS junction with a single barrier in the N region with two cases of the SNS junction with $w=0$ and $w=0.4$. Here, $w$ is defined by $w=2 U_0/ \sqrt{\nu}$. In this case, we have taken $\Delta_0 = 0.01 \times \nu$. Here, the distance between two SN edges is taken as 6 and the width of the barrier is taken as 1. In the inset of (g) and (i) we show the intermediate chiral edge states which contribute to the fluctuation in conductivity. the red and blue dot denotes their electron like or hole like nature. The corresponding table of the hole probabilities are given in appendix. We have shown a schematic diagram of the model system that we are considering in (h). In (h) we have shown $2$ rectangular barriers in the N region.
• Figure 2: In (a), (b) and (c) we show the dispersion calculated from Method II, from Eq. \ref{['nbarrierbloch']} in the main text for $n_{B}=100$ barriers in the QH region of a SQHS junction. The $x$ and $y$ axes are same as in FIG. \ref{['disp1']} (a)-(d). In (a), (b) and (c) we have taken $n=35$, $n=45$ and $n=50$. $n=50$ gives same result as Method-I, shown in Eq. \ref{['nbarrier']} in the main text. In (c) we show the number of points in the $E$-$X$ dispersion plots (bound states) obtained from the method II Eq. \ref{['nbarrierbloch']} for the case of $n_{B}=24$, $40$, $60$ and $100$. The solid line denotes the fitted curve. The last points in each curve is where we do not have Bloch condition at all. This is same as the results obtained using Method-I.
• Figure 3: In (a), (b), (e) and (f) we show the dispersion plot for SN(BN)$^n_{B}$S junction with $n=30$. For (a) $\nu/2= 2.75$ and $V_0=0.2$, (b) $\nu/2= 2.75$ and $V_0=0.8$, (e) $\nu/2= 1.125$ and $V_0=0.2$ and (f) $\nu/2= 1.125$ and $V_0=0.8$. The with of the barriers are taken as $d=0.3$ and the separation between them is taken as $D=0.3$. In (c) and (d) we show the conductivity, calculated using Eq. \ref{['condbtk']} as a function of both $V_0$ and $\nu/2$. The $x$ axes of the dispersion plots (a), (b), (e) and (f) are $X$ and the $y$ axes are energy $E$. The $E$ vs $X$ dispersions (a), (b), (e) and (f) correspond to 4 points A, B , E and F in the conduction plots (c) and (d). For the dispersion we have taken $\Delta=2$ and the conduction plots we have taken $\Delta= 0.01 \nu$. The dispersion plots are calculated with same values of $\nu$ and $V_0$. However as we have changed the $\Delta$ for the conductance calculation the window of energy in which Andreev bound states are formed get reduced to $-0.01 \nu$ to $+0.01 \nu$.
• Figure A1: In (a) we show the dispersion(energy vs guiding center) plot for an SNS junction for the case of $\nu=5.5$. These are the solutions of $\det(M_{SN})\det(M_{NS})=0$. We denote the LLs from $f_X$ by $n_e$ and LLs from $g_X$ by $n_h$. In (b) we show the dispersion for the case of a single barrier present in the QH region. We have taken $\nu=2.1$ and $d=2$. The maroon curve shows the effective potential $V_{eff}= (x-X)^2 /2 + V_0 \Theta (-x+d/2) \Theta(x+d/2)$ acting on the LLs for $X=0$. The value of the y axis in this curve is scaled appropriately to display with the dispersion plot and the $x$ axis is exact. The dotted red (blue) lines show the positions of the start of the lifting of degeneracy in $f_X$ ($g_X$) LLs due to barrier potential in the QH region. $n_e$ and $n_h$ are the same as in (a). $E_{b,e}^{(1)}$ corresponds to the eigenvalue inside the barrier region, and we have given its corresponding expression in the text. In (c), we show the dispersion of the monolayer graphene-based SQHS junction with the separation between the two superconductors $2 L=8$, $\mu=0.4$ and superconducting gap $\Delta_0=10$. Here we have measured the lengths in the units of magnetic length $l$ and energies in the units of $\frac{\hbar v_F}{l}$.
• Figure C1: (a) Edge current, (b) Andreev current and (c) Supercurrent calculated from Eq. \ref{['chcurr']} for the first three electron-like LL states for an NS junction with a barrier in the N region. In (d), we show the charge current from Eq. \ref{['chcr']} for an SNS junction with two barriers of $d=4$ and $D=4$. In (e), we show the charge current for $d=1$ and $D=1$ in an NS junction with two barriers in the N region.