Orbital Josephson interference in a nanowire proximity-effect junction

A semiconductor nanowire based superconductor-normal-superconductor (SNS) junction is modeled theoretically. A magnetic field is applied along the nanowire axis, parallel to the current. The Bogoliubov-de Gennes equations for Andreev bound states are solved while considering the electronic subbands due to radial confinement in the $N$ section. The energy-versus-phase curves of the Andreev bound states shift in phase as the $N$-section quasiparticles with orbital angular momentum couple to the axial field. A similar phase shift is observed in the continuum current of the junction. The quantum mechanical result is shown to reduce to an intuitive, semiclassical model when the Andreev approximation holds. Numerical calculations of the critical current versus axial field reveal flux-aperiodic oscillations that we identify as a novel form of Josephson interference due to this orbital subband effect. This behavior is studied as a function of junction length and chemical potential. Finally, we discuss extensions to the model that may be useful for describing realistic devices.

Figure 1

(a) Schematic of the nanowire SNS junction of length $L$. It is modeled as a cylinder with a nanoscale diameter $d$ smaller than both the London penetration length and the $S$-section phase coherence length. An axial magnetic field $\mathit{B}=B_{\parallel }\widehat{\mathit{x}}$ penetrates the cylinder. (b) The superconducting order parameter has the magnitude ${\mathrm{\Delta }}_0$ in the $S$ section, and is zero in the normal section, with a jumplike variation at the boundaries.

Figure 2

Subband energies for the electrons in the $N$ section [i.e., the eigenenergies of $H_0$ in Eq. (1) at $k=0$] vs the normalized magnetic flux $\mathrm{\Phi }=\left(\pi B_{\parallel }{R}^2\right)/(h/e)$. The $l=0$ and $l=\pm 1$ subbands with $n=0$ are shown. The energies are parabolic because of the $\frac{{\hslash }^2}{2m^{*}{R}^2}{(l-\mathrm{\Phi })}^2$ contribution to the energy by $H_{\theta }$ [see Eqs. (1c) and (4a)]. The upper dashed line is the Fermi energy. The chemical potential $\mu$ is the Fermi energy measured from the bottom of the lowest subband, ${\zeta }_{0,0}$ (lower dashed line). We have assumed ${\zeta }_{0,\pm 1}={\zeta }_{0,0}$, so the bottoms of the $n=0$ subbands have the same energy. The effective chemical potential for electrons at flux $\mathrm{\Phi },{\mu }_{n,l}^e\left(\mathrm{\Phi }\right)$, is the difference between the subband energy and the Fermi energy at that flux. This is shown for the subband $(n,l)=(0,-1)$ at $\mathrm{\Phi }=1$. For the holelike states (not shown), the subband energies [Eq. (4b)] are inverted (mirrored) with respect to the $E=E_F$ line, and the $l$ quantum number negated $\left(l\to -l\right)$. The effective chemical potential for holes is given by ${\mu }_{n,l}^h\left(\mathrm{\Phi }\right)={\mu }_{n,-l}^e\left(\mathrm{\Phi }\right)$.

Figure 3

Eigenenergies of the Andreev bound states of a short $\left(L=25\mathrm{nm}\ll {\xi }_{n,l}^0\sim 200\mathrm{nm}\right)$, cylindrical SNS junction with no barriers at the $S-N$ interfaces, vs the superconducting phase difference $\chi$ of the leads. Two values of normalized magnetic flux, $\mathrm{\Phi }=\pi R^2{B}_{\parallel }/(h/e)=0,2.5$, are shown. We concentrate on one subband $(n,l)$ with $n=0$ and $l=1$. (a) Zero magnetic field, $\mathrm{\Phi }=0$. The energies correspond to the four allowed wave functions (defined in the main text), and the states are pairwise degenerate for all $\chi$. (b) $\mathrm{\Phi }=2.5$, where the degeneracies are lifted, and there is a phase shift with opposite directions for each pair of states. The phase shift is small because of the short length of the junction. The following parameters were used for both panels: $\mu =200\mathrm{meV},R=30\mathrm{nm},\text{and}T=100\mathrm{mK}$.

Figure 4

(a) bound-state current $I_{n,l}$, continuum current $J_{n,l}$, and the sum $I_{n,l}+J_{n,l}$ for the subband $(n,l)=(0,1)$ vs the superconducting phase $\chi$ at zero magnetic field, $\mathrm{\Phi }=0$. Since the junction is long, $L=500\mathrm{nm}\gtrsim {\xi }_{n,l}^0$, the CPR is triangular. The kinks in $I_{n,l},J_{n,l}$ at $\chi =0.05,0.95$ are due to Andreev bound states crossing the gap edge into the continuum levels, but the total subband current is a smooth function of $\chi$. (b) Total subband current $I_{n,l}+J_{n,l}$ vs the superconducting phase as a function of the normalized axial magnetic flux $\mathrm{\Phi }$. At zero flux the maximal value occurs near $\chi =\pi$. At finite flux, the bound-state and continuum currents are phase shifted. For $\mathrm{\Phi }=0.1$, two discontinuities can be seen in $I_{n,l}+J_{n,l}$ because of the phase shifts, and the maximal value no longer occurs near $\chi =\pi$. At $\mathrm{\Phi }=0.35$, the phase shifts amount to $2\pi$ and the zero-field curve is recovered. The maximal current at this flux is slightly smaller than the zero field case, due to a decrease in the average momentum of the Andreev quasiparticles with increasing flux. The following parameters were used in both panels: $L=500\mathrm{nm},\mu =8.5\mathrm{meV},R=30\mathrm{nm},\text{and}T=100\mathrm{mK}$.

Figure 5

(a) Critical current $I_c$ vs normalized axial magnetic flux $\mathrm{\Phi }$ of a 500-nm junction with $\mu =8.5\mathrm{meV}$. The subbands with $l=-1,0,1$ are occupied and contribute to $I_c$. Oscillation of $I_c$ with $\mathrm{\Phi }$ is observed, which is not periodic in $\mathrm{\Phi }$: the positions of the peaks get closer together as $\mathrm{\Phi }$ is increased. At flux points indicated with vertical dotted lines, individual subband currents are plotted in (b)–(f) vs the superconducting phase $\chi$. (b)–(f) CPR for $l=0$ (solid lines) and $l=\pm 1$ (dash-dotted lines) subbands. The current due to the $l=1$ subband is equal to that of the $l=-1$ subband for all $\chi$ and $\mathrm{\Phi }$ values. Here, the contribution due to only one of the two subbands is shown for clarity. The vertical dotted lines indicate the phase $\chi$ at which the critical current occurs. Note the difference in the $y$-axis scale between panel a and the other panels. The following parameters were used in all plots: $L=500\mathrm{nm},\mu =8.5\mathrm{meV},R=30\mathrm{nm},\text{and}T=100\mathrm{mK}$.

Figure 6

(a) Critical current $I_c$ vs normalized axial magnetic flux $\mathrm{\Phi }$ of a 500 nm junction with $\mu =20\mathrm{meV}$. The subbands with $\left|l\right|\le 3$ are occupied at zero flux. The subbands with $\left|l\right|=3,2$ depopulate at $\mathrm{\Phi }=0.2,1.2$, respectively. An aperiodic oscillation of $I_c$ vs $\mathrm{\Phi }$ is observed. At flux points indicated with vertical dotted lines, individual subband currents are plotted in (b)–(f) vs the superconducting phase $\chi$. (b)–(f) CPR for individual subbands, displaying different configurations which can lead to a peak in $I_c$ vs $\mathrm{\Phi }$. The vertical dotted lines indicate the phase $\chi$ at which the critical current occurs. Note the difference in the $y$-axis scale between panel a and the other panels. The following parameters were used in all panels: $L=500\mathrm{nm},\mu =20\mathrm{meV},R=30\mathrm{nm},\text{and}T=100\mathrm{mK}$.

Figure 7

Normalized critical current of the junction vs the normalized magnetic flux $\mathrm{\Phi }=\left(\pi B_{\parallel }{R}^2\right)/(h/e)$, for $\mu =8.5\mathrm{meV}$ [(a)–(d)] and $\mu =20\mathrm{meV}$ [(e)–(h)], and several values of the junction length $L$. The following parameters were used: $R=30\mathrm{nm}\text{and}T=100\mathrm{mK}$. In (a)–(d), the vertical line at $\mathrm{\Phi }=1.0$ indicates the depopulation of the $\left|l\right|=1$ subbands. Similarly in (e)–(h), the vertical lines at $\mathrm{\Phi }=0.2,1.2,2.2$ indicate the depopulation of the $\left|l\right|=3,2,1$ subbands, respectively.