Odd-frequency pairing in a superconductor coupled to two parallel nanowires

We study the behavior of Cooper pair amplitudes that emerge when a two-dimensional superconductor is coupled to two parallel nanowires, focusing on the conditions for realizing odd-frequency pair amplitudes in the absence of spin-orbit coupling or magnetism. In general, any finite tunneling between the superconductor and the two nanowires induces odd-frequency spin-singlet pair amplitudes in the substrate as well as a substantial odd-frequency interwire pairing, both of which vanish locally. Interestingly, in the regime of strong superconductor-nanowire tunneling, we find that the presence of two nanowires allows for the conversion of nonlocal odd-frequency pairing to local even-frequency pairing. By studying this higher-order symmetry conversion process, we are able to identify a notable effect of the odd-frequency pairing in the superconductor on local quantities accessible by experiments. Specifically, we find that the odd-frequency pairing plays a direct role in the emergence of certain subgap features in the local density of states and, importantly, it is responsible for a reduction of the maximum Josephson current between the two nanowires, measurable using Josephson scanning tunneling microscopy. We discuss ways to control the sizes of these effects induced by odd-frequency superconductivity by tuning the parameters describing the nanowires.

Figure 1

Schematic of the system described by the Hamiltonian in Eq. (1): two parallel nanowires, separated by a distance $d$, coupled to the 2D surface of a conventional spin-singlet $s$-wave superconductor. Throughout this work the nanowires are assumed to be infinitely long 1D objects in the $y$ direction and the superconductor is assumed to span the entire 2D plane.

Figure 2

Absolute magnitude of the ratio of odd-$\omega$ to even-$\omega$ pair amplitudes in the interwire channel $|\delta F_{o:e}|$, computed using Eq. (11), with $k_y$ fixed and units chosen such that ${\xi }_L=1$. (a) $|\delta F_{o:e}\left(\omega \right)|$ for ${\xi }_R=-3{\xi }_L$, with vertical dotted lines indicating frequencies for which the ratio equals unity: $\omega =\{\pm {\xi }_{L,k_y},\pm {\xi }_{R,k_y}\}$. (b) $|\delta F_{o:e}\left(\omega \right)|$ for ${\xi }_R=3{\xi }_L$, with vertical dotted lines indicating frequencies with strictly odd-$\omega$ pairing: $\omega =\pm \sqrt{{\xi }_{L,k_y}{\xi }_{R,k_y}}$.

Figure 3

Magnitude of even-$\omega$ (left column) and odd-$\omega$ (right column) terms in the retarded pair amplitude of the superconducting substrate, $|F_s^{\left(L\right)}(x-\frac{d}{2},-\frac{d}{2},z=\omega +i\eta )|$, plotted as a function of the distance from the nanowire $x$, and the frequency above the Fermi level $\omega$. Amplitudes are computed by numerically integrating the exact expression in Eq. (18) over $k_y$. We set the parameters of the substrate so that $\mathrm{\Delta }={\mu }_s/100$, and we report distances in $k_F^{-1}$, where $k_F=\sqrt{2m_s{\mu }_s}$. For all plots $m_L=m_s$ and ${\mu }_L=\mathrm{\Delta }/2$, while the three rows represent different tunneling parameter choices: $t_L=0$ [(a), (b)], $t_L={\mu }_s/100$ [(c), (d)], and $t_L={\mu }_s/10$ [(e), (f)].

Figure 4

Magnitude of even-$\omega$ (left column) and odd-$\omega$ (right column) terms in the retarded pair amplitude of the superconducting substrate, $|F_s^{\left(L\right)}(x-\frac{d}{2},-\frac{d}{2},z=\omega +i\eta )|$, plotted as a function of frequency, $\omega$, and evaluated at the positions at which the pair amplitude peaks, $x=0$ and $x=\pi /\left(2k_F\right)$, respectively, where $k_F=\sqrt{2m_s{\mu }_s}$. The amplitudes are computed by numerically integrating the exact expression in Eq. (18) over $k_y$. We set the parameters of the substrate so that $\mathrm{\Delta }={\mu }_s/100$, and we have fixed $t_L={\mu }_s/10$ to obtain a large odd-$\omega$ component. (a), (b): Fixed ${\mu }_L=\mathrm{\Delta }/2$ for different masses, $m_L=m_S/4$ (solid/red), $m_L=m_S/2$ (dash-dotted/green), $m_L=m_S$ (dash-dotted/cyan), $m_L=2m_S$ (dotted/blue), and $m_L=4m_S$ (dotted/magenta). (c), (d): Fixed $m_L=m_S$ for different chemical potentials, ${\mu }_L=\mathrm{\Delta }/4$ (solid/red), ${\mu }_L=\mathrm{\Delta }/2$ (dashed/green), ${\mu }_L=\mathrm{\Delta }$ (dash-dotted/cyan), ${\mu }_L=5\mathrm{\Delta }$ (dash-dotted/blue), and ${\mu }_L=10\mathrm{\Delta }$ (dotted/magenta). In each panel we also plot the results for $t_L=0$ (solid/black).

Figure 5

Density plots in the $x,\omega$ plane of real (a) and imaginary (b) parts of the retarded local pair amplitude of the superconducting substrate for the $\mathrm{L}+\mathrm{SC}+\mathrm{R}$ system, $F_s(x,z=\omega +i\eta )$, given by Eq. (26); real (c) and imaginary (d) parts of the even-$\omega$ contributions, $F_e(x,z=\omega +i\eta )$, given by Eq. (27); and real (e) and imaginary (f) parts of the odd-$\omega$ contributions, $F_o\left(x\right)$, given by Eq. (27). We use the same units and parameters as in Figs. 3, 3: ${\mu }_L={\mu }_R=\mathrm{\Delta }/2,m_L=m_R=m_s$, and $d=\pi k_F^{-1}$, where $k_F=\sqrt{2m_s{\mu }_s},\mathrm{\Delta }={\mu }_s/100$, and $t_L=t_R={\mu }_s/10$.

Figure 6

Density plots in the ${\mu }_R,\omega$ plane of the absolute magnitude of $|F_s(x=0,\omega +i\eta )|$, given by Eq. (26), (a) and (b); $|F_e(x=0,\omega +i\eta )|$, given by Eq. (27), (c) and (d); and $|F_o(x=0,\omega +i\eta )|$, given by Eq. (27), (e) and (f). In each case, we have fixed the model parameters such that $m_L=m_s,{\mu }_L=\mathrm{\Delta }/2,\mathrm{\Delta }={\mu }_s/100,d=\pi k_F^{-1}$, where $k_F=\sqrt{2m_s{\mu }_s}$, and plotted each function for two different values of $m_R$, as indicated in each panel.

Figure 7

Density plots in the $m_R,\omega$ plane of the absolute magnitude of $|F_s(x=0,\omega +i\eta )|$, given by Eq. (26), (a) and (b); $|F_e(x=0,\omega +i\eta )|$, given by Eq. (27), (c) and (d); and $|F_o(x=0,\omega +i\eta )|$, given by Eq. (27), (e) and (f). In each case, we have fixed the model parameters such that $m_L=m_s,{\mu }_L=\mathrm{\Delta }/2,\mathrm{\Delta }={\mu }_s/100,d=\pi k_F^{-1}$, where $k_F=\sqrt{2m_s{\mu }_s}$, and plotted each function for two different values of ${\mu }_R$, as indicated in each panel.

Figure 8

Density plots in the $d,\omega$ plane of the absolute magnitude of $|F_s(x=0,\omega +i\eta )|$, given by Eq. (26), (a) and (b); $|F_e(x=0,\omega +i\eta )|$, given by Eq. (27), (c) and (d); and $|F_o(x=0,\omega +i\eta )|$, given by Eq. (27), (e) and (f). In each case, we have fixed the model parameters such that $m_R=m_L=m_s,{\mu }_L=\mathrm{\Delta }/2,\mathrm{\Delta }={\mu }_s/100$; distances are given in units of $k_F^{-1}=1/\sqrt{2m_s{\mu }_s}$, and each function is plotted for two different values of ${\mu }_R$, as indicated in each panel.

Figure 9

Density plots of the contributions to the LDOS in the ${\mu }_R,\omega$ plane. (a) and (b) the total LDOS, $\mathcal{N}(x=0;\omega )$, given by Eq. (29); (c) and (d) the even-$\omega$ contribution, ${\mathcal{N}}_e(x=0;\omega )$, given by Eq. (31); and (e) and (f) the odd-$\omega$ contribution, ${\mathcal{N}}_o(x=0;\omega )$, given by Eq. (30). In each case, we have fixed the model parameters to the same values as Fig. 6.

Figure 10

Density plots of the contributions to the LDOS in the $m_R,\omega$ plane. (a) and (b) the total LDOS, $\mathcal{N}(x=0;\omega )$, given by Eq. (29); (c) and (d) the even-$\omega$ contribution, ${\mathcal{N}}_e(x=0;\omega )$, given by Eq. (31); and (e) and (f) the odd-$\omega$ contribution, ${\mathcal{N}}_o(x=0;\omega )$, given by Eq. (30). In each case, we have fixed the model parameters to the same values as Fig. 7.

Figure 11

Density plots of the contributions to the LDOS in the $d,\omega$ plane. (a) and (b) the total LDOS, $\mathcal{N}(x=0;\omega )$, given by Eq. (29); (c) and (d) the even-$\omega$ contribution, ${\mathcal{N}}_e(x=0;\omega )$, given by Eq. (31); and (e) and (f) the odd-$\omega$ contribution, ${\mathcal{N}}_o(x=0;\omega )$, given by Eq. (30). In each case, we have fixed the model parameters to the same values as Fig. 8.

Figure 12

Maximum value of the LJC ($\mathrm{\Phi }=\pi /2$) (solid/black) given by Eq. (33) and separated into even-$\omega$ contributions (dotted/red) given by $I_e$ and odd-$\omega$ contributions (dashed/blue) given by $I_o$ in Eq. (34). The horizontal dashed line represents the LJC in the absence of the nanowires. We have fixed the model parameters such that $m_L=m_R=m_s,{\mu }_L=\mathrm{\Delta }/2,\mathrm{\Delta }={\mu }_s/100,d=\pi k_F^{-1}$, where $k_F=\sqrt{2m_s{\mu }_s}$, and use two different values of ${\mu }_R$, as indicated in each panel.

Figure 13

Maximum value of the LJC ($\mathrm{\Phi }=\pi /2$) (solid/black) given by Eq. (33) and separated into even-$\omega$ contributions (dotted/red) given by $I_e$ and odd-$\omega$ contributions (dashed/blue) given by $I_o$ in Eq. (34). The horizontal dashed line represents the LJC in the absence of the nanowires. We have fixed the position $x=0$ to focus on the suppression of $I_J$ relative to $I_e$, and we have set $m_L=m_s,{\mu }_L=\mathrm{\Delta }/2,\mathrm{\Delta }={\mu }_s/100$, throughout, and present all distances in units of $k_F^{-1}=1/\sqrt{2m_s{\mu }_s}$. In each panel we plot these functions with respect to one of the model parameters: $m_R,{\mu }_R$, and $d$. In each panel the fixed values of the other two parameters are indicated.

Figure 14

Maximum value of the LJC ($\mathrm{\Phi }=\pi /2$) (solid/black) given by Eq. (33) and separated into even-$\omega$ contributions (dotted/red) given by $I_e$ and odd-$\omega$ contributions (dashed/blue) given by $I_o$ in Eq. (34). The horizontal dashed line represents the LJC in the absence of the nanowires. We have fixed the position $x=0$ to focus on the suppression of $I_J$ relative to $I_e$, and, to highlight the effect of the odd-$\omega$ pairing, we have set $m_L=8m_s$; all other parameters are the same as Fig. 13 except where specified. In each panel we plot these functions with respect to one of the model parameters: $m_R,{\mu }_R$, and $d$. The fixed values of the other two parameters are indicated.