Nonlinear resonance of superconductor/normal metal structures to microwaves

We study the variation in the differential conductance $G=dj/dV$ of a normal metal wire in a superconductor/normal metal heterostructure with a cross geometry under external microwave radiation applied to the superconducting parts. Our theoretical treatment is based on the quasiclassical Green’s functions technique in the diffusive limit. Two limiting cases are considered: first, the limit of a weak proximity effect and low microwave frequency and second, the limit of a short dimension (short normal wire) and small irradiation amplitude.

Figure 1

Structure under consideration.

Figure 2

(Color online) Bias voltage dependence of the normalized variations in the resistance contributions $\delta R_{nN}/R_{nN}$ and $\delta R_n/R_n$. Parameter values: $\phi =\pi /3$, $L_y/L_x=1$, ${\epsilon }_N/\Delta =2.5\times {10}^{-2}$, ${\epsilon }_S/\Delta =5\times {10}^{-3}$, and $R_n/R_{nN}=1$.

Figure 3

(Color online) Bias voltage dependence of the normalized conductance variation $\delta G/G_n$. Parameter values: $\phi =\pi /3$, $L_y/L_x=1$, ${\epsilon }_N/\Delta =2.5\times {10}^{-2}$, and ${\epsilon }_S/\Delta =5\times {10}^{-3}$. Different cases: (a) $R_n/R_{nN}=0.5$, (b) $R_n/R_{nN}=1$, and (c) $R_n/R_{nN}=2$.

Figure 4

(Color online) Bias voltage dependence of the normalized time-averaged conductance variation $\delta G_{av}/G_n$. Parameter values: ${\phi }_0=\pi /3$, $L_y/L_x=1$, ${\epsilon }_N/\Delta =2.5\times {10}^{-2}$, ${\epsilon }_S/\Delta =5\times {10}^{-3}$, $eV/\Delta =5\times {10}^{-2}$, and $R_n/R_{nN}=1$.

Figure 5

(Color online) Normalized stationary differential conductance ${\tilde{G}}_0$ versus bias voltage $V$. Parameter values: $T/\Delta ={10}^{-2}$, ${\epsilon }_S/\Delta =0.1$, and ${\epsilon }_N/\Delta ={10}^{-2}$. Different cases: (a) ${\phi }_0=\pi /8$, (b) ${\phi }_0=\pi /4$, and (c) ${\phi }_0=3\pi /8$.

Figure 6

(Color online) Normalized second-order correction of differential conductance $\delta \tilde{G}$ versus bias voltage $V$. Parameter values: $T/\Delta ={10}^{-2}$, ${\epsilon }_S/\Delta =0.1$, ${\epsilon }_N/\Delta ={10}^{-2}$, and $\Omega /\Delta =5\times {10}^{-2}$. Different cases: (a) ${\phi }_0=\pi /8$, (b) ${\phi }_0=\pi /4$, and (c) ${\phi }_0=3\pi /8$.

Figure 7

(Color online) Normalized second-order correction of differential conductance $\delta \tilde{G}$ versus ac frequency $\Omega$. Parameter values: $T/\Delta =2\times {10}^{-3}$, ${\epsilon }_S/\Delta =0.1$, ${\epsilon }_N/\Delta ={10}^{-2}$, and $eV/\Delta ={10}^{-2}$. Different cases: (a) ${\phi }_0=\pi /6$, (b) ${\phi }_0=\pi /4$, and (c) ${\phi }_0=\pi /3$.

Figure 8

(Color online) Normalized second-order correction of differential conductance $\delta \tilde{G}$ versus ac frequency $\Omega$. Parameter values: $T/\Delta =2\times {10}^{-3}$, ${\epsilon }_S/\Delta =0.1$, $eV/\Delta ={10}^{-2}$, and ${\phi }_0=\pi /3$. Different cases: (a) ${\epsilon }_N/\Delta =5\times {10}^{-3}$, (b) ${\epsilon }_N/\Delta ={10}^{-2}$, and (c) ${\epsilon }_N/\Delta =2\times {10}^{-2}$.

Figure 9

(Color online) Normalized second-order correction of differential conductance $\delta \tilde{G}$ versus ac frequency $\Omega$, Parameter values: ${\epsilon }_S/\Delta =0.1$, ${\epsilon }_N/\Delta ={10}^{-2}$, $eV/\Delta ={10}^{-2}$, and ${\phi }_0=\pi /3$. Different cases: (a) $T=0$, (b) $T/\Delta =2\times {10}^{-3}$, (c) $T/\Delta =6\times {10}^{-3}$, and (d) $T/\Delta ={10}^{-2}$.

Figure 10

(Color online) Normalized second-order correction of differential conductance $\delta \tilde{G}$ versus stationary phase difference ${\phi }_0$. Parameter values: $T/\Delta =2\times {10}^{-3}$, ${\epsilon }_S/\Delta =0.1$, ${\epsilon }_N/\Delta ={10}^{-2}$, and $\Omega /\Delta ={10}^{-2}$. Different cases: (a) $V=0$, (b) $eV/\Delta =2\times {10}^{-2}$, and (c) $eV/\Delta =4\times {10}^{-2}$.