Half-Integer Shapiro Steps at the $0\mathrm{\text{−}}\pi$ Crossover of a Ferromagnetic Josephson Junction

We investigate the current-phase relation of $\mathrm{S}/\mathrm{F}/\mathrm{S}$ junctions near the crossover between the 0 and the $\pi$ ground states. We use $\mathrm{Nb}/\mathrm{CuNi}/\mathrm{Nb}$ junctions where this crossover is driven both by thickness and temperature. For a certain thickness a nonzero minimum of critical current is observed at the crossover temperature. We analyze this residual supercurrent by applying a high frequency excitation and observe the formation of half-integer Shapiro steps. We attribute these fractional steps to a doubling of the Josephson frequency due to a $\sin (2\varphi )$ current-phase relation. This phase dependence is explained by the splitting of the energy levels in the ferromagnetic exchange field.

Figure 1

(a) Energy of Andreev levels versus phase difference in a one-dimensional ballistic junction ($E_{Th}=\hslash v_F/d\ll \Delta$) [4]. Each state of the normal case (thin lines) is split by the exchange energy $E_{\mathrm{ex}}$ of the ferromagnet (thick lines). The states in solid (dashed) lines carry positive (negative) currents. (b) Current-phase relation for this model at zero temperature in the normal case (thin line), in $\pi$ junctions (dotted line), and at the crossover (thick line). (c) Current-phase relation in diffusive regime for the same situations.

Figure 2

Temperature dependence of the critical current for two $\mathrm{Nb}/{\mathrm{Cu}}_{52}{\mathrm{Ni}}_{48}/\mathrm{Nb}$ junctions. The upper graphs are expanded views near the crossovers between the 0 and $\pi$ states: the critical current has a nonzero (zero) minimum for the 17 nm (19 nm) thick junction.

Figure 3

Integer Shapiro steps in the voltage-current curve of a 19 nm thick junction with an excitation at 800 kHz (amplitude about $18\mu \mathrm{A}$). The steps disappear at the $0\mathrm{\text{−}}\pi$ crossover temperature $T^{*}$. Curves at 5.21 and 6.80 K are shifted by 10 and $20\mu \mathrm{A}$ for clarity.

Figure 4

Shapiro steps in the voltage-current curve of a 17 nm thick junction with an excitation at 800 kHz (amplitude about $18\mu \mathrm{A}$). Half-integer steps ($n=1/2$ and $n=3/2$) appear at the $0\mathrm{\text{−}}\pi$ crossover temperature $T^{*}$. Curves at 1.10 and 1.07 K are shifted by 10 and $20\mu \mathrm{A}$ for clarity.

Figure 5

(a) Shapiro steps in the voltage-current curves of the 17 nm thick junction at 1.12 K with an excitation at 800 kHz. Excitation amplitudes are about 18, 9, 5, and $0\mu \mathrm{A}$. The respective curves are shifted by 0, 5, 10, and $15\mu \mathrm{A}$ for clarity. (b) Width of the integer and half-integer steps versus normalized amplitude of the alternative current. Experimental values (symbols) are compared with numerical simulations for two current-phase relations containing a dominant $\sin (2\varphi )$ component (lines). All graphs have the same axis and scales.