Experimental evidence of a Josephson junction
We demonstrate experimentally the existence of Josephson junctions having a doubly degenerate ground state with an average Josephson phase . The value of can be chosen by design in the interval . The junctions used in our experiments are fabricated as 0- Josephson junctions of moderate normalized length with asymmetric 0 and regions. We show that (a) these Josephson junctions have two critical currents, corresponding to the escape of the phase from and states; (b) the phase can be set to a particular state by tuning an external magnetic field or (c) by using a proper bias current sweep sequence. The experimental observations are in agreement with previous theoretical predictions.
Josephson junctions (JJs) with a phase shift of in the ground stateBulaevskiĭ et al. (1977) attracted a lot of interest in recent yearsBaselmans et al. (1999); Ryazanov et al. (2001a); Kontos et al. (2002); Weides et al. (2006a); van Dam et al. (2006); Gumann et al. (2007). In particular, these JJs can be used as on-chip phase batteries for biasing various classicalOrtlepp et al. (2006) and quantumFeofanov et al. (2010) circuits, allowing for removing external bias lines and reducing decoherence. Currently, it is possible to fabricate simultaneously both 0 and JJs using various technologies such as superconductor-ferromagnet heterostructuresRyazanov et al. (2001b); Weides et al. (2006b); Frolov et al. (2008); Weides et al. (2007); Pfeiffer et al. (2008) or JJs based on d-wave superconductorsVan Harlingen (1995); Tsuei and Kirtley (2000); Smilde et al. (2002); Gürlich et al. (2009).
It would be remarkable to have a phase battery providing an arbitrary phase shift , rather than just 0 or . The simplest idea is to combine 0 and JJs to obtain a JJ. However, this is not as straightforward as it may seem. The balance between 0 and parts is complicated as shown in the pioneering workBulaevskii et al. (1978), where conditions for having a non-trivial -state were derived. Long artificialBuzdin and Koshelev (2003); Pugach et al. (2010) and naturalMints (1998); Il’ichev et al. (1999); Mints et al. (2002) arrays of -0--0-- JJs with short segments were analyzed in detail and suggested as systems, where a JJ could be realized. More recently, only one period of such an array, i.e., one 0- JJ, was analyzed in an external magnetic fieldGoldobin et al. (2011). In these works, the Josephson phase is considered as a sum of a constant (or slowly varying) phase and a deviation from the average phase. Then, for the average phase one obtains an effective current-phase relation (CPR) for the supercurrentGoldobin et al. (2011)
where the averaged value of the critical current . The CPR (1) exactly corresponds to a JJ at zero normalized magnetic field , if , cf. Fig.1(a). Here is the total length of JJ, while and are the lengths of 0 and parts, accordingly, is the width of JJ.
It is worth noting that the term “ JJ”, introduced in Ref. Buzdin and Koshelev, 2003, refers to a JJ with a degenerate ground state phase . In the particular case of Eq. (1) at one has . The coefficients and are defined asLipman et al. (2012)
where , are the lengths normalized to the Josephson length and , are the critical current densities of 0 and parts, respectively. Here is calculated using the average value of the critical current .
The physics of JJs with a CPR given by Eq. (1) is quite unusualGoldobin et al. (2007). In particular, one should observe two critical currentsGoldobin et al. (2007, 2011) at , corresponding to the escape of the phase from the left () or the right () well of the double-well Josephson energy potential
where . The critical currents are different because the maximum slope (maximum supercurrent in Fig. 1(a)) on the rhs (positive bias) of the well is smaller than the maximum slope (maximum supercurrent in Fig. 1(a)) on the rhs of the well, see Fig. 1(b).
In this letter we present experimental evidences of a JJ made of one 0 and one segment, see Fig. 2(a).
The samples were fabricated as NbAl-AlONiCuNb heterostructuresWeides et al. (2006b, 2010). These superconductor-insulator-ferromagnet-superconductor (SIFS) JJs have an overlap geometry as shown in Fig. 2(b). Each junction consists of two parts, a conventional 0-segment and a -segment. It is well knownKontos et al. (2002); Oboznov et al. (2006); Weides et al. (2006a); Buzdin (2005) that the critical current in SFS or SIFS JJs strongly depends on the thickness of the F-layer and can become negative within some range of ( junction). Therefore, to produce the 0 and the segments, the F-layer has different thicknesses and , as shown in Fig. 2(a). To achieve this, the F-layer of thickness , corresponding to ( JJ) was fabricated first. Then the area indicated in Fig. 2(b) was etched down to , corresponding to (0 JJ). Usually one obtains asymetric valuesKemmler et al. (2010), i.e. .
We have studied 3 samples. One of them has very little asymmetry of about 5%, and , which corresponds to . In this case even 5% of asymmetry brings the sample out of the domain, see Fig. 3 for a qualitative picture. The other two JJs have and are deep inside the -domain in parameter space. Both samples show similar results. Here we present the results obtained on one of them. Its parameters are summarized in Tab. 1 and its position within the domain is indicated in Fig. 3. The values of and cannot be measured directly. In our case we have measured the dependence (see below) and then simulated it numerically using and as fitting parameters. The best fitting was obtained for the values specified in Tab. 1.
|, , ()|
According to theoretical predictionsGoldobin et al. (2007, 2011) for a JJ at zero magnetic field one expects two critical currents (for each bias direction), corresponding to the escape of the phase from and wells of , respectively for bias current and vice versa for . The current is always observed. To observe one has to have low damping so that retrapping in the well is avoided (for positive ). The – characteristic (IVC) of the investigated 0- JJ at measured at show only one critical current. Therefore the experiments were performed in a He cryostat at down to where the damping in SIFS JJs reduces drasticallyPfeiffer et al. (2008). In the temperature range from down to both and are clearly visible in the IVCs as shown in Fig. 4.
EarlierGoldobin et al. (2007) we proposed a technique that allows to choose which critical current one traces in the IVC, i.e. from which well the phase escapes. The control is done by choosing a proper bias sweep sequence. For example, if the junction is returning from the positive voltage state, the potential is tilted so that the phase slides to the right. When the tilt becomes small enough the phase will be trapped, presumably in the right well, cf. Fig. 1(c). However, this natural assumption is not always true (see below). Then, if the phase is trapped at , we can sweep the bias (a) in the positive direction and will observe escape from (to the right) at or (b) in the negative direction to observe escape from (to the left) at .
In experiment, at , when the damping is very low, the currents and are traced in random order. Recording one IVC after the other, we were able to obtain IVCs with all 4 possible combinations: (a) , (b) , (c) , (d) . Choosing a specific sweep sequence as described above does not make the outcome ( or ) predictable. We believe that in this temperature range the damping is so low that, upon returning from the positive voltage state, the phase does not simply stop in the well, but can also reflect from the barrier and find itself in a well, cf. Fig. 1(c). The absence of determinism suggests that most probably we are dealing with a system exhibiting chaotic dynamics. This issue will be investigated elsewhere. Nevertheless, at we managed to achieve deterministic behavior as described above, see Fig. 4.
Another fingerprint of a JJ is its dependence, which (a) should have the main cusp-like minima shifted off from point-symmetrically with respect to the origin, see Fig. 4 of Ref. Goldobin et al., 2011; (b) should show up to four branches in total (for both sweep polarities) at low magnetic fieldGoldobin et al. (2011). In essence, the latter feature results from the escape of the phase from two different energy minima in two different directions and is an extension of the two-critical-currents story to the case of non-zero magnetic field. Instead, the feature (a) alone cannot serve as a proof of a JJ as it is a common feature of every asymmetric 0- JJ even if its ground state is 0 or Goldobin et al. (2011).
The experimentally obtained dependence at is shown in Fig. 5. First, in the whole temperature range – we observe the main minima shifted point-symmetrically to a finite field, which is . Second, at one observes the crossing of two branches and two critical currents for each bias current polarity. The left hand branches correspond to the escape of the phase out of the -well, while the right branch corresponds to the escape from the -well. Note, that at the crossing of branches is not visible. One just observes a cusp-like minimum. In this case, both cannot be seen together and the existence of two states, as should be present for a -JJ, cannot be proven.
To trace the intersection of the branches better we were applying a special value of the magnetic field during reset from the voltage state back to state, to “prepare” a specific state ( or ) of the system. Then the field was set back to the “current” value and the bias was ramped up to trace . By doing this for different values of , one is able to trace the intersection of the branches much better than without this technique, see Fig. 5. Still, in experiment we were not able to trace the branches for positive and negative up to the point where they meet, as shown in Fig. 4(a) and (b) of Ref. Goldobin et al., 2011, probably because of retrapping.
To extract some parameters of our JJ from the curve, we performed numerical simulations, by solving the full sine-Gordon equation for a 0- JJ, using the normalized length and the critical currents and as fitting parameters. The simulations were performed using StkJJGoldobin (2011). The objective was to obtain the best fit close to the origin, especially the point where the branches of cross. The best fit is shown in Fig. 5 and was obtained for and the values and from Tab. 1. Measured data in Fig. 5 are shifted along axis by to compensate for average remanent magnetization of the F-layer. To obtain an almost perfect fit we have also assumed a difference in the constant remanent magnetizations in the 0 and in the parts of , similar to earlier studiesKemmler et al. (2010); Weides et al. (2010). One can also see that one of the experimental branches after the main maximum runs parallel to the simulated curve (the experimental is not point symmetric). This shift stays the same even if we cycle the sample through of Nb, but changes if we cycle through the Curie temperature of the F-layer. We conclude that this is related to the non-uniform remanent magnetization of the F-layer.
The parameters obtained from the fit allow to see the location of our JJ on the plane, see Fig. 3. One sees that the JJ is not really short and lays quite deep in the domain. In this region the analytic resultsGoldobin et al. (2011); Lipman et al. (2012) and, in particular, Eqs. (1) and (3) are valid only qualitatively, cf., Fig. 3 in Ref. Goldobin et al., 2011. Therefore we calculated the CPR of our JJ numerically. We started with a 0- JJ with parameters , , and obtained from the fit, assumed some applied bias current and found all static solutions numerically. Then for each of these solutions we calculated and plotted all those on a plot. By repeating this procedure for different , we obtain an effective CPR shown in Fig. 1(a) together with the curve produced by the analytical formula (1). One can see that the exact effective CPR calculated numerically and the approximate CPR calculated analytically are qualitatively similar. From the numerical CPR the value of for our particular JJ is extracted.
In conclusion, we have demonstrated experimentally the realization of a Josephson junction based on a - SIFS junction. We have observed experimentally a shift of the minimum according to the effective CPR (1), as predictedGoldobin et al. (2011), as well as two critical currents close to zero magnetic field for each bias current polarity. We showed that one can choose between the states of the system using an externally applied magnetic field, which removes their degeneracy. We also showed that one can bring the system into one of the two states by properly tilting the potential using the bias current. Depending on the damping (temperature) one can achieve deterministic behavior (when damping is small enough to see the lower critical current, but large enough to trap the phase in a particular well) as well as random behavior at very low damping. The obtained JJ has at ).
We acknowledge financial support by the German Israeli Foundation (Grant No. G-967-126.14/2007) and DFG (via SFB/TRR-21, project A5 as well via project Go-1106/3). H.S. gratefully acknowledges support by the Evangelisches Studienwerk e.V. Villigst
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