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^{1,a)}, Myung-Ho Bae

^{1}and A. Bezryadin

^{1}

### Abstract

We expose superconductingnanowires to microwave radiation in order to study phase lock-in effects in quasi-one-dimensional superconductors. For sufficiently high microwave powers a resistive branch with Shapiro steps appears in the voltage-current characteristics. At frequencies in the range of 0.9–4 GHz these steps are of integer order only. At higher frequencies steps of 1/2, 1/3, 1/4, and even 1/6 order appear. We numerically model this behavior using a multivalued current-phase relationship for nanowires.

The authors thank R. Giannetta for useful discussions. This work was supported by the U.S. Department of Energy under Grant No. DE-FG02–07ER46453. This work was carried out in part in the Frederick Seitz Materials Research Laboratory Central Facilities, University of Illinois, which are partially supported by the U.S. Department of Energy under Grant Nos. DE-FG02–07ER46453 and DE-FG02–07ER46471. M.-H.B wants to thank the Korea Research Foundation Grant No. KRF-2006-352-C00020.

### Key Topics

- Nanowires
- 17.0
- Superconductivity
- 15.0
- Superconducting wires
- 8.0
- Critical currents
- 4.0
- Numerical modeling
- 4.0

## Figures

(a) Positive bias, , curves for sample A taken at with the switching current, , return current, , JNS, and PSC regimes indicated with arrows. Curve 1 (black) is measured at zero MW power and curves 2 (black) and 3 (red) are measured at and output MW powers at 3 GHz frequency. Inset: SEM image of sample A. (b) Normalized voltage, eV/hf vs. curves for the PSC regime in sample A for applied MW powers decreasing from top to bottom curves as follows (in decibels): , , , , and . These curves end where the wire switches to JNS. (c) Numerical simulations using the CPR given in the text for , , , , and , for decreasing from top curve to bottom curve as follows: 2.4, 2.0, 1.6, 0.8, and 0.6. The simulated curves show a similar shape to the experimental ones.

(a) Positive bias, , curves for sample A taken at with the switching current, , return current, , JNS, and PSC regimes indicated with arrows. Curve 1 (black) is measured at zero MW power and curves 2 (black) and 3 (red) are measured at and output MW powers at 3 GHz frequency. Inset: SEM image of sample A. (b) Normalized voltage, eV/hf vs. curves for the PSC regime in sample A for applied MW powers decreasing from top to bottom curves as follows (in decibels): , , , , and . These curves end where the wire switches to JNS. (c) Numerical simulations using the CPR given in the text for , , , , and , for decreasing from top curve to bottom curve as follows: 2.4, 2.0, 1.6, 0.8, and 0.6. The simulated curves show a similar shape to the experimental ones.

(a) for sample B taken at 9.5 GHz and 500 mK showing clear resonances. The boxed region represents the portion of the curve located between steps and . (b) The boxed region of the curve is shown here for various MW frequencies. The horizontal axis is (in arbitrary units). Each curve is shifted to line up with the MW frequency at which it was measured. The vertical axis is normalized voltage, 2 eV/hf. The frequencies starting from the left are (in gigahertz) as follows: 0.9, 2.7, 2.9, 5.4, 6.2, 8.2, 9.5, and 15. (c) Numerical simulation for all the resonances between the and steps using the CPR given by Eq. (2) as a function of the normalized frequency, , taken at and .

(a) for sample B taken at 9.5 GHz and 500 mK showing clear resonances. The boxed region represents the portion of the curve located between steps and . (b) The boxed region of the curve is shown here for various MW frequencies. The horizontal axis is (in arbitrary units). Each curve is shifted to line up with the MW frequency at which it was measured. The vertical axis is normalized voltage, 2 eV/hf. The frequencies starting from the left are (in gigahertz) as follows: 0.9, 2.7, 2.9, 5.4, 6.2, 8.2, 9.5, and 15. (c) Numerical simulation for all the resonances between the and steps using the CPR given by Eq. (2) as a function of the normalized frequency, , taken at and .

Schematic of the CPR used in numerical simulations. The CPR is shown as the black line, which represents supercurrent, , plotted versus the phase difference between the ends of the wire. When reaches , a phase slip occurs (indicated by the red arrows) and changes abruptly to a value given by the CPR at the resulting phase. Inset: Schematic multivalued representation of the same CPR. The solid curves are the stable branches which are separated by a phase difference of . The dotted line shows the unstable branches which are not used in our simulation. Here the red arrows show how a phase slip is equivalent to moving to the adjacent stable branch.

Schematic of the CPR used in numerical simulations. The CPR is shown as the black line, which represents supercurrent, , plotted versus the phase difference between the ends of the wire. When reaches , a phase slip occurs (indicated by the red arrows) and changes abruptly to a value given by the CPR at the resulting phase. Inset: Schematic multivalued representation of the same CPR. The solid curves are the stable branches which are separated by a phase difference of . The dotted line shows the unstable branches which are not used in our simulation. Here the red arrows show how a phase slip is equivalent to moving to the adjacent stable branch.

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