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arXiv:0910.1337 · Physical Review B

Current-phase relation and Josephson inductance in a superconducting Cooper-pair transistor

Antti Paila, David Gunnarsson, Jayanta Sarkar, Mika A. Sillanpää, Pertti Juhani Hakonen

Published Oct 21, 2009• 11 citations
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Abstract

We have investigated the Josephson inductance ${L}_{J}$ of a superconducting Cooper pair transistor (SCPT). We traced the inductance using microwave reflection measurements on a tuned resonance circuit in which a SCPT was mounted in parallel to a $\ensuremath{\sim}200\text{ }\text{pH}$ strip line inductance. When the inverse of the Josephson inductance, determined on the charge-phase bias plane for a SCPT with a Josephson to Coulomb energy ratio of ${E}_{J}/{E}_{C}=1.75$, is integrated over the phase, we obtain a current-phase relation, which is strongly nonsinusoidal near the charge degeneracy point.

DOI
10.1103/physrevb.80.144520

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We have investigated the Josephson inductance LJ{L}_{J}LJ​ of a superconducting Cooper pair transistor (SCPT). We traced the inductance using microwave reflection measurements on a tuned resonance circuit in which a SCPT was mounted in parallel to a \ensuremath∼200 pH\ensuremath{\sim}200\text{ }\text{pH}\ensuremath∼200 pH strip line inductance. When the inverse of the Josephson inductance, determined on the charge-phase bias plane for a SCPT with a Josephson to Coulomb energy ratio of EJ/EC=1.75{E}_{J}/{E}_{C}=1.75EJ​/EC​=1.75, is integrated over the phase, we obtain a current-phase relation, which is strongly nonsinusoidal near the charge degeneracy point.
We have studied dynamic properties of a special Josephson charge qubit. The device consists of a single-Cooper-pair transistor (with capacitive gate) closed by a superconducting inductive loop. The phase bias is implemented by an external flux that threads the loop. Restricting ourselves to a two-band model, we have derived analytical expressions for the oppositely circulating ring currents in terms of gate charge and phase bias. They permit the determination of important device characteristics such as critical current, loop parameters, and Josephson inductance. For the purpose of measurements in connection with macroscopic quantum coherence, we investigate the qubit's microwave response within the frame of Bloch equations with phenomenological relaxation rates. We find the (nonlinear) qubit impedance in terms of quality, detuning, and the quantum resistance unit, which reflects the competition between irradiation and dephasing.
We have studied the microwave response of a single Cooper-pair transistor (CPT) coupled to a lumped-element microwave resonator. The resonance frequency of this circuit, fr{f}_{r}fr​, was measured as a function of the charge ng{n}_{g}ng​ induced on the CPT island by the gate electrode and the phase difference across the CPT, ${\ensuremath{\varphi}}_{B}$, which was controlled by the magnetic flux in the superconducting loop containing the CPT. The observed ${f}_{r}({n}_{g},{\ensuremath{\varphi}}_{B})$ dependencies reflect the variations of the CPT Josephson inductance with ng{n}_{g}ng​ and ${\ensuremath{\varphi}}_{B}$ as well as the CPT excitation when the microwaves induce transitions between different quantum states of the CPT. The results are in excellent agreement with our simulations based on the numerical diagonalization of the circuit Hamiltonian. This agreement over the whole range of ng{n}_{g}ng​ and ${\ensuremath{\varphi}}_{B}$ is unexpected, because the relevant energies vary widely, from 0.1 to 3 K. The sensitivity of the CPT as an electrometer is peaked when the CPT excitation level approaches that of the microwave resonator.
We measure the high-frequency emission of a single Cooper pair transistor (SCPT) in the regime where transport is only due to tunneling of Cooper pairs. This is achieved by coupling on chip the SCPT to a superconductor-insulator-superconductor junction and by measuring the photon assisted tunneling current of quasiparticles across the junction. This technique allows a direct detection of the ac Josephson effect of the SCPT and provides evidence of Landau-Zener transitions for proper gate voltage. The emission in the regime of resonant Cooper pair tunneling is also investigated. It is interpreted in terms of transitions between charge states coupled by the Josephson effect.
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The rf impedance of superconducting point contacts has been measured as a function of the quantum-mechanical phase difference $\ensuremath{\varphi}$ across the point contact. By representing the point contact as a parallel combination of an inductor $\mathfrak{L}(\ensuremath{\varphi})=(\frac{\ensuremath{\hbar}}{2e}){(\frac{\ensuremath{\partial}{I}_{p}}{\ensuremath{\partial}\ensuremath{\varphi}})}^{\ensuremath{-}1}$ and a resistor $R\ensuremath{\equiv}\frac{1}{G(\ensuremath{\varphi})}$ the current-phase relation ${I}_{p}(\ensuremath{\varphi})$ and the phase-dependent conductance $G(\ensuremath{\varphi})$ have been determined from measurements at 30 MHz on phase-biased niobium point contacts. For point contacts with sufficiently small critical current Ic{I}_{c}Ic​, the inductance was $\mathfrak{L}(\ensuremath{\varphi})=\frac{\ensuremath{\hbar}}{2e{I}_{c}cos\ensuremath{\varphi}}$ yielding the expected negative inductance branch for $\frac{\ensuremath{\pi}}{2}\ensuremath{\le}\ensuremath{\varphi}\ensuremath{\le}\ensuremath{\pi}$ and a sinusoidal current-phase relation. For larger critical currents there were departures from the sinusoidal form for the measured ${I}_{p}(\ensuremath{\varphi})$. There was a phase-dependent conductance that is an increasing function of $\ensuremath{\varphi}$ corresponding to a negative coefficient for the $cos\ensuremath{\varphi}$ term in the Josephson current.
We investigate the switching current of a Cooper pair transistor with tunable Josephson energy. The junctions are fabricated in a superconducting quantum interference device (SQUID) geometry which allows for an in situ tunable effective Josephson energy by application of a magnetic field. We find a 2e2e2e-periodic switching current versus gate charge. As the magnetic field is increased the switching current stays 2e2e2e-periodic but the magnitude is suppressed. At a magnetic field of half a flux quantum through the SQUID's the switching current is minimum. We can theoretically model the experimental data by assuming a switching current which is proportional to the ideal critical current squarred. We show that such a dependence is expected in the limit where the effect of thermal fluctuations on the system is strong.