arXiv:cond-mat/0502657 · Physical Review B

Quantum charge fluctuations in a superconducting grain

Manuel Houzet, D. A. Pesin, Anton V. Andreev, L. I. Glazman

Published Sep 9, 2005• 12 citations

Abstract

We consider charge quantization in a small superconducting grain that is contacted by a normal-metal electrode and is controlled by a capacitively coupled gate. At zero temperature and zero conductance $G$ between the grain and the electrode, the charge $Q$ as a function of the gate voltage ${V}_{g}$ changes in steps. The step height is $e$ if $\ensuremath{\Delta}<{E}_{c}$, where $\ensuremath{\Delta}$ and ${E}_{c}$ are, respectively, the superconducting gap and the charging energy of the grain. Quantum charge fluctuations at finite conductance remove the discontinuity in the dependence of $Q$ on ${V}_{g}$ and lead to a finite step width $\ensuremath{\propto}{G}^{2}\ensuremath{\Delta}$. The resulting shape of the Coulomb blockade staircase is of a novel type. The grain charge is a continuous function of ${V}_{g}$ while the differential capacitance, $dQ∕d{V}_{g}$, has discontinuities at certain values of the gate voltage. We determine analytically the shape of the Coulomb blockade staircase also at nonzero temperatures.

DOI: 10.1103/physrevb.72.104507

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arXiv:cond-mat/0502657 · Physical Review B

Quantum charge fluctuations in a superconducting grain

Manuel Houzet, D. A. Pesin, Anton V. Andreev, L. I. Glazman

Published Sep 9, 2005• 12 citations

arXiv HTML PDF

Abstract

G

between the grain and the electrode, the charge

Q

as a function of the gate voltage

{V}_{g}

changes in steps. The step height is

e

\ensuremath{\Delta}<{E}_{c}

, where

\ensuremath{\Delta}

and

{E}_{c}

are, respectively, the superconducting gap and the charging energy of the grain. Quantum charge fluctuations at finite conductance remove the discontinuity in the dependence of

Q

{V}_{g}

and lead to a finite step width

\ensuremath{\propto}{G}^{2}\ensuremath{\Delta}

. The resulting shape of the Coulomb blockade staircase is of a novel type. The grain charge is a continuous function of

{V}_{g}

while the differential capacitance,

dQ∕d{V}_{g}

, has discontinuities at certain values of the gate voltage. We determine analytically the shape of the Coulomb blockade staircase also at nonzero temperatures.

DOI: 10.1103/physrevb.72.104507

Related papers

Coulomb effects in a ballistic one-channel S-S-S device

Dmitri A. Ivanov, M. V. Feigel’man

We develop a theory of Coulomb oscillations in superconducting devices in the limit of small charging energy $E_C \ll Δ$ . We consider a small superconducting grain of finite capacity connected to two superconducting leads by nearly ballistic single-channel quantum point contacts. The temperature is supposed to be very low, so there are no single-particle excitations on the grain. Then the behavior of the system may be described as quantum mechanics of the superconducting phase on the island. The Josephson energy as a function of this phase has two minima which become degenerate at the phase difference on the leads equal to $π$ , the tunneling amplitude between them being controlled by the gate voltage at the grain. We find the Josephson current and its low-frequency fluctuations and predict their periodic dependence on the induced charge $Q_x=C V_g$ with period $2e$ .We develop a theory of Coulomb oscillations in superconducting devices in the limit of small charging energy $E_C \ll Δ$. We consider a small superconducting grain of finite capacity connected to two superconducting leads by nearly ballistic single-channel quantum point contacts. The temperature is supposed to be very low, so there are no single-particle excitations on the grain. Then the behavior of the system may be described as quantum mechanics of the superconducting phase on the island. The Josephson energy as a function of this phase has two minima which become degenerate at the phase difference on the leads equal to $π$, the tunneling amplitude between them being controlled by the gate voltage at the grain. We find the Josephson current and its low-frequency fluctuations and predict their periodic dependence on the induced charge $Q_x=C V_g$ with period $2e$.

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Attractive correlation and charge skipping in a metallic grain capacitively coupled to a superconducting island

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Journal of Experimental and Theoretical PhysicsAug 1, 199818 citations

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Frank Neumann, Hermann Grabert

The charge on a superconducting island coupled to a superconducting electrode by a Josephson junction is determined in the region where the gap energy and the single-electron charging energy of the island are of comparable magnitude. The Coulomb staircase describing the incremental charging of the island in response to a biasing voltage is studied. The rounding of the staircase corners due to thermal and quantum fluctuations is found to depend on the parity of the island charge in the adjacent tread. This should allow for an experimental disentanglement of the effects of quantum charge fluctuations from thermal fluctuations.The charge on a superconducting island coupled to a superconducting electrode by a Josephson junction is determined in the region where the gap energy and the single-electron charging energy of the island are of comparable magnitude. The Coulomb staircase describing the incremental charging of the island in response to a biasing voltage is studied. The rounding of the staircase corners due to thermal and quantum fluctuations is found to depend on the parity of the island charge in the adjacent tread. This should allow for an experimental disentanglement of the effects of quantum charge fluctuations from thermal fluctuations.

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Rüdiger Bauernschmitt, Jens Siewert, Yuli V. Nazarov +1 more

The transport properties of a small superconductor--normal-metal--superconducting tunnel junction can be controlled by a gate electrode coupled capacitively to the central island. We evaluate the critical Josephson current ${\mathit{I}}_{\mathit{c}}$ through such a system as a function of the gate voltage ${\mathit{V}}_{\mathit{g}}$ taking into account parity effects in the superconductors. The dependence ${\mathit{I}}_{\mathit{c}}$ ( ${\mathit{V}}_{\mathit{g}}$ ) shows resonant singularities and has a qualitatively different character for \ensuremath{\Delta} ${\mathit{E}}_{\mathit{C}}$ and \ensuremath{\Delta}> ${\mathit{E}}_{\mathit{C}}$ , \ensuremath{\Delta} being the superconducting gap and ${\mathit{E}}_{\mathit{C}}$ = ${\mathit{e}}^{2}$ /2C being the charging energy. Due to the sweeping of the gate voltage, the system can be driven to states which are metastable with respect to a change of the number of electrons on the central island. Since the lifetime of the metastable states can be macroscopically large, one can observe a remarkable bistability of the Josephson current. A weak magnetic field suppresses the interference of the electrons propagating through the central electrode, and, consequently, the magnitude of the critical current.The transport properties of a small superconductor--normal-metal--superconducting tunnel junction can be controlled by a gate electrode coupled capacitively to the central island. We evaluate the critical Josephson current ${\mathit{I}}_{\mathit{c}}$ through such a system as a function of the gate voltage ${\mathit{V}}_{\mathit{g}}$ taking into account parity effects in the superconductors. The dependence ${\mathit{I}}_{\mathit{c}}$(${\mathit{V}}_{\mathit{g}}$) shows resonant singularities and has a qualitatively different character for \ensuremath{\Delta}${\mathit{E}}_{\mathit{C}}$ and \ensuremath{\Delta}>${\mathit{E}}_{\mathit{C}}$, \ensuremath{\Delta} being the superconducting gap and ${\mathit{E}}_{\mathit{C}}$=${\mathit{e}}^{2}$/2C being the charging energy. Due to the sweeping of the gate voltage, the system can be driven to states which are metastable with respect to a change of the number of electrons on the central island. Since the lifetime of the metastable states can be macroscopically large, one can observe a remarkable bistability of the Josephson current. A weak magnetic field suppresses the interference of the electrons propagating through the central electrode, and, consequently, the magnitude of the critical current.

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Influence of Controlled Quantum-Mechanical Charge and Phase Fluctuations on Josephson Tunneling

Marion Matters, W. J. Elion, Johan E. Mooij

We study the interplay of controlled quantum-mechanical charge and phase fluctuations by measuring their influence on the supercurrent through a small superconducting grain. The grain is connected to leads by two small Josephson tunnel junctions. The phase fluctuations are controlled by the magnetic flux threading a SQUID that connects the grain to a large superconductor. The charge fluctuations are modulated by a voltage on a gate capacitor. As a function of gate voltage we observe a broadening of the Coulomb resonance of the supercurrent for decreasing phase fluctuations, in agreement with theoretical predictions.We study the interplay of controlled quantum-mechanical charge and phase fluctuations by measuring their influence on the supercurrent through a small superconducting grain. The grain is connected to leads by two small Josephson tunnel junctions. The phase fluctuations are controlled by the magnetic flux threading a SQUID that connects the grain to a large superconductor. The charge fluctuations are modulated by a voltage on a gate capacitor. As a function of gate voltage we observe a broadening of the Coulomb resonance of the supercurrent for decreasing phase fluctuations, in agreement with theoretical predictions.

Physical Review LettersJul 24, 199538 citations

G

between the grain and the electrode, the charge

Q

as a function of the gate voltage

{V}_{g}

changes in steps. The step height is

e

\ensuremath{\Delta}<{E}_{c}

, where

\ensuremath{\Delta}

and

{E}_{c}

are, respectively, the superconducting gap and the charging energy of the grain. Quantum charge fluctuations at finite conductance remove the discontinuity in the dependence of

Q

{V}_{g}

and lead to a finite step width

\ensuremath{\propto}{G}^{2}\ensuremath{\Delta}

. The resulting shape of the Coulomb blockade staircase is of a novel type. The grain charge is a continuous function of

{V}_{g}

while the differential capacitance,

dQ∕d{V}_{g}

, has discontinuities at certain values of the gate voltage. We determine analytically the shape of the Coulomb blockade staircase also at nonzero temperatures.

We develop a theory of Coulomb oscillations in superconducting devices in the limit of small charging energy

E_C \ll Δ

. We consider a small superconducting grain of finite capacity connected to two superconducting leads by nearly ballistic single-channel quantum point contacts. The temperature is supposed to be very low, so there are no single-particle excitations on the grain. Then the behavior of the system may be described as quantum mechanics of the superconducting phase on the island. The Josephson energy as a function of this phase has two minima which become degenerate at the phase difference on the leads equal to

π

, the tunneling amplitude between them being controlled by the gate voltage at the grain. We find the Josephson current and its low-frequency fluctuations and predict their periodic dependence on the induced charge

Q_x=C V_g

with period

2e

+2e

) skipping odd charge numbers are found to be followed by negative ones (

-e

) signaling the occurrence of a negative differential capacitance. The condition for charge skipping is analyzed as a function of the ratio between the Josephson (

E_J

) and charging energies. The non-monotonous charging curves are calculated in the limits of large and small

E_J

{\mathit{I}}_{\mathit{c}}

through such a system as a function of the gate voltage

{\mathit{V}}_{\mathit{g}}

taking into account parity effects in the superconductors. The dependence

{\mathit{I}}_{\mathit{c}}

(

{\mathit{V}}_{\mathit{g}}

) shows resonant singularities and has a qualitatively different character for \ensuremath{\Delta}

{\mathit{E}}_{\mathit{C}}

and \ensuremath{\Delta}>

{\mathit{E}}_{\mathit{C}}

, \ensuremath{\Delta} being the superconducting gap and

{\mathit{E}}_{\mathit{C}}

{\mathit{e}}^{2}

/2C being the charging energy. Due to the sweeping of the gate voltage, the system can be driven to states which are metastable with respect to a change of the number of electrons on the central island. Since the lifetime of the metastable states can be macroscopically large, one can observe a remarkable bistability of the Josephson current. A weak magnetic field suppresses the interference of the electrons propagating through the central electrode, and, consequently, the magnitude of the critical current.