Measurement of the even-odd free-energy difference of an isolated superconductor

Abstract

We have measured the difference between the free energies of an isolated superconducting electrode with odd and even number of electrons using a Coulomb blockade electrometer. The decrease of this energy difference with increasing temperature is in good agreement with theoretical predictions assuming a BCS density of quasiparticle states, except at the lowest temperatures where the results indicate the presence of an extra energy level inside the gap.PACS numbers: 74.50.+r, 73.40.Rw, 74.25.Bt The key concept of the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity [1] is the pairing of electrons. A surprising feature of the theory appears when one considers a macroscopic piece of superconducting metal with a fixed number of electrons TV. If TV is even, all the electrons can condense in the ground state. If TV is odd, however, one electron should remain as a quasiparticle excitation. In principle, if one would measure the energy required to add one electron to the superconductor, there should be a difference between the cases of even and odd TV. This fundamental even-odd asymmetry, which might vanish due to sample imperfections [2], does not manifest itself in conventional experiments on superconductors because these experiments are only sensitive to a finite fraction of quasiparticles. In this Letter, we report a new experiment based on singleelectron tunneling [3] with which we measured the evenodd free energy difference introduced by Tuominen et al. [4].Consider a superconducting-normal (SN) tunnel junction in series with a voltage source U and a capacitor C s (see Fig. 1), a basic Coulomb blockade circuit whose normal-normal junction version has been nicknamed the electron "box" [5,6]. The superconducting electrode which is common to both the junction and the capacitor is surrounded everywhere by insulating material. When the junction tunnel resistance R t is such that R t^> R[(=h/e 2 J the number n of excess electrons on this "island" is a good quantum number [3,7L The w-dependent part of the ground-state energy of the circuit, including the work done by the source £/, is given by En^Ecin -C s U/e) 2 + <£", where E c = : e 2 /2Cz is the electrostatic energy of one excess electron on the island, Cz the total capacitance of the island, and &" is the nonelectrostatic part of the energy of the island. For a normal island G n = 0 [ Fig. 2(a)], whereas for a superconducting island, one has S n -DoPn where Do is the energy difference between the odd-n and even-/? island ground states, and p n =nmod2[ Fig. 2(c)]. The BCS theory yields D 0 =A where A is the superconducting gap of the island. In equilibrium at zero temperature, n will be determined by the lowest E n and is therefore given by a staircase function of U [Figs. 2(b) and 2(d)]. In the normal case, the steps are of equal size, whereas in the superconducting case even-AZ steps are longer than odd-Ai steps. For Do> E c , the odd-n steps disappear, while for Do<E c , the ratio p between the length of the odd-and even-« steps is related to Do throug...

Spectroscopy of discrete energy levels in ultrasmall metallic grains

“…Here N eff (T ) is the effective number of states available for quasiparticle excitations at temperature T , and for d ≪∆ is given by N eff (T ) = 8πT∆/d 2 [84]. Below the corresponding crossover temperature where δF = 0, determined by k B T * cr =∆/ ln[N eff (T * cr )] and roughly equal to∆/ ln 8π∆ 2 /d 2 , the single unpaired electron begins to matter: it causes a crossover from e-periodicty to 2e-periodicity in the I-V characteristics of mesoscopic superconducing SET's [84][85][86][87][88], due to the ground state energy difference E 1/2 − E 0 ≃∆. Since T * cr becomes of order∆ in nanoscopic grains with d ≃∆, parity effects should survive to temperatures as high as the (bulk) superconducting transition temperature T c itself.…”

“…Q 0 can of course be determined reasonably accurately by studying the largescale Coulomb oscillations of the I-V curve that occur as functions of V g at fixed V , a procedure that is well-established for mesoscopic SETs, for which indeed it has been possible to measure the ground state energy difference between a superconducting island with an even or odd number of electrons [62,[84][85][86][87][88]. However, a complication arises for the nanoscopic grains of present interest, due to the smallness of their gate capacitances (typically ≃ 0.1 aF): to sweep V g through one Coulomb oscillation, the gate voltage V g must be swept through a range so large (namely e/C g ≃ 1V) that during the sweep, RBT routinely observed small "rigid" shifts of the entire tunneling spectrum at random values of V g .…”

“…To be observable [84][85][86][87][88], they only require the temperature to be smaller than the free energy difference δF ≃∆ − k B T ln[N eff (T )] between an odd and even grain. Here N eff (T ) is the effective number of states available for quasiparticle excitations at temperature T , and for d ≪∆ is given by N eff (T ) = 8πT∆/d 2 [84].…”

Tunability of a2eperiodic single Cooper pair box

“…14 The charge of the ground state of the SCB thus increases in a steplike manner as the external voltage is increased, which results in the so called Coulomb staircase. 12,13 The tunneling due to E J will smear the Coulomb staircase around the Cooper pair transition. The shape of the staircase thus depends strongly on E C and E J , so that the SCB's characteristic energies can be obtained from the shape, or vice versa.…”

“…In this paper we address the issue of controllability of one possible qubit system, namely, the single Cooper pair box (SCB). [8][9][10][11] We demonstrate controllability of both the charging energy and the Josephson coupling energy and demonstrate how the quantum smearing of the Coulomb staircase 12,13 is controlled by an external magnetic field.…”