# nondivergent calculation of unwanted high-order tunneling rates in single-electron devices

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## The normal/superconducting electron box.

In this section, we consider the case of a superconducting electron box where the island is made of a superconducting metal and connected to a non-superconducting lead through a superconducting/normal tunnel junction. We can regard the island as a small piece of superconductor free to exchange charges with an electron reservoir. According to the BCS theory of superconductivity, electrons are paired in the ground state of a superconductor .
The pairing of electrons clearly breaks the invariance of the ground state with respect to the parity of the total number N of conduction electrons. Since, in the island of the electron box, this number N is fIxed, we must distinguish two cases depending on the parity of N. If N is  even, all the electrons can be paired in the island and there is a unique superconducting ground state. If N is odd, one electron should remain unpaired as a quasiparticule excitation with an energy at least equal to the BCS energy gap .1 and the superconducting ground state is degenerate. As pointed out by Averin and Nazarov , this odd-even asymmetry should result in a parity dependence of the ground state energy of the box. We will discuss in this section how the electron box experiment can reveal this odd-even asymmetry. We will also show under which conditions the macroscopic charge on a superconductor is quantized in units of2e.

### Odd-even symmetry breaking and 2e-quantization in the normal/superconducting electron box at T=0.

For the sake of simplicity, we fIrst assume that the total number N of conduction electrons in the island and the number n of excess electrons have the same parity. At T = 0 and in absence of magnetic fIeld, the total energy E of the superconducting box is given by (40).
where Do is the energy difference at T = 0 between the odd-n and the even-n ground states of the system, and Pn = n mod 2. The first term in the right-hand side of Eq. (40) is the usual total electrostatic energy En of the circuit of the non-superconducting case, hereafter referred to as  the normal case. The second term is a parity dependent energy which corresponds to the fact that an unpaired electron must remain when the number of electrons stored in the island is odd. IfNand n have opposite parities, Pn is given by Pn = (n+ l)mod2. The BCS theory predicts that in zero fIeld the excitation energy Do is equal to /1, the superconducting energy gap of the island. Nevertheless, a fInite magnetic fIeld or the presence of paramagnetic impurities inside the sample can induce pair-breaking effects . As we shallsee in section 2.2.4, these effects modify the quasiparticles energy spectrum of a superconductor. Thus the excitation energy Do involved in the ground state energy of the box is not necessarily equal to the pair potential /1 in the superconducting island. In Fig. 2.11, we plot the energy E versus CsUIe and we get a set of parabolas indexed by n. The odd-n parabolas are shifted up with respect to the even-n parabolas by an amount equal to Do. Therefore, at T =0, the energy cost of adding one extra electron in the island will depend crucially on the relative magnitude of the charging energy Ec and the excitation energy gap Do. Two cases must be distinguished.

#### Observability of the 2e-quantization of the macroscopic charge.

The above calculation shows that in the T ® H plane there are three concentric domains corresponding to three different behaviors of the superconducting electron box: i) when D(T,H) = 0, the staircase is symmetric with an incremental charge equal to e, ii) when  0< D(T, H) < Ec ‘ the staircase is asymmetric with an incremental charge equal to e, iii) finally, when D(T,H) > Ec ‘ the staircase displays the 2e-quantization of the island charge. The 2e-periodicity of the symmetric or asymmetric staircase originates in the pairing of electrons in the island but an e-periodicity of the staircase does not mean that the island is in the non-superconducting state. The cross-over temperature To, which determines the boundary of the asymmetric staircase domain at H = 0, depends only logarithmically on the island volume. For an aluminum island fabricated by nanolithographic techniques, To will be always of the order 200-300 mK. Provided Ec »kBT, the observation of an assymetric staircase is actually not constrained by the size of the sample or the junctions.
This is not the case for the 2e-symmetric staircase. From Eq. (57), one can show that the boundary of the 2e-quantization domain intersects indeed the T-axis at a threshold temperature T2e given by (67).
We already know that To weakly depends on the sample parameters, and thus T2e is fixed essentially by the charging energy Ec and hence by the island capacitance C~. Eq. (12) and Eq.  67) show that the sharpness of the staircase and the area of the 2e-quantization domain have opposite variations with the charging energy Ec ‘ In contrast with the usual charging effects, the smaller the junction size, the lower is the temperature required to observe the 2equantization of the island charge. This is due to the fact that this phenomenon is subject to the double inequality kBT« Ec < D(T,H).
In order to observe the 2e-quantization of the macroscopic charge one must therefore find a compromise between these two opposite effects. Note that the thermal rounding of the staircase is not as important as in the normal case because a carrier with charge equal to 2e yields a staircase four times sharper than in the normal case at the same temperature. Finally Eq. (58) predicts that only one quasiparticle state inside the energy gap can strongly diminish the odd-even free energy difference. It can go below the charging energy and completely suppres the 2e-quantization even at very low temperature. In that sense, the 2e-quantization of the island charge constitutes a sensitive test of the ideality of the superconductivity of an isolated superconductor.

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Josephson coupling between the charge states of the box

We now consider an electron box in which both sides of the junction are superconducting (Fig. 2.14). The tunnel junction establishes a Josephson coupling between the island and the lead. We assume that T =0 and 1:1 > Ec’ We also assume that there is no out of equilibrium quasiparticle in the island and in the lead attached to the junction. Under these conditions, the island only contains an even number of electrons and the states of the system are characterized by the number of excess Cooper pairs in the island.
We restrict our analysis to the interval 0 < CsUIe < 2. Since T =0, we can consider only the Hilbert space spanned by the two states 10) and 11), which correspond respectively to the ground state of the superconducting island with zero and one excess Cooper pair. It is convenient to measure the energy of these states relatively to a reference set at Ec(l- CsUle)2. The energies of the states 10) and 11) are then respectively equal to -E/2 and +E/2, where E=4EAI-CsUle). These two energies are represented on Fig. 2.15 by two straight lines which cross at the threshold value CsUIe = l.

Effect of the Electromagnetic Environment

The question now arises as to whether the coherent quantum superposition of charge states leading to (72) and (73) will survive in the presence of dissipation in the leads which has been neglected so far. We will thus evaluate the effect of the electromagnetic environment of the junction on the box considered as an effective two-state system. We model the electromagnetic environment as an impedance Z(oo) in series with the superconducting  element in series with an effective impedance Zt (00) and an effective voltage source. Zt (00) is the total impedance seen by the pure tunnel element of the junction and its the real part is given by (74) where 1C = C/Cr.. The impedance Zt(OO) is equivalent to a set of L-C oscillators of frequency 0)j = 1/~LjCj (see Fig. 2.17c) such that: Re[Z/(oo)]= L1t/Cj O(oo-ooJ.

1. INTRODUCTION
2. REVIEW OF THEORETICAL PREDICTIONS ON MACROSCOPIC CHARGE QUANTIZATION IN THE SINGLE ELECTRON BOX
2.1 The nonnal electron box
2.1.1 The Coulomb staircase and the Coulomb sawtooth at T=0
2.1.2 Macroscopic charge quantization at finite temperature
2.1.3 Quantum fluctuations of the island charge
2.1.4 Tunneling rate in the electron box
2.2 The normaVsuperconducting electron box
2.2.1 Odd-even symmetry breaking and 2e-quantization in the normaVsuperconducting electron box at T=0
2.2.2 Effect of finite temperature
2.2.3 Calculation of the odd-even free energy difference D(T,H)
2.2.4 Influence of the magnetic field
2.2.5 Observability of the 2e-quantization of the macroscopic charge .
2.3 The superconducting electron box
2.3.1 Josephson coupling between the charge states of the box .
2.3.2 Effect of the electromagnetic environment
3. EXPERIMENTAL RESULTS ON THE ELECTRON BOX
3.1 Normal case
3.1.1 Paper 1: direct observation of macroscopic charge quantization
3.2 Superconducting case
3.2.1 Paper 2: measurement of the even-odd free energy difference of an isolated superconductor
3.2.2 Paper 3: 2e-quantization of the charge on a superconductor
4. CHARGE TRANSFER ACCURACy
4.1 Theoretical Predictions
4.1.1 Paper 4: passing electrons one by one: is a 10-8 accuracy achievable?.
4.1.2 Paper 5: nondivergent calculation of unwanted high-order tunneling rates in single-electron devices
4.2 Experimental results
4.2.1 Paper 6: Direct observation of macroscopic charge quantization:
a Millikan experiment in a submicron solid state device
5. CONCLUSION
APPENDIX 1: total energy of a general tunnel junctions circuit.
APPENDIX 2: fabrication of the superconducting/normal tunnel junctions
APPENDIX 3: the single electron transistor..

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