Charge sensitivity of superconducting single-electron transistor

It is shown that the noise-limited charge sensitivity of a single-electron transistor using superconductors (of either SISIS or NISIN type) operating near the threshold of quasiparticle tunneling, can be considerably higher than that of a similar transistor made of normal metals or semiconductors. The reason is that the superconducting energy gap, in contrast to the Coulomb blockade, is not smeared by the finite temperature. We discuss also the increase of the maximum operation temperature due to superconductivity and a new peak-like feature on the I − V curve of SISIS structures. Typeset using REVTEX 1 Electron transport in the systems of small-capacitance tunnel junctions shows a variety of single-electron effects. The simplest and most thoroughly studied circuit revealing these effects is the so-called Single Electron Transistor (SET) which consists of two tunnel junctions connected in series. At low temperatures (T ≪ e2/CΣ, CΣ = C1 + C2 where C1 and C2 are the junction capacitances) the current through this structure depends on the background charge Q0 of the central electrode (the dependence is periodical with a period equal to the electron charge e). Hence, controlling Q0 (for example, by a capacitive gate) it is possible to control the current I through the circuit. The possibility to use the SET as a highly-sensitive electrometer has been confirmed in numerous experiments. The most developed technology of the SET fabrication uses the overlapping narrow aluminum films with a typical junction capacitance about few times 10 F (see, e.g. Refs. 3–5). Consequently, the operation temperature is typically less than 1 K, and the electrodes are in the superconducting state unless the superconductivity is intentionally suppressed by the magnetic field. It has been noticed that the superconductivity of electrodes improves the performance of the SET (operating near the threshold of quasiparticle tunneling) as an electrometer in comparison with the normal-state operation. However, we are not aware of any attempts of quantitative theoretical analysis of this issue, which will be the subject of the present paper. There are two major characteristics of the SET operation as an electrometer. The first one is the amplitude of the output signal modulation for Q0 variations larger than e. It was found experimentally that the use of superconducting electrodes increases the modulation amplitude of current I (for fixed bias voltage V ), especially at temperatures comparable to e2/CΣ, thus increasing the maximum temperature. The theoretical results of the present paper confirm this statement for both NISIN and SISIS structures. The other, even more important characteristic of the SET operation is the noise-limited sensitivity (ability to detect variations of Q0 much smaller than e). The best achieved sensitivity so far (by the normal state SET) is 7 × 10e/ √ Hz at 10Hz. In the presentday technology this figure is limited by 1/f noise which is most likely caused by random 2 trapping-escape processes in nearby impurities. So, in some sense, the sensitivity does not depend much on the parameters of the SET, but rather on the purity of the sample. It is unlikely that superconductivity of electrodes can significantly affect these processes. Hence, the present-day sensitivities of superconducting and normal SETs with similar parameters should not differ much for reasonably low temperatures when both SETs show sufficient modulation amplitude. However, with the technology improvement one can expect the reduction of the noise due to impurities. Then the charge sensitivity of the SET would achieve the limit determined by the intrinsic noise of the device caused by random electron jumps through tunnel junctions (this “white” noise has been recently measured in experiment). Though the theory of the “classical” thermal/shot intrinsic noise of the SET is applicable to the general case of oneparticle tunneling (normal metals, semiconductors, quasiparticle current in superconductors, etc.), most numerical results in Refs. 6 and 7 as well as in a number of subsequent papers on this subject (see, e.g. Refs. 9–12) were obtained only for SETs made of normal metals. (Recently some generalization was done to include the possibility of two-particle tunneling which can be important in the superconducting case.) In the present paper we apply the theory of Refs. 6 and 7 to the cases of capacitively coupled superconducting SISIS and NISIN SETs (the analysis of a resistively coupled SET can be done in a similar way see Ref. 6). We show that the noise-limited sensitivity of a SET-electrometer can be considerably improved by the use of superconducting electrodes. We consider only the quasiparticle tunneling, neglecting the Josephson current, resonant tunneling of Cooper pairs, Andreev reflection, and cotunneling. This assumption is appropriate when the Josephson coupling is negligible and the normal state resistances R1 and R2 of tunnel junctions are well above the resistance quantum RQ = πh̄/2e . We use the “orthodox” theory of the SET and the BCS theory for the calculation of the tunneling rates. Figure 1 shows the I − V curves at different temperatures for (a) the normal metal NININ case, (b) NISIN case (which is equivalent to SINIS case), and (c)–(d) SISIS