Thermopower of a superconducting single-electron transistor

Abstract

We present a linear-response theory for the thermopower of a single-electron transistor consisting of a superconducting island weakly coupled to two normal-conducting leads. The thermopower shows oscillations with the same periodicity as the conductance and is rather sensitive to the size of the superconducting gap ⌬. In particular, the previously studied sawtooth-like shape of the thermopower for a normal-conducting singleelectron device is qualitatively changed even for small gap energies. DOI: 10.1103/PhysRevB.71.220503 PACS number͑s͒: 72.15.Jf, 73.23.Hk, 74.45.ϩc The transport properties of small conducting grains in the Coulomb blockade ͑CB͒ regime have extensively been studied during the past years. This regime is characterized by a unique energy scale, the so-called charging energy E c of the grain ͑see below͒. The most prominent phenomenon is the occurence of CB oscillations in the low-temperature conductance of a small grain weakly coupled to the leads. 1 Recently, thermoelectric effects in single-electron devices such as the thermopower have attracted growing interest.2-11 The thermopower is related to the current that arises due to a finite temperature difference between the two leads.14 It yields additional information about the kinetics of the system as it measures the average energy of the electrons carrying the current through the system. Therefore, some type of electronhole asymmetry in the system is necessary in order to observe a nonvanishing thermopower.In analogy to the CB oscillations of the conductance, the thermopower of a small grain shows oscillations of the same periodicity but with sawtooth-like shape. 2,3 In contrast to the conductance this dependence on the external gate voltage is very sensitive to the conditions under which the thermoelectric transport occurs. This sensitivity has been demonstrated, e.g., for the transition from the sequential tunneling regime to the cotunneling regime. 4,5 Recently, the thermopower of open quantum dots with strong coupling to the leads was investigated.6-8 Further, the influence of Kondo correlations in ultrasmall quantum dots on the thermoelectric effects was studied in Refs. 9 and 10, while the thermopower of a molecule with internal degrees of freedom and weakly coupled to the leads was discussed in Ref. 11.It is surprising that, despite the enormous interest in superconducting single-electron transistors ͑SETs͒, the thermopower of such structures has not been investigated yet. In this work we study theoretically the thermopower of a normal-superconducting-normal SET ͑NSN SET͒, i.e., a small superconducting island that is weakly coupled to normal-conducting leads ͑cf. Fig. 1͒, in an experimentally accessible regime. We show that even for rather small superconducting gaps ͑compared to the charging energy of the island͒ the functional dependence of the thermopower on the gate voltage is qualitatively changed while its amplitude remains on the same order of magnitude. This is in clear contrast to the corresponding results for the conductance w...

Colloquium: Heat flow and thermoelectricity in atomic and molecular junctions

“…This would naively suggest that for these types of systems, the above single-particle picture accounts for most of the thermopower. However, more recent investigations which include effects of interactions, show that in both types of systems interactions may induce deviations from the Wiedmann-Franz law, thus reducing the agreement with experiments (Kubala et al, 2008;Lunde et al, 2006;Turek and Matveev, 2002;Turek et al, 2005;Zhang, 2007), suggesting that the agreement in the singleparticle case may be the result of cancelation of errors.…”

Quantum-fluctuation effects on the thermopower of a single-electron transistor

“…19,20,21,22,23,24,25 Thermoelectric effects in various mesoscopic systems have also been studied theoretically. 26,27,28,29,30,31,32,33,34,35 Quantum dots with discrete 20 (single-particle) energy level spectrum as well as dots closer to the metallic (quasi-continuous) limit 19 have been compared to sequential-tunneling theory. For increased coupling and lower temperatures, systems of discrete level structure showed signatures of cotunneling 21 and also of Kondo physics.…”

Thermopower and thermoelectric power factor ofZkparafermion quantum dots

“…(3), (5), (9) and (8) the plot of the thermopower for r = 1 might be unrealistic, as experiments and numerical calculations suggest that v n < v c , it is instructive for the analysis of the characteristics of the thermopower in general, which are similar to those for metallic islands: the thermopower grows linearly with the gate voltage V g and the edges are smoothened at finite temperature; it is always 0 at the maximum of a conductance peak expressing the fact that the tunneling energy is zero at the conductance peaks; there are sharp jumps of thermopower (discontinuous at T = 0) in the middle of the CB valleys between neighboring conductance peaks, expressing particle-hole symmetry [40]. The plot of the thermopower for the Z 3 parafermion FQH state is similar to that of the Moore-Read (Pfaffian) state [10], and to that of the superconducting SET [41] as well, only the number of small-amplitude oscillations of S has been increased form one to two, corresponding to the number of short-period conductance peaks. This allows us to anticipate the general pattern of the oscillations of the thermopower in the Z k parafermion states without quasiparticles localized in the bulk: it naturally confirms the results obtained in Eq.…”