PHYSICAL REVIEW B VOLUME 52, NUMBER 7 15 AUGUST 1995-I Effect of polarization screening length on electron-pump cotunneling errors R. W. Rendell* Code 6813, Naval Research Laboratory, 8'ashington, D. C. 20375 (Received 11 April 1995) We have calculated cotunneling errors occurring in the electron pump, including the e6'ects of polar- ization screening length as determined by the gate-to-junction capacitance ratio. Previous analyses have been done in the limit of very large screening length, or equivalently in the limit of a very small ratio of gate-to-junction capacitance, where cotunneling errors are at most of order (N — 1) for an N-junction pump for a restricted range of end-to-end bias. Within this same range, we find that the calculated co- tunneling errors can increase significantly with increasing capacitance ratios 0. 15 — 0. 3, which encompass those reported for several experimental pumps in the literature. With realistic parameters, the calculat- ed first-, second-, third-, and fourth-order cotunneling errors increase the error of a five-junction pump by factors of between 2 and 10 over much of the range of end-to-end bias and by up to a factor of 5 X 10 toward the negative end of the bias range. We discuss the implications of these results for recent experi- mental observations which are at least two orders of magnitude larger than calculations based on very small capacitance ratios. In recent years there has been a rapid development in the area of "single-electron" devices, in which the motion of single electrons are controlled by the Coulomb blockade of tunneling. ' ' The success of single-electron devices for applications in metrology and computation depends crucially on the control of errors in the electron motion. The use of the electron pump or turnstile for the development of a new dc current standard ' requires er- rors as low as one part in 10 . The most extensive experi- mental and theoretical efforts of error analysis ' have been carried out for the electron pump, which is the lead- ing candidate for metrological applications. In the elec- tron pump, the motion of a single electron is clocked in and out of the islands located between the tunnel junc- tions by time-dependent modulation of the gate biases ap- plied successively to the gate capacitors. In the ideal behavior of the electron pump, the transitions between is- lands occur by tunneling across a single junction during each cycle. The accuracy of this current source may be prevented from reaching that of the frequency source for the gates by several additional physical effects. ' These include undesired transitions of the electron due to thermally assisted tunneling, missing the desired tunnel- ing process due to pumping at too fast a rate, and un- desired transitions due to the simultaneous tunneling through more than one junction known as "cotunnel- ing. "' There can also be complications in an experimen- tal circuit, which will deviate from the ideal one such as the existence of background charges located in the dielec- tric of the junctions, cross capacitance between gate capacitors and islands, and noise due to the finite im- pedance of the leads. The most complete experimental and theoretical study of the electron pump to date was carried out by Martinis, Nahum, and Jensen' (MNJ) for metrological applications. They analyzed the operation of a five-junction electron pump in great detail and found that the observed error is at least two orders of magni- tude greater than predicted from existing theory. They concluded that the present understanding of the Coulomb blockade is incomplete. In a recent theoretical study' of the computational capabilities of locally interconnected tunneling arrays operating in the Coulomb blockade regime, it was found that a key element for achieving reliable operation is con- trol of the polarization screening length A, of the electrical potential surrounding an electron on an island. For a long chain of Coulomb islands with identical gate capaci- tances C~ and identical junction capacitances C&, the po- tential on island i due to a single excess electron located at island j varies approximately exponentially as y; =exp( — ~i — j~ /A. ), where the screening length is given b 14 1/2 C 1 C Cg ' =— ln 1+ —— +4 (1) 2C~ 2 CJ CJ For Cz/Cz =0. 01, A, = — 10 islands; for C /CJ = 1.0, A, =1.04 islands. Since the Coulomb blockade is an elec- trostatic effect, the size of the screening length affects the probability of tunneling to an adjacent island. It also influences the threshold voltages and tunneling rates for additional tunneling and cotunneling processes that can occur within the array. For an electron pump, these ad- ditional events can lead to errors from the ideal pump operation, wherein the single excess electron moves ahead one island during each gate bias cycle. If C /Cz « 1, the expressions governing the tunneling transition rates can be obtained analytically and they do not depend explicitly on A, . In this limit, the most impor- tant undesired tunneling events in an X-junction pump consist of cotunneling of order (X — 1) (in inverse tunnel- ing resistance) provided the end-to-end voltage bias is kept suf6ciently small. The more general computational arrays studied in Ref. 12 have excess electrons on more than one island and complicated biasing schemes and this allows more types of undesired tunneling and cotunneling processes than in the simple electron pump, including large first-order tunneling errors. Numerical simulation of the computational arrays showed that the errors were sensitive to the size of the screening length and a regime was found in which the arrays could operate within error tolerance. This raises the question of how the screening length also affects the errors in the simple electron pum. p 4684