Some outlines of circuit applications for a single-electron 2-island subcircuit

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1 Some outlines of circuit applications for a singleelectron 2island subcircuit Jaap Hoekstra and Janaina Guimaraes Electronics Research Laboratory, Subfaculty of Electrical Engineering, Delft University of Technology J.Hoekstra@ITS.TUDelft.nl www:nanocom.et.tudelft.nl Abstract To benefit from the reduction of the devices feature sizes new circuit concepts can be introduced. These new circuits will require new devices, such as singleelectronic devices. Singleelectronics devices are capable of controlling the transport of only one electron. In this manner, the charge transfer through the device is quantized. However, singleelectronics is still a highly experimental technology. As an example of singleelectronics we discuss circuits based on the socalled singleelectron tunneling (SET) device, including tunnel junctions. One of the basic issues is the development of circuits able to exploit the device performance. In this paper we analyze some potential applications of the singleelectron 2island subcircuit. A special example of this subcircuit, namely the electron pump, use the oulomb blockade to control the electrons through a sequence of ultrasmall tunnel junctions. The device is essentially an electron counter, and with proper design it can be highly accurate. I. Introduction Nanoelectronics is a generic term for all kind of emerging and very promising microelectronic circuits containing devices with critical dimensions of nanometer scale. According to the SIA (Semiconductor Industry Association) roadmap the evolution of the current semiconductor devices can still reach smaller dimensions than nowadays. In the year 200, a transistor shall have feature sizes bellow 70 nm. However, to benefit from the reduction of sizes new circuit concepts can be introduced [], [2], [3]. Nanoscale electronics is still in its startup phase. Developing integrated circuits using nanometer scaled devices seems to be a good perspective. First of all, the basic devices can be small, extremely small. Second, it has the potency to operate with very low supply power. And third, quantum properties that appear at nanoscale in principle represent a huge increase in signalprocessing power. The nanoscale circuits will require new device concepts, such as singleelectronics [4]. Several new nanoelectronic devices have already been proposed, among which are: single electron transistors (SETs); resonant tunneling diodes (RTDs); quantum dots and quantum cellular automata (QA); carbon nanotubes; rapid single flux quantum logic (RSFQs) and molecular nanoelectronic devices. Singleelectronic devices are able to control the movement of individual electrons. Therefore, the transport of charge through the device can be quantized. Actually, singleelectrons appear from investigation of a device known as a tunnel junction, formed by two metal electrodes separated by a thin insulator. According to quantum mechanics, one electron has a small probability of passing through the thin insulator. This phenomena is called tunneling. As an example of singleelectronics we discuss circuits based on the singleelectron tunneling device. In this paper, in Section 2 we will try to give an overview of the most important new circuit concepts that are necessary when entering the field of nanoelectronics [], [5], [2], [6]. In Section 3 we describe some circuit models of the devices, which form the elements used in the (sub)circuits; both the circuit models of the orthodox theory of singleelectronics [7], [8], [9], [4] and the impulse model we developed at Delft [0], [], [2], [3], [4], [5] are presented. Based on the SET junction, some simple subcircuits are described in Section 4. Section 5, then, discusses circuits based on the 2island subcircuit. We will discuss briefly the singleelectron pump circuit [6], [7] and the on the singleelectron pump structure based singleelectron digital logic (SED) [8]. Finally, we conclude in Section 6. II. New circuit concepts To discuss the challenges and threats of nanoelectronics in general, we have to consider the design complexity. The fundamental design aspect is that individual electrons can be manipulated instead of currents. The complexity arises if we realize that we 44

2 2 have to cope with, or even to exploit, the typical properties like inaccuracies and stochastic behavior, that inherently go hand in hand with decreasing dimensions. The design complexity has to be tackled from two approaches: a topdown structured design and a bottomup structured design. A. Topdown design issues The topdown design focuses on the interdependencies of the design choices made at different functional levels. Especially the application of appropriate signal definition, the choice of what functions at which level to implement, and the use of redundancy or adaptivity or neural networks are the most important nanoelectronic issues []. The necessity of a topdown approach comes from a number of reasons. First there are the uncertainties and inaccuracies caused by quantum effects or just by the imperfection due to the nano fabrication technology. In general these uncertainties and inaccuracies have to be tackled at different levels in the design. We have the choice to try to avoid them or the cope with them. Then there is the interconnection problem. Because nano devices are small many could be placed on a single chip. The interconnections between all those devices demands smart topologies and architectures. Besides this the capacitance of all those wires will significantly influence the small nano devices. B. Bottomup design issues A design methodology only following the topdown approach will result in functional blocks that have to be implemented with nano devices without exploiting the specific properties of the devices. The choice of functional blocks is based on existing circuit design paradigms that do not fully take into account the new possibilities of the nanoelectronic devices. A bottomup approach, has just to exploit from the lowest level the capabilities of the nano devices. Such an approach can try to use the discrete character of the singleelectron tunneling process and its stochastic behavior. As is usually done in a bottomup design strategy we start with the basic physics equations describing the device. From this levels we are able to propose circuit elements, or equivalent subcircuits that approach these physics equations in certain domains of signal processing. Those circuit elements and subcircuits can form the basis for circuit analysis, circuit synthesis, and SPIElike transient simulation. In the sequel of this section the important physics equations describing the tunnel condition and describing the stochastic behavior are briefly discussed.. Tunnel condition The quantum description of tunneling between two metal plates of the tunnel junction is depicted in Fig.. The description envisages electrons held in the metal plates by a potential which, to a first approximation, may be described by a box of finite height. The electrons are stacked up in dense or close spaced energy levels since the box is very wide. Levels from which tunneling can occur Fig.. E F, W eu s Well I Well 2 Tunneling through a barrier Because no more than two electrons can occupy any given energy level as mandated by the Pauli exclusion principle, the lowest energy state of the metal pictures a configuration where all levels up to the Fermi level are filled at T = 0K. When the temperature is above 0K, a few electrons are excited to higher levels. The difference in energy between the Fermi level and the top of the barrier is the work function, W. Basic quantum mechanics shows that, in case of many electrons, an approximate expression for the barrier transmission coefficient depends on the work function W and the voltage across the tunnel junction. Together with the number of available states, the transmission coefficient acts as a conductance G t : E F,2 i = G t U s () The formula holds for small currents at small voltages across the junction. Important is to note that the conductance only comes from the stochastic behavior of the tunnel junction; and in fact is not a real conductance in terms of circuit analysis. In singleelectronics this conductance cannot play a role. The probability for a single electron to tunnel is generally described with a Poisson distribution. To stress the importance of G t not being a standard conductance, 45

3 3 we want to point out that the energy released at the junction is proportional to u j but not proportional to the square of u j, as would be expected if G t is a conductance. The question of a singleelectron will actually tunnel does also depend on thermodynamic conditions. In the orthodox theory this condition is expressed in system energy before and after the tunnel event; and contrary, in the impulse model this condition is expressed in the local voltage across the tunnel junction immediately before and immediately after the tunnel event. In the last model there will be a blockade, called oulomb blockade (that is, electrons cannot tunnel) if: u jb u ja < 0 (2) All singleelectron circuits can be analyzed by calculating, at different times, the voltages across all junctions and to evaluate wether a tunnel event can occur. The critical voltage for tunneling is given by: u cr j u jb = u ja (3) III. SET circuit elements In this section we consider the circuit analysis parts: devices (components), circuit elements, and their equivalent subcircuits. A. Devices(omponents) In the case of circuits including metallic SET tunnel devices the u i relation of the tunnel junction is strongly determined by the remainder of the circuit. For example, see Fig. 2 the SET junction, the two u i relations seen belong to, respectively, a junction excited by a current source (the dashed line) [5], and a junction excited by a voltage source (the solid line). We can immediately conclude that tunnel junction is a nonlinear device. Besides the nonlinear tunnel junction there is also a nonlinear island device. An island is created, for example, as soon as two or more junctions or capacitors are connected. On the floating node, charge can be stored on or released from. To see the nonlinear character of the island device consider, for example, two capacitors in series. The voltage over a single capacitor can be expressed as a function of the voltage over both capacitors, u s, and the value of the charge on the island, q i : u = u 2 = 2 u s q i 2 2 (4) u s q i 2 2 (5) Symbol Symbol SET junction SET transistor u g (t) i i different u g different environments Fig. 2. Two examples of devices with their symbols and possible ui relations (dependent on the environment of the devices, like for instance the applied sources and their values) As can easily be verified, the superposition principle doesn t hold. B. Nonlinear elements In both the orthodox theory and the impulse model circuit elements have been proposed. Fig. 3 shows the elements. In the orthodox theory the island is not always seen as a device, and consequently not always a specific element is used. The junction in the orthodox theory is always modelled as a pure capacitance as long as the junction is in blockade, and as a capacitor in series with a resistance as soon as electrons tunnel. In our impulse model, for singleelectronics, the junction is modeled as a capacitance, and the tunneling is described by λ, a parameter determining the Poisson distribution. Equivalent subcircuits modeling these circuit elements are treated in the next subsection.. Equivalent subcircuits Recently, we published several equivalent subcircuits that makes it possible to embed the nonlinear tunnel junction and the nonlinear island in a linear circuit analysis [0], []. Fig. 4 shows the equivalent subcircuits. Due to the use of the generalized delta function the subcircuits can be dealt with in conventional linear circuit analysis. The model for the tunnel junction find its origin u u 46

4 4 t,r t t, =0 q i or functional blocks. Before going into a more detailed description of the 2island subcircuit we want to emphasize that not all possible circuit with the circuit elements are able to show oulomb blockade (no tunneling). (a) (b) Fig. 3. The circuit elements in the orthodox theory (a) and in the impulse theory (b); all elements are nonlinear. t, t e (tt b ) (a) A. Minimal subcircuits To obtain blockade phenomena it is necessary that the tunneling electron tunnels from one potential well to another. The wells determine the position of the elctron and thus its charge. The tunneling phenomena as described in the second section shows such a distinction, see fig., one metal side of the junction forming well, the other well 2. Figure 5 shows some minimal circuit architectures within a blockade is possible. It shows the single junction excited by a current source; the double junction excited by a voltage source (the single junction excited by a voltage source will always tunnel!); and a possible SET transistor architecture. =0 q i q i δ(t) 2 I 2 (b) I Fig. 4. Equivalent subcircuits that can be embedded in a linear circuit analysis: (a) the SET junction and (b) the island device. in modeling the energy released at the junction during a tunnel event; the model for a charged island originates from considering all the charges that are stored on all capacitances forming an island due to external island charges. The stochastic behavior of the tunnel junction is modeled in time t b, the time just before the actual tunneling, taking into account a wait time after the point in time that tunneling may occur. The equivalent subcircuits allow us to make use of superposition in the design trajectory; this enable us to analyse and design many circuits that could not be analysed or designed upto now. IV. Examples of subcircuits based on SETs In this section we show some subcircuits that can form a starting point in the design of useful circuits Fig. 5. I 2 B. an 2island subcircuit Some minimal circuit acrchitectures In Fig. 6 the 2island subcircuit is drawn. The voltage across a tunnel junction is determined by all the independent sources: current sources, voltage sources, and island charges. A 2island subcircuit works confining the electron in one node (potential well) and shifting this electron under the influence of appropriate clocking waveforms applied to the gate voltages, just like in a chargecoupled device (D). 47

5 5 Fig. 6. 2island subcircuit pump is simulated both with SIMON [9], a device simulator based on the orthodox theory, and with our own SPIE model based on the impulse model [2]. Both simulators give the same results. Fig. 8 shows the voltage across junction 3 versus time. otunneling and offset charges are not considered. To make this 2island subcircuit work properly it is necessary to supply some operational conditions. First of all the subcircuit needs excitation. Voltage or current sources can be applied to the circuit supplying electrons to or removing electrons from the substructure. The sources will determine the direction which the charge will flow, because this kind of structure is reversible. An electron can be transferred if waveforms are applied to the gate voltages. V. Some circuits based on the 2island subcircuit We now describe two circuit examples both based on the 2island subcircuit. The first deals with the counting of single electrons, the socalled singleelectron pump, the second with the possibility to use the structure for digital logic. A. Singleelectron pump We can excite the 2island subcircuit with an extra voltage V b, this is shown in Fig. 7, the circuit is called the singleelectron pump [8]. V b Fig g g2 V 2 V Singleelectron pump The basic idea of this kind of circuit consists in transferring only one electron, during one cycle of an external frequency source f. The dc current can be expressed as: i = ef (6) A clocked control of charge flow, electron by electron, requires at least three tunnel junctions and two gates. Although a singleelectron pump can be described using the orthodox theory, we describe the pump in terms of circuit theory using the impulse model. The Uc3 4e05 3e05 2e05 e05 0 e05 2e05 3e05 4e Fig. 8. Simulation of the singleelectron pump, the voltage across junction 3 versus time For the simulation the following values are chosen [7]: = 2 = 3 =.5fF. g = g2 = 0.02fF. Using the above values for the capacitors, the critical voltages are: U cr = U2 cr = U3 cr = 35.7µV. We applied a asymmetrical clock. What can be seen in the simulation results are three tunnel events. The first one is a tunneling through junction 3. The second and third are the tunneling through junction 2 and junction. time B. Singleelectron digital logic As another circuit example we will briefly mention the circuits based on digital chargecoupled logic (DL) [20]. As can be concluded from the previous subsection the electron pump behavior can be compared with the behavior of chargecoupled devices. Based on this equivalence singleelectron digital logic circuits (SEDs) were proposed [8]. In figure 9 the circuit performing the AND/OR function is shown. A remark on the proposed SED circuits could be made. First, as is also valid for DL logic circuits, the fanin and fanout of the circuits is limited, besides this the structure and design is topology restricted. The fanin fanout problem is not the major problem for use in singleelectronics, because the nanoelectronic circuits will always have to deal with limited fanin and fanout due to the limited capacitance that is, in general, necessary in those circuits. This kind of problem has to be solved at a higher design level. A "uc3" 48

6 6 A AND/OR 3 4 A.B 23 B 3 4 AB Fig. 9. AND/OR function realized in singleelectron digital circuit second remark is that the SED circuits can not cope with (random) background charges and cotunneling. Still the SED and JL field is very interesting to get design ideas from. VI. onclusions Singleelectronics will become feasible due to the ongoing downsizing of the device dimensions. We showed the bottomup approach to design circuits based on the metallic singleelectron tunneling device. ircuit elements and subcircuits were described. As circuit examples we discussed circuits based on the 2 island subcircuit. Especially, circuit for the electron pump and digital singleelectron logic were shown. Simulation results indicate the expected behavior of the devices. VII. Acknowledgement We gratefully acknowledge the financial support of the European Melari/NID ANSWERS project, and the Delft Interfaculty Research enter Novel omputational Structures based on Quantum Devices. References [] A. van Roermund and J. Hoekstra. Design philosophy for nanoelectronic systems, from sets to neural nets. International Journal of ircuit Theory and Applications, 28(6): , [2] J.. Da osta, J. Hoekstra, M.J. Goossens,.J.M. Verhoeven, and A.H.M. van Roermund. onsiderations about nanoelectronic gsi processors. Analog Integrated ircuits and Signal Processing, 24:59 7, [3] K.F. Goser,. Pacha, A. Kanstein, and ML. Rossmann. Aspects of systems and circuits for nanoelectronics. Proceedings of the IEEE, 85(4): , april 997. [4] K.K. Likharev and T. laeson. Single electronics. Scientific american, pages 50 55, June 992. [5] A. van Roermund and J. Hoekstra. From nanotechnology to nanoelectronic systems, for sets to neural nets. In IEEE international Symposium on ircuits and Systems, pages I 8 I, Geneva, Switzerland, May ISAS [6] Eelco Rouw, Rudie van de Haar, Arthur van Roermund, Roelof Klunder, and Jaap Hoekstra. Neural nets using set technology. In Proc. IEEE/ProRIS 99 workshop, number ISBN: , pages STW, 999. [7] D.V. Averin and K.K. Likharev. Single electronics: A correlated transfer of single electrons and cooper pairs in systems of small tunnel junctions. In mesoscopic phenomena in solids, volume 30, chapter 6, pages Elsevier Science Publihers BV, ISBN: , Department of physic, Moscow state university, Moscow, USSR, 99. [8] Edited by H. Grabert and M.H. Devoret. Single charge tunneling oulomb blockade Phenomena in Nanostructures, volume 294 of NATO ASI series B. Plenum Press, New York, physics edition, 992. [9] K.K. Likharev. correlated discrete transfer of single electrons in ultrasmall tunnel junctions. IBM Journal of Research and Development, 32():44 58, january 988. [0] R.H. Klunder and J.Hoekstra. Energy conservation in a circuit with single electron tunnel junctions. In The IEEE international Symposium on ircuits and Systems, pages I 59 I 594, Sydney, Australia, May 200. ISAS 200. [] R. Klunder, K. van Hartingsveldt, and J. Hoekstra. Modelling of independent node charges in metallic single electron tunneling circuits. In Proceedings of the ninth workshop on Nonlinear Dynamics of Electronic Systems, pages , Delft, The Netherlands, 223 June 200. NDES, ISBN [2] R. van de Haar, R.H. Klunder, and J. Hoekstra. Spice model for the single electron tunnel junction. In IES 200, volume 3, pages , Malta, September 200. IEEE International onference on Electronics, ircuits and Systems, ISBN: [3] J. Hoekstra. On the origin of energy loss in singleelectron tunneling devices. In Proceedings of the ninth workshop on Nonlinear Dynamics of Electronic Systems, pages 7 20, Delft, The Netherlands, 223 June 200. NDES, ISBN [4] R.H. Klunder and J. Hoekstra. Extracting the component values of single electron tunneling transistors from measurement results. In SAFE/IEEE 2000, pages STW, ISBN: , November [5] R.H. Klunder and J. Hoekstra. Different environments in single electron tunneling circuits. In ProRIS/IEEE 2000, pages STW, ISBN: , November [6] J. Guimaraes, R. van de Haar, R. Klunder, and J. Hoekstra. ircuit analysis of a singleelectron 2island set subcircuit. In accepted for: SBMIRO, Brasil, 200. [7] J.G.Guimaraes and J..da osta. Basic circuit structures using singleelectron tunneling devices. In International onference on Microelectronics and Packaging, August 999. [8] M.G. Ancona. Design of computationally useful singleelectron digital circuits. J. Appl. Phys., 79(): , January 996. [9]. Wasshuber, H. Kosina, and S. Selberherr. Simon a simulator for single electron tunnel devices and circuits. IEEE transactions on computer aided design of integrated circuits and systems, 6(9): , September 997. [20] R.A. Allen et al. hargeoupled Devices in Signal Processing Systems, volume V. U.S. Navy Final Report,