Maxwell11.2 March 2008 2 Due to the ongoing downsizing of
microelectronic circuit components, many new nanoelectronic devices have been proposed and manufactured in the past years. These nanoelectronic devices have critical dimensions of several nanometers and take advantage of quantum phenomena that appear at nanometer scale. To understand the possible utilization of the nanoelectronic devices, useful and competitive circuits have to be designed. New circuit ideas must be developed exploiting the quantum character, the small feature size, and the low power operation of nanoelectronics.
There are a couple of different definitions of nanoelectronics. From the physics point of view, nanoelectronics often deals with circuits including nanoelectronic devices which dimensions have reached such a small length that the wave nature of the electrical carriers cannot be neglected, and that device and circuit simulations for essentially classical device structures are confronted with a real quantum mechanical description rather than with classical models. From the electrical engineering point of view, nanoelectronics is understood merely as an electronics based on nanoelectronic devices which utilise quantum mechanical phenomena; they have to be described with semi-classical models to make circuit synthesis possible.
The topics discussed in this article are part of on-going research on single-electron single-electronic circuit design issues and single-electron electronics; that is, nanoelectronic circuits in which information and signal processing are
Considering modeling of nanoelectronic devices, it is argued that
four really different modeling levels exist. These are the
quantum-physics level, the (semi)classical-quantum-physics level, the circuit level, and
the system level. The circuit level is best suited for predicting the
utilization of newly proposed nanoelectronic devices. Possibilities
of and restrictions on this level are examined. Furthermore, a small
historical overview is given of the equivalent circuit models that
have been proposed for the single-electron tunneling junction up
to today. This overview shows that, even for the “simple”
metal-insulator-metal junction, there is no consensus on the correct or
best equivalent circuit. However, that such equivalent circuits can be
used to successfully analyse and design nanoelectronic circuits is
shown by the comparison of a SPICE simulation of a single-electron
pump with experimental measurements.
Author: dr. Jaap Hoekstra, Electronics Research Laboratory/DiMES
The single-electron tunneling junction
Modeling nanoelectronic devices in circuit theory
d
w
h
(a)
(b)
TJ
Figure 1: (a) Metallic single-electron tunneling (SET) junction: two metal leads separated by a thin layer of insulating material; (b) symbol for the SET junction circuit element.
0
Maxwell11.2 March 2008considered by means of one single or only a few electrons. Especially, as signals that carry the information are considered: the value of a voltage as a function of time, the value of a current as a function of time, or the number of additional electrons on an isolated island as function of time. To explain the important aspects of single-electron single-electronics, circuit design with metallic single-electron tunneling (SET) junctions is examined. Schematically, the metallic tunnel junction consists of two metal conductors separated by a very thin (typically: 5<d<10 nm) insulator, see Figure 1. Due to the extreme small insulator thickness tunneling of electrons through this insulator becomes possible.
Different Levels in Modeling
To analyse the possible application in circuits the nanoelectronic devices, and so also the single-electron tunneling junction, have to be modeled. There are various domains for doing this modeling. Take a look at Figure 2, it shows different levels in modeling that can be distinguished when treating the various topics that are involved in nanoelectronic circuit design. It is necessary to understand that any modeling effort of a real device involves a theoretical framework on a specific level. Sometimes, the same device or device property can be modeled at different levels. For example, a transistor can be fully described by semi-classical physics on the classical-physics level; but for circuit design we often can suffice with a description of the transistor’s characteristics (e.g., in the v-i plane) onthe circuit-theoretical level. In other cases, a typical phenomenon can be modeled correctly at only one level, for example, the tunneling of electrons can only be described with quantum physics. The important message in Figure 2 is that the various levels really differ in the way they describe the physics and in the way
they analyse circuits. They have different assumptions and use different tools. The quantum-physics level differs from the classical-physics level by considering the electron not only as a particle but also as a wave; the classical-physics level differs from the circuit-theory level because it includes the release of electromagnetic radiation in circuits; and, the circuit-theory level differs from the system-theory level by taking much more details into consideration and by using different simulation tools.
The basic physical phenomenon under consideration is the quantum mechanical tunneling of electrons through a small insulating gap between two metal leads. The metal-insulator-metal structure through which the electrons tunnel is called a tunnel(ing) junction. This tunneling is considered to be stochastic, that is, successive tunneling events across a tunnel junction are uncorrelated, and is described by a Poisson process. Tunneling through a potential barrier is considered to be non-dissipative (the tunneling process through the barrier is considered to be elastic).
As quantum mechanics is described in terms of energy, it seems obvious to describe, that is to analyse, the behavior of SET devices and SET circuits with energies. This is what the so-called orthodox theory of single electronics does. In this semi-classical-physics theory an electron will tunnel if the free (electrostatic) energy in the circuit after tunneling is equal or lower than the free energy in the circuit before tunneling. An example of such a theory is the so-called orthodox theory of single-electronics [1]. A more circuit design oriented theory has been developed by us and is called the impulse circuit model [2], [3]. The strength of both theories is that it successfully explains the Coulomb blockade phenomenon.
However, to design, that is to synthesise circuits with SET devices, we need a circuit theory. It is based on Kirchhoff’s voltage and current laws. In contrast, as we will see in the next subsection, these laws ensure energy conservation in circuits: any energy dissipation in the circuit is delivered by sources; and vice versa, all energy delivered by sources is either stored either dissipated in the circuit. It is because of this that we cannot follow the orthodox theory for designing circuits, but have to follow the impulse circuit model.
Electron transport in the nanoelectronic devices can best be described by quantum mechanics or approximated by a semi-classical-physics formulation; nanoelectronic circuits can best be described by using Kirchhoff’s voltage and current laws, which are only an approximation of the Maxwell’s equations in classical physics. This creates an area of tension that is taken for granted; experiments with circuits have to approve whether we can successfully include the quantum character of the current in a circuit theory for single-electron electronics.
DIFFERENT LEVELS IN MODELING
SYSTEM THEORY CIRCUIT THEORY CLASSICALPHYSICS QUANTUMPHYSICS MODELING EXAMPLE: NANOELECTRONIC NEURAL NETWORK NANOELECTRONIC CIRCUIT COULOMB BLOCKADE TUNNELING
Figure 2: Different levels in modeling topics involved in nanoelectronic circuit design.
Maxwell11.2 March 2008 1
Modeling at the circuit-Theory
Level: Tellegen’s Theorem
The best approach to finding a circuit theory for nanoelectronic devices is based on treating them as “black boxes”, that is, elements in which the specific behavior such as tunneling is hidden from the outside. If such elements can be found, and we will soon see that they can be found, then general circuit theorems on these black boxes can reveal the possibilities and restrictions that a circuit theory imposes on them. For tunnel(ing) junctions a general one-port circuit element, as shown in Figure 3, is a suitable black box. The element embodies the passive sign convention, that is, the power is directed towards the element: the positive current reference is going into the element at the terminal with the positive voltage reference.
New nanoelectronic devices, such as the tunneling junction, are devices having “strange” properties in terms of conventional circuit theory. So, obviously, the best approach is to start with the
most general circuit theorem that exists: Tellegen’s theorem. Tellegen’s theorem is general in that it depends solely upon Kirchhoff’s laws and the topology of the network. It applies to all (sub)circuits and elements, whether they be linear or non-linear, invariant or time-variant, passive or active, hysteretic or
non-hysteretic. The excitation might be arbitrary; the initial conditions are also arbitrary. It describes elements as black boxes only assuming a topology and the validity of Kirchhoff’s laws.
Tellegen’s theorem (1952) can be formulated as follows. Suppose we have two circuits with the same topologies, and all N elements are represented by a one-port as in Figure 4. For each element k in one circuit (the unprimed circuit), there is a corresponding element k’ in the other circuit (the primed circuit). If we now take the voltage across element k and multiply it with the current through the corresponding element k’ and sum this for all elements then:
To fully understand this theorem we consider the next example (2), Figure 4. We have a unprimed and a primed network consisting of three elements a,b and c. Using Kirchhoff’s voltage law the voltages v across the elements are expressed in potentials u with respect to the datum (ground). Now we can easily check that:
because according to Kirchhoff’s current law the currents at node 1 and 2 sum up to zero.
It is clear that our black boxes meet the theorem. The importance, however,
comes from the fact that, if we take the same circuit for both the primed and the unprimed circuit the theorem states the conservation of power and after integration it states the conservation of energy. Thus, if we describe our nanoelectronic devices as standard one-ports, then the conservation of energy must hold in the circuit. The next question is can we describe the tunnel(ing) junction as a one-port. The answer is yes. There even exist various proposals. A historical review is given.
SET-Junction circuit Models
An overview will show the development of circuit ideas to obtain an equivalent circuit for the tunnel(ing) junction. We notice that even for the most simple nanoelectronic device, such as the tunnel(ing) junction, many proposals already exist. Obviously, even the simplest device is not so simple to model.A good starting point is the modeling of two metals with a larger than nanome-ter separation, by a capacitor Figure 5(0). This is, of course, how electronics star-ted two centuries ago. The discovery of the tunnel effect by Esaki and Giaever in 1958-1959, led to a model in which the tunneling junction was represented by a “tunnel resistance”. This is a reasonable model, Figure 5(1), because the v-i cha-racteristic, of a tunneling junction excited by a voltage source, shows a linear rela-tion for small voltages.
Q
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kFigure 3: Schematic representation of a one-port network element k. The arrows follow the passive sign convention, so the current and the power are directed towards the element. 0 0 . 0 . ) ( ) ( ) ( 2 1 ' ' 2 ' ' 1 ' 2 ' 2 1 ' 1 ' ' ' = + = + − + + = + − + = + + u u i i u i i u i u i u u i u i v i v i v c b b a c b a c c b b a a
+
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+
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+
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_
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a a' b'v'
c c' u'1 u'2Figure 4: Example of the Tellegen’s Theorem
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.
2
Maxwell11.2 March 2008Based on research of Giaever and Zeller on small metal particles embedded in an oxide film, and of their own research on what we would now call a single-electron box, Lambe and Jaklevic proposed the parallel capacitor/resistance model Figure 5(2). In both studies, tunneling is represented by a resistor, and anomalies in the characteristics are explained by introducing a capacitor.
The next important step had to be taken when Fulton and Dolan measured the Coulomb blockade phenomenon in 1987. Due to the existence of a metal island the current through a metal-insulator-metal-insulator-metal is blocked for small values of the applied voltage (the circuit consist of a M-I-M-I-M structure excited by a voltage source). For values of the source larger than a critical voltage the resistive behavior is shown again. The in this way obtained characteristic is an affine relation between the voltage across and the current through the device. To avoid dominating noise from the influence of temperature, very small junctions and low temperatures were (and still are) needed. The equivalent circuit they proposed was a parallel capacitor/current source combination. In fact the current source they propose is voltage controlled, so in fact it is a nonlinear resistance; but the idea of modeling a tunnel current by a current source is an interesting and, in fact, a natural idea.
While developing the orthodox theory of single-electronics, more or less at the same time, Averin and Likharev proposed the description in which the tunneling junction was described by a parallel combination of a capacitor, if the junction is in blockade, and a resistor, if the junction is tunneling. There ideas can be best described by including a switch in the parallel combination, Figure 5(3). The resistor is equal to the “tunneling resistance” Rt. The fact that a current source disappeared was a consequence of
the assumption that the orthodox theory neglects the tunneling time and the energy contribution of current sources.
Up to now, the parallel combination of a capacitor and a nonlinear resistor, Figure 5(4,5), is still the dominant equivalent circuit for the tunnel(ing) junction. If we include, however, the predicted behavior of the tunnel(ing) junction excited by a current source the parallel capacitor/ nonlinear resistor combination does not correctly describe the predicted behavior.
A solution for this is again based on the parallel combination of a (charged) capacitor and a (impulsive) current source as shown in Figure 6. In this model the tunnel current is described by a impulsive current source, in which the Dirac delta-function is multiplied by the elementary electron charge, and includes the time the actual tunneling starts. The model is called the impulse circuit model for the single-electron tunnel(ing) junction.
A
d ? 10 nm d Al Al Giaever 1959 (1) Lambe & Jaklevic 1969 (2)Fulton & Dolan 1987 (3) A B RTJ= Rt CTJ CTJ RTJ= Rt i TJ(v) τ = 0 ( τ ) d? 10 nm d Al Al (4) Averin & Likharev 1986 (3) CTJ RTJ = Rt general in u , from 1986 upto now (5) CTJ RTJ Devoret et al. 1990 CTJ Rt A ±1800 (0) d>> 10 nm d Al Al CTJ e s
Maxwell11.2 March 2008 ( ) stra . d? 10 nm d Al Al (6 B Hoek et al 2001, 2004 + _ CTJ eδ(t-t0) q(t0-)δ t-t0) t= t0 t= t0
Figure 6: Proposed equivalent circuits for the single-electron tunneling junction
Figure 7: Photograph of the single-electron pump; fabricated at the PTB, the German Institute of Standards
Figure 8: The single-electron pump circuit
Figure 9: Spice simulation of the single-elec-tron pump
Figure 10: Measurements on the single-elec-tron pump. The measurements were done in cooperation with the NMI, the Dutch Institute of Standards.
A Simulation Example
To shown that it is really possible to use the equivalent circuits to simulate circuits including single-electron tunneling junctions, we simulated a so-called single-electron pump and compare the results with measurements. The equivalent circuit of the impulse model from Figure 6, is therefore implemented in SPICE.
The SET single-electron pump is a practical circuit that can be used as a current standard in metrology institutes. The circuit consists of three tunnel junctions in series coupled to an off chip voltage source by two resistors (typically 500 kOhm). Between any two junctions a capacitor with a control voltage source is placed. If a proper clocking scheme is applied to the control voltage sources the electron pump circuit is able to pump one electron at the time through the loop formed by the three junctions and the voltage source. From a so-called state diagram, a proper clocking scheme for single electron transport can be obtained. In such a state diagram the stable operation regions are shown as a function of the two control voltage sources. Such a state diagram can also be measured. Figure 7 shows a photograph of the single-electron pump. Figure 8 the circuit and Figure 9 shows the Spice simulation of a
single-electron pump. Figure 10 shows the measurements. To compare both Figures 9 and 10 one has to check the position of the triangles. The triangles are stable states in the pumping operation; the pumping cycle can be viewed as transitions through three adjacent triangles.
conclusion
The article showed that it is possible to analyse and synthesise circuits containing nanoelectronic single-electron tunneling junction devices. A circuit theory can be applied on equivalent circuits for those nanoelectronic devices if and only if energy is conserved in the circuits. Various equivalent circuits were proposed last years. An overview shows the development of ideas as a consequence of progress in experiments. Based on the equivalent circuit called the impulse circuit model it is shown that circuit simulators, such as Spice, can be successfully used for nanoelectronic circuit design enabling the possibility to investigate utilisation of new nanoelectronic devices.
A
[1] K.K. Likharev. Single-Electron Devices and Their Applications. In Proceedings of the IEEE, volume 87, pages 606-632, 1999.
[2] J. Hoekstra. On Circuit Theories for Single-Electron Tunneling Devices. In IEEE Transactions on Circuits and Sy-stems 1: Regular Papers, volume 54, pa-ges 2352-2360, 2007.
[3] J. Hoekstra. Towards a circuit theory for metallic single-electron tunnelling de-vices. International Journal of Circuit Theory and Applications, volume 35, pa-ges 213-238, 2007.
