Quantum-Effect and Single-Electron Devices

Transcription

1 368 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003 Quantum-Effect and Single-Electron Devices Stephen M. Goodnick, Fellow, IEEE, and Jonathan Bird, Senior Member, IEEE Abstract In this paper, we review the current status of nanoelectronic devices based on quantum effects such as quantization of motion and interference, and those based on single electron charging phenomena in ultrasmall structures. In the first part, we discuss wave-behavior in quantum semiconductor structures, and several device structures based on quantum waveguide behavior such as stub tuners, Y-branches, and quantum ratchets. Discussion is also given of proposals for use of interference phenomena in quantum computing followed by the issue of quantum decoherence which ultimately limits utilization of quantum effects. In the second part, we discuss single electron effects such as Coulomb blockade, and associated devices such as the single electron transistor and single electron charge pumps. This is followed by an overview of some recent work focusing on Si based single electron structures. We conclude with a discussion of proposals and realizations for single-electron circuits and architectures including single electron memories, single electron logic, and single electron cellular nonlinear networks. Index Terms Ballistic devices, nanoelectronics, quantum devices, single-electron transistors and circuits, single-electron tunneling. I. INTRODUCTION AS SEMICONDUCTOR dimensions continue to shrink in accordance with, and sometimes in contradiction to, the International Technology Roadmap for Semiconductors [1], an eventual end to the roadmap is anticipated which precludes further scaling of CMOS technology. Alternative technologies are desired which will allow continued increase in the density of memory and logic into the terabit regime. At the same time, there is a realization that the architectures necessary at ultra-high densities may have to be quite different than those currently employed in for example microprocessor design, in order to accommodate new device concepts, and the increasing need of fault-tolerance as device to device fluctuations become larger at the ultimate limits of integration. It is in fact likely that conventional digital architectures will co-exist with special purpose applications using nanoscale devices which perform parallel analog processing at much greater speeds than digital ones based on algorithms such as those based on quantum computation, and biologically inspired cellular architectures. Hence, there is a strong motivation for understanding and designing new and functional device structures at the nanoscale. In the present paper, we review some of the basic physics and technology of nanoelectronic devices, with a particular focus on quantum effect structures and single electron phenomena. In Section II, we discuss quantum effect devices, starting with analogies between quantum wave phenomena and Manuscript received September 30, The authors are with the Department of Electrical Engineering, Arizona State University, Tempe, AZ USA ( goodnick@asu.edu). Digital Object Identifier /TNANO electromagnetics, and the implications for coherent transport at the nanoscale. Several proposals and related experiments are discussed relative to quantum interference devices, followed by discussion of quantum decoherence, and the limitations on quantum coherent transport in semiconductor systems. In Section III, we review single electron tunneling phenomena, and the single electron transistor, which is a basic element in various implementations of single electron logic. We discuss some recent results on single electron device structures with a focus on Si based realizations, followed by a review of various proposed circuit and architecture proposals for single electron memories, logic, and analog array processing. II. QUANTUM-EFFECT NANOSTRUCTURED DEVICES A. Electron-Wave Phenomena in Quantum Devices As the size of transistors in integrated circuits is shrunk into the deep-submicron regime, it becomes important to consider the point at which the wave-mechanical characteristics of the electron become important for the discussion of device behavior. In MESFET s implemented in GaAs/AlGaAs heterojunctions, for example, typical electron densities are of the order cm, corresponding to a Fermi wavelength of nm. Since this length scale is comparable to the critical dimension for transport in many nanostructures, we expect that their electrical properties should be strongly influenced by the wave-mechanical nature of the electron. A striking demonstration of electron-wave behavior in a semiconductor nanostructure is provided in Fig. 1. Here, we compare the transmission properties of a GaAs/AlGaAs quantum dot, in which electrons are confined to a region a few-hundred nanometers in size, to those of a microwave cavity resonator, with macroscopic dimensions but the same basic geometry as the quantum dot. The electromagnetic-field distributions inside the resonator are well known to be described by the usual Helmholtz equations [2] (where is the usual wavenumber). These equations are functionally identical with that of the Schrödinger equation, for a freely-moving electron confined in a quantum dot That is, (1) and (2) indicate that electrons in a quantum dot should exhibit exactly analogous properties to microwaves in a cavity resonator. This analogy is apparent in the results of Fig. 1 [3], in which we show the variation with frequency of the microwave transmission probability, between the two antennas (1) (2) X/03$ IEEE

2 GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES 369 Fig. 1. Microwave-analogue study of quantum-dot transport. Shown top left are the dimensions of the microwave cavity resonator that is used to model the quantum dot, whose lithographic dimensions are shown in the top-right figure. The dotted lines indicate schematically the shape of the actual quantum dot that is formed in the 2-D electron gas. The bottom-left figure shows the variation of microwave transmission with frequency, while the bottom-right figure shows the measured magneto-resistance of the quantum dot. To allow a quantitative comparison, the two sets of data have been rescaled onto a so-called Weyl axis, which essentially involves converting the magnetic field, or microwave frequency, into an equivalent energy that is normalized in terms of the average level spacing (1). Further details on this may be found in [3]. (Figure provided courtesy of Y.-H. Kim and H.-J. Stöckmann.) of the cavity. We also show, in the same figure, the variation of the resistance of the quantum dot with magnetic field, applied perpendicular to the plane of the dot. The dot is essentially a split-gate [4] MESFET, in which electron-beam lithography has been used to define a pair of metal gates, with the lithographic pattern indicated in the upper part of Fig. 1. With a negative voltage applied to the gates, electrons underneath them are depleted, and electrical current is forced to flow through the quantum dot. The magneto-resistance of the dot clearly exhibits a similar fluctuating structure to that observed in the transmission of the cavity resonator, and this similarity is a consequence of the correspondence between (1) and (2). That is, the resistance fluctuations exhibited by the quantum dot are a direct manifestation of electron-wave behavior in this structure. B. Transmission and the Landauer Büttiker Approach Transport in nanostructures is often ballistic, involving a process in which electrons are transmitted through the active region of the device without undergoing any energy relaxation. The excess energy of these electrons is only lost after they arrive at the source, and equilibrate to the electrochemical potential of this reservoir. To describe transport in this regime, a flux-based description, known as the Landauer Büttiker formalism [4], [5], has been developed. The key features of this approach are nicely illustrated by considering its application to the problem of ballistic transport in a quantum point contact, which is a narrow constriction connected to macroscopic reservoirs that are biased to different electrochemical potentials. Since the constriction width is comparable to the electron Fermi wavelength, transport through it is mediated by a fixed number of one-dimensional subbands. The subbands are distinguished by the value of their momentum component, perpendicular to the axis of the constriction. The strong confinement of motion in this direction quantizes the transverse momentum into a discrete set of values, which in turn may be used to identify the different subbands. With a small applied bias across the reservoirs, the current carried by each of the subbands may be easily calculated [5], [6] (3)

3 370 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003 Fig. 2. Conductance quantization in the quantum point contact. The upper inset shows a schematic of the device geometry. (Figure reproduced with the permission of the author, from K. J. Thomas et al.) where is the applied voltage across the reservoirs. This important result is known as the equi-partition of current, and indicates that the same current is carried by each subband, independent of the subband index. In a constriction in which subbands are occupied at the Fermi level, the conductance of the constriction will be quantized according to The first demonstrations of this conductance quantization were provided by studies of split-gate quantum point contacts, which were implemented in the high-mobility two-dimensional (2-D) electron gas of GaAs/AlGaAs heterojunctions [7], [8]. In this approach, metal gates are formed on the surface of the heterojunction and are used to define the constriction in the 2-D electron gas, by the application of suitable, depleting voltages (Fig. 2). The size of the constriction that forms in the electron gas is determined by the range of the fringing fields that develop around the gate edges, and may be tuned continuously in experiment by variation of the gate voltage. As the width of the constriction is reduced in this manner, successive subbands are depopulated and the conductance of the point contact decreases in integer steps of (Fig. 2). Conductance With Imperfect Transmission: The quantization of conductance in the quantum point contact may be viewed as arising from a bottleneck effect, which arises as current flows between the reservoirs and the narrow constriction [5]. The reservoirs themselves may be viewed as one-dimensional conductors, which support an infinitely-large number of modes. As current flows between the reservoirs and the constriction, a redistribution of current among the different subbands must therefore occur. With an adiabatic transition between the different regions, and in the absence of disorder within the constriction, scattering between the subbands is quenched, and it is this property that ultimately leads to the conductance quantization. If we now consider some device in which, possibly due to the presence of disorder, or some geometrical feature, intersubband scattering can occur, then (4) Fig. 3. (a) Experimentally measured conductance characteristic of an electron waveguide featuring a double-bend structure. The device structure is shown in the inset, in which the lithographic width of the waveguide is 100 nm. (b) Numerically calculated conductance variation for a hard-walled waveguide structure. For further details, we refer the reader to [10]. the principle of current equipartion no longer holds and the conductance quantization breaks down. In this situation, the conductance may instead be expressed as where is the total transmission probability of the device, and may be expressed as a sum of individual probabilities ( ), due to each of the different subbands. Among the possible sources of scattering in nanostructures, the simplest is the presence of unintentional disorder [9]. Of more interest, however, is scattering that arises from the presence of intentionally-induced discontinuities in the device profile. These features disrupt adiabatic transport, giving rise to device-specific features in the transmission, and so in the conductance. As an illustration of these ideas, consider the device structure shown in Fig. 3, which consists of an electron waveguide that incorporates a double-bend discontinuity. In Fig. 3(a), we show the measured variation of the waveguide conductance as a function of the gate voltage [10]. Rather than exhibiting quantized plateaus, the conductance instead shows slow oscillations, on top of which finer resonant features are superimposed. The nonquantized conductance of this structure is a consequence of the interference of electron partial waves, which are multiply scattered from the waveguide geometry. As the gate-voltage is (5)

4 GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES 371 Fig. 4. Computed variation of conductance with energy for an open, stadium-shaped, quantum dot. Note how the energy in this figure is normalized to the average level spacing of the dot 1 = 2 =m A, where A is the dot area. The geometry of the corresponding closed dot is shown in the bottom-right corner of the figure. The different triangles represent the calculated energy levels of this closed dot, and the 134th, 138th, and 142nd eigenstates are indicated with filled symbols. At the top of the figure, we show computed probability densities for the closed (left panel) and open (other two panels) dots. For further details, we refer the reader to [12]. used to change the shape of the waveguide, the interference is modulated between constructive and destructive in nature, giving rise to the nonmonotonic conductance variations seen in experiment. In Fig. 3(b), we show the computed [10] variation of the conductance with the width of an ideal, hard-walled waveguide structure. The conductance here has been obtained by application of the Landauer Büttiker formula, beginning with a calculation of the energy-dependent transmission probability. Striking resonances are seen in the calculated conductance, and the general behavior shows some similarity to the results of Fig. 3(a). The computed resonances are much sharper than those seen in experiment, but this difference can be attributed to the finite temperature at which the latter is performed, and the softwalled nature of the confining profile in real devices [10]. Another type of device whose conductance is strongly influenced by the scattering of electron waves from its boundaries is provided by the quantum dot. The confinement of charge within such structures quantizes the electronic energy into a set of discrete levels, the details of which may be probed in transport measurements [11]. Of particular interest here, is the behavior exhibited by open quantum dots, such as that shown in Fig. 1. These structures are connected to their reservoirs by means of waveguide leads that support one or more subbands. This coupling gives rise to a highly nonuniform [12] broadening of the original states of the isolated dot, which has important implications for the conductance of this structure, as we illustrate in Fig. 4. Here, we show the calculated variation of the conductance with energy, for a stadium-shaped open dot. The triangles in the figure indicate the computed energy levels of the closed dot, whose geometry is shown in the lower-right-hand inset of the plot. The conductance shows three resonances, which occur at energies correlated to three of the known levels of the isolated dot (indicated by the filled triangles). These three states aside, however, we see that the remaining energy levels (open triangles) do not give rise to any noticeable signature in the conductance. In other words, when the dot is opened to the reservoirs, Fig. 5. Magneto-conductance fluctuations in an open quantum dot. The different curves were obtained at 0.01 K with different gate voltages applied to the quantum dot, whose geometry is shown in the electron micrograph at the top-left corner. The lithographic size of the central dot is 0.4 microns. For further details, we refer the reader to [15]. many of its states become broadened due to their short lifetime in the dot, while a smaller subset show a much longer lifetime and survive to give rise to signatures in the conductance [12]. A special feature of these few robust states, revealed in the numerical studies, is that the probability density in their wavefunctions tends to be built up away from the leads. Consider, for example, the wavefunctions shown at the top of Fig. 4. The panel on the left shows the 142nd eigenstate of the isolated dot, while the other two panels show the wavefunction of the equivalent open dot, for two different sizes of lead opening. There is clearly a strong correspondence between the open- and closed-dot wavefunctions, which both exhibit a buildup of probability density at the center of the dot. The crucial point here is that the states in this figure show only weak coupling to the leads of the dot, and it is this property of the eigenfunctions that allows them to remain resolved, even with the leads configured to support several propagating modes. In fact, this behavior is a manifestation of the effect of resonance trapping, which is well known from studies of nuclear systems [13], and microwave cavities [2]. Strong experimental support for this theoretical picture has now been found in transport studies of open quantum dots [14], [15]. Here, the magnetic field or gate voltage is actually varied, rather than the energy of the carriers (which is set by the doping level in the semiconductor). Nonetheless, the effect of either of these parameters is to sweep the discrete states of the open dot past the Fermi level, giving rise to the observation of fluctuations in the conductance [14], [15] (Fig. 5). While it is clear from the preceding discussion that the Landauer Büttiker formalism provides a powerful tool for analyzing the characteristics of quantum devices, for completeness its limitations should also be pointed out. In particular, (5) is only expected to be valid at low temperatures, where the filling of the one-dimensional subbands in the wire can be considered

5 372 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003 to be close to that expected in the ground state. Also, in deriving (5), we have neglected the energy dependence of the transmission probability, which limits the validity of our approach to the linear source-drain limit. Nonetheless, the Landauer Büttiker formalism can also be extended to the case of incoherent transport, provided that there is no net loss of energy from the electron system [5]. A more significant problem is that this model ignores completely the influence of electron electron interactions on the transmission characteristics. In spite of these various shortcomings, however, there are many situations in which the Landauer Büttiker formalism does provide a good understanding of the electrical characteristics of small devices. C. Quantum-Effect Devices In this section, we review some of the basic concepts for device operation that utilize quantum phenomena in mesoscopic devices. Our interest here, in particular, will focus on device implementations that exploit the wave-mechanical properties of the electron, rather than single-electron phenomena, which will be discussed in Section III of this review. The Stub Tuner: One of the simplest quantum devices, which exploits the wave-mechanical properties of the electron, is the electron analog of the microwave stub tuner. The electronic implementation of this structure consists of a narrow quantum wire, which widens over some short distance to form a small cavity structure (Fig. 6). Transmission through the stub structure is determined by the interference between electron partial waves that undergo multiple reflections from the stub geometry. Using Schottky gates to modulate the geometry of the stub, the interference between these waves may be tuned from constructive to destructive in nature, giving rise resonant features in the conductance of the device [16], [17]. In the case where the quantum wire supports just a single propagating mode, 100% resonant modulation of the transmission is expected [18], [19], as we illustrate in Fig. 6. Experimental implementations of this structure typically exhibit a smaller effect, however, as can be seen from a comparison of the two curves in Fig. 6. While the trend in the experimental data is clearly reminiscent of the theoretical curve, the effective conductance modulation is no larger than 40% in the former case. This discrepancy can basically be ascribed to two different origins. Firstly, realization of the stub-tuner by the split-gate technique inevitably results in a device with much smoother walls, than the hard-walled structure assumed theoretically. This softness tends to suppress reflections within the device, promoting more adiabatic transport, as we have discussed already. Secondly, the well-defined phase-relationship, required between the differently reflected waves to give rise to the conductance modulation, can be disrupted at finite temperatures by various sources of electron decoherence [20]. Among these sources include electron electron and electron-phonon scattering, as well as thermal smearing of the Fermi energy. Since the decoherence rate generally increases with increasing temperature, effective operation of the stub tuner is typically washed out at temperatures in excess of a few [19]. The Y-Branch Switch: While the stub tuner is a two-terminal device, another important implementation of one-dimensional electron waveguides is provided by the three-terminal, Y-branch switch [21] [23] (Fig. 7). The basic idea of the Y-branch struc- Fig. 6. The electron stub tuner. The basic geometry of the stub-tuner is shown as an inset to the lower figure. The upper figure shows the computed response of the stub tuner, with a single mode present in the electron waveguide. The parameter a=a that is varied on the horizontal axis of this figure essentially corresponds to a variation of the voltage applied to the stub gates. The lower figure shows the experimentally-measured conductance traces of an electron stub tuner at three different temperatures: 0.09 K (thick solid line), 1 K (thin line), and 2.5 K (dotted line). In this figure, the conductance is measured as the voltage (V ) applied to the upper gate of the stub tuner is varied, while a fixed voltage is applied to the lower gate. For further details, we refer the reader to [18]. (Figure reproduced with the permission of the authors. Copyright 1999 by the American Institute of Physics.) ture is to deflect electrons incident from the center branch, into either the left or right branch, by the application of a suitable electric field. An important advantage of the Y-branch is that its switching action is achieved by deflecting electrons from one branch into another, and so therefore does not require that we stop the incident carriers completely, at some suitable barrier, as is more typical in conventional devices. Recent theoretical studies [24], [25] have suggested that a number of novel applications may arise from exploiting the ballistic transport in these structures. In one approach to its operation, a small bias is applied to the center terminal of the Y-branch and the currents that flow through its other two terminals are measured, as a push-pull

6 GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES 373 Fig. 7. The Y-branch switch. The basic geometry of the device is shown schematically in the top-left figure. The bottom-left figure shows the actual structure studied in by Worschech et al. in [28]. Figure reproduced with permission of the authors. Copyright 2001 by the American Institute of Physics. The right-figure shows the self-gating action of the Y-branch at room temperature. (Figure reproduced from [29] with permission of the authors. Copyright 2001 by the American Institute of Physics.) voltage is applied to gates that modulate the width of the left and right waveguides. While this allows for switching of current between the two waveguides [22], [26], self-gating of the device can be achieved by operating it in the nonlinear regime [23], [24]. In this alternative approach, the voltage of the floating center branch is measured, with fixed voltages ( ) applied to the left and right waveguides. For a Y-branch structure in which the electron motion is diffusive, the voltage should simply correspond to the average of the terminal voltages and. By modeling the Y-branch as a ballistic cavity that is adiabatically coupled to three waveguides [24], however, it may be shown that the floating stem tends to follow the higher electrochemical potential of the two reservoirs. With push-pull voltages ( ) applied to the left and right waveguides, the voltage of the stem will therefore always be negative, rather than taking the classically-expected value of zero. This property is basically a consequence of the ballistic nature of motion in the junction, and it may furthermore be shown [24] that the stem voltage Here, is a positive constant, which varies with temperature, the reservoir chemical potential, and the conductance of the center waveguide. The parabolic variation of the center voltage with source voltage has now been observed in a number of experiments [27] [29]. The only requirement for the observation of this effect is that electron transport in the center cavity should be ballistic, and it is for this reason that clear signatures of the nonclassical voltage variation, predicted in (6), have been observed at room temperature [27], [29] (Fig. 7). The nonclassical response of the Y-branch switch has also been predicted [24], [25] to allow for rectification and basic transistor action, second-harmonic generation, and logic operation. In particular, with biases applied to left and right reservoirs, the output voltage (6) of the center waveguide will only be positive when a positive voltage is applied to both the left and right reservoirs, indicating that the Y-branch may be used as a compact AND gate. With this motivation, there has been much interest in the development of novel circuit architectures, based upon the properties of the Y-branch [30]. Classical and Quantum Ratchets: Another interesting example of nonlinear ballistic transport is provided by devices that feature deliberately-introduced asymmetries in their confining profile. One such device consists of a cross junction, formed by two electron waveguides, that includes a triangular antidot at its center [31] (Fig. 8). Before discussing the characteristics of this device, we recall that, for current flow in the linear regime, the Landauer Büttiker formalism predicts that the measured resistances should obey the following reciprocity relation [32] Here denotes the resistance measured with current flowing between probes and in Fig. 8, and voltage measured between probes and. In the nonlinear regime, however, the device of Fig. 8 shows very different behavior. Regardless of the sign of the drive current ( ), we see that the measured voltage ( ) isalways negative, indicating that the reciprocity relation of (7) is violated. Physically, this effect is easily understood to arise from the symmetry-breaking properties of the scatterer at the center of the junction. Regardless of the direction of current flow, this directs incident carriers into the lower contact, causing to always be negative. What is perhaps more surprising is that such behavior is not predicted by the Landauer Büttiker formalism, which suggests that (7) should always hold. The reason for this discrepancy is that, in the Landauer Büttiker approach, the current dependence of the subband transmission probabilities is neglected. Song has shown, however, that the formalism can be extended to the nonlinear regime, and that the resulting predictions are quite (7)

7 374 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003 Fig. 8. Rectification in a cross junction with a triangular antidot at its center (see inset). (Figure reproduced from [31] with permission of the authors. Copyright 1998 by the American Physical Society.) consistent with the experiment of Fig. 8 [33]. Another feature of Fig. 8 is significant asymmetry in the two curves, around, and this has been attributed [31] to irregularities in the antidot geometry that are unintentionally introduced during the fabrication of the device. The current-voltage characteristic of Fig. 8 is similar to that of a bridge rectifier, and these junctions have been used to provide rectification of applied voltages in the GHz regime [34]. Due to the ballistic motion of carriers in the junction, it has been suggested that the rectification should persist to very high frequencies, and operation at 50 GHz has actually been demonstrated, in a modified structure consisting of planar arrays of the triangular antidots [35]. We emphasize that the main requirement for the observation of these effects is the ballistic motion of carriers in the material, for which reason room-temperature operation has also been successfully achieved [35]. Particles that move in an asymmetric potential can exhibit net motion in a particular direction, even in a situation where the time averaged force that acts on the particles is equal to zero. The rectifying junction considered in Fig. 8 represents a classical manifestation of such a ratchet device, since the ballistic scattering of electrons from the antidot is responsible for the rectification. Quantum-mechanical ratchets have also recently been implemented, however, by exploiting the nonlinear transport characteristics of quantum dots [36] [38]. Linke et al. have shown that asymmetric quantum dots can be used to generate a rectified direct current, when a suitable alternating voltage is applied between the source and drain [36], [37]. In contrast to the behavior in ballistic junctions [31], the rectified current decreases strongly with increasing temperature, and is typically no longer observed above a few Kelvin. This indicates that the origin of this effect is a wave-mechanical interference phenomenon. Electron Waveguides for Quantum Computing: There have recently been several interesting proposals to exploit the unique properties of electron waveguides, in solid-state implementations of quantum computing [39], [40]. In one scheme [39], a structure consisting of two single-mode waveguides, coupled to each other by means of a short potential barrier (Fig. 9), has been proposed as a realization of a quantum-mechanical bit (qubit). A wave-packet launched in one of the waveguides will oscillate back and forth between the two branches, as it passes through the region where the potential barrier is located. Dependent upon the length of the barrier, and the initial energy of the wavepacket, it is therefore possible to switch the electron between the two waveguides, which are therefore used to represent the two logic states of the qubit. An alternative implementation of this approach, based on the use of plane waves, rather than wave packets, has also been proposed [40]. In Fig. 9, we show the results of calculations for such a structure, and see that the electron-wave probability can be switched back and forth between the two waveguides, by controlling the length of the coupling window. While there have been some investigations of transport in coupled quantum wires [41], [42], practical implementations of the waveguide-based qubit remain to be demonstrated. D. Decoherence and its Implications for Quantum Devices Many of the devices we have discussed thus far rely for their operation on interference effects, involving electron partial waves that are scattered from different regions of the device. In an ideal system, at zero temperature, the wave-propagation is completely coherent, and the interference is determined exclusively by the energy of the electrons, and the geometrical details of the device. In real structures, however, which operate at nonzero temperature, coherence of the electron waves is not preserved indefinitely, but is instead disrupted by phase-randomizing scattering [20]. Among the possible sources of this scattering, include the electron-phonon and the electron electron interaction. To quantify the influence of this scattering, we define a so-called phase-breaking time ( ), which may be viewed as the average time over which the electron may propagate without losing its wave coherence. Experimental investigations suggest that the value of this time scale is of order a nanosecond at mk temperatures, but that its decreases rapidly with increasing temperature [20]. Another effect of increasing temperature is thermal smearing of the electron population near the Fermi level, and this effect combines with the scattering-induced decoherence discussed above to suppress quantum signatures in the conductance (Fig. 10). III. SINGLE ELECTRON DEVICES In Section II, we focused on phenomena resulting from size scaling in terms of phase coherent transport and quantization of the electronic spectrum. Another consideration in ultrasmall structures is the granularity of charge itself in terms of the finite number and charge of electrons. Single electron phenomena refers to effects manifest in the injection and extraction of individual electrons from a mesoscopic system such as the quantum dot structure discussed in Section II-B, and the corresponding change in energy of the system. These include Coulomb blockade, the Coulomb staircase, Coulomb oscillations, turnstile behavior, etc. associated with the interaction of single electrons with their nanoscale system. Such single

8 GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES 375 Fig. 9. Qubit operation based on coupled quantum wires. A schematic illustration of the structure is shown top left, while the bottom-left figure shows the computed variation of the conductance, as a function of the length of the coupling window between the wires. The variation of the transmission was obtained by using a lattice discretization of the Schrödinger equation and the two curves shown here correspond to different grid sizes (2.0 and 2.5 nm) in the discretization. The finer oscillations on top of the large transmission variation should therefore be regarded as numerical artifacts. (Figures reproduced from [40] with permission of the authors. Copyright 2001 by the American Institute of Physics. The right images show calculated probability density in the wires for different coupling lengths, indicating the switching action of the waveguides. Figures provided courtesy of M. J. Gilbert and D. K. Ferry.) and devices (see for example [4], [5], [43], and [44]). In the following, we briefly review single electron phenomena with a focus on semiconductor systems in Section III-A, including more recent results in Si nanoelectronic technologies. This is followed by a discussion of single electron circuits and systems, and some of the proposals, and preliminary demonstrations, of functional applications of single electron phenomena. A. Single Electron Phenomena The phenomenological understanding of single electron behavior is most easily provided in terms of the capacitance,, which relates the charge to the potential difference between two conductors, and the corresponding electrostatic energy,, stored in the two conductor system Fig. 10. Influence of finite temperature on the conductance fluctuations in a 1-micron split-gate quantum dot. Unpublished data. electron effects have been the basis for a number of device and architectural proposals and demonstrations such as single electron memories, single electron transistors, quantum cellular automata, as well as many others. The interested reader is referred to detailed reviews of single electron phenomenon (8) When physical dimensions are small, the corresponding capacitance (which is a geometrical quantity in general) is correspondingly small, and a change in charge,, corresponding to a single electron, can result in a nonnegligible change in the electrostatic energy. As an example, the capacitance of a conducting sphere of radius above a ground plane is approximately. For a 5-nm radius nanocluster (for example, Au forms stable cluster shells of even smaller dimensions), the change of energy associated with removing an electron from an initially charge neural cluster corresponds to approximately 145 mev, which is much larger than the thermal energy, even at room temperature. The basic building block of single electron devices and circuits is the tunnel junction, illustrated by the circuit elements

9 376 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003 where the two junctions is. The electrostatic energy stored in (11) The free energy corresponds to the difference of the electrostatic energy and the work done in delivering charges from the source to the SET system,. The work done in delivering charge to the system is given by the time integral of the power delivered to the SET from the external sources as (12) Fig. 11. Equivalent circuit for a single electron transistor (SET). shown in Fig. 11, which illustrates the schematic of a single electron tunneling transistor (SET). A tunnel junction is characterized by its capacitance,, and tunnel resistance,, the latter of which corresponds in the usual generic way to the height and width of the potential barrier between electrodes. Tunnel junctions are actually representative of a broad range of permeable device technologies including ultrasmall metal-oxidemetal junctions (oxidized Al for example), the quantum point contact structure discussed in Section II, sidewall constrictions in an etched Si on insulator or GaAs/AlGaAs structure, and even the contacts to carbon nanotubes. The SET transistor, first realized experimentally by Fulton and Dolan [45] and Kuzmin and Likharev [46], consists of a pair of tunnel junction separated by an island with an applied source-drain bias as shown in Fig. 11. The island itself (which represents an isolated conducting region) is capacitively coupled through to a gate bias, making a three-terminal structure. To understand the behavior, we consider the energy change when electrons tunnel back and forth across the two tunnel junctions. Here we let be the net number of electrons that tunneled through the first junction onto the island, the number of electrons that tunneled through the second junction exiting the island, and is the net number of excess electrons on the island. With a bias voltage applied across the two junctions the charges on the junctions and island can be written as where is the net charge on the island, is the background charge induced by stray capacitances associated with material imperfections and fabrication induced defects, and the effect of the gate electrode is to contribute and additional controllable polarization charge on the island. In terms of the applied bias,, we can rewrite the junction potentials as (9) (10) where is the total charge transferred from the drain or gate voltage sources, including the integer number of electrons that tunnel into and out of the island, as well as the continuous polarization charge that builds up in response to the change in electrostatic potential on the island. We can now look at the change in free energy of the entire circuit due to electrons tunneling across junctions 1 and 2 separately by considering the total free energy before and after tunnel events which decrease or increase the net number of electrons tunneling across junctions 1 and 2 as (13) (14) We can now argue that the only high-likelihood tunneling events are those that result in transitions to final states of lower energy, i.e., negative change in the free energy above. Coulomb Blockade: Assume for simplicity that we have a double junction system without an external gate, i.e.,, and that the stray polarization charge is zero. Further, assume that the two tunnel junctions are identical, i.e.,. If we start from a condition in which the island is initially charge neutral, i.e.,, then its clear from (13) and (14) that there is a minimum applied drain voltage,, necessary in either direction before the change in free energy is negative, i.e., for, tunneling cannot occur. This phenomena is referred to as Coulomb blockade (CB), or the suppression of tunneling due to the effective charging energy barrier to adding or removing an electron from the island. An illustration of this phenomenon in terms of the energy band diagram of the system, and the expected - characteristics are shown in Fig. 12. Essentially, the Coulomb charging energy opens a gap in the continuous spectrum of energy states associated with the island, which forbids tunneling until this barrier is surmounted with an applied bias. Once an electron enters the dot, (i.e., ), a new Coulomb blockade exists until the electron tunnels out the other side. Hence, tunneling in this idealized situation also corresponds to the correlated tunneling of one electron at a time for biases just above this threshold. Higher thresholds exist in which it is energetically favorable for 2 electrons, 3 electrons, etc. to be injected, which for asymmetric barriers, results in a Coulomb staircase corresponding to a series of

10 GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES 377 Fig. 12. Energy band diagram for a double junction systems illustrating the band diagram in the (a) Coulomb blockade regime and (b) tunneling regime. (c) Coulomb gap in the current-voltage characteristic for 0e=2C <V <e=2c. Fig. 13. Stability diagram for the single electron transistor of Fig. 11. current plateaus of successively high integer numbers of electrons tunneling across the double junction. Coulomb Oscillations and the Single Electron Transistor: If we now consider the effect of the gate capacitance and bias in (13) and (14), we see that the CB may be lifted with appropriate combination of positive or negative gate bias and number of excess electrons on the dot,. Therefore, as a function of gate and source-drain bias, there are going to be regions where the free energy change is positive, corresponding to little current flow, and regions where tunneling is allowed energetically. This may be conveniently represented by a stability diagram as shown in Fig. 13. There the shaded regions correspond to combinations of the two biases where CB occurs, which forms the diamond pattern shown for integral values of the electron number on the dot. For successive changes of the effective gate charge,, the source-drain conductance goes through successive oscillations or resonances where CB is lifted. This oscillatory behavior can be better understood looking at the energy band diagram on and off resonance as shown in Fig. 14 and the corresponding current-voltage characteristics. The effect of the gate is to tune the Coulomb gap in the density of states through the Fermi energies on the left and right (which are nearly coincident for small source-drain bias). As the gap is pulled below the Fermi energies, electron tunneling onto the island can occur, increasing the number by one, and resulting in a new CB regime, hence the successive diamonds along the axis. The corresponding conductance then exhibits a series of peaks spaced periodically in, which are sometimes referred to as Coulomb oscillations. The picture above is relevant for the idealized case of a perfectly conducting metallic island consisting of many electrons with quasicontinuous energy spectrum. In the case that the island is a semiconductor dot formed by artificial confinement as discussed in Section II, the quantized energy of the dot states can be appreciable compared to the Coulomb energy, and must be accounted for in the description. If we label the energy levels in the dot as, then successive oscillations in the conductance with gate bias include both the Coulomb contribution plus the energy spacing of successive quantized levels as (15) Since in general the energy spacing of the quantized levels of an artificial molecule are nonuniform, the overall spacing of the Coulomb oscillations with gate bias will no longer be strictly uniform. Within the picture presented above of Coulomb blockade and Coulomb oscillations, we have implicitly assumed that the electrons are well localized on the dot, i.e., that we can talk about the electron as residing inside or outside the dot, which is not the case in the open dot structures discussed in Section II-B. Hence, the probability of tunneling in and out of the dot should be sufficiently large that the electron is localized on the dot, which is usually satisfied if the tunneling resistance itself is much larger than the fundamental resistance corresponding to a single conducting channel, i.e.,. B. SET Modeling and Simulation To actually formulate the current-voltage characteristics of SET structures, a kinetic equation approach was generalized by Averin and Likharev [47] now referred to as the orthodox theory of single-electron tunneling. Within a kinetic equation approach, tunneling processes are considered as random scattering events that instantaneously change the energy of the system. Using time dependent perturbation theory, the usual theory of tunneling via the tunneling Hamiltonian approach (which treats tunneling via perturbation theory) can be generalized to include the change in free energy discussed above before and after tunneling. Hence, the tunneling rate for the th tunnel junction in an junction system is given by (16) where is the change in free energy, which as defined for the two junction SET system by (14) and (15). Within the kinetic equation framework, we can define the distribution function for the island occupancy,, which is the probability of the system having electrons on island 1, electrons on island 2, etc. A kinetic or master equation can

11 378 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003 Fig. 14. Energy-band diagram and I-V characteristics as a function of gate bias of a SET transistor under small source-drain bias (a) in the Coulomb blockade regime and (b) in resonance. then be derived which represents an equation of motion for through a detailed balance of tunneling events onto and off of each island See equation (17) at the bottom of the page, where here refers to the net tunneling onto each island from all possible junctions using the junction rate defined in (17). Once is calculated, averages may be calculated for quantities of interest such as the total energy, or current flow through a particular junction. Direct solution of this master equation has been successfully used to model the - characteristics of the single electron transistor discussed in the previous section (see for example [48]). The derivation of (16) is based on first order time dependent perturbation theory (i.e., Fermi s golden rule), however, higher order tunneling processes may in fact be important, particularly when the tunnel resistance approaches that of the fundamental conductance,. Higher order processes, or co-tunneling, represent tunneling processes that occur through multiple junctions, such as for example resonant tunneling in a double barrier resonant tunneling diode. The theory of co-tunneling has been developed by Averin and Nazarov [49], which gives corrections to second order in the inverse tunnel resistance, and gives rise to additional power law dependencies on the voltage and temperature. The tunnel resistance dependence has been studied in detail experimentally using quantum point contact structures (where the tunnel resistance can be tuned) [50]. While direct solution of the master (17) is feasible, for arbitrary large SET circuits, it has become increasingly popular to utilize Monte Carlo techniques for the simulation of single electron tunneling [44], [51] [53]. In Monte Carlo simulation, basically the stochastic tunneling events across all possible junctions based (16) (and extensions to higher order and nonlinear tunneling resistances) are simulated in time using the computer random number generator to generate the time between tunneling events. Commercial simulators are available such as SIMON2.0 [54] which provide schematic capture for design and simulation of single electron circuits, applications of which are discussed later in Section III.D. C. Recent Experimental Studies An exhaustive review of experimental work on single electron phenomena is beyond the scope of this paper, and the interested reader is referred to several of the reviews and books listed here (e.g., [4], [43], [44]). Basically, Coulomb blockade and associated single electron behavior such as the Coulomb staircase were first observed in metal-oxide tunnel junction systems in the 1980s (see [43] and references therein). As mentioned earlier, Fulton and Dolan [45] fabricated the first successful single electron transistor, in which the CB regime and Coulomb staircase could be controllably modified by a gate. Following this, (17)

12 GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES 379 Fig. 15. MOS single electron transistor. a) Illustration of the double oxide split gate structure and b) associated conductance-voltage characteristics with inversion layer bias. For a detailed description of the device structure and its electrical characterization, we refer the reader to [63]. (Figure reproduced with the permission of the authors.) researchers were able to realize single electron turnstiles and pumps [55], [56] in which single electrons could be systematically clocked through an array of tunnel junctions by periodic modulation of the gate potential at rf frequencies, and the resulting current is given quite accurately by, where is the ac frequency. Such turnstile devices are still an active area of investigation for accurate metrological standards. The first definitive demonstration of Coulomb blockade in semiconductor structures was reported by Meirav et al. using a pair quantum point contact structures to form a double tunnel junction system over a high mobility 2DEG GaAs/AlGaAs heterostructure to form a quantum dot as the island, and using substrate bias as the gate potential [57]. Clear periodic oscillations of the source-drain conductance with gate bias were observed. Subsequent demonstration of the Coulomb staircase, and turnstile behavior in more elaborate gate geometry QPC quantum dots in high mobility 2DEG material were reported by the Delft group [58]. The flexibility of the QPC structure has led to increasingly more complicated geometries to investigate single electron tunneling through multiple dots, where molecular hybridization of the states in coupled dots is observed when the dots are allowed to interact (see for example [59]). Si Nanoelectronic Devices: More recently, interest has focused on the development of single electron devices in Si that are compatible with Si CMOS processing for potential realization of Si nanoelectronic circuits. The advantages of such Si based nanoelectronic systems is that Si process technology is more mature, and the high quality of the native oxide as well as improvements in Si on insulator (SOI) technology, provide more flexibility in design of single electron devices than III-V compound technologies. The main disadvantage is the higher effective mass, lower mobility, and generally higher density of defects in Si compared to high purity epitaxial growth techniques used for the 2DEG structures discussed earlier, which tend to mask quantum and single electron behavior. High-resolution nanofabrication techniques such as STM/AFM lithography have led to considerable reduction in the size of single electron devices. Sub-10-nm tunnel junctions have been realized using in-situ anodization of Ti films with an AFM tip [60]. Similar technology was used by Matsumoto to fabricate ultra-small double barrier SET structures of anodized Ti on an oxidized Si substrate, which exhibited strong evidence of Coulomb staircase with 150 mv period at room temperature [61]. Conventional CMOS technology has been used to realize single electron structures as well [62] [64]. Fig. 15 shows the schematic of a split-gate quantum point contact structure similar to those discussed already, but fabricated in a double oxide, Si MOS transistor structure [63]. As shown in Fig. 15(b), distinct Coulomb oscillations are observed at 4 K as a function of the top inversion gate bias, superimposed on a background of rising conductance as the channel forming the QPCs opens up with increasing gate bias. A third terminal or plunger adjacent to the dot acts like the gate of a SET, while at the same time changing the shape of the dot itself. A plot of the location of the peak conductance as a function of both plunger and inversion-gate bias is shown in Fig. 16. As can be seen, there are several sets of peaks that evolve with different slope, and which exhibit anti-crossing behavior which appears somewhat analogous to that of atomic levels. In fact, it appears that in these dots, that conductance oscillations are dominated by the energy spectrum of the dot itself as much as by the Coulomb charging energy as given by (15). Silicon on Insulator technology (SOI) has gained increasing acceptance in recent years as a technology for scaling conventional CMOS technology below the 10 nm gate length

13 380 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003 Fig. 16. Conductance peak positions as function of both inversion gate and plunger gate bias exhibiting crossing and anti-crossing behavior of apparent level structure of dot [63]. The closed symbols in the figure denote the position of well-defined peaks in the conductance, while the open symbols correspond to ranges of the voltages where shoulder-like features, rather than clear peaks, were found in the conductance. (Figure reproduced from [63] with the permission of the authors.) node. SOI technology has also proved promising in realizing SET structures with potential for room temperature operation. Zhuang et al. first demonstrated a quantum dot structure fabricated on an SOI wafer [65], in which the SOI layer is etched down forming a corrugated Si wire over which a poly-si gate is deposited. The NTT group demonstrated SETs using a so called vertical pattern-dependent oxidation (V-PADOX) technique on SOI [66]. Multigate SETs were demonstrated in this technology with functional logic behavior up to 40 K [67]. An illustration of a recent SET transistor fabricated using an SOI process is shown in Fig. 17(a) [68]. As in earlier work, a narrow Si wire is fabricated above the buried oxide of an SOI wafer, and then sidewall depletion gates are fabricated to control the left and right tunnel barriers into an island defined in the inversion layer of the Si layer. Coulomb oscillations with gate bias are observed as shown in Fig. 17(b), which are strongly correlated with the effective size of the dot (defined by the depletion gates) and its corresponding capacitance. D. Single Electron Circuits Single electron devices have long been proposed as candidates for a variety of nanoscale circuit applications due to their inherently small size (and improving performance as the size decreases), and low power dissipation, at least for certain applications. The inherent robustness of single electrons as a quanta of charge is also attractive for potential circuits applications like memories. It is difficult to do full justice to the extensive number of proposals for various single electron circuit technologies, so in the following, we touch on just a few relative to memory applications, SET logic elements, and analog signal processing. A substantial review of single electron circuits and applications is given in elsewhere (see for example [44], [69]. Here we defer on any discussion of the so-called Quantum Cellular Automata, which is inherently a single electron structure based on field coupling between QCA cells [70], as this is discussed in detail elsewhere in the article by G. Snyder in the present issue. (a) (b) Fig. 17. SOI single electron transistor, (a) schematic of basic structure, and (b) measured I-V characteristics at 15 K for various depletion gate biases and for devices with various distances (S ) between the sidewall gates. We refer the reader to [68] for further details. Single Electron Memory: Memory is an obvious candidate for single electron devices, since a single electron memory represents the ultimate scalability of current semiconductor memory technology, with potential memory storage densities on the order of cm. Various proposals include the single electron flip-flop proposed by Korotokov et al. [71] composed of cross-coupled SETs, the single electron trap proposed by Nakazato and Ahmed [72], and single-island memories which represent the single electron limit of floating gate memory technology (see for example [73]). Probably the most commercialized single electron memory is that developed by Yano and co-workers based on naturally occurring islands formed in single grains in a poly-silicon film [74], which depends on conductance through a single percolation path, which in turn is controlled by the charge stored in a grain in the vicinity of the conducting channel (Fig. 18). This technology has been successfully employed by Hitachi to fabricate a 128 Mb memory [75] Single Electron Logic: One of the most widely cited single electron logic elements is SET inverter proposed by Tucker [76] shown in Fig. 19(a), together with an electron micrograph of

14 GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES 381 Fig. 18. Si polycrystalline few-electron memory structure. (Figure reproduced from [74].) the experimental realization of the same structure by Heij et al. [77] based on metallic tunnel junctions where the functionality was experimentally demonstrated. The circuit consists of two single electron transistors as indicated, which play the role of the pmos and nmos transistors in a conventional CMOS inverter. Similar to a standard CMOS circuit, the switch will be on and the switch will be off when the gate voltage is high, and conversely when the gate voltage is low. Unlike the nmos and pmos devices, the SET junctions are physically identical. Also, in contrast to conventional CMOS inverters, the output characteristics are actually periodic functions of the input rather than truly bi-stable behavior, due to the periodicity of the Coulomb oscillations discussed in Section III.A. Fig. 19(b) shows the measured - characteristics by Heij et al., which evidence the expected transfer curve between input and output in comparison to master equation solution for this structure. As can be seen, the gain (as measured by the maximum slope) of the inverter is low compared to conventional CMOS, less than 1. More recently, Ono et al. [78] fabricated a Si complementary single-electron inverter on an SOI substrate in which a gain larger than unity, was demonstrated at 27 K. CNN Cell Design Using SETs: As one final circuit application, we discuss SET proposals for analog neural networks using cellular networks. Some applications of neural networks that have been proposed include optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision. Of recent interest is a circuit architecture referred to as Cellular Nonlinear Networks (CNN), which was first proposed by Chua and Yang [79]. The CNN is a cellular parallel computer network which has demonstrated broad applications in the area of image and video signal processing, robotic and biological visions. The basic circuit unit of a CNN is a cell, whose input and output is locally connected in a weighted fashion to a neighborhood of identical cells as illustrated in Fig. 20. Each cell is a dynamical system that has an input, output, and a state evolving in relation to dynamical laws. For a 2-D CNN array, there are four variables associated with a cell: the input, threshold, state, and output. The state equation of a standard CNN may be written as (18) (a) (b) Fig. 19. (a) Implementation of a single electron inverter circuit and (b) measured transfer curve at 25 mk compared with orthodox model. The notation 1Q in this figure is used to denote the difference between the charge stored on the two islands of the inverter. (Figure reproduced from [77]. Copyright 2001 by the American Institute of Physics.) where are the synaptic feedback weights of the output, and the are the synaptic feedforward weights of the input of cells in the neighborhood as illustrated in Fig. 20. The output, is related in a nonlinear fashion to the state variable (which represents a charge or voltage on the cell), which typically has a bi-stable behavior as for example the following (19) The CNN template defines the interconnection weights among the cells and determines the functionality of the CNN. The cloning template consists of the matrix that determines the feedforward weights, the feedback operator that defines the weighted feedback of the outputs into adjacent cells, and that describes the threshold. The building block of a CNN in CMOS implementations of such architectures [80] is a summing node coupled to a transconductance element (voltage-to-current-transducer) that switches between two stable operating points. The transconductance element resembles that of a CMOS inverter and because of its simplicity it can be integrated in the CNN cell circuit. A summing node

15 382 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003 Fig. 20. (a) Illustration of a 2-D CNN array showing (a) feedforward synapses, and (b) feedback synapses. (b) (a) (b) Fig. 21. (a) Schematic structure of a SET summing node/transducer element for application in CNN circuits, (b) Simulated results for different input conditions for the circuit using SIMON. For further details, we refer the reader to [81]. couples the outputs and inputs in adjacent neighborhood cells to the input of the transducer. The SET realization of this neuron is illustrated schematically in Fig. 21 [81], which is based on circuit proposed by Goosens et al. [82]. It consists essentially of the Tucker SET inverter discussed in Section III-D, which provides a bistable behavior (at least within one period of the input), similar to a CMOS transducer element, and it has multiple capacitive inputs to the inverter to form a voltage summing node such that the input voltage is the weighted sum of the external voltages, with weights determined by the capacitance value of each input signal. The transfer characteristics of the neuron circuit as a function of various input voltages, obtained using SIMON2.0, is shown in Fig. 21, simulated at 300 K. For the SET transistor to operate at room temperature, the tunnel capacitances were chosen on the order of, such that the single electron charging energy, is greater than the thermal energy,. As can be seen, a bi-stable operation is obtained which is dependent on the weighted sum of the input voltages to the inverter node. As an example of CNN functionality, a shadowing CNN was synthesized using the circuit Fig. 21 as the cell of a linear CNN array. The shadowing CNN represents a basic image processing applications, in which cells to the left or right of a high cell in the input are shadowed in the output, i.e., all low or all high depending on the orientation of the light source. A shadowing CNN has only positive control and feedback connections and requires only one inter-cell connection as described by the template: ; ;. The implementation of a shadowing SET-CNN circuit is shown in Fig. 22, where the SET network with only one inter-cell connection. The capacitance template is and. In the three-bit example of Fig. 22, whenever a 1 occurs in the input, all the cells to the right (including the high input cell) are high in the output, corresponding to shadowing of the output bits by the input bits. Since no negative feedback is required in the template, the SET implementation of the shadowing CNN functions gives the proper truth table between input and output as shown in the inset of Fig. 22. The transient simulation results using SIMON for two different input conditions in the shadowing SET-CNN are shown in Fig. 23. As can be seen, he output evolves dynamically to a stable condition corresponding to the truth table of Fig. 22 after an instantaneous change of the input. IV. SUMMARY In the preceding, we reviewed several aspects of phenomena at the nanoscale in semiconductors including quantum interference and single electron effects. In terms of coherent transport through ultrasmall semiconductor structures, transport is due to the quantum mechanical transmission through and reflection from the structure, which is a sensitive measure of the electronic states in such artificially confined systems. A number of device ideas have been set forth in the past including waveguide type

16 GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES 383 Fig. 22. A 3-cell shadowing SET-CNN. The inset shows the capacitive template values as well as the truth table exhibiting shadowing behavior. For further details, we refer the reader to [81]. (a) (b) Fig. 23. Transient response for the 3-cell shadowing SET-CNN. The curves show the cell output voltages. For further details, we refer the reader to [81]. devices, ratchets, and coupled waveguide structures for quantum computing. The ultimate limitations of such quantum effect devices are the characteristic time and length scales for phase coherence, which set upper limits on the dimensions of such structures. We also reviewed single electron phenomena, and the single electron transistor which are due to the discrete nature of the charge of an electron, and the associated energy change due to tunneling into and out of a nanoscale system. Several proposed applications of SETs were discussed, including very accurate current standards, single electron memories, single electron logic, and finally cellular nonlinear network applications. There are presently a number of issues to overcome in realizing functional nanoelectronic circuits based on the devices and concepts discussed in the present paper. Present limits today on the minimum lithographic feature sizes limits the observation of quantum interference and single electron effects primarily to low temperatures, although increasing reports of room temperature behavior are appearing. Disorder and device to device fluctuations are also a dominant problem at present, which have relegated most investigations to that of individual devices, with relatively little reported to date on more complicated circuit implementations. Hence, there are challenging material and fabrication problems to surmount. Based on the tremendous technological achievements over the past three decades in addressing such issues as they have arisen in the scaling of semiconductor devices to their present level of integration, the future appears hopeful in this respect for developing the requisite technologies to address nanoelectronic circuits and systems in the future. ACKNOWLEDGMENT The authors would like to thank students and colleagues whose work contributed to this manuscript, including R, Akis, C. Gerousis, D. K. Ferry, A. Gunther, M. Khoury, J. Wu, and M. N. Wybourne. The authors would like to acknowledge the support of the Office of Naval Research MURI program in this research. REFERENCES [1] (1997) The National Technology Roadmap for Semiconductors. The National Semiconductor Association, San Jose, CA. [Online]. Available: [2] H.-J Stöckmann, Quantum Chaos An Introduction. Cambridge: Cambridge Univ. Press, [3] Y.-H. Kim, M. Barth, H.-J. Stöckmann, and J. P. Bird, Wavefunction scarring in open quantum dots: A microwave analog study, Phys. Rev. B, vol. 65, p , [4] D. K. Ferry and S. M. Goodnick, Transport in Nanostructures. Cambridge, U.K.: Cambridge Univ. Press, [5] S. Datta, Electronic Transport in Mesoscopic Systems. Cambridge, U.K.: Cambridge Univ. Press, [6] C. W. J. Beenakker and H. van Houten, Quantum transport in semiconductor nanostructures, Solid State Physics, vol. 44, pp , 1992.

17 384 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003 [7] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel, and C. T. Foxon, Quantized conductance of point contacts in a two-dimensional electron gas, Phys. Rev. Lett., vol. 60, pp , [8] D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, and D. C. Peacock, One-dimensional transport and the quantization of the ballistic resistance, J. Phys. C, vol. 21, pp. L , [9] G. Timp, When does a wire become an electron waveguide?, in Semiconductors and Semimetals, M. A. Reed, Ed. New York: Academic, 1992, vol. 35, pp [10] J. C. Wu, M. N. Wybourne, A. Weisshaar, and S. M. Goodnick, Waveguide effects in quantum wires with double-bend discontinuities, J. Appl. Phys., vol. 74, pp , [11] J. P. Bird, Ed., Electron Transport in Quantum Dots. Norwell, MA: Kluwer-Academic, [12] R. Akis, J. P. Bird, and D. K. Ferry, The persistence of eigenstates in open quantum dots, Appl. Phys. Lett., vol. 81, pp , [13] I. Rotter, A continuum shell model for the open quantum mechanical nuclear system, Rep. Progr. Phys., vol. 54, pp , [14] J. P. Bird, R. Akis, D. K. Ferry, D. Vasileska, J. Cooper, Y. Aoyagi, and T. Sugano, Lead orientation dependent wavefunction scarring in open quantum dots, Phys. Rev. Lett., vol. 82, pp , [15] J. P. Bird, R. Akis, and D. K. Ferry, Magneto-probing of the discrete level spectrum of open quantum dots, Phys. Rev. B, vol. 60, pp , [16] F. Sols, M. Macucci, U. Ravaioli, and K. Hess, Theory for a quantum modulated transistor, J. Appl. Phys., vol. 66, pp , [17] S. Datta, Quantum devices, Superlatt. Microstruct., vol. 6, pp , [18] P. Debray, O. E. Raichev, M. Rahman, R. Akis, and W. C. Mitchell, Ballistic transport of electrons in T-shaped quantum waveguides, Appl. Phys. Lett., vol. 74, pp , [19] P. Debray, O. E. Raichev, P. Vasilopoulos, M. Rahman, R. Perrin, and W. C. Mitchell, Ballistic electron transport in stubbed quantum waveguides: Experiment and theory, Phys. Rev. B, vol. 61, pp , [20] J. J. Lin and J. P. Bird, Recent experimental studies of electron dephasing in metal and semiconductor mesoscopic structures, J. Phys.: Condens. Matter, vol. 14, pp. R , [21] T. Palm and L. Thylén, Analysis of an electron-wave Y-branch switch, Appl. Phys. Lett., vol. 60, pp , [22] T. Palm, Prediction of sawtooth oscillations in an electron Y-branch switch, Phys. Rev. B, vol. 52, pp , [23] J.-O. J. Wesström, Self-gating effect in the electron Y-branch switch, Phys. Rev. Lett., vol. 82, pp , [24] H. Q. Xu, Electrical properties of three-terminal ballistic junctions, Appl. Phys. Lett., vol. 78, pp , [25], Diode and transistor behaviors of three-terminal ballistic junctions, Appl. Phys. Lett., vol. 80, pp , [26] L. Worschech, B. Weidner, S. Reitzenstein, and A. Forchel, Investigation of switching effects between the drains of an electron Y-branch switch, Appl. Phys. Lett., vol. 78, pp , [27] K. Hieke and M. Ulfward, Nonlinear operation of the Y-branch switch: Ballistic switching mode at room temperature, Phys. Rev. B, vol. 62, pp , [28] L. Worschech, H. Q. Xu, A. Forchel, and L. Samuelson, Bias-voltageinduced asymmetry in nanoelectronic Y-branches, Appl. Phys. Lett., vol. 79, pp , [29] I. Shorubalko, H. Q. Xu, I. Maximov, P. Omling, L. Samuelson, and W. Seifert, Nonlinear operation of GaInAs/InP-based three-terminal ballistic junction, Appl. Phys. Lett., vol. 79, pp , [30] S. Kasai and H. Hasegawa, A single electron binary-decision-diagram quantum logic circuit based on Schottky wrap gate control of a GaAs nanowire hexagon, IEEE Electron Device Lett., vol. 23, pp , [31] A. M. Song, A. Lorke, A. Kriele, J. P. Kotthaus, W. Wegscheider, and M. Bichler, Nonlinear electron transport in an asymmetric microjunction: A ballistic rectifier, Phys. Rev. Lett., vol. 80, pp , [32] M. Büttiker, Four-terminal phase-coherent conductance, Phys. Rev. Lett., vol. 57, pp , [33] A. M. Song, Formalism of nonlinear transport in mesoscopic conductors, Phys. Rev. B, vol. 59, pp , [34] A. M. Song, S. Manus, M. Streibl, A. Lorke, and J. P. Kotthaus, A nonlinear transport device with no intrinsic threshold, Superlatt. Microstruct., vol. 25, pp , [35] A. M. Song, P. Omling, L. Samuelson, W. Seifert, and I. Shorubalko, Room-temperature and 50-GHz operation of a functional nanomaterial, Appl. Phys. Lett., vol. 79, pp , [36] H. Linke, W. Sheng, A. Löfgren, H. Xu, P. Omling, and P. E. Lindelof, A quantum-dot ratchet: Experiment and theory, Europhys. Lett., vol. 44, pp , [37] H. Linke, T. E. Humphrey, A. Löfgren, A. O. Sushkov, R. P. Taylor, R. E. Newbury, and P. Omling, Experimental tunneling ratchets, Science, vol. 286, pp , [38] M. Switkes, C. M. Marcus, K. Campman, and A. C. Gossard, An adiabatic quantum electron pump, Science, vol. 283, pp , [39] A. Bertoni, P. Bordone, R. Brunetti, C. Jacoboni, and S. Reggiani, Quantum logic gates based on coherent electron transport in quantum wires, Phys. Rev. Lett., vol. 84, pp , [40] J. Harris, R. Akis, and D. K. Ferry, Magnetically switched quantum waveguide qubit, Appl. Phys. Lett., vol. 79, pp , [41] C. C. Eugster and J. del Alamo, Tunneling spectroscopy of an electron waveguide, Phys. Rev. Lett., vol. 67, pp , [42] J. S. Moon, M. A. Blount, J. A. Simmons, J. R. Wendt, S. K. Lyo, and J. L. Reno, Magnetoresistance of one-dimensional subbands in tunnel-coupled double quantum wires, Phys. Rev. B, vol. 60, pp , [43] H. Grabert and M. H. Devoret, Eds., Single Charge Tunneling, Coulomb Blockade Phenomena in Nanostructures. New York: Plenum, 1992, vol. 294, NATO ASI Series B. [44] C. Wasshuber, Computational Single-Electronics. New York: Springer, [45] T. A. Fulton and G. J. Dolan, Observation of single-electron charging effects in small tunnel junctions, Phy. Rev. Lett., vol. 59, no. 1, pp , [46] L. S. Kuz min and K. K. Likharev, Direct experimental observation of discrete correlated single-electron tunneling, JETP Lett., vol. 45, pp , [47] 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, B. L. Altshuler, P. A. Lee, and R. A. Webb, Eds. Amsterdam, Oxford, New York, Tokyo: Elsevier, 1991, pp [48] K. Likharev, Correlated discrete transfer of single electrons in ultrasmall tunnel junctions, IBM J. Res. Develop., vol. 32, pp , [49] D. V. Averin and Y. V. Nazarov, Virtual electron diffusion during quantum tunneling of the electric charge, Phys. Rev. Lett., vol. 65, pp , [50] C. Pasquier, U. Meirav, F. I. B. Williams, D. G. Glattli, Y. Jin, and B. Etienne, Quantum limitations on coulomb blockade observed in a 2D electron system, Phys. Rev. Lett., vol. 70, pp , [51] R. H. Chen, A. N. Korotkov, and K. K. Likharev, Single-electron transistor logic, Appl. Phys. Lett., vol. 68, pp , [52] M. Kirihara, N. Kuwamura, K. Taniguchi, and C. Hamaguchi, Monte Carlo study of single-electronic devices, in Proceedings of the Int. Conf. on Solid State Devices and Materials, Yokohama, 1994, pp [53] C. Wasshuber, H. Kosina, and S. Selberherr, SIMON: A simulator for single-electron tunnel devices and circuits, IEEE Trans. CAD, vol. 16, pp , [54]. [Online]. Available: [55] L. J. Geerligs, V. F. Anderegg, P. A. M. Holweg, J. E. Mooij, H. Pothier, D. Esteve, C. Urbina, and M. H. Devoret, Frequency-locked turnstile device for single electrons, Phys. Rev. Lett., vol. 54, pp , [56] H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M. H. Devoret, Singleelectron pump based on charging effects, Europhysics Lett., vol. 17, pp , [57] U. Meirav, M. A. Kastner, and S. J. Wind, Single-electron charging and periodic conductance resonances in GaAs nanostructures, Phys. Rev. Lett., vol. 65, no. 6, pp , [58] L. P. Kouwenhouven, A. T. Johnson, N. C. van der Vaart, C. J. P. M. Harmans, and C. T. Foxon, Quantized current in a quantum-dot turnstile using oscillating tunnel barriers, Phys. Rev. Lett., vol. 67, no. 12, pp , [59] F. R. Waugh, M. J. Berry, D. J. Mar, R. M. Westervelt, K. L. Campman, and A. C. Gossard, Single-electron charging in double and triple quantum dots with tunable coupling, Phys. Rev. Lett., vol. 75, pp , [60] E. S. Snow and P. M. Cambell, AFM fabrication of sub-10-nanometer metal-oxide devices with in situ control of electrical properties, Science, vol. 270, pp , 1995.

18 GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES 385 [61] K. Matsumoto, Room temperature operated single electron transistor made by STM/AFM nano-oxidation process, Physica B, vol. 227, pp , [62] H. Matsuoka and S. Kimura, Transport properties of a silicon single electron transistor at 4.2 K, Appl. Phys. Lett., vol. 66, no. 5, pp , [63] M. Khoury, M. J. Rack, A. Gunther, and D. K. Ferry, Spectroscopy of a Si quantum dot, Appl. Phys. Lett., vol. 74, pp , [64] F. Simmel, D. Abusch-Magder, D. A. Wharam, M. A. Kastner, and J. P. Kotthaus, Statistics of coulomb-blockade peak spacings of a Si quantum dot, Phys. Rev. B, vol. 59, no. 16, pp. R10441 R10444, [65] L. Zhuang, L. Guo, and S. Y. Chou, Silicon single-electron quantum-dot transistor switch operating at room temperature, Appl. Phys. Lett., vol. 72, no. 10, pp , [66] Y. Ono, Y. Takahashi, K. Yamazaki, M. Nagase, H. Namatsu, K. Kurihara, and K. Murase, IEEE Trans. ED, vol. 47, p. 147, [67] Y. Takahashi, A. Fujiwara, K. Yamazaki, H. Namatsu, K. Kurihara, and K. Murase, Multigate single-electron transistors and their application to an exclusive-or gate, Appl. Phys. Lett., vol. 78, no. 5, pp , [68] D. H. Kim, S.-K. Sung, K. R. Kim, J. D. Lee, B.-G. Park, B. H. Choi, S. W. Hwang, and D. Ahn, Silicon single-electron transistors with sidewall depletion gates and their application to dynamic single-electron transistor logic, IEEE Trans. ED, vol. 49, no. 4, pp , [69] K. K. Likharev, Single-electron devices and their applications, Proc. IEEE, vol. 87, pp , [70] C. S. Lent, P. D. Tougaw, and W. Porod, Bistable saturation in coupled quantum dots for quantm cellular automata, Appl. Phys. Lett., vol. 62, pp , [71] A. N. Korotkov, R. H. Chen, and K. K. Likharev, Possible performance of capacitively coupled single-electron transistors in digital circuits, J. Appl. Phys., vol. 78, pp , [72] K. Nakazato, R. J. Blaikie, and H. Ahmed, Single electron memory, J. Appl. Phys., vol. 75, pp , [73] L. Guo, E. Leobandung, and S. Y. Chou, A room-temperature silicon single-electron metal-oxide-semiconductor memory with nanoscale floating-gate and ultranarrow channel, Appl. Phys. Lett., vol. 70, pp , [74] K. Yano, T. Ishii, T. Hashimoto, T. Kobayashi, F. Murai, and K. Seki, Room-temperature single-electron memory, IEEE Trans. Electron Devices, vol. 41, no. 9, pp , [75] K. Yano, T. Ishii, T. Sano, T. Mine, F. Murai, T. Kure, and K. Seki, 128 MB early prototype for gigascale single-electron memories, Proc. IEEE ISSCC, pp , [76] J. R. Tucker, Complementary digital logic based on the coulomb blockade, J. Appl. Phys., vol. 72, p. 4399, [77] C. P. Heij, P. Hadley, and J. E. Mooij, A single-electron inverter, Appl. Phys. Lett., vol. 78, no. 8, pp , [78] Y. Ono, Y. Takahashi, K. Yamazaki, M. Nagase, H. Namatsu, K. Kurihara, and K. Murase, Si complementary single-electron inverter with voltage gain, Appl. Phys. Lett., vol. 76, no. 21, pp , [79] L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., vol. 35, pp , [80] I. Baktir and M. M. Tan, Analog CMOS implementation of CNN, IEEE Trans. Circuits Syst. II, vol. 40, pp , [81] C. Gerousis, S. M. Goodnick, and W. Porod, Toward nanoelectronic cellular neural networks, Int. J. Circ. Theor. Appl., vol. 28, no. 6, pp , [82] J. Goossens, J. Ritskes, C. Verhoeven, and A. van Roermund, Learning single electron tunneling neural nets, in Proc. ProRISC Workshop on Circuits, Systems, Signal Processing, 1997, pp Stephen M. Goodnick (M 87 SM 92 F 03) received the B.S. degree in engineering science from Trinity University, San Antonio, TX, in 1977, and the M.S. and Ph.D. degrees in electrical engineering from Colorado State University, Fort Collins in 1979 and 1983, respectively. From 1986 to 1997, he was a Faculty Member with the Department of Electrical and Computer Engineering at Oregon State University, Corvallis, OR, and is presently Chair and Professor of Electrical Engineering at Arizona State University, Tempe. He has coauthored over 130 journal articles, books and book chapters related to transport in semiconductor devices and microstructures. Dr. Goodnick was an Alexander von Humboldt Fellow at the Technical University of Munich, Germany and the University of Modena, Italy in 1985 and 1986, respectively. Jonathan Bird (M 97 SM 03) received the B.Sc. (Hons.) and the D.Phil. degrees in physics, from the University of Sussex, Falmer, Brighton, UK, in 1986 and 1990, respectively. From 1991 to 1992, he was a Visiting Fellow with the Japan Society for the Promotion of Science at the University of Tsukuba, Japan. From 1992 to 1997, he was with the Frontier Research Program of the Institute for Physical and Chemical Research (RIKEN), Saitama, Japan. In 1997, he joined the Department of Electrical Engineering at Arizona State University, Tempe, where he is currently an Associate Professor. He is coauthor of more than 150 peer-reviewed publications. His research interests include nanoelectronics. Prof. Bird is a Fellow of the Institute of Physics, a Member of Sigma Xi and the American Physical Society.