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.
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.
Index Terms—Ballistic devices, nanoelectronics, quantum
devices, single-electron transistors and circuits, single-electron
tunneling.
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
cm , corresponding to a Fermi wavelength of
order
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]
I. INTRODUCTION
A
S 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, 2003.
The authors are with the Department of Electrical Engineering, Arizona State
University, Tempe, AZ 85287 USA (e-mail: goodnick@asu.edu).
Digital Object Identifier 10.1109/TNANO.2003.820773
II. QUANTUM-EFFECT NANOSTRUCTURED DEVICES
A. Electron-Wave Phenomena in Quantum Devices
(1)
(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
(2)
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
1536-125X/03$17.00 © 2003 IEEE
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 ( ).
Further details on this may be found in [3]. (Figure provided courtesy of Y.-H. Kim and H.-J. Stöckmann.)
1
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)
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
(4)
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
(Fig. 2).
in integer steps of
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
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
(5)
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
GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES
Fig. 4. Computed variation of conductance with energy for an open,
stadium-shaped, quantum dot. Note how the energy in this figure is normalized
=m A , where A is
to the average level spacing of the dot
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].
371
1 = 2
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
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
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 voltand . By modeling the Y-branch as a ballistic cavity
ages
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
) applied to the left and right waveguides,
voltages (
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
(6)
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
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]
(7)
denotes the resistance measured with current
Here
and
in Fig. 8, and voltage
flowing between probes
and . In the nonlinear regime,
measured between probes
however, the device of Fig. 8 shows very different behavior.
), we see that
Regardless of the sign of the drive current (
) is always negative, indicating that
the measured voltage (
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
to always be negative. What is
the lower contact, causing
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
374
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.)
IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003
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
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
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
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
(8)
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
When physical dimensions are small, the corresponding capacitance (which is a geometrical quantity in general) is correspond, corresponding to
ingly small, and a change in charge,
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 approxi. For a 5-nm radius nanocluster (for example,
mately
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
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
. The work done in
to the SET system,
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 couto a gate bias, making a three-terminal strucpled through
ture. To understand the behavior, we consider the energy change
when electrons tunnel back and forth across the two tunnel juncbe the net number of electrons that tuntions. Here we let
the number
neled through the first junction onto the island,
of electrons that tunneled through the second junction exiting
is the net number of excess electhe island, and
trons on the island.
With a bias voltage applied across the two junctions the
charges on the junctions and island can be written as
(9)
is the background
where is the net charge on the island,
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.
, we can rewrite
In terms of the applied bias,
the junction potentials as
(10)
is the total charge transferred from the drain or
where
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
, then its clear from (13) and (14) that there
neutral, i.e.,
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
), a new Coulomb blockade exists until the
the dot, (i.e.,
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
GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES
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)
2
2 .
Coulomb gap in the current-voltage characteristic for
377
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
, then successive oscillations in the conductance
in the dot as
with gate bias include both the Coulomb contribution plus the
energy spacing of successive quantized levels as
0e= C < V < e= C
(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 con.
ducting channel, i.e.,
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
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)
is the change in free energy, which as defined for
where
the two junction SET system by (14) and (15). Within the kinetic equation framework, we can define the distribution func, which
tion for the island occupancy,
electrons on island 1,
is the probability of the system having
electrons on island 2, etc. A kinetic or master equation can
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
refers to the net tunneling onto each island from all
here
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
. Higher order processes, or co-tunneling,
conductance,
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)
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 re, where is
sulting current is given quite accurately by
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
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
cm . Various proposals include the single
on the order of
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
GOODNICK AND BIRD: QUANTUM-EFFECT AND SINGLE-ELECTRON DEVICES
381
Fig. 18. Si polycrystalline few-electron memory structure. (Figure reproduced
from [74].)
(a)
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
, threshold ,
four variables associated with a cell: the input
, and output
. The state equation of a standard CNN
state
may be written as
(18)
(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
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.)
1Q
where
are the synaptic feedback weights of the output, and
are the synaptic feedforward weights of the input of cells
the
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,
that describes the threshold. The building block of a
and
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
382
IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003
(a)
Fig. 20.
(b)
Illustration of a 2-D CNN array showing (a) feedforward synapses, and (b) feedback synapses.
(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
, such that the single electron
chosen on the order of
is greater than the thermal energy,
.
charging 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
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)
Fig. 23.
(b)
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:
www.public.itrs.net
[2] H.-J Stöckmann, Quantum Chaos – An Introduction. Cambridge:
Cambridge Univ. Press, 1999.
[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. 165 317, 2002.
[4] D. K. Ferry and S. M. Goodnick, Transport in Nanostructures. Cambridge, U.K.: Cambridge Univ. Press, 1997.
[5] S. Datta, Electronic Transport in Mesoscopic Systems. Cambridge,
U.K.: Cambridge Univ. Press, 1995.
[6] C. W. J. Beenakker and H. van Houten, “Quantum transport in semiconductor nanostructures,” Solid State Physics, vol. 44, pp. 1–228, 1992.
384
[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. 848–851, 1988.
[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.
L209–214, 1988.
[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. 113–190.
[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. 4590–4597, 1993.
[11] J. P. Bird, Ed., Electron Transport in Quantum Dots. Norwell, MA:
Kluwer-Academic, 2003.
[12] R. Akis, J. P. Bird, and D. K. Ferry, “The persistence of eigenstates in
open quantum dots,” Appl. Phys. Lett., vol. 81, pp. 129–131, 2002.
[13] I. Rotter, “A continuum shell model for the open quantum mechanical
nuclear system,” Rep. Progr. Phys., vol. 54, pp. 635–682, 1991.
[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. 4691–4694, 1999.
[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.
13 676–13 681, 1999.
[16] F. Sols, M. Macucci, U. Ravaioli, and K. Hess, “Theory for a quantum
modulated transistor,” J. Appl. Phys., vol. 66, pp. 3892–3906, 1989.
[17] S. Datta, “Quantum devices,” Superlatt. Microstruct., vol. 6, pp. 83–93,
1989.
[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. 768–770, 1999.
[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. 10 950–10 958,
2000.
[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. R501–596, 2002.
[21] T. Palm and L. Thylén, “Analysis of an electron-wave Y-branch switch,”
Appl. Phys. Lett., vol. 60, pp. 237–239, 1992.
[22] T. Palm, “Prediction of sawtooth oscillations in an electron Y-branch
switch,” Phys. Rev. B, vol. 52, pp. 13 773–13 775, 1995.
[23] J.-O. J. Wesström, “Self-gating effect in the electron Y-branch switch,”
Phys. Rev. Lett., vol. 82, pp. 2564–2567, 1999.
[24] H. Q. Xu, “Electrical properties of three-terminal ballistic junctions,”
Appl. Phys. Lett., vol. 78, pp. 2064–2066, 2001.
, “Diode and transistor behaviors of three-terminal ballistic junc[25]
tions,” Appl. Phys. Lett., vol. 80, pp. 853–855, 2002.
[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. 3325–3327, 2001.
[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.
16 727–16 730, 2000.
[28] L. Worschech, H. Q. Xu, A. Forchel, and L. Samuelson, “Bias-voltageinduced asymmetry in nanoelectronic Y-branches,” Appl. Phys. Lett.,
vol. 79, pp. 3287–3289, 2002.
[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. 1384–1386, 2001.
[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. 446–448,
2002.
[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. 3831–3834, 1998.
[32] M. Büttiker, “Four-terminal phase-coherent conductance,” Phys. Rev.
Lett., vol. 57, pp. 1761–1764, 1986.
[33] A. M. Song, “Formalism of nonlinear transport in mesoscopic conductors,” Phys. Rev. B, vol. 59, pp. 9806–9809, 1999.
[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. 269–272, 1999.
IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 2, NO. 4, DECEMBER 2003
[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. 1357–1359, 2001.
[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. 341–347, 1998.
[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. 2314–2317, 1999.
[38] M. Switkes, C. M. Marcus, K. Campman, and A. C. Gossard, “An adiabatic quantum electron pump,” Science, vol. 283, pp. 1905–1908, 1999.
[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. 5912–5915, 2000.
[40] J. Harris, R. Akis, and D. K. Ferry, “Magnetically switched quantum
waveguide qubit,” Appl. Phys. Lett., vol. 79, pp. 2214–2215, 2001.
[41] C. C. Eugster and J. del Alamo, “Tunneling spectroscopy of an electron
waveguide,” Phys. Rev. Lett., vol. 67, pp. 3586–3589, 1991.
[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.
11 530–11 534, 1999.
[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, 2001.
[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.
109–112, 1987.
[46] L. S. Kuz’min and K. K. Likharev, “Direct experimental observation of
discrete correlated single-electron tunneling,” JETP Lett., vol. 45, pp.
495–497, 1987.
[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. 173–271.
[48] K. Likharev, “Correlated discrete transfer of single electrons in ultrasmall tunnel junctions,” IBM J. Res. Develop., vol. 32, pp. 144–158,
1988.
[49] D. V. Averin and Y. V. Nazarov, “Virtual electron diffusion during
quantum tunneling of the electric charge,” Phys. Rev. Lett., vol. 65, pp.
2446–2449, 1990.
[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. 69–72, 1993.
[51] R. H. Chen, A. N. Korotkov, and K. K. Likharev, “Single-electron transistor logic,” Appl. Phys. Lett., vol. 68, pp. 1954–1956, 1996.
[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. 328–330.
[53] C. Wasshuber, H. Kosina, and S. Selberherr, “SIMON: A simulator for
single-electron tunnel devices and circuits,” IEEE Trans. CAD, vol. 16,
pp. 937–944, 1997.
[54] . [Online]. Available: http://home1.gte.net/kittypaw/
[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. 2691–2694,
1990.
[56] H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M. H. Devoret, “Singleelectron pump based on charging effects,” Europhysics Lett., vol. 17, pp.
249–254, 1992.
[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. 771–774, 1990.
[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.
1626–1629, 1991.
[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.
705–708, 1995.
[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. 1639–1641, 1995.
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.
92–94, 1996.
[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.
613–615, 1995.
[63] M. Khoury, M. J. Rack, A. Gunther, and D. K. Ferry, “Spectroscopy of
a Si quantum dot,” Appl. Phys. Lett., vol. 74, pp. 1576–1578, 1999.
[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, 1999.
[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. 1205–1207, 1998.
[66] Y. Ono, Y. Takahashi, K. Yamazaki, M. Nagase, H. Namatsu, K. Kurihara, and K. Murase, IEEE Trans. ED, vol. 47, p. 147, 2000.
[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. 637–639,
2000.
[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. 627–635, 2002.
[69] K. K. Likharev, “Single-electron devices and their applications,” Proc.
IEEE, vol. 87, pp. 606–632, 1999.
[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. 714–716, 1993.
[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. 2520–2530, 1995.
[72] K. Nakazato, R. J. Blaikie, and H. Ahmed, “Single electron memory,”
J. Appl. Phys., vol. 75, pp. 5123–5134, 1994.
[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.
850–852, 1997.
[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. 1628–1638, 1994.
[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. 344–345, 1996.
[76] J. R. Tucker, “Complementary digital logic based on the coulomb
blockade,” J. Appl. Phys., vol. 72, p. 4399, 1992.
[77] C. P. Heij, P. Hadley, and J. E. Mooij, “A single-electron inverter,” Appl.
Phys. Lett., vol. 78, no. 8, pp. 1140–1142, 2001.
[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. 3121–3123, 2000.
[79] L. O. Chua and L. Yang, “Cellular neural networks: Theory,” IEEE
Trans. Circuits Syst., vol. 35, pp. 1257–1272, 1998.
[80] I. Baktir and M. M. Tan, “Analog CMOS implementation of CNN,”
IEEE Trans. Circuits Syst. II, vol. 40, pp. 200–206, 1993.
[81] C. Gerousis, S. M. Goodnick, and W. Porod, “Toward nanoelectronic
cellular neural networks,” Int. J. Circ. Theor. Appl., vol. 28, no. 6, pp.
523–535, 2000.
[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. 179–186.
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.