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Quantum-dot cellularautomata: computing with

coupled quantum dotsWOLFGANG POROD , CRAIGS LENT , GARY H.

BERNSTEIN , ALEXEI O. ORLOV , ISLAMSHA

HAMLANI , GREGORY L. SNIDER & JAMES L.

MERZ

Version of record first published: 09 Nov 2010.

To cite this article: WOLFGANG POROD , CRAIGS LENT , GARY H. BERNSTEIN ,

ALEXEI O. ORLOV , ISLAMSHA HAMLANI , GREGORY L. SNIDER & JAMES L. MERZ

(1999): Quantum-dot cellular automata: computing with coupled quantum dots,

International Journal of Electronics, 86:5, 549-590

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INT. J. ELECTRONICS, 1999, VOL . 86, NO. 5, 549590

Quantum-dot cellular automata: computing with coupled quantum dots

WOLFGANG POROD , CRAIG S. LENT,

GAR Y H. BERNSTEIN, ALEXEI O. ORLOV,

ISLAMSHAH AMLANI, GREGORY L. SNIDER ,

and JAMES L. MER Z

We discuss novel nanoelectronic architecture paradigms based on cells composedof coupled quantum-dots. Boolean logic functions may be implemented in specicarrays of cells representing binary information, the so-called quantum-dot cellularautomata (QCA ). Cells may also be viewed as carrying analogue information andwe outline a network-theoretic description of such quantum-dot nonlinear net-

works (Q-CNN). In addition, we discuss possible realizations of these structuresin a variety of semiconductor systems (including GaAs/AlGaAs, Si/SiGe, andSi/SiO2), rings of metallic tunnel junctions, and candidates for molecular imple-mentations. We report the experimental demonstration of all the necessaryelements of a QCA cell, including direct measurement of the charge polarizationof a double-dot system, and direct control of the polarization of those dots viasingle electron transitions in driver dots. Our experiments are the rst demon-stration of a single electron controlled by single electrons.

1. Introduction

Silicon technology has experienced an exponential improvement in virtually any

gure of merit, following Gordon Moores famous dictum remarkably closely for more

than three decades. However, there are indication s now that this progress will slow, or

even come to a standstill, as technologica l and fundamental limits are reached. This

slow-down of silicon ULSI technology may provide an opportunity for alternative

device technologies. In this paper, we will describe some ideas of the Notre Dame

NanoDevices Group on a possible future nanoelectronic computing technology

based on cells of coupled quantum dots. M ore specically, we envision nanostructureswhere information is encoded by the arrangement of single electrons. Now, 100 years

after the discovery of the electron, it has become feasible to manipulate electrons one

electron at a time, and to engineer device structures based on individual electrons.

Among the chief technological limitation s responsible for this expected slow-

down of silicon technology are the interconnect problem and power dissipatio n

(Ferry et al. 1987, 1988, Keyes 1987, Bohr 1996). As more and more devices arepacked into the same area, the heat generated during a switching cycle can no longer

be removed and may result in damage to the chip. Interconnections do not scale in

concert with device scaling because of the eect of wire resistance and capacitance,giving rise to a wiring bottleneck. It is generally recognized that alternate approaches

are needed to create innovativ e technologies that provide greater device and

International Journal of Electronics ISSN 00207217 print/ISSN 13623060 online

1999 Taylor & Francis Ltd

http://www.tandf.co.uk/JNLS/etn.htm

http://www.taylorandfrancis.com/JNLS/etn.htm

Received 1 June 1997. Accepted 30 October 1998. Department of Electrical Engineering, University of Notre Dame, Notre Dame,

IN 46556, USA. Corresponding author.

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interconnect functionality, or to utilize innovative circuit and system architectural

features that provide more function per transistor (Semiconductor Industry

Association 1994). However, these obstacles for silicon circuitry may present anopportunity for alternative device technologies which are designed for the nanometer

regime and which are interconnected in an appropriate architecture.

In this paper, we describe our ideas of using nanostructures (more specically,quantum dots) which are arranged in locally-interconnected cellular-automata-likearrays. We will demonstrate that suitably designed structures, the so-called `quan-

tum-dot cellular automata (QCA ) (Lent et al. 1993 b), may be used for computationand signal processing. The fundamental idea for QCA operation is to encode infor-

mation using the charge conguration of a set of dots. This is an important break

with the transistor paradigm. From the electro-mechanical relays of Konrad Zuse to

the modern CMOS circuit, binary information has been encoded by current switches.

It works well so long as the switched current from one element can be transformed

into the control voltage for another element. As device sizes shrink to the molecularlimit, the current-switching paradigm falters for three primary reasons:

(1 ) the nano-switch becomes leaky, making a hard OFF-state dicult to main-tain;

(2 ) the switched current is so small in magnitude that it is hard for it to activatethe next device; and

(3 ) the interconnects which carry current from one device to another begin todominate the performance.

The QCA approach eliminates these problems by adopting an approach to

coding information which is more naturally suited to nanostructures.

Our work is based on the highly advanced state-of-the-art in the eld of nano-

structures and the emerging technology of quantum-dot fabrication (Capasso 1990,

Weisbuch and Vinter 1991, Kelly 1995, Turton 1995, Montemerlo et al. 1996). Asschematically shown in gure 1, several groups have demonstrated that electrons

550 W. Porod et al.

Figure 1. Schematic diagram of articial `quantum-dot atoms and `quantum-dot molecules

which are occupied by few electrons.

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may be completely conned in semiconductor nanostructures, which may then be

thought of as articial `semiconductor atoms (Reed et al. 1988, Meirav et al. 1990).Controllable occupation of these quantum dots has been achieved in the few-electron

regime (Meurer et al. 1992). One may speak of `quantum-dot hydrogen, `quantum-dot helium, `quantum-dot lithium, etc. (Kastner 1993, Ashoori 1996). Ver y

recently, coupling between quantum-dot atoms in close proximity has been observed,thus realizing articial `quantum-dot molecules (Homann et al. 1995, Waugh et al.

1995, Blick et al. 1996).Note that our scheme is not a quantum computer in the sense of the `quantum

computing community, as reviewed by Spiller (1996). QCAs do not require quan-tum mechanical phase coherence over the entire array; phase coherence is only

required inside each cell, and the cellcell interactions are classical. This limited

requirement of quantum mechanical phase coherence makes QCAs a more attractive

candidate for actual implementations.

The use of quantum dots for device applications entails a need for new circuitarchitecture ideas for these new devices. The nanostructures we envision will contain

only few electrons availabl e f or conduction . It is hard to imagine how devices based

on nanostructures could function in conventional circuits, primarily due to the

problems associated with charging the interconnect wiring with the few electrons

available. Therefore, we propose to envision a nanoelectronic architecture where the

information is contained in the arrangement of charges and not in the ow of

charges (i.e. current). In other words, the devices interact by direct Coulomb coup-ling and not by currents through wires. We envision to utilize the existing physical

interactions between neighbouring devices in order to directly produce the dynamics,

such that the logical operation of each cell would require no additional connections

beyond the physical coupling within a certain range of interactions. We are led to

consider cellular-autom ata-like device architectures (To oli and Margolus 1987,

Biafore 1994) of cells communicating with each other by their Coulombic inter-action.

Figure 2 schematically shows a locally-interconnected array consisting of cells of

nanoelectronic devices. The physical interactions together with the array topology

determine the overall functionality. What form must a cellular array take when itsdynamics should result directly from known physical interactions? If we simply

Quantum-dot cellular automata 551

Figure 2. Schematic picture of a cellular array where the interconnections are given by

physical law. The underlying physics determines the overall functionality of the array.

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arrange nanometer-scale devices in a dense cellular array, the device cells may inter-

act, but we have given up all control over which cells interact with which neighbours

and when they interact. In general, the state of a cell will depend on the state of its

neighbours within a certain range. The main questions now are: ` What f unctionalit y

does one obtain for a given physical structure? and `Given a certain array beha-

viour, there a physical system to implement it?In the f ollowing sections we will develop these ideas in detail and we will present

a concrete example of a quantum-dot cell with an appropriate architecture, the so-

called quantum-dot cellular automata. We will discuss how one may construct QCA

cells that encode binary information and how one can thus realize Boolean logic

functions. We will also discuss that one may view these arrays as quantum-dot

cellular neural (or, nonlinear) networks (Q-CNNs). A key question, of course, areimplementations. We will discuss ideas (and on-going work) for attempting to imple-ment these structures in a variety of semiconductor systems (including GaAs/

AlGaAs, Si/SiGe, and Si/SiO2) and also metallic dots. A lternative implementationsinclude molecular structures. We will call attention to a specic molecule which

appears to be particularly promising since it possesses a structure similar to a

QCA cell. One of the most promising material systems appears to be Si/SiO 2, mostly

due to the excellent insulating properties of the oxide. Note that our search for a

technology beyond silicon may bring us back to silicon!

Exciting as the vision of a possible nanoelectronics technology may be, many

fundamental and technological challenges remain to be overcome. We should keep in

mind that this exploration has just begun and that other promising designs remain

yet to be discovered. This exciting journey will require the combined eorts

of technologists, device physicists, circuits-and-systems theorists and computer

architects.

2. Quantum-dot cellular automata

Based upon the e merging technology of quantum-dot fabrication, the Notre

Dame NanoDevices Group has developed the QCA scheme for computing with

cells of coupled quantum dots (Lent et al. 1993 a), which will be described below.To our knowledge, this is the rst concrete proposal to utilize quantum dots for

computing. There had been earlier suggestions that devicedevice coupling might be

utilized in a cellular-automata scheme, alas, without an accompanying proposal for a

specic implementation (Ferry and Porod 1986, Grondin et al. 1987).What we have in mind is the general architecture shown in gure 3. The coupling

between the cells is given by their physical interaction, and not by wires. The physical

mechanisms available for interactions between nanoelectron ic structures are the

Coulomb interactio n and quantum-mechanical tunnelling .

2.1. A quantum -dot cell

The quantum-do t cellular automata (QCA) scheme is based on a cell whichcontains four quantum dots (Lent et al . 1993 a), as schematically shown ingure 4( a). The quantum dots are shown as the open circles which represent theconning electronic potential. In the ideal case, each cell is occupied by two elec-

trons, which are schematically shown as solid dots. The electrons are allowed to

`jump between the individual quantum dots in a cell by the mechanism of quantum

552 W. Porod et al.

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mechanical tunnelling . Tunnelling is possible on the nanometer scale when the elec-

tronic wavefunction suciently `leaks out of the conning potential of each dot, and

the rate of these jumps may be controlled during f abrication by the physical separa-

tion between neighbouring dots.

This quantum-dot cell represents an interesting dynamical system. The two elec-trons experience their mutual Coulombic repulsion, yet they are constraine d to

occupy the quantum dots. If left alone, they will seek, by hopping between the

dots, the conguration corresponding to the physical ground state of the cell. It is

clear that the two electrons will tend to occupy dierent dots because of the

Coulomb energy cost associated with bringing them together in close proximity on

the same dot. It is easy to see that the ground state of the system will be an equal

superposition of the two basic congurations with electrons at opposite corners, as

shown in gure 4( b).


Figure 3. Each cell in the array interacts with the `environment which includes the Coulombinteraction with neighbouring cells.

Figure 4. Schematic of the basic four-site cell. ( a) The geometry of the cell. The tunnellingenergy between two neighbouring sites is designated by t, while a is the near-neighbourdistance. ( b) Coulombic repulsion causes the electrons to occupy antipodal sites withinthe cell. These two bistable states result in cell polarizations of P = + 1 and P = - 1.

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We may associate a `polarization with a specic arrangement of the two elec-

trons in each cell. Note that this polarization is not a dipole moment, but a measure

for the alignment of the charge along the two cell diagonals. If cells 1 and 3 are

occupied, we call it a polarization of P = + 1, while electrons on sites 2 and 4 giveP = - 1 (compare gure 4). Any polarization between these two extreme values is

possible, corresponding to congurations where the electrons are more evenly`smeared out over all dots. The ground state of an isolated cell is a superposition

with equal weight of the two basic congurations, and therefore has a net polariza-

tion of zero.

As described in the literature, this cell has been studied by solving the

Schro dinger equation using a quantum mechanical model Hamiltonian (Tougaw

et al. 1993). We do not need to concern ourselves here with the details, but suceit to say that the basic ingredients to the equation of motion are:

(1 ) the quantized energy levels in each dot;(2 ) the coupling between the dots by tunnelling;

(3 ) the Coulombic charge cost for a doubly-occupied dot;

(4 ) the Coulomb interaction between electrons in the same cell and also withthose in neighbouring cells.

The solution of the Schro dinger equation, using cell parameters for an experi-

mentally reasonable model, conrms the intuitive understanding that the ground

state is a superposition of the P = + 1 and P = - 1 states. In addition to the groundstate, the Hamiltonian model yields excited states and cell dynamics.

2.2. Cellcell coupling

The properties of an isolated cell were discussed above. The two polarization

states of the cell will not be energetically equivalent if other cells are nearby. Here, we

study the interactions between two cells, each occupied by two electrons. The elec-

trons are allowed to tunnel between the dots in the same cell, but not between

dierent cells. Since the tunnelling probabilities decay exponentially with distance,this can be achieved by having a larger dotdot distance between cells than within

the same cell. Coupling between the two cells is provided by the Coulomb interaction

between the electrons in dierent cells.

Figure 5 shows how one cell is inuenced by the state of its neighbour. The inset

shows two cells where the polarization of cell 1 ( P1) is determined by the polarizationof its neighbour (P2). The polarization of cell 2 is presumed to be xed at a givenvalue, corresponding to a certain arrangement of charges in cell 2, and this charge

distribution exerts its inuence on cell 1, thus determining its polarization P1 . Theimportant nding here is the strongly nonlinear nature of the cellcell coupling. As

shown in the gure, cell 1 is almost completely polarized even though cell 2 might

only be partially polarized. For example, a polarization of P2 = 0.1 induces almostperfect polarizatio n in cell 1, i.e. P1 = 0.99. In other words, even a small asymmetryof charge in cell 2 is sucient to break the degeneracy of the two basic states in cell 1

by energetically favouring one conguration over the other.

The abruptness of the cellcell response function depends upon the ratio of the

strength of the tunnellin g energy to the Coulomb energy f or electrons on neighbour -

ing sites. This reects a competition between the kinetic and potential energy of the

554 W. Porod et al.

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electron. For a large tunnelling energy, an electron has a tendency to spread out

more evenly over the available dots, and the nonlinearity becomes less pronounced .

Stronger Coulomb coupling tends to keep electrons apart, and the nonlinearity

becomes more pronounced. Properly designed cells will possess strongly nonlinear

coupling characteristics.This bistable saturation is the basis for the application of such quantum-dot cells

for computing structures. The nonlinear saturation plays the role of gain in conven-

tional circuitsrestoring signal levels after each stage. Note that no power dissipa-

tion is required in this case.

These general conclusions regarding cell behaviour and cellcell coupling are not

specic to the four-dot cell discussed so far. Similar behaviour is also found for

alternate cell designs, such as cells with ve dots (four in the corners and one in

the centre), as opposed to the four discussed here (Tougaw et al. 1993).

2.3. QCA logic

Based upon the bistable behaviour of the cellcell coupling , the cell polarizatio n

can be used to encode binary information. We have demonstrated that the physical

interactions between cells may be used to realize elementary Boolean logic functions

(Lent et al. 1994, Lent and Tougaw 1994).Figure 6 shows examples of simple arrays of cells. In each case, the polarization

of the cell at the edge of the array is kept xed; this is the so-called driver cell and it is

plotted with a thick border. We call it the driver since it determines the state of the

whole array. Without a polarized driver, the cells in a given array would be un-

polarized in the absence of a symmetry-breaking inuence that would f avour one of

the basis states over the other. Each gure shows the cell polarization s corresponding

to the physical ground state conguration of the whole array.

Figure 6( a) shows that a line of cells allows the propagatio n of information, thusrealizing a binary wire (Lent and Tougaw 1993). Note that only in information butno electric current ows down the line, which results in low power dissipation.

Information can also ow around corners, as shown in gure 6( b), and fan-out is


Figure 5. The cellcell response. The polarization of cell 2 is xed and its Coulombic eecton the polarization of cell 1 is measured. The nonlinearity and bistable saturation ofthis response serves the same role as gain in a conventional digital circuit.

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possible, shown in gure 6(c). Cells which are positione d diagonally from each othertend to anti-align. This feature is employed to construct an inverter as shown in

gure 6( d). In each case, electronic motion is conned to within a given cell, but notbetween dierent cells. Only information, and not charge, is allowed to propagate

over the whole array.

These quantum-dot cells are an example of quantum-functional devices. Utilizing

quantum-mechanical eects for device operation may give rise to new functionality.Figure 7 shows the fundamental QCA logical device, a three-input majority gate,

from which more complex circuits can be built. The central cell, labelled the device

cell, has three xed inputs, labelled A, B and C. The device cell has its lowest energy

state if it assumes the polarization of the majority of the three input cells. The output

can be connected to other wires from the output cell. The dierence between input

and output cells in this device, and in QCA arrays in general, is simply that inputs

are xed and outputs are free to change. The inputs to a particular device can come

from previous calculations or be directly fed in from array edges. The gure also

556 W. Porod et al.

Figure 6. Examples of simple QCA arrays showing ( a) a binary wire, ( b) signal propagationaround corners, ( c) the possibility of fan-out, and ( d) an inverter.

Figure 7. Majority logic gate. The basic structure simply consists of an intersection of lines.

Also shown are the computed majority logic truth table and its logic symbol.

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shows the majority logic truth table which was computed as the physical ground

state polarizations for a given combination of inputs. Using conventional circuitry,

the design of a majority logic gate would be signicantly more complicated, being

composed of some 26 MOS transistors. The new physics of quantum mechanics gives

rise to new functionality, which allows a rather compact realization of majority logic.

It is possible to `reduce a majority logic gate by xing one of its three inputs inthe 1 or 0 state. In this way, a reduced majority logic gate can also serve as a

programmable AND/OR gate. Inspection of the majority-logic truth table reveals

that if input A is kept xed at 0, the remaining two inputs B and C realize an AND

gate. Conversely, if A is held at l, inputs B and C realize a binary OR gate. In other

words, majority logic gates may be viewed as programmable AND and OR gates, as

schematically shown in gure 8. This opens up the interesting possibility that the

functionality of the gate may be determined by the computation itself. Combined

with the inverter shown above, this AND/OR functionality ensures that QCA

devices provide logical completeness.As an example of more complex QCA arrays, we consider the implementation of

a single-bit full adder. A schematic of the logic device layout for an adder imple-

mented with only majority gates and inverters is shown in gure 9. A full quantum


Figure 8. Reduction of the majority logic gate to AND and OR gates by xing one of theinputs.

Figure 9. Schematic diagram of the QCA cell layout necessary to implement the single bit

full adder.

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mechanical simulation of such an adder has been performed, and veries that it

yields the correct ground state output for all eight possible combination s of the

three inputs (Tougaw and Lent 1994). The general principles for computing withsuch QCA arrays will be discussed below.

2.4. Computing with QCAs

A QCA array like the adder discussed above works because the layout of the

quantum-dot cells has provided a mapping between the physical problem of nding

the ground state of the cells and the computational problem. The physical problem

can be stated as follows: given the boundary conditions imposed by the input, what

is the lowest energy conguration of the electrons in the cellular array? It is the

ability to make this mapping between the physical ground state and the unique

logical solution state that is at the heart of the QCA approach. This is illustrated

schematically in gure 10.Without getting too far into implementation-specic features, let us briey

address the question of input and output in a QCA array. Setting an input wire

requires coercively setting the state of the rst cell in the wire. This can be accom-

plished very simply by charging nearby conductors to repel electrons from one dot

and attract them to another. In quantum dots made in semiconductors or metals,

this has become a standard experimental technique, usually called a `plunger elec-

trode, to alter electron occupancy of a dot. Reading an output state is more dicult.

We require the ability to sense the charge state of a dot without having the measure-

ment process alter the charge state. Since the local charge produces a local electro-

static potential, this is really a question of constructing a small electrometer.

Fortunately, electrometers made from ballistic point-contacts and from isolated

558 W. Porod et al.

Figure 10. Schematic representation of computing with a QCA array. The key concepts are

`computing with the ground state and `edge-driven computation described in the text.

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dots themselves have already been demonstrated, both in metallic and semiconduc-

tor systems (Baza n et al. 1996, Bernstein et al. 1996). These electrometers can non-invasively measure the charge state of a single dot. Note that input and output are

only performed at the edges of the array; no information or energy need ow to

interior cells.

Computation in a QCA array proceeds along the following three basic steps:(1) First, the initial data is set by xing the polarization of those cells at the edge,

which represent the input information (edge-driven computation).

(2) Next, the whole array is allowed to relax (or is adiabatically transformed) tothe new ground state, compatible with the input cells kept xed (computing

with the ground state).

(3) Finally, the results of the computation are read by sensing the polarization ofthose cells at the periphery which represent the output data.

Computing is thus accomplished by the mapping between the physical ground

state of the array and the logical solution state of the computational problem. The

two key features which characterize this new computing paradigm are `computing

with the ground state and `edge-driven computation which we discuss in further

detail below.

2.4.1. Computing with the ground state. Consider a QCA array before the start of

a computation. The array, left to itself, will have assumed its physical ground

state. Presenting the input data, i.e. setting the polarization of the input cells, will

deliver energy to the system, thus promoting the array to an excited state. The

computation consists in the array reaching the new ground state conguration,

compatible with the boundary conditions given by the xed input cells. Note that

the information is contained in the ground state itself, and not in how the ground

state is reached. This relegates the question of the dynamics of the computation to

one of secondary importance; although it is of signicance, of course, for actual

implementations. In the following, we will discuss two extreme cases for this

dynamics, namely one where the system is completely left to itself, and another

where exquisite external control is exercised.

2.4.1.1. L et physics do the computing. The natural tendency of a system to assume

the ground state may be used to drive the computation process, as schematically

shown in gure 11( a). Dissipative processes due to the unavoidabl e coupling tothe environment will relax the system from the initial excited state to the new

ground state. The actual dynamics will be tremendously eomplicated since all the

details of the system-environment coupling are unknown and uncontrollable.

However, we do not have to concern ourselves with the detailed path in which the

ground state is reached, as long as the ground state is reached. The attractive fea-ture of this relaxation computation is that no external control is needed. However,

there also are drawbacks in that the system may get `stuck in metastable states

and that there is no xed time in which the computation is completed.

2.4.1.2. Adiabatic computing. Due to the above diculties associated with meta-

stable states, Lent and co-workers have developed a clocked adiabatic scheme for

computing with QCAs. The system is always kept in its instantalleou s ground

state which is adiabatically transformed during the computation from the initial

state to the desired nal state, as schematically depicted in gure 11( b). This is


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accomplished by lowering or raising potential barriers within the cells in concert

with clock signals. The modulation of the potential barriers allows or inhibits

changes of the cell polarization. The presence of clocks makes synchronized opera-

tion possible , and pipeline d architectures have been proposed (Lent et al. 1994,

Lent and Tougaw 1996, 1997).

2.4.2. Edge-driven computation . Edge-driven computation means that only the per-

iphery of a QCA array can be contacted, which is used to write the input and toread the output of the computation. No internal cells may be contacted directly.

This implies that no signals or power can be delivered from the outside to the in-

terior of an array. All interior cells interact only within their local neighbourhood.

The absence of signal and power lines to each and every interior cell has obvious

benets for the interconnect problem and heat dissipation.

The lack of direct contact to the interior cells also has profound consequences for

the such arrays can be used for computation. Since no power can ow from the

outside, interior cells cannot be maintained in a far-from-equilibrium state. Since no

external signals are brought to the inside, internal cells cannot be inuenced directly.

These are the reasons why the ground state of the whole array is used to represent

the information, as opposed to the states of each individual cell. In f act, edge-driven

computation necessitates computing with the ground state!

Conventional circuits, on the other hand, maintain devices in a far-from-equili-

brium state. This has the advantage of noise immunity, but the price to be paid

comes in the form of the need for wires to deliver power (contributing to the wiring

bottleneck) and the power dissipated during switching (contributing to the heat

dissipation problem).

560 W. Porod et al.

Figure 11. Schematic representation of (a) relaxation computing and (b) adiabatic computing.

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2.5. Switching of QCA arrays

As discussed in the previous section, quantum-dot cellular automata take advan-

tage of the concept of computing with the ground state, which means that the

physical ground state of the system is mapped directly to the logical solution of

the problem that the device is designed to solve. This emphasis on the ground

state is one of the strengths of the QCA architecturethe details of the evolutionof the system, which may be hard to control, are not essential in getting the compu-

tation right. The dynamics of the system are doing the computing only in the sense

that they move the system to its new ground state. This view of the computational

process has also made it appropriate to rst study the steady-state behaviour of these

devices before looking at the dynamic behaviour.

The dynamics of the system, however, cannot be completely neglected. The

dynamics of the system are relevant for two reasons. The rst is that an analysis

of the systems dynamics is necessary to quantify the switching speed of QCA arrays.

Second, as has been pointed out by Landauer, the presence of metastable statescould cause a signicant delay in the system reaching its new ground state, so the

identication of such states is important.

We have considered two approaches to switching the array from the solution of

one problem to another. The rst approach involves switching the input cells sud-

denly and allowing dissipative coupling to the environment to relax the array to the

new solution state. The inputs are kept xed during this relaxation. The second

method involves switching the array gradually by smoothly changing the input states

while simultaneously modulating the inter-dot barriers over the whole array. In this

way the array can be switched adiabatically, keeping the system at all times in the

instantaneous ground state. This adiabatic approach thus removes any problems

associated with possible metastable states and enables clocked control of switching

events. An analogous regime for metal cells, for which barriers cannot be lowered

but occupancy can be changed, has been developed.

In the adiabatic switching approach described in the previous section, it was

always assumed that the interdot potential barrier was modulated simultaneously

for all cells in the array. From the point of view of fabrication complexity, this is an

important f eature. It permits one conductor, typically one gate electrode, to controlthe barriers of all cells. If each cell had to be separately timed and controlled, the

wiring problem introduced could easily overwhelm the simplication won by the

inherent local interconnectivity of the QCA architecture itself.

We can gain signicant advantage , however, by relaxing this requirement

slightly. If we subdivide an array of cells into subarrays, we can partition the com-

putational problem and gain the advantages of multi-phase clocking and pipelining.

For each sub-array a single potential (or gate) modulates the inter-dot barriers in allthe cells. This enables us to use one sub-array to perform a certain calculation , then

freeze its state by raising the inter-dot barriers and use the output of that array as the

input to a successor array. During the calculation phase, the successor array is kept

in the unpolarized state so that it does not inuence the calculation. An analogous

regime for metal cells, for which barriers cannot be lowered but occupancy can be

changed is also possible.

The adiabatic pipelining scheme has several benets. The most obvious benet is

that the clocking cycles of the cells are interlaced so that as soon as information is no

longer necessary for further calculations, it is released to free up room for new

information. This allows the device to be in the process of carrying out several


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calculation s at once, especially if the pipeline is long. Such simultaneous calculation

stages maximize the throughpu t of each total system. A second benet of this system

is that the number of cells in each sub-array can be kept well below the thermo-

dynamic limits to it size. Finally, this clocked approach clearly demonstrates that, at

least from an architectural standpoint, general purpose computing with QCA arrays

is feasible.

2.6. Thermodynamic considerations

Thermal uctuations are of concern for ground state computing. Thermal noise

may excite the system from its ground state to a higher-energy state, and thereby

interfere with the computation. The probability for the occurrence of such errors is

basically given by the relative magnitude of a typical excitation energy to the thermal

energy, k T. Excitation energies become larger as the size of the system becomes

smaller, thereby providing increasing noise immunity. A typical QCA cell fabricated

with current state-of-the-art lithograph y (for minimum feature sizes of 20 nm) isexpected to operate at cryogenic temperatures, whereas a molecular implementation

(for feature sizes of 2 nm) would work at room temperature, as illustrated ingure 12.

Thermodynamic considerations also are of concern for large arrays, and entropy

needs to be taken into account. The tendency of a system to increase its entropy

makes error states more favourable. As the size of the system increases, so do the

number of possible error congurations. This sets an upper limit on the size of the

allowed number of cells in an array before entropy takes over. We have shown that

this limit depends in an exponential fashion on the ratio of a typical excitationenergy to the thermal energy (Lent et al. 1994). For example, arrays with about20 000 cells are feasible if this typical error energy is 10 times larger than the thermal

energy.

3. Quantum-dot cellular neural networks

In addition to employing QCA cells to encode binary information as described

in the previous section, these cells may also be used in an analogue mode. As

562 W. Porod et al.

Figure 12. Cellcell response f unctions fo r various temperatures. (a) Semiconductor cell withminimum lithographic feature sizes of 20 nm and ( b) molecular implementation withdimensions of 2 nm.

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schematically shown in gure 13, each cell interacts with its neighbours within a

certain range, thus forming what we call a quantum-dot cellular nonlinear network

(Q-CNN) (Toth et al. 1996). This way of viewing coupled cells as an analogue non-linear dynamical system is similar to cellular nonlinear (or, neural) networks (CNN),which are locally-interc onnected structures implemented using conventiona l circuitry

(Chua and Yang 1988). Each cell is described by appropriate state variables, and thedynamics of the whole array is given by the dynamical law for each cell, which

includes the inuence exerted by the neighbours on any given cell.In the paragraphs below, we develop a simple two-state model for the quantum

states in each cell and show how the quantum dynamics of the array can be trans-

formed into a CNN-style description by choosing appropriate state variables. The

general features of this model are:

(1) each cell is a quantum system, characterized by both classical and quantumdegrees of freedom;

(2) the interactions between cells only depends upon the classical degrees of

freedom; the precise form of the `synaptic input is determined by the physicsof the intercellular interactions; and

(3) the state equations are derived from the time-dependent Schro dinger equa-tion; one state equation exists for each classical and quantum degree of

freedom.

For the case of a two-dimensional array, each Q-CNN cell possesses an equiva-

lent CNN-cell model described by the dierential equations given below. We may

thus think of such a quantum-dot cell array as a special case of cellular nonlinear

networks (Nossek and Roska 1993). The equivalent circuit describing a cell is com-posed of two linear capacitors, four nonlinear controlled sources and eight linear

controlled sources representing the interactions between the cell and its eight neigh-

bours (Csurgay 1996).Our quantum model is a special case of a general formalism for `quantum net-

works developed by Mahler ( 1995). A s schematically shown in gure 14, a quantumnetwork consists of subsystems which are special quantum objects denoted as

`network nodes and the interaction channels between them are denoted as `net-

work edges. For the study of such systems, Mahler has developed a density


Figure 13. Schematic view of a locally-connected cellular QCA array. The circle indicates therange of interaction for the central cell.

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matrix formalism for the theoretical study of small (coherent) quantum networks:the network node is taken as a nite local state space of dimension n. The network

might be a regular lattice or an irregular array of nodes. The network is coupled to

external driving elds and dissipative channels, which are required for measurement.

This approach provides a system-theoretic tool adaptable to situations where a nite

quantum mechanical state space is controlled by a classical environment. This form-alism is suciently general to also include dissipation, but we have not yet incorpo -

rated that in our Q-CNN models; this work is in progress.

3.1. Quantum model of cell array

Following the work of Toth et al. (1996), we describe the quantum state in eachcell using the two basis states ju1 i and ju2 i which are completely polarized.

jC i = aju1 i + bju2 i

Within this two-state model, each property of a cell is completely specied by the

quantum mechanical amplitudes a and b. In particular, P, the cell polarization isgiven by

P = jaj2

- jbj2

The Coulomb interaction between adjacent cells increases the energy of the

conguration if the two cell polarizations dier. This can be accounted for by an

energy shift corresponding to the weighted sum of the neighbouring polarizations,

which we denote by PE. The cell dynamics is then given by the Schro dingerequation

ihd/dtjC i = HjC i

where H represents the cell Hamiltonian . Once the Hamiltonian is specied, the cell

dynamics is completely determined. jC i represents the state of q given cell, and not

the state of the whole array; quantum entanglement is not accounted for in this

formulation.

564 W. Porod et al.

Figure 14. Quantum network: the subsystems of the network are special quantum objects

denoted as `network nodes and the interaction channels are denoted as `network edges(after Mahler 1995).

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3.2. Formulating CNN-like quantum dynamics

In order to transform the quantum mechanical description of an array into a

CNN-style description, we perform a transformation from the quantum-mechanical

state variables to a set of state variables which contains the classical cell polarization,

P, and a quantum mechanical phase angle, u

jC i = (a, b) -! jC i = (P, u)

This transformation is accomplished by the relations

a = [(1 + P) / 2]1/ 2

b = [(1 - P) / 2]1/ 2

eiu

With this, the dynamical equations derived from the Schro dinger equation can be

rewritten as CNN-like dynamical equations for the new state variables P and u

hd/dt P = - 2g sin u(1 - P2)

1/ 2

hd/dtu = - PE+ 2g cos uP/(1 - P2) 1/ 2

The term PEaccounts for the cellcell interaction and g is the tunnelling matrixelement between dots. For a more detailed derivation, we refer to the origina l paper

by Toth and co-workers (1996).It can be shown that the resulting dynamics for each cell is governed by a

Liapounov function V(P, u) which is given by Porod et al. (1997)

V(P, u) = 2g cos u(1 - P2)

1/ 2+ PPE

3.3. Cellular network model of quantum -dot array

This above network model simulates the dynamics of the polarization and the

phase of the coupled cellular array. If the polarization of the driver cells of an array

in equilibriu m is changed in time, a dynamics of the polarization s and phases f or all

cells in the whole array is launched. In the framework of the CNN model, ground-

state computing by the quantum cellular array corresponds to transients between

equilibrium states.It is well known for CNN arrays that the dynamics may give rise to interesting

spatio-temporal wave-phenomena (Chua 1995). A signicant literature exists on thissubject, and dierent classes of wave behaviour and pattern formation have been

identied (Zuse 1969, Porod et al. 1996).In complete analogy, spatio-temporal wave-behaviour also exists for the

dynamics of Q-CNN arrays. We have begun to study these phenomena and we

present a few examples below.

Figure 15 shows wave front motion in a linear Q-CNN array. The driver cell on

the left-hand-side is switched at t = 0, thereby launching such a soliton-like wavefront. The gure shows snapshots of the classical polarization P and the quantummechanical phase angle u at various times. Note that the information about the

direction of propagatio n is contained in the sign of the phase angle. Just from

the polarization alone, one could not tell whether the wave front would move to

the right or left.

Figure 16 shows examples of wave-like excitation patterns in two-dimensional

Q-CNN arrays. The top panel shows an example of wave behaviour induced by a

periodic modulation of the boundaries. Note that a xed cell block is also included


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566 W. Porod et al.

Figure

15.

WavefrontmotioninalinearQ-CNNarray.

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Figure16.

Examplesof

wave-likeexcitationpatternsintwo-dimensionalQ-CNN

arrays.Thetoppane

lshowsconcentricwavesduetoaperiodic

modulationoftheboundaries.

The

bottompanelshowsaspir

alwaveduetocyclicalmo

dulationoftheboundaries.

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near the upper left-hand corner. Several snapshots are shown (time increases from

left to right) which display concentric wave fronts. The bottom panel shows anexample of cyclical excitations at the boundaries which give rise to a spiral wave.

The excitation at each of the four boundaries are 90 out of phase with respect to

their neighbouring edge.

4. Possible quantum-dot cell implementations

In this section, we discuss possible implementations of the coupled-quantum-dot

cells discussed so far. Based upon the reported studies in the literature on single-

electron charging and dotdot coupling, experimental e orts are underway at Notre

Dame to realize a QCA cell in semicondutors using split-gate technology and in rings

of metallic tunnel junctions (Bernstein et al. 1996). We give a brief review of thatwork, including the results and implications of our numerical modelling. We also

discuss a candidate molecule which might serve as a prototype molecular electronicsimplementation of a QCA cell.

4.1. Gate-controlled quantum dots

The fabrication of a QCA cell by split-gate technology is a challengin g problem,

yet appears to be within reach of current lithographic capability (Bernstein et al.

1996). Figure 17 shows a possible physical realization which is based on electrostaticconnement provided by a top metallic electrode (Taylor 1994). The key imple-mentation challenges are (i) to gain sucient gate control in order to dene quantumdots in the few-electron regime, and (ii) to place these dots suciently close toeach other in order to make coupling possible. Using these techniques, it is con-

ceivable that coupled-dot cells may be realized in a variety of materials systems,

such as IIIV compound semiconductors, Si/SiGe heterolayers, and Si/SiO2 struc-

tures.

In order to achieve a crisp conning potential, it is important to minimize the

eects of fringing elds, which may be accomplished by bringing the electrons as

close as possible to the top surface. This design strategy of `trading mobility versusgate control by utilizing near-surface 2DEGs has been pioneered by Snider et al.

(1991). However, the resultant proximity of the quantum dot to the surface raises thequestion of the eect of the exposed surface on the quantum connement. To study

these questions, we have undertaken extensive numerical modelling of such gate-

controlled dots, and we have explictly included the inuence of surface states which

are occupied, in a self-consistent fashion, according to the local electrostatic poten-

tial (Chen and Porod 1995).

568 W. Porod et al.

Figure 17. Possible physical realization of gate-controlled quantum dots by top metallic gates.

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We have performed numerical simulations for the design of quantum dot struc-

tures in the few-electron regime, both in the AlGaAs/GaAs and Si/SiO2 material

systems. The conning potential is obtained from the Poisson equation within a

Thomas-Fermi charge model. The electronic states in the quantum dot are then

obtained from solutions of the axisymmetric Schro dinger equation. Our model takes

into account the eect of surface states by viewing the exposed surface as the interface

between the semiconductor and air (or vacuum). Figure 18 schematically shows thesimulation strategy, wher e we employ a nite-element technique for the semiconductor

domain and a boundary element method for the dielectric above (Chen et al. 1994).Both domains are coupled at the exposed surface, taking into account the eect of

charged surface states. This is particularly important for modelling the IIIV

material system, where surface states are known to be signicant.Our modelling shows that the single most critical parameter for the design of

gate-controlled dots is the proximity of the 2DEG to the top metallic gates. This

distance is limited in the IIIV material system by the `leakiness of the layer separ-

ating the electronic system from the electrodes. Distances as close as 25 nm have been

achieved, but more typically 40 nm are being used. The silicon material system

appears to be particularly promising candidate because of the extremely good insu-

lating properties of its native oxide. SiO2 layers can be made as thin as 10 nm, and

even less (4 nm appears to be about the limit). This allows for extremely crisp con-ning potentials.

We have explored various gate congurations and biasing modes. Our simula-

tions show that the number of electrons can be eectively controlled in the few

electron regime by the combined action of depletion and enhancement gates,

which we will illustrate below.

4.1.1. AlGaAs/GaAs material system. Figure 19 shows an example of the occupa-

tion of quantum dots for combined enhancement/depletion mode biasing on an

AlGaAs/GaAs 2DEG. The main idea is to negatively bias the outer electrode


Figure 18. Solution strategy used in the numerical simulations. Finite elements are used inthe semiconductor domain, and a boundary element technique for the dielectric above.

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(gate 2) such that the 2D electron density is depleted, or near depletion; a positivebias on the inner electrode (gate 1) is then utilized to induce the quantum dot andto control its occupation. In this example, we have chosen a radius of rG1 = 6 nm

for the centre enhancement gate, and a radius of rG2 = 50 nm for the surrounding

depletion gate. The resulting num ber of electrons induced by three dierent voltages

on the depletion gate, VG2 , is plotted as a function of the enhancement gate bias

voltage, VG1 . We see that variations of the depletion-gate bias of 10 mV will result

in threshold-voltage variations of as much as 80 mV. This biasing mode appears

to be an eective way of controlling the quantum-dot threshold voltage in thefew-electron regime.

The use of a single, negatively-biased depletion gate would not suce for our

purposes. Even though it would be possible to obtain few-electron dots, the resulting

potential variations are too gradua l to allow fabricatio n of a QCA cell with closely-

spaced dots (Chen and Porod 1995).

4.1.2. Si/SiO 2 material system. We have also performed numerical simulations for

the design of gated few-electron quantum dot structures in the Si/SiO2 material

system. The motivation for this work has been to investigate the feasibility of

transferring the emerging technology of quantum dot fabricatio n from the IIIV

semiconductors, where it was pioneered over the past few years, to the technologi-

cally more important Si/SiO2 material system. Silicon appears to be a promising

candidate due to the excellent insulating behaviour of thin SiO2 lms which yields

the required crisp gate-control of the potential in the plane of the two-dimensional

electron gas at the Si/SiO2 interface. Another advantage of silicon for quantum

dot applications appears to be the higher eective mass, as compared to the IIIV

materials, which reduces the sensitivity of the energy levels to size uctuations.

570 W. Porod et al.

Figure 19. Example of dot occupation using combined enhancement (G1) and depletion (G2)gates in the AlGaAs/GaAs material system.

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Quantum dots may be realized by applying a positive bias to a metallic gate on

the surface, as schematically shown in the inset to gure 20. The positive voltage

induces an inversion layer underneath the biased gate, which may lead to the for-

mation of an `electron droplet at the silicon/oxide interface, i.e. a quantum dot.

Figure 20 shows, for an applied gate bias of 1.7 V, the corresponding potential

variations along the Si/SiO2 interface. An electronic system is induced when the

silicon conduction band edge at the oxide interface, indicated by the solid line,

dips below the Fermi level (indicated by the thin horizontal line). We see that the

formation of a quantum dot critically depends upon the thickness of the oxide layer.Our modelling shows that for a 10 nm gate radius, an oxide thickness around (or

below) 10 nm is required. Further modelling shows that these dots are occupied inthe few-electron regime (Chen and Porod 1998).

Our modelling suggests a design strategy as schematically shown in gure 21.

Two metallic electrodes are being used, one (the bottom one) as a depletion gate, andthe top one as an enhancement gate. We envision to fabricate openings in the bottom

electrode by electron beam lithography with dimensions of about 20 nm. The top

electrode is then evaporated onto the patterned and oxidized metallic layer, whichresults in a top electrode which may reach the semiconductor surface inside the

openings, providing the enhancement gates.

4.2. Rings of metallic tunnel junctions

In addition to the semiconductor systems discussed above, single-electron tunnel-

ling phenomena may also be observed in metallic tunnel junctions (Fulton and

Dolan 1987, Grabert and Devoret 1992). Consider a ring of metallic tunnel junc-

tions, schematically shown in gure 22( a). The tunnel junctions are represented by


Figure 20. Potential variation at the silicon/silicon dioxide interface as a f unction of oxidethickness.

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the crossed capacitor symbols, indicating that these junctions are characterized by

capacitance and tunnel resistance. The metallic droplets themselves are the `wires

between the tunnel junctions. Consider now that two extra electrons are added to

such a cell, as schematically shown in part (b)

of the gure. It can be shown that this

cell exhibits precisely the same two distinct ground state congurations as the semi-

conductor cell discussed above. In addition, the cellcell coupling also shows the

same strongly nonlinear saturating characteristic (Lent and Tougaw 1994). Note that

572 W. Porod et al.

Figure 21. Design strategy for a quantum-dot cell in the silicon material system using thecombined action of two metallic electrodes.

Figure 22. Possible QCA implementation using rings of m etallic tunnel junctions. ( a) Basic

cell, (b) cell occupied by two additional electrons, (c) line of capacitively coupled cells.

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cellcell coupling is purely capacitive, as schematically shown for the line of cells in

gure 22( c). The metallic tunnel-junction cell may be used as a building block formore complicated structures, in a fashion completely analogous to the semiconduc-

tor implementations.

Using the technique of shadow evaporation, coupled-dot structures have recently

been fabricated and tested in the aluminium-aluminium oxide material system. Theseexperiments will be described in more detail in a separate section below.

4.3. Possible molecular implementation

As discussed above, QCA room temperature operation would require molecular-

scale implementations of the basic cell. Molecular chemistry promises to oer the

versatility for the desired miniaturization. The requirements for a molecular QCA

technology include:

(1) cells made of a rigid array of identical clusters with inter-cell interactions thatare insulating (e.g. a square arrangement of four clusters);

(2) two-electron occupation of each cell with distinct arrangements of the twocharges;

(3) charge interchange between these distinct arrangements, which are energeti-cally equal in the absence of a polarizing eld;

(4) patterning of cells into predened array geometries on a substrate; and

(5) connections to the periphery of an array for inputs and outputs.In previous work by Fehlner and co-workers, a candidate for such a prototypical

molecular cell has been synthesized and crystallographically characterized (Cen et al.

1992, 1993). As schematically illustrated in gure 23, these molecular substances withthe formula M2 {(CO)9 Co3 CCO2}4, where M = Mo, Mn, Fe, Co, Cu, consist ofsquare arrays of transition metal clusters; each containing three cobalt atoms. It is

remarkable that the four clusters are arranged in a (at) square, as opposed to a(three-dimensional) tetrahedron, which one might have expected. The reason for this

behaviour lies in the two metallic atoms at the centre which form a `spindle and theclusters attach themselves in the plane perpendicular to this axis.

It has been demonstrated that these compounds may be obtained as pure crystal-

line molecular solids in gram quantities. In spite of high molecular weights, these

substances are soluble and most dissolve without dissociation, which makes it

possible to disperse them on a substrate. Each cell has an edge-to-edge distance

of about 20 nm, which is precisely the desired dimension for QCA room tempera-

ture operation. The spectroscopic properties demonstrate intra-cell cluster-core

electronic communication and inter-cell interactions are insulating.

5. Experimental demonstration of QCA elements

As described above, a basic QCA cell can be built of two series-connected dots

separated by tunnelling barriers and capacitively coupled to identical double-dots. If

the cell is biased such that there are two excess electrons within the four dots, these

electrons will be forced to opposite corners by Coulomb repulsion. Experimentally, it

is necessary to both set and detect the desired cell polarization. Operation of a

realistic QCA cell should be demonstrated in an environment comparable to that


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574 W. Porod et al.

Figure23.

Candidatefor

amolecularQCA

cellconsistingof4outer(CO)9Co3Cclusterswhicharearra

ngedinaplanearounda`s

pindleconstituted

bytheM2c

ore.

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in which it will function. The control of a QCA cell must be by single electrons, since

in a QCA system, polarization of cells is inuenced totally by the single electron

polarizatio n of nearest neighbour-cells. In short, we must have single electrons

controlled by single electrons.

Critical to any device or system whose operation depends upon the motion of

single electrons is a means of detecting their positions. It has been shown that a singleelectron transistor (SET) can be used to detect charge variation in a nearby dot(Lafarge et al. 1991, Bazan et al. 1996). In previous experiments, the Coulombinteraction of electrons within a double-dot has been inferred exclusively from

their series conductance (Pothier et al. 1992, Sakamoto et al. 1994, Waugh et al.

1995). A detection scheme that can probe the polarization state of the double-dotexternally, and with high sensitivity, has not heretofore been developed.

5.1. Measurement of double-dot polarizationIn this section, we present direct measurement of the internal charge state of a

metal/oxide double-dot system. Specically, our charge detection technique is sensi-

tive not only to the charge variation of individual dots, but also to the exchange of

one electron between the two dots. This important property of our detection scheme

makes it suitable for sensing the polarization state of a QCA cell.

Metal dot structures are constructed of small aluminium wires separated by

extremely thin aluminium oxide barriers which allow electrons to tunnel between

the sections. Figure 24 shows a scanning electron micrograph (SEM)

of an SET. The

area of the oxide barriers (i.e. the overlap of lines) is small (about 60 60 nm2),resulting in correspondingly small capacitances between the sections. A centre island


Figure 24. Field emission scanning electron micrograph of a single electron tunnelling tran-sistor fabricated at Notre Dame. The gate electrode is to the left, and the two tunneljunctio ns are at bo th ends of the `d ot . The sizes of the tun nel junctions are approxi-

mately 60 60 nm2

.

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of metal, referred to as the `dot, is therefore isolated from the rest of the system by

`leaky barriers. These dots assume the property of accepting charge by units of

single electrons only.

Fabrication of metal dots connected by leaky tunnel junctions is accomplished

through a series of carefully controlled processing steps, due to Fulton and Dolan

(1987). First, a double layer of resist is exposed by electron beam lithograph y ( EBL)and developed to form the desired pattern. (The resist layers are chosen to result in alarge degree of undercut, allowing the tilting procedure described below.) Second,the wafer is coated in a high-vacuum chamber by a thin aluminium lm which

reaches the surface of the wafer only where the resist has been developed , i.e.

removed. (The remaining aluminium rests on the surface of the resist to be removed

in the nal step. ) Next, the aluminium is exposed in situ to a controlled amount ofoxygen in order to oxidize its surface. Since this is a self-limiting process, the nal

thickness of the oxide is well-controlled. The wafer is then tilted so that a second

evaporated aluminium pattern is translated laterally from the rst, but overlappingin very tiny areas which form the leaky junctions. The presence of the metal over the

oxide then protects the junction from further oxidation in ambient. In our case, the

bottom electrode metal was 25 nm thick with 50 nm of Al forming the top electrode.

In the nal lift-o step, the resist is dissolved in a suitable solvent and all extraneous

metal residing on the surface of the resist is removed.

Figure 25 is a schematic diagram of our metal dot system, consisting of two

islands in series connected by a tunnel junction with each island capacitively coupled

576 W. Porod et al.

Figure 25. Schematic diagram of the device structure. Capacitance parameters of dierentparts of the device are listed in table 1. The capacitances of the coupling capacitors C11and C22 are approximately 10% of the total capacitances of the electrometers. Thecircuit used to compensate for parasitic capacitance between the driver gates VA /VB

and the electrometer islands E1/E2 is not shown.

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to an SET electrometer. An interdigitated design is used to increase the coupling

capacitance, enhancing electrometer sensitivity. The two dots labelled D1 and D2

between the three tunnel junctions were 1.4mm long. In the vicinity of each dot are`driver gate electrodes A and B used to change the electron populations of their

respective dots. Each dot of the double-dot system is also capacitively coupled to an

SET (labelled E1 and E2) with an island length of 1.1mm. Figure 26 is an SEM of thecell depicted in gure 25. The typical tunnel resistance of a junction, based on IVmeasurements of the electrometers at 4.2 K, was approximate ly 200 kV . The total

capacitance of the electrometers, CS , extracted from the charging energy(EC 80 meV), was approximately 1 fF.

Measurements were carried out at the base temperature of our dilution refrig-

erator ( 10 mK) using standard ac lock-in techniques. A 4 mV excitation voltage ata frequency of 20 Hz was used to measure the conductance of the double-dot and

the electrometers. A 1 T magnetic eld was applied to suppress the superconductivity

of Al.Initial experiments were performed to extract the lithographic and parasitic

capacitance parameters of the dierent parts of the circuit. Capacitances between

each gate and island were determined from the period of the Coulomb blockade


Figure 26. Scanning electron micrograph of fabricated QCA cell test system. On the left isthe double-dot structure, and on the right are the two SET electrometers. The electro-

meters are coupled to the double-dot structure via interdigitated capacitors.

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oscillations (CBO) (Averin and Likharev 1991). Table 1 shows the extracted systemcapacitances.

In subsequent experiments, the charge on the double-dot structure was varied by

sweeping gates A and B. Conductances through the double-dot and both SET elec-

trometers were simultaneously measured. The sensitivities of the electrometers, as

expected from Lafarge et al. (1991), were proportional to CD1 - E1 and CD2- E2. Also,the operating points of the electrometers were set to be equal. Coupling capacitors

CD1 - E1 and CD2 - E2 were designed to be relatively large in order to increase the

sensitivity of the electrometers (table 1), yet constitute a suciently small fractionof the total capacitance of the electrometers for them to act as non-invasive probes

(Bazan et al. 1996).Our external circuitry was more involved than that shown in gure 25 in order to

compensate for parasitic capacitances between the gates and the islands, which are

non-negligible, as can be seen from table 1. To suppress the inuence of the parasitic

capacitances CA- E1 , CA- E2, CB- E1 , and CB- E2 , we applied inverted compensationvoltages proportional to VA and VB to gates C and D. Using this charge compensa-tion technique, we were able to reduce the eect of the parasitic capacitance by at

least a factor of 100.

Figure 27 is a contour plot of the conductance through the double-dot as a

function of driver gate voltages, VA and VB . The resulting charging diagram ofsuch a measurement forms a `honeycomb structure rst observed by Pothier et al.

(1992). The honeycomb boundarie s ( solid lines) represent the regions where a changein electron population (n1 , n2) occurs on one or both of the dots, with n1 and n2representing excess population s of D1 and D2, respectively. Each hexagonal cell

marks a region in which a given charge conguration is stable due to Coulomb

blockade. In the interior of the cell there is no charge transport through the

578 W. Porod et al.

ApproximateCapacitance Type capacitance (aF)

Cjunction Lithographic 400CA-D1 Lithographic 47.7CB- D2 Lithographic 49.5CC-E1 Lithographic 29

CD- E2 Lithographic 26.6CD1- E1 Lithographic 106CD2- E2 Lithographic 106CA-E1 Parasitic 21CA-E2 Parasitic 8CB- E1 Parasitic 9.7CB- E2 Parasitic 21.3CC-E2 Parasitic 7.5CD- E1 Parasitic 7.5CE1- E2 Parasitic 36.8

CA-D2 Parasitic 6.4CB- D1 Parasitic 6.4CD1- E2 Parasitic 16CD2- E1 Parasitic 16

Table 1. Lithographic and parasitic capacitancesbetween various gates and islands shown ingure 24, measured from the period of theCoulomb blockade oscillations.

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580 W. Porod et al.

Figure 28. Charging diagram of the electrometers as a function of the same driver gatevoltages shown in gure 2 with the honeycomb boundaries of gure 4 superimposed.( a) Charging diagram of E1. Sharp transitions in the horizontal direction indicate achange in the population of D1. ( b) Charging diagram of E2. Sharp transitions in the

vertical direction reect a change in the population of D2.

(b)

(a)

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of an electron on D1. Similarly, sharp variation in conductance of E2 in the vertical

direction [gure 28(b)] indicates discrete variation of charge on D2. Hence, thesharpest variations in the conductances of each electrometer can be used to sense

the charging of their capacitively coupled dots.

Sensing the state of a QCA cell requires detection of charge redistribution in the

double-dot shown by direction III in gure 27, but as seen in gure 28, the transitionalong this direction is weaker than the others. During charge redistribution in the

double-dot (line III in gure 27), the populatio n of each dot simultaneou sly changesby one electron with one dot losing an electron and the other gaining one, so the

signals from the two dots are out of phase by 180. The cross capacitance between

D1 (D2) and E2 (E1) causes charging of each dot of the series double-dot to be seenin both electrometers, and since the signals are out of phase, the net response is

reduced. For instance, the signal detected in E1 (E2) along direction III is about 30%weaker than that along the direction I (II).

Figure 29 shows honeycomb borders (solid lines taken from gure 27 ) overlaidon a grey-scale contour plot of a dierential signal, ( G1- G2), where G1 and G2 are theconductances of E1 and E2, respectively. Along the directions I and II in gure 27,

the signals of E1 and E2 have the same phase, giving a suppressed dierential signal.

The most conspicuous transition, represented by a higher density of contour lines,


Figure 29. Dierential signal obtained from the charging diagrams of the individual electro-meters. The most salient transition is in the direction of charge redistribution indicated

by a higher density of contour lines.

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occurs at the boundary between the ( 0, 1) and (1, 0) states, indicating movement ofan electron from one dot to the other. As mentioned above, this is due to the phase

dierence (180) in the signals of the individual electrometers, yielding a dierentialsignal which is approximately twice as strong as the one detected by a single electro-

meter.

Our device architecture allows us to directly observe the internal state of adouble-dot system by detecting all possible charge transitions of a single electron.

A dierential detector that utilizes the signals from both electrometers is most

sensitive to the charge redistribution in the double-dot. With this investigation, we

demonstrate that our dierential detector can be used as the output of a QCA cell.

5.2. Single electron switch, controlled by a single electron

As described above, the operation of a QCA cell requires a change in the cell

polarization controlled by a single electron transition. Changing the polarization ofthe cell is accomplished using gate electrodes to force one electron to switch from one

dot to the other within a double-dot (the input double-dot) which in turn induces aswitch of the other electron in its double-dot (the output double-dot). This is onlypossible when the change of the electrostatic potential on input double-dot due to the

charge redistribution is strong enough to `drag the charge state of the output

double-dot through the corresponding honeycomb border (thick line) of gure 30,thus switching its electron conguration.

Therefore, to produce a functioning QCA cell, the gate biases for each double-

dot must set the working point of the charging state as closely as possible to the

centre of the transition border. Then the cell is in its most symmetrical state, and in

the absence of an input signal the two polarizations are equally probable.

Calculations show that the charge state of the output double-dot can switch under

the inuence of electrostatic potential from a single electron in the input double-dot.

This switching will be reected in a movement of the honeycomb border back and

forth along the diagonal direction E. Switching takes place each time the border

crosses the working point.

Here we report the rst experimental observation of a honeycomb border shift ina double-dot caused by single electron charging of adjacent dots. We compare these

experimental results with the results of the dierential detector of the previous sec-

tion, and with theoretical calculation s to conrm that this shift causes electron

exchange within the double-dot.

To perform this experiment we use the same circuit as in the previous section,

shown schematically in gure 25. Now we use the SET transistors as the input dots

of a QCA cell and investigat e how these single electron drivers aect the output

double-dot charge state. Since we no longer have direct-coupled electrometers, we

must determine the charge state of the output double-dot indirectly by measuring its

conductance, which requires a sweep of gates A and B. To prevent these voltages

from aecting the input dots, we apply compensating voltages to gates C and D as

described in the previous section. It is important to note that without such cancella-

tion, the simple honeycomb pattern of the output double-dot changes dramatically,

since charge states of the input dots will depend on the voltages of gates A and B.

If we carefully control the charge on the SET islands E1 and E2, they can mimic

the input double-dot of a f ull f our-dot QCA cell. Gates A and B, as before, control

the charge state of the double-dot, while the purpose of gates C and D is now very

582 W. Porod et al.

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dierent. Gates C and D are now the input drivers of the QCA cell, which are used to

`switch an electron between the input dots, inducing an electron switch within the

output double-dot. We cannot actually switch an electron between the input dots

since they are not congured as a double-dot. However, if biases with opposite

polarities are applied to gates C and D, each time an electron is removed from

one input dot, another is added to the other, mimicking an electron exchange in

an input double-dot. In our experiment, special care is taken to compensate for

background charges to ensure that the electron transitions occur simultaneously.

A switch in electrons between the input dots will cause a shift in position of the

conductance peaks of the output double-do t (and the entire honeycomb) seen ingure 30. In order to observe this honeycomb border shift, `snapshots of the honey-

comb are taken for di erent ` push-pull (+ VC = VD) settings on the driver gates. It iswell known (Lafarge et al. 1991) that the potential on a metal dot in the Coulombblockade regime changes linearly as a function of gate voltage with an abrupt shift

when the electron population of the dot changes, resulting in sawtooth oscillations.

We indeed observe a slow shift of the honeycomb border corresponding to the

gradual increase of the potential on the input dots, followed by an abrupt `reset.


Figure 30. Charging diagram of the output double-dot. QCA operation will be shown by themotion of this honeycomb along the diagonal E. In particular we will concentrate onthe border between the (0, 1) and (1, 0) states which is marked with a heavy line.

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584 W. Porod et al.

Figure31.

Chargingdiag

ramoftheoutputdouble-dotwith(a)VC

=

-

VD

=

-

0.6

7mV,and(b)VC

=

-

VD

=

+

0.6

7mV.

Thesepre

sentthemaximum

shift

oftheborderbetweenthe

(0,

1)and(1,

0)states.

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Figure 31 shows two snapshots taken at the extremes of the shift, with a measured

shift of approximately 440mV.To demonstrate that the observed shift is due to single electron charging of the

input dots, and not to parasitic coupling from the driver gates, we directly measure

the shift of the honeycomb border while sweeping gates C and D over sucient

voltage that several electron transfers occur in the input dots. The shift from electroncharging will be periodic, since the potential on the input dots resets at each electron

transfer, while parasitic coupling will add a monotonic term to the border shift. To

detect the honeycomb border, we sweep the double-do t gates A and B along the

diagonal E in gure 30. At nite temperature, the conductance of a double-dot

shows a peak as we sweep VA and VB , and the position of this peak marks theborder. Here we dene the `diagonal voltage as the change in voltage along

the direction E, with D VA = -D VB . For each setting of the driver gates we sweepthe diagonal voltages to nd the position of the honeycomb border. These data are

assembled as a 3D contour plot.Figure 32 shows a contour plot of one border as a function of diagonal voltage,

the displacement from the working point (D VA and -D VB ) and driver (+ VC and


Figure 32. Contour plot of the conductance through the double-dot as a function of thedierential driver voltages VC = - VD and the diagonal voltage. The honeycomb bor-der is marked by the peak in conductance, and the black circles mark the calculated

position of the border.

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VD ) voltages, representing the position of the honeycomb border with respect to axed working point dened as Vdiagonal = 0. The solid circles in gure 32 plot thetheoretical position of the border calculated using the measured capacitance values

given in table 1. The experiment matches theory extremely well, considering that

the only tting parameter used is background charge. The background charge can

randomly displace the border, but cannot change the magnitude or period of anobserved shift. The somewhat smaller amplitude of the theoretical honeycomb shift

is most likely due to uncertainty in the measured junction capacitance ( 15% ). Thesawtooth oscillations of gure 32 occur each time an electron is added to one and

removed from the other input dot, conrming that the shift is due to a single electron

transfer. If the working point on the honeycomb pattern of gure 30, ( VA , VB), ischosen close to a border, the shift will cause a border crossing when electron

exchange occurs within the input dots.

To conrm that the border shift is indicativ e of an electron transfer in the output

double-dot, we compare the border shift observed in gure 32 with the data from thedierential detector in gure 29, where a diagonal voltage change of approximately

600 mV is required for an electron to complete the switch from one dot to the other.As mentioned earlier, the transition is not abrupt due to nite temperature, and for a

functional QCA cell the border shift must be sucient to overcome this broadened

transition. The observed shift of 440mV is nearly sucient to completely switch theelectron, but a larger shift would be desirable. An increase of the coupling capaci-

tance between the input and output dots will increase the shift.

We also compare the observed border shift to the results of our simulations,

which include the complete circuitry of the system, assume a temperature of 0 K,

and include all experimentally determined parasitic capacitances. Figure 33(a) showsthe calculated potential on input dot E1, which, as expected, is a sawtooth pattern

with abrupt transitions when an electron enters or leaves the dot. Figure 33( b) showsthe calculated potential on the output dot, D1, and gure 33(c) shows the calculatedshift of the honeycom b border as a f unction of the driver gate voltages. Figure 33(b)and (c) demonstrate that with a proper setting of the working point of the doubledot, and a temperature of 0 K, the input dots are able to produce sucient shift to

switch an electron from one output dot to the other, as evidenced by the abruptpotential changes in gure 33( d). Since the match between the calculated and experi-mental honeycomb shift is very good, we believe that the data of gure 33( b), inconjunction with the dierential detector data, conrms that we are able to control

the position of the electron on the output double-dot with a single electron on the

input dots. However, if the voltage range of the drivers is too large, the parasitic

coupling from the drivers will pull the working point away from the border so that

the shift from the electron transfer is unable to cause a border crossing, and the

electron population on the double-dot remains unchanged. This eect corresponds

to the monotonic tilt of the sawtooth curve in gure 33( c).We have experimentally demonstrated the rst switching of a single electron

between dots controlled by a single electron. This control is shown by a honeycomb

border shift induced by changes in the charge conguration of the coupled input

dots. The observed shift is in excellent agreement with calculations, and a compari-

son of the shift with the electrometer measurement shows that the border shift

induces an electron switch which is nearly complete. Future devices will be optimized

by an increased coupling of the input and out

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