Quantum-dot cellular automata: computing with coupled ... - CiteSeerX

INT. J. ELECTR ONICS ,

1999,

VOL .

86,

NO.

5, 549± 590

Quantum-dot cellular automata: computing with coupled quantu m dots WOLFGANG POROD ² ³ , CR A IG S. LENT ² , GAR Y H. BER NSTEIN ² , A LEX EI O. ORLOV ² , ISLA MSHA H A MLA NI² , GREGORY L. SNIDER ² , and JA MES L. MER Z² We discuss novel nanoelectronic architecture paradigms based on cells composed of coupled quantum-dots. Boolean logic f unctions may be implemented in speci® c arrays of cells representing binary information, the so-called quantum-dot cellular automata ( QCA ) . Cells may also be viewed as carrying analogue information and we outline a network-theoretic description of such quantum-dot nonlinear networks (Q-CNN ) . In addition, we discuss possible realizations of these structures in a variety of semiconductor systems ( including GaA s/A lGaA s, Si/SiGe, and Si/SiO2 ) , rings of metallic tunnel junctions, and candidates f or molecular implementations. We report the experimental demonstration of all the necessary elements of a QCA cell, including direct measurement of the charge polarization of a double-dot system, and direct control of the polarization of those dots via single electron transitions in driver dots. Our experiments are the ® rst demonstration 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 Moore’ s famous dictum remarkably closely for more than three decades. However, there are indications now that this progress will slow, or even come to a standstill, as technological 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 NanoDevice s Group on a possible future nanoelectron ic computing technology based on cells of coupled quantum dots. More speci® cally, we envision nanostructur es where inf ormation is encoded by the arrangement of single electrons. Now, 100 years af ter 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. A mong the chief technological limitations responsible for this expected slowdown of silicon technology are the interconnect problem and power dissipation (Ferry et al. 1987, 1988, Keyes 1987, Bohr 1996 ) . A s more and more devices are packed 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 e ect of wire resistance and capacitance, giving rise to a wiring bottleneck. It is generally recognized that alternate approaches are needed to create innovative technologies that provide greater device and R eceived 1 June 1997. A ccepted 30 October 1998. ² Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA . ³ Corresponding author. International Journal of Electronics ISSN 0020± 7217 print/ISSN 1362± 3060 online http://www.tandf .co.uk/JNLS/etn.htm http://www.taylorandfrancis.com/JNLS/etn.htm

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interconnect functionality, or to utilize innovative circuit and system architectural features that provide more function per transistor (Semiconductor Industry A ssociation 1994 ) . However, these obstacles for silicon circuitry may present an opportunity 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 nanostructu res (more speci® cally, quantum dots ) which are arranged in locally-interc onnected cellular-autom ata-like arrays. We will demonstrate that suitably designed structures, the so-called `quantum-dot cellular automata’ (QCA ) (Lent et al. 1993 b) , may be used for computation and signal processing. The fundamental idea for QCA operation is to encode inf ormation using the charge con® guration 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 inf ormation has been encoded by current switches. It works well so long as the switched current from one element can be transf ormed into the control voltage for another element. A s device sizes shrink to the molecular limit, the current-switching paradigm falters for three primary reasons: (1 ) the nano-switch becomes leaky, making a hard OFF-state di cult to maintain; (2 ) the switched current is so small in magnitude that it is hard for it to activate the next device; and (3 ) the interconnects which carry current from one device to another begin to dominate the performance. The QCA approach eliminates these problems by adopting an approach to coding inf ormation which is more naturally suited to nanostructu res. Our work is based on the highly advanced state-of -the-art in the ® eld of nanostructures and the emerging technology of quantum-do t fabrication (Capasso 1990, Weisbuch and Vinter 1991, Kelly 1995, Turton 1995, Montemerlo et al. 1996 ) . A s schematically shown in ® gure 1, several groups have demonstrated that electrons

Figure 1.

Schematic diagram of arti® cial `quantum-dot atoms’ and `quantum-dot molecules’ which are occupied by few electrons.

Q uantum -dot cellular autom ata

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may be completely con® ned in semiconductor nanostructu res, which may then be thought of as arti® cial `semiconductor atoms’ (R eed 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, A shoori 1996 ) . Very recently, coupling between quantum-dot atoms in close proximity has been observed, thus realizing arti® cial `quantum-dot molecules’ (Ho mann 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 quantum mechanical phase coherence over the entire array; phase coherence is only required inside each cell, and the cell± cell 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 circuit architecture ideas for these new devices. The nanostructur es we envision will contain only few electrons available for conduction. It is hard to imagine how devices based on nanostructu res 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 nanoelectron ic architecture where the inf ormation is contained in the arrangem ent of charges and not in the ¯ ow of charges (i.e. current ) . In other words, the devices interact by direct Coulomb coupling 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 interaction. Figure 2 schematically shows a locally-interc onnected array consisting of cells of nanoelectro nic devices. The physical interactions together with the array topology determine the overall functionality. What form must a cellular array take when its dynamics should result directly from known physical interactions? If we simply

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 interact, 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 functionality does one obtain for a given physical structure?’ and `Given a certain array behaviour, there a physical system to implement it?’ In the following sections we will develop these ideas in detail and we will present a concrete example of a quantum-do t cell with an appropriate architecture, the socalled quantum-do t cellular automata. We will discuss how one may construct QCA cells that encode binary inf ormation 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, are implementation s. We will discuss ideas (and on-going work ) for attempting to implement these structures in a variety of semiconductor systems (including GaA s/ A lGaA s, Si/SiGe, and Si/SiO 2 ) and also metallic dots. A lternative implem entations include molecular structures. We will call attention to a speci® c 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 nanoelectron ics technology may be, many fundam ental 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 e orts of technologists , device physicists, circuits-and-systems theorists and computer architects.

2.

Quantum-dot cellular automata

Based upon the emerging technology of quantum-do t fabrication, the Notre Dame NanoDevice s 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 device± device coupling might be utilized in a cellular-autom ata scheme, alas, without an accompanyin g proposal for a speci® c implem entation (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 interaction and quantum -mechanical tunnelling. 2.1. A quantum -dot cell The quantum-do t cellular automata (QCA ) scheme is based on a cell which contains four quantum dots (Lent et al. 1993 a ) , as schematically shown in ® gure 4( a ) . The quantum dots are shown as the open circles which represent the con® ning electronic potential. In the ideal case, each cell is occupied by two electrons, 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


Figure 3.

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Each cell in the array interacts with the ènvironment’ which includes the Coulomb interaction with neighbouring cells.

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

mechanical tunnelling. Tunnelling is possible on the nanometer scale when the electronic wavef unction su ciently `leaks’ out of the con® ning potential of each dot, and the rate of these jumps may be controlled during fabrication by the physical separation between neighbouring dots. This quantum-dot cell represents an interesting dynam ical system. The two electrons experience their mutual Coulombic repulsion, yet they are constrained to occupy the quantum dots. If left alone, they will seek, by hopping between the dots, the con® guration corresponding to the physical ground state of the cell. It is clear that the two electrons will tend to occupy di erent dots because of the Coulomb energy cost associated with bringing them together in close proxim ity 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 con® gurations with electrons at opposite corners, as shown in ® gure 4( b ) .

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We may associate a `polarization’ with a speci® c arrangement of the two electrons 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 give P = 1 (compare ® gure 4 ) . A ny polarization between these two extreme values is possible, corresponding to con® gurations 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 con® gurations, and therefore has a net polarization of zero. As described in the literature, this cell has been studied by solving the SchroÈ dinger equation using a quantum mechanical model Hamiltonian (Tougaw et al. 1993 ) . We do not need to concern ourselves here with the details, but su ce it 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-occup ied dot; (4 ) the Coulomb interaction between electrons in the same cell and also with those in neighbouring cells. The solution of the SchroÈ dinger equation, using cell parameters for an experimentally reasonable model, con® rms the intuitive understandi ng that the ground state is a superpositio n of the P = + 1 and P = 1 states. In addition to the ground state, the Hamiltonian model yields excited states and cell dynamics.

2.2. Cell± cell 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 electrons are allowed to tunnel between the dots in the same cell, but not between di erent cells. Since the tunnelling probabilities decay exponentially with distance, this can be achieved by having a larger dot± dot distance between cells than within the same cell. Coupling between the two cells is provided by the Coulomb interaction between the electrons in di erent cells. Figure 5 shows how one cell is in¯ uenced by the state of its neighbour. The inset shows two cells where the polarization of cell 1 ( P1 ) is determined by the polarization of its neighbour ( P2 ). The polarization of cell 2 is presumed to be ® xed at a given value, correspondin g to a certain arrangement of charges in cell 2, and this charge distribution exerts its in¯ uence on cell 1, thus determining its polarization P1 . The important ® nding here is the strongly nonlinear nature of the cell± cell coupling. A s 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 almost perfect polarization in cell 1, i.e. P1 = 0.99. In other words, even a small asymmetry of charge in cell 2 is su cient to break the degeneracy of the two basic states in cell 1 by energetically favouring one con® guration over the other. The abruptness of the cell± cell response function depends upon the ratio of the strength of the tunnelling energy to the Coulomb energy for electrons on neighbouring sites. This re¯ ects a competition between the kinetic and potential energy of the


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Figure 5. The cell± cell response. The polarization of cell 2 is ® xed and its Coulombic e ect on the polarization of cell 1 is measured. The nonlinearity and bistable saturation of this response serves the same role as gain in a conventional digital circuit.

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 conventional circuitsÐ restoring signal levels after each stage. Note that no power dissipation is required in this case. These general conclusions regarding cell behaviour and cell± cell coupling are not speci® c 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. Q CA logic Based upon the bistable behaviour of the cell± cell coupling, the cell polarization can be used to encode binary inf ormation. 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 unpolarized in the absence of a symmetry-breaking in¯ uence that would favour one of the basis states over the other. Each ® gure shows the cell polarizations correspondin g to the physical ground state con® guration of the whole array. Figure 6( a ) shows that a line of cells allows the propagation of inf ormation, thus realizing a binary wire (Lent and Tougaw 1993 ) . Note that only in information but no electric current ¯ ows down the line, which results in low power dissipation. Inf ormation can also ¯ ow around corners, as shown in ® gure 6( b ) , and fan-out is

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

possible, shown in ® gure 6(c) . Cells which are positioned diagonally from each other tend to anti-align. This feature is employed to construct an inverter as shown in ® gure 6( d) . In each case, electronic motion is con® ned to within a given cell, but not between di erent cells. Only informatio n, and not charge, is allowed to propagate over the whole array. These quantum-do t cells are an example of quantum-f unctional devices. Utilizing quantum-m echanical e ects 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 di erence 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

Figure 7. Majority logic gate. The basic structure simply consists of an intersection of lines. A lso 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 signi® cantly 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 in the 1 or 0 state. In this way, a reduced majority logic gate can also serve as a programmable A ND/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 A ND 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 A ND 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 A ND/OR functionalit y ensures that QCA devices provide logical completeness. A s 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 implemented 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 the inputs.

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 veri® es that it yields the correct ground state output for all eight possible combinations of the three inputs (Tougaw and Lent 1994 ) . The general principles for computing with such QCA arrays will be discussed below . 2.4. Com puting with Q CAs 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 computation al problem. The physical problem can be stated as follows: given the boundary conditions imposed by the input, what is the lowest energy con® guration 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 -speci® c features, let us brie¯ y 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 accomplished 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 electrode’ , to alter electron occupancy of a dot. R eading an output state is more di cult. We require the ability to sense the charge state of a dot without having the measurement process alter the charge state. Since the local charge produces a local electrostatic potential, this is really a question of constructing a small electrometer. Fortunately, electrometers made from ballistic point-contac ts and from isolated

Figure 10. Schematic representation of computing with a QCA array. The key concepts are `computing with the ground state’ and èdge-driven computation’ described in the text.


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dots themselves have already been demonstrated, both in metallic and semiconduc tor systems (BazaÂ n et al. 1996, Bernstein et al. 1996 ) . These electrometers can noninvasively measure the charge state of a single dot. Note that input and output are only performed at the edges of the array; no inf ormation 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 inf ormation (edge-driven computation ) . (2 ) Next, the whole array is allowed to relax (or is adiabatically transf ormed ) to the 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 of those 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 computation al problem . The two key features which characterize this new computing paradigm are `computing with the ground state’ and èdge-driven computation ’ which we discuss in further detail below. 2.4.1. Com puting 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 con® guration, compatible with the boundary conditions given by the ® xed input cells. Note that the inf ormation is contained in the ground state itself , and not in how the ground state is reached. This relegates the question of the dynam ics of the computation to one of secondary importance; although it is of signi® cance, 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 com puting. 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 unavoidable coupling to the environm ent will relax the system from the initial excited state to the new ground state. The actual dynam ics will be tremendously eomplicated since all the details of the system-environm ent 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 feature 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 com puting. Due to the above di culties associated with metastable 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 transf ormed 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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Schematic representation of (a ) relaxation computing and (b ) adiabatic computing.

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 operation possible, and pipelined architectures have been proposed (Lent et al. 1994, Lent and Tougaw 1996, 1997 ). 2.4.2. Edge-driven com putation. Edge-driven computation means that only the periphery of a QCA array can be contacted, which is used to write the input and to read 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 interior of an array. A ll interior cells interact only within their local neighbourhoo d. The absence of signal and power lines to each and every interior cell has obvious bene® ts for the interconnect problem and heat dissipation. The lack of direct contact to the interior cells also has prof ound 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-f rom-equilibr ium state. Since no external signals are brought to the inside, internal cells cannot be in¯ uenced directly. These are the reasons why the ground state of the whole array is used to represent the inf ormation, as opposed to the states of each individual cell. In fact, edge-driven computation necessitates computing with the ground state! Conventional circuits, on the other hand, maintain devices in a far-f rom-equilibrium 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 ) .


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2.5. Switching of Q CA arrays A s discussed in the previous section, quantum-do t cellular automata take advantage 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 architectureÐ the details of the evolution of the system, which may be hard to control, are not essential in getting the computation 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 system’ s dynamics is necessary to quantif y the switching speed of QCA arrays. Second, as has been pointed out by Landauer, the presence of metastable states could cause a signi® cant delay in the system reaching its new ground state, so the identi® cation 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 suddenly 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 simultaneous ly 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 assum ed that the interdot potential barrier was modulated simultaneous ly for all cells in the array. From the point of view of fabrication complexity, this is an important feature. It permits one conductor, typically one gate electrode, to control the barriers of all cells. If each cell had to be separately timed and controlled, the wiring problem introduced could easily overwhelm the simpli® cation won by the inherent local interconnectivity of the QCA architecture itself . We can gain signi® cant advantage, however, by relaxing this requirement slightly. If we subdivide an array of cells into subarrays, we can partition the computational 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 all the 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 in¯ uence the calculation. A n 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 bene® ts. The most obvious bene® t 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 inf ormation. This allows the device to be in the process of carrying out several

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Figure 12. Cell± cell response f unctions f or various temperatures. (a ) Semiconductor cell with minimum lithographic feature sizes of 20 nm and (b ) molecular implementation with dimensions of 2 nm.

calculations at once, especially if the pipeline is long. Such simultaneou s calculation stages maximize the throughput of each total system. A second bene® t of this system is that the number of cells in each sub-array can be kept well below the thermodynamic 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. Therm odynam ic consideration s 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 interf ere 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 lithography (for minimum feature sizes of 20 nm ) is expected to operate at cryogenic temperatures, whereas a molecular implementation (for feature sizes of 2 nm ) would work at room temperature, as illustrated in ® gure 12. Thermodynamic consideration s 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. A s the size of the system increases, so do the number of possible error con® gurations. 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 excitation energy to the thermal energy (Lent et al. 1994 ) . For example, arrays with about 20 000 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 informatio n as described in the previous section, these cells may also be used in an analogue mode. A s


Figure 13.

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Schematic view of a locally-connected cellular QCA array. The circle indicates the range of interaction f or the central cell.

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 nonlinear dynamical system is similar to cellular nonlinear (or, neural ) networks (CNN ) , which are locally-interc onnected structures implemented using conventional circuitry (Chua and Y ang 1988 ) . Each cell is described by appropriate state variables, and the dynamics of the whole array is given by the dynamical law for each cell, which includes the in¯ uence 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 transformed 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 quantum degrees 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 physics of the intercellular interactions; and (3 ) the state equations are derived from the time-dependent SchroÈ dinger equation; one state equation exists for each classical and quantum degree of freedom . For the case of a two-dim ensional array, each Q-CNN cell possesses an equivalent CNN-cell model described by the di erential equations given below. We may thus think of such a quantum-dot cell array as a special case of cellular nonlinear networks (Nossek and R oska 1993 ) . The equivalent circuit describing a cell is composed of two linear capacitors, four nonlinear controlled sources and eight linear controlled sources representing the interactions between the cell and its eight neighbours (Csurgay 1996 ) . Our quantum model is a special case of a general formalism for `quantum networks’ developed by Mahler (1995 ) . A s schematically shown in ® gure 14, a quantum network consists of subsystems which are special quantum objects denoted as `network nodes’ and the interaction channels between them are denoted as `network edges’ . For the study of such systems, Mahler has developed a density

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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’ ( af ter Mahler 1995 ) .

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 environm ent. This formalism is su ciently general to also include dissipation, but we have not yet incorporated that in our Q-CNN models; this work is in progress.

3.1. Q uantum m odel of cell array Following the work of Toth et al. (1996 ) , we describe the quantum state in each cell using the two basis states j u 1 i and j u 2 i which are completely polarized. j

C i

= a ju

+ b ju

1i

2i

Within this two-state model, each property of a cell is completely speci® ed by the quantum mechanical amplitudes a and b . In particular, P, the cell polarization is given by

P= ja

2 j

j

b j

2

The Coulomb interaction between adjacent cells increases the energy of the con® guration if the two cell polarizations di er. This can be accounted for by an energy shif t corresponding to the weighted sum of the neighbouring polarizations, which we denote by PE. The cell dynam ics is then given by the SchroÈ dinger equation

ihd / d tj C i

= Hj C i

where H represents the cell Hamiltonian. Once the Hamiltonian is speci® ed, the cell dynamics is completely determined. j C 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.

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Q uantum -dot cellular autom ata 3.2. Form ulating CNN-like quantum dynam ics

In order to transf orm the quantum mechanical description of an array into a CNN-style description, we perform a transform ation from the quantum-m echanical state variables to a set of state variables which contains the classical cell polarization, P, and a quantum mechanical phase angle, u j

C i

= (a , b )

! j

C i

= ( P, u )

This transf ormation is accomplished by the relations 1/ 2

a = [( 1 + P) / 2]

1/ 2 iu

b = [( 1 P) / 2]

e

With this, the dynamical equations derived from the SchroÈ dinger equation can be rewritten as CNN-like dynamical equations for the new state variables P and u 1/ 2 h d / d t P = 2 g sin u ( 1 P2 ) 1/ 2 h d / d t u = PE + 2g cos u P/ ( 1 P2 )

The term PE accounts for the cell± cell interaction and g is the tunnelling matrix element between dots. For a more detailed derivation, we refer to the original 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 ) 1/ 2 V ( P, u ) = 2 g cos u ( 1 P2 ) + PPE

3.3. Cellular network m odel 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 equilibrium is changed in time, a dynam ics of the polarizations and phases for all cells in the whole array is launched. In the framework of the CNN model, groundstate 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-pheno mena (Chua 1995 ) . A signi® cant literature exists on this subject, and di erent classes of wave behaviour and pattern formation have been identi® ed (Zuse 1969, Porod et al. 1996 ) . In complete analogy, spatio-temporal wave-behavio ur 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 wave front. The ® gure shows snapshots of the classical polarization P and the quantum mechanical phase angle u at various times. Note that the inf ormation about the direction of propagation 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

Figure 15.

Wavefront motion in a linear Q-CNN array.

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Figure 16.

Examples of wave-like excitation patterns in two-dimensional Q-CNN arrays. The top panel shows concentric waves due to a periodic modulation of the boundaries. The bottom panel shows a spiral wave due to cyclical modulation of the boundaries.

Q uantum -dot cellular autom ata 567

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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 an example of cyclical excitations at the boundaries which give rise to a spiral wave. The excitation at each of the four boundaries are 90ë out of phase with respect to their neighbouring edge.

4.

Possible quantu m-dot cell implementations

In this section, we discuss possible implementations of the coupled-quan tum-dot cells discussed so far. Based upon the reported studies in the literature on singleelectron charging and dot± dot 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 that work, including the results and implications of our numerical modelling. We also discuss a candidate molecule which might serve as a prototype molecular electronics implementation of a QCA cell. 4.1. Gate-controlled quantum dots The fabrication of a QCA cell by split-gate technology is a challenging 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 electrostatic con® nement provided by a top metallic electrode (Taylor 1994 ) . The key implementation challenges are (i ) to gain su cient gate control in order to de® ne quantum dots in the few-electron regime, and (ii ) to place these dots su ciently close to each other in order to make coupling possible. Using these techniques, it is conceivable that coupled-dot cells may be realized in a variety of materials systems, such as III± V compound semiconductors, Si/SiGe heterolayers , and Si/SiO 2 structures. In order to achieve a crisp con® ning potential, it is important to minimize the e ects of fringing ® elds, which may be accomplished by bringing the electrons as close as possible to the top surf ace. This design strategy of `trading mobility versus gate control’ by utilizing near-surf ace 2DEGs has been pioneered by Snider et al. (1991 ) . However, the resultant proxim ity of the quantum dot to the surface raises the question of the e ect of the exposed surf ace on the quantum con® nement. To study these questions, we have undertaken extensive numerical modelling of such gatecontrolled dots, and we have explictly included the in¯ uence of surf ace states which are occupied, in a self -consistent fashion, according to the local electrostatic potential (Chen and Porod 1995 ) .

Figure 17.

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


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

We have performed numerical simulations for the design of quantum dot structures in the few-electron regime, both in the A lGaA s/GaA s and Si/SiO 2 material systems. The con® ning 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 SchroÈ dinger equation. Our model takes into account the e ect of surf ace states by viewing the exposed surf ace as the interf ace between the semiconductor and air (or vacuum ) . Figure 18 schematically shows the simulation strategy, where we employ a ® nite-element technique for the semiconduc tor domain and a boundary element method for the dielectric above (Chen et al. 1994 ) . Both domains are coupled at the exposed surf ace, taking into account the e ect of charged surf ace states. This is particularly important for modelling the III± V material system, where surface states are known to be signi® cant. Our modelling shows that the single most critical parameter for the design of gate-control led dots is the proximity of the 2DEG to the top metallic gates. This distance is limited in the III± V material system by the `leakiness’ of the layer separating 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 insulating 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 con® gurations and biasing modes. Our simulations show that the number of electrons can be e ectively 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 m aterial system . Figure 19 shows an example of the occupation of quantum dots for combined enhancem ent/depletion mode biasing on an A lGaA s/GaA s 2DEG. The main idea is to negatively bias the outer electrode

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Figure 19.

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

(gate 2 ) such that the 2D electron density is depleted, or near depletion; a positive bias on the inner electrode (gate 1 ) is then utilized to induce the quantum dot and to control its occupation. In this example, we have chosen a radius of rG1 = 6 nm for the centre enhancem ent gate, and a radius of rG2 = 50 nm for the surrounding depletion gate. The resulting number of electrons induced by three di erent voltages on the depletion gate, V G2 , is plotted as a function of the enhancement gate bias voltage, V G1 . We see that variations of the depletion-gat e bias of 10 mV will result in threshold-vol tage variations of as much as 80 mV. This biasing mode appears to be an e ective way of controlling the quantum-do t threshold voltage in the few-electron regime. The use of a single, negatively-bia sed depletion gate would not su ce for our purposes. Even though it would be possible to obtain few-electron dots, the resulting potential variations are too gradual to allow fabrication of a QCA cell with closelyspaced dots (Chen and Porod 1995 ) . 4.1.2. Si/S iO 2 m aterial system . We have also performed numerical simulations for the design of gated few-electron quantum dot structures in the Si/SiO 2 material system. The motivation for this work has been to investigate the feasibility of transf erring the emerging technology of quantum dot fabrication from the III± V semiconductors, where it was pioneered over the past few years, to the technologically more important Si/SiO 2 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/SiO 2 interf ace. A nother advantage of silicon for quantum dot applications appears to be the higher e ective mass, as compared to the III± V materials, which reduces the sensitivity of the energy levels to size ¯ uctuations.


Figure 20.

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Potential variation at the silicon/silicon dioxide interface as a function of oxide thickness.

Quantum dots may be realized by applying a positive bias to a metallic gate on the surf ace, 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 formation of an èlectron droplet’ at the silicon/oxide interface, i.e. a quantum dot. Figure 20 shows, for an applied gate bias of 1.7 V, the correspondin g potential variations along the Si/SiO 2 interf ace. A n electronic system is induced when the silicon conduction band edge at the oxide interf ace, 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 in the 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, and the 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, which results in a top electrode which may reach the semiconductor surf ace inside the openings, providing the enhancement gates. 4.2. Rings of m etallic tunnel junctions In addition to the semiconductor systems discussed above, single-electron tunnelling phenomena may also be observed in metallic tunnel junctions (Fulton and Dolan 1987, Grabert and Devoret 1992 ) . Consider a ring of metallic tunnel junctions, schematically shown in ® gure 22( a ). The tunnel junctions are represented by

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Figure 21.

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

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 con® gurations as the semiconductor cell discussed above. In addition, the cell± cell coupling also shows the same strongly nonlinear saturating characteristic (Lent and Tougaw 1994 ) . Note that

Figure 22. Possible QCA implementation using rings of metallic tunnel junctions. ( a) Basic cell, (b ) cell occupied by two additional electrons, (c) line of capacitively coupled cells.


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cell± cell coupling is purely capacitive, as schematically shown for the line of cells in ® gure 22( c) . The metallic tunnel-juncti on cell may be used as a building block for more complicated structures, in a fashion completely analogous to the semiconduc tor implem entations. Using the technique of shadow evaporation, coupled-dot structures have recently been fabricated and tested in the alum inium-alum inium oxide material system. These experim ents will be described in more detail in a separate section below. 4.3. Possible m olecular im plem entation A s discussed above, QCA room temperature operation would require molecularscale implementations of the basic cell. Molecular chemistry promises to o er the versatility for the desired miniaturizatio n. The requirements for a molecular QCA technology include: (1 ) cells made of a rigid array of identical clusters with inter-cell interactions that are insulating (e.g. a square arrangement of four clusters ); (2 ) two-electron occupation of each cell with distinct arrangements of the two charges; (3 ) charge interchange between these distinct arrangements, which are energetically equal in the absence of a polarizing ® eld; (4 ) patterning of cells into prede® ned 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 crystallograp hically characterized (Cen et al. 1992, 1993 ) . A s schematically illustrated in ® gure 23, these molecular substances with the formula M 2 {(CO) 9 Co 3 CCO2 }4 , where M = Mo, Mn, Fe, Co, Cu, consist of square 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 the clusters attach themselves in the plane perpendicula r to this axis. It has been demonstrated that these compounds may be obtained as pure crystalline 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 temperature operation. The spectroscopic properties demonstrate intra-cell cluster-core electronic communication and inter-cell interactions are insulating.

5.

Experimental demonstration of QCA elements

A s 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

Figure 23.

Candidate for a molecular QCA cell consisting of 4 outer (CO) 9 Co 3 C clusters which are arranged in a plane around a `spindle’ constituted by the M2 core.



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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 in¯ uenced totally by the single electron polarization of nearest neighbour-ce lls. 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 single electron transistor (SET ) can be used to detect charge variation in a nearby dot (Laf arge et al. 1991, Bazan et al. 1996 ) . In previous experim ents, the Coulomb interaction of electrons within a double-dot has been inf erred 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-dot externally, and with high sensitivity, has not heretof ore been developed. 5.1. Measurem ent of double-dot polarization In this section, we present direct measurement of the internal charge state of a metal/oxide double-dot system. Speci® cally, our charge detection technique is sensitive 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 alum inium oxide barriers which allow electrons to tunnel between the sections. Figure 24 shows a scanning electron micrograph (SEM ) of an SET. The 2 area of the oxide barriers (i.e. the overlap of lines ) is small (about 60 60 nm ) , resulting in correspondin gly small capacitances between the sections. A centre island

Figure 24. Field emission scanning electron micrograph of a single electron tunnelling transistor f abricated at Notre Dame. The gate electrode is to the left, and the two tunnel junctions are at both ends of the `dot’ . The sizes of the tunnel junctions are approximately 60 60 nm 2 .

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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 lithography (EBL) and developed to form the desired pattern. (The resist layers are chosen to result in a large degree of undercut, allowing the tilting procedure described below. ) Second, the waf er is coated in a high-vacuum chamber by a thin alum inium ® lm which reaches the surf ace of the waf er only where the resist has been developed, i.e. removed. (The remaining aluminium rests on the surf ace of the resist to be removed in the ® nal step. ) Next, the aluminium is exposed in situ to a controlled amount of oxygen in order to oxidize its surf ace. Since this is a self-limiting process, the ® nal thickness of the oxide is well-controlle d. The wafer is then tilted so that a second evaporated aluminium pattern is translated laterally from the ® rst, but overlapping in 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 A l 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 surf ace 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

Figure 25. Schematic diagram of the device structure. Capacitance parameters of di erent parts of the device are listed in table 1. The capacitances of the coupling capacitors C11 and C22 are approximately 10% of the total capacitances of the electrometers. The circuit used to compensate f or parasitic capacitance between the driver gates V A / V B and the electrometer islands E1/E2 is not shown.


577

to an SET electrometer. A n 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 1.4 m m 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 1.1 m m. Figure 26 is an SEM of the cell depicted in ® gure 25. The typical tunnel resistance of a junction, based on I± V measurements of the electrometers at 4.2 K, was approximate ly 200 k V . The total capacitance of the electrometers, CS , extracted from the charging energy ( EC 80 meV) , was approximate ly 1 fF. Measurements were carried out at the base temperature of our dilution refrigerator ( 10 mK ) using standard ac lock-in techniques. A 4 mV excitation voltage at a 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 superconduc tivity of A l. Initial experiments were performed to extract the lithographic and parasitic capacitance parameters of the di erent 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 is the double-dot structure, and on the right are the two SET electrometers. The electrometers are coupled to the double-dot structure via interdigitated capacitors.

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W . Porod et al. Capacitance C junction C A - D1 C B- D2 C C- E1 C D- E2 C D1- E1 C D2- E2 C A - E1 C A - E2 C B- E1 C B- E2 C C- E2 C D- E1 C E1- E2 C A - D2 C B- D1 C D1- E2 C D2- E1 Table

Type

A pproximate capacitance (aF )

Lithographic Lithographic Lithographic Lithographic Lithographic Lithographic Lithographic Parasitic Parasitic Parasitic Parasitic Parasitic Parasitic Parasitic Parasitic Parasitic Parasitic Parasitic

400 47.7 49.5 29 26.6 106 106 21 8 9.7 21.3 7.5 7.5 36.8 6.4 6.4 16 16

1. Lithographic and parasitic capacitances between various gates and islands shown in ® gure 24, measured from the period of the Coulomb blockade oscillations.

oscillations (CBO) (A verin and Likharev 1991 ) . Table 1 shows the extracted system capacitances. 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 electrometers were simultaneous ly measured. The sensitivities of the electrometers, as expected from Laf arge et al. (1991 ) , were proportional to CD1 - E1 and CD2- E2 . A lso, 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 su ciently small fraction of 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 in¯ uence of the parasitic capacitances CA - E1 , CA - E2 , CB- E1 , and CB- E2 , we applied inverted compensation voltages proportional to V A and V B to gates C and D. Using this charge compensation technique, we were able to reduce the e ect 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, V A and V B . The resulting charging diagram of such a measurement forms a `honeycomb’ structure ® rst observed by Pothier et al. (1992 ) . The honeycomb boundaries (solid lines ) represent the regions where a change in electron population ( n1 , n2 ) occurs on one or both of the dots, with n1 and n2 representing excess populations of D1 and D2, respectively. Each hexagonal cell marks a region in which a given charge con® guration is stable due to Coulomb blockade. In the interior of the cell there is no charge transport through the


579

Figure 27. Charging diagram of the double-dot as a function of driver gate voltages. Charge con® gurations ( n1 , n2 ) , which represent the number of extra electrons on D1 and D2, respectively, are arbitrarily chosen. Lines labelled I, II and III show a f ew directions in which charge can shift between di erent con® gurations of the double-dot.

double-dot; conductance through the double-dot peaks only at the `triple points’ , depicted by a convergence of contour lines, where the Coulom b blockade is lif ted for both dots. The charge con® guration of the double-dot can be varied by sweeping gate voltages A and B along any of the three directions shown in ® gure 27, and will not result in a current ¯ ow through the double-dot. For instance, along directions I and II, charge is added to only one of the dots in units of single electrons, while the population of the other dot stays constant. Charge redistributio n in the double-dot (line III) takes place when the driver gates are swept in opposite directions. A long direction III, electrons are shif ted from one dot to the other while the total charge on the double-dot remains unchanged. Figure 28 shows grey-scale contour plots of the conductance through SET electrometers E1 [® gure 28( a)] and E2 [® gure 28( b )] as a function of the driver gate voltages, where light areas represent higher conductance . Superimposed on each plot are the solid lines that de® ne the honeycomb structure of ® gure 27. The change in the conductance of each electrometer re¯ ects the variation of the electrostatic potential in the dot capacitively coupled to it. A sharp change in the conductance of E1 from light to dark in the horizontal direction [® gure 28( a)] represents addition

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(a )

(b )

Figure 28. Charging diagram of the electrometers as a f unction of the same driver gate voltages shown in ® gure 2 with the honeycomb boundaries of ® gure 4 superimposed. ( a) Charging diagram of E1. Sharp transitions in the horizontal direction indicate a change in the population of D1. ( b ) Charging diagram of E2. Sharp transitions in the vertical direction re¯ ect a change in the population of D2.


581

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, the sharpest 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 redistributio n in the double-dot shown by direction III in ® gure 27, but as seen in ® gure 28, the transition along this direction is weaker than the others. During charge redistributio n in the double-dot (line III in ® gure 27 ), the population of each dot simultaneou sly changes by 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 ë . The cross capacitance between D1 (D2 ) and E2 (E1 ) causes charging of each dot of the series double-dot to be seen in 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 ) overlaid on a grey-scale contour plot of a di erential signal, ( G1 - G2 ) , where G1 and G2 are the conductances of E1 and E2, respectively. A long the directions I and II in ® gure 27, the signals of E1 and E2 have the same phase, giving a suppressed di erential signal. The most conspicuous transition, represented by a higher density of contour lines,

Figure 29. Di erential signal obtained f rom the charging diagrams of the individual electrometers. 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 of an electron from one dot to the other. A s mentioned above, this is due to the phase di erence (180 ë ) in the signals of the individual electrometers, yielding a di erential signal which is approximate ly twice as strong as the one detected by a single electrometer. Our device architecture allows us to directly observe the internal state of a double-dot system by detecting all possible charge transitions of a single electron. A di erential detector that utilizes the signals from both electrometers is most sensitive to the charge redistributio n in the double-dot. With this investigation, we demonstrate that our di erential 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 of the 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 a switch of the other electron in its double-dot (the output double-dot ) . This is only possible 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 honeycom b border (thick line ) of ® gure 30, thus switching its electron con® guration. Therefore, to produce a functioning QCA cell, the gate biases for each doubledot 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 in® uence of electrostatic potential from a single electron in the input double-dot. This switching will be re¯ ected 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 shif t in a double-dot caused by single electron charging of adjacent dots. We compare these experimental results with the results of the di erential detector of the previous section, and with theoretical calculations to con® rm that this shif t 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 investigate how these single electron drivers a ect 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 a ecting 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 cancellation, 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 full four-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


583

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

di erent. 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 con® gured 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 simultaneous ly. A switch in electrons between the input dots will cause a shift in position of the conductance peaks of the output double-dot (and the entire honeycomb ) seen in ® gure 30. In order to observe this honeycomb border shif t, `snapshots’ of the honeycomb are taken for di erent `push-pull’ (+ V C = V D ) settings on the driver gates. It is well known (Laf arge et al. 1991 ) that the potential on a metal dot in the Coulomb blockade regime changes linearly as a function of gate voltage with an abrupt shif t when the electron population of the dot changes, resulting in sawtooth oscillations. We indeed observe a slow shif t of the honeycomb border correspondin g to the gradual increase of the potential on the input dots, followed by an abrupt `reset’ .

Figure 31.

Charging diagram of the output double-dot with (a ) V C = VD = 0.67 mV, and ( b) V C = V D = + 0.67 mV. These present the maximum shif t of the border between the (0, 1 ) and ( 1, 0) states.



585

Figure 31 shows two snapshots taken at the extremes of the shif t, with a measured shif t of approxim ately 440 m V. To demonstrate that the observed shif t is due to single electron charging of the input dots, and not to parasitic coupling from the driver gates, we directly measure the shif t of the honeycomb border while sweeping gates C and D over su cient voltage that several electron transf ers occur in the input dots. The shif t from electron charging will be periodic, since the potential on the input dots resets at each electron transf er, while parasitic coupling will add a monotonic term to the border shift. To detect the honeycomb border, we sweep the double-dot gates A and B along the diagonal E in ® gure 30. A t ® nite temperature, the conductance of a double-dot shows a peak as we sweep VA and V B , and the position of this peak marks the border. Here we de® ne the `diagonal voltage’ as the change in voltage along the direction E, with D V A = D V B . For each setting of the driver gates we sweep the 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 V B ) and driver (+ V C and

Figure 32. Contour plot of the conductance through the double-dot as a function of the di erential driver voltages V C = V D and the diagonal voltage. The honeycomb border is marked by the peak in conductance, and the black circles mark the calculated position of the border.

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V D ) voltages, representing the position of the honeycomb border with respect to a ® xed working point de® ned as V diagonal = 0. The solid circles in ® gure 32 plot the theoretical 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 an observed shif t. The somewhat smaller amplitude of the theoretical honeycomb shif t is most likely due to uncertainty in the measured junction capacitance ( 15% ) . The sawtooth oscillations of ® gure 32 occur each time an electron is added to one and removed from the other input dot, con® rming that the shif t is due to a single electron transf er. If the working point on the honeycomb pattern of ® gure 30, ( V A , V B ) , is chosen close to a border, the shif t will cause a border crossing when electron exchange occurs within the input dots. To con® rm that the border shif t is indicative of an electron transf er in the output double-dot, we compare the border shif t observed in ® gure 32 with the data from the di erential detector in ® gure 29, where a diagonal voltage change of approxim ately 600 m V is required for an electron to complete the switch from one dot to the other. A s mentioned earlier, the transition is not abrupt due to ® nite temperature, and for a functional QCA cell the border shif t must be su cient to overcome this broadened transition. The observed shift of 440 m V is nearly su cient to completely switch the electron, but a larger shif t would be desirable. A n increase of the coupling capacitance between the input and output dots will increase the shif t. We also compare the observed border shif t 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 ) shows the 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 ) shows the calculated potential on the output dot, D1, and ® gure 33( c) shows the calculated shif t of the honeycomb border as a function of the driver gate voltages. Figure 33( b ) and (c) demonstrate that with a proper setting of the working point of the double dot, and a temperature of 0 K, the input dots are able to produce su cient shif t to switch an electron from one output dot to the other, as evidenced by the abrupt potential changes in ® gure 33( d ) . Since the match between the calculated and experimental honeycomb shif t is very good, we believe that the data of ® gure 33( b ) , in conjunction with the di erential detector data, con® rms 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 shif t from the electron transfer is unable to cause a border crossing, and the electron population on the double-dot remains unchanged. This e ect 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 shif t induced by changes in the charge con® guration of the coupled input dots. The observed shift is in excellent agreement with calculations, and a comparison of the shif t 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 output dots to fully overcome the ® nite temperature smearing.


587

Figure 33. Simulated operation of the QCA cell as a f unction of the di erential driver voltages V C = V D . ( a) The potential of the input dot E1, showing a sawtooth behaviour with sharp transitions corresponding to single electron charging events. ( b ) The potential of dot D1 of the double-dot, where the sharp transitions occur when an electron exchanges position within the double-dot. (c ) The displacement f rom the working point ( V A = VB = 1.5 mV ) of the honeycomb border along the diagonal E. Electron exchange occurs when the border crosses zero.

6.

Conclusion

We have developed a novel nanoelectron ic scheme for computing with coupled quantum dots, where inf ormation is encoded by the arrangement of single electrons. We have shown that such structures, the so-called quantum-dot cellular automata, may be used for binary inf ormation processing. In addition, an analogue version is also possible, the so-called quantum-do t cellular nonlinear networks, which exhibit

588

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wave phenomena. We have discussed possible realizations of these structures in a variety of semiconductor systems (including GaA s/A lGaA s, Si/SiGe, and Si/SiO 2 ) , rings of metallic tunnel junctions and candidates for molecular implementations. Finally we have experimentally realized the key elements of QCA operation. Our studies are the ® rst experiments to demonstrate the control of a single electron by single electrons. Acknowledgements

The work described in this paper was made possible by the hard and dilligent work of the postdocs and students in Notre Dame NanoDevices Group, including Greg BazaÂ n, Y uriy V. Brazhnik, Minhan Chen, Henry K. Harbury, Geza Toth, P. Doug Tougaw, Weiwen Weng and X iaoshan Zuo. This research was supported in part by DAR PA , ONR, and NSF. References

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